Exploring Mathematical Modeling with Young Learners (Early Mathematics Learning and Development) 3030638995, 9783030638993

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Early Mathematics Learning and Development

Jennifer M. Suh Megan H. Wickstrom Lyn D. English Editors

Exploring Mathematical Modeling with Young Learners

Early Mathematics Learning and Development Series Editor Lyn D. English Queensland University of Technology Kelvin Grove, Brisbane, QLD, Australia

This series provides international educators and researchers with a platform with which to discuss the active fields of childhood learning and mathematics learning. The series looks to answer questions such as what constitutes mathematics learning in the prior-to-school and the early school years, what innovative early mathematics programs are being implemented around the globe, what is the role of play in early mathematics learning and many others. It will explore in detail many perspectives including, but not limited to, research, curriculum development, and policy issues. Together, the collection of volumes spans a broad range of issues that extends beyond just mathematics education. This series is aimed at university researchers, curriculum developers, policy makers, educational organizations, early childcare professionals, and parental associations. • • • •

The first series to target the field of early mathematics learning; A significant and welcome outlet for academics in the field; Brings together experts from all over the world; and Will contribute to improved early mathematics learning.

Book proposals for this series may be submitted to the Editor: Lyn English l. [email protected] or the Publishing Editor: Melissa James. e-mail: Melissa. [email protected] More information about this series at http://www.springer.com/series/11651

Jennifer M. Suh  •  Megan H. Wickstrom Lyn D. English Editors

Exploring Mathematical Modeling with Young Learners

Editors Jennifer M. Suh College of Education and Human Development George Mason University Fairfax, VA, USA

Megan H. Wickstrom Department of Mathematical Sciences Montana State University Bozeman, MT, USA

Lyn D. English Queensland University of Technology Brisbane, QLD, Australia

ISSN 2213-9273     ISSN 2213-9281 (electronic) Early Mathematics Learning and Development ISBN 978-3-030-63899-3    ISBN 978-3-030-63900-6 (eBook) https://doi.org/10.1007/978-3-030-63900-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Acknowledgments

We wish to extend our sincere thanks to the following reviewers. Without their time, expertise, and insightful feedback, this book would not be what it is today. Dr. Elizabeth Arnold – Colorado State University, USA Dr. Cynthia Anhalt – University of Arizona, USA Dr. Jonas Bergman-Ärlebäck – Linköping University, Sweden Dr. Jill Brown – Deakin University, Australia Dr. Elizabeth Burroughs – Montana State University, USA Dr. Mary Alice Carlson – Montana State University, USA Dr. Joe Champion – Boise State University, USA Dr. Marta Civil – University of Arizona, USA Dr. Mary Foote – City University of New York, USA Dr. Elizabeth Fulton – Montana State University, USA. Dr. Melissa Gallagher – University of Maine at Farmington, USA Dr. Benjamin Galluzo – Clarkson University, USA Dr. Andrew Gilbert – George Mason University, USA Dr. Jennifer Green – Michigan State University, USA Dr. Mairead Hourigan – Mary Immaculate College, University of Limerick, Ireland Dr. Hyunyi Jung – University of Florida, USA Dr. Elham Kazemi – University of Washington, USA Dr. Kathleen Kavanagh – Clarkson University, USA Dr. Cynthia Langrall – Illinois State University, USA. Dr. Aisling Leavy - Mary Immaculate College, University of Limerick, Ireland Dr. Rachel Levy – Harvey Mudd College, USA Dr. Kathleen Matson – George Mason University, USA Dr. Amy Roth McDuffie – Washington State University, USA Dr. Helena Osana – Concordia University, Canada Dr. Matt Roscoe – University of Montana, USA Dr. Gloria Stillman – Australian Catholic University, Australia. Dr. Erin Turner – University of Arizona, USA Dr. Rose Mary Zbiek – Penn State University, USA v

Contents

Part I The Nature of Mathematical Modeling in the Early Grades 1

Mathematical and Interdisciplinary Modeling in Optimizing Young Children’s Learning��������������������������������������������������������������������    3 Lyn D. English

2

Synthesizing Research of Mathematical Modeling in Early Grades����������������������������������������������������������������������������������������   25 Hyunyi Jung and Sarah Brand

3

 Mathematical Modeling Thinking: A Construct for Developing Mathematical Modeling Proficiency������������������������������������������������������   45 Cynthia O. Anhalt, Ricardo Cortez, and Julia M. Aguirre

4

Data Modelling and Informal Inferential Reasoning: Instances of Early Mathematical Modelling������������������������������������������   67 Aisling Leavy and Mairéad Hourigan

5

 Development in Mathematical Modeling����������������������������������������������   95 Corey Brady and Richard Lesh

Part II Identifying the Knowledge of Content and Pedagogy Needed for Mathematical Modeling in the Elementary Grades 6

Elementary Teachers’ Enactment of the Core Practices in Problem Formulation through Situational Contexts in Mathematical Modeling����������������������������������������������������������������������  113 Jennifer Suh, Kathleen Matson, Sara Birkhead, Samara Green, MaryAnne Rossbach, Padmanabhan Seshaiyer, and Spencer Jamieson

7

 Teaching Practices to Support Early Mathematical Modeling������������  147 Mary Alice Carlson

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viii

Contents

8

 Teachers’ Use of Students’ Mathematical Ideas in Mathematical Modeling ��������������������������������������������������������������������������������������������������  169 Elizabeth Fulton

9

Teaching and Facilitating Mathematical Modeling: Teaching, Teaching Practices, and Innovation��������������������������������������  195 Rose Mary Zbiek

Part III Mathematical Modeling and Student Experiences as Modelers 10 Mathematical  Modeling: Analyzing Elementary Students’ Perceptions of What It Means to Know and Do Mathematics������������  209 Megan H. Wickstrom and Amber Yates 11 Upcycling  Plastic Bags to Make Jump Ropes: Elementary Students Leverage Experiences and Knowledge as They Engage in a Relevant, Community-­Oriented Mathematical Modeling Task������������������������������������������������������������������������������������������  235 Erin E. Turner, Amy Roth McDuffie, Julia M. Aguirre, Mary Q. Foote, Candace Chappelle, Amy Bennett, Monica Granillo, and Nishaan Ponnuru 12 A  Window into Mathematical Modeling in Kindergarten ������������������  267 Robyn Stankiewicz-Van Der Zanden, Stacy Brown, and Rachel Levy 13 The  Genesis of Modeling in Kindergarten��������������������������������������������  311 Helena P. Osana and Katherine Foster 14 Mathematical  Modeling with Young Learners: A Commentary ��������  337 Elham Kazemi Part IV Interdisciplinary and Community-Based Modeling 15 Convergent  Nature of Modeling Principles Across the STEM Fields: A Case Study of Preservice Teacher Engagement��������������������������������  345 Andrew Gilbert and Jennifer M. Suh 16 Supporting  Students’ Critical Literacy: Mathematical Modeling and Economic Decisions ��������������������������������������������������������  373 Melissa A. Gallagher and Jana P. Jones 17 Culturally  Relevant Pedagogy and Mathematical Modeling in an Elementary Education Geometry Course������������������������������������  389 Emily J. Yanisko and Laura Sharp Minicucci 18 Learning  from Mothers as They Engage in Mathematical Modeling ��������������������������������������������������������������������������������������������������  413 Marta Civil, Amy Been Bennett, and Fany Salazar 19 Insights  Regarding the Professional Development of Teachers of Young Learners of Mathematical Modeling��������������������������������������  437 Elizabeth A. Burroughs

Contributors

Julia M. Aguirre  University of Washington, Tacoma, WA, USA Cynthia O. Anhalt  The University of Arizona, Tucson, AZ, USA Amy Bennett  University of Arizona, Tucson, AZ, USA Amy Been Bennett  University of Nebraska-Lincoln, Lincoln, NE, USA Sara Birkhead  George Mason University, Fairfax, VA, USA Corey Brady  Department of Teaching and Learning, Peabody College, Vanderbilt University, Nashville, TN, USA Sarah Brand  Columbia University, New York City, NY, USA Stacy Brown  California State Polytechnic University, Pomona, CA, USA Elizabeth A. Burroughs  Montana State University, Bozeman, MT, USA Mary Alice Carlson  Department of Mathematical Sciences, Bozeman, MT, USA Candace Chappelle  Washington State University, Pullman, WA, USA Marta Civil  University of Arizona, Tucson, AZ, USA Ricardo Cortez  Tulane University, New Orleans, LA, USA Lyn D. English  Queensland University of Technology, Brisbane, QLD, Australia Mary Q. Foote  Queens College, CUNY, New York, NY, USA Katherine  Foster  Department Montreal, Canada

of

Education,

Concordia

University,

Elizabeth  Fulton  Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA Melissa A. Gallagher  University of Houston, Houston, TX, USA

ix

x

Contributors

Andrew Gilbert  College of Education and Human Development, George Mason University, Fairfax, VA, USA Monica Granillo  University of Arizona, Tucson, AZ, USA Samara Green  Fairfax County Public Schools, Fairfax, VA, USA Mairéad Hourigan  Department of STEM Education, Mary Immaculate College, University of Limerick, Limerick, Ireland Spencer Jamieson  Fairfax County Public Schools, Fairfax, VA, USA Jana P. Jones  LeBlanc Elementary School, Abbeville, LA, USA Hyunyi Jung  University of Florida, Gainesville, FL, USA Elham Kazemi  College of Education, University of Washington, Seattle, WA, USA Aisling  Leavy  Department of STEM Education, Mary Immaculate College, University of Limerick, Limerick, Ireland Richard  Lesh  School of Education (Emeritus), Indiana University, Blooming ton, TN, USA Rachel Levy  Pomona Unified School District, Pomona, CA, USA Kathleen Matson  George Mason University, Fairfax, VA, USA Amy Roth McDuffie  Washington State University, Pullman, WA, USA Laura Sharp Minicucci  Baltimore City Public Schools, Baltimore, MD, USA Helena P. Osana  Department of Education, Concordia University, Montreal, Canada Nishaan Ponnuru  University of Arizona, Tucson, AZ, USA MaryAnne Rossbach  Fairfax County Public Schools, Fairfax, VA, USA Fany Salazar  University of Arizona, Tucson, AZ, USA Padmanabhan Seshaiyer  George Mason University, Fairfax, VA, USA Jennifer M. Suh  College of Education and Human Development, George Mason University, Fairfax, VA, USA Erin E. Turner  University of Arizona, Tucson, AZ, USA Robyn Stankiewicz- Van Der Zanden  Harvey Mudd College, CA, USA Megan  H.  Wickstrom  Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA Emily  J.  Yanisko  Urban Teachers at Johns Hopkins University School of Education, Baltimore, MD, USA

Contributors

Amber  Yates  Department Bozeman, MT, USA

xi

of

Education,

Montana

State

University,

Rose Mary Zbiek  Department of Curriculum and Instruction, The Pennsylvania State University, University Park, PA, USA

Part I

The Nature of Mathematical Modeling in the Early Grades

Chapter 1

Mathematical and Interdisciplinary Modeling in Optimizing Young Children’s Learning Lyn D. English

This chapter explores mathematical and interdisciplinary modeling within the early and elementary school years, where children’s learning potential often remains untapped. The terms, models, modeling, and modeling processes, have been used in various ways in the literature (e.g., English, 2013; English, Arleback, & Mousoulides, 2016; Gainsburg, 2006; Lesh & Doerr, 2003a, 2003b; Lesh & Zawojewski, 2007), with debate over whether these are components of the broad spectrum of problem solving or entities in their own right. There are also different approaches to when and how modeling is introduced in school curriculum documents (e.g., Common Core State Mathematics Standards, 2013; Californian Mathematics Framework: Appendix B, 2018). With the differing opinions on what constitutes a modeling task and when modeling should be introduced in the curriculum, implementation within the classroom can be particularly challenging (as noted in Carlson’s Chap. 7). Consideration is first given to the importance of establishing mathematical and interdisciplinary modeling in the elementary school years, beginning with the earliest grades. Interdisciplinary modeling as used here encompasses the STEM disciplines as well as aspects of the humanities. Next, different forms of modeling for grades K-6 are reviewed including model-eliciting activities (MEAs e.g., Lesh & Doerr, 2003a, 2003b), data modeling (e.g., Lehrer & English, 2018), and STEM-­ based modeling (e.g., English, 2018a; Fulton, Chap. 8). Modeling contexts with a focus on cultural and community issues are also explored (e.g., Anhalt, Staats, & Cortez, 2018; Greer, Verschaffel, & Mukhopadhyay, 2007; Turner et al., Chap. 11). To illustrate how elementary students can work effectively with modeling problems, an MEA and a follow-up STEM-based modeling problem implemented in fourth-­ grade classrooms (9-year-olds) are presented. L. D. English () Queensland University of Technology, Brisbane, QLD, Australia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_1

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L. D. English

1.1 Early Modeling The elementary grades are the ideal time to introduce young learners to the foundations of mathematical and interdisciplinary modeling, where they already have the basic competencies for developing important modeling skills (English, 2014; Leavy & Hourigan, 2018; Lehrer & Schauble, 2002). Although recent years have seen a greater recognition of young learners’ modeling competencies, as evident in the present chapters, the inclusion of modeling in early years curricula is still lacking. Even the Common Core State Standards for School Mathematics does not include mathematical modeling as a specific domain in the younger grades; rather, it is listed as one of the Standards for Mathematical Practice (MP4: “Model with Mathematics”). On the other hand, the Mathematics Framework for California Public Schools (2018) does highlight the importance of modeling during these informative years. Likewise, the authors of the GAIMME report (2nd edn., 2019) maintain that “mathematical modeling should be taught at every stage of a student’s mathematical education” (Garfunkel and Montgomery (2016) p. 7). Well over 10 years ago, Carpenter and Romberg (2004) noted that their research had shown how “children can learn to model, generalize, and justify at earlier ages than traditionally believed possible, and that engaging in these practices provides students with early access to scientific and mathematical reasoning” (p. 4). One has to question why mathematical modeling has been reserved largely for the secondary school years (Greer et al., 2007). With confusion over the nature of modeling, inadequate teacher professional development coupled with insufficient classroom resources and limited awareness of young children’s modeling capabilities, it is perhaps not surprising that elementary school modeling is lacking (English, Arleback, & Mousoulides, 2016; Fulton, Chap. 8; Jung & Brand, Chap. 2).

1.2 Defining Modeling for the Elementary Grades How we define mathematical and interdisciplinary modeling naturally governs the many issues we investigate such as task design, key competencies to be developed (e.g., Anhalt et al., Chap. 3), the thinking and reasoning processes to be fostered, and how we assess students’ developments (e.g., Turner et  al., Chap. 11). It is beyond the scope of this chapter to explore the different ways in which modeling has been interpreted. Suffice to say that modeling is not simply completing standard number operations or basic word problems, as the chapters in this book demonstrate. Modeling is different to “traditional” mathematics children complete in school, in which speed and accuracy are often considered all important, where the mathematics is of a cognitively low level and where children’s thinking is not challenged. As Wickstrom and Yates indicate in their chapter (Chap. 10), mathematical modeling is structured differently, in that it is more challenging yet manageable for students, is more motivating and rewarding for learners, and encourages students to generate their own mathematical ideas (in contrast to being given the mathematics

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upfront). Furthermore, modeling problems are set within contexts that are authentic and relevant to students, with opportunities for their cultural backgrounds to play a key role (Anhalt, Turner, Aguirre, Foote, & Roth MCDuffie, 2018; Suh et al., 2018). From this real-world perspective, Anhalt, Staats, et al. define mathematical modeling as “a process in which students use their knowledge of an everyday situation to engage in cycles of mathematical inquiry” (p. 307). Similarly, the GAIMME report (2nd edn., 2019) defines mathematical modeling as “a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-­ world phenomena” (p.  8). As noted in the report, most definitions of modeling emphasize the relationship between modeling and the world in which we live and the use of mathematics to “explore and develop our understanding of real world problems” (p. 8). A number of modeling problem categories have been identified in the literature (Suh & Seshaiyer, 2019; Turner et  al., 2018). The four basic categories below describe the most common forms of modeling problems, although a given problem can display features of more than one category as indicated in this chapter. Descriptive modeling: Using real-world data to describe/present/analyze a situation; using modeling to describe possible outcomes, taking into consideration assumptions and/or outcomes. Predictive modeling: Using modeling to analyze relationships or trends in a set of data to predict further data or outcomes. Optimizing modeling: Using modeling to determine the “best” option or plan to achieve a desired or given goal. Rating and ranking modeling: Using modeling to rate and rank different options based on given (or determined) criteria and data. Students decide on how to weight criteria and data and apply their ranking to make a selection or a decision.

1.3 Model-Eliciting Activities Perhaps the most well-known perspective on modeling is that of Lesh and his colleagues, namely, model-eliciting activities (MEAs), which are supported by clearly defined principles (Lesh & Doerr, 2003b). MEAs can fall within one or more of the foregoing modeling categories. Numerous publications have emerged from the foundations Lesh established (e.g., Arleback, Doerr, & O’Neil, 2013; English, 2010; Lesh & Doerr, 2003a, 2003b; Lesh, English, Sevis, & Riggs, 2013; Lesh & Zawojewski, 2007; Suh, Matson, & Seshaiyer, 2017). Descriptions of modeling stemming from Lesh’s work include English and Sriraman’s (2010) definition, namely, “modeling problems are realistically complex situations where the problem solver engages in mathematical thinking beyond the usual school experience and where the products to be generated often include complex artifacts or conceptual tools that are needed for some purpose, or to accomplish some goal” (p. 273). Two key components of such a definition are that of mathematical thinking beyond school and the generation of complex products designed to serve a particular

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purpose. Mathematical modeling from this perspective can be applied readily to the elementary school—it provides substantial scope for designing problems of varying levels of sophistication where children of all school achievement levels, from the earliest years, can experience learning “success.” For such success to occur, however, the modeling problem must be carefully structured to facilitate learning. As Zawojewski (2010) argued, a key distinction between traditional notions of problem solving and modeling activities is how the activity is designed. MEAs focus on the processes of problem interpretation and reinterpretation of problematic information and on the iterative development of mathematical ideas as children form initial models, test them, and then refine their models according to given specifications. MEAs frequently entail statistical information of varying complexity, which even very young children can handle (e.g., Lehrer & English, 2018; Leavy & Hourigan, 2018). As such, MEAs provide a valuable foundation for the development of skills that students require as consumers of data in the real world. For example, as they mature, students will have to interpret and understand the implications of insurance documents, financial agreements, phone and Internet plans, and frequently deceptive opinion polls, to name but a few. As students work with the data presented in these modeling problems, they can be seen to generate their own mathematical concepts and processes (“mathematization”). For example, students develop notions of mode, median, frequencies, and ranks and apply operations involving sorting, organizing, selecting, quantifying, weighting, and transforming large data sets—often before all of these have been formally introduced (English, 2014). Integral to the foregoing mathematizing processes are the various representational forms needed to display the models that students produce, such as a variety of graphs, tables, lists, diagrams, and oral and written reports. These representations provide further insights into the growth of students’ thinking and learning as they work the modeling activities (Lehrer & English, 2018; Lehrer & Schauble, 2000). Figure 1.1 presents one diagrammatic representation of the modeling processes I have been describing, but of course it is just one of many that researchers have proposed over the years. Other examples can be found in the present chapters, such as that of Turner et al. (Chap. 11), who make the important point that students’ progression through the modeling processes does not necessarily follow the direction indicated by the arrows. Although such clockwise movement is typically what is observed, students can move back and forth between the processes. An example of an MEA implemented in fourth-grade classrooms is presented in a later section.

1.4 Data Modeling Drawing on elements of the MEA approach, activities involving data modeling engage children in statistical investigations where major outcomes include drawing inferences from models generated (English, 2014; Leavy & Hourigan, Chap. 4;

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INTERPRET AND UNDERSTAND PROBLEM PARAMETERS · Context · Purpose · Data · Criteria/Constraints COMMUNICATE TO PEERS POSE QUESTIONS

DRAW INFERENCE/S · Make decisions · Recognize uncertainty · Generalize

REFINE MODEL · Review problem parameters · Confirm/modify

ORGANIZE DATA · Select · Eliminate · Prioritize · Generate

DEVELOP MODEL · Structure and represent · Interpret and analyze model

Fig. 1.1  Representation of modeling processes

Lehrer & English, 2018; Lehrer, Jones, & Kim, 2014; Lehrer & Schauble, 2005). Making informal inferences includes recognizing uncertainty, detecting variation, and making predictions (Makar, 2016; Lehrer & Romberg, 1996; Lehrer & Schauble, 2002, 2004). Data modeling typically entails a number of processes including (a) posing statistical questions within meaningful contexts; (b) designing investigations to address the posed questions, (c) generating, selecting, and measuring attributes; (c) organizing, structuring, and representing data; and (d) developing a model and drawing informal inferences. The importance of these processes is evident in the mathematical modeling perspective adopted in the GAIMME report (2nd Edn., 2019, cited previously). Engaging young students in data modeling is increasingly important in today’s world where they readily encounter a range of diverse media. However, as Lehrer and English (2018) emphasize, the media rarely give insights into how the claims reported have been derived or any modeling processes used in drawing inferences

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and conclusions, such as predictions about economic growth. Claims are frequently made without a consideration of variability and uncertainty. Data modeling introduced early provides the important foundations for dealing effectively with statistics that govern our world. In a similar vein, Leavy and Hourigan (Chap. 4) stress the importance of early experiences in informal inference, where students are provided with initial steps in accessing a statistical culture and growing statistically literate. Communicating final models to peers is an important component of modeling, for it is here that students explain and justify their models, conclusions, and inferences and receive critical, but constructive feedback from their peers. Students need opportunities to develop skills in both explaining and justifying their models, as well as asking critical questions about their peers’ models. Articulating mathematical ideas and arguments effectively, listening and offering constructive and critical feedback, and discovering and conveying new insights during problem solving are recognized as important twenty-first-century skills; such skills are inherent in mathematical modeling (Suh et al., 2017). As I indicate in the next section, modeling that incorporates philosophical inquiry provides rich opportunities for developing these twenty-first-century competencies.

1.5 Modeling Within Cultural and Community Contexts The use of meaningful contexts is a critical feature of modeling problems; indeed, the wide range of multidisciplinary contexts applicable to modeling are invaluable in “bringing mathematics to life” for students. As Wickstrom and Yates (Chap. 10) indicate, students can be more motivated to engage with a modeling problem because they see meaning in its context or because they find it rewarding to make connections between the mathematics they are working with and the real world. Examples of many such modeling contexts appear throughout this book. One such example is Turner et al.’s jump rope context (Chap. 11), which provided a common ground for students across elementary grade levels to create models for making jump ropes from plastic bags. The students’ lived experiences informed their learning as they formed assumptions, made decisions, and applied mathematical operations in their model creations, analysis, and refinements. Studies that address modeling within cultural and community contexts are responding to the many calls for making mathematics more relevant to students from a range of backgrounds. Numerous studies have indicated the learning affordances provided by cultural and community contexts (e.g., Anhalt, Staats, et  al., 2018; Greer et al., 2007; Poling, Naresh, & Goodson-Espy, 2018; Suh et al., 2018; Turner, Aguirre, Foote, & Roth McDuffie, 2018; Turner et  al., 2018). Modeling experiences designed within situations that are relevant to students’ lives, cultures, and communities are likely to be more motivating (Anhalt, Cortez, & Smith, 2017; Anhalt, Turner, et al., 2018) and lead to a better appreciation of how mathematics shapes our world. Furthermore, modeling within such contexts helps children see

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that mathematics is not simply a means of calculating answers but is a vehicle for social justice (e.g., Cirillo, Bartell, & Wager, 2016; English, 2013; Greer et  al., 2007; Wickstrom & Yates, Chap. 10; Yanisko & Minicucci, Chap. 17). There appear a limited number of studies, however, which implement modeling as a vehicle for social justice. One underrepresented approach to modeling that facilitates awareness of community and cultural issues is through philosophical inquiry (English, 2013). There has been little research from this perspective, despite the close relationship between philosophy and mathematics. Establishing a community of philosophical inquiry within the classroom can broaden children’s mathematical learning to incorporate issues of social justice (English, 2013). I have defined such a community as a classroom that embraces philosophical inquiry where a spirit of cooperation, trust, and ease is evoked. In such a classroom: … there is a willingness to share, respect, question, and critique one another’s ideas on issues that are relevant, meaningful, and considered worthy of investigation…..for such communities to thrive, both students and teachers must be open and committed to the sharing of alternative ideas, to the critical questioning of mathematical and contextual assumptions, and to the continued enrichment of their thinking. (English, 2013, p. 46)

The increasingly important field of critical mathematics education addressing concerns of social justice also encompasses philosophical inquiry (Kennedy, 2012; Lipman, 1988), yet specifically establishing such communities seems to be rarely explored. Mathematical modeling within interdisciplinary, authentic contexts is ideal for incorporating philosophical inquiry; indeed, as I have argued, “philosophical inquiry is an inbuilt component” of such modeling (English, 2013, p.  49). Furthermore, both philosophical inquiry and mathematical modeling develop shared thinking and reasoning skills that are essential to both mathematical problem solving and philosophy. These include, but are not limited to, creative and flexible thinking, critical and reflective thinking, deductive and inductive reasoning, investigative inquiry, and drawing informal and formal inferences. Philosophical inquiry plays a significant role in modeling involving data, where students should be encouraged to think critically about the information presented. Mukhopadhay and Greer (2007) point out how mathematics education should “convey the complexity of mathematical modeling social phenomena and a sense of what demarcates questions that can be answered by empirical evidence from those that depend on value systems and world-views” (p. 186). In a similar vein, Gallagher and Jones (Chap. 16) report on integrating mathematical modeling and economics, where teaching interns were presented with a task involving a community issue, a short while after a school shooting. With many such shootings occurring in the USA and elsewhere in the world, numerous courses of action have been proposed for addressing this growing problem. Not surprisingly, various community viewpoints exist on such proposals, giving rise to valuable contexts for modeling problems where data are to be considered. With the escalation of statistical data from the mass media, it is imperative to commence the foundations of critical thinking early. Students’ skills in asking

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critical questions as they work with data in constructing and improving a model, reflect on what their models convey, and justify and communicate their conclusions should be nurtured early.

1.6 STEM-Based Modeling Numerous interpretations of STEM education and integrating the STEM disciplines have appeared in the literature, with respect to the number of disciplines being addressed and the nature and extent of their integration (e.g., English & King, 2018; Honey, Pearson & Schweingruber, 2014; Moore & Smith, 2014). Ranging from a single discipline approach to multidisciplinary and transdisciplinary approaches (Vasquez, Sneider, & Comer, 2013), the notions of STEM and STEM integration remain vague and often contentious (Bybee, 2013). One perspective that is especially germane to STEM-based modeling is that of Shaughnessy (2013), who refers to STEM education as “…solving problems that draw on concepts and procedures from mathematics and science while incorporating the teamwork and design methodology of engineering and using appropriate technology” (p. 324). Drawing on the works of Bryan, Moore, Johnson, and Roehrig (2015), I proposed a STEM integration matrix that includes a focus on both disciplinary content and problem context (English, 2017). Table 1.1 presents one example of this matrix, which shows disciplinary content as either the primary domain/s being learned or the supporting one/s, together with a range of problem contexts in STEM-based modeling problems. The importance of both content and context was evident in the 2015 PISA mathematics framework (OECD, 2015), with the proposed 2021 framework including mathematical reasoning with the subscale of “Mathematical modeling as a lens to the real world” (Carr, 2018). A STEM-based approach to modeling that I have advanced recently is that of modeling with design (English, 2018b). Such an approach has the potential to promote independent learning and student knowledge generation since the very nature of design requires recognizing how prior knowledge might be applied under new and unexpected circumstances (McKenna, 2014). The notion of design adopts different forms in different domains and contexts (e.g., Crismond & Adams, 2012; Wright & Wrigley, 2017) and is increasingly being used across the Table 1.1  Sample of STEM integration matrix Content Primary Supporting Context Disciplinary Background

Science √

Technology

Science √ Personal

Technology √ Societal √

Engineering √ Engineering √ Occupational

Mathematics √ Mathematics Historical √

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STEM disciplines and beyond (as noted by Kelley & Sung, 2016). A common description of engineering design incorporates the iterative nature of mathematical modeling problems and involves (a) defining a problem by identifying criteria and constraints for acceptable solutions, (b) generating a number of possible solutions and assessing which of these best meets the problem parameters, and (c) improving the solution by systematically testing and refining, including overriding less significant aspects for the more important (English & King, 2015; Lucas, Claxton, & Hanson, 2014; Next Generation Science Standards, 2014). The research of Zawojewski, Hjalmarson, Bowman, and Lesh (2008) summarized succinctly the links between engineering design processes and the cyclic processes of modeling, namely: A problem situation is interpreted, initial ideas (initial models, initial designs) for solving the problem are brought to bear; a promising idea is selected and expressed in a testable form; the idea is tested and information from the test is analyzed and used to revise (or reject) the idea; the revised (or a new) idea is selected and expressed in a testable form; etc. (p. 6)

Despite the obvious link between engineering design processes and those of modeling, this important connection remains underutilized in STEM-based experiences in the elementary and middle school years. This is despite the increased recognition that younger learners can engage effectively in engineering design processes and can apply multiple ideas and approaches to innovative and creative problems (Dorie, Cardella, & Svarovsky, 2014; English & King, 2015; English & Moore, 2018; Portsmore, Watkins, & McCormick, 2012).

1.7 Examples of Modeling Problems 1.7.1 Model-Eliciting Activity The following example is the first of two modeling problems from a set of problem activities (the Sweets Pantry) implemented in fourth-grade classes. The first example is mainly a descriptive and optimizing MEA (with a preliminary rating/ranking component). 1.7.1.1 The Sweets Pantry (Part A) The MEA was implemented in the second half of 2018  in four grade 4 classes (9-year-old students) in a Brisbane state primary school. All classes had been participating in the program, Philosophy for Children (described in Splitter, 1995), since their early school years. The students referred to the program in their group and class interactions. The MEA was the students’ first experience with mathematical modeling and served as a foundation for the second problem, namely, a STEM-based modeling

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example involving the design and construction of sweets packs. Packaging is the first impression that consumers receive when purchasing food products. A recent study (Gunaratne et al., 2019) on chocolate packaging found significant differences in liking a chocolate based on how it was packaged. Consumers’ perception of a chocolate’s taste was influenced by its package, highlighting the importance of attractive and appealing packaging in food manufacturing. For both problems the students were supplied with individual workbooks comprising a number of questions to be answered, together with blank pages for recording their data, their plans and designs, and their models. Students were videotaped as they worked both problems in their groups (three to five students), participated in whole-class discussions, and presented group reports on their models to their peers. Students’ interactions were transcribed with a focus on their application of mathematical disciplinary knowledge, reasoning processes, and peer questioning as they worked the problems and shared their models. Children were initially presented with an introductory task as follows: Imagine you have to choose a pack of sweets from a display stand. There are many different colorful packs to choose from. What four things would you look for? List them in order, from the most important to the least important. Explain how you decided on your order.

Following this introductory task, children were presented with a table of data comprising eight different sweets (e.g., M&M’s, Marshmallow Drops), costs of the sweets (ranging from $4.00 to $17.50), packaging materials (e.g., tissue, cellophane, cardboard, plastic), weight (ranging from 30  g to 110  g), and packaging decorations (e.g., purple and pink ribbon, stars and spots). The following problem was then presented: HELP THE SWEETS PANTRY: MAKE YOUR SELECTIONS A new sweets store, the Sweets Pantry, has opened in Hillsdale. It needs your help in filling some mysterious orders it has received from neighborhood children. They did not say exactly what they wanted, not all children wanted the same packs of sweets, and not all children had the same amount of money to spend. Here is one order the Sweets Pantry received: Dear Sweets Pantry, We would like to purchase some packs of sweets but we are unsure which to choose. Can you please help us?

• • • •

We are taking the sweets on a school trip. We wish to impress our friends on the trip. We do not want to spend more than $50 and we want value for money. We would like to buy at least 400 grams of sweets. Here is what the Sweets Pantry has to offer the children. In your group, use the table to create a model or way in which the Sweets Pantry could fill this order and use your model to decide which sweets packs your group would select. Record the packs you choose on page 5. The Sweets Pantry should be able to use your model to fill other similar mysterious orders.

It is interesting to note how one group generated three models and evaluated each to determine which was closest to meeting their criteria. The group was trying to

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optimize their sweets selections while keeping within the budget and weight limits. The three possible models appear in Fig. 1.2. Sam explained that “The bottom one (model) is the best option because we used the least amount of grams.” While observing the group working, I asked why they chose that model and what they would tell others who might apply their model. I was keen to learn more about their model generation and how they would apply their model to problem solution: Sam: You tell them to look at the types of packaging, if it’ll be stable enough. You tell them to look at the cost. You tell them to look at how it tasted, if you had it before. And you’d also think about the weight, if… Researcher: What would you say about the weight? Sam: You’d have to have all for 400 grams… Researcher: Are there different things you can come up with, like, using those [models]? Sam: We’d do a philosophy (Sam was referring to the Philosophy for Children program implemented in his school). Terry: One of them is, there may be no single way to answer… Sam: And when we’re doing that, we have to consider the weight of which sweet we’re choosing. Because if it’s too heavy for the bag then it’s going to break. Researcher: What procedure did you use to determine that? Terry: Adding things up.

Fig. 1.2  Sam’s documentation for three possible models

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Sam: What we did is… We wanted to use less priced things so we could get more of things that could support an entire class. Yeah, and when we had like $17 left, but we still had a lot of weight left. We also got to use something else, in addition to that (an item selection), which brought us to the lowest possible total we could get of weight 420 grams. Various other forms of model documentation were observed across the classes, with tables and grids common, as illustrated in Figs. 1.3 and 1.4. In the other class, children’s peer reporting on their models elicited a number of critical questions from the class, which indicated application of their learning from the Philosophy for Children program. When describing their model, Frank explained that they wanted something that they can take to a field trip, easy to carry; that was one of our sections; so we chose choc drops, the things that are easy to carry, choc drops, marshmallow twirls, rock candy bites, the crazy twirls. So we chose choc drops; they are more [sic] smaller and they will be able to carry on the field trip. They also want to impress their friends; we have a category that looks fancy. We only had three fancy lookers… so we said, well, they want to impress their friends so we chose choc chops because they are easy to carry and they look fancy, so we chose four of them, 240 grams, and we chose lolly pops because they are only $5; we went a lot over the limit of grams.

He explained that his group used a “graph/grid” and “wrote down on the side all types of sweets” and “write down the qualities (labeling the columns) and tick each box depending on what it has.” Figure 1.3 displays the grid that one member of his group created. Students from the class critically questioned Frank’s group on their category of “fancy”: Student A: How do you know it’s fancy? Frank: Look at the description: purple and pink ribbons. We guess that was kind of fancy as they will be fluttering everywhere. Floral top black ribbon; first of all, black is an extremely cool color and floral means flowery I think … (provides more details on the rest of the colors and ribbons that comprise their selection). Student B: First of all, when you said black is a very cool color, I might not think that. How do you know those ribbons flutter, that makes it fancy? How do you know that?

1.7.2 STEM-Based Modeling Problem Part B of the Sweets Pantry problem is presented next. It is primarily a descriptive and optimizing example. The problem is considered a STEM-based modeling example because it incorporates a science component (materials science), mathematics (measurement, geometry, number sense, computation), and engineering

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Fig. 1.3  Creation of a grid for model documentation

Fig. 1.4  Creation of a table for model documentation

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(packaging engineers, design processes, construction from designs). In terms of the STEM matrix (Table 1.1), science and mathematics may be regarded as the primary content and engineering as the supporting content. The background context pertains mainly to the personal category, but society/community could apply also. The activity commenced with the students viewing video snippets of packaging engineers to learn about their roles and responsibilities. Following discussion, students were introduced to sweets packaging and invited to reflect on sweets packs with which they were familiar; they were to describe or illustrate these packs. Next, the students were to design and construct a model for a sweets pack given the instructions and constraints, as displayed below: The Sweets Pantry wishes to improve the packaging of its different sweets products. Some customers have complained that the packs are boring, don’t have enough sweets in them, and often break if they are not careful with them. The owners of the Sweets Pantry have therefore decided to have a competition to see who can create a new sweets pack that customers will like. These are the rules for the competition: The packs must be attractive. They must hold between 250 grams and 450 grams. The packs can be any shape and size. They can be bags, boxes, cylinders, or whatever you like. You must choose materials from those shown but you cannot spend more than $15.00 in buying your materials. Imagine you are entering the Sweets Pantry competition. Here is the range of sweets the Pantry sells.

The students were given a table comprising eight different sweets and the weight per item (e.g., “Chocolate Drops,” 30 grams per item; “Chocolate Twists,” 25 grams; M&M’s, 20 grams), together with a list of supplied materials and their cost per sheet (e.g., paper at $1.00 per sheet; vinyl, $6.00; cellophane, $3.00; cardboard, $4.00; fabric, $8.00). The students were to decide the type of pack to design and subsequently the types of sweets their pack would hold and the number of each type of sweet to be included. Next, the weight in grams the pack would hold was to be calculated, and the types and quantities of materials their pack would comprise were to be determined. Following the group’s labeled design sketches of their proposed sweets packs, the students were to choose and justify the design they would use. On constructing their models, students were to test them, provide a rating for their designs, and identify ways in which they might address problematic components, that is, improve on their designs and subsequent models. Prior to considering some of the students’ responses, it is worth noting how they described the responsibilities of packaging engineers at the beginning of this activity. Their responses included: • They have to test to see if they [packages] are durable; They have to design and check the package; They have to know what the best material is to make it; They need to know the best shape and size for what they would be packaging. • [they] are responsible for designing, testing, and implementing different packaging for food products. Packaging engineers are specialized. They try to solve problems. Use different materials for packaging not just plastic. Packaging effects products. Work with different companies that make packaging materials. Test packaging to see if it is good quality so they save money and the quality of the packaging. Engineer is someone that wants to know why and how things work. There are heaps of types of engineers like electrical engineers and macineal

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[mechanical] engineers. All engineers solve problems. They need to be organized. Make packaging to match the product. Students documented their responses to a range of questions in their workbooks including their explanation and justification for their chosen design and subsequent model, as well as ways in which they would improve their model after having tested it. Students were cognizant of the criteria they had to meet including the budgetary and weight constraints. As can be seen in the following excerpts, students applied their science knowledge (properties of materials) and mathematics understandings (measurement pertaining to weight, linear measures, capacity, and geometry) in justifying their group design selection and model construction. Celine:  Why I chose it is because, cellophane and cardboard are both strong, durable. And cellophane is waterproof. And, I know that cardboard is not waterproof, but when I wrap cardboard in cellophane, it’s now waterproof. Also, cellophane is clear so we can have like a viewing area. Also—I didn’t actually mention this—but this is a viewing area, so you can see the candies because it’s clear. Francis: We think our design is a winner because it’s really bright and it stands out. It also has a cellophane square bit cut out so you can see inside and you can see the lollies. And, it has 6 sections with different types of lollies (Fig. 1.5) and that’s why I think it’s a winner. James:  We tested our design using 450 grams of marbles and tried to carry it around and that worked. We also dropped it from a meter high—1.5 meters and it kinda broke and the lid fell off. Rigina:  I also think it’s a winner because it’s quite big—the top and bottom. When you drop it, it spreads the impact around its wide bottom. And—because it’s different sections—it’s able to spread out the weight so it seems lighter.

Fig. 1.5 Model construction displaying compartmentalized sweets pack with cellophane lid

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It is interesting to note the actions of another group who were also planning to use compartments in their pack. In developing their design, the children referred to a book’s structure as analogous to that of their planned pack, as illustrated in the photos of Fig. 1.6 below.

1.8 Concluding Points This chapter has illustrated different forms of modeling that young learners can undertake competently in the elementary school years, beginning with the earliest grades. Mathematical modeling is a topic that has received widespread global attention for several decades (e.g., Blum & Borromeo Ferri, 2009; Blum & Niss, 1991), but mainly in the upper middle grades and high school years. As Blum and Borromeo Ferri indicate, modeling is challenging for both students and their teachers. Part of this challenge for teachers is what is meant by mathematical modeling. Numerous interpretations and definitions of modeling have been advanced over the years. This chapter has examined just a few forms of mathematical and interdisciplinary modeling including the foundational model-eliciting activities of Lesh and his colleagues (e.g., Lesh & Doerr, 2003a, 2003b), from which many other research studies have emanated. Other forms of modeling addressed in this chapter include data modeling and STEM-based modeling. As indicated in the examples provided, modeling is ideal for drawing on interdisciplinary contexts, such as cultural and community contexts, which present students with meaningful and appealing problem-solving settings. Furthermore, modeling caters for all students from varying school achievement levels and social and cultural backgrounds. The modeling problems I have implemented across the elementary school grades feature low floors and high ceilings (Gadanidis & Hughes, 2011). The low floor feature enables students to tackle the activity at their entry or readiness level of disciplinary content knowledge. The high ceilings afford students opportunities to extend their thinking and learning through exploring mathematical and STEM-based ideas that are beyond their grade level. The focus of the activity then becomes one of learning or idea generation, rather than just the application of routine procedures or problem-solving strategies (English, 2017). In generating their “new” mathematics, students display conceptual surprises (Gadanidis & Hughes, 2011) for both themselves and their teachers, that is, students who do not normally display high outcomes on standard mathematical tests reveal their talents in working these problems. It is rewarding to see the excitement of these children as they create novel and effective solutions beyond what is “typically expected” of them. As the chapters in this book demonstrate, the foundations of mathematical modeling do not begin in the high school years—they commence long before then during children’s earliest school experiences and even before this time, in their home and community environments. We need to recognize these children’s modeling capabilities and capitalize on them. The present chapters provide many examples of

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Fig. 1.6  Using a book to explain proposed structure of sweets pack

approaches to doing so, in contrast to the limited, if any, guidance provided by curriculum documents. Stankiewicz, Brown, and Levy (Chap. 12) rightly question the limited attention to early modeling as evident in MP4 (mathematical practice 4, Model with Mathematics) of the Common Core State Standards for Mathematics: The authors of the CCSS-M suggest that in the early grades, ‘this practice, model with mathematics, might be as simple as writing an addition equation to describe a situation.’ However, one might wonder whether students could engage in all of the activities described in MP4 while doing something as ‘simple as writing an addition equation.’

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Stankiewicz et al. present rich examples of how they worked with teachers and students in their IMMERSION project to establish mathematical modeling “from the ground up” (p. xx). The final words of their chapter are worth citing in our efforts to promote greater recognition of young children’s capacities for modeling. In providing evidence of the modeling potential of young students, including those in the highest need settings, Stankiewicz et al. express the sentiment: We hope the reader will walk away from our chapter with images of the ways in which our youngest students’ mathematical wonderings might serve to open the mathematical terrain of the classroom as students work to identify and articulate their assumptions, propose constraints and boundaries, query into margins of error, examine quantities and the ways those quantities might be unitized, be in proportion or covary, and, lastly, revisit and refine their mathematizing of situations. (p. XX)

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Moore, T. J., & Smith, K. A. (2014). Advancing the state of the art of STEM integration. Journal of STEM Education, 15(1), 5–10. Mukhopadhyay, S., & Greer, B. (2007). How many deaths? Education for statistical empathy. In B.  Sriraman (Ed.), International perspectives on social justice in mathematics education (pp. 169–189). Charlotte, NC: Information Age Publishing. Next Generation Science Standards: For States by States (2014). https://www.nextgenscience.org/ search/node. Accessed 21 Feb 2019. OECD. (2015). PISA 2015 draft mathematics framework. Paris, France: OECD Publishing. OECD. (2018). PISA for development mathematics framework. In In PISA for development assessment and analytical framework: Reading, mathematics and science. Paris, France: OECD Publishing. https://doi.org/10.1787/9789264305274-­5-­en Poling, L. P., Naresh, N., & Goodson-Espy, T. (2018). Empowering mathematics through modeling. Mathematics Teaching in the Middle School, 24(3), 138–146. Portsmore, M., Watkins, J., & McCormick, M. (2012). Planning, drawing and elementary students in an integrated engineering design and literacy activity. Paper presented at the 2nd P-12 Engineering and Design Education Research Summit. Washington, DC. Shaughnessy, M. (2013). By way of introduction: Mathematics in a STEM context. Mathematics Teaching in the Middle School, 18(6), 324. Splitter, L. (1995). Teaching for better thinking: The classroom community of inquiry. Melbourne, Australia: The Australian Council for Educational Research. Suh, J., Britton, L., Burke, K., Matson, K., Ferguson, L., Jamieson, S., & Seshaiyer, P. (2018). Every penny counts: Promoting community engagement to engage students in mathematical modeling. In I.  Goffney & R.  Gutierrez (Eds.), Rehumanizing mathematics for black, indigenous, and Latinx students (pp. 63–76). Reston, VA: National Council of Teachers of Mathematics. Suh, J.  M., Matson, K., & Seshaiyer, P. (2017). Engaging elementary students in the creative process of mathematizing their world through mathematical modeling. Educational Sciences, 7(62). https://doi.org/10.3390/educsci7020062 Suh, J. M., & Seshaiyer, P. (2019). Co-designing and implementing PBL through mathematical modeling in STEM contexts. In M. Moallem, W. Hung, & N. Dabbagh (Eds.), Handbook of problem based learning (pp. 529–550). Hoboken, NJ: Wiley-Blackwell Publishing. Turner, E. E., Aguirre, J. M., Foote, M. Q. Anhalt, C. O., Roth McDuffie, A., Civil, M. (2018). Mathematizing the world: Routines and tasks that foster mathematical modeling with cultural and community contexts. Presentation at the bi-annual meeting of TODOS: Mathematics for all, Scottsdale, AZ. Turner, E. E., Aguirre, J. M., Foote, M. Q., & Roth McDuffie, A. M. (2018, April) Learning to leverage mathematical resources of elementary Latinx children through community-based mathematical modeling tasks. In M.  Civil (Chair), Foregrounding cultural ways of being in mathematics teacher education: Cases From Latinx and Pāsifika communities. Presentation as part of a symposium at the annual research conference of the annual meeting of the American Educational Research Association, New York, NY. Vasquez, J., Sneider, C., & Comer, M. (2013). STEM lesson essentials, grades 3–8: Integrating science, technology, engineering, and mathematics. Portsmouth, NH: Heinemann. Wright, N., & Wrigley, C. (2017). Broadening design-led education horizons: Conceptual insights and future research directions. International Journal of Technology and Design Education, 27(4). Zawojewski, J. (2010). Problem solving versus modeling. In R.  A. Lesh, P.  L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 237–244). Dordrecht, The Netherlands: Springer. Zawojewski, J. S., Hjalmarson, M. A., Bowman, K. J., & Lesh, R. (2008). A modeling perspective on learning and teaching in engineering education. In J. S. Zawojewski, H. Diefes-Dux, & K. Bowman (Eds.), Models and modeling in engineering education: Designing experiences for all students (pp. 1–16). Rotterdam, The Netherlands: Sense Publishers.

Chapter 2

Synthesizing Research of Mathematical Modeling in Early Grades Hyunyi Jung and Sarah Brand

2.1 Introduction Research has shifted its attention from discussing models, modeling, and applications to conducting studies around the idea of mathematical modeling, “a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena” (Consortium for Mathematics and Its Applications & Society for Industrial and Applied Mathematics [COMAP & SIAM], 2016, p.  8). The process requires students to make interpretations and assumptions about real-life phenomena, use mathematics to represent and solve a problem, and validate results in the context of a real-life problem situation. While this process is often depicted as a one-way cycle in relevant literature, learners progress through multidirectional cycles of these phases in practice (e.g., Czocher, 2016; Zbiek & Conner, 2006). Further, the multidirectional cycles embedded in mathematical modeling distinguish it from word problems or mere applications of mathematical concepts (COMAP & SIAM, 2016). Unlike nonsensical problems, such as painting houses at consistent speeds for several days, many students report that engaging with mathematical modeling is one of their most memorable and meaningful mathematics learning experiences due to its authenticity (Gann, Avineri, Graves, Hernandez, & Teague, 2016). An assumption has emerged that mathematical modeling is complex and therefore ought to be reserved for the secondary school curriculum (e.g., Galbraith, H. Jung () University of Florida, Gainesville, FL, USA e-mail: [email protected] S. Brand Columbia University, New York City, NY, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_2

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Blum, Booker, & Huntley, 1998). However, many research studies have proven elementary students capable of engaging with mathematical modeling (e.g., English, 2006; English & Watters, 2005). In review of the International Group for the Psychology of Mathematics Education (PME) proceedings from 2005 to 2015, English, Ärlebäck, and Mousoulides (2016) found three research reports that focused on young students’ learning of mathematical modeling. Two reports showed that without teacher’s direct instruction, students were able to identify patterns and relationships in the data and develop and improve their own models through several modeling cycle iterations (Mousoulides & English, 2008; Mousoulides, Pittalis, & Christou, 2006). The authors emphasized that young learners were able to successfully solve mathematical modeling problems when they collaboratively engaged with meaningful situations. In another PME report, Bonotto (2009) discussed how the student-centered nature of a mathematical modeling problem created a motivating atmosphere where students effectively communicated with one another. In addition, reviews of the PME proceedings revealed that a comparatively small number of the proceedings focused on early mathematical modeling. In light of this, more research with primary grades is needed as young learners benefit from opportunities to explore informal notions of mathematical concepts through mathematical modeling (English, 2006). A recent emphasis on mathematical modeling at all school levels delineates an expectation that instruction on modeling should begin early in the curriculum and continue throughout the secondary years (Blum & Niss, 1991; COMAP & SIAM, 2016; National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010). While many studies emphasized that young learners are capable of successfully engaging with mathematical modeling, few research reports focused on the nature and scope of mathematical modeling in early grades. Schukajlow, Kaiser, and Stillman (2018) summarized the history of mathematical modeling in secondaryand upper-level mathematics teaching and synthesized research related to the teaching and learning of mathematical modeling in general. There are recent summaries of studies regarding specific topics of mathematical modeling, such as competencies (Kaiser & Brand, 2015), theoretical development (Geiger & Frejd, 2015), or assessment (Frejd, 2013), but none of them focused solely on early mathematical modeling. A study conducted by Stohlmann and Albarracín (2016) synthesized 29 research articles at the elementary level in terms of mathematical contents embedded in mathematical modeling activities (e.g., ratios and proportional reasoning, measurement), data collection methods, population, and units of analysis. Our study extends this study by reviewing “high-quality” research journals (Williams & Leatham, 2017), in addition to the practitioner journal Teaching Children Mathematics (TCM), an official peer-reviewed journal of the National Council of Teachers of Mathematics (NCTM). TCM often publishes newly created real-world-­ based activities and resources intended for early elementary school teachers and teacher educators. While other high-quality NCTM practitioner journals like Mathematics Teaching in the Middle School or The Mathematics Teacher may include activities and resources for Grades 5–8, their audience tends to be teachers and teacher educators seeking resources for secondary school students. Reviewing

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TCM provides insights into the types and nature of recently published early-grade mathematical modeling activities that research articles may not offer. We also present patterns found in these articles and structure our paper with these as common themes while avoiding categories previously used by recent review studies (Frejd, 2013; Geiger & Frejd, 2015; Kaiser & Brand, 2015; Stohlmann & Albarracín, 2016).

2.2 Purpose of the Chapter This book chapter addresses the need to investigate current mathematics education research and synthesize knowledge shared in studies of mathematical modeling in the early grades. We examined research articles about mathematical modeling with Grades K-8 that were published in eight mathematics education journals from recent years. We aim to present the critical commonalities of these studies that may stimulate a conversation regarding newly-established research on mathematical modeling and guide future research on early mathematical modeling. A review and synthesis of these common features will allow for effective use of research findings about mathematical modeling in Grades K-8 and will also provide timely resources and guidelines for early-career researchers exploring mathematical modeling.

2.3 Selection Criteria We selected the journals Education Studies in Mathematics (ESM), Journal for Research in Mathematics Education (JRME), Journal of Mathematical Behavior (JMB), For the Learning of Mathematics (FLM), Mathematical Thinking and Learning (MTL), Journal of Mathematics Teacher Education (JMTE), and ZDM, the International Journal on Mathematics Education. These seven journals are commonly considered “high-quality” (Toerner & Arzarello, 2012; Williams & Leatham, 2017) or “first tier” (Nivens & Otten, 2017) mathematics education journals and are highly regarded by mathematics education researchers. Additionally, we included the practitioner journal Teaching Children Mathematics (TCM) to broaden the scope of our analysis and to provide a wealth of resources for teaching mathematical modeling in early grades. Although there are many other journal articles and book chapters that include novel and valuable findings related to early mathematical modeling, we focused on these highly cited journals to explore how early modeling studies are visible by mathematics education communities and distinguish our search of literature from a previous synthesis shared by other researchers (e.g., Stohlmann & Albarracín, 2016).  By examining recent research articles published in these journals, we hope to invite more modeling researchers to share their work and make it apprent to many other  readers in mathematics education communities.

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Both authors read the titles and abstracts of the articles published by the eight journals from the years 2000 to 2017 to screen for their potential to discuss mathematical modeling. If papers appeared borderline or had inconclusive abstracts, the authors skimmed them to determine their relation to mathematical modeling. We started the search process by reading titles and abstracts, instead of using search terms, to avoid excluding papers that used unconventional terms to describe the modeling process. Indeed, some articles described mathematical modeling activities but used alternative terms (e.g., mathematical investigations, interdisciplinary project). The authors then used search terms such as “modeling,” “model,” “real world,” “realistic,” “authentic,” “real life,” and “application” to verify that relevant articles were not omitted. In our analysis of TCM, we excluded a short paper (one to two pages) that included a set of problems or tasks without evidence of student learning. Papers that discussed altered notions of modeling (e.g., models of student learning, using manipulatives, presenting different representations, use of pre-­ mathematized problems) were also excluded. Upon completion, we identified 28 papers to examine for our synthesis (see Table 2.1 for the selected papers for each time period and journal). We also added an asterisk next to articles in the references to indicate that they were surveyed. We started our review by listing the characteristics of studies that had the potential to define patterns across articles (Cooper, 2010). Our initial characteristics included study objectives, grade level(s) of the study population, the number of students and/or teachers in the study, design framework of the study, real-life context of mathematical modeling activities used in the study, major findings, implications, and future recommendations. A spreadsheet was created to include data from the articles for each of the characteristics. We then individually coded a few randomly selected articles on the spreadsheet and met to resolve discrepancies. After reaching a consensus on how to code each article using the characteristics, we individually coded the rest of the articles and met to discuss the best approach to organize the patterns of coded data in a Word document. After the discussion, the second author documented the patterns; for example, she distinguished whether each study focused on student learning or teacher learning and made a table listing author references as documentation of the type of learning present in each. The first author integrated the mathematical modeling activities with relevant findings and distinguished each article by the discipline on which it focused (e.g., engineering, science). Based on the patterns shown in the spreadsheet and the Word document, the first author finalized the common themes and created a summary document for each study. A draft result was developed by the first author and verified by the second author. Table 2.1  Number of papers included in the synthesis

Research journals TCM

2000– 2002 1

2003– 2005 5

2006– 2008 3

2009– 2011 1

2012– 2014 1

2015– 2017 1

Total 12

0

2

4

0

3

7

16

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Regarding the 12 selected articles from the research journals (five in MTL, three in ESM, two in JMTE, one in JRME, one in ZDM, and zero in JMB and FLM), four common themes of these studies were identified as (a) young learners’ progress in developing mathematical models, (b) features of mathematical modeling activities that supported student learning, (c) teachers’ development through the implementation of mathematical modeling, and (d) features of mathematical modeling activities that supported teacher learning. In analyzing the 16 selected articles published in TCM, three additional themes became apparent, namely, (e) engineering-based mathematical modeling, (f) science-based mathematical modeling, and (g) socially and personally relevant mathematical modeling. Although these seven themes are not the only way to classify these studies, they appeared multiple times in the selected articles and aided with the organization of our results. Regarding the research articles, alternative themes—such as the view of modeling, mathematical modeling competency, or data collection methods—could be used to structure our paper. We avoided these alternative themes because previous studies had already organized mathematical modeling research articles using them (English et al., 2016; Kaiser & Brand, 2015; Stohlmann & Albarracín, 2016). We separately identified themes for the articles from TCM because their nature and scope differed from those of the seven research journals. TCM articles are often centered around a newly created activity and its implementation. We thought analyzing the modeling activities with a disciplinary lens (e.g., engineering-based, science-based) would enable us to identify gaps in the types of mathematical modeling activities that have been recently published by the practitioner journal.

2.4 Results We aimed to provide an overview of the manuscripts published in the seven research journals and practitioner journal during the period of 2000 and 2017 which focused on mathematical modeling in Grades K-8. This section features the summary of the manuscripts around the previously identified common themes, so readers can quickly ascertain the necessary resources and follow-up readings for early mathematical modeling from these leading journals in mathematics education.

2.4.1 Summary of Studies Published in Research Journals Young Learners’ Progress in Developing Mathematical Models  In the field of mathematics education, the word “model” has diverse meanings. Major uses of the word refer to models as demonstrations, manipulatives, or frameworks for learning or teaching (CMA & SIAM, 2016). However, these uses are different from the use of mathematical models developed by students when engaging in mathematical modeling activities. A mathematical model refers to “purposeful mathematical

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descriptions of situations” (Lesh & Lehrer, 2003, p. 109) or “an instrument, like a microscope or a telescope, allowing us to see things previously hidden, and a predictive tool” (Pelesko, 2014, p.  150). Although comparatively little opportunities have been provided to younger students to develop mathematical models in the past, research shows that elementary-grade students are capable of developing mathematical models and improve their abilities to do so (English, 2012). In our review, five studies in JRME, MTL, ESM, and ZDM focused on the progress of young learners’ mathematical model development as they engaged with mathematical modeling. The first study was conducted by Lesh and Harel (2003), who examined students’ progress of solving mathematical modeling problems, especially those who previously showed low academic performance. When the authors identified similarities and differences across video transcripts of students engaging in four modeling activities, they found prominent trends. Students’ processes of developing models were somewhat aligned with corresponding stages of human development, as in Piaget’s and Van Hiele’s theories of conceptual development. At the same time, a single student’s stage of development often varied across activities and across mathematical modeling cycles for a particular activity. Overall, students’ conceptual tools were gradually extended and refined when they were encouraged to develop their own models based on personal experiences, introduced to representational systems for connecting relevant models, and challenged to go beyond developing sharable and reusable tools. Contrary to the ladderlike stages of conceptual development, the analysis of the transcripts showed that students often alternated between various interpretations of a problem-solving situation. That is, regardless of the similarities between the theoretical stages of development and the stages of students’ conceptual development during mathematical modeling activities, it was too simplistic to identify a student as belonging to a certain stage in the development theories. This research found that most of the models developed by students are in transitional stages and students’ apparent stages of development varied across time within a particular activity and across activities. Lesh and Lehrer (2003) also emphasized that mathematical modeling activities led to children’s development of mathematical models, including students who were previously considered incapable of engaging in such activities. Seventh-grade students, for example, were tasked with developing a procedure to assess paper airplanes for flight characteristics using each flight’s distance flown, landing distance from the target, and duration. The analysis of the lesson transcript showed that students were involved in iterative modeling cycles during which their initial solutions were tested and revised. Students also used important mathematical ideas and skills often not addressed in textbooks or tests, including multimedia representational fluency and skills needed to plan and monitor processes. Since this type of activity required a broad scope of ideas and skills, it often allowed diverse students to succeed. To compare students’ model development from different types of activities, Mousoulides, Christou, and Sriraman (2008) examined students’ models and the progression of students’ problem solving over time. One experimental group (194 sixth and eighth graders) and one control group (209 sixth and eighth graders)

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participated in this research. For 3 months, the experimental group engaged with six mathematical modeling activities, while the control group engaged with their regular mathematics curricula. Both groups completed a modeling ability test three times. Analysis of the tests showed that the experimental group significantly outperformed the control group for both sixth- and eighth-grade students. The authors also found that the mathematical modeling activities supported all students’ learning over time, especially students who previously showed low performance. A reason for this result was identified: the social interactions in the classroom created a safe environment for students to present and discuss their ideas and improve their performance in solving relevant problems, even without direct instruction. Related to data modeling, English (2012) focused on young children’s opportunities to engage with data. In the study, first-grade students in three classes selected attributes of interest, re-represented their collected data, and considered informal inferences when solving mathematical modeling problems. Students showed progress in generating new attributes and representations by focusing their attention from one feature to another. With respect to the students’ re-representations of data, students displayed distinct results, including pictorial representations, written text, and symbols. Students also drew on their informal inference skills, such as considering variations and predicting the solution based on the data and problem context. The results demonstrated the promising learning outcomes of young children through data modeling with realistic problem situations. Lehrer and Schauble (2000) also described the progress of students’ learning with data and classification. In this study, students developed a model to categorize a set of students’ drawings by grade level, using only the characteristics of these drawings as indicators of the student artists’ grades. Students in first, second, fourth, and fifth grades collaboratively categorized drawings made by other students and mathematized their categorization procedures. Except for a few students, most first and second graders demonstrated evolving lists of grade-level attributes but failed to use these systems to guide classification. In contrast, fourth and fifth graders made greater progress in developing data structures. They negotiated and evaluated categories by combining and revising attributes and treated their category systems as models to justify their choices of grade level for each artist. Some of the fifth graders even considered weighing diagnostic attributes in their solution methods. These studies reveal patterns in students’ progress in mathematical model development. Students’ models were continuously refined and revised during mathematical modeling activities (English, 2012; Lesh & Harel, 2003; Lesh & Lehrer, 2003), and therefore, it is too simplistic to define students’ modeling development with the ladderlike stages of conceptual development. (Lesh & Harel, 2003). Another pattern across the studies was young learners’ abilities to develop mathematical models (English, 2012; Lehrer & Schauble, 2000; Lesh & Harel, 2003; Lesh & Lehrer, 2003; Mousoulides et al., 2008), especially students who previously showed low academic performance (Lesh & Harel, 2003; Lesh & Lehrer, 2003; Mousoulides et al., 2008). Features of Mathematical Modeling Activities That Supported Student Learning  While the previous theme focuses on the evolution of learners’ devel-

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oped models, this theme emphasizes the characteristics of mathematical modeling activities that promoted students’ learning. Studies in JRME, MTL, ESM, and ZDM emphasized several features of mathematical modeling activities that supported student learning in different ways. English (2009), for example, discussed the interdisciplinary nature of mathematical modeling activities and provided examples of students’ models in the domains of science and literacy. As they completed the activities, students interpreted the problem context and data sets, quantified the data, and documented their processes using a multitude of representations. The author further detailed two opportunities—one which addressed society and environmental issues and one which involved the study of engineering. Because of their interdisciplinary potential, modeling problems provide a rich basis for producing research projects within the regular curriculum without overloading an already-packed curriculum. Such problems provide enriching learning experiences for students in diverse classrooms. Beneficial features of a sequence of mathematical modeling activities, specifically model-eliciting activities (MEAs), were discussed by Doerr and English (2003). The authors focused on the development of one group of students’ mathematical reasoning across one task sequence and a macrolevel analysis of patterns in student thinking for two task sequences. Based on their study, several features of MEAs were found to promote middle school students’ development of ranked quantities, transformations on their ranks, and understanding of relationships between quantities. The features included the activity’s ability to elicit students’ systems of generalizable and reusable relationships. Further, the activity required students to engage in iterative cycles of interpreting the problem context to find connections among quantities. English (2006) displayed other valuable characteristics of MEAs. The activities in her study fostered an environment where students could develop their own mathematical constructs. While creating a consumer guide to choose the best snack chip, for example, students partook in cycles of sharing ideas, outlining relevant factors, and revising their created models, all without a teacher’s direct instruction. This characteristic of the MEA—that students develop their own models—allowed for various solution pathways. Doerr and English (2003) further found that MEAs allowed students to supply their diverse thought processes as they reached conclusions. Due to MEAs’ authenticity, students devised multiple procedures for usefully interpreting data as they made decisions about realistic problem situations. The inherent opportunities for students to communicate their mathematical thinking constituted another effective feature of MEAs. Doerr and English (2003) observed the ways in which students explained how their initial models led to revisions of key mathematical ideas. English (2006) also discussed that students communicated throughout the process of sharing their different representations, testing models with group members, and reporting their final verbal and written solutions with the whole class. This communication allows for formative assessment of student learning (English, 2006). Additionally, students assessed their own work as they communicated with group members and reported results to the class. Teachers and researchers were also able to formatively assess students’ verbal and written

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reports. They gained insights into the students’ thinking and, ultimately, continued to test and revise these insights. This opportunity for formative assessment provided a platform for examining how students’ mathematical constructs can be increasingly fostered and refined. In sum, the activities that best supported student learning required students to develop generalizable and reusable relationships among quantities, provided opportunities for students to develop their own models, and expected students to communicate throughout the problem-solving and revision process. MEAs used in these studies were designed to incorporate these features (English, 2006; Doerr & English, 2003). In addition, the rich contexts of these mathematical modeling activities allow for the development of diverse interdisciplinary projects that can be integrated into existing curricula (English 2009). Teachers’ Development Through the Implementation of Mathematical Modeling  Research published in the MTL, ESM, and JMTE presented ways in which teachers developed when they implemented mathematical modeling activities in their classrooms. Based on the multitiered teaching experiment (Lesh & Kelly, 2000), English (2003) documented fifth- through seventh-grade students’ engagement with two modeling activities that involved students’ development of mathematical constructs like data relationships, proportional reasoning, the notion of rate, and ranking systems. Her research focused on how classroom teachers and preservice teachers identified the features of student learning as they implemented these two tasks. As their students interacted with these tasks, the teachers were able to identify features of student learning and aspects of the tasks which supported this learning. These observations included a recognition that (a) the authentic characteristics of the mathematical modeling problem supported students’ mathematical learning, (b) students’ perspectives on problem solving had changed throughout the sequence of tasks, (c) students who didn’t normally demonstrate high mathematical achievement were able to provide important insights to their teams, and (d) students’ autonomous and diverse thinking was notable and encouraged. This research reveals how the design of research-teaching experiences can maximize the learning of teachers while positively influencing their students’ learning. Schorr and Koellner-Clark (2003) had also documented that teachers developed their knowledge of teaching mathematics through the implementation of mathematical modeling in their classrooms. Twelve middle school teachers had participated in a 4-month mathematical modeling project in which they considered their own approaches to teaching, made predictions about student learning during planning, tested these predictions during implementations, and collaboratively reflected on teaching. The research focused on one teacher’s fundamental changes in his practice. This teacher made the shift from a traditional teaching methodology to a reform-based approach. In the beginning of the study, he did not use students’ thinking to spark discussion, nor did he help students make connections to additional mathematical constructs. Throughout the project, he began to look for students’ different solutions, have students explain their thinking to other students, and use their diverse thinking to make instructional decisions. He also made the transition

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from focusing whole group discussions on procedures to concentrating on larger conceptual mathematical ideas. To do this, he revised all aspects of his teaching, including his goals, lesson plans, chosen problems, and the degree to which he utilized students’ input. He noticed that as he modified his pedagogical approach, his students also developed the ability to debate mathematical ideas and build on one another’s ideas. The authors argued that these adjustments were possible due to a critical feature of the multitiered teaching experiment: the opportunity for teachers to describe, implement, and refine their approaches to the teaching of mathematics. Another example of teacher learning was described by Jung and Brady (2016). The authors used the multitiered teaching experiment as a way to maximize the authentic partnership between a researcher and teachers. During in situ professional development, the researcher worked closely with the teachers in their classrooms to co-plan, co-teach, and reflect on three MEAs. The research focused on the roles of both the researcher and teachers in achieving an equitable relationship and detailed how their collaborations resulted in teacher change. Topics of the researcher-teacher meetings had been determined with consideration of the diversity of students, classroom environments, and teachers’ input from previous meetings. Teachers had the opportunity to modify mathematical modeling lessons to tailor them to their students. Further, teachers developed assessment tools that could be shared with other teachers. One of the teachers’ reflections featured factors that influenced student learning: (a) the iterative process of mathematical modeling implementations for both teachers and students, (b) changes in the teacher’s perspectives of problem solving which molded her actions, and (c) changes in her students’ views toward collaborative learning. The authors contrasted their research to traditional research, noting that traditional research often treated teachers as the research objective, but the multitiered teaching experiment invited teachers to act as integral collaborators in the research process. These three studies (English, 2003; Jung & Brady, 2016; Schorr & Koellner-­ Clark, 2003) used multitiered teaching experiments (Lesh & Kelly, 2000), which involved students, teachers, and researchers in the classrooms. At the student level, students collaboratively engaged with rich mathematical problems. They explained, represented, and refined their mathematical models, which were shared with all other participants. At the classroom teacher level, teachers worked collaboratively with researchers to plan, implement, and reflect on student experiences. Teachers then explained and refined their interpretations of student learning. At the researcher level, researchers collaborated with preservice teachers and classroom teachers to plan, implement, and reflect on the learning experiences of students and teachers. They also developed models to describe the knowledge development of all participants. Such collaboration empowered teacher learning as they observed student changes when engaging with mathematical modeling activities. Features of Mathematical Modeling Activities That Supported Teacher Learning  Studies published in MTL and JMTE highlighted the features of mathematical modeling activities, especially MEAs, that supported teacher learning. Doerr and English (2006) used a sequence of five mathematical modeling activities

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as a means of examining the nature of teacher learning. The authors specifically focused on teachers’ subject matter knowledge, pedagogical content knowledge, knowledge of supporting student learning, and the characteristics of the activities that promoted these types of teacher knowledge. Teachers developed mathematical content knowledge as they observed students’ sensemaking of the problems’ content. For example, a teacher’s uncertainty and her original focus on mathematical computation transformed into conceptual understanding as she prepared for the lesson and observed students present their solutions. Another teacher gained new pedagogical content knowledge regarding the range of potential solution approaches. She used her newfound knowledge to facilitate a classroom discussion on the similarities and differences among students’ solution procedures. The main features of the mathematical modeling activity that promoted such teacher development concerned self-evaluation and model documentations. The activity offered the opportunity for students to self-evaluate the reasonableness of their solution approaches, which allowed teachers to analyze student understanding rather than their work. The activity also required students to develop their own representations, which exhibited new solution approaches for teachers to adopt. MEAs granted room for teachers to entertain new roles in the teaching practice as they learned about tasks and interacted with their students. Lesh and Lehrer (2003) also proposed that the thought-revealing feature of mathematical modeling activities offered teachers the opportunity to develop tools that other teachers and researchers could use to understand student learning. One example was an observation checklist that teachers could use to gather information about student progress during small group activities. Also, documentation tools would help teachers identify strengths and weaknesses of models developed by students. The authors provided assessment guides as an example of tools to evaluate the quality of alternative student models. As shown in these two studies (Doerr & English, 2006; Lesh & Lehrer, 2003), MEAs allowed for students’ diverse thinking, which led to learning opportunities for teachers who observed students’ mathematical representations that were different from their own. Because of the opportunities for students to self-evaluate when engaging with MEAs, teachers were able to focus on students’ diverse thinking instead of correcting their work. Moreover, the use of observation checklists, documentation tools, or assessment guides (Lesh & Lehrer, 2003) provides opportunities for other teachers and researchers to understand student learning.

2.4.2 Summary of Studies Published in TCM Engineering-Based Mathematical Modeling  Four articles from TCM described mathematical modeling activities with an engineering context. These activities tasked students with comparing three insulation packages or designing a paper airplane, building, or clubhouse. English, King, Hudson, and Dawes (2014) provided

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an opportunity for fourth graders to design, construct, test, and reconstruct a paper plane that would stay in the air for the longest period of time. Students applied their learning of measurement, geometry, force, and technology to solve the real-world problem. With the same engineering design model that involved the process of brainstorming, designing, building, testing, redesigning, and rebuilding, English and King (2016) introduced an activity, Designing an Earthquake-Resistant Building, which encouraged sixth graders to construct a structure that would remain steady during an imitated earthquake. Students showed their ability to use multiple approaches and solutions while applying their geometric and scientific learning. Another task related to civil engineering was designing a “Dream Clubhouse” (Suh, Moyer, & Sterling, 2003). The authors engaged third graders in designing their clubhouses as they explored attributes of 2D and 3D shapes and considered budget constraints. Students developed problem-solving strategies and procedural fluency by measuring geometric shapes, identifying the perimeter and area of constructs, and planning the cost of the design. Yanik and Memis (2015) also highlighted the context of building, engaging fifth and sixth graders with comparing three insulation packages, and determining which option would be most economical for a client. The students discussed how insulation affects heating and cooling a building, as well as the electricity bill. They used mathematics to make decisions as they attempted to reduce costs over time. These learning opportunities meet the growing demand for fostering within students the skills of engineering and show a potential to successfully incorporate engineering-based mathematical modeling activities in elementary mathematics curricula. Science-Based Mathematical Modeling  Seven articles incorporated science-­ based mathematical modeling activities. Contexts for these articles involved physics, encryption, ergonomics, or environmental science. Focusing on physics, Yanik and Karabas (2014) asked fifth graders to develop a user guide for a lever. Upon encountering the tool for the first time, students explored the relationships between balance, distance, and weight. Through this activity, students had opportunities to develop, validate, and assess mathematical models that they constructed to share with a new user of the tool. Similarly, Durmus and Karabork (2016) introduced fifth and sixth graders to Caesar’s encryption method, one of the foundations of computer science that focuses on changing data into secure formats as protection from unauthorized users. The students designed a new encryption device by integrating the fundamental concepts of encryption and decryption, testing the accuracy of their conceptions, and revising their original ideas. In another manuscript, Orona, Carter, and Kindall (2017) aimed to develop second graders’ basic awareness of ­ergonomics. The students’ teacher introduced the problem by saying that a giant had visited the classroom the previous night and left handprints behind. The students were subsequently tasked with designing a 2D model of the giant’s head with dimensions that corresponded with those of his handprints. Students explored the proportional relationships between the sizes of human body parts and critically considered key mathematical concepts which supported continued discussion of proportional relationships and 3D measurement.

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Four of the science-based articles involved a real-life situation related to environmental science. Poth (2006) described the “Bird Station Investigation,” in which third graders aimed to answer the question, “Does the height of a bird feeder affect the amount of birdseed that wildlife eats?” (p. 175). The students made predictions, planned and completed their investigation over the course of 4 weeks and finally reflected on their measurements and graphs to reach a conclusion. Throughout the investigation, students used mathematical skills such as metric measurements, considered the meaning of operations and estimations in context, and employed patterns, tables, and graphs to assist with data interpretation. They also explored the patterns of natural phenomena and made connections between mathematics and science. In another manuscript, Wickstrom, Nelson, and Chumbley (2015) engaged second and third graders with constructing a garden for Earth Day. Students conceptualized the ideas of area and perimeter beyond formulaic reasoning by varying the dimensions of a rectangular garden using the same amount of fencing and exploring the relationship between each new layout and the number of possible garden plots. Students predicted, measured, drew, and built their own gardens and discussed a series of related questions about garden structure. The activity allowed for 3 days of captivating instruction while also supporting student understanding of spatial reasoning. With another environmental context, Isabelle and Bell (2007) challenged fourth, fifth, and sixth graders by asking them to measure the surface area of leaves to determine the average collective surface area of a tree’s leaves. They used various estimation skills, including leaf symmetry, groups of squared centimeters, and the average of small, medium, and large leaves’ surface areas from one tree. Additionally, the students learned that a larger surface area of leaves would collect a larger amount of solar energy. They also developed problem-solving and communication skills and made connections between various mathematical ideas and science. Lastly, English, Fox, and Watters (2005) introduced Australia’s cyclone disasters to fourth and fifth graders. The students were commissioned with determining which geographical locations to avoid when building vacation resorts. To complete the task, students considered data on cyclones that affected Western Australia over the previous 12 years. Students communicated with their group members as they brainstormed ideas, made assumptions and predictions, and developed and refined their own models. The activity allowed students to build collaborative learning skills, as well as their problem-solving and problem-posing abilities. Socially and Personally Relevant Mathematical Modeling  While the activities introduced above involved relevant contexts for students, manuscripts identified under this theme concerned a real-life problem that was socially or personally relevant to students’ current interests. For example, the context of the problem involved a topic of student interest, such as an amusement park or events happening in students’ classrooms or communities. Bleiler-Baxter, Stephens, Baxter, and Barlow (2017) presented the “Theme Park” task to teachers and introduced the context of Universal Studios Park as an example of Standards for Mathematical Practice 4: Model with Mathematics (NGA &

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CCSSO, 2010). The task required students to establish the best model for determining missing wait times for the amusement park rides using already-listed wait times. Students identified important quantities, made assumptions and approximations, analyzed and mapped relationships across the consecutive wait times in the data table, and reflected on their results. Through this task, students improved their problem-­solving abilities and made connections to the real world. In another article, English (2008) discussed how mathematical modeling activities can be used to introduce complex systems to young children. The author presented two activities, both of which were situated in the students’ community. One activity involved an upcoming sporting event, and the other discussed a gardening service. The first activity asked students to choose a swimming team for the next Olympic games given swimmers’ data from the last ten competitions. The second activity involved selecting four people to be hired for a lawnmowing service based on the workers’ data collected from the past summer. These activities allowed students to develop, test, and refine shareable models. Further, students formed conceptual connections across multiple disciplines. English (2008) noted that the complex systems embedded in the activity should reflect a real-world situation and connect to students’ prior knowledge and experiences. Both activities were tailored to relate to students’ lives; in fact, the Olympic games and lawnmowing service activities were created around the students’ curricular themes. Lomax, Alfonzo, Dietz, Kleyman, and Kazemi (2017) discussed three-act tasks (Meyer, 2011) as activities which can engage second-grade students in mathematical modeling. The teacher recorded a video of her daughter, who often visited the classroom, approaching a pile of crayon boxes. As she took crayons from one of the boxes and then from another, the second graders were inspired to pose mathematical questions. Students shared their solution strategies and used a variety of representations to answer their own questions. The task enabled students to share their interests, explore mathematical concepts, and experience a broader set of entry points to learning—specifically, through videotaped action and sound. English (2013) engaged first graders with data modeling, presenting three relevant mathematical modeling activities. For the first activity, students received a set of familiarly shaped cutouts, such as the silhouettes of a dog biscuit and a toy teddy bear. They were asked to collaboratively classify these cutouts using any attributes they deemed appropriate. For the second activity, students were introduced to a storybook about a character who cleans up a dirty town. Then, given a collection of waste and reusable items, students sorted them using different representations, such as drawings and graphs. The third activity allowed students to read another storybook about a creature living in a pile of trash. This story acted as a segue for introducing students to tables which listed items that the “Litter Police” collected daily. Students then predicted missing data in these tables using variations in the previously listed data. These rich activities motivated students and scaffolded students’ appreciation of statistics. Unlike the shorter lessons published in TCM, Turner and Strawhun (2007) introduced a 5-week mathematics unit that increased student engagement and positively altered students’ attitudes toward mathematics. A sixth-grade teacher implored

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students to brainstorm and share their concerns about their school or community. One of these concerns was the overcrowding of the school. As students calculated the current area of their school’s spaces and collected and analyzed data from another school for comparison, they learned about key mathematical ideas, including area measurement and operations with rational numbers.

2.5 Discussion and Future Direction Although previous research has demonstrated that young learners are capable of engaging with mathematical modeling (e.g., Bonotto, 2009; Mousoulides & English, 2008; Mousoulides et al., 2006), comparatively few research studies have focused on younger students’ learning of mathematical modeling or their learning of mathematics through mathematical modeling. This chapter aimed to provide an overview of the research and practitioners’ recommendations presented in journals highly regarded by mathematics education researchers, with a particular focus on the teaching and learning of mathematical modeling in the early grades. To deepen our understanding of young students’ opportunities to learn about mathematical modeling, this chapter provided a synthesis of past research published in eight selected journals from 2000 to 2017. Based on this synthesis, we conclude the chapter by discussing a few possibilities for future research. In reviewing the articles published in research journals (i.e., ESM, JRME, MTL, JMTE, and ZDM) and TCM, patterns emerged with respect to the frameworks of the articles, the mathematical topics presented, and the number of publications in these two types of journals. From the research journals, several papers (English, 2003, 2016; Jung & Brady, 2016; Lesh & Harel, 2003; Lesh & Lehrer, 2003; Schorr & Koellner-Clark, 2003) specifically adopted models and modeling perspectives (MMPs) that consequently shifted students’ attention beyond computing and instead toward interpreting real-world situations mathematically (Lesh & Doerr, 2003). Along with the MMPs, many studies engaged students with MEAs (English, 2006, 2009, Doerr & English, 2003, 2006; Lesh & Harel, 2003; Lesh & Lehrer, 2003) and multitiered teaching experiments (English, 2003; Jung & Brady, 2016; Schorr & Koellner-Clark, 2003). Results of these studies show the potential of using these perspectives, activities, and frameworks in research to support young learners’ modeling development. In addition, significantly more research articles were published in the 2000s than in the 2010s. Many of these articles addressed young learners’ modeling related to interpreting and representing data. For example, the real-life contexts incorporated in the mathematical modeling activities of these articles included determining the most important factors for purchasing shoes (e.g., English, 2003; Doerr & English, 2006), organizing attributes of student drawings (e.g., Lehrer & Schauble, 2000), ranking restaurant preferences (Doerr & English, 2003), ranking drugs, and choosing a city or workers based on data (e.g., Mousoulides et  al., 2008). Only a few research studies incorporated mathematical modeling activities with other

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mathematical topics, such as scales, measurement, and symmetry through the production of a quilt template (Lesh & Harel, 2003) and developing geometric reasoning through the design of a model of a new house or painting a car (Mousoulides et al., 2008). On the other hand, more TCM articles had been published after 2010 than in the 2000s. Many of them included activities which emphasized more diverse mathematical topics than those included in the research articles, such as subtraction (Lomax et al., 2017), area and perimeter (Wickstrom et al., 2015), and Caesar’s encryption method (Durmus and Karabork, 2016). These recently published mathematical modeling activities could be used with more students and teachers and have potential for future publication in research journals. Furthermore, mathematical modeling activities which address broader mathematical topics should continue to be developed and employed with young students to more thoroughly document the impacts of implementing mathematical modeling activities in mathematics classrooms. Recommendations for future studies have been highlighted by articles published in research journals. More research related to students’ modeling behavior is needed by trialing diverse types of mathematical modeling activities (Mousoulides et al., 2008). In addition, the data modeling competencies of young learners require more attention in early mathematics curricula, as well as research examining its development (English, 2012). Based on the analysis of TCM, more articles discussed young students’ learning through science-based activities, especially environmental science, compared to other topics, such as engineering or social studies. Most engineering-­based activities published in TCM incorporated a civil engineering context related to designing buildings. Hence, future research is required to investigate young students’ learning of mathematical modeling through diverse interdisciplinary contexts. Several studies in our investigation incorporated the use of technology to introduce the problem context or the data collection and operations. Lomax et al. (2017), for example, featured a short video developed by the classroom teacher to engage and encourage students to pose context-based mathematical questions. This example shows that multimedia can provide diverse entry points into a problem, instead of solely introducing the problem context with text. In another article, Turner and Strawhun (2007) had students brainstorm concerns about their school and community, in addition to collecting data from another school for comparison. Without pre-collected data, students gained data collection experience, thereby producing a more authentic problem. By utilizing technology, many aspects of the mathematical modeling process—including the presentation of real-life context, data collection and analysis, and operations—can be displayed in more novel and engaging ways. While research has discussed the critical role of technology in mathematical modeling (e.g., Hamilton, Lesh, Lester, & Brilleslyper, 2008; Lagrange & Hoyles, 2009), few research studies published in the journals of our investigation highlighted the significant role of technology in mathematical modeling for young learners. Overall, our analysis indicated that further effort is essential to increasing the visibility of mathematical modeling in the early grades in research journals, as there were few publications on early modeling in the leading mathematics education

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journals from 2000 to 2017. Regarding research methods, all the contributions were qualitative studies, with the exception of one article (e.g., Mousoulides et al., 2008). This shortage of quantitative research methods in mathematical modeling research is also discussed by Schukajlow et al. (2018), who surveyed mathematical modeling articles focused on all grade levels from 2012 to 2017. Our analysis found that the lack of quantitative research is especially apparent in early mathematical modeling studies. Given that the manuscripts reviewed for this chapter demonstrated young learners’ capabilities of engaging with mathematical modeling, especially with those who previously showed low academic performance (e.g., Lesh & Harel, 2003; Lesh & Lehrer, 2003; Mousoulides et al., 2008) and in settings with support from researcher-teacher collaborations (e.g., English, 2003; Jung & Brady, 2016; Schorr & Koellner-Clark, 2003), more research conducted with young learners and teachers is promising and encouraged in the field of mathematics education.

References *Bleiler-Baxter, S.  K., Stephens, D.  C., Baxter, W.  A., & Barlow, A.  T. (2017). Modeling as a decision-making process. Teaching Children Mathematics, 24(1), 20–28. Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects – State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68. Bonotto, C. (2009). Artifacts: Influencing practice and supporting problems posing in the mathematics classroom. Proceedings of PME, 33(2), 193–200. Consortium for Mathematics and Its Applications & Society for Industrial and Applied Mathematics. (2016). Guidelines for assessment and instruction in mathematical modeling Education. Retrieved from http://www.siam.org/reports/gaimme.php Cooper, H. (2010). Research synthesis and meta-analysis: A step-by-step approach (4th ed.). Thousand Oaks, CA: Sage. Czocher, J. A. (2016). Introducing modeling transition diagrams as a tool to connect mathematical modeling to mathematical thinking. Mathematical Thinking and Learning, 18(2), 77–106. *Doerr, H. M., & English, L. D. (2003). A modeling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 1, 10–136. *Doerr, H. M., & English, L. D. (2006). Middle grade teachers’ learning through students’ engagement with modeling tasks. Journal of Mathematics Teacher Education, 9(1), 5–32. *Durmus, S., & Karabork, M. A. (2016). Redesigning Caesar’s wheel with a modeling perspective. Teaching Children Mathematics, 23(4), 252–257. *English, L. D. (2003). Reconciling theory, research, and practice: A models and modelling perspective. Educational Studies in Mathematics, 54(2–3), 225–248. *English, L. D. (2006). Mathematical modeling in the primary school: Children’s construction of a consumer guide. Educational Studies in Mathematics, 63(3), 303–323. *English, L. D. (2008). Introducing complex systems into the mathematics curriculum. Teaching Children Mathematics, 15(1), 38–47. *English, L.  D. (2009). Promoting interdisciplinarity through mathematical modelling. ZDM, 41(1–2), 161–181. *English, L.  D. (2012). Data modelling with first-grade students. Educational Studies in Mathematics, 81(1), 15–30. *English, L. D. (2013). Surviving an avalanche of data. Teaching Children Mathematics, 19(6), 364–372.

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English, L. D., Ärlebäck, J. B., & Mousoulides, N. (2016). Reflections on progress in mathematical modelling research. In Á. Gutiérrez, G.  C. Leder, & P.  Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 383–413). Rotterdam, The Netherlands: Sense. *English, L. D., Fox, J. L., & Watters, J. J. (2005). Problem posing and solving with mathematical modeling. Teaching Children Mathematics, 12(3), 156–163. *English, L.  D., & King, D.  T. (2016). Designing an earthquake-resistant building. Teaching Children Mathematics, 23(1), 47–50. *English, L. D., King, D. T., Hudson, P., & Dawes, L. (2014). The aerospace engineering challenge. Teaching Children Mathematics, 21(2), 122–126. English, L.  D., & Watters, J.  J. (2005). Mathematical modeling in third-grade classrooms. Mathematics Education Research Journal, 16, 59–80. Frejd, P. (2013). Modes of modelling assessment. A literature review. Educational Studies in Mathematics, 84(3), 413–438. Galbraith, P. L., Blum, W., Booker, G., & Huntley, I. D. (1998). Mathematical modeling: Teaching and assessment in a technology-rich world. West Sussex, UK: Horwood Publishing Ltd.. Gann, C., Avineri, T., Graves, J., Hernandez, M., & Teague, D. (2016). Moving students from remembering to thinking: The power of mathematical modeling. In C.  R. Hirsch & A.  R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp.  97–106). Reston, VA: The National Council of Teachers of Mathematics. Geiger, V., & Frejd, P. (2015). A reflection on mathematical modelling and applications as a field of research: Theoretical orientation and diversity. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research and practice (pp.  161–171). Cham, Switzerland: Springer. Hamilton, E., Lesh, R., Lester, F., & Brilleslyper, M. (2008). Model-eliciting activities (MEAs) as a bridge between engineering education research and mathematics education research. Advances in Engineering Education, 1(2), 1–25. *Isabelle, A., & Bell, K. (2007). Sun catchers. Teaching Children Mathematics, 13(8), 414–423. *Jung, H., & Brady, C. (2016). Roles of a teacher and researcher during in situ professional development around the implementation of mathematical modeling tasks. Journal of Mathematics Teacher Education, 19(2–3), 277–295. Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research and practice. Cultural, social and cognitive influences (pp. 129–149). Cham, Switzerland: Springer. Lagrange, J.-B., & Hoyles, C. (Eds.). (2009). Mathematical education and digital technologies: Rethinking the terrain. New York, NY: Springer. *Lehrer, R., & Schauble, L. (2000). Inventing data structures for representational purposes: Elementary grade students’ classification models. Mathematical Thinking and Learning, 2(1–2), 51–74. Lesh, R., & Doerr, H. (Eds.). (2003). Beyond constructivism: A models and modeling perspective. Mahwah, NJ: Lawrence Erlbaum Associates. *Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking and Learning, 5(2–3), 157–189. Lesh, R., & Kelly, A. (2000). Multitiered teaching experiments. In A.  Kelly & R.  Lesh (Eds.), Research design in mathematics and science education (pp. 197–230). Mahwah, NJ: Lawrence Erlbaum Associates. *Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5(2–3), 109–129. *Lomax, K., Alfonzo, K., Dietz, S., Kleyman, E., & Kazemi, E. (2017). Trying three-act tasks with primary students. Teaching Children Mathematics, 24(2), 112–119. Meyer, D. (2011). The three acts of a mathematical story. Tasks posted to http://blog.mrmeyer. com/2011/the-­three-­acts-­of-­a-­mathematical-­story/

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Mousoulides, N., & English, L. D. (2008). Modeling with data in Cypriot and Australian primary classrooms. Proceedings of PME 32 and PME-NA, 34(3), 423–430. Mousoulides, N., Pittalis, M., & Christou, C. (2006). Improving mathematical knowledge through modelling in elementary schools. Proceedings of PME, 30(4), 201–208. *Mousoulides, N. G., Christou, C., & Sriraman, B. (2008). A modeling perspective on the teaching and learning of mathematical problem solving. Mathematical Thinking and Learning, 10(3), 293–304 National Governors Association Center for Best Practices and Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: NGA Center and CCSSO. Nivens, R. A., & Otten, S. (2017). Assessing journal quality in mathematics education. Journal for Research in Mathematics Education, 48(4), 348–368. *Orona, C., Carter, V., & Kindall, H. (2017). Understanding standard units of measure. Teaching Children Mathematics, 23(8), 500–503. Pelesko, L. (2014). Initial thoughts on the mathematical perspective. Proceedings of PME 38 and PMENA, 36(1), 149–152. *Poth, J. (2006). Bird Station investigation. Teaching Children Mathematics, 13(3), 174–180. *Schorr, R. Y., & Koellner-Clark, K. (2003). Using a modeling approach to analyze the ways in which teachers consider new ways to teach mathematics. Mathematical Thinking and Learning, 5(2–3), 191–210. Schukajlow, S., Kaiser, G., & Stillman, G. (2018). Empirical research on teaching and learning of mathematical modelling: A survey on the current state-of-the-art. ZDM, 1–14. Stohlmann, M. S., & Albarracín, L. (2016). What is known about elementary grades mathematical modelling. Educational Research International. https://doi.org/10.1155/2016/5240683 *Suh, J. M., Moyer, P. S., & Sterling, D. R. (2003). Junior architects: Designing your dream clubhouse using measurement and geometry. Teaching Children Mathematics, 10(3), 170–180. Toerner, G., & Arzarello, F. (2012, December). Grading mathematics education research journals. Newsletter of the European Mathematical Society, 86, 52–54. *Turner, E. E., & Font Strawhun, B. T. (2007). Posing problems that matter: Investigating school overcrowding. Teaching Children Mathematics, 13(9), 457–463. *Wickstrom, M.  H., Nelson, J., & Chumbley, J. (2015). Area conceptions sprout on earth day. Teaching Children Mathematics, 21(8), 466–474. Williams, S. R., & Leatham, K. R. (2017). Journal quality in mathematics education. Journal for Research in Mathematics Education, 48(4), 369–396. *Yanik, H. B., & Karabas, C. (2014). Promoting fifth graders’ mathematical modeling. Teaching Children Mathematics, 20(7), 458–462. *Yanik, H. B., & Memis, Y. (2015). Making insulation decisions through mathematical modeling. Teaching Children Mathematics, 21(5), 314–319. Zbiek, R. M., & Conner, A. (2006). Beyond motivation: Exploring mathematical modeling as a context for deepening students’ understandings of curricular mathematics. Educational Studies in Mathematics, 63(1), 89–112.

Chapter 3

Mathematical Modeling Thinking: A Construct for Developing Mathematical Modeling Proficiency Cynthia O. Anhalt, Ricardo Cortez, and Julia M. Aguirre

Mathematical modeling is in the unique position of being content that teachers at all levels are expected to teach but have little exposure to. Many studies have documented that prospective teachers without significant modeling experience face obstacles when solving open-ended modeling tasks, in part because proficiency in mathematical modeling requires time and practice to develop. We propose that documented difficulties associated with open-ended mathematical modeling tasks can be reduced by developing dispositions and ways of thinking necessary for modeling. We introduce the term mathematical modeling thinking, which is intended to promote the development of those dispositions and lay the foundation for success in modeling. This approach is analogous to algebraic thinking, in which certain dispositions such as identifying patterns and making generalizations target the development of foundational concepts for eventual success in algebra. We describe mathematical modeling thinking and discuss activities that promote it, providing guidance to teachers and teacher educators to develop such thinking, which we consider necessary for success in mathematical modeling. We highlight work with elementary prospective and practicing teachers that support and illustrate the development of mathematical modeling thinking.

C. O. Anhalt (*) The University of Arizona, Tucson, AZ, USA e-mail: [email protected] R. Cortez Tulane University, New Orleans, LA, USA e-mail: [email protected] J. M. Aguirre University of Washington, Tacoma, WA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_3

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3.1 Introduction The National Council of Teachers of Mathematics (NCTM, 1989, 2000) standards and the Common Core State Standards for Mathematics (CCSSI, 2010) have recommended that the curriculum includes mathematics to solve realistic problems. The Common Core explicitly included Model with mathematics as a standard for mathematical practice in K-12, generating intensive activity in teacher preparation programs, professional development for teachers, and research in mathematical modeling in K-12 education. Much of this activity is being developed with the evidence-based realization that most teachers have had few opportunities, if any, to experience and become proficient in mathematical modeling as learners (Anhalt & Cortez, 2016; Cai et al., 2014; Ng, 2013) and they have not been required to teach it under prior standards. This places mathematical modeling in the unique position of being content that teachers at all levels are expected to teach but have not been exposed to. Multiple studies show evidence that proficiency in mathematical modeling requires repeated experiences to develop (Blum & Borromeo-Ferri, 2016), and numerous studies have concluded that engaging in mathematical modeling without a proper foundation can lead to frustration and obstacles to successful modeling (Bleiler-Baxter, Barlow, & Stephens, 2016; Ng, 2013; Zawojewski, Lesh, & English, 2003). In this chapter we propose that a proper foundation for mathematical modeling includes specific ways of thinking that need to be developed and practiced. We view these ways of thinking as analogous to algebraic thinking, in which noticing, categorizing, and making generalizations are used in processes to solve mathematical problems (Seeley, 2004). The goal of facilitating algebraic thinking is to develop foundational concepts and skills for eventual success in algebra. Based on this analogy, we refer to mathematical modeling thinking (MMT) as a way of thinking, including dispositions and attitudes, for eventual success in modeling. Like the development of algebraic thinking, the development of mathematical modeling thinking is a process, not an event. And, like algebraic thinking, it can start in early elementary grades and will take time and many opportunities to develop (Carpenter, Franke, & Levi, 2003; Kaput, Carraher, & Blanton, 2008). In order to define MMT, it is necessary to unpack the requirements for proficiency in mathematical modeling. We base our discussion on modeling competency frameworks available in the literature (Blomhøj & Jensen, 2003; CCSSI, 2010, p.73; Maaß, 2006). A starting point that is relevant to all of K-12 is the Standards for Mathematical Practice (SMP) 4 – Model with Mathematics (CCSSI, 2010), which is one of eight SMPs describing productive dispositions and “expertise that mathematics educators at all levels should seek to develop in their students” (CCSSI, 2010, p.6). The analysis of SMP4, supplemented by elements of a comprehensive modeling competency framework (Maaß, 2006), shed light on particular ways of thinking and the type of awareness needed for modeling. These are at the core of MMT.

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Once MMT is defined and characterized, we turn to the question of what are classroom activities and individual practices that promote MMT, paying attention to ways in which these activities can make productive use of current curricula. In some cases, problem-solving tasks already developed can be extended to include a discussion on assumptions that are implicit in the solution approach or some other aspect relevant to the modeling process. In other cases, new classroom activities that promote MMT are needed. We adopt the point of view that the continuous development of increasingly sophisticated mathematical modeling reasoning can provide connections between the world around us and mathematics, which we consider one in the same. By this we mean that all aspects of the world we experience can be investigated using mathematics. The Mathematics Teaching Practices outlined by the NCTM (2014) can be organically incorporated in activities that promote MMT, and we integrate this discussion later in the chapter. We also describe activities, modeling tasks, and other types of problems that build and promote MMT and report on some of the activities that have been implemented with prospective and practicing elementary teachers as new modelers.

3.2 Unpacking the Meaning of Proficiency in Mathematical Modeling Engaging in mathematical modeling successfully can be viewed from the standpoint of modeling competency, which is defined as “someone’s insightful readiness to carry through all parts of a mathematical modelling process in a given situation” (Blomhøj & Jensen, 2003). This involves proficiency in a number of techniques, skills, and approaches as well as knowing how and when to apply them to multiple situations to be modeled. While there are several mathematical modeling competency frameworks in the literature, we focus on the work by Maaß (2006). Our goal is to use this framework only as a vehicle to identify the type of knowledge that an individual needs to be successful in mathematical modeling. A summary of Maaß’s competency framework is: (1) make assumptions, identify key variables, and construct relation between variables; (2) mathematize and simplify quantities and their relations and choose appropriate representations; (3) use mathematical knowledge to solve the problem and utilize heuristics, such as rephrasing, dividing, and varying; (4) interpret results in context and generalize solutions; (5) critically check and reflect on solution, review model and iterate, reflect on other solution approaches, and question the model; and (6) report out (communicate). Some elements of this framework, such as “make assumptions,” “review model, and iterate,” are specific to modeling. On the other hand, “choose appropriate representations” and “communicate the solution” apply more broadly to mathematics (Niss, 2003). We note that the Common Core (CCSSI, 2010), Standards for Mathematical Practice 4, Model with Mathematics, contains similar information in narrative form.

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The description implies that the modeling process is iterative and calls for proficiency in mathematical content central to modeling: SMP4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. (p.7)

SMP4 states that to model with mathematics, students must develop proficiencies summarized in Table 3.1 and gives grade-appropriate examples of what it means for students to apply mathematical concepts to solve contextual problems. The paragraph does not discuss how to develop proficiency or particularly useful ways of thinking that are needed for successful mathematical modeling. This absence opens up opportunities to introduce a set of classroom activities and practices that promote MMT. Table 3.1  A detailed look at the Common Core K-12 Standards for Mathematical Practice 4, Model with Mathematics Identify and make assumptions for a purpose. This includes identifying assumptions that are implicit in the solution of a problem as well as making explicit assumptions that will facilitate the solution of a problem. Make approximations for calculating and estimating quantities. This may involve making decisions about appropriate units, level of precision, rounding or estimation in measurement, or representing irregular shapes with common polygons. Simplify complex situations and mathematical structures. Since problems arising in everyday life are usually too complex to mathematize directly, it is necessary to make choices and assumptions that reduce the complexity of the situational context and of the resulting mathematical model. Identify, relate, and represent ideas and quantities of interest. This involves identifying factors believed to impact the solution more significantly than other factors and determine how those factors are related and how the relations may be represented in ways that help the understanding of the situation. Interpret results in context and reflect on solution. This involves contextualizing mathematical solutions and conclusions drawn from them. After interpreting mathematical conclusions in context, there is a need to determine if the solution is adequate for the purpose of the model or if there is a need to revise choices and iterate.

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3.2.1 Representation and Communication as Central to Mathematical Modeling Maaß’s (2006) mathematical modeling competency framework embodies representation and communication as central to mathematical modeling, yet the SMP4 does not explicitly focus on the underpinnings and influence of representation and communication in mathematical modeling (CCSSI, 2010). We elaborate on representation and communication because they cut across all areas of teaching and learning mathematics and have far-reaching implications for understanding mathematics in general and modeling in particular, connecting different approaches to problem solving and other aspects of MMT. Mathematical representation and communication have been identified as important resources for learning mathematics and thus underscored as process standards (NCTM, 2000) prior to the call for mathematical modeling in the Common Core (CCSSI, 2010). And more recently, the NCTM’s (2014) mathematics teaching practices, Use and connect mathematical representations and Facilitate meaningful mathematical discourse, recognize the importance of mathematical representation and communication in all areas of mathematics teaching, which includes mathematical modeling. Teachers’ and students’ use of representation to communicate mathematically allows for their conceptualization of ideas to become transparent for others to observe while engaging in problem solving. Thus, representation allows students to think critically and communicate mathematically, which are fundamental to problem solving (Huinker, 2015). Similarly, communicating in, with, and about mathematics is critical in all areas of mathematics, and especially important in mathematical modeling, since throughout the modeling process, it is important to provide justification for choices being made and communicate the interpretation of the results and the conclusions reached. Yet, learning to communicate mathematically takes time. In working with eighth graders in mathematical modeling, findings were reported by a teacher that described students’ communication of the models, assumptions and justifications, and conclusions as the competency that was least developed (Anhalt, Cortez, & Smith, 2017). While the complete description of modeling competency seems daunting, especially for any one particular grade level, it captures the full modeling process. Thus, it is reasonable to expect that mathematical modeling competency is developed over time provided that students are given appropriate experiences as they mature mathematically. We propose that mathematical modeling thinking, that is, the development of mathematical dispositions and specific ways of thinking in mathematical modeling, is part of the foundation upon which mathematical modeling competency is built.

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3.3 Conceptualization of Mathematical Modeling Thinking We view recent research findings in mathematical modeling education as an opportunity to propose activities that can engage certain ways of thinking necessary for the eventual development of modeling competency. Toward this end, we introduce the concept of mathematical modeling thinking (MMT). We are inspired by the description of algebraic thinking by Seeley (2004) which includes “recognizing and analyzing patterns, studying and representing relationships, making generalizations, and analyzing how things change” (p. 1). She explains that “facility in using algebraic symbols is an integral part of becoming proficient in applying algebra to solve problems. But trying to understand abstract symbolism without a foundation in thinking algebraically is likely to lead to frustration and failure” (p. 1). Research supports that “children can successfully develop critical components of algebraic thinking skills that are foundational to the successful study of algebra in the later grades” (Blanton et  al., 2015, p.71). This finding has a parallel in mathematical modeling. Several studies have shown that engaging in mathematical modeling without critical preparation can lead to frustration and obstacles that prevent completing modeling activities.

3.3.1 Specialized Ways of Thinking Required for Mathematical Modeling Proficiency Multiple studies have recognized that certain modeling competencies require repeated opportunities to develop and put them into action. Blum and BorromeoFerri (2016) write, “Students and teachers need to have continuous opportunities to acquire and to practice the required competencies and in particular modeling” (page 71). Making assumptions, choosing to leave out some variables during an initial attempt at a solution and other competencies associated with modeling can be challenging before the students and teachers gain experience with them. Zawojewski et al. (2003) report that students experience frustration when a solution approach is not readily identified. Bleiler-Baxter et al. (2016) describe challenges that students face in modeling related to simplifying the situation, mapping relationships, and validating conclusions. These authors observed that on a task related to finding the optimal location of a fire station, students were hesitant to ignore any contextual information and wanted to consider all given information. This extreme focus on contextual issues became an obstacle to simplifying the situation and to making progress in the modeling process. The findings are not restricted to students. Reports indicate that prospective teachers are not receiving sufficient training in the modeling process to describe its typical elements (Siller & Kuntze, 2011) and that prospective teachers have difficulty developing a solution of open-ended tasks that require making assumptions and choices (Anhalt, Cortez, & Been Bennett, 2018; Ng, 2013; Thomas & Hart, 2010). Other

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researchers reported obstacles to successful modeling in studies that aimed to identify the occurrence or removal of blockages during the modeling process (Galbraith & Stillman, 2006; Galbraith, Stillman, & Brown, 2006; Schaap et al., 2009; Stillman et al., 2010). These examples suggest that the lack of dispositions specific to modeling, such as the awareness that some information in a given task can be initially omitted in order to pose a simpler scenario, prevents success in mathematical modeling. They show that modeling proficiency is not immediate and requires practice and familiarity with certain approaches and thinking that are effective in modeling. The awareness, dispositions, and approaches are part of MMT.

3.3.2 Characterization of Mathematical Modeling Thinking The NCTM (1989) defined mathematical dispositions as attitudes that extend learning beyond concepts, procedures, and applications and include tendencies to view mathematics as a powerful way to analyze situations. Dispositions are tendencies to think and act in productive ways that are manifested in the approaches to tasks, whether with confidence in using mathematics to solve problems, to communicate ideas, and to reason, or the willingness to explore alternatives, perseverance, and interest, curiosity, and inventiveness in doing mathematics and in their tendency to reflect on their own thinking. We characterize MMT by considering NCTM’s (1989) mathematical dispositions in connection to mathematical modeling. The central idea of MMT is the realization or the awareness that mathematics can be used to investigate contextual situations arising in everyday life. This awareness is a disposition that includes the curiosity and desire to understand and explain the world around us using mathematical concepts, symbols, and other representations as well as multiple forms of communication. MMT can be characterized by: • The awareness that because realistic situations are not prefabricated mathematical problems, there is a need to introduce information that provides the structure for the contextual situation to become a mathematical problem posed in a familiar way. • The understanding that the modeler’s own experience and background knowledge influence the modeling process, including the choices of mathematical concepts and representation used. • The awareness that starting with a simple case and making revisions with increasing complexity is an appropriate and effective way to approach open-ended problems. As an example, the ability to simplify a complex situation in order to transform it into a manageable task is an integral part of mathematical modeling. However, in order to initiate this strategy, one has to have the awareness that appropriately modifying the original situation is an acceptable and useful approach to the task mathematically. This way of thinking is also related to other dispositions associated with

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the iterative nature of mathematical modeling. If we are able to solve a simplified version of the situation, we may wonder if we can solve it without simplifying it or at least with a lesser degree of simplification (and therefore closer to reality). This way of thinking requires an awareness that starting with a simple case and making revisions with increasing complexity is an appropriate way to approach open-ended problems. Without this awareness and willingness to pursue it, it may not be possible to engage fully in mathematical modeling.

3.4 Practices that Target Mathematical Modeling Thinking We propose that students at all levels must have opportunities to develop the necessary awareness and dispositions for modeling over time by engaging in targeted ways of thinking associated with modeling. In support of this notion, the GAIMME Report (Garfunkel & Montgomery, 2016) advocates for mathematical modeling be taught at every stage of a student’s mathematical education. Preparing teachers with facility in MMT is therefore of paramount importance to their students’ success in developing MMT. We have identified six concrete practices that can be incorporated into teacher education and the k-12 classroom for the purpose of developing MMT. The six practices listed here were identified largely through the studies that reported obstacles to successful modeling. Activities were created to provide the type of preparation aimed at eliminating the obstacles, and some have been tested with teachers and prospective teachers. The results of working with teachers and prospective teachers are discussed later in the chapter. We also considered those skills, dispositions, and techniques that are commonly used in mathematical modeling by professionals outside mathematics education. This literature is valuable to understand how modeling complex phenomena requires detailed analysis of assumptions (Rubin & Wenzel, 1996), model validation based on data (Cisneros, Cortez, Dombrowski, Goldstein, & Kessler, 2010), and how the iterative nature of the modeling process plays out in time (Swanson, Brigde, Murray, & Alvord, 2003). Figure 3.1 displays these six interrelated MMT practices that are also useful more broadly in mathematics and therefore some of them may be familiar. The figure has been created to imply that these practices have no particular order. Furthermore, the open spaces represent additional practices that successfully promote MMT to be identified in the future as we learn more about the process of building the foundation for modeling and about the effectiveness of the practices.

3.4.1 Recognize Assumptions This includes both implicit and explicit assumptions in contextual problem statements and in solution approaches, routinely making appropriate assumptions in order to solve problems. MMT includes recognizing when implicit assumptions

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Fig. 3.1  Six practices that develop mathematical modeling thinking (MMT). The cells, written in no particular order, intersect to represent the interrelated nature of the practices. The open cells are spaces for additional practices that might be identified in the future

have been made in the solution of problems and how these assumptions influenced the answer. We posit that most solutions of problems have underlying assumptions that usually go undetected and unmentioned. This presents an opportunity to develop awareness of implicit assumptions and making explicit assumptions, both of which are necessary throughout the mathematical modeling process (Anhalt et al., 2018). This practice can be introduced in elementary grades as part of a discussion following many mathematics problems. For example, an elementary problem from illustrativemathematics.org (5.NF.B.3) is the following: Alex, Bryan, and Cynthia are about to eat lunch, and they have two sandwiches to share. Draw a picture to show how they could equally share the sandwiches. How much of a sandwich does each person get?

The classroom discussion will involve operations with fractions. To promote MMT, one can include an added piece about implicit assumptions that are made when engaging in this problem by asking the questions, Are the sandwiches equal in size? How does your solution change if one sandwich is bigger than the other one? Are they the same or different type of sandwich and how does this matter? Teachers can take advantage of these opportunities to leverage common problems in the curriculum to discuss implicit assumptions in explicit ways. It is important for teachers to reflect on how a discussion on assumptions has the potential to influence student thinking and build MMT.

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Fig. 3.2  Approximating the figure of a piñata for estimating the number of candy that could fit inside

3.4.2 Approximate and Estimate to Reason Quantitatively Every situation can be approximated by simpler versions of itself. Depending on the goal of an activity, certain objects may be approximated by more familiar shapes that facilitate estimating the length, the area, or something else about the object. For example, Anhalt et al. (2018) describe a task that asked for an estimate of the amount of sunlight the leaves of a tree absorb, which required the area of the leaves. The area of one leaf may be approximated by the area of a triangle of certain proportions. This practice of approximation is useful as a systematic approach to estimation, rather than an unmeasured guess. This practice is also closely connected to the mathematical modeling competency of simplifying a complex situation since the latter often involves approximating unknown quantities with estimates. For example, in Fig. 3.2, while we do not know how many pieces of candy are in a piñata, we can estimate it systematically by considering the size of the piñata initially through approximations of the shapes that compose the piñata.

3.4.3 Prioritize Factors that Affect the Solution as a Means of Simplifying the Problem As reported earlier, the competency of simplifying complex situations is challenging because there are many acceptable ways to do it. The modeler must determine ways of simplifying that are effective for a given task. We have identified and tested a practice that eases the development of this competency. When faced with a complex situation, it is helpful to make a list of all possible factors that can affect the solution of the task. This list can be long which reveals the complexity of the situation. As a second step, we prioritize the list based on how important we perceive

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each factor to be. Once the factors on the list are ranked in order of importance, we choose the top three or four factors on the list as the only ones that will be included in the solution; the rest of the factors are omitted, leading to a simplified situation. Later in this chapter we report on teachers’ work using this practice during professional development.

3.4.4 Use Multiple Representations to Express Mathematical Ideas This practice is useful in learning mathematics in general and is especially important in mathematical modeling, which requires the use of abstract notation, language, concrete objects, or diagrams, to express mathematical ideas when “mathematizing” a situation. Students can express their mathematical ideas through diagrams, pictures, numbers, equations, language, and concrete objects and thus should be asked to explain their thinking using more than one representation and make mathematical connections between the representations expressed. This practice of asking students to express mathematical ideas is not new, yet it can be done with a more purposeful goal of developing MMT in preparing students for mathematical modeling.

3.4.5 Reflect on a Solution, Meaning, and Reasonableness Within the Original Context Reflecting on the meaning of a solution helps develop general mathematical skills for success, and it is fundamental to mathematical modeling. By reflecting on a solution, we mean developing the practice of making sure the units are consistent, that parameters and variables are in reasonable ranges of values, and that the computations are correct. In contextual problems, this practice also involves interpreting the meaning of the mathematical solution in context. For example, in a typical problem about preparing apples for making a pie, “If a baker peels and slices 3 apples in 15 minutes,” the problem could ask, “how many minutes would it take for the baker to peel and slice twelve apples?” or “how long would it take 5 bakers to peel and slice 15 apples?” While the focus of the activity might be on proportional reasoning, it is worthwhile to engage in a discussion of whether a constant rate of peeling and slicing is reasonable or if the bakers might slow down as they work for a long time. These discussions connect the mathematics to everyday life, can stimulate creativity, and develop an awareness of the reasonableness of a mathematical solution. It is expected that students (or teachers) will draw on their background knowledge and lived experiences to reflect on the solution in context.

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3.4.6 Reconsider, Revise, and Refine the Solution Most people do not solve a mathematics problem perfectly in the first attempt (depending on the complexity of the problem), such as the word problem above about bakers peeling and cutting apples. Most would start and get only so far, go back to change something and get further, and so on. It is not a clean process but rather full of human intuition and imperfection. However, the solution is finally presented cleanly and without evidence of the process that took place. The work that is done toward finding a solution is valuable and presents opportunities for learning since it reveals student thinking. MMT acknowledges that solving open-ended problems is a process of incremental improvements and asks that the modeler be open to learning from the intermediate attempts in order to formulate improvements. This notion is common in many disciplines and aspects of life. For example, in writing we typically edit several drafts before the final product. In fact, the Common Core Writing standard (CCSS.ELA-Writing Standard LITERACY.W.9-10.5) calls to “Develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach, focusing on addressing what is most significant for a specific purpose and audience.” MMT involves reconsidering, revising, and refining the solution to all problems and recognizing that this iterative process is natural in modeling.

3.5 Situating Our Work with Teachers as Modelers We worked with 20 elementary grades 3–5 teachers as part of an NSF-funded professional development project that focused on mathematical modeling in community and cultural contexts. We also worked with 21 elementary prospective teachers in a mathematics content course in an elementary teacher preparation program. Both groups had no prior exposure to mathematical modeling in their mathematics education backgrounds or in their professional preparation. And as a result, we were able to leverage their novelty in the experiences and notice their awareness and development in dispositions for MMT. We provide examples from the elementary prospective teachers’ and practicing teachers’ work to illustrate their engagement in MMT. In general, the mathematical modeling activities were structured around a launch/ explore/summarize approach to engage the prospective and practicing teachers in activities to facilitate the development of dispositions in MMT. The launch phase included eliciting background knowledge of the context of a situation, followed by a discussion about noticing and wondering (NCTM, 2015) when prompted with a visual stimulus on the context of a problem. This activity integrated individual think time, verbalization of noticings and wonderings that led to posing of a task, collaborative small group exploration, and communication about their work and solutions.

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3.5.1 Snapshots of Mathematical Modeling Thinking During the professional development institute with the practicing elementary teachers, we posed a problem that was relevant to the community of their schools. A local restaurant in the community has pupusas on the menu, a Salvadoran cultural food that is made of corn masa, meat, cheese, and other ingredients. These are customarily sold freshly made on the streets in El Salvador. We launched the task with asking the teachers which customs and traditions they had in their families followed by introducing pupusas through a short video of pupusas being made. We asked the teachers to discuss things they noticed in the video and things they wondered about related to pupusas. Ingredients for making pupusas were an important thing to notice in the video, which the teachers discussed. We then asked them to consider the cost of making one pupusa to help the owner of the restaurant decide the price of pupusas for her menu. We posed the following task: Part 1: How could you use mathematics to estimate the cost of making one pupusa? Part 2: Decide what price the owner should charge for each pupusa, so that she makes a profit, but also keeps prices low. Part 3: Write a letter to the owner explaining your menu pricing recommendations and include your assumptions and reasons. Include how she could use your method for pricing other items on the menu.

In the discussion below, we focus specifically on Part 1 of the task and describe the way we targeted the MMT practice of prioritizing factors that affect the solution. We developed and implemented a routine to guide the teachers through the process of simplifying a complex situation in a systematic way. The routine began by asking the teachers to make a list of all possible factors they can think of that may affect the cost of making one pupusa. This part served as a brainstorming activity designed to reveal the complexity of the situation. The list of factors can include things the teachers know about making pupusas and the ingredients required as well as things that may need to be researched. A second part of the routine asked the teachers to prioritize the list they just made by the relative importance of each factor on the list. They made decisions about which factors to keep and which factors to no longer consider. The collective knowledge of the group of teachers working together helped them make decisions about each item on the list. Figure 3.3 shows the prompts for discussion around the consideration and prioritization of factors that impact a solution. After going through this MMT activity, the teachers had in fact simplified a complex situation by first considering all the details that might affect the solution and then negotiating within their small groups to reduce the list to its essential factors. This is an effective strategy to address the challenges of getting started with a solution to the task and removing the blockages to successful mathematical modeling discussed earlier. Figure 3.4 shows examples of teachers’ work in initially coming up with individual lists of factors that would potentially impact the solution to open up the discussion for prioritizing the factors that will be included in their group solutions.

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What do we know about the topic?

What information do we need to research?

What information do we use to solve the problem?

Let’s make a list of all possible factors that affect the solution. State the reasons for each.

Let’s find out what else we need to know.

What is the relative importance of each factor? Are there factors that you will not consider further?

These promote a representation of the real, complex situation.

This leads to a simplified situation with essential variables.

Fig. 3.3  Routine prompts for considering and prioritizing factors that impact a solution

Fig. 3.4  Sample teacher work showing lists made by two teachers and the priority they placed on the factors. The left poster shows a list of seven factors identified as affecting the answer in the task. It also shows (labeled with an asterisk) four prioritized factors. The right poster shows a list of nine factors and the three (labeled with a circle) that were considered essential for a model

Once the teachers had gone through the MMT activity, they used their prioritized lists to determine the cost of making one pupusa which informed their formulation of a model for recommending the price of pupusas on the menu. Figure 3.5 shows sample work from teachers in determining the cost of one pupusa taking into consideration their prioritized list of factors. After working on the pupusa task, the teachers discussed how prioritizing their factors individually initially helped them enter the small group conversation to discuss and justify their choices in prioritizing their factors that would influence their solution in determining the cost of making one pupusa. This practice of prioritizing factors serves as a preface for identifying and determining variables that will impact their models. In working with a different group, elementary prospective teachers, our goal was to identify implicit and explicit assumptions that are made when solving problems.

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Fig. 3.5  Snapshot of teachers’ work in determining the cost of making one pupusa

They were given a task about the weight of oranges in a crate (mathplayground. com) which is designed to practice fraction arithmetic, but we included questions about assumptions that are made when solving the problem. Initially, the prospective teachers were asked to solve the problem. A crate filled with oranges weighs 10 lbs. The track team ate two-fifths of the oranges. The remaining oranges and crate together weighed 7 lbs. What is the weight of the empty crate?

Following a discussion on the agreed-upon solution to the problem, the prospective teachers were asked to discuss the following questions in small groups: (1) Did you make any assumptions when solving this problem? If so, what were the assumptions? (2) Do you think you were expected to make those assumptions? (3) How would you solve this problem if you remove the assumptions you made? We explicitly focused on the importance of assumptions and the impact that the assumptions have on the solution. This promotes MMT although it is not often discussed or even acknowledged. The elementary prospective teachers commented that they had never thought about assumptions, implicit or explicit, in solving word problems in their own mathematics education background.

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Following a discussion on assumptions, the elementary prospective teachers were given an open-ended task about estimating the dimensions of a bookcase to hold books that were newly donated to a classroom: Book Worms, a publisher company, is donating 283 new books to your classroom. Your class would like to build a bookcase for the new books. Determine the dimensions of a bookcase to hold all of the new books.

The task aligns with Common Core standards appropriate at the upper elementary grades. Our goal was to focus on multiple MMT practices. The elementary prospective teachers worked in small groups to develop a solution of the task. They created and presented a poster with the problem statement, the information needed, their assumptions, and their solution, including representations considered useful. They also reflected on their experience and the reasonableness of their solutions. The poster presentations provided an opportunity to communicate mathematically and to listen to the other groups’ description of their work. Figure 3.6 shows the poster of one group describing the information they chose to use, the assumptions they made, their calculations including representations in terms of equations in figures. The poster also includes a statement of the validity and contextualization of their solution. While the poster does not include the justification for their assumptions, the elementary prospective teachers used their knowledge of elementary school books as a guide for the dimensions of a typical book. This is an example of how elementary prospective teachers were able to use assumptions in their models. Several mentioned or implied that their assumptions represent average values. In the bookcase task, it is important to realize that each assumption must be considered individually. Since the books are placed side by

Fig. 3.6  Elementary prospective teachers’ solution

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side, an average thickness is reasonable to assume. However, for the height of the books, a maximum value seems more appropriate because a shelf of average book height cannot hold slightly taller books, only average height or shorter. The elementary prospective teachers were asked to mention in their reflections how this type of task was different from traditional word problems and how the solution is influenced by the assumptions made. Their reflections show two themes, that the bookworm problem was different from math problems they were used to because not all the information needed to solve the problem is given in the statement. Most participants commented on the fact that the task gives very little information in contrast to traditional word problems where all information needed is given. The second theme is that the assumptions necessary to solve the problem have an effect on the solution, and therefore more than one solution may be considered correct. The fact that they recognized that the task is different from traditional problems facilitates discussion of the dispositional differences and difference in thinking that might be required for modeling.

3.6 Discussion We have discussed research-based evidence that when teachers and students who have had little to no exposure to mathematical modeling are presented with a modeling task, they often have difficulty getting started with a solution approach and develop frustration. We believe that a major reason for this is that certain dispositions and strategies, such as introducing appropriate assumptions or beginning with a simpler version of the problem, that allow a modeler to develop an initial approach to the problem and to surpass information gaps in the problem statement, are not yet part of the mathematics education of teachers and students. We have introduced the concept of mathematical modeling thinking (MMT), which includes awareness and dispositions that are involved in successful mathematical modeling. Because mathematical modeling is a part of mathematics, other mathematical dispositions for mathematical thinking also apply to modeling. By targeting awareness and dispositions routinely, we build MMT, which is part of the foundational basis for success in modeling. We also have identified six practices for the development of MMT. The practices listed in Fig. 3.1 are not independent of one another, and more than one are often experienced together. Additional practices are likely to surface in future studies. Importantly, our research suggests that some activities that target MMT do not require engaging in a full modeling tasks; they can be discussions about assumptions or approximations associated with tasks designed for a different purpose. These are important discussions as they reveal aspects of the solution that often go unnoticed or unmentioned. These kinds of experiences in MMT allow new modelers to engage with tasks in flexible ways allowing for building a repertoire of dispositions toward solving problems whether they are traditional textbook problems or more open-ended problems closer to mathematical modeling.

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The pupuseria example task invited teachers to bring their background knowledge and lived experiences to brainstorm important factors that influenced their decision making. This is one way to enter into an open-ended problem and removing the struggle of not knowing how to get started with an open-ended task. Through critical discussions, decisions were made to help prioritize the important information needed. This prioritizing process pushed mathematical discussions, provided a systematic way to simplify the situation, and make progress on making sense of the situation. This entry approach to the problem allowed the teachers to recognize the need for researching information and for making necessary assumptions to move forward in creating a solution. The elementary prospective teachers also recognized that individual culture plays a role in mathematics. One member of the group that worked on the bookworm task wrote in the reflection: The solution varied greatly when one assumes because everyday life styles can be considered differently depending on the person, meaning that every person will assume something differently. This comment shows that MMT activities including reflections can have the additional benefit of allowing the teachers to experience how an individual’s lived experiences are important and valuable in mathematical modeling.

3.6.1 Implications for Mathematics Education: Developing Mathematical Modeling Thinking The Standards for Preparation of Teachers of Mathematics (SPTM) developed by the Association of Mathematics Teacher Educators (AMTE) (2018) call for elementary teachers to understand how to engage learners in the use of mathematical modeling. Among the standards for math teacher preparation, the AMTE SPTM state that “well-prepared beginning teachers of mathematics have solid and flexible knowledge of core mathematical concepts and procedures they will teach, along with knowledge both beyond what they will teach and foundational to those core concepts and procedures” (C.1.1. Know Relevant Mathematical Content, p.  8). According to Ball, Thames, and Phelps (2008), this points to the need for teachers to develop mathematical knowledge for teaching (MKT) including how the mathematics they teach is related to the mathematics students will learn in a later grade to be able to set the mathematical foundation for what will come later. In the context of mathematical modeling, teachers at the elementary level must develop in their students the foundational ways of thinking that will prepare them for mathematical modeling in which they will engage beyond elementary school years. Teachers need support finding effective ways to develop in their students a way of thinking that mathematical modeling requires. The practices introduced for developing MMT offer teachers a kind of progression in teaching toward mathematical modeling competency building. Through activities designed for development of MMT, elementary prospective and practicing teachers can explore powerful ways of thinking that allow for entry into mathematical modeling. It would be beneficial for

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teachers to examine their current curriculum through the lens of MMT as a pathway to developing competencies for mathematical modeling. When teachers design lessons targeting MMT, students can begin to develop necessary ways of thinking for mathematical modeling gradually over time as preparation for building mathematical modeling competency. The six practices for developing MMT add new dimensions to the curriculum that teachers are expected to teach. MMT has the potential to build knowledge, skills, and ways of thinking that advance students toward becoming critical and proactive mathematical thinkers. For example, when approaching a problem, whether it is a close- or open-ended problem, teachers trained in the kind of thinking that involves approximating and estimating, recognizing implicit and explicit assumptions, prioritizing factors involved, the use of multiple representations, reflecting, reconsidering, and revising solutions will develop knowledge and skills for teaching beyond the current curriculum provided to them. The practices proposed for the development of MMT can have the added benefit of improving mathematics teaching and learning by developing critical and creative ways of looking at everyday situations mathematically. By approaching problems through the lens of MMT practices, teachers will remain aligned with NCTM’s (2014) Mathematics Teaching Practices, which provide a foundation for improving mathematics teaching. For example, to develop MMT, teachers can guide students to discuss their process that leads to simplification of a situation through making lists of what they know about the context of the problem, deciding what information they need to research, determining which factors are important in the problem, and explaining their choices and necessary assumptions. All of these practices involve communication and are aligned with facilitating meaningful mathematical discourse. In addition, teachers can explicitly promote the use and connections of multiple representations of mathematical concepts as well as non-mathematical ideas in different ways and pose purposeful questions that create awareness of mathematization as an approach to understand the world. These Mathematics Teaching Practices (NCTM, 2014) provide practical guidelines for teachers to use their specialized expertise and professional knowledge, including knowledge of mathematical modeling and the flexibility to understanding ways that make it useful for helping students develop MMT.  We encourage the teacher education community to support teachers in developing MMT as professional knowledge so that teachers are able to unpack MMT and examine MMT “through the eyes of their learners, as well as to be able to work with many learners simultaneously, each with unique backgrounds, interests, and learning needs” (NCTM, 2014, p.12).

3.7 Conclusion The aim of this chapter is to bring to the forefront the concept of mathematical modeling thinking (MMT) to position students to succeed in mathematical modeling in later grades and beyond. This is analogous to developing algebraic thinking

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for eventual success in algebra. MMT is about generalizing problem-solving strategies to include iteration, using judgment in determining the completeness of a problem statement, and using reflection on that judgment. In addition, MMT involves recognizing information given and missing in contextual situations and that assumptions are used to extract a complete statement from the contextual situation. We argue that by incorporating MMT into early elementary grades and continuing in upper elementary, middle and high school grades can have a powerful impact on preparing students to incorporate ways of thinking for solving mathematical modeling problems. Our aim is to focus on teacher education in MMT to strengthen mathematical modeling education and to make breakthroughs in teaching and learning mathematics inclusive of mathematical modeling. By identifying practices that develop mathematical modeling thinking (MMT), we have established a framework that allows teachers, teacher educators, and curriculum designers to imagine practices that support mathematical modeling not only through complete modeling tasks but by targeting specific competencies. This approach can provide a way to weave modeling into the curriculum in different content areas with repeated opportunities to experience mathematical modeling and develop the associated competencies. Acknowledgment  This project was partially supported by the Mathematical Modeling with Cultural and Community Contexts (M2C3), funded by the National Science Foundation, Award number 1561304.

References Association of Mathematics Teacher Educators (2018). Standards for Preparing Teachers of Mathematics. https://amte.net/sptm. Anhalt, C., & Cortez, R. (2016). Developing understanding of mathematical modeling in secondary teacher preparation. Journal of Mathematics Teacher Education, 19(6), 523–545. https:// doi.org/10.1007/s10857-015-9309-8. http://link.springer.com/article/10.1007/s10857-0159309-8?wt_mc=alerts.TOCjournals Anhalt, C., Cortez, R., & Been Bennett, A. (2018). The emergence of mathematical modeling competencies: An investigation of prospective secondary mathematics teachers. Mathematical Thinking and Learning International Journal, 20(3), 1–20. https://doi.org/10.1080/10986065 .2018.1474532 Anhalt, C., Cortez, R., & Smith, A. (2017). Mathematical modeling: Creating opportunities for participation in mathematics. In A. Fernandez, S. Crespo, & M. Civil (Eds.), Access and equity: Promoting high quality mathematics in grades 6–8 (pp.  105–119). Reston, VA: National Council of Teachers of Mathematics. Ball, D.  B., Thames, M.  H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. https://doi. org/10.1177/0022487108324554 Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87. Bleiler-Baxter, S. K., Barlow, A. T., & Stephens, D. C. (2016). Moving beyond context: Challenges in modeling instruction. In C.  Hirsch (Ed.), Annual perspectives in mathematics education:

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Mathematical modeling and modeling mathematics (pp. 53–64). Reston, VA: National Council of Teachers of Mathematics. Blomhøj, M., & Jensen, T. H. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning. Teaching Mathematics and its Applications, 22, 123–139. Blum, W., & Borromeo Ferri, R. (2016). Advancing the teaching of mathematical modeling: Research-based concepts and example. In C. Hirsch (Ed.), Annual perspectives in mathematics education: Mathematical modeling and modeling mathematics (pp. 65–76). Reston, VA: National Council of Teachers of Mathematics. Cai, J., Cirillo, M., Pelesko, J. A., Borromeo Ferri, R., Borba, M., Geiger, V., … Kwon, O. N. (2014). Mathematical modeling in school education: Mathematical, cognitive, curricular, instructional and teacher education perspectives. In P.  Liljedahl, C.  Nicol, S.  Oesterle, & A. Darien (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (pp. 145–172). Vancouver, Canada: PME-NA. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann. Cisneros, L.  H., Cortez, R., Dombrowski, C., Goldstein, R.  E., & Kessler, J.  O. (2010). Fluid dynamics of self-propelled microorganisms, from individuals to concentrated populations. In G. K. Taylor, M. S. Triantafyllou, & C. Tropea (Eds.), Animal Locomotion. Berlin, Heidelberg: Springer. Common Core State Standards Initiative (CCSSI), (2010), National Governors Association Center for Best Practices and Council of Chief State School Officers. http://www.corestandards.org/ assets/CCSSI_Math%20Standards.pdf. Galbraith, P., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. ZDM, 38(2), 143–162. Galbraith, P., Stillman, G., & Brown, J. (2006, July). Identifying key transition activities for enhanced engagement in mathematical modelling. In Identities, cultures and learning spaces. Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia (pp. 237–245). Garfunkel, S. & Montgomery, M. (2016) Guidelines for assessment and instruction in mathematical modeling education (GAIMME) report. Boston/Philadelphia: Consortium for Mathematics and Its Applications (COMAP)/Society for Industrial and Applied Mathematics (SIAM). Huinker, D. (2015). Representational competence: A renewed focus for classroom practice in mathematics. Wisconsin Teacher of Mathematics, 4–8. Illustravtive Mathematics, (https://www.illustrativemathematics.org/content-standards/tasks/2074, retrieved December 2018). Kaput, J., Carraher, D., & Blanton, M. (Eds.). (2008). Algebra in the early grades. New York, NY: Routledge. Maaβ, K. (2006). What are modeling competencies? Zentralblatt für Didaktik der Mathematik, 38(2), 113–142. Mathplayground.com, (http://www.mathplayground.com/wordproblems.html, retrieved May 2017). National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Richmond, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2014). Principles to action: Ensuring success for all students. Reston, VA: Author. Ng, K.  E. D. (2013). Initial perspectives of teacher professional development on mathematical modelling in Singapore: A framework for facilitation. In G.  A. Stillman, G.  Kaiser, W.  Blum & J.  P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 415–425). Singapore, Singapore: Mathematics and Mathematics Academic

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Group, National Institute of Education, Nanyang Technological University. https://doi. org/10.1007/978-94-007-6540-5_28, Springer Science+Business Media Dordrecht. Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project, 3rd Mediterranean conference on mathematical education, 115–124. Rubin, D., & Wenzel, A. (1996). One hundred years of forgetting: A quantitative description of retention. Psychological Review, 103(4), 734–760. Seeley, Cathy L. (2004). A journey in algebraic thinking, NCTM News Bulletin (September). (https://www.nctm.org/uploadedFiles/News_and_Calendar/Messages_from_the_President/ Archive/Cathy_Seeley/2004_09_journey.pdf, retrieved January 2019). Siller, H-S. & Kuntze, S. (2011). Modelling as a big idea in mathematics – knowledge and views of pre-service and in-service teachers. Journal of Mathematical Modelling and Application, 1(6), 33–39. Swanson, K. R., Bridge, C., Murray, J. D., & Alvord, E. C. (2003). Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion. Journal of the Neurological Sciences, 216(1), 1–10. Thomas K., Hart J. (2010) Pre-service Teachers’ Perceptions of Model Eliciting Activities. In: Lesh R., Galbraith P., Haines C., Hurford A. (eds) Modeling Students’ Mathematical Modeling Competencies. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0561-1_46. Zawojewski, J. S., Lesh, R. A., & English, L. D. (2003). A models and modeling perspective on the role of small group learning activities. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 337–358). Mahwah, NJ: Lawrence Erlbaum Associates.

Chapter 4

Data Modelling and Informal Inferential Reasoning: Instances of Early Mathematical Modelling Aisling Leavy and Mairéad Hourigan

Data modelling develops the skills and competencies necessary for real-world problem solving and the evaluation of evidence. In the early years, data modelling involves posing questions, identifying attributes of phenomena, measuring and structuring these attributes and then composing, revising, making inferences and communicating the outcomes. Many of these processes, particularly making inference and predictions, are fundamental to mathematical modelling. In this chapter, we focus on the latter stages of the data modelling process – making informal inferences about data. We explore the approaches used by 5–6-year-old children when presented with a situation requiring them to make informal inferences about data presented within the context of a data modelling activity. We identify the strategies young children use to make inferences about data and discuss what these strategies communicate about early understandings of statistical inference. The findings suggest that making inferences can be challenging for younger students primarily due to the powerful influence of their developing understandings of number. However, there is evidence that children possess some of the building blocks of informal inference most notably in the approaches that point to a pre-aggregate view of data. We present evidence suggesting that situating data modelling activities within interesting and relevant contexts, alongside good teacher questioning and opportunities to listen to the reasoning of their peers, contributes to the creation of environments that support and develop early understandings of inference.

A. Leavy (*) · M. Hourigan Department of STEM Education, Mary Immaculate College, University of Limerick, Limerick, Ireland e-mail: [email protected]; https://www.mic.ul.ie/faculty-of-education/department/ stem-education © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_4

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4.1 Introduction Examination of the mathematical modelling literature reveals different perspectives and approaches on the use of mathematical modelling in mathematics teaching and learning. Common across all perspectives is the emphasis on students mathematising a complex real-world situation, representing and documenting their work and striving for generalizability. In doing so, students must organise, sort, select and quantify information in an effort to arrive at the solution (English & Sriraman, 2010; Lesh & Doerr, 2003). While these features are common across many modelling perspectives, nonetheless, variation in thinking pervades in relation to the processes of modelling and how to conceptualise mathematical modelling (Zawojewski, 2013). However, Lesh and Fennewald (2013, p. 5) point out that in the early stages of theory development: A certain amount of diversity in thinking is as healthy for research communities as it is for (for example) engineers who are at early “brainstorming” stages in the design of space shuttles, sky scrapers, or transportation systems. Furthermore, we believe that the mathematics education research community in particular has suffered from more than enough pressure for premature ideological orthodoxy.

In Chap. 1 of this book, English presents a useful organising framework within which to conceptualise different forms of modelling for the early years of K-6. We situate our work, and perspective on modelling, as falling within one of the forms of modelling identified by English – data modelling – specifically, a particular component of data modelling that reveals young children’s emerging models of statistical understanding, informal inferential reasoning. While the concept of informal inference has been developing traction in the field of statistics education over the past decade, the reasoning skills and thought processes that underpin inferential reasoning are fundamental to mathematical modelling and across all aspects of everyday life. Young children are frequently presented with information from which they make predictions and inferences. Such situations are many and range from daily decisions regarding whether to wear a raincoat based on weather observations to making predictions regarding the actions of a character in a gaming environment based on previous observations of their behaviour. Inferential reasoning is also critical in the school setting when evaluating evidence emulating from scientific inquiry investigations. In this sense, informal inference is important as it constitutes the beginning of evidence-based reasoning. Promoting this type of reasoning in the early years provides young learners with experiences that serve as the initial steps to accessing a statistical culture and becoming statistically literate.

4.1.1 Theoretical Perspective Statistical inference involves drawing conclusions that extend beyond a given set of data. This may involve inferring properties of a population based on a random sample selected from that population or using inferential statistics to ascertain whether differences between groups are due to some systematic influence other than chance.

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In recent years, the importance of introducing younger students to the fundamental notions of statistical inference has been advocated (Leavy and Hourigan, 2018a, 2018b; Ben-Zvi, 2006; English 2010, 2012; Makar & Rubin, 2009). In this study, efforts are made to characterize the nature of statistical understandings prior to formal work with statistical inference, i.e. the foundational ideas, more recently referred to as ‘informal inference’ (Rubin, Hammerman, & Konold, 2006, p. 1). 4.1.1.1 Definition of Informal Inferential Reasoning (IIR) While many different definitions of informal inference have been posited, a useful definition is ‘the way in which students use their informal statistical knowledge to make arguments to support inferences about unknown populations based on observed samples’ (Zieffler, Garfield, delMas, & Reading (2008, p.  44). Zieffler et al. (2008) identify three components of an IIR framework as making judgements or predictions, using or integrating prior knowledge and articulating evidence-based arguments. It is their contention that by engaging young learners in making inferences that we provide experiences in reasoning about events that will underpin later notions of formal inference. Zieffler et al. (2008) also identified three types of tasks that incorporated each of the three components of IIR; however, these tasks were more targeted towards middle and high school students. Arising from research with primary students, Makar and Rubin (2009, p.  85) propose three principles that are considered essential for informal statistical inference as ‘(1) generalization, including predictions, parameter estimates, and conclusions, that extend beyond describing the given data; (2) the use of data as evidence for those generalizations; and (3) employment of probabilistic language’, in describing the generalization, including informal reference to levels of certainty about the conclusions drawn. The first principle, referred to as generalisation, involves ‘making a claim about the aggregate that goes beyond the data’ (p. 86). When looking beyond the given data and towards a population, the population may be defined as a ‘future population not yet existing’ (p. 86). In that sense, the generalizations consider not just the data that is presented but take into account the context within which the data are situated. The second principle, using data as evidence, acknowledges the fundamental (and often overlooked) importance that recognising the need for data within any framework of statistical thinking (cf. Wild & Pfannkuch, 1999). In the case of young children, in particular, the use of abductive reasoning constitutes as inferential as it represents an effort to provide explanations for data based on informed guesses arising from an individual’s knowledge base. The third principle, probabilistic language, acknowledges the uncertainty involved in making generalisations beyond the data. It is evidenced in the use of language that communicates varying levels of confidence and uncertainty and also in the use of ranges, rather than single data values, in predictions. These three principles which form the basis for IIR rely on a number of core skills and understandings during their implementation. Ben-Zvi (2006) makes the case for the relationship between informal reasoning and argumentation when he

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states, ‘Deriving logical conclusions from data – whether formally or informally – is accompanied by the need to provide persuasive arguments based on data analysis’ (p. 2). The ability, then, to express uncertainty and the use of probabilistic language are important components of argumentation. Multiplicative reasoning, solving problems arising from proportional situations, has also been identified as a component of mathematical reasoning central to inferential reasoning (Ben-Zvi, 2006); multiplicative reasoning underpins many other core ideas such as proportional reasoning and aggregate reasoning all of which support learners in drawing conclusions from evidence presented (Hurst & Hurrell, 2016; Konold, Pollatsek, Well, & Gagnon, 1997). The ability to view data as an aggregate (Makar & Rubin, 2009; Rubin et  al., 2006) has been identified as a critical understanding when making inferences and generalisations that extend beyond the data. 4.1.1.2 Reasoning About Aggregates Properties of aggregates are patterns that emerge from attending to features of distributions rather than features of individual data values. Discerning the properties of an aggregate involves a focus on propensities and trends in the data (Konold et al., 1997) and the ability to differentiate signal from noise (Konold & Pollatsek, 2002; Rubin et al., 2006). Thinking about aggregates, while possible, has been shown to be challenging for children (Cobb, 1999; Hancock, Kaput, & Goldsmith, 1992). However the ability to reason about aggregates is essential in order to see patterns and trends and in turn make inferences from a given set of data. Recently, Konold, Higgins, Russell and Khalil (2015) identified four perspectives that students use when working with data. These perspectives include data as pointer (focus on the event rather than the data), data as case value (focus on individual data values or cases), data as classifier (identifying subsets of data values that may be the same or similar) and data as aggregate (view all the data values in aggregate as an ‘object’ or a distribution). They describe the perspectives as forming a loose hierarchy where ‘a high level subsumes or encapsulates lower ones’ (p. 309). Thus very young children’s idiosyncratic reasoning about data is reflective of the ‘data as pointer’ perspective wherein they refer to the particular event from which the data were collected rather than the actual data. It is anticipated that students who hold the ‘data as case value’ perspective may only focus on individual data values or cases and may not see that a group of cases might be related. In contrast the ‘data as classifier’ perspective involves identifying subsets of data values that may be the same or similar (i.e. a category or a cluster). Finally, those holding the ‘data as aggregate’ perspective view all the data values in aggregate as an ‘object’ or a distribution. They consider statistical properties of a distribution (e.g. statistics about spread or measures of centre; proportions of data values found in different sections of a distribution; overall shape of distribution) that are not features of individual cases or groups of data values. The use of these perspectives as a way to analyse an individual’s particular interpretation of data may provide valuable insights into their aggregate reasoning

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and in turn the extent to which they possess the necessary building blocks for informal inference. 4.1.1.3 Data Modelling Environments and Informal Inferential Reasoning (IIR) Designing tasks that engage young children in informal inferential reasoning is challenging for reasons including the limits of their counting skills and the identification of meaningful contexts that lend themselves to inference. There is evidence however of the success of data modelling environments in promoting informal inferential reasoning. Within these data modelling environments, some studies report on the benefits accruing from the use of picture books with young learners to support inferences (English, 2010, 2012; Kinnear, 2013; Kinnear & Clarke, 2016), while other studies have designed contextually rich problem situations (Leavy & Hourigan, 2018) and inquiry-based learning environments (Makar, Bakkar, & BenZvi, 2015) which require the construction of inferences to inform decision making. Data visualisation environments which use innovative statistical software programs designed for use with children have been shown in studies (Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008) to support informal inference by engaging young children in solving ill-structured problems within data modelling environments. English (2012) utilised an environmental science context in her examination of the data modelling approaches of 6-year-olds. Children recognised patterns and trends in the environmental data presented to them and made predictions and informal inferences about the situations presented. There was evidence of the presence of a case value lens used by those children who focused on the values of particular cases. However, the presence of a classifier lens was demonstrated by those who showed the ability to consider the frequency of cases with a particular value. A new category of inferential reasoning was also identified by English (2012) as a preaggregate lens which incorporated those approaches that considered all the data, compared the frequencies and had some attention to overall trends. Evidence for the presence of a pre-aggregate view of data has also been presented in two other studies by Kinnear (2013) and Leavy (2017). When making predictions based on their knowledge of a storybook, 6-year-old children in a study by Kinnear (2013) generalised beyond the data and used the data to support their generalisations thus demonstrating some elements of aggregate reasoning. The approaches used by 5–6-year-old children while working in a rich data modelling activity also indicated the presence of a pre-aggregate view of data. While the aforementioned children who demonstrated a pre-aggregate lens did not demonstrate reasoning as sophisticated as learners identified as using an aggregate lens (Konold, Higgins, Russell, & Khalil, 2015), it is a strong indicator of the potential of young children to engage in informal inference. Another indicator of the presence of inferential reasoning in the use of abductive reasoning by young children. Abductive reasoning refers to situations in which

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young children based their predictions on the context but not on the data presented. Indeed, familiarity with the data context has been identified as a critical component supporting the development of statistical understandings of young children (Leavy & Hourigan, 2016; Watson, 2018). Both Kinnear (2013) and Makar and Rubin (2009) report on the use of abductive reasoning by young children when observing patterns in data. As the young children in these studies have limited life experiences to draw from, abductive reasoning is usually contextually based and the predictions arising from this type of reasoning are not as accurate as databased reasoning. Nonetheless, abductive reasoning represents an effort to provide explanations for data based on informed guesses arising from an individual’s knowledge base and as such can be considered as inferential.

4.2 Research Approach This three-tiered teaching experiment (Lesh & Kelly, 2000) explored understandings of researchers, pre-service teachers and children as they worked together in a classroom setting. In this chapter, we focus on the first tier which is aimed at ‘investigating the nature of students’ developing knowledge and abilities’ (Lesh & Kelly, 2000, p. 197), in this case the developing understandings of 5–6-year-olds in relation to informal inference.

4.2.1 Participants The 24 participants were 5–6-year-old children in a multigrade class; all were in the second or third year of primary school. They had diverse gender, socio-economic and ethnic backgrounds. Participation in the research was voluntary, and ethical approval was obtained from the College Research Ethics Committee.

4.2.2 Method The research took place within a 10-week university course in initial teacher education focusing on classroom inquiry in mathematics. The authors worked with 25 pre-service primary teachers on the design and implementation of a four-lesson data modelling investigation for children in the early primary years. Pre-service teachers were assigned to one of four groups; each group focused on the design and implementation of one of the four lesson foci. The investigation was set within the context of a zoo, and each of the four designed lessons focused on developing understandings relating to a big idea in data modelling and statistics (see Table 4.1). Lessons

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Table 4.1  Lesson order and focus Lesson order 1 2 3 4

Focus of lesson Data generation and collection Identifying attributes Representing data Informal inference

# Pre-service teachers 6 6 7 6

which lasted approximately 60 min were taught on consecutive days to the same class of children. Details of lesson 1 which focuses on data generation and collection (Leavy and Hourigan, 2018a) and lesson 2 which examines attribute selection (Leavy and Hourigan, 2018b) have been outlined elsewhere. This chapter focuses exclusively on lesson 4 – informal inference. The research process had two distinct phases spanning the 10-week semester: (1) research and preparation and (2) teaching and reflection. As part of the research and preparation phase, which took place in the first 5 weeks, the researchers and a subgroup of six pre-service teachers researched the literature relating to informal inferential reasoning. Discussions and considerations relating to the lesson context, task design and pedagogies led to multiple revisions to the lesson design. The phase culminated with the design of a research lesson following the lesson note format presented by Ertle, Chokshi and Fernandez (2001). This format was supportive in providing a coordinated focus on the task and associated teacher questions alongside consideration of possible student responses. The 5-week teaching and reflection phase initially involved the lesson being taught to the class of 5–6-year-old children. Both researchers and the six pre-service teachers were present for the lesson; a professional film crew video recorded the lesson. Children were placed in six groups (four to five children per group) and a pre-service teacher assigned to each group. Pre-service teachers had a very defined purpose and role within the groups; this was to capture (via digital audio recorder) and support the development of conversations within the group. One of the groups was also video recorded. A post-lesson discussion, immediately following the lesson, focused on observations of children’s inferences and data modelling efforts. 4.2.2.1 Framing the Lesson For the 3 days prior to this lesson, children had been engaged in a data modelling investigation situated within the context of a zoo (Leavy and Hourigan, 2018a, 2018b) which focused on a character called Zac the zookeeper. This lesson on informal inference consisted of a series of tasks designed to gain access to children’s abilities to make informal inferences about data. While designing these tasks, we were mindful of the recommendations made by Makar and Rubin (2009) to support children in engaging in informal inferential reasoning, notably the importance of posing driving questions, selecting engaging contexts and ensuring sufficient complexity in the data.

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4.2.2.2 Task Design We used the Zieffler et al. (2008) framework during the task design in an effort to challenge children to make judgements or predictions about a population based on a sample, to draw on prior knowledge (informal knowledge, in the case of these young children) and to articulate evidence-based arguments. Thus, we provided a data set representing animal births in a zoo across 4  years and asked children to make a prediction regarding the number of births for the next year. The zoo and animal birth context supported children in drawing on informal knowledge they may hold about the context. Alongside attending to the engaging context and ensuring sufficiently complex data, careful consideration was made to curricular experiences of children and the concomitant implications for task design. Most notably, 5–6-year-olds in Ireland only develop thorough number sense of numbers up to 10 in their formal schooling thus placing limitations on the magnitude of numbers that could be used and amount of data that we could present. At the start of the lesson, children received a phone call from Zac the zookeeper during which he posed the following problem: Hi everyone. Do you remember me – I am Zac the zookeeper. Thanks for all your help over the past 3 days in counting and sorting my animals and in making a map of the zoo. I was wondering if you could help me with another problem I have? [pause] Well, the elephant’s home in the zoo is getting a little bit crowded. I think we need to make it a bit bigger. But, I don’t know how many elephants will be in the zoo next year which makes it difficult to plan ahead. I was hoping that you children could look at the numbers of elephants that have been born in the zoo for the last four years, and predict how many will be born next year. I’m sending you a video of the elephants with their babies in their home and a picture of the data we have of the number of elephants born here over the last four years. Good luck and thanks!

4.2.2.3 Presenting the Tasks Children then viewed a 3-minute clip from a documentary about Dublin zoo focusing on the elephant living area. It informed them that some new baby elephants had been born in the zoo. The teacher (one of our pre-service teachers) explained that the zookeeper needed to know how many elephants would be born next year. She outlined the number of elephants that had been born in years 1, 2, 3 and 4 (see Fig. 4.1). Children then worked in their groups, alongside a pre-service teacher, to make predictions for the number of elephants that would be born in year 5. The preservice teacher working with each group asked probing questions in an effort to identify what was informing children’s decisions, i.e. idiosyncratic responses, guesses and data-based reasoning. The process was repeated for data sets relating to wolf, giraffe and monkey births (see Table 4.2, Appendix).

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Fig. 4.1  The elephant task Table 4.2  Data sets presented to children Animals Elephants Giraffes Wolves Monkeys

Animal births in years 1–4 Year 1 Year 2 3 4 5 6 8 8 3 5

Year 3 7 2 5 0

Year 4 6 3 5 2

4.2.2.4 Role of the Pre-service Teacher Pre-service teachers supported the development of conversations through posing questions that aided children in articulating their predictions and in providing a rationale to account for their predictions (thus uncovering models of their thinking). For these purposes, we developed and used a ‘question protocol’ consisting of a list of suggested questions, probes and prompts that were to be used during the small group interactions. Thus the child was placed to the forefront in all interactions – the purpose of the pre-service teacher was only to facilitate the children in presenting their ideas, predictions and strategies when making informal inferences.

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4.2.2.5 Analysis of the Data Conversations within each small group were captured digitally, downloaded and transcribed. The first phase of the data analysis focused on attributing strategies to the birth rate predictions and involved two waves of analysis focusing initially on individual (i.e. child) and then group (i.e. animal) level data. One researcher examined all transcribed data and recorded each child’s prediction for the four animal groups. If a child changed their prediction, arising from conversation with group members or a probe by the pre-service teacher, all predictions for that child were recorded. A list of 84 birth predictions, linked to individual children, was generated across the data corpus from this process. While at times it is difficult to ascertain the extent to which the predictions are stochastically motivated, it can be said with certainty that all predicted values fell within, or close to, the range of the data presented indicating that children were attending to, and cognizant of, the presented data. Once the list of all predictions was generated, our focus was to ascertain the types of strategies children were using to generate the predicted birth rate data. The second wave of analysis, carried out by one researcher, focused on the group (animal  – elephant, giraffe, wolf, monkey) level data. For each group (i.e. elephant, giraffe), all predictions were collated, children’s justifications and conversations pertaining to each prediction were examined and a strategy was then linked to each predicted value. Children’s conversations and explanations were central to linking the predicted value to the strategy; a list of nine strategies was initially established. The second researcher then examined the list of strategies and associated birth predictions. Two of the strategies were reconsidered and a final list of seven different strategies identified. The prevalence of strategies differed across the groups and were closely tied to the values presented in the data sets. The strategies used were context (drawing on real-world experiences of animal), earlier/next counting number, filling a gap in the sequence of counting numbers, a high frequency number, repeating/extending a perceived number pattern, the sum of all numbers presented and identifying some trend in the data. At times, children used a combination of strategies, and other times it was unclear what strategy was being used to inform the prediction. Video data from one group were then examined by both researchers in an effort to verify the established strategies. The second phase of data analysis considered children’s statistical thinking and reasoning, as captured in the transcripts and observations, within a data modelling framework in an effort to ascertain the extent to which they served as models of the children’s informal inferential reasoning. We used the three principles of IIR (Makar & Rubin, 2009, see Sect. 4.1.1.1) considered essential for informal statistical inference as a structuring framework as we re-examined and subsequently reported on the data corpus. The third phase of data analysis examined the extent to which children reasoned about aggregates (see Sect. 4.1.1.2).

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4.3 Findings Makar and Rubin’s (2009) three principles of informal statistical inference were used as a structuring framework to both analyse and present the findings.

4.3.1 Principle 1: Generalisations, Including Predictions and Conclusions, that Extend Beyond Describing the Given Data Across all four tasks, children demonstrated the ability to make informed predictions about the number of births for the fifth year. While, at times, those predictions constituted generalisations, this was not always the case. There were two strategies where the counting numbers were used as a mechanism to produce a predicted number of births (earlier/next counting number, filling a gap in the sequence of counting numbers); hence the predicted values on these occasions were more descriptive that inferential. 4.3.1.1 Using the Sequence of Counting Numbers as a Mechanism to Generate an Estimation In one third (33%) of situations, children drew on their knowledge and experience of counting numbers as both a process for generating predictions and as a means of justification to support their predictions. When explaining how they generated their predictions, their responses revealed the tendency to examine the presented data values, order them numerically and compare this ordering to the sequence of counting numbers. Based on this, the predicted number of births for year 5 demonstrated an effort to either close or extend the sequence of counting numbers within the data set. The former situation, closing the number sequence, is not unlike the notion of closure in Gestalt psychology wherein an individual perceives an object as a whole even though it is not complete. When presented with the births for the first 4 years, children identified a gap in the presented data set and ‘filled in the gap’ by suggesting a number that did not appear in the presented data. This strategy constituted 18% of the generated predictions (16 of the 84 strategies) and was strongly influenced by the small range of numbers presented to children (as a result of their limited number knowledge). For example, as can be seen in the transcript below, when discussing the elephant data [3, 4, 7, 5], both Barbara and Peter try to close the set

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of numbers by providing a missing number that falls within the sequence of the ordered data values. Further insights into children’s reasoning can be seen in Peter’s effort to explain the reasoning. He justifies his selection of six births as correct ‘if you want the data to go the right way’. Thus they both seem motivated by a logical ordering of the counting numbers. Teacher: Barbara:

Barbara have you had any ideas how many elephants were born? I think maybe 6 because here is 3 4 7 and 5 and there’s no 6 so it will have to be 6.

Teacher: Peter:

Why do you think 6 Peter? Because 3 4 7 5 and the number that’s kind of missing is 6. They go like … 3 4 5 6 7. Okay. If you want it to go the right way it must be 6. What do you mean by the right way? You can see that if it was a different way it would be 3 4 5 6 7. But they made it to 3 4 7 5 then nothing so they should have 6.

Teacher: Peter: Teacher: Peter:

In other situations, children provided a predicted number of births that was ‘one more’ than the last value. What constituted the last value was usually one of two possibilities. On approximately half of the occasions, as in the situation with Anna below, the last value was the value listed in the sequence of births as presented (i.e. year 4). Thus Anna identified 5 as the last number in the elephant births [3, 4, 7, 5] and based her prediction as one birth more than five. For the other half of children using this strategy, children ordered the presented values [from (3, 4, 7, 5) to (3, 4, 5, 7)] and then made a predicted number of births based on the last value following this reordering. What is interesting about this approach used by Aidan, and others, is that children did not seem unduly perturbed by the gaps in the sequence of counting numbers; in many cases they appeared to mentally close the set and then identify the next highest counting number. Thus these predictions represented an effort to extend the sequence of numbers to the next counting number and indicated a focus on pattern in the sense of ordered counting numbers. This strategy occurred 15% of the time (13 of the 84 strategies). Anna: Teacher: Anna: Aidan: Teacher: Anna:

I think it might be 6. Why do you think that? Because you have 5 here (pointing to the last number in the elephant data) and one more is 6. I think it will be 8. 8, why do you think 8 Aidan? Because 1 2 3 4 5 6 7 and one more is 8.

The earlier rationale provided by Peter that data go ‘the right way’ was also evident when Matthew and Olivia were predicting the wolf births [5, 6, 2, 3]. Interestingly,

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they both extend the counting numbers at the lower end of the sequence of counting numbers. Olivia: Teacher: Olivia: Teacher: Matthew:

I put in number 1. Ok why do you think number 1? Because it’s the fifth year and they are all mixed up. 1 will make it 1, 2, 3, 4, 5, 6. What do you think Matthew? Ya one. There is just … Yeah one. There is no four either. If you have no four, instead you can change it for a one. One is a good number. It is the start.

This tension demonstrated by Matthew in choosing between closing and extending the number sequence is evident in the conversation between Moira and Caitlynn in their discussion of the wolf births [5, 6, 2, 3]. We can see the influence of the sequence of counting numbers in informing predictions when Caitlynn argues that you have to fill in (close) the missing numbers first before you can continue on (extend) the sequence of counting numbers. Moira: Teacher: Moira: Teacher: Caitlynn: Teacher: Caitlynn:

Seven baby wolves. Why do you think that Moira? That’s very interesting. Because, they are six (pointing to 6) and they are gonna be seven. Okay, so in the second year there was six. I think four. Four and why do you think four? Because you can’t count seven without a four.

4.3.1.2 Moving Beyond the Sequence of Counting Numbers to Make Generalisations However, it may be remiss to assume that just because children were relying on recitation of counting numbers that they were not attending to the data as an entity or to the context. In situations where the pre-service teacher provided possible numbers of births that were quite high or low, children rejected these possibilities. These situations indicate some element of inferential reasoning as their arguments were not based on the sequence of counting numbers – instead their argument referred to the data as an entity. Hence it could be argued that while the counting numbers informed predictions, there was at the same time an awareness of the aggregate which could be drawn on when evaluating the suitability of other predictions. The conversation with Robin around the wolf birth data [5, 6, 2, 3] illustrates Robins’ flexible thinking in relation to the data. We can see him move from an effort to close the counting numbers to then extend the counting numbers. When generating both these values, he appears not to attend to the context; however, when the

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teacher suggests 15 as a possible number of wolves born, Robin rejects the value as being ‘too big’ as a result of comparing the value to the birth data for the first 4 years. Robin: Teacher:

Robin: Teacher: Robin: Teacher: Robin: Teacher: Robin:

4 wolves … cause there isn’t a 4 there. What if we knew there was a 4 here before year 1? If an old zoo keeper remembered that before this zoo keeper started counting that there was 4 wolves born. Look I’ll write them in here. How many do you think would be born in the fifth year? Then I’d pick 5. No 7. Why 7? Cause 2, 3, 4, 5, 6, 7. Might there have been 15 born? No that’s too big a number. They are all smaller (pointing to the data values). It couldn’t happen. What about zero wolves born in the fifth year? Is that possible? Laughs. Well no! Then there wouldn’t be any wolves. That doesn’t make sense. There was never none born. Look (pointing to the data values).

There were other situations where the primacy of the counting numbers appeared less evident. On 13% of occasions (11 of 84 strategies), children appeared to be making generalisations that extended beyond the data set. In these situations, the predictions indicated an awareness of trends in the data or indicated an effort to coordinate understanding of the context with the values presented in the data set. This was seen in the elephant task with Fia and Otille below where their predicted number of births took into account trends in the occurrence of data values rather than continue or complete a sequence of counting numbers. Teacher: Otille: Teacher: Fia: Teacher: Fia:

Otille can you tell me what you are thinking about the number of baby elephants born? We are thinking 9 because that could be 3, 4, 7, 5. And do you agree Fia? Yes. So, so we think it’s going up. We also think it could be 8. Could it be 1? No it couldn’t because it is getting bigger every year. Well not the 5. The rest are … it’s sort of going up all the time.

Similarly, Caitlynn and Polina provided values that demonstrated an ability to move beyond the data at hand. They drew on the context to predict a larger number of births (8) due to the cumulative increase in number of elephants over time as a result of the animal births.

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Polina: Teacher: Polina: Teacher: Polina:

Teacher: Polina:

Caitlynn: Teacher Caitlynn

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These elephants (pointing to baby elephants born in year 1) would be grown up. So there’d be a lot more here (pointing to year 5). How many do you think would be in year 5? 10. Ok 10. Tell me again why you think that. Because I think that these are going to grow up (pointing to the values for years 1–4) so there’s going to be more. And these ones (pointing to her predicted value for year 5) will be in their tummies. There will always be more. There will always be more. What do you mean by that Polina? Yes I think so, I think there will be babies born from these ones (pointing to the values for years 1–4). These ones are going to be all grown up, they will be adults. Ten altogether. Why did you pick ten? Because there is so much elephants already. They would all be having babies.

Other strategies drew on perceived algebraic trends and repeating patterns and resulted in children adding doubles, repeating and extending patterns and adding totals. These are more fully discussed in the section Reasoning about Aggregates. In summary, in situations where the sequence of counting numbers was used as a mechanism to generate a prediction, it can be concluded that the estimates were informed by counting rather than by the data or the context. While this resulted in the production of estimates that are reasonable and fall closely within the parameters of the presented data sets, the reasoning was located and justified within the world of counting numbers. In one sense, these predictions were predominantly descriptive and generalised within the data presented to children. However, it cannot be concluded that these situations constitute evidence of the ability to generalize beyond the given data, at least not in a stochastical sense. However, Makar and Rubin (2009) argue that the ability to make descriptive generalisations may better prepare children to make inferential generalisations at a later stage. While the use of counting numbers, and thus the production of descriptive generalisations, was by far the most common strategy used by children, there were situations where there was evidence of inferential reasoning. The strongest evidence was those situations where the predicted number of births extended beyond the data at hand and moved beyond a focus on the data at hand. In other cases, where children argued that certain values were not reasonable estimates, their reasoning strongly indicated the ability to make generalisations that were inferential.

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4.3.2 Principle 2: The Use of Data as Evidence for Those Generalisations Analysis of strategies used to generate birth predictions reveals that in the majority of occasions (92%) children used data as evidence to support the predictions. In the remaining 8% of occasions, children presented guesses that were not informed by data or it was not possible to ascertain the genesis of the prediction. The evidence used predominantly referred back to the data at hand and was in many cases based on a justification relating to the sequence of counting numbers. While the generalisation may not always be categorised as inferential, the justifications presented were certainly data-based. Makar and Rubin (2009) refer to abductive reasoning as a form of data-based reasoning employed by young children when observing patterns in data. In the case of young children who have limited life experiences to draw from, the reasoning is usually contextually based. There were many occasions in this study where the inferences, or abductions, made by children drew on their knowledge of the zoo/ animal context when trying to account for or provide explanations for patterns in data. Knowledge and consideration of the context appeared to drive and inform the data-based reasoning when forming predictions for birth rates. When reasoning about the elephant [3, 4, 7, 5] and monkey [3, 5, 0, 2] births, the context of having babies was the primary rationale for the predicted births presented by several children. As can be seen in the transcripts below, explanations of fatigue were used to justify a prediction of a smaller number of births by Eva (3, elephant data) and Maura (3, wolf data). Teacher: Eva: Teacher: Eva:

Eva and Paul, what were you thinking? 3. Because some of them would have to have a rest. Could she have no babies then in the fifth year? No. They always have some babies. Look (pointing to the numbers for years 1–4).

Teacher: Maura: Teacher: Maura:

Now what are you thinking Maura about the number of monkeys born? 3. Can you tell me why Maura? Because, if she had too many babies all those times, then the mommy wolf would have to have a rest all day.

Knowledge of the birth rates in the animal kingdom also formed the basis to support abductive reasoning. Hannah predicted 14 for the number of monkeys born because ‘monkeys have lots of babies I think’. In the transcript below, we see children adjust predicted monkey birth rates downward from 100 by taking into consideration the birth rates of humans and the relationship between human and monkey birth rates.

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Kate: Teacher: Kate: Jack: Kate: Teacher: Ayesha: Teacher: Ayesha: Kate: Teacher: Kate: Jack:

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Maybe its 4 because here’s 3 and here’s 2. So it counts 2, 3, 4. It is not always about counting. It’s a guess and it can help by looking at the other numbers. Could it be 100? Well no not 100, would a monkey have 100 babies? Spiders would. Ya, they would actually but this is monkeys. It’s like our moms having babies. 2, 3, 4 or 5. Is that your guess? No that’s how many babies a human might have. Yeah monkeys are like human beings. Jack what do you think? Monkeys invented humans. It will be higher than the others.

The ability of children to accept or reject values posed by the pre-service teachers as possible birth rates shows their flexibility in reasoning and their ability to consider a value within the context of the prior data values. When reasoning about the wolf data, children respond to the pre-service teacher’s suggestion that zero animals might be born. While the children varied in their responses with some believing it could be possible (Natasha) and others concluding it was impossible (Barbara and Paul below, Eva previously), they all provided coherent and reasoned responses which took the data context into account. Teacher: Polina: Teacher: Barbara: Teacher: Barbara: Polina: Teacher: Anna: Teacher: Anna: Natasha:

Could there be 100 wolves born? No that’s too many actually. Do you think there could be 0 born? No. Why not? Because there should be some born in the fifth year. Yes. Because there’s some born on the other years. The mothers all need their babies that’s why. Okay. Natasha and Anna do you think there could be 0 born? No because then they won’t have any babies. But maybe that’s possible? No. I think there could be 0. Like there might not be any more babies.

These estimates remain close to or within the range of births already demonstrated indicating that some children (Polina, in particular) are coordinating understanding of context in conjunction with consideration of existing data values when making predictions.

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4.3.3 Principle 3: Employment of Probabilistic Language in Describing the Generalisation, Including Informal Reference to Levels of Certainty About the Conclusions Drawn The importance of expressing uncertainty when making inferences is an important component of IIR (Makar & Rubin, 2009) and can be identified in efforts to avoid deterministic claims and in the use of probabilistic language. Analysis of the transcripts reveals widespread use of probabilistic language as children drew conclusions about birth rates based on the data presented to them and used this data to make predictions beyond the data. While many words may indicate uncertainty, for example, colloquial terms such as ‘I think’, we considered only probabilistic terms previously used in the literature (maybe, could, might). We counted their occurrence across the data corpus, and because terms were used more than once by the same child, we also counted the usage across different children, of the terms maybe (n = 48, 13 children), could (n = 29, 10 children) and might (n = 11, 7 children) across the four tasks. Almost half of the children did not use probabilistic language while discussing animal births, while seven children used all three terms and six children used two terms. In the conversation with Matthew around the monkey birth data, we see that when engaged in a conversation, he is able to express certainty around particular predictions not being reasonable; where interestingly, he uses data-based reasoning to support his belief (i.e. 25 were never born before). At the same time, his use of the terms ‘maybe’ indicates his ability to consider the possibility of certain values. Whereas he used data-based reasoning to reject 25 as a possibility, he draws on the context to provide support for the possibility that zero animals could be born. As can be seen, Matthew used the term ‘maybe’ four times in the conversation, however, did not use any other probabilistic terms across the other animal data sets. Teacher: Matthew: Teacher: Matthew: Teacher: Matthew:

Could there be 25 born there? No. I never seen 25 (pointing to the other years). How about 11? Well maybe 11. Zero? Maybe. Cause maybe you had no brothers and sisters that year. Maybe there was just a mom and a dad there and no new kids.

4.3.4 Reasoning About Aggregates As the ability to reason about aggregates is an important building block for informal inference, we were interested in locating any tendencies towards aggregate reasoning. Thus we categorised children’s explanations, to support their predictions,

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within the four perspectives on data posited by Konold et al. (2015). Two general findings emerged from this analysis. 4.3.4.1 A Case Value Lens Was Limited to Specific Scenarios The use of a case value lens was evident in two particular situations: (1) when zeros occurred in the data sets (i.e. years when no animals were born) and (2) when the data displayed a strong algebraic pattern. When presented with the monkey data [3, 5, 0, 2], children frequently referred to the presence of zero births in year 3 in their utterances ‘there are none there’, ‘there are no monkey faces there’ and ‘on year three there is nothing’. Thus it appears that zero was a sufficiently distracting data value to overpower the focus on the aggregate. While this case value lens, stimulated by the presence of zero, indicates a lack of focus on the aggregate, it is interesting to note that zero was not considered in isolation of the data context. In many cases the opposite was true. The children searched for meaning for zero; thus their reasoning was situated within the greater data context. For example, Sian drew on her knowledge of the context in her efforts to explain why no monkeys were born in year 3 when she stated ‘there was none born that year cause all the mommies were pregnant’, and Paul also drew on the context in his comment ‘maybe all the monkeys fell out of the trees that year and so they had no babies’. Another indicator of the use of a case value lens is evidenced in approaches involving summing data values and calculating totals (English, 2012). These approaches were limited in response to one particular type of data set – where the patterns of births reflected a strong algebraic pattern. The algebraic pattern evident in the giraffe births [8, 8, 5, 5, _] seemed to shift the focus away from the heretofore strong focus on context and the sequence of counting numbers and towards a focus on individual data values. This was evidenced by children’s focus on individual numbers occurring in AABB algebraic pattern, in particular the appearance of doubles. In fact, across the four tasks, the only place where the strategy of adding doubles (subtotals) occurred was in the giraffe tasks (n = 7) resulting in predictions of 10 (5 + 5) and less often a prediction of 16 (8 + 8). In the transcript below, we see Leah and Sian generate subtotals to inform birth predictions in year 5: Sian: Teacher: Sian and Leah: Teacher

Sian Teacher Sian Leah Teacher

10!! There’ll be 10 born. 10. Why? 5 + 5 equals 10. Okay. Keep thinking… So 8 giraffes were born the first year, 8 as born the second year, 5 the third year and 5 again the fourth year. What do you think will happen the fifth year? Now what are you thinking girls? 10! 10! 10 Sian. So you think 10 because 5 + 5 = 10. 10 or 16? 16 … because 8 + 8 = 16. Why are you adding them?

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Sian and Leah Leah

I don’t know. Because 5 + 5 = 10. We even wrote a plus.

In the following conversation with one of the groups, a variety of strategies emerge. We see an algebraic approach informing a prediction of eight births, and a doubles strategy results in a prediction of ten births. However, it is also one of the few situations were we see a lapse in (contextual) sensemaking as revealed by predictions informed by a random combining of the presented numbers. This resulted in predictions of 85 births from Thomas. It is interesting to note that this random combining of numbers also occurred in three other groups of children when discussing the giraffe data: a prediction of 85 from Leah, two births from Sheena (because 5 s and 8 s come in 2 s) and 109 births from Julia and Mia (effort to add the outcome of 5 + 5 + 8 + 8). In the transcript below, Thomas appears to randomly combine the two digits to predict 85 births. However, Polina mediates this and refers back to the context and uses the context as an argument to adjust the prediction. Matthew: Teacher: Matthew: Teacher Polina Teacher Polina Teacher Thomas Teacher Thomas: Polina Teacher Polina Thomas Teacher Thomas

Eight because it might go in patterns. What kind of patterns? Like 8 8 5 5 8. Thomas and Polina, what do you both think? I think it is going to be 10. Tell me why. Because it’s 8 over here and 5 over here and 5 again and 5 and 5 makes 10. What do you think Thomas? I think 85. Why? Because there is 8 here and 5 here. They wouldn’t fit into the box. They are definitely not going to fit into the zoo also. Really? Why do you think that? Because giraffes are very big. Yeah. 11. Oh you changed your number. Why? Because it’s a smaller number. Because it’s smaller. You can’t have 85 because there would be a lot of giraffes.

4.3.4.2 Awareness of Trends in the Data Provides Evidence for the Presence of a Pre-aggregate Lens Responses that suggest an awareness of overall patterns and trends in the data were categorised as a pre-aggregate lens by English (2012). Twenty percent of the responses in our study indicate awareness of trend and may point to some emerging

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sense of distribution. Mia’s response to the wolf births indicated an awareness of pattern when she described the births as ‘going up and going down’. Melios’ prediction of eight births for the giraffe data ‘Cause it’s a pattern: 8, 8, 5, 5, 8’ reveals his recognition and subsequent extension of the repeating pattern in the giraffe data set. During a whole class discussion about the number of monkeys that would be born in year 5, awareness of patterns was evident in the comments from Otille and Kate below: Teacher: Otille: Kate:

How many monkeys [3, 5, 0, 2] did you think were born in year 5? I think 1 maybe. Because it goes down, up, down, up, down. 5. Cause 5 here [points to 5] and then low [points to 0 and 2] so it would go back to high.

As can be seen from the transcripts above, some of the justifications, particularly those relating to the giraffe data, may be indicative of an algebraic rather than statistical perspective on data. However, another indicator of the presence of a preaggregate lens was in the reasoning of those children who married an awareness of trends in the data with their understandings of the context in constructing their predictions. This was evident in Polina and Caitlynn’s earlier discussion of the trends in elephant births [see Sect. 4.3.1.2]. In this extract, Polina’s predictions were informed by her recognition of the variation in the data and ensured she always predicted beyond the upper range of the presented data as evidenced in her statement ‘there will always be more’.

4.4 Conclusions This study provides classroom-based insights into one instance of mathematical modelling with young children – data modelling. We provide evidence of the emerging modelling competencies of young learners in data modelling environments. Specifically, this study provides examples of young learners engaging in processes that are fundamental to the mathematical modelling perspective; such processes included dealing with messy questions, analysing and making predictions (Garfunkel & Montgomery, 2016). This study also demonstrates that some young children can engage in relatively sophisticated reasoning about patterns in data and can make predictions about birth rate data that serve as models of their nascent inferential reasoning capabilities. For instance, the 5–6-year-old children in this study provided reasonable predictions for the number of animal births. In making their predictions, they drew on what they knew about the context (the animal kingdom) and also drew on their mathematical knowledge (in this case, their understanding of counting numbers). Consideration of Makar and Rubin’s (2009) three principles for informal inference provides valuable indicators of situations that suggest children are making informal inferences. For example, references to trends in the data as a justification for predicted births point to children making generalisations.

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Data-based reasoning was evident in two situations. Firstly, in situations where teachers suggested possibilities for numbers of births, almost akin to a random element, children rejected these values on the grounds that they were not similar to the presented actual data (i.e. data-based reasoning). Secondly, the instances of abductive reasoning, where children drew on their knowledge of the animal kingdom to inform predictions about birth rates, also constitute data-based reasoning (Kinnear, 2013; Makar & Rubin, 2009). Finally, while at times children made deterministic statements, they also made many references to uncertainty through their use of probabilistic language. In relation to aggregate reasoning, we found no evidence of children holding a ‘data as pointer’ perspective. Perspectives involving ‘data as case value’, while limiting, were strongly driven by the structure of the data sets presented to children wherein algebraic patterns and the presence of zero appeared to almost override their ability to view the data set as an entity. Similar to the work of English (2012), there was evidence of some children applying a pre-aggregate lens and thus pointing to the nascent potential for these young children to engage in aggregate reasoning. This study also reveals the challenges associated with exploring informal inference in the early years. When analysing young children’s responses, it can be challenging to distinguish between data-based and number-based reasoning. This occurs due to limitations in children’s expressive language and restrictions in the magnitude of numbers and amount of data that can be presented to young children. Furthermore it can be challenging to create contexts that foster children’s engagement in data-based reasoning and argumentation. Such contexts generally require that we build in variability in the data. Thus there is the need to develop contexts and situations that are engaging for children and accessible while at the same time steer them towards statistical reasoning. For a number of reasons, making inferences from the presented data sets in this study was not a trivial task for these young children. It can be argued that children were not provided with a lot of evidence from which to make predictions – they had only four data values (4 years) and did not have contextual information on the number of female animals, their health and so on. The limitations posed by the structure of the data, highlighted by Makar and Rubin (2009) in their reporting of a statistical investigation with 9-year-old children, were also evident in this study. Due to the children’s limited knowledge of number, the data presented had limited variability, and all data values were less than 10. Nonetheless, children in our study used mathematical justifications and logical reasoning when providing their predicted births; whether these responses constituted informal statistical inferences was not always clear. Our work builds on previous work in data modelling in the elementary years by providing in-depth insights in children’s strategies when making inferences about a situation presented within a data modelling environment. Furthermore, this study

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identifies factors (in this case context and number understandings) that impact the efficacy of such tasks in early-years classrooms. Our data confirms the critical role played by context, and we provide insights into how young learners engaged meaningfully with and drew on their knowledge of the context when making informal inferences. Unique insights are also provided into two ways that early number development impacts design considerations, and subsequent statistical reasoning, arising from data modelling tasks. Specifically, we refer to the hegemony of number in children’s early mathematical worlds and identify instances of where this number dominance limited opportunities to engage in making informal statistical inferences. We also identify two mathematical situations that received priority when making statistical inferences, notably the presence of zero and repeating algebraic patterns, resulting in the subjugation of attention to contextual considerations. These same contextual considerations had, in other situations, served to anchor the informal inferences and support children in making meaningful inferences. While the inferences presented by some of the young children in this study may be considered primitive models (Lesh, 2010) as they focus on a subset of relevant characteristics and present unintegrated and partial representations of the situation; this study points to a combination of factors (including the aforementioned limitations of children’s number reasoning and the concomitant restrictions placed on the task design), which contribute to these primitive models. Further studies may be able to utilise new and emerging technologies to support the representation of data in ways that overcome the challenges placed by limited number understandings. Finally, there are a number of limitations relating to this research. As this is a study of just one class of 5–6-year-olds, the results cannot be generalised to all children of this age. Secondly, while individual children provided a similar pattern of responses across some tasks, they were influenced by the data sets we presented and possibly by the representations used. While the study undoubtedly contributes to the area, the findings point to many crucial questions and considerations which need to be addressed in order to move the field forward. For example, the important questions of how to move young children from a deterministic to a stochastical system? How do we elicit a statistical perspective in the questions we ask? Certainly, we need to help young children understand that statistics is a different kind of activity than mathematics, and a fundamental component is using data as evidence. We need to develop appreciation for the types of questions being asked in statistical situations and develop an understanding that the nature of the answers to these questions might be a range of values rather than one specific (deterministic) number. The study also highlights that the language of uncertainty underpins the relationship between informal inference and probability and poses a question regarding the extent of the role of uncertainty in informal inferential reasoning. Funding  This work was supported by Mary Immaculate College Faculty Seed Funding.

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Appendix

The giraffe task

The wolf task

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The monkey task

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English, L.  D. (2018). Young children’s statistical literacy in modelling with data and chance. In A.  M. Leavy, M.  Meletiou-Mavrotheris, & E.  Paparistodemou (Eds.), Statistics in early childhood and primary education: Supporting early statistical and probabilistic thinking. Singapore, Singapore: Springer Nature. https://doi.org/10.1007/978-981-13-1044-7 Ertle, B., Chokshi, S., & Fernandez, C. (2001). Lesson planning tool. Retrieved from www. tc.columbia.edu/centers/lessonstudy/doc/Lesson_Planning_Tool.pdf on November 28, 2016. Garfunkel, S., & Montgomery, M. (Eds.). (2016). Guidelines for assessment and instruction in mathematical modeling education (GAIMME) report. Boston, MA: Consortium for Mathematics and Its Applications (COMAP)/Society for Industrial and Applied Mathematics (SIAM). Hancock, C., Kaput, J. J., & Goldsmith, L. T. (1992). Authentic inquiry with data: Critical barriers to classroom implementation. Educational Psychologist, 27(3), 337–364. Hurst, C., & Hurrell, D. (2016). Investigating children’s multiplicative thinking: Implications for teaching. European Journal of STEM Education, 1(3), 56. Kinnear, V. (2013). Young children’s statistical reasoning: A tale of two contexts (Unpublished PhD dissertation). Queensland University of Technology. Kinnear, V., & Clarke, J. (2016). Young children’s abductive reasoning about data. In Proceedings of the13th International Congress on Mathematical Education. Hamburg, July 24–31. Konold, C., Higgins, T., Russell, S. J., & Khalil, K. (2015). Data seen through different lenses. Educational Studies in Mathematics, 88, 305–325. Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33(4), 259–289. Konold, C., Pollatsek, A., Well, A., & Gagnon, A. (1997). Students analyzing data: Research of critical barriers. In J. B. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics: 1996 Proceedings of the 1996 IASE Round Table Conference (pp. 151–167). Voorburg, The Netherlands: International Statistical Institute. Retrieved from http://www.dartmouth.edu/~chance/teaching_aids/IASE/IASE.book.pdf Leavy, A., & Hourigan, M. (2016). Crime Scenes and Mystery Players! Using interesting contexts and driving questions to support the development of statistical literacy. Teaching Statistics, 38(1), 29–35. https://doi.org/10.1111/test.12088 Leavy, A.  M., & Hourigan, M. (2018a). Inscriptional capacities of young children engaged in statistical investigations. In Leavy, A.M. Meletiou-Mavrotheris, M & Paparistodemou, E. (Editors). Statistics in Early Childhood and Primary Education: Supporting early statistical and probabilistic thinking. Springer. ISBN 978-981-13-1043-0 Leavy, A., & Hourigan, M. (2018b). The role of perceptual similarity, data context and task context when selecting attributes: Examination of considerations made by 5–6 year olds in data modelling environments. Educational Studies in Mathematics, 97(2), 163–183. https://doi. org/10.1007/s10649-017-9791-2 Leavy, A. M. (2017). Insights into the approaches of young children when making informal inferences about data. In Paper presented at the Congress of European Research in Mathematics Education (CERME10). Dublin, Ireland. Lesh, R. (2010). Tools, researchable issues & conjectures for investigating what it means to understand statistics (or other topics) meaningfully. Journal of Mathematical Modeling and Application, 1(2), 16–48. Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 3–34). Mahwah, NJ: Lawrence Erlbaum. Lesh, R., & Fennewald, T. (2013). Introduction to part I modeling: What is it? Why do it? In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 5–10). New York, NY: Springer.

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Lesh, R., & Kelly, A. (2000). Multitiered teaching experiments. In A.  Kelly & R.  Lesh (Eds.), Research design in mathematics and science education (pp. 197–230). Mahwah, NJ: Lawrence Erlbaum Associates. Makar, K., Bakker, A., & Ben-Zvi, D. (2015). Scaffolding norms of argumentation-based inquiry in a primary mathematics classroom. ZDM  – The International Journal on Mathematics Education, 47, 1107. https://doi.org/10.1007/s11858-015-0732-1 Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference. Statistics Education Research Journal, 8(1), 82–105. [Online: http://www.stat.auckland. ac.nz/~iase/serj/SERJ8(1)_Makar_Rubin.pdf]. Paparistodemou, E., & Meletiou-Mavrotheris, M. (2008). Enhancing reasoning about statistical inference in 8 year-old students. Statistics Education Research Journal, 7(2), 83–106. Rubin, A., Hammerman, J. K. L., & Konold, C. (2006). Exploring informal inference with interactive visualization software. In A. Rossman & B. Chance (Eds.), Working cooperatively in statistics education: Proceedings of the Seventh International Conference on Teaching Statistics. Salvador, Brazil. [CDROM]. Voorburg, The Netherlands. Watson, J. (2018). Variation and expectation for six-year-olds. In A.  M. Leavy, M.  MeletiouMavrotheris, & E.  Paparistodemou (Eds.), Statistics in early childhood and primary education: Supporting early statistical and probabilistic thinking (pp.  55–73). Springer Nature: Singapore, Singapore. https://doi.org/10.1007/978-981-13-1044-7_3 Wild, C., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry (with discussion). International Statistical Review, 67(3), 223–265. Zawojewski, J.  S. (2013). Problem solving versus modeling. In R.  Lesh, P.  L. Galbraith, C.  R. Haines, & A.  Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 237–243). New York, NY: Springer. Zieffler, A., Garfield, J., delMas, R., & Reading, C. (2008). A framework to support research on informal inferential reasoning. Statistics Education Research Journal, 7(2), 40–58. [Online: http://www.stat.auckland.ac.nz/~iase/serj/SERJ7(2)_Zieffler.pdf].

Chapter 5

Development in Mathematical Modeling Corey Brady and Richard Lesh

The chapters in this section represent noteworthy steps in a research agenda with implications far beyond traditional conceptions of models and modeling, addressing key questions such as: How does the K-12 mathematics curriculum need to adapt to prepare students for the rapidly changing nature of “mathematical thinking” outside of school? What does it mean to “understand” the most important “big ideas” in elementary (K-16) mathematics? How do these ideas and understandings develop? How can these developments be documented and assessed, in their earliest manifestations? How can assessments of students’ most important conceptual achievements be based on operational definitions that do not simply reduce them to checklists of factual and procedural knowledge? Our goal here is to briefly describe how these chapters are situated within this larger context.

5.1 Three Traditions of Modeling-Related Research At least three very different traditions have shaped modern research on mathematical models and modeling. One, focusing on applications, aims at the objective articulated in Hans Freudenthal’s phrase, “to teach mathematics so as to be useful” (Freudenthal, 1968). A second, focusing on heuristics, seeks to formulate generalizable strategies that can support problem-solvers across domains (Polya, 1945). C. Brady (*) Department of Teaching and Learning, Peabody College, Vanderbilt University, Nashville, TN, USA e-mail: [email protected] R. Lesh School of Education (Emeritus), Indiana University, Bloomington, IN, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_5

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Finally, a third, focusing on the development of children’s mathematical thinking, considers modeling as a rich context for studying idea development as it occurs in response to real-world situations (Lesh & Doerr, 2003). This third tradition, known as the Models and Modeling Perspective, or MMP, is our primary focus. In common with the applications tradition, the MMP values the intersection of math and lived experience in the increasingly designed, systemic, and connected world. However, the MMP feels greater pressure to move beyond simplified mathematizations of isolated features of that world and to engage instead with the embroiled messiness of problems as they confront real people in life and work outside school. Similarly, the MMP shares with the heuristics tradition a commitment to bringing higher-order thinking to classroom mathematics. However, in privileging realistic problem settings, the MMP aims to place students in the role of bricoleurs, making sense of settings by “making-do”—innovating by piecing together known mathematical structures to fit the bill. For the MMP, “model” is a central term, both as a noun and a verb. To model is to develop conceptual innovations in response to needs imposed by the world; the result (a model) is a way of thinking about that world, which enables interpretation and action. Inspired by Piaget’s work on the development of children’s mathematical thinking, MMP research seeks to document and understand the active construction of mathematical concepts in response to the demands of real-life situations. These historical roots account for some of the MMP’s distinctive features: in particular, its focus on mathematical models and modeling arises, as it did for Piaget, because mathematics is the study of structure and systems, and mathematical models are the means humans use to describe and understand such structures and systems in the world. MMP researchers also attend to modeling as a very situated practice, involving local conceptual development (Lesh & Doerr, 2003), which then needs to be refined and extended to a variety of other situations. (Here, connections to applications research are helpful.) The scale and grain size of local conceptual development enables the MMP to study acts of mathematical creativity analogous to the everyday mathematical work of professionals such as entrepreneurs, business managers, and engineers.

5.2 Contrasts Between the Traditions An applications-oriented perspective studies how previously learned mathematical concepts and skills are adapted when attempts are made to use them in “real-life” situations. In contrast, the MMP’s focus is on identifying situations that create the need for mathematically significant ways of thinking (Lesh, Middleton, Caylor, & Gupta, 2008). Rooted in the real world, these ways of thinking almost inevitably integrate constructs from multiple topic areas—first, because real-life situations often involve complex systems that resist reduction to their components and, second, because they involve goals (such as optimizing a system feature) and/or constraints (such as achieving goals while maximizing or minimizing another system

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feature) that necessitate trade-offs. Simplifying real-life situations to the point that they center on a single mathematical construct or topic most often removes these systems’ aspects: whereas this can be desirable from an applications perspective, it is unacceptable from a models and modeling perspective. Similarly, a heuristics perspective drives toward generalizable strategies (Schoenfeld, 1992), aiming to articulate typologies—types of problems and learnable categories of productive response to those problems. Ideally, it is possible to identify separable task variables that magnify difficulty and teachable heuristic strategies that lead to success. These typologies are seen as general, and the power of the heuristics analysis comes from this generality. In contrast, for the MMP, modeling is an interpretive and partly subjective act, so that while learners do interpret new situations in terms of familiar ones, this is a personal form of “aspect seeing” (or seeing-as, cf. Wittgenstein, 1958), rather than reflecting objective truths about the situation in the world that learners are mathematizing. The resulting models are “better” or “worse” based on their efficacy in addressing the situated needs of a client, rather than their correctness or match with preconceived optimal solutions. Moreover, heuristics researchers may often be interested in modeling activities as treatments—which move learners forward measurably in their ability to recognize and solve the types of problem in question. For the MMP researcher, in contrast, activities are valuable as contexts—which offer opportunities to engage in processes of mathematizing in both new and familiar ways. Finally, and relatedly, heuristics research tends to view problem-solving as a topic to be learned, with the goal of developing critical thinking skills, while the MMP is interested in modeling as a lens on the variety of situated ways humans create mathematics and make sense of aspects of their world by doing so. These differences shed further light on the rationale behind distinctive features of MMP research. For instance, conceptual development in response to complex, real-world situations often begins with a hodgepodge of partly conflicting and overlapping intuitions that need to be gradually sorted out, integrated, pruned, and formalized. Far from “designing away” this messiness to isolate mathematical structures (applications) or potentially productive strategies (heuristics), MMP research seeks to capture and concentrate this messiness as a provocation for modeling. Other core features of MMP designs provide essential conditions for the evolution of ideas. By assembling learners in groups, they emulate the team-centered reality of workplaces, but they also amplify the range of perspectives that learners need to contend with. This strengthens the diversity component of a set of Darwinian conditions for idea development. The MMP’s focus on a client’s needs, understood clearly and adopted as problem-defining, positions learners to continually assess the fit of emerging solutions. From this feature, a selection component is added. And because models as tools for human thought and action are syncretic, multilayered, case-based constructions, they thrive through combination and refinement, adding the component of reproduction and propagation. For MMP researchers viewing modeling in terms of idea development along a Darwinian paradigm, many of the problems, questions, tools, and directions of research are radically different from

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those that make sense within applications and heuristics traditions. We argue that some of the confusion among educators and researchers about the purpose and value of “modeling activities” for classroom learning can be mitigated with clarity about how their work aligns with and draws upon these three very different research traditions.

5.3 Grounds for Complementarity In spite of their differences, we do not argue for isolating these research traditions from one another. In fact, we believe that MMP perspectives can complement applications and heuristics research, addressing some of the key challenges that arise internally to those traditions. A fundamental drive of applications research is to find ways to take mathematical constructs that have already been learned and make them useful in real-life situations. One challenge that arises internally to this enterprise is the question of how those constructs were learned in the first place. MMP research provides the means to provoke the need for constructs in contexts where their formation can be studied. Moreover, if we believe that the means and processes of learning affect the nature of knowledge (the viewpoint of genetic epistemology (cf, Piaget, 1970)), then this approach can be informative for the questions that applications research will wish to investigate. Likewise, a fundamental drive of heuristics research is to create a learnable set of “higher-order” condition-action rules. (“If I encounter a situation of this type, I should take that action.”) One challenge that arises internally to this enterprise is the question of “transfer,” about which many volumes have been written. The fine texture of the “condition” half of higher-order condition-action rules is difficult to specify: How do learners see and interpret the situation? But this is a question about the mental structures that learners use or construct to interpret their world—models—the core of the MMP tradition. Understanding how learners draw upon prior experiences to construct, adapt, extend, and combine models is a live question for both learning and teaching within the MMP. We see such MMP work as potentially complementing the heuristics tradition by enriching the notion of what it really means to understand the constructs that are the basis for desired interpretations of the world in problem-solving.

5.4 Considering Development We view the chapters of this section as making significant contributions to the MMP, extending and enhancing a rich and connected tradition. Our goal is to place these authors’ work in conversation with one another and with trends in the international field of modeling research. In particular, we consider the notion of “competencies”

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and the relationships between modeling among young children and modeling among older learners, as part of a developmental treatment of models and modeling.

5.5 Children’s Capabilities as Modelers A shared view across the chapters is that elementary-school children are mathematically capable and that as researchers and teachers we are more likely to underestimate than to overestimate their capabilities. English (this volume) looks for resources that these learners have developed in their everyday experiences and that may prepare them as potential modelers. Jung and Brand (this volume) catalogue research that documents young learners’ mathematical innovations, examines features of activities that offer them opportunities to do so, and describes settings for involving teachers in becoming connoisseurs of their students’ sophisticated ways of thinking. Finally, both Anhalt et al. (this volume) and Leavy and Hourigan (this volume) are committed to asset-based views of young learners as capable modelers with resources that can be cultivated through carefully designed activities. Modeling in the elementary grades has been a powerful setting for demonstrating that young learners can engage with developmentally appropriate versions of the full range of fundamental mathematical ideas treating K16. This view contrasts with standards documents that confine young learners to engaging with numbers as counts of objects. For instance, Lesh and colleagues produced and tested activities that engage kindergarten-aged learners in age-appropriate activities that maintain the full range of uses of number, seen across K16 (Lesh & Nibbelink, 1978). It seems to us that all four of the chapters in this section express a spirit consistent with this viewpoint. This then raises the challenge to identify relationships between elementary school modeling and later modeling—that is, to fashion a developmental view.

5.6 A Modeling Competencies Lens and Its Entailments In the international research literature on the teaching, learning, and assessment of modeling as it develops in students, the construct of modeling competencies has played a dominant role (cf Blum, 2015; Kaiser & Brand, 2015; Maass, 2006; Schukajlow, Kaiser, & Stillman, 2018). The value of the construct does not appear to be in its intrinsic content. (As seen in the Danish KOM project, competence in a domain means merely mastery of “essential aspects of life in that domain” (Niss, 2003) and a competency is “a clearly recognizable and distinct, major constituent” of competence.) Rather, it is the indeterminacy of a term like “modeling competencies” that partly enables it to organize this community’s research efforts, creating conversations that may produce an emergent, shared meaningfulness for the construct. For this reason, it may be all the more important to connect MMP research with that community.

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Debates (see, e.g., Blomhøj & Højgaard Jensen, 2003, 2007; Stillman, 2019) have engaged whether modeling competencies are essentially “holistic” (i.e., comprehensive; best or only engaged in the full experience of modeling; cf. Blum & Leiß, 2007) or, alternatively, whether an “atomistic” approach (involving sub-­ competencies or even sub-sub-competencies) can be taken toward developing or assessing them (e.g., Djepaxhija, Vos, & Fuglestad, 2017; Frejd & Ärlebäck, 2011; Zöttl, Ufer, & Reiss, 2011). Moreover, under either approach (or hybrids), there have been discussions about whether modeling competencies are purely cognitive or whether they also include dispositional or affective features (Jankvist & Niss, 2015) and what role metacognition has in developing or supporting these competencies (Vorhölter, 2017, 2018). Each of the four chapters in this section engages with matters directly or indirectly associated with modeling competencies, as used in this international research community. We may thus identify potentially fruitful connections and complementarities by considering their relations and contrasts with this body of work. First, an absolutely fundamental commitment shared across the chapters is a serious attention to young learners. In contrast, the international competencies work above tends to foreground secondary-school and older learners, suggesting either that modeling is appropriate only for older students or that secondary-level versions of modeling can be reverse engineered to guide understandings of modeling among younger learners. The chapters of this section seem unified in articulating a view that young learners’ ways of thinking are both powerful resources and qualitatively different from those of older learners. If one takes an MMP view of modeling problems, as contexts to generate the need for constructing important mathematics, this difference does not pose fundamental difficulties. However, we argue that the modeling competencies literature adopts features of the applications and heuristics traditions toward modeling, which, in contrast, do create conceptual challenges to bridging between young learners’ and older learners’ modeling activity. We see these challenges as opportunities and entry points for the MMP tradition to contribute. The vigorous German school of modeling, led by Blum, Kaiser, and colleagues, has strong ties on one hand to the applications tradition. The theory of modeling (or the model of modeling) underlying this work attempts to schematize modeling practice. With a consistency that we view as admirable, just as this tradition looks for applications of simplified mathematical models in real-world situations, it also attempts to apply a simplified, structural model of modeling to the real-world practice of modeling itself. A common foundation for work is a sequential diagram: for example, the “seven-step” diagram shown in Fig. 5.1 (Blum & Leiß, 2007). This community acknowledges limitations of its own model—that it is at best an analytical reconstruction, versus a description of modeling as actually experienced (Niss, 2019). Moreover, several studies demonstrate that in fact modelers do not typically follow its steps in order (Borromeo Ferri, 2006, 2007) or that their progress is not uniform or unidirectional (Ärlebäck, 2009; Czocher, 2016). In contrast, an MMP approach concerns itself with whether essential features of modeling are lost or backgrounded, as such a scheme is imposed. Fortunately, this

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Fig. 5.1  A seven-step modeling cycle. (Blum, 2015)

question can be pursued empirically rather than dogmatically. For instance, we might investigate what conditions create a very high degree of “fit” between modelers’ actual experience and behavior and this model of modeling. One conjecture might be that the fit increases with the experience and sophistication of the modeler. This would offer a basis for viewing the diagram as capturing an “expert model” of modeling, to which learners are converging as they “learn modeling.” However, several existing studies point away from this conjecture (Lesh & Zawojewski, 2007; Lester & Kehle, 2003). Outside of familiar territory, even members of mathematics-­ intensive professions exhibit much more messy, idiosyncratic, and situated modeling cycles (diSessa, Hammer, Sherin, & Kolpakowski, 1991; Gainsburg, 2006; Hall, 1999). An alternative conjecture would be that the modeling cycle best describes actual modeling practice in situations where modelers see the problem situation as related to ones they have encountered in the past, that is, when they interpret a problem as an application-with-adaptation of a familiar problem type. Such a conjecture, quite testable, would point to findings that delimit the utility of modeling cycle diagrams to modeling contexts emphasized by the applications tradition. This would make such diagrams less applicable to settings where young learners feel pressed to develop mathematical constructs that they see as innovative responses to the world. A second, related, feature of the international work on modeling competencies, also associated with cycle diagrams, creates another difficulty in adopting it with young learners and research in the MMP. The competencies articulated in this work use the cycle diagram itself to derive categories of modeling competencies and sub-­ competencies. For instance, Galbraith and Stillman’s (2006) study of “blockages” that students experience at different stages and transitions in the cycle suggests an approach to removing these by giving students practice in the stages or promoting increasing approximation of the cycle’s description in their modeling activity.

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Such approaches align with the heuristics tradition, in taking modeling as a topic area, rather than as an organic practice. A more “situated” perspective like the MMP would argue that, rather than separating modeling as a distinct topic for study and development, we must instead accept that modeling only fully comes to life in rich disciplinary and real-world contexts. Accounts of modeling in terms of a cycle diagram may offer descriptions that have utility for categorizing work or for providing mnemonic accounts for the modelers themselves—just as post hoc accounts of problem-solving offer valuable descriptions of heuristic strategies. But an MMP perspective would conjecture that these would be useful more for modelers (individuals or communities) to convert their own work into resources for the future than as schemes to be taught in the abstract or descriptions of modeling “in general.”

5.7 Perspectives on Modeling Competencies from Chapters in This Section Returning to the chapters of this section, we read them as contending with important developmental questions, in the context of this literature. They provide evidence that one should not (at least exclusively) use the idea of modeling competencies as a directive to “pre-teach” or to create gatekeepers to modeling that have the purpose of preparing the ground. Such a plan would have the effect of deferring young learners’ access to authentic mathematical modeling “until they were ready.” At which point, some proportion of learners will have already decided that mathematics is not a subject for them. Instead, we read the chapters of this section as celebrating young learners’ capacities to engage in rich, generative, and creative modeling work, given appropriate conditions. The research, then, seeks to uncover, foster, and honor students’ competencies as resources for learning, as well as to identify the conditions under which they can receive full exercise. This perspective is particularly evident in English’s and Jung and Brand’s (this volume) citations across a range of allied research, identifying and characterizing students’ capabilities (e.g., Bonotto, 2009; Mousoulides & English, 2008; Mousoulides, Pittalis, & Christou, 2006), when supported by teachers’ expectations (English, 2003) and activity design (Gadanidis & Hughes, 2011). English’s chapter (this volume) also contributes to a competencies-as-resources perspective, in each of her themes. These researchers describe young learners’ progress in developing models, illustrating concretely the kinds of reasoning that are within reach of students at the elementary level. Together they identify design features that produce evidence of students’ ways of thinking—promising directions for the enterprise of assessing modeling. Likewise, the teacher-level studies that Jung and Brand (this volume) review provide further basis for both supporting and assessing modeling among young learners.

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Anhalt et al. (this volume) come closest to aligning with a modeling-cycle-driven view of competencies, in developing mathematical modeling thinking (MMT) as a foundation for modeling. Their goal of identifying “activities that can engage certain ways of thinking necessary for the eventual development of modeling competency” (emphasis added) threatens to push modeling to older grades. However, this statement is intended to be nonexclusive—that MMT might be developed in a focused way alongside engagement in modeling activities. Thus, their work can be seen as integral to a developmental MMP approach. In particular, though they cite research on sources of frustration with modeling activities (Bleiler-Baxter, Barlow, & Stephens, 2016; Ng, 2013; Zawojewski, Lesh, & English, 2003), the chapter does not advocate eschewing modeling activities in favor of “precursor” MMT activities. Instead, on the strength of the democratizing theme (cf Kaput 1994, 1995) of the early algebra work that inspired Anhalt et al., they support a “both-and” approach— engaging in age-appropriate modeling activities while also taking a modeling lens on a broader set of activities that foster a modeling approach to situations in the world. Finally, we see Leavy and Hourigan (this volume) as suggesting a potentially complementary approach to Anhalt et al. The framework for informal inference they adopt (Konold, Higgins, Russell, & Khalil, 2015; Makar & Rubin, 2009; Zieffler, Garfield, delmas, & Reading, 2008, among others) is built upon ways of thinking about data that young children have been observed to engage. They aim to design activities that give learners opportunities to activate these ways of thinking and to move beyond their limits toward ever more powerful patterns of mathematical activity (here, inference) along a continuum where milestones have been identified. Specifically, Leavy and Hourigan (this volume) base their designs on a developmental view of the subject matter of data modeling and inference, providing learners with situations that create the need for concepts that may be within or just beyond their current grasp. Having such opportunities can be valuable to learners, even when they come short of a full solution. For one thing, such situations provide researchers and teachers rich examples of ways of thinking that students do have, suggesting designs that may support their growth. These perspectives are complementary—on one hand, Anhalt et al. (this volume) analyze challenges and obstacles experienced by novices in modeling (including pre- and in-service teachers as well as children) to develop their six practices and MMT constructs; on the other, Leavy and Hourigan (this volume) observe the capacities of children in different settings and interpret these capacities as seeds of modeling practices. Further, Anhalt et al.’s six practices may have the right nature and “grain size” to support teachers in being on the lookout for opportunities for their students to exercise and reflect on emerging modeling competencies. Those opportunities can appear in many places—not only in full modeling activities,but also in “MMT activities” designed specifically for those purposes.

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5.8 Directions for Future Research The chapters of this section outline promising future research directions for modeling research. In terms of methods, those we see as most promising cast mathematics education research as a design science (Collins, 1992; Lesh & Sriraman, 2005). A range of research products other than findings that “x works” are valuable results in such an enterprise. These include tools for capturing and documenting experiences and for supporting reflection by modelers. And of particular value are approaches that can enable a range of design experiments in the MMP tradition to be accumulative. We have seen that the placeholder of “competencies” has power for convening conversations: How can these conversations build a growing foundation of research findings about modeling and idea development? Such accumulation depends on a vision of the field’s knowledge. The larger structures of knowledge in design sciences reflect a balance between adaptability and method, between heterogeneity and analogy. With this in mind, we briefly describe three appearances of this balance—in tiers, timescales, and tools for modeling—and suggest how they might frame future work in modeling research. Modeling exhibits heterogeneity and analogy across tiers. It is noteworthy that much of the work reviewed in the chapters of this section organize their thinking around the construct of three-tiered teaching experiments (Lesh & Kelly, 2000), involving students, teachers, and researchers in parallel modeling activity. Students make sense of problems and engage in modeling activity; teachers make sense of students’ work and build models of their modeling, and researchers make sense of the emerging system. Each tier is unique but analogies can be discerned. We note just two lines of thought among many that could drive future work here. First, the work that English (this volume) describes in her account of modeling in cultural and community contexts might make significant contributions by exploring ways to stabilize a fourth tier of modeling (communities), as well as construct new relations across tiers of modelers. Such efforts not only increase the power and relevance of modeling in students’ experience, but they also provide new, diverse settings for studying how individuals and groups of different kinds interpret the world to formulate coherent strategies for effecting change. A second line of work associated with tiers of modeling might pursue studies taking groups of learners as the unit of analysis. In particular, understanding the growth of classrooms as modeling communities is an important area for both research and practice. The norms and microcultures of classrooms have enormous effects on their members’ actual experience of modeling. For instance, it is possible (and even common) for a highly traditional classroom not only to experience frustration with a modeling task but actually to convert it into an application problem or other more familiar activity type. However, the reverse can also be true, as shown in our reading of Anhalt et al.’s chapter (this volume). Researchers, teachers, and students may learn to “see the modeling potential” in activities that were not designed as modeling activities; and with a supportive classroom culture, children may begin to seek and find spontaneous opportunities for interpreting the world with mathematics.

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Modeling also exhibits heterogeneity and analogy across timescales. Extending the reach of modeling to course-sized (or larger) scales of idea development opens up new questions for research (Eames, Brady, Jung, Glancy, & Lesh, 2018), as well as new means for gauging the power of a modeling approach (Lesh, Carmona, & Moore, 2009). A line of work in this direction in the last decade has focused on Model Development Sequences (MDSs) (Ärlebäck, Doerr, & O’Neil, 2013; Brady, Eames, & Lesh, 2015, Doerr & English, 2006; Hjalmarson, Diefes-Dux, & Moore, 2008; Lesh, Cramer, Doerr, Post, & Zawojewski, 2003; Zawojewski, Magiera, & Lesh, 2013). Space does not permit a full sketch of this research, but MDSs and alternative structures aim to create comparability across MEA-centered studies to support both coherence and adaptability in implementations, and to extend the temporal reach of MMP research in ways that support developmental perspectives. Finally, modeling exhibits heterogeneity and analogy across tools and representations. A strong trend in mathematical work outside of school is the emergence of computational thinking and modeling. Jung and Brand (this volume) note a trend and future direction toward incorporating technology into modeling, and the pace is accelerating. Across all disciplines, approaches to making sense of the world increasingly depend on models that are nuanced, conditional, and based on numeric patterns rather than simple, uniform, and based on analytic functions. To be prepared to reason about real-world problems in science and society, a complexity and emergence lens is needed—and one that enables generative and creative reasoning, rather than treating simulations of complexity as authoritative black boxes. The result is that accessible, “glass box” computational models of complex systems are increasingly powerful and essential representational forms. Learning-centered agent-based and systems-dynamics modeling environments need to be integrated into research in the MMP tradition, building on analogies with other representational media while attending to distinctive features. With careful design, technology-­ enhanced modeling environments can increase students’ agency and representational expressivity, rather than foreclosing the interpretive and expressive dimensions of modeling activity. Data modeling activities using TinkerPlots (Konold & Miller, 2005) provide a good historical inspiration. And work with young learners is a particularly good setting to nurture designs that avoid a “technocentric” approach and continue to see the computer as a children’s machine (Papert, 1988), where learners’ constructions are truly thought-revealing artifacts (Lesh et al., 2000).

5.9 Conclusion Seeking to support and understand modeling, MMP research foregrounds fundamental questions, such as What kind of thing-to-learn is modeling? We have suggested modeling should not be thought of as a topic area; it is instead rooted in a stance toward mathematics, its relations to classroom work, and its connections to worlds consequential to students. If we can arrange environments in which learners construct mathematical descriptions and tools as strategic means to illuminate their

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world or effect change within it, we will have gone a long way toward addressing the challenges of making modeling accessible to all students. In the face of messy, realistic modeling activity, schematic descriptions that reduce easily to lists of components or to simple diagrams seem to fall short. Following the arguments of the chapters in this section, MMP research on modeling with young learners can focus instead on (a) characterizing the kinds of accessible situations that create the need for powerful mathematical ideas and, then, on (b) the kinds of connections that young learners have made and are prepared to make among these powerful ideas. Further, if we take seriously the proposition that models are interpretive systems, we can view modeling activity as making strategic but also possibly “playful” choices about how to look at the world. We suggest that playfulness can be a powerful entry point for all learners and perhaps especially for young learners. For example, consider playful approaches to: • Problem finding and problem-posing (and the relation to introducing structure to turn a “mess” into a “problem” (Schön, 1983)). • Strategic assuming (introducing assumptions, where the end goal may be to simplify a problem, but where there is value simply in seeing the effects of different assumptions). • Constructing connections (including making connections to one’s own experience). Students may notice how a problem changes or deepens based on such connections. • Flexibly representing (describing different aspects of the problem in different mathematical ways). This can reveal new features of both the problem and forms of representation. When we ask students to “play with a situation” mathematically, to see it in new lights, we open opportunities for students to be innovative and creative in modeling and to experience interpreting situations in many different mathematical ways. In closing, we assert that MMP research on modeling provides compelling paths for progress on what are debatably four of the most important problems facing mathematics educators today. These involve: 1. How should mathematics curriculum change in order to provide better foundations for the future in an increasingly technology-based age of information? 2. How should assessment and accountability practices change in order to deal more effectively with issues of equity, diversity, and opportunity for all? 3. How can an emphasis on models and modeling facilitate teacher development by turning everyday in-class teaching experiences into learning experiences for teachers? 4. How should learning activities change in situations where modern technologies are available? The chapters in this section indicate ways of advancing the field’s response to these and other questions.

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References Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Montana Mathematics Enthusiast, 6(3), 331–364. Ärlebäck, J. B., Doerr, H. M., & O’Neil, A. H. (2013). A modeling perspective on interpreting rates of change in context. Mathematical Thinking and Learning, 15(4), 314–336. Bleiler-Baxter, S. K., Barlow, A. T., & Stephens, D. C. (2016). Moving beyond context: Challenges in modeling instruction. In C.  Hirsch (Ed.), Annual perspectives in mathematics education: Mathematical modeling and modeling mathematics (pp. 53–64). Reston, VA: National Council of Teachers of Mathematics. Blomhøj, M., & Højgaard Jensen, T. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning. Teaching Mathematics and Its Applications, 22(3), 123–139. Blomhøj, M., & Højgaard Jensen, T. (2007). What’s all the fuss about competencies? In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (pp. 45–56). New York, NY: Springer. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In The proceedings of the 12th international congress on mathematical education (pp. 73–96). Cham, Switzerland: Springer. Blum, W., & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling: Education, engineering, and economics (pp. 222–231). Chichester, UK: Horwood. Bonotto, C. (2009). Artifacts: Influencing practice and supporting problems posing in the mathematics classroom. Proceedings of PME, 33(2), 193–200. Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM— International Journal on Mathematics Education, 38(2), 86–95. https://doi. org/10.1007/BF02655883 Borromeo Ferri, R. (2007). Modelling problems from a cognitive perspective. In C.  Haines, P.  Galbraith, W.  Blum, & S.  Khan (Eds.), Mathematical modeling: Education, engineering, and economics (pp. 260–270). Cambridge, UK: Woodhead Publishing Limited. Brady, C., Eames, C., & Lesh, R. (2015). Connecting real-world and in-school problem-solving experiences. Quadrante: Revista de Investigaçião em Educaçião Matemática [Special Issue, Problem Solving], 26(2), 5–38. Collins, A. (1992). Toward a design science of education. In New directions in educational technology (pp. 15–22). Berlin, Heidelberg: Springer. Czocher, J. (2016). Introducing modeling transition diagrams as a tool to connect mathematical modeling to mathematical thinking. Mathematical Thinking and Learning, 18(2), 77–106. diSessa, A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-­ representational expertise in children. Journal of Mathematical Behavior, 10(2), 117–160. Djepaxhija, B., Vos, P., & Fuglestad, A. B. (2017). Assessing mathematizing competences through multiple-choice tasks: Using students’ response processes to investigate task validity. In G. Stillman, W. Blum, & G. Kaiser (Eds.), Mathematical modelling and applications: Crossing and researching boundaries in mathematics education (pp.  601–611). Cham, Switzerland: Springer. Doerr, H. M., & English, L. D. (2006). Middle grade teachers’ learning through students’ engagement with modeling tasks. Journal of Mathematics Teacher Education, 9(1), 5–32. Eames, C., Brady, C., Jung, H., Glancy, A., & Lesh, R. (2018). Designing powerful environments to examine and support teacher competencies for models and modelling. In Lehrerkompetenzen zum Unterrichten mathematischer Modellierung (pp.  237–266). Wiesbaden, Germany: Springer Spektrum. English, L. D. (2003). Reconciling theory, research, and practice: A models and modelling perspective. Educational Studies in Mathematics, 54(2–3), 225–248.

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Part II

Identifying the Knowledge of Content and Pedagogy Needed for Mathematical Modeling in the Elementary Grades

Chapter 6

Elementary Teachers’ Enactment of the Core Practices in Problem Formulation through Situational Contexts in Mathematical Modeling Jennifer M. Suh, Kathleen Matson, Sara Birkhead, Samara Green, MaryAnne Rossbach, Padmanabhan Seshaiyer, and Spencer Jamieson

6.1 Introduction Problem formulation is one of the critical first steps in mathematical modeling (MM). This important step requires modelers to identify a problem that requires modeling. Our chapter examines how elementary teachers and students learn to be problem posers as they learn to enact mathematical modeling in their own classrooms. We collected data from teachers implementing MM in their classroom starting from their planning process to the enactment phases of mathematical modeling tasks. Data was collected from lesson studies and teacher interviews exploring the processes of mathematical modeling. We focused on better understanding how teachers involve elementary students in the problem formulation process as they co-construct the problem and how they work through making assumptions, building a solution or a model, and revising their model as they relate back to their problems. We also analyzed the context in which the problem formulation occurred: community-­based, school-based, or curriculum-based contexts and how it impacted authentic learning. 1. How do elementary teachers develop the core practices of problematizing and formulating mathematical modeling tasks and design early modeling experiences to engage elementary students in this process? J. M. Suh () Mathematics Education, George Mason University, Fairfax, VA, USA e-mail: [email protected] K. Matson · S. Birkhead · P. Seshaiyer George Mason University, Fairfax, VA, USA S. Green · M. Rossbach · S. Jamieson Fairfax County Public Schools, Fairfax, VA, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_6

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2. How do teachers plan and orchestrate the process of moving students towards making assumptions, defining variables, and providing constraints to advance meaningful mathematical learning?

6.2 Identifying Practices that Support Problem Formulation in Mathematical Modeling Mathematical modeling describes an iterative and dynamic cyclic process that supports the translation of real-world problems into a mathematical language (Blum, 2011; Bliss, Fowler, & Galluzzo, 2014; Lesh & Doerr, 2003a, 2003b; Pollak, 2003; Suh & Seshaiyer, 2017). The MM process involves posing a problem, making assumptions, establishing relationships or identifying rules; solving the problem using multiple approaches which could be symbolic, physical, or abstract; and then using these solutions to compare and validate the real-world system that was approximated to start with (Blum, Galbraith, Henn, & Niss, 2007; Cirillo, Pelesko, Felton-Koestler, & Rubel, 2016). Although there have been a number of researchers documenting modeling at the elementary levels (e.g., English, Fox, & Watters, 2005; English, 2009; Verschaffel, De Corte, & Borghart, 1997) more of the research on modeling has been in secondary mathematics education or at the tertiary levels (Pollak, 2016). Recently, with the release of the Common Core Standards for Mathematical Practices (NGA & CCSSO, 2010) which includes Standard 4: Model with mathematics, elementary teachers have become more aware of what mathematical modeling entails (Pollak, 2016). However, teaching through mathematical modeling is ambitious instruction for many elementary teachers (Lampert et  al., 2013) especially because traditional mathematics methods courses have not included mathematical modeling as part of the elementary curriculum (ACEI, 2007) for preparing elementary teachers. In many mathematics curriculum, models and modeling are used to describe concrete materials to help students develop abstract mathematical thinking (Lesh, Cramer, Doerr, Post, & Zawojewski, 2003b) or as mathematical representations such as the area model or multiplication array that illustrate a mathematical concept (Cirillo et al., 2016; Hirsch & McDuffie, 2016). Concurrent with the Common Core Standards for Mathematical Practice (CCSMP), the GAIMME Report, Guidelines for Assessment & Instruction in Mathematical Modeling Education (COMAP & SIAM, 2016), strongly encourages introducing modeling to young children to ensure the development of modeling specific competencies across all levels of schooling. Internationally, modeling has been a central topic for mathematics application in school mathematics (Blum, 2002; Pollak, 1997). The notion of mathematical modelling has historically been a cornerstone of the Programme for International Student Assessment PISA framework for mathematics (e.g. OECD, 2018). The International Commission on Mathematics Education and the International Community of Teachers of

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Mathematical Modelling and Applications have been focused on research topics related to teaching and applying modeling in school mathematics and exploring teachers’ and students’ mathematical modeling competencies (Blum et al., 2007; Blum & Ferri, 2016; Lesh, Galbraith, Haines, & Hurford, 2010; Niss, 2013; Kaiser, 2007; Kaiser, Blum, Borromeo Ferri, & Stillman, 2011; Stillman, Kaiser, Blum, & Brown, 2013; Stillman, Blum, & Biembengut, 2015). According to PISA’s definition, Mathematical literacy is an individual’s capacity to formulate, employ and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena. It assists individuals to recognize the role that mathematics plays in the world and to make the well-founded judgements and decisions needed by constructive, engaged and reflective citizens. (OECD, 2018, p.4)

The PISA definition links mathematical literacy to MM and the indicators outlined under each of the mathematical processes “formulate”, “employ”, and “interpret” connect to the processes of mathematical modeling (OECD, 2018). The formulating process involves identifying opportunities to use mathematics in problem situations by recognizing mathematical structure in the situation and representing the contextualized problem in a mathematical form. The employing process involves performing computations and manipulations and applying concepts and facts to arrive at a mathematical solution to a problem formulated. The interpreting process involves reflecting upon mathematical solutions, interpreting them in the context of a real-­ world problem, and determining whether the results or conclusions are reasonable (see Table 6.1). One of the first steps in the modeling process is problem formulation. This step requires students to use higher-order, critical thinking skills, beyond just computational and arithmetic skills, to analyze problems that can be addressed through MM. We contend that in order for elementary teachers to provide modeling experiences for early learners, they need professional development focused around formulating situations mathematically. In fact, a deeper look at the PISA Mathematics Framework (OECD, 2018) helps decompose the practice of formulating Table 6.1  Indicator of mathematical literacy Mathematics process Formulate

Employ

Interpret

Indicators 1. Identify the mathematical aspects of the problems contained in the real context situation and the important variables. 2. Translate the problem into mathematical language and representation. 1. Design and implement strategies to find mathematical solutions. 2. Use and switch between different representations in the process of finding solutions. 1. Interpret the results of mathematical answers to the initial problem. 2. Evaluate the reasonableness of mathematical solutions in the context of the real-world problems.

Adapted from the PISA Framework (2015)

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mathematical modeling problems into several key skills and areas of knowledge required of the modeler. For early modeling experiences, it is the teacher who must have these skills in order to present opportunities for young mathematicians to encounter situations that can lead to successful MM. The necessary knowledge and skills include: identifying the mathematical aspects of a problem situated in a real-world context and identifying the significant variables; recognizing mathematical structure (including regularities, relationships and patterns) in problems or situations; simplifying a situation or problem in order to make it amenable to mathematical analysis; identifying constraints and assumptions behind any mathematical modelling and simplifications gleaned from the context; recognizing aspects of a problem that correspond with known problems or mathematical concepts, facts or procedures; representing a situation mathematically, using appropriate variables, symbols, diagrams and standard models; translating a problem into mathematical language or a representation. (OECD, 2018, p. 53).

All of these indicators require identification of a situation that has the potential for productive problem formulation from which a modeling task can be formulated. Galbraith (2007a.) outlined a criteria for evaluating modeling problems using six principles to determine whether a situation is suitable for model development. The criteria includes six principles that help evaluate if a problem can be posed: 1 . has relevance and is motivating with a genuine link to the real world of students; 2. is accessible in that one can identify and specify mathematically tractable questions from a general problem statement; 3. has feasibility of approach involving (a) the use of mathematics available to students, (b) the making of necessary assumptions within the context, and (c) the assembly of necessary data; 4. has feasibility of outcome, in that students can arrive at the solution and have opportunities to iterate and refine the model; 5. can be validated, in that there is an evaluation procedure available that enables solution(s) to be checked for (a) mathematical accuracy and (b) appropriateness with respect to the contextual setting; 6. has flexibility, in that the developed model can be useful and reusable, while keeping the integrity of the real situation. (p. 55) We used this criteria to help teachers in the design process and to evaluate their modeling tasks. In examining research on how mathematical modeling problems are formulated, Galbraith, Stillman and Brown (2013) reported on turning ideas into modeling problems. They describe that “a nucleus of an idea” is developed into a modeling problem and extended into related problems closer to the “personal experience of adolescents” (p.  133). This focus on contexts that interest students relates to Galbraith’s (2007a, 2007b) first principle of relevance and motivation. Mathematical modeling should be used “to motivate students to study mathematics by showing them the real-world applicability of mathematical ideas” (Zbiek & Connor, 2006, p. 89). Julie (2007) points out that what students might perceive as “personally relevant to them” is also “transitive and time-dependent”, but it is often argued “that

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personal ownership of problem situations can be fostered by contexts found desirable by learners” (p. 201). In other words, when posing a mathematics modeling problem, it is important to choose a context familiar to students because “the formulation of the real-world situation itself requires demanding a priori knowledge” (Caron & Bélair, 2007, p. 127). Galbraith, Stillman, and Brown (2013) also emphasized the importance of the practice of making assumptions, “traditionally in the formulation phase much is made of assumptions that need to be made in setting up the model for solution. However, it is often overlooked that assumptions need to be invoked at all stages of the modeling process” (p. 142). Suh and Seshaiyer (2019) conducted a case study of two lesson study teams enacting modeling in elementary classrooms and identified the critical norms necessary for successful enactment of modeling by teachers and students. They found that there was an interplay between the pedagogical norms and student participation norms that promoted students’ engagement in the process and the teacher’s skillful orchestration of the modeling cycle. They found five important pedagogical normed practices critical for teachers including being able to (a) provide an authentic and personally relevant context for students to see mathematics with multidisciplinary connections; (b) promote mathematics discussions to elicit the use of mathematical modeling to describe, optimize, predict, and make decisions using mathematics; (c) plan for appropriate scaffolds to help students revise and refine math ideas in the iterative modeling process; (d) introduce open-ended math problem posing with multiple entry points and creative solutions; and (e) establish socio-mathematical norms for collaboration, participation, and promoting productive struggle. In addition, they found five complementary participation norms for students including, being able to (a) make connections between real world mathematics situations and the mathematics learned in school; (b) use mathematics they know and “new” math to solve real-world problems that may require more diverse skills of mathematics; (c) expect to be asked questions to revise, refine, and reflect on mathematical ideas; (d) embrace and be comfortable with open-ended problem posing and its complexities and being creative and critical problem solvers; (e) contribute in the mathematics classroom by listening, commenting on, and questioning their classmates respectfully and persevering through complexity. In addition, findings from successful implementation of mathematical modeling by Suh, Matson and Seshaiyer (2017) revealed effective pedagogical practices teachers used in enacting mathematical modeling with elementary students. Matson (2018) found teachers need to balance affordances and constraints by accepting and normalizing the tensions created by the openness and messiness of mathematical modeling and by having tools to help navigate these tensions. Skillful teachers demonstrated facilitative moves during the enactment of modeling finding the “sweet spot” along the continuum of openness and providing a focus for the mathematical learning; seizing “teachable moments” using their deep curricular knowledge; using mathematical modeling process tools and student-generated artifacts to thoughtfully orchestrate the MM Process and mathematical discourse; and used pedagogical moves that promoted students’ twenty-first century skills (including communication, critical thinking, creativity and collaboration). These facilitative

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Fig. 6.1  The interplay between teacher facilitator and student modelers during math modeling

moves that the teachers killfully orchestrated in the modeling cycle were critical in students’ engagement in the process. As shown in Fig. 6.1, teachers had to anticipate the potential mathematics that would be elicited by a modeling task and how the model could connect across the mathematics curricular learning progression. In addition, teachers had to unpack the problem situation with students considering the information needed to frame the problem as well as asking students to define the variables that mattered to the problem situation, which all happened as students were in the problem formulation phase of the cycle. Model-Eliciting Activities (MEAs) were first developed as tools to help researchers understand students’ conceptual models of cognition and problem-solving behavior and were later expanded as an approach to support students’ problem-­ solving competencies (Lesh & Zawojewski, 2007). Lesh, Doerr, Carmona, and Hjalmarson (2003a) compared problem solving and modeling. Problem solving, they explained, produces “one-word, one-number, or one-sentence answers” (p.  225). Modeling, in contrast, develops models through an iterative process of interpreting a problem situation and assessing the product by its usefulness and generalizability. As the modeling perspective took hold in the education community, educators engaged in professional development and research found benefits in co-planning and designing MM tasks with teachers (Jung & Brady, 2016). Considering the research on the importance of problem formulation in maximizing deep learning and the importance of personally-relevant and familiar contexts in motivating students in the modeling process (Suh et  al., 2018), we propose that teachers who know their students the best, are ideal designers of modeling experiences. Addressing the call from mathematics education to define core practices to ambitious teaching, (McDonald, Kazemi, Kavanaugh, 2013) with a call for a

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common language and collective activity, we embarked on this study to start to define the core teaching practices and routines that support early mathematical modeling experiences. We recognize that mathematical modeling has an open and non-routine nature with problem posing and formulation, however, we believe that teachers as designers could benefit from structured support when they are novices “trying on” modeling in their classrooms. The aim of this chapter is to better understand how teachers learn to formulate mathematical modeling tasks, problematize these tasks for their students, and the support they need as they take on the role of designers of early mathematical modeling experiences.

6.3 Research Questions This current study delves deeper into the core practice of problem formulation for teachers which focuses primarily on the first two phases in the MM cycle: (a) posing a problem, which we referred to as the act of “problematizing,” and (b) making assumptions and defining variables and constraints, which we referred to as the act of “mathematizing.” In mathematizing, features of the real-world situation become central to the mathematical learning and determine the mathematical structures that will be applied to solve the problem. It includes identifying situational features that require the use of mathematics, locating data and information needed to solve the problem, and making assumptions and defining variables to solve the problem. In order to decompose the core practice of problem formulation, we focused on the following two research questions: 1. How do elementary teachers develop the core practices of problematizing and formulating mathematical modeling tasks and design early modeling experiences to engage elementary students in this process? 2. How do teachers plan and orchestrate the process of moving students towards making assumptions, defining variables and providing constraints to advance meaningful mathematical learning?

6.4 Context for Our Study This study was part of a 3-year research project focused on researching and evaluating the effects of professional development (PD) in mathematical modeling for elementary mathematics teachers. The data collected for this study comes from 3 years between August 2015 and August 2018. Using a networked improvement community model, we intentionally designed our professional development using a university-school partnership and organized it with leadership from both communities to develop a colleagueship of expertise (Bryk, Gomez, Grunow, & Le Mahieu, 2015, p. 195), a community of academic and clinical experts deliberately assembled to address a specific improvement problem. According to Bryk, Gomez, Grunow,

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and LeMahieu (2015), the colleagues involved are improvers seeking to generate strong evidence about how to achieve better outcomes more reliably (Byzk et al., 2015). With this colleagueship of expertise, we focused on a well-specified common aim to develop elementary teachers as modelers who can then learn and lead elementary students and other teachers through the modeling process. In fact, some of these teacher designers became our subsequent year’s instructional leaders – creating a system for growing instructional leadership within the district. Using design research as a method of inquiry, we first worked with teacher leaders to develop a deep understanding of what we meant by mathematical modeling in the early grades and developed a PD module delivered during a design institute. A focus of the institute was to share how implementing mathematical modeling can bring rigor to the mathematics curriculum being taught in elementary grades. Networked improvement communities (Byzk et al., 2015) use design research and improvement science to develop, test, and refine interventions and to accelerate their diffusion out into the field. The nature of these communities supports effective integration into varied educational contexts. During phase one of the project (see Fig. 6.2), a cohort of 25 elementary teachers was identified in collaboration with the school district. These elementary teachers were then immersed in a weeklong summer institute to experience mathematical modeling firsthand and improve their content-­specific knowledge of mathematical modeling. Teachers were recruited in school teams which allowed for collaboration beyond the design institute with ongoing semimonthly design meetings in the fall. The second phase of the project included a collaborative coaching and lesson study cycle to support teachers as they

Fig. 6.2  Design institute cycle with lesson study and teacher research

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designed and implemented MM in their elementary classrooms. In lesson study (Lewis, 2015), educators choose an aim for improvement (i.e., implementing mathematical modeling), agree on how they will recognize improvement (i.e., student engagement and rigorous mathematics content), identify the changes that might produce improvement (i.e., teaching through modeling process), and test these changes in the lesson study cycle using the core framework of improvement science – the plan-do-study-act (PDSA) cycle. Lesson study involves studying the curriculum, planning a research lesson, conducting and observing the research lesson with a focus on student learning, and debriefing and making improvements (Lewis, 2002; Suh & Seshaiyer, 2014). During the 1-week summer design institute (5 full-day sessions for a total of 40 h), our teacher participants gained knowledge about the mathematical modeling process as they became modelers themselves, directly engaging in the process of posing a problem, making assumptions, defining variables, building a solution, analyzing their model and relating it back to the real world phenomenon, and revising their model based on needs (see Fig. 6.1. Mathematical Modeling Process). After experiencing the modeling process as learners, teacher participants also reflected on the MM task and discussed the teacher moves necessary to facilitate the task and elicit important mathematical ideas through classroom discourse. The second phase of the study, the school-based lesson study, allowed teacher designers to test this innovative approach by implementing the designed lessons in their own classrooms. Teachers collaborated on designing a MM task, and the professional developers/ researchers provided support and assistance. A host teacher taught the first cycle of the lesson while other teacher participants observed student learning. The debrief allowed teachers to discuss the designed task, the enactment of the modeling process, and the related student mathematical thinking elicited through the lesson activity. Based on their debrief, teachers adapted the lesson for the second iteration in their own respective classrooms. After the fall lesson study, the teacher teams presented the outcome of their first attempt at mathematical modeling at a lesson study symposium, where they shared their excitement and voiced their challenges in implementing this innovation. This symposium allowed for networked communities of schools to share their lesson study outcomes to better understand how implementation of mathematics modeling worked across different school teams, for whom, and under what conditions. We were able to recruit school teams from diverse communities that served high populations of English language learners (ELLs), classrooms that served advanced academic (gifted) students, heterogeneous general education classrooms, and inclusive classrooms with high numbers of special needs students. Lesson study with school teams allowed for us to study the nature of mathematical modeling implemented in a variety of settings while treating “variation in implementation and setting as important sources of information and [providing] tools to grasp and learn from variation in order to redesign both the intervention and the system” (Lewis, 2015, p, 55). Finally, we conducted teacher research with six teachers each spring where we deeply engaged teachers in researching their implementation of modeling to better understand how implementation varied for diverse settings and populations.

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6.5 Data Sources and Analysis The design of our study involved continual monitoring and recording of teachers as they enacted multiple cycles of mathematical modeling in their classrooms. For each implementation year, we collected data sources including the lesson study artifacts (planning materials, lesson plans, classroom observations, debrief notes, student artifacts) and a post survey assessing teachers’ confidence in enacting mathematical modeling in their classrooms with both a Likert scale and open-ended responses. In addition, each spring, we conducted case studies of six teachers which included detailing their continuation with modeling lessons through field notes, researcher memos, and interviews. Data analysis included content analysis examining documents and student artifacts in addition to interview data with teachers reflecting on the planning and enactment of mathematical modeling. We analyzed the open-ended responses that asked about each phase of the modeling process to look for themes related to how teachers formulated and orchestrated modeling with their students. We also analyzed all lesson study plans to identify how teacher designers formulated modeling problems and how different problem types offered a variety of pathways for mathematical learning in the elementary grades. Through our analysis, we wanted to identify how teachers designed meaningful mathematical modeling experiences for elementary students to deepen their learning. We examined how elementary teachers and students learn to be problem posers as they learn to enact mathematical modeling in their own classrooms. Over the three years of the project, we collected survey data from 50 teachers and conducted case studies of 12 teachers implementing MM in their classroom starting from their planning process to the enactment phases of mathematical modeling tasks through lesson studies and interviewed teachers on their process of problem formulation. To focus our study, we examined the aspects of the modeling process which teachers felt the most confident or most challenged by as they planned and orchestrated math modeling in the elementary grades. To do this, we analyzed our descriptive data from the survey that measured the levels of confidence teachers reported during the modeling process. In addition, we used the qualitative data to delve deeper into teachers’ successes and challenges. The teacher self-reported survey on their confidence in orchestrating mathematical modeling in their own elementary classrooms was on a 5-point Likert scale. Results of the means were not statistically significant across the items but indicated that teachers were generally feeling a high-level confidence (see Table 6.2). In fact, when we collapsed the confident and very confident categories (levels 4 and 5), 75% of teachers reported feeling confident about posing mathematical modeling questions. While in general, teachers indicated confidence in problem posing, they found that making assumptions and defining variables to narrow and focus on the mathematical problem was the most challenging part of problem formulation. This phase had the lowest mean on the teacher survey with only 58% of the teachers feeling confident or very confident in allowing students to make assumptions to narrow

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Table 6.2  Teachers self-report on level of confidence orchestrating mathematical modeling Survey questions specifically targeted to assess teachers’ confidence in orchestrating mathematical modeling Posing a problem – How confident do you feel posing math modeling questions? Making assumptions and defining variables – How confident do you feel allowing students to make assumptions to narrow their investigation to a focused problem? Building a solution – How confident do you feel facilitating the building solution phase without doing the math for the students? Analyze and create a model – How confident do you feel facilitating discussion with students around the model they developed? Revise, refine, and report – How confident do you feel supporting students as they critique and revise their models to maximize learning?

Mean and SD (n = 52) Mean = 3.94 (SD = 0.67) Mean = 3.51 (SD = 0.76)

% of teachers feeling confident 75%

Mean = 3.77 (SD = 0.70)

66%

Mean = 3.72 (SD = 0.94)

62%

Mean = 3.76 (SD = 0.81)

61%

58%

their investigation to a focused problem. This contrast between teachers’ confidence with problem posing and their relatively lower confidence with guiding students mathematizing through making assumptions led to our focus on the core practices at the problem formulation phase. To delve deeper and decompose the core teaching practices that supported problem formulation, we analyzed the qualitative responses from each Likert item as well as selected a purposeful sampling of exemplar teacher designers to better understand the interplay between the teacher and the students in creating productive mathematical modeling experiences. The three teacher designers showcased in this chapter came from the case study teachers from the 3-year project. Anne, a sixth-­ grade teacher, was from our first year’s cohort and created a unit around service learning that focused on creating a school store that generated funds for a local food shelter. Susan, a fourth grade teacher, was from our second year’s cohort, and Katie, a second-grade teacher, was from our third year’s cohort; both taught a mathematical modeling unit called the Field Trip Problem.

6.6 Results 6.6.1 Core Practice of Posing a Modeling Problem While Engaging Students in the Process For our first research question, we examined how elementary teachers developed their ability to problematize and formulate MM tasks to design early modeling experiences for elementary students. Analysis of the open-ended responses on the survey, interviews with our case study teachers, lessons, and researcher memos

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revealed four themes related to posing mathematical modeling problems, which included: 1. Leveraging Problem Posing Routines: When posing a MM problem, teacher designers adopted instructional routines for problem posing and worked on developing teacher and student questioning competence. 2. Connecting Familiar Context that Engages Students: Teachers, as designers, looked for situational features that warranted mathematizing and searched for contexts that were relevant and important to support student engagement in modeling. 3. Considering Categories of MM tasks: The modeling tasks tended to fall into four general categories where a mathematical solution or model could be used to describe, predict, optimize, and make decisions about real-world situations. In addition, teachers pushed students to think about how their solution was shareable, reuseable, or generalizable in order to evaluate whether a systematic model was created. 4. Connecting with Curricular Content: Teachers connected the need for mathematics in a modeling task with the curricular objectives for their grade level. Leveraging Problem Posing Routines  Through our analysis of the open-ended survey responses and interviews, we found that teachers became competent with searching for problems that had mathematical modeling potential. Several common routines (see Table 6.3) emerged from teachers that facilitated this process of problem formulation through problem posing. These routines included brainstorming and thoughtful questioning around topics with MM potential, using sentence starters Table 6.3  Core practices and problem posing routines Core practices and instructional routines (e.g., strategies or repertoires) Generating multiple mathematical questions to rehearse problem posing – Photo elicitation routine promotes students generating as many questions as they can think of. “Where’s the math” focuses on recognizing the math they know to apply to a real-world problem. Brainstorming all the questions related to the problem and marking the questions that mathematics can be used to solve the problem centralizes the mathematics. Facilitating classroom discourse using “mini” modeling tasks like Three-Act Math allows for a shorter modeling experience using video elicitation of a real-world situation – Problem is posed; estimate is made. Data is revealed in Act Two and solution revealed in Act Three Authoring and designing problems from personal and local contexts allows teachers to design problems from the local context. Selecting a school or community event that became a math happening allowed for authenticity. For example, the principal came in asking for the students to help in solving a problem related to an upcoming school event or community function.

Available resources to support routines 101 QS https://www.101qs. com/ Notice and Wonder (I-­Notice-­I-­Wonder) Brainstorm with “Where’s the math?” Three Act Math, (http://blog. mrmeyer.com/ category/3acts/)

Authoring problem posing through “Math Happenings” (Suh, 2007) http:// mathhappenings.onmason. com/

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to elicit and formulate student ideas, and using a structure presented in the PD institute such as a Notice and Wonder (I-­Notice-­I-­Wonder) graphic organizer, the format of Three-Act Math (http://blog.mrmeyer.com/category/3acts/), “Where’s the Math,”, or “Math Happenings” (Suh, 2007). In all cases, teachers used a photo or video to elicit a mathematical question and then revealed pieces of information about the situation until students felt that they knew enough to find a solution. Teachers used their “pedagogical license” to either keep tight control over the information they intended to share or to let the students’ ideas and interests drive the direction of the MM situation depending on the goal they set for the particular lesson. Brainstorming was a part of every MM lesson at some point in the process, often coming into play while figuring out the assumptions that would be necessary to make in order to reach a solution to the problem. However, teachers also employed brainstorming to find problems that could be addressed through MM. In formulating the problem that led to the creation of a sixth-grade school store, Anne invited students to brainstorm ways they could help their community and, with the list of activities, asked students to place an “M” where mathematics could be used. Anne started with a problem context and used that to inspire mathematical pathways. Inspired by the question, “How can we impact our school and community?”, she supported her students in identifying ideas which could be solved with mathematics and used her students’ ideas and her knowledge of the curriculum to develop authentic and relevant MM projects for her class. Anne described the importance of this authenticity saying that, “students are empowered by knowing they can really do something. It is unlike contrived math that comes from a book. I think it is important that they know it is something they can really do.” In this way, she implies that the act of problem posing through brainstorming and the subsequent MM experience can lead to improved math learning and engagement for students. Similarly, when modeling the field trip modeling task, Susan prompted the fourth-grade students to pose questions about their class field trip and marked the ones that contained mathematics potential. In this way, generating many different related questions became a key practice in these classrooms to get students comfortable with problem posing. In the second-grade case, Katie also asked her students to brainstorm, but they were focused on generating the information that they would need to find out how many buses were needed to take their school on a field trip rather than generating possible problems. Similar to the Three-Act Math structure which starts with a photo or video, she used a photo elicitation routine called “Where’s the math?” in which students looked at an image and identified elements of mathematics that could be elicited by what they saw. Another routine that helped make MM personally relevant to students was called “Math Happenings” (Suh, 2007). In this structure, students or the teacher shares a real-life occurrence in which they used mathematics, thereby, becoming authors and designers of their own modeling task. Eliciting students to share math that happened to their family, school, or their community was a way to inspire a modeling activity that bridged a productive relationship between one’s self and the mathematics. As students engaged with real-world situations, teachers noticed that familiar context

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made assumptions easier, whereas, unfamiliar contexts made it difficult for students to make relevant assumptions. Teachers selected a school or community event as a math happening, which allowed for authenticity and a common experience for all. In addition, introducing this routine to design problems from the local context allowed students to take ownership in posing modeling problems. Students started bringing in math happenings. Connecting Familiar Context that Engages Students  Analysis of the lesson study plans allowed us to distinguish the types of contexts teachers used to plan for math modeling experiences. These included (a) community-based, (b) school-­ based, and (c) family and personal contexts. Categorizing modeling tasks used for collaborative lesson study revealed that teachers chose contexts that were largely situated in the school, such as field trip planning, creating a system for lunch counts, running a class store, and planning school supply distribution. The other problems were situated in community-based contexts such as fundraiser events, helping an animal shelter, and building a community garden. Soon students made connections from class modeling activities to family events like planning a family trip, finding the best bargain, or planning a family party. Teachers described the value of having authentic and familiar contexts embedded in the mathematics tasks. Responding to the qualitative items in our exit survey, teachers reported that students’ engagement was extremely high as a result of the use of authentic contexts. Teachers described students as “excited” and “invested in their learning.” They described seeing “emotion and empathy” and “ownership and investment” from students engaged in MM.  They also described that students “became risk takers” as a result of their participating in MM. One teacher described her assessment of the change in her students’ attitudes towards math class this way, “My students are excited about Math. I asked parents how their children have changed since the beginning of the year. More than half of them have said that their child is excited about math!” Our participants found that they began to see modeling opportunities everywhere in their contexts. A common sentiment from teachers was that, “the more I incorporate math modeling into my classroom, the​easier it seems to be to develop authentic MM tasks,” recognizing that the best way to improve their teaching of MM was to continually engage in MM activities. One coach stated, “As I observe teachers in their classrooms, opportunities present themselves out of real needs.” Another teacher described an example of how MM began to emerge from her daily interactions: My school’s secretary and I were in the office one day discussing a new type of desks the school is considering purchasing for when we finish renovating. We realized I could lead the students in a math modeling task in which they decide if these desks would be a good fit for our school (the desks have triangular surfaces instead of rectangular). We discussed having the students plan room layouts centered around this type of desk. They would need to research which types of furniture are generally found in classrooms, as well as measure the furniture. They would then need to develop different table group arrangements to determine the possibilities for teachers.

With experience, modeling tasks emerged all around their schools and communities.

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When we asked teachers the survey question, “What were some successes you encountered in posing authentic problems?”, there were several MM connections that mattered to teachers as they formulated math modeling tasks. These included: Curricular Content: Degree to which the MM task connected to curricular content Connected to Lived Experiences: Degree to which the MM task connected to students’ lived experiences (temporal and physical proximity- recent time and local problems) Affect and Empathy: Degree to which the MM task connected to students’ feelings, interests, emotions, and motivation to learn (social and emotional learning) Process Skills: Degree to which the MM task would connect to critical thinking, collaborative skills, and twenty-first-century PBL-like skills being promoted in districts as career and college readiness Civic Engagement and Awareness of Social Issues: Degree to which the MM task would heighten awareness of social issues and civic duty (service learning) (e.g., using math modeling to come up with solutions to food insecurities, recycling, community gardens) Teachers felt that modeling tasks that met one or more of these criteria were more likely to be successful and to support student engagement and success with MM. Modeling tasks that were connected to students’ lived experiences tended to be tasks that were local and current having both temporal and physical proximity. Describing why these contexts support MM with elementary students, a third-grade teacher explained: We found success when using our focus lesson – ​planning for pizza in the school lunchroom and another task relating to animals in a pet shelter. Students had some familiarity with both of these contexts – most had a pet or knew of someone with a pet so they really possessed another layer of knowledge that they could apply to problem solving.

This idea that familiarity allowed students to more productively engage with a problem was evidenced in many teachers’ experiences. This recognition of relevant contexts that would support MM in the classroom led many of the teachers to develop what we called “math modeling lens eyes and ears.” Teachers started to see many more contexts that were fruitful and connected to the mathematics objectives they were teaching the more they engaged in MM problem posing. Blanton and Kaput (2005) described how teachers started to develop algebraic “eyes and ears” after professional development focused on algebraic reasoning (AR) where they planned for AR but noticed spontaneous AR happening as teachers became immersed in this habit of mind. Similarly, as Anne’s students continued to work on their school store, further spontaneous modeling opportunities emerged. The students found that they had a problem of long lines within the constraints of short store hours before and after school. To address this, Anne asked students to dig deeper into the problem of long lines and with the help of her students posed the modeling question, “How long

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does it take each customer to get through the line?”. This modeling task led students in collecting data to understand the rate of flow in order to make decisions to optimize the flow of people through the store. Anne’s questions varied from the main driving question of the MM task that kept students focused on the mathematics of the problem to day-to-day questions that pressed students to explain and justify their assumptions, choices, and reasoning (pressing questions). Anne focused her questioning on working with large amounts of data (focusing questions). She led her students, overwhelmed by large amounts of data, to arrive at use of representative samples to process 600+ surveys. Her questioning helped highlight how this data analysis approach could apply to other situations, maximizing her students’ learning. We found that asking the right types of questions was an important element of successful MM tasks. Teachers planned purposeful questions to accompany their modeling situations, but others emerged as spontaneous questions during the process. Teachers became more comfortable with allowing questions to emerge authentically as they became more comfortable with the process of MM. When interviewing Katie, we asked her why she chose the context of field trip planning for a MM task. She responded, “Each year, we take a school-wide field trip. I began to think about all of the math that goes into planning our trip. I decided to have my students use Math Modeling to figure out how many school buses we would need to go on our school-wide field trip. We even had our principal come in to ask for their help.” This highlights the way that a school context can ensure that students are comfortable with the problem and also how they can be motivated to engage, in this case being asked, by the principal, to help solve a problem. Because students were familiar with matters involved in taking a field trip, the class could successfully begin thinking about what information they knew and needed to know to solve the problem. This collective thought was recorded by the teacher on the chart (see Fig. 6.3).

Fig. 6.3  Math modeling knowledge map – planning for potential math modeling pathways

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Teacher designers found a sweet spot when certain modeling tasks hit all of the MM connections. For example, the school supply modeling task included all of the MM connections, and because they were among the consumers of the products they sold, this experience connected deeply with their ‘lived experiences’. It also heightened their awareness of conserving resources and using mathematics skills to collect data on usage of materials in order to predict the consumption of supplies. The field trip task met many of the components but did not include civic engagement and awareness of social issues. An example of how another teacher started to consider modeling problems that were related to awareness of social issues and civic duty was captured in this statement: I ​found myself looking at everyday life with the math modeling lens.​​During the past [state] election [there] were articles about voting rights and how there are less now then in the 1960s for certain groups of people. It got me thinking of the political lines and gerrymandering.… I want the kids to explore funding and the distribution of supplies, teachers, class sizes, funding, etc. I was thinking it could tie into ratios or proportions.

One common challenge with problem posing and content alignment was in finding age-appropriate, relevant, and authentic contexts that aligned to mathematical content. When contexts were too complex, teachers found that, “students not connecting with the context leads to struggle with assumptions.” Another teacher expressed, “I think finding a problem that they find interesting and have enough background knowledge to pose questions about [is challenging].” These comments reveal the difficulties that teachers encounter when interesting modeling situations require mathematics beyond students’ zone of proximal development or the context was so distant and unfamiliar to students’ lives that making assumptions was challenging. The Use of a Taxonomy of MM Tasks to Identify Mathematical Modeling Potential and Situational Features  Teachers became keen on the use of models to make generalizations and discuss how their models were reusable to describe, predict, optimize, and make decisions about real-world issues. During our design institutes, we had identified five main types of modeling categories (see Table 6.4) that were productive in elementary grades. Table 6.4  Taxonomy of mathematical modeling problem situations Descriptive Modeling – Using math to describe, represent, and analyze a situation or a phenomenon. Describe situations and quantify like finding the cost of the field trip. Optimization Modeling – Using data to find the “best” by maximizing or minimizing some variable (i.e., cost, space) in a situation. Determine optimal time, budget, resources, and logistics based on relevant data. Decision Making Rating and Ranking – Using a criteria where one assigns weights or mathematical measures as a way to rate and rank options to make decisions and prioritize. Predictive and Prescriptive Modeling – Using trends, data analysis, and probability to predict an outcome or use patterns (data analysis, probability, and algebra). STEM Data Modeling – Using the analysis of patterns in data to explain a phenomenon like in STEM lessons, science, or probability experiments to create a model for the behavior of data or phenomenon. See http://completemath.onmason.com/math-­modeling/ for Math Modeling Category Cards

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Analysis of the MM lesson study tasks developed by our teacher designers found that they fell into the first four types of models (see Table 6.5): descriptive, optimization, decision making, and predictive models. For example, in the school store task, as students compiled their surveys, Anne asked them to predict the first week’s sales in their store. This led students to compute unit rates, calculate estimated profit, differentiate between wholesale and retail costs, and draw graphs to represent their predictions (predictive). When they had an overstock of pencils, she saw potential to apply linear functions by asking students to find the best sale option to get rid of stock and promote bulk sale (predictive), and with the wait-time dilemma, students computed rate and flow issues and collected data to reduce wait-time in line (optimization). In Susan’s field trip task, students had to optimize the time they could spend at sites and maximize fun but minimize cost (optimization). All of the modeling tasks began with some descriptive modeling where mathematics was used to describe a situation or a pattern of mathematical behavior. Teachers in the upper elementary grades asked students to consider replacing the numbers with variables, for example, the cost of tickets for the field trip as the variable t and the cost of the buses for b. The tasks that were categorized as “optimization models” were posed as finding the “best,” which in some cases translated as the most affordable (meal planning), the most profit (school store), or the most area for collaboration (classroom design). Rating and ranking models came into play when the class had to make decisions among competing models for the “best.” For example, the class had to think about the criteria for what makes a field trip the best and assign values, such as cost, education value, distance, and fun factor, to rate and rank each option. Predictive models came into play in many of the modeling tasks where the class wanted to look at data and trends to predict outcomes. Modeling tasks like the school supply dilemma, the school store project, and the snack corner setup (see Table 6.5) called for data collection and analyzing trends, such as attendance, sales, and consumption in order to make predictions about how much supply they would need. Categorizing these tasks revealed that many of the teachers choose similar situations like planning and logistics, finding the “best,” predicting based on real-time data, and making decisions using mathematics by quantifying criteria for decision making. Oftentimes, the modeling task led to numerous solutions that satisfied the problem without always deriving a generalizable model. Facilitators and coaches for the teacher designers, at times, pushed teachers to think of ways student solutions could be created as a model that could be generalizable and reusable to describe, predict, optimize, and make decisions about real-world situations. More specifically, we asked teacher to consider if the solution is shareable, could the solution be communicated as a systematic model that can be shared with others? If it is reusable, could the solution be applied again to a task with slightly different variables? Or if it is generalizable, could it be generalizable, or can we “zoom out” and state that we have a model that can be used for other similar situations? Connecting to the Curricular Content on Grade Level  Teachers connected the need for mathematics in a modeling task with their grade-level curricular objectives. They chose authentic tasks that could be implemented with real data collection and

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Table 6.5  MM task analysis – problem potential, pathways, and production of models Mathematical modeling problems in local contexts *Connection to math content Lunch count (Gr. 2–4) How can we figure out the number of pizzas to order? (School) This MM task engaged students in determining the appropriate pizza order. They began with their own classroom and then expanded their model to the entire school. *Number sense and computation School store project (Gr. 6) What supplies should we sell in the school store? How do we set prices for each item? How can we get rid of dead stock? What is the “best” sale deal? (school and community based) This MM task involved a service learning project where students started up their school store to raise money for a “Fight Hunger” campaign. *Data analysis, computation with fractions, decimals, and percent Coin harvest: fundraiser for meal baskets (Gr. 2–6) How can we get the “best” meal basket for the families? How soon will we reach our goal for the coin harvest? (school and community-based) This MM task involved a service learning project where students did a coin harvest as a fundraiser and planned for meal baskets for families in their community. They looked at the statistics in their county relative to the one in five kids go hungry statement. *Computation and data analysis Back to school: classroom supply conservation (Gr. 1–6) How soon will we run out? (school) This MM task launched at the start of school where students became more aware of waste and ways to conserve as they described the rate of use of school supplies and how they could predict running out of supplies. *Pattern and algebra; data analysis Designing our new classroom (Gr. 3–4) How can we arrange our new classroom to “best” meet our needs?(school) This MM task launched when a teacher saw a MM potential in the school reconstruction. They were getting a new classroom and wanted students to determine the best way to use the space for learning. * Geometry, measurement, computation skills

Situational features that afford math modeling potential, pathways, and production of useable models Descriptive, predictive Determine the most accurate count by surveying classmates and collecting and describing the data. Finally, students predict and prescribe how much pizza to order. *Produced model for determining the number of pizzas to order Descriptive, predictive, optimization, rating and ranking Describe and predict rate of change; find optimal price/profit; make decisions about inventory and sales. *Produced model for determining sale promotion and how to improve slow checkout lines using rate of flow

Descriptive, predictive, rating and ranking Plan logistics; determine budget *Produced model for determining trend for donation through the drive; model for planning best meal based on food groups and food preference and cost

Descriptive, predictive Describe and predict rates of usage by first describing how much inventory of school supplies they have and determining the rate in which they used the supplies. *Produced a model for supply use to ensure inventory to last the year Descriptive, optimization, rating and ranking Design a class layout with the arrangement of furniture to optimize space. Use rating and ranking to determine priority for classroom space. * Produced a model for determining optimal space for learning (continued)

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Table 6.5 (continued) Mathematical modeling problems in local contexts *Connection to math content Planning the “best” trip (Gr. 3–6) What is the “best trip”? (school and family) This MM task was inspired by a group of college students who had a travel blog called “America in a Day.” They visited a class and the third-grade class all designed their own “best” family/school field trips. *Computation; elapsed time School-wide reading challenge (Gr. 2–6) How long will it take for our class to read a million words? How do we determine how fast we read and when we will reach our challenge?(school) This MM task was inspired by a school-wide reading challenge *Computation; pattern and algebra; data analysis Snack corner setup for school basketball BB game (Grades 4–6) How do we determine how much food we need? (school and community-based) This MM task was inspired by a school-wide BB tournament, and students were planning with their teacher the menu and supplies for the snack corner. *Computation pattern and algebra; data analysis

Situational features that afford math modeling potential, pathways, and production of useable models Descriptive, predictive, optimization, rating and ranking Plan logistics, determine optimal time, budget, ordering buses, figuring out the cost of tickets for exhibits. *Produced a model for determining cost of field trip/family trips Descriptive, predictive, optimization Plan logistics, collect, describe data, and predict rate of reading for each class. *Produced a model for how many words each class read each week Descriptive, predictive Determine the most accurate count of attendees, and make assumptions of most popular snacks. Collect, describe data, and predict and prescribe snacks to purchase for a fundraiser. *Produced a model for predicting how much snack to buy for a school event

mathematics with a focus on building a solution and a usable mathematical model that was appropriate for young learners. Some teachers started by identifying contexts and found ways to incorporate grade-level mathematics content. Two of our case study teachers began their MM process by choosing a relevant, local context. While Anne allowed the content to emerge spontaneously as the project developed, knowing that the central mathematics concept would be focused on data analysis with algebraic connections, Katie applied a very focused context-to-­ content progression that we saw among many of our teachers. The context of planning a field trip was something that they felt certain would be familiar and engaging for their students. Katie saw that her second-grade students could employ many of the grade-level standards including addition, subtraction, and skip counting but also knew that some of her students began to explore above grade-level mathematics concepts such as multiplication and division. Other teachers began with the curriculum and worked backward. For teachers concerned about the curriculum pacing guide, this approach supported them in adopting MM.  Teachers who used a “backward design” approach laid out their quarterly math objectives and connected the mathematics concepts with relevant situations and events happening at their school. For example, Susan began with

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content objectives in her planning process using the district pacing guide. She knew that her students needed practice with decimal operations and then saw the potential in the upcoming grade-level field trip to explore the cost of buses. She started from the content that she wanted to cover and found a context that would allow her students to engage in MM while employing the desired content. A difficulty that emerged with content alignment was whether the mathematics content within a MM situation was appropriate to grade-level standards. Teachers identified difficulties in ensuring students would be able to access the grade-level math within the context of the question and also balancing the demands of covering required content while allowing the MM to develop naturally in the classroom. Teachers also expressed difficulty in making decisions and the tension they felt as they facilitated mathematical learning. There was concern about providing the “just right” guidance and scaffolding while maintaining mathematical rigor in the MM situation. One teacher commented, “I also felt like I had to pose questions when I saw the kids going way off track or getting stuck. I think questioning is an important part of the process, but I feel like I’m not allowing true authenticity,” highlighting her struggle with finding how much support was needed and what was too much. In addition to the struggle of fitting grade-level content into the MM situations, some teachers found that the open-ended nature of the modeling process sometimes led both the teacher and students to content areas outside their comfort zone or too advanced for their grade level. This relates to teachers’ Curricular Horizon Knowledge, as defined by the knowledge of the mathematics curriculum that comes before, after the grade that they teach. In our survey, teachers assessed their confidence with 96% of them stating that they were confident or very confident about the curriculum they currently teach and 76% of the teachers stating that they were confident or very confident about the curriculum that precedes the grade they teach but only a little more than half, 55% of the teachers feeling confident about the curricular knowledge that follows the grade they teach. Another major area of challenge in problem posing was curriculum alignment. Teachers had to look at the real-world problem to focus on at least one of the situational features that would connect with their content. This was critically important to teachers because many were constrained by the reality of the curriculum and their district pacing guides.

6.7 Collective Activities that Supported Students as They Mathematized, Made Assumptions, and Defined Variables Related to the Problem Posed For our second research question, we examined how teachers planned and orchestrated the process once they posed the problem and moved them towards making assumptions, defining variables, and providing constraints to advance meaningful mathematical learning. As mentioned from the survey results, teachers struggled with making assumptions and defining variable phases of the modeling process.

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One teacher stated, “I didn’t provide enough initial guidance to allow them to feel confident about the assumptions they made. They guessed rather than base assumptions on facts. I stated the problem so broadly that they were not able to narrow the focus.” Another teacher reflected that, “my struggle was that I allowed my students to continue with vague assumptions. I let them keep things too open.” To better understand the core practices that supported the problem formulation phase of the modeling tasks, we examined lesson plans, video enactments of classroom vignettes, and teacher reflections. Teachers Design Tools, Scaffolds, and Assessment Questions to Advance Collective Knowledge  One of the challenges encountered by teachers was their struggle with how much direction and what type of support to provide to students during the modeling process. Teachers wanted students to have more ownership in the problem, but faced a challenge when encouraging students to address the mathematizing aspect of the project. For example, when a team first implemented a task called “Design a Butterfly Garden,” the first iteration left students focused more on designing than focusing on the mathematics. To overcome this challenge in future lessons, teacher designers and lesson study facilitators planned for mathematical modeling pathways. Anticipating mathematical modeling pathways (see Fig. 6.3), before the lesson, allowed for teachers to consider all the productive mathematics that can be encountered in the task before enacting it. This allowed the task to stay open yet provide some productive pathways for the teacher to guide students where they can focus on pursuing the important mathematics. For example, for the modeling tasks of planning the field trip, the lesson study team mapped out several productive modeling pathways. These included a descriptive modeling experience of determining the number of school buses needed for a class or a school trip; an optimization modeling experience in determining the “best” deal for the mode of transportation between ordering a charter bus or a school bus, rating and ranking model to create criteria (education, distance, fun factor, cost) and decide among many proposed field trip sites, or using elapsed time to maximize the time during their field trip. Teacher designers also worked collectively to put together what we called “Math Modeling Content Tools.” Pedagogical Content Tools was coined by Rasmussen and Marrongelle (2006) to refer to student-generated models that are used in the classroom to push the mathematics forward. In our case, Math Modeling Thinking Tools included both teacher-created tools like brainstorm sheets, data tables, and selected websites as well as student-created tools, like models of organizing the data, solution strategies including number sentences, drawings, and student-created mathematical models. These authentic MM thinking tools provided teachers and students scaffolds as they moved through the collective activity, a joint endeavor towards fruitful mathematical learning. A common thinking tool like the KWA (What do you Know, Want to know, and What will you Assume) and Data Charts helped students think about all their assumptions and important variables and constraints to consider. Another tool that supported the elicitation of student assumptions and ideas was the use of sentence starters. To help students in elaborating

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Table 6.6  Core practices and Math Modeling Thinking Tools with repertoire for mathematizing Core practices that advanced collective knowledge Inviting students to break down the problem and consider relevant data to the problem situation Facilitating the organization and interpretation of presented data Eliciting assumptions to connect to mathematical ideas central to the problem situation Focusing on key variables that can enhance the model or solution

Mathematical Modeling Thinking Tools with instructional repertoire (e.g., scaffolds, strategies, or routines) Through a KWAS chart – what do you Know, Want to know/ need to Assume to Solve, the teacher orchestrates class discussion charting what do you know about the problem, want to know, or need to assume to solve the problem By providing data through charts, infographics, and/or websites, the teacher prepares information relevant to the problem that students manipulate for sensemaking Encourage brainstorming through think-aloud strategies using sentence stems like “if I knew … I could.…” For example, “if I knew how many people are going on the field trip, I could figure out how many buses to order” Teachers prepare “focusing” questions to focus on information to solve the problem. What variable/ information can change in your model? How might you improve your model? What if…?

possible modeling experiences, teachers used think-aloud strategies with a sentence starter, “If I knew … I could….” For example, “If I knew how many people attend our pancake breakfast fundraiser, I could make an estimate of how many pancakes we need.” This allowed teachers to facilitate students in the experience of making assumptions and defining variables that mattered in the task. In addition, teachers collaborated with lesson study facilitators during lesson planning to pose purposeful questions that would scaffold students without spoon-feeding them information. In order to monitor how students were making assumptions, defining variables, several lesson study teams created rubrics to assess students’ modeling competencies. To evaluate students’ strengths in making assumptions, teachers rated competencies by asking, “can the student clearly identify data/information that is known and important to the problem while making mathematical observations related to the problem situation?” In the criteria for defining variables, teachers rated competencies by assessing if a “student can generate lists of variables and could distinguish between parts of the problem situation that can be changed and those that cannot be changed.” (Table 6.6).

6.8 Vignettes to Illustrate the Interplay Between Teacher and Students in Problem Formulation To best illustrate the interplay between the teacher and students in the process of problematizing and mathematizing, we examined classroom vignettes of MM lessons. Teachers wanted to encourage students to lead the modeling process by posing problems, making assumptions, and defining variables. However, they knew that modeling was something many of the students never experienced in their previous

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mathematics classroom. Therefore, teachers modeled the problem posing and making assumptions and defining variables by skillfully planning for student participation. We will briefly share two teaching vignettes to illustrate how one modeling problem, planning a field trip, allowed students to experience the usefulness of modeling in solving a problem for their school. Analysis of two math modeling lessons at different grade levels revealed ways teachers used a local context and modeling routines, to effectively elicit grade-level content to bring out a descriptive model. As we zoom in on a vertical modeling task, it will be evident how one context can lead to multiple modeling pathways and connect to a variety of concepts within the learning progression. This vertical modeling task also demonstrates how teachers’ intentional planning with tools, scaffolds, and focusing questions can help pave the way for a fruitful modeling experience for both second and fourth grade. Teaching Vignette of the Second-Grade Field Trip Modeling Task  This lesson was launched with the principal coming to a second-grade classroom and asking the class to help her decide how many buses to order for their school-wide field trip. This authentic problem automatically had the buy-in from all the second graders. They gathered around the teacher to map out a KWAS chart, what do they Know, what do they Want to know/Need to Assume, and what to Solve for (see Fig. 6.4). The Math Modeling Thinking Tool that the teacher provided after their KWAS talk was a data table with the teachers’ name and total students in each class. As students wrestled with the problem-solving stage, observers noted two particular modeling

Fig. 6.4  Use of a graphic organizer for defining the problem parameters in Katie’s field trip modeling lesson (left). A student uses the data chart and one’s own tally chart with the “extras box” to create a model for determining the number of buses (right)

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experiences as noteworthy. These two student-created models are provided here as examples of how the students were able to formulate, identify the mathematical aspects of the problems contained in the real context situation, identify important variables, and translate the problem into mathematical language and representation. The first mathematical model emerged during an observed lesson where a group of second graders determined how many people would be going on a field trip from each grade. Students verbalized that they would need to add the number of students and the chaperones. Then another student chimed in, “You forgot one more for the teacher.” This dialogue captured an emergent model that could be generalized to all classrooms using number of students + number of chaperones + number of teachers. Although one may think that this is a simple model, it adheres to Galbraith’s (2007a) six principles, meeting all six principles by being connected to students’ lived experiences, accessible, feasible in that it uses math available to students, valid, and flexible in that it can be used to determine passengers not just for the class or grade, but for the entire school. In the second episode, we captured another discussion that we defined as an emergent mathematical model for determining the number of buses. One observer noticed a table that had an interesting system for figuring out the number of buses. A boy who was leading the group work asked his friends to tell him how many people were in sixth grade. A partner yelled out “103!” The leader then proceeded to put a tally mark next to the number 70 on his paper and put 33 in a box on top of his paper. He then said, “Okay, that’s 33  in the extra box.” He proceeded in this fashion, pulling 70 out of the total number of people for each grade level and then placing the leftovers in the extra box. Then he tackled the extra box and grouped 33 with 36 and then added a one from 21 to add another tally mark next to the 70. This student was using repeated subtraction, making a tally mark for every 70 passengers including recognition that the extras from each grade level would be combined to make further groups of 70 until all people had a spot on a bus. These two episodes from one lesson proved to be a great example of how second graders were capable of making assumptions and used the data table to create a “useable” and “shareable model.” Here the students also provided an example of employing mathematics to design and implement strategies to find mathematical solutions and used and switched between different representations (numbers, tallies, extra box for remainders) in the process of finding solutions. Teaching Vignette of the Fourth-Grade Field Trip Modeling Task  In the case of the fourth-grade classroom, the teacher used scaffolds while keeping the task open and providing students choice to pursue different math modeling pathways. As mentioned above, she and her team had anticipated the potential pathways of students: determining the number of school buses needed for the entire fourth-grade class (descriptive); determining the “best” deal for the mode of transportation between ordering a charter bus or a school bus and planning an itinerary that made the “best” use of their time (optimization); or deciding on the ideal field trip by rating and ranking the proposed field trip based on created criteria (education, distance, fun factor, cost).

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Fig. 6.5  Use of “catch and release” to build collective knowledge and scaffold the learning

Susan scaffolded the modeling experience by providing students with an information sheet that had different bus options and with questions to consider. She used a teacher move which we coined a “catch and release” when she sensed that students were lost in the data (Fig. 6.5). This allowed her to use focusing questions in their math talk and also bring up pedagogical content tools (e.g., student-generated contributions and discoveries to move the mathematics forward). Different groups of students asked questions that were related to the important variables they needed to consider: • What time does the location open? What time do we need to be back by? • How far can we travel? How do we find the cost of the bus if they are using mileage? • How many chaperones do we need? Is there a cost for the entrance fee? These conversations allowed Susan to focus on the important variables and the assumptions they needed to make. She also realized that students needed a mini lesson on using rates to find the cost for transportation. Susan used think-aloud strategies as she talked through some of the assumptions to engage students in a discussion on how to calculate the field trip expenses by using the given information. This led the whole class in the discussion of what variables to include in their model. As she monitored the discussion in small groups, she engaged in important mathematical discourse including questions like: • If mileage is $2.40 per mile and our destination is 10 miles away, what is the total cost for the distance? • Do we need to return by a set time? How will we show this for our costs? • If we leave at 9:30 and return at 1:30, how many hours do we have to pay the bus driver? • How much will we need to pay the bus driver if they get $38/hour? • Why would I add the mileage and the hourly bus driver expenses?

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Susan recounted that based on observing the mathematics the students engaged in, scaffolding with purposeful and focusing questions through quick mini lessons seemed crucial for the class to move forward with the mathematics. Despite having to provide some support through these timely mini lessons, Susan was able to maintain openness, as she states in her reflection, “Though students were provided with a cost sheet, they were responsible for deciding when they would leave, how long they would spend at site, what type of bus they would use, etc.” In addition, she recounts how the real-world connection allowed students to use sensemaking to discover their own errors. She states: Students discovered their own mistakes with the decimal operation. Jordan mentioned that, ‘My answer wasn’t reasonable, I realized I had to do it again.’ If it were just a number problem, I am not sure if that student would have cared if the product was $13,500 but he realized, based on the context, that could not be the cost of the bus ride- which made him realize that he was multiplying decimals and it was only $135.

Student-reported accomplishments during wrap-up included discovering their own mistakes with decimals; working individually to confirm the correctness of their group’s mathematics; finding out the mileage costs for one, two, and three buses; and determining the distance to their location. After eliciting students’ thinking about the important variables to consider, the teacher and the students came up with a model that would help them figure out the cost of the transportation. They found that there was the cost of the driver and mileage, along with the rental of the bus. They wanted to find how much transportation cost would figure into the cost of the field trip for each person. The class was considering either hiring a school bus or a charter bus because the charter bus would give them more flexibility with departure and arrival time. The teacher and students co-created a model that would account for the “school bus cost” versus a “charter bus cost.” The figure above (Fig. 6.6) is from one student group’s presentation showcasing the mathematical model they created for the charter bus cost which included[(flat rate fee * number of hours) + (driver fees * number of hours)] * number of buses = cost per person number of students

Fig. 6.6  Student group presentation showcasing a mathematical model for cost per person

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Fig. 6.7  Learning to critiquing a model using student mathematics knowledge

If other 10-year-olds saw this formula, they would probably balk at the length of this equation. However, since this equation was derived collectively as students considered all the important variables and they participated in the research aspect of funding a bus, the class created a model for finding the cost per person for transportation that made sense to them. To extend students’ thinking, Susan also created tasks that allowed students to critique other models. In the task below (Fig. 6.7), it states “Adam believes that he has come up with an equation that summarizes all the expenses for this field trip using order of operations. Is he accurate?” After performing the calculation, the student responded that it was inaccurate and did not match up with their group’s calculation. This led the class into a rich conversation about the importance of the order of operations and how to critique and revise a model to make it better and more precise.

6.9 Discussion Results indicated specific core practices that teachers as designers engaged in to get students to participate in the problem formulation phase at different grade levels and different problem types that offered a variety of pathways for mathematical learning

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in diverse grades within the elementary grades. The aim of this chapter was to better understand the core teaching practices that support teachers in designing and problematizing MM tasks to maximize student learning. We highlight the important role of elementary teachers as designers of early mathematical modeling experiences for elementary students. We acknowledge that mathematical modeling in the early grades will be qualitatively different than what is recognized as MM at the secondary level, but we have evidence from teaching episodes that students are capable of embracing the cyclic nature of modeling, posing questions, making assumptions, building solutions, evaluating the solution, and making adjustments when necessary. We posit that this is a natural part of mathematical modeling in the elementary grades and that these early experiences with MM serve to develop student understanding of and comfort with MM that will eventually lead them to develop generalizable models. Thus, for elementary grades capitalizing on the usable model looks different than it would in upper grades. Exposing elementary students to early mathematical modeling is a way of developing their mathematical modeling habits of mind. These habits of mind develop students’ productive dispositions towards mathematics, including the inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy (NRC, 2001, p. 116). We saw that students seemed to develop this disposition as a result of the excitement evidenced in all our teacher interviews where they saw even reluctant learners engaged in the mathematical modeling tasks because they cared about the work and it “pulled at their heart strings.” We witnessed teachers, as modelers and as designers, feel empowered by the pedagogical license they felt to create relevant and engaging learning experiences for their students. Teachers shared their excitement as they saw students genuinely engaged in the problem and making sense of mathematics. Classroom teachers are well-positioned to be designers of MM tasks because they have intimate knowledge of what students care about and this allows for the context to matter more and be customized to connect to important math in the curriculum. Teachers’ knowledge of students, their interest, their readiness, and the curriculum make it ideal for teachers to be designers of modeling tasks. Teachers immersed in mathematical modeling shared their excitement for developing their eyes and ears for mathematical modeling. Teacher designers embraced choice and negotiated a balance between curricular pressures and providing meaningful modeling experiences. One of the benefits of working within a networked improvement community using lesson study over 3  years with teacher leaders, coaches, teachers as designers, district leaders, and university researchers was that we were able to test out and create usable tools to support teachers. In addition, video vignettes from classrooms allowed us to decompose the complexity around the art of problem formulation in action. These tools, routines, and teaching repertoires helped us better understand the core practices that supported productive modeling experiences for elementary students that were developed through collective knowledge building activities. These enactments and video vignettes allowed us to disseminate “pictures of practice” to other teachers who wanted to try out this ambitious teaching. Many of our teacher designers in year one became teacher leaders and co-instructors during our design institutes in

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subsequent years. In addition, these teacher leaders who had intimate knowledge of the challenges our teachers experienced in enacting these ambitious practices offered many teacher-created tools and support. In our study, our teacher designers gained confidence in enacting mathematical modeling and working on problem formulation with their young mathematicians. Using research-based practices for effective PD (sustained, regular meetings, peer collaboration, content-focused lesson study), we immerse teachers as students and later as designers of mathematical modeling and provided experiences to discover the joy of mathematical modeling with their students in the early grades. In addition, we were able to leverage the PD experiences of teachers to create new teacher leaders and teacher designers who could create and share more modeling tasks with their colleagues to, in turn, deepen the nodes and hubs in their networked improvement communities to bring many more modeling experiences to students in the elementary grades. Funding  This project was supported by the National Science Foundation, Award number STEM C # 1441024.

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Lesh, R., & Zawojewski, J. S. (2007). Problem Solving and Modeling. In: Lester, F. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–802). Greenwich, CT: Information Age Publishing. Lewis, C. (2002). Lesson study: A handbook of teacher-led instructional change. Philadelphia, PA: Research for Better Schools. Lewis, C. (2015). What is improvement science? Do we need it in education? Educational Researcher, 44(1), 54–61. Matson, K. M. (2018). Teachers perspective on how they learn mathematical modeling. Doctoral dissertation, George Mason University, Fairfax, VA. McDonald, M., Kazemi, R., & Kavanaugh, S. S. (2013). Core practices and pedagogies of teachereducation: A call for a common language and collective activity. Journal of Teacher Education, 64(5), 378–386. National Governors Association & Council of Chief State School Officers. (2010). Common core state standards. Washington, DC: NGA Center and CCSSO. Retrieved from http://www.corestandards.org/Math/Practice/#CCSS.Math.Practice.MP4 National Research Council. (2001). Adding it up: Helping children learn mathematics (Kilpatrick, J., Swafford, J., & Findell, B. Eds.). Washington, DC: National Academy Press. NCTM. (n.d.). Beginning to problem Solve with “I Notice, I Wonder”. Retrieved from https:// www.nctm.org/Classroom-­Resources/Problems-­of-­the-­Week/I-­Notice-­I-­Wonder/ Niss, M. (2013). Modeling a crucial aspect of students’ mathematical modeling. In R.  Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 43–59). New York, NY: Springer. Notice and wonder. (n.d.). Retrieved from http://blog.mrmeyer.com/category/3acts/ OECD. (2018). PISA for development mathematics framework. In PISA for development assessment and analytical framework: Reading, mathematics and science. Paris, France: OECD Publishing. https://doi.org/10.1787/9789264305274-­5-­en Pollak, H. (1997). Solving problems in the real world. In L. A. Steen (Ed.), Why numbers count: Quantitative literacy for tomorrow’s America (pp. 91–105). New York, NY: The College Board. Pollak, H. (2003). A history of the teaching of modeling. In G.  M. A.  Stanic & J.  Kilpatrick (Eds.), A history of school mathematics (Vol. 1, pp. 647–669). Reston, VA: National Council of Teachers of Mathematics. Pollak, H. (2016). Forward. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. vii–viii). Reston, VA: National Council of Teachers of Mathematics. Rasmussen, C., & Marrongelle, K. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Journal for Research in Mathematics Education, 37(5), 388–420. Retrieved from http://www.jstor.org/stable/30034860 Stillman, G. A., Blum, W., & Biembengut, M. S. (2015). Mathematical modelling in education research and practice: Cultural, social and cognitive influences. Cham, Switzerland: Springer. Stillman, G. A., Kaiser, G, Blum, W. & Brown, J. P. (2013). Teaching mathematical modelling: Connecting to research and practice, Springer Dordrecht, The Netherlands. Suh, J. M. (2007). Tying it all together: Building mathematics proficiency for all students. Teaching Children Mathematics, 14(3), 163–169. Suh, J.  M. (n.d.). Modeling category cards. Retrieved from http://completemath.onmason.com/ math-­modeling/ Suh, J. M., Burke, L., Britton, K., Matson, K., Ferguson, L., Jamieson, S., & Seshaiyer, P. (2018). Every penny counts: Promoting community engagement to engage students in mathematical modeling. In R. Gutierrez & I. Goffney (Eds.), Annual perspectives in mathematics education: Rehumanizing mathematics for students who are black, indigenous, and/or Latin@ (pp. 63–78). National Council of Teachers of Mathematics: Reston, VA. Suh, J. M., Matson, K., & Seshaiyer, P. (2017). Engaging elementary students in the creative process of mathematizing their world through mathematical modeling. Education Sciences, 7, 62. https://doi.org/10.3390/educsci7020062 Suh, J. M. & Seshaiyer, P. (2014). Examining teachers’ understanding of the mathematical learning progression through vertical articulation during Lesson Study. Journal of Mathematics Teacher Education, 18(3), 217–229.

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Chapter 7

Teaching Practices to Support Early Mathematical Modeling Mary Alice Carlson

Attention to mathematical modeling in K-12 settings has demonstrated its potential to help students develop adaptive expertise (Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009), facilitate opportunities for meaningful mathematics learning (Bonotto, 2005; De Lange, 2003), and engender a positive disposition toward mathematics (Kaiser & Maß, 2007). Mathematical modeling has received attention internationally for some time and, due at least in part to its role in standards documents and assessments, is receiving increased attention in the United States. The Program for International Student Assessment’s (PISA) definition of mathematical literacy has long been reflective of mathematical modeling (OCED, 2017), and the Common Core State Standards for School Mathematics (NGA/CCSSO, 2010) include mathematical modeling as both a content standard and a standard for mathematical practice. In spite of its growing importance in school mathematics, investigating classroom instruction around modeling “remains a challenge” (English, Ärlebäck, & Mousoulides, 2016, p. 10), and issues related to teaching mathematical modeling are underexplored (Kaiser, 2018). The purpose of this chapter is to understand the practices teachers use when engaging students in mathematical modeling. Using data from a teacher study group I facilitated in 2015, I explore practices teachers used to facilitate three processes fundamental to mathematical modeling: working with the real-world context that motivates the modeling task, using mathematics to build solutions, and deciding what might represent a finished model.

M. A. Carlson (*) Department of Mathematical Sciences, Bozeman, MT, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_7

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7.1 The Modeling Cycle Unpacking the practices of teaching mathematical modeling begins with understanding the demands of modeling. Broadly described, mathematical modeling involves taking a real-world situation or scenario, translating that scenario into the mathematical world where the modeler pursues a mathematical solution, and then interpreting results in light the original, real-world situation (Pollak, 1979). Depictions of the modeling cycle vary and “highlight different aspects of the modeling process depending on the purpose and focus of the particular research” (English et al., 2016, p. 2). Though there is general agreement that mathematical modeling is a cyclical process that relates the real and mathematical worlds, the cycle itself has been broken down varying phases (Kaiser, 2018; Perrenet & Zwaneveld, 2012) and sometimes includes subprocess that describes the cognitive work of modelers as they transition from one phase to the next (Blomhøj & Højgaard-Jensen, 2003; Blum & Leiß, 2007; Galbraith & Stillman, 2006; Zbiek & Conner, 2006). Descriptions of the modeling cycle’s subprocesses are useful in classroom research because they give insights into what modelers accomplish as they work. These descriptions illuminate how modelers make “relevant decisions, and perform appropriate actions in situations where those decisions and actions are necessary to enable success” (Stillman, Galbraith, Brown, & Edwards, 2007, p. 690). For example, moving from a messy, real-world situation to a problem statement requires modelers to make assumptions, formulate, and then mathematize the situation (Blomhøj & Højgaard-Jensen, 2003; Galbraith & Stillman, 2006). When working with the model itself, modelers may combine multiple mathematical objects into a single entity (Zbiek & Conner, 2006). Other actions like interpreting (Zbiek & Conner, 2006) or exposing (Blum & Leiß, 2007) involve linking work done in the mathematical world to its real-world context. Applied to classroom settings, the subprocesses of mathematical modeling align with student, rather than teacher, activity. Teachers must understand the work students are doing, but their aim is not to engage in modeling directly. Thus, modeling cycles may be descriptive of the work done by students in classrooms, but they are limited when it comes to understanding teaching. The field does not yet understand how teachers create the curricular and social conditions for classroom work, support the development of productive dispositions among students, and interpret and use students’ mathematical ideas as they move through the modeling cycle.

7.2 Teaching Practices and Mathematical Modeling Knowledge of content, students, and pedagogy, while critical, is not sufficient for meeting the demands of teaching (Darling-Hammond, 2008). Because teacher knowledge is used in practice, “much of what they have to learn must be learned in and from practice” (Ball & Cohen, 1999, p. 12). It follows that supporting teachers

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in implementing mathematical modeling involves investigating what they actually do when teaching and how their decisions create or constrain opportunities for students to learn mathematics for understanding. The National Research Council (2001) defined understanding as five interwoven strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and a productive disposition. Practices related to teaching for understanding are sorted into categories including, but not limited to, noticing and responding to students’ mathematical thinking (e.g., Jacobs & Empson, 2016; Jacobs, Lamb, & Philipp, 2010; van Es & Sherin, 2002); supporting student discourse (e.g., Chapin, O’Conner, & Anderson, 2013; Hufferd-Ackles, Fuson, & Sherin, 2004; Smith & Stein, 2011); maintaining high cognitive demand (e.g., Henningsen & Stein, 1997; Stein, Smith, Henningsen, & Silver, 2009); and drawing on students’ cultural and community knowledge bases (e.g., Civil, 2002; Turner et al., 2012). The aim of this chapter is not to provide another list of teaching practices. It is to consider how teachers describe their efforts to engage students in modeling and, in doing so, create space to explore how teaching mathematical modeling is similar to and different from teaching mathematics in other areas. Though it is reasonable to expect that some, if not many, practices of teaching mathematics already explored in the research base will translate to modeling lessons, mathematical modeling includes processes not typical in most mathematics classrooms (Niss, Blum, & Galbraith, 2007). This makes modeling “quite difficult for teachers because real-world knowledge about the context for modeling is needed, and because teaching becomes more open and less predictable when students engage in more open-ended modeling situations” (Cai et  al., 2014, p.  148). For example, modeling tasks are not only open-ended, like many tasks defined as having high cognitive demand, they are also open at the beginning. The modeler has to pose a mathematical question and decide what mathematics might be useful answering that question. To maintain high cognitive demand during a modeling lesson, teachers likely need to notice students’ mathematical thinking and sustain rich mathematics discussions (Smith & Stein, 2011). They may also need to help students access and then translate their real-world knowledge about the modeling context to the mathematical world. A critical task for those who would like to see mathematical modeling successfully integrated into K-12 classrooms is to consider what teachers do when facilitating mathematical modeling with their students.

7.2.1 Developing Competency in Modeling Much of the research on teaching mathematical modeling has focused on preservice and in-service teachers as modelers themselves (Cai et al., 2014). Preservice and in-service teachers alike have limited experiences in and low self-efficacy toward mathematical modeling (Kuntze, Siller, & Vogl, 2013; Ng, 2013; Tan & Ang, 2013). Their belief structures, built on prior experiences with mathematics and problemsolving, do not always translate well to the open, cyclical nature of mathematical

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modeling (Ng, 2013). Preservice and in-service teachers face similar “blockages at the beginning of the task as well as feelings of frustration and loss during the task” (Ng, 2013, p. 346), view modeling as a linear rather than cyclic process, and face difficulty using known mathematical or statistical tools in novel modeling situations (Doerr, 2007). Differences between preservice and in-service teachers’ approaches to modeling exist, suggesting that knowledge for teaching mathematical modeling is distinct from knowing mathematical modeling. Preservice teachers are more likely to apply college-level mathematics to modeling tasks, whereas in-service teachers may rely more on the strategies they expect their students to use (Ng, 2013). Preservice teachers are also more comfortable with technologies that support mathematical modeling than their in-service counterparts but less likely to recognize its potential for use in classrooms (Tan & Ang, 2013). Incorporating guided reflection on modeling along with novel modeling tasks may help teachers “explicate aspects and nuances of the modeling process” (Tan & Ang, 2013, p. 382), but teachers also need to have deep understanding of content, consider the role children’s thinking plays in learning, and develop new instructional practices (Borko, 2004). Growing as modelers is an important part of teacher development, but teachers of modeling face demands that are unique to classrooms and need additional knowledge and skill sets to meet those demands.

7.2.2 Developing Teachers of Modeling Though the modeling cycle and its subprocesses do not describe practices necessary for teaching modeling directly, they have been used to describe ways teachers might support their students. As discussed above, elaborated depictions of the modeling cycle provide a framework for understanding students’ cognitive work during different phases of the modeling cycle. In studies that rely on the modeling cycle to unpack the work involved in teaching modeling, a window into students’ cognitive work is presumed to help teachers target modeling competencies that are specific to phases in the modeling cycle, such as mathematizing or analyzing models (Blomhøj & Højgaard-Jensen, 2003). Focusing on these competencies helps teachers identify the blockages students face (Galbraith & Stillman, 2006) and provide appropriate “adaptive interventions” (Kaiser & Stender, 2013, p. 288) in the form of scaffolding that is responsive to both the phase of the modeling cycle and the perceived source of the student’s challenges. However, the tasks of teaching any cognitively demanding task involve more than targeting specific competencies and intervening when students are stuck (Smith, Stein, Henningsen, & Silver, 2009). A more complete picture of the practices teachers are engaged in and the decisions they are making while their students are modeling is needed. Studies of teachers engaged in or reflecting on teaching mathematical modeling reveal that it is challenging. The open nature of mathematical modeling tasks leaves teachers with questions related to how they should structure student work, ways

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they might tie modeling tasks to curricular mathematics, and how to best respond to the variety of approaches students might use (Doerr, 2007; Manouchehri, 2017). These findings echo those from studies of teachers working in mathematical environments where students’ strategies and solutions are less predictable. For example, Henningsen and Stein (1997) found that high cognitive demand tasks were prone to decline into unsystematic exploration, lose mathematical substance, or shift to a focus on procedures when teachers did not engage in practices that maintained student engagement in doing mathematics. Research on teacher noticing suggests learning to respond to students’ mathematical ideas is just one part of a set of practices that begins with attending to students’ mathematical thinking (Jacobs et al., 2010; van Es, 2011). A lack of curricular materials focused explicitly on mathematical modeling compounds the challenges teachers face, as it leaves them with few resources and forces them to be task designers as well as teachers (Blum, 2015; Ikeda, 2007). Navigating the challenges of teaching mathematical modeling involves implementing instructional practices that support individual and collective inquiry; knowing how to hear, represent, and connect student ideas and representations; and a willingness to grapple with the ambiguous and the unexpected (Cai et al., 2014; Doerr, 2007). Not all of the work on teaching mathematical modeling highlights its challenges. English’s (2003) study found that teachers who engaged students in mathematical modeling constructed their own understanding of “modeling activities and what they considered to be effective ways of implementing them” (English, 2003, p. 235). Her study provides a unique and important perspective on the work of teaching modeling and the ways teachers navigate the challenges of that work. Teachers in her study emphasized the importance of understanding students’ perspectives on problem-solving and attending to the way individual learners and the whole group engaged in modeling tasks. Teachers also viewed “an awareness of the student modeling activities, as well as the ways in which the students engaged with the mathematics of the activities” (p. 236) as important. Still missing from the literature is an emergent account of the ways elementary school teachers navigate planning and implementing mathematical modeling in their own classrooms. In this study, I report on an analysis of discourse data from a teacher study group focused on mathematical modeling. The purpose of this study is to explore teachers’ understandings of their own practice when teaching mathematical modeling. It addresses the following research question: What practices do teachers describe using when they facilitate mathematical modeling in elementary schools?

7.3 The Current Study Data for this study are drawn from a larger project focused on mathematical modeling in the elementary and middle school years. Teachers participated in week-long summer professional development where they learned about the modeling cycle,

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engaged in modeling tasks themselves, and considered how modeling might fit in their own curriculum. In the fall, teachers participated in study groups as they incorporated modeling in their own classrooms. I was a member of a team that included mathematics teacher educators, statisticians, applied mathematicians, and classroom teachers who designed and led the professional development. During professional development, we placed more emphasis on teachers learning the processes and practices of modeling than on using mathematical modeling to teach specific content. We also emphasized the importance of using mathematical modeling to address situations that are real to the students and understood as problems outside the mathematical world (Barbosa, 2006). For our work with teachers, we developed a representation of the modeling cycle intended to capture its fundamental components and, at the same time, depict a process that could represent the work of even the youngest modelers (Fig. 7.1) Modeling begins with a problem from outside the mathematical world. To translate a real-world problem to the mathematical world, the modeler poses a mathematical question. Learning to ask a mathematical question is central to the modeling cycle. It is also part of what distinguishes mathematical modeling from the mathematics teaching and learning students typically experience. During professional development, we stressed student involvement in finding and posing a mathematical question. Next, the modeler builds a mathematical solution. When discussing this phase with teachers, we deemphasized finding a function or other mathematical representation that could stand alone as a model. Both the mathematical tools to which students have access and the representations they may use change significantly throughout the elementary school years. We did not want to suggest that a certain level of mathematical sophistication is a prerequisite for mathematical modeling. Third, the modeler evaluates the solution in light of the real-world situation where it began. In this phase, we discussed opportunities for students to assess their work through experiences that could illuminate the strengths and limitations of their solutions, rather than formal processes used to evaluate mathematical models.

Fig. 7.1  The modeling cycle. The research team developed and used this depiction of the modeling cycle for teacher professional development

Mathematical Problem

Real-World Problem

Build Mathematical Solutions

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7.3.1 Participants and Setting This paper focuses on a single teacher study group I facilitated in 2015. All of its members participated in the summer professional development ahead of the study group and taught in the same school district in a small city in the Rocky Mountain West. The primary aim of the study group was to create a collaborative space where teachers could talk about the opportunities and challenges they experienced when teaching mathematical modeling for the first time. The study group included two third-grade classroom teachers, Amy1 and Maria; one fourth-grade classroom teacher, Nora; and Carol, a district mathematics coach. Because Carol did not have her own classroom, she acted as a visiting third-grade instructor in another teacher’s classroom. Amy, Maria, and Nora each had between 5 and 10  years of teaching experience. Carol spent more than 20 years in the classroom and was in her second year as a mathematics coach. When university professors collaborate with teachers, they can adopt a variety of roles (Yeh, Hung, & Chen, 2012). I viewed my role as both a facilitator and an encourager. As a facilitator, I sought to focus the discussion on teaching and learning mathematical modeling. As an encourager, I worked to create an atmosphere that promoted transparency about the challenges and opportunities created when trying something new. We all acknowledged ourselves as learners who were wrestling with feelings of uncertainty. We also acknowledged that teachers cannot always decide what they will say and do during instruction ahead of time. As a group, we anticipated student responses and discuss instructional options but did not view plans as definitive. Rather, we identified points in the lesson where teachers might have to make in-the-moment decisions as they responded to students’ ideas about the modeling task. The study group met seven times, beginning with a meeting during the summer professional development. Sessions 1–6 focused on planning, implementing, and reflecting on mathematical modeling. We spent the seventh session preparing for a modeling symposium. At the symposium, teacher study groups shared their modeling process and results with other professional development participants, school administrators, and other invited guests. Each study group session lasted approximately 75 min. Study group sessions 2–6 were loosely structured around three activities: reporting, reviewing, and planning. During the reporting phase, teachers shared their progress on the modeling task and, if they chose to do so, discussed any challenges or questions they were facing. During the reviewing phase, we discussed teacherselected classroom artifacts. Classroom artifacts included short videos of students, to individual student notebooks, and anchor charts created during class discussions. The artifacts provided a window into students’ understanding of the modeling task, students’ mathematical thinking, and the relationships between instructional decisions and students’ experiences. In most of the study group sessions, discussion of  All teacher names are pseudonyms.

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Table 7.1  Study group sessions, dates, and purposes Session Date 1 July 31, 2015 2 September 21, 2015 3 October 1, 2015 4 October 13, 2015 5 October 29, 2015 6 November 5, 2015 7 November 15, 2015

Purpose Initial planning meeting during summer PD Planning, implementing, and reflecting on a multi-week modeling task

Preparing for the modeling symposium

artifacts transitioned directly to planning for upcoming modeling lessons. Teachers discussed how they might advance to the next phase of the modeling task and anticipated students’ potential questions, mathematical strategies, and challenges. I documented our decisions using an online lesson plan template that captured the progress of the group. Teachers kept track of their own adaptations to the plan in whatever way they chose. The study group I facilitated developed and implemented a “Fieldtrip Problem.” Students identified a class fieldtrip, selected activities at the fieldtrip site, investigated transportation by school bus, and used mathematics to predict the fieldtrip cost for their entire class or grade. Students presented the results of their work to the school principal, who acted as the client for the modeling task. Table 7.1 details the study group sessions, dates, and major activities.

7.3.2 Data and Analysis I recorded and transcribed study group sessions 2–6. These transcripts became the primary data source for this study. I also kept a researcher log and study group artifacts (e.g., lesson plans and brainstorming documents). I analyzed the transcripts using multiple cycles of process coding (Miles, Huberman, & Saldana, 2013). First, I labeled segments of text as reporting, reviewing, or planning. Because I wanted to focus on teachers’ descriptions of their practice, I focused on those study group segments identified as “reporting” and assigned codes to teachers’ descriptions of their teaching practices. A second round of reviewing and consolidating codes reduced the number of distinct teaching practices to 15. I compared these tasks to my field notes, the rest of the transcripts, and study group artifacts to look for any practices of teaching mathematical modeling not represented in the transcript. I identified three: identifying contexts important to students, predicting mathematical pathways, and deciding what counts as a model. I grouped the codes into categories and identified four themes: working with student ideas, interacting with the real-world context, monitoring and supporting mathematics learning, and linking student ideas with the mode (Table 7.2).

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Table 7.2  Teachers’ reported practices when teaching mathematical modeling Themes Teacher Practices Working with student Eliciting student ideas ideas

Operational Definition Making students’ background knowledge and ideas about the real-world problem context visible Focusing student ideas Helping students focus on those aspects of the real-world problem that may be important when mathematizing the situation Considering Taking unanticipated student suggestions under unexpected ideas consideration Building consensus Facilitating processes through which students agree on an idea or direction for their next step in the modeling task Facilitating direct interactions with, or asking Facilitating students to consider the perspective of, the connections to the person or group who has decision-making client authority over the real-world situation Interacting with the Identifying contexts Finding and learning about real-world real-world context important to students situations that are interesting and accessible to students for modeling tasks Facilitating student Creating opportunities for students to build research their knowledge of the real-world situation Dealing with setbacks Responding to changes in the real-world context that influence students’ work on the modeling task Anticipating what mathematics students might Predicting Monitoring and use when developing mathematical solutions mathematical supporting mathematics learning pathways Discerning when mathematics required by the Recognizing task is difficult for students and deciding mathematical whether such difficulties are leading to challenges unproductive mathematical work Modifying contextual Changing features of the real-world context to adjust the mathematical demands of the task features of the modeling task Linking to curricular Making implicit or explicit connections to mathematics mathematics students have learned, are learning, or will learn Facilitating in-thePausing to explicitly teach or review a moment mini-lessons mathematical topic needed for the modeling task Asking students to explain and justify their Pressing for mathematical work, either orally or in writing explanation and justification (continued)

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Table 7.2 (continued) Themes Linking student ideas with the mathematical solution

Teacher Practices Facilitating transitions between the context and mathematics Connecting student ideas and the model Supporting students evaluating their work Deciding what counts as a model

Operational Definition Drawing students’ attention to the ways contextual features of the problem are represented in the mathematical solution Highlighting student contributions that may become part of the mathematical solution Asking students to consider whether their work is reasonable and accurate or answers the mathematical question Articulating an endpoint or product for the task, especially when the task concludes with something other than a formula or function

7.4 Results I organized the results section around four exchanges from the study group sessions. Each exchange depicts the study group discussing a different part of the modeling process and includes practices related to one or more of the four themes revealed during analysis. Although these exchanges do not illustrate every teaching practice in Table 7.2, they offer a window into the ways the teachers made sense of their efforts to engage students in mathematical modeling. In the first exchange, Nora describes navigating challenges related to working with students and their ideas about the real-world context. She discusses what happened when student preferences became a driving force behind the modeling task and how she sought to balance student ideas with boundaries imposed by the school. In the next two exchanges, teachers discuss what happened as their classes built mathematical solutions. Amy and Maria discuss facilitating student research and using research results in the modeling task. Then, Amy describes practices that supported students when the modeling task demanded mathematics students had not studied formally. Finally, I share an exchange during which Amy, Maria, and Nora consider how to bring closure to the fieldtrip problem.

7.4.1 Working with Student Ideas Real-world problems involve multiple stakeholders whose perspectives play a role in how the modeler defines, works on, and solves the problem. For the teachers in the study group, acknowledging others’ perspectives was especially important as students made choices regarding where to go and what to include when predicting the cost. Planning a fieldtrip took place in a system that included the students themselves, the school principal, and the teachers. Budget limitations district policies also played a role in determining what made a destination realistic. The teachers had to decide how students could authentically participate in decisions typically made by adults.

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Teachers’ practices for working with student ideas involved getting access to students’ thinking and then responding in ways that both valued students’ contributions and facilitated progress on the modeling task. Nora initially decided to allow individual students and small groups to pursue their own unique fieldtrip ideas. She wanted her students to use their model as part of an argument for what fieldtrip was “the best” when they shared their ideas with the principal. Nora acknowledged that allowing this degree of choice might complicate her work with students but wanted to give her students a chance to pursue their own ideas. The exchange below took place during study group session 4. Teaching practices are indicated by the brackets in the right hand column. 126 Nora:

Okay. So I’ve been trying to get them to come up with their own idea, not necessarily with any in mind for myself. And so the end of last week going into today, I was like we really need to figure out how to narrow down our choices because we have all of these and we can’t focus in. So my question to them was, “How should we narrow down our choices?” and my goal then was to have them come up with some kind of, I guess, constraints for what would make a fieldtrip realistic. So we had a conversation about what is realistic in our case, and I was actually really impressed when they were talking in their groups, they came up with cost, distance, and safety as the constraints to narrow down our list. So then I already had kids jumping into wanting to figure out how much their choice would cost, and they were looking on their computers a little bit more. What I’m running into is that they’re all so bought into their idea of fieldtrip or the one that they think is best that they’re not really listening to other people’s ideas. 127 MAC: Oh, no. 128 Nora: So today we had a long conversation about what we should do and how we can narrow this down even further. And yesterday, I had them choose, in their groups, just one that they thought would be great so we can narrow it down to that. And then I took those, and I hung them up on the wall, and they did the dot thing where they basically voted without making a big deal out of it. Well it backfired because now they want to have a classroom sleepover 129 Maria: One of my kids said that. 130 Nora: I have no idea… 131 Maria: We did that in Wilbur with third and fifth graders. 132 Nora: Did you? 133 Maria: Yeah, for I Love to Read Month.

Eliciting student ideas

Focusing student ideas

Building consensus

158 134 Nora:

M. A. Carlson And I’m open to it, but I’m like we have to figure out what our client is gonna want. Like what are we gonna have to do to make this. And we talked about learning. Learning had to be greater than or equal to the fun. [laughs] Yeah. That’s what we talked about today. So pretty much now we’ve narrowed it down to and I actually even went around and had a conversation with each kid because I felt peer pressure was really coming in about what they wanted. And so we finally narrowed it down to Gold Town, Parkville, and a classroom sleepover.

Considering unexpected ideas Facilitating connections to the client Building consensus

For Nora, an important part of eliciting student ideas was encouraging students to suggest fieldtrips without concern for what she might want to hear. Her efforts to “to get them to come up with their own ideas and not necessarily having any in mind myself” (Turn 126) indicated that she set aside her own agenda and focused on taking her students’ suggestions seriously. As students worked, Nora realized a long list of potential fieldtrips made it harder to focus on building mathematical solutions. The teaching practice, focusing student ideas, helped Nora’s students both shorten the list and focus on features of fieldtrips that might eventually inform their solutions. Rather than tell her students what fieldtrips were or were not realistic, Nora asked the class to focus on features that made fieldtrips more or less plausible. Nora also found she needed to build consensus among her students. She reported students having high investment in their own ideas but less interest in listening to others. Encouraging the class to focus their ideas meant some students would invest in predicting the cost of a fieldtrip they did not nominate, and Nora was concerned that the process was becoming competitive, rather than cooperative. For Nora, building consensus was not only about shortening the list of potential fieldtrips. It was also about engaging students in a process they viewed as fair. Nora’s commitment to working with student contributions was also evident in the way she considered unexpected ideas. All of the teachers reported students suggesting fieldtrips that were not plausible. Students eliminated suggestions like faraway theme parks when teachers asked them to focus on locations the client was likely to approve. The classroom sleepover created a different challenge because from Nora’s perspective it was possible, just unanticipated. Nora acknowledged the sleepover as important to students, considered whether or not she would be willing to chaperone such an event, and decided how to integrate the new idea with her planned trajectory for the modeling task.

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To integrate the sleepover into the modeling task, Nora facilitated connections to the client. All of the teachers named the principal as the client for the task. As client, the principal would review and approve or disapprove of students’ recommendations. Nora quickly assessed that having a class sleepover was subject to the principal’s approval. She was honest with her students about the need to consider the client’s perspective. Although the principal found the sleepover was against district policy, passing the decision-making authority to the client let Nora take her students’ ideas seriously and preserved task’s authenticity. Nora reported that her students accepted the principal’s decision as part of developing a solution to a real-world problem that included a client.

7.4.2 Interacting with the Real-World Context Teaching practices related to working with the real-world context involved building teacher and student knowledge about the situation that motivated the modeling task. Sometimes, as when identifying contexts important to students or dealing with setbacks, the teachers carried out this work as they planned the modeling lesson. More often, working with the real-world context involved facilitating opportunities for students to find and use information in their solutions. Deciding when and how to have students interact with the real-world context created challenges and opportunities for the teachers. Student research could be time-consuming and distract from building mathematical solutions, but the teachers also felt it played an important role in giving students ownership of the task. The teachers were strategic when they facilitated student research. Sometimes, they simply gave information to students. At other times, they asked students to find information themselves. The teachers anticipated what students would need to find, previewed websites, and tracked down additional information as needed. Teachers also decided if and how the class would incorporate results from research into their mathematical solutions. In the exchange below, Amy and Maria discuss helping students find and use information about Gold Town in their mathematical solutions.

160 And then I let them research on the computer. Now we’re done with transportation cost, now we’re moving to activities, so brainstormed the amount. Then we talked about the trains. So, we came up with these, and maybe I shouldn’t have done it this way, I don’t know. But we had, the train would be $8 per person, if we went in groups of six or more, and then the combo for the kids, for the student deal was $10 for the gold panning. 112 Maria: So do you think, Amy, on that note, shouldn’t the train be included in that cost? 113 Amy That’s what I was hoping. 114 Maria: So I’m hoping that it’s included on it, because some of the kids asked about that too, and I was like “I need to call.” 115 Amy: That’s what I told them too. [crosstalk, inaudible]

M. A. Carlson

111 Amy:

Facilitating student research

Amy and Maria’s students used the Gold Town tourism website to find information about activity costs. The students’ apparent success in finding usable information was due in part to teacher preparation. In an earlier study group meeting, the teachers previewed Gold Town’s tourism website. The number of activity options was small, and the prices were in whole-dollar amounts. The teachers decided the website was navigable for third-grade students and saw that its information would be useful when building mathematical solutions. Because the information provided on the website was not complete, the teachers also had to supplement student research by contacting the vendor (Turn 114).

7.4.3 Monitoring and Supporting Mathematics Learning The teachers could not always predict the mathematics students would use when building solutions. Monitoring and supporting students’ mathematics learning involved ongoing attention to the mathematics required by the task and the support students might need to advance their mathematical work. In the exchange below, Amy discusses finding the number of buses needed for the fieldtrip. The fieldtrip context gave students several opportunities to apply multiplicative reasoning. For early third graders, this content was challenging. Students had informal strategies for multiplication, but the class would not formally cover the topic until later in the year. Amy describes facilitating students’ mathematical work when the demands of the modeling task involve content that is new to students.

7  Teaching Practices to Support Early Mathematical Modeling 104 Amy:

My kids are in the middle of their stuff, but they really worked on persevering today because it was really tricky for them because we haven’t gotten into multiplication very much, just a little bit. 105 Maria: They’re still on addition and subtraction and place value, rounding. We talked about, they figured out, the scenarios with my class, with Jane’s class, Mrs. Anderson, and with Mrs. Neil’s class. She has the three/four [combination class]. We looked at, “How many kids would we have if we just go?” We’d have 56. If we have Neil’s thirds go we’d have nine more. So we planned the bus for 56, we planned the bus for us and her thirds, and then we planned the bus for all of us. We all go with that in total. So, we did that, and then we figured we’d need one bus for us. We could fit us and Neil’s thirds in one bus, plus the107 Maria: Very easily.1 108 Amy: And with the, I said seven kids per volunteer… 109–110 [brief interruption due to background noise] 111 Amy: …just because, the 7 will go into the 56 nicely. So I tried to make groups and we talked about division too when making groups. But anyhow, they figured this out. They figured what the total buses would be, they figured up what the total amount of people would be for all of us.

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Recognizing mathematical challenges Linking to curricular mathematics

106 Amy:

Facilitating transitions between context and mathematics

Modifying contextual features of the task

For Amy, supporting students who needed to use new mathematics on a modeling task involved first recognizing the mathematical challenge and its link to curricular mathematics (Turns 104–105). The practices facilitating transitions between the context and mathematics (Turn 106) and modifying contextual features of the task (Turn 111) helped Amy adjust the cognitive demands of the task. Amy did not simply give students data from all three classes. She helped them decompose the task into smaller parts. In addition, Amy chose the ratio of volunteers to students carefully. The context fixed the number of students and teachers who would go on the fieldtrip, but it afforded some flexibility with respect to volunteers. Asking students to assign one volunteer for every seven students meant the class did not have to account for “extra” students or different sized groups.

7.4.4 Linking Student Ideas and the Mathematical Solution Linking student ideas and the model involved identifying and creating opportunities for students to either mathematize the real-world situation or translate their mathematical solutions back to the real-world context. Teachers used the first two practices in this category, facilitating transitions between the context and mathematics

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and connecting student ideas and the model, as students moved from the real-world problem to the mathematical problem and as students built their mathematical solutions. Two practices, supporting students in evaluating their own work and deciding what counts as a model, came up when teachers discussed bringing closure to the task. During study group session 1, we discussed potential products for the modeling task. We wondered if students should come up with a formula or function that predicted the cost of the fieldtrip for any class or if we would be satisfied with a solution that only applied to their group. We revisited this topic during session 6, as teachers planned to bring closure to the modeling task. The teachers decided students would present solutions to the client and justify their conclusions with mathematics. However, the teachers remained undecided with respect to asking students for a general solution. 94 Nora:

No, I think that maybe what you said, that whole typical, what would the typical group look like? 95 Maria: Yeah, I liked that. 96 Amy: That’ll be a fun place to take it, for sure. I like that, because it’s just more. Then, I think that’s where the formulas can come out. How do we figure out how many buses we need and figure out a formula for the bus or 97 Maria: The formula for the cost. 98 Amy: An equation for the cost, that kind of thing.

Deciding what counts as a model

Eventually, Amy and Maria decided they might revisit the task and develop a general model as an extension. Nora recognized a potential link to fourth-grade mathematics content and encouraged her students to develop a formula. Later, in the same study group session, she explained: 170 Nora: …then, I made this for the next step. I said, “I’ve already heard other classes want to know too what you did, and your cost plan.” So, I gave this, for a class of blank students, it would cost blank, to go to blank. So I said, “That’s our last hurdle with this.” So, I don’t know if they’ll get there. I think some of them will. You putting that table up there, that’s what made me think of it, because we talk in fourth grade about rules with those input output tables, and so if they are able to see the relationship, then I would think they could turn that into a formula.

Deciding what counts as a model

Linking to curricular mathematics

Although our study group did not meet after they finished the modeling task, Amy and Maria shared their students’ final products with their colleagues at the modeling symposium. Students worked in small groups to prepare a slide show presentation for their clients. Their slides included the total cost of the fieldtrip and

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an explanation, with supporting mathematical work, for each component of the total cost. At the time of the symposium, Nora’s students were still developing their presentations, and Carol’s class had not yet finished the task.

7.5 Discussion and Conclusion I begin this section by acknowledging two important limitations of this study. First, the study relies on teachers’ descriptions of their work. The activities we focused on during the study group meetings may have limited or influenced by the practices teachers shared. Second, the study is descriptive. My intent, both as I facilitated the study group and in my analysis and reporting of the data, was to understand the practices teachers described as they made sense of their own work. I did not evaluate whether or not the practices were effective in engaging students in mathematical modeling or in teaching mathematics generally. Understanding the strengths and limitations of teaching practices for mathematical modeling will require direct observation of multiple teachers in concert with analysis of student work. The aim of this chapter was to understand the practices teachers use when engaging students in mathematical modeling. Drawing on discourse data from a teacher study group, I suggested teaching practices for mathematical modeling fell into four categories: working with student ideas, interacting with the real-world context, monitoring and supporting mathematics learning, and linking student ideas with the mathematical model. I also offered descriptions of the ways teachers’ practices helped them facilitate students’ work with the real-world context, support students when building mathematical solutions, and decide how to bring closure to the modeling task. The teachers’ descriptions suggest that teaching mathematical modeling is ambitious work (Lampert, Beasley, Ghousseini, Kazemi, & Franke, 2010). The students in Amy, Maria, Nora, and Carol’s classrooms performed authentic tasks related to solving a real problem. In order to support students’ work, the teachers needed to “observe and listen and adjust both content and methods to what they observe in those performances to enable diverse learners to succeed in doing high-quality academic work” (Lampert et al., 2010, p. 130). Many of the practices teachers described intersect with those already in the literature. Teachers in this study discussed eliciting students’ ideas about the problem’s real-world context and using those ideas during instruction. Turner et  al. (2012) included eliciting, attending, and awareness as important initial practices for engaging with children’s multiple mathematics knowledge bases during instruction. Although the fieldtrip problem built on knowledge students likely constructed in their school, rather than home, communities, it was not explicitly mathematical. The teachers sought to integrate students’ ideas about the problem context with their mathematical solutions. They were aware of what contexts might be interesting and accessible to students, elicited and attended to students’ mathematical and contex-

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tual knowledge, and sought meaningful ways to integrate students’ multiple knowledge bases. The teachers in this study also reported using practices that support teachers’ “skillful improvisation” (Smith & Stein, 2011, p. 7). Smith and Stein identified five practices that give teachers some control over how a lesson develops when teaching is not predictable. Three are especially salient to this study: • Anticipating likely student responses to challenging mathematical tasks • Monitoring students’ actual responses to the tasks • Connecting different students’ responses and connecting the responses to key mathematical ideas (p. 8) A key difference between the work Smith and Stein described and the work the teachers in this study discussed is that the study group teachers’ practices applied to students’ nonmathematical, as well as their mathematical work. Anticipating and connecting played a critical role in linking the real and mathematical worlds. Before teachers could anticipate student mathematics, they had to consider what students might say about fieldtrips, how students’ knowledge of fieldtrips could inform their mathematical solutions, and decide what new information students might need. For Smith and Stein (2011), connecting was especially important at the end of the task when the class discusses student solutions. In the current study, practices related to connecting (facilitating connections between the context and connecting student ideas and the model) were less about working with students’ solutions and more about linking students’ ideas about the context to their emergent mathematical work. Helping students “draw connections between their solutions and other students’ solutions as well as the key mathematical ideas in the lesson” (Smith & Stein, 2011, p. 11) may be an effective practice when facilitating mathematical modeling, but the teachers in this study did not describe doing so. Teachers’ practices related to monitoring students’ actual responses to the modeling task also had similarities to and differences from Smith and Stein’s (2011) work. The teachers’ comments during the study group indicated that they were certainly “paying attention to the thinking of students during the actual lesson as they work[ed] individually and collectively” (p.  37), but their focus was not only on facilitating future discussions. Instead, the teachers considered how the mathematics unfolding during the task connected with the school curriculum. When monitoring students’ work on the modeling task, the teachers seemed to be drawing on their own horizon knowledge. That is, “the way mathematical topics are related over the span of mathematics included in the curriculum” (Ball, Thames, & Phelps, 2008, p.  403). In order to respond to students’ mathematical ideas, the teachers in this study made quick assessments of whether students were applying concepts they already learned, encountering content that intersected with current mathematics, or developing strategies they would formalize later. These assessments helped the teachers decide what supports students might need, identify opportunities to use the modeling task as a launch point for other lessons, and determine what a model could or should look like in their classrooms.

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The opportunities mathematical modeling affords classroom communities are promising, and the increased attention it is receiving is encouraging. Continued progress in making mathematical modeling a part of elementary students’ mathematics experience will hinge at least in part on the field developing a deeper understanding the demands it places on teachers. Future studies could investigate practices of teaching mathematical modeling directly through classroom observation, develop images of more and less effective use of these practices, and examine professional development that helps teachers learn to engage their students in mathematical modeling. Acknowledgments  This work is partially supported by the National Science Foundation under Grant No. 1141024.

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Chapter 8

Teachers’ Use of Students’ Mathematical Ideas in Mathematical Modeling Elizabeth Fulton

8.1 Introduction Researchers argue it is critical that students understand the usefulness of mathematics in order to become successful problem-solvers and critical thinkers (Middleton, Lesh, & Heger, 2003). While learning the skills taught in mathematics is important, students also must be taught how those skills are useful and be given opportunities to apply what they learn to situations around them (Verschaffel, De Corte, & Vierstraete, 1999). Mathematical modeling is an authentic way to address this need because modelers use mathematics as a tool to answer real-world questions (Wolf, 2015). Proponents of mathematical modeling argue that mathematical modeling should occur in elementary school (Kaiser & Maass, 2007) because modeling helps students learn to think mathematically (Lesh & Doerr, 2003; Pollak, 2012; Zbiek & Conner, 2006). However, little research and few descriptions of teachers implementing mathematical modeling in elementary school exist (English, 2006). Few teachers have engaged in mathematical modeling tasks as students and few teachers themselves have learned how to engage students in mathematical modeling tasks as part of their initial teacher education (Doerr, 2007). There is little research on how the teaching of mathematical modeling at the elementary level is enacted (Tam, 2011, p. 32). Part of the difficulty of teaching mathematical modeling is that it requires students to make assumptions and value judgments. This allows for students’ input and perspectives to shape the direction of the modeling cycle. Some teachers may be unfamiliar or uncomfortable with teaching lessons based on students’ input because they cannot plan the entire lesson ahead of time or know exactly how the task ought to culminate and close when modeling tasks do not have a singular correct solution. E. Fulton (*) Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_8

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This study helps researchers to understand how successful teachers of mathematical modeling interact with and incorporate their students’ contributions and mathematical thinking into their teaching of mathematical modeling.

8.2 Literature 8.2.1 Modeling Cycle and Mathematics’ Role in Modeling Mathematical modeling is a process that uses mathematics as a tool to answer authentic real-world questions. This process comprises a nonlinear sequence steps – form a mathematical question, define assumptions, formulate a model, determine results, and interpret and test the solution in context of the real-world scenario (Mooney & Swift, 1999). The Society for Industrial and Applied Mathematics (SIAM) defines a mathematical model as “a representation of a system or scenario that is used to gain qualitative or quantitative understanding of some real-world problems and to predict future behavior” (Bliss, Fowler, & Galluzzo, 2014, p. 3). A model is a product, and modeling is a mathematical process. Mathematics plays a valuable role in mathematical modeling by distinguishing it from word problems and applications often used in mathematics classes illustrated below (Fig.  8.1). Word problems and applications in mathematics curricula have idealized settings in mathematical terms, and mathematics is clearly present in a given problem; there is little, if any, interpretation from the real-world to the mathematical world, the mathematics solution strategy is predetermined or fairly clear, and there is often one solution (Tran & Dougherty, 2014). Modeling tasks are described in the real world, and the modeler must formulate a mathematical problem (Pollak, 2012), the solution strategy is not straightforward, and there is not one correct solution. In contrast, real-world exploration activities such as problem-­based

Fig. 8.1 Problem-solving continuum adaption. Problem-solving continuum (Tran and Dougherty, 2014)

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learning or STEM design activities are often set in complex, real-world situations, but the curricular purpose of the activity is not necessarily mathematical (Jolly, 2014). Because how mathematics is used distinguishes modeling from other mathematical and real-world exploration activities, it is important to study the role of mathematics in cases of elementary mathematical modeling.

8.2.2 Teachers’ Interactions with Student’s Thinking In mathematical modeling, students are the mathematical modelers. Because students are the mathematical modelers, they will be asking mathematical questions, making assumptions, building models, and testing and refining models. Since students are contributing many mathematical ideas, researching how teachers use student thinking in classroom instruction can contribute to understanding how teachers might use student thinking in instruction of mathematical modeling. The National Council of Teachers of Mathematics (NCTM) advocates that teachers use their students’ reasoning in lessons (NCTM, 2000). Students benefit from this because it provides them opportunities to express their thinking, to learn from their classmates reasoning, and to value others’ thinking (Jacobs & Spangler, 2017). Teachers can guide students’ attention to student thinking through facilitating group and classroom discussions on mathematical ideas. Research has shown that students have high engagement in mathematical discussions and higher achievement (Jacobs & Spangler, 2017). When leading mathematical discussions and guiding cognitively demanding tasks, students will produce many differing mathematical responses; it can be challenging for teachers to guide the class discussion toward developing deep mathematical understanding. Smith and Stein (2011) suggest that teachers may work toward running a smooth, meaningful, and engaged classroom that focuses on students’ mathematical contributions to further develop mathematical understanding by practicing the following actions: • Anticipating likely student responses to challenging mathematical tasks • Monitoring students’ actual responses to the tasks • Selecting particular students to present their mathematical work during the whole-class discussion • Sequencing the student responses that will be displayed in a specific order • Connecting different students’ responses and connecting the responses to key mathematical ideas (Smith & Stein, 2011, p. 8) Much of the literature on mathematical discussions and strategies that teachers use to focus on students’ reasoning is situated in mathematics lessons that have a predetermined mathematical goal. This is in direct contrast to mathematical modeling, in which the mathematical content is emergent even if the teachers have particular mathematical goals and expectations. In these situations where the mathematical activity is positioned in a real-world context, it is even harder to guide the lesson to a mathematical point (Sleep, 2012).

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8.2.3 Teaching Mathematical Modeling Carlson, Wickstrom, Burroughs, and Fulton (2016) developed a Teaching Framework for Modeling in the K-5 Setting1 (also referred to as the Teaching Framework for Modeling) to support teachers in creating modeling opportunities for their elementary students (Fig.  8.2). The Teaching Framework for Modeling considers how teachers might successfully prepare to engage their students in a modeling task and what teachers might do as their students work through the modeling process. Teachers spend time developing and anticipating a modeling task, they then enact the modeling lesson, and finally follow up with revisiting the task. In the enactment stage, the teacher and students both have roles. Students work through the modeler’s cycle of pose mathematical questions, build mathematical solutions, and validate conclusions in the context of the task; these steps are condensed modeling steps described from literature. The teacher moves through the steps of organizing the students to the modeling step, monitoring student work, and regrouping students to share ideas and work. The Teaching Framework for Modeling provides questions and points for teachers to consider for each stage. The Teaching Framework for Modeling offers a tool for investigating the emerging research area of teaching modeling in grades K-5. The Teaching Framework for Modeling is informed by literature describing teaching practices for facilitating mathematical discussions on cognitively demanding tasks (Doerr, 2007; Smith & Stein, 2011; Stein, Smith, Henningsen, & Silver, 2009). The framework is theoretically based and thus should be empirically studied to determine how it describes the realities of teaching mathematical modeling in the elementary grades.

Fig. 8.2  Illustration of the teaching framework for modeling in the K-5 setting (Carlson et al., 2016). Terms written in black indicate teacher actions; terms in white indicate student actions

 K-5 refers to elementary grades. Students are typically ages 5–11 years old.

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8.2.4 Research Question This study examined how a teacher incorporates mathematics when teaching mathematical modeling to elementary students. I address the following research question: “How do teachers interact with their students’ mathematical thinking throughout a mathematical modeling lesson?”

8.2.5 Theoretical Perspectives This research is guided by the underlying philosophy of social constructivism: the notion that knowledge is actively constructed by the learner in a social setting (Smith & Stein, 2011; Vygotsky, 1978). Teachers provide materials, opportunities, and environments to best facilitate learning, but it is the students who create their own knowledge (Schneider & Stern, 2010). Through the process of learning in a social setting, learners take their prior knowledge and experiences and integrate new information, resources, motivation, and ideas. In the context of mathematical modeling, teachers build their understanding of modeling through their work and collaboration with colleagues as well as in their work of implementing modeling with their students. Guided by this philosophy, it is valuable to study teachers’ interactions with students’ mathematical ideas. The interactions give a view on how the students and teachers are working together throughout the modeling process for the students to create their understanding of the modeling task. The teachers may guide their students’ experience, but it is the students who are the modelers in a modeling task. This first lens of analysis examines the teaching moves that the teachers use throughout the task. This will allow us to understand the ways that teachers guide their students in a modeling lesson. Because mathematics is fundamental to the mathematical modeling process, this research analyzes the mathematics of the task. This second lens of analysis examines the mathematics specific to mathematical modeling and influenced data collection and data analysis by focusing on teachers’ and students’ use of mathematics. While collecting and analyzing the data, I focused on who introduced mathematical ideas and how the mathematics contributed to the mathematical modeling task. This lens is informed by social constructivism by considering not just what mathematical ideas were considered but who introduced the ideas and the relationship of the students and teacher with the mathematical ideas. Using the lens of mathematics ensures that analysis of teachers’ interactions with students’ mathematical thinking is focused on the mathematics that students contributed and not their nonmathematical contributions. This study comes out of my dissertation where I analyze the tasks as mathematical modeling tasks in addition to teachers’ interactions with students’ mathematical ideas and the mathematics in the tasks. For analysis of the tasks as modeling tasks, please reference my dissertation, The Mathematics in Mathematical Modeling (Fulton, 2017). In this chapter I extend work from my dissertation by extending analysis on teachers’ interactions with their students’ mathematical ideas by examining the instructional practices used in teaching mathematical modeling.

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8.3 Methodology 8.3.1 Setting and Participants The two teachers in this study participated in professional development on mathematical modeling for elementary teachers sponsored by an NSF-funded grant.2 The professional development had a similar stance as the theoretical framework that I use in this study: knowledge is constructed in social settings, mathematical opportunities exist in real-world situations, and teaching mathematical modeling requires teaching best practices. The professional development asked the teachers to learn about mathematical modeling from two perspectives (Fulton, Wickstrom, Carlson, & Burroughs, 2019). First, the teachers worked in groups as mathematical modelers learning to address authentic questions with mathematics to arrive at a solution. Second, teachers studied mathematical modeling from the perspective of a facilitator. The teachers studied the Teaching Framework for Modeling and considered pedagogical practices associated with high cognitive demand tasks (Stein et  al., 2009) such as differentiating classroom instructions, facilitating classroom discourse, and managing group work. The professional development consisted of a week-long summer course and a fall semester teacher study group which 24 teachers in a district elected to participate. The study took place in the school year following the teachers’ participation in the professional development course. The teachers were selected because of their successful implementation of modeling during the professional development course and their interest to continue using mathematical modeling in their classrooms. The two teachers that participated in this case study taught first and fifth grade at an elementary school in the Rocky Mountain West. Table 8.1 describes the teachers, Rebecca and Amy, and the modeling tasks that they taught their class. Rebecca and Amy agreed to allow me to watch them teach mathematical modeling when they taught modeling in their classrooms. The teachers determined when they

Table 8.1  Description of the teachers’ modeling tasks Rebecca Grade First Years teaching 6 Number of students 21 Task Ritz Cracker Length 1 day – 35 min School year September Developed by Rebecca

Amy Fifth 20 28 Pringles Challenge 7 days – 8 h February Amy and her colleagues

 This material is based upon work supported by the National Science Foundation under Grant No. 1441024. 2

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would teach modeling and which modeling tasks they would teach. I was not present for non-modeling teaching, and thus the analysis is on teachers’ interactions with their students’ mathematical ideas of the observed modeling tasks rather than analysis of interactions in modeling compared to interactions in non-modeling mathematical instruction.

8.3.2 Data Collection In this case study, data are video and audio recording of modeling tasks carried out in classrooms, classroom observations, teacher interviews before and after each observation, lesson plans, and student work samples. I used multiple sources of data and multiple methods of data collections to triangulate the data and support validity (Merriam, 2009). I describe data collection methods as illustrated in the timeline displayed in Fig. 8.3. Before and after each lesson, I conducted interviews (Merriam, 2009; Stake, 1995). The preinterviews were often conducted the day before or the day of the lesson. The post-interviews occurred after each lesson, either the day of the lesson or the next day. Since Amy’s task was several lessons long, some of Amy’s interviews combined a post-interview from a previous lesson with a preinterview from an upcoming lesson. For both pre- and post-interviews, I conducted semi-structured interviews; preplanned questions were designed to understand what the teacher planned for the lesson, the anticipated mathematics, how the implemented lesson compared to the planned lessons, and the mathematics of the implemented lesson. The additional, spontaneous questions were follow-up questions to teachers’ responses or addressed instances from the observation. Classroom lesson observations in this study (nonparticipant observer) were used to document how the process of modeling unfolded in the classroom; I directly witnessed the implementation of modeling lessons and observed teacher interactions with their students while modeling (Merriam, 2009). I recorded each lesson with a video camera and a small microphone worn by the teacher. The video captured the whole-class discussion and some group activity. The video was unable to

Teacher

Pre-Interview

* Audio Recordings Data Collection

* Lesson Plans

Lesson One * Recordings audio video * Observation Notes * Photographs of Student Work

Fig. 8.3  Timeline of data collection surrounding a lesson

Post-Interview * Audio Recordings * Lesson Plans

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record every conversation the teacher had with students during group work, so teachers wore an audio recorder to record the conversations between teachers and individuals. During the lesson, I took observation notes; I recorded observations on student and teacher roles, teacher and student questions and ideas, interactions between students, and student-teacher interactions. My focus was on the mathematical activity of the class and all interactions surrounding the mathematical activity. From the lessons, I also took photographs of the board work and of student work.

8.3.3 Data Analysis Following the completion of each task, I wrote a summary of the task including how the teacher created the modeling task, how the task was introduced, how the class worked on the task, and the final result. I sent the summaries to each teacher and asked them to review for accuracy and any mischaracterizations. After each observation, I reviewed my observation notes, video, and audio recordings of the lessons. I supplemented the observation notes by adding dialogue and details missed during the observation. These supplemented observation notes were used as the primary data source for analysis. I analyzed the data in two stages: first through the lens of mathematics and second through the lens of teachers’ interactions with students’ mathematical ideas. To begin analysis though the lens of mathematics, I marked each mathematical occurrence in the supplemented observation notes. A mathematical occurrence, adapted from Leatham, Peterson, Stockero, and Van Zoest (2015), is defined as actions, or a collection of actions, in which students or the teacher introduces or discusses a mathematical strategy or idea. A “mathematical idea” means the mathematics embedded in the mathematical occurrence and an occurrence could be as short as a phrase or as long as a conversation (Leatham et al., 2015). I documented the time of the occurrence, if the comment was made in a whole-class or group setting, and who made the comment(s). I also analyzed the actions that resulted because of the mathematical occurrence: if the teacher asked further questions about the idea, if the class used the idea, or if the idea was ignored. To generate an overarching understanding of the mathematical ideas used over the course of the task, I also created a graphic analysis of the timeline of mathematical ideas (adapted from Remillard, 2016). This graphic helped to illustrate how mathematical ideas were connected and how time was spent investigating ideas. Figure 8.4 gives an example of a graphic timeline. This example shows the individual ideas and connected mathematical ideas discussed in a whole-class discussion. The arcs in the graphic illustrate mathematical ideas discussed by the whole class. Some mathematical ideas are related to one another; these are illustrated by embedded arcs. Individual mathematical ideas are singular arcs. Many mathematical ideas arise during group work, but because ideas may be present in some groups, but not others, these are not illustrated. Instead, group work is illustrated with a rectangle.

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Fig. 8.4  Example of a mathematical timeline created to illustrate how mathematical ideas are connected to investigate the mathematical modeling task. Adapted from (Remillard, 2016)

After analyzing the mathematics within the lessons, I returned to the data and reexamined it through the lens of teachers’ interactions. To the table of mathematical occurrences, I added a column for teacher actions that described teachers’ actions related to the occurrence. I used descriptive coding of the “Teacher Action” by looking for overarching categories in the description of teacher actions. For each teacher, I considered the best way to categorize their types of interactions to most clearly present the interactions and teaching practices that occurred over the course of the task. I refined the codes by comparing the categorization of the teacher interactions looking for commonalities and differences. I then analyzed teacher actions in the timeline with the Teaching Framework for Modeling (Carlson et al., 2016). Examining the graphic analysis of each task (see Fig.  8.4), I questioned if the lesson could be broken into the stages of organize, monitor, and regroup. I considered each teacher’s interactions and questioned whether there were types of interactions that were common for both Rebecca and Amy within each stage. I also looked for examples of teachers’ actions that contradicted the teacher actions codes. The codes of questioning, selecting and presenting mathematical reasoning, and connecting will be discussed in results. In Sect. 8.4, I describe the tasks taught by Amy and Rebecca, how mathematical ideas were introduced to the class, common teacher actions, and how the teacher actions fit within the Teaching Framework for Modeling.

8.4 Results 8.4.1 Nature of the Modeling Tasks The problems that Rebecca and Amy presented to their students were situated in a school setting through snack time and class competition. Both tasks were modeling tasks – students were presented with questions that are not traditional mathematics questions but with questions that are situated in the real world. Students had to apply their mathematical skills to address the questions. The presented problems do

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not have one correct solution nor have set solution paths that student must use to solve the questions (Fulton, 2017). The following sections describe the tasks, how the teachers implemented the tasks, and how students engaged with the tasks. 8.4.1.1 Rebecca’s Task Rebecca developed the Ritz Cracker task for her first-grade class after she recognized an opportunity for a modeling task when she thought about distributing crackers as a class snack. Families in Rebecca’s class bring containers of snacks for Rebecca to distribute at snack time, and one of those snacks was a box of short tubes of Ritz crackers. Because of the short, nontypical size of the packaging, Rebecca found herself questioning how many tubes she would need to distribute. Rebecca stated she was aware that this question had mathematical opportunities, and those were appropriate mathematics for her students early in the school year because it could be solved with counting. She scheduled the task during the class’s snack time and told her students that she had a problem that they could use mathematics to help her solve. She showed her class the box of Ritz crackers (Fig. 8.5) and said we have “all of these [tubes] of crackers, and what I wonder is, how are we going to do snack today with all of these different containers of snack?” One student suggested they “give a pack to each student,” and the class investigated the feasibility of this idea. The class found that the suggestion would not work after determining that there are more students than packs of crackers. Rebecca then asked partners to brainstorm other ways of solving the problem. After students talked with the person sitting next to them, Rebecca told the class that one student had an idea but could not yet prove it. Rebecca suggested that students might want to use white boards to justify their ideas. Students worked in groups investigating how they would distribute the crackers. Afterward, the students met back at the carpet to share their ideas. Groups shared their ideas one at a time, and Rebecca questioned individual groups, rephrased their ideas, and compared ideas among the groups. After the groups presented and justified their mathematical work, Rebecca summarized the three main ideas: each student gets two crackers, each student gets

Fig. 8.5  The box of crackers with small packaging that was the focus of the Ritz Cracker Task

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three crackers, or pairs of students get one pack of crackers (six and one-half crackers per student). Students voted by raising their hands, and there was a clear majority for six and one-half crackers. Rebecca then gave pairs of students one packet of crackers and told them that they needed to figure out “how to give six crackers to each person and how to split the remaining cracker.” The students then ate their snack after making sure that each partner had the same amount. 8.4.1.2 Amy’s Task Amy worked collaboratively with other fifth-grade teachers in the district to develop the Pringles Challenge task. The challenge was to design a box that was small and light and protected a single Pringle potato chip in travel via inner-district mail to another elementary school; seven fifth-grade classes competed against each other to design the best box. Amy began the task with an introductory lesson about modeling in which the class brainstormed the meaning of mathematical modeling, mathematical questions involved in a seemingly nonmathematical situation, and how a mathematical modeler might solve problems. Having defined mathematical modeling, Amy then introduced the Pringles Challenge to her students. The class spent 15 min reading the rules, asking questions, and determining the important variables of the challenge. They finished the day working individually to design a box and were instructed to think about the mass, dimensions, and materials of their boxes. For the third lesson, Amy asked her students to consider how to “mathematize the problem.” The class discussed how to mathematize the variables of mass, volume, and chip intactness; students had the mathematical tools to easily calculate the mass and volume but found that mathematizing the chip intactness was difficult. Amy led a discussion to consider and describe the best-case scenario and worst-case scenario for chip intactness upon arrival. After describing the situation, the class got into groups and worked to assign values to describe how their chips might arrive. Amy paused group work twice to share students’ thinking and scoring systems. Groups submitted their ideas at the end of class, which Amy said she would present to the other fifth-grade teachers to determine an “official” scoring system based on students’ ideas. Amy began the next day’s lesson by sharing students’ work, and the class compared several scoring systems, which were different versions of scoring rubrics with different scales. Then Amy presented the official scoring system chosen by the teachers. Students noted similarities between the official scoring system and the systems they created. Next, Amy asked her class how they should combine their variables of volume, mass, and chip intactness to create a relationship to produce a final score. Students worked in groups to develop equations to find a final score. Amy selected one group’s equation, V − M + C, for the class to discuss together. Amy showed the class how they could test this equation by substituting test values – reasonable numbers for mass, volume, and chip intactness that were also easy to work with. They tried three examples with the equation, changing only one variable at a time. The class discussed if a high score or low score was a better result and the meaning of

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the final score in relationship to the tested box. After discussing this equation, groups continued to develop their own equations and tested their equations. Students submitted their equations at the end of class, with the understanding that Amy would take their proposals to the other fifth-grade teachers for the teachers to determine the final equations. The next day, Amy shared a few groups’ equations, and the class discussed why the equations might work. Comparing these equations, Amy introduced the final equation that the teachers decided to use for the Pringles Challenge, C = Final Score , and students noted similarities with the equations their class MV produced. The class used the final equation with a test package to understand how the scoring worked. Each group built their box prototype and scored their box. Amy directed her students to revise their packages based on the equation. Most groups made their boxes smaller, while some took out material to lower mass; the groups then predicted the score of their revised boxes. Amy sent her classes’ boxes via intra-district mail to another elementary school participating in the Pringles Challenge. Later the class scored the boxes they received with Amy checking their work. To conclude, the fifth-grade teachers made a presentation with a picture and score for each box that was shared with participating classes.

8.4.2 Organizing Students to Introduce and Use Mathematical Ideas The research question for this study is “How do teachers interact with their students’ mathematical thinking throughout a mathematical modeling lesson?” Throughout the lesson, students and teachers focused on mathematical ideas while working as a class. Both Rebecca and Amy developed and organized their tasks so that students would be able to investigate their questions with mathematics. They both explained their respective problems and communicated the expectation of using mathematics to solve those tasks. Overwhelmingly, students in both Rebecca’s and Amy’s class introduced most of the mathematical ideas discussed. Rebecca introduced very few mathematical ideas in her class. She began the Ritz Cracker task by posing a real-world question and asked her students if they could use mathematics to help her answer the question. One student offered a solution, and other students used mathematics to explain how to determine if that solution worked. Though all mathematical ideas introduced were generated by students, Rebecca’s work guided students toward using their mathematical ideas to investigate the problem. By doing so, students appeared prepared to develop strategies to hand out snacks, such as distributing one pack to a table or one pack to a pair of students, and to use mathematics to determine if their solutions would succeed. Students then worked in groups justifying their ideas with mathematics. Afterward, groups presented their solutions to the class with mathematical explanations and justifications.

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Fig. 8.6  Amy’s directions for her lesson

In only one instance during group work did Rebecca introduce a mathematical idea; she suggested that a group use a doubling strategy to find the number of crackers needed for all students in the class. “One pack is two people, so two packs are what?” she asked. The student responded, “Four people. Or we can count by twos! 2, 4, 6, 8, …” Even in this instance, by saying “or we can…” it seems as though the student believes he is using a different strategy than the one suggested by his teacher. Amy’s lessons were structured such that each day had a clear objective. For instance, on the second day, students knew they needed to draw a box design, determine the materials they needed, and estimate the volume and the mass of the box (see Fig.  8.6). Amy organized several conversations throughout the Pringles Challenge task around mathematical ideas she wanted the class to investigate. On one occasion, she had students consider mathematizing the Pringles chip intactness. Later, Amy described a need for a common evaluation equation and discussed the necessary variables before the students created their own evaluation equations. Finally, Amy led the class through evaluating another school’s package using the evaluation equation before having students evaluate and revise their own packages. For each of these instances, Amy had mathematical ideas she wanted to pursue but structured the lesson around her students’ ideas rather than directly introducing her mathematical ideas. This approach allowed students to create and share their own mathematical ideas. Within each portion of the task, students were given freedom to explore the posed question with mathematics. For example, groups created unique equations with different operations when creating evaluation equations such as V + M – C and V − M + C; some gave good boxes high scores, some gave good boxes low scores, and some proposed equations failed to meaningfully evaluate boxes.

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Most of the mathematical ideas from Amy’s task were introduced by the students, and Amy would lead conversations around these mathematical ideas. For instance, after the class spent a day creating their own scoring systems or equations, Amy would display students’ scoring systems, equations, or the teacher’s scoring system or equation and ask the students, “What do you notice?” Every time Amy began this conversation, she gave students time to look at the displayed work before having students share. Students often responded with, “I notice…” and always commented on mathematical features of the work. The features that students commented on were often differences or similarities with their own work, and many students in the class contributed mathematical ideas in the conversations that followed. In the instance below, Amy presented two students’ solutions of

VM and V + M − C. C

Amy: What do you notice? Student A: The operations are, like, similar. Student B: Inverse. Student C: Not inverse, but close to inverse, maybe inverse. Amy: Inverse? These are inverse. (Amy points to × and ÷ symbols on the board.) Student D: Division is taking away, and subtraction is taking away. And addition is adding, and multiplication is adding. Amy: Agree or disagree? Multiplication is repeated addition, and division is repeated subtraction. (Reading from a student solution) [The students wrote], “We want a small V and M, and when we add them, if we designed a small package, we’ll still get a small amount.” And that’s what some of you were saying. So, if we have a small V and M that means we have a small package. So then “If we protected the chip, we’ll take more away.” Is that what we are doing with this (points to division), in a sense? Are we taking more away by having a better chip?

By initially directing the students to note mathematical observations, Amy could use students’ mathematical ideas to focus the discussion toward particular mathematical ideas. While students worked, Amy generally did not introduce mathematical ideas. The only time Amy introduced mathematical ideas occurred when she introduced the task of needing to mathematize the Pringles chip by telling the students to start mathematizing the chip score by describing the best chip and the worst chip. Using her students’ descriptions, she wrote qualities of the best Pringles chip and the worst Pringles chip on the board (see Fig. 8.7). She then asked her students to get into groups and to mathematize the chip intactness score.

8.4.3 Teachers’ Interaction with Student’s Ideas By analyzing teachers’ interactions with student’s ideas with the Teaching Framework for Modeling and using the graphic analysis tool (Fig. 8.8), most of the class time was spent with students doing group work where the teacher monitored (rectangles) and in whole-class conversation when the teachers regrouped (arcs

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Fig. 8.7  Amy’s record of students’ descriptions of “best chip” and “worst chip”

indicate mathematical ideas in a whole-class setting). Figure  8.8 illustrates how Rebecca went through the organize-monitor-regroup cycle once with class discussion, group discussion, and class discussion. It illustrates that Amy’s class iterated between group discussion and class discussion many times  – which was largely monitoring and regrouping. The next three sections describe Rebecca’s and Amy’s interactions in the following stages: • Interaction with students and their mathematical ideas during the monitor stage • Selection of ideas in the monitor stage to share with the class in the regroup stage • Interactions with students and their mathematical ideas during the regroup stage 8.4.3.1 Monitor Rebecca’s students work in groups once in her 1-day task, and Amy’s students worked in groups many times in her 7-day task (See Fig.  8.8). While students worked, both Rebecca and Amy walked around the room listening to groups work, asking questions, and observing the mathematics each group developed. Both teachers asked their students questions that encouraged them to justify or clarify their work. To elicit mathematical justification in students’ work, Rebecca commonly asked, “How do you know?” For example, a student had an idea for how to share the crackers. When asked how she knew, the student used a picture to explain why her idea worked. Amy also asked her students to justify and explain their mathematical work. An example of a line of Amy’s questioning that prompted students to justify their work occurred while groups where revising their boxes. After groups had constructed boxes and tested their box with the evaluation equation, groups then revised

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Fig. 8.8  Graphic analysis tool illustrating the monitor and regroup stages for the two teachers

their box and evaluated again. In the dialogue below, one group mathematically justifies why they revised their box with shorter sides. Amy: Student: Amy: Student: Amy: Student:

Why would you cut those parts off? To make it even. Okay, mathematically why? To make it small. What do you want to make smaller? You want the smallest volume and smallest mass possible.

Amy often asked her students, “Mathematically, why?” in response to a posed mathematical idea or solution; this common occurrence usually elicited mathematical justification from her students. Rebecca and Amy also posed questions prompting students to clarify their ideas. Examples of Rebecca’s questions of this type were “How many people will share the 13 crackers?” and “How many crackers should each person get?” These questions led students to think about ways to clarify their work by scaffolding their reasoning. Though these questions guided students, they did not provide answers or

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“give away” a procedure. Similarly, Amy asked her students questions to clarify their mathematical work. She regularly asked her students what numbers on the page meant, prompting students to explain their scoring systems, equations, measurements, and conversions. When students created equations, they tested their equations with various mass, volume, and chip intactness scores. Amy suggested that groups only vary one variable at a time and then asked groups to tell her what their numbers meant and what the change in score meant. Sometimes students could not initially answer these questions, and as a result, Amy asked questions to encourage students to clarify their ideas. Below is an example dialogue of this type of conversation. Student: I did another one [test of the equation with one changed variable]. I made the chip 100. I just changed the chip. Amy: Do we want the lower score or the higher score? Student: Bigger. Amy: Do the better packages have higher scores? Can you tell yet?

The students then indicated that they could not yet tell and continued working. 8.4.3.2 Moving from Monitor to Regroup Groups shared different and varied mathematical ideas and questions with their teacher during group work. Both Amy and Rebecca made decisions about which student ideas and questions to make public to the class following group work. In some instance, the teachers would pause group work to state one student’s idea or question. For instance, Rebecca paused group work to share a student’s thought that drawing would help her explain her work. These interruptions generally took only a few seconds. Largely, however, when Rebecca and Amy wanted to share student’s mathematical ideas and questions as a class, they asked groups to stop their work and return to the carpet or their desks. Sometimes, the teachers had a clear idea of a mathematical idea they wanted students to consider and listened to groups work, waiting for students to identify the mathematical idea. Other ideas that teachers shared were not written in lesson plans and may have been decisions made in the moment; for example, Rebecca did not plan on sharing the idea of using drawings to help explain their work. Because Amy’s class worked in groups and regrouped as a class many times, Amy used different tactics toward choosing mathematical ideas to share with the class. Some of her tactics included the following: • Find several different ideas. • Use one rich idea to explore. • Have students volunteer while also identifying individuals to share. When students mathematized the Pringle problem, several groups struggled to define a scoring system. One group found a scoring system much earlier than other

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groups, but Amy waited until three groups had an idea to share. Only when she saw three different ideas, she had the groups present to the class. After each group presented, she asked the class to compare differences between ideas. Amy stated that she did not want to have the first group share earlier because she didn’t want the first groups’ work to be interpreted as the only way to mathematize the intactness of a Pringle chip. Amy’s tactic was to select a few different ideas to make public in effort to scaffold for some struggling groups, not to direct class work toward one specific idea. There were several occasions where Amy wanted to share a variety of student ideas. Some classes she would look for ideas to share while she walked around the room working with groups. For a few lessons, she ended class with no regrouping and only asked students to turn in their work. Before the next class period, Amy selected student work to share that had features she wanted to address as a class. Often, these were features that were present in the teachers’ scoring system and equations that she wanted students to be familiar with. Selecting group work allowed Amy to make chosen ideas public to the class while also letting the students have ownership over the mathematical process. At times, Amy decided to publicly share only one group’s idea. In one instance, when the class was working on creating and testing evaluation equations for the Pringles Challenge, one group created an equation, S = V − M + C (score = volume  – mass  +  chip intactness). Amy realized that due to the number of students needing help with testing their various equations, it was time to consider this topic as a class. Amy said she wanted to use this equation because there were problems with the equation the whole class would benefit from discussing. Amy said the equation, S = V − M + C, was a good attempt, and the group would not mind being wrong in front of the class. In this regrouping strategy, Amy looked for an example that provided rich mathematics that the class could investigate in depth. Another sharing strategy that teachers used was to not select groups to share but to have students volunteer their work. Rebecca and Amy both used this strategy. Because the teachers had already questioned and observed the groups, the volunteered ideas were not surprises. During work time, Amy would sometimes quietly ask a few students to share their work during share-out time. Then during share time, Amy would ask for volunteers to share their reflections or ideas. Both preselected and nonselected students would share their ideas. This tactic brought forth a wide variety of ideas including the ideas that Amy wanted the class to consider. 8.4.3.3 Regrouping When the teachers led conversations around mathematical ideas in whole-class settings, Amy and Rebecca used several tactics. First, they asked questions eliciting their students to justify and clarify their work just as they did in group work. Secondly, they focused the classroom conversation around particular students’ mathematical ideas or questions. Thirdly, they made connections between ideas. Lastly, they made the decision to not pursue all ideas presented by students. Each strategy is presented below.

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Justify and Clarify Rebecca used similar questioning in group presentation as she did while students worked; her questions prompted groups to clarify and justify their ideas. In one case, a student said that 13 is odd so they need to “take down one,” in the context of two students sharing one package of crackers, and Rebecca asked the student to explain what he meant. The student then explained how 12 was even and 2 students could evenly share the crackers, but 13 is odd, so there will be one leftover. The class then discussed what to do with the leftover cracker. Amy also often asked her students “Why?” As with Rebecca’s students, asking why sometimes caused students to justify their work. Other times, asking why required students to do more work and group discussion to clarify their idea. In the following example, the class was discussing the expression V ÷ 10 × 14, and one student suggested simplifying the equation: Student A: Instead of dividing by 10 and multiplying by 14, you should just divide the volume by 4. Amy: Why would you do that? Student A: You can take the 10 away because you multiply by 10. Amy: You’ll just take 10 away from 14? Student A: Yeah. Student B: I don’t think you can do that. After more discussion Amy: Can you try both ways and see what you get? Show me [your work]. I want you to try, and we’ll see if it works.

The student who suggested simplifying V ÷ 10 × 14 to V × 4 could not explain why the two expressions were equal. Rather than telling the students that the two equations are not equal or explaining why, Amy gave the students more time to continue working to clarify their thinking. Using Students Ideas to Guide and Teach As Rebecca introduced her modeling problem, the first idea given by a student was to give each student a package of crackers. Rebecca used this idea to guide students to explore if this was a mathematically reasonable solution. Students offered mathematical ideas about how to determine if the idea was feasible, and Rebecca guided the class discussion by comparing ideas and choosing the order of ideas to investigate. After the class determined that one pack per student was not a feasible approach, students worked to generate, investigate, and justify solutions. Amy also used students’ ideas to guide her lessons. Notably, Amy used students’ mathematical questions to conduct mini-lessons where she helped to explain mathematics or give students guided practice. When her students were calculating the volume of their box, they realized that they needed to multiply and divide decimals. The students asked Amy for help because they knew what math they needed to do but had questions about how to do so. Amy explained the mathematics only after

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students realized that they needed the mathematical skills. Even while Amy taught multiplying decimals, she asked students many questions; her mini-lesson was not a mini-lecture. Connecting Ideas Both Rebecca and Amy asked students to connect their mathematical ideas with their classmates. The teachers also made connections between the students’ ideas. In examples shared in earlier sections, Rebecca’s class shared their ideas of how to distribute the Ritz crackers, and Rebecca asked students how their strategies related to the other students’ strategies. In the same spirit, Amy’s students compared chip VM scoring systems, such as versus V + M − C, and equations to evaluate chip-­ C shipping boxes with their classmates as well. Not Pursuing all Mathematical Ideas Students generated many mathematical ideas throughout the modeling tasks. A common feature in both cases was that each teacher made choices about which student-­generated ideas to bring to the class in the regrouping periods. This meant that some mathematical ideas introduced by students in groups went unpursued. In the whole group discussion, the teachers helped guide the students toward investigating certain mathematical ideas and away from others. For example, during the students’ time to share possible approaches of dispensing Ritz crackers, one student suggested handing out goldfish crackers. Rebecca responded by saying, “Okay, so another solution is to get another snack.” Once all groups presented and the class voted on the best option, Rebecca did not include goldfish as an option. Likewise, Amy redirected students’ mathematical ideas when guiding mathematical discussion to pursue other students’ ideas. On several occasions when an important issue would emerge which Amy wished to address but did not consider the timing appropriate as it would remove the focus from the current issue. In the VM following dialogue, students explained why the equation (where V stands for C volume, M for mass, and C for chip intactness) makes sense. Amy begins to question the approach and then suggests they revisit the idea later. Student A: We want the smallest amount to divide so we have to multiply small numbers, like 5 × 2, mass 5 and volume 2, then it would be 10. If you have a high chip score like 50, then you’d have 10 ÷ 50 would go in the negatives, it’d be really low. Student B: No, just a remainder. Student C: It’s okay to have remainder, if you have low number and a super good chip, you’ll get a remainder. Amy: What did you do with the remainder? Student A: It’s just with the number, like if you have 2 leftover then—

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Amy:  So, are you thinking fractions as the remainder, or decimals, or…[what?]. I want to come back to that later. Do you have it written down? Student A: Yes.

While Amy initially wanted to discuss remainders and how the student calculated and represented the remainder, she decided to keep the focus of the conversation on VM making sense of the equation . I do not know if Amy and the student later C discussed remainders, but it was not discussed in class during the Pringles Challenge Task. Each teacher guided the class to investigate mathematical ideas in their regrouping period. During class discussion, teachers pushed their students to justify and clarify mathematical ideas. Teachers guided the conversations to pursue some mathematical ideas and questions but also chose not to pursue other mathematical ideas. Because students engaged in modeling, it appears that not pursuing all ideas does not interfere with the progress of modeling.

8.4.4 Modeling Instructional Practices Analysis of the instructional practices as teachers’ interacted with their student’s ideas was examined within the organize, monitor, and regroup stage from the Teaching Framework for Modeling (Carlson et al., 2016). Several practices were common throughout the three stages. In this section, I present modeling instructional practices as they were used throughout the modeling cycle, rather than isolated to specific stages. I reexamined the practices by considering all of the teachers’ interactions as a whole, looking for shared practices and common purposes for each practice. These modeling instructional practices are synthesized in Table 8.2 as four categories of methods for purposefully responding to students’ mathematical ideas. Throughout the modeling task teachers (1) elicited mathematical ideas and questions throughout the modeling task; (2) illuminated mathematical processes and content; (3) coordinated students ideas to move the class forward in the modeling process; and (4) maintained student’s focus on mathematics. The entries of each column are specific examples of modeling instructional practices under each of the categories. 8.4.4.1 Teaching to Elicit Student’s Mathematical Ideas In this study, students contributed most of the mathematical ideas investigated in the modeling tasks. This was possible because teachers expected the mathematics to come from their students, and they used practices to elicit students’ mathematical input throughout the modeling cycle. The modeling instructional practices (e.g., ask

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Table 8.2  Modeling instructional practices Eliciting Asks students for mathematical observations

Illuminating Shares different strategies at the same time Investigates one student’s idea in-depth

Coordinating Pauses group work to share idea

Asks for mathematical explanations, justifications, and clarifications

Compares students’ solutions

Guides class through investigating one student’s idea before asking groups or individuals to investigate

Emphasizes mathematical ideas or questions Holds mini-lesson Decides which ideas to make public

Connects students ideas (in the moment or collect work at end of class and displays work the next day) Chooses not to pursue all ideas in the lesson

Explicitly tells students to use mathematics

Asks students to connect ideas

Maintaining Asks scaffolding questions Investigates ideas from student proposals Gives additional time to clarify ideas Refers to mathematical reasoning throughout task

Reiterates original question

students for mathematical observations, tell students to use mathematics, or ask for mathematical explanations) that elicit students’ mathematical ideas (as listed in Table 8.2) were used throughout the task, not just used to introduce the task. This enables students to be the mathematical modelers throughout the task as well as gain the benefits associated with investigating student reasoning (Jacobs & Spangler, 2017). 8.4.4.2 Teaching to Illuminate Mathematical Processes and Content While the students introduced the mathematical ideas, the teachers’ work was in guiding the students’ ideas. The way in which Rebecca and Amy guided the tasks reflected their teaching goals. Rebecca wanted to introduce modeling and observe her students. From the task, students learned how to use mathematics to solve their problem, and Rebecca learned how her students used mathematics. Amy had curricular goals in addition to the goal of giving her students an experience to model with mathematics. From her task, students learned how to use mathematics to answer their question, but they also explored and developed fifth-grade mathematics standards. By using the modeling instructional practices (e.g., share different strategies at a time, compares students’ solutions), both Amy and Rebecca were able to illustrate the mathematical process and content standards that they hoped to use and investigate.

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8.4.4.3 Coordinating Student’s Mathematical Ideas In both cases, students engaged in the process of mathematical modeling either despite or because their teachers used different modeling instructional practices at various points within the modeling process. This lends evidence to the conjecture that when teachers are faced with many decisions while they teach modeling, there are multiple reasonable decisions they can make that will each encourage students to model. For example, in some instances teachers chose not pursue all mathematical ideas shared and at other times connected students’ ideas (see Table 8.2 for more examples). Perhaps the specific modeling instructional practice itself is less important than the purpose and intent of the instructional decisions made (Stein, Engle, Smith, & Hughes, 2008). The complexity of teaching modeling mirrors the complexity of the mathematical modeling process (English, 2006). 8.4.4.4 Maintaining High Cognitive Demand Tasks in the Midst of Emergent Mathematics Both Rebecca and Amy began with high cognitive demand tasks by having modeling tasks. Students introduced mathematical ideas to investigate the real-world questions, and the teachers maintained students’ focus on using and developing mathematical reasoning (e.g., investigates mathematical ideas in students’ proposals and gives additional time for students to clarify their ideas). There are strong connections to Smith and Stein’s (1998) work describing strategies to implement and maintain high cognitive demand tasks. This is not surprising because their work, along with the work of Stein et al. (2009), was drawn on in the professional development that the teachers participated in. This study offers two cases of strategies used by teachers to engage students in high cognitive demand tasks where mathematical content is emergent throughout the task. 8.4.4.5 Relationship Between Categories Teachers make many in-the-moment decisions and choices critical to the teaching of mathematical modeling. These decisions may involve several modeling instructional practices together, and the practices certainly may overlap with each other. For example, a teacher may decide to pause the group work to share an idea in order to move the whole class forward. They could then follow up by asking students to mathematically justify the presented idea to elicit student reasoning. The teacher might then emphasize a mathematical point of the student’s idea in order to illuminate a mathematical content standard. Though spending time investigating the students’ idea at a deep level, the class maintains focus using mathematical reasoning. The multitude and variation of teaching choices and decisions made by Rebecca and Amy during the implementation of modeling exemplifies the complexity of teaching inquiry-based learning in a student-centered classroom (Stein et al., 2008).

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8.5 Discussion The findings of this study are instructive for teachers who want to learn how to teach mathematical modeling and how teachers handle the uncertainty of students’ mathematical contributions to a modeling lesson. Findings show that teaching mathematical modeling is complex; teachers guide their students in the mathematical modeling cycle such that their students introduce mathematical ideas; teachers guide the student’s mathematical ideas to move the class forward in the modeling process; teachers make many differing decisions throughout the process depending on their goals and intentions. This case study analyzes the teaching of two mathematical modeling tasks in elementary grades. The teachers in this study are not typical because they participated in 50 h of professional development focused on mathematical modeling. They were chosen for this case study because they were likely to provide cases of teaching mathematical modeling because they had already implemented modeling. Therefore, this is an instrumental case study (Stake, 1995) because it describes optimal cases of teaching mathematical modeling so that teachers and researchers may understand and envision what is possible with mathematical modeling in elementary grades. This leads to questions about what is possible for teachers who do not have access to professional development on mathematical modeling. What supports and resources do teachers need to implement mathematical modeling? The aim of this article is to describe who introduces mathematical ideas in modeling tasks and how teachers interact with their students’ mathematical ideas. Results show that teachers elicit students’ mathematical ideas and questions throughout the modeling cycle. By encouraging students to introduce mathematical ideas and by giving students the freedom to investigate their own ideas, teachers provided significant opportunities for their students to develop mathematical agency (Boaler, 2002). Further research on students of mathematical modeling could study how students develop mathematical agency through modeling. This study also describes modeling instructional practices teachers use when making mathematical and instructional decisions so that students focus on using mathematical tools to answer real problems. I only followed the participants of the study in their mathematical modeling lessons, but through conversations I believe that their modeling instructional practices were similar to their typical mathematical instructional practices. Additional research could study how teachers’ modeling instructional practices compare to their general instructional practices and decisions in their mathematical curriculum. Considering this question would provide insight for developing teacher practices on mathematical modeling. Focusing on teaching practices that enable teaching modeling could help to improve teaching practices overall in addition to instructional practices. Teaching mathematical modeling is an ambitious (Kazemi, Franke, & Lampert, 2009) and important teaching practice. From introduction to conclusion, by its open nature, mathematical modeling presents many choices in planning and teaching. Teachers need to facilitate the initial step of mathematical interpretation of a

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real-world problem given its open nature. Students use multiple and varied mathematical strategies while investigating their modeling task, and thus teachers need to be prepared for productive mathematical strategies and potential questions students generate in the middle of the task. Multiple reasonable solutions exist that students may present in modeling tasks. Teaching mathematical modeling is complex and takes class time, but from engaging in mathematical modeling, the class addresses both mathematical content and process standards in engaging and motivating contexts (Zbiek & Conner, 2006).

References Bliss, K. M., Fowler, K. R., & Galluzzo, B. J. (2014). Math modeling: Getting started and getting solutions (1st ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics. Boaler, J. (2002). Learning from teaching: Exploring the relationship between reform curriculum and equity. Journal for Research in Mathematics Education, 33(4), 239–258. Carlson, M. A., Wickstrom, M. H., Burroughs, B. A., & Fulton, E. W. (2016). A case for mathematical modeling in the elementary classroom. In Annual perspectives in mathematics education (APME) 2016: Mathematical modeling and modeling mathematics (pp. 121–129). Doerr, H. M. (2007). What knowledge do teachers need for teaching mathematics through applications and modeling? In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (1st ed., pp. 69–78). New York, NY: Springer. English, L. D. (2006). Mathematical modeling in the primary school: Children’s construction of a consumer guide. Educational Studies in Mathematics, 63(3), 303–323. Fulton, E. W. (2017). The mathematics in mathematical modeling. Montana State University. Fulton E.W., Wickstrom M.H., Carlson M.A., Burroughs E.A. (2019). Teachers as Learners: Engaging Communities of Learners in Mathematical Modelling Through Professional Development. In G. Stillman & J. Brown (Eds.) Lines of Inquiry in Mathematical Modelling Research in Education. ICME-13 Monographs. Springer, Cham. Jacobs, V., & Spangler, D. (2017). Research on core practices in K-12 mathematics teaching. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 766–792). Reston, VA: National Council of Teachers of Mathematics. Jolly, A. (2014). Six characteristics of a great STEM lesson – Education week. Retrieved February 2, 2016, from http://www.edweek.org/tm/articles/2014/06/17/ctq_jolly_stem.html Kaiser, G., & Maass, K. (2007). Modelling in lower secondary mathematics classroom – Problems and opportunities. In W. Blum, P. L. Galbraith, Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (1st ed., pp. 99–108). New York, NY: Springer. Kazemi, E., Franke, M., & Lampert, M. (2009). Developing pedagogies in teacher education to support novice teachers’ ability to enact ambitious instruction. In Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia, vol. 1, pp. 11–30. Leatham, K.  R., Peterson, B.  E., Stockero, S.  L., & Van Zoest, L.  R. (2015). Conceptualizing mathematically significant pedagogical opportunities to build on student thinking. Journal for Research in Mathematics Education, 46(1), 88–124. Lesh, R., & Doerr, H. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (1st ed., pp. 3–34). Mahwah, NJ: Lawrence Erlbaum Associates.

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Merriam, S. B. (2009). Qualitative research: A guide to design and implementation. San Francisco, CA: Jossey-Bass. Middleton, J., Lesh, R., & Heger, M. (2003). Interest, identity and social functioning: Central features of modeling activity. In R.  Lesh & H.  M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching2 (pp. 405–431). Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Mooney, D., & Swift, R. (1999). A course in mathematical modeling. The Mathematical Association of America. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Pollak, H. (2012). Introduction. In H. Gould, D. R. Murray, & A. Sanfratello (Eds.), Mathematical modeling handbook (pp. viii–xi). Consortium for mathematics and its applications. Retrieved from http://www.comap.com/modelingHB/Modeling_HB_Sample.pdf Remillard, J. (2016). Teachers’ design decisions and the role of instructional resources. In International Congress of Mathematics Educators. Schneider, M., & Stern, E. (2010). The cognitive perspective on learning: Ten cornerstone findings. In H. Dumont, D. Istance, & F. Benavides (Eds.), The nature of learning: Using research to inspire practice (pp.  69–90). Organisation for economic co-operation and development. Retrieved from http://www.educ.ethz.ch/pro/litll/oecdbuch.pdf Sleep, L. (2012). The work of steering instruction toward the mathematical point: A decomposition of teaching practice. American Educational Research Journal, 49(5), 935–970. Smith, M. S., & Stein, M. K. (2011). 5 Practices for orchestrating mathematics discussion. NCTM. Stake, R. E. (1995). The art of case study research (1st ed.). Thousand Oaks, CA: Sage. Stein, M. K., Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3(4), 268–275. https://doi. org/10.5951/mtms.3.4.0268 Stein, M.  K., Engle, R.  A., Smith, M.  S., & Hughes, E.  K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standardsbased mathematics instruction: A casebook for professional development (2nd ed.). New York, NY: Teachers College Press. Tam, K.  C. (2011). Modeling in the common core state standards. Journal of Mathematics Education at Teacher College, 2(1), 28–33. Tran, D., & Dougherty, B. J. (2014). Authenticity of mathematical modeling. The Mathematics Teacher, 107(9), 672–678. Verschaffel, L., De Corte, E., & Vierstraete, H. (1999). Upper elementary school pupils’ difficulties in modeling and solving nonstandard additive word problems involving ordinal numbers on JSTOR. Retrieved April 22, 2015, from http://www.jstor.org/stable/749836?origin=crossref&s eq=1#page_scan_tab_contents Vygotsky, L.  S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press. Wolf, N. B. (2015). Modeling with mathematics. Portsmouth, NH: Heinemann. Zbiek, R. M., & Conner, A. (2006). Beyond motivation: Exploring mathematical modeling as a context for deepening students’ understandings of curricular. Educational Studies in Mathematics, 63(1), 89–112. https://doi.org/10.1007/sl0649-005-9002-4

Chapter 9

Teaching and Facilitating Mathematical Modeling: Teaching, Teaching Practices, and Innovation Rose Mary Zbiek

Each of the three chapters in this section targets one or more aspects of mathematical modeling (MM) and strives to understand the teaching of mathematical modeling in the early years through collaborations with classroom teachers. Based on her inquiry of teachers and a coach in a study group, Carlson describes the practices that teachers report using when they plan and facilitate MM in elementary school classrooms. Fulton analyzes classroom lessons led by teachers as she investigates how teachers interact with their students’ mathematical thinking during MM lessons. Casting teachers within a school–university partnership as designers and implementers of MM experiences, Suh and colleagues study how elementary school teachers formulate MM problems and engage their students in mathematizing situations with attention to assumptions, variables, and constraints. The three chapters individually and collectively contribute to the field’s understanding of MM teaching from a U.S.-centric perspective. The works also fall within a broader conversation about teaching, teaching practices, and innovation.

9.1 Teaching and Facilitating Mathematical Modeling The three chapters share a common conception of MM as a cyclical process with both an initial question and a final result that are clearly situated in the real world. To teach MM, then, is to facilitate students’ progress within and across the phases of the process as students wrestle with the messiness of the real world and as they R. M. Zbiek (*) Department of Curriculum and Instruction, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_9

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apply and enhance their limited but growing mathematical repertoires. This work involves broad and unpredictable student responses, positions teachers as designers and implementers of MM activities, and draws on teachers’ agility as modelers. Teachers of MM work in classrooms in which student responses are often unpredictable especially, as Carlson underscores, at the start of the MM cycle. This observation seems to make teachers’ ability to understand and facilitate students’ modeling work both important and challenging. Variability in student responses underscores why Carlson’s attention to how teachers interact with student ideas is valuable. The ideas elicited, the interpretations and assumptions made, the features of the real context foregrounded, and the variables and constraints taken as relevant by the teacher both reflect the breadth of students’ responses and limit or expand the extent to which subsequent student responses might be predictable. Student responses are deselected or ignored as the teacher reacts to the students’ thinking during the MM work. Similar focusing of students’ responses happens as teachers react to students’ suggestions and choices of mathematical tools. Teacher use of student ideas can also affect how the problem is interpreted, including the extent to which it fits with the school’s mathematics curriculum. Based on observations of the work of teaching of MM, I do wonder about the claim that teachers in the classroom do not engage in modeling directly (e.g., Carlson, this volume). Perhaps effective teachers are indeed engaging in modeling but doing so at a faster pace or in a more tentative way than their students. If so, the use of “directly” in the claim is crucial. As teachers plan lessons, it seems teachers need to engage in some parts of the modeling process in order to anticipate possible student responses. In the classroom, teachers react to student ideas to some extent as a modeling team member would react to the suggestion of a teammate. Teachers engaging perhaps not with students but alongside students have implications for how the lesson unfolds and what mathematics students encounter. For example, Suh and colleagues describe how one teacher, Anne, saw the potential for her students to apply linear functions if the students were asked to develop a sales option in order to eliminate products in their overstocked school store. The teacher might not follow all of the modeling paths suggested or taken in the classroom to completion. As students work, teachers are processing the work and looking for options and opportunities. It seems that some of the work that teachers do is in the spirit of advancing the modeling work in implicit ways that are perhaps unrecognized by them and by us. Importantly, the work of teaching is much more than classroom instruction (Ball & Forzani, 2009; Kilpatrick et al., 2015). In particular, planning is essential. It is honored in Carlson’s attention to teachers as they plan MM lessons and is realized in the lesson study group work done by teachers with Suh and colleagues. It seems that the way in which teachers, teacher educators, and researchers can most productively influence teaching practices is not simply to identify, sort, or label practices but to share the thinking that underlies them and revise that thinking so that the practices are more powerfully employed in the future. This is a direct parallel to the work that teachers do with their students in classes that elicit, share, and revise authentic and alternative strategies for solving mathematics problems—something that is celebrated in Carlson’s description of what Amy’s students could have done with multiplication strategies.

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Teaching as a collaborative endeavor is a delightful experience. I find Carlson’s work with a study group to be an important kind of research to inform us about the work of teaching as it can be and as it is viewed by teachers. Suh and colleagues similarly honor teachers’ lenses on MM teaching practices by attending to the percentages of teachers in their group who self-reported feeling confident or very confident about different aspects of MM. They then target professional development to address teacher concerns about how to support students who are in the process of making assumptions in their MM work. The extent to which these chapters acknowledge teachers and researchers as co-conspirators in designing, implementing, and improving early MM experiences is a signal that teaching MM is a challenge and that the ways in which the field moves forward should resonate with a growing body of empirical results, careful application of robust theory, and harmony with classroom experience.

9.2 Contributions of Chapters The chapters contribute to our understanding of teaching of MM in the early grades in multiple ways. Some draw our attention to how teachers might engage in particular elements of modeling cycles. Carlson makes progress in terms of connecting the teaching practices she articulates directly to aspects of mathematical modeling activity. Fulton identifies and articulates instructional practices arising from her data with statements that do not mention any aspects of modeling, though general phases of MM are used to frame the grouping of the stated teaching practices. The difference between Carlson’s and Fulton’s perspectives on teaching practices is intriguing. One explanation might draw on the role of MM in the classrooms that the two researchers observed. Carlson cared about how teachers facilitated MM in the interest of teaching MM. MM was incorporated in their classrooms for the purposes of improving students’ ability to do MM, and Carlson documented the teachers’ perspectives on the practices they employed in MM lessons. In contrast, Fulton seemed to focus on teacher actions or moves that would further a mathematics lesson that involved MM activity. She seemed to allow for the centrality of school mathematics curriculum topics in the teachers’ work and for the teachers’ perceived need that students should do MM work in addition to learning required mathematics content. The extent to which researcher- or teacher-identified MM teaching practices focus on elements of MM or on more general teacher moves seems to reflect a distinction between a focus of the study on teaching and facilitating MM or a focus on teaching school mathematics through MM, respectively. A somewhat related question worth pursuing is to what extent do teaching practices that support use of the real world in MM work differ from those that support mathematical work during MM activity. As others previously elaborated (e.g., Cirillo, Bartell, & Wager, 2016), one of the benefits of MM is the extent to which it can be pursued in concert with social ­justice. Problem formulation and classroom discussions of real-world situations can move into topics that can be socially or ethically uncomfortable for teachers even if they do not seem to be troublesome to students. The challenge of these conversations is

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not unrelated to the observation in Carlson’s chapter that teachers decide how to engage students authentically in decision-making that is typically done by adults. In general, supporting teachers in their implementation of MM activities includes supporting them in their ability to talk with young students about what might otherwise be viewed as adult issues. Researchers are early in their work to understand the variety and interrelationships of teaching practices for MM. Teachers are similarly early in their establishment and refinement of their teaching of MM.  Carlson’s chapter is especially important as she probes how teachers perceive their practices as they seriously engage in this work. The stories of the teachers in her study are reminiscent of early voices of teachers who attempted to connect classroom mathematics to the real world and to engage students in what now seems to be limited experiences with modeling and models. For example, Lynch, Fischer, and Green (1989) note the challenge of open-ended discussions that arose when “asking students to think of the important factors in a quantitative, decision making situation” (p.  691). Their description of classroom discussion corresponds to considered unexpected ideas as a teaching practice evident in Carlson’s teacher, Nora. In Carlson’s account of Nora’s perspective on facilitating her students’ modeling work with the field trip problem, Nora honored the principal’s and school’s limitations on field trips and other sponsored events. In consulting with the principal, the teacher seems to slide into a modeling role—that of interfacing between the client and the modeling team. Nora engaged in what Carlson forwarded as the teaching practice of facilitated connections to client. Perhaps recognizing this role would lessen angst about students not being able to pursue their ideas, as happened when Nora’s fourth-grade class wanted the field trip to be a sleepover. The interface role allows constraints imposed on students’ work to be part of a rather natural modeler– client relationship. This conception of teacher can also address concerns about engaging students in decisions that are typically made by adults. Use of the interface role might also provide an opportunity for the teacher to model listening to others’ ideas, which is a teacher concern noted by Carlson. A blending of MM and social justice has interesting implications for how a client beyond the classroom might be conceptualized and then engaged and perceived by teachers and children. The three chapters also offer tools that could be useful to researchers, teachers, and teacher educators who are interested in providing MM learning experiences for young children and perhaps others. For example, Carlson mentions, but does not describe, an online lesson plan template that might be productive for planning MM lessons. Fulton employs a graphic analysis tool and framework from a previous study, both of which might be useful in analyzing how mathematical ideas arise and connect in a classroom. Suh and colleagues explicitly note the potential of particular tools to support teachers in formulating MM problems and in supporting students in their mathematization of real-world situations. Among the tools used by teachers in the studies to elicit students’ assumptions and thoughts about variables and constraints were KWA (What do you Know, What do you Want to know, What will you Assume) charts and sentence starters of the form “If I knew … I could….” The efficacy of these and other tools to support students and teachers would be a useful research topic.

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9.3 Teaching Practices in a Broader Field Reflection on the notable differences between the practices identified or valued in each of the three chapters draws attention to the question of why such differences would exist. One factor might be the extent to which the context of the studies naturally privileged different aspects of MM teaching. Three of the four kinds of practices that the teachers in Carlson’s chapter perceived they used seem to foreground students: working with student ideas, monitoring and supporting [students’] mathematics learning, and linking student ideas with the mathematical solution. Her fourth kind of practice highlights perhaps the factor that the teachers find most novel or challenging: interacting with the real-world context. The focus on the real world as both source of problems and domicile of solutions is perhaps the one feature of MM that separates it from much of the mathematics that students experience in schools. The practices identified and classified by Fulton through observation of teachers are expressed and grouped in categories with names that convey a focus on teacher actions: eliciting, illuminating, coordinating, and maintaining. The difference between focus on students and focus on teacher actions might reflect a difference between Carlson’s teachers viewing teaching with a focus on students and Fulton’s researcher/teacher educator view of teaching with a focus on teachers. Differences could also reflect the contexts—teaching as planning and teaching as instruction. Alternatively, the differences might depend on the specific MM tasks or students involved. Despite the differences, the perspectives are not in conflict. They are different lenses on the same phenomenon, and the pairing of them might be useful in understanding teacher–researcher/teacher–educator conversations. What more might we learn if we applied both approaches, teacher perception and researcher observation, to the same MM lessons? Not only might highlighted teaching practices and the values placed on them differ when viewed through the eyes of teacher and the eyes of teacher educator as observer, teachers also might differ due to the purpose and positioning of MM in the school curriculum and classroom. Different sets of practices might be foregrounded depending on whether the class is learning mathematics through MM or learning to do MM. Practices employed could be dependent on the teacher’s balance of attention to students’ engagement in MM for the purpose of learning MM and attention to MM for the purpose of learning school mathematics. Differences might also arise between settings in which attention is on school mathematics across the span of the curriculum and settings in which attention is on specified curricular mathematics (Zbiek & Connor, 2006). The former draws on a teacher’s horizon knowledge, as Carlson and Suh and colleagues suggest; the latter privileges curricular mathematics as the specific mathematics to be learned at this point in the students’ school curriculum. The strongest example of the influence of curricular mathematics is the work of some of Suh and colleagues’ teachers, who laid out the mathematics objectives for several weeks and connected the topics of the objectives to situations and events in their schools. Fulton’s descriptions of the two teachers in her study exemplify difference related to teachers’ intentions for

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MM in the classroom: Rebecca wanted to introduce MM and to learn how her students used mathematics, while Amy wanted to give her students MM experiences and to meet curricular mathematics goals. MM teaching practices might be similarly named but play out differently, depending on whether MM is the focus of instruction or MM is the venue for learning mathematics. For example, the practice of linking to curricular mathematics, illuminated by Carlson, might play a greater role in the latter than in the former. Concurrently, considering unexpected ideas might be important in both settings, but the teachers’ decisions might be very different. The effects of teacher goals on MM activities are noted explicitly by Suh and colleagues. Depending on the goals that the teachers had for the particular lesson, teachers in their projects employed their pedagogical license to control information shared with their students and to decide when student ideas and interests drive the MM work. The three chapters did not seem to explore whether the identified MM practices were indeed effective. Carlson noted explicitly that this piece was missing. It seems the teachers’ practices were assumed worthy of study based on the teachers’ participation in professional development projects with which the researchers affiliated or the preparation or dispositions that the teachers were known to have. While the teaching practices of teachers believed to be committed, informed, or accomplished teachers of MM are valuable, more is needed to understand which of the various practices or types of practices are effective to help young students learn to be better modelers and/or to help students learn curricular mathematics through MM. A bit of insight, however, might be gained by exploring the connection of the findings in the reported studies to prior research on teaching practices.

9.4 Effective, Equitable, and High-Leverage Teaching Practice Within the last two decades, a number of scholars and organizations came forward with lists of teaching practices, such as effective (National Council of Teachers of Mathematics, 2014), high-leverage (Ball & Forzani, 2009), and equitable (Bartell et  al., 2017) teaching practices. They have also honored practices to enact tasks (Smith & Stein, 2011) and practices indicative of ambitious teaching (Kazemi, Franke, & Lampert, 2009; Lampert, Beasley, Ghousseini, Kazemi, & Franke, 2010). How then do the practices that support student learning of and engagement in MM differ from the practices that support doing mathematics and learning mathematics in general? Insights come from looking at the three chapters through a lens of innovation, including the discomfort, effect of dispositions, and focus on particular aspects of practice that come with nontrivial classroom innovations. One underlying question at which all three papers hint is whether teaching practices for mathematics modeling are necessarily different from those for mathematics in general. As Carlson noted explicitly, the intent of the contributions in this section is not to simply add another list of teaching practices to the literature. We

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should unpack how the practices identified across the three chapters relate to each other. In a later section we extend this idea to how the identified practices compare with practices identified in other strands of research on teaching. Carlson identified a difference between observations in her study and the teacher practices promoted by Smith and Stein (2011). She noted how attending to student thinking was present but differs in the two settings. In Smith and Stein’s case, the purpose was to structure and facilitate subsequent classroom discussions. In one teacher’s case, according to Carlson, the purpose was to connect what students are doing with the school mathematics curriculum. Carlson further noted that teachers working in the spirit of Smith and Stein’s use of the term are working at the horizon knowledge as they attend to mathematics across the span of the curriculum. The type of knowledge used to connect mathematics with the real world is not quite as easily identified or labeled. Additional insights into different aspects of teacher knowledge and how it is leveraged to carry out a practice that differs based on whether MM is present are critical to those who wish to (better) prepare current and prospective teachers of MM to facilitate MM in their classrooms.

9.5 Innovations in Teaching MM is an innovation in teaching for most elementary school teachers, which is why engagement of the teachers in professional development projects would be valuable. The role of the real world in MM is one of its unique features. A key question lurking across the three chapters is how do and can elementary school teachers help students to probe the real world and apply mathematics to solve problems. Carlson broached the subject by noting that teachers need space to explore how teaching MM is similar to and different from teaching mathematics in other areas. The need to interact with the real world in reactive, unpredictable, and intimate ways suggests teachers need dispositions that allow for MM to unfold in messy ways. However, productive dispositions toward modeling and teaching seem to be counter to strong beliefs about mathematics and mathematics teaching (Zbiek, 2016b). For example, the messiness and iterative nature of MM counters common beliefs that a mathematics problem should have one correct answer and that good students quickly produce those correct answers. Arguably a primary contribution of professional development, such as that in which the participating teachers in the studies engaged, is to develop and sharpen productive dispositions toward MM.  Carlson points out that the field needs yet to know how to support the ­development of productive dispositions among students. It also needs to know better how to influence disposition among teachers and teacher educators. Perhaps part of the dispositional impact comes with the feelings of experienced teachers faced with significant innovation. Teachers of MM needing to teach as they have not previously done is like the experiences of teachers as technology entered their classrooms and curricula in the 1990s. For example, teachers in these chapters felt the need to prepare explicitly. In reading the concerns and actions of Amy and

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Rebecca in Fulton’s chapter, one can be reminded of LeeAnn—a teacher who sought to bring technology and function models into her classroom more than 25  years ago. She planned intently and was clear about what mathematics could come out of the experience. In fact, she planned so well and so intently that the classroom experience of her students became not an exploration of their mathematics in service of real-world situations but rather a march through LeeAnn’s previously completed mathematical work (Zbiek, 1995). Carlson’s description of her work with new and experienced teachers in MM suggests that teachers who are learning and experiencing new things and are pressed by a need to address the school curriculum in a timely way might narrow opportunities and impose preplanned MM paths on their students. In both MM and technology cases, it seems that the tension between engaging students in mathematical work possible due to the innovation is readily at odds with engaging them with the prescribed topics of an established school curriculum.

9.6 Framing Teaching Practices for Mathematical Modeling The mention of LeeAnn’s response to technology in her classroom corresponds to a musing about how the three chapters frame teaching mathematics in the early grades. The chapters offer seemingly related and alternative or perhaps competing choices of how to frame teaching practices for MM. One might be intrigued with various ways in which scholars frame any given area, especially those involving teacher knowledge and teaching. Different framing tools appropriately serve very different purposes in general, which is clear in the three chapters in this section. As an example, consider how teachers grow in their use of mathematics technology. Technology in the teaching of school mathematics is perhaps more empirically elaborated in the literature than is the teaching of mathematics modeling. Zbiek (2018) described three different tools used to frame this phenomenon: Technological, Pedagogical, and Content Knowledge (TPACK) (Mishra & Koehler, 2006); Mathematical Understanding for Secondary Teaching (MUST) (Heid & Wilson, 2015); and Play, Use, Recommend, Incorporate, and Assess (PURIA) (Beaudin & Bowers, 1997; Zbiek & Hollebrands, 2008). The details of these rich framing tools are not necessary for current purposes, but some information is needed. Concisely described, TPACK expands on Shulman’s (1986) construct of pedagogical content knowledge (PCK) by incorporating technology with pedagogy and content and then using combinations of the three facets as knowledge areas for teachers. Developed as a framework from analysis of mathematical responses to curious questions that arose in the work of mathematics teachers, MUST considers useful understandings for secondary mathematics teachers in terms of three areas: mathematical proficiency, mathematical activity, and mathematical context of teaching. The first two areas are similar to the strands of mathematical proficiency (National Research Council, 2001) and mathematical processes or practices (e.g., National Governors Association Center for Best Practices and Council of Chief

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State School Officers, 2010). One might see some parallels between MUST and frameworks for teaching mathematics in the early grades, such as Mathematical Knowledge for Teaching (MKT) (Ball, Thames, & Phelps, 2008). PURIA (Play, Use, Recommend, Incorporate, and Assess) began in the context of bringing high-­ powered mathematics software into education yet fits the patterns of teacher growth suggested in the literature on incorporating technology into classroom practice. TPACK highlights the addition of technology to teaching, MUST prioritizes mathematics content knowledge, and PURIA allows pedagogy to be the novelty. The choice of framework depends on which is a useful tool for understanding teaching and effecting improvement. Whether one uses TPACK, MUST, or PURIA as a framing tool might be based on whether it is technology, mathematics content, or pedagogy, respectively, that is novel in teaching practice or that one wishes to foreground in teaching practice. In various ways, the three chapters in this section seem to offer and consider different ways of framing teaching practices and teaching of MM to elementary students.

9.7 To Be Continued The studies reported in the three chapters of this section are part of an overall early attempt to understand what teaching—and especially effective teaching—of MM in the early grades should be and how to support teachers in this work. Teachers in general, and especially those in the three chapters, are themselves learners in professional development settings. This phenomenon fits with the notion of teachers as lifelong learners. The observation also raises a question of what is the general trajectory, if any, of teachers’ growth and changes in practice. A challenge shared by the three chapters is the nature of the teachers in whose classrooms the teaching occurs. The teachers are not only participants in professional development focused on MM but also seemingly very successful in that context. Suh and colleagues assert that elementary school teachers, and to which let’s add also mathematics teacher educators, need professional learning experiences if they are going to be asked to facilitate productive modeling experiences for their students. Like Carlson and Suh and colleagues, many have facilitated professional learning experiences that focused on teachers learning to do MM rather than on their teaching of it. Researchers need to complement studies involving MM professional development participants with studies of teachers less connected to MM and perhaps less prepared to implement MM in the classroom. To do so, however, remains a challenge as long as MM is relatively new and essentially unfamiliar to most elementary school teachers, leaving little opportunity to collect data on sufficiently effective practice. The field is ripe for seeing how MM teaching practices are understood and embraced by teachers who do not have the benefit of extensive participation in funded workshops or of direct connection with expert facilitators of MM.

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Teachers in Carlson’s chapter suggest there is not a MM curriculum, at least not one that they found available to them. The notion of bridging the gap between MM activities and current curricula—at least in the United States—is a serious concern. How does the curriculum evolve and to what extent does it continue to flourish under the same MM practices? The use of activities that capture some but not all elements of the MM process, such as the Three-Act Problems used by teachers in Suh and colleagues’ work, would be one example of a bridging tool. An atomistic approach to MM is a related option. Geiger, Ärlebäck, and Frejd (2016) suggested that students within a holistic approach engage in the complete MM process, while students in an atomistic approach attend to particular elements of MM at different times. When atomistic experiences are distributed over time—and periodically connected within single problems (Zbiek, 2016a)—students have the opportunity to learn to model regardless of curricular demands. Interestingly, the three chapters illustrate the promise of an atomistic approach to learning to teach MM.

9.8 Concluding Thoughts Based on reflections on the chapters by Carlson, Fulton, and Suh and colleagues, learning to teach MM seems similar to other experiences in learning to teach differently, such as learning to teach with mathematics technology. The comparison is not to minimize the task of learning to teach MM but rather to celebrate the ways in which teachers learn and the ways in which teacher practices can be challenged and enhanced as well as celebrated and refined as innovation in curriculum materials occurs. The comparison also suggests the trajectory of empirical and theoretical work related to design and facilitation of early MM experiences. Consistent with the understandings gained from this section, the work of teaching MM and facilitating early MM experiences is difficult. We should not leave this section without articulating arguably the most important message implicit in these chapters: young children can engage in MM experiences, and their teachers can design and implement early MM experiences in their classrooms.

References Ball, D. L., & Forzani, F. M. (2009). The work of teaching and the challenge for teacher education. Journal of Teacher Education, 60(5), 497–511. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. Bartell, T., Wager, A., Edwards, A., Battey, D., Foote, M., & Spencer, J. (2017). Toward a framework for research linking equitable teaching with the standards for mathematical practice. Journal for Research in Mathematics Education, 48(1), 7–21. Beaudin, M., & Bowers, D. (1997). Logistics for facilitating CAS instruction. In J.  Berry, J. Monaghan, M. Kronfellner, & B. Kutzler (Eds.), The state of computer algebra in mathematics education (pp. 126–135). Lancashire, UK: Chartwell-York.

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Cirillo, M., Bartell, T. G., & Wager, A. A. (2016). Teaching mathematics for social justice through mathematical modeling. In C.  Hirsch (Ed.), Annual perspectives in mathematics education 2016: Mathematical modeling and modeling mathematics (pp. 87–96). Reston, VA: National Council of Teachers of Mathematics. Geiger, V., Ärlebäck, J. B., & Frejd, P. (2016). Interpreting curricula to find opportunities for modeling: Case studies from Australia and Sweden. In C.  Hirsch (Ed.), Annual perspectives in mathematics education 2016: Mathematical modeling and modeling mathematics (pp. 207– 215). Reston, VA: National Council of Teachers of Mathematics. Heid, M. K., & Wilson, P. W. (with Blume, G. W.). (Eds.). (2015). Mathematical understanding for secondary teaching: A framework and classroom-based situations. Charlotte, NC: Information Age. Kazemi, E., Franke, M., & Lampert, M. (2009). Developing pedagogies in teacher education to support novice teachers’ ability to enact ambitious instruction. Crossing Divides: Proceedings of the 32nd Annual Conference of the Mathematics Education Research Group of Australasia, 1 (pp. 11–30). Kilpatrick, J., Blume, G., Heid, M. K., Wilson, J., Wilson, P., & Zbiek, R. M. (2015). Mathematical understanding for secondary teaching. In Heid, M. K., & Wilson, P., with Blume, G.W. (Eds.), Mathematical understanding for secondary teaching: A framework and classroom-based situations (pp. 9–30), Charlotte, NC: Information Age. Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., & Franke, M. (2010). Using designed instructional activities to enable novices to manage ambitious mathematics teaching. In M. K. Stein & L. Kucan (Eds.), Instructional explanations in the disciplines (pp. 129–141). Boston, MA: Springer. Lynch, J. K., Fischer, P., & Green, S. F. (1989). Teaching in a computer-intensive algebra curriculum. Mathematics Teacher, 82(9), 688–694. Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A new framework for teacher knowledge. Teachers College Record, 108, 1017–1054. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. National Governors Association Center for Best Practices and Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved September 1, 2019, from http://www.corestandards.org National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. Shulman, L.  S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4–14. Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics. Zbiek, R.  M (1995). Her math, their math: An in-service teacher’s growing understanding of mathematics and technology and her secondary school students’ algebra experience. In D. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the seventeenth annual meeting, North American chapter of the international group for the psychology of mathematics education (pp. 214–220). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. [ED 389 610] Zbiek, R.  M. (2016a). Developing mathematical modelers (Research into practice collection). Boston, MA: Pearson Education. Zbiek, R. M. (2016b). Supporting teachers’ development as modelers and teachers of modelers. In C. Hirsch (Ed.), Annual perspectives in mathematics education 2016: Mathematical modeling and modeling mathematics (pp. 263–272). Reston, VA: National Council of Teachers of Mathematics. Zbiek, R.  M. (2018). Contemporary framing of technology in mathematics teaching. In M.  E. Strutchens, R. Huang, D. Potari, & L. Losano (Eds.), Educating prospective secondary mathematics teachers (pp. 109–124). Dordrecht, The Netherlands: Springer.

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Zbiek, R. M., & Connor, A. (2006). Beyond motivation: Exploring mathematical modeling as a context for deepening students’ understandings of curricular mathematics. Educational Studies in Mathematics, 63(1), 89–112. Zbiek, R. M., & Hollebrands, K. (2008). A research-informed view of the process of incorporating mathematics technology into classroom practice by inservice and prospective teachers. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Volume 1 (pp. 287–344). Charlotte, NC: Information Age.

Part III

Mathematical Modeling and Student Experiences as Modelers

Chapter 10

Mathematical Modeling: Analyzing Elementary Students’ Perceptions of What It Means to Know and Do Mathematics Megan H. Wickstrom and Amber Yates

Mathematical modeling is a process that encourages students to use mathematics to investigate real-world questions by interpreting data, engaging in investigations, and representing information mathematically. In other words, mathematical modeling can be defined as “a system of conceptual frameworks used to construct, interpret and describe a situation” (English & Watters, 2005, p.  59). Typically, mathematical modeling is different than traditional mathematics instruction in that students are not taught concepts and given practice problems. Instead, they engage in thinking “beyond the usual school experience” and “create complex artifacts or conceptual tools that are needed for some purpose, or to accomplish some goal” (English & Sriraman, 2010). Many different sources, such as the Common Core Standards for School Mathematics and the GAIMME Report (Garfunkel & Montgomery, 2016), have illustrated the modeling cycle in detail. Most agree that modeling begins when students have identified a problem drawn from a real-world situation, a situation they want to understand and explore on a deep, mathematical level. From there, students select important variables and determine the relationships between them considering which information is important and which does not relate to the task and formulating different ways of reaching a solution. Lastly, they must make sense of their solution in light of the scenario to see what new insights and results were found.

M. H. Wickstrom (*) Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA e-mail: [email protected] A. Yates Department of Education, Montana State University, Bozeman, MT, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_10

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10.1 Benefits of Mathematical Modeling Researchers have argued the importance of modeling across several different domains including mathematical literacy, productive dispositions toward mathematics, agency and ownership, mathematical content knowledge, and perspective. Modeling tasks are not trivial and students must incorporate several different mathematical competencies to make judgments about a real-world situation. Many have argued that the competencies involved in modeling are the same needed to promote mathematical literacy (de Lange, 2003; Niss, 2003; Steen, Turner, & Burkhardt, 2007). Mathematical literacy is defined as “an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments, and to engage in mathematics in ways that meet the needs of that individual’s current and future life as a constructive, concerned and reflective citizen” (OECD, 2012, p. 25) Through the modeling cycle, researchers argue that students, at any age, are gaining productive practice using mathematics to make sense of situations and, hopefully, will be able to carry these sense-making strategies into adulthood (Carlson, Wickstrom, Burroughs, & Fulton, 2016; Lesh, English, Sevis, & Riggs, 2013). Modeling tasks also allow students to consider their personal values and judgments in relation to mathematics. Researchers have argued that modeling may promote productive views of and dispositions toward mathematics because students see modeling as relevant and useful (Bonotto, 2007; Lesh & Yoon, 2007), particularly in investigating topics that are personally meaningful to them (Greer, Verscheffel, & Mukhopadhyay, 2007). They may be motivated to work with the problem because they see meaning in the context or because they enjoy making connections between mathematics and the real world (Zbiek & Conner, 2006). Anhalt, Cortez, and Smith (2017) discussed that mathematical modeling allows for greater participation in mathematics because modeling tasks leverage students’ backgrounds. In addition, making connections with personal knowledge motivates students to make sense of the problem, engage in challenging tasks, and persevere. Mathematical modeling utilizes the various interests of the students to provide a sense of agency and ownership, both personally and mathematically (Anhalt et al., 2017; Anhalt, Staats, Cortez, & Civil, 2018; Suh, Matson, & Seshaiyer, 2017; Turner, Aguirre, Foote, & Roth McDuffie, 2018; Wickstrom, 2017). Mathematical modeling looks different from traditional mathematics instruction in that there is no one correct solution to a problem. Instead of the teacher posing a problem, students brainstorm to determine what question best fits the situation at hand. In addition, students draw on their own background and experiences in considering the problem (English & Watters, 2005), and this allows for explicit connections between the mathematical world and students’ lived experiences (Anhalt et  al., 2018; Turner et al., 2018). There is also a sense of creativity and freedom that arises during the modeling process (Suh et al., 2017). In devising a solution, students choose what tools to use and how to use them. Wickstrom, Carr, and Lackey (2017) found that students found joy in the freedom of choosing their own path, and it sparked further

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curiosity toward related modeling tasks. Students have choices in determining their final solution and whether or not to make revisions. As they choose tools and solution strategies, they are actively making connections with known mathematics to determine if there is a better path forward (Muller & Burkhardt, 2007). Although research is limited, many have argued that mathematical modeling may deepen understanding of mathematics. Some have argued that the act of using mathematics in a real-world situation encourages connection and reflection with the mathematics employed (Lehrer & Schauble, 2007; Schoenfeld, 2013). In contrast, others have argued that the act of using mathematics to solve real-world problems may encourage students to pursue more mathematical knowledge (Zbiek & Conner, 2006). Because modeling can draw on several different aspects of mathematical topics and is usually social in nature, it may naturally encourage students to make connections between mathematical topics and engage in varied mathematical practices. Lastly, modeling may provide students with the opportunity to consider the world from another’s perspective. Several groups (Anhalt et  al., 2017; Cirillo, Bartell, & Wager, 2016; Turner et al., 2018) have argued that mathematical modeling can be directly connected to issues of social justice in that it aligns well with the features of teaching mathematics for social justice: engaging students in ill-defined problems, leveraging students’ real-world knowledge, and raising students' interests. Through the lens of mathematical modeling, students are encouraged to consider a problem from multiple perspectives and to consider the needs of the person posing the problem.

10.2 The Child’s Voice Although there has been research on the proposed and actual benefits of the incorporation of mathematical modeling in K-12 classrooms, there has been very little research, primarily anecdotal, investigating mathematical modeling from the student’s perspective. It is important to assess if proposed benefits of modeling are evident from the student perspective and if there are other benefits or limitations we have not yet considered. We became interested in this work because many of the teachers we have worked with in designing modeling tasks have discussed their students’ perceptions of mathematical modeling. Anecdotally, they described that students who do not typically enjoy or engage in traditional mathematics instruction have enjoyed mathematical modeling tasks and vice versa. They also discussed that students tend to view mathematics instruction and modeling tasks as separate activities and do not typically recognize modeling as mathematics instruction. Considering the students' perspective may shed light on these informal observations to help us understand students’ perceptions of the modeling process in relation to typical mathematics instruction.

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10.3 Theoretical Framework: Mathematical Identity and Construction of the Discipline The mathematics classroom is a space in which students develop a mathematical identity as well as an understanding of mathematics as a discipline (Cobb & Hodge, 2002). Identities are not stagnant or stable. They can be envisioned as a set of beliefs, behaviors, and ideas (Holland, Lachicotte Jr., Skinner, & Cain, 2001) that change over time (Martin, 2006) and are often shaped by other’s perceptions (Sfard & Prusak, 2005) and the individual’s other identities. Several researchers (Aguirre, Mayfield-Ingram, & Martin, 2013; Boaler & Greeno, 2000; Boaler & Selling, 2017) have made strides in formalizing mathematical identity and describe it in the following way: Mathematical identity involves the ways in which students think about themselves in relation to mathematics and the extent to which they have developed a commitment to, are engaged in, and see value in mathematics and in themselves as learners of mathematics. (Boaler & Selling, 2017, p. 82)

Not all mathematics classrooms are the same, and students’ commitment to mathematics and their understanding of the discipline vary based on the constructed learning environments (Boaler & Greeno, 2000). Factors such as task features, interest, and utility matter in fostering mathematical engagement and productive identity (Middleton, Jansen, & Goldin, 2017). Multiple researchers have shown that students develop an engaged and positive identity when they participate in mathematical tasks as a dialogue that allows them to draw on their own experiences and mathematical ideas and feel that their contribution is valued (Boaler, 2015; Boaler & Greeno, 2000; Boaler & Staples, 2008; Cobb, Gresalfi, & Hodge, 2009). Students develop agency when they see mathematics as something they participate in rather than information they receive (Aguirre et al., 2013; Gresalfi, 2009). In their compendium chapter on engagement, Middleton et al. (2017), discuss interest and utility as factors in mathematical engagement. They defined interest as “activities or topics that students tend to seek out and find enjoyable and toward which they orient their identities” (p.  678). In addition, they discuss that interest is one of the strongest predictors of students’ positive affective experiences in mathematics (Schiefele & Csikszentmihalyi, 1995). Lastly, when students see mathematics as useful to them, they display greater interest (Middleton, 2013). Social support, classroom interactions, and teacher expectations also play a fundamental role in shaping students’ understanding of mathematics and themselves in relation to mathematics (Middleton et al., 2017). When classroom interactions and teacher expectations foster the idea that mathematics is about being quick and correct with little perceived effort (Bartholomew, 2000; Berry III, Thunder, & McClain, 2011), many students do not identify as being successful at mathematics. Unless the classroom environment emphasizes other competencies in determining effectiveness, students rely on extrinsic recognition from teachers and fellow classmates (Seymour & Hewitt, 1997) as well as test scores (Berry III et al., 2011) to shape their mathematical identity and willingness to participate.

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Reflecting on this literature, we theorize that mathematical modeling provides opportunities for engagement and development of mathematical identity that traditional instruction often lacks. Mathematical modeling problems are interesting and useful. Students draw from their lived experiences and multiple mathematical competencies to make sense of a problem that does not have a clear-cut solution. They are no longer following routine procedures, so there are opportunities for multiple forms of valued expertise. In addition, they are asked to work on a task in ways that may not mirror mathematics instruction on a day-to-day basis. Through the perspectives of the students, we will address how students’ perceive and engage in typical mathematical lessons as compared to mathematical modeling lessons. We will answer the following questions: • How do second-grade students conceptualize what it means to know and do mathematics? • How do second-grade students conceptualize what it means to know and do mathematical modeling? • How do second-grade students experience and perceive mathematical modeling tasks in ways that are similar or different to traditional instruction? We recognize that modeling is mathematics, but, for the children and for the teacher, mathematics and mathematical modeling were designated as two different activities.

10.4 Methodology 10.4.1 Setting and Participants This study took place at Washington Elementary School in Ms. Applegate’s second-­ grade classroom in the months of September and October. Washington Elementary is a neighborhood school with 246 students and 2 classrooms per grade level. Of the students at Washington, 70% identify as white, 10% identify as Hispanic, 7% identify as Native American, 6% identify as multiracial, 4% identify as African American, and 3% identify as Asian. The demographics of Ms. Applegate’s classroom were similar to those of the school overall. Ms. Applegate is an experienced second-grade teacher. At the time of the study, she had 18 years of experience teaching at the elementary level. In addition, she participated in 3 years of professional development and became a teacher leader in mathematical modeling through a grant funded by the National Science Foundation. The authors recruited Ms. Applegate’s classroom for this study because Ms. Applegate regularly implemented modeling tasks in her classroom and also taught mathematics using a reform-based curriculum. On a typical day, Ms. Applegate’s mathematics lesson involved discussing a new concept or problem together as a class on the carpet. Students then worked on a

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small set of problems in pairs and then went to their seats to complete practice problems, independently. Students were pulled out to work on additional problems if they needed help, and there was a subset of students designated as gifted who were pulled out once a week to work on additional problems. Ms. Applegate integrated open-ended problems and application problems once a week, typically on Fridays and not during the designated mathematics time.

10.4.2 The Modeling Task The authors worked as a team with Ms. Applegate to develop a 3-day modeling task that was appropriate for second-grade students’ mathematical content knowledge and was relevant and engaging. Washington Elementary has a field day at the end of the school year in which each student receives a small prize. Our team discussed that we anticipated students would explore questions surrounding the number of prizes needed, cost, and what prize would be appealing to all students. These questions would guide each day of instruction. When working with modeling tasks, Ms. Applegate preferred introducing a setting or context and allowing students to pose mathematical problems rather than giving them a problem. Ms. Applegate introduced the task by telling the students the principal needed their help planning the prizes for field day. She asked them what questions they thought should be answered to help the principal make a decision. The students responded with the following: • • • • •

What toy should we buy? What constraints should we consider as we decide which toy is best? How much can we spend? How many toys will we need? Will everyone receive a toy?

On the first day of the lesson, Ms. Applegate focused on how many toys the principal should buy, and she asked students what they needed to know. Students worked in groups of four. Many of the students asked if they could go to each classroom and count the number of students because that would determine the number of toys. One student remembered that Ms. Applegate has a classroom roster that she uses to take attendance and wondered if they could use rosters to determine the total number of students. To determine the total number of prizes needed, students either added the number of students in each class using different addition strategies or rounded and then added. Figure  10.1 highlights this strategy. One student, George, wondered what would happen if students came to school or left before field day. The class discussed that it might be better to over-order and have extra prizes. On the second day of the task, Ms. Applegate focused on budget and told students that it is typical for the principal to spend about $250 on prizes. Ms. Applegate also gave students the website that the principal typically uses to purchase prizes. Prizes are sold in packages of 4, 6, or 12. Students were asked to research which

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Fig. 10.1  Students’ work to determine the number of prizes needed

Fig. 10.2 Students’ calculations

prizes they might pick and why. Students prepared a list of potential prizes, number of packages needed, and total cost. Students realized early on that if they calculated the number of packages of 4, 6, and 12 needed for the whole school, those calculations could apply across prizes with the same number in each package. Figure 10.2 illustrates one group’s list and how they crossed out items that were too expensive after doing calculations. To calculate the number of packages needed, most students used repeated addition or doubling to get close to the total number of students. One group thought about each class, determined the number of dozens for that class, and then summed the number of dozens together as shown in Fig. 10.3. Once the number of packages was determined, students still needed to check the price. Students did this by rounding to the nearest dollar or adding directly. In Fig. 10.4, one student used a doubling strategy to get close to the number of packages needed. Once each group had successfully determined the number and price of their prize, Ms. Applegate brought the class together to have groups present what they found and determine which prize would best fit the needs of the entire school. As students presented prizes, Ms. Applegate had them consider the attributes of a good prize. Students discussed that the prize should be fun but also age appropriate. Many

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Fig. 10.3  Students’ work with dozens of prizes Fig. 10.4 Student’s doubling strategy for price

students worried that a prize for a kindergartner might not be fun for a fifth grader and a prize for a fifth grader might be dangerous for a kindergartner. They also discussed that the prize should be related to the school or mascot, the owls. Some students also discussed that getting medals as prizes might be appropriate. As a class, the students narrowed down the list, and Ms. Applegate had them vote to determine which prize they thought best fit the constraints. The winner, as shown in Fig. 10.2, was the owl squirts because they are fun for all children, not dangerous, and related to the school mascot. In this results section of the paper, we discuss students’ perceptions of the task further.

10.4.3 Data Collection For this study, our purpose was to document students’ perceptions of mathematics and mathematical modeling, and we wanted to interview students with varied experiences learning mathematics. Our two main data sources were one-on-one interviews with students: one related to mathematics and one related to mathematical modeling. Figure 10.5 shows the timeline of events involved in the study. To select students, we used a Likert-scale survey for young learners developed by Kutaka (2013). Kutaka found that students in the early elementary grades had already begun to construct a narrative of what it means to know and do mathematics in relation to other students in the classroom. We believed this would be a

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Fig. 10.5  Timeline of study events

reasonable tool to select interview participants because we could capture students who held positive beliefs about mathematics as well as negative. To respond to survey questions, students are shown a shape that becomes larger from left to right. The largest shape indicates strong agreement, while the smallest shape indicates strong disagreement. Ms. Applegate asked the students the following questions: • How good in mathematics are you? • If you were to list all the students in your class from best to worst, where would you put yourself? • Compared to other school subjects, how good are you at mathematics? • How well do you expect to do in mathematics this year? • How good would you be at learning something new in mathematics? To use survey data, we needed consent from both parent and child. Of the students in Ms. Applegate’s classroom, only about half had parental consent. Following the survey, we sorted through student responses and chose four students with positive responses to all questions, three students with negative responses to all questions, and three students with varied responses. We did not select based on race or gender because our pool was very small. Of the ten students we identified, seven agreed to participate in interviews. Figure 10.6 lists the questions given to students as well as where they placed themselves on the Likert scale. Grace, Sven, Layla, and Henry provided primarily positive responses, Libby and Wayne provided varied responses, and George provided primarily negative responses. It is interesting to note that of the three students that did not participate, two provided primarily negative responses, and one had varied responses. Of the students, Sven identified as African American, while the other children identified as white. Once students were selected, they each participated in an individual, audio-­ recorded interview that lasted approximately 30 min with both researchers present. We alternated posing questions and taking notes with each student and used various studies about beliefs (Leder, Pehkonen, & Toerner, 2002) to inform appropriate

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Q1: How good in mathematics are you?

Layla Wayne Henry Sven Libby

Grace

George

Layla Grace Wayne Henry Libby Sven Q2: If you were to list all the students in your class from the worst to the best in mathematics, where would you put yourself? George

Wayne Libby

Layla Grace Henry Sven Q3: Compared to most of your other school subjects, how good are you at mathematics? George

Layla Libby

Wayne Grace Henry Sven

Layla Grace

Wayne Henry Sven

Q4: How well do you expect to do in math this year? George

Libby

Q5: How good would you be at learning something new in mathematics?

Fig. 10.6  Results of mathematics survey developed by Kutaka (2013)

interview questions considering the age of the students. In the first two questions, students were asked to describe their understanding of what mathematics is (i.e., What do you use mathematics for? How do you see other people using mathematics?) and what it means to be good at mathematics (i.e., if you are good in mathematics, what can you do?). The other interview questions related to students’ responses on the survey instrument and why they chose a particular size shape in response to a question. Following the first interview, Ms. Applegate taught her modeling lesson which lasted approximately 3 days, as described earlier in the methods section. During the lesson, researchers took notes and monitored interview students’ behaviors and mathematical thinking during instruction. Following the modeling task, the selected students were asked to participate in a second interview. We asked students to describe mathematical modeling, the task, and how modeling was similar or different to what they usually did during mathematics instruction. In addition, students were asked to describe parts of the activity that they liked more than typical mathematics instruction and parts that they liked less.

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10.4.4 Data Analysis We began analysis by transcribing the student interviews verbatim, which resulted in 34 pages of text. Both of us read the transcripts, independently, and took descriptive notes regarding each student’s responses to each interview question about mathematics and mathematical modeling. We attended to the following questions (Fig. 10.7). Following the first read through, we met to discuss our initial perceptions of the data. After discussion, we independently engaged in a descriptive coding process (Miles, Huberman, & Saldana, 2014) which entails “assigning labels to data to summarize in a word or short phrase – most often a noun- the basic topic of a passage of qualitative data” (p.  74). Each student’s response to each interview question was considered a chunk of data that could be assigned multiple codes. After each question was coded loosely, each event was coded again, and codes were collapsed or separated to create more concrete definitions of what a code represented. For example, when coding the question “what does it mean to be good at mathematics?”, we had initial codes of efficiency, able to solve difficult problems, mathematics is not challenging for you, do more problems, do more problems correctly, teacher recognition, and speed. We looked across to make sure these codes captured our students’ responses and if any of the codes could be collapsed together. For example, we had several conversations to determine if and how efficiency, speed, doing more problems, and doing more problems correctly fit together or separately as codes. As new codes emerged or were redefined, the data was reanalyzed. After each interview question had been analyzed and codes and themes had been defined, we looked across the cases for similarities and differences in how students experienced traditional mathematics instruction and mathematical modeling.

Mathematics Questions

Modeling Questions

1. What is mathematics?

1. What is mathematical modeling?

2. What do you use it for?

2. What mathematics is involved in

3. How do you see people using it? 4. What does it mean to be good at mathematics? 5. How does mathematics compare to other subjects? 6. How well can you learn something new in mathematics?

mathematical modeling? 3. How does mathematical modeling compare to typical mathematics instruction (both challenges and affordances)? 4. What does it mean to be good at mathematical modeling?

Fig. 10.7  Qualitative questions guiding interview analysis

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10.5 Results The results are organized by the two data points: Interview 1 and Interview 2. In Interview 1, we describe students’ perceptions of what it means to know and do mathematics, what they enjoy about mathematics, and what it means to excel at mathematics. In Interview 2, we share findings related to students’ perceptions of what it means to know and do mathematical modeling, affordances and limitations of modeling, and how modeling compares to traditional instruction.

10.6 Interview 1 10.6.1 Students Descriptions of Mathematics In Ms. Applegate’s classroom, students participated in a variety of mathematical activities, but a typical day often involved the teaching of a particular topic followed by practice problems. Six out of the seven students described mathematics as what they were asked to solve and learn about in class including addition, subtraction, and multiplication using numbers and equations, with one student including fractions. This is not surprising, considering the students’ descriptions fit within the second-grade standards for operations and algebraic thinking. Below is Sven’s description of mathematics: Interviewer: What is math? Sven: It’s adding numbers. Interviewer: Ok, is it anything else? Sven: Fractions, numbers.

One student, George, did not define mathematics as a set of equations or numbers, like the other students, but rather felt it was a practice that would improve overall intelligence. On the survey, George identified as having negative perceptions of mathematics and discussed that by doing the mathematics problems, he hoped to improve. Below is George’s description of mathematics. Interviewer: Ok, so first question. Can you tell us what is math? George: Math is a way to help you get smarter.

It is interesting to note that none of the students connected the word mathematics to activities outside of the classroom or to the real world.

10.6.2 Doing Mathematics After providing descriptions of mathematics, we asked students to consider how they used mathematics. Five of the seven students were able to provide an example of how they used mathematics, while one was unsure and one made up a

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hypothetical situation. Four of the students used examples from outside of the classroom such as building, measuring, playing games, or counting toys. For example, one student, Wayne stated: I use it (mathematics) for like adding up how many Lego pieces I put into this creation and how many things I put into the other and then if I put them together, how much will I have and stuff like that.

Two of the students, including one from above, referenced classroom activities for how they used mathematics. Sven referenced mathematical games he plays on the computer as well as an apple and plum application task Ms. Applegate had recently taught. George referenced doing problems on a worksheet as doing mathematics. Lastly, Grace discussed a hypothetical situation that someone might be able to count the trees in their yard as a way to use mathematics. Beyond their personal experiences, we also asked students to consider how they saw other people using mathematics. Five out of the seven students were unable to provide an example of where they had seen other people using mathematics. One of the students who provided an example used a nonspecific scenario like “counting stuff you have,” while another student, Wayne, discussed witnessing mathematics in the after school problem. He stated, “I see, sometimes, the after school care teachers are using math to add up how many kids are in afterschool.”

10.6.3 Students’ Beliefs About Their Mathematical Ability When we looked across students’ discussion about their mathematical ability, we identified eight themes that emerged as indicators when describing what it meant to be good at mathematics: speed, little perceived effort, efficiency, number of correct problems, number of problems solved, solving difficult problems, teacher/parent recognition, and knowing content before it is taught. Three of the students referenced speed in relation to being good at mathematics. To them, this meant being able to see or hear a problem and know an answer right away. Wayne described it in the following way: Interviewer: What does it mean to be good in math? Wayne: It means to be able to like solve like two times twelve quickly. Interviewer: So like how fast you can solve? Wayne: Well, like quickly and get the answer right.

Two students expanded on the ideas of speed to add that not only are good mathematics students fast, but they also solve problems with ease, know how to do the problem ahead of time, and know the best way to solve the problem. For example, in the excerpt below, Grace discussed efficiency when solving a mathematical task. Interviewer: What does it mean to be good at math? Grace: I think it’s to be good at adding and subtracting and, um, multiplication but also being able to do it efficiently and knowing the right ways to do it without having to make it too hard for yourself

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Almost all of the students indicated the number of problems solved and/or correct as well as the difficulty level of the problems as indicators of mathematical ability. Students discussed that when they were able to solve many practice problems or get more of the problems correct than incorrect, they felt they were doing well at mathematics. In addition, they discussed that if a person is good at mathematics, they should be able to solve difficult problems and a lot of them. George, below, discussed this theme when describing why he does not feel good at mathematics. Interviewer: When we asked you “how good in math are you,” you filled out this circle here. Can you tell us why you filled out this circle? George: I don’t think I’m very good at math. Interviewer: Why? George: Because I barely get any questions right.

Three students discussed feedback from their teacher, parents, and assessments as ways of knowing if they were good at mathematics. This might include being called back to talk to the teacher about specific problems indicating a student needed extra help, receiving praise for the number of correct questions, doing well on a mathematics computer game, or receiving general praise from parents about their performance in class. Wayne discussed that, in first grade, he knew he was good at mathematics and not good at English because he never got called to the back of the room to meet with the teacher about mathematics problems. He stated: And, I also was just feeling good about it because I never had to get called back because of math. I had to get called back because I missed a bunch of writing because I’m not that good at writing, but I never got called back because I missed math. I feel like that’s a signal that if you don’t have to do that then you’re pretty good at math.

It is also interesting to note that none of the students felt that they were the best mathematics student even though a few had indicated this on their surveys. Most discussed that they were somewhere in the middle.

10.6.4 Students’ Beliefs About Learning New Mathematics In thinking about mathematical ability, we also asked students to consider how well they would do at learning something new in mathematics. Five of the seven students discussed a process of not understanding the mathematics the first time around. Two discussed that this often gets better with practice, two discussed that the confusion could sometimes result in understanding, and one discussed that the confusion persists. Sometimes, with the teacher’s help and practice, the students understood, and other times they did not. Henry describes this in the interview below: Interviewer: When we asked how well do you expect to do in math this year, you filled out this rectangle. Can you tell me why you filled out this rectangle? Henry: Because I think I am going to get better. Interviewer: How do you know you are going to get better? Henry: Because when I just start something and I do the second thing on it, I get better at it.

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Henry believed he could improve as the year went on because, even when he made mistakes, he would get better each time he tried. George, on the other hand, believed he would not be able to improve because of the difficulties and confusion he had faced so far. Interviewer: Ok, when we asked “how well do you expect to do in math this year” you picked this one. Will you tell us why you picked that one? George: Because I’m not doing very good at math right now. Interviewer: So you expect in the future that will be the same thing? George: Mhm.

George did describe that he does very well learning mathematics on the computer and indicated that he would do very well in that regard. In addition to practice, two of the students discussed external resources such as classmates, teachers, and parents to help as supports in helping them become better at mathematics.

10.6.5 Mathematics in Relation to Other Disciplines In addition to understanding students’ beliefs about mathematics, we hoped to understand how they felt about mathematics in relation to other subjects and to understand reasons for and against mathematics. Only one student stated that she liked mathematics more than other topics, while the other six students chose another discipline. Students discussed themes surrounding difficulty, enjoyment, physical activity, and social activity. In terms of difficulty, five students referenced that a particular subject was easier than another subject. Grace stated, “I think I’m better at math than some other subjects because some other subjects are a lot harder for me to understand and do.” In contrast, Sven discussed feeling really good at physical education because he was the best runner at the game Tag and had a strategy not to get caught. Two students, Libby and George, discussed social and physical aspects of science as to why they enjoyed it more than mathematics. They both discussed that they liked that science involved hands-on experiments and working together in groups with classmates. Interviewer: It was science? So what can you do in science that makes you better at doing science than what you can do in math? Libby: It’s more fun and you do more physical activity, and you have lots of groups. In math you just do most of it on your own.

Lastly, all students referenced being better at something in terms of how much they enjoyed or did not enjoy that particular topic. Looking across students and responses, where students marked themselves on the survey instrument did not seem to influence how they conceptualized mathematics and themselves in relation to mathematics, except for George. Students primarily perceived mathematics as something they did in the classroom but could identify situations outside of the classroom when pressed. Students conceptualized being good at mathematics as answering questions quickly and correctly and with little

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perceived effort. They relied on feedback from teachers, parents, and peers to confirm their ability. Even when students marked themselves as good in mathematics, they did not consider themselves as the best in the class. Students believed they would become better at mathematics across the year, either through practice or help from teachers. Mathematics was also conceptualized as something that was not a social or active endeavor. George was the only student who marked low responses for all survey questions, and his responses were different from the group. When defining mathematics, George discussed that it is a way to help you become smarter and if he could understand it he would be viewed as smart. When pressed for examples of mathematics, George referenced mathematics worksheets. Lastly, when asked how he thought he would do this year, George was despondent and did not think he would get better.

10.7 Interview 2 The purpose of interview 2 was to gauge the students’ understanding of what it meant to engage in modeling, how students saw modeling in relation to mathematics, and how they saw modeling as related to their mathematical selves. This interview was administered a week after students completed the field day modeling task.

10.7.1 Students’ Descriptions of Modeling When asked to describe modeling, all students described actions that took place during the modeling lesson such as determining cost of items that come in different packages, determining how many items you need based on the number of students, and determining what item best fits the needs of the target audience. In addition, all students recognized that modeling was more than answering one question and instead represented a series of tasks or mathematical questions. Some students described the modeling process exactly in terms of what happened during the task while others were able to consider the scenario more broadly. For example, in the transcript below, Wayne described the task broadly and described that modeling might be good for determining how to buy items for more than one person. Interviewer: So my first question to you is what is math modeling? Wayne: It’s kind of like saying here’s something and it costs this much and it comes in this much and you need this many. How many do you need? How much will you have to pay? How many would you get for how many packs of things you have to get? Interviewer: What do you use math modeling for? Wayne: Figuring out like, if you want to buy something for more people than just one. That would be a pretty smart way to use it.

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In addition, one of the students, Henry, was able to consider mathematical modeling on a broader level and discussed that mathematical modeling is combining different types of mathematics to answer a larger question. Interviewer: So can you describe what math modeling is? Henry: Math modeling is basically like it’s lots of different types of math in one where you’re trying to figure out like a big thing.

As students described the modeling process, they often reflected on the types of mathematics they engaged in. Seven of the eight students discussed adding large numbers either to determine the number of students in the school or to determine the number of packages of items needed, and four of the students described voting or coming to a consensus as to which item to choose.

10.7.2 Challenges and Affordances of Modeling in Relation to Mathematics In two of the interview questions, we asked students to consider mathematical modeling in relation to typical mathematics instruction both in terms of challenges and affordances. Across the eight students, we identified four themes that emerged as both challenges and affordances depending on the students. Their survey responses were not indicative of whether they saw themes as challenges or affordances. The themes are shown below (see Fig.10.8) and described in detail, with students quotes in the following section. Responsibility, to the students, was the idea that they were given the opportunity to make an important decision about what prize should be given to the entire school. Four of the eight students discussed that this was something they really enjoyed about mathematical modeling and described it as fun or exciting to make choices on behalf of the entire school. For example, when asked why she enjoyed mathematical modeling, one student, Grace, stated: Grace: Because it’s fun, it sort of makes you feel responsible, picking out something for the rest of the school.

Affordance

Challenge Responsibility Difficult Mathematics Group Dynamics Real World Application

Fig. 10.8  Mathematical modeling challenges and affordances spectrum

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Two students discussed that responsibility was a challenging aspect of mathematical modeling. It was difficult for them to determine, in their groups, what prize they would put forth for consideration. Often, each student advocated for his or her own favorite prize, and it was difficult to abandon personal choice for group preference. Interviewer: Were there parts of the activity that you liked less than math? Wayne: Yes, it was having to make the decision on what to buy. It was so hard. Interviewer: Why was it hard? Wayne: Because there was so much stuff we could choose, like who knows… Layla wanted to buy 80s stuff, George wanted to buy animal stuff, and I wanted owl stuff. We ended up picking owl stuff, owl squirts.

Students also discussed difficult mathematics as a predominant feature of mathematical modeling. Difficult, to them, meant that the numbers were larger than they had encountered before, and there were more mathematical questions involved in the solution than one. For example, several of the students discussed using different strategies to add strings of larger numbers or to determine how many packs of toys they needed to buy. Other students discussed that there were a lot of mathematical questions that needed to be addressed. Five of the students discussed this aspect of modeling as fun. They enjoyed wrestling with larger questions and numbers and simplifying a complex situation. Grace described it in the following way: Interviewer: Was there anything else that you liked about it? Grace: It was fun writing down all the big equations. Interviewer: That is really fun. How is the activity different than what you usually do in math? Grace: I don’t know. It just feels special. Interviewer: Ok, well what was special about it? Grace: Usually in math we work alone or in partners, and in that we had to learn how to work together in big groups, and we were working on really big math problems instead of just little math problems at different times.

Three students described the mathematics involved in the task as challenging or stressful, either because there were many things to keep track of or because the numbers involved were challenging to work with. For example, in the transcript below, Libby discussed that mathematical modeling was stressful because she had to add numbers into the hundreds place, and that was challenging for her. Libby: Because sometimes it stresses me out and sometimes it’s just really fun because you get to do fun stuff. Interviewer: What about math modeling stressed you out, what was stressful about it? Libby: Like the numbers we had to add were really hard. Interviewer: It was really hard? Libby: Like adding all the students together because there might have been like 60 or like big numbers, and it was in the hundreds and whenever I get to the hundreds it’s really hard for me.

Group dynamics was the ability to work through the task together, communicate ideas, and listen to one another. For three of the students, this aspect of modeling was very challenging. They described that they liked working on mathematics problems independently, it was difficult to listen to others, and it was challenging to

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reach consensus across group members. One student, Libby, discussed group work as both a challenge and an affordance of modeling. She enjoyed working together as a team to reach a common goal but found it challenging to listen to her groupmates while making decisions. Libby: I think I’m bad at not listening to the other people about what they think. Interviewer: Sometimes it’s hard. (Later in the interview) Interviewer: You said there were some parts that were fun. What was fun about math modeling? Libby: What was fun about it was that we were all working as a team and that we all figured it out together and that we could do it.

Real-world application was the fact that the problem was situated in the real world and included actual people, toys, and prices. Many of the students found this to be a motivating aspect of the modeling process. They enjoyed that the numbers that they were working with represented dollar amounts and quantities of toys that existed. Here are two excerpts from Layla and Henry in which they compared how modeling might be better than what they do during a typical day of instruction. Layla: We were talking about money instead of just numbers. Henry: Normally in math we just, we do like story problems and stuff, and this we actually had to figure out how much money with like cents and dollars, and we also had to figure out what to get and figure out how much of it. Like if it came in dozens, how many dozens.

In both accounts, the students describe the connection to the real world as a motivating factor in solving the problem. One student, Sven, commented that he could see uses for mathematical modeling beyond this task. When describing what he liked about mathematical modeling, he stated, “I like math modeling, because you can use it anywhere you need to,” and followed with an example of constructing a snow fort and determining the number of snowballs needed. In contrast, one student, George, wrestled with the idea of the problem being set in the real world. He described that the answer they found in September for the number of toys might not be accurate at the end of the school year. He was worried that some students would not receive a prize. He describes his ideas in the excerpt below: Interviewer: And were there parts that you liked less than what you usually do? George: Yeah, because if you have a small number (of prizes) and there’s a big amount of people in the class, I mean just in the school, and if there’s more kids that come to the school, I kind of think we won’t get enough (prizes) because new people are coming to the school and old people are leaving.

10.7.3 Mathematical Modeling and Student Identity In several of the interview questions, we asked students to consider themselves in relation to other students and consider their learning during a modeling situation in relation to typical mathematics instruction. Students did not really know how to

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compare themselves in relation to each other in terms of mathematical modeling. They did describe ways in which modeling challenged their identity and understanding of classroom culture and mathematics. All of the students described modeling in relation to enjoyment or “fun,” but their reasons for enjoyment varied. For some students, modeling was enjoyable because it connected to their personal identity and they were able to find success in mathematics for the first time or purpose in mathematical activity. For other students, modeling connected to the discipline of mathematics, and students found enjoyment in the calculations themselves. Lastly, students did comment on lack of enjoyment in relation to group dynamics and mitigating disputes. One of the students, George, who typically felt bad at mathematics, commented that he enjoyed modeling because he felt success and was able to contribute in a productive way. George enjoyed being able to show other students how to use a computer and calculator. He used the calculator to add large sums together for his group. Here is an excerpt from his interview: George: I just think math modeling is kind of easier for me than math. Interviewer: What makes it easier? George: Most of it on the computer and I’m really good at computers but I’m not very good with math. George: First, I thought I didn’t like math but then I realized that math is kind of fun. Interviewer: Oh ok. George: Even though it’s hard sometimes.

Students (Sven and Grace) discussed that modeling was enjoyable because it made mathematics useful and purposeful. Sven described that he recently used modeling on the playground to help make sense of who to help build a snow fort and how many snow chunks he would need. (In mathematics) I don’t really get that much done that fast, and I don’t get that good on my grades for my math. I like math modeling because you can use it anywhere you need to and probably if I didn’t know that I would never figure out which fort to help (build)… Outside it’s like how many snow chunks do we need to make a fort, and we probably need maybe 20 more. You have to figure out how much, because there are big pieces that can go right on top.

Grace discussed that she enjoyed mathematical modeling because the problem mattered and they were actually going to buy the toy that the second graders decided on. Five of the students discussed that they enjoyed mathematical modeling because it involved mathematics in new and different ways, and they were able to apply what they had learned to wrestle with a difficult question. Students commented that they liked adding more than two numbers together and using different strategies to add. Lastly, two students described that they did not enjoy the social nature of mathematical modeling. They associated their success in modeling to being able to work with group mates and did not feel successful with the task because of communication issues. Looking across students and interview responses, students conceptualized mathematics in new and different ways when doing mathematical modeling compared to traditional instruction. The students were able to see and appreciate that

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mathematics tasks can be purposeful and challenging. They also saw that they could connect and apply previously learned ideas together to solve a larger task. They also perceived mathematics as a social activity that involved listening to others and valuing different perspectives. In terms of mathematical identity, it is interesting that students were still thinking about the task or freely applying ideas about the task to other situations. For example, George, who viewed himself as not good at mathematics, was still wrestling with the idea of the number of prizes being appropriate. In addition, the students did not seem to have a way to discuss ability with modeling. We wonder if this is because there was not one way to solve the problem and students had the autonomy to choose which solution worked best. This was markedly different than their typical instruction that included answering problems for accuracy. Even though students did not always appreciate all of the attributes of the modeling lesson, they were valuable because they expanded notions of what it means to know and do mathematics.

10.8 Discussion In this section we consider modeling in relation to students’ perceptions of what it means to know and do mathematics. In considering mathematics, as a discipline, most of the students thought about mathematics as what they did within the classroom, and only about half were able to identify ways in which they used mathematics outside of the classroom. As evident through themes of real-world application and responsibility, modeling allowed for connections between the real world and mathematics. The modeling task gave the mathematics a reason and purpose that was evident to the students. Modeling also challenged the nature of what it meant to do mathematics inside the classroom. As most of the students discussed, mathematics was often an independent activity with correct and incorrect answers. Modeling allowed for complexity in problem-solving as well as social interaction that was not typically present. Students had to take previously learned mathematical ideas and determine how to interconnect them together to best solve the task. In addition, they had to learn to work with one another to determine paths forward in the process. There was also not one correct answer. It was up to the class to determine which prizes best fit multiple needs. It is important to note that the themes in modeling were both challenges and affordances depending on the student. Modeling added new dimensions to mathematics that students had to wrestle with, and we argue that these dimensions are beneficial to students. The themes as challenges encourage students to engage in mathematics and practices in new ways that they might not have had the opportunity to do in a typical lesson. They are learning additional skills that benefit them as mathematical problem-solvers.

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The themes as affordances connect directly to student identity. Modeling provided a space where students could experience mathematics in a different way. For some of them, the affordances provided connections between mathematics and their lived experiences, academic strengths, or aspects of learning they enjoy. We witnessed the promise of modeling firsthand in the case of George. He saw himself as not good at mathematics, but his disposition and engagement were remarkably different during the modeling task, when he felt he could contribute to solving the task.

10.9 Conclusion In this study, we considered the research on the benefits of mathematical modeling in the classroom but extended it to the child’s perspective. Much of the previous research related to creating positive identities and experiences in the mathematics classroom was also evident in our findings surrounding mathematical modeling. Researchers have argued that when students are able to draw on their own experiences and feel involved in the problem-solving process, they are more likely see it as relevant and useful (Boaler, 2015; Boaler & Greeno, 2000; Boaler & Staples, 2008; Cobb et al., 2009). In several cases, we saw students demonstrate mathematical literacy by connecting mathematics to their lived experiences, finding purpose and enjoyment in mathematics, and continuing their thinking about modeling beyond the classroom. This was seen with Sven who described a task he constructed while playing with snow on the playground. Multiple researchers have also stated that mathematical modeling encourages positive dispositions toward mathematics (Bonotto, 2007; Lesh & Yoon, 2007). Although there were students who preferred the traditional way of learning mathematics, our research indicates that the mathematical modeling task provided students an effective opportunity to experience mathematics in new ways that, for some, better aligned with their perceived strengths and learning preferences. In addition, modeling contributed to the idea that mathematics is more than solving problems quickly and correctly. Students had to grapple with group dynamics, social interactions, and the various forms of technology utilized during the task. Students were also able to see the tasks as relevant and purposeful. This aligned with other research that mathematical modeling tasks encourage student agency because they are able to see meaning in the problem and form connections between the real world and the mathematics they are using in the classroom (Zbiek & Conner, 2006). In this task, students worked with money, toys, and technology that directly related to the students' classroom experiences. In addition, their decision, once finalized, was carried out. We wonder if the nature of the modeling task helped to create a sense of responsibility and ownership in this scenario. Many researchers have argued that the mathematical processes that students work through in a mathematical modeling task are the same needed to build content knowledge (de Lange, 2003; Niss, 2003; Steen et  al., 2007). Our research gave students a chance to connect mathematical ideas together and build content knowledge. We observed students approaching the task using various strategies at different levels of competencies as

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they reached for a common goal. Students made a point to mention the difficulty of approaching larger numbers and having to take different paths to determine the number of toys needed. That being said, we did not formally measure student’s mathematical understanding, so there is no way to know if this area of content knowledge was fully developed for these students. Finally, this task allowed students to understand multiple perspectives. Students were asked to make choices on behalf of the entire school. Similar to Greer and colleagues’ findings (Greer et  al., 2007), students became invested in the problem because it was personally meaningful (Greer et al., 2007). They were able to give a voice to their classmates in a situation where they might not typically have a choice. This increased ownership and motivation. The students worked hard, explored different paths, and edited their model, if necessary, to ensure that they were making a choice that benefited not only themselves but the school as a whole. This was a slow process as students initially advocated for their own personal choice but gained practice in putting group needs above individual wants. Our initial research has highlighted that modeling has both challenges and affordances from the student’s perspective, but we are also left with new questions. Moving forward, we wonder how a student’s definition of mathematical modeling would change as he/she comes in contact with more tasks. This was the first mathematical modeling task of the school year, and it would be interesting to see how definitions shift as students are faced with new tasks. In relation, we noticed students did not have markers to describe their modeling skills in relation to other students. Further studies would yield a better understanding of how students’ perceptions of their modeling abilities change over time. Lastly, we wonder if the challenges and affordances noted by students are task specific or if they would arise across all modeling tasks.

References Aguirre, J.  M., Mayfield-Ingram, & Martin, D.  B. (2013). The impact of identity in K-8 mathematics: Rethinking equity-based practices. Reston, VA: National Council of Teachers of Mathematics. Anhalt, C.  O., Cortez, R., & Smith, A. (2017). Mathematical modeling: Creating opportunities for participation in mathematics. In A. Fernandes, S. Crespo, & M. Civil (Eds.), Access and equity: Promoting high quality mathematics in grades 6-8 (pp. 105–119). Reston, VA: National Council of Teachers of Mathematics (NCTM). Anhalt, C.  O., Staats, S., Cortez, R., & Civil, M. (2018). Mathematical modeling and culturally relevant pedagogy. In Y.  J. Dori, Z.  R. Mevarech, & D.  R. Baker (Eds.), Cognition, metacognition, and culture in STEM education (pp.  307–330). Springer. https://doi. org/10.1007/978-­3-­319-­66659-­4_14 Bartholomew, H. (2000). Negotiating identity in the community of the mathematics classroom. Paper presented at the British Educational Research Association annual conference, September 7–10. Berry, R. Q., III, Thunder, K., & McClain, O. L. (2011). Counter narratives: Examining the mathematics and racial identities of black boys who are successful with school mathematics. Journal of African American Males in Education, 2(1), 10–23. Boaler, J. (2015). What’s math got to do with it? How teachers and parents can transform mathematics learning and inspire success (Rev. ed.). New York, NY: Penguin.

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Boaler, J., & Greeno, J.  G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171–200). Westport, CT: Ablex. Boaler, J., & Selling, S. K. (2017). Psychological imprisonment or intellectual freedom? A longitudinal study of contrasting school mathematics approaches and their impact on adults’ lives. Journal for Research in Mathematics Education, 48(1), 78–105. Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of Railside School. Teachers College Record, 110(3), 608–645. Bonotto, C. (2007). How to replace word problems with activities of realistic mathematical modeling. In W. Blum, P. L. Galbraith, H. W. Henn, & M. Niss (Eds.), Modeling and applications in mathematics education: The 14th ICMI study (1st ed., pp. 185–192). New York, NY: Springer. Carlson, M. A., Wickstrom, M. H., Burroughs, E. A., & Fulton, E. W. (2016). A case for mathematical modeling in the elementary classroom. In C.  Hirsch (Ed.), Annual perspectives in mathematics education: Mathematical modeling and modeling mathematics (pp.  121–129). Reston, VA: National Council of Teachers of Mathematics. Cirillo, M., Bartell, T. G., & Wager, A. A. (2016). Teaching mathematics for social justice through mathematical modeling. In C.  Hirsch (Ed.), Annual perspectives in mathematics education: Mathematical modeling and modeling mathematics (pp. 87–96). Reston, VA: National Council of Teachers of Mathematics. Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education, 40(1), 40–68. Cobb, P., & Hodge, L. L. (2002). A relational perspective on issues of cultural diversity and equity as they play out in the mathematics classroom. Mathematical Thinking and Learning, 4(2), 249–284. de Lange, J. (2003). Mathematics for literacy. In B. L. Madison & L. A. Steen (Eds.), Quantitative literacy: Why numeracy matter for schools and colleges (pp. 75–89). Princeton, NJ: National Council on Education and the Disciplines. Retrieved from. http://www.maa.org/sites/default/ files/pdf/QL/pgs75_89.pdf English, L.  D., & Sriraman, B. (2010). In B.  Sriraman & L.  D. English (Eds.), Theories of mathematics education: Seeking new frontiers (Advances in mathematics education series, pp. 263–285). Berlin/Heidelberg, Germany: Springer. English, L. D., & Watters, J. (2005). Mathematical modeling in the early school years. Mathematics Education Research Journal, 16(3), 59–80. Garfunkel, S., & Montgomery, M. (Eds.). (2016). Guidelines for assessment and instruction in mathematical modeling education (GAIMME) report. Boston, MA: Consortium for Mathematics and Its Applications (COMAP)/Society for Industrial and Applied Mathematics (SIAM). Greer, B., Verscheffel, L., & Mukhopadhyay, S. (2007). Modeling for life: mathematics and children’s experience. In Modeling and applications in mathematics education: The 14th ICMI study (pp. 89–98). Gresalfi, M. S. (2009). Taking up opportunities to learn: Constructing dispositions in mathematics classrooms. Journal of the Learning Sciences, 18(3), 327–369. Holland, D., Lachicotte, W., Jr., Skinner, D., & Cain, C. (2001). Identity and agency in cultural worlds. Cambridge, MA: Harvard University Press. Kutaka, T. (2013). Young children’s beliefs about the self as a learner and producer of mathematics: A mixed methods study. ETD collection for University of Nebraska – Lincoln. AAI3558618. http://digitalcommons.unl.edu/dissertations/AAI3558618 Leder, G., Pehkonen, G., & Toerner, G. (Eds.). (2002). Beliefs: A hidden variable in mathematics education? Dordrecht, The Netherlands: Kluwer Academic Publishers. Lehrer, R., & Schauble, L. (2007). A developmental approach for supporting the epistemology of modeling. In W. Blum, P. L. Galbraith, Henn, & M. Niss (Eds.), Modeling and applications in mathematics education: The 14th ICMI study (1st ed., pp. 155–160). New York, NY: Springer. Lesh, R., English, L. D., Sevis, S., & Riggs, C. (2013). Modeling as a means for making powerful ideas accessible to children at an early age. In S. Hegedus & J. Roschelle (Eds.), The SimCalc vision and contributions: Democratizing access to important mathematics (pp.  419–436). Dordrecht, The Netherlands: Springer.

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Lesh, R., & Yoon, C. (2007). What is distinctive in (our views about) models and modeling perspectives on mathematics problem solving, learning, and teaching? In W. Blum, P. L. Galbraith, Henn, & M.  Niss (Eds.), Modeling and applications in mathematics education (1st ed., pp. 161–170). New York, NY: Springer. Martin, D.  B. (2006). Mathematics learning and participation as racialized forms of experience. African American parents speak of the struggle for mathematics literacy. Mathematical Thinking and Learning, 8, 197–229. Middleton, J., Jansen, A., & Goldin, G. (2017). The complexities of mathematical engagement: Motivation, affect, and social interactions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 667–699). Reston, VA: NCTM. Middleton, J.  A. (2013). More than motivation: The combined effects of critical motivational variables on middle school mathematics achievement. Middle Grades Research Journal, 8(1), 77–95. Miles, M.  B., Huberman, A.  M., & Saldana, J. (2014). Qualitative data analysis: A methods sourcebook. Thousand Oaks, CA: Sage. Muller, E., & Burkhardt, H. (2007). Applications and modeling for mathematics overview. In W. Blum, P. Galbraith, H. Henn, & M. Niss (Eds.), 3.4.0 (1st ed., pp. 267–274). New York, NY: Springer. Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM Project (eds. A.  Gagtsis & S.  Papastavrisis, pp.  115–124). Athens, Greece: The Hellenic Mathematical Society and Cyprus mathematical society. Organization for Economic Co-operation and Development (OECD). (2012). PISA 2012 mathematics framework, pp. 1–42. https://doi.org/10.1787/9789264190511-­3-­en. Schiefele, U., & Csikszentmihalyi, M. (1995). Motivation and ability as factors in mathematics experience and achievement. Journal for Research in Mathematics Education, 26(2), 163–181. Schoenfeld, A.  H. (2013). Mathematical modeling, sense making, and the common core state standards. In B.  Dickman & A.  Sanfratello (Eds.), Conference on mathematical modeling (pp.  13–25). New  York, NY: Program in Mathematics and Education Teachers College Columbia University. Seymour, E., & Hewitt, N. (1997). Talking about leaving: Why undergraduates leave the sciences. Boulder, CO: Westview. Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytical tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 14–22. Steen, L. A., Turner, R., & Burkhardt, H. (2007). Developing mathematical literacy. In W. Blum, P. L. Galbraith, Henn, & M. Niss (Eds.), Modeling and applications in mathematics education: The 14th ICMI study (1st ed., pp. 285–294). New York, NY: Springer. Suh, J. M., Matson, K., & Seshaiyer, P. (2017). Engaging elementary students in the creative process of mathematizing their world through mathematical modeling. Educational Sciences, 7, 62. https://doi.org/10.3390/educsci7020062 Turner, E.  E., Aguirre, J.  M., Foote, M.  Q., & Roth McDuffie, A.  M. (2018, April). Learning to leverage mathematical resources of elementary Latinx children through community-based mathematical modeling tasks. In M.  Civil (Chair), Foregrounding cultural ways of being in mathematics teacher education: Cases From Latinx and Pāsifika communities. Presentation as part of a symposium at the annual research conference of the annual meeting of the American Educational Research Association, New York, NY. Wickstrom, M. H. (2017). Mathematical modeling: Challenging the figured worlds of elementary mathematics. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th annual meeting of the north American chapter of the international group for the psychology of mathematics education (pp. 685–692). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators. Wickstrom, M. H., Carr, R., & Lackey, D. (2017). Exploring Yellowstone National Park with mathematical modeling. Mathematics Teaching in the Middle School, 22(8), 462–470. Zbiek, R. M., & Conner, A. (2006). Beyond motivation: Exploring mathematical modeling as a context for deepening students understandings of curricular. Educational Studies in Mathematics, 63(1), 89–112. https://doi.org/10.1007/sl0649-­005-­9002-­4

Chapter 11

Upcycling Plastic Bags to Make Jump Ropes: Elementary Students Leverage Experiences and Knowledge as They Engage in a Relevant, Community-­ Oriented Mathematical Modeling Task Erin E. Turner, Amy Roth McDuffie, Julia M. Aguirre, Mary Q. Foote, Candace Chappelle, Amy Bennett, Monica Granillo, and Nishaan Ponnuru

11.1 Mathematical Modeling in the Elementary Grades Mathematical modeling is a high-leverage topic, critical for college and career readiness, participation in STEM education, and civic engagement (Aguirre, Anhalt, Cortez, Turner, & Simi-Muller, 2019). While researchers have offered various definitions of mathematical modeling, most concur that mathematical modeling is “a process that uses mathematics to represent, analyze, make predictions and otherwise provide insight into real-world phenomena” (Garfunkel & Montgomery 2016, p. 8). Consistent with this definition, others have described modeling as involving formulating, testing, and validating mathematical models to analyze real-world situations and inform decision-making (Blum & Borromeo Ferri, 2009; Lesh & Fennewald, 2010). Unlike typical textbook word problems that require students to map information onto specific quantities and operations, often disregarding realistic considerations (Greer, 1997; Verschaffel, De Corte, & Borghart, 1997), modeling E. E. Turner () · A. Bennett · M. Granillo · N. Ponnuru University of Arizona, Tucson, AZ, USA e-mail: [email protected]; [email protected] A. R. McDuffie · C. Chappelle Washington State University, Pullman, WA, USA e-mail: [email protected]; [email protected] J. M. Aguirre University of Washington Tacoma, Tacoma, WA, USA e-mail: [email protected] M. Q. Foote Queens College, CUNY, New York, NY, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_11

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tasks invite students to consider real-world contexts as well as real-world solutions (Anhalt, Cortez, & Been Bennett, 2018, Anhalt, Staats, Cortez, & Civil, 2018). In modeling, students engage in a cyclical process of (a) analyzing situations; (b) constructing models that represent the situation, based on information and assumptions; (c) using models to perform operations, and reason about results in terms of the original situation; (d) validating or revising the model; and (e) reporting conclusions (CCSSM, 2010). Although mathematical modeling has a well-established research base in secondary and undergraduate education (Doerr & Tripp, 1999; Gainsburg, 2006), it has been underemphasized and under-supported at the elementary level, with a few notable exceptions (e.g., Carlson, Wickstrom, Burroughs, & Fulton, 2018; English & Watters, 2004; Suh, Matson, & Seshaiyer, 2017). This may be due to limited attention to modeling in elementary teacher preparation and professional development and an absence of rigorous mathematical modeling tasks in elementary curricula (Burkhardt, 2006). However, young children have “the foundational competencies on which modeling can be developed,” and thus instruction in modeling can begin earlier (English & Watters, 2004, p. 336). In fact, research conducted with elementary grade students and teachers demonstrates that mathematical modeling is accessible to children in elementary grades, including students with limited prior experience with modeling (Chan, 2009; English, 2006) and students from a diverse range of mathematical and cultural backgrounds (Turner, Gutiérrez, Simic-­ Muller, & Díez-Palomar, 2009). Moreover, modeling supports elementary students’ understandings of core content including quantitative reasoning, algebraic thinking, and representing and interpreting data. For example, English and Watters (2005) reported on a third-grade modeling task about a farmer who grew various types of beans; students analyzed data for weights of plants after 6, 8, and 10 weeks of growth under different conditions. Students engaged in a modeling cycle by determining the best growth conditions, writing a letter to the farmer with recommendations, and explaining how their method could be used to predict growth in the future. Carlson et al. (2018) described a fourth-grade modeling task connected to an annual dinner fundraiser at the school. Students proposed “the best meal” for the fundraiser and considered both which meals would be popular among family and friends and profitable for the school. As students moved through the modeling process, students realized that some of their assumptions about costs and prices were inaccurate, and they revised their profit estimates accordingly. Studies have also shown that the authentic, real-world nature of mathematical modeling promotes critical thinking and deepened student engagement and understanding among elementary school students (English & Watters, 2004; Carlson et  al., 2018; Garfunkel & Montgomery 2016). Suh et  al. (2017) focused on how modeling activities promoted 21st century skills among elementary grade students, namely, critical thinking and problem-solving, creativity and innovation, and communication and collaboration (P21 Partnership for 21st Century Learning, 2017). In one modeling activity, third-grade students planned a 1-day family vacation given budget, time, and distance constraints and a requirement to include meals and

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tourist sites. Students shared their initial vacation plans with family members, and in many instances families provided insights that helped students to refine their models (e.g., considerations related to younger siblings). In short, a growing body of evidence indicates that elementary students can and do engage in mathematical modeling tasks using their own experiences as part of the sense making process.

11.2 Mathematical Modeling with Cultural and Community Contexts Research suggests that culturally responsive, community-based approaches to teaching mathematics have added benefits, particularly for students from underrepresented groups in STEM fields (Aguirre & del Rosario Zavala, 2013; Civil, 2007; Lipka et al., 2005; Turner, Celedón-Pattichis, & Marshall, 2008). Grounding mathematics in contexts that are community-based and connected to students’ experiences (rather than generic or assumed-to-be-relevant scenarios) can enhance student engagement and learning (Ladson-Billings, 2009; Lipka et  al., 2005; Sembiring, Hadi, & Dolk, 2008; Turner et al., 2009) and help to counter students’ tendency to ignore real-world considerations in problem-solving (Greer, 1997; Palm, 2008; Verschaffel & De Corte, 1997). Mathematical modeling tasks do just this by “recogniz[ing] and reward[ing] a broader range of mathematical abilities than those traditionally emphasized” (Lesh & Doerr, 2003, p. 23). For example, the realistic mathematics education (RME) initiative has argued that contexts that are meaningful in relation to students’ experiences allow students to draw upon situational knowledge and real-world considerations as they engage in mathematical modeling, instead of “cutting bonds with reality” (Bahmaei, 2011). Well-chosen contexts offer “students opportunities to develop informal, highly context-specific models and solving strategies,” which serve as entry points to more abstract mathematics (Doorman & Gravemeijer, 2009). These connections help students understand how mathematics matters in personal and socially meaningful contexts. In research with middle school students, Anhalt and colleagues outlined a collection of modeling tasks related to authentic issues in students’ borderland community such as border crossing, produce distribution, water shortages, and media use (Anhalt, Cortez, & Smith, 2017). During one activity related to declining television viewing rates, the students drew upon their understandings of media usage among their peers and family members to make sense of data trends and anomalies. Anhalt and colleagues noted that the experiential knowledge that students brought to the task shaped their modeling activity. Moreover, by connecting modeling to complex societal issues such as water quality and contamination (Aguirre et  al., 2019), food distribution (Anhalt et  al., 2017), weather patterns and natural disasters (English, Fox, & Watters, 2005), or recycling and waste (English, 2010), students had opportunities to articulate and justify their positions and consider the ideas of others. In short, mathematical modeling is increasingly recognized as a powerful way not only to deepen students’

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understanding but also students’ appreciation of mathematics as a critical tool to investigate and act upon important situations in their schools, communities, and in the world (Aguirre et  al., 2019; Asempapa, 2015; Carlson et  al., 2018; Suh et al., 2017). In our work with the Mathematical Modeling with Cultural and Community Contexts (M2C3) research project, we build on this prior research to investigate culturally responsive, community-based approaches that support mathematical modeling with elementary students, particularly those historically marginalized in STEM education. Specifically, we collaborated with elementary teachers to develop modeling lessons that build on students’ mathematical thinking (Carpenter, Fennema, Peterson, Chang, & Loef, 1989; Carpenter, Franke, Jacobs, & Fennema, 1998) and their cultural and community-based knowledge and experiences (Civil, 2007; González, Andrade, Civil, & Moll, 2001). We developed modeling tasks grounded in home and community contexts, such as planning family events (e.g., birthdays), and operating a local business (e.g., arrangement of booths at a swap meet), as well as contexts relevant to the school community (e.g., gardening, play activities, school celebrations). As teachers enact these tasks in their classrooms, our research focused on how students’ contextual knowledge and experiences inform, support, and guide their modeling activity. Specifically, we investigated how students drew upon their experiences as they engage in the various phases of the mathematical modeling process. While prior work contributed general or single teacher-based descriptions of how students’ experiences impacted their modeling activity, the field lacked more systematic, detailed analyses across multiple teachers and across each phase of the modeling process. In this study we contribute important insights related to the gap in the extant literature, via an analysis across teachers and across-modeling phases of student engagement in a common modeling task  – Upcycling Plastic Bags to Make Jump Ropes – enacted in four elementary grade classrooms.

11.3 The Mathematical Modeling Process Figure 11.1 is a representation of the mathematical modeling process that we used to frame our project work and to organize the findings reported in this chapter. This representation is informed by other research studies (Anhalt, Cortez, & Been Bennett, 2018), as well as the mathematical modeling cycle depicted in the CCSSM’s (2010) high school mathematical modeling standard. More specifically, the five phases reflected in our representation of the modeling process mirror the phases included in the Common Core State Standards (problem (Phase 1), formulate (Phase 2), compute (Phase 3), interpret (Phase 4), validate (Phase 5), and report (reporting out) CCSM, 2010). In our representation of the model, we elaborated the title of each phrase “phase” not “phrase” to clarify the activities and processes involved. Our elaborations built on similar adjustments made by Anhalt, Cortez, et al. (2018) in their work with prospective teachers. We suggest that these elaborations may be

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Fig. 11.1  Mathematical modeling process for elementary students and teachers

particularly useful for teachers who are new to mathematical modeling, as was the case with the teachers in our study. In addition, our representation includes three smaller rectangles (positioned between Phase 1 and Phase 2) that reflect key model-­ building activities that occur across these two phases. The salience of these model-­ building activities has been noted in other research (Maaß, 2006), and similar to Anhalt, Cortez, and Been Bennett, we chose to draw attention to these activities in our representation of the modeling process. In the next section, we briefly describe each phase and then outline possible pathways through the modeling process.

11.3.1 Phase 1: Make Sense of a Situation or Problem Students begin the mathematical modeling process by making sense of a situation or problem presented by a teacher or identified by students based on their experiences and knowledge. As part of making sense of the problem, students consider questions such as the following: • • • •

What do I know about this situation? What experiences have I had related to the situation? What additional information do I need? How might I find or generate that information?

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11.3.2 Phase 2: Construct a Model As students begin to construct a model for the situation, they consider what quantities are relevant and important, and how those quantities relate to one another. Students also consider what information is provided or available, what information might need to be collected or researched, and what they will need to assume or decide. These model-building considerations tightly connect to the context, so as students construct a model, they often continue to make sense of the situation or problem. In other words, students tend to move iteratively between Phase 1 and Phase 2, as the two arrows indicate and as is reflected in the way we have positioned the different modeling-building considerations (listed in the smaller, lightly shaded rectangular boxes) between the two phases. We highlighted these considerations in our model because they reflect points in the modeling process where students’ experiences and knowledge may be particularly salient.

11.3.3 Phase 3: Operate on Model Next, students create a solution for the problem, perform computations, and check for precision both in their results and in their labels and/or explanation of their work. At this point, students might have one or more solutions relevant to the original problem, and these solutions represent an important step toward creating a generalized model (Phase 5).

11.3.4 Phase 4: Interpret/Analyze Solutions and Refine Model In Phase 4, students interpret their solutions in relation to the original situation or problem from Phase 1. They ask whether their solution makes sense based on what they know from their experiences and knowledge about the context, draw conclusions about what the solutions imply for the situation, and refine and revise their model (if needed). Refinements might include adjusting their model based on new assumptions to fit a wider range of situations.

11.3.5 Phase 5: Validate and Generalize Model The next phase in the mathematical modeling process is an evaluation phase that involves validating and generalizing the model so that it is reusable and allows for applications to similar scenarios. If students are not satisfied with their model, or if they want to redefine the problem or situation and construct a new model to reflect

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this redefined situation, they iterate within the cycle and return to Phase 1. If they are satisfied with the accuracy and comprehensiveness of their model, they progress out of the cycle and report out their solution (i.e., describe their model so that it can be shared with others). This branching point in the cycle is reflected by the two arrows from Phase 5 that indicate either reporting out or returning to Phase 1 to reengage the modeling process.

11.3.6 Movement Across the Phases As shown in Fig. 11.1, students enter the modeling process at Phase 1. Although we have represented a progression as occurring predominantly clockwise, students can move between any of the nodes in the process of modeling. The arrows represent several possible pathways through the modeling process. For example, to interpret and analyze their solutions (Phase 4), students may return to the real-world situation or problem (Phase 1) to determine what their solution means in relation to the real-­ world context and whether it provides a viable and satisfactory solution. If students determine that adjustments are needed, they may construct a new model (Phase 2) and then operate on this refined model (Phase 3). In this study we focused on how students draw upon their knowledge and experiences across multiple contexts (school, home, community) as they engage in the various phases of the mathematical modeling process. Specifically, the following research question guided this study: How do students draw upon their knowledge and experiences both to influence and engage in each phase of the mathematical modeling process during the Upcycling Jump Rope task?

11.4 Context of the Study As part of the broader M2C3 project, teachers participated in professional development to learn about mathematical modeling. In the sections that follow, we briefly describe the professional development experiences. In addition, we provide a description of the Upcycling Jump Rope task, developed for this project, and the focus of the analysis in this chapter.

11.4.1 M2C3 Teacher Professional Development Summer Institute  Approximately 30 teachers from 2 districts participated in 30-h summer institutes focused on teaching mathematical modeling in grades 3–5. The districts were in two different regions of the USA (northwest and southwest); both districts served multiracial, multilingual, multicultural, and working-class

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c­ ommunities, consistent with the project goals to improve instruction for diverse and historically marginalized students. Teachers attended the institute in schoolbased teams which included classroom teachers from grades 3 through 5 and in some instances an instructional coach. During the institutes, teachers engaged in modeling tasks developed by the research team, explored key phases of mathematical modeling, and discussed critical features of meaningful, relevant modeling tasks. Additionally, teachers shared information about students’ and families’ interests and experiences. Teachers then visited community locations frequented by students and interviewed community members (e.g., business owners, patrons, employees) about their practices, including ways that they used mathematics. Academic Year Teacher Study Groups  During the academic year, teachers met on a monthly basis to plan, discuss, and reflect on mathematical modeling tasks enacted in their classrooms. These discussions included identifying and analyzing the different kinds of knowledge and experiences that students leveraged to support modeling activities. Twice per year, researchers presented modeling tasks and related lesson materials (photos, handouts, lesson launch slides) designed by members of the research team. We designed these tasks to address content standards reflected in district quarterly curriculum maps and to connect to relevant contexts in schools and communities. The teachers enacted these shared tasks, to facilitate analysis of a common lesson enacted across different community and cultural contexts. The Upcycling Jump Rope Modeling Task  During the 2017–2018 academic year, one shared modeling task focused on upcycling single-use plastic bags to make jump ropes. This task was inspired by teachers’ and students’ interest in environmental issues, including recycling and upcycling to reduce waste. Students had responded positively to other modeling activities related to environmental issues, such as tasks that explored water use in showers versus baths (Felton, Anhalt, & Cortez, 2015). Additionally, in several schools, students participated in jump rope activities, either in physical education class, morning exercise routines, or “jump for the heart” fitness challenges, which provided other relevant experiences that students could bring to the task. Finally, in one community, a local Girl Scout troop launched a service project related to making jump ropes from plastic bags. The girls produced an instructional video documenting their process for making jump ropes, and this video became a key resource in the task design. We selected the Upcycling Jump Rope task for analysis not only because it was taught by all participating teachers but also because we designed the task to facilitate connections to students’ experiences and knowledge and thus provided a compelling context to explore our research questions. Figure 11.2 displays one version of the task, in which students planned a set of jump ropes for physical education classes and calculated the number of plastic bags needed to make all the jump ropes in the set. Other versions of the task varied the target audience for the jump rope set, including (a) students’ classroom, (b) a community center, and (c) a school-wide jump-athon event. We encouraged teachers to select task options that would be most

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Fig. 11.2  Task: jump rope set for PE class

Fig. 11.3  Three plastic bags, braided, make 1 foot (12 inches) of jump rope

relevant and mathematically appropriate for their students and to adapt the task materials as needed. These optional materials included images and video clips to introduce the task context, a chart of recommended jump rope lengths for people of various ages and heights, and an instructional video that outlined a specific method for making jump ropes with plastic bags (https://youtu.be/znWQmjJ5gyo, adapted from the video made by the Girl Scouts). The method was based on braiding bags together, and in the video, the girls explained that three plastic bags were needed to make one foot of jump rope. Figure 11.3 includes images that show this relationship. Notably, all teachers used this video in their lessons.

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Finally, the Upcycling Jump Rope task reflected key design principles of model eliciting activities (Lesh, Hoover, Hole, Kelly, & Post, 2000), drawing on prior research that found that such activities support students in developing and testing mathematical models, and they promote deep understanding of mathematical ideas (Lesh & Doerr, 2003). Specifically, the Upcycling Jump Rope task reflected the design principles as follows: • The situation was authentic and included opportunities to execute the proposed plans (reality principle). • The task required generating and representing a model that could be used to figure out the number of bags needed to make sets of jump ropes of varied lengths (model construction and documentation principles). • Students considered and justified how their plans would work (self-evaluation principle). • In some instances, students generalized their plan for use in other similar situations (generalization principle).

11.5 Methods In the following sections, we describe the case study design and the classrooms selected as case studies for this analysis. We then describe our data collection and analysis methods.

11.5.1 Case Study Design Using a qualitative case study design (Creswell, 2013; Stake, 1995), we investigated elementary students (third and fifth graders) in four different classes as they engaged in a mathematical modeling lesson using the Jump Rope task. We considered each lesson as a case and the students who participated in the lesson as the primary focus of study. Lessons lasted between one and three class periods and typically included students’ engagement across all phases of the mathematical modeling process. Case study research lends itself to investigating issues and questions, wherein issues provide “conceptual structure to force attention to complexity and contextuality” (Stake, 1995, p. 16). In case study research, the goal is the deep understanding of happenings, guided by holistic interpretation that attends to particular contexts and circumstances. This study is suited for case study because it seeks to understand a complex phenomenon: In mathematical modeling lessons, how do students draw on their knowledge and experiences to influence the ways they engage in the mathematical modeling process? Correspondingly, we examined patterns within each lesson and across lessons (Stake, 2006).

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11.5.2 Participants We purposefully selected four classrooms from our broader data set of Jump Rope task lessons. We applied several criteria to identify those classrooms, (a) a robust and complete data set for the jump rope lesson (specific data sources are described in the next section), (b) high rates of parent permission and student assent to support a comprehensive analysis, and (c) diversity of grade levels. We ultimately selected one third-grade (8–9-year-olds) and one fifth-grade classroom (10–11-year-olds) from each of two partner districts. A brief introduction to each classroom follows. Ms. S is a White, third-grade teacher with 9 years of teaching experience. She teaches in District A at a neighborhood school. Her class of 20 students mirrored the demographics of the broader school, with several English learners and 87% Latinx, 9% Native American, and 2% White students. Ms. E is a Latina, fifth-grade teacher with 8 years of teaching experience. She teaches in District A at a project-based learning school. Her class of 29 students mirrored the demographics of the broader school, with 5 students learning English and 65% Latinx, 20% White, and 6% African American students. Ms. W is a White, third-grade teacher with 3 years of teaching experience. She teaches in District B. In her class of 22 students, 9 students were new learners of English. The racial/ethnic demographics mirrored the school with 30% Latinx, 19% White, 15% African American, 14% of 2 or more races, 9% Asian, 3% Native Hawaiian/Pacific Islander, and 1% American Indian/Alaskan Native students. Mr. H is a White fifth-grade teacher with 13 years of teaching experience; he also teaches in District B.  In his class of 27 students, 6 students were new learners of English. The racial/ethnic demographics mirrored that of the school with 30% Latinx, 19% White, 15% African American, 14% of 2 or more races, 9% Asian, 3% Native Hawaiian/Pacific Islander, and 1% American Indian/Alaskan Native students. All classrooms included a typical range of student backgrounds in mathematics, including students receiving special education services, and others who achieved at high levels.

11.5.3 Data Sources Data sources included video-recorded observations of mathematics lessons, post-­ observation teacher interviews, teacher reflections on lesson enactments during a subsequent teacher study group, student work, and other lesson artifacts (e.g., images of board work). The four lessons analyzed in this study ranged in length from 2 to 2.5 h. During video-recording, we followed the teacher to capture instructional decisions and moves. Correspondingly, when students worked in small groups, the video captured the teacher’s interactions with small groups and/or individual students. Videos also captured all whole group interactions, including s­ tudents’ questions and comments as teachers introduced tasks, intermittent check-ins when students shared emerging ideas and understandings, and students’ presentations of models and solutions and typo “at” not “and” the end of each lesson.

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This decision to follow the teacher meant that we did not capture some small group interactions on video. However, a benefit of this videoing decision was that we documented sustained interactions between teachers and small group of students in these recordings which provided important data for students’ ideas and understandings. All teacher interviews were audio-recorded and transcribed. Videos were selectively transcribed with a focus on teacher questions and prompts and examples of students’ thinking, reasoning, experiences, and knowledge that they brought to the task.

11.5.4 Data Analysis and Analytical Framework Through multiple and iterative cycles of analysis, we conducted within-case and cross case analyses for these lessons (Creswell, 2013; Stake, 1995). First, we developed preliminary coding categories based on (a) our research foci in the larger project and (b) key ideas identified in the literature, especially ideas related to modeling and connecting to students’ experiences and knowledge (categorical aggregation). These categories included teacher moves to elicit and/or connect with students’ experiences and knowledge; ways students’ experiences/knowledge connected with each of the five phases of the mathematical modeling cycle; and ways that connections to students’ experiences supported sense-making and learning. Next, we engaged in iterative cycles of sorting the data for each lesson within these categories and writing analytic case memos to identify and refine emerging themes (Miles, Huberman, & Saldaña, 2014). To achieve interpretive convergence and ensure consistency, multiple researchers were involved in developing, reviewing, and adding interpretive and analytic comments to the memos. After categorizing, interpreting, and identifying emerging themes in each lesson, our focus shifted to cross case analysis. We found that references to the phases of the modeling cycle appeared frequently in the analytic memos for each lesson, and we decided to use these phases to further identify and refine emerging themes across the four lesson cases. Connecting themes to phases in the lesson helped to clarify and sharpen the emerging themes. While viewing the data through the lens of the phases of the modeling cycle, we generated a narrative compilation (Creswell, 2013) of preliminary findings across lessons. This compilation included representative and compelling examples, along with non-examples, from all data sources to test emerging themes under each of the phases. Two researchers iteratively constructed, reviewed, and refined these findings.

11.6 Findings Consistent with our final phase of analysis, we structure our findings according to the five phases of the mathematical modeling process. In each section of the findings, we draw on illustrative examples from across the four lessons analyzed.

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11.6.1 Phase 1: Making Sense of the Situation or Problem In each of the four lessons, students drew upon understandings about plastics, recycling, and upcycling as they made sense of the task context. All four teachers used images and videos included in the lesson materials to launch the activity with an extended conversation about the environmental impact of plastic waste and alternatives to single-use plastic shopping bags. 11.6.1.1 Students Connecting to Family Practices with Plastic Shopping Bags One way students made sense of the situation in the Upcycling Jump Rope task was to connect to family practices related to plastic bags. For example, one of the third-­ grade teachers, Ms. S, asked students to share whether their families used plastic bags for other purposes. Student 1: Me and my papá whenever we go shopping, we use plastic bags, but we don’t throw them away -we just put them in a bigger bag. Ms. S: Okay, and what do you do with them after that? Student 1: I made my own kite out of three plastic bags, … and we also used the plastic bags for other stuff. Ms. S: So it sounds like that when you made the kite, that’s called upcycling. And …this project is going to be focusing on that idea of making something out of something else.

As students shared knowledge about the task context, they not only supported one another in making sense of the situation, they also provided their teachers with opportunities to learn more about students. For example, one of the fifth-grade teachers, Ms. E, noted that while she anticipated that students would bring knowledge about jump ropes to the task (given the school’s morning jump rope routine), she was surprised by students’ experience with recycling. She reflected: I didn’t realize that they [students] have this amazing background knowledge of recycling and upcycling and repurposing things. … This was something that they could connect to.

Students also shared understandings about the environmental impact of plastic bags, particularly when bags end up in waterways or landfills. The optional materials that accompanied the task included photos and video clips related to plastic bag pollution, and these played a key role in eliciting students’ experiences and understandings. For example, third-grade students in Ms. W’s class noted that plastic bag pollution harms birds, fish, and other animals because “animals could think there is food in the bag, and then they go in and get stuck.” Students also expressed their desire to address plastic bag pollution (e.g., “If I saw a bag on the side of the road or on a tree I would pull over and stop and pick it up”). Students’ care for the environment was an important resource that guided their work on the Jump Rope task, and in some classrooms, this interest led to lengthy discussions about plastic bag consumption and pollution. We found teachers made different decisions about how to invite students to share experiences with overarching problems of plastic bags, while also focusing the conversation on the specific context of the task.

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11.6.1.2 Students Connecting to Experiences with Play Students also leveraged experiences jumping rope as they made sense of the task scenario. We found that as students shared their experiences, teachers skillfully focused their attention on key features of the situation, such as different kinds of ropes, and jumping activities. In the following excerpt from Ms. E’s fifth-grade classroom, students viewed various images of children jumping rope. Ms. E asked them to share their understandings. Ms. E: Is there one way to jump rope? Student 1: There are different tricks you can do and different speeds you can jump. Student 2: You can jump with two ropes. Student 3: You can jump backwards and you can double dutch. Ms. E: And what does it mean to double dutch? Student 3: It’s where two people are holding the ropes on both sides and then turning the ropes like this [moves hands in a circular motion] while the person is jumping. Ms. E: So when do you think it would be best to do double dutch versus waiting your turn to have one person use their own jump rope? Let’s say we’re all going outside to jump rope and there were only a limited number of jump ropes, is double dutch something you can do with other people? Students: Yes! Ms. E: How many people does it take to run the ropes and to jump? Students: Two! Three! Student 4: It takes two people to hold the ropes and then it takes one to jump. Ms. E: … Keep that in your head. You’re going to use that information that you know.

We documented similar conversations in the other three classrooms; in each instance, students shared experiences related to jumping rope and teachers used targeted probes and revoicing to connect students’ understandings to key features of the task, especially quantities that could be important in building a model. For example, Ms. W shared similar images of children jumping rope and asked her third-grade students to share what they noticed about the images. When students commented on the different jumping arrangements, Ms. W called attention to this idea, noting that the student was observing that jump ropes come in various sizes. Student 1: I notice that when you look at the jump ropes [student gets up and frames the images of jump ropes with her hands to emphasize lengths] from here you can see it’s like that – and then when you go like that [student moves to another image with a smaller rope], it’s not that big compared to it, so it’s a little bit bigger in each [image]. Ms. W: So [student name] is telling me that the sizing is different. Student 1: There’s one [size] for like three [students], there’s one for two, there’s one for one. There’s one [jump rope] for certain people, like if you’re small, there might be a small jump rope. So at recess there’s a very long one [rope] that tons of people go on, and there’s also one that only two people do. … one person is holding the handle and the other person [is holding the other side] and they’ll both jump inside the rope together.

Discussions such as this helped students to begin to identify important quantities and assumptions and, thus, support students in preparation for Phase 2.

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11.6.2 Phase 2: Constructing a Model As students continued to make sense of the situation, they began to identify relevant quantities and to discuss information that was known, that they needed to decide, or that they could assume. In this section, we focus on students’ reasoning about two quantities in particular – the length of the jump ropes and the number of each type of jump rope to be included in a jump rope set – as these quantities were central to students’ model-building. 11.6.2.1 Reasoning About the Length of Jump Ropes In all four classrooms, students drew on experiences jumping rope to inform assumptions and decisions about appropriate jump rope lengths for different students and activities. For example, in Ms. S’s third-grade class, students shared instances when ropes hit their legs as they tried to jump, as well as different methods to determine if a jump rope was the right size. In the excerpt below, Ms. S revoiced students’ descriptions of their experiences and then asked focusing questions to encourage students to connect these experiences with key quantities and relationships in the task. In this case, Ms. S aimed to support students’ emerging understandings about relationships between jump rope lengths and height. Student 1: I notice … when you jump rope fast some can hurt you because when you jump really fast you can mess up and it will slap your legs. Ms. S.: Ah it might slap your leg because if it’s too short – you might hit it on your leg. Remember when we did the Jump [Rope] for Heart? Did any of you have trouble  there should not be a line break here jumping over it? Yeah, what would have made it so that we didn’t hit our legs? Student 2: Bigger. Ms. S: If we had a longer jump rope, if we jumped higher. … Student 3: What happens if you have short legs? Ms. S: What happens if you have short legs? You might want what kind of jump rope? Student 3: Small Ms. S: A smaller jump rope or a shorter jump rope. Student 4: To measure it you put it underneath your feet and you hold it up and if it goes to your shoulder on each side that’s perfect. Ms. S: Oh, I didn’t know this, so say that one more time. Student 4: You hold it up and if it goes to your shoulder on each side that’s perfect. … Ms. S: That’s a great use of measurement tool. So everyone would be different in terms of measurement …. let’s stand up for a moment and think about that. … Are we all going to need the same size jump rope? Students: No!

As the lesson continued, Ms. S’s students focused on finding appropriate length jump ropes for each jumper in their classroom. They assumed that every student would want an individual-sized jump rope and built models that included ropes of different lengths to accommodate taller and shorter students in their classroom. While Ms. S provided students with a table that displayed recommended jump rope

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lengths for students of different heights and ages (see Table 11.1), most students based their estimates on a theory proposed by a classmate (Student 4). This student claimed that a jump rope should measure approximately double the distance from one’s feet to one’s shoulders – so that the rope could extend up to the shoulders on both sides of the body. Student 4: To help me because I’m about four something feet and [my jump rope should be] like 8-feet since that’s how I said put that underneath your foot and like all the way up [to your shoulders] (referencing his measuring strategy) like half of eight is four. Ms. S: Okay so I heard you say that you’re four feet from about here to here (referencing shoulders to feet) or here to here (referencing head to feet)? Student 4: From my toes to my shoulder. Ms. S: You’re about 4 feet…How would you use that fact to build your jump rope?... Student 5: He said that like half of him [meaning his height is half of the length of his jump rope] … that sounds right.

Interestingly, many classmates appropriated an adapted, simplified version of this model – “your height is half of your jump rope length” to determine an appropriate jump rope length for each student. This method led to plans that included very long jump ropes (i.e., a 10-foot rope for a 5- foot person), which some students later refined. (More details on models from Ms. S’s classroom are included in subsequent findings sections, Phase 3 and Phase 4.) In Mr. H’s fifth-grade class, students shared similar methods for determining if a given jump rope was the “right size” for a particular individual. For example, one student explained, “At [school name] when we used to check if a rope is big enough for me we would stand up in the very middle of the rope and we would hold it up and if it was right here to your [points to armpits and shoulders].” Most students spoke explicitly about relationships between height and jump rope length –“How long [the jump rope] is depends on their height”  – and constructed models that included jump ropes of various lengths to accommodate jumpers of different heights. Yet interestingly, other students in Mr. H’s class drew on experiences jumping with ropes that were seemingly “too long” to reason that jump ropes do not have to be a specific size for individuals of a certain height. These students argued that longer jump ropes would work for everyone as the extra length can be wrapped around one’s hands to make the rope shorter. In the following excerpt, the students discuss their model-building work with the teacher. Table 11.1  Recommended jump rope lengths for students of various heights and ages Jump rope length 7 feet 8 feet

Good jump rope for Kinder, first and some second graders Second graders, third graders, fourth graders, some fifth graders 9 feet Some fifth graders and adults 10 feet Tall adults 14 feet or longer 2 people swing the rope, double dutch

Height of jumper Up to 4′ 6″ tall 4′ 6″–5′ 5″ tall 5′ 5″–6′ 2″ tall 6′ 2″–6′ 7″ tall

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Mr. H: What did you guys come up with? Student 1: We did 9 foot ropes. Mr. H: For everybody? Student 1: Yeah Mr. H: …So you did 28 [number of students], 9 foot ropes? No variation? No change? Student 1: We did one size fits all. If it’s too long you can wrap around their hand with it. Mr. H: So you assumed that if it is too long you can wrap it around your hand? Student 1: Yes.

We find this interaction notable. Students’ reasoning about key quantities in their models was shaped by their own experiences using jump ropes that were not the “recommended” length. Furthermore, what on one hand seems like an oversimplification of the task (i.e., a one-size-fits-all approach) also reflects a sensibility for flexible solutions that would accommodate a range of children in a PE class. Mr. H revoiced and inserted the word “assumed,” helping students to recognize the role of assumptions, grounded in their experiences, in constructing a model. While most conversations about jump rope length focused on the relationship between the height of the jumper and length of the rope, students also considered how different jump rope activities required ropes of different lengths. As one student in Mr. H’s class noted, “We have to figure out the length of the ropes first. … not everybody is going to do the same [activity]. So, we have to figure out different kinds of ropes.” We elaborate these ideas in the next section. 11.6.2.2 Reasoning About What Constitutes a Set of Jump Ropes (How Many and What Kinds) As students continued to construct their models, they considered the number of jump ropes needed for a jump rope set and how many of those ropes should be individual length ropes (for individual jumping) versus group length ropes (for group jumping activities such as double dutch). In the two fifth-grade classes, students constructed models for jump rope sets that included ropes for different jumping activities; we focus on those lessons in this section. (In the two third-grade classes, all students pursued models based on one rope per individual student.) In Mr. H’s classroom, students drew on their experiences to assume that a jump rope set needs to include ropes that allow students to engage in jumping activities together. They also decided that while individual ropes are important, and need to account for the height variance among students in the class, a jump rope set does not need to include individual jump ropes for each student as some students will jump rope individually while others jump together with longer ropes. In the excerpt below, several students explain their group’s emerging plan to Mr. H. Student 1: So, we have best of both worlds. Mr. H: So, tell me best of both worlds … Student 2: We came up with the idea that we could split people into different categories based on their height because some people are different sizes, and some are like almost the same size. Student 3: Yeah, estimate their sizes. …

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Mr. H: How many 8-foot ropes did you have? Student 2: Six Mr. H: How many 9-foot ropes did you have? Student 2: Five Mr. H: How many 10-foot ropes did you have? Student 2: Three. One 14-foot rope Mr. H: Let me actually ask you a question. Are you guys assuming then …. some people will be doing individual jump rope practice [and] other people would be doing at the same time group practice? Student 2: Yeah, Lot of people can play with one double-dutch [14-foot] rope. So we wouldn’t need that much [individual ropes]. Mr. H: So a classroom set for you then doesn’t mean that everybody would have to be doing it individually at the same time. It just means everybody is jump roping. … Student 2: We felt like one long rope holds a lot of people.

In Mr. H’s classroom, students drew on their experiences with different jump rope activities and understandings about the relationships between jump rope length and height as they determined key elements of their model – numbers of jump ropes, of specific lengths, for specific activities. We found a similar pattern in Ms. E’s classroom. In fact, Ms. E explicitly encouraged students to draw on their own understandings of jump rope sets as they planned a set for a third-grade class at another district school where jump ropes were not readily available. Ms. E: You have this great advantage of having this experience [in our PE class], you know that we share jump ropes. You know that [the PE teacher] doesn’t have a [rope] for every single person. … So using what you know, I want you to make sure that when you’re making a set for a real class, a real third grade class- it’s not pretend.

In the excerpt below, Ms. E’s students discuss their understandings of jumping rope with children of different ages to propose a set of jump ropes of varied lengths. They consider how the set might be usable not just by the third-grade class but also by other students at the school. Student 1: We’re thinking of doing the 7-foot, the 8-foot, and the 14-foot or longer. Student 2: Yeah because if we did the 9-feet, only some people can use it and that wouldn’t be fair. So we don’t know what grades there are in their school. Student 3: Yeah, but the 9-feet would be better though. Student 2: I know, but what if there are little kids and they are trying to learn and stuff and they trip over the jump rope.

Ultimately, this group decided on a set of eight jump ropes that included each of the lengths discussed above. Figure 11.4 displays a portion of the group’s final poster.

Fig. 11.4  Ms. E’s students list the jump ropes included in their model

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Reflecting on the lesson, Ms. E noted that the salience of jumping rope in her school community and the multiple experiences with jump ropes that students brought to the task impacted the models that students generated in significant ways. Ms. E: The community here at [our school] has a lot of experience with jump ropes. … Everybody does it. It’s not just for kinders, it’s not just for the older kids, Not just for girls or boys.... I think they were able to bring all of their prior knowledge for that and just throw it into the task and I think that really helped them determine like what a class set looks like.… So they were able to say, ‘OK well there’s not a jump rope for every single student here. We share.…[So] we can make a few and just have a plan for them to share it because we know you can use a 14-foot rope and like four people can jump.’

In summary, not only did this task provide opportunities for students to leverage their knowledge and experiences, students’ experiences determined the direction of the discussion and supported various groups in the class in constructing a model. Students used their experience with individual jump ropes to reason about the length of ropes needed for various students (depending on their heights) and for group jumping activities, and in the two fifth-grade classes, students also reasoned about the combination of lengths of ropes that would be appropriate for a class set.

11.6.3 Phase 3: Operating on a Model As students determined relevant quantities, they began to relate and operate on those quantities to calculate the number of plastic bags needed for a jump rope set for their intended audience. In this section, we discuss several examples of how students operated on the models they constructed in Phase 2. In many cases, just as students first constructed models for a single rope, their initial operations focused on calculating the number of bags needed to make one rope of a given length. In Ms. S’s third-grade class, many students created visual models to represent the three plastic bags needed for each foot of jump rope (the specific rate provided in the instructional video from the Girl Scouts). They then used repeated addition, skip counting, and understandings about multiplication to find the total number of bags needed for one rope. Figure 11.5 displays two different ways that students represented the relationship between key quantities in their model (i.e., the length of a jump rope in feet and the unit rate of three plastic bags per foot) and ways that they operated on those quantities to find the number of bags needed. In other instances, students used actual jump ropes, and mathematical tools such as rulers, to represent the relationships between quantities and to support their calculations. For example, several students in Ms. W’s third-grade class stretched out manufactured jump ropes on the floor, and then lined up one-foot rulers alongside the jump rope using each ruler to represent the three bags needed to make one foot of jump rope (see Fig. 11.6).

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Fig. 11.5  Ms. S’s students calculate the number of bags needed to make a single jump rope

Fig. 11.6  Ms. W’s students measure jump ropes with rulers that represent three bags/foot

The student explained that each ruler represented the three bags needed to make one foot of jump rope and that he iterated the ruler along the length of the rope to figure out how many bags were needed in all. Student 1: It’s 27 bags! Ms. W: How did you figure out it’s 27 bags? Student 1: So each ruler is 3 bags and I was using two rulers and I was doing it over and over again. … I was doing this, and this [sets rulers down, along the length of the jump rope], and I kept on doing this until I got to the end. It was nine times. Ms. W: and then what did you multiply it by? Student 1: Three times each ruler. Times nine times [there were 9 rulers] Ms. W: And you had – so you had 27 bags for this sized jump rope? Student 1: Yeah … I did nine times three and I got 27.

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We found that the third-grade students in particular drew upon physical tools and visual representations to illustrate the relationships between key quantities in their models – the length of the jump rope and the number of bags needed per foot – and then to operate on those quantities to reach a solution. This approach connected repeated addition (through the iterative measuring) with multiplication (i.e., “three times each ruler”), an important connection for students in understanding multiplicative relationships. These representations not only supported students’ multiplicative thinking, but they also evidenced the various ways that students leveraged prior mathematical understandings as they operated on the models they constructed. As students moved beyond calculations related to single jump ropes, to calculate the number of bags needed for a group of jump ropes of a given length, or for the entire jump rope set, they represented the relationships among quantities in their models in various forms, including equations, written descriptions, and tables. For example, some students grouped classmates by height and used these grouping to calculate the number of ropes of a given length they needed. Next, they used the rate of three bags per foot of jump rope to determine the number of bags needed for one rope of each length. Finally, they multiplied the number of bags per rope times the number of ropes of that length and then added the number of bags needed for each type of rope to find the total number of bags for the jump rope set. The student poster in Fig. 11.7 illustrates how students in Mr. H’s classroom operated on their model in this way. As this group explained their model to the class, they highlighted the ropes of different lengths and how they operated on those quantities to find the total number of bags needed. Student 1: We started with 27 kids Mr. H: 27 kids and then what? Student 1: We said, we have eight 7-foot ropes, Teacher: Eight 7-foot ropes, ok? Student 1: Thirteen 8-foot ropes, and then -….six 9-foot ropes

Fig. 11.7  Mr. H’s students operate on a model based on grouping students by height

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Teacher: So that’s 27 (referring to the total number of ropes so far), Student1: And one 14-foot long rope Teacher: And one for just a big jumping situation? Student1: Yeah … Teacher: And how did you come up with that [the total number of bags]? Student1: …we thought of ideas and we combined all our ideas, so we thought if one foot is 3 bags, we did 3 times 7 is 21 [bags for one 7-foot rope], 21 times 8 is 168 bags [for eight 7-foot ropes], and 13 times 24 cause 3 times 8 is 24 [bags for one 8-foot rope], that is 8-feet, we got 312 bags [for 13, 8-foot ropes], and 3 times 9 we got 27 [bags for one 9-foot rope], and [27] times 6 is 162 bags [for six 9-foot ropes]. … And we have one 14 feet long Student 2: 14 times 3 Student 1: is 42 bags [for a 14-foot rope], we added all that together which is 684 [bags].

In summary, as students operated on quantities in their models to calculate the number of plastic bags needed for their jump rope set, they drew on a range of understandings, tools, and representations to support their work. They leveraged representations of multiplication, such as skip counting and repeated addition, to relate the number of bags per foot of jump rope to the number of bags for the length of the rope. We also found that physical tools, such as jump ropes and one-foot rulers, supported students in understanding and operating on these relationships.

11.6.4 Phase 4: Interpret/Analyze Solutions and Refine Model After students operated on their models to determine the number of bags needed to make the jump ropes in their set, they analyzed their solutions, and in some instances, students refined their solutions. In this section, we discuss several examples of how students’ interpretations and analysis prompted refinements, with attention to how students’ experiences informed this phase of their modeling activity. We found that just as students drew upon physical artifacts and tools to support their modeling activity in Phases 2 and 3, physical artifacts were important to students’ model refinement activity. For example, as students in Ms. S’s classroom shared initial models for their jump ropes, other students realized that their proposed jump ropes were not an appropriate length. A few students had proposed jump ropes that were the same length as students’ heights (i.e., a 5-foot rope for a 5-foot tall person), without considering how such a rope might work. In the following exchange, one student (Student 1) proposed a 4-foot rope because he measured about 4 feet from his feet to his shoulders. Ms. S prompted him to consider how this rope would be used, using a measuring tape to represent a jump rope. Collectively, students reasoned that a 4-foot rope would only fit on “one side” of his body, and thus his model for jump rope lengths (length equals the distance from one’s feet to one’s shoulder) needed refinement. Ms. S: I have a question about your model. So … you’re four feet [tall] right? … When you jump rope do you jump rope on one side of your body or both sides of your body? Student 1: Both?

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Ms. S: So your jump rope is going to be four feet long and I heard you say – that you were thinking about this measurement right from here to here (Ms. S references a measuring tape she has placed under her feet and stretched to her shoulder). So if your jump rope is this long, will it fit over both sides of your body or just one? Student 1: Oh, one. Ms. S (to class): What do you guys think? Student 2: It has to go on both sides Ms. S: So how might you adjust or adapt your model or revise your model? Student 1: Four times four Ms. S: Four times four? … Would that be a good revision to your model? How many sides of you do you have? Student 1: Two Ms. S: Two.… So, what might he do to revise his model? Which means -what might he do to change his model because right now his model is including only one side of him.

In the class discussion that followed, students proposed various suggestions, including doubling the distance from one’s feet to one’s shoulders to find an appropriate jump rope length and a further refinement, doubling the feet-shoulder distance and then adding “a few inches” to account for the length of jump rope that passes under one’s feet while jumping. In the excerpt below, Student 2 interprets Student 1’s proposed plan and argues that a model needs to account for “both sides” of one’s body and also the length on “the bottom.” Student 2: Well,… you said use four feet, so then – it goes under your foot like when you’re about to start jump roping when it goes under your foot. Does he count where you, like this (touching to show space under his feet)? Ms. S: Good question Student 2: It’s going to have that extra to go all the way through. Ms. S: That’s a good point so you’re not just accounting for just the side of your body.… You also have what [other] part of your body? Student 2: The bottom Ms. S: The bottom.… So you have to add maybe a few inches for the bottom part Student 2: I did. That’s why I put nine feet [instead of 8 feet]. Student 1: Well what if I just like bend my knees to jump? Ms. S: Oh so maybe … when you jump you bend your knees, so then do you have to account for that?

Interestingly, while Student 1 accepted his classmates’ suggestion to double the feet-shoulder distance so that the jump rope would be long enough to go along both sides of his body, he drew upon his experiences jumping (i.e., what if I just like bend my knees to jump?) to reject Student 2’s suggestion to add additional inches to the length. Early in the modeling process, students drew on their experiences to construct a model. In these examples, we found that students again leveraged their experiences as they interpreted and refined their models by considering more nuanced aspects (e.g., how the jump action affects the length of rope needed). In other instances, students’ interpretations and analysis of their emerging models led to revisions in how they defined an appropriate set of jump ropes for their target audience. For example, one group of students in Ms. E’s class initially proposed a jump rope set that included only 14-foot ropes. Ms. E encouraged students to draw on their own experiences jumping rope to consider whether the proposed set would satisfy all students’ needs and interests.

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Ms. E: What if [physical education teacher] only got 14-foot jump ropes? What would that look like? Student 1: They would all be too long. Not everyone wants to hold the handle and take turns. Ms. E: You could jump in a group and take turns. What’s another way you could jump? …Could you jump by yourself? Student 2: Yeah Ms. E: Could you jump by yourself with a 14-foot rope? Student 2: No Ms. E: So what is a way that we can kind of fix and make adjustments to your class set so that it’s a little bit more doable, where people have a choice in what rope they want to use? Student 1: Make less 14-foot jump ropes Ms. E: So if we’re going to make less 14-foot jump ropes, what about the other ones? Student 2: Maybe we could do all the different lengths of jump ropes in different amounts.

This example shows how students’ experiences in play (e.g., taking turns, playing individually or in a group) influenced how they interpreted and refined their solution. These students went beyond considering rope length to decide what combination of rope lengths would be best for the way they wanted to play with the ropes and with their classmates. Students in Mr. H’s classroom engaged in a similar process of interpretation and refinement related to the specific jump ropes included in their set. For example, one group proposed a jump rope set with only one 14-foot rope, but when prompted by a classmate (Student 1), they considered whether that configuration would allow double-dutch style jumping. Student 1: (responding to the initial model shared by peers) For double-dutch you have to have two [jump ropes]. Mr. H: Oh, for double-dutch you have to have two? (to group of students who shared their model) Did you mean actual double-dutch or big jumping? [cross talk between the students who presented] Student 2: [We meant] double dutch Student 3: I just want to jump rope Mr. H: If actual double-dutch was a requirement of our class, would that change our outcome? What would we need [to change]? … So if you are doing double-dutch what do we have to do for the bags for the 14 foot [rope] if we are doing double-dutch? Student 2: We have to double it [double the number of bags] Mr. H: We could double it to get a double dutch rope [to have enough for two ropes].

This example again illustrates how students’ prior experience and understandings related to jumping rope informed their analysis and interpretation of their own solutions (and those of peers) and their suggested refinements. Our findings for Phase 4 focused on how students analyzed, interpreted, and refined quantities (e.g., jump rope length, the number of jump ropes in a set) and relationship among quantities (i.e., between height and rope length). In other words, we provided examples of students analyzing and refining their ideas from Phase 2 and discussed how these refinements were informed by the knowledge and experiences students brought to the task. Students also interpreted and revised their work from Phase 3, with a focus on correcting calculations, operations on units, and so

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forth. While revising calculations is important, these refinements mirrored the kind of work that we have long encouraged students to do as part of looking back in problem-solving. For this reason, we did not report on these refinements in this section and instead focused on how students interpreted and refined ideas related to the models they constructed, as such refinements are critical to the mathematical modeling process and less prevalent in other problem-solving experiences.

11.6.5 Phase 5: Generalizing the Model to Other Contexts Across the four classes, we found that students remained focused on planning a set of jump ropes for a specific context in their classroom or school. We suspect that the limited attention to generalizing models so that they would be sharable or reusable in other related situations was in part a reflection of time constraints; by the time students determined specific solutions for their class or school context, teachers had already devoted two to three class periods to the task. Moreover, developing a generalized, reusable model is complex as it requires reasoning beyond a specific situation to consider how the quantities and relationships involved might vary or change in other situations. That said, in one of the fifth-grade classrooms, students did move toward a more generalizable model for a jump rope set, in part because of how their teacher (Ms. E) framed the task. We focus on examples from Ms. E’s lesson in this section. In a step toward generalization, Ms. E first helped students shift from considering themselves (in the warm-up task, they figured out how many bags they would need to make one jump rope, for themselves) to a broader context – designing a set of jump ropes for a third-grade class at another school [referred to as Mr. O’s class in student’s work]. Ms. E encouraged students to consider what aspects of their initial model would need refinement or reconsideration, given the change in context. She mentioned that since they were designing a set of jump ropes for a different class at a different school, they would need to reconsider some of their initial assumptions. Ms. E: What are some things we need to reconsider? Student 1: That they’re third graders. Student 2: How tall they are. Student 3: How many students are in his class Student 4: How long the jump ropes are going to be.

The chart below (Fig. 11.8) illustrates one group’s record of known information that would support them in the task, as well as the information that they would need to know or assume as they developed a model for Mr. O’s classroom. During the whole group discussion that followed, students concurred that they needed to know how many students were in the third-grade classroom at the other school [Mr. O’s class]. They used information about the number of students in each third-grade class at their school, treating those classes as typical examples, to make

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Fig. 11.8  Small group recording of information that is known and information that is needed

an assumption about the number of third-grade students in Mr. O’s class. Based on the information gathered, some students assumed that Mr. O had 26 students (as explained in the work sample that follows), while others assumed a class size of “about 30 students.” One student explained their process, in writing, in the following way: How do we know that there will be enough jump ropes without a lot of extras is because we went and asked each third grade class what the number amount of students there are and the amounts were 25, 26, 17 and the class that had 17 was a third and fourth grade [combo class] so that class didn’t count as a third grade class so it was between 26 and 25 so we went with the higher amount of students which was 26. Not each kid will have to have their own jump rope because the 14-foot jump ropes can be used by 3 people. (written explanation on student poster)

As she reflected on the lesson, Ms. E highlighted students’ movement toward a model that was usable in other situations. She emphasized the ways that students used understandings that they brought to the task to help them reason about quantities that were unknown. Ms. E: [My students] went around and they checked out third grade classes here and they talked about what the average was. … Because they knew they had to make an assumption about the class size because there was at this point in the year there was no way to get that information [from the other school]. … So they really have learned that process of identifying what is actually given to you, what they need to figure out mathematically, and what are some assumptions or decisions they have to make to proceed with the task instead of getting stuck and stopping if they don’t have enough information.

Ms. E’s reflection points to the ways that students drew upon their experiences and understandings to inform their engagement in each phase of the modeling process, including their movement toward Phase 5.

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Our findings suggest that engaging in the final phase of mathematical modeling that leads to a shareable and reusable model was a challenge for most classrooms. However, Ms. E’s fifth-grade students evidenced some traction toward generalization because they made specific assumptions and identified key variables (class size; children’s heights) needed for a similarly situated context. Ms. E’s reframing of the task to plan a set of jump ropes for a different group of students and another school pushed the fifth graders to determine the importance and relevance of each component of their model. Although the set of jump ropes they created were to be used by a specific third-grade class at another school, one could argue that the set would be used by any third-grade class at any school, and in this way they moved toward a generalizable model.

11.7 Discussion This study focused on how students drew on their experiences and prior knowledge as they engaged in the various phases of the mathematical modeling process. Via an analysis of student engagement in a common modeling task – across teachers and across-modeling phases  – our findings contribute important insights related how students’ experiences inform their modeling activity. Following this discussion, we conclude with implications for teachers and teacher educators and questions for future research. Across the four lessons, we found that students’ experiences and knowledge related to jump ropes and jump rope activities supported their engagement in each phase of the modeling process. Our findings illustrated different ways students leveraged their experiences to identify important quantities and relationships, to make assumptions, and to analyze and interpret the reasonableness of their solutions. We also documented how students drew on mathematical understandings, strategies, and tools to represent relationships and to operate on quantities in their models and how they leveraged their experiences to revise their models when needed. We found connections to students’ experiences to be so salient across the various phases of the modeling process that students often privileged their own perspectives over information from other sources (e.g., their teacher, peers). We are encouraged by the sense-making evidenced in students’ stances. Students did not accept data without question; they considered the reasonableness of the information against their own experiences and accepted or refuted the suggestions accordingly (i.e., one student rejected the suggestion to add additional length to his rope because he reasoned that he would bend his knees while jumping). Students’ reliance on their own sense-making highlights the potential of relevant mathematical modeling lessons to support students’ empowerment as mathematical learners, as well as their reasoning about real-world situations (Bahmaei, 2011). Yet, students also benefitted from opportunities to weigh their own perspectives against information from other sources (such as the chart with recommended jump rope lengths). This underscores the importance of opportunities to test and discuss conjectures in mathematical

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modeling tasks (as part of Phases 4 and 5), as this allows students to confirm or confront ideas that may still be in development. Perhaps the salience of student experience reflected the relevance of the Upcycling Jump Rope task. As other scholars have noted, a well-chosen context supports students in developing “informal, highly context-specific models and solving strategies” (Doorman & Gravemeijer, 2009). Yet we also suspect that teachers’ frequent invitations for students to share their experiences played a key role. All four teachers elicited students’ insights related to the task and explicitly marked moments when students were contributing new and important ideas (e.g., “Oh, I didn’t know this, so say that one more time.”) Consistent with other research that has noted the positive impact of assigning competence to students’ ideas, we found that students responded to teachers’ invitations with interest. These discussions not only provided opportunities for students to share their experiences but also considered and responded to the perspectives of others (Ladson-Billings, 2009; Turner et al., 2008). A detailed analysis of how specific teacher moves support connections to students’ experiences during modeling lessons would be a productive focus for future research. Our findings also highlighted how sharing and discussing solutions helped students to interpret and analyze their own models and to identify areas in need of refinement (Phase 4). In some cases, as students shared solutions, they identified mistakes in calculations or limitations of assumptions (e.g., revising the assumption that the rope should be the same length as a student’s height). In other cases, when students listened to the solutions of other groups, they recognized revisions or refinements needed in their own models (e.g., realizing that two ropes are needed for double-dutch jumping). Thus, sharing solutions was an important part of the modeling process not just a final reporting out of a product. In fact, the peer interactions and group discussion that occurred frequently in Phase 4 suggested that not only did students draw on their own experiences to inform model analysis and revision, but they also considered and learned from the perspectives of peers. Furthermore, the teacher questioning during this sharing also played a key role in supporting this analysis and refinement of models. While students in all four classrooms leveraged knowledge and experiences from across multiple contexts as they engaged in mathematical modeling, we noted interesting contrasts between the grade levels. For example, in the two third-grade lessons, students focused on their own classrooms as they planned a jump rope set and generated straightforward models that included one individual length rope per person. We wondered if this focus reflected the complexity of concepts and operations involved in the task (multiplicative relationships, rates, changing units). Indeed, for these young students, calculating the number of bags needed for one rope per person involved significant work. In contrast, in the two fifth-grade lessons, students generated more complex models that included both considerations about height and jump rope activities. Although students’ models varied in complexity across grade levels, they shared a common grounding in students’ experiences. Generalizing is essential to the mathematical modeling process, but in the reality of classroom implementation, it has remained challenging, in part due to time constraints. In the four lessons included in this analysis, students spent between two and three class periods moving across Phases 1 through 4 of the modeling process.

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Thus, students were left with minimal time to explore how their models could be reusable to others in similar situations. Only one classroom evidenced movement toward a generalizable model. In this case, the teacher’s reframing of the task to focus on planning a set of jump ropes for students at a different school was pivotal, as it implicitly suggested that the model could be generalized. Future research should focus on ways to facilitate generalization in elementary mathematical modeling, both through task development and lesson implementation.

11.8 Implications Our findings have important implications for mathematics modeling instruction. First, given the salience of children’s knowledge and experiences across all phases of the modeling process, teachers should explicitly elicit students’ experiences and perspectives and position these experiences as resources to support meaningful engagement in mathematical modeling. Another important consideration for teachers is selecting mathematical modeling tasks that connect to relevant contexts in their schools and local communities and that connect to the knowledge and experiences that their students bring to the classroom. Second, our findings highlight pedagogical decisions in mathematical modeling lessons that demand further investigation. For example, while students readily shared experiences related to the broader task context (plastic consumption, recycling, and pollution), teachers had to determine when to encourage this sharing and when to redirect the conversation to key features of the specific modeling task. In some classes, discussions about the environmental impact of plastic were brief and provided a clear entry point into the modeling process. In other instances, students engaged in lengthy discussions of many aspects of plastic bag use and consumption, most of which were unrelated to the Jump Rope task. Thus, when enacting modeling tasks that connect to authentic, meaningful contexts in students’ lives and communities, teachers will need to negotiate a balance between capturing students’ wide range of interests during broad discussions of the topic and focusing attention on mathematics learning and engaging in the modeling process. A third implication is related to more effective engagement with the final phase of the modeling process – generalizing. While generalization of models remained elusive for three of the four teachers, emerging evidence suggests that some teachers can extend tasks to facilitate shareable and reusable models for similar situations, but class time and teacher commitment are needed to support this possibility.

11.9 Conclusion In conclusion, meaningful contexts that connect to authentic experiences in students’ homes, schools, and communities matter in the mathematical modeling process. Across four different lessons of the Upcycling Jump Rope task, elementary

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students connected to their knowledge and experiences with plastic bags, recycling, and jump rope activities to make sense of the modeling task. Student knowledge and experiences informed their assumptions, decisions, and mathematical operations used to formulate their models, as well as the analysis and refinement of their models. This, in turn, enabled teachers to recognize students’ knowledge and experiences as resources for mathematics learning.

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Doorman, L., & Gravemeijer, K. (2009). Emergent modeling: Discrete graphs to support the understanding of change and velocity. ZDM Mathematics Education, 41, 199–211. English, L. (2010). Young children’s early modeling with data. Mathematics Education Research Journal, 22(2), 24–47. English, L., & Watters, J. (2004). Mathematical modeling in the early school years. Mathematics Education Research Journal, 16(3), 59–80. English, L. D. (2006). Mathematical modeling in the primary school: Children’s construction of a consumer guide. Educational Studies in Mathematics, 63(3), 303–323. English, L. D., Fox, J. L., & Watters, J. J. (2005). Problem posing and solving with mathematical modeling. Teaching Children Mathematics, 12(3), 156–163. English, L.  D., & Watters, J.  J. (2005). Mathematical modelling in the early school years. Mathematics Education Research Journal, 16(3), 58–79. Felton, M., Anhalt, C., & Cortez, R. (2015). Going with the flow: Challenging students to make assumptions. Mathematics Teaching in the Middle School, 20(6), 342–349. Gainsburg, J. (2006). The mathematical modeling of structural engineers. Mathematical Thinking and Learning, 8, 3–36. Garfunkel, S. A., Montgomery, M. (Eds.) (2016) GAIMME: Guides for instruction and assessment in mathematical modeling education. Bedford, MA/Philadelphia, PA: Consortium for Mathematics and its Application [COMAP, Inc.]/Society for Industrial and Applied Mathematics [SIAM]. Available online: http://www.siam.org/reports/gaimme-­full_color_for_ online_viewing.pdf. Accessed 15 Oct 2017. González, N., Andrade, R., Civil, M., & Moll, L. (2001). Bridging funds of distributed knowledge: Creating zones of practices in mathematics. Journal of Education for Students Placed at Risk, 6(1&2), 115–132. Greer, B. (1997). Modeling reality in mathematic classrooms: The case of word problems. Learning and Instruction, 7(4), 389–397. Ladson-Billings, G. (2009). The Dreamkeepers: Successful teachers of African American children (2nd ed.). San Francisco, CA: Jossey-Bass Publishers. Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem-solving. In R.  Lesh & H.  Doerr (Eds.), Beyond ­constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3–33). Mahwah, NJ: Earlbaum. Lesh, R., & Fennewald, T. (2010). Introduction to part I modeling: What is it? Why do it? In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 5–10). Boston, MA: Springer. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-­ revealing activities for students and teachers. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 591–645). Mahwah, NJ: Lawrence Erlbaum Associates. Lipka, J., Hogan, M., Webster, J. P., Yanez, E., Adams, B., Clark, S., & Lacy, D. (2005). Math in a cultural context: Two case studies of a successful culturally based math program. Anthropology and Education Quarterly, 36(4), 367–385. Maaß, K. (2006). What are modelling competencies? ZDM, 38(2), 113–142. Miles, M., Huberman, A. M., & Saldaña, J. (2014). Qualitative data analysis: A methods sourcebook (3rd ed.). Thousand Oaks, CA: Sage. P21 Partnership for 21st Century Learning. (2017). Available online: http://www.p21.org. Accessed 5 Jan 2019. Palm, T. (2008). Impact of authenticity on sense making in word problem solving. Educational Studies in Mathematics, 67(1), 37–58. Sembiring, R., Hadi, S., & Dolk, M. (2008). Reforming mathematics learning in Indonesian classrooms through RME. ZDM Mathematics Education, 40, 927–939. Stake, R. (1995/2013). The art of case study research. Thousand Oaks, CA: Sage. Stake, R. (2006). Multiple case study analysis. New York, NY: Guilford Press.

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Suh, J., Matson, K., & Seshaiyer, P. (2017). Engaging elementary students in the creative process of mathematizing their world through mathematical modeling. Education Sciences, 7(62). Turner, E., Celedón-Pattichis, S., & Marshall, M. E. (2008). Cultural and linguistic resources to promote problem solving and mathematical discourse among Hispanic kindergarten students. In Promoting high participation and success in mathematics by Hispanic students: Examining opportunities and probing promising practices (vol. 1, pp. 19–42). Turner, E. E., Gutiérrez, M. V., Simic-Muller, K., & Díez-Palomar, J. (2009). “Everything is math in the whole world”: Integrating critical and community knowledge in authentic mathematical investigations with elementary Latina/o students. Mathematical Thinking and Learning, 11(3), 136–157. Verschaffel, L. & De Corte, E. (1997). Teaching realistic mathematical modeling in the elementary school: A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28(5), 577–601. Verschaffel, L., De Corte, E., & Borghart, I. (1997). Preservice teachers’ conceptions and beliefs about the role of real-world knowledge in mathematical modeling of school word problems. Learning and Instruction, 7(4), 339–359.

Chapter 12

A Window into Mathematical Modeling in Kindergarten Robyn Stankiewicz-Van Der Zanden, Stacy Brown, and Rachel Levy

12.1 Introduction Over the past two decades as countries have added mathematical modeling to their education standards, much remains unknown about how mathematical modeling might unfold in elementary classrooms (Stohlmann & Albarracín, 2016). In the United States, the Common Core State Standards for Mathematics (CCSS-­ M)1 were introduced in 2010. These standards not only included mathematical content standards but also eight standards for mathematical practice: “ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years” (CCSS-M, p.  8). Among these practice standards was, for the first time, a standard on mathematical modeling, MP4 Model with mathematics (Fig. 12.1): This standard, MP4, discusses students’ ability to apply mathematics to problems arising in everyday life, in society, and in the workplace when making “assumptions and approximations to simplify a complicated situation” as well as when students (1) “identify important quantities in a practical situation and map their relationships” using appropriate tools and (2) “analyze those relationships mathematically to draw conclusions.”

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MP4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Fig. 12.1  US Common Core Mathematical Practice 4 (MP4)

Speaking at the level of specific grade bands, however, the CCSS-M authors suggest that in the early grades, “this practice, model with mathematics, might be as simple as writing an addition equation to describe a situation.” This raises the question, “In which ways might all of the mathematical modeling activities, as described in MP4, be enacted in elementary classrooms?” Is it as simple as the CCSS-M documents suggest? Turning to the literature, one finds that research on mathematical modeling in the elementary grades has increased over the past two decades as countries’ standards for mathematics began to include explicit statements about mathematical modeling practices and both national and international assessments placed more emphasis on mathematical modeling. However, in a review of what is known about elementary grades mathematical modeling, Stohlmann and Albarracin (2016) note that despite increased attention to mathematical modeling in schools, “more research is needed at the elementary level.” Specifically, Stohlmann and Albarracin reviewed 29 articles and book chapters on elementary grade mathematical modeling and found that between 1991 and 2015 only 4 of the studies addressed “considerations for the teachers” and involved teachers who had received professional development on implementing mathematical modeling (English & Watters, 2005; English, 2007, 2009; Watters, English & Mahoney, 2004). These studies, which involved grade 3 and grade 5–7 teachers and students, explored students’ engagement in model-­eliciting activities (MEAs). This work is important to understanding the potential for young children to engage in elementary mathematical modeling. Indeed, as teachers of our youngest students face specific challenges, research is needed to better understand the ways in which kindergarten children might engage in mathematical modeling activities. In addition to these four studies of mathematical modeling in the classroom, a body of recent research examines elementary students’ representational and conceptual competence as indicated by students’ responses to mathematical modeling tasks, the types of models students create when engaging in modeling tasks, and students’ beliefs and persistence before and after engaging with mathematical

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modeling tasks. This body of research advances knowledge of elementary mathematical modeling; however, it alone provides little in terms of support for teachers who seek to understand the ways in which teachers and students might co-construct modeling activities. Work situated in classrooms and focused on the co-construction of modeling activities is critical to advancing the practice of mathematical modeling, since rather than point to what children cannot do, it sheds light on the types of learning environments that foster students’ modeling capacities. Furthermore, our review of the research on elementary mathematical modeling indicates that a great deal of existing research focuses on the upper elementary grade students, that is, grades 3, 4, and 5 (for some examples, see Cyrino & Oliveira, 2011; Peter-Koop, 2004; Petrosino, Lehrer, & Schauble, 2003; Verschaffel & De Corte, 1997; Verschaffel, De Corte, & Lasure, 1994). Hence, little is known about young children’s engagement in modeling activities. Indeed, in our review of the literature, we found only two studies (English, 2012; Lesh, English, Sevis, & Riggs, 2013) that focused on grade 1 students’ engagement in model-eliciting activities. Thus, there is a need for research that addresses the question, “How might our youngest school-age children engage in mathematical modeling?” To answer this question, we formed a collaborative team involving an experienced kindergarten teacher, who teaches in a high-need school within a high-need school district (as indicated by the number of children participating in the free and reduced-fee breakfast and lunch program); a mathematics education researcher, who studies teaching and learning in K-6 classrooms; and an applied mathematician with a degree in education, who has taught mathematical modeling in grades 5–16. Together we endeavored to understand the work teachers engage in when planning, facilitating, and revising mathematical modeling tasks in a kindergarten classroom and the mathematics children might bring to bear on such tasks. We believe this work provides an important glimpse at one of the many ways in which mathematical modeling might develop from the ground up, that is, from students’ earliest formal experiences in classrooms. Moreover, we believe that such work should be conducted through collaborative efforts with teachers. As is argued in NCTM position statement on “Linking Mathematics Education Research to Practice”: Researchers and school personnel must work together to address and examine issues pertinent to the teaching and learning of mathematics. Together they need to consider ways to implement strategies across contexts that build on findings from research. Collaboration between researchers and school personnel provides integrated perspectives for addressing critical issues in mathematics teaching and learning.2

Indeed, teaching mathematics in the elementary classroom is a complex endeavor. Working as a collaborative team, we aimed to form a community in which we could bring our collective knowledge to bear on the question of how children might engage in the practice of mathematical modeling in the kindergarten classroom and to do so in ways that speak to the needs of practitioners. To meet these aims, we present three classroom vignettes drawn from a semester-­ long case study of mathematical modeling in a kindergarten classroom. In each  Retrieved from https://www.nctm.org/uploadedFiles/Standards_and_Positions/Position_ Statements/Linking%20Research%20and%20Practice.pdf, November 1, 2018 2

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vignette, the kindergarten teacher (the first author) shares (1) her anticipations for the lesson; (2) her efforts to facilitate children’s engagement in the practice of mathematical modeling, with transcripts of the classroom discussion; and (3) her reflections on the lesson having completed the mathematical modeling activity. We then collectively analyze and discuss the children’s mathematical reasoning elicited, as well as the question of why each activity constitutes an instance of mathematical modeling in the kindergarten classroom. We conclude by discussing the work entailed with enacting mathematical modeling in the kindergarten classroom, when mathematical modeling tasks are co-constructed by teachers and students.

12.2 The IMMERSION Project The IMMERSION project (NSF-1441024) was a 4-year study of how targeted professional development with follow-up lesson study might influence teachers’ implementation of mathematical modeling in K-8. The research took place through school district/higher education collaborations in Virginia, Montana, and California and explored mathematical modeling across a wide range of educational settings.

12.2.1 Our Perspective on Modeling The perspective of mathematical modeling that informed the IMMERSION project was founded on modeling principles similar to those described in the GAIMME report,3 which discusses ways students can engage in context-rich real-world problems that are open at the beginning, middle, and end. Specifically, the GAIMME report defines mathematical modeling as “a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena”. In our project, the kindergarten teacher was seeking to develop students’ capacity to engage in mathematical modeling as a process by engaging the students in a sequence of activities: whole class discussions about the mathematics in their literature (and in the world around them); the analysis of the situations arising in the stories and the articulation of their assumptions and constraints on the quantities involved; and individual-level work during which students could select relevant constraints and pose and represent solutions using age-appropriate mathematical tools, such as pictures, words, manipulatives, and symbols. Our decision to draw on children’s literature as a context for mathematical modeling with very young children aligns with the position taken by Lesh et al. (2013), who argue children’s “real-world” experiences differ from adults and that these differences should be taken into consideration when designing modeling activities. Like Lesh et  al., we posit that children’s stories provide appropriate contexts for   The GAIMME report is available at https://www.siam.org/Publications/Reports/Detail/ Guidelines-for-Assessment-and-Instruction-in-Mathematical-Modeling-Education 3

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young children’s modeling because these stories provide contexts for “simulations of situations in which some important type of mathematical thinking is useful beyond school.” In particular, when selecting stories for children’s modeling activities, we applied criteria similar to those articulated by Lesh et al. (2013): (a) do the children try to make sense of the problem using their own “real life” experiences—instead of simply trying to do what they believe some authority (e.g., their teacher) considers to be correct (even if it doesn’t make sense to the children)? and (b) when the children are aware of several different ways of thinking about a given problem, are they themselves able to assess the strengths and weaknesses of these alternatives— without needing to ask their teacher or some other authority? (Lesh et al., 2013, p. 240)

Specifically, in our work we did not sidestep the point raised by Manouchehri and Lewis (2017); namely, that “greater attention may need to be devoted to unpacking what might constitute as real” in educational settings. Instead, in our case study the teacher purposely attended to this issue and selected stories from the children’s literature that would (a) enable children to draw on their funds of knowledge to make sense of the situations; (b) serve as simulations of “real-world” situations that children could imagine or might have experienced; and (c) afford opportunities for children to develop intellectual autonomy in ways that were important to the practice of modeling.

12.2.2 Teachers Professional Development Cycle In IMMERSION’s yearly professional development cycle, teachers engaged in an intensive 1-week professional development institute followed by several months with a small teacher study group who planned mathematical modeling lessons, observed each other’s teaching, revised and retried the lessons, and presented their experiences and observations at a culminating fall conference. The kindergarten teacher whose classroom is the subject of this chapter engaged in the summer and fall professional development activities. She then returned the second year, to participate in the summer professional development institute and served as a lead teacher for a fall study group. In the spring of that year (Year 2), she took the initiative to continue engaging her class in weekly mathematical modeling cycles. The data used for the case study (i.e., the student work samples and classroom video recordings) discussed in this chapter are from the spring of Year 2.

12.3 Theoretical Framing of Mathematical Modeling in the Classroom In the IMMERSION project professional development, teachers engaged in mathematical modeling activities as learners and reflected on these activities to better understand the practice of mathematical modeling. The project leadership also

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Fig. 12.2  A framework for teachers’ roles in modeling. (Carlson et al., 2016, p. 122)

introduced teachers to a mathematical modeling in the classroom framework (Carlson, Wickstrom, Burroughs, & Fulton, 2016) (see Fig. 12.2). The framework was introduced to provide teachers with a potential trajectory for classroom modeling activities, rather than a strict sequence of steps. Indeed, the IMMERSION leadership repeatedly emphasized that the framework was meant as a resource, that much still needed to be learned about mathematical modeling in the elementary classroom, and that teachers could and should contribute to revise and/or restructure the framework. Turning to the framework itself, we see that the inner circular diagram reflects the work of students: posing questions, building solutions, and validating conclusions. The outer part reflects the work of the teacher: developing and anticipating student questions and solution approaches; organizing activities and discussions; monitoring and facilitating students’ solution building; revisiting students’ initial questions and ideas; and supporting students’ efforts to validate conclusions. Thus, it illustrates the ways in which teachers and students might co-construct mathematical modeling activities. We include the framework here, as it influenced our exploration of mathematical modeling in the elementary classroom and served as a framework for the vignettes of our case study.

12.4 The Case Study Our aim in this chapter is to provide a case study (Creswell, 1997) of one kindergarten classroom in which mathematical modeling was implemented periodically throughout an academic year.

12.4.1 Methodology To our knowledge, the literature does not yet contain detailed accounts of mathematical modeling in kindergarten classrooms, though researchers have begun to address the question of how mathematical modeling might unfold, generally, at the

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elementary level (cf. English, 2012). We contend, however, that kindergarten classrooms warrant special consideration. Children at this stage of development are often learning to read and write, to count, to symbolize, and to socialize. They are still learning to speak English (the language of instruction), especially when the classroom includes English language learners. Most importantly, many kindergarten children are experiencing a formal learning environment for the very first time. Consequently, students are learning about school norms and practices. It is due to these reasons and the lack of prior research on mathematical modeling in the kindergarten classroom that we decided a case study was warranted. Specifically, this qualitative methodology was selected because it provides a means to understand a phenomenon as it unfolds over time in a specific context (Creswell, 1997). Moreover, case studies provide the basis for later comparisons across cases and the identification of themes which transcend context and provide a foundation for large-scale studies. As is typical of case studies, our focus is on the provision of a thick description, which draws on multiple data sources (Creswell, 1997). In our work, these sources included but were not limited to video recordings of classroom activities, classroom transcripts, audio recordings of individual students describing their solutions, students’ written work samples, teacher reflections of classroom lessons, and teacher and researcher field notes. Collectively, the research team (the authors) reviewed the video recordings and transcripts of classroom discussions and students’ individual modeling work within the Seesaw™ electronic platform, a platform which records students' verbal and written remarks in real time.4 We then selected three lessons for an in-depth analysis. The selected lessons were reviewed by the research team to help clarify and understand the children’s and teacher’s activities, after which the classroom teacher (the first author) provided a detailed description of the anticipation, navigation, and reflection phases for each mathematical modeling activity. Drawing on these resources, the classroom transcripts and individual student data, the research team furthered this analysis by collectively examining the children’s mathematics5 and funds of knowledge (Moll, Amanti, Neff, & Gonzalez, 1992) elicited during the activities. This dual exploration – that of the teacher’s anticipation, navigations, and reflections and of the students’ mathematics and funds of knowledge – is reflective of our perspective of enacted classroom lessons; namely, that lessons are co-constructed by teachers and students. We turn now to this perspective, after which we present the three classroom vignettes.

 For more information about the electronic platform see: https://web.seesaw.me  Our use of the phrase “children’s mathematics” draws on Steffe and Thompson’s (2000) notion of “student’s mathematics.” Thus, by “children’s mathematics” we mean the mathematics we, as observers, attribute to children and that is indicated by “what they say and do as they engage in mathematical activity” (p.  268). Children’s mathematics can differ from the standard canon of mathematics. 4 5

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12.4.2 A Perspective of Enacted Classroom Lessons Our exploration of mathematical modeling was guided by both a shared perspective of teaching and an emergent collective understanding of what it means to enact mathematical modeling lessons. To describe this perspective of classroom teaching, we refer to the instructional triangle found in Cohen, Raudenbush, and Ball (2003). The figure shows the interconnectedness of the teacher, content, students, and environments. As this figure highlights, classroom lessons cannot be described by sets of tasks or content standards, by teacher remarks or instructions, and they are not context-free. Instead classroom lessons develop through an interplay between teachers, students, and content within classrooms within communities. Indeed, this point was made by Ball (2017) who notes, “the instructional triangle makes visible that teaching is co-constructed in classrooms through a dynamic interplay of relationships, situated in broad socio-political, historical, economic, cultural, community, and family environments … through the interpretations and interactions of teachers, students, and content” (p. 15). Of particular importance to us is the fact that this perspective recognizes that lessons are enacted, that is co-constructed by teachers and students (Remillard & Bryans, 2004). As a result of this perspective, we recognized that our goal of understanding how mathematical modeling might unfold in the kindergarten classroom could not be achieved via an a priori analysis of content within classrooms or written curricular tasks or by focusing solely on either the teacher or the students. Instead, enacted mathematical modeling lessons should be investigated with attention to the dynamic interplay between teachers and students situated within classrooms. We recognize, as did Ball (2017), that “All of this complexity could make learning highly improbable” (p. 15). Indeed, recognizing the ways in which classrooms function as dynamic systems means recognizing that there are many moving parts in classroom learning environments – that classrooms are inherently chaotic. It also means recognizing that teaching entails work (Ball, 2017) in the form of a constant attending to and managing of interactions and the ways in which tasks, practices, and resources are constraining or affording learning. Thus, two of our primary aims in carrying out the case study were to understand the ways in which the classroom teacher (1) prepared for and anticipated children’s mathematical modeling activities and (2) attended to and managed interactions so as to afford children’s mathematical modeling. This approach is distinct from the cognitive approach to studying modeling employed by others (cf. Kaiser, 2017), for it is focused on the co-construction of mathematical modeling activities rather than the ways children’s reasoning might diverge from intended or anticipated curricular aims. We turn now to the specific context of our study.

12.4.3 Classroom Context The three mathematical modeling activities that are the focus of our analysis took place between December and March in a kindergarten classroom in the Pomona Unified School District, a school district located in southern California. The

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Table 12.1  School demographics Student group Black or African American Hispanic or Latino White Socioeconomically disadvantaged Students with disabilities English learners

Percent of total enrollment 16.2 74.1 7.9 80.9

CAASPP results in mathematics – grades 3–6 percent met or exceeded 36 38 – 33.93

10.1 23.7

16.67 38.46

kindergarten class had 16 students (the school set caps at 24 students per class in grades K-1). Pomona is a working class, urban-industrial community in Los Angeles County, California. The school is a designated Title 1 school with a majority of students receiving free and reduced lunch and is a district-dependent charter, which means it operates under the umbrella of the district, but has autonomy about how funding is used. Specific demographics for the school site, according to the 2016–2017 School Accountability Report Card (SARC), are provided in Table 12.1. As indicated in the table, the case study occurred in a school in which the majority of students are categorized as socioeconomically disadvantaged. This facet of our case is important for this characteristic is commonly linked to students’ academic outcomes in American school achievement data (cf. https://www.nationsreportcard.gov). The classroom teacher had experience with and professional development in Number Talks and Cognitively Guided Instruction (CGI). As both Number Talks and Cognitively Guided Instruction involve pedagogies in which teachers honor children’s strategies and work from students’ ways of reasoning, we saw these initiatives as compatible with (but distinct from) the teacher’s efforts to incorporate mathematical modeling. In particular, through these initiatives the teacher had practice establishing a classroom culture in which students shared their thinking, considered each other’s reasoning, and at times voiced judgments by agreeing or disagreeing. The classroom in which we conducted the case study was selected for several reasons. As noted earlier, the teacher had participated in week-long summer professional development programs in teaching mathematical modeling for two consecutive years. After the first summer professional development, she was a participant of a teacher study group in the fall and then after the second summer professional development served as a lead teacher of a fall study group. Additionally, the teacher had been observed teaching modeling, as part of the larger research project. Several lead teachers attended an NCTM Innov8 conference to present their work teaching mathematical modeling, including the kindergarten teacher (the first author) and another teacher from her school. After the kindergarten teacher observed this third-grade teacher from the same school, she experienced a sense of urgency to prepare her students for the work they could be capable of in third grade; a goal which echoes points made by Manouchehri and Lewis (2017). To build her own professional competencies in mathematical modeling, the teacher designed and facilitated modeling activities every week in her classroom the following academic year. It is three of these lessons which we will analyze and discuss in this chapter.

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Of note is the fact that our case study was assisted by several practices, which the teacher had developed to afford her own professional growth. She was recording her classroom lessons. Additionally, in her classroom she was employing a technological tool, Seesaw, an electronic platform which simultaneously records student verbal remarks, drawings, and inscriptions. She also had an established practice of analyzing and reflecting on the results of classroom observations and of sharing and analyzing her own lessons with colleagues. Drawing on these practices and the data sources they provided, we met as a collaborative team to better understand how mathematical modeling might unfold in a kindergarten classroom.

12.4.4 Mathematical Modeling Lesson Sequence For each mathematical modeling lesson sequence, the teacher typically followed a week-long cycle in which the first 3 days covered standard curricular tasks (and activities) which then served as support for the mathematical modeling activity. The sequence started with the teacher reading aloud a story on Monday. The students then worked with manipulatives based on an idea in the story to build up strategies for later use. On Tuesday and Wednesday, the class did Cognitively Guided Instruction (CGI) problems that involved characters from the same story. These took the form of story problems with differentiated number sets that the teacher selected based on the strategies she saw students using when they had worked with the manipulatives. On Thursday (or the fourth day of the sequence), the teacher would start the mathematical modeling activity, with a prompt developed around characters and an event connected to the story. This was what we called the “dive in” part of the modeling process. The “dive in” is intended to launch the children into a discourse that would support their efforts to engage in a subsequent modeling problem. During this part of the process, the teacher would work to facilitate children’s efforts to consider what information mattered in the problem or what information was needed. Specifically, the teacher worked to facilitate discussions in which the children could bring their understanding of the world to bear on the situation as they considered (a)  relevant quantities, (b) data needed, (c) assumptions about the problem, or (d) constraints (reasonable limitations on the quantities). As such, the dive-in portion of the process provided an environment for what García and Ruiz-Higueras (2013) refer to as “meaningful” and “generating” questions, in particular, the inclusion of opportunities to “question the world” through inquiry. After the dive in on Thursday, the teacher designed a modeling task to pose on Friday, which was based on prior discussions with the students. After posing the task to the group, students worked individually and justified their answer using the Seesaw computer platform. The aim of this chapter is to provide examples of how mathematical modeling became an emergent practice in the kindergarten classroom through a collection of vignettes. Specifically, we seek to understand how the teacher and students

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co-­constructed the mathematical modeling activities. We worked toward this aim in each vignette by considering the teacher’s planning, facilitation of the Thursday conversations, and her reflections on the classroom activities. Collectively we explored the mathematical ways of reasoning elicited from students during the Thursday “dive-in” classroom discussions and observed in the sample of students’ work from the following Fridays, when students individually produced their Seesaw responses. The vignettes are presented in the teacher’s voice and include three parts: (1) teacher’s anticipations, (2) navigating as a teacher-facilitator, and (3) teacher’s reflections. When transcripts are included, the time elapsed is presented in brackets [minutes:seconds] and we use the following abbreviations: S = student, SS = multiple students, S1 = first student, and SN = new student (to distinguish that a new student is talking). Each vignette is followed by our collective analysis of the mathematical ways of reasoning elicited during the mathematical modeling activity.

12.4.5 Vignette 1: Mouse Count

Teacher’s Anticipation Mouse Count is a story that is often used by teachers for math lessons in order to reinforce counting concepts, as well as adding and subtracting. In the story, the main character, snake, is very hungry so he gathers groups of mice in a jar. Since he is hungry he gathers more and more groups of mice. Eventually the mice trick him and get away. Working from the story, the students and I worked with counting, focusing on cardinality and the strategy of counting on. Opening the story to a modeling task was a challenge, mostly because this was new to me (and to the students). This was only the second modeling task for us, and

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the first was rather difficult for the students to understand what kinds of questions they should ask. Fortunately, the story did not have the snake eating any mice at the end, so I could set up a problem that had them determine how many mice would be enough. So I created a task that showed a jar with five mice and posed this task: Snake was VERY hungry! He was collecting mice for a meal. He had 5 in the jar. It wasn’t enough! How many will be enough? Show how many more he needs. Write how many in total below. I chose to put five mice in the jar since we had been working on subitizing the quantity of 5 and we had also been practicing counting on from 5. I wanted the students to consider how hungry the snake could be, how big are the mice, and how many would be enough or too many. Transcript Excerpt 1 (Holding up the worksheet showing the jar after listening to the story on Youtube) [6:08] T: You have a problem to solve today. In our problem today I am going to show you…we have a jar and there are some mice in there. How many mice are in the jar? S: 1..2..3..4..5 [Other students show how they counted using different counting strategies] [7:35] T: Here’s our problem today. Snake was very hungry. He was collecting mice for his dinner. He had 5 in the jar but it was not enough. So we have to figure out how many will be enough. Oh oh oh! Don’t start shouting out numbers. We got to think now. How many will be enough? But we don’t know. Before we start thinking of numbers, what things do we need to know? Hands down…we need thinking time. (Spent some time remembering the things that were asked to solve last week’s task, such as how big are the animals’ heads and how big is the water hole?) [8:50] T: How are we going to know when it’s enough? S1: Maybe he has a lot of family. T: So maybe that’s our first question… are the [mice] just for him or does he have to feed his family? SS: His family! T: We are just going to put the question down okay because we don’t know the answer quite yet. (writes the question “Are the mice for him? Or his family too?”) Because what would happen if he was trying to get mice for his family too? S: That would be helpful. T: It would be helpful but what would that mean? How many would he have to get? S: Maybe a lot because he maybe has a lot of family. SN: And maybe 10! T: Right! Maybe a lot because…how big is his family? Should we put that? (writes the question “How big is his family?). (Students start chiming in thoughts) T: So let’s pretend that we know that these mice are just for snake. What other questions should we ask?

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S: How many snakes are in his family? T: Okay…so we said that (points to the screen with questions) “How big is his family?” But what if it was just for him? What if his family has to get their own mice? What kinds of questions would we ask if the mice are just for him? S: I think it’s for his neighbors. T: Listen to what I’m saying. I’m saying let’s pretend that he is just getting mice for him. What other questions would we have? Yes… S: He has all of them and his family has to get them by themselves. T: Okay. SN: Maybe they’re having a race and whoever gets… T: But…okay. But what other things do we have to know to be able to answer this? (points to worksheet). S: We have to know that they have to get more or maybe for themselves. T: So we are wondering how big is his family. Are his family getting any mice? (Students chime yes and no as teacher writes question “Are his family helping?”) SN: We don’t know that yet! T: We don’t. So if the family is helping, does he have to get as many? (Students chime yes/no) He does? S: Some of his family can get half, half, and half. T: Okay, so if they are sharing the job, maybe mom can get 6, and he can get 6, and then dad can get 6, and then he doesn’t have to get so many…because they are sharing. [13:13] (after conversation about deciding how much to eat) T: What about that? [Student] just…How many will give him a stomachache? (writes question on the board “How many will give snake a tummy ache?”) S: I agree! T: Okay! So we are wondering how many mice will be too many and he will get a tummy ache? What else? S: That would be a lot. T: That WOULD be a lot. Navigating as a Teacher-Facilitator Navigating this task posed similar struggles to the first task that we did: the challenge of getting the kids to understand what kinds of questions they should be asking. This requires a lot of wait time, strategic questioning, and a great deal of scaffolding to bring their ideas out and turn them into a question (kindergarteners do struggle with the task of asking a specific kind of question, despite questioning coming quite naturally in the context of learning). A student started off our “wonderings” with the suggestion that maybe snake has a lot of family (this was something I had not even considered in my anticipatory thoughts toward this task). I flipped the thought into a question, “Are the mice for him? Or his family too?” at which the students were responding with shouts of “His family!” I had to remind them that we did not know that information, and that is why we were writing the question down as a wondering.

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We proceeded a bit further with wonderings of how big his family might be. However, knowing that the task that I had put together on their worksheet did not involve the family, I attempted to steer the thinking back to just snake. I posed the thought “So let’s pretend that we know that these mice are just for snake. What other questions should we ask?” Immediately the response was “How many snakes are in his family?” I honored the question by pointing where we had already recorded the similar thought “How big is his family?”, and then I tried again to bring the students back to considering the task in relation to snake only. The next response involved the neighbors of the snake, and when I again tried to get the students to consider that the mice were JUST for snake, a student replied that all of the mice would be for him and the family would have to get their own. At this point, I began to realize that students were set in considering the family and I needed to let the questions flow in that direction. Interestingly enough, as I tried to guide the questioning to get the kids to consider how hungry snake was, a student brought up the fact that too many mice would give him a stomachache. This limitation really resonated with the students and unknown to me at the time, would be something that students would refer to in future modeling tasks.

Teacher’s Reflection My students were absolutely resolved that family had to be involved in the answer to this problem, and really it was an assumption that I needed to have room to give them. Because I had already composed the task in a worksheet and put in place that the mice were only for snake, this posed a “textual” problem. To keep the task (worksheet) focused only on snake took away a lot of ownership from the students and gave this math time the feel of a typical textbook problem versus a “real-world” problem that did not provide such “complete” information. It was after this task that I started separating my modeling tasks over 2 days  – the first day involves the “messy” wonderings and deals with the recording of students’ ideas and assumptions. For the second day, I develop a more defined task derived from Day 1

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thinking. This gives students a sense of ownership as they can recognize their collaborative efforts of thinking in the task posed for them to solve. This interaction with students also gave me insight that although I need to anticipate what students might ask when I pose the task, I have to be open to what may come as wonderings depending on their funds of knowledge. Although I may want them to consider certain things, I cannot push them to go down that path of thinking if it’s not what is on their mind right now. Interestingly enough, a peer’s thinking often will trigger another student to consider something different and important (that I had tried to get them to consider prior).

Student Sample 1

This student added 6 mice to get a total of 11. Interestingly, he wrote an equation (not shown) that 10 + 1 = 11. When he counted his mice, he did start at 5 and then counted up to 11. As I look at the work through the lens of modeling, I notice that many questions from our wonderings were addressed. This student wrote: “The mice are small. His family is helping. The mice is for his family. The family is big. They are.” I was pleased that this math task gave this student an opportunity to practice counting on, play with equations, and demonstrate critical thinking and justification skills.

Student Sample 2

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When my students complete a modeling task, I require them to post it using Seesaw so that they have an opportunity to explain their model. From there, I am able to see where they are at in their thinking and what my next steps are. This student drew a snake (on the left) and a mouse (on the right). Although it does not look like much, it is the thinking that is impressive. The student explains: “How much can he eat? He needs too much…but if he gets a big one he doesn’t need to get more.” This student did not show counting or equations but is critically thinking about the size of the mice and justifies that getting one more big one is enough for snake.

Student Sample 3

One thing I appreciate about modeling is that it gives students a safe place to think how they want to solve the problem and attempt to represent it. This student explained that ten mice were needed. The family helped the snake. The mice were for the family and him. He had ten and the family had ten. You can see that this student struggled with writing an equation to show his thinking, but the representation of the mice is accurate, at least the beginning of the explanation in which ten mice were needed. At this point, I am not sure if the student is thinking that the 10 will be shared or if indeed 20 are needed. Nevertheless, this student addressed and justified the wonderings of how many will be enough, if the mice are for the snake and the family, as well as the consideration of the family helping.

12.4.6 Our Discussion of Vignette 1 Our aim in this discussion is to explore the work entailed co-constructing the Mouse Count activity as an instance of mathematical modeling in the kindergarten classroom. Many kindergarten teachers use Mouse Count to develop students’

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understanding of ten as a composite unit (i.e., the ability to move flexibly between 1 unit of ten and 10 units of one) and to provide students with opportunities to create and recognize partitions of ten. Working toward the latter goal, teachers often provide students with ten objects and ask students questions such as “How else might snake have gathered the mice?” Such questions are important, as they create an opportunity for the children to generate their own partitions of ten via an imagined alternative storyline. Our aim here is not to dismiss these goals, for they are important to students’ development of arithmetic algorithms and number sense. Instead, our goal is to ask: How is the Mouse Count activity in Vignette 1 different from these numeracy lessons? How is it an instance of mathematical modeling in the kindergarten classroom? What work was entailed as the modeling activity was co-­ constructed by the teacher and students? To begin to answer this question, we must go back to the beginning of the activity. Rather than have students create their own partitions of ten, the teacher asked, “Snake was VERY hungry! He was collecting mice for a meal. He had 5 in the jar. It wasn’t enough! How many will be enough? Show how many more he needs. Write how many in total below.” This question is different from the question “How else might have snake gathered the mice?” for the question of “How many will be enough” opens the mathematical floor for students to engage in the mathematical practices that are essential to modeling. Indeed, the question leaves open what it means to have “enough.” We know snake is hungry but we do not know how much snake requires. Looking to the students’ responses, we see that the students respond by further opening the mathematical floor, essentially inquiring into who snake is gathering mice for. Drawing on their own experiences as young children who essentially never eat alone, the children shift the question (and therefore, the activity) from one focused on “How many mice should snake gather if he is hungry” to the question of “How many mice does snake need to feed his family?” As there is no mention of family in the original story, we see this shift in focus as evidence of students bringing their experiential worlds to bear on the question considered and in so doing, reframing the mathematical activity to align with that experiential world. This observation echoes the observations of Czocher (2016) and of Manouchechri and Lewis (2017), who noted that learners transformed problems into more realistic versions. It also highlights García and Ruiz-Higueras’ (2013) point, “Questions do not live in isolated, but embodied in systems (biological, economical, social…)” (p.  424). Indeed, though the teacher had assumed the children would focus on snake’s needs, the children’s experiences draw them to a different assumption: snake’s needs will include the needs of his family. The teacher navigates this unanticipated assumption by further opening the task through the remark, “So maybe that’s our first question…are the [mice] just for him or does he have to feed his family?” The teacher includes these wonderings in the questions collected on the whiteboard and attempts to move the students toward her original question but the students have fixated on snake being a member of a family and seek to clarify their assumption by asking, “How many snakes are in his family?”

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Such furthering is important, for it indicates an implicit recognition that the mice needed will be dependent on the number of family members. It demonstrates an emerging form of mathematical thinking in which one attends to how quantities might covary in relation to each other; that is, we see in the students’ remarks the seeds of what will grow to be students’ proportional reasoning as students seek to describe the relationship between the total needed and number of family members provided for. Evidence of this emergent way of reasoning is also seen as the students raise questions about how a family of snakes might gather mice when the students wonder, “He has all of them and his family has to get them by themselves…” and “Maybe they’re having a race and whoever gets…?” The teacher honors the students’ wonderings by recording the question about whether or not his family is helping. In so doing, the teacher not only honors the children’s emergent proportional reasoning but also furthers their modeling activities, as these recordings serve to clarify assumptions – an activity critical to modeling. They also demonstrate how teachers might address Manouchehri and Lewis’s (2017) concern that the implementation of mathematical modeling tasks will necessarily require attending to “learner’s intuitions regarding legitimate key variables based on their real life experiences” (p. 108). To be sure, the children’s considerations likely arise from their experiential knowledge of the world, a world in which during preschool and kindergarten class time, children might have observed or been taught, for example, that “clean up” works best if they all help pick up the toys together. And, while it might not be reasonable to expect snake to collect a large amount of mice, a team of snakes might be able to do so – a point implicitly raised through the students’ remarks. Once the discussion moves from clarifying the assumptions regarding who snake is gathering mice for and who will gather mice, the teacher and children begin attending to the number of mice snake will need to manage his own hunger. And it is here that we see the first instance of students introducing what might be called the “constraints” or “bounds” on the amount of mice. Indeed, the students’ concerns regarding snake getting sick from overeating lead the teacher to record the wondering, “How many will give him a stomachache?” This question is, in essence, the question “How many mice are too many?” It is a question about bounds. Again, we should not be surprised by such a query. Young children are often warned not to eat too much or they will get a “tummy ache.” They extend the same consideration to snake and in so doing, wonder about the constraints they need to consider if they are to generate a reasonable answer to the question “How many will be enough?” Reflecting on the “dive-in” portion of the Mouse Count lesson, we note that the teacher’s initial prompt “How many will be enough?” provided an avenue by which the students’ wonderings could emerge and serve as a basis for describing, reconsidering, and elaborating on the situational factors that might serve as assumptions or constraints on the amount required. These considerations emerged interactively as the teacher probed the children’s reasoning and elicited their ideas – ideas that essentially result in the children’s reframing the question. Moving to the children’s individual written work and verbal descriptions, we see further evidence of this reframing and of the children’s progress toward mathematical modeling. Indeed, the students’ solutions and their discussions of those solutions illustrate how each child attended to and articulated their underlying assumptions and constraints about relevant quantities and reasoned with those quantities.

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Turning to Student 1’s work sample, we see Student 1 shares an initial assumption regarding the unit of measurement by remarking “The mice are small,” an assumption illustrated by the student’s drawing small mice uniform in size. Student 1 also articulates other assumptions when the student reasons that many mice will be needed and collected because of “His family is helping” and “The mice [are] for his family,” assumptions employed to warrant the collecting of more mice than in the original story. Further assumptions are articulated about the total amount needed and potentially collected as Student 1 remarks, “The family is big” before Student 1 shares that the amount needed is 11 mice. Student 3, like Student 1, has snake gather food for the entire family with the family. The number gathered for snake is still ten but the context of snake’s work has been expanded beyond that of the original scenario. This student’s response, like that of Student 1, is seen as an instance of mathematical modeling because the student clearly articulates the assumptions employed regarding who will gather mice and who will receive mice, as the student draws mice similar in size to the five provided in the diagram. Thus, we not only hear Student 3 articulate assumptions about who will collect and receive mice, we see evidence of the student’s attention to the types of units selected and hear the student rationalize a solution. Student 2, unlike Students 1 and 3, does not assume that the unit of measurement is uniform or that snake will gather mice for his family. Instead, Student 2 argues that snake is gathering mice for himself. Considering what is needed for snake, Student 2 remarks “if he gets a big one he doesn’t need to get more.” The reasoning given demonstrates an emergent understanding that many small things will be the same as one large thing. Since a standard unit was not part of the task design, Student 2 is free to discuss and generate a new unit and to use unequal-sized units. His response highlights not only the assumption employed but also another way the Mouse Count mathematical modeling task is open; namely, that the student has the autonomy to select units as he or she feels is most appropriate. This is noteworthy. In standard mathematical tasks, students are often working in contexts with preselected units. Indeed, rarely do we find students choosing appropriate units even though such reasoning is often important when applying mathematics outside of school. Moreover, questions of “how much” are often measurement questions which require one to consider not only an amount of “stuff” but also the size of the “stuff” amounted. Consequently, the questions are inherently proportional because the amount of units required is dependent on the magnitude of the unit; e.g., we have different heights depending on whether our height is measured in inches or centimeters (Thompson, Carlson, Byerley, and Hatfield, 2014). Student 3 attends to these underlying issues as he argues that one large mouse can be used in place of many small mice. Looking across the students’ individual Seesaw responses to the mathematical modeling task, we see the issues considered by the individual students echo ideas that arose during the classroom discourse, for many reframed the task to include the snake’s family. We also see students articulating assumptions, using age-appropriate inscriptions and describing ways of reasoning best described as a form of emergent proportional reasoning, a form of reasoning that lives in a mathematical domain beyond basic numeracy. Specifically, we see evidence of students recognizing that the measure employed impacts the measurement of an amount and considering the size of the mice as an important factor, and we see them taking into consideration

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the idea that the viability of a solution will depend not only on who the mice will feed but the number of snakes collecting the mice. This finding echoes those of English (2012), who also found that young learners were capable of developing and describing concepts such as variation during mathematical modeling activities. And, it stands in contrast to the findings of Verschaffel and De Corte (1997), who found that elementary students tend to neglect real-world knowledge and other appropriate considerations when solving modeling problems. Lastly, and perhaps most importantly, we see the students creating age-appropriate representations and inscriptions during their efforts to mathematize the situation and to communicate their solutions to others – an important aspect of mathematical modeling in the kindergarten classroom.

12.4.7 Vignette 2: Cupcakes for a Party Teacher’s Anticipation This was the fourth modeling task presented to my students. The task was pulled from the story How Many Snails (Paul Giganti Jr.) in which numerous items were presented for students to count, such as clouds, snails, sea stars, and cupcakes. What was fun about this book was the affordances to define what would be counted. Would we count all the clouds? What happens if we counted just the big, puffy clouds? What happened to our total if we defined the “constraints” even further to only count the big, gray, puffy clouds? I was pleasantly surprised at how much the children enjoyed the book, and it was delightful to have them notice the total amount decrease as we added the descriptive words to more finely define what we were counting. For the task itself, I decided to present a problem to the students involving cupcakes. At this point I am no longer preparing a worksheet beforehand with the problem, but instead I create an open-ended verbal task that I present and expect the students together to consider what they need to know to solve it. I posed the task: Faith had a birthday and Mr. Pete had to buy some cupcakes for her party. How many cupcakes should he buy so there is enough? I was hoping the kids would think about number of people and size of the cupcakes. Transcript Excerpt 2 [0:00-0:21] Teacher setting up the stage to get mindsets to solve Modeling task [0:22] T: So you guys know that, um, in our book that we looked at all week there was a page full of cupcakes, right? Yummy, delicious cupcakes…some with sprinkles and some with frosting. And you guys also know that Faith just had a birthday and she just turned 2. And so Mr. Pete had to go get cupcakes for her birthday. How many cupcakes should he get? S: Ummm T: Think first…

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[1:00-1:14] Teacher repeats question and sets up the need for justification (how would you know it would be enough?) Gives significant wait time. [1:15] S: 13 T: 13. Okay. Why 13? S: Because there’s a lot of kids at parties. T: Okay. Do we know that he’s at a party? SN: No. T: What do we need to know to figure out a good number for this? Ss: Uhh…16….20…. T: Okay. But what do you need to know to know how to solve this problem? [1:38-3:07] Teacher takes more number suggestions and keeps challenging the thinking with “How do you know?”. The question keeps being repeated “What do we need to know?” and is still met with more number suggestions. [3:08] T: Okay. So how does Mr. Pete decide on how many to get? What does he need to know? He goes to the store and he goes to buy some cupcakes. How does he know how many to get? S: Maybe because…maybe Grace can tell how much she wants. T: Well he knows he needs one for Grace (starts to write on the board). SN: (shouting out) How many kids were there? T: Oh! Do we need to know how many kids there were? SS: Yes we do! T: Oh…so before we can decide on a good number we need to know (writes on the board) how many kids. [3:52-4:29] Teacher and students discussed about different numbers of cupcakes suggested compared to possible number of kids there will be. [4:30] T: So Mr. Pete needs to find out how many kids are there. Is there anything else? S: If Grace eats some more, she will get a belly-flu. T: Oh…a tummy ache. [4:42-8:29] Teacher and students discuss how many cupcakes each child should get. The discussion had some children changing their thinking from as many as they want to just 2 for little kids and 3 for big kids. [8:30] T: So what other ideas are we thinking? So we are thinking we need to know how many kids and how many cupcakes are they going to eat…what else? S: But they can share. T: Oooh. Is anyone going to share? Oh, I don’t know. We have to find out. We can cut one in half right? (writes on the board). Okay, what else? S: They can cut that one in half, and they will have two (student pointing to a drawing on the board of cupcakes to kid ratio). SN: Cutting them in half means they will have a lot of cupcakes…cutting more and more like…

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T: That’s true! So if we have one cupcake and we cut it in half it would make enough for two people. Right? SN: Yeh but if they have two cupcakes they can cut them in half and then they will feel happy and then they can eat it. SN: But that would be a little piece and they would want more and more. T: Oh…so does it kind of depend on how big the cupcake is? Because you know sometimes you go to the store and they have those teeny cupcakes, right? (Writes question on the board) [9:50- 10:35] Discussion ensues about different size cupcakes and cutting them and wondering if anyone will want to share (depending on the size of the cupcake) [10:36] T: But what if the cupcakes are really big? SS: They will want to share! T: They might want to share. MIGHT…I don’t know. I know Grace wouldn’t want to share. But she does get full. Right? She does get full. S: Oh! The amount that the person is hungry or full. T: Uh huh…yeh. How hungry…should we put that on there? How hungry are people? (Starts writing on the board.) Is anyone going to want to share…how much do they want? Or how hungry are they? S: They might need to go to the restroom and they will go to the restroom and they will feel better. T: Oh…hey I’m wondering something…you guys only have kids on here. Is there a problem with that or no? Do just kids eat cupcakes? S: Grown ups too. T: Yeh…don’t you think Mr. Pete? SN: And kids… T: So should Mr. Pete get a cupcake? S: Yah!!! T: Or does he just buy them? S: He gets one too! T: (Student name), what do you think? S: He might steal one! T: Oh oh! You know my husband! Yes!!! If he doesn’t have a cupcake he might try to steal Grace’s. So should we give him a cupcake too? S: I think we should give him 100 because he steals! T: Oh yes! So how about… S: You should text him that he can get some for you! T: OH me too! I get a cupcake?! SS: Yah!! T: So wait…so should we add on to this question? (pointing at the board) “How many kids…?” How many….what? SS: Adults! T: How many adults do we need? Okay. Huh. Is there anything else about cupcakes that we should find out?

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[12:16-15:10] Discussion about how many extras should be bought and how many to eat at the party and what to do with the leftovers. Class decided that we need to have enough and it’s ok if we have extra…but not too many extra. Navigating as a Teacher-Facilitator The struggle again was getting the questions started. Students have a lot of ideas of quantities that make sense to them, but when posed the question “Can you prove that that is a good number?” or “Why do you say that?” there is little to no response. This task went three whole minutes before I even had an idea or wondering to record on the board. This is a lot of time, but again necessary wait time for their thinking and to create some cognitive space. What is so fascinating to me is that with this “divein” process, it only takes that ONE idea from a student to get the momentum going. Once a student suggested that a cupcake was needed for my daughter, the kids began thinking about how many kids, the size of the cupcakes, would they be sharing, etc. There was even a mini-conversation regarding how many cupcakes would cause a tummy ache. This led to a conversation about how many cupcakes will each child eat? Although some initially thought kids should eat as many as they wanted, they changed their thinking when they were reminded that kids can get sick with too much sugar. Once the group decided that two cupcakes should be given to little kids and three to bigger kids, I noticed that they struggled with conceptualizing how many would be a good amount. Even though they had agreed to have each little kid eat two, they still thought two cupcakes would be enough for two kids. I tried to diagram it and show them that there would need to be four cupcakes for two kids, but I could see that they were not grasping the notion. So we concluded that segment with “This can be really confusing.” We moved on to consider some other ideas. The following day, I presented the problem with some constraints, considering all that we had discussed prior (I added the detail about chocolate frosting wondering if any kids would consider it in their solutions):

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Teacher’s Reflection I walked away from this reflection with two considerations. The first was for me to consider and wonder about the fact that the task was centered on my family. It seemed to create an ability for the students to visualize and be engaged more with the problem. They have heard me share stories of Mr. Pete (my husband) and my girls (Grace and Faith) throughout the year, and I often used pictures of them for my CGI problems. Was the access to so many significant wonderings on their end a result of the problem centering around my family or more so because this was not as new to them? I do know that their funds of knowledge make a dramatic difference in the accessibility to the problem. I did have the experience the following week with a task that involved an aquarium that resulted in far less considerations of constraints and wonderings on the board. When I compared it to the cupcake task, it occurred to me that most of my students do not have personal experiences with aquariums to ask significant questions (even though that task, too, centered around my family). The second consideration involved the way I posed the problem initially. Why was it when I first presented the task I couldn’t get any wonder statements, but after 3 minutes when I rephrased the problem and put Mr. Pete at the store, wondering what he needed to know to buy the right amount of cupcakes, I got a wondering that sparked the rest of the 12-minute math discussion?

Student Sample 1

This student walked into my classroom in August having no English. In this recording, he is able to explain that two cupcakes are for Mr. Pete, two are for the big kids, two are for the little kids, and two are for the…moms (he got stuck for a word here and was a bit frustrated). I am wondering if he meant to say the last two were for the grown-ups/adults since it was the one group he did not address in his justification.

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Student Sample 2

Even though this student did not give a total number of cupcakes, it pleased me greatly that he considered that each group would want a different amount. In his response, he reported that the little kids get one cupcake each, the big kids get two cupcakes each, and the grown-ups get three each. He didn’t count them all up, but he ended up with a total of 12 cupcakes, different from the response from Student 1 who decided 8 would be enough.

Student Sample 3

This student did total the amount of cupcakes and notated the amount to be seven. This student decided everyone at the party will get one cupcake, circling two for the little kids, two for the big kids, and two for the grown-ups. This student even

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went further to consider getting an extra for auntie – because auntie always shows up to the party. As I further reflect on the student work and thinking of the task, I am both impressed but acknowledge the work ahead in this journey. The questioning during the dive in surpassed my expectations of them simply asking how many people would there be and what are the size of the cupcakes. My students thought further and considered sharing, how hungry are people, will adults eat more than kids or the opposite, would we need some for the party and some separate for the family to have the next day for dessert. I loved participating with them in a very natural feel of a conversation that also had the weight of a serious problem that needed to be solved. The diversity of their answers makes my kindergarten teacher heart melt. Modeling is an art in which they can express their thinking their way. So maybe we have ways to go in making sure we write down a total quantity, making sure we share our “why” thinking, and addressing all parts of the constraints. But at this stage of the game, I am more pleased that my students are learning to think critically about the problem posed, showing perseverance, and are not afraid to dive in.

12.4.8 Our Discussion of Vignette 2 As in our prior discussion, our aim in this discussion is to understand what is entailed with co-constructing mathematical modeling activities and, more specifically, to explore what makes the Cupcakes for a Party activity an instance of mathematical modeling in the kindergarten classroom. To understand this, we must return to the initial discussion of the task: Faith had a birthday and Mr. Pete had to buy some cupcakes for her party. How many cupcakes should he buy so there is enough? Looking to the students’ reactions, we see that even at this early stage of development, the students focus initially on providing a numeric answer – the perceived question is essentially, what number do we need? Indeed, for 3 minutes of the discussion, the teacher reiterates and rephrases questions such as “But what do you need to know to know how to solve this problem?” and “How do we know?” Since mathematical modeling tasks are not about singular numeric answers, the teacher pushes until the students begin to wonder. Doing so entails engaging in what Garcia et al. (2013) refer to as “a collaborative and shared study process, looking for good answers as well as for good questions” (p. 423). And, as Garcia et al. note, during this process “the classical distribution of responsibilities between X (students) and Y (teachers) will be continually renegotiated … Y avoids being the question-poser, the unique source of information (media) and the ultimate source of validity” (p. 423). As was the case with the Mouse Count activity, the students’ wonderings begin with some basic questions that are, essentially, aimed at clarifying the assumptions that will impact the viability of their solutions. For instance, the students ask, “How many kids will get cupcakes?” And then, the students’ wonderings move toward the

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constraints as the class considers letting the children have as many as they want (i.e., setting no bounds on the amount per child) and then rejects this idea out of concerns for “belly flu” (i.e., recognizing the contextual warrant for a bound). As the classroom discussion continues, students begin to articulate assumptions: small children will get two cupcakes and big kids will get three. These assumptions about the amount of cupcakes per child not only refine the students’ inquiry but also lead to other important wonderings. Will children share cupcakes? How hungry are they? These wonderings open the mathematical floor for a rich discussion of which unit is best suited for their situation. Responding, the children note, in their own language, that if a child is given half a cupcake they may want two halves and that if they use really little pieces, children will want a lot of them. As was the case with the first vignette, these wonderings highlight the ways in which modeling tasks not only recurrently warrant children’s unitizing activities but also provide a context for their emerging proportional and measurement thinking. Indeed, the students’ remarks indicate a tacit understanding of the ways in which the size of the portion might be inversely related to the amount children wish to consume and the fact that the total amount needed will vary in relation to the unit selected either through the partitioning of wholes (e.g., using half cupcakes) or through the purchasing of different sized cupcakes (e.g., “tiny cupcakes”). As the discussion progresses, the students engage in yet another facet of mathematical modeling, when they are given the opportunity to reconsider and revise initial assumptions about who gets cupcakes and what might constitute a reasonable solution. Indeed, as the students consider the possibility that not only children but also adults might participate in the cupcake party, they introduce and consider the idea that leftovers might be okay – an idea that produces further wondering about how many leftovers are “too many.” Turning to the students’ individual written work, we see that though the task included many of the constraints the students had discussed in class, the task remained opened. To be sure, the students have the autonomy to decide how many cupcakes each person (or each type of person) will receive and, consequently, the total amount needed. For Student 1, the task becomes an equal sharing task. For Student 2, the problem is one in which each person receives a portion that is relative to their size; e.g., adults receive more than big kids. For Student 3, the problem is initially a question of one-to-one correspondence but then includes remarks regarding an acceptable margin of error. Hence, the task remained open to the students, for they could attend to the constraints of the task in the ways they felt “best fit” the situation. Reflecting on these classroom activities, we see that the Cupcakes for a Party activity engaged the children in the following: articulating and analyzing assumptions; recognizing, describing, and evaluating constraints; unitizing; and determining reasonable boundaries on quantities. We see them engaging in a whole class discussion that opens the mathematical floor to new wonderings and affords the children the opportunity to revisit and revise assumptions, essentially, reframing and revising the task. And, we see for the first time, students’ emergent reasoning related to the acceptability of a margin of error.

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This last aspect of the Cupcakes for a Party activity must not be ignored. It is critical to mathematical modeling. Yet, for much of students’ K-12 education, we overemphasize exactness and precision even though it is often the case in the world outside of the classroom that we need to plan for contingencies or leave room for error. Planning for contingencies and leaving room for error require one to ask questions such as, “how much extra is reasonable?” and it is exactly this question that the Cupcakes for a Party activity elicited from the kindergarteners, as a result of their efforts to revisit and reconsider their basic assumptions.

12.4.9 Vignette 3: Two Ways to Count to Ten: A Liberian Folktale

Teacher’s Anticipation This task was our tenth task and landed in late March. It evolved from the story Two Ways to Count to Ten: A Liberian Folktale written by Ruby Dee. I was noticing that the students were getting better at the dive in, even though they still would initially shout out numbers instead of considering what information they needed. I thought a lot about what task to pose; sometimes creating a task from the literature poses a greater challenge than not. I was thinking about the wedding that was going to happen in the story, and how the students and I had talked a lot about it during the week with the posed CGI problems that involved grouping.

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I wanted to find a task that would both be open for students to create models and possibly incorporate the grouping computations we had been doing with our CGI work. I decided to pose a task that involved the king serving meatballs at the wedding; the meatballs gave a greater amount of openness than a food option like pizza. I personally made the assumption that the students had eaten meatballs before and had some ideas about the size, how to eat them, and how many would be too many. Additionally, due to time constraints on a related project, in this case the worksheet was created before the brainstorm, which constrained my ability to build a problem in response to the student’s ideas. On the worksheet I had posed that the king had 20 meatballs to share with his daughter, the gazelle, and himself. Seesaw allows me to post work digitally, and students solve using a stylus to draw on the screen. For this assignment, students could use the digital version or just draw their solution on a blank paper. My work with tasks prior led me to design worksheets that have the task represented in a rebus form. In the past I found that the students would forget what the constraints were and I wanted them to have access to all the information throughout their work on the problem. I also record myself telling the problem on Seesaw, and students are able to listen to it as often as needed. Transcript Excerpt 3 [0:00-0:39] Teacher reviewed the storyline and students counted by 2s. [0:40] T: So the king was VERY excited about this wedding for his daughter because he liked the gazelle because he was clever. AND the king also had a love for meatballs. He loved to eat meatballs so he decided at this wedding he was going to give some meatballs for dinner. Okay? So how many meatballs should everybody get? S: 5? SN: No I said 5. S: 2? [1:16-2:21] Teacher poses the question again and pairs students to turn and talk for a minute in a pair/share about what they think and what they are wondering. Teacher joins a group of students. [2:22] T: So you guys think that 2 wouldn’t be enough because it’s a little number. What do you guys think of that idea? S: I think 6. T: You think 6? Why do you think 6? S: Because that’s a bigger number. T: It’s a bigger number. Okay. SN: I think 10…I think umm….20. T: 20 for each person? SN: I disagree. T: Why do you disagree with that? S: Because they could get a stomachache.

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T: Oh that might be too many and they could get a stomachache…(students in group all responding). Wait a minute…what do you mean they would all get 1. So if there’s 20 meatballs and they all get one, what does that mean? SN: That means…they have…ummm…they have like a lot of animals and they give 20 meatballs and then they cut it in half… T: Okay so I’m going to stop you there. So you think there’s going to be a lot of animals? S: No like they’re… SN: There is a lot of animals in the jungle. S: There IS a lot of animals in the jungle. And we have 20 meatballs and they cut them it in half and share… T: So they have to even share the 20 meatballs because you’re saying there’s a lot of…so is there a question that we have that we need to answer? S: There would have to be a lot of meatballs because there’s a lot of animals in the jungle. T: There is a lot of animals in the jungle. How about at the wedding? Do you think there will be a lot of… S: There will be a lot of animals. T: Are we wondering how many animals will be at the wedding? SN: I know how much because one time I went to the zoo. There are umm…90 animals. T: There were 90 animals. Do you think all the animals that would be at the zoo are going to be at the wedding? SS: No. SN: Since in the jungle there’s a lot of animals then there would have to be a lot of meatballs. T: So you think there would have to be… S: They would have to share them. T: They would have to share them…all those meatballs. So I’m wondering when we come back together do we know how many animals are at the wedding? SS: Yes! No. T: Do we know? SN: 6. T: Do we know? We have ideas… SN: Well 6 is a little number too, SN: 96. T: How do you know there’s 96? S: I know there’s 96 because when we read the…because when I went to the zoo one time…every time I go I counted them. T: So you’re thinking there’s 96, but do you know for sure? (waited for response from student) Umm….not really. So you guys, (student name) is really thinking there’s going to be 96 because when she went to the zoo there was 96 animals. SN: Well that’s what (student name) thinks. And um…there’s not that many animals at a party because that would be WAY too much. T: Okay. So are we…so do we know how many will be at this party or is that something we need to find out?

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S: It’s something we need to find out because there’s a lot of animals at a wedding and there’s a lot of humans at a wedding. T: Okay so we think there’s going to be a lot but we don’t know how many yet. So should we put that question on the board? (students nod) Okay, let’s do it. [5:25-5:52] Teacher regroups the students and shares the thinking of the group. [5:53] T: If there’s a lot of animals at the wedding that would be too many. SS: Yeh! T: And so… S: Well there would have to be too much meatballs. T: Way too many meatballs because if there’s 96 animals… S: Well, it’s because there’s a lot of animals in a jungle. T: Okay. SN: And it can’t be a lot of animals because they might spend all their time wasting all their money and they need to buy flowers and chairs. T: Okay so you’re also thinking you can’t invite a lot of people because the more people you invite the more money it will cost. So let me ask you…should we put the question up here “How much money will it cost?” [6:45-8:30] Teacher writes the question and helps students navigate discussion about sharing the meatballs, if there would be knife at the wedding because it’s in a jungle, and adds the question “Will there be a knife to cut meatballs to share?” Questions previously written are reviewed. [8:31] S: If there’s 4 animals, then we can get 4 meatballs. T: Oh so… SN: That’s just a small number because there’s a lot of animals in the jungle. T: You’re thinking…so we are back to (circling the question on the board) “How many animals will there be? Right? (Student name) is really stuck on there’s a lot of animals and the wedding will probably have a lot of animals because there’s so many. Is that what you’re thinking? S: Yes. T: So (student name) I am going to say “How many is each animal going to get?” Is that what you’re thinking? (writes question on the board). Okay (student name), what are you thinking? SN: I wonder how many meatballs are there? T: Oh! Do we need to know how many meatballs are there? Right because if we are going to share them we need to know… SN: That was our question! SN: Well there has to be 100 because 100 is more than the animals because there’s not 100 animals in the jungle. [9:55-15:04] The topic of eating too many came up and how many will give a stomachache. The question of whether some animals will eat too many was also charted. Leftovers and extras were also discussed and questions charted. It was determined that if there are a lot of animals at the wedding, there will need to be a large number of meatballs and if not a lot of animals, then fewer meatballs are needed. Discussion led to cutting the meatballs into slices.

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[15:05] S: What my grandma does is gives me 3 or 2 tortillas then put it in the microwave with cheese then cuts them. T: She cuts them so you guys are thinking maybe people…does it matter how big the meatballs are? SS: Yes! No! S: It doesn’t even matter because meatballs are just a medium size. T: Oh, they’re usually medium. S: They’re not the same size as humans’. SN: Oh like the fish! When we were at the table right there (pointing to teacher table)! How big is it? T: So do you think we should put how big are the meatballs? SN: And how small are their meatballs? T: So maybe it’s maybe are the meatballs big or small? Is that what we are wondering? (writes the question). SS: agreeing with thoughts S: That’s a lot of words! T: I know. Because we are amazing. Right? So if they’re big meatballs (student name)… S: Well that would be a lot to cut. T: We would cut them right?…so they don’t shove the whole thing in their mouths. What would happen if they shoved the whole thing in their mouth? S: They would …choke. SN: Maybe…maybe… T: And probably get a stomachache. S: Maybe there will be a hippopotamus there. T: Oh! Does that matter? What if there’s a hippopotamus there? SS: (laughing) S: That’s a big person. He could eat like a ….LOT! SN: He could eat 100. T: And yesterday we had a monkey in our story. Do you think a monkey would eat the same as a hippopotamus? SS: No! S: Because a monkey is skinny and a hippopotamus is way too fat. T: (laughing) He is going to eat a lot, huh. So do we need to know what animals will be there? [16:57-25:53] Teacher and students further discuss the size of animals and how it will relate to how much they will eat. “What kinds of animals are coming?” is added. Student made a connection to a large potato in another story we read and wondered what would happen if there was just one big meatball. All the questions were reviewed again and the task was presented to end the dive in session. Navigating as Teacher-Facilitator Navigating this task was interesting. Where in the past the struggle was to get students to generate questions and ideas toward the posed problem, a new struggle

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came up. What do you do as a teacher when students get “stuck” on an assumption? I had one student insist that there were 96 animals because that was how many were at the zoo. I tried to pose multiple questions to get the student to understand just because that was how many at the zoo does not mean that the same amount will be at the wedding. I had another student always referring to there being “a lot of animals in the jungle” so there would probably be a lot of animals at the wedding (even though this same student brought up that you shouldn’t have too many people at a party). This also was an assumption that this student held onto throughout our class discussion and often posed itself as a lens that she looked through when commenting on other student ideas. Any time a smaller number was suggested, this student would reply that it would not be enough. Another new struggle was the number of student hands that were raised to share thinking and participate. This was difficult since usually ideas shared evolve into short discussions of agreeing or disagreeing and ultimately ending in the class deciding what question should be put on the board as a result. The nature of the dive in as it is set up makes it difficult to call on all hands that are raised to share.

Teacher’s Reflection I have watched the recording of this task multiple times for varying reasons, which has forced me to do quite a bit of reflection on what happened and things to improve on. Before I even began the task, I should have had a discussion with the students and inquired who has eaten meatballs before and who has been to a wedding. I made some assumptions about both of these and I am wondering if it may have helped with some of the “stuckness” on large quantities for the number of animals at the wedding. Noticing how stuck some students were with their assumptions, I have

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begun to wonder how I may begin to honor this thinking and allow students to work with their own assumptions. This will take serious consideration since one can foresee how that could get messy really quick with 24 5-year-olds creating models with their own chosen constraints. I’m also wondering how much ownership students felt with the task since they were focused on such large quantities of animals and the task posed to them only had three animals.

Student Sample 1

This student drew the 20 meatballs and divided them up “equally.” He counted five in each group and shared that the two leftover would be for the one more animal coming. The animal can eat the two meatballs and then they can leave. Even though the counting is inaccurate (possibly due to the digital tool usage – this student is usually very accurate in math work), I noticed decisions were made about how many each animal would get and how to share the leftovers.

Student Sample 2

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The student shared that the king gets ten and the queen gets nine and seven for the gazelle. When the king runs out of meatballs, he will go to the zoo store for more. The gazelle was given less because it wouldn’t eat as much as the cheetahs.

Student Sample 3

This student uses manipulatives to visualize their modeling choices. The student gathered 20 linker cubes and then proceeded to divide them between the drawn images of the cheetah king, his daughter, and the gazelle. The student shared that the king got six meatballs, but the gazelle and the princess cheetah each were given seven because the king wanted them to have more. What I really took away from my multiple opportunities to reflect on this modeling task is the important consideration of student voice. I had so many hands that were raised in the hopes to share their thinking, and I failed to get to each student for sharing. I have brainstormed how I might track who has shared and who hasn’t and at the end of the session have the students who haven’t spoken yet share at least an idea they agree with. Also, I found it surprising that as I watched the video and some students seemed disengaged, they would surprise me by sharing an excellent thought toward the class conversation. Engagement does not always equal still bodies with eyes focused on the teacher. A final thought as I watched the video is the number of times I misunderstood what the student was sharing in the moment. Even though I repeated the idea and asked the child “Is that what you’re thinking?” it seemed that many answered yes because they weren’t sure what else to answer. It really is a tricky thing at this age to navigate the pacing of the conversation to ensure it is going fast enough to keep engagement high and also try to understand in that quick-paced moment what exactly a 5-year-old is trying to explain. I am not sure what the solution is, except at this point to be more mindful to really listen. Voices being heard and accurately represented are vital.

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12.4.10 Our Discussion of Vignette 3 Like the Mouse Count and Cupcakes for a Party activities, the Two Ways to Count to Ten activity demonstrates how mathematical modeling can emerge in the kindergarten classroom through a constructive interplay between teachers’ and students’ reasoning. Indeed, the student wonderings, which were prompted, refined, and reiterated by the teacher, explored who would attend the wedding and what would be a reasonable amount of food for the attendees; which unit would be appropriate for the food being portioned; the reasonableness of bounds on those portions; what amount of guests would be too many; and whether or not a guest’s physical size should determine a guest’s portion size. The Two Ways to Count to Ten activity also progressed past their prior modeling activities, for we saw the emergence of wonderings in which multiple quantities must be considered for they jointly covary. This is evident in the students’ discussions when we see their considerations related to needing (or wanting) a large wedding (assuming the number of animals in the jungle mirror the large quantities in a nearby zoo), while they also recognize that a large wedding party will require a large amount of meatballs and, therefore, a large amount of money. Thus, their wonderings progress from considering how two quantities might covary (e.g., cupcakes and people or snakes and mice) to genuine efforts to consider how multiple quantitative relationships affect each other. And, while we recognize that it would be extremely difficult (and, most likely, developmentally inappropriate) for the students to symbolize and inscribe their thinking – their emerging proportional, measurement, and covariation reasoning – we also believe that activities like Two Ways to Count to Ten are critical to students’ mathematical development. Indeed, if students are to see a reason to inscribe (or symbolize) proportions, to mathematize linearity and rates of change, then they should be afforded opportunities to develop the ways of reasoning that warrant such mathematics (Smith & Thompson, 2007). This reasoning does not develop spontaneously but rather emerges over time. To be sure, what we see in the Two Ways to Count to Ten activity is an instance of a mathematical modeling activity affording students’ use of these forms of reasoning in developmentally appropriate ways. And, in so doing, these activities not only serve as a foundation but also provide warrants for students’ future mathematical work.

12.5 Discussion: The Work of Teaching Mathematical Modeling in Kindergarten Our goals in conducting a case study were to explore, “How might children engage in mathematical modeling in kindergarten classrooms?” and to provide a thick description of the work of teaching kindergarten mathematical modeling. As

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illustrated in the vignettes, the enactment of mathematical modeling lessons in the kindergarten classroom evolved over time and involved building a classroom environment in which the tasks and activities are co-constructed as the teacher elicits and scaffolds students’ mathematical wonderings. This work required that the teacher (1) attend to and notice the mathematical facets of students’ wonderings in the moment, while also providing the necessary supports so that these wonderings could emerge into genuine acts of mathematical inquiry; (2) situate this inquiry with respect to the mathematics the children were exploring in their classroom, while leaving open the possibility that the children’s mathematics might diverge from (or extend far beyond) the classroom’s curricular mathematics; and (3) draw on her own pedagogical and mathematical knowledge to anticipate students’ mathematics in the modeling activity, while avoiding limiting the children’s mathematics to that which was anticipated. This latter aspect of the work of teaching kindergarten mathematical modeling is particularly noteworthy, for it highlights the ways in which modeling activities are open not only with respect to the products one might produce but also with respect to the processes and tools children might employ. Thus, the vignettes illustrate a case where the work of teaching departed from what Winsløw, Matheron, and Mercier (2013) described as situations where “students are only given the option to work with narrow questions and only when the teacher has provided the students with all elements needed to answer” (p. 282), while at the same time providing further support for the claim that specific skills are required of teachers, such as “scaffolding (Schukajlow, Kolter, & Blum, 2015; Stender & Kaiser, 2015), and attending to student validating and metacognition (Czocher, 2014; Goos, Galbraith, & Renshaw, 2002; Stillman & Galbraith, 1998)” (see Czocher, 2019). Furthermore, while viewing teaching as an activity that entails continuously interpreting students’ mathematics, our analyses suggest that interpreting students’ mathematics in relation to mathematical modeling tasks is recognizably distinct from the mathematical work required to interpret students’ engagement with structured mathematical tasks in the moment. Indeed, the former is necessarily more complex since the mathematical terrain of children’s wonderings is neither predetermined nor closed. Instead, the teacher is tasked with surveying the mathematical terrain to situate students’ mathematics rather than steering students toward or through a particular path in that terrain. This is why mathematical modeling tasks are open not only with respect to the type of solution to be developed but also with respect to the mathematics the children can bring to bear – mathematics that might extend far beyond what one expects of children at this stage of development. This latter observation echoes the remarks of Stohlmann and Albarracin (2016), who noted in their call for further research on elementary mathematical modeling that “teaching mathematical modeling can be difficult for teachers and studies should take this into consideration … when students are engaged in modeling activities, teachers are likely to encounter substantial diversity in thinking” (p. 6). And, Lesh et  al. (2013) found that when given this opportunity young children are able to

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pursue these questions while drawing on “some remarkably advanced ‘big ideas’ from much later in the K-12 mathematics curriculum” (p. 420). Lastly, we see teaching kindergarten mathematical modeling as involving distinct types of work, because teachers must work to grow students’ capacity to mathematize their world. Our case study illustrates one way such work might unfold. We observe that when presented with a relatable situation, the children brought their knowledge of the world to bear on questions of which constraints are relevant, which boundaries must be considered, which quantities we need to know, how those quantities might be unitized, and what it might mean to reasonably achieve one’s goals. These wonderings reframed and reshaped “the question” considered. Thus, the question was co-constructed by the teacher and students as the teacher purposefully elicited, reframed, corralled, and integrated the children’s ideas so that “the task” – a mathematical question – emerged from the children’s wonderings.

12.6 Revisiting Mathematical Modeling Frameworks The GAIMME report provides a continuum from a “bare bones” math problem to a modeling problem (see Fig. 12.3). Reflecting on this continuum, we see the mathematical modeling activities that emerged in the kindergarten classroom are more than problems with labels and words. We see them as more than application problems, for the tasks that emerged through the classroom discussions afforded the kindergarteners the opportunity to work from their own interpretation of the problems. Moreover, the activities emerged in ways that required students to impose additional assumptions, constraints, and interpretations for the children were not provided all of the necessary information but instead gathered that information through their inquiry. The GAIMME report also characterizes mathematical modeling as a process, which might unfold in a multitude of ways. Indeed, it argues that there might be as many modeling cycles as there are modelers. Nevertheless, to support readers in their efforts to understand the process of mathematical modeling, they presented the diagram in Fig. 12.4.

Fig. 12.3  GAIMME continuum from math problem to modeling problem

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Fig. 12.4  GAIMME math modeling process

Looking to this description of the process of mathematical modeling and our observations of mathematical modeling in a kindergarten classroom, we see many parallels. Indeed, we see the vignettes as illustrating the ways in which kindergarten children can engage in the process of mathematical modeling through those processes described in the circles of Fig. 12.4 (Fig. 1.2 of the GAIMME report). Below we describe a kindergarten modeling process, with references to the figure above in italics. Modeling Day 1 1. To identify and specify the problem to be solved, the teacher based the problems in recently read literature, which provided the context accessible to young children. Note that while the world of these stories is fictional, they can be quite real to very young students. They also provide a shared context of reference for classroom discussions. Thus, during the “dive in” on the first day, the teacher and students interactively began to identify and specify the problem to be solved during their whole class discussion (group work). 2. To make assumptions and define essential variables, the teacher elicited the students’ wonderings and queried into their reasonings. This work entailed encouraging students to identify important information (but did not call it a variable) and to state assumptions about the needed information (e.g., how many animals live in the jungle?). The teacher also asked students to justify their claims by

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asking “how do we know?” Sometimes the assumptions took the form of constraints, such as when students noted that the amount of food eaten should be less than the amount that would cause a stomach upset (group work). Modeling Day 2 3. On the second day, as the students individually did the math and got a solution, they brought information from their experience and the context of the story to further specify (sometimes tacitly) the question they would answer and to identify constraints considered in their solution. For example, Auntie will come to the party even if she is not invited or snake will want to feed his family, even if the question is only about him. Kindergarteners selected age-appropriate mathematical tools to describe and discuss the quantities noticed, as evidenced by the variation in individual student solutions to the same prompt, and to rationalize their selection of quantities (individual activity using Seesaw). 4. While the analysis and assessment of the models and solutions didn’t look the same as it might with more mature modelers (e.g., considering how changing a parameter value changes the output of a model), such work could arise by asking kindergarten students to extend their thinking. For example, the teacher might ask students to consider how many cupcakes would be needed if people wanted to eat two cupcakes instead of one or what would happen if there were more guests. Kindergartners were able to verbalize when a solution presented by another student made sense to them and when it did not seem to reconcile with their own thinking about the problem (group discussion). 5. Kindergarteners did not engage in iteration and model extension for their particular solutions to a task, but they did apply ideas from previous modeling problems to new ones. We also saw them sometimes want to revise assumptions/ constraints presented in the task, such as considering belly aches or feeding snake’s family instead of just snake (group discussion). 6. As their form of implementing the model and reporting their results, the kindergarteners individually choose their method of communicating and justifying their solutions, which included developmentally appropriate pictures, manipulatives, and spoken language (individual activity using Seesaw and some presentations to the class). Indeed, from this analysis, we can see much resonance between the mathematical modeling activity that emerged in the kindergarten classroom and the types of activities described in the GAIMME framework for modeling, which was originally designed for high school students. We note however that to recognize these connections one must be open to the fact that the mathematical tools, inscriptions, and resources employed by children will be reflective of the child’s stage of development and that, at the same time, children’s reasoning may far outstrip these tools, inscriptions, and resources (e.g., as seen in children’s verbal characterizations of covariation and proportionality). Figure 12.5 contains a summary of the process as developed by the teacher and described in the vignettes.

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Fig. 12.5  A window into mathematical modeling in kindergarten based on case studies

12.7 Final Remarks How might kindergarten children engage in mathematical modeling in classrooms? What work might be entailed in teaching mathematical modeling? These are the questions our collaborative team sought to explore through our discussion of three classroom vignettes. Moving beyond descriptions that may ultimately inhibit mathematical modeling in the early grades (e.g., “this practice, model with mathematics, might be as simple as writing an addition equation to describe a situation”), we have provided evidence of the potential not only for young students, but our youngest students in our highest need settings, to do much more. Indeed, we hope the reader will walk away from our chapter with images of the ways in which our youngest students’ mathematical wonderings might serve to open the mathematical terrain of the classroom as students work to identify and articulate their assumptions, propose constraints and boundaries, query into margins of error, examine quantities, and explore the ways those quantities might be unitized, be in proportion or covary, and revisit and refine their mathematizing of situations. For us such images are important. Indeed, we were recurrently surprised by the students as we grew in our knowledge of their capacity to mathematically wonder and reason. By engaging in mathematical modeling with kindergarten students, we

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learned that young children can begin to see that math is everywhere and that everyday problems can be mathematized. To be certain, we learned that teachers can use mathematical modeling to provide students a mathematical space to be creative, while also practicing the math they know, and to explore the math that comes naturally to them, as students draw on their funds of knowledge and wonder about their world. And, more importantly, that our ability to do so relies on us recognizing that children’s representations of this mathematics will take developmentally appropriate forms, even when the ideas children share and pursue may lie far beyond that being represented. In other words, mathematical modeling in the kindergarten classroom becomes feasible when one is willing to recognize and build on children’s mathematics – a point that aligns well with that of Czocher (2019): The path forward is to find ways to lead students to mathematics content that allows them to model the world as they see it, rather than constraining them to see the world as curricular mathematics allows.

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Chapter 13

The Genesis of Modeling in Kindergarten Helena P. Osana and Katherine Foster

Success in mathematics, both in school and out, is dependent on a strong start in early childhood. Indeed, a considerable body of research has demonstrated that children’s early numeracy competence has been found to be highly predictive of their future mathematical success in school (e.g., Mazzocco & Thompson, 2005; Romano, Babchishin, & Pagani, 2010). For example, kindergarteners’ counting skills, their knowledge of symbolic representations of number (e.g., reading and interpreting written numerals), and their judgments about the relative magnitude of quantities (e.g., knowing that 7 is larger than 5 but smaller than 10) are predictive of their achievement at the end of the first grade (Jordan, Kaplan, Locuniak, & Ramineni, 2007; Martin, Cirino, Sharp, & Barnes, 2014); other studies have shown that early numeracy skills predict achievement in the third grade (Jordan, Glutting, & Ramineni, 2010; Jordan, Kaplan, Ramineni, & Locuniak, 2009) and beyond (Duncan et al., 2007; Geary, Hoard, Nugent, & Bailey, 2013; Watts, Duncan, Siegler, & Davis-Kean, 2014). In our research, we focus on supporting kindergarten teachers who teach in lowSES communities prepare young children for the mathematics they will encounter in the first grade. Our overarching aim is to provide teachers with various types of professional development support, with an emphasis on enhancing the intentional interactions they have with their students about numeracy in the classroom (National Research Council, 2009; Stipek, 2013). Part of our recent work with the teachers involved incorporating mathematical modeling in their classrooms. Modeling provides a learning environment that is contingent on inquiry and mathematical discourse: Indeed, an important design principle for incorporating modeling in the mathematics classroom is to place emphasis on productive teacher-student converH. P. Osana (*) · K. Foster Department of Education, Concordia University, Montreal, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_13

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sations about constructing and refining models to solve problems (e.g., Zawojewski, Lesh, & English, 2003). In this chapter, we describe our collaboration with two kindergarten teachers to incorporate mathematical modeling in their classrooms. The goal of the modeling unit in each of the teacher’s classrooms was to support the development of the students’ foundational numeracy skills, principles such as cardinality and order irrelevance, and strategies for enumerating objects and representing quantities when solving problems. When modeling is the focus of classroom activity, students use mathematics to construct models to find solutions to real-world problems. In such environments, students use external representations to model solutions, which are then revised and extended through inquiry and communication (Doerr & English, 2003; Lesh, Hoover, Hole, Kelly, & Post, 2000; Stillman, Blum, & Kaiser, 2017). Students are required to “create their own mathematical constructs” through meaning-making (English, 2010, p. 26), rather than reproduce demonstrated procedures for certain classes of traditional “textbook” problems. Teachers need specific knowledge and skills to create and sustain modeling environments in the classroom (e.g., Anhalt & Cortez, 2016; Doerr, 2006), including, but not limited to, understanding the role of models in mathematical activity, the nature of children’s mathematical thinking, and how to interact with children to support their learning through modeling. Little is known, however, about the types of professional supports teachers need to design and orchestrate such modeling contexts, particularly in the early years. Too often, early years educators do not receive sufficient training, either at the preservice or inservice level, on mathematical modeling, nor on how to engage young children in meaningful mathematical activity in the context of solving problems through modeling (Cetinkaya et al., 2016; Doerr, 2007; Ginsburg, Lee, & Boyd, 2008; Henn, 2010). To the extent that modeling is an appropriate activity for young learners of mathematics, teachers need specialized professional development to support modeling in their classroom practice (Blum, 2015; Kaiser, Schwarz, & Tiedemann, 2013; Lingefjärd, 2013). The objective of this chapter is to describe a modeling activity that we implemented in two kindergarten classrooms and to use our observations about the teachers’ perceptions of the activity and the students’ mathematical thinking to draw conclusions about the features of professional development that might support early childhood teachers’ efforts to incorporate modeling into their practice. The central focus of the chapter is not to present a portrait of children’s learning in the modeling environments we created, nor to document the pedagogical practices that occurred during the activity. Rather, our goal is to document the teachers’ reactions to our modeling unit and to their students’ work as indicators of the types of knowledge that may provide a foundation for teachers’ modeling practices. In so doing, we propose recommendations for professional development initiatives for modeling in early childhood. At the same time, the observations described in this chapter contribute to the scant literature on teachers’ perceptions of modeling, particularly in the early years, and the professional foundations for modeling in early childhood classrooms (Doerr, 2007).

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13.1 Conceptual Framework for Mathematical Modeling A central goal of mathematics education is to support students’ ability to solve realworld problems. Many textbooks aim to help students apply key mathematical concepts to situations that in some ways simulate the outside world. Part of the problem with this approach is that these “application” problems tend to be contrived and often focus on isolated concepts and principles within limited contexts of application (English, 2010). Encouraging students to engage in mathematical modeling, in contrast, encourages them to construct models to support the mathematization of genuine problem situations (Lesh et  al., 2000); modeling requires a broad set of mathematical skills and processes, including interpretation, the generation of conjectures, representation, justification, and revision of their problem-solving strategies (Thomas & Hart, 2010), through which mathematical concepts emerge (English, 2006). Several theoretical models of mathematics modeling can be found in the mathematics education literature (e.g., Blum & Leiss, 2007; Lesh & Yoon, 2007; Stender & Kaiser, 2015). In our research, we have adopted the components that are common to some of the most well-known modeling frameworks. As such, we designed the task for the kindergarten classrooms in the present study to elicit key modeling processes, namely, interpreting a real-world phenomenon, representing it by constructing models with external representations, using the models to mathematize the situation, solving the problem and evaluating the solution, and then rejecting or revising the models so that more appropriate solutions can be generated, if necessary (Anhalt, Cortez, & Bennett, 2018; Blum, 2015; English, 2003). Successfully incorporating modeling into the classroom is dependent on a number of factors, including the teachers’ conceptions about modeling and their mathematical knowledge for teaching, as well as the students’ skills, previous experience, and beliefs (Blum, 2015). English (2010) claimed, however, that the defining feature of modeling hinges more on the nature of the task than on the student. Consistent with this view, Lesh et al. (2000) presented a list of principles for designing tasks to elicit model construction in the mathematics classroom. Adhering to four of these principles when designing the modeling activity for this research, we ensured that the activity allowed for students’ thinking to be elicited and for it to be made visible through model construction, solution representation, and the sharing of solution strategies and justifications. The task, creating an “ideal” kindergarten classroom, was meaningful to the students and allowed them to assess their own work and revise it if necessary.

13.2 Research on Teachers’ Knowledge and Perceptions of Modeling Whereas a large portion of research on mathematical modeling has been concerned with the benefits to student thinking and learning (e.g., Lesh & Lehrer, 2003) and student affect (e.g., Verschaffel & De Corte, 1997), a smaller, but growing, body of

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literature addresses modeling from the teacher’s perspective. This research can be placed into two broad categories. The first group of studies emphasizes the knowledge and skill teachers need to productively engage their students in mathematical modeling (e.g., Anhalt et al., 2018; Bautista, Wilkerson-Jerde, Tobin, & Brizuela, 2014; Kaiser et  al., 2013). The second group of studies has examined teachers’ beliefs and perceptions about modeling, including their beliefs about modeling itself and its role in the mathematics classroom (e.g., Kaiser & Maass, 2007; Thomas & Hart, 2010).

13.2.1 Teachers’ Knowledge of Mathematical Models and Modeling Borromeo Ferri and Blum (2009) argued that for teachers to engage their students in mathematical modeling, they must have knowledge that can be characterized by four dimensions. First, teachers must understand the nature of modeling and why it is important for the development of students’ mathematics (see also Lesh & Lehrer, 2003). Second, teachers must understand the modeling tasks themselves and know what mathematical knowledge can emerge from specific tasks (see also Krauss et al., 2008). In addition, teachers must have the pedagogical knowledge to encourage students to engage in modeling activities, which involves knowledge about what questions to ask and when and how to offer appropriate support and feedback (Schukajlow, Kolter, & Blum, 2015). Teachers’ knowledge of how to engage students in modeling goes hand in hand with knowing how to assess student thinking in modeling environments and provide appropriate feedback for students to refine and extend their knowledge Diefes-Dux, Zawojewski, Hjalmarson, & Cardella, 2012). All four of these dimensions have been examined in recent research with both preservice and inservice teachers. Bautista et al. (2014) studied the extent to which practicing teachers from fifth to ninth grade understood what models are and the relationship between a model and empirical data. In particular, the authors found that there was considerable variability in the teachers’ understanding and that their academic backgrounds predicted their knowledge of models. In general, teachers with mathematics, mathematics education, or science backgrounds held deeper understandings – that models are “abstractions” of the data and that their main function is to predict real-world phenomena. This stood in contrast to those teachers with no mathematics or science backgrounds, who were primarily focused on the “exactness” of models and whether the models could be used to reproduce the actual data set. Anhalt et al. (2018) studied secondary preservice teachers’ ability to engage in mathematical modeling themselves. In the teachers’ work, the authors identified the presence of key elements of mathematical modeling, such as making assumptions about the problem (in their case, about the oxygen-producing abilities of trees), creating simplified models of the situation, and using appropriate tools. They also

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found that the teachers struggled to accomplish various components of the task, however, which often resulted in frustration. The authors concluded that preservice teachers’ modeling capacities can be extended through training, but that more attention on modeling is needed in teacher preparation. In another study by Paolucci and Wessels (2017) with elementary preservice teachers, the authors found other weaknesses related to the content knowledge necessary to understand modeling and how it aligns with the school curriculum. The authors asked the teachers to create their own problems that would encourage mathematical modeling. Their findings have several implications for teacher preparation. First, although the teachers were given a choice to write a problem for either the first, second, or third grade, most of the teachers did not write problems for the younger grades, perhaps because of the lack of resources available for modeling in the early years. The authors also found that the teachers were generally unable to see what mathematical content could emerge from the problems they themselves created, revealing a shallow understanding of the content in the school curriculum. In other research, the focus rests on teachers’ pedagogical knowledge related to modeling, including the extent to which teachers can evaluate their students’ thinking during a modeling activity and support their learning with questions and feedback. Examining teacher data from a number of their own projects, Blum and Borromeo Ferri (2009) identified some of the challenges experienced by teachers attempting to incorporate modeling in their mathematics classrooms. First, the authors found that teachers often imposed their preferred solution strategies on the students rather than give them time to work through the task before receiving feedback. The challenge was in maintaining a balance between intervening and allowing students the space to generate their own approaches to the tasks. The authors claimed that the teachers’ practices can be attributed to their lack of awareness of the complexity and richness of the modeling “problem space.” Given the research describing the state of teachers’ knowledge for mathematics modeling, Paolucci and Wessels (2017) maintained that greater attention must be paid to preparing teachers to support modeling in the classroom.

13.2.2 Teachers’ Beliefs and Perceptions About Modeling A second focus in the literature focuses on educators’ beliefs and perceptions about modeling. Thomas and Hart (2010) asked a group of elementary preservice teachers to engage in a model-eliciting activity in small groups. In a subsequent focus group session, the teachers indicated that although their experiences were “frustrating” because they were not comfortable without a specific procedure for completing the task, they could see clear benefits to modeling activities with elementary students. They indicated that modeling would be engaging and fun for children, that it would encourage them to construct meaning for the underlying mathematics, and that modeling would foster higher-level thinking as opposed to procedural knowledge gained through worksheets.

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In another study, Handal and Bobis (2004) examined practicing secondary teachers’ perceptions of “teaching mathematics thematically.” Although the authors did not describe modeling per se, their description of teaching mathematics thematically implied many similarities, such as solving problems using mathematical representations of real-world phenomena and placing the student in the position of making sense of the mathematics embedded in an authentic problem. The teachers participated in a professional development course on teaching thematically and were asked to share their thoughts and perceptions through questionnaires and individual interviews. What emerged from the data is what the teachers perceived as obstacles to incorporating thematically based teaching. Some of the obstacles were instructional; for example, several teachers mentioned that the problems were too difficult for students, who they believed lacked the motivation to engage in ­mathematically-rich activity. Teachers also lamented their loss of control in the classroom and the shift in focus away from basic skills. The teachers also mentioned curricular obstacles: The objectives of thematically-based teaching were not, in their minds, aligned with the standard mathematics curriculum or in line with commonly-administered student assessments. Finally, the teachers claimed that theme-based teaching was time intensive and expensive. Centikaya et  al. (2016) delivered a modeling course for preservice secondary mathematics teachers with the objectives of expanding the teachers’ knowledge of mathematical modeling and equipping them with the knowledge and skills to engage in modeling in their own classrooms. The preservice teachers worked in small groups on six model-eliciting ideas over the course of the semester and provided the researchers with their written work, questionnaire responses, and written reflections and engaged in interviews and focus group sessions about modeling itself and their own work in the course. The authors observed that the teachers, who began the course by equating modeling with problem solving using concrete materials, ended the course with more comprehensive and nuanced conceptions. For example, they gained more appropriate notions about how modeling gives meaning to real-world problems, could better distinguish between modeling and “traditional” school mathematics, and could see the potential of model-eliciting activities for developing conceptual understanding of mathematics. In sum, investigations of modeling from the teacher’s perspective, and its implications for teacher preparation, are gaining traction in the research community. Several weaknesses were identified in teachers’ knowledge of modeling and on ways to engage students in modeling activity. In terms of their beliefs and conceptions about modeling, the research has shown that while teachers often see the value in modeling, including that it is fun and motivating for students, they have identified various obstacles to implementing it in their own classrooms. Further, the focus of the research has been primarily on middle-school and secondary preservice teachers, with little attention on modeling in early childhood. In one of the few studies on modeling in young children, English (2010) found that first graders were able to work collaboratively to reach consensus on assumptions and definitions related to the task, constructed useable models that were tailored to the problem at hand, created a number of representations to display their models, and even exhibited aware-

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ness of which types of representations would be more appropriate than others. In another study, Lesh, English, Riggs, and Sevis (2013) found that first graders are capable of extending their mathematics knowledge through modeling and working collaboratively on solving real-world problems, thereby concluding that “basic skills” should be acquired through modeling rather than for modeling. Our focus in kindergarten is not on the children’s mathematics per se, but rather on what the teachers noticed as they observed a modeling unit and the types of foundational knowledge for teaching one can glean from their observations to support early childhood modeling practices. Our focus on the teacher in early childhood modeling is currently an underrepresented area of research.

13.3 Present Study In each of two kindergarten classrooms, we implemented a four-day activity to engage children in key processes of mathematical modeling. The “Ideal Classroom” activity was centered on a narrative (an idea borrowed from Lesh et al., 2013) that pertained to creating the “ideal” kindergarten classroom, which included a variety of constraints, including the principal’s directives, social components of classroom design, and assigned changes to the physical setup of the classroom. The mathematical objective was to provide a context in which students would develop counting skills and numeracy concepts. The activity provided considerable opportunities for the students to develop and extend their understanding of central counting principles, including oneto-one correspondence, cardinality, order irrelevance, and stable order. The students also practiced and refined strategies for counting collections, including subitizing and keeping track of counts. The repeated counting opportunities also supported and extended students’ knowledge of the counting sequence. In terms of numeracy concepts, the Ideal Classroom activity allowed for repeated opportunities to compare quantities, work with part-whole relations, and create different representations of quantities, including written numerals. From a modeling perspective, the activity provided several opportunities for model construction and revision; mathematizing and evaluating solutions; and conjecturing, communication, and justification. Because the two participating teachers had never been explicitly exposed to mathematical modeling, nor did they have experience implementing modeling activities in their classrooms, we decided to work with the students ourselves, in effect “modeling modeling” for the teachers. The second author, KF, an experienced preschool and elementary teacher, worked with the students on the Ideal Classroom activity in both classrooms. We asked the teachers to be present for each of the four modeling sessions in their classrooms primarily to observe KF and her interactions with the students, but also to assist KF with classroom management and by working with the students in their small groups, if needed. We also asked the teachers to participate in short interviews after each modeling session to capture their observations, perceptions, and beliefs about mathematics modeling in general, and in particular, with the students in their specific schools.

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Using audio-recorded interviews with both teachers, we were able to address the following research questions: 1. What were the teachers’ conceptions of the modeling unit? What aspects of modeling were salient to them? 2. What mathematical knowledge for teaching appears to be prerequisite, or foundational, for the teachers to productively engage their students in modeling activities? 3. What constraints, institutional or other, did the teachers indicate might render modeling difficult to incorporate in their teaching? The conclusions we draw based on the data we collected will contribute to the literature on modeling in early childhood settings and, in particular, will augment current understandings of the needs and perspectives of the early childhood educator. We acknowledge that the data do not directly address the types of knowledge that is required for teachers to incorporate specific modeling practices in their classrooms; as such, our data are not definitive in this regard. We do, however, argue that to incorporate modeling successfully in early childhood classrooms, teachers must begin with an understanding of children’s numeracy and their developmental trajectories, and it is in this light that we conclude the chapter with recommendations for professional development in mathematical modeling for teachers of young children.

13.4 Method The data for this study were part of an ongoing, larger project in four schools in a low-SES community in a large cosmopolitan area in Canada. Our involvement in the larger project entailed a year-long professional development initiative on children’s early numeracy with all the pre-K and kindergarten teachers in the participating schools. Our overarching objective was to support teachers in providing rich numeracy experiences in their classrooms. For the present study on modeling, we worked in one kindergarten class in each of two participating schools. We provided three day-long workshops over the course of the year, which were coupled with one or two visits to each of five classrooms during the year to provide support during numeracy activities. In the workshops, we did not introduce the term “modeling,” nor did we provide explicit descriptions of elements commonly found in modeling processes and environments. The emphases in the workshops were on (a) explicit discussions of core numeracy skills, such as cardinality, subitizing, and one-to-one correspondence, and children’s numeracy through the development of such skills, and (b) effective ways to interact with students in the classroom about numbers and quantities in the context of concrete activities and everyday routines. With respect to the focus on classroom interactions, lengthy discussions were held with the teachers about the types of questions and prompts that would move children’s thinking forward. As such, although there was no explicit emphasis on modeling, our aims were entirely consistent with some of its core elements, such as processes of mathematizing, communication, and justification.

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13.4.1 Participants The two teachers who participated in the study, Pam and Lauren, were seasoned kindergarten teachers. Pam had previously taught for 17 years, with 10 years as a kindergarten teacher. Lauren had 18 years of elementary teaching experience, with 16 years at the kindergarten level. At the start of the professional development in the fall of 2017, we observed that, in general, the teachers did not focus on numeracy in their classrooms, and when they did, their practices emphasized low-level skills, such as rote counting and single-digit number recognition. Having worked with both Pam and Lauren over the course of the year, we were aware of the evolving learning objectives for their students, which included cardinality and subitizing by the end of the year. Despite Pam and Lauren having little to no explicit knowledge of modeling, both were receptive to learning more about mathematical modeling and welcomed us in their classrooms to deliver the Ideal Classroom activity. The second author (KF), a practicing classroom teacher with 10 years of experience (6 years at the preschool level and 4 at the elementary level), delivered the Ideal Classroom activity in each class at the end of the academic year (May in Pam’s classroom and June in Lauren’s classroom). The activity took place over four class periods, which ranged from 30 to 60  minutes each, depending on the children’s focus and attention level. There were 16 students in Pam’s classroom and 17 students in Lauren’s classroom. Students worked in small groups with about four students in each group. In Pam’s classroom, the students worked collaboratively on the modeling activity on two consecutive days in the first week and on two consecutive days the following week. In Lauren’s classroom, the students worked on the activity on two consecutive days in 1 week and then two consecutive days 2 weeks later. After each day of activity, KF met with the teacher to discuss the day’s activity and to see if they felt adjustments were necessary for the next modeling session.

13.4.2 The Ideal Classroom Activity Characteristics and Objectives  The objective for the students working on the Ideal Classroom activity was to design their ideal kindergarten classroom, taking into account various constraints, such as the number of students in each classroom and the configuration of children at tables of various sizes. The mathematical objectives of the modeling task were distinct, but interrelated, numeracy skills that have been shown to be predictive of children’s later success in school mathematics. These skills included the knowledge of the number sequence, strategies for keeping track of counted objects, cardinality, subitizing, magnitude comparison, and interpreting writing numerals and using them to represent quantities (Carpenter, Franke, Johnson, Turrou, & Wager, 2016). We designed the modeling task with four key “modeling-eliciting” principles (Lesh et al., 2000) in mind. First, the task adhered to the model construction principle by prompting the students to use manipulatives and other tools (i.e., number

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lines, classroom calendar) to build models of their ideal classroom that were successively refined over the course of the activity. As such, students described their models, explained their reasoning, and assessed alternate solutions through observation, discussion, and teacher feedback. The task was also consistent with the Reality principle because the students all had experience in kindergarten and could make sense of the elements of their models (e.g., classrooms, students, desks). They also had prior numeracy knowledge that would allow them access to the task, which provided considerable opportunities for refinement and extension. The Ideal Classroom task was also designed to allow students to assess their models and the solutions they were generating. Following the self-assessment principle, students were able to judge for themselves or with feedback from the teacher, whether or not the constraints of the task were met. They were able to count the number of bears at each table, for example, and assess whether their models needed revising as they interacted with each other or shared their thinking with the teacher. Finally, the task adhered to the construct documentation principle because it required students to make their interpretations, explanations, and justifications explicit as they created their models and represented their solutions. Delivery of Modeling Task  The tasks on each of the four days of the modeling activity are described in Table 13.1. On Day 1, once the students were placed in their groups, KF introduced the activity to the students in the context of a story. The first task for the students was for each group to name its school, the objective of which was to get the students used to working in their groups and discussing ideas out loud. It also helped the students identify personally with their school and take ownership of its creation. Once each school had a name, each group received a sandwich bag of small, plastic bear counters in three different colors. Each bag contained either 27, 28, 29, 31, or 33 bears. The bears represented the children in each school and the students were asked to work collaboratively to count the bears they were given (from this point, the children in the schools will be referred to as “bears”). Once this task was completed, the students were asked to write down the number of bears they had in their respective schools. If writing was a challenge, they could use stickers to represent the bears, and if they wanted to, they could use any combination of numbers and stickers. Day 2 began with a whole group discussion led by KF, who described the groups’ models and representations (using stickers or written notation) from the previous day to all the students in the class. KF invited the students to share their work with their peers and explain their thinking. This encouraged them to ask questions as well as to see the different ways each group performed the same task. Certain students were reluctant to speak, so KF would verbalize aspects of the children’s counting and representing that she wished to highlight. The excerpt below from the start of the second day of the unit in Lauren’s class illustrates the ways in which KF encouraged the students to reflect on their counting. In this conversation, KF placed the students’ representations of their models from the previous day at the front of the class and invited students from each group

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Table 13.1  Components of the Ideal Classroom activity across the four day unit Day Task 1 a Name your school b Count the numbers of kids in your school c Find a way to write down (or otherwise represent) how many kids are in your school 2 a Make our classes; can make 2 or 3 classes with the following rules: Each class must have a minimum of 10 and a maximum of 17 students b Write down (or otherwise represent) the number of kids in each class 3 a Place tables in your classrooms b Place kids at the tables (no constraints) 4

a Rearrange kids using the following constraints: No more than 4 kids can be at a big table and no more than 1 kid at a small table. b Modifications to the classrooms (additional students; fewer tables)

Materials Teddy bear manipulatives; pencil and paper; stickers Construction paper for the classes; teddy bear manipulatives; paper and pencil; number lines (only in Pam’s class)

Teddy bear manipulatives; single blocks (small tables); 2 × 2 blocks glued together (large tables) Teddy bear manipulatives; single blocks (small tables); 2 × 2 blocks glued together (large tables)

to describe how they could use the representations to find out how many bears were in their schools. KF encouraged students to demonstrate their counting strategies and to reflect on the types of visual representations that make counting easier (C = Child). KF: [Speaks to a child]: Come up and show me and I’ll know exactly what you are talking about. C: [Points to two horizontal rows of stickers at the top of the page and two stickers placed horizontally on a third line in the middle of the page.] KF: Do you think these ones are easier to count? Why are these easiest to count for you? C: Because they’re all lines. KF: Because they’re all lines, so you like it when it’s in lines. Can you tell me how many pumpkins are in this line [points to a row of stickers at the top of the page]? C: [Points to each sticker]: 1, 2, 3, 4, 5, 6, 7, 8, 9. KF: And how many are in this line [points to bottom line]? C: [Points to each sticker]: 1, 2. KF: Nice! Thank you very much. So, you’re putting things in lines to count them. [Addresses another student]: You have something you want to say? Come on up…What did you want to tell me about? C: Because this is so easy! KF: Why is this so easy? C: Because we have to count them. KF: And which one is easiest? And can you tell me how many are here?

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C: [Points to each sticker in rows spread out on the paper]: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. KJF: Now [addresses another student] … I have a really interesting question for you. Somebody was up here … and he said that this [points to the representation with five stickers in each corner] was easiest and you just looked it and said there are 20 there. I want to know, how did you know that? C: ’Cause 10 plus 10 is 20. KF: But how did you know there were 10 and 10? C: [Counts the stickers in one corner]: 1, 2, 3, 4, 5. KF: Oh, there’s 5 in each corner, so then you told me 5 plus 5 equals… C: 10. KF: Plus 5, [which is] another 5, [and] 5, and … C: 20. After the whole class discussion, KF then gave the students their task for the day, which was to separate the bears into two classes following two rules: (a) There could be no more than 17 students in a class, and (b) each class had a minimum of 10 students. KF reviewed the concept of “more than” and “less than” with the students as a whole class, and then they were asked to return to their tables to create their own classrooms. Once the students had separated their bears into two classes, they were asked to write down the number of bears in each class. Pam gave the students in her class laminated number lines, approximately 60 cm in length, with whole numbers written in order from 1 to 20. The students were encouraged to count the numbers on the number line by tagging them so they could make the connection between the number words and the symbols. When they reached the numbers they needed for their totals, they were encouraged to write them down. In Lauren’s class, the students used the calendar displayed in the classroom to find out how to represent quantities in numerals. The students were encouraged to use the calendars in the same way as the number lines in Pam’s class – that is, to start counting at 1 and copying the numbers they needed. At the end of the class, all the students came together to share their work and to comment on how the activity compared to that of the previous day. Day 3 began again with a whole class conversation led by KF who shared the work of each group with the rest of the class. Students were encouraged to tell their peers about their models from the day before and explain how and why they created their models. Again, the discussion was prompted by KF’s questions when the students had difficulty verbalizing their thinking and when she wished to share important aspects of the children’s mathematical activity. The students were then placed in their small groups and given the task to create classrooms where the bears were all sitting at tables getting ready to eat their lunches. The students were provided two pieces of construction paper to represent the two classrooms, single blocks to represent the small tables, large blocks consisting of four single blocks glued in a 2 × 2 configuration to represent the large tables, and the teddy bear manipulatives. The students were asked to place their tables in the classroom in such a way that there were enough places at the tables for each bear in the class.

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After each group assembled their classrooms, the students were invited to walk from table to table taking note of the similarities and differences between the setups created by the other groups. The objective was to allow the students to present their models to their peers without requiring presentations to the whole class. Using prompts and questions, KF encouraged the students to share their decisions for how they had constructed their classrooms. The students’ explanations sometimes involved using cardinal numbers (“I have four bears here because…”), and at other times, the reasons they offered involved discussions about other concepts, such as area and quantity comparison, as illustrated by one of the students in Pam’s class below. C: Our school is … it’s all just a play school, so if you go to this table you want to paint [large table]. If you want to play doctor, you go to this table [three small tables together]. If you want to go to this table, you are a pretend teacher [two small tables together]. If you want to go to this table, you are in the kitchen right there [three small tables together]. KF: And why do you need more space to paint than to be a pretend teacher? C: You know why? Because all the other people like to paint and so the teacher thought if she made big tables they would stay together. Students then returned to their own groups and revised their own configurations if they wished. Some examples of the students’ models are presented in Fig. 13.1. On Day 4, an additional constraint was added to the task: The students were told they were allowed to place a maximum of four bears at the big tables and one bear at the small tables. The students were given the same materials as the previous day and were free to place the tables and bears in the classroom in whatever configurations they wished as long as they attended to the new constraints. Once the students had placed the bears around the tables, the researcher photographed the classrooms and asked the students how many bears were at the small tables, the large tables, and altogether. KF then introduced a few small changes to the situation by continuing the narrative: For example, some schools had new bears enter the classrooms and the students were required to revise their models while continuing to take the constraints into account. In the next part of the narrative, some classrooms had broken tables, so the students would need to revise their models using fewer tables. The objective was to encourage students to make design changes and to have them justify which designs worked best for their specific classrooms. For instance, if students decided to build a classroom where one bear was placed at a big table by himself and another big table had four bears sitting together, KF would ask the students if any of the bears might be upset by this arrangement. The objective was to show students that a setup may be mathematically correct (i.e., it followed the rule of no more than four bears per table) but not necessarily a good choice practically. This was intended to help students grasp the real-life consequences of their designs and the connections between the mathematics inherent in the situation and the potential social ramifications of the choices they made. Once the final classroom tables and chairs were counted, KF brought the groups together to share and reflect on the projects they had created.

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Fig. 13.1  Sample models of the ideal classroom

Although the basic modeling task was the same in the two classrooms, KF made slight adjustments to the implementation in each because of different student behaviors and prior knowledge. For example, the students in Pam’s class had considerably more difficulty counting to 30 unassisted, and KF therefore spent more time practicing counting skills with this group. The stickers distracted the students in Pam’s class, so they were not used after the first day, but the students in Lauren’s class worked productively with them, so the stickers were used on all four days of the activity. The greater focus in Lauren’s class gave the second group time for an extra lesson on subitizing as they discussed different pattern formations that could be made with stickers that made counting easier.

13.4.3 Researcher and Teacher Roles Because mathematical modeling was new to both Pam and Lauren, the teachers’ primary role was to observe the interactions between the researcher and the students. Before the unit, the teachers decided on the composition of the small groups and through conversations with KF after each session, ensured that subsequent

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activities were appropriate given the numeracy knowledge their students developed over the course of the year. The teachers were not given any specific framework to use during their observations, nor were they asked to take notes, but because they knew their students better than we did, they were invited to step in at any time to assist with the small group work and provide classroom management support. KF led the modeling activity in both classrooms, taking responsibility for the creation and execution of all the components of the task. She maintained a collaborative relationship with both teachers; KF worked with them to plan the modeling activities from day to day, which were sometimes modified on the fly while observing students’ responses. For example, as KF moved between groups interacting with the students when they were counting the number of bears in their schools, she noticed many different types of counting errors. Some students had problems with cardinality, some had difficulty reciting the correct number sequences, and still others had not yet mastered the one-to-one principle. One of KF’s objectives was to draw the teachers’ attention to these difficulties, so they could see how mathematical modeling provided an opportunity to isolate individual students’ strengths and weaknesses. When she worked with the students, KF used questions not only to identify their difficulties but also to support their numeracy development. She would often ask the students to justify their strategies and solutions but also provided corrective feedback. For instance, she noticed many students were able to count the stickers and count the bears, but they were unable to make a one-to-one correspondence between the stickers and the bears, indicating difficulties providing representations of their work. In this case, KF asked, “how can I be sure that the number of stickers you have is the same amount as the bears? How can I figure that out?” The students started to generate a variety of answers including, “count both of them,” “make two lines and then count the first line and then the second line,” and “put the yellow bear on the sad face sticker and the other yellow bear on the sad face sticker.” KF shared these suggestions with the other students in the class and then encouraged them to revise their representational strategies in their small groups. As well, the researcher encouraged students to think about how their problemsolving impacted the consequences of their models. For example, KF noticed that one group had created two classrooms, one with 20 students and one with 6 students. KF asked the students which class they would rather be in. One student chose the large class whereas another student wanted to be in the small one. When asked why, the student who chose the small group answered, “because it’s easy to count and it’s quieter.” KF’s questions were designed to encourage students to think deeply about their models and to show students how mathematics plays a role in their everyday lives.

13.4.4 Data Sources All classroom modeling activities were videotaped. The students who received permission from their parents to be video recorded were placed in the same small group. In Pam’s class, two small groups were video recorded (there were four stu-

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Table 13.2  Protocol for semi-structured teacher interviews Interview question Tell me what you noticed about the class today. What math do you think your students learned/practiced? What specific aspects of counting did you see your students learn/practice today? Were there any specific challenges you noticed? Why do you think I had the students build classrooms? Why do you think I had the students share their work with the class? Why do you think I asked the questions I did? How do you think this activity can help the students in other aspects of the curriculum? How easy would it be for you to deliver activities like this in the future? Please explain. What might be some of the barriers to incorporating activities like this into your classroom on a regular basis? What specific aspects would not work for you or your students? Please explain.

dents in each of these small groups). In Lauren’s class, two small groups were video recorded, with three students in one group and four in the other. KF’s voice was recorded even when she was interacting with students who were not being video recorded. Many of the students’ physical models were photographed or captured on video, and KF collected all their written work. The video recordings allowed us to document the ways in which the students used their developing numeracy knowledge to work through the modeling unit, which in turn gave us a context for conducting the interviews with each teacher and for interpreting their responses. Individual interviews took place after each day of modeling. Each interview lasted between 10 and 15 minutes, and all interviews were audio recorded. The interviews were semi-scripted and targeted four general issues: (a) general observations of the day’s activities, (b) observations about student numeracy and learning, (c) perceptions of the task itself, and (d) perceived student interest and affect. The questions that were asked the most frequently across all eight interviews are in Table 13.2. Additional questions and prompts were generated during the interviews depending on the direction and content of the conversation. All recordings of the classroom activity (eight lessons across both classrooms) and the teacher interviews (four interviews per teacher) were transcribed verbatim by a research assistant. The transcripts of the teacher interviews were then coded using three main codes that corresponded to each research questions. That is, each idea present in the transcripts was classified as conception, constraint, or knowledge, with some ideas placed in more than one category. We coded all comments by the teachers that related to the students’ numeracy or modeling activity, the students’ behavior and reactions to the unit, as well as those that described the unit itself, the tools used, and KF’s interactions with the students. Because the teachers had had no previous training in modeling, we did not expect or search for specific modeling terms in the transcripts. Rather, we coded for descriptions of modeling principles expressed in their own words. Once this round of coding was completed, we coded the ideas in each category one more time identifying themes and subthemes using grounded theory techniques, such as constant comparative analysis (Glaser & Strauss, 2014).

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13.5 Findings 13.5.1 Teachers’ Conceptions of Modeling Three themes emerged from the data regarding teachers’ conceptions of modeling. The first theme relates to the teachers noticing social elements of the modeling activity. The second theme entailed the teachers’ observations of the affective components of the activity, such as students’ levels of enjoyment as they engaged with the manipulatives. Finally, we note that what was salient to the teachers was the concrete nature of the activity: They noticed the students’ work with the manipulatives, with relatively less attention paid to the learning affordances of concrete activity. Social Aspects of Modeling  Pam referred to a specific student, who was often absent and usually very quiet, taking a leadership role in her small group. During the interview, Pam said, “…there was one little girl, is very weak. She just started school here at the end of February and she is absent very often so I find she is a little bit behind and she was the one taking leadership and saying ‘no no no no we have to count them and make sure we have the same stickers!’ and I thought wow!” Pam added that she was able to notice the level of independence her students had achieved since the beginning of the year: “I love [to see] the progress. When they come in September and they don’t know anything, and then at the end you let them go and they’re so independent.” Lauren also commented on the social benefits of the activity, particularly for the shy students: “I like that because they get to see what other people are doing. But they may not say anything but they go back and say ‘oh that’s how you do it.’” Affective Components of Modeling  The second theme that emerged from the data was the teachers’ observation of the students’ level of enjoyment during the activity. Pam indicated that the children did not consider the activity “work,” adding that young children learn best through play. Pam elaborated by pointing out that students can be engaged in important mathematical activities, but to an outsider, it may just look like “playing around.” She also pointed out that the story that provided the context for the activity grabbed the students’ interest and would be an approach she would try to incorporate in her practice in the future: “I find if you give them … the story you did with them, they are more engaged because they were their story … instead of saying you have two bears and take one away and then how many do you have left … it’s a bit more engaging.” Both teachers focused on the students’ enjoyment of the concrete activities. Lauren stated that the students loved working with the manipulatives, and “I liked that [the activity] was hands-on and … it’s everything they like - stickers and bears, and you made it fun for them and even the kids that have trouble.” Concrete Nature of Modeling  An additional finding that emerged was that the teachers’ observations focused on the concrete nature of the modeling activity; they were particularly drawn to the students’ use of manipulatives and other tools. When asked

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how she would incorporate modeling in her future practice, Lauren indicated that she already engaged in similar kinds of activities to those in the Ideal Classroom. When KF prompted her to elaborate, Lauren explained that she also engaged her students in counting activities with manipulatives: “[I put] bears in the middle with dice … then they have to take a turn with the dice. We start off with one [die] to get them to know 1 to 5 and they roll a 5, they have to take 5 [bears from the middle] … As they get better with numbers, I’ll introduce two dice.” In contrast, neither teacher mentioned the specific counting or numeracy skills that were being applied by their students as they constructed and revised their models. Of course, this does not imply that the teachers were not aware of the cognitive aspects of their students’ numeracy during the modeling activity, only that they did not offer any such observations during the interviews. Pam also focused on the students’ physical work with the concrete objects. When asked what aspects of her own practice are similar to the modeling activity she had observed, she described specific counting activities, one of which involved splitting up oranges and counting the sections as they were being distributed during snack time. Clearly, such activities can be highly effective for the development of children’s numeracy, and indeed, the professional development we provided underscored the importance of embedding counting and discussions about quantities in classroom routines. As such, it is not surprising that Pam identified this feature of modeling in her own practice. Similar to Lauren, however, Pam described specific, concrete activities without elaborating on the number concepts that the students used, practiced, and likely developed during the modeling unit we delivered in her classroom. Another example that illustrates the tendency of the teachers to focus on concrete aspects of modeling occurred when Pam described one group of students who placed the bears at single tables in rows in their classroom models instead of in groups around tables. She suggested that the arrangement of the desks looked more like a sixth-grade classroom than a kindergarten classroom, indicating to her an unanticipated level of maturity. Array configurations, such as the rows of teddy bears in this instance, are often used to introduce students to equal groups and skip counting, but Pam did not make a connection from the physical arrangement – and its meaning on a social level – to multiplicative concepts.

13.5.2 Mathematical Knowledge for Teaching The teachers’ observations of the social and affective nature of the modeling unit may have concealed the extent to which they noticed nuances in their students’ numeracy knowledge and subtleties in KF’s targeted interactions. In one example, a boy in Pam’s class consistently skipped over the number 16 when counting the bears in his school. On the fourth day of the modeling unit, KF noticed the same error and asked the boy to count the bears again in front of her. KF asked him to count the set of bears two more times, each time providing assistance on the number sequence as he tagged the objects. When asked specifically about this boy during the interview after the class, however, Pam focused on the boy’s social behavior during the class. Again, because Pam did not mention the boy’s counting does not mean she did not

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notice or reflect on the lengthy exchanges between him and KF. Because such interactions were frequent during the modeling unit, however, we were surprised that they were not elicited more often in the teacher interviews. In Lauren’s class, we made similar observations. In one small group, KF worked with a group of children who were counting the 29 bears in their school. The first time the children counted, they correctly counted the 29 objects. KF asked them to count again to be sure, even though they had arrived at the correct number. The second time, the children counted to 30. The third time, they counted to 24, and finally, the fourth time they counted to 29. It was only by asking children to count repeatedly that KF noticed specific counting skills in this small group. All the children used the correct counting sequence to 30, but many of them, and one in particular, had difficulties with one-to-one correspondence. KF’s interactions with the children were deliberate and focused: The students’ counting required careful observations and goal-directed questions. Yet, neither KF’s questions nor the students’ specific counting skills arose during the interviews with Lauren. A final example from Lauren’s class serves as an additional illustration to suggest that the teachers rarely noted the types of interactions KF had with the students to support their numeracy development. On the first day of the activity, KF was working with a group of children who had counted 33 bears in their school. When KF asked the children to write down the number “33,” the students indicated that they did not know how and that the calendar (displayed in the classroom) went only as high as 31, which they said was “easy” to write. KF prompted the students to think about the counting sequence past 31 to see if this would support their written representation of 33 (i.e., “What number comes after 31? What number comes after 32?”). One of the conclusions we drew from this interaction is that there were no written representations of numbers larger than 31 in the classroom, and the students would likely have benefitted from seeing more numerals displayed (e.g., number lines). Lauren did not come to the same conclusion, however. When asked about this interaction between KF and the students in the follow-up interview after the class, Lauren had noticed that the children had some difficulty writing numbers: “I liked the challenge of writing the numbers because as soon as I do it, they go uhhh. I also noticed that [child], he had to write 27, but the first time he wrote 72, so I said go check on the calendar. So that was challenging.” While Lauren acknowledged that the calendar can be a useful tool for the development of transcoding skills, she did not comment on the types of interactions that KF had with the students when a more suitable tool for writing numbers above 31 was not available.

13.5.3 Constraints to Incorporating Modeling During the interviews, we asked the teachers what barriers might exist to incorporating similar modeling activities in their classrooms. In their responses, Pam and Lauren focused on the students’ cognitive readiness for this type of activity, but mentioned few pedagogical or institutional impediments that might make modeling difficult to implement. Pam provided several reasons that modeling would be a challenge in her classroom, with classroom management at the top of the list. She was

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eager to try modeling with her students, but indicated it would be more feasible with two educators in the classroom or by working on modeling with half the class while the other half engaged in free play: “As soon as I sit down, or when they’re done, they are getting out of hand … so, I find that working in a smaller group while the others do free play or something like that would be easier than as a whole group.” Related to the notion of managing the classroom, both teachers mentioned that it would be difficult to keep the students focused on the activity. Pam believed that the story of creating an ideal classroom would keep the students’ attention longer than individual, unrelated word problems. Lauren proposed to make sure the students are “calm” before any modeling activity can take place in small groups. Although the teachers rarely pointed to how specific prior numeracy skills might facilitate or impede modeling activities at this age, both Pam and Lauren alluded in general terms to the students’ cognitive readiness. Both teachers indicated that a significant challenge to implementing modeling is that the students are all at “different levels.” Lauren expanded to say that a related impediment would be that some students would not understand the task, particularly at the beginning of the year: “Some of them wouldn’t understand what you told them to do today. They would be like, ‘what are you talking about?’ Maturity… time at the beginning… when they come in, some of them know their numbers 1 to 10 and some of them don’t … so, they’re all at different levels.” Pam also stated that a related barrier would be the challenge in assessing the students’ knowledge given their disparate backgrounds, personalities, and facility with the language of instruction. Both teachers saw clear benefits to the open-ended nature of the Ideal Classroom activity. For example, they believed that open-ended activities allowed the students to develop independence and creativity, and Pam noticed that the mathematics emerged from the students’ work and not from KF exclusively, which was something she wanted to incorporate in her teaching practice more frequently in the future. Despite these important observations, both teachers appeared to struggle with the notion of relinquishing control of their classrooms if they were to incorporate modeling. Pam, for example, enjoyed observing her students helping each other in their small groups and added that allowing them this freedom would be challenging. Lauren, also indicating discomfort with the open-ended nature of the modeling activity she observed, shared that she would have started the activity with more rules about what is permissible: “Like I would [have] given them what you can do with the [bears], what you can’t do with the bears.” She objected to the open-ended nature of working with the manipulatives, particularly at the beginning of the process, and said that she would show them exactly “what they can’t do and what they can do.”

13.6 Discussion The objective of the current study was to describe a modeling activity implemented in two kindergarten classrooms and to use two teachers’ observations to make suggestions for professional development in early childhood numeracy. Because nei-

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ther participating teacher was familiar with the theoretical foundations of modeling nor how to create and sustain modeling in the classroom, we decided to deliver the unit ourselves and report the teachers’ reactions to the ways in which we engaged their students in modeling. In a real sense, then, we engaged in “modeling modeling,” which allowed us to gain the teachers’ perspective on classroom activity in which modeling actually took place, rather than document any challenges we may have observed in their own implementation. The modeling unit took place over four days in each classroom. We asked the children to create an “ideal” kindergarten classroom that adhered to various constraints, including the number of children in each class and the number that can be seated at tables of various sizes. The mathematical objectives for the students included counting skills and principles, as well as other key numeracy skills, such as number identification, transcoding, and set comparisons. One of our main findings was that the teachers focused on the social and affective dimensions of the modeling unit they observed, such as their students’ personalities as they worked with their peers, and the enjoyment the children experienced during the unit. In fact, the arbiter of a “successful activity” for the teachers was the amount of fun the children were having and their perceptions of our interactions with the students were positive because the questions and prompts kept them engaged. We suggest that the social and affective dimensions were primary because of the relatively low priority placed on numeracy in early childhood settings, despite the necessary foundation it provides for students’ future mathematical success (e.g., Duncan et al., 2007; Jordan et al., 2007; Romano et al., 2010). Indeed, it has been well documented that on the whole, little numeracy activity takes place in preschool and kindergarten settings relative to activities that target children’s literacy and socio-emotional development (Hindman, 2013; LeFevre et al., 2009; Stipek, 2013). Another finding was that the concrete aspects of modeling were particularly salient to the teachers. Both highlighted two specific features of the unit: that it was built around a story and that the students worked with concrete objects. We speculate that these aspects of the unit stood out because they reflected what the teachers practiced in their own classrooms. On the other hand, several broader modeling components, such as model construction, model revision, and mathematization (e.g., Sriraman & Lesh, 2006)  – and the learning and consolidation of students’ numeracy skills that emerged as a result of such activity – were rarely mentioned. Modeling does not have any currency in the local context in which we work. The early childhood educators with whom we collaborate have not been exposed to modeling, either as a framework for characterizing mathematical activity or as a vehicle for mathematics instruction. As such, we are fully aware that there was no reason for us to expect that the teachers would notice or comment on the specific modeling principles that we used to design the Ideal Classroom activity. In fairness to the teachers, the professional development we had provided during the year did not introduce modeling in any explicit way, but rather focused on the development of children’s numeracy skills. Therefore, one of the implications of the present study is that early childhood educators need direct and explicit exposure to mathematical modeling, how it can be elicited in the classroom, and its role in children’s numeracy development.

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Rich and extended analyses of children’s mathematics are a pillar upon which effective professional development in mathematics should rest. This notion is not new; several successful professional development initiatives in mathematics have been based on the idea that teachers are better equipped to move their students’ thinking forward if they have knowledge about children’s mathematics and can use that knowledge to productively assess their own students’ strengths and weaknesses (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Jacobs, Lamb, & Philipp, 2010). Indeed, the literature on modeling more specifically has generated similar claims (e.g., Doerr, 2007). Because the teachers in the present study rarely initiated discussions about the specific numeracy skills we had discussed during the workshops, we speculate that the professional development we provided fell short of its primary goal: to support teachers’ noticing of students’ numeracy knowledge and the application of such knowledge during classroom activity. Had we provided the teachers with more prolonged exposure to children’s thinking, and supplemented the workshops with more sustained support in the classroom, we hypothesize that the teachers would have been better equipped to notice the nuances in students’ numeracy in the context of a modeling activity. We also propose that classroom modeling activity may look messy, disorganized, and perhaps aimless to the “uninitiated”; for our teachers, this may have further prompted them to focus on students’ social interactions and their affective responses. As such, while professional development should include explicit descriptions of models and individual elements of modeling processes (e.g., model construction, mathematization, problem-solving, revision), a view of modeling in situ is also necessary so that teachers can learn to identify and reflect on the central features of model-eliciting activities, even when they are “hidden” by the movement and discussion that are unavoidable when implemented in the classroom. Indeed, our observation that the teachers did not view the modeling activity as qualitatively different from what they already practiced underscores the notion that they need support in identifying critical modeling components as they occur in “real time.” This may be achieved by analyzing videos of classrooms that incorporate various modeling components, for example, and by providing in-class support as teachers finesse their ability to create and sustain modeling activity with their own students. The present study was an initial foray into the investigation of early childhood modeling from the educator’s perspective. Despite our contributions to the modeling literature, there are ways in which the research can be strengthened. First, to increase the external validity of the findings, larger samples of teachers from different school settings and with different background experiences are needed. Second, for the teachers’ observations to be more informative for professional development, it might be beneficial to advise them about modeling ahead of time and prepare them in advance to watch for and reflect on specific modeling activities. Better preparing teachers in this way could take place in professional development contexts that are closely aligned with actual classroom practices and on-the-ground support. Moreover, in terms of study design, inviting teachers to observe a modeling unit in a classroom other than their own could reduce the level of distraction the teachers

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may have experienced because of the detailed knowledge they had acquired about their own students. Finally, future research could examine the effects of specific elements of professional development on teachers’ conceptions and understandings of modeling in early childhood settings. For this, an assessment of teachers’ knowledge and conceptions of modeling would be necessary both before and after the professional development, and a control group would be required to eliminate at least some alternate explanations. Despite these limitations, we nevertheless propose that our observations can be used as a springboard for the design of early numeracy professional development. As indicated earlier, our research motivates us to base future professional development squarely on children’s thinking and how it can be deepened and extended through modeling activity. This is not enough, however, as we argue that the teachers also need what Ball, Thames, and Phelps (2008) have called “horizon knowledge,” which is “an awareness of how mathematical topics are related over the span of mathematics included in the curriculum” (p.  403). Professional development should include meaningful and situated reflections on how children’s numeracy develops over the course of the preschool and elementary years. As such, we ­propose that horizon knowledge entails an awareness not only of curricular topics, but also about how children’s thinking about mathematical topics can impact their thinking about mathematics in future years. Our recommendations amount to a tall order, of course, but our contribution to the literature is in explicating the nature of that order, which is required before any small steps can be made to enhance teaching, and modeling practice in particular, in the early years. Acknowledgments  This research was made possible by funding from Concordia University.

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Borromeo Ferri, R., & Blum, W. (2009). Mathematical modelling in teacher education – experiences from a modelling seminar. In V.  Durand-Guerrier, S.  Soury-Lavergne, & F.  Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education: CERME 6 (pp. 2046–2055). Lyon, France: INRP. Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Education Research Journal, 26, 499–531. https://doi.org/10.3102/00028312026004499 Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3–20. Carpenter, T. P., Franke, M. L., Johnson, N. C., Turrou, A. C., & Wager, A. A. (2016). Young children’s mathematics: Cognitively guided instruction in early childhood education. Portsmouth, NH: Heinemann. Cetinkaya, B., Kertil, M., Erbas, A. K., Korkmaz, H., Alacaci, C., & Cakiroglu, E. (2016). Preservice teachers’ developing conceptions about the nature and pedagogy of mathematical modeling in the context of a mathematical modeling course. Mathematical Thinking and Learning, 18, 287–314. https://doi.org/10.1080/10986065.2016.1219932 Diefes-Dux, H. A., Zawojewski, J. S., Hjalmarson, M. A., & Cardella, M. E. (2012). A framework for analyzing feedback in a formative assessment system for mathematical modeling problems. Journal of Engineering Education, 101(2), 375–406. Doerr, H. M. (2006). Teachers’ ways of listening and responding to students’ emerging mathematical models. ZDM, 38, 255–268. https://doi.org/10.1007/BF02652809 Doerr, H. M. (2007). What knowledge do teachers need for teaching mathematics through applications and modelling? In W. Blum, P. Galbraith, W.-G. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education. The 14th ICMI study (pp. 69–78). New York, NY: Springer Science & Business Media. Doerr, H. M., & English, L. D. (2003). A modeling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34(2), 110–136. Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., … Japeli, C. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428–1446. https://doi.org/10.1037/0012-1649.44.1.217 English, L. D. (2003). Reconciling theory, research, and practice: A models and modelling perspective. Educational Studies in Mathematics, 54(2–3), 225–248. https://doi.org/10.1023/B:E DUC.0000006167.14146.7b English, L. D. (2006). Mathematical modeling in the primary school: Children’s construction of a consumer guide. Educational Studies in Mathematics, 63(3), 303–323. English, L.  D. (2010). Young children’s early modelling with data. Mathematics Education Research Journal, 22, 24–47. Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2013). Adolescents’ functional numeracy is predicted by their school entry number system knowledge. PLoS One, 8, e54651. https://doi. org/10.1371/journal.pone.0054651 Ginsburg, H. P., Lee, J. S., & Boyd, J. S. (2008). Mathematics education for young children: What it is and how to promote it. Social policy report. Society for Research in Child Development, 22, 1–24. Glaser, B., & Strauss, A. (2014). Applying grounded theory. The Grounded Theory Review, 13(1), 46–50. Handal, B., & Bobis, J. (2004). Teaching mathematics thematically: Teachers’ perspectives. Mathematics Education Research Journal, 16, 3–18. Henn, H. W. (2010). Modelling pedagogy: An overview. In R. Lesh, P. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modelling competencies (pp. 321–324). New York, NY: Springer. Hindman, A. H. (2013). Mathematics instruction in Head Start: Nature, extent, and contributions to children’s learning. Journal of Applied Developmental Psychology, 34, 230–240. https://doi. org/10.1016/j.appdev.2013.04.003

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Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41, 169–202. Jordan, N.  C., Glutting, J., & Ramineni, C. (2010). The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences, 20, 82–88. https://doi.org/10.1016/j.lindif.2009.07.004 Jordan, N. C., Kaplan, D., Locuniak, M. N., & Ramineni, C. (2007). Predicting first-grade math achievement from developmental number sense trajectories. Learning Disabilities Research & Practice, 22, 36–46. https://doi.org/10.1111/j.1540-5826.2007.00229.x Jordan, N.  C., Kaplan, D., Ramineni, C., & Locuniak, M.  N. (2009). Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology, 45, 850–867. https://doi.org/10.1037/a0014939 Kaiser, G., & Maass, K. (2007). Modelling in lower secondary mathematics classrooms – problems and opportunities. In W. Blum, P. Galbraith, W.-G. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education. The 14th ICMI study (pp. 99–108). New York, NY: Springer Science & Business Media. Kaiser, G., Schwarz, B., & Tiedemann, S. (2013). Future teachers’ professional knowledge on modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies. ICTMA 13 (pp. 433–444). New York, NY: Springer. Krauss, S., Brunner, M., Kunter, M., Baumert, J., Blum, W., Neubrand, M., & Jordan, A. (2008). Pedagogical content knowledge and content knowledge of secondary mathematics teachers. Journal of Educational Psychology, 100, 716–725. https://doi.org/10.1037/0022-0663. 100.3.716 LeFevre, J. A., Fast, L., Sowinski, C., Osana, H. P., Skwarchuk, S. L., & Manay Quian, N. (2009). Who’s counting? Numeracy and literacy practices of early learning and child care practitioners. Canadian council on learning final report. Ottawa, ON: Canadian Council on Learning. Lesh, R., English, L. D., Chanda, R., & Serife, S. (2013). Problem solving in the primary school (K-2). The Mathematics Enthusiast, 10, 35–59. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thoughtrevealing activities for students and teachers. In A.  Kelly & R.  Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 665–708). Mahwah, NJ: Erlbaum. Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5, 109–129. https://doi.org/10.1080/1098 6065.2003.9679996 Lesh, R., & Yoon, C. (2007). What is distinctive in (our views about) models and modelling perspectives on mathematics problem solving, learning, and teaching? In W. Blum, P. Galbraith, W.-G.  Henn, & M.  Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (pp. 161–170). New York, NY: Springer Science & Business Media. Lingefjärd, T. (2013). Teaching mathematical modeling in teacher education: Efforts and results. In X.-S. Yang (Ed.), Mathematical modeling with multidisciplinary applications (pp. 57–80). Holboken, NJ: Wiley. Martin, R. B., Cirino, P. T., Sharp, C., & Barnes, M. (2014). Number and counting skills in kindergarten as predictors of grade 1 mathematical skills. Learning and Individual Differences, 34, 12–23. https://doi.org/10.1016/j.lindif.2014.05.006 Mazzocco, M.  M., & Thompson, R.  E. (2005). Kindergarten predictors of math learning disability. Learning Disabilities Research & Practice, 20, 142–155. https://doi. org/10.1111/j.1540.5826.2005.00129.x National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: National Academies Press. Paolucci, C., & Wessels, H. (2017). An examination of preservice teachers’ capacity to create mathematical modeling problems for children. Journal of Teacher Education, 68, 330–344. https://doi.org/10.1177/0022487117697636 Romano, E., Babchishin, L., & Pagani, L.  S. (2010). School readiness and later achievement: Replication and extension using a nationwide Canadian survey. Developmental Psychology, 46, 995–1007. https://doi.org/10.1037/a0018880

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Schukajlow, S., Kolter, J., & Blum, W. (2015). Scaffolding mathematical modelling with a solution plan. ZDM, 47(7), 1241–1254. Sriraman, B., & Lesh, R. A. (2006). Modeling conceptions revisited. ZDM, 38(3), 247–254. Stender, P., & Kaiser, G. (2015). Scaffolding in complex modelling situations. ZDM, 47(7), 1255–1267. Stillman, G. A., Blum, W., & Kaiser, G. (Eds.). (2017). Mathematical modelling and applications: Crossing and researching boundaries in mathematics education. New York, NY: Springer. Stipek, D. (2013). Mathematics in early childhood education: Revolution or evolution? Early Education & Development, 24, 431–435. https://doi.org/10.1080/10409289.2013.777285 Thomas, K., & Hart, J. (2010). Pre-service teachers’ perceptions of model eliciting activities. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies. ICTMA 13 (pp. 531–530). New York, NY: Springer. Verschaffel, L., & De Corte, E. (1997). Teaching realistic mathematical modeling in the elementary school: A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28, 577–601. https://doi.org/10.2307/749692 Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). What’s past is prologue: Relations between early mathematics knowledge and high school achievement. Educational Researcher, 43, 352–360. https://doi.org/10.3102/0013189X14553660 Zawojewski, J. S., Lesh, R. A., & English, L. D. (2003). A models and modeling perspective on the role of small group learning activities. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 337–358). New York, NY: Erlbaum.

Chapter 14

Mathematical Modeling with Young Learners: A Commentary Elham Kazemi

These four chapters paint a vision of the possibilities and challenges we face in opening up rich modeling activities in the primary grades. Across them we see how young children and their teachers engage with modeling tasks. As I read across these chapters, I was interested in the nature of tasks that students engaged in, what counted as good modeling activities for elementary-aged students, what students did inside the situations they were modeling, how teachers responded, and finally what teacher educators might need to consider as they support teachers to take up modeling. I have organized my response in this commentary around a set of questions: 1 . Why engage in modeling activities with young children? 2. What kinds of modeling tasks were used in these classrooms? 3. What were the consequences for students of engaging in these tasks? 4. What can we learn from these studies about supporting teachers?

14.1 Why Modeling? Mathematics educators continue to search for ways to make the learning of mathematics in school contexts to be about meaning-making and intellectually rewarding problem-solving work. We posit that school mathematics should be empowering and identity-affirming work and that doing mathematics should contribute to

E. Kazemi (*) College of Education, University of Washington, Seattle, WA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_14

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students’ ability to engage and transform our world toward more just ends (see, e.g., Aguirre, Mayfield-Ingram, & Martin, 2013). Francis Su (2017) said in his presidential address to the Mathematical Association of America, “Mathematics is for human flourishing.” Mathematics, he argued, engages our desire for play, beauty, truth, and justice. If we can create learning environments where these desires are embraced, then we might more successfully enable students to experience mathematics as a means by which to flourish. Unfortunately, what is still more normative in mathematics instruction in the USA, and likely across the globe, are didactic approaches to teaching mathematics. In the classrooms described in these chapters, teachers are working to change this too-typical student experience and in the process open their classrooms to more play, beauty, truth, and justice. While engaging in the process of mathematical modeling, students engage in practices that may enrich how they know and do mathematics, their beliefs about themselves as mathematical learners, and their experience of the aims of doing mathematics.

14.2 What Kinds of Modeling Tasks Were Used in These Elementary Classrooms? The practice standards in the Common Core Mathematics Standards (2010) rightly state that the act of writing an equation to model a problem-solving situation is one way young children engage in modeling. These four chapters go well beyond this observation and create a much more vivid and inspiring view of what modeling can look like for young children, embracing the desires that Su described in his speech. Some of the tasks students pursued were playful, such as in Stankiewicz and colleagues’ chapter where stories were constructed from well-loved books read in the classroom or in Osana and Foster’s chapter, in which kindergartners designed an “ideal” kindergarten classroom. Some of the tasks were linked to students’ concerns with environmental change and justice. In Turner and her colleagues’ chapter, upper elementary students engaged in a modeling challenge inspired by their study of the environmental impact of plastic waste and the possibilities of upcycling plastic. The second-grade students in Ms. Applegate’s class described by Wickstrom and Yates were seeking truth in a way because they had the chance to use mathematical modeling to advise their principal about what toys she should actually buy for the school’s big field day at the end of the year. Modeling is defined in all of the articles through a cycle. The GAIMME (2016) report is a common citation. The modeling cycle begins with observing and interpreting a phenomenon, representing it through models of various forms, working with models to pursue and evaluate a solution, and finally rejecting or revising models based on an evaluation of how the modeling process produced a viable solution. What I found interesting was listening to how children explained this whole process. Across the papers, it appears that the cyclical nature of the process was not the

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most salient aspect to young children. What we hear in children’s ideas fits more with Francis Su’s ideas about desires. In Osana and Foster’s chapter, children found their modeling work to be fun and play-like and not at all like work. Wickstrom and Yates document a range of affective dimensions children experience while modeling including joy, creativity, freedom, connection, and reflection. And I was taken by children in that study who said: “Usually in math, we work alone or in partners and in that [sic] we had to learn how to work together in big groups and we were working on really big math problems instead of just little math problems at different times.” In a similar way, another student said, “Math modeling is basically like, it’s lots of different types of math in one where you’re trying to figure out like a big thing.” The children perceived their experience with modeling to be “big.” Their comments suggest that they were intellectually engaged because they got to put together smaller ideas and problems in service of a bigger one. As a reader, I sensed that they were motivated by and proud that they got to work on such big problems. These comments push us to consider the expectations that we have of children. They are capable of much more than what they experience in the typical school curriculum.

14.3 What Were the Consequences for Students of Engaging in These Tasks? Working on “really big math problems” as the students did in these classrooms invites us to appreciate children’s capabilities. The chapters revealed various ways of engaging students in these big math problems. The ideal kindergarten classroom task unfolded over 4 days of instruction, and on each day children tackled new constraints that were introduced by the teacher into the problem space. In contrast, the storybook inspired tasks began with a fairly simple premise. For example, one problem situation was, “Snake is hungry and he has 5 mice. How many more does he need?” In this classroom, we see that students generated the constraints themselves. How many more does snake need for what? The problems children created were based on the way they elaborated the problem situation and the constraints they themselves imagined. Similarly, in challenging students to make a classroom set of jump ropes by upcycling plastic bags, we observed various ways that children considered what it meant to have a classroom set. Does each child need a jump rope? Can different amounts of different lengths produce a set that can be used by a whole class even if there isn’t one jump rope per student? How do children actually use jump ropes in the PE classes at this particular school? Finally, the students who advised the principal on prizes to buy for field day drew on their actual experiences and what they imagined to be the ideal situation in figuring out what prizes to buy. Seeking their own truth for these varied situations created a different kind of working environment among students, and we have glimpses of the kind of collaboration and tensions that students navigated. It makes sense then that social themes were one prominent outcome of engaging in modeling tasks in the descriptions of students’ collaboration across these chapters. It was not always easy, for example, for

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students who were selecting the kind and quantity of prizes for field day to listen to each other and reach consensus. Citing Bahmaei (2011), Turner et  al. state in their chapter, “contexts that are meaningful in relation to students’ experiences allow students to draw upon situational knowledge and real-world considerations as they engage in mathematical modeling, instead of ‘cutting bonds with reality.’” Students, whether they were designing their ideal classroom, choosing prizes for field day, or creating a set of jump ropes, trusted their own knowledge and expertise. Turner et  al. observed, “Students did not accept data without question; they considered the reasonableness of the information against their own experiences, and accepted or refuted the suggestions accordingly…” While not all mathematics needs to be anchored in real-life phenomenon to be relevant and important to learn, it is still valuable that when students engaged in mathematical modeling, they did so with a feeling of ownership, authority, and voice. Across these chapters, while modeling as a practice is newly introduced to teachers, we can see how the design and enactment of the modeling tasks naturally connect to other important mathematical practices. The consideration of the context and its varied constraints invites sensemaking. As students explore the various dimensions of the problem, they have to make use of mathematical tools appropriately and attend to mathematical structure. As they listen and argue for various pathways through the problem, they are developing and evaluating mathematical arguments. So, while the modeling cycle is not prominent in the way we typically talk about the content of elementary mathematics, these chapters illuminate the opportunities and promise for children’s mathematical learning of putting more energy into designing and using more ambitious modeling projects.

14.4 What Were the Consequences for Teachers of Engaging in These Tasks? Children were more active sensemakers in these classrooms, and again the use of these open-ended and complex modeling tasks seemed to have opened up opportunities for teachers’ own implicit biases to be challenged. Much of the emphasis in school mathematics on speed and accuracy curtails our ability to see what children can do. As I read one teacher’s comment in Osana and Foster’s chapter, I found myself reflecting on the power of creating experiences for teachers that enable them to see children better “..there was one little girl, [who] is very weak. She just started school here at the end of February and she is absent very often so I find she is a little bit behind and she was the one taking leadership and saying ‘no no no no we have to count them and make sure we have the same stickers!’ and I thought wow!” I paused on the teacher’s characterization of the girl as “weak” and was hopeful that her observations of the child digging into a more complex task and taking on leadership role challenged this deficit perspective. There is so much power in centering children’s ideas and voices when teachers allow themselves to notice what children can do.

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14.5 What Can We Learn from These Studies About Supporting Teachers? Attuning teachers’ professional vision to notice what students are doing mathematically, socially, and affectively when they are engaging in the modeling process certainly takes work. We are reminded through the accounts in these chapters about the complexity of developing teachers’ knowledge about children’s thinking, mathematical content, and mathematical teaching practice in relationship to one another. The teachers in Osana and Foster’s chapter had begun some professional learning about the concept of number in the early elementary grades but in their interviews did not seem to notice or talk much about the nuanced ways in which the researchers had interacted with students in order to better understand as well as press on their mathematical understanding. The researchers could have asked more directed questions so that teachers would reflect on pedagogical choices in instructional conversations, and their observations made me wonder about the need to develop explicit frameworks that braid together content, student thinking, and practice in professional learning experiences. Some of the research activities in these articles may in fact be fodder for professional learning activities. Observing another teacher interact with one’s own students can challenge one’s perceptions. Two observations from Osana and Foster’s chapter sit with me: (1) the teacher who is surprised by a “weak” child’s leadership actions (2) and a comment that children’s unruly behavior has to be contained. Designing professional learning opportunities for teachers to witness children’s brilliance and capabilities is vital. I was interested in the teacher’s view of her students when she said that a constraint she experienced in implementing modeling activities is that, “Students had to be ‘calm.’ As soon as I sit down, or when they’re done, they are getting out of hand..so I find that working in a smaller group while the others do free play or something like that would be easier than as a whole group.” Creating a respectful and joyful learning community takes much intentionality and skill. Supporting teachers to develop the norms for children to be leaders, boisterous, and active in sharing their ideas is going to produce different kinds of challenges than managing classrooms where children do independent work at their seats. Another data collection strategy used in these chapters that could be useful in professional development is asking students to define mathematics and reflecting on their responses. We learn a lot about children’s perceptions and developing understandings when they define mathematics and mathematical modeling. If they characterize them in opposition to one another, then teachers would know that their classroom experiences are not yet coherent for students. Teachers themselves will need more experiences engaging in mathematical modeling. Creating tasks that work with their students will take time and intentionality. Stretching instruction over a number of days will require materials and new skills in how to carry conversations from one day to the next rather than working on discrete skills in bite-­ sized pieces.

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14.6 What Implications Do These Studies Suggest for Future Work? As a set, these chapters provide complementary views into the possibilities of engaging elementary-aged children in complex meaning-making through mathematical modeling. We learn that young children are quite capable of taking on these tasks. We see the mathematical, social, and affective dimensions of this work as students have to engage each other’s ideas over multiple connected days of instruction. Even the idea that a problem can be worked on over time is not a small matter given that much of what children typically get to work on involves just a few minutes of work. We see that children can have sustained interaction with a big problem as new constraints are considered. Modeling as an entry into other mathematical practices such as developing arguments, attending to precision, making use of structure, learning to use tools appropriately, and of course sensemaking is clearly apparent across all of these chapters. Importantly, the children’s voices tell us important things about how they experience modeling as more authentic. Their views of themselves and of mathematics are more expansive. Given the clear benefits outlined in these chapters, more collaborations will be needed to generate contextually specific and rich modeling tasks that will enable young learners to engage critically with real-world phenomena of direct importance to them. We need to continue to study the range of instructional decisions that are made over the course of the modeling cycle that enables students to benefit from each phase. Teachers too will need to deepen their understanding of interpreting and revising solutions to modeling situations based on varied constraints.

References Aguirre, J., Mayfield-Ingram, K., & Martin, D.  B. (2013). The impact of identity in K-8 mathematics: Rethinking equity-based practices. Reston, VA: National Council of Teachers of Mathematics. Bahmaei, F. (2011). Mathematical modeling in primary school: Advantages and challenges. Journal of Mathematical Modeling and Application, 1(9), 3–13. Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME). (2016). Consortium of Mathematics and Its Applications (COMAP), Bedford, MA, and Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Retrieved from http://www. siam.org/reports/gaimme.php National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards mathematics. Washington D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers. Su, F. (2017, January 8). Mathematics for human flourishing [Blog post]. Retrieved from https:// mathyawp.wordpress.com/2017/01/08/mathematics-­for-­human-­flourishing.

Part IV

Interdisciplinary and Community-Based Modeling

Chapter 15

Convergent Nature of Modeling Principles Across the STEM Fields: A Case Study of Preservice Teacher Engagement Andrew Gilbert and Jennifer M. Suh

15.1 Introduction: Issues Facing Elementary Teachers in STEM A recent report by the Committee on STEM Education of the National Science and Technology Council (NSTC) outlined the strategies for STEM education entitled Charting a Course for Success where one pathway for success was “Engaging Students Where Disciplines Converge.” It stated that STEM concepts are best learned at the K-12 level where: We make STEM learning more meaningful and inspiring to students by focusing on complex real-world problems and challenges that require initiative and creativity. It promotes innovation and entrepreneurship by engaging learners in transdisciplinary activities such as project-based learning, science fairs, robotics clubs, invention challenges, or gaming workshops that require participants to identify and solve problems using knowledge and methods from across disciplines. (p. vi, White House, 2018)

This strategic pathway connects with many of the national educational initiatives promoting the connections between STEM disciplines and real-world problem solving as described in the Principles and Standards for School Mathematics, imploring that students “solve problems that arise in mathematics and other contexts” (NCTM 2014, p. 402). The Common Core State Standards (2010) corroborate this notion with a standard on modeling and suggest that mathematics programs Gilbert, A. & Suh, J.M. (In press). Convergent nature of modeling principles across the STEM fields: A case study of preservice teacher engagement. In J.M Suh, M. Wickstrom & L. English (Eds.), Exploring the Nature of Mathematical Modeling in the Early Grades. New York: Springer. A. Gilbert (*) · J. M. Suh College of Education and Human Development, George Mason University, Fairfax, VA, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_15

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should prepare future teachers that can help children to “solve problems arising in everyday life, society, and the workplace” (p.  7). Similarly, the Next Generation Science Standards (NGSS, 2013) address an urgent need to transition from traditional science approaches composed of disconnected content areas toward a more realistic enactment of science. These are designed to not only build content knowledge but also develop the skills of communication, collaboration, and problem solving, which are the hallmarks of innovation in STEM. However, future teachers in the USA have had inadequate preparation in STEM (PCAST, 2010). There exist complicating factors for elementary teachers in STEM, mainly surrounding the well-documented fears and anxiety freighted with learning math and science content and the prospect of having to teach within those content areas. STEM education has multiple facets and nuances depending on the context with seemingly infinite combinations of possibilities. In order to provide clarity for our own use of this term, we utilize Vasquez et al. (2013) definition for STEM education, “STEM education is an interdisciplinary approach to learning that removes the traditional barriers separating the four disciplines of science, technology, engineering and mathematics and integrates them into real-world, rigorous and relevant learning experiences for students” (p. 4). Current discussions regarding STEM typically operate on a presumed interest of the student, where if the content is well structured, accurate, and focused on real-world issues, the students will be able to easily engage in meaningful STEM learning. This assumption discounts many serious contextual issues that directly relate to the preparation of elementary teachers across the STEM fields. These issues represent a broad range of challenges faced by both students and the preservice teachers. For instance, if we focus solely on the science facet of STEM, those challenges could include: problematic access to scientific discourse (Kelly, 2016), individual identity formation (Brown et al., 2017), distinct lack of confidence in science content (Weld & Funk, 2005), an incomplete understanding for the nature of science (Akerson et al., 2006), low levels of science self-efficacy (Kazempour, 2014), and well-developed fears of teaching science (Akerson & Volrich, 2006). These issues push many preservice elementary teachers to opt out of rigorous science preparation during their academic careers (Tytler, 2007), which impacts both their understanding of science content and the process of science itself (Marcum-Dietrich et al., 2011). Lastly, many future elementary teachers will choose not to teach science in their classrooms, because of these fears and lack of confidence concerning content understanding (Kenny et al., 2014). These issues frame just one content area (science) that those wishing to engage future teachers in STEM practice, there are also similar challenges faced in mathematics. Researchers have investigated math anxiety faced by prospective elementary math teachers through personal narratives revealing feelings of mathematics anxiety that range from the struggles to understand mathematical content, dealing with continuous competition with others, avoiding risk-taking, and experiencing feelings of embarrassment that made them believe they were unsuccessful in mathematics (Drake & Sherin, 2006; Stoehr & Carter, 2011; Stoehr, 2012). Despite the math anxiety faced by prospective elementary teachers, Swars et al. (2006) found that

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teachers asserted that using real-life situations in mathematics was important for motivating students to learn mathematics and would be a crucial aspect of their mathematics instruction. This pedagogical dilemma is an important problem that educators need to tackle so that efficacious views that preservice teachers have toward using authentic situations in mathematics, which align with the reform vision of NCTM, can be enacted in more classrooms. An important implication from these studies is that prospective teachers value the use of real-­world situations in mathematics, but they need more support from teacher educators and greater attention paid to these ideas in methods courses in order to enact teaching through authentic problem solving. One of the solutions to alleviating these fears is to use modeling and model-­ eliciting activities (MEAs) as a form of pedagogical instruction to build teachers’ content understanding and problem-solving approaches often employed by scientists, engineers, and mathematicians. These professionals are faced with problems where solutions are not apparent and answers are almost never composed of simple manipulation of a memorized algorithm. Rather, problems tend to contain complex interactions of content-related principles. Current reforms reflect this complexity calling for twenty-first-century learners who can communicate, collaborate, as well as think critically and creatively (Partnership for 21st Century Skills, 2011). We argue that more fully integrated STEM approaches offer elementary preservice teachers the opportunity to demonstrate these twenty-first-century skills.

15.2 Literature Review Modeling as a Practice That Connects STEM Discipline  In the STEM fields, mathematical modeling becomes necessary (Levy, 2015, para. 2) because mathematics is essential in decision-making, and needed to “optimize a limited resource such as time, money, energy, distance, safety or health.” In many school districts STEM Problem-Based Learning (PBL) modules are being implemented without much vetting for rich content or sound pedagogy. We contend this could be due in part to the lack of high-quality professional development for developing efficacy in knowledge and pedagogies associated with STEM instruction. Research has indicated that the most effective professional development is rooted in experiences that allow teacher time to dig into content, analyze instructional decision-making, reflect on their practices, and formulate responses (Darling-Hammond, 2006; Desimone, 2009). Researchers further argue that when learners are provided opportunities to enact the community practices of science (and STEM more broadly), they are able to gain content knowledge and participate in productive inquiry (Calabrese-Barton & Tan, 2009; Varelas et al., 2006). Teachers and children rarely get meaningful opportunities to problematize and hypothesize about science content which is central to the disciplined thinking in science (Phillips et al., 2018). We also contend that problematizing and hypothesizing about content-related phenomena is essentially important in both engineering and mathematics.

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Integrating across the broad swath of content involved in STEM, particularly given elementary teachers’ anxieties in terms of mathematics and science, still remains a serious issue for those teaching both inservice and preservice teachers. Recent reforms in science (NGSS, 2013) and in mathematics (Common Core Standards, 2010) have created broad visions for how these contents can be actualized within these content areas. For instance, the National Research Council (2014) articulated a vision that moves beyond a single content-area context toward a framework of STEM integration that included: • Goals  – STEM literacy, twenty-first-century skills, work readiness, interest in STEM • Nature of integration – type and disciplinary emphasis within STEM • Outcomes – STEM achievement, learning, interest, and building STEM identity • Implementation – pedagogy and the creation of the STEM learning environment Our work encompasses these aspects of the STEM framework and focuses closely on how the nature of integration and implementation can directly impact the outcomes and goals of integrated STEM learning. In particular, we argue for the primacy of modeling to inform both the nature of integration and the implementation of PBL STEM approaches. This is a direct attempt to address NRC’s (2014) call to build cross-sector collaborations to spur innovation into preservice and inservice teacher professional learning and the associated impact on children. The Next Generation Science Standards also articulated that unlocking meaningful science instruction must focus on three organizing concepts: (a) science and engineering practices, (b) disciplinary core ideas, and (c) crosscutting concepts. These dimensions represent the organizational framework for scientific engagement and developing twenty-first-century science knowledge. Embedded within the science and engineering practices are a myriad of key components that comprise a range of traits to engage learners in science, which include: (1) asking questions and defining problems, (2) planning and carrying out investigations, (3) analyzing and interpreting data, (4) developing and using models, (5) constructing explanations and designing solutions, (6) engaging in argument from evidence, (7) using mathematics and computational thinking, and (8) obtaining, evaluation, and communicating information (NRC, 2012, p. 42). These practices framed the activities that we designed, but we also considered where we might be able to focus on one particular area that would allow us to delve deeply into integration of mathematics and science. Research Regarding Modeling in Science and Engineering  Modeling in the sciences is generally defined as a representation for structure and behavior of a scientific principle while simultaneously providing a framework to explain, communicate, and predict the mechanisms that determine the nature of those phenomena (Cheng & Brown, 2015; Schwarz et al., 2009; Windschitl et al., 2011). Modeling as a teaching tool in science education provides a mechanism to construct understanding of science content, communicate that understanding, and visualize abstract scientific principles. Schwarz et al. (2009) contend there are four key elements to enacting the process of modeling in the science teaching contexts: construction, utilization, eval-

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uation, and revision. This process is aligned with both design-based practices and procedures associated with mathematical modeling. These processes are essential to help students construct understanding and provide evidence for the explanations of natural phenomena. Cheng and Brown (2010) argue that when trying to engage non-scientists in the construction, use, and revision of models, they must develop specialized forms of knowledge in order to recreate abstract concepts. When given ample scaffolding and support, children were able to revise their models to better represent the visualization of both the phenomena and their explanations of those phenomena in more coherent and complex ways (Cheng & Brown, 2015; Harlow, 2010). This is not just important for children, but will be especially important for future teachers that have had continued lack of success in mathematics and science as well as a general misunderstanding for what the nature of mathematics and science teaching should entail in the classroom. This vision for the utilization of modeling to frame science teaching practice requires “places high demand on teachers” (Schwarz et al., 2009, p. 633) and the resulting activities planned by our team worked to address those high demands in preparing teachers to carry out exceptional modeling practices. Similarly, designing, building, and refining models (to explain scientific phenomena) through observation of the natural world, documenting those observations with data, identifying patterns within data, and testing explanations for those apparent patterns are processes at the very core of the scientific endeavor (Hubber & Tytler, 2013; Mulder et al., 2010). These parameters also directly address the “science and engineering practices” articulated by NGSS, where creating a representation or model leads to powerful thinking (Prain & Tytler, 2012) while simultaneously providing avenues for discussing and framing content-related ideas (Nielsen & Hoban, 2015). Furthermore, processes not only provide opportunities for learners to observe phenomena, but also to build reasoning and explanations for how those phenomena operate (Braaten & Windschitl, 2011; Kim et al., 2007). Modeling in engineering requires design thinking where a problem solution is prototyped and becomes a model and encourages iterative thinking as students apply their discipline specific knowledge to solving that problem. The design thinking process involves understanding the problem through empathy, defining the issue at hand through problem formulation, generating many possible solutions through ideation, prototyping several solutions, and testing the solutions and getting feedback (Suh & Seshaiyer, 2014). Consequently, these scientific and engineering practices have a great deal in common with the mathematical practices outlined in CCSSM with mathematical modeling (NGACBP & CCSSO, 2010). Research Regarding Modeling in Mathematics  As stated in the Common Core Standards for Mathematical Modeling, “Real-world situations are not organized and labeled for analysis; formulating tractable models, representing such models, and analyzing them is appropriately a creative process” (p. 72). These real-world problems tend to be messy and require multiple mathematics concepts, creativity, and involves a cyclical process of revising and analyzing the model.

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Modeling links classroom mathematics and statistics to everyday life, work, and decision-­ making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. (p. 72)

English (2006) described several task features that are important to consider when designing modeling-based classroom experiences which includes planning for (a) multiple interpretations, approaches, and models; (b) motivating and challenging multidisciplinary contexts that support the existing curriculum; (c) incorporation of different forms of data; (d) modeling as a springboard for further statistical investigations; (e) group collaboration and communication; and (f) opportunities for peer reporting (English, 2006). For modeling to become part of the curricular practice and content, it is important to link modeling activities to the school’s current curriculum so that they become integral components of students’ studies and are not seen as add-ons. In many model-eliciting activities, students are presented with situations where they are confronted with the need to develop a model, followed by an opportunity to revise or refine their current ways of thinking about the given problem situation. A unique characteristic of a model is that it can be usable and shareable with others as well as generalizable in that it can often be applied in other problem situations (Lesh & Yoon, 2004). In MEAs, problem solving is included in the mathematical activity (i.e., the processes of problem posing, abstracting, computing, and checking the solution against a real context) of solving an issue in the real world. Model-eliciting activities focus on developing a conceptual system (a model) that is useful for some purpose that is understood by the learner (Lesh et al., 2000). We chose to hone in on the notion of modeling since it covers a variety of the approaches called for in reforms and policy documents across the STEM fields. For instance, in the description of “developing and using models,” NGSS states: A practice of both science and engineering is to use and construct models as helpful tools for representing ideas and explanations. These tools include diagrams, drawings, physical replicas, mathematical representations, analogies and computer simulations. Modeling tools are used to develop questions, predictions and explanations; analyze and identify flaws in systems; and communicate ideas. (NGSS, 2013, n.p.)

Reform initiatives (NGSS and NCTM) are recognizing the importance of modeling and the powerful learning opportunities associated for teachers and children.

15.3 Conceptual Framework Modeling Instruction as Perfectly Positioned to Take on the Challenge of STEM Teaching  Examination of national standards and teaching practices in the STEM disciplines reveals a parallel process (see Table  15.1) where modeling is used in mathematics, science, engineering, and computing. These standards promote students collecting real-world data and organizing, displaying, analyzing, and

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Table 15.1  Parallel processes where modeling is used in mathematics, science, engineering, and computing Inquiry through scientific modeling (NGSS, 2013) Students as scientists explore phenomena by asking questions, conducting experiments, collecting and analyzing data, arguing about evidence, and communicating findings

Problem posing through mathematical modeling (NGACBP & CCSSO, 2010) Students, as mathematics modelers, pose problems in a real-world context and consider what data/ information is necessary to answer the question. Mathematics is used to make a decision on the best, optimal choice or make predictions

Design thinking through engineering (Stanford D-School) Students, as designers, use the key process of design thinking that encourages iterative thinking to solve real problems. They pose a problem, brainstorm ideas, rank them, and prototype, test, and refine the design

Computing and technology (CSTA, 2017) Students, as data scientists, organize and present collected data visually to highlight relationships and use data to propose cause-and-­ effect relationships, predict outcomes, or communicate an idea/ model

interpreting data in real time using the engineering design and scientific processes (NGSS, 2013). In addition, NCTM promotes developing mathematical ideas and solving problems (NCTM, 2014) where students formulate questions, design studies, and collect data; select, create, and use appropriate graphical representations of data; and analyze and interpret data. The science and mathematics education community advocate a common goal of developing quantitative thinking and scientific reasoning as children interpret real-world contextual problems. In fact, modeling scientific and engineering practices overlap. Schwarz et al. (2009) operationalized the practice of modeling to include four elements: • Students construct models consistent with prior evidence and theories to illustrate, explain, or predict phenomena. • Students use models to illustrate, explain, and predict phenomena. • Students compare and evaluate the ability of different models to accurately represent and account for patterns in phenomena and to predict new phenomena. • Students revise models to increase their explanatory and predictive power, taking into account additional evidence or aspects of a phenomenon (p. 635). Interpreting Modeling in Inquiry Contexts  The authors each came to this teaching journey from their own content and teaching locations that were steeped in inquiry, but framed mainly within our own areas of expertise (science and mathematics, respectively). One of our first noticings was that the language of modeling in mathematics and science shared common elements that we were interested in exploiting to help teachers make sense of how to interpret STEM integration. We each mapped our own vision for how modeling gets operationalized within the content and pedagogical expectations that are expected within our respective fields. Suh’s interpretation for the operationalization of modeling in mathematics education is below. Using seminal research on modeling (Lehrer & Schauble, 2010), we

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Fig. 15.1  Mathematics modeling in inquiry contexts

Fig. 15.2  Scientific modeling in inquiry contexts using the 5E framework

used this model to engage our teachers in identifying and deconstructing the problem, designing an approach to gathering information about the problem, mathematizing the problem, building a model, refining a model, using the model to address the problem, predicting from the model, and analyzing and interpreting the solution (Fig. 15.1). Gilbert noticed that this conception of modeling in mathematics was quite similar to how scientists conceptualize modeling as a means to problem solving in the sciences. He shared his interpretation with our preservice teachers regarding scientific modeling with the context of a 5E pedagogical framework. The 5E starts with engagement through “wonder” prompts, designing an investigation and gathering data to develop an explanatory model, elaborating and extending by connecting it back to the big ideas or scientific phenomenon, and sharing the results (Fig. 15.2).

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Based on these assumptions regarding the overlapping nature of modeling across mathematics and science, we conceptualized a modeling process across mathematics, science, computing, and engineering. These frameworks for modeling informed our integrated efforts during the content delivery of our methods course.

15.3.1 Goals for the Chapter We argue that MEAs offer powerful and accessible means to engage teachers in the process for how mathematicians, scientists, and engineers work to solve real-world authentic problems (English, 2006). To this end, we articulated an approach that actively combined modeling frameworks from both the science and mathematics education fields to build an integrated problem-solving approach. The specific research questions investigated in this instrumental case study included: 1. How does immersing preservice teachers in integrated STEM modeling approaches impact their interest and understanding in STEM integration? 2. How do preservice teachers begin to plan and enact in ways that support STEM modeling instruction and see how the STEM disciplines converge? Through the exploration of these two research questions, we describe the preservice teachers’ journey as STEM learners and modelers as they immerse in the process of modeling and sense making and detail how they began to notice how the STEM disciplines converged while enacting their own STEM MEAs. The chapter describes the context where this work took place, articulates researcher conjectures regarding the convergence of scientific and mathematical modeling, highlights the teaching process, and discusses the overarching implications for integrated STEM approaches. Through these descriptions, other STEM educators will witness the learning and teaching journey of two veteran elementary teacher educators as they worked to integrate key tenets of modeling across mathematics and science. The goal was to tease out important overlaps and differences in order to highlight a detailed vision for the convergence of modeling across science and mathematics as well as building trusting relationships needed to teach with courage that integration demands.

15.4 Methods Case Study Approach  The structure and approach of this qualitative study was defined as an instrumental case study (Stake, 1995, 2000). Instrumental case study differs from the traditional notion of case study research because the questions of the researcher are paramount as opposed to the case itself. This research method was chosen because we wanted to better understand the specific impact of modeling on both preservice teacher interest in STEM and how they understood and enacted

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those processes. We created our research design to best capture specific events (Schram, 2006) that were steeped in STEM integrated modeling tasks as a means to shed light on the specific questions that framed the instrumental case. The main value of utilizing this approach was to study the complex situations that impacted the participants’ thinking toward STEM engagement and science learning and to cast a light on what we can learn from these cases (Flyvbjerg, 2006; Stake, 1995). Case study research allows for the methodological freedom (Stake, 1995) to utilize ethnographic data collection and analysis that were most appropriate for the questions that were investigated within this project. Context and Participants  This case study took place during an integrated math and science methods course over the summer of 2018. The case itself included 13 preservice teachers, all of whom were enrolled in a Master’s in Education program designed to provide initial licensure for elementary classrooms. The preservice teachers had completed their first year in the program that also included field-based experiences across that first year and they began a year-long internship in the fall of 2018. The course itself was conceptualized and carried out by the authors as a way to provide future teachers with examples of integrated mathematics and science practices. This approach rose from our frustration with trying to navigate the sometimes rigid traditions that frame university teacher preparation courses that are structured as discrete subject areas that do not easily allow for truly integrated experiences. Future teachers are often expected to integrate across content areas when they enter the professional classroom, particularly given many new STEM initiatives in schools. To this end, we crafted a six-credit course embedded in a diverse local school and taught the course on-site during the last month of the public school year, which coincided with the first summer session of the university. We leaned on the strength of our university’s professional development school program that allowed us to teach on-site and place our preservice teachers into school classrooms to carry out the integrated STEM tasks. This summer program was designed as a testing ground for how we will consider integrated approaches during the standard 15-week semester in the future. The participants engaged in numerous integrated math and science approaches across the first two weeks of the course (6 h per day, Monday through Thursday, with both Fridays as reading days). In week three, the participants designed integrated STEM mini-units. These included three connected lessons that enacted a 5E approach (Bybee, 2002) and taught in teams within elementary classrooms on-site of the school. Participants were supported by faculty before, during, and after the teaching. Lastly, participants designed (with the help of university faculty) a STEM fair open to all students of the school across grades K-6. We designed the course using what we termed the “parallel learning” model for teachers and students. This Envisioning STEM Through Learning and Teaching Cycle (Fig. 15.3) involved teachers immersed as learners along with children. We designed STEM PBL where our preservice teachers experienced firsthand the activity as learners and modelers and then have them work directly with children, classroom teachers, and faculty.

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Fig. 15.3  “Parallel learning” – Envisioning STEM Through Learning and Teaching Cycle

Fig. 15.4  Aluminum boats  – sample STEM MEA activity  – using mathematics to look at the relationships between the design of their boats and the optimal weight it can hold

Using Lesh, Hoover, Hole, Kelly, and Post’s six design principles for MEA Lesh et al., (2000), we immersed our teachers with activities that required 1. The model construction – activity that required to use mathematics to describe, explain, or justify a prediction; 2. The reality – connect to situations that happen in real life; 3. The self-assessment – activities that allow students to determine if their solution can be improved, refined, or extended; 4. The documentation  – students report on their thinking about the situation/ phenomenon; 5. Shareability and reusability – solution can be shared and modified for other situation; and 6. The effective prototype – the model can be used to relate to other STEM phenomena or mathematical situations. In one of the STEM modeling lessons, we utilized the construction of aluminum boats (Fig.  15.4) as a means to begin hypothesizing how ships made of material denser than water can float and used it as a means to provide teachers with powerful avenues toward thinking about science phenomena, engineering, and a means to

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Fig. 15.5  Designing a wind-powered car and using a CBR to measure the speed at (m/s)

meaningfully integrate connections to mathematics and mathematical modeling. In addition, preservice teachers designed a wind-powered car and tested their vehicles using the CBR (Calculator-Based Ranger) which allowed them to compare the data and the graphical representation as they compared who created the “best” wind-­ powered vehicle (see Fig. 15.5). This exposed our preservice teachers to collecting data in real time and describing patterns in data visualizations, through charts or graphs, to make predictions (CSTA, 2017). After experiencing both STEM modeling activities as learners, we asked teachers to debrief with the instructors. Next, they delivered these activities with students to experience the teaching of STEM MEAs with elementary students. Finally, for their final clinical practice assignment, we asked preservice teachers to collaboratively plan and teach a STEM MEA lesson and debriefed with the instructors. The preservice teachers were then placed in classrooms within the school where the STEM course was embedded during the final few weeks of the school year. The elementary school itself has a large and diverse population with approximately 860 students representing a broad range of ethnicities including: 39% White, 24% Asian, 22% Hispanic, 10% Black, and 5% representing two or more ethnicities. In addition, 25% are identified as economically disadvantaged and approximately 36% are English language learners (ELL). The preservice teachers were the focus of this study, but the school context is provided so that readers can understand the context from which the course and participant data was carried out. Data Gathering  In order to best understand preservice teacher experiences, we collected key data artifacts associated regarding STEM modeling. We focused these artifacts on two specific integrated content approaches that the authors felt were best suited to STEM modeling tasks, namely, designing wind-powered cars and aluminum foil boats. These data collected during these elements of the course included preservice teacher open-ended journal entries and responses to course writing prompts. The reasoning for focusing on these particular tasks was because they were best suited to demonstrate strong elements of integrated STEM modeling, because

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not all model-eliciting activities and/or guided inquiry approaches are well suited for STEM integration. These journal entries were examined to assess preservice teachers’ conceptions of learning through STEM modeling activities and what they noticed as learners that would impact them as teachers. One of the entries asked them to reflect on how the hands-on math and science activities from the STEM fair gave diverse learners access to the content: Describe your station and give us a vignette of how you differentiated for diverse learners both in math and in science concepts shared. In addition, we examined lesson plans within their STEM modeling units and individual reflections to examine how the second-phase teaching through STEM modeling was conceptualized and experienced by our preservice teachers. Data Analysis  The research team utilized online qualitative analysis tool Dedoose© to organize, reduce, and analyze all aspects of the data set. In an effort to build credibility, data sets were subjected to multiple complete readings where one researcher generated a list of preliminary emergent themes and coded all data into those categories (Miles & Huberman, 1994). We subjected these codes to testing inter-rater reliability using Cohen’s kappa and found our agreement to be 0.82, which is considered to be within the range of a high-degree of agreement. Since we demonstrated strong agreement across the codes, we proceeded to work together to negotiate these initial codes into larger more robust themes (Strauss & Corbin, 1998). As such, we met and discussed themes to examine commonalities and differences that provided key insights into the preservice teacher experience and visions for STEM modeling. These efforts were predicated on Holliday’s “principle of emergence” (Holliday, 2007, p. 93), where the entire data set is viewed thematically and changed and evolved as part of the emergent process.

15.5 Findings The first research question sought to understand if immersing preservice teachers in integrated STEM modeling approaches would impact their interest and understanding in STEM integration. Three main themes emerged regarding how preservice teachers described their views on STEM modeling and integration as learners, which included: characterizing STEM modeling, engaging diverse learners, and recognizing and valuing STEM convergence. Characterizing STEM Modeling  Preservice teachers articulated that STEM modeling and understanding STEM processes were essential aspects in their building content understanding. In other words, the process drove the content. This was an important notion particularly in the context of elementary teaching, where so much emphasis is placed solely on memorizing content or finding the one solution (Milne, 2010). This was also depicted as preservice teachers worked through ­elements of engineering when solving small-scale engineering design issues that directly reflected on how they were using science and mathematics concepts to improve their designs.

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But as we were doing our activities I found myself choosing to do the more precise math work in order to be able to accomplish a task or evaluate data to help my understanding of a concept. For instance, when I tested my wind powered car I noticed that it kept veering to one side which I thought made it less likely that it was travelling as far as it would if it was straight. I realized that when I constructed the sail part of my car I didn’t measure to make sure it was the same size on both sides of the mast. When I went back and measured to make sure both sides were the same length the car stayed straight and nearly doubled the distance it travelled.

This brings the role of engineering to the fore as a means to make sense of mathematics and science content in real-world questions or problems. Here this preservice teacher saw the power in her figuring out the issue she was facing. She highlights the importance of precise mathematics work and how that precision helped accomplish a task and evaluate data to help better understand a concept. This counters the over-utilized approach of taking children through a lock-step series of procedures to ensure that all students achieve “success” in following the same path to one single content-based answer. The notion of interest was also at the heart of this theme, where students’ intense need to know drove them to continually strive for solutions to the design and/or content dilemma they were facing. Preservice teachers also mentioned notions related to freedom of thought and not overly constraining student ideas through overly structured methods. For example, in the car/wind turbine activity, learners get to think for themselves in building their models. They get to come up with their own questions and figure out their own answers. This access to unlimited creativity forms genuine results and data that learners are more motivated to work with. Answering the big picture question, “So what?” after their experiments gets learners to think critically about what their data means.

The “unlimited creativity” is often something that new teachers fear because it can lead to so many big questions. However, engaging in the multiple possible pathways offered by STEM integration provides preservice teachers with a measure of trust that students will continue to engage in finding solutions to the overlying challenge and that those processes are also valuable learning journeys for their students. In some ways, it can be said our preservice teachers began to envision their practices with new eyes. Something I’m beginning to realize with our STEM integrated activities is the role that math plays in science particularly in design, data gathering /representation/analysis and how that helps fine-tune the concepts we learn or teach. I think it’s critical to find activities that allow access without minimizing content.

Engaging Diverse Learners  An essential aspect of current STEM-related research foci and educational reform efforts are squarely focused on creating more opportunities and access for diverse learners in STEM.  Consequently, engaging learners from diverse backgrounds was an essential aspect of the STEM goals within the study and course approach. Therefore, the course itself was held at a school site with a highly diverse student population where preservice teachers were placed in classrooms to teach the integrated STEM lessons that they designed. These were direct efforts to counter prevailing teacher attitudes that often exist concerning diverse students, where children are often seen as less capable to engage with challenging content even when they demonstrate effort, desire, and ability (Gilbert, 2013). By having future teachers deliver active and challenging approaches, we provided an

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opportunity for future teachers to make the road by walking and build pathways for understanding how to bring integrated STEM teaching into diverse contexts. The following excerpt highlights how preservice teachers described STEM as a means to connect diverse students to content. Many students will need additional support or stimulation beyond the “basic” instruction. Additionally, instructional methods need to consider the audience. For example, if there are a high number of ELLs in a classroom, using lengthy word problems will likely be an ineffective method. Methods should also consider culture, giftedness, learning differences, and socioeconomic status to ensure that the math is presented within a familiar context. You may not want to use an example of an airplane if none of the students have ever been on one, as that is not an experience with which they can relate.

The key here is that the preservice teacher is placing the student as not deficient because they have not traveled on an airplane or that they are actively learning English. These are seen more as factors for how teachers need to adjust and rework their own practice. This directly carried into how they also envisioned similar means to directly connect children to STEM content and processes. Students can work within the level of math that is appropriate, whether that is measuring (weight, time, distance, materials), cost of building the car, collecting data as they perform their time trials, comparing data with other classmates, elaborating on the activity by changing a variable to see how it impacts another variable, etc. All of these are valuable math opportunities and the activity – which students can participate in regardless of their focus – should keep students engaged and motivated while showing them how math can be used in a meaningful way.

This theme of finding motivation and interest through engagement within the process and content of STEM ran through all of the examples within this theme. As such, the power of STEM integration in diverse settings was a catalyst for how they began to operationalize what STEM modeling entailed. The following section highlights how these future teachers were able to articulate and identify key elements of integrated STEM modeling during this experience. Recognizing and Valuing STEM Convergence  The questions became would future teachers envision these more drawn out and less predictable integrated approaches as valuable in the classroom? In addition, would they see how they inform one another and how these processes might converge in the classroom? The following excerpts provide insights for the extent to which the participants wrestled with the notion for how these aspects of STEM content and processes converged within this experience. Surprisingly, the preservice teachers did not conceptualize this as something to fear or to shy away from, but rather it was a conduit to building a number of connections to content. When it comes to science, it is easy to pick out a lesson. Flotation, volume, physics. When picking out the math, the boat experiment can really be used to attack problems like volume and area, and other units of measurement. It can also be used for figuring out regrouping methods as well. I had to regroup and try different combinations of numbers to be able to keep my boat floating. The car experiment is good for speed, distance and length traveled. I could also try and make a lesson on guessing and predicting future outcomes since they will be using different trials. These could both be differentiated for different students by changing up the materials and the difficulty and extent of the data collected for each.

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These future teachers maintained a strong vision for the value of converging these ways of knowing, thinking, and acting across STEM.  This takes on even greater importance in the crowded curriculum of the elementary classroom, where teachers are under ever-increasing pressure to reach more and more content standards with children. In addition, these processes better match calls for integrated practice across crosscutting themes of the NGSS (2013). This notion of truly seamless convergence across mathematics and science because of the overlapping nature for modeling processes also came through in many future teachers’ comments, which is best exemplified by the following excerpt: Something I really started to notice this week was how I have a hard time remembering which lessons and activities we have done are science and which are math. I think that is so cool! I think I used to have a binary way of thinking about the two subjects, and through seeing the integration of them in this class I can see how they are really one in the same!

Seeing beyond the “binary” notion that mathematics and science are completely discrete entities will facilitate preservice teachers in the development of integrated practice. This brings integrated practice within reach for them as something that is possible as opposed to something only veteran or super-talented teachers are capable of achieving. This notion of grappling with developing meaningful STEM approaches but still seeing themselves as capable of becoming a teacher able to enact converged STEM practices came through from numerous participants. All of the activities we have done so far have really made me think hard about how the math is incorporated. Once we are told the different ways that math standards are incorporated into all of the activities then it seems so obvious. However, I still struggle to think of how math is incorporated in the beginning. I think familiarizing myself with the standards more will definitely help out with this, but it is slowly getting easier. I love how measurement and data collection are both easy ways to incorporate math because those two components are also important for most science topics.

This vision for STEM convergence represents a promising avenue for developing preservice teacher content connections.

15.5.1 Analysis of STEM Modeling Instruction Through Lesson Vignettes To address our second research question on how preservice teachers planned and enacted ways that support STEM modeling instruction, we analyzed their lesson plans that were part of their STEM units. As preservice teachers moved from experiencing mathematics and science as learners toward having to teach their own lesson with elementary children, they found opportunities to bring out the state mathematics and science standards through their STEM units. The following lesson vignettes provide detailed insights for how these preservice teachers both developed and enacted their visions for integrated STEM that enacted modeling principles.

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15.5.2 Use of Mathematical Modeling to Help Make Decisions, Describe, Predict, and Explain Phenomena About Science, Engineering, and Technology Preservice teachers experienced the power of mathematical modeling while enacting their own lessons that were embedded within their integrated STEM units. The following findings highlight three types of STEM modeling lessons created by our preservice teachers: (a) mathematical modeling used to describe, predict, and explain and connect to a scientific phenomenon; (b) mathematical modeling used to evaluate and make decisions on the optimal design of an engineering challenge; and (c) mathematical modeling used to determine a cost/benefit analysis on a science topic. The following sections highlight the best examples of preservice teacher lessons depicting each of these categories. Mathematical modeling used to describe, predict, and explain and connect a scientific phenomenon. As an example, for a preservice teacher-designed STEM modeling activity, we offer the following static electricity lesson. The preservice teachers launched the lesson and engaged students by making connections to real life by asking, “Where in your life have you seen static electricity?” Students investigated how they can make static electricity occur by rubbing a balloon on given objects such as a paper towel, felt, shower cap, and their own hair. Students rubbed the balloon on objects and counted how many pieces of newspaper squares it picked up, keeping track of which object attracted the most paper in the graph and which attracted the least while making hypotheses about what was happening. Each student collected data using a fraction form, how many pieces of the 25 newspaper squares it picked up, n/25. From the data collected from the students on day one, they combined the data and created a classroom graph to show what objects collected the most pieces of paper. They also found the averages for each object. Using the data, the class connected the result to the big ideas around static electricity and used the mathematics to connect to explaining the scientific phenomenon of how rubbing different materials releases negative charges, electrons, which can build up on one object to produce a static charge. Mathematical modeling used to evaluate and make decisions of the optimal design in the engineering process. One of the lessons, the preservice teacher designed, was a STEM design activity called airplane alley. We showcase the following STEM design activity, Airplane Alley, because it was the best example for utilizing design to develop a model and in this case determine the “best airplane” (Fig. 15.6). The question posed was, how could one determine the accuracy of the flight of the airplane? There were many criteria the class could have used to determine the “best,” but the teachers and students decided on determining accuracy by measuring the distance between the center of the target to where the plane landed, using the radius as a measurement of accuracy. They asked students to design an airplane, launch it, and measure the distance from the target. Then they co-created a stem and leaf chart with students to graph the data. The “teacher-created” model provided an experience for the students to appreciate a systematic way to determine

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Fig. 15.6  STEM modeling – using mathematics to help make decisions about the optimal “best” airplane based on accuracy

accuracy by measuring the distance from the center to where the plane landed (radius) and allowed them to evaluate and make decisions of the optimal design in the engineering process. Mathematical Modeling Used to Determine a Cost/Benefit Analysis on a Science Topic  In this section, we highlight two lesson examples focused on using mathematical modeling to determine a cost-benefit analysis. One lesson explored animal adaptation where the teachers asked students to create a creature with new features. There were prices set for each feature/adaptation, and the students needed to consider the cost/benefit tradeoff of purchasing a new feature that would give the animal an advantage for survival. The other STEM modeling activity that we showcase next was a model-eliciting activity that preservice teachers designed where students had to determine a cost/benefit tradeoff in designing a national park with different amenities and having a dilemma of drilling for oil on their land. They used mathematics to make decisions weighing the cost and benefit and the value of adding amenities like a zip wire adventure, a trail, and other priceless items such as esthetics of nature. The National Park Unit (Fig. 15.7) had heavy emphasis on science with conservation concepts as well as language arts skills such as researching, interpreting, and debating on public policy decisions relating to the environment. Key concepts include management of nonrenewable resources and cost/benefit tradeoffs in conservation policies. The unit also has a heavy emphasis on mathematics because students have to find the area and perimeter of land that was often an irregular polygon and work with a constraint of the problem, and worked with a given budget to

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Fig. 15.7  Rating and ranking and cost-benefit analysis for their national park

make decisions about what features they can afford in their parks, compute the new area of their parks given an amount of land the oil company wants to lease, and factor the money offered by the oil company into their budgets. Students also had to wrestle with and balance the notion of preservation/conservation principles related to protecting the environment and weigh those decisions on resource exploration within the park. There is emphasis on engineering as students are required to plan and build/design a park given constraints/budget and later engage in a cost-benefit analysis. The deeper analysis of the successfully implemented STEM modeling lessons allowed us to identify four key attributes. We describe each attribute with an example from the National Park lesson. 1. Launch with a meaningful problem-based scenario. Inquiry drove the STEM modeling activities with the need for students to better understand the problem at hand. Students researched the problem, made assumptions, and defined variables important to solve the problem or collected data from the investigation or problem. In the case of the National Park, students grappled with the complexities of conservation policies by using cost-benefit analysis. Students created and took on responsibility for a piece of land that would become a new National Park. 2. Connect to curricular standards. The STEM modeling lessons were closely tied to the mathematics and science standards. In the case of the national park unit, teachers focused on Science SOL 6.9 (Earth Resources)  – investigating and understanding public policy decisions relating to the environment. Key concepts include a) management of nonrenewable resources and b) cost/benefit tradeoffs in conservation policies. The student solved practical problems involving area and perimeter (Math SOL 6.10 – Measurement). There were many more curricular standards addressed and many more STEM modeling pathways teachers considered, but based on the time allotted for the unit, they focused primarily on these mathematics and science learning objectives.

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3. Iterate to revise and refine solutions/models as a habit of mind. Iterative process of creating a model/prototype with testing and revision. As students were designing their national park, the students were given some parameters and constraints: Decide on shape and size – it cannot be a square or a rectangle and between 300 and 500 square miles. As they drew the plot of land, many used irregular shapes, some that were too small or too big and had to revise their thinking. In addition, students received a budget and supplies to get started and had to revise their design based on their budget constraints. The revision through iterative refinement is a critical habit of mind for all students to develop early in their schooling. 4. Analyze the solution and promote systemic thinking. Solution involved a systemic process and the use of data to make decisions or explain a phenomenon; STEM modeling lessons allowed students to make assumptions and define variables then to build a solution/prototype while constantly analyzing considering the real-world context. In the case of the National Park lesson, the teacher explained the concept of cost-benefit analysis, highlighting the idea that costs and benefits consist of more than financial tradeoffs. The third lesson concluded the unit with the Elaborate and Evaluate phases. In the Elaborate phase, student groups used a cost-benefit analysis approach to debate an offer from an oil company that wants to drill within their park in exchange for money to improve or maintain other areas of it. The Evaluate phase consisted of a whole class discussion of the choices made within groups, as well as the choices they could make as individuals, and connect those choices to existing real-world policy.

15.6 Discussion Modeling in STEM focuses on the need for modelers to go through a process of iteration and refinement. In our STEM lessons, our preservice teachers used the modeling process to have student modelers go through an iterative process of design and redesign as they used mathematics to refine their airplane and their aluminum boats or to collect data and test their hypothesis and refine their understanding of a scientific phenomenon. Students also optimized their design for the national park as they evaluated using the “student created” criteria for “best” which placed more weight on the attributes that they cared about the most (i.e., aesthetics, fun factor, financial, health/well-being, and ecological features). The importance of this iterative and refinement of ideas are echoed by Zawojewski (2013): When engaged in the modeling process, modelers go through iterations of expressing, testing, and revising the trial model. In doing so, they simultaneously improve their model and also develop deeper understandings of the constraints and limitations that still exist at each stage of model development, and learn to articulate (to group members) the trade-offs and benefits of a particular model. Therefore, a very important component of individuals’ development of modeling processes is learning to interpret and eventually produce different points of view in order to facilitate the model revision process. (p. 240)

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15.6.1 Converging STEM Practices as a Means to Articulate “STEM Modeling” Teachers in the classroom attended to important standards put forth by each of the disciplines and finding where these practices converge can be a useful framework. From our work together teaching methods courses that support science and mathematics and integrating engineering practices and the more recent computer science standards, we created a framework where the modeling is the practice that brings all the disciplines together. More specifically, the National Research Council’s A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012) identifies modeling as an important practice too often “underemphasized in the context of science education.” According to the Framework, “engaging in the practices of science helps students understand how scientific knowledge develops; such direct involvement gives them an appreciation of the wide range of approaches that are used to investigate, model, and explain the world” (p.  42). Mathematical Modeling according to the Common Core State Standards in Mathematics (CCSSM, 2010) “links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions” (p. 72). In addition, design thinking in engineering converges with these practices through the design process, having students research and define a problem, ideating, prototyping, testing, and refining the solution design through an iterative process. Weintrop et al. (2016) alluded to the notion of how convergence across STEM practices can produce new ways of thinking and created a taxonomy broken down into four major categories: data practices, modeling and simulation practices, computational problem-solving practices, and systems thinking practices. As we considered Weintrop’s taxonomy (Weintrop et  al., 2016) and our work through this research project, we developed a vision of STEM Modeling Practices that teachers can use that brings together different disciplinary processes together. We first overlapped the scientific investigation, mathematical modeling, engineering design, and computing processes to look for recurring process skills that converged and were important practices that engage students in describing, explaining, investigating, and modeling a real-world phenomenon. Figure 15.8 illustrates the convergent nature of the scientific investigation process, the mathematical modeling process, the engineering and computing processes, and where these disciplinary processes converge into these five STEM modeling practices: Questioning Practices, Data and Investigative Practices, Modeling Practices, Analytic and Interpretive Practices, and Systems Thinking Practices. For instance, modeling in STEM begins with the open process of posing and solving real-world problems and using data to represent, analyze, make predictions, or otherwise provide insight into real-world phenomena. More specifically, when looking across the processes of how scientific investigation, mathematical modeling, and more recently computing and engineering processes are taught in schools, we find

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Fig. 15.8  Researcher-conjectured model for the convergences of STEM practices

that there is convergence in that all of these processes begin with an inquiry stance with the Questioning Practices. They begin with posing a question and defining the problem, which leads to the Data and Investigative Practices. In each of these STEM disciplines, this means doing background research, formulating hypotheses, designing an experiment in science, or making assumptions, defining variables, doing the mathematics and solving the real-world problem in mathematical modeling, or specifying the requirement, creating alternative solutions, and choosing the best one in engineering. In computing, there is an emphasis on developing algorithmic thinking where students decompose a problem and create a sequence of steps to solve a problem. From this point, we get into Modeling Practices, where we are able to test our hypotheses or our scientific models by doing an experiment, building a new mathematical model, creating a simulation model that imitates real-world processes, or building a prototype. Once we have a workable model, we moved into the Analytic and Interpretive Practice which allowed us to analyze our results and draw conclusions in science, or analyze and critique a mathematical model, or test and redesign as needed in engineering. Finally, all disciplines value the practice of communicating these results back into the holistic notion of Systems Thinking Practices, where in science we communicate results based on how it relates to the big ideas within the context of the scientific phenomenon; in mathematics we refine our solution/model and report on how this solution or model works/fits in with reality; in computing, we present data in visual formats to communicate (CSTA, 2017); in engineering, we communicate how the design improves the problem and brings about a solution to a big and complex problem. In conceptualizing the convergence

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of these STEM practices, elementary teachers who teach across disciplines can see how mathematics, science, engineering, and computing processes all converge and see the wondrous benefit of inquiry-based teaching.

15.6.2 The Value of Modeling for Elementary Teaching Practice One of the most insightful aspects of this case study was that preservice teachers seemed to internalize and articulate the value of STEM modeling for their future practice. This is a powerful considering the issues faced by science education, where the amount of time spent on teaching elementary science is declining (Blank, 2013), this includes a lack of science content confidence (Weld & Funk, 2005), and an incomplete understanding for the nature of science (Akerson et  al., 2006). Traditionally, elementary science instruction works to amass science vocabulary and the accumulation of science facts that are disconnected facts from the actual processes of science and provide students little chance to experience inquiry itself (Forbes & Skamp, 2014). Some researchers have articulated these issues have resulted in a crisis of interest in science education (Chubb, 2013; Tytler, 2007). Furthermore, the President’s Council of Advisors on Science and Technology has cited that there is inadequate teacher preparation in STEM (PCAST, 2010). These also coincide with concerns regarding decreasing creativity and imagination involved in school contexts, particularly as students progress through the grades, which strongly correlates with waning interest in school (Egan, 2005). STEM modeling offers in-roads to managing several of these challenges through addressing interest and creativity and explicitly building content interest. Engaging future teachers in the work of STEM modeling both provided a vision for what is possible in elementary practice and also built excitement and connection during the modeling process. This in turn provided positive impacts on preservice teacher interest in both science and mathematical content and processes. In this way, STEM modeling became an avenue for future teachers to envision new possibilities of practice, and those new ways of thinking were built through their own content engagement and deepened content understanding. Our analysis provided insight into how STEM modeling approaches can also impact similar issues facing mathematics. We were also intrigued by the ways in which preservice teachers saw STEM modeling as a pathway to engage diverse learners. This is imperative as a means to address both the needs of children and also the overarching goals of twenty-first-­century STEM initiatives.

15.7 Conclusions We feel that our own journey in terms of developing integrated mathematics and science practices may be indicative of the same triumphs and struggles that our future teachers face when expected to powerfully integrate mathematics and science. In

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many cases, institutional structures do not often fully support integrated practices where courses are taught as discreet “silo/siloed” entities. These challenges to integration occur across K-16 contexts, for instance, in university methods courses designed to focus on one content area where state licensing mandates, course structure, available teaching spaces, common assessments, and other institutional expectations stand in the way of true integration efforts. As university teacher researchers, we had to work to create the spaces for STEM integration, and we were able to provide an example to future teachers that this work is both possible and valuable. These institutional structures are also entrenched in elementary contexts where elementary teachers will need to understand both how to create these opportunities for children by integrating across predetermined content schedules, curricular approaches, and mandated writing/reading blocks. Teachers should work along with administrators to conceptualize possibilities toward STEM integration. STEM modeling provides a context for meaningful pedagogy, but there will need to be consideration for how to best make these approaches a reality in elementary classrooms.

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Chapter 16

Supporting Students’ Critical Literacy: Mathematical Modeling and Economic Decisions Melissa A. Gallagher and Jana P. Jones

16.1 Introduction Students often find the mathematics they encounter in school to be irrelevant because of the typical focus on rote calculations and word problems with very little connection to the real world (Peterson, 2013). Mathematical modeling, using mathematics to represent the real world, is perhaps the most real and relevant application of mathematics. Applied mathematicians partner with scientists to model the impact of building roads on the habitats and growth of different animal species (Goddard, Morris, Robinson, & Shivaji, 2018). Engineers model how precipitation will impact different watersheds (Haile, Rientjes, Habib, Jetten, & Gebremichael, 2011). Economists model how countries’ budgets will be impacted by different tax laws (Kitain et al., 2017). Most real-life applications of mathematics involve mathematical modeling and using these models to make decisions. If educators seek to make mathematics relevant to their students, they must help students use mathematics to model problems from the real world and use those models to make decisions. The purpose of this chapter is to discuss the integration of mathematical modeling with economics and discuss how mathematics as a tool for decision-making supports students’ critical literacy skills.

M. A. Gallagher (*) University of Houston, Houston, TX, USA e-mail: [email protected] J. P. Jones LeBlanc Elementary School, Abbeville, LA, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_16

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16.2 Why Mathematical Modeling Most definitions of mathematical modeling highlight connections between mathematical representations and the real world. We use the definition of mathematical modeling presented in the seminal report, Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME; Garfunkel & Montgomery, 2016): “Mathematical modeling is a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena” (p. 8). The authors of the GAIMME report describe the modeling process as made up of the following components: (a) identifying the problem, (b) making assumptions and identifying variables, (c) doing the math, (d) analyzing and assessing the solution, (e) iterating (i.e., revisiting past steps to revise the model), and (f) implementing the model and reporting the results (Fig.  16.1). This modeling process aligns with the Common Core Standards’ description of mathematical modeling, wherein students make assumptions, model relationships, and analyze and interpret their results: Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, CCSS.MATH.PRACTICE.MP4 Model with mathematics)

Fig. 16.1  The mathematical modeling process. (Garfunkel & Montgomery, 2016, p. 13)

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Mathematical modeling can be a key aspect of instruction beginning as early as prekindergarten because it is grounded in students’ personal experiences and empowers them to make decisions and see the relevance of mathematics to their everyday lives (English & Watters, 2004; Garfunkel & Montgomery, 2016). Mathematical modeling in the elementary classroom has many benefits, including supporting students’ conceptual understanding, procedural fluency, adaptive reasoning, strategic competence, and productive dispositions about mathematics (Suh & Seshaiyer, 2017). If students have productive dispositions about mathematics, they are inclined to believe in their own efficacy for doing mathematics as well as to “see mathematics as sensible, useful, and worthwhile” (National Research Council, 2001, p.  116). Additionally, students who are interested in mathematics have higher achievement in mathematics (Middleton, Jansen, & Goldin, 2017; Schiefele & Csikszentmihalyi, 1995). Mathematical modeling also has the potential to challenge students’ ideas about what it means to be successful in mathematics and who is capable of doing mathematics (Wickstrom, 2017). Students who are not exposed to mathematics in real-life contexts tend to rely on procedures rather than putting themselves in the situation, and this may limit their view of mathematics as a tool to critically evaluate different options and make decisions. For instance, in a 2017 study completed by Ulu, fourth grade students were presented with a mathematical modeling problem whereby they had to evaluate two options for buying cardboard for a technology design class and choose the best one. Option A offered cardboard for 6 Turkish liras and was located at the school. Option B offered cardboard for 3 Turkish liras, but required carfare due to its location. Of the 22 students in the study, only five questioned the researcher about the location of the options and price of the carfare. The authors concluded that the students “who could not provide a realistic solution solved the problem with their mathematical procedural knowledge, those who could provide a realistic solution solved the problem with their conceptual knowledge” (Ulu, 2017, p.  574). Engaging students in mathematical modeling problems on a regular basis can help shift their ideas of what it means to do mathematics, from a perspective that mathematics is a collection of skills to the perspective that mathematics is a tool which can be used to evaluate real-world options (Wickstrom, 2017). Mathematical modeling lends itself to integration with other content areas and is often integrated with other STEM fields (e.g., English, 2009; Maiorca & Stohlmann, 2016; Yanik & Memis, 2015). Peterson (2013) argues that although science and mathematics are often integrated, to truly bring mathematics alive for students, it should be integrated across the curriculum. Among German educators there was a discussion regarding the role that mathematical modeling should play in public primary schools. Advocates of modeling wanted a “stronger connection between arithmetic and social studies” (Greefrath & Vorhölter, 2016, p. 2). Greefrath and Vorhölter (2016) argued that students were not seeing numbers outside of school and modeling helps to make that connection before adulthood. Geiger, Ärlebäck, and Frejd (2016) present two social science mathematical modeling tasks used in Australia and Sweden. In the Australian classroom, students in grade 9 were tasked with finding the most cost-efficient route for a highway between their town and a larger town

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116 km away. The teacher provided several realistic constraints for the students and allowed them to work in teams to determine the most cost-effective route. After testing multiple models, students decided on the best routes and presented their solutions to the class. The other students critiqued the reasoning of their peers and together they discussed the merits of the different models. In the Swedish classroom, a seventh grade teacher asked her students to consider the pricing of candy at different stores (Geiger et al., 2016). Data from the four stores were each presented through a different representation: words, graphs, tables, and algebraically. Students worked in groups to determine which store gave the best value for their money. After this discussion, the teacher demonstrated how to graph all the different forms of data on one coordinate plane. Students then engaged in a whole class discussion about the affordances of the different representations and how to transition between them. Although there are examples of mathematical modeling tasks integrated with social sciences, these integrations are less common in the elementary classroom and less common than the STEM integrations. In this chapter we will describe two lessons in which mathematical modeling was integrated with economics—where mathematical computations were used to support making economic decisions—and discuss how using mathematics as a tool for evaluating different options supports students’ critical literacy. First, we will describe the connection between economics and mathematical modeling.

16.3 Economics and Mathematical Modeling Economics, one of the four core disciplines in the Social Studies State Standards (National Council for the Social Studies [NCSS], n.d.), provides a natural integration between social studies and mathematics. According to Merriam-Webster, economics is “a social science concerned chiefly with description and analysis of the production, distribution, and consumption of goods and services” (Economics, n.d.). Economists use and create theories to explain how different variables impact one another, using these theories to inform decision-making, often at the policy level. Because of the complexity of the real economy, economists create models, simplified analytical frameworks, which focus on the relationships of interest for the understanding of particular economic concepts (Chiang & Wainwright, 2005). Some economists, such as Chiang and Wainwright (2005), distinguish between mathematical and nonmathematical economics, arguing that nonmathematical economics outlines assumptions using words and sentences rather than symbols, and considering geometric models, limited to 2-dimensional space, to be nonmathematical. Other economists are more inclusive in their definition of mathematical models in economics: A mathematical model of the economy is a formal description of certain relationships between quantities [emphasis added], such as prices, production, employment, saving, investment, etc., with the purpose to analyze their logical implications [emphasis added]. Some of those relationships derive from empirical observation; others are deduced from

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theoretical axioms concerning the assumed behavior of a “rational” economic agent, the so-called homo economicus. (Medio, 2009, p. 222)

For the purposes of this chapter, and our connections to elementary mathematics, we adopt Medio’s (2009) conception of what constitutes a mathematical model of the economy, further simplifying to the elementary level by considering mathematical modeling in economics to be any use of mathematics, including arithmetic, to describe relationships between quantities and to use that description to analyze the implications of different options in order to make decisions. For instance, with regard to the NCSS grades K-2 standard: “D2.Eco.2.K-2. Identify the benefits and costs of making various personal decisions,” students could use an inquiry approach to investigate the costs and benefits of various decisions using arithmetic and/or comparing the values of different decisions. They could then analyze the logical implications of these different decisions and engage in a whole class debate about these different options, justifying their positions with mathematical and logical arguments. In this chapter, we describe how mathematical modeling was used in two lessons: (a) to focus prospective teachers on the financial implications of a proposal to address gun violence in schools and (b) to teach the economic concept of opportunity cost in a third grade classroom. In both these lessons, the learners had to investigate different options, engage in small group debate regarding the costs of the different options, and present and justify their decisions to the whole class.

16.4 Theoretical Framework The tasks presented here, and particularly the task presented to the prospective teachers, are embedded in economics, as described above, and were informed by our belief in the importance of mathematics as a tool for evaluating and analyzing multiple options to support critical literacy. “Critical literacy means to approach knowledge critically and skeptically, see relationships between ideas, look for underlying explanations for phenomena, and question whose interests are served and who benefits” (Gutstein, 2006, p. 5). In the present post-truth era wherein media outlets are largely funded by private interests, where people in power claim any unflattering facts are “fake news,” and Russian propaganda is spread widely through social media (Mueller, 2019), critical literacy is imperative. Mathematics is inextricably tied to critical literacy, as many hotly debated ideas omit numbers entirely, leaving people to argue on the basis of opinions rather than facts and feasibility. Mathematics is a tool that can be leveraged to evaluate the feasibility of such proposals, helping people to make decisions. Evaluating and modeling different options using mathematics, as in the two tasks shared here, can start to build students’ critical literacy skills and help them see mathematics as a tool to aid decision-making.

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16.5 Our Context In order to support prospective teachers’ understanding of mathematical modeling and its use as a pedagogical tool, the first author included a modeling lesson as part of an elementary mathematics methods course taught alongside the first semester of a teaching internship. The task was enacted in consideration of the mathematical modeling components outlined in the GAIMME report (Garfunkel & Montgomery, 2016). We describe the enactment of the task through the following components: (a) identifying the problem, (b) making assumptions and identifying variables, (c) doing the math, and (d) implementing the model. The components “analyzing and assessing the solution” and “iterating” are presented within the making assumptions and doing the math components. Additionally, we use italics to denote where the second author’s (the teaching intern’s) voice presents a different perspective than our collective voice.

16.5.1 Identifying the Problem The following task was presented to teaching interns (prospective teachers in their first semester of a yearlong teaching residency). The task was presented just a few weeks after the Parkland school shooting, and so the class began with a discussion of the sensitivity of the topic and the need for respectful dialogue before we dug into the task. For this task, the prospective teachers were asked to evaluate a proposal for a National School Protection Force, which would recruit military veterans to patrol schools (Fig. 16.2). The prospective teachers were given time to read the blog post and consider the argument. As a class, we unpacked what the author was saying and posed questions we had regarding the blog’s proposal that armed military veterans should volunteer to patrol public schools to keep them safe from active shooters.

There have been a large number of school shootings in the USA. People disagree widely about what should be done about it. Any proposed solution could be considered from a mathematical standpoint. Take a look at the following proposal: https://www.stevereicherttraining.com/blog/national-school-protection-force-nspf What questions do you have? What information would you need to assess the feasibility of this proposal?

Fig. 16.2  Mathematical modeling task presented to prospective teachers

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The prospective teachers were then encouraged to discuss in small groups how they might go about solving this problem, but were told that they must keep their personal opinions aside, using only mathematics to evaluate the feasibility of this proposal. Analyzing the feasibility of this proposal requires investigating the supply of veterans and costs of the program, among others, which are both economic principles. As an intern, I experienced my first mathematical modeling task in a math methods course specific to elementary education majors. In groups of four, we were asked to research and evaluate a proposal aimed at controlling gun violence by having retired veterans monitor schools during their operating hours. There was no procedure to follow; the only instructions were to decide if the program was possible from a mathematical standpoint.

In the second author’s reflection, she highlighted the openness of the task, remembering that there were no procedures to follow and that the only instruction was to determine the program’s feasibility. Creating an open task and focusing prospective teachers on the mathematics in order to make a decision were two intentional pedagogical moves to help the prospective teachers critically evaluate the blog post without fear of judgment.

16.5.2 Making Assumptions and Identifying Variables The groups began eagerly searching the Internet, trying to determine how many schools and veterans there were in the USA, but soon became overwhelmed with the task, unable to agree on what numbers were needed and where to find them. At this point, the first author told the class that a practice that mathematicians regularly engage in, especially when modeling a problem from the real world, is to simplify the complexity by making assumptions. Each group began to pose different constraints and search the Internet for different data they needed in order to pose a solution. After many disputes over assumptions being made and calculations of costs and veterans, I remember thinking, “Did we just get tricked into doing math?”

Determining assumptions when critically evaluating proposals, such as the National School Protection Force, pushed the prospective teachers to debate in their groups and present their points using data. Some groups decided to impose the assumption that they were just evaluating this program for the state of Louisiana, thereby narrowing the data they would need to research. Other groups made assumptions about how many guards would need to be posted at schools, which led them to debate student-to-guard ratios and reasonable shifts the guards would take, as well as to research the costs of different trainings and psychological evaluations the volunteers would need to pay for.

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16.5.3 Doing Math At this point in the mathematical modeling task, after making assumptions and finding the necessary data, the prospective teachers had to do math. One group chose to remove 20% of veterans from the pool of available veterans because they read that 1:5 had post-traumatic stress disorder. Another group calculated the cost of psychological testing to ensure veterans were mentally healthy enough to patrol schools armed with guns. They used the constraints they had imposed and computed whether the proposal to put armed veterans in schools would be feasible. In a sense, we were tricked into doing math, but because the topic was extremely controversial at the time and there was no right or wrong answer, it felt necessary to come to an answer. It was not like completing a set of problems with symbols, digits, and no context, or even a word problem. It was a real problem that someone would actually be facing, which was worth establishing a plan for the program and evaluating its effectiveness.

The reality of the problem and the importance of mathematics as a tool to evaluating this proposal were relevant to the prospective teachers, especially given the recency of the Parkland school shooting. Moreover, the task required them to put aside their personal opinions and evaluate the proposal from a purely mathematical perspective. Many of the prospective teachers appeared to have an opinion on whether or not this proposal was a good idea, but as they were not allowed to voice their opinion, they looked for mathematical rationales that supported their opinions. The relevancy to their own lives and the desire to support their opinions using mathematics motivated them to want to do the math.

16.5.4 Implementing the Model This component of mathematical modeling refers both to the reporting of a solution and the implementation of that solution. The implementation of this model was a reporting out of groups’ assumptions, data collected, and final decisions regarding whether or not this proposal was feasible. Groups had the opportunity to question each other’s data, assumptions, and conclusions. Because not all groups made the same assumptions nor identified the same variables, they did not all come to the same conclusion. Some groups decided to restrict the proposal to the state of Louisiana to determine its feasibility. Because of the large number of veterans in Louisiana and small number of schools, relative to other states, these groups found that within the state of Louisiana the proposal would be feasible, assuming (a) only a fraction of the veterans would be eligible and would want to volunteer, (b) reasonable shift hours, and (c) a reasonable veteran to school or veteran to student ratio (different groups chose different ratios). Other groups decided to evaluate the proposal for the entire USA.  These groups found that the proposal was not feasible given the same assumptions about eligibility and shift hours, because the ratio of veterans to students was too low. Groups that evaluated the proposal’s feasibility for

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the entire country also found that it would be difficult to provide the coverage necessary for many rural schools. Lastly, one group decided to evaluate the costs of the psychological and gun safety courses and, based on these findings, determined that the cost per veteran would be about $2000, either in actual cost or if these services were donated then this would be an opportunity cost to the psychologists and gun safety training centers. In order to engage students in a meaningful discussion about their findings, the first author made a few purposeful pedagogical moves: (a) she frontloaded expectations during the presentations of findings, (b) she allowed the presenting group to call on peers to respond to their findings, and (c) she sat down, so as not to be standing over the prospective teachers in a position of authority. At the beginning of the presentation of findings, the first author set clear expectations, stating that the groups presenting would share their assumptions, their findings, and their conclusions, and that the groups listening would be listening to see if they agreed with the assumptions, if the findings aligned with the conclusions, and if there were any differences between the model being presented and their own model. She then sat down for the remainder of the discussion, allowing the presenting teachers to take charge of the conversation and interjecting as needed with reminders about the expectations. By removing herself from a standing position of power, and by putting the prospective teachers in charge of moderating the discussion, she gave them ownership of the discussion. The class was able to engage in a whole group discussion analyzing the logical implications of the different models as the prospective teachers explored thinking that differed from their own. Groups that had evaluated the feasibility for the entire USA began to think about the implications for Louisiana, shifting their focus to their home state, and many became excited about the feasibility of this proposal. One prospective teacher, whose brother is a veteran, exclaimed that she wanted to take action by implementing this proposal in reality. However, when the last group shared the high costs of the trainings and evaluations, the other groups realized that they had overlooked a key aspect of the feasibility of this proposal: the costs. Working in small groups had allowed the prospective teachers to navigate and decide on assumptions together, but the whole class discussion allowed everyone to hear the conclusions reached by models created with different assumptions than their own.

16.5.5 Implications for Teacher Education If teacher educators want teachers to engage their K-12 students in mathematical modeling tasks, they must first engage their prospective teachers in those tasks (Phillips, 2016). By engaging prospective teachers in such tasks, teacher educators can demonstrate the impactfulness of mathematical modeling tasks as well as demonstrate effective pedagogical skills for implementing such tasks, including: choosing an ill-defined and relevant real-world problem, helping students to unpack the

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nuances of the problem, teaching them how to make assumptions, fostering collaboration, supporting research, encouraging analysis and revision of early models, creating opportunities for sharing solutions, and engaging students in respectful debate of different solution models. Following the enactment of the Gun Violence task, the first author unpacked these pedagogical moves for her prospective teachers.

16.6 Effects of Opportunity Costs: Mathematical Modeling with Third Graders Shortly after participating in the Gun Violence task, the second author was planning for a social living lesson on opportunity cost, the idea that every choice requires giving up other possibilities, the most valuable of which is considered the opportunity cost. Unbeknownst to the first author, she decided that a mathematical modeling task would present students with authentic reasons to make choices involving opportunity cost. My decision to incorporate mathematical modeling seemed like a perfect way to show my students how math is not only present in other academic areas, but also in everyday tasks. Third graders in my classroom were able to experience opportunity cost through mathematical modeling during an economics lesson.

We weave a vignette depicting how this lesson went in the second author’s class throughout this section to illustrate the kind of questioning that teachers need to engage in during a task, as well as the kind of thinking that third graders are capable of when given a relevant problem with which to engage.

16.6.1 Identifying the Problem The term opportunity cost had been introduced at the start of the lesson as the intern read Dr. Seuss’ “What Pet Should I Get?” The story follows siblings through a pet store where they are instructed to choose one pet to bring home. The siblings struggle with the thought of leaving some of the pets at the store and encounter the concept of opportunity cost. In the previous 2 weeks, the students learned about basic concepts of economics, such as goods and services and supply and demand. After making sure her students understood these key terms, she presented her third graders with the following task, which she adapted from EconEdLink (Kehler, 2003/2010). The task corresponded to a third grade standard from the College, Career, and Civic Life Framework, 3.7.3 “Explain the benefits of comparative shopping when making economic decisions.” The following task was presented: You are now in charge of supplying your classroom. The principal has prepared a budget for you and I have come up with a list of items you may buy. It is your job to decide what your classroom does and does not need to function effectively.

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In order for students to understand how different content areas impact their lives, they must have experiences with individual skills in real-life contexts. The teaching intern sought to make visible to students the numbers that impacted their everyday lives by giving them an authentic task in which to engage. “As you know, sometimes we are faced with decisions that cause us to turn down options. I’m sure you have faced a time when you were only allowed to pick one option in a group of two or more. Today, I am challenging you to make some important decisions of your own. You are now in charge of supplying your classroom!” I watched as students exchanged glances that said, “How are we supposed to do that?!” Some grew very eager for the challenge, asking what they could choose from. “I will give you a list of common classroom supplies and their average prices. You and your partner will decide which items you think are most necessary for your classroom while not exceeding your budget of $5,400. Remember, you and your partner must agree on the decisions being made.”

The teaching intern gave students a list of supplies which exceeded their budget of $5400, thereby forcing the students to choose items from the list which they could logically argue would best support their classroom. Having students work in partners forced students to voice their arguments and debate which items their classroom needed most. By not being able to choose all items on the list, the students experienced opportunity cost and had to determine how to minimize opportunity cost by choosing the most valuable items to their class.

16.6.2 Making Assumptions and Identifying Variables The students got to work immediately, reading through the list and deciding which of the items were definitely necessary and which they could do without. As the intern walked around the room, she heard students defending themselves to their partners. “We don’t HAVE to buy a TV because we can use the computer and screen projector to watch videos. The TV and computer would be really expensive if we bought both.” They took turns explaining why either the price or the purpose of the item deemed it to be unnecessary for the classroom.

At the elementary level, and in particular with students who have never engaged in mathematical modeling tasks, more assumptions may need to be established by the teacher at the outset. The second author chose to limit her students by giving them a set budget and a limited number of items from which to choose. However, students still had to make decisions regarding the value to the class of different items and negotiate in their groups the relative value of different items. As new arguments were made and different items moved up the list in terms of their relative value and usefulness to the class, the budget needed to be revised to accommodate these changes.

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16.6.3 Doing Math Students may be surprised when mathematical modeling is integrated with another content area, such as economics, and in particular, when it is taught during the economics period. One particular pair of students created a list of the items they planned to purchase and called the teaching intern over to their desk. “Ms. Jana, what would the total be for these items? Would it go over $5,400?” “Well, do you think you could add them together and find out?” I responded. “But we don’t do math in social living!” One of the students exclaimed. “Economics and math have a strong connection. When you go to the store and buy something, you’ve actually done math AND economics. I know that you can add multi-digit numbers, so give these a try and find out their total for yourself.” Up to this point in their time as students, they had seen no obvious cross-curricular connections and assumed that different content areas were not to intertwine.

Reasoning through economic decisions and budgeting wisely not only gave the students experience with a realistic task, but it also allowed them to further understand how mathematics is applied in the real world. They were given an option of adding up prices or subtracting from the budget. There were no equations written on the page waiting for them to apply a procedure. They were asked to solve a problem, which was not, from the students’ perspective, obviously a mathematics problem. Students then continued to iterate through making assumptions and doing math as they continued to add up prices, eliminate options, and repeat the process until they had agreed on their list.

16.6.4 Implementing the Model Unfortunately, the principal had not allocated $5400 for the second author’s classroom and so the students were not able to implement their solution. However, they did present their solutions and thought processes to the class. After students completed their lists, I asked that a spokesperson from each pair stand and state their total expenses, their opportunity costs, and their reasons for eliminations. The first pair said, “We spent $5,000. We didn’t buy the writing utensils because that can go on the students’ supply list. We didn’t pay for the classroom aide because we are not in Pre-K or kindergarten. We didn’t buy the class pet or the field trip because we don’t think those are that important. And we didn’t have enough money left over for maps, a laminator, or reading materials.” “Very good!” I praised their thought process. “I appreciate how the two of you made decisions that you both agreed on.” “We couldn’t agree on the class pet!” a little girl blurted out. “I think a class pet would be cool but there are other things we need more than a pet!” Her partner gave her a look of annoyance. “But we can use our class pet to learn science when we talk about animals! She thinks a pet is pointless but I know it would help us with science!” “It seems like the two of you still need some time to make a decision.” I chimed in. “Why don’t you each come up with a list of reasons why and why not to get a class pet. Sometimes when making economic decisions, we need to list the pros and cons in order to see if our decision is worthwhile. We’ll come back to you two to hear your decision.” As time went on,

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many students asked for my opinion when faced with a difficult decision. “You’re the teacher! What do you think is the most important?” I reminded them that these materials were not only for the benefit of the teacher, but for the entire class. Students began to categorize items used daily, weekly, and rarely. I saw much easier eliminations being made once the students thought of themselves as the ultimate decision makers. They had a sense of independence and took on a new role of leadership.

As pairs of students shared their solutions and their thought processes, the intern was able to highlight and make real to them the opportunity costs of their decisions. Every so often, the intern would ask, “Would you consider your opportunity cost a want or a need?” As students built the vocabulary into their plan, it became evident that they grew more confident in their choices. The intern’s opinion was left out of the discussion, which left the students in full control. Students were able to respectfully disagree with any of the opportunity costs stated so long as they provided a reason for its use or effectiveness. With the task, I felt like they truly saw math in action.

For young students it may be hard to showcase economics in the world around them considering that they lack the responsibilities of adulthood and rarely deal with financial tasks, but they were eager to role play and take on the “responsibility” of adhering to a budget and making tough decisions. Without the mathematical modeling task, they may have only learned what opportunity cost is rather than the effects it has and how it relates to their lives.

16.7 Discussion Mathematical modeling is a tool that people can use to understand the world, with which to “see math in action.” As such, it can also help them to evaluate the costs and benefits of different options they may be faced with, helping them to critically evaluate information that options that are presented to them. Although there are many definitions for critical literacy, Gutstein’s (2006) definition has three main components: (a) approaching knowledge critically, (b) examining relationships and underlying explanations, and (c) questioning whose interests are served. All three of these components can be achieved through mathematical modeling, particularly through mathematical modeling of economic situations. In the National School Protection Force task, the prospective teachers critically evaluated the proposal to put volunteer veterans in schools by gathering data and examining relationships between how many veterans might be able to support this program, costs associated with the program, and reasonable schedules, among other factors. Furthermore, allowing the prospective teachers to take ownership of the discussion during the reporting of findings seemed to encourage them to engage with one another’s ideas, questioning and learning from one another’s models. Although the prospective teachers were not pushed to question whose interests were served, that could be an extension to this task. In the Opportunity Cost task the third graders examined relationships between costs of various items, classroom needs, and their total budget, providing explanations for choosing some items over others. Students took part in

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what would be described as a debate rather than an argument as the intern took on the role of facilitator. Students were listening to the ideas of others instead of seeing their own as superior. Although there is room to expand this task to be more critical, we believe it was a good introduction to mathematical modeling for the third graders. If a goal of mathematics education is for elementary teachers to enact mathematical modeling tasks in their classrooms, teacher educators need to prepare them to do so (Sevinc & Lesh, 2018). One way to prepare prospective teachers is to engage them in mathematical modeling tasks (Karali & Durmus, 2015) and, in debriefing the task, highlight the pedagogical moves necessary to effectively enact this type of task in the classroom. Engaging prospective teachers in mathematical modeling tasks can help them to see the benefits of using such tasks in their classroom as they consider their own engagement with the task. More research is needed regarding how best to implement mathematical modeling tasks in mathematics methods courses and if experiencing a mathematical task influences prospective teachers to implement such tasks in their own classrooms. Mathematical modeling tasks provide rich opportunities for interdisciplinarity in the elementary mathematics classroom (English, 2009). Economics provides a natural integration between social studies and mathematics. Prospective teachers should be exposed to interdisciplinary mathematical modeling tasks and encouraged to find opportunities to implement mathematical modeling in their fieldwork or internship classrooms.

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Haile, A. T., Rientjes, T. H., Habib, E., Jetten, V., & Gebremichael, M. (2011). Rain event properties at the source of the Blue Nile River. Hydrology and Earth System Sciences, 15, 1023–1034. https://doi.org/10.5194/hess-15-1023-2011 Karali, D., & Durmus, S. (2015). Primary school pre-service mathematics teachers’ views on mathematical modeling. Eurasia Journal of Mathematics, Science & Technology Education, 11(4), 803–815. Kitain, A., Gallagher, M., Khadka, R., Carpenter, S., Wondim, T., Byamukama, G., & Swaray, S.  M. A. (2017). Benchmarking the tax system in Liberia. Washington, DC: United States Agency for International Development. Kehler, A., (2003/2010). You decide! EconEdLink. https://www.econedlink.org/resources/ you-decide/ Maiorca, C., & Stohlmann, M. (2016). Inspiring students in integrated STEM education through modeling activities. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 153–161). Reston, VA: National Council of Teachers of Mathematics. Medio, A. (2009). Mathematical models in economics. In J. A. Filar & J. B. Krawczyk (Eds.), Mathematical models, encyclopedia of life support systems (Vol. III, pp.  222–237). Oxford, UK: Eolss Publishers/UNESCO. Middleton, J., Jansen, A., & Goldin, G. A. (2017). The complexities of mathematical engagement: Motivation, affect, and social interactions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 667–699). Reston, VA: National Council of Teachers of Mathematics. Mueller, R. S., III. (2019). Report on the investigation into Russian interference in the 2016 presidential election. Washington, DC: U.S. Department of Justice. National Council for the Social Studies. (n.d.). College, career, and civic life: C3 framework for social studies state standards. Silver Spring, MD: National Council for the Social Studies. Retrieved from https://www.socialstudies.org/sites/default/files/2017/Jun/c3-framework-forsocial-studies-rev0617.pdf National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington DC: Author. National Research Council. (2001). Adding it up: Helping children learn mathematics. In J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Peterson, B. (2013). Teaching math across the curriculum. In E. Gutstein & B. Peterson (Eds.), Rethinking mathematics: Teaching social justice by the numbers (2nd ed., pp.  9–18). Milwaukee, WI: Rethinking Schools Ltd.. Phillips, E. D. (2016). Supporting teachers’ learning about mathematical modeling. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 249–251). Reston, VA: National Council of Teachers of Mathematics. Schiefele, U., & Csikszentmihalyi, M. (1995). Motivation and ability as factors in mathematics experience and achievement. Journal for Research in Mathematics Education, 26(2), 163–181. Sevinc, S., & Lesh, R. (2018). Training mathematics teachers for realistic math problems: A case of modeling-based teacher education courses. ZDM, 50(1–2), 301–314. https://doi.org/10.1007/ s11858-017-0898-9 Suh, J. M., & Seshaiyer, P. (2017). Modeling mathematical ideas: Developing strategic competence in elementary and middle school. Lanham, MD: Rowman & Littlefield. Ulu, M. (2017). Examining the mathematical modeling processes of primary school 4th-grade students: Shopping problem. Universal Journal of Educational Research, 5(4), 561–580. Wickstrom, M. H. (2017). Mathematical modeling: Challenging the figured worlds of elementary mathematics. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp.  685–692). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators. Yanik, H. B., & Memis, Y. (2015). Making insulation decisions through mathematical modeling. Teaching Children Mathematics, 21(5), 314–319.

Chapter 17

Culturally Relevant Pedagogy and Mathematical Modeling in an Elementary Education Geometry Course Emily J. Yanisko and Laura Sharp Minicucci

17.1 Overview of the Literature 17.1.1 Culturally Relevant Pedagogy Culturally relevant pedagogy (CRP), as first conceptualized by Ladson-Billings (1995), included three central components: academic success, cultural competence, and critical consciousness. The underpinnings of CRP were that success in school mathematics for Black and Latin@ students can be supported and actualized through grounding mathematics experiences in the culture of the students (Ladson-Billings, 1997). However, it is not sufficient to simply add students’ names into word problems – the mathematics has to be deeply interrelated with students’ real lived experiences. In order for this to be successful, the teachers must have deep knowledge of the students’ community and the mathematics content (Ladson-Billings, 1997). CRP integrates students’ cultural and personal experiences into the content curriculum in order to help students not only maintain their culture but also provide a counternarrative to the impact of hegemony in school mathematics (Leonard & Guha, 2002). When teaching mathematics in a culturally relevant way, teachers can build empowering relationships in which students are seen as powerful knowers and doers of mathematics (Ladson-Billings, 1997; Matthews, 2003). Matthews, Jones, and Parker (2013) proposed a definition of what can be called culturally relevant cognitively demanding (CRCD) tasks. CRCD tasks are built on a

E. J. Yanisko () Urban Teachers at Johns Hopkins University School of Education, Baltimore, MD, USA L. S. Minicucci Baltimore City Public Schools, Baltimore, MD, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_17

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framework of cognitive demand theorized by Stein, Henningsen, and Silver (2000). Matthews et al. (2013) define CRCD tasks as being mathematically rich and embedded in activities that provide opportunities for students to experience personal and social change. The context of the task may be drawn from students’ cultural knowledge and their local communities. But, the use of context goes beyond content modification and explicitly requires students to inquire (at times problematically) about themselves, their communities, and the world about them. In doing so, the task features an empowerment (versus deficit or color-blind orientation) toward students’ culture, drawing on connections to other subjects and issues. CRCD tasks ask students to engage in and overcome the discontinuity and divide between school, their own lives, community and society, explicitly through mathematical activity. The tasks are real-world and focused, requiring students to make sense of the world, and explicitly to critique society – that is, make empowered decisions about themselves, communities and world. (p. 132)

We theorize that CRCD tasks can be a vehicle to implement culturally relevant pedagogy in mathematics – high achievement through engaging students in deep and rigorous math tasks, cultural competence through grounding tasks in cultural and community knowledge, and critical consciousness through tasks that ask students to question the world around them. Furthermore, we argue that these culturally relevant mathematics tasks can be mathematical modeling tasks.

17.1.2 Culturally Relevant Pedagogy and Mathematical Modeling Mathematical modeling is defined in the Common Core State Standards for Mathematics (CCSSM) in two ways: as a content standard and as a practice standard (NGA, 2010). The content standard of modeling, according to the CCSSM, only applies to high school, but the practice standard – model with mathematics – is to be a way of thinking that is embedded in all levels of mathematics education. The description of modeling with mathematics, according to the CCSSM, is that: mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation…Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later….They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. (p. 7)

This is not the same thing as using concrete models to show and investigate mathematical concepts, but instead it is the iterative process of applying the mathematics that one knows to a real-life, often messy, situation in order to make sense of it (Lerman, 2000; NGA, 2010). Anhalt, Staats, Cortez, and Civil (2018) argue that CRP and mathematical modeling can be naturally integrated, and that mathematical modeling can be a tool that bolsters weaker implementations of CRP. They state that everyday situations that are relevant to students’ lived experiences can be the basis for mathematical

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modeling tasks, and that this integration will drive the learning of mathematics content through building student investment and promoting discussion between students that will require them to draw on their own cultural resources to explain their modeling process to others. Developing these mathematical modeling tasks requires teachers to draw on students’ lived experiences and knowledge. Therefore, mathematical modeling can be a mechanism for teaching rigorous mathematics in a culturally relevant way, as mathematical modeling allows for growth of mathematical knowledge and habits of mind that can be applied in an out of school. Students and teachers can build cultural competence through the act of drawing on students’ lived experiences to drive task construction. Also, critical consciousness – the component of CRP that seems to be often neglected – can be supported while students engage in discussion about mathematics through the lens of social issues that may be important to them, their families, and their community (Anhalt et al., 2018). Integrating mathematical modeling and CRP can potentially be a way to make mathematics accessible to students who may feel their identities are not consistent with the practice of school mathematics and have thus been excluded from its practice. This way mathematics cannot be just for all, but by all (Volmink, 1994). Anhalt et al. (2018) propose that culturally relevant pedagogy and mathematical modeling are easily integrated, especially in the context of social justice. This modeling cycle (p. 314) demonstrates the iterative nature of mathematical modeling (Fig. 17.1). This implies that when students are modeling with mathematics, they would make sense of a situation or problem from their community context, then engage in some evidence gathering and decision making to formulate a model,

Fig. 17.1  Modeling cycle

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apply this model to the situation, and interpret the solution. In this chapter, we are describing curricula that focus on geometry content and pedagogy for elementary resident teachers (ERTs), so in this context it is helpful to think about applying shape, size, transformations, area, perimeter, and related mathematics concepts to a problem of social justice. We propose that teachers learn to develop tasks that encourage students to build mathematical models that are embedded in cultural activity and issues of social justice.

17.1.3 Using Community Knowledge to Build Social Justice Tasks In order to support ERTs’ development of powerful and impactful mathematics teaching practice, it is necessary to build on the out-of-school knowledge in which their students and the members of their community are experts. Through this integration, students will be able to see how mathematics knowledge and practice is useful in making sense of their world and working toward social justice. These learnings are well aligned with CRP as actualized in a mathematics classroom (Aguirre & del Rosario Zavala, 2013). It is important that these social justice mathematics (SJM) tasks be situated within community practices that exist in students’ lives so that the outcomes of these pasts are relevant and impactful for students, and based in their actual cultural realities – not stereotypes based on their race, culture, socioeconomic status, or location (Aguirre & del Rosario Zavala, 2013; Anhalt et al., 2018). Teachers sometimes conceptualize CRP as bringing real-life situations into traditional word problems in an attempt to make the mathematics more relatable (González, 2009). Instead, the goals of the types of mathematical experiences pursued here are to help students use mathematics to develop critical consciousness, agency, and identity while also meeting mathematics goals (Gutstein, 2003). In order to promote the use of this kind of mathematics in elementary classrooms, we propose that it is necessary that prospective teachers, through their preparatory programs, are engaged in learning mathematics through experiencing, planning, rehearsing, implementing, and reflecting (McDonald, Kazemi, & Kavanagh, 2013; Lampert et al., 2013) on these types of tasks for use in their classrooms. Gutstein argues that SJM intertwines three types of knowledge, which he refers to as the “3 Cs (a) community knowledge, knowledge of one’s reality experiences, culture, language, neighborhood, and informal ways of relating to and interpreting the world, (b) classical (mathematical) knowledge, academic mathematics, and (c) critical knowledge, reading the world with mathematics” (p.  458). In order to attempt to interconnect the 3 Cs in an authentic manner, it is useful to uncover issues of relevance in the community, rather than creating them from whole cloth (Gutstein, 2016, 2018; Leonard & Guha, 2002). In one study, a teacher had students take pictures in the community. The teacher then shared their own pictures from a recent trip and looked at geometric shapes and other mathematical features of the pictures

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before asking the students to do the same with their own pictures (Leonard & Guha, 2002). Gutstein (2016) asked students to generate their own ideas about social issues to be addressed in class (Gutstein, 2016). Later, Gutstein (2018) and a teacher collaborated to develop SJM tasks that were based in the teacher’s own experiences of working for social justice in the community. There are ways in which the tasks can be truly culturally relevant mathematics modeling tasks.

17.1.4 Mathematical Modeling and Critical Consciousness Critical consciousness is an aspect of CRP that corresponds well to the concept of mathematical modeling. Issues of social justice are not “nice” problems with “neat” equations that describe them – they are multifaceted and multivariate and have room for interpretation and application. The CCSSM argues that students should be college and career ready (NGA, 2010), but mathematics instruction should also make sure that students are ready to be full participants in society. For students to be able to use mathematics in a way that matters, students must know more than how to “play the game of school” (Gutiérrez, 2013, p. 12). Problems of social justice are those that students should be most ready to solve in reality. Mathematics as completely decontextualized and abstract becomes useless when students cannot apply what they know in situations that matter to them (Volmink, 1994). Furthermore, students often feel mathematics is useless when it is presented as a collection of arbitrary rules. Students may develop a more productive disposition (Kilpatrick, Swafford, & Findell, 2001) toward mathematics if they are presented with opportunities to make sense of the world around them, specifically in terms of situations and injustices that directly affect them and their communities (Gutstein, 2003). However, it is not sufficient for students to simply use mathematics to describe the milieu in which they exist, but they must also feel they have the power to change it. Gutstein (2003) argues that “teaching for social justice…has three components: helping students develop sociopolitical consciousness, a sense of agency, and positive social and cultural identities” (p. 66). If students are only provided with opportunities to use mathematics to analyze existing injustice without concurrently being provided with ways to use that same mathematics to work toward social change, they may feel disempowered (González, 2009). When the first author initially was introduced to SJM, she implemented a task where students could analyze racial biases in income. However, without incorporating opportunities for students to apply mathematics in an effort to make change in this reality as presented, the students reacted in ways that showed that they felt defeated by the new information uncovered by mathematizing the situation. This is not uncommon with teachers new to SJM  – teachers often embed problems in situations that allow connections between mathematics and society, but they often do not include opportunities for students to make change (Bartell, 2013). Students can use mathematics to elevate

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their awareness of sociopolitical realities, but in order to be empowered to become change agents in their world, they must also use mathematics to have an impact on those same sociopolitical realities (Gutstein, 2016). Through SJM and mathematical modeling, both students and teachers should be developing critical consciousness together while students are developing their own political agency (Anhalt et al., 2018). In order to develop a course for ERTs that integrates geometry content, mathematical modeling, and culturally relevant SJM, we applied a conceptual framework that guided our use of student, family, and community knowledge and assets to uncover culturally competent social justice issues (Aguirre & del Rosario Zavala, 2013; Gutstein, 2016; González, 2009) in order to develop culturally relevant, cognitively demanding (CRCD) assignments and instructional experiences (Matthews et al., 2013) that required the use of mathematical modeling (Anhalt et al., 2018).

17.1.5 Conceptual Framework of Mathematics Modeling for Social Justice When approaching the development of this course, we integrated the concepts of mathematical modeling, high cognitive demand tasks, culturally relevant pedagogy, CRCD tasks, and social justice mathematics (SJM). The interaction between these concepts is shown in Fig. 17.2. We envision mathematical modeling as an overarching concept that can be integrated with mathematics in multiple ways. We argue that tasks that involve mathematical modeling as described in Fig. 17.1 are high cognitive demand tasks in their very nature. What can be considered a subset of high cognitive demand tasks (Stein et al., 2000) are those that are grounded in the lived experiences and cultural and community assets of the students in the K–6 classroom and/or tasks that elicit critical consciousness – two pillars of culturally relevant pedagogy that are left unaddressed by looking at high cognitive demand mathematical modeling alone. We feel that CRCD tasks represent the intersection that the name implies – that the tasks are both culturally relevant and cognitively demanding. Finally, we view social justice mathematics tasks as a subset of CRCD tasks. We believe that mathematical modeling is essential to developing social justice math tasks that are CRCD – because of the messy nature of these real-world social justice contexts, mathematical modeling is the authentic way to frame and approach these tasks. These tasks are designed specifically to use mathematical modeling to address issues of social justice, or mathematical modeling for social justice (MMSJ), that are relevant to the students and their communities. In this chapter, we will be discussing how we used this framework to develop a geometry course that is one course in a trajectory to prepare ERTs.

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Fig. 17.2  Conceptual framework

17.2 Goals of the Chapter In order to prepare teachers to teach in urban settings, it is not sufficient to merely teach them mathematics content and methods for teaching that content. For novice teachers preparing for urban settings, it is especially important to prepare teachers to teach in culturally relevant ways. We define urban settings as cities or metropolitan areas with a large percentage of minoritized populations (racially, ethnically, and/or socioeconomically). Culturally relevant pedagogy promotes students’ academic success by grounding the learning of mathematics content in the cultural identities of students and their communities while empowering them to analyze injustice and math in ways that feel authentic to them (Matthews, 2003). Although knowledge of culture and its diversity should be considered key foundations for teacher preparation, these concepts are often relegated to a single “diversity” course and student teaching experiences in diverse settings (Ladson-Billings, 2000). This also suggests that cultural knowledge and cultural pedagogy are often treated as add-ons  – distinct knowledge that is not integrated into the content or methods courses in most teacher preparation programs.

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Gutiérrez (2013) argues that teaching is in its very nature political, especially in settings that have a high population of minoritized students. Therefore, teacher candidates should be prepared with political knowledge for teaching. In her study, a traditional teacher education program is supplemented by experiences not only in rigorous mathematics, but also teaching with and for social justice. These supplemental supports provided teachers with the knowledge to serve students of color in schools in which teachers often had to teach their students as well as negotiate the politics of school, district, state, and governmental education policies that are often detrimental to student learning. These experiences helped pre-service teachers build the political knowledge needed for teaching in urban contexts. Our teacher preparation program places teachers exclusively in urban settings, so when we were tasked with developing a new geometry content class for elementary school teachers, we took this as an opportunity to integrate CRP into the overall concept of the course. The overarching design goals for development of new courses for the elementary program of study included that K–6 students feel empowered by learning mathematics to be change agents in their world, and that the ERTs recognize the culture and community assets of their students to improve the effectiveness of mathematics instruction. Toward this end, we looked to the true integration of culturally relevant pedagogy into elementary mathematics teacher education courses on content and pedagogy so the K–6 students would be able to see themselves as doers of mathematics as well as be able to use mathematics to further their own aims and the aims of their communities. Through MMSJ, we wanted ERTs to experience geometry tasks that are CRCD and that integrated the CCSS for geometry and the practice standard “model with mathematics” so they could experience tasks that represented this stance as a learners so they could then apply that knowledge to plan their own lessons for their students. The goals of this chapter are to describe the curriculum design process that went in to developing this geometry course, our evaluation of how well the course met our goals of immersing ERTs in mathematical modeling for social justice, and how well we designed assessments of their ability to apply these principles in developing and implementing their own instruction.

17.3 Methods The curriculum we were designing was a master’s-level education course for ERTs in the content and teaching of geometry (K–6). The ERTs take the majority of their coursework during a 14-month residency while they are simultaneously placed in a clinical setting. The coursework is intended to be clinically based, and often they enact the assignments they plan in their clinical setting and then reflect upon them. This geometry course falls in the third semester (of six) of their program of study. The research question we asked was: How can we design a geometry course that immerses ERTs in and assesses their internalization of MMSJ? When engaging in curriculum design, designers should focus on the essential questions found in Fig. 17.3 (van den Akker, 2010, p. 4).

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Rationale Aims & Objectives Content Learning activities Teacher role Materials & Resources Grouping Location Time Assessment

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Why are they learning? Toward which goals are they learning? What are they learning? How are they learning? How is the teacher facilitating learning? With what are they learning? With whom are they learning? Where are they learning? When are they learning? How far has the learning progressed?

Fig. 17.3  Essential questions for curriculum design

The curricular design discussed here is part of a larger lesson study of iterative curriculum development. As such, this chapter discusses the first four questions and the last question. The data sources for this study were the products of development, which include the curricular goals (Why are they learning?), course goals (Toward which goals are they learning?), course assignments (How far has learning progressed?), session outline (What are they learning?), and the instructional activities (How are they learning?) for each session. The unique feature of this particular course development is that the syllabus and instructional activities for the whole course are developed a priori by us for implementation by multiple instructors in multiple cities. However, when instructors do implement the course, they will be able to engage in lesson study, either formally or informally, to revise and refine the syllabus and instructional activities to meet the needs of the ERTs, either generally or in specific. Those revisions will influence future iterations of the course on the road to a more robust course design. This chapter is intended to give a window into the initial design. Data analysis was conducted through evaluation of course goals, assignments, and session activities to determine if the instructors were given materials and session suggestions that would engage the ERTs in the experiences necessary to build their capacity as culturally relevant mathematics teachers and their implementation of MMSJ. Our first focus was how the students’ learning would be assessed. We then focused on how the students would be learning and how the instructor would be facilitating that learning and with what. We analyzed our goals and outcomes, scope and sequence, and course assignments through the lens of CRCD and mathematical modeling. To evaluate the task as CRCD or not, we used a rubric developed by Matthews et al. (2013), which is a way to evaluate tasks to make sure they are interdisciplinary, grounded in community knowledge, cognitively demanding, mathematical in nature, and culturally relevant. For purposes of analysis, we have adapted this rubric to apply to teacher education – namely, the potential products of course assignments or the tasks to be used during course instructional activities as a model for future ERT practice (Fig. 17.4). Practice that integrates cognitively demanding tasks with culturally relevant pedagogy can be described on a trajectory (Matthews, Personal Communication 2018; Fig. 17.5).

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Description Course assignment or instructional activity explicitly either requires prospective teachers to, or gives prospective teachers a model of how to, use math tasks to elicit student inquiry (at times problematically) about themselves, their communities, and the world about them. Course assignment or instructional activity may either require prospective teachers to or give prospective teachers a model of how to draw from connections to other subjects and issues through math tasks. Course assignment or instructional activity either requires prospective teachers to or gives prospective teachers a model of how to develop math tasks that draw from students’ community and cultural knowledge. Course assignment or instructional activity may either require prospective teachers to or gives prospective teachers a model of how to explicitly seek to engage students in math tasks that add to this knowledge through mathematical activity. Math task developed in a course assignment or used during instructional activity is mathematically rich and cognitively demanding and embedded in cultural activity. Course assignment or instructional activity explicitly either requires prospective teachers to or gives prospective teachers a model of how to use math tasks to ask students to engage the discontinuity and divide between school and their own lives – home and school. Tasks developed by prospective teachers as a result of assignments, or tasks used in instructional activity are real-world focused, requiring students to make sense of the world through mathematics. The explicit goal of the task developed by prospective teachers as a result of assignments, or tasks used in instructional activity is to critique society – that is, make empowered decisions about themselves, communities, and world.

Degree in Task Structure Exemplary Developing

Emerging

Fig. 17.4  Adapted CRCD rubric

Emerging

Requires considerable cognitive effort in mathematics

Developing

Requires considerable effort AND embedded in cultural/self/community inquiry and activity

Exemplary

Fig. 17.5  CRCD rubric

Requires considerable cognitive effort embedded in cultural inquiry and activity and targets cutural/self/community empowerment and social

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We used these definitions to assign a rubric level for each of the parts of the curriculum under analysis. Furthermore, we analyzed our instructional activities both more broadly and more deeply  – focusing on whether instructional experiences designed in the course immersed ERTs in mathematical modeling in general and MMSJ in specific.

17.4 Findings 17.4.1 Course Goals and Outcomes Our first step in the design process was creating the syllabus. The course learning objectives were developed prior to our engagement in this research and were very generally tied to the rubric that we use to evaluate our teachers. Those course learning objectives were to: • Design lesson plans that incorporate high-quality teaching methods • Design or select math tasks aligned to state standards • Collect formative assessment data to measure students’ level of understanding of mathematical concept or skills aligned to geometry and measurement standards • Analyze and use student performance data to inform future instruction These learning goals were directly aligned to a learning cycle we call the Clinically Based Learning Theory (Urban Teachers, 2016), found in Fig.  17.6, which builds on the idea of the reflective teaching cycle (e.g., McDonald et al., 2013). ERTs were supposed to be able to, as a result of this (and every) course, plan, enact, and analyze their teaching in their residency clinical setting. Where this course had the most leverage was in the “introducing and learning about the activity.” Therefore, the course description was crafted to be: Fig. 17.6 Clinically-based learning theory

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In this course, participants will be immersed in the progression of the standards for geometry in grades K-6. Participants will acquire knowledge about the theoretical model of geometric understanding (van Hiele) and use this to select a math task aligned to standards, develop lesson plans, and analyze and use student performance data to inform future instruction. Emphasis will be placed on identifying community assets and resources and leveraging those assets to plan geometry instruction that is real-world, authentic, and meaningful for elementary learners [emphasis added]. Participants will demonstrate their understanding of geometric progression in the standards, levels of geometric understanding, and community assets by developing a week-long unit

.This course description connected to the idea of MMSJ needing to be grounded in the personal, cultural, and community knowledge and assets of the K–6 students that our ERTs would be serving. This course description focused on the need for ERTs to be drawing on the resources and the knowledge of the community to plan their mathematics lessons. During our writing of this course, we noticed that we had to balance the programmatic design of the learning cycle, the incorporation of CRP, and the mathematical content when it came to course outcomes. We found that this was a tension between our ideal course description (that would have been continually refined during course design) and the more general one that was developed and submitted for review before we began our iterative process. However, we were hopeful that we could focus that description into more specific action items for students through revision of course assignments.

17.4.2 Assignments In order to get students thinking about CRCD, we developed a sequence of assignments. We considered the assignment descriptions to be the beginning of a trajectory toward CRCD, whereas the instructional experiences in the course would drive the connections to MMSJ. We analyzed these course assignments with an eye to our CRCD rubric and MMSJ; however, we understood these assignments (a) built upon each other on a trajectory and (b) would be supplemented and bolstered by in-class instructional activities that would help ERTs learn to enact MMSJ. The intent of the first assignment was to orient students toward standards-aligned tasks that attended to developmental and standards progressions toward geometric understanding. For this assignment, ERTs are asked to select a math task that is aligned to a CCSS content standard for geometry. They are to do the math involved in the task and then write a paper on how the math task connects to the standards, fits into the geometry progressions (Common Core Standards Writing Team, 2013), and can assess students’ place on the developmental learning progression in geometry. This assignment, in and of itself, was marked as emerging in all of the indicators of CRCD assignments and instructional activities. However, this was designed as a launch point for subsequent assignments that we intended to build toward tasks that were CRCD and MMSJ. Matthews et al. (2013) suggest that having a cognitively demanding task is necessary (but not sufficient) to building CRCD tasks.

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The second assignment asked ERTs to adapt their task based on community exploration and knowledge, and the final assignment was to implement their task with their students, reflect on their and their students’ learning, and adapt the task based on their reflection. The second assignment was intended to build upon the first assignment, in that ERTs would be asked to modify the task to be more culturally relevant. The assignment description reads: Participants will go into the community to collect information and realia about school, community, and student assets that could be used in adapting the selected task to be more relevant to students’ lived experiences. Participants will then adapt the math task based on both the community assets and realia and feedback from the previous assignment, write a lesson plan for implementation, and write a 2-3-page reflection, including copies of the relevant realia, that describes what was learned from the community walk and realia collection and how that influenced the adaptation of the math task.

Students were asked to do community walks and collect information and realia about the school community in order to have a more authentic interpretation of the student community so they could adapt their tasks based on clearer notions of culture, and use of community knowledge and assets (Civil, 2016) rather than stereotypes of urban students and their families (Anhalt et al., 2018; Aguirre & del Rosario Zavala, 2013). In this way, the adaptations of the tasks would be, in a sense, generative. Though the tasks would not be built on student-generated ideas (Gutstein, 2016), they would at least be grounded in community knowledge. The final assignment was a culmination of the previous two assignments, and its connection to CRCD, and MMSJ would be entirely determined by the ERTs’ internalization of instructional experiences and the submission and feedback from the second assignment. However, the final assignment required ERTs to modify their lessons to enhance their cultural relevancy, enact those lessons with students, analyze student work, and reflect on their instruction moving forward. We evaluated the assignments on the above adapted CRCD rubric. Our evaluation is shown in Fig. 17.7. The indicators that were rated developing on this rubric were rated thusly because the course assignment was not explicit in its requirements for those specific things, but a student could have chosen to adapt a modeling task that was oriented toward social justice and critical consciousness and would have met the requirements of the assignment. The assignment was rated as exemplary in the requirement for ERTs to draw from students’ community and cultural knowledge, and requirement for tasks to be real-world focused because students are explicitly asked to embed their previously selected cognitively demanding task within the community knowledge that was collected during their exploration. The assignment was rated exemplary on the requirement to be cognitively demanding and embedded in cultural activity because it built on the first submission, and instructor feedback from that submission, of the selection or development of a cognitively demanding task, and the teachers are asked to adapt this task based on cultural knowledge from their community walk. With regard to any of the indicators of critical consciousness or MMSJ, the assignment made no requirement of the teacher to adapt the math task to allow

402 Description Course assignment or instructional activity explicitly either requires prospective teachers to or gives prospective teachers a model of how to use math tasks to elicit student inquiry (at times problematically) about themselves, their communities, and the world about them. Course assignment or instructional activity may either require prospective teachers to or give prospective teachers a model of how to draw from connections to other subjects and issues through math tasks. Course assignment or instructional activity either requires prospective teachers to or gives prospective teachers a model of how to develop math tasks that draw from students’ community and cultural knowledge. Course assignment or instructional activity may either requires prospective teachers to or gives prospective teachers a model of how to explicitly seek to engage students in math tasks that add to this knowledge through mathematical activity. Math task developed in a course assignment or used during instructional activity is mathematically rich and cognitively demanding and embedded in cultural activity. Course assignment or instructional activity explicitly either requires prospective teachers to or gives prospective teachers a model of how to use math tasks to ask students to engage the discontinuity and divide between school and their own lives – home and school. Tasks developed by prospective teachers as a result of assignments, or tasks used in instructional activity are real -world focused, requiring students to make sense of world through mathematics. The explicit goal of the task developed by prospective teachers as a result of assignments, or tasks used in instructional activity is to critique society – that is, make empowered decisions about themselves, communities, and world.

E. J. Yanisko and L. S. Minicucci Degree in Task Structure Exemplary Developing

Emerging

Fig. 17.7  Evaluation of assignments with the CRCD rublic

students to develop agency in critiquing and potentially changing their world with mathematics. As for mathematical modeling, there was no reference to it at all in the assignment description. However, assignment descriptions can often be broad, and the expectations of those assignments can be refined during instruction through instructional experiences and models and instructor direction. Therefore, it was determined that ERTs should have experiences with MMSJ tasks through instructional experiences during course sessions. When organizing the course outline for the syllabus, we felt that the progression of learning was important. There were three factors at play: the progression of geometry content learning through the grade levels and how the concepts were organized in the geometry progressions (Common Core Standards Writing Team, 2013), the timing of the assignment due dates, and the timing of the educational experiences (including MMSJ). The course sessions were first mapped out by content focus, and then the due dates for assignments were determined. The course had ten sessions, and it was important not only to spread the assignments over the course of the semester but also to make sure that enough instructional experiences occurred between the assignments and the instructors had time to give feedback for improvement. The first, second, and final assignments were determined to occur in sessions 3, 6, and 9, respectively, and therefore the MMSJ sessions were designed to occur in sessions 4 and 7.

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17.4.3 Instructor Notes In this teacher preparation program, since courses are being implemented by many different instructors in many different cities, we develop instructor notes to accompany the syllabi. These instructor notes suggest lesson structure and instructional experiences to be implemented by those teaching the course. These instructor notes are not intended to be prescriptive but are intended to support instructors in their planning. During the course implementation, instructors across cities will meet on a regular basis using a virtual platform in order to discuss and plan for course sessions. Therefore, these notes were generated as a basis on which to begin the planning discussion. This course was grounded in K–6 geometry content. In this course, ERTs were to learn and unpack geometry through both a content and pedagogy lens. The pedagogical features of this course included an investigation into developmental and content trajectories, as well as the standards for mathematical practice (NGA, 2010). The mathematical practices demonstrated and unpacked within this course included making sense of problems and persevere in solving them, look for and make use of structure, and model with mathematics. The first two practices were investigated through engagement in Routines for Reasoning (Kelemanik, Lucenta, & Creighton, 2016) – specifically the routines Connecting Representations and Three Reads. The practice of modeling with mathematics was addressed first through a three-act task (Meyer, 2010) that involved only knowledge of shapes and space from early geometry, and then elaborated upon through a MMSJ (a modified three-act task) that required geometry math content as well as content from other domains. Both of these tasks were conceptualized through the lens of community knowledge and social justice (Civil, 2016; Gutstein, 2018). The instructional experiences analyzed here included sessions that engaged ERTs in mathematical modeling tasks in contexts of social justice. 17.4.3.1 Task Design The earlier sessions of the course focused on conceptions of shapes in elementary geometry. Through discussion we decided to design a three-act task (Meyer, 2010). The ERTs would be shown pictures of murals in the city and finish with the mural outside of a community garden. They would read an article about food injustice and the community garden, and from there, the class would suggest questions about what they wanted to know. The instructor would guide them to a question about the construction of a community garden. Through this task, the ERTs would then be able to use shape and space to model the construction of a community garden – all the while generating their own assumptions and constraints. This was considered MMSJ (Anhalt et al., 2018), as the ERTs would have a problem that had few constraints (e.g., the size of the lot, space needed for different fruits and vegetables) but

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would be able to use shapes to use the space in an efficient way to build their community garden. They would then be able to present their plans to their classmates and discuss their solution. Finally, they would be able to see how the actual community garden was designed. Through this task, ERTs would be able to engage in the iterative process of mathematical modeling to develop a plan, evaluate that plan, and then change the plan if necessary. This task was at the “doing mathematics” level of cognitive demand (Stein et al., 2000) because it was open ended and required the teachers to use their spatial sense and knowledge of shape to solve a messy problem with no procedural solution. This task was considered to be culturally relevant because it allowed students access to academic success in applying geometrical knowledge to a space, cultural competence by grounding the problem in community murals and ideas of beautification, and critical consciousness in addressing food injustice (i.e., food deserts). The connection to critical consciousness was through a lens of social justice in the community. This session would immediately follow their first submission of a cognitively demanding task and would precede the second submission that required them to adapt a task based on community assets. The instructors would direct ERTs to go on investigatory walks in their own school community, so they could adapt their math task to use geometry to make models to apply to issues of social justice in the school community. In the later sessions, the content focused more on shape construction and deconstruction and geometric measurement, and much of the geometry content was aligned to the upper elementary grades. Therefore, in session 7, we decided to build on the original modeling task and expand the area of modeling and exploration beyond the original constraints. In this task, ERTs were asked to apply knowledge from geometry, measurement, and number and operations to design their garden, decide what to plant, determine what was needed, and decide how to sell their products. In this modeling task, they would need, for example, to determine the space and other resources needed to grow different kinds of fruits and vegetables and determine the optimal ones to plant based on space, resources, demand, initial cost, potential sale price, and cost to maintain. We designed this particular instructional activity in order for ERTs to experience MMSJ grounded in community knowledge, but also as a way for ERTs to reflect upon how to increase or decrease constraints on a situation based on students’ place in their progression of school mathematics knowledge and how to make a mathematical modeling for social justice task connect different domains of mathematical knowledge. This second experience was situated in the course directly after their second submission, so they could use this experience and the feedback of the instructor to make adjustments to their task before implementing it in their classroom. The instructional activities would include how to engage students in a real-world modeling task with a content focus of area and perimeter. Students would plan and rehearse their modeling task.

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17.4.3.2 CRCD Analysis The two tasks were analyzed as one, since they build off of each other from the same initial context (Fig. 17.8). Half of the indicators were rated as exemplary because the tasks gave ERTs a way to connect geometry, measurement and data, number and operation, and mathematical modeling in a real-world task that was grounded in the prevalent urban community issue of community beautification and food deserts. The first indicator was rated as developing because the task, as designed in the instructor notes, did not arise from ideas and issues generated from the community walks of the ERTs, or better yet, their K–6 students. This was one of the difficulties that arose from developing this course ahead of time. The fourth indicator was rated as developing because the task did not require ERTs or their students to fully add to community knowledge, as the task was grounded in an issue that had already been raised and partially addressed within the community. The sixth indicator was rated as developing because the task did not explicitly connect to the home lives of ERTs’, or their students’, personal experiences at home or in the community. The final indicator was rated as developing because the solution to the task at hand was pre-determined, and it did not fully give ERTs, or their students, the agency to create their own solutions to social justice issues in the world around them.

Description Course assignment or instructional activity explicitly either requires prospective teachers to, or gives prospective teachers a model of how to, use math tasks to elicit student inquiry (at times problematically) about themselves, their communities, and the world about them. Course assignment or instructional activity may either require prospective teachers to or give prospective teachers a model of how to draw from connections to other subjects and issues through math tasks. Course assignment or instructional activity either requires prospective teachers to or gives prospective teachers a model of how to develop math tasks that draw from students’ community and cultural knowledge. Course assignment or instructional activity may either requires prospective teachers to or gives prospective teachers a model of how to explicitly seek to engage students in math tasks that add to this knowledge through mathematical activity. Math task developed in a course assignment or used during instructional activity is mathematically rich and cognitively demanding and embedded in cultural activity. Course assignment or instructional activity explicitly either requires prospective teachers to or gives prospective teachers a model of how to use math tasks to ask students to engage the discontinuity and divide between school and their own lives – home and school. Tasks developed by prospective teachers as a result of assignments, or tasks used in instructional activity are real-world focused, requiring students to make sense of world through mathematics The explicit goal of the task developed by prospective teachers as a result of assignments, or tasks used in instructional activity is to critique society – that is, make empowered decisions about themselves, communities, and world

Fig. 17.8  Evaluation of tasks with the CRCD rubric

Degree in Task Structure Exemplary Developing

Emerging

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With regard to MMSJ, the ERTs were asked to participate in the cycle of mathematical modeling, especially with session 7. In session 4, there was less of a need to research information needed and make assumptions and choices. Furthermore, due to the content specificity and grade level, there were fewer ways to analyze the model or validate conclusions. However, since there were more parameters, higher grade-level content, and more cross-cutting content in the task in session 7, ERTs would have to do a great deal of research and had to make many assumptions and choices (e.g., Were they going to sell their produce?, What vegetables would have the best profit margin?, What amount of space would they need?). They would also have to continually analyze their model to make sure they were making the best choices to meet their goal of addressing food injustice as well as providing an avenue for sustainability within the community. Whether the ERTs internalized this experience and applied to their planning and implementation will remain to be seen when the course is taught, and the assignments are collected.

17.5 Discussion This course development was iterative in nature – building through course descriptions, assignment descriptions, course outlines, and proposed instructional activities. There were three themes that arose during the analysis of our development of the course: tensions between content, pedagogy, and critical pedagogy; the interdisciplinary nature of modeling in geometry on different grade levels; and ideas around authenticity of modeling task in a course that was developed in this a priori way separate from the settings, instructors, ERTs, and their students.

17.5.1 Tensions Between Content, Pedagogy, and Critical Pedagogy In this course, we had three assignments. Out of those three, only one of the assignments was explicitly stated as related to developing math tasks that were rooted in culture. However, it should be noted that this assignment built off the work in the first assignment where students created or adapted a high cognitive demand geometry task. Additionally, in the final assignment, students refined their tasks to enhance their cultural relevancy and enacted these lessons with students, analyzed student work, and reflected on their instruction moving forward. It was not explicitly stated in the assignment descriptions that these tasks were required to be MMSJ tasks. González (2009) found that, initially, teachers may consider SJM as simply grounding mathematics tasks in cultural or social contexts. Although we supported those assignment descriptions with in-class experiences with MMSJ experiences, it

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remains to be seen how much of that in-class experience with the task is taken up by the ERTs and demonstrated in the assignment submissions. Furthermore, out of the ten sessions, only two of them were spent in immersive experiences with MMSJ. Although two sessions cannot be considered as a sufficient immersion in MMSJ tasks, this geometry course was designed for ERTs to experience, plan, rehearse, implement, and reflect upon lessons centered around rich CRCD and MMSJ tasks. At the outset of the course, students critically examined mathematical content and practice standards, determined and defended the level of cognitive demand for various geometry tasks, and analyzed student work to assess K–6 students’ level of geometric thinking. In each session, ERTs engaged in high cognitive demand geometry tasks that required them to make sense of and reflect on the cross-sections between mathematics and meaningful real-world situations. As the course progressed, ERTs engaged in deeper examination and discussion of what makes a mathematical task both culturally rich and cognitively demanding. While only two course sessions specifically included MMSJ tasks, CRCD tasks were at the heart of this course, and ERTs had the opportunity to refine their own CRCD tasks to be implemented with students during multiple course sessions. Course assignments were intentionally scaffolded to support ERTs in developing and refining a lesson centered around rich CRCD tasks to be implemented with students, then to analyze students’ work to determine their levels of geometric thinking. It is necessary for ERTs to be immersed in MMSJ in order for them to make connections. It is important that instructors of the course build toward and continually refer to and develop experiences around CRCD mathematical modeling. The other sessions focused on content and pedagogy were not intentionally designed to incorporate culturally relevant pedagogy – although the instructors could certainly make decisions about tasks and implementation of those sessions. Although there was more work on developing modeling tasks, those problems were less “messy” (or not messy at all), and although the tasks still included references to culture – as there is no mathematics that is “culture-free” (Ladson-Billings, 1997) – these experiences were not intentionally designed for cultural relevance. There seemed to be an emphasis on the same things as mathematics education courses everywhere: content knowledge and pedagogy (Gutiérrez, 2013). However, it is important to note that content knowledge and pedagogy must include knowledge of mathematical modeling, CRP, and SJM. This seems to be a tension that is not uncommon when classroom teachers are developing courses around MMSJ (Bartell, 2013). Gutstein (2016) argues that even when a teacher educator is implementing K–12 classes designed for SJM, there is always a tension on time spent in contextualized mathematics, decontextualized mathematics, and non-mathematical talk. It should have come as no surprise that we would also struggle in our endeavors to marry mathematical modeling and culturally relevant pedagogy in geometry. This seems to imply that treating content like geometry as stand-alone and separate from other aspects of elementary mathematics may hamper an instructor’s ability to incorporate mathematical modeling for social justice into coursework.

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17.5.2 Interdisciplinary Nature of Modeling and Grade-Level Appropriateness During these first few iterations of design, we noticed it was difficult to (1) develop robust mathematical modeling tasks that were specific only to geometry standards and (2) develop culturally relevant mathematics modeling tasks that would be accessible to K–2 students in an authentic way. The latter we addressed during the first instructional experiences – using modeling to plan the use of a space using nothing but knowledge of shapes. However, when it came to the former, we eventually agreed that the best way to engage ERTs in authentic, iterative, mathematical modeling in a geometry course was to abandon any designs on keeping the content of geometry separate from the other domains in K–6 mathematics. The task became more robust and authentic when we integrated measurement and number and operations. 17.5.2.1 Setting Constraints and Open-Ended Problems Another concern that arose around grade-level appropriateness was around the level of constraints that we as teacher educators should and would set for these modeling tasks. It is very easy for teachers – and, arguably, teacher educators – to fall short of truly culturally relevant mathematical modeling because of our comfort level with “nice,” convergent, closed problems that can be modeled simply without students making assumptions or engaging in the iterative cycle of revision that is the essence of modeling (Anhalt et al., 2018; Leonard & Guha, 2002). However, we continuously reminded each other to keep the tasks as open as possible so that students could impose their own constraints during the modeling process. In the end, we included a framing discussion after the second instructional experience that hopes to engage the ERTs in a conversation around what levels of constraints are appropriate and when they are appropriate.

17.5.3 Tension in Authenticity in Planning a Curriculum That Is Static A final theme that arose was a tension that we felt we could not resolve during this stage of design. The basis of culturally relevant pedagogy is that one must have deep knowledge of not only the mathematics content but of the students as well (Ladson-­ Billings, 1997). Although we designed assignments and experiences that drew on CRCD and MMSJ as a driving force, these experiences were not truly grounded in knowledge of the ERTs that would be enrolling in the class in the future, or the K–6 students they would be teaching. In that sense, we are only simulating cultural relevance. As with any curriculum, it must be adapted for the students in the room, and

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the hope is that the instructors engage in further iterative design, formally or informally, to make sure the course is truly culturally relevant for ERTs. Perhaps grounding those MMSJ tasks in themes suggested from the lived experiences of the ERTs themselves would be a way that instructors could take a pre-designed teacher preparation course and reform it for their students. Since the same ERTs are enrolled in the coursework throughout the trajectory of the program, insights about the ERTs’ lived experiences could be shared between instructors in order to build coursework content that is more responsive and relevant to the people in the room. Furthermore, content from coursework could be built upon during instructional coaching were the ERTs could be supported to make meaningful connections to their students which would allow them to build MMSJ that were CRCD. This represents only one course in an elementary mathematics teacher program of study. Issues of “culture” and “diversity” should not be relegated to a single course (Ladson-Billings, 2000). Although the ideas of cultural relevance and social justice are embedded alongside the mathematics throughout the vehicle of modeling in this course, it will not be sufficient unless this continues across the entire program of study. Culturally relevant pedagogy has been recognized as important for the education of children, especially minoritized children (Ladson-Billings, 1997). Although it has often been found to be a struggle to connect culturally relevant pedagogy and mathematics (Bartell, 2013; González, 2009), we used mathematical modeling as a vehicle for that integration (Anhalt et  al., 2018). Although there were tensions between MMSJ and the traditional topics of the course, and between open-ended modeling problems and the more constrained “typical” modeling problems, we found this course development process to be a useful first step in designing a course (and eventually a program of study) that seamlessly embeds tasks that are CRCD and MMSJ, and educates teachers how to seamlessly teach mathematics in a way that is culturally relevant. With the development of the Standards for Preparing Teacher Educators by the Association of Mathematics Teacher Educators ([AMTE], 2017), there is a push for teacher preparation to integrate issues of equity into teacher preparation. However, the CCSSM is largely devoid of connections to issues of social justice that could be approached from a mathematical modeling perspective. Often, standards for mathematics instruction do not take into account issues of equity – even in their very development (Martin, 2015). Teacher educators must work to build in opportunities for pre-service and in-service teachers to experience and learn how to engage students in mathematical modeling, especially through MMSJ tasks. We suggest questions for future study – what would the continuation of this design process entail, and would there ever be a fully completed end product? How can teacher educators collaborate to iteratively revise the course? Specifically, since MMSJ tasks should be grounded in contexts and purposes that are relevant to the students in the classroom (Gutstein, 2018), how can we as teacher educators build CRCD tasks and experiences that demonstrate to our pre-service or in-service teachers that central tenant?

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References Aguirre, J. M., & del Rosario Zavala, M. (2013). Making culturally responsive mathematics teaching explicit: A lesson analysis tool. Pedagogies: An International Journal, 8(2), 163–190. Anhalt, C. O., Staats, S., Cortez, R., & Civil, M. (2018). Mathematical modeling and culturally relevant pedagogy. In Y. J. Dori, Z. R. Mevarech, & D. R. Baker (Eds.), Cognition, metacognition, and culture in STEM education (pp. 307–330). Cham, Switzerland: Springer. Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. Retrieved from: amte.net/standards Bartell, T. G. (2013). Learning to teach mathematics for social justice: Negotiating social justice and mathematical goals. Journal for Research in Mathematics Education, 44(1), 129–163. Civil, M. (2016). STEM learning research through a funds of knowledge lens. Cultural Studies of Science Education, 11(1), 41–59. Common Core Standards Writing Team. (2013). Progressions for the common core state standards in mathematics (draft). Grades K–5, geometry. Tucson, AZ: Institute for Mathematics and Education, University of Arizona Delpit, L. (2003). Educators as “seed people” growing a new future. Educational Researcher, 32(7), 14–21. Delpit, L. D. (2012). “Multiplication is for white people”: Raising expectations for other people’s children. New York, NY: The New Press. González, L. (2009). Teaching mathematics for social justice: Reflections on a community of practice for urban high school mathematics teachers. Journal of Urban Mathematics Education, 2(1), 22–51. Gutiérrez, R. (2013). Why (urban) mathematics teachers need political knowledge. Journal of Urban Mathematics Education, 6(2), 7–19. Gutstein, E. (2003). Teaching and learning mathematics for social justice in an urban, Latino school. Journal for Research in Mathematics Education, 34, 37–73. Gutstein, E. (2016). “Our issues, our people—Math as our weapon”: Critical mathematics in a Chicago neighborhood high school. Journal for Research in Mathematics Education, 47(5), 454–504. Gutstein, E.  R. (2018). The struggle is pedagogical: Learning to teach critical mathematics. In The philosophy of mathematics education today (pp. 131–143). Cham, Switzerland: Springer International Publishing. Kelemanik, G., Lucenta, A., & Creighton, S. J. (2016). Routines for reasoning: Fostering the mathematical practices in all students. Portsmouth, NH: Heinemann. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. Ladson-Billings, G. (1995). But that’s just good teaching! The case for culturally relevant pedagogy. Theory Into Practice, 34(3), 159–165. Ladson-Billings, G. (1997). It doesn’t add up: African American students’ mathematics achievement. Journal for Research in Mathematics Education, 28(6), 697–708. Ladson-Billings, G. (2000). Fighting for our lives: Preparing teachers to teach African American students. Journal of Teacher Education, 51(3), 206–214. Lampert, M., Franke, M. L., Kazemi, E., Ghousseini, H., Turrou, A. C., Beasley, H., … Crowe, K. (2013). Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching. Journal of Teacher Education, 64(3), 226–243. Leonard, J., & Guha, S. (2002). Creating cultural relevance in teaching and learning mathematics. Teaching Children Mathematics, 9(2), 114. Lerman, S. (2000). The social turn in mathematics education research. In Boaler J. (Ed.) Multiple perspectives on mathematics teaching and learning (pp. 19–44). Westport, CT: Ablex. Martin, D. B. (2015). The collective black and principles to actions. Journal of Urban Mathematics Education, 8(1), 17–23.

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Matthews, L. E. (2003). Babies overboard! The complexities of incorporation culturally relevant teaching into mathematics instruction. Educational Studies in Mathematics, 53(1), 61–82. Matthews, L.  E., Jones, S.  M., & Parker, Y.  A. (2013). Advancing a framework for culturally relevant, cognitively demanding mathematics tasks. In J. Leonard & D. B. Martin (Eds.), The brilliance of Black children in mathematics: Beyond the numbers and toward new discourse (pp. 123–150). Charlotte, NC: Information Age Publishing. McDonald, M., Kazemi, E., & Kavanagh, S. S. (2013). Core practices and pedagogies of teacher education: A call for a common language and collective activity. Journal of Teacher Education, 64(5), 378–386. Meyer, D. (2010). TEDxNYED: Math class needs a makeover. Retrieved from https://www.ted. com/talks/dan_meyer_math_curriculum_makeover?language=en#t-­586113 National Governors Association Center for Best Practices, Council of Chief State School Officers [NGA]. (2010). Common core state standards mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of chief State School Officers. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-­ based mathematics instruction: A casebook for professional development. New  York, NY: Teachers College Press. Teachers, U. (2016). Clinically based learning theory. Baltimore, MD: Author. van den Akker, J. (2010). Curriculum perspectives: An introduction. In J. van den Akker & W. Kuiper (Eds.), Curriculum landscapes and trends (pp. 1–10). Dordrecht, the Netherlands: Springer Science and Business Media. Volmink, J. (1994). Mathematics by all. In S. Lerman (Ed.), Cultural perspectives on the mathematics classroom (pp. 51–67). Dordrecht, the Netherlands: Springer.

Chapter 18

Learning from Mothers as They Engage in Mathematical Modeling Marta Civil, Amy Been Bennett, and Fany Salazar

This chapter focuses on a study of mathematical modeling with a group of Mexican American mothers. As mathematical modeling becomes more present in elementary classrooms, engaging parents in this work can help support their children’s learning, especially given that modeling can have rich connections to everyday life situations. Our research is driven by the potential of mathematical modeling to promote culturally sustaining teaching for non-dominant students. The work we present here is part of a larger project with parents and teachers in working-class, Latinx communities aimed at developing a two-way dialogue about mathematics education between home and school. In this chapter we discuss two related components of the work we have done with a group of eight mothers. One component looks at how the mothers approached a mathematical modeling task based on an activity (making paper flowers) that was familiar to some of the women in the group. For the second component we invited the mothers to contribute contexts for potential modeling tasks that reflect their everyday experiences and knowledge. The chapter begins with a brief overview of relevant research in mathematics modeling with an eye on equity implications. We then present the main theoretical constructs that guide our work, namely, funds of knowledge (González, Moll, & Amanti, 2005) and parents as intellectual resources (Civil & Andrade, 2003). After describing the context for this research study, we turn to the two components that form the core of this chapter. We conclude with implications for developing culturally sustaining mathematical modeling tasks.

M. Civil () · F. Salazar University of Arizona, Tucson, AZ, USA e-mail: [email protected] A. B. Bennett University of Nebraska-Lincoln, Lincoln, NE, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_18

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18.1 Review of Literature on Mathematical Modeling and Funds of Knowledge Mathematical modeling has been defined as a process for connecting the real world to the world of mathematics (Blum & Borromeo Ferri, 2009; Blum & Leiss, 2007), as an “applied problem-solving process” (Blum & Niss, 1991, p. 38) and as a subset of sense-making (Schoenfeld, 2013). Our definition of mathematical modeling aligns with Anhalt, Staats, Cortez, and Civil (2018) who summarized that “mathematical modeling is an iterative process whereby we use mathematics to understand or analyze some situation that often comes from outside mathematics” (p.  313). More specifically, this process breaks down into “posing authentic, open-ended problems, making assumptions, identifying constraints and variables, building mathematical solutions and, finally analyzing and interpreting these solutions” (Suh, Matson, & Seshaiyer, 2017, p. 2). Mathematical modeling is often represented by a cyclic diagram (see Fig. 18.1) to illustrate the iterative nature of the modeling process (Anhalt & Cortez, 2016; Anhalt, Cortez, & Bennett, 2018; Blum & Leiss, 2007; Mousoulides, Christou, & Sriraman, 2008), each one a variation on the overarching concept of “a process in which students … make sense of an everyday situation” (Anhalt, Cortez, & Bennett, 2018, p. 202). The modeling steps highlighted in the diagram in Fig. 18.1 include (1) making sense of the problem, (2) formulating a model, (3) solving or analyzing the model, (4) interpreting the solution and drawing conclusions, (5) validating conclusions, and (6) reporting out. These six steps, or variations of them, often comprise what is meant by the modeling process and are consistently used in research

Fig. 18.1  Mathematical modeling cycle from Anhalt et al. (2018)

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and policy documents (Aguirre, Anhalt, Cortez, Turner, & Simic-Muller, 2019; CCSSI, 2010; Garfunkel & Montgomery, 2016). The open-ended structure of modeling tasks allows for students to make their own mathematical decisions and choose which strategies and solutions are useful and validated (Carmona & Greenstein, 2010). Similarly, modeling tasks challenge the perception of who is capable of doing mathematics and what it means to know and do mathematics (Wickstrom, 2017). Rather than confirming the task developer as the authority, authentic modeling tasks designate the students as powerful and necessary agents of change, as well as provide opportunities for equitable positioning of students (Wickstrom, 2017). Recent research has shown that introducing mathematical modeling concepts in early grades has multiple benefits for students. Not only are students better able to reason and think critically about realistic situations, they also gain understanding about mathematical concepts in general, many of which are assessed on standardized tests (Mousoulides et  al., 2008; Stohlmann & Albarracín, 2016; Suh et  al., 2017). Additionally, engaging in modeling tasks with elementary and middle school students leads to improved classroom discourse and development of life skills required for real-world situations (Asempapa, 2015; Stohlmann & Albarracín, 2016). Since modeling tasks are often created out of authentic situations using real data (Aguirre et al., 2019; Anhalt & Cortez, 2016; Anhalt, Cortez, & Bennett, 2018; English, 2012; English & Watters, 2005; Lesh & English, 2016) and can be adapted to fit the demographics and location of specific classrooms (Asempapa, 2015), they are rich in mathematics content and allow teachers to draw on cultural funds of knowledge as well as multiple mathematical knowledge bases (see Turner et  al., 2012). In some cases, researchers work with teachers to design modeling tasks that are relevant to their school or community contexts (Suh et al., 2017; Wickstrom, 2017), creating rich examples of the relationship between mathematics found inside and outside of the classroom. Along these lines, we present in this chapter some aspects of work done with a group of Mexican American mothers on mathematical modeling with a funds of knowledge orientation. The larger project that serves as the context for the study presented in this chapter draws on the concept of parents as intellectual resources (Civil & Andrade, 2003), which means that parents have knowledge and experiences that can strengthen children’s learning of mathematics both at home and in school. At the basis of this work is the concept of funds of knowledge, a term coined by Vélez-Ibañez and Greenberg (1992) to refer to the “strategic and cultural resources, which we have termed funds of knowledge, that households contain” (p. 313). The idea is that all communities, families, and households have knowledge, resources, experiences, and skills that allow them to get ahead and thrive (Moll, Amanti, Neff, & Gonzalez, 1992). Prior research (Civil, 2002, 2007, 2016; Sandoval-Taylor, 2005) has shown the potential for students’ engagement in rich mathematical tasks through building on students’ and their families’ funds of knowledge. Past research (Civil, 2007, 2016) has also explored the relationship between school mathematics and real-world situations. In particular, this research discusses dilemmas around how much academic mathematics is really needed to

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solve everyday problems. In this chapter we continue this line of work by looking at the intersection of funds of knowledge with mathematical modeling.

18.2 Context and Methods The study presented here was conducted at an elementary school that serves a majority of Mexican American children. The school principal helped recruit a small group of parents to be part of the project. The only condition for participation was that each parent had at least one child in grades K-3. While the invitation was directed to fathers and mothers, only mothers joined the project. This reflects our experience in prior work with parents and mathematics in the same community (Civil & Bernier, 2006; Quintos, Civil, & Bratton, 2019). The eight mothers in our study are of Mexican background and have Spanish as their home language. One of the components of this project were the Math for Parents (MFP) sessions, which are short courses, usually three 2-hour sessions on a theme (e.g., fractions; different algorithms with whole numbers; measurement). Spanish was the dominant language in the MFPs. The study presented here took place during a series of three consecutive Math for Parents (MFP) sessions on mathematical modeling. Prior to the tasks that serve as the focus on this chapter, five of the eight mothers had worked on two mathematical modeling tasks adapted from M2C3.1 While these tasks were not necessarily connected to these mothers’ funds of knowledge, they were based on contexts that are likely to be familiar to many people (siblings deciding how to split the cost of a present for a family member; exploring how much popcorn to buy for an event at school). Thus, at the moment of this study several of the mothers were familiar with some of the key ideas of modeling, in particular the concept of assumptions and the open-endedness of modeling tasks (as compared to other mathematical tasks). The high level of engagement and richness of the mathematical discussions around these two prior modeling tasks prompted us to develop this line of inquiry further. From funds of knowledge-based interviews with the mothers and conversations in the MFPs, we learned that some of the mothers knew how to make paper flowers. We decided to devise a task grounded on this idea and then use that experience to engage the mothers in the co-development of other modeling tasks based on their funds of knowledge. This chapter focuses on these two aspects, the mothers’ work on the Paper Flowers task and the mothers’ contribution to the development of modeling tasks. The mothers worked in two groups of three to four women. They usually chose where to sit and the only stipulation was to have groups of similar size. Lidia, Julieta, Marcela, and Esmeralda were in one group (though Esmeralda was not there

 M2C3: Mathematical Modeling with Cultural and Community Contexts (Turner, Aguirre, Foote, and Roth McDuffie, https://sites.google.com/qc.cuny.edu/m2c3/) 1

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for the first MFP); we refer to this group as Lidia’s group. The other group was formed by Sandra, Alondra, Magali, and Esther (though Esther was not there for the first MFP); we refer to this group as Sandra’s group. We set up an audio recorder and a video camera at each of the two tables. We also had a video camera for the whole group discussion. We collected their written work at the end of each session, scanned it, and brought it back for the following session. So, our sources of data include: artifacts (their posters, notes, pictures of flowers, actual flowers), the video and audio recordings, and our field notes. We had two MFP sessions on the Paper Flowers task and one MFP session on the development of a modeling task. After the first MFP session, we looked at the videos and their notes, and in the second MFP session, we used respondent validation (Maxwell, 2013). This feedback ensured that we understood what they had written and said. After the three MFP sessions, two of us (who know Spanish well) viewed the videos individually with a general focus on aspects related to the mathematical approaches to the tasks, as well as any connections to everyday knowledge. After this first view, we each focused on one group of mothers and looked for salient themes that related to modeling and funds of knowledge in the previously noted instances and the rest of the data. In this way, we used a priori or concept-driven codes to guide our categorization of themes (see Gibbs, 2007). The categories were based on the six phases of the modeling cycle (see Fig. 18.1). We isolated segments that we transcribed and brought up for discussion in our research meetings. During this discussion, we agreed on the significance of each segment based on two main goals: aspects of the mathematical modeling process and connections to the mothers’ funds of knowledge.

18.3 The Paper Flowers Task As we said earlier, the Paper Flowers task built on some of the mothers’ knowledge of either making or decorating with paper flowers. This is consistent with our position of viewing parents as intellectual resources. Since the mothers in our group are very involved in the school community, we created a realistic yet fictitious context for the task, based on decorating a wall at the school for a display. We clarify that the setting of the task was realistic, in that it could happen, but fictitious in that we did not carry it out. The Paper Flowers task states: Our school wants to decorate a wall with paper flowers for a back to school display. To help with the decorating, you volunteer to design the display and buy tissue paper for the flowers. How many packages of tissue paper do you need to buy to decorate the wall with paper flowers? Our goal was to make the task as realistic as possible by using real data from stores where the mothers had mentioned shopping: Store 1 sold tissue paper packets of 8 sheets with dimensions 16 × 24 in. at $1.49, and Store 2 sold tissue paper packets of 24 sheets with dimensions 20 × 20 in. at $1.00. We provided this information in our initial launch of the lesson, as well as a task handout to help the mothers keep

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track of their ideas and calculations. The task sheet prompted the mothers to clearly address specific aspects of the task, as described below: Your plan to decorate the wall should include: –– –– –– –– ––

The number of flowers you plan to make of each size A diagram or picture of how the flowers will decorate the wall An example of how you use the individual sheets of tissue paper The amount (if any) of paper that is wasted How your plan could be used to decorate for other events

In our presentation of the task, we provided dimensions for the wall that the mothers would be decorating (9 feet by 12 feet), as well as options and prices for buying the packages of tissue paper. We showed a video that briefly demonstrated one way to make flowers out of five-layered rectangles of tissue paper. We emphasized that this was only one way to make paper flowers, and we asked the mothers who had experience making flowers to compare this method to their own. We also brought two physical examples of the flowers that we made by following the steps in the video. The task was intended to address K-6 mathematics concepts such as measurement (in particular, area), multiplication, and ratios and proportional reasoning. The complexity of the task also necessitated a certain level of spatial reasoning in order to fit the rectangle cutouts on each sheet of tissue paper and to arrange the flowers of various sizes on the wall. We anticipated these challenges and created scaffolds to guide the mothers’ thinking as they engaged in the task. For example, we expected that the mothers would want to make flowers of various sizes for their display. Therefore, we created a table with four sizes (small, medium, large, extra large) and their corresponding dimensions. Since each size of flower required different sizes of rectangles to cut out, we provided an example template of how to arrange various rectangles on a single sheet of tissue paper. In alignment with open-ended tasks, we constantly emphasized that our ideas and examples were only one approach to solve the task. As mentioned above, the mothers had previously completed two modeling tasks. During both of these tasks, we briefly explained the modeling process and focused on the step of making assumptions. We discussed what an assumption is and why they are necessary when solving open-ended tasks, where not all the information is given. During the Paper Flowers task, we presented a diagram of the modeling cycle (Anhalt, Staats, et al., 2018) and discussed each step in the modeling process (see Fig. 18.1) with the goal that the mothers would consider each of these steps while creating a plan for their paper flowers display. At the end of the first two sessions, we projected the modeling cycle diagram again, and the mothers reflected on how the modeling elements were apparent (or not) in their work and solution. The atmosphere of our sessions was casual and friendly; the mothers had been working with Marta and Fany for 2 years and felt very comfortable around them. In a lighthearted conversation, we joked that we were going to “hire” a team to create the flowers and decorate, which created an (unintended) friendly competition between the two groups (teams). The mothers were motivated to formulate a plan

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that highlighted the most important aspects to them, such as saving money or creating the most attractive display. As the reader will see in the cases below, the mothers were very enthusiastic and treated the task as an authentic scenario.

18.3.1 The Work of Lidia’s Group For this group we focus on the initial design of the flowers. This team reflected Lidia’s expertise in the business world (as a buyer-seller for a company) through their emphasis on the cost (including taxes) and the businesslike approach. For example, they talked in terms of adjusting the design, number, and size of flowers to their clients’ requests. They seemed to take the setting of the task (i.e., hiring one of the two teams to decorate the school wall) seriously in that they mentioned how many volunteers they would need to make the flowers and who would do what (e.g., cutting the paper, putting up the string). They also considered the time involved in making flowers and the need to be efficient: bigger flowers will cover more of the wall and take about the same time to make as smaller flowers. Also guiding their thinking was an interest in minimizing paper wasted. A key aspect of this team was Julieta’s expertise with making and decorating with paper flowers (even though it was a different type of flower). Thus, Julieta was positioned as an expert in this team. For example, though the video in the introduction to the task showed five layers of tissue paper per flower, this team decided on many more sheets to make them fuller, arguing that with five layers they would not look as nice. Thus the decisions in this team were guided by Lidia’s business experience and most markedly by Julieta’s knowledge of how to make paper flowers. The team decided on having flowers in three different sizes. They reached a decision on the larger size quite quickly. They would be using twenty-four 20 × 20 in. sheets of tissue paper for one flower. So, it would cost $1.08 for one flower, and since they wanted seven flowers, that would be $7.56. The discussion for the medium size was more involved and in fact was further revised the second day they worked on the task. They wanted to minimize waste, and since the dimensions of the sheets are 20 × 20 in. (or 16 × 24 in. from a different store, which was more expensive), they thought about doing medium size flowers that would be 10 × 20 in. But they were concerned that the flowers would not look nice, as they explain in the following excerpt: [Julieta and Marcela are looking at the 10 × 20 in. rectangle and are frowning] Julieta: For a flower, it has to be a square Marcela: Yes, and here [pointing to the 10 × 20 in.] it would be rectangular, and if you fold it [10 × 10 in.], it would be too small. Julieta: If we fold [starts folding the 20 in. side along the 10 in. side], it’s going to be fewer folds and it’s going to come out very long and it’s not going to have the shape of a flower. Marcela: Yes, right, we are going to have to cut. [Julieta folds to the 10 × 10 in.]

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Marcela: But it’s very small; it’s smaller than the one she [Amy] made [pointing to the first one on the table which is 8 × 12 in.]. Lidia: Maybe we need to buy a different size paper.

[Marta comes over and asks the group what seems to be the issue with the 10 × 20 in.] Marta: But with the 10 × 20 in. you don’t think it will look nice. What makes you think that it won’t look nice? I’m sure you are right, I’m only asking. [Julieta is thinking and looking at the paper.] Julieta: Ah, but yes, we can do it… we fold in the other direction. Marcela: But it’s going to look small; we wanted to come up with a size that would follow the large one [which is 20 × 20 in.] [Marcela gestures from a big flower to a small flower]. Lidia: So what’s the difference if we fold it this way? Julieta: Because then there will be more folds and it will be fuller (más tupida). Lidia: Aha, with less paper. Julieta: But if you fold like this [in the other direction], it is going to have fewer folds and it is not going to have the shape, it will be very pointy (picuda) and very sad (triste). Marcela: Very sad (laughs).

This group was very quick at settling on the size for the larger flower, using 24 sheets of 20 × 20 in. (see Fig. 18.2 for one of Julieta’s large flowers), which is considerably more sheets than the examples discussed in the launching of the task. There was no discussion on the overall appearance of this larger flower. They seemed to be in agreement that it would be the right level of fullness and that it would look good. Yet, for the medium-size flower, they seemed to be constrained by the sizes of the sheets available and their desire to not waste paper while making sure that the flower looked good. The 10  ×  20  in. flower folded in one direction would look “sad,” and if folded in the other direction, it may end up being too small. In what follows we see Lidia working on making sense of why the 10 × 20 in. may

Fig. 18.2  Julieta’s large paper flower

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not look right. Lidia looked at the sizes given in the table (6 × 8 in.; 8 × 12 in.; 12 × 16 in.; 16 × 24 in.) and said: The difference in inches in this rectangle (pointing the 6 × 8 in.) is 2, here it’s 4 (for 8 × 12 in.), here it’s 8 (for the 16 × 24 in.) but they are in proportion to the size, and here (for the 10 × 20 in.), it’s 10 inches difference between the sides, so that’s why she [Julieta] says that it would be very irregular, and it would not look round like a flower, it would look like a bird of paradise [gesturing what such “flower” may look like and they all laugh].

While Julieta and Marcela seemed to see the “problem” with the 10 × 20 in. based on their ability to visualize the flowers, Lidia analyzed the dimensions provided in the table and gave an argument hinting to proportional reasoning (“they are in proportion to the size”) by comparing the lengths of the sides of the different rectangles (although she was looking at the difference between the sides, an additive approach). This seemed to help convince her that the 10 × 20 in. would not work as well. This exchange points to a potentially interesting line of further discussion between intuitive and experiential knowledge and the need for a more “mathematical” justification. The group continued working on the task and eventually settled on three sizes for the flowers (5 × 5 in.; 16 × 12 in.; 20 × 20 in.). What we want to highlight with this vignette was the interplay of the range of expertise of the group members as they worked on the task. The three women were actively engaged in the task and took on different roles. Julieta brought in her experience with making paper flowers; Lidia brought in her business experience; and Marcela acted as a link in the group, by making observations at key moments in the discussion. Both Julieta and Marcela were comfortable with the idea of making paper flowers, while Lidia clearly deferred to them and throughout the exchanges often mentioned that she was not good at doing crafty things. Lidia’s genuine questions to her teammates about how folding the flower in different ways would impact the aesthetic outcome highlighted her appreciation for their crafting expertise. From a mathematical modeling point of view, there were several interesting moments, which we discuss in a later section.

18.3.2 The Work of Sandra’s Group For this group, we focus on the mathematical connections that the women made while engaging in the Paper Flowers task. This group had little experience making paper flowers. Thus, they followed quite closely the information we provided and ultimately chose a plan that minimized cost and adhered to the flower sizes we suggested. To understand how to divide each sheet of tissue paper, Sandra’s group originally drew rectangles of the same size on the same sheet. Alondra remarked that “doing it like this, on the paper, it’s easier to understand.” They were not overly concerned about wasting paper since the total cost of the tissue paper for their proposal was $2.00 plus tax. They even mentioned that wasted paper could be converted into confetti and spread around the flower wall. But we challenged them to

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Fig. 18.3  Sketch of possible rectangles on a sheet of tissue paper

be efficient with the use of tissue paper, which prompted them to draw rectangles of different sizes on the same sheet (Fig. 18.3 is their sketch of what they did on tissue paper). In a 20  ×  20 inch (“pulgadas”) sheet, they drew a 16  ×  12  in. rectangle (“grande”), a 12 × 8 in. rectangle (“mediana”), and a 6 × 8 in. rectangle (“chicas”). Also, from a 20 × 20 in. sheet, they drew six rectangles of dimension 6 × 8 in. By the end of the first day, the team had calculated how many flowers could be made using two packets of tissue paper with only five layers of tissue paper per flower. At the second session, Sandra and Alondra, who are sisters-in-law, shared about the family project they had done over the weekend. As part of their Mexican culture, some families in the neighborhood have an altar with religious images where they pray and worship during festivities. Sandra and Alondra decided to decorate this altar with paper flowers since they had just learned how to make them. Sandra explained that they decided to do it because the flowers were easy to make; the whole decoration, as they had calculated in class, would be inexpensive; the altar would look prettier; and it was a nice opportunity for their families to make something that also involved the children. Everyone in their families helped to construct the altar. For the construction of the frame, they used the same measurement provided in the flowers task, 9 × 12 ft.; their husbands were in charge of cutting the wood, measuring, and putting it together. We should mention that Sandra’s and Alondra’s husbands work as handymen and possess several skills in woodworking, plumbing, construction, and general home repairs. Thus, constructing the frame was something that presented no challenge for them. Some of their children helped paint the frame while Sandra, Alondra, and the rest of the family made the paper flowers. As shown in Fig. 18.4, the frame has flowers also on the diagonal pieces that make the frame sturdier. This led to a discussion related to the theorem of Pythagoras as we explain in what follows. Pythagorean Theorem Discussion  On the second MFP session for the Paper Flowers task, Sandra’s team worked on drawing a mathematical diagram to match the arrangement of flowers that Alondra, Sandra, and their families had made for the altar. Sandra drew the diagram to scale with one inch representing one foot. Alondra,

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Fig. 18.4  The altar constructed by Alondra’s and Sandra’s families

who frequently helped her husband make diagrams for his work projects, completed the diagram by drawing the different size flowers across the frame. Since Sandra and Alondra did the project at home, they already had an idea of how to place the flowers; however, they came across a challenge when trying to place the flowers evenly spaced on the frame in their drawing. On the actual frame at home, they placed the flowers by approximating distances between them, but in the mathematical task, they had to calculate and write exact measurements. Their frame’s design included two right triangles, one in each top corner (similar to the frame in Fig. 18.4). The two legs of each right triangle were three feet long. The arrangement of the flowers in their diagram shared some similarities but also differed from the arrangement of the flower on the altar constructed. For example, they wanted to put a flower in each of the vertices of the triangles. They also wanted to have three more flowers along the hypotenuse (instead of just one as in the altar), a medium one in the center and two small ones (one at each side), equally spaced. This posed a challenge as they needed to know the length of the hypotenuse. As noted earlier, in the constructed altar the flowers were positioned without measuring the distances between them. But in wanting to draw their design on paper, they needed to have exact measurements. Thus, we can notice a shift in focus from the real-world experience to the mathematical problem at hand. Besides using their ruler to measure the length of the hypotenuse in their diagram, they did not recall any other way of calculating the length of this side. During a pivotal mathematical exchange, Sandra’s group organically began to explore the relationship between the

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sides of a right triangle. In this conversation, Fany challenged the mothers to analyze their rationale for the length of the diagonals in their display. The team started playing with numbers and checking if their guesses made sense with the dimensions of the triangle. They knew that the measure of the diagonal could not be 6 in. (since the lengths of the other two sides of the triangle were 3 in. each), and as Alondra explained, walking around the corner (as in streets in a town) would be longer than walking along the diagonal. After some time, Alondra decided to measure it with a ruler. She knew the measurement would be very close to the real one since their diagram was drawn to scale (1 in:1 ft.). She measured 4.5 in. (the actual measurement should be less than 4.5 in. since 18 is less than 4.5), which in the actual frame would be 4.5 ft. When the rest of the team heard this measurement, they assumed that the measure of the diagonal was 4.5 ft. because 3 times 3 was equal to 9, and half of this was 4.5 ft. Fany challenged their claim: could they calculate the hypotenuse length by multiplying the side lengths and dividing it by two? She asked the team what would happen if each side of the triangle was 4 inches long. The mothers started analyzing and discussing this question. Alondra transferred the problem to her real-life experiences while trying to make sense of it. Alondra: If we use four and four, we would be walking eight feet [while talking Alondra is making hand gestures that simulate walking around the corner]. But she [Fany] is saying, what would the triangle ruler (escuadra) be, what would the measure of this be [pointing to the hypotenuse]?

In order to make her point clear, Alondra used her everyday experience and related this mathematical problem to walking on the street. Her experiences made her doubt the method they had proposed to find the diagonal length. At this point, each of the mothers had a different answer to the problem and did not seem to reach an agreement. Thus, Fany suggested drawing the triangle with two sides of length 4  cm and measuring its hypotenuse. After measuring the hypotenuse of the new right triangle, the team realized that multiplying the two shorter sides and dividing by two did not work. Alondra said, “it would not be multiplication because 4 cm times 4  cm would be 16  cm and half of 16  cm is 8  cm. So, I would just measure here.” Using conceptual reasoning and real-world examples, they convinced themselves that the diagonal of a four-inch square cannot be eight inches. Throughout this discussion, the mothers went back and forth between the decontextualized numbers and the real-world context, interpreting what the numbers meant, analyzing each other’s conjectures, and eventually concluding that their initial method for finding the hypotenuse, multiplying the shorter sides and dividing by two, does not always work. This exchange inspired Alondra to keep learning and asking questions beyond the classroom setting. Several weeks later, Alondra shared with us that she was so intrigued by the discussion that she had asked her daughter, who is a university student, how she could calculate the length of the hypotenuse of a right triangle without actually measuring or estimating. She proudly explained that her daughter taught her the Pythagorean Theorem and showed her how to use it. At the end of this conversation with her

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daughter, she said, “next time I go to the mathematics workshop for parents, I’ll tell Fany that I know how to do it.” While Alondra was sharing this, her face reflected confidence, success, and pride which displayed her interest for learning mathematics and applying it to her daily life. This exchange also highlights the bidirectional nature of learning mathematics in parent-child relationships. Sandra and Alondra invested time in this task during and outside of the sessions. Their level of interest led them to share the task with their families and used it as a family project where their children could help. Alondra went even further by asking her older daughter how to find the hypotenuse of a right triangle. Overall, the four women in this group displayed their mathematical skills and experiences throughout the Paper Flowers task. They used their everyday experiences to understand the mathematics and to verify their mathematical claims about the hypotenuse in right triangles. This back and forth between real-world experiences and the mathematical task made evident their progression through the modeling process, which we explore more in depth in the following section.

18.3.3 Elements of Mathematical Modeling As we engaged in the Paper Flowers task with both groups of mothers and later analyzed their work, it became clear to us that making assumptions, typically one of the first steps in formulating a mathematical model, was a foundational aspect of their plans. In previous sessions with the mothers, Marta and Fany emphasized the importance of making reasonable assumptions while working on a mathematical modeling task. Internalizing this step of the modeling process, the mothers continually referred back to their assumptions throughout the Paper Flowers task. For example, Lidia’s group assumed that they were being hired by a client who wanted the display to look nice and be efficient (i.e., not waste paper or time). These assumptions were a driving force of many of their decisions throughout the task, such as adding more layers to make the flowers “más tupidas” (“fuller”), making more of the larger flowers because it saved time, and choosing dimensions of paper that were “more square-like” so that the flowers were less “picuda” (“pointy”). Both groups relied on their assumptions to create and act on their mathematical model. In Sandra’s group, the initial model was a sketch illustrating how they would arrange and cut the rectangles of various sizes (chicas, mediana, grande) out of a sheet of paper (see Fig. 18.3). This diagram allowed them to see that making an extra large flower (16 × 24 in.) out of a 20 × 20 in. sheet would not be possible, so they needed to revise their plan. As shown in Fig. 18.3, this group realized that six small flowers (6 × 8 in.) would fit on a single sheet of tissue paper, but there would be some wasted space. Although their background knowledge was not apparent in their assumptions, Alondra drew on her experiences in carpentry to measure and draw the final diagram of the flower display. The mothers in both groups quickly understood the underlying structure of the arithmetic mathematical model for the total number of packages: for each size of

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flower, multiply the number of flowers times the number of rectangles (layers) per flower; then, divide that product by the number of rectangles that fit on one sheet of paper and divide that quotient by the number of sheets per package. Lidia’s group systematically wrote out these operations for each size of flower that they wanted. Using this as their model, they substituted numbers for each parameter to determine how many packages of tissue paper were required for each size of flower. In addition to the arithmetic model for the total number of packages, finding a geometric model, both for the number and sizes of rectangles that fit on a sheet of tissue paper and for the number of flowers of various sizes needed to create the final display, involved mathematical reasoning and problem solving. We initially anticipated that the design of the flower display would be the most mathematically challenging component of the task, but again would allow the women to validate their work through a tangible product. In our design of the task, we predicted that participants would need to record the diameter of each flower as a linear measurement and also possibly consider the estimated area of each flower to cover some portion of the wall. Lidia’s group had to redesign their flower display multiple times because they realized they would not have enough length along the perimeter to fit all the flowers (see Fig. 18.5). (The words naranja (orange), morado (purple), and azul (blue) refer to the colors of the flowers.) In both groups, this led to a rich discussion about the Pythagorean Theorem, as both teams eventually decided on designs that followed the hypotenuse created by the corner angles. In particular, Lidia’s group had to

Fig. 18.5  Lidia’s drawing of her group’s flower design on a 9 × 12 ft. wall

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rethink their design in Fig. 18.5, which was not drawn to scale, so that it maximized the linear space without giving up the desired aesthetic. Another challenging aspect for both groups was figuring out how many rectangles of each size would fit on a single sheet of paper. Of course, this number changed based on the size and orientation of the rectangles and the size of the tissue paper, which is why the sketch shown in Fig.  18.3 was crucial to the validation of the group’s earlier work of drawing the rectangles on the actual sheets of tissue paper. Due to the hands-on nature of this task, the modeling steps of analyzing and validating were oftentimes more apparent, both to the mothers and to us as researchers. The final product was tangible, so it was clear when the plan needed to be revised to create a more appealing design. Before validating their final plan, Lidia’s group made repeated revisions and had several exchanges of ideas. Julieta and Marcela were convinced by their experience with making paper crafts that a 10  ×  20  in. rectangle would make a “picuda” (“pointy”) and unsatisfactory flower. However, Lidia could not visualize this and instead relied on an informal understanding of proportional reasoning to confirm their claims. Using words like “picuda” (“pointy”) and “triste” (“sad”) to describe a 10  ×  20  in. rectangle of paper demonstrated that the mothers could translate between the numeric mathematical decisions and the physical aesthetic outcome. The interpretation of the numbers in the real-world context illustrated a fluency that mathematical modeling tasks attempt to develop: the ability to simultaneously speak about abstract mathematical reasoning and concrete realistic applications. This process of analyzing, interpreting, and drawing conclusions is crucial to creating a realistic model or plan that addresses the main idea of the task. For both groups in the Paper Flowers task, the process led to an iteration of the mathematical modeling cycle, a step that is encouraged by the cyclic nature and bidirectional arrows in the diagram. Lidia’s group made several small analyses throughout the task, each leading to an interpretation of how their mathematical decisions influenced the appearance of the flowers and ultimately to a revision in their plan. Even after they created a poster and presented their work at the end of the first session, they later revised their sizes of flowers and the design of their display. One of the most exciting and unexpected elements of this task occurred outside of the classroom. To further validate their design, Sandra and Alondra involved their families in a weekend project to build and decorate a frame for an altar based on their plan in the Paper Flowers task. It was rewarding to see not only their engagement in the task, but also how they validated their model by actually constructing the display, using both their mathematical work from the task and the funds of knowledge from their families. Alondra continued the modeling process by learning about the Pythagorean Theorem from her daughter and sharing it with us. We considered this the final step in the modeling cycle, because Alondra kept revising her thinking until she was satisfied with her understanding of the mathematics. This example illustrates how the mathematical modeling process extended outside the MFP sessions; the reporting out phase occurred when the mothers shared their knowledge and experience with others.

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Both groups enhanced elements of the mathematical modeling process by bringing in personal experiences and building on their different expertise, such as knowing how to make paper flowers or diagrams to scale. The key is that these women saw their funds of knowledge as valid and necessary during the mathematical modeling process. This idea of building on the participants’ funds of knowledge is what guided the second part of our work with the mothers, as we describe next.

18.4 Co-developed Modeling Tasks The second phase of our module was the co-development of modeling tasks with the mothers in our project. Viewing them as intellectual resources, we asked the women to draw from their funds of knowledge to design a mathematical task for others to solve. The Paper Flowers task served as an example of an activity that connected their knowledge of crafts to the school community while highlighting relevant mathematical content for elementary grade children. We likewise encouraged them to consider their interests, family traditions, or daily routines for possible ideas. Both groups of women spent most of the third session discussing topics and formulating a task. In the session, we acknowledged that creating mathematical tasks is challenging and we did not expect a completely polished product. The goal of our co-development phase was to generate ideas and ultimately choose a specific context for a modeling task. We encouraged both groups to discuss different elements in the modeling process, such as potential assumptions, relevant mathematics content, and opportunities for revising. In this section, we focus on the work of Lidia, Julieta, and Esmeralda. After spending time brainstorming activities and hobbies, this group decided to focus on a fundraising opportunity at their school. The following case describes the “Cupcake task,” aptly named because the fundraiser consisted of making and selling cupcakes. After the case description, we offer suggestions and modifications for formulating a more complete modeling task from the generated context.

18.4.1 The Case of the Cupcake Task Lidia, Julieta, and Esmeralda started by listing several familiar activities as contexts for mathematics tasks; many of their ideas were related to food, including cooking and adjusting recipes for food allergies, totaling daily nutritional values and calories based on the labels, and buying food at the grocery store. They also discussed craft ideas, such as painting large rocks to put in a garden or on a porch for decoration. Lidia shared pictures from her phone of painted rocks: some had designs and spirals, others looked like ladybugs or frogs on lily pads. All three women liked this idea and discussed questions like, how much paint would you need to decorate

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these, and how much would it cost? They ultimately decided that it was a nice idea for a craft activity but may be difficult to make into a mathematical task. All three women fixated on the authenticity and feasibility of the task. They wondered if they were creating a fictitious scenario, or if they would actually try to implement the result of their work once our session ended. This was a crucial point that determined the topic and structure of their task. They saw several limitations and difficulties if they chose the rock painting idea, because it involved details and materials that would not be easy to find. On the other hand, if they chose a food-­ related idea, such as baking cupcakes, they eliminated many of the unknown factors and hard-to-find supplies. Lidia offered to bring packaged cake mix from home, and Esmeralda knew where to get the frosting. Julieta noticed that a box of cake mix (found online) listed the recipe and ingredients on the back. The three women decided that a task based on making and selling cupcakes would be easier to accomplish in real life and thus would be a worthwhile mathematical exploration. One of the group’s top priorities was creating a task that included their children; they wanted an activity that was interesting and accessible to elementary school children. Esmeralda first suggested the cupcake theme because she knew her children would not get bored by making them. They decided to plan a fundraiser for their school based on making and selling cupcakes. Julieta and Esmeralda had experience with school fundraisers, such as selling paletas (fruit popsicles) after school, and realized that they could design a similar fundraiser around the cupcakes. Lidia deferred to the two of them when it came to making assumptions and decisions about the fundraiser, even though she had extensive business experience. The women discussed what role their children would play in the making and selling of cupcakes. The following exchange illustrates how intentional they were about including their children in the entire process: Lidia: And, for example, the children, what can they do? They can…. Esmeralda: Well you can give them the cup and the child is going to… Julieta: They can read the measurements Esmeralda: “Look here, what does it say? Three and a half cups of water,” and the child can go get three and a half cups of water. Lidia: And then, for example, they can put different toppings, right? Esmeralda: Yes, that too. Lidia: They can put different things on top, like fruit. Esmeralda: Yes, once they cool down, then they can do it. Lidia: Then they can decorate them.

The team based many of their decisions on this discussion. Once they noticed that making cupcakes had more potential for the children to get involved in the preparation and decoration, they agreed on proceeding with the cupcake topic for their task. While discussing the mathematics involved in their task, the group referred to the list of ingredients from the cake mix box and their experience with scaling recipes for large groups. Esmeralda brought knowledge about baking and frosting cupcakes. She knew that a box of cake mix would make 24 mini cupcakes and that each box costs $1.00 plus tax. For the decoration, she knew the price of a container of

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frosting and that half of one container would be enough for all 24 mini cupcakes. With these mathematical relationships in mind, she felt confident making and selling cupcakes at a fundraiser similar to others at her school. In addition to their knowledge of baking and recipes, the three women relied on their knowledge of money and budgets. Lidia connected to the buying and selling aspect of the fundraiser, because it was similar to her job in industry. She meticulously added sales tax whenever the group calculated the cost of materials, as she did when working on the Paper Flowers task. The mothers agreed that the Cupcake task could help them design a fundraiser for the school. But one may wonder what the actual mathematical task was. The group did not have a formal mathematical question in mind (although both Marta and Amy prompted them to “come up with a task” rather than solve one), but they had an authentic event that they wanted to plan for their school. Based on their mathematical work and decisions, we can assume that their task was to figure out how much money they could raise from the cupcake fundraiser. They discussed what worked best for the school and made realistic considerations accordingly. For instance, they decided to sell the cupcakes on Wednesdays, since this is an “early release” day for students, and they estimated the number of cupcakes they should make each week. In the next section, we discuss how their practical and experience-based considerations could lead to a modeling task (and a profitable fundraiser).

18.4.2 Commentary on the Cupcake Task We now comment on our co-development exercise from an educator and researcher perspective. We also offer suggestions and modifications for other educators and researchers who wish to implement a similar task co-development activity with parents of elementary children. We called this part of our module the co-development phase, but it was interesting that the women in this group “solved” the task before they (formally) created it. As mentioned, Marta and Amy each prompted the mothers (Lidia, Julieta, and Esmeralda) to first brainstorm a context and then formulate a question or task that represented the situation. In fact, that was their entire objective: pose a contextualized task that could be addressed with mathematics. We know that posing interesting, complex mathematics problems is challenging (Crespo & Sinclair, 2008), so perhaps we could have built up to this request or provided scaffolds for the mothers to use, but we were intentional in leaving the request very open-ended and even vague. Our goal was not to have the mothers develop a full-fledged modeling task, but rather to have them help us develop a funds-of-knowledge context for a potential modeling task. We used the activity of them working towards developing a possible modeling task as a way to learn more about and from their funds of knowledge. If we, as researchers and educators, are to develop modeling tasks that reflect the experiences and knowledge of the participants, we need to make sure that they have

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a say in that. Thus, by inviting them to develop a task, we were able to gain an understanding of what may be topics of interest to them. So, what we argue for here, is to engage the participants as co-developers of modeling tasks by working with contexts that draw on their knowledge and experiences. The situation they chose was raising money for the school by baking and selling cupcakes; thus a reasonable mathematical modeling task could be phrased as, “How much money can we raise from selling cupcakes for a month?” or “How many cupcakes do we need to bake and sell to meet our fundraising goal?” or even “Can we raise more money from selling cupcakes than from selling paletas?” The mothers never explicitly asked a question such as these, but their mathematical work suggests that their goal was to find out “how much money they could make from selling cupcakes for a school year.” If this implicit statement is understood as their task, then they were in fact quite successful with formulating a mathematical task that represented the situation. Since Lidia’s group chose a situation that was familiar to them (making and selling food as a fundraiser), there were not many unknown factors. In other words, they did not have to make many assumptions in order to solve their task. The mothers in this group were so involved in their school community that they could provide precise amounts based on their experiences. One could argue that their key assumption was that the cupcake fundraiser would proceed exactly like the previous ones; thus they could use their prior knowledge to provide any unknown pieces. For example, the mothers discussed how many cupcakes they might sell each Wednesday. Rather than making an estimation based on the number of students and parents who might stop by each Wednesday, they decided it would be exactly 192 cupcakes, since they would make eight boxes of cupcakes with 24 mini cupcakes per box. The decision to make 192 cupcakes (rather than one member’s suggested estimate of 200 cupcakes) is again due to Julieta’s experience: she recalled in a previous fundraiser that they bought exactly eight boxes of cake mix each week, which made 192 mini cupcakes, and they generally sold all of the cupcakes each week. Essentially, all of their assumptions were implicit: they would make exactly 192 cupcakes (assuming none would break, fall, or be eaten prior to selling) and they would sell all of their cupcakes each week (assuming none would be left). By assuming that they knew how many cupcakes they would sell each week, they simplified the task so that it was solvable in a short amount of time, but still accurately modeled an authentic situation. As modeling task developers, we would eliminate some of the known information to make the task more open-ended. Upon further reflection, we realized that one way to encourage assumptions is to ask how they could plan a similar fundraiser at another school in their community. The variability in the location and school (e.g., a middle or high school) encourages a generalized model with parameters rather than known quantities. We also considered asking, “How do you think your children (or students in your child’s class) would solve this task? Would they make different estimates? Would they have chosen a different selling price for the cupcakes?” These questions hint at the variability of authentic modeling tasks and encourage mathematical decision-making and assumptions.

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We were struck by how engaged Lidia, Julieta, and Esmeralda were as they worked on the Cupcake task. As they planned for this school fundraiser, it became clear to us that this was not a superficial assignment, but rather an authentic event that they wished to carry out. The women took complete control of the task, deciding on a topic that interested them and would make a difference in their school community. They prioritized including their children, both in the planning of the fundraiser and decorating of the cupcakes and in the mathematical work. While our co-development objectives were to create tasks that elementary children could solve, the mothers taught us that the children’s ideas and participation could be incorporated into every aspect of the task development and implementation process.

18.5 Some Closing Considerations In this chapter we have described two aspects of our work on mathematical modeling with a group of Mexican American women. Our goal in sharing this work is to illustrate the immense potential of using tasks that connect to the participants’ funds of knowledge. Because the tasks drew on the participants’ interests and experiences, they highlighted specific contexts (e.g., crafts, cooking). With a different group of participants (e.g., if fathers had also joined these MFP sessions), the contexts would probably be different. But that is to be expected if we want to develop tasks that build on the participants’ funds of knowledge. It is worth noting that Alondra and Sandra took the Paper Flowers task home and found ways to involve their families. This expanded context of making paper flowers and making an altar could lead to other possibilities for modeling tasks to include a more diverse group of participants. As we saw in the work of these women, there was deep engagement (two of the three sessions ran much past the two hours), persistence, and enjoyment. In the Paper Flowers task, there was a sense of this being a real task, that is, one in which Julieta (who had expertise making a different kind of paper flower) could lead a team in making the required flowers to decorate the wall. They talked about adjusting to the client’s requests, about the practicality of having larger versus smaller flowers since these would take longer to make and they may need more volunteers to help out. They brought in aesthetic aspects (e.g., wanting the flowers to be fuller and not pointy), but also concerns about not wanting to waste paper, and of course budget considerations. Sandra and Alondra made it real by taking the task home and engaging their families in making paper flowers to decorate their altar. That was certainly an unexpected and gratifying outcome. As they went back and forth between the physical structure of the altar and the task on paper during the sessions, there were several opportunities to explore mathematical ideas both in context and at a more abstract level. The connection between mathematics and real-world tasks brings up an issue that we have discussed elsewhere (Civil, 2007), which is, how much mathematics do we really need to solve certain real-life problems?

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In the placing of the flowers for the altar (see Fig. 18.4), they did not need to measure exactly; they could approximate their placement. They had the physical contraption (the frame) to help them mediate their decision on where to put the flowers. When they tried to make a design on paper, with exact measurements on where to put the flowers, the mathematics became more involved. In particular, their wanting to place several flowers along the hypotenuse of the two triangles on the corners of the frame led to a discussion of how to find the length of the hypotenuse. While we did not discuss the theorem of Pythagoras with them at that time, it is situations like these that can serve as strong motivators for mathematical explorations. An exploration of the theorem of Pythagoras grounded on their physical experience building the frame for the altar and making its scale drawing is likely to develop a strong conceptual understanding of the theorem. The story of Alondra asking her college daughter how to figure out the length of the hypotenuse is a powerful reminder of how this group of mothers, like the mothers in prior work (Civil, Bratton, & Quintos, 2005), want to engage as adult learners. Throughout our work, we maintain that parents are intellectual resources and have a wealth of knowledge about various topics that interest their children. In alignment with the larger project’s overarching goal to bring together parents and teachers, the co-development exercise intertwined the roles of those inside and outside of the classroom, fusing researchers’ knowledge of mathematical modeling with parents’ funds of knowledge about the school community. Additionally, the mothers’ insistence to involve their children signaled to us the importance of collaboration between parents and children in curriculum development. In the Cupcake task, the mothers kept bringing up ways in which their children could participate. In the Paper Flowers task, we saw their involvement first-hand when Alondra and Sandra shared how their children had helped out when making the altar. As others have noted, the context and relevance of the mathematics task matter (Suh et al., 2017; Wickstrom, 2017). For this group of mothers, the context was of interest to them and they could bring in their expertise. Involving their children in the tasks made them even more relevant. The mothers’ work on these tasks underscores the importance of bringing together multiple types of expertise, beyond only “academic” mathematics knowledge. Even though Lidia had extensive business experience and arguably a stronger background in academic mathematics, she knew that Julieta’s and Esmeralda’s knowledge of school fundraisers was critical to the formulation of an authentic task. As a learner, Lidia deferred to her team’s funds of knowledge when estimating costs and quantities, and as facilitators, we did not privilege Lidia’s mathematical knowledge over her group’s shared community knowledge. Similarly, the various mothers’ expertise with making paper flowers was key to how they approached the modeling task. And the experiential knowledge of making the altar served as a good context and motivation for the mothers to further explore mathematical ideas. Whereas typical, closed-ended word problems may underscore only one type of mathematical knowledge, rich modeling tasks allow for solvers with varying types of knowledge to engage in the context and learn from each other (Carmona & Greenstein, 2010; Drake et al., 2015; Turner et al., 2012).

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Furthermore, our co-development task phase emphasized that mathematical modeling tasks are not typical (i.e., textbook) problems with given information leading to one correct answer (e.g., Aguirre et  al., 2019). Consistent with past research with pre-service and in-service teachers, the mothers in our project wrestled with the notion of open-ended tasks and potentially multiple solutions, likely due to their lack of exposure to this type of task (Anhalt & Cortez, 2016; Doerr, 2007). When formulating a mathematical question, the mothers involved in the Cupcake task made assumptions and decisions that affected the complexity of their model (see Anhalt, Cortez, & Bennett, 2018); namely, they chose a context with factors that were very familiar to them. The group’s insistence on solving the task to ensure it was a possible and worthwhile fundraising idea demonstrated to us (as researchers) the importance of creating realistic mathematical tasks that are based on situations that the solvers see as significant and relevant (Anhalt, Staats, et al., 2018; Carmona & Greenstein, 2010), not only to their own lives but also to their families and communities. This opens up a potentially powerful approach to parental engagement efforts in schools, one in which parents bring their expertise, their knowledge, and their experience, in short, their funds of knowledge to the co-development of mathematical modeling tasks for (and with!) their children. It causes us to wonder: would children be more engaged in mathematics tasks if they knew that their mothers (parents; other significant adults) were directly involved in the development of such tasks? Acknowledgments  This work was funded by the Heising-Simons Foundation, Grant #2016-065. The views expressed here are those of the authors and do not necessarily reflect the views of the funding agency.

References Aguirre, J.  M., Anhalt, C.  O., Cortez, R., Turner, E.  E., & Simic-Muller, K. (2019). Engaging teachers in the powerful combination of mathematical modeling and social justice: The Flint water task. Mathematics Teacher Educator, 7(2), 7–26. Anhalt, C., & Cortez, R. (2016). Developing understanding of mathematical modeling in secondary teacher preparation. Journal of Mathematics Teacher Education, 19(6), 523–545. https:// doi.org/10.1007/s10857-­015-­9309-­8 Anhalt, C.  O., Cortez, R., & Bennett, A.  B. (2018). The emergence of mathematical modeling competencies: An investigation of prospective secondary mathematics teachers. Mathematical Thinking and Learning, 20(3), 202–221. Anhalt, C. O., Staats, S., Cortez, R., & Civil, M. (2018). Mathematical modeling and culturally relevant pedagogy. In Y. J. Dori, Z. Mevarech, & D. Baker (Eds.), Cognition, metacognition, and culture in STEM education (pp. 307–330). New York, NY: Springer. Asempapa, R.  S. (2015). Mathematical modeling: Essential for elementary and middle school students. Journal of Mathematics Education, 8(1), 16–29. Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58.

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Blum, W., & Leiss, D. (2007). How do students and teachers deal with modeling problems? In C. R. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Modeling (ICTMA–12): Education, engineering and economics (pp. 222–231). Chichester, UK: Horwood Publishing. Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modeling, applications, and links to other subjects– State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68. Carmona, G., & Greenstein, S. (2010). Investigating the relationship between the problem and the solver: Who decides what math gets used? In R. Lesh, P. Galbraith, C. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp.  245–254). New  York, NY: Springer. Civil, M. (2002). Culture and mathematics: A community approach. Journal of Intercultural Studies, 23(2), 133–148. Civil, M. (2007). Building on community knowledge: An avenue to equity in mathematics education. In N. Nasir & P. Cobb (Eds.), Improving access to mathematics: Diversity and equity in the classroom (pp. 105–117). New York, NY: Teachers College Press. Civil, M. (2016). STEM learning research through a funds of knowledge lens. Cultural Studies of Science Education, 11(1), 41–59. https://doi.org/10.1007/s11422-­014-­9648-­2 Civil, M., & Andrade, R. (2003). Collaborative practice with parents: The role of researcher as mediator. In A.  Peter-Koop, A.  Begg, C.  Breen, & V.  Santos-Wagner (Eds.), Collaboration in teacher education: Working towards a common goal (pp. 153–168). Boston, MA: Kluwer. Civil, M., & Bernier, E. (2006). Exploring images of parental participation in mathematics education: Challenges and possibilities. Mathematical Thinking and Learning, 8(3), 309–330. Civil, M., Bratton, J., & Quintos, B. (2005). Parents and mathematics education in a Latino community: Redefining parental participation. Multicultural Education, 13(2), 60–64. Common Core State Standards Initiative. (2010). National Governors Association Center for Best Practices and Council of Chief State School Officers. http://www.corestandards.org/assets/ CCSSI_Math%20Standards.pdf Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11(5), 395–415. https://doi.org/10.1007/s10857-­008-­9081-­0 Doerr, H. M. (2007). What knowledge do teachers need for teaching mathematics through applications and modeling? In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (pp. 69–78). New York, NY: Springer. Drake, C., Land, T. J., Bartell, T. G., Aguirre, J. M., Foote, M. Q., Roth McDuffie, A., & Turner, E. E. (2015). Three strategies for opening curriculum spaces. Teaching Children Mathematics, 21(6), 346–353. English, L.  D. (2012). Data modelling with first-grade students. Educational Studies in Mathematics, 81(1), 15–30. English, L.  D., & Watters, J.  J. (2005). Mathematical modeling in third-grade classrooms. Mathematics Education Research Journal, 16, 59–80. Garfunkel, S. A., Montgomery, M. (Eds.). (2016). GAIMME: Guides for instruction and assessment in mathematical modeling education; consortium for mathematics and its application [COMAP, Inc.] Bedford, MA: USA. Society for Industrial and Applied Mathematics [SIAM] Philadelphia, PA: USA.  Available online: http://www.siam.org/reports/gaimme-­ full_color_ for_online_viewing.pdf. Accessed on 15 Oct 2017). Gibbs, G. R. (2007). Analyzing qualitative data. London, UK: SAGE Publications, Ltd.. González, N., Moll, L., & Amanti, C. (Eds.). (2005). Funds of knowledge: Theorizing practice in households, communities, and classrooms. New York, NY: Routledge. Lesh, R., & English, L. D. (2016). Case studies for kids! Purdue University. https://engineering. purdue.edu/ENE/Research/SGMM/CASESTUDIESKIDSWEB/case_studies_table.htm Maxwell, J. A. (2013). Qualitative research design: An interactive approach (3rd ed.). Thousand Oaks, CA: SAGE Publications.

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Moll, L., Amanti, C., Neff, D., & Gonzalez, N. (1992). Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms. Theory Into Practice, 31(2), 132–141. Mousoulides, N., Christou, C., & Sriraman, B. (2008). A modeling perspective on the teaching and learning of mathematical problem solving. Mathematical Teaching and Learning, 10, 293–304. Quintos, B., Civil, M., & Bratton, J. (2019). Promoting change through a formative intervention: Contradictions in mathematics education parental engagement. Mind, Culture, and Activity, 26(2), 171–186. https://doi.org/10.1080/10749039.2019.1602656 Sandoval-Taylor, P. (2005). Home is where the heart is: A funds of knowledge-based curriculum module. In N. González, L. Moll, & C. Amanti (Eds.), Funds of knowledge: Theorizing practice in households, communities, and classrooms (pp. 153–165). Mahwah, NJ: Lawrence Erlbaum. Schoenfeld, A. H. (2013). Mathematical modeling, sense making, and the common core state standards. Journal of Mathematics Education at Teachers College, 4, 13–25. Stohlmann, M. S., & Albarracín, L. (2016). What is known about elementary grades mathematical modelling. Education Research International, 2016, 1–9. Suh, J.  M., Matson, K., & Seshaiyer, P. (2017). Engaging elementary students in the creative process of mathematizing their world through mathematical modeling. Education in Science, 7(62), 1–21. Turner, E.  E., Drake, C., McDuffie, A.  R., Aguirre, J., Bartell, T.  G., & Foote, M.  Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15(1), 67–82. Vélez-Ibáñez, C.  G., & Greenberg, J.  B. (1992). Formation and transformation of funds of knowledge among U.S.-Mexican households. Anthropology and Education Quarterly, 23(4), 313–335. Wickstrom, M. H. (2017). Mathematical modeling: Challenging the figured worlds of elementary mathematics. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp.  685–692). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators.

Chapter 19

Insights Regarding the Professional Development of Teachers of Young Learners of Mathematical Modeling Elizabeth A. Burroughs

This section has addressed the preparation of others to teach mathematical modeling to young learners. Each of the four chapters examines how adults, either preservice teachers or parents, learn to teach mathematical modeling to young children. There does not yet exist a robust tradition of elementary grade students engaged in the process of mathematical modeling in the USA, where the studies in these chapters are situated. Then how do elementary teachers learn to teach modeling? The chapter authors describe their research about and experiences in teaching modeling to those who are new to formal modeling. These chapters anticipate and examine answers to the questions: What challenges and opportunities can teacher educators face as they engage teachers or parents in learning about mathematical modeling? How can teacher educators support teaching practices that give teachers and parents the best chance at engaging young in mathematical modeling?

19.1 Articulating Values in Order to Make Decisions All of the chapters in this section highlight an essential affordance of engaging young students in modeling: the opportunity to consider mathematics alongside the consideration of human values (Carlson, Wickstrom, Burroughs, & Fulton, 2016). In using mathematical tools to represent, understand, and address authentic problems, mathematical modeling provides learning opportunities in mathematics that involve making value judgments. The translation of a situation from an authentic, lived experience into mathematics necessarily includes an articulation—either E. A. Burroughs (*) Montana State University, Bozeman, MT, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Suh et al. (eds.), Exploring Mathematical Modeling with Young Learners, Early Mathematics Learning and Development, https://doi.org/10.1007/978-3-030-63900-6_19

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explicit or implicit—of what values are important to the modeler. Sometimes, this articulation results in modeling that is aimed toward understanding issues of social justice (Cirillo, Bartell, & Wager, 2016), and sometimes it is aimed toward understanding routine, lived experiences (Civil, 2016). Gilbert and Suh make the discussion of what is “best” an integral part of the STEM modeling tasks they describe. Gallagher and Jones describe how third graders began their modeling task with the explicit instruction to decide what was important for the classroom to function effectively—that is, the students had to start with a consideration of what they, as a classroom community, value. Yanisko and Minicucci are explicit in their aim to engage teachers of modeling in developing critical consciousness by studying mathematics through the lens of social issues. With this as an aim, they were purposeful in selecting the situation of the design of a community garden so that it would call on geometry content and provide opportunities to highlight the social injustice of food deserts. Civil, Been Bennett, and Salazar engaged the mothers in their program in modeling in order to tap into their community funds of knowledge, and intended that the mothers would identify what is important to them. The mothers valued aesthetics, conservation, budgetary efficiency, and family involvement, all of which extend what might otherwise be considered something to which mathematics should apply.

19.2 Emphasizing Agency for Modelers The chapters in this section offer insight into the importance of teaching modeling through the use of prescriptive models, as opposed to relying on descriptive models only. A descriptive model looks at the state of things; a prescriptive model makes recommendations (Stillman, 2019). The agency inherent in creating a model that will be or might be used to make a decision or change an injustice is at the forefront of mathematical modeling with young children (Niss, 2015). Yanisko and Minicucci address this crucial difference in noting that when pursuing a descriptive model examining racial injustice, “students reacted in ways that showed they felt defeated by the new information uncovered by mathematizing the situation”; the authors advocate for engaging students in modeling that results in an actionable model. Gallagher and Jones note that they presented a situation to students as though the students were recommending to the principal how to spend $5400; this was a fictitious but realistic situation, highlighting the classroom usefulness of a prescriptive stance toward a descriptive reality. Civil, Been Bennett, and Salazar indicate their intention in the paper flower task was a fictitious but realistic situation of designing paper flowers for a client, and yet some of the mothers found a real situation in which to use their model, in decorating a family altar. Gilbert and Suh embedded prescriptive modeling in their STEM tasks, in that the modeling often resulted in something to be designed or built.

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19.3 Engaging in the Practice of Modeling in Anticipation of Teaching Others to Model The need to engage teachers in the practice of modeling as modelers is reinforced with these examples and is common in professional development for mathematical modeling. As students, few teachers have engaged in mathematical modeling tasks, and in teacher preparation, few teachers have learned how to engage students in mathematical modeling tasks (Doer, 2007). In each case, the authors chose to engage the pre-service teachers or the parents in a modeling task as modelers, even though they eventually would be the ones to provide modeling opportunities for students to engage in as modelers. Gilbert and Suh describe how pre-service teachers engaged in STEM modeling with an interdisciplinary focus, first as modelers and later in teaching modeling. Gallagher and Jones describe how pre-service teachers engaged in an examination of gun violence as modelers before teaching students to use modeling in a lesson focused on the opportunity cost involved in choosing classroom supplies. Yanisko and Minicucci describe pre-service teachers engaging in planning a community garden with an eye toward addressing food insecurity, before teaching the lesson to others. Civil, Been Bennett, and Salazar describe mothers engaged in a modeling task of designing paper flower displays and how that experience led to some of the mothers enacting a variation of that task with their families, even though the teaching of modeling was not an explicit expectation given to the mothers. In all cases in this section, the emphasis on the experiential in influencing understanding is highlighted, whether those experiences are simulated (e.g., imagine you have a budget of $5400 to spend), orchestrated for the classroom (e.g., build an aluminum boat or design a community garden), or naturally occurring (e.g., use mathematics to design and build a family altar).

19.4 Learning to Develop Modeling Tasks The examples of mathematical modeling professional development described in this section address how to prepare parents or teachers to design modeling tasks. Teachers do not have explicit curricular support for implementing mathematical modeling (Gould, 2016). The result is that in practice, modeling tasks are left to be developed by teachers, designed and modified in the moment as students engage in a task. Civil, Been Bennett, and Salazar demonstrate that the context of a task is an integral part of engagement in it. When the mothers chose the context of a bake sale, they relied on their expertise in baking and knowledge of school fundraisers. Yanisko and Minicucci address the intersection of content and task design. They attempted to formulate a task that would target particular content in geometry, and found that targeting to be unnecessarily constraining. They recommend that the rich, social

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justice context of the task is the appropriate focus, and that content covered is a flexible part of the expectations of a modeling task. The programs described herein provide different avenues by which the pre-service teachers or parents were prompted to design modeling tasks. Gallagher and Jones report how, for one pre-service teacher, engagement as a modeler in coursework led to her initiating a modeling task for elementary students. That is, it was not an explicit aim of the professional development to follow up with a modeling task implemented with students, but the pre-service teacher saw the applicability of modeling in teaching and embraced the opportunity. Gilbert and Suh explicitly expected the pre-service teachers to teach the modeling tasks that they had engaged in as learners, so the tasks were already designed and it was the implementation that was left up to the pre-service teachers in their internships. Likewise, Yanisko and Minicucci include task design as an integral part of the work they expect of preservice teachers. Civil, Been Bennett, and Salazar asked the mothers to design a task that their children could implement, but did not expect the task to be implemented. Gilbert and Suh explore the interdisciplinary nature of STEM modeling and land squarely in advocacy of the use of modeling practices, whether explicitly mathematical or integrated across STEM disciplines, as that which defines modeling. This diversity of examples illustrates the multifaceted approach different projects take toward fundamental questions about what characterizes modeling tasks in the elementary grades, none of which need precise answers, but each of which is worth asking. Is it student engagement in the modeling process that makes a task a modeling task? Or, is it the formulation of a model during a task that makes something a modeling task? Who should create tasks, and how are they to be created?

19.5 Does Professional Development for Pre-service Audiences Differ from That for In-service Audiences? The collection of work in this section does not feature any professional development designed for in-service teachers. This prompts a question about the difference between modeling with pre-service teachers (or parents) and modeling with in-service teachers. What is the difference between professional development in mathematical modeling for those who are experienced at teaching mathematics but new to teaching mathematical modeling, and professional development for those who are new both to teaching and to modeling? I have been part of a team that provided a professional development course for in-service teachers designed to engage teachers in experiences as modelers in order to support them as teachers of modeling (Fulton, Wickstrom, Carlson, & Burroughs, 2019), and I have taught pre-service teachers about teaching modeling in an undergraduate mathematics and methods courses. Through these experiences, I have relied on a definition of teaching modeling that rests on an inquiry-based view of teaching mathematics. Mathematics teaching that is open-ended, focused on student

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engagement with mathematical concepts, and puts students’ mathematical ideas at the center of instruction provides an environment in which mathematical modeling can flourish. This work with in-service teachers has revealed that mathematical modeling shares many of the challenges that persist across the teaching of mathematics, for example, facilitating productive discussions, and especially of teaching mathematical tasks that are of high cognitive demand (Stein, Smith, Henningsen, & Silver, 2009). Wrestling with the openness of modeling is difficult for teachers, whether in-service or pre-service, and this difficulty exists both in learning to model and in learning to teach modeling. Professional development that addresses management of these in-the-moment decisions teachers make while teaching modeling is useful for all those learning to teach modeling (Kaiser, 2017). Although the chapters featured in this section on professional development focus on examples that come from either coursework for pre-service teachers or community-based informal education, the insights presented here about facilitating others’ learning to teach modeling will persist whether the audience is pre-service teachers, in-service teachers, parents, or community members.

19.6 Conclusion In these chapters, modeling emerges as a process that integrates traditional disciplines (mathematics and social studies or mathematics and science), combines inschool and out-of-school experiences (home life and modeling experiences in the mathematics for parents’ initiative), and blends the roles of learner and teacher. The nature of modeling is multifaceted, and these chapters have highlighted different aspects of mathematical modeling. What does it take to teach modeling to young children? Teachers (or parents) who ask students questions that require students to make decisions about what is important in a given situation, with attention to quantitative or geometric aspects of the situation. Teachers (or parents) who establish an environment where students are well-situated to make supported claims about what is probable or desirable or optimal in a situation. Teachers (or parents) who allow students to make modeling decisions so that they learn to make good modeling decisions. Teachers (or parents) who embrace the opportunity to make explicit the values that underlie quantitative decisions. Modeling requires developing an understanding of mathematics as a tool for making good, well-reasoned decisions that make a difference in the lives of students and their communities. At the core, working on modeling professional development, whether with preservice teachers, in-service teachers, or parents, sets up a lifelong disposition and stance toward modeling. Preparing teachers and parents to teach modeling to young children provides the opportunity to address the role of values in mathematics teaching and learning.

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References Carlson, M. A., Wickstrom, M. H., Burroughs, E. A., & Fulton, E. W. (2016). A case for mathematical modeling in the elementary school classroom. In C. R. Hirsch (Ed.), Mathematical modeling and modeling mathematics (pp. 121–129). Annual perspectives in mathematics education 2016. Reston, VA: National Council of Teachers of Mathematics. Cirillo, M., Bartell, T. G., & Wager, A. A. (2016). Teaching mathematics for social justice through mathematical modeling. In C. R. Hirsch (Ed.), Mathematical modeling and modeling mathematics (pp. 87–96). Annual perspectives in mathematics education 2016. Reston, VA: National Council of Teachers of Mathematics. Civil, M. (2016). STEM learning research through a funds of knowledge lens. Cultural Studies of Science Education, 11(1), 41–59. https://doi.org/10.1007/s11422-014-9648-2 Doer, H. M. (2007). What knowledge do teachers need for teaching mathematics through applications and modeling? Modelling and Applications in Mathematics Education, 10, 69–78. Fulton, E. W., Wickstrom, M. H., Carlson, M. A., & Burroughs, E. A. (2019). Teachers as learners: Engaging communities of learners in mathematical modelling through professional development. In G.  A. Stillman & J.  P. Brown (Eds.), Lines of inquiry in mathematical modelling research in education (pp. 125–142). Cham, Switzerland: Springer Open. Gould, H. (2016). What a modeling task looks like. In C. R. Hirsch (Ed.), Mathematical modeling and modeling mathematics (pp. 179–186). Annual perspectives in mathematics education 2016. Reston, VA: National Council of Teachers of Mathematics. Kaiser, G. (2017). The teaching and learning of mathematical modeling. In The compendium for research in mathematics education (pp. 267–291). Reston, VA: National Council of Teachers of Mathematics. Niss, M. (2015). Prescriptive modelling – Challenges and opportunities. In G. Stillman, W. Blum, & M.  S. Biembengut (Eds.), Mathematical modelling in education research and practice. International perspectives on the teaching and learning of mathematical modelling (pp. 67–79). Cham, Switzerland: Springer. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standardsbased mathematics instruction: A casebook for professional development (2nd ed.). New York, NY: Teachers College Press. Stillman, G. A. (2019). State of the art on modelling in mathematics education-lines of inquiry. In G. A. Stillman & J. P. Brown (Eds.), Lines of inquiry in mathematical modelling research in education (pp. 1–20). Cham, Switzerland: Springer Open.