Elementary Mathematics Curriculum Materials (Research in Mathematics Education) 3030385876, 9783030385873

Citation preview

Research in Mathematics Education Series Editors: Jinfa Cai · James A. Middleton

Janine T. Remillard Ok-Kyeong Kim

Elementary Mathematics Curriculum Materials Designs for Student Learning and Teacher Enactment With contributions by Kirsti Hemmi, Rowan Machalow, Luke Reinke and Hendrik Van Steenbrugge

Research in Mathematics Education Series Editors Jinfa Cai Newark, DE, USA James A. Middleton Tempe, AZ, USA

This series is designed to produce thematic volumes, allowing researchers to access numerous studies on a theme in a single, peer-reviewed source. Our intent for this series is to publish the latest research in the field in a timely fashion. This design is particularly geared toward highlighting the work of promising graduate students and junior faculty working in conjunction with senior scholars. The audience for this monograph series consists of those in the intersection between researchers and mathematics education leaders—people who need the highest quality research, methodological rigor, and potentially transformative implications ready at hand to help them make decisions regarding the improvement of teaching, learning, policy, and practice. With this vision, our mission of this book series is: (1) To support the sharing of critical research findings among members of the mathematics education community; (2) To support graduate students and junior faculty and induct them into the research community by pairing them with senior faculty in the production of the highest quality peer-reviewed research papers; and (3) To support the usefulness and widespread adoption of research-based innovation. More information about this series at http://www.springer.com/series/13030

Janine T. Remillard • Ok-Kyeong Kim

Elementary Mathematics Curriculum Materials Designs for Student Learning and Teacher Enactment With contributions by Kirsti Hemmi, Rowan Machalow, Luke Reinke and Hendrik Van Steenbrugge

Janine T. Remillard Graduate School of Education University of Pennsylvania Philadelphia, PA, USA

Ok-Kyeong Kim Department of Mathematics Western Michigan University Kalamazoo, MI, USA

ISSN 2570-4729     ISSN 2570-4737 (electronic) Research in Mathematics Education ISBN 978-3-030-38587-3    ISBN 978-3-030-38588-0 (eBook) https://doi.org/10.1007/978-3-030-38588-0 © Springer Nature Switzerland AG 2020 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

This book is dedicated to our fathers, Bruce M. Remillard and Yung Kim.

Author and Contributor Bios

This volume presents curriculum analysis undertaken as part of the ICUBiT research project, directed by principal investigators Janine Remillard and Ok-Kyeong Kim. Remillard and Kim conceptualized the themes and structure of this volume and served as the lead authors on the majority of chapters. The volume also benefits from the work of several authors who contributed chapters to Parts II and III.

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Acknowledgments

Curriculum materials are ubiquitous resources in today’s educational landscape. Still, the work of analyzing curriculum materials can be an uncertain and arduous journey, involving numerous decisions along the way. The frameworks and findings presented in this book represent years of analysis and refinement to which many people contributed. We are deeply appreciative of all who have been part of this journey. We first thank the members of the Improving Curriculum Use for Better Teaching (ICUBiT) project team, who worked with us to develop an extensive coding scheme and conducted portions of the analysis, which took several years. The team members included Napthalin Atanga, Nina Hoe, Shari McCarty, Luke Reinke, Dustin Smith, Joshua Taton, and Hendrik Van Steenbrugge. Many of the ideas presented in this book were influenced by members of the team. We appreciate their valuable insights, hard work, and patience as we deliberated over methodological decisions and struggled to achieve intercoder reliability. We count ourselves as fortunate to have received funding for this work from the National Science Foundation between 2009 and 2015 (grants Nos. 0918141 and 0918126). Without this funding, our collaborative work on the project and this book would not be possible. We are grateful to the members of the project advisory board, Deborah Ball, Matthew Brown, Mary Kay Stein, and Sarah Sword, who offered us sage and direct advice on every aspect of the project. Our early work investigating curriculum materials was supported by the Center for the Study of Mathematics Curriculum (CSMC), an NSF-funded Center for Learning and Teaching. CSMC provided a valuable space and engaged community of scholars, educators, and doctoral students with whom we could try out our ideas. In many ways, the ICUBiT project was hatched out of the conversations that took place within the center. We are grateful to all members and especially to P.I. Barbara Reys for her leadership of CSMC. We thank the curriculum authors who agreed to be interviewed by us, sharing their goals and vision, the development process they underwent, the reasoning behind their decisions, and the challenges they faced along the way. These conversations were extraordinarily enlightening. We spoke with Andy Isaacs, Debbie Leslie, ix

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Acknowledgments

and another author of Everyday Mathematics, Catherine R.  Kelso and Jennifer Mundt Leimberer of Math Trailblazers, and Susan Jo Russell and Karen Economopoulos of Investigations in Number, Data and Space. We are grateful for their expertise, insights, and honest reflections on the complex work of curriculum design. We appreciate the authors who worked with us to conduct additional analysis and contribute to the writing of several chapters: Rowan Machalow, Luke Reinke, and Hendrik van Steenbrugge. We also thank Kirsti Hemmi, who wrote a commentary after reading draft chapters of our analysis of the five curriculum programs. Many scholars and colleagues assisted us by reviewing draft chapters and providing invaluable feedback, which prompted us to refine our thinking and writing. We thank Scott Block, Jeff Choppin, Jon Davis, Corey Drake, Daniel Heck, Mariana Levin, Jane-Jane Lo, Lorraine Males, Kelsey Quigley, and Steve Ziebarth for their generosity. We are especially indebted to Steve Ziebarth, who served as the external evaluator of the ICUBiT Project and read and provided feedback on all of the chapters. We are grateful to Kelsey Burns, Jennifer Moore, and Hope Smith who edited draft chapters. Kelsey Burns, especially, provided detailed and substantive editing support on almost all of the chapters. She also supported the painstaking process of procuring permission to reprint excerpts and images from the five curriculum publishers. We are also grateful for the support and prayers of several colleagues and friends who patiently encouraged us and rallied us as we drew closer to the finish line: Annette Lareau and Caroline Ebby (Janine); Inyol Kwon-Woo and Suna Cho (Ok-Kyeong). Finally, we are grateful for the support and patience of our families (David and Alexander Tristano; Young-Seon, Joshua, and Chloe) while we were working on this book. Janine T. Remillard Ok-Kyeong Kim

Contents

1 A Framework for Analyzing Elementary Mathematics Curriculum Materials������������������������������������������������������������������������������    1 Janine T. Remillard and Ok-Kyeong Kim 1.1 Introduction��������������������������������������������������������������������������������������    2 1.1.1 Clarifying Terms ������������������������������������������������������������������    3 1.1.2 Introduction to the ICUBiT Project��������������������������������������    4 1.2 The Evolution of Curriculum Materials in the USA Between 1990 and 2019 ������������������������������������������������������������������    4 1.3 Understanding Opportunities for Learning in Curriculum Materials������������������������������������������������������������������������������������������    8 1.3.1 Core Assumptions Guiding Curriculum Analysis����������������    8 1.3.2 A Framework for Analyzing Opportunities to Learn in Curriculum Programs ����������������������������������������   10 1.4 Situating Our Work in the Field ������������������������������������������������������   12 1.4.1 Purpose of Analysis��������������������������������������������������������������   13 1.4.2 Focus of Curriculum Analysis����������������������������������������������   16 1.4.3 Methodological Issues����������������������������������������������������������   17 1.5 Curriculum Program Selection and Other Methodological Decisions������������������������������������������������������������������������������������������   17 1.5.1 Curriculum Program Selection ��������������������������������������������   18 1.5.2 Methodological Decisions����������������������������������������������������   20 1.6 Overview of the Book����������������������������������������������������������������������   21 References��������������������������������������������������������������������������������������������������   22 Part I Designing Opportunities for Student Learning 2 Examining the Mathematical Emphasis in Five Curriculum Programs��������������������������������������������������������������������������������������������������   29 Ok-Kyeong Kim and Janine T. Remillard 2.1 Introduction��������������������������������������������������������������������������������������   30 2.2 Theoretical Foundations ������������������������������������������������������������������   30 xi

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2.2.1 The Nature of Mathematics Knowledge ������������������������������   31 2.2.2 Cognitive Demand����������������������������������������������������������������   31 2.2.3 Representation����������������������������������������������������������������������   32 2.2.4 Learning Pathways����������������������������������������������������������������   33 2.3 Methods��������������������������������������������������������������������������������������������   34 2.3.1 Cognitive Demand and Representation��������������������������������   34 2.3.2 Scope and Sequence of Number and Operations, and Ongoing Practice������������������������������������������������������������   37 2.4 Scope and Sequence of Number and Operations in the Five Programs������������������������������������������������������������������������   38 2.4.1 Size of Whole Number����������������������������������������������������������   38 2.4.2 Whole Number Addition and Subtraction����������������������������   38 2.4.3 Whole Number Multiplication����������������������������������������������   41 2.4.4 Whole Number Division ������������������������������������������������������   43 2.4.5 Whole Number Operations in the Five Programs����������������   45 2.4.6 Fraction and Decimal Concepts��������������������������������������������   49 2.4.7 Operations with Fractions and Decimals������������������������������   50 2.5 The Nature of Mathematical Work��������������������������������������������������   52 2.5.1 Cognitive Demand of Lesson Activities ������������������������������   52 2.5.2 The Nature of Ongoing Practice ������������������������������������������   54 2.6 Visual and Physical Representations������������������������������������������������   56 2.6.1 Common Visual/Physical Representations ��������������������������   57 2.6.2 Unique Visual/Physical Representations and Variations������������������������������������������������������������������������   57 2.6.3 Notable Approaches to Using Visual/Physical Representations ��������������������������������������������������������������������   60 2.7 Discussion: Implications for Teachers and Teaching ����������������������   60 2.7.1 There Are Substantial Variations in the Scope and Sequence������������������������������������������������������������������������   61 2.7.2 Some Programs Place Emphasis on Strategies and Relationships������������������������������������������������������������������   62 2.7.3 Many Tasks Start with Connections, But Some Connections Are More Superficial Than Others������������������   62 2.7.4 Ongoing Practice Is Not Just for Mastery of Skills��������������   63 2.7.5 The Kind of Representation Matters, and Yet How It Is Used Needs More Attention������������������������������������������   63 References��������������������������������������������������������������������������������������������������   64 3 Examining the Pedagogical Approaches in Five Curriculum Programs��������������������������������������������������������������������������������������������������   67 Janine T. Remillard, Ok-Kyeong Kim, and Rowan Machalow 3.1 Introduction��������������������������������������������������������������������������������������   68 3.2 Pedagogical Approach in Curriculum Materials?����������������������������   68 3.2.1 The Emergence of Pedagogical Approach in Mathematics Curriculum Materials����������������������������������   69

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3.2.2 Analysis of Pedagogical Approach in Curriculum Materials ������������������������������������������������������������������������������   70 3.2.3 Analysis of Pedagogical Approach Outside of Curriculum Materials��������������������������������������������������������   72 3.3 Analytical Framework: Surfacing Pedagogical Approach in Teacher’s Guides��������������������������������������������������������������������������   74 3.3.1 Explicit Perspective on Mathematics Learning��������������������   74 3.3.2 Lesson and Participant Structure������������������������������������������   74 3.3.3 Nature of Student Learning and Work����������������������������������   76 3.3.4 Teacher’s Role����������������������������������������������������������������������   78 3.4 Methods��������������������������������������������������������������������������������������������   79 3.4.1 Coding����������������������������������������������������������������������������������   79 3.4.2 Analysis��������������������������������������������������������������������������������   80 3.5 Pedagogical Emphasis: Comparison of Five Programs��������������������   81 3.5.1 Investigations: Students as Agents of Their Own Learning ����������������������������������������������������������������������   81 3.5.2 Math Trailblazers: An Integrated Approach��������������������������   86 3.5.3 Everyday Mathematics: Combining Meaning and Repeated Practice ����������������������������������������������������������   91 3.5.4 Math in Focus: Scaffolding Student Learning����������������������   95 3.5.5 Scott Foresman–Addison Wesley: A Mathematics Textbook Approach ��������������������������������������������������������������  100 3.6 Looking Comparatively at Pedagogical Approach��������������������������  103 3.6.1 Considering a Dialogic Versus Direct Lens��������������������������  103 3.6.2 From a Constellation of Features Lens ��������������������������������  104 3.6.3 Remaining Curiosities and Limitations��������������������������������  105 References��������������������������������������������������������������������������������������������������  105 4 Authors Retrospective Reflections on Designing Opportunities for Student Learning ������������������������������������������������������������������������������  109 Ok-Kyeong Kim and Janine T. Remillard 4.1 Introduction��������������������������������������������������������������������������������������  110 4.1.1 The Author Interviews����������������������������������������������������������  110 4.2 The Curriculum Authors’ Design Decisions������������������������������������  112 4.2.1 Overarching Program Goals and Principles��������������������������  112 4.2.2 Communicating Mathematical Goals to the Teacher������������  116 4.2.3 Sequencing the Mathematics Content����������������������������������  118 4.2.4 Pedagogical Approach and Communicating It to the Teacher��������������������������������������������������������������������  124 4.3 The Nature of the Work of Curriculum Development����������������������  130 4.4 Challenges of Developing Curriculum Resources ��������������������������  133 References��������������������������������������������������������������������������������������������������  136

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Part II How Curriculum Authors Communicate with Teachers 5 Beyond the Script: How Curriculum Authors Communicate with Teachers as Curriculum Enactors��������������������������������������������������  141 Janine T. Remillard and Ok-Kyeong Kim 5.1 Introduction to Part II ����������������������������������������������������������������������  142 5.2 An Approach to Examining How Curriculum Authors Communicate with Teachers������������������������������������������������������������  143 5.2.1 Conceptual and Empirical Background��������������������������������  143 5.2.2 Analytical Framework����������������������������������������������������������  146 5.3 Methods��������������������������������������������������������������������������������������������  147 5.3.1 Curriculum Programs and Sampling������������������������������������  147 5.3.2 Primary Coding and Analysis ����������������������������������������������  149 5.4 Results of Primary Coding ��������������������������������������������������������������  151 5.4.1 Quantity and Location of Communication ��������������������������  151 5.4.2 What Curriculum Authors Communicate About �����������������  153 5.4.3 How Curriculum Authors Communicate with Teachers ������  154 5.5 Discussion and Next Steps ��������������������������������������������������������������  156 5.5.1 Understanding Different Approaches to Communication����������������������������������������������������������������  156 5.5.2 Need for Further Analysis����������������������������������������������������  159 References��������������������������������������������������������������������������������������������������  159 6 Examining Communication About Mathematics in Elementary Curriculum Materials������������������������������������������������������������������������������  161 Ok-Kyeong Kim and Janine T. Remillard 6.1 Introduction��������������������������������������������������������������������������������������  162 6.2 Theoretical Background ������������������������������������������������������������������  163 6.2.1 Teachers’ Content Knowledge for Teaching������������������������  163 6.2.2 Content Support for Teachers ����������������������������������������������  164 6.3 Approaches to Communicating Mathematics to Teachers and Analytic Framework������������������������������������������������������������������  165 6.4 Methods��������������������������������������������������������������������������������������������  167 6.4.1 Compilation of the Data Set��������������������������������������������������  168 6.4.2 Data Analysis������������������������������������������������������������������������  168 6.5 Extent and Location of Mathematical Explanations������������������������  171 6.5.1 Extent and Location of Mathematical Explanations in Individual Lessons������������������������������������������������������������  171 6.5.2 Extent and Location of Mathematical Explanations Beyond Individual Lessons��������������������������������������������������  175 6.6 How Mathematics is Communicated in Individual Lessons������������  176 6.6.1 Mathematical Explanations in Individual Lessons ��������������  177 6.6.2 Components of Lessons��������������������������������������������������������  183 6.7 Types of Mathematics Communicated to Teachers��������������������������  186

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6.8 Discussion����������������������������������������������������������������������������������������  188 6.8.1 Mathematics Teachers Need to Know����������������������������������  188 6.8.2 Seeking Better Communication About Mathematics������������  190 References��������������������������������������������������������������������������������������������������  192 7 How Curriculum Materials Support Teachers’ Noticing of Student Thinking ��������������������������������������������������������������������������������  195 Rowan Machalow, Janine T. Remillard, Hendrik Van Steenbrugge, and Ok-Kyeong Kim 7.1 Introduction��������������������������������������������������������������������������������������  196 7.2 Conceptualizing Student Thinking in Teacher’s Guides������������������  198 7.2.1 Teacher Noticing������������������������������������������������������������������  198 7.2.2 Supporting Multiple Strategies ��������������������������������������������  200 7.2.3 Building Foundational Knowledge ��������������������������������������  201 7.3 Analytical Framework����������������������������������������������������������������������  201 7.4 Methods��������������������������������������������������������������������������������������������  203 7.4.1 Unit of Analysis��������������������������������������������������������������������  204 7.4.2 Coding����������������������������������������������������������������������������������  204 7.4.3 Analysis��������������������������������������������������������������������������������  207 7.5 Results����������������������������������������������������������������������������������������������  207 7.5.1 Anticipating/Attending to Student Work: Possible Student Responses��������������������������������������������������  207 7.5.2 Analyzing and Evaluating Students and Student Work��������  211 7.5.3 Foundational Guidelines that Support Noticing ������������������  219 7.5.4 Teacher’s Guide Profiles ������������������������������������������������������  221 7.6 Discussion����������������������������������������������������������������������������������������  223 References��������������������������������������������������������������������������������������������������  224 8 Examining Design Transparency in Elementary Mathematics Curriculum Materials������������������������������������������������������������������������������  227 Luke T. Reinke, Janine T. Remillard, and Ok-Kyeong Kim 8.1 Introduction��������������������������������������������������������������������������������������  228 8.2 Conceptualizing Design Transparency in Teacher’s Guides������������  229 8.2.1 Mathematical Knowledge for Teaching��������������������������������  230 8.2.2 Communicating Rationale����������������������������������������������������  231 8.2.3 Attending to the Development of Curriculum Over Time ����������������������������������������������������������������������������  233 8.3 Analytical Framework����������������������������������������������������������������������  234 8.4 Methods��������������������������������������������������������������������������������������������  235 8.4.1 Unit of Analysis and Coding������������������������������������������������  235 8.4.2 Analysis��������������������������������������������������������������������������������  236 8.5 Results����������������������������������������������������������������������������������������������  237 8.5.1 Communicating Rationale����������������������������������������������������  237 8.5.2 Storyline Support������������������������������������������������������������������  245 8.5.3 Other Design Transparency��������������������������������������������������  251 8.6 Discussion����������������������������������������������������������������������������������������  254 References��������������������������������������������������������������������������������������������������  255

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Part III Synthesis and Commentary 9 Complexity of Curriculum Materials as Designed Artifacts: Implications and Future Directions��������������������������������������������������������  259 Janine T. Remillard and Ok-Kyeong Kim 9.1 Introduction��������������������������������������������������������������������������������������  260 9.2 Key Findings: Trends Within Variation��������������������������������������������  260 9.3 Consequences of Methodological Decisions������������������������������������  262 9.4 Curriculum Materials as Complex and Layered������������������������������  264 9.5 Curriculum Materials as Artifacts of Design Decisions������������������  266 9.5.1 Conceptual Model of Curriculum Materials as Artifacts of Design������������������������������������������������������������  267 9.5.2 Interpretation of Designed Curriculum Artifacts������������������  269 9.6 Implications for the Curriculum–Teacher Relationship ������������������  275 9.6.1 Implications for Teachers: Developing Design Capacity������  275 9.6.2 Implications for Curriculum Authors: Supporting Teachers Design Capacity����������������������������������������������������  279 9.6.3 Implications for Research and Innovation����������������������������  281 References��������������������������������������������������������������������������������������������������  283 10 Commentary��������������������������������������������������������������������������������������������  287 Kirsti Hemmi 10.1 My Perspective��������������������������������������������������������������������������������  287 10.2 A Reflection on the Studies Seen with an Outsider’s Eyes������������  289 10.3 Who Knows Best?��������������������������������������������������������������������������  292 10.4 Contribution and Future Visions ����������������������������������������������������  293 References��������������������������������������������������������������������������������������������������  295 List of Appendices��������������������������������������������������������������������������������������������  297 Author Index����������������������������������������������������������������������������������������������������  313 Subject Index����������������������������������������������������������������������������������������������������  317

Lead Authors

Ok-Kyeong  Kim  is a Professor of mathematics education at Western Michigan University, USA. Kim has taught elementary school in South Korea and preservice teachers in elementary and middle school levels in the United States. She has conducted research on teaching and learning of mathematics in elementary and middle school classrooms. She has also investigated developing and using mathematical thinking and reasoning in school and nonschool settings. Dr. Kim’s current research centers on the role of teacher and curriculum resources in mathematics instruction and the relationship among teacher, curriculum, and instruction that supports students’ learning of mathematics. She is particularly interested in teacher knowledge and capacity needed for using curricular resources productively to teach mathematics and curricular support for mathematics teaching and learning. Currently, she is designing and examining systematic ways that support preservice and inservice teachers to develop their knowledge and capacity to use curricular resources productively. Janine  T.  Remillard  is a Professor of mathematics education and the Faculty Director of Teacher Education at the University of Pennsylvania’s Graduate School of Education. Her research interests include teachers’ interactions with mathematics curriculum resources, mathematics teacher learning in urban classrooms, and teacher education. She is co-editor of the volume, Mathematics Teachers at Work: Connecting Curriculum Materials and Classroom Instruction. Remillard has undertaken research and development projects on teacher learning and development, curriculum use, and formative assessment. She is active in the mathematics education community in the USA and internationally, having chaired the U.S. National Commission on Mathematics Instruction, a commission of the National Academy of Sciences, Engineering, and Medicine, and served two terms on the board of the AERA SIG-Research in Mathematics Education. She is also involved in international, comparative research on mathematics curriculum, currently collaborating with researchers from Belgium, Sweden, and Finland.

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Contributing Authors

Kirsti Hemmi  is a Professor of mathematics and science education at Åbo Akademi University in Finland and visiting professor at Uppsala University in Sweden. Her research interests include teaching and learning of proof, development of algebraic thinking, cultural aspects of mathematics education, curricula and textbooks, and teachers’ interaction with materials. She is currently leading a research project financed by Swedish Research Council focusing on teaching and learning of algebra in Sweden and a project on the use of artificial intelligence and social robots in mathematics classrooms in Finland. She is also involved in international comparative research on mathematics curriculum. Rowan  Machalow  is a PhD candidate in Teaching, Learning, and Teacher Education at the University of Pennsylvania’s Graduate School of Education. Their research interests are mathematics curriculum and teaching, early childhood education, urban education, teaching for social justice, and science education. Machalow is currently involved in international, comparative research on mathematics curriculum, collaborating with researchers from Belgium, Sweden, and Finland, as well as comparative mathematics curriculum research within the USA. Their prior experience includes teaching in elementary and middle schools, founding and directing a nature-based preschool, designing and implementing professional development workshops across the USA, and developing and overseeing development of mathematics and science curriculum for several educational publishers. Luke  T.  Reinke  is an Assistant Professor of mathematics education at the University of North Carolina at Charlotte. His research examines how contextualized problems are used to develop students’ mathematical understanding, how teachers implement curriculum materials in the classroom, and how teachers develop the capacity for enacting ambitious mathematics instruction. Reinke teaches courses on instructional design and elementary mathematics teaching methods and has taught middle and high school mathematics. He also has an interest in curriculum design and has experience developing technology-rich middle school mathematics curriculum materials, as well as locally relevant lessons for urban contexts. xix

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Hendrik  Van Steenbrugge  is a senior lecturer at Stockholm University and Mälardalen University in Sweden. His research focuses primarily on the curriculum resources teachers use and how they are being used by teachers. He frequently studies these topics from cross-cultural perspectives. He is currently co-directing a comparative study of teachers’ use of print and digital mathematics resources in Sweden, Finland, the USA, and Flanders (Belgium). He also studies how resources can support teachers’ collective learning while preparing for and reflecting on teaching mathematics lessons. He has previously been involved in designing curriculum resources that are educative for teachers.

Chapter 1

A Framework for Analyzing Elementary Mathematics Curriculum Materials Janine T. Remillard and Ok-Kyeong Kim

Abstract  This chapter serves as an introduction to this volume. It introduces the idea of curriculum analysis and situates the work of this volume in the Improving Curriculum Use for Better Teaching (ICUBiT) project. After providing a brief historical overview of the evolution of mathematics curriculum materials in the USA, beginning in 1990, we detail the framework guiding our analysis of five elementary mathematics programs, along with a set of assumptions that undergird our approach. We then locate our approach to curriculum analysis in the wider body of such work, identifying several key methodological questions that researchers consider when undertaking analysis of curriculum materials. Finally, we introduce the five curriculum programs that are the focus of our analysis and provide an overview of key aspects of our methodological approach. We conclude with an overview of the remaining chapters in the volume. Keywords  Curriculum analysis · Mathematics curriculum materials · Elementary mathematics · History of U.S. mathematics curriculum materials · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman-Addison Wesley Mathematics · Improving Curriculum Use for Better Teaching (ICUBiT) Project · Teacher’s guide

J. T. Remillard (*) Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] O.-K.Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_1

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1.1  Introduction Textbooks, curriculum materials, and instructional resources have long been a mainstay in classrooms around the world (Valverde, Bianchi, Wolfe, Schmidt, & Houang, 2002), especially in mathematics education. They are viewed as m ­ essengers or mediators of curriculum policy (Valverde et al., 2002), tools to support teachers’ instructional decision-making (Stein, Remillard, & Smith, 2007), and mechanisms to ensure curricular coherence across classrooms and schools (Schmidt et al., 2001). Over the last several decades, as public attention to what is taught in school mathematics has risen, curriculum materials have solidified their prominent place in educational debates and policies, supporting a multibillion dollar, cross-­ national industry. Over the same period of time, the forms and formats of these resources have evolved considerably. Currently, the field is experiencing an uptick in research on curriculum programs and their impact on teachers and students. This surge is primarily a response to a host of new curriculum programs entering the market due to new curriculum initiatives. Some researchers have studied the impacts of curriculum programs on student learning (Stein et al., 2007) and teacher learning (Collopy, 2003; Remillard, 2000), while others have studied how teachers use materials: including how they read, interpret, and modify them (e.g., Atanga, 2014; Choppin, 2011; Kim, 2018, 2019), and the extent to which they follow them with fidelity (e.g., Brown, Pitvorec, Ditto, & Kelso, 2009; Chval, Chávez, Reys, & Tarr, 2009; Kim, 2019). Others, still, have examined the materials themselves, attending to their presentation of the content, approach to teaching and learning, or alignment with externally developed frameworks (e.g., Pepin & Haggarty, 2001; Stein & Kim, 2009). An underlying assumption of the extensive body of research on curriculum materials and teachers’ use of them is that these materials matter for the types of opportunities made available to students’ learning (Cai & Cirillo, 2014). The analysis in this book begins with this assumption. It is also guided by the assumption that teachers and teaching matter for the types of learning opportunities students have to learn mathematics through curriculum materials. This idea reflects an assertion made by Jerome Bruner many years ago: “A curriculum is more for teachers than it is for pupils. If it cannot change, move, perturb, inform teachers, it will have no effect on those whom they teach” (Bruner, 1977, p. xv). In this book, we present a comparative analysis of five mathematics curriculum programs focusing on the potential opportunities for student learning designed into the curriculum programs, which includes how teachers are guided to support student learning. In this introductory chapter, we lay the groundwork for this analysis by clarifying the language and terms we use throughout and detailing background on the study, the evolution of curriculum materials since 1990, our approach to analyzing them, and other approaches to analyzing curriculum materials. We also introduce the five curriculum programs analyzed and provide an overview of the book.

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1.1.1  Clarifying Terms A broad array of terms has emerged to describe curriculum materials and related resources, reflecting changes in both available resources and in the way scholars and educators conceptualize these tools. Many use the term instructional resources to refer to the wide array of programs and tools, print and digital, available to teachers, or generated by them to support classroom instruction. We use the term curriculum materials to refer to a narrower set of resources, those designed to support a program of instruction and student learning over time (Remillard, 2019). In other words, curriculum materials are one type of instructional resource. The term curriculum refers to a course or pathway for learning. The feature of sequencing in curriculum materials is an important component, as it proposes an intended learning progression for particular mathematical domains. Choppin (2011) has identified these learning sequences as a critical element of many curriculum programs, which are not always made visible to the teacher. Sleep (2009) names identifying learning sequences as an important feature of content-specific curriculum knowledge. Our focus in this book is on curriculum materials. Still, curriculum materials remain a fairly general term. We use the term curriculum program to refer to a package of resources assembled by developers for the purpose of guiding instruction and student learning. Curriculum programs include both student-facing materials or texts and teacher-facing material, such as teacher’s guides (Remillard, 2019). Student texts and tools are typically designed for the students’ consumption (Fan, Zhu, & Miao, 2013) and interaction and often include exercises, problems, and other tasks for students to undertake, along with examples, illustrations, and other notes. Student-facing materials also include digital tools designed for students to interact with. In the USA and a number of other countries, student-facing materials are accompanied by a teacher’s guide, which is written to communicate with teachers and support them in shaping lessons, monitoring student progress, and providing additional support (Remillard, 2018). Teacher’s guides are organized into daily lessons that typically align with the student textbook. Digital capabilities have expanded the nature and purpose of curriculum support for teachers. In addition to supporting their instructional decision-making and classroom enactments, digital resources might include tools to track student progress through online tasks or modules, analyze features of student work over time, provide digital professional learning, and connect teachers to others. Teacher’s guides and textbooks, the primary components of curriculum programs, are often accompanied by additional resources, both print and digital, which are also part of the full curriculum program. Curriculum programs are designed with both students and teachers in mind. Some include parents, support staff, and administrators among the intended audience. In the case of elementary mathematics, the teacher plays a critical role in shaping students’ learning opportunities, even when enacting lessons originally designed by curriculum developers. As we allude to in the description of the study

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that follows, our analysis of curriculum programs acknowledges the complexity of these resources and is premised on the belief that using them requires skill.

1.1.2  Introduction to the ICUBiT Project The analysis presented in this book is part of the work on the Improving Curriculum Use for Better Teaching (ICUBiT) project.1 The primary goal of the project was to study the capacity required for teachers to use curriculum materials productively and effectively to design instruction. Our work was guided by the view that curriculum materials are complex and multifaceted; they are also a tool routinely used by teachers on a daily basis. Using these resources is an aspect of teachers’ work that is not well understood or developed in teacher education or professional development. The project included three overlapping phases of work: a) Analyzing a set of five elementary curriculum programs in order to understand the mathematical and pedagogical visions they offered and what is required of teachers when using them; b) Developing a tool to measure an individual teacher’s ability to read the mathematics embedded in designed curriculum resources; and c) Analyzing teachers’ uses of the five programs. This book presents findings from the curriculum analysis, which was the first phase of the study. In the following section, we contextualize our analysis by, first, sketching a brief history of mathematics curriculum material development in the USA between 1990 and 2019, when this volume was completed. We then detail the perspectives and framework guiding our analysis before situating our approach within other curriculum analyses.

1.2  T  he Evolution of Curriculum Materials in the USA Between 1990 and 2019 The last 30 years have seen tremendous transformation in what curriculum materials look like, and how they are accessed and used by teachers. By many accounts, mathematics led the way, when the National Council of Teachers of Mathematics (NCTM) published the Curriculum and Evaluation Standards for School

 The ICUBiT project was funded by the National Science Foundation under grants No. 0918141 and 0918126 and led by P.I.s Janine Remillard and Ok-Kyeong Kim. Over the 6 years that the ICUBiT team members worked together, all members contributed valuable insights and time to the project. Many of the ideas presented in this book were influenced by members of the team, which included Napthalin Atanga, Nina Hoe, Shari McCarty, Luke Reinke, Dustin Smith, Joshua Taton, and Hendrik van Steenbrugge. That said, with the exception of Chaps. 7 and 8, which were coauthored, Remillard and Kim are responsible for authoring this text. Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. 1

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Mathematics (NCTM, 1989). This document, although underspecified in many ways, called for substantial changes in what mathematics students should be expected to learn and how they should learn it, including a decreased emphasis on rote procedures taught in isolation and an increased emphasis on mathematical ­reasoning, conceptual understanding, and problem solving in realistic contexts. The publication of the NCTM Standards and their general embrace among many mathematics educators and local leaders in the USA sparked fundamental changes in mathematics curriculum materials. Inspired by the NCTM Standards, advocates for change in mathematics curriculum and pedagogy initiated other efforts to move these ideas into practice. The National Science Foundation (NSF), an independent U.S. federal agency that issues limited-term grants, funded the development of Standards aligned instructional materials in 1991, under a call for proposals entitled, Instructional Materials Development (IMD). As was required by NSF, the 15 curriculum programs supported by the IMD funding stream, referred to as either Standards-based2 or NSF funded, were eventually marketed by commercial publishers (Senk & Thompson, 2003) and found their way into the mainstream curriculum market in the mid-1990s.3 As several Standards-based programs increased in market share and became more widely used, they were also met with staunch critiques and considerable backlash by some parents and mathematicians, who protested both the content and the pedagogical approach of these materials. In his accounting of what has become known as “the Math Wars,” Schoenfeld (2004) proposed that the debates underlying these disagreements reflect unresolved questions about what it means to learn mathematics and who should learn it that have existed for over 100  years. Moreover, these debates proceeded despite the absence of any reliable data about what students were or were not learning in math classrooms. Instead, they took a political turn. By 2001, when the No Child Left Behind (NCLB) Act was passed by the U.S. Congress, states had already begun to develop their own standards. Those that had not, were required to do so. Political disagreements fueled by the Math Wars led, in many cases, to substantial variations across state standards. The singular focus on high-stakes tests tied to state standards, required by NCLB, led states to demand curriculum programs closely aligned to their standards. During this period of time, the curriculum developers and publishers scrambled to keep up with these divergent demands. Materials sold in each state needed to be modified to align with its standards. Given the diversity of state standards, the Standards-based programs were able to persist alongside more conventional programs, sometimes sold by the same publisher. Around the time that the Math Wars were heating up, another phenomenon caught the attention of many concerned about the U.S. mathematics curriculum. Findings from the Third International Mathematics and Science Study (TIMSS)

 We use the term Standards-based throughout this book.  For more information about the 15 (4 elementary, 5 middle school, 6 high school) curriculum programs funded by the IMDP, see Senk and Thompson (2003). 2 3

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were released to much interest and fanfare. Like the First and Second international comparison studies before it, TIMSS included assessment results of students of specified age groups from a number of different countries. But TIMSS was also more robust than the two previous versions. In addition to testing students in grades 3, 4, 7, and 8, and the final year of secondary school in 41 educational systems, researchers gathered video data of classrooms and analyzed mathematics textbooks. In fourth and eighth grade mathematics, Singapore ranked at the top by a substantial margin, followed by South Korea, Japan, and Hong Kong (US DOE, 2000). These results directed the gaze of many seeking to improve mathematics learning in the USA, especially those dissatisfied with the Standards-based curriculum programs, to these countries. Analysis of the way the curriculum was structured in these countries revealed a tendency to spend more time on fewer topics each year, in contrast to the approach of touching briefly on many topics each year in the USA (Schmidt, Houang, R.T., & Cogan, 2002). Because Singapore held the top rank and because English was its primary language of instruction, Primary Mathematics, a version of its mathematics curriculum found its way into the U.S. system. By 2010, Houghton Mifflin Harcourt, a U.S. publisher was marketing Math in Focus, describing it as “the U.S. edition of Singapore Math.” Some of the characteristics of the Singapore approach to teaching mathematics were shared by other nations that consistently scored at the top of international comparisons in the first decade of the twenty-first century. These characteristics included a scope and sequence that dedicated extensive time to focal topics at each grade level, allowing for deeper learning, and sequencing that coherently developed concepts over several years. In 2008, when the National Governors’ Association and Council of Chief State School Officers initiated the development of the Common Core State Standards for Mathematics (CCSSM), these characteristics were at the center of the call to adopt “a common core of internationally benchmarked standards” (NGA et al., 2008, p. 24). These ideas were captured in three words—focus, coherence, and rigor—characteristic features of the new standards (McCallum, 2015). Additionally, state officials and curriculum developers viewed the CCSSM as having the potential to reduce the variation in curriculum standards across the states. Once released in 2010, CCSSM brought renewed attention to the need for new curriculum materials across the U.S. Commercial publishers updated their conventional textbooks, as did authors of Standards-based programs, to align to the CCSSM. By this time, many of the features previously unique to Standards-based materials, like the inclusion of cognitively demanding tasks, an emphasis on multiple representations and strategies, and student-generated strategies were adopted by mainstream publishers (Remillard & Reinke, 2017). It is important to note that many publishers and material developers were quick to affix a “Common Core Aligned” label to the cover of their materials regardless of the depth of the alignment. Polikoff (2015), for example, analyzed three grade 4 mathematics textbooks published in 2012 and claiming to be “Common Core aligned.” All texts generally covered the content of the standards, but “systematically overemphasize[d] procedures and memorization and underemphasize[d] more conceptual skills relative to their emphasis in the standards” (p. 1185).

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By this time, the development and distribution of digital instructional resources were beginning to pick up steam, as educators, would-be entrepreneurs, and tech companies began to see the Internet as a way to gain direct access to teachers and schools. The larger, more common market, created by the CCSSM also fueled more potential players to get into the instructional development game As a result, some of the earliest “Common Core aligned” resources came not through major curriculum publishers, but other sources. In many cases, the Internet as a distribution mechanism allowed developers to circumvent the powerful textbook publishing industry. States and school districts compiled and disseminated resources to schools via the web and funders seeking to back the Common Core initiative supported material development that could be published online and made freely available to educators and schools wishing to use them. This approach to disseminating resources, both supplemental and eventually more comprehensive, became known as Open Educational Resources (OER). It gained traction during a period of time in the USA when states were cutting back on educational funding, as a result of conservative economic policies and depleted budgets after the market crash of 2008. New York State provides a noteworthy example. The state used Federal Department of Education funds to partner with external agencies to develop a comprehensive set of materials for both mathematics and English language arts (Remillard & Reinke, 2017). Initially called EngageNY, these materials became among the most widely used in states that had adopted the Common Core State Standards (Kaufman et al., 2017). As a result of the developments described above, and despite the unifying intentions behind the Common Core State Standards, the curriculum material market in the USA has become highly diverse in form and content. In addition to updated commercially developed and Standards-based programs, one can find new comprehensive programs and supplemental resources developed in response to CCSSM or adapted from other countries. Some are fully digital, while others have print components. Due to digital publishing, all are able to make changes or add additional components fairly quickly, compared to programs of the past. This rapidly shifting landscape places many new demands on teachers and school leaders who must navigate and use these new options. The field is in need of a framework and approach for examining the variation among curriculum resources with an eye toward how they support teachers to create opportunities for student learning and the nature of these opportunities. Using a comparative approach to analyzing five elementary mathematics programs, this book offers such a framework. The five programs were published just prior to the release of the CCSSM. They represent a moment in time. Nevertheless, the framework offers a lens for examining curriculum programs across many periods of time. Moreover, our findings shed light on critical ways that curriculum programs developed at similar periods of time or with common goals might differ. These differences, as we explore in the chapters of this book, have implications for the potential opportunities for learning they support. In the following section, we elaborate on this guiding focus of our analysis.

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1.3  U  nderstanding Opportunities for Learning in Curriculum Materials We use the term opportunities for learning to characterize the mathematical and pedagogical vision offered by curriculum designers and their support for teachers to enact this vision. Opportunity to learn (OTL) is a widely used concept to characterize aspects of classroom instruction that are consequential for student learning. OTL, in fact, is seen as the primary predictor of student learning outcomes (National Research Council, 2001). Students cannot learn if they do not have opportunities to do so. Traditionally, OTL referred to the specific topics covered in the classroom and the time devoted to each (Floden, 2002). This approach treats quantity of treatment as the primary variable of learning opportunities. The goal of our analysis was to characterize the potential opportunities for learning available in curriculum programs from a more multifaceted perspective, attending to the nature of learning opportunities for students, the role of the teacher in facilitating them, the support for the teacher in this role, and the presence and treatment of particular mathematics topics (Remillard, Harris, & Agodini, 2014). This perspective draws on other analyses of mathematics instruction, which suggest that learning opportunities are not only influenced by the topics covered and time devoted to them, but also by the nature and quality of that time and the structures that frame engagement with specific tasks (Hiebert et al., 2005; Hiebert & Grouws, 2007; Munter, Stein, & Smith, 2015; Stein, Grover, & Henningsen, 1996). Examining opportunities to learn intended by curriculum developers, then, involves considering a number of curriculum features in relation to one another, including: how mathematics is treated; the types of mathematical work students are expected to do; the role that the teacher is encouraged to play in promoting student learning; and how teachers are supported in these roles. This perspective on opportunities to learn designed into curriculum materials was one of the foundational assumptions that guided the questions we asked and the analytical framework we developed to guide our work. In the following section, we offer four additional guiding assumptions about curriculum materials, their development, and their use that are central to our work. These assumptions also underline the importance of curriculum analysis to understand opportunities to learn. We then provide an overview of the framework that guided our analysis.

1.3.1  Core Assumptions Guiding Curriculum Analysis Although the findings presented in this volume offer analyses of the opportunities to learn in five mathematics curriculum programs used at a particular point in time in the USA, the approach we use, including the framework guiding the analysis, has relevance beyond these specific programs. Our approach, which rests on core assumptions about what curriculum materials are (and are not), illuminates their

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complexity as resources for teachers. The first assumption, outlined above, characterizes the factors that contribute to opportunities to learn that can be identified in the design of curriculum materials. Four additional assumptions, briefly summarized below, further elaborate our position and help to explain our approach to curriculum analysis. Our first assumption is about what curriculum materials represent in relation to designers’ intentions. We draw on Brown et al.’s (2009) distinction between curriculum authors’ ideas and written lessons in curriculum materials. Specifically, they argue that curriculum authors’ ideas about specific content to be learned and their ideas about the activities students might engage in in order to learn the content are “coupled together” to “form a set of ideas that are the conceptual basis of a written lesson” (p. 369). Importantly, however, the written lesson represents the authors’ ideas, but is not the same thing as the authors’ ideas. When designing curriculum materials, authors translate their ideas into tasks and instructional guidance, sequenced in a particular way; these decisions form the written lesson. Our second assumption builds on the first: Translating ideas into designed curriculum components or artificial objects, as Wittmann (1995) called designed teaching units, involves many layers of decisions. These decisions include how to organize the content at macro- and microlevels, how to structure learning opportunities, how to balance different types of learning demands, such as problem solving, reasoning, and automaticity with basic skills, the types of representations and models to introduce, how to guide teachers and how to communicate their design decisions to teachers, and even how to layout each page of the teacher- and student-facing material, such that it is accessible and usable by many different teachers and students. It goes without saying that curriculum developers make different decisions, even when they share similar mathematical or pedagogical goals. For instance, the development of three of the programs featured in our analysis was funded by NSF, following the NCTM Curriculum Standards. As such, each of these programs was designed to reflect the content and pedagogical goals spelled out by the NCTM Standards. Our interviews of curriculum authors, described in Chap. 4, illustrate how a similar set of goals can be taken up differently. One goal of the analysis in the book is to highlight the different types of decisions curriculum authors make and to illustrate how these decisions play out in curricular designs. Third, just as authors’ ideas and written lessons are not the same thing, written and enacted curriculum are ontologically different things. Written lessons represent ideas and guidance for the enacted curriculum, but they are not the enacted curriculum written down. And they are not self-enacting. Decades ago, Ben-Peretz (1990) described curriculum materials as constituting an “expression of educational potential” (p. 45). M. Brown (2009) developed this idea further when he compared curriculum materials to sheet music. Both are “static representations” of intended activity, but they are not the activity itself (p. 17). Curriculum materials, Brown goes on to explain, “come alive only through interpretation and use by practitioners” (p. 22). This idea echoes Ben-Peretz’s view that when using curriculum materials, teachers are doing curriculum development work themselves. In other words, translating curriculum from its written to an enacted form involves real work on the part

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of the teacher. From this perspective, we can gain greater insight into the potential of curriculum materials when we consider how the teacher’s role is conceptualized and supported in teacher’s guides. Our final assumption speaks to our perspective on the nature of teachers’ work as users of curriculum materials. Using curriculum materials to guide one’s teaching is not a straightforward process of implementation; it involves reading, interpreting, selecting, modifying, and bringing to life the ideas in the written materials (Remillard & Kim, O.K., 2017). This assumption is rooted in a participatory perspective, which views curriculum use as an interactive and dynamic process that is concomitantly influenced by the teacher and features of the resource itself (Remillard, 2005, 2019). Brown’s (2009) sheet music metaphor helps to illustrate this point. Sheet music conveys rich ideas through “succinct shorthand” (p.  21). Performing the music requires individual musicians to interpret the intended score. It is work that involves specialized knowledge and abilities. Importantly, different musicians will read and perform the same song in very different ways, influenced by their unique talents, preferences, and histories. Similarly, mathematics curriculum materials utilize a type of shorthand. Lessons are primarily comprised of tasks and representations with which students should engage. The key mathematical ideas or learning goals are not always made explicitly (Remillard, Reinke, & Kapoor, 2019; Sleep, 2009). Even in settings where teachers feel compelled to follow their adopted curriculum programs closely, they bring their own meanings to these tasks, influenced by their local knowledge and commitments. This assumption leads us to consider how curriculum authors communicate with and position teachers, who are responsible for bringing their designed potential to life. A primary goal of the analysis in this volume is to surface the range of design choices authors have and illustrate the variety of decisions they might make. We also consider the implications of these decisions for teachers who use the curriculum materials to guide their teaching. In the section that follows, we describe the analytical framework that we used to look comparatively at five curriculum programs. The framework reflects this goal as well as the assumptions detailed above.

1.3.2  A  Framework for Analyzing Opportunities to Learn in Curriculum Programs The analytical framework used in this book considers three primary ways that opportunities to learn mathematics are designed into curriculum materials: a) what and how mathematics is emphasized; b) the pedagogical approaches used to structure learning opportunities for students; and c) how the authors communicate their intentions with teachers. The first two components of the framework are overlapping and, together, characterize the nature of the intended opportunities to learn mathematics designed into the curriculum program. To varying degrees, these aspects of mathematics instruction have been analyzed and characterized in the

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l­ iterature (e.g., Hiebert et al., 1997; Munter et al., 2015). The third component, communication with the teacher, offers a novel approach to curriculum analysis and one, we argue, that is underappreciated in research on curriculum materials. The development of our analytical framework was largely informed by contemporary discussions and debates about mathematics education in the USA between 1990 and 2019 (Hiebert, 2003; Schoenfeld, 2004) and the role of curriculum materials in furthering reform initiatives (Ball & Cohen, 1996; Senk & Thompson, 2003; Stein et al., 2007). It was well understood that the NCTM Standards and the curriculum programs developed in response to them attended to both what mathematics students had opportunities to learn and the nature of those opportunities. These expectations had substantial implications for teaching practices. In fact, the Professional Standards for Teaching Mathematics (NCTM, 1991) were produced as a companion document to the curriculum Standards (NCTM, 1989), offering a “vision of what a teacher at any level of schooling must know and be able to do to teach mathematics as envisioned” by the Curriculum and Evaluation Standards for School Mathematics (p. 5). Thus, in developing our framework, we were most interested in the different ways mathematics was represented in the programs and how the teacher’s role was conceptualized. We looked to research that characterized important differences in how mathematics is organized and represented and the types of mathematical work students are asked to do (Hiebert et al., 2005; Hiebert & Grouws, 2007; Stein et  al., 1996) to specify the first two dimensions of the framework. Being well aware of the fact that most of the programs we were analyzing offered novel approaches to mathematics and mathematics teaching, we were also interested in the extent to which curriculum authors made their commitments and expectations accessible to teachers through various forms of communication (Remillard, 1999). Ball and Cohen (1996) and Davis and Krajcik (2005) offer frameworks characterizing the types of communication likely to support teachers when using these novel approaches. This work informed the development of the third dimension of our framework. The following is a high-level overview of the three-part analytical framework guiding our analysis of opportunities to learn mathematics in elementary curriculum materials. More detail on each dimension can be found in subsequent chapters. Mathematical Treatment and Emphasis refers to aspects of mathematics emphasized and how mathematics is organized and presented for student learning. Analysis of this dimension focused on three major indicators: a) scope and sequence of key topics in the Number and Operations strands across grades 3 through 5; b) the nature of mathematical work expected, including cognitive demand of core tasks during each lesson (Stein et al., 1996) and the opportunities for ongoing practice; and c) how representations are used to communicate mathematical ideas and support student learning of mathematical concepts. Pedagogical Approach refers to both explicit and implicit messages about how students should interact with mathematics, one another, the teacher, the textbook, and other learning tools around mathematical ideas and the teacher’s role in leading and supporting students’ interactions and learning. Analysis of this dimension

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focused on four types of indicators: a) pedagogical positions and aims explicitly stated by the authors; b) how lessons are structured, including how students are expected to participate; c) how student learning is conceptualized in the lessons, based on the types of opportunities students have to engage with mathematics concepts and do mathematical work; and d) the role the teacher is expected to play in supporting student learning opportunities. Communication with the Teacher refers to how authors communicate with teachers and what they communicate about. This analysis focused on the communication in each daily lesson guide intended for the teacher’s consumption. We compared the quantity and location of the types of communication provided, including the differences between directing teacher action and speaking directly to teachers (Remillard, 1999). Drawing on Davis and Krajcik’s (2005) framework, we compared how authors communicated with teachers about a) mathematics, b) attending to student thinking, c) design decisions and transparency, and d) teacher decision making.

1.4  Situating Our Work in the Field Our comparative analysis of mathematics curriculum materials is situated within a long tradition of work, which uses tools of document analysis to scrutinize, and often compare, curriculum authors’ intentions, underlying philosophies or assumptions, and the implicit messages embedded in curriculum documents. In a survey of research on mathematics textbooks, Fan et  al. (2013) identified 105 studies published between 2008 and 2012 and found that 61% of these studies involved analyses of features of textbooks and in many cases included comparisons of features of different series of textbooks from the same country or different countries. Even though document analysis cannot replace research on the enacted curriculum (Stein et al., 2007), there is strong evidence and general agreement within the field that curriculum materials matter for what, when, and how students have opportunities to learn (Agodini, Harris, Seftor, Remillard, J.T., & Thomas, 2013; Cai, J., & Cirillo, 2014; Stein et al., 2007). Further, curriculum analysis provides the field opportunities to consider, prospectively or retrospectively, “how different conceptualizations of mathematics play out in the teaching and learning of mathematics” (Cai & Cirillo, 2014, p. 134). At the same time, there is substantial variation in how those who analyze curriculum documents undertake the work. They focus on different components or features of curriculum materials and they are guided by different purposes. Further, they adopt a wide range of methods and approaches. A brief review of the body of analyses of mathematics textbooks and curriculum material documents reveals the variety of decisions researchers face. In the following discussion, we consider two key types of decisions made by those who analyzed curriculum documents, the purpose for curriculum analysis and its focus. We also discuss some of the most critical methodological issues that have been raised by those undertaking this work.

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1.4.1  Purpose of Analysis Looking across decades of analyses of mathematics textbooks and curriculum materials, we identified four major purposes, several of which often overlap. We categorize them as aiming to a) uncover approaches and trends, b) make comparisons, c) consider relationships, and d) surface aspects of culture. Table 1.1 provides a collection of illustrative examples of curriculum analyses published between 1996 and 2019. We refer to these examples to illustrate the different purposes and foci most commonly adopted in mathematics curriculum analysis. The most general purpose, which seldom stands alone, was to uncover approaches taken by curriculum authors, identifying key elements or critical features and, when analyzing multiple documents, surfacing trends. Mesa (2004) analyzed how functions are taught in 24 middle school mathematics textbooks from 15 different countries. Through the analysis, she highlighted different practices associated with functions and algebra across textbooks. Herbel-Eisenmann’s (2007) analysis of a single student textbook provides another example of uncovering approaches and critical features in curriculum materials. She used conventions from discourse analysis, such as pronouns, imperative verbs, and modality, to examine the authors’ approach to communicating with students. Based on this analytical approach, Herbel-Eisenmann asserted that although students were expected to actively engage in constructing ideas, the voice of the textbook tended to be authoritative, positioning mathematics as being absolute and without human involvement. Most curriculum analyses that focus on approach do so along with an additional purpose of making comparisons between different textbooks or curriculum programs. In fact, making comparisons between textbooks within or across contexts is one of the most common purposes adopted by those who analyze textbooks. Because textbooks and curriculum materials, to a large extent, represent opportunities to learn in school systems, there is considerable interest in making comparisons among them. One of the most comprehensive comparative analyses of textbooks was undertaken by Valverde et al. (2002), using documents collected through the 1995 TIMSS study. These researchers analyzed features of 400 textbooks from 13 TIMSS countries, comparing macro features, such as size and length, content inclusion and sequencing, as well as micro features, including how the content was presented, the complexity of expectations for students, and types of student activities in the lessons. Most curriculum comparisons take a cross-national approach, however, some analyses compare different curriculum programs within one context. Stein and Kim (2009), for example, compared two Standards-based curriculum programs in the USA. A third purpose guiding some curriculum analyses is to enable researchers to examine relationships between written curriculum materials and other forms of curriculum, such as national frameworks or the curriculum enacted by teachers. Over time, many researchers have been interested in examining how close classroom enactments are to the written curriculum, referred to by some as “fidelity of implementation.” In order to measure fidelity, researchers must identify key features or

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Table 1.1  Illustrative examples of curriculum analyses published between 1996 and 2019 Author and year Reys, Reys, and Koyama (1996)

Summary Analyzed three Japanese textbooks grades 1–3, comparing them to U.S. textbooks. Considered a number of topics, repetition across grades, types of problems, and expectations for student work. Dowling Analyzed types of problems in high school (1998) textbooks series in England, comparing editions written for high achieving students with those intended for students with lower abilities to uncover differences in contexts used. Parker (1999) Analyzed images in a German mathematics textbook for representation of gender, comparing the depiction of male and female children and adults. Analyzed written text of middle school Seah and mathematics textbooks from Australia and Bishop Singapore for values about the nature of (2000) mathematics, how it should be used, and who should use it. Analyzed cultural traditions of the role of Pepin and textbooks, how they are used, and how they Haggarty present the nature of mathematics in three (2001) countries. Valverde Analyzed features of 400 textbooks from 13 et al. (2002) TIMSS countries from macro (size and length, content inclusion, and sequencing) to micro (representation of content, complexity of expectations for students, and types of student activities included). Mesa (2004) Analyzed how functions were taught in 24 middle school mathematics textbooks from 15 different countries, highlighting different practices associated with functions and algebra across textbooks. Used discourse analysis to characterize the Herbel-­ voice of Standards-based middle school Eisenmann textbook through examining pronouns (I, (2007) you, we), imperative verbs (consider, explain, draw, solve, etc.), and modality (could, should, might, etc.). Vincent and Compared Australian grade 8 mathematics textbooks with enacted lessons from the Stacey TIMSS Video Study, focusing on approaches (2008) to teaching fraction operations, solving linear equations, and geometry with triangles and quadrilaterals.

Purpose Comparison, culture

Focus Math, learning

Culture

Social

Culture

Social

Culture, comparison

Learning Social

Culture, relationship

Comprehensive

Comparison, culture

Math, learning

Approach

Math

Approach

Learning, social

Approaches, comparison

Math, learning

(continued)

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Table 1.1 (continued) Author and year Stein and Kim (2009)

Summary Analyzed two U.S. curriculum programs, considering the demands they placed on teachers and the level of support they provided teachers to use them. Stylianides Analyzed tasks within a series of U.S. (2009) mathematics textbooks for opportunities for students to engage in reasoning-and-proving. Analyzed and compared how U.S. and Son and Korean textbooks represent computation of Senk, S. L., fractions. (2010) Century et al. Propose a framework for characterizing the (2010) critical components of the intended curriculum in a curriculum program, with the intention of measuring fidelity of implementation (FOI) in enacted curriculum. Pepin et al. Analyzed and compared textbook intentions, (2013) learning approaches, and relationship to national curriculum documents in three countries. Jones and Compared treatment of geometry in Fujita (2013) Japanese and English textbooks, considering number of topics covered, types of tasks, and presentation of the mathematics.

Purpose Comparison

Focus Learning, teacher use

Approach

Math

Comparison, culture

Math, learning

Relationship

Learning Comprehensive

Relationship, comparison, culture

Learning, comprehensive

Comparison, culture

Math, learning

components that characterize the written or intended curriculum. With this aim in mind, Century, Rudnick, and Freeman (2010) proposed a framework for characterizing the critical components of the intended curriculum in a text, with can then be used to assess fidelity of implementation to those intentions. Researchers have also used curriculum analysis to examine the relationship between written curriculum and national standards or frameworks. Pepin, Gueudet, and Trouche (2013) examined the relationship between textbook intentions and national curriculum documents, as well as how these intentions were enacted. By undertaking this analysis within the French and Norwegian school systems, they were also able to make comparisons across these two cultural contexts. Comparisons across cultural contexts frequently allow researchers to adopt a fourth purpose in curriculum analysis, identifying cultural traditions or assumptions as they play out in the intended curriculum. In their comparative analysis of French and Norwegian curriculum documents and practices, Pepin et al. (2013), for example, also identified influential cultural traditions that pervaded each school system, such as strong egalitarian influences (Norway) versus reason and rationality (France), and could be detected across all three forms of curriculum. As with many examples in Table 1.1, Pepin et al.’s analysis had more than one purpose.

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1.4.2  Focus of Curriculum Analysis Even when researchers adopt a similar purpose for undertaking curriculum analysis, they often make decisions to focus their analysis on different aspects of the curriculum. As we noted earlier, curriculum materials are complex and multi-faceted and it is challenging, and often not appropriate, to examine all components at once. Guided by particular research questions, researchers find themselves identifying different foci. In our review of the body of curriculum analyses, we have identified five primary foci adopted by researchers when analyzing mathematics textbooks and curriculum materials. (See Table  1.1, the far right column.) The most commonly adopted focus is the treatment of particular mathematical topics or domains (Fan et  al., 2013). Researchers identified a topic, such as geometry (Jones & Fujita, 2013), algebra (Mesa, 2004), computation of fractions (Son & Senk, S. L., 2010), or reasoning and proof (Stylianides, 2009), to focus their analysis. A second focus, which can overlap with the first, is the treatment of learning. This focus includes both the nature of mathematics that is emphasized and how learners are encouraged to engage with the content. To what extent is procedural knowledge emphasized, as opposed to conceptual understanding? And to what extent are students expected to solve problems and justify their answers, as opposed to follow procedures offered by the teacher? A third focus, which we found to be uncommon in the literature, examines curriculum materials from the teacher’s perspective, focusing on what might be involved in using them or how the materials are designed to support teachers. Stein and Kim’s (2009) comparative analysis of two Standards-based programs, considers the level of demand the approach places on teachers and the extent to which authors support teachers by helping them anticipate student thinking or explaining the rationale behind features of the design. Some approaches to curriculum analysis adopt a more holistic focus, seeking to characterize the curriculum program in a comprehensive way. This focus is illustrated by Century et al.’s (2010) framework, which seeks to offer an approach to characterizing the “intended program model” (p. 202) based on surfacing the written curriculum’s critical components and organizing them in a set of categories. The framework identifies structural components and instructional components and attends to organization and treatment of the content, pedagogical features, and elements of student engagement. A final focus of curriculum analysis that we found occurring as either the central focus or a secondary focus of analysis is the treatment of social relations, such as gender, social class, ethnicity, or various forms of power relationships that might be implicit in the design of materials. Dowling’s (1998) analysis of a mathematics textbook series in England, compared the problem contexts used in different editions of the text. Problems in the edition for lower achieving students tended to reference every day or manual work, like making repairs or shopping, whereas problem contexts included in the edition for higher achieving students tended to reference academic or intellectual activities. Dowling argued that these differences reinforce existing social orders and serve as gatekeepers.

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1.4.3  Methodological Issues The different approaches to curriculum analysis detailed in the previous sections also surface the role that methodological decisions play in curriculum analysis. In their discussions of textbook or curriculum analysis, both Fan et al. (2013) and Cai and Cirillo (2014) point to a number of methodological challenges and related decisions. Cai and Cirillo, in fact, offer the following list of key methodological decisions: • • • • •

How many textbooks should we analyze? Which textbook(s) should we analyze? What text in the textbook(s) should we analyze? The exposition? The exercises? How much of that text should we analyze? How should we analyze it (i.e., what framework will we use to conduct our analysis)? • What research questions should guide our analysis? (p. 136) How researchers answer these questions in the process of designing curriculum analysis, as Cai and Cirillo point out, will influence the findings. Similar questions pursued using different analytical frameworks will necessarily lead to different claims. The last two decisions on the list arguably should be the starting point of any analysis. Earlier in this chapter, we discussed the guiding questions and analytical framework that guided our analysis, along with the assumptions that supported it. These decisions hone the lens of the researcher, but still leave a number of further questions unanswered. The four questions at the beginning of Cai and Cirillo’s list, which concern determining which documents within a textbook series or curriculum program, how much and how many, and which particular components to analyze, are equally critical, but less frequently scrutinized in discussions of curriculum analysis. As with all data collection, researchers make decisions about sampling and then apply findings from the sample to a fuller set of documents. The question of what counts as a representative sample of a curriculum program is a complex one. What types of claims can be made based on what types of subsamples? In most cases, it depends on the question being asked. Is it sufficient to examine one chapter or grade level? Once the findings of a curriculum analysis are produced, they are typically applied to the entire program or series, regardless of the subsample analyzed. As we discuss in the following section, our team grappled with many of these questions when designing our methods.

1.5  C  urriculum Program Selection and Other Methodological Decisions The curriculum analysis was the first step in a three-step research project (although components of it went on throughout the life of the study). Nevertheless, the curriculum programs we selected were critical to all three components of the study.

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Thus, our selection of the five programs to analyze would shape the entire study. In a less direct way, the other methodological decisions we made for the curriculum analysis also shaped our findings, in that these decisions influenced the claims we would make about the nature of these programs and what is involved in using them. In the following sections we detail both types of decisions.

1.5.1  Curriculum Program Selection A primary focus of the ICUBiT project was to understand teacher capacity involved in using curriculum materials, particularly those unfamiliar to teachers. Thus, our intention was to select programs that reflected a diversity of the options on the market and that might offer some challenges to teachers using them, especially for the first time. The five programs we selected are summarized in Table 1.2. Below, we provide our rationale for selecting them and follow with a brief description of each. Our rationale for selecting these five programs is as follows. At the outset of the project, in 2009, we identified three curriculum programs that had initially been developed through NSF funding, under the IMDP initiative in the early 1990s: Everyday Mathematics; Investigations in Number, Data, and Space; and Math Trailblazers. It was well understood that each of these programs offered a vision of mathematics instruction that reflected the NCTM Standards (1989; 2000) and was challenging for teachers to enact (Stein & Kim, 2009). In 2009, these programs were in their second or third edition, since initial funding in the early 1990s and accounted for at least 25% of the market share at the time (These were considered Standards-based and accounted for at least 25% of the market share at the time) (Malzahn, 2013). In addition, we selected a publisher-developed program (Stein et al., 2007), Scott Foresman-Addison Wesley Mathematics, which we saw as representing the mainstream approach to elementary curriculum materials at the time. We also selected a program developed by Marshall Cavendish, the publisher responsible for the most commonly used mathematics curriculum program in Singapore. Table 1.2  The five curriculum programs analyzed Abb. EM

MIF

Curriculum title (edition) Everyday Mathematics (3rd Edition) Investigations in Number, Data, and Space (2nd Edition) Math in Focus

MTB

Trailblazers (3rd Edition)

INV

SFAW Scott Foresman–Addison Wesley Mathematics (2008 Edition)

Developers/authors University of Chicago School Mathematics Project Education Research Collaborative at TERC Singapore Ministry of Education Marshall Cavendish International TIMS at the University of Illinois at Chicago Scott Foresman–Addison Wesley

Publisher Wright Group/ McGraw-Hill Pearson Houghton Mifflin Harcourt Kendall Hunt Pearson

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This program, Math in Focus, was distributed by Houghton Mifflin Harcourt and was a new arrival in the U.S. curriculum market at the beginning of the study. Given the interest in the Singapore approach to teaching mathematics rippling across the USA, Math in Focus seemed like a good choice. It was also substantially different from the others and offered a number of approaches that appeared to influence the development of the CCSSM, which were in the works at the time. Everyday Mathematics Everyday Mathematics (EM) was packaged as a comprehensive curriculum program for pre-kindergarten through sixth grade mathematics instruction, developed by the University of Chicago School Mathematics Project. Work on the first edition began in 1985, prior to the 1989 NCTM Standards. Further development was supported by NSF, through an Instructional Materials Development (IMD) grant. The 2012 National Survey of Science and Mathematics Education (Malzahn, 2013) identified it as one of the two most commonly used elementary mathematics materials in 2012. Investigations in Number, Data, and Space (INV) Investigations, a K-5 series, was developed by the staff of the Education Research Center, TERC.  The first edition was published between 1994 and 1998 and was funded as one of NSF’s IMD funding streams. Math in Focus Math in Focus (MIF) was adapted from one of the elementary mathematics programs developed and used in Singapore. It was developed by Marshall Cavendish International in collaboration with the Singapore Ministry of Education. The curriculum was adapted for the U.S. market by Marshall Cavendish and distributed by Houghton Mifflin Harcourt. Math Trailblazers Math Trailblazers (MTB), an elementary (K-5) mathematics program, was developed by the Teaching Integrated Mathematics and Science Project (TIMS) at the University of Illinois at Chicago. Funding for its development came from NSF’s IMD funding stream.

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Scott Foresman–Addison Wesley Mathematics (SFAW) Mathematics was originally developed and marketed by the publishing company Scott Foresman before it merged with Addison Wesley. The company and its products were then purchased by Pearson publishing.

1.5.2  Methodological Decisions The methodological decisions we made in order to undertake the analysis were guided by our analytical framework, informal knowledge of U.S. curriculum programs, and several logistical issues. In order to surface the opportunities for learning encapsulated in the five programs, we focused our analysis on components that teachers would have access to and use on a regular basis. For this reason, the daily lessons in the teacher’s guide became the primary focus of our analysis. These lessons included images of the main student-facing material, along with information and guidance written for the teacher, all of which were intended to be used by the teacher on a daily basis. We believe these lessons offered the best representation of the authors’ intentions for students on a daily basis. That said, we also referred to other components of the curriculum programs, such as introductory material in the teacher’s guide, implementation guides, or assessment resources intended to provide the teacher with additional insights. At the time of the analysis, most of the programs we analyzed included online resources, however, these were supplementary to the lessons in the teacher’s guides, which were available in both print and a static, PDF format. Once we had determined the components of the curriculum programs to analyze, we still needed to identify a sampling strategy (Cai & Cirillo, 2014). At the outset of the study, the ICUBiT project had decided to focus on the Numbers and Operation strands, including whole and non-whole numbers, within grades 3–5 for all aspects of the study. For this reason, we constrained our curriculum analysis to these grades and topics as well. The number of total lessons varied across the five programs and per grade, ranging from 52 (MIF, grade 4)4 and 164 (SFAW, grade 5). Between 60 and 70% of the lessons in each program and grade were devoted to Numbers and Operations. We randomly selected 30 lessons from each program, 10 from each of the three grades, to analyze closely, giving us a total of 150 lessons. These formed the corpus of our data. The particular analytical approach we used varied, depending on the questions we were pursuing. Details of the various approaches are discussed in the subsequent chapter. We provide an overview of these chapters in the following section.

 As is explained in Chap. 3, MIF contained a number of lessons designated for more than one instructional day, and thus had the smallest numbers of total lessons per grade. 4

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1.6  Overview of the Book This book is organized into three parts, following this introductory chapter, which establishes the analytical and curricular context of the book. Part I, which contains three chapters, focuses on the mathematical emphasis and pedagogical approach in the elementary curriculum programs, and provides insight into perspectives of curriculum authors on developing curriculum programs. Chapter 2 focuses on how mathematics content is presented in each program in terms of the scope and sequence, cognitive demand of problems and tasks, representations and connections, and skill practice and concept reinforcement in daily routine activities. This chapter highlights differences in the nature of mathematical work and learning trajectories. Chapter 3 focuses on the pedagogical approaches of each program, elaborating on how mathematical learning is structured, the intended instructional models, and the expected teacher role during instruction. This chapter illuminates multiple ways to support students’ mathematical thinking with regard to pedagogical approaches, rather than the direct or dialogic approaches, often used to analyze the pedagogical aspect of mathematics instruction. Chapter 4 presents the retrospective reflections of curriculum developers for three of the five programs. Through interviews, the authors shared their original hopes, design decisions, and learning about the process over time. This chapter presents key themes that emerged from these interviews, using direct quotations to honor their voices. We explain different choices the authors made despite shared goals, the evolving nature of the curriculum, and the challenges of developing curriculum programs. In Part II, we attend to how curriculum authors communicate with teachers, building on arguments by Ball and Cohen (1996), Davis and Krajcik (2005), and Remillard (2000) that curriculum authors might support teachers’ work enacting curriculum materials. Chapter 5 provides an overview of the analyses conducted in Part II. It describes the rationale and methodology for the approach and presents findings on overall patterns in how different curriculum authors communicate with teachers about key aspects of teaching. Chapter 6 examines how authors communicated about the mathematics they teach, including explanations and illustrations. This chapter highlights variations across the five programs in the kinds and extent of mathematics explained to teachers and the location of these explanations. Chapter 7 explores what and how authors communicate about student thinking. Using a noticing framework (Jacobs, Lamb, & Philipp, 2010; van Es & Sherin 2002, 2008), the analysis examines the way authors support teachers to attend to and anticipate student responses, interpret them, and respond. Chapter 8 focuses on ways in which curriculum authors make their design decisions transparent to teachers by communicating rationales for curricular features and making the learning sequence (or a storyline) explicit to teachers. This chapter finds that most authors attended to the need for design transparency, but primarily in superficial ways. In Part III, we draw on the findings of our analyses to consider implications for teaching and present a commentary on our analyses. Chapter 9 synthesizes findings across the analytical chapters and discusses implications for teachers and teaching.

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This chapter also considers implications for curriculum and teacher development and calls for future work and research. Chapter 10, an invited chapter written by Kirsti Hemmi, discusses what the book adds to the understanding in the field and what questions it raises for future curriculum development and research.

References Agodini, R., Harris, B., Seftor, N., Remillard, J.T., & Thomas, M. (2013). After two years, three elementary math curricula outperform a fourth. (NCEE 2013–4019). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Atanga, N. A. (2014). Elementary school teachers’ use of curricular resources for lesson design and enactment (Unpublished dissertation in Western Michigan University). Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is—Or might be—The role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–8. Ben-Peretz, M. (1990). The teacher-curriculum encounter: Freeing teachers from the tyranny of texts. Albany, NY: State University of New York Press. Brown, M.  W. (2009). Toward a theory of curriculum design and use: Understanding the teacher-tool relationship. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 17–36). New York: Routledge. Brown, S.  A., Pitvorec, K., Ditto, C., & Kelso, C.  R. (2009). Reconceiving fidelity of implementation: An investigation of elementary whole-number lessons. Journal for Research in Mathematics Education, 40(4), 363–395. Bruner, J. (1977). The process of education. Cambridge, MA: Harvard University Press. Cai, J., & Cirillo, M. (2014). What do we know about reasoning and proving? Opportunities and missing opportunities from curriculum analyses. International Journal of Educational Research, 64, 132–140. Century, J., Rudnick, M., & Freeman, C. (2010). A framework for measuring fidelity of implementation: A foundation for shared language and accumulation of knowledge. American Journal of Evaluation, 31(2), 199–121. Charles, R. I., Crown, W., Fennell, F., et al. (2008). Scott Foresman–Addison Wesley Mathematics. Glenview, IL: Pearson. Choppin, J. (2011). Learned adaptations: Teachers’ understanding and use of curriculum resources. Journal of Mathematics Teacher Education, 15(5), 331–353. Chval, K. B., Chávez, Ó., Reys, B. J., & Tarr, J. (2009). In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 70–84). New York: Routledge. Collopy, R. (2003). Curriculum materials as a professional development tool: How a mathematics textbook affected two teachers’ learning. Elementary School Journal, 103(3), 287. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Dowling, P. (1998). The sociology of mathematics education: Mathematical myths/pedagogic texts. London: Falmer. Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: Development status and directions. The International Journal on Mathematics Education (ZDM), 45(5), 633–646. Floden, R. (2002). The measurement of opportunity to learn. In A. C. Porter & A. Gamoran (Eds.), Methodological advances in cross-national surveys of educational achievement. Board on

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International Comparative Studies in Education (pp. 231–236). Board on Test on Testing and Assessment, Center for Education, Division of Behavioral and Social Sciences and Education). Washington, DC: National Academy Press. Herbel-Eisenmann, B. A. (2007). From intended curriculum to written curriculum: Examining the “voice” of a mathematics textbook. Journal for Research in Mathematics Education, 38(4), 344–369. Hiebert, J. (2003). What research says about the NCTM standards. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to the principles and standards for school mathematics (pp. 5–24). Reston, VA: NCTM. Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F.  K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Greenwich, CT: Information Age Publishing. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Hannley, M., Oliver, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, ME: Heinemann. Hiebert, J., Stigler, J. W., Jacobs, J. K., Garnier, H., Smith, M. S., Hollingsworth, H., Manaster, A., Wearne, D., & Gallimore, R. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 video study. Educational Evaluation and Policy Analysis, 27(2), 111–132. Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. Jones, K., & Fujita, T. (2013). Interpretations of National Curricula: The case of geometry in textbooks from England and Japan. The International Journal on Mathematics Education (ZDM), 45, 671–683. Kaufman, J. H., Davis, J. S., II, Wang, E. L., Thompson, L. E., Pane, J. D., Pfrommer, K., & Harris, M. (2017). Use of open educational resources in an era of common standards: A Case Study on the Use of EngageNY. Santa Monica, CA: RAND Corporation. Kim, O. K. (2018). Teacher decisions on lesson sequence and their impact on opportunities for students to learn. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Recent advances in research on mathematics textbooks and teachers’ resources (pp. 315–339). New York: Springer. Kim, O. K. (2019). Teacher fidelity decisions and the quality of enacted lessons. A manuscript submitted for publication. Malzahn, K. A. (2013). 2012 National Survey of Science and Mathematics Education: Status of elementary school mathematics. Chapel Hill, NC: Horizon Research. Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. McCallum, W. (2015). The common core state standards in mathematics. In S.  J. Cho (Ed.), Selected regular lectures from the 12th international congress on mathematical education (pp. 547–561). Springer: Switzerland. Mesa, V. (2004). Characterizing practices associated with functions in middle school textbooks: An empirical approach. Educational Studies in Mathematics, 56, 255–286. Munter, C., Stein, M. K., & Smith, M. S. (2015). Dialogic and direct instruction: Two distinct models of mathematics instruction and the debate(s) surrounding them. Teachers College Record, 117(11), 1–32. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards mathematics. Washington, DC: Author. National Governors Association, the Council of Chief State School Officers, & Achieve, Inc. (2008). Benchmarking for success: Ensuring U.S. students receive a world-class education (A report by the NGA, CCSSO, and Achieve. Washington, D.C.) National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

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NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. NCTM. (2000). Principles and standards for school mathematics. Reston, VA: Author. No Child Left Behind [NCLB] Act. (2001). Public Law 107–110. 107th Congress. Retrieved from http://www2.ed.gov/policy/elsec/leg/esea02/107-110.pdf Parker, K. (1999). The impact of the textbook on girls; perceptions of mathematics. Mathematics in Schools, 28(4), 390–407. Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French, and German classrooms: A way to understand teaching and learning cultures. The International Journal on Mathematics Education (ZDM), 33(5), 158–175. Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teachers’ work and interactions: A collective perspective on resources, their use and transformation. The International Journal on Mathematics Education (ZDM), 45(7), 929–943. Polikoff, M. S. (2015). How well aligned are textbooks to the common core standards in mathematics? American Educational Research Journal, 52(6), 1185–1211. Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers’ curriculum development. Curriculum Inquiry, 29(3), 315–342. Remillard, J. T. (2000). Can curriculum materials support teachers’ learning? Elementary School Journal, 100(4), 331–350. Remillard, J. T. (2005). Examining key concepts of research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Remillard, J.  T. (2018). Examining teachers’ interactions with curriculum resource to uncover pedagogical design capacity. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Recent advances in research on mathematics teachers’ textbooks and resources (pp. 69–88). New York: Springer. Remillard, J. T. (2019). Teachers’ use of mathematics resources: A look across cultural boundaries. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), The ‘resource’ approach to mathematics education. New York: Springer. Remillard, J.  T., & Kim, O.-K. (2017). Knowledge of curriculum embedded mathematics: Exploring a critical domain of teaching. Educational Studies in Mathematics, 96(1), 65–81. Remillard, J.  T., & Reinke, L.  T. (2017). Mathematics curriculum in the United States: New challenges and opportunities. In D.  R. Thompson, M.  A. Huntly, & C.  Suurtamm (Eds.), International perspectives on mathematics curriculum (pp.  131–162). Greenwich, CT: Information Age Publishing. Remillard, J. T., Harris, B., & Agodini, R. (2014). The influence of curriculum material design on opportunities for student learning. ZDM, The International Journal on Mathematics Education, 46(5), 735–749. Remillard, J. T., Reinke, L. T., & Kapoor, R. (2019). What is the point? Examining how curriculum materials articulate mathematical goals and how teachers steer instruction. International Journal of Educational Research, 93, 101–117. Reys, B. J., Reys, R. E., & Koyama, M. (1996). The development of computation in three Japanese primary-grade textbooks. The Elementary School Journal, 96(4), 423–437. Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H., Wiley, D. E., Cogan, L. S., & Wolfe, R. G. (2001). Why schools matter: A cross-national comparison of curriculum and learning. San Francisco, CA: Jossey-Bass. Schmidt, W. H., Houang, R., & Cogan, L. (2002). A coherent curriculum: The case of mathematics. American Educator (Summer), 1–17. Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253–286. Seah, W.  T., & Bishop, A.  J. (2000). Values in mathematics textbooks: A view through two Australasian regions. In 81st Annual meeting of the American Educational Research Association. New Orleans, LA. Senk, S. L., & Thompson, D. R. (2003). Standards-based school mathematics curricula: What are they? What do students learn? Mahwah, NJ: Lawrence Erlbaum Associates. Sleep, L. (2009). Teaching to the mathematical point: Knowing and using mathematics in teaching. Unpublished doctoral dissertation. University of Michigan, USA.

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Son, J.-W., & Senk, S. L. (2010). How reform curricula in the USA and Korea present multiplication and division of fractions. Educational Studies in Mathematics, 74(2), 117–142. Stein, M. K., & Kim, G. (2009). The role of mathematics curriculum materials in large-scale urban reform: An analysis of demands and opportunities for teacher learning. In J.  T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 37–55). New York: Routledge. Stein, M. K., Grover, B. W., & Henningsen, M. A. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classroom. American Educational Research Journal, 33(2), 455–488. Stein, M. K., Remillard, J. T., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–369). Greenwich, CT: Information Age Publishing. Stylianides, G. J. (2009). Reasoning-and-Proving in School Mathematics Textbooks. Mathematical Thinking and Learning, 11(4), 258–280. TERC. (2008). Investigations in Number, Data, and Space (2nd edition). Glenview, IL: Pearson Education Inc. TIMS Project (2008). Math Trailblazers (3rd Edition). Dubuque, IA: Kendall/Hunt Publishing Company. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill. U.S. Department of Education. (2000, December). Pursuing excellence: Comparisons of international eighth-grade mathematics and science achievement from a U.S. perspective, 1995 and 1999 (PDF). Jessup, MD: U.S. Department of Education. Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht: Kluwer. van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions. Journal of Technology and Teacher Education, 10, 571–596. van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education, 24, 244–276. Vincent, J., & Stacey, K. (2008). Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS video study criteria to Australian eighth-grade mathematics textbooks. Mathematics Education Research Journal, 20(1), 82–107. Wittmann, E.  C. (1995). Mathematics education as a design science. Educational Studies in Mathematics, 29, 355–374.

Part I

Designing Opportunities for Student Learning

Chapter 2

Examining the Mathematical Emphasis in Five Curriculum Programs Ok-Kyeong Kim and Janine T. Remillard

Abstract  In this chapter, we examine mathematical treatment and emphasis in the five curriculum programs—what aspects of mathematics are emphasized and the way in which the mathematics is represented and organized for student learning. In doing so, we attend to mathematics content presented for daily instruction, in particular, the scope and sequence of number and operations. In addition, the cognitive demand of instructional activities, the nature of ongoing practice, and the visual and physical representations used are examined in order to discern the mathematical emphasis in the programs. The results show that there are substantial variations in the scope and sequence of number and operations, cognitive demand of tasks and problems, ongoing practice, and representations used across the five programs. Such variations reveal different mathematical treatments and emphasis in the programs. We summarize important findings of the study and discuss implications for teachers and teaching for each of the findings. Keywords  Curriculum analysis · Mathematics curriculum materials · Elementary mathematics · Mathematical emphasis · Cognitive demand · Sequencing · Ongoing practice · Representations · Whole numbers · Fractions and decimals · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman–Addison Wesley Mathematics

O.-K. Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA J. T. Remillard (*) Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_2

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2.1  Introduction We begin our analysis of the five curriculum programs by examining and comparing their mathematical emphasis. Researchers have examined mathematics curriculum materials (textbooks) in terms of the nature of mathematics presented in them. Although using some different terms, such as mathematical intentions (Pepin & Haggarty, 2001), mathematical practices (Pepin, Gueudet, & Trouche, 2013), and mathematical emphasis (Remillard, Harris, & Agodini, 2014), they analyzed curriculum materials from different countries by attending to what kind of mathematics was valued for teaching and learning. With a similar focus, in this chapter we examine mathematical treatment and emphasis in the five curriculum programs1—what aspects of mathematics are emphasized and the way in which the mathematics is represented and organized for student learning. In doing so, we attend to mathematics content presented for daily instruction and the ways in which it is laid out for teachers and students. In particular, the scope and sequence of the programs are examined in the content strand of number and operations. In addition, the cognitive demand for instructional activities, the nature of ongoing practice, and the representations used are examined in order to discern ways in which the mathematics content is presented in the programs. Specific questions for this analysis are as follows: 1. Scope and sequence: What mathematics content is presented in the five curriculum programs, especially in the strand of number and operations? How are they organized within and across grades? What similarities and differences are present among the five programs? 2. Cognitive demand, ongoing practice, and representations: How is the mathematics content (number and operations) presented in the five curriculum programs, in terms of cognitive demand of tasks, the nature of ongoing practice, and representations used? What similarities and differences occur across the five programs?

2.2  Theoretical Foundations In order to examine the mathematical emphasis in the five curriculum programs, we focused on the scope and sequence of the mathematics content, cognitive demand, ongoing practice, and representations. The rationale for attending to the nature of ongoing practice, cognitive demand, and representations is closely related to the nature of mathematics knowledge.

 The five programs are Everyday Mathematics (EM), Investigations in Number, Data, and Space (INV), Math in Focus (MIF), Math Trailblazers (MTB), and Scott Foresman–Addison Wesley Mathematics (SFAW). See Chap. 1 for more details about the programs. 1

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2.2.1  The Nature of Mathematics Knowledge What does it mean to know, do, or learn mathematics? This is an important question for mathematics teaching and learning. It is also an important question for the development of curriculum resources that are used for daily mathematics teaching and learning. Overall, the field, education in general, and mathematics education in particular, has accepted the position that knowledge is multifaceted and thus learning involves multiple aspects. For example, Hiebert (1986) emphasized the relationship between conceptual and procedural knowledge. Students need to develop a conceptual understanding and procedural competence of the mathematics content they learn. Moreover, Sfard (1998), using metaphors of acquisition and participation, highlighted the importance of learning both propositional knowledge (e.g., concepts, skills, and facts) and processes (actions). Given the multifaceted nature of mathematics, engaging in mathematical processes became an essential part of learning mathematics. For example, the National Council of Teachers of Mathematics (NCTM) (2000) put forth five process standards: problem solving, reasoning and proof, communication, connections, and representation. The National Research Council (2001) defined mathematical proficiency as a reasonable demonstration of the following aspects: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Similarly, the Common Core State Standards of the United States (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010) elaborated on eight key mathematical practices: (1) Make sense of problems and persevere in solving them, (2) Reason abstractly and quantitatively, (3) Construct viable arguments and critique the reasoning of others, (4) Model with mathematics, (5) Use appropriate tools strategically, (6) Attend to precision, (7) Look for and make use of structure, and (8) Look for and express regularity in repeated reasoning. As such, in our analysis of mathematical emphasis in the curriculum programs, we not only attended to mathematics content but also examined the kinds of the tasks and problems posed in the materials in order to see how the mathematics content was presented to students in these problems and tasks, and whether the tasks and problems required students to engage in mathematical processes, or just produce desired answers.

2.2.2  Cognitive Demand Tasks serve as a context in which students explore the content, and they indicate what students are required to do (Doyle, 1988; Hiebert & Wearne, 1993). Doyle claimed, “Tasks influence learners by directing their attention to particular aspects of content and by specifying ways of processing information … the processing of information for meaning versus the processing of information for surface features” (Doyle, 1983, p. 161). Doyle and Carter (1984) examined levels of academic tasks and ways in which one English teacher and her students implemented those tasks.

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In doing so, they distinguished low and high levels of tasks based on cognitive requirement of the student, i.e., whether the tasks can be done by simply reproducing information and applying algorithms, procedures, or routines, and whether they require students to “construct products by assembling information and making executive level decisions about its use” (Doyle & Carter, 1984, p. 144). The high-­ level tasks described here contain many characteristics similar to mathematical processes mentioned above (Doyle, 1988). Hiebert and Wearne (1993) argued that tasks requiring different cognitive processes (e.g., comprehension, strategy development, and procedural skills) induce different kinds of learning, and therefore understanding the tasks given to students is essential for understanding instruction. Stein and her colleagues developed the Mathematical Tasks Framework (e.g., Smith & Stein, 1998; Stein, Grover, & Henningsen, 1996) and the task analysis guide elaborating characteristics of mathematical tasks at each of the four levels of cognitive demand (Stein, Smith, Henningsen, & Silver, 2000). The four levels of tasks include two low levels (i.e., memorization and procedures without connection) and two high levels (i.e., procedures with connections and doing mathematics). Stein and her colleagues also analyzed mathematical tasks in different phases (i.e., (1) tasks that appeared in instructional materials, (2) tasks set up by teachers, and (3) tasks implemented by students) in the Mathematical Tasks Framework and ways in which mathematical tasks were transformed during instruction (e.g., Stein & Kim, 2009; Stein & Smith, 1998). Researchers recognize the Mathematical Tasks Framework as one of the important frameworks for mathematics teaching (e.g., Sztajn, Confrey, Wilson, & Edgington, 2012), and this framework has been widely used in the analysis of mathematics curriculum and instruction (e.g., Son & Kim, 2015). In our analysis of mathematical emphasis in the five curriculum programs, we examine tasks as they appear in the lessons randomly selected, which are in the first phase of the Mathematical Tasks Framework (Stein et al., 2000). Stein et al. (2000) note that low-level tasks can have some features of high-level tasks, such as requiring an explanation, having a real-world context, and involving multiple steps. Although low-level tasks may appear to be high-level tasks, these tasks focus on producing the answer without requiring a connection to meaning or concepts in the end. This is an important point one needs to keep in mind when analyzing tasks presented in the curriculum resources. In our analysis, we included one additional category of cognitive demand labeled procedures with superficial connections because we noticed some tasks exhibit an attempt to support students’ understanding of the procedures and related concepts along with illustrations and representations, and yet the connections attempted were superficial, not leading to the deep understanding expected.

2.2.3  Representation As mentioned above, the National Council of Teachers of Mathematics (NCTM, 2000) proposed representation as one of the five process standards for school mathematics. Representations take various forms in mathematics, such as manipulatives,

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pictures, diagrams, tables, graphs, words, and equations. In particular, visual and physical representations can concretize mathematical ideas and concepts by using images and manipulations. Elementary mathematics textbooks include many of these representations to illustrate the mathematical concepts, ideas, and procedures in their lessons in a way that students can make sense of them, reason about them, and solve problems using them. Representations are viewed as tools for problem solving, not just for the illustration of the concepts. When interviewing mathematicians with nonstandard calculus problems and conducting a teaching experiment with middle school students, Stylianou (2011) identified important roles of representation in problem-solving activities as a means to understand information, a recording tool, a tool that facilitates exploration, and a monitoring and evaluating device. In discussing tasks, Doyle (1983) claims, “The resources available to students also affect the nature of academic tasks” (p. 161). He also includes resources as one important component of a task (Doyle, 1988). We posit that representations are resources that students use to make sense of and complete tasks. Lesh and Jawojewski (2007) argue, “Representations and the tools to produce them are among the most important artifacts that students project into and encounter in the world” (p. 791). They also emphasize the importance of representational fluency and insist that students not just use representations by teachers and textbooks, but also develop and use their own representation system. According to Lesh and Jawojewski, translations between representations, i.e., the relationships among the representations rather than isolated idiosyncratic tools, are important in students’ understanding of mathematics and flexible thinking. To summarize, representations play an important and versatile role in teaching and learning mathematics. They are tools for problem solving, modeling, reasoning, explanation, generalization, justification, application, and many more processes. Students need to use representations flexibly in various aspects of learning mathematics. In our analysis of the mathematics presented in the five elementary curriculum programs, we attended to kinds of visual and physical representations, connections among the representations used, and their purpose (i.e., students’ exploration versus teacher demonstration or illustration).

2.2.4  Learning Pathways Learning trajectory (e.g., Clements & Sarama, 2004; Simon, 1995), learning pathway (e.g., Remillard & Kim, 2017), learning progression (e.g., Corcoran, Mosher, & Rogat, 2009), and mathematical storyline (e.g., Sleep, 2009, 2012) are a few different expressions that indicate a sequence of student learning that outlines the development of related concepts in a structured order. Clements and Sarama (2004) use the term learning trajectory to consider both psychological developmental progressions and instructional sequences. They conceptualize learning trajectory as:

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Elaborating on Simon’s (1995) notion of hypothetical learning trajectory, Clements and Sarama (2004) argue that it includes three aspects: “the learning goal, developmental progressions of thinking and learning, and sequence of instructional tasks” (p. 84). They highlight the importance of the trajectory in learning, teaching, and curriculum. Similarly, Confrey, Maloney, and Corley (2014) see learning trajectories as boundary objects connecting teaching and learning, curriculum, assessment, and teacher education. Developing and implementing a curriculum, one must carefully consider the sequence of instructional tasks (Confrey et al., 2014; Gelman & Brenneman, 2004; Sztajn et al., 2012). There is a particular sequence of student learning in a single activity, task, or lesson (micro level), and also the sequence involves multiple activities, tasks, and lessons within and across topics (macro level). Curriculum developers consider both the lesson level and unit/grade level, and also across grades when they determine the content for each lesson, unit, and grade (see Chap. 4 of this volume). A sequence of tasks, activities, lessons, and units within and across grades is carefully laid out for students’ mathematical growth. Curriculum frameworks, such as Principles and Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM 2000) and Common Core State Standards Mathematics (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010) in the USA, provide detailed guidance on specific mathematics content within and across grades. In our analysis, we examined not only the scope of the content but also ways in which a specific content area (e.g., whole number operations) is sequentially organized within and across grades 3–5. This will reveal individual curriculum programs’ anticipated learning pathways along with the content expected for students’ learning. The different choices curriculum authors made regarding learning pathways will be explored in Chap. 4, through the comments made by authors of three of the programs.

2.3  Methods 2.3.1  Cognitive Demand and Representation For the analysis of cognitive demand of tasks and representations in the five curriculum programs, we randomly selected 30 lessons in grades 3–5 (10 lessons each grade) from each curriculum program, a total of 150 lessons,2 in the strands of

 These lessons are also used for the analysis of pedagogical approaches in Chap. 3.

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number and operations, and algebra. Our unit of analysis was a lesson segment, which indicates a chunk of a lesson for a particular activity or task, which was usually specified by the headings of the lesson. Mostly, each of the lessons selected was divided into two segments, but occasionally they were divided into one or three. Focusing on main lesson segments from each lesson, we did not include daily routines, ongoing practice problems, or optional activities. In each lesson segment, the level of cognitive demand of the main task was determined using the task analysis guide by Stein and her colleagues (e.g., Stein et al., 2000). As indicated previously, in addition to the four levels of cognitive demand by Stein and her colleagues, we included one level (procedure with superficial connections) in order to acknowledge the curriculum developers’ attempt to make the procedure meaningful while ultimately failing to promote a desired level of cognitive demand. In addition, we elaborated on the five levels of cognitive demand in terms of ultimate goals that the tasks appeared to accomplish (see Table  2.1). We also incorporated ideas from the Mathematics Tasks Framework interpreted by Otten and Soria (2014), Smith and Stein (1998), and Boston and Candela (2018) in our analytic framework. As we tested out the analytic framework with a few lessons from each program, we refined the framework. Memorization tasks involve recalling facts, rules, and definitions without meaning. Procedure without connections tasks have no meaning or connection pursued, and no intention for connections with underlying concept. The ultimate goal of these tasks is to master procedures. Procedure with superficial connections tasks include some connections at the beginning with representations and illustrations, but these are touched briefly and the procedure is not fully developed in relation to the underlying concept. Often, the connections are not consistently pursued. In such a case, the connections attempted revealed only limited opportunities for students’ work on procedures, and the ultimate goal of these tasks is to master the procedures having connections as possible scaffolding. Procedure with extended connections tasks requires students to understand and use the procedure along with underlying mathematical meaning. Doing math tasks make students develop and use strategies for solving unstructured problems in order to learn and use mathematical concepts, which often require students to explain, justify, and critique their and others’ reasoning and strategies. Attending to the characteristics and ultimate goal of the task, we coded its level of cognitive demand in each lesson segment. In some rare ­incidences, two levels of cognitive demand were identified when the lesson segment included characteristics of both kinds of tasks. Also, in each lesson segment, visual and physical representations used were identified and recorded in order of appearance (i.e., representation 1 and representation 2). When a lesson segment included more than one visual or physical ­representation, it was further examined to see whether in the segment students are engaged in making a connection between the representations in terms of the mathematical idea represented. When a representation included in a lesson segment was only for an illustration of a concept/procedure, or teachers’ use, and not meant for students’ exploration, it was also indicated. Once the documentation of representations from the lesson segments was completed, common and unique representa-

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Table 2.1  An analytic framework for cognitive demand of tasks Level Memorization

Ultimate goals and characteristics To recall facts, rules, and definitions without meaning • Involves either producing previously learned facts, rules, formulae, or definitions OR committing them to memory. Procedures without To master procedures connections • Tasks are algorithmic. Use of procedures is either specifically called for or their use is evident based on instruction. • The primary purpose of the task is to learn a procedure. • Procedures do not need to be computational. • Require limited cognitive demand for successful completion. There is little ambiguity about what needs to be done and how to do it. • Have no [or limited] connection to the concepts or meaning underlying the procedure used. To master the procedures using connections as possible scaffolding Procedures with • Tasks make initial connections to underlying meaning, but this superficial connection is not maintained or limited and the primary focus becomes on connections the procedure. • Seemingly doing math or procedure with connections tasks in which structured steps direct attention away from meaning entirely. • Tasks make underlying meaning visible. Students’ use of these connections is superficial, involving filling in blanks or using a predetermined outline or structure; Presentation of task allows students to follow a pattern or use a shortcut to solve the problem, often without understanding why. • Students can complete the tasks without fully engaging the mathematical meanings. To understand and do the procedure in relation to underlying mathematical Procedures with meaning extended • Focus students’ attention on the use of procedures for the purpose of connections developing deeper levels of understanding of mathematical concepts and ideas. • Doing math tasks made more procedural through structured steps. • Tasks make underlying meaning visible. Students are expected to use the meanings to solve problems and, in particular, make connections to mathematical procedures or notations. • Students must engage the mathematical meanings to complete the task. • Can include memorization of facts, as long as it is with meaning. Doing mathematics To develop and use strategies (including previously learned procedures) for solving unstructured problems in order to learn and use mathematical concepts as they apply; explain and justify their reasoning, determine whether different solution strategies make sense. • Students need to impose their own structure and procedure. • Tasks go beyond doing or using operations.

tions, and approaches to using the representations were further examined to account for the nature of mathematical work in the programs. We focused on representations that were either visual or physical in our analysis. Although story contexts and equations are useful representations, these are already commonly used in mathematics programs; so, we did not attend to those kinds of representations.

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2.3.2  S  cope and Sequence of Number and Operations, and Ongoing Practice In order to analyze the content coverage and the sequence of the content in the strand of number and operations, we started with the sequence of the entire mathematics content in each curriculum program. Each program provided a large chart or extended table summarizing the content of individual chapters or units within each grade. These charts and tables were provided on a separate piece of thick paper, and/or embedded in the implementation guides by each program. Also, the tables of contents included in the teacher’s guides were examined to trace the content in order of the lesson. Once overall contents in order of appearance were identified from the sequence charts/ tables and the tables of contents, individual lessons from chapters and units containing the strand of number and operations were further examined in order to find the details of the sequence outlined in the charts and tables of contents and to make sure the content and sequences identified were accurate. In the content strand of number and operations, two specific areas were documented and traced in sequence: (1) whole number and operations, and (2) rational numbers (fractions and decimals) and operations. In each of these two areas, common content and its placement, and required computation methods and emphasis in them were compared across the programs. In addition, some unique content or unusual placements of specific content were recorded for further comparison of the five curriculum programs. Then, in each program, components of daily routines or ongoing practice were identified along with the purpose and nature of these components, examining problems in each component and explanations about them in the implementation guides or other teacher resources. Given the importance of practice problems, researchers have attended to how practice can better support students’ learning. Elaborating on the notions of substantial teaching units (i.e., resources for instruction) and substantial learning environment, Wittmann (1995, 2001) described that practice needs to be a rich mathematical activity, beyond basic skills. Designing tasks for students’ engagement and sense-making, Barzel, Leuders, Prediger, and Hußmann (2013) pursued a similar idea to develop tasks for practice which they called “productive exercises.” In a similar vein, we examined ongoing practice routines in individual lessons, characterizing and comparing practice problems/tasks in the five programs, in order to see their ultimate goals and emphasis in them. Table 2.2 summarizes the four dimensions we attended in order to examine mathematical treatment and emphasis in the elementary mathematics curriculum programs. Examining the scope and sequence across grades in the programs revealed the kind of mathematical content each program valued and the learning expectations each program had for students. Cognitive demand, ongoing practice, and visual/ physical representations revealed the nature of mathematical work expected of students, along with specific aspects of mathematical work (main tasks and practice) that created different potentials for students’ learning opportunities. In all analyses above, the codes, summaries, and records were compared in order to see similarities and differences among the programs. The findings from these

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Table 2.2  Four dimensions to examine mathematical treatment and emphasis Dimension Scope and sequence of number and operations

Description Content and its placement across grades, patterns (common and specific) in and approaches to presenting content for student learning Cognitive demand of student Types of main mathematical tasks for students, ultimate goals of tasks student learning Ongoing practice Types of practice problems, expectations Representations (visual and Types of representations, connections among representations, physical) approaches to using representations

analyses revealed specific aspects of the mathematics presented in the curriculum programs and various ways in which curriculum developers provided the mathematics content for everyday instruction.

2.4  S  cope and Sequence of Number and Operations in the Five Programs 2.4.1  Size of Whole Number The five curriculum programs vary greatly in the size of whole numbers students learn in each grade (see Table 2.3). For example, in grade 3, MTB, and SFAW cover thousands, whereas EM does millions, INV covers numbers up to 1000 and MIF numbers up to 10,000. SFAW covers numbers up to 999,999,999 in grade 4, and does not address the concept of whole numbers in grade 5. EM addresses larger numbers in each grade than the other programs. All programs require students to read and write numbers in millions by the end of grade 5, although some programs expect students to know numbers much larger than millions, such as trillions.

2.4.2  Whole Number Addition and Subtraction Table 2.4 summarizes the size of numbers used in addition and subtraction in each grade in the five programs. By the end of grade 5, students are expected to be able to add and subtract 4- and 5-digit numbers, except for MIF. Table 2.5 summarizes computational methods used for addition and subtraction across grades in each program, which reveal a diversity of computation methods across the five programs. All programs except for EM introduce the standard algorithms for addition and subtraction by the end of grade 5. In MIF, students add and subtract whole numbers up to 10,000 using the standard algorithms in grade 3, and the program does not include any lessons on addition and subtraction of whole numbers beyond grade 3.

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Table 2.3  Size of whole number in the five programs Curriculum EM INV MIF MTB SFAW

Grade 3 Millions Up to 1000 Up to 10,000 Thousands Thousands

Grade 4 Billions Up to 10,000 Up to 100,000 Millions Up to 999,999,999

Grade 5 Trillions Up to 100,000; million, billion, trillion Up to ten millions Trillions –

Table 2.4  Number of digits in numbers used in addition and subtraction in the five programs EM INV MIF MTB SFAW

Grade 3 2, 3, 4 2, 3 3, 4 2, 3, 4 2, 3

Grade 4 3, 4, 5 3, 4

Grade 5 3, 4, 5 3, 4, 5

4, 5 4, 5

4, 5

3 indicates 3-digit numbers

SFAW does not have any lessons focusing on addition and subtraction of whole numbers in grade 5. Students using the program are expected to add and subtract 4-digit numbers using the standard algorithms by the end of grade 4 (Addition and subtraction with 5-digit numbers can be done in different ways, such as using a calculator). In EM, INV, and MTB, adding and subtracting whole numbers are concentrated in grades 3 and 4, and revisited for review and practice in grade 5. In these programs, addition and subtraction of whole numbers move gradually from 2-digit numbers to a larger number of digits, encouraging students to use various methods to add and subtract whole numbers. INV and MTB introduce the standard algorithms for addition and subtraction as students develop ways to compose and decompose numbers in place value (e.g., tens and ones) in adding and subtracting numbers. Although the standard algorithms are introduced as one of the ways to add and subtract in INV, students are still encouraged to use their own strategies for addition and subtraction with large numbers (4- and 5-digit numbers). MTB and EM extensively use base-ten blocks and shorthand of base-ten blocks for addition and subtraction with whole numbers (see Fig. 2.1 for an example from MTB). EM uses alternative algorithms for addition and subtraction extensively (the partial-­ sums method and the column-addition method for addition, and the trade-first method and the partial-differences method for subtraction).3 Although EM reveals  The partial-sums method for addition is adding numbers in places (e.g., tens and ones) first and then adding the partial sums to find the answer; the column-addition method for addition is adding the digits in each place first and then adjusting digits in each place to find the answer. The tradefirst method for subtraction is similar to the standard algorithm for subtraction, but all the trading needed (“borrowing”) is completed and then subtracting each place to find the answer; the partialdifferences method for subtraction is finding differences for each place and then adding them to find the answer. 3

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Table 2.5  Addition and subtraction methods in the five programs

EM INV MIF MTB SFAW

Grade 3 Addition Alt Stst US Stst/US US

Subtraction Alt Stst US Stst/US US

Grade 4 Addition Alt Stst/US

Subtraction Alt Stst

US Mix

US Mix

Grade 5 Addition Alt Stst

Subtraction Alt Stst/US

US

US

US stands for the US standard algorithms, Alt for alternative algorithms, Stst for student strategies, Mix for various methods including the standard algorithms and using a calculator

Fig. 2.1  Base-ten Blocks for Addition in MTB (GR3 6.3, p. 45). Excerpt from Math Trailblazers, Grade 3, Unit 6 Resource Guide by TIMS.  Copyright © 2008 by Kendall Hunt Publishing Company. Reprinted by permission

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the idea of the standard algorithms through the alternative algorithms, the program neither formally introduces nor uses the standard algorithms for addition and subtraction by the end of grade 5.

2.4.3  Whole Number Multiplication Overall, multiplication is introduced in grade 3, developed in grade 4, and refined in grade 5. Only EM starts multidigit multiplication (i.e., both the multiplicand and the multiplier have the number of digits higher than one) in grade 3 and the other programs do in grade 4 (see Table 2.6). Students are expected to multiply 2-digit by 3-digit numbers by the end of grade 5, although EM, MIF, and SFAW include more challenging computations, such as multiplying 3-digit by 3-digit, and 4-digit by 2-digit numbers. The expectations and the sequence of specific numbers used across grades vary in the five programs (see Tables 2.6 and 2.7). Multiplying by powers of ten is important content to understand place value and multiplication with large numbers. This topic is placed in each grade pretty similarly in EM, MIF, and SFAW across grades, whereas it appears in grades 3 and 4 up to multiples of 10,000  in MTB and in grades 4 and 5 up to multiples of 10 and 100 in INV (see Table 2.7). Table 2.6  Multidigit multiplication in the five programs EM INV MIF MTB SFAW

Grade 3 2 × 2, 3 × 2

Grade 4 2 × 2, 3 × 2 2 × 2 2 × 2, 3 × 2 2 × 2 2 × 2, 3 × 2, 4 × 2

Grade 5 2 × 2, 3 × 2, 3 × 3 2 × 2, 3 × 2 2 × 2, 3 × 2, 4 × 2 3 × 2 3 × 2, 3 × 3

Note that 3x2 denotes multiplying a 3-digit number by a 2-digit number

Table 2.7  Multiplying by multiples of ten in the five programs EM

Grade 3 Multiples of 10, 100, and 1000

INV MIF

Multiples of 10 and 100

MTB

Multiples of 10 and 100

SFAW Multiples of 10, 100, and 1000

Grade 4 Multiples of 10, 100, and 1000 Multiples of 10 Multiples of 10, and 100 Multiples of 10, 100, 1000, and 10,000 Multiples of 10, 100, and 1000

Grade 5 Multiples of 10, 100, and 1000 Multiples of 10 and 100 Multiples of 10, 100, and 1000

Multiples of 10, 100, and 1000

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Computational methods used for multiplication are as diverse as those for addition and subtraction in the five programs (see Table 2.8). EM uses alternative algorithms for multiplication extensively (the partial-products method and the lattice method, see Figs. 2.2 and 2.3) in grades 3–5. INV encourages students to use their own strategies and introduces the standard algorithm for multiplication as one method in grade 5. MTB uses student strategies early on, and then the partial-­ products method (“all-parts algorithm”) with/without base-ten blocks. The program also introduces the lattice method for multiplication as an option as well as the standard algorithm (“the compact method”). Both MIF and SAFW use the standard algorithm for multiplication from early on.

Fig. 2.2  Partial-products method for 869 × 6 = 5214 in EM. Excerpt from Everyday Mathematics, Grade 4, Teacher’s Lesson Guide, Volume 1, p. 339. (University of Chicago School Mathematics Project). Copyright 2007 by McGraw Hill. All rights reserved. Reprinted by permission of the publisher Table 2.8  Multiplication and division methods in the five programs

EM INV MIF MTB SFAW

Grade 3 Multiplication Stst/Alt

Division Stst

Grade 4 Multiplication Alt Stst US Stst/Alt/US US

Division Alt Stst US Alt US

Grade 5 Multiplication Alt Stst/US US Alt /US US

Division Alt Stst US Alt US

US stands for the US standard algorithms, Alt for alternative algorithms, Stst for student strategies Fig. 2.3  Lattice method for 473 × 16 = 7568 in EM. Excerpt from Everyday Mathematics, Grade 4, Teacher’s Lesson Guide, Volume 1, p. 351. (University of Chicago School Mathematics Project). Copyright 2007 by McGraw Hill. All rights reserved. Reprinted by permission of the publisher

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2.4.4  Whole Number Division As with whole number multiplication, whole number division is introduced in grade 3, developed in grade 4, and advanced with larger numbers in grade 5. In grade 3, students work on the division in relation to multiplication facts (up to 12 × 12), and division as computation begins in grade 4 mostly. Students are expected to divide 4-digit numbers by 2-digit numbers by the end of grade 5 in all five programs (see Table 2.9). However, the pathways to get to this target goal and the emphasis on different computation methods vary greatly across programs, as shown in Table 2.9. INV covers division by using multiplication combinations up to 12 × 12 in grade 3. In grades 4 and 5, INV consistently asks students to solve division problems using strategies, often in relation to multiplication (see Fig.  2.4, for example). Although INV introduces the standard algorithms for addition, subtraction, and multiplication, the program does not cover the standard algorithm for division. EM asks students to divide 2-digit by 1-digit numbers with and without a remainder in story contexts and divide multiples of 10, 100, 1000 by 1-digit numbers using equal groups in grade 3. Then, students are asked to use the partial quotients algorithm for whole number division (see Fig. 2.5) in grades 4 and 5. As with the other operations, the program does not introduce the standard algorithms for division. Table 2.9  Whole number division in the five programs EM

INV

MIF

Grade 3 Divide 2-digit by 1-digit divisors with/without a remainder; divide multiples of 10, 100, 1000 by 1-digit divisors Divide by using multiplication combinations Divide 2-digit numbers by 1-digit divisors

Grade 4 Divide 3-digit numbers by 1- or 2-digit divisors

Grade 5 Divide 3- or 4-digit numbers by 1- or 2-digit divisors

Divide 2-digit or 3-digit numbers by 1-digit or 2-digit divisors by using strategies Divide by a 1-digit divisor with a remainder

Divide 3-digit and 4-digit numbers by 2-digit divisors by using strategies

Divide by tens, hundreds, or thousands, 2-digit, 3-digit, and 4-digit numbers by 2-digit divisors Divide by 1-digit divisor MTB Divide 2-digit numbers (no Divide 2- and 3-digit numbers by 1-digit divisors (review and practice); divide paper-pencil algorithm); (using the partial-quotients 4-digit by 1-digit divisors; interpreting remainders divide whole numbers by method) 2-digit divisors (the partial-quotients method) Divide 3- or 4-digit dividends SFAW Divide with 2, 5, 3, 4, 6, 7, Divide multiples of 10, by 1-digit divisor, then 8, 9, 1, 0, 10, 11, and 12 100, 1000 by 1-digit 2-digit divisors divisors, 2- and 3-digit by 1-digit divisors; divide with 2-digit divisors with the standard algorithm

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Fig. 2.4  An example of student strategies for division in INV. Excerpt from INV GR5 7.3.1, p. 72 illustrating a sample student strategy. From Investigations 2008 Curriculum Unit (Grade 5, Unit 7) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

Fig. 2.5  Partial quotient method for division in EM (EM GR4 6.10) Excerpt from Everyday Mathematics, Grade 4, Teacher’s Lesson Guide, Volume 1, p. 458. (University of Chicago School Mathematics Project.) Copyright 2007 by McGraw Hill. All rights reserved. Reprinted by permission of the publisher

MIF covers division of 2-digit by 1-digit numbers in grade 3, and division by 1-digit numbers with a remainder in grade 4. Then, in grade 5, the program requires students to do division with larger numbers. MTB covers dividing 2-digit numbers with and without a remainder in grade 3. Then, students are expected to use the partial-quotient algorithm (“forgiving method”) in grades 4 and 5. Although some solutions using the partial-quotients algorithm look similar to the standard algorithm for division “if the best estimate is made at each step” (TIMS Project, 2008, p. 10), notations clearly indicate partial

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Fig. 2.6  Partial quotients method for division similar to the standard algorithm in MTB (GR4 13.3, p. 75). Excerpt from Math Trailblazers, Grade 4, Unit 15 Resource Guide by TIMS. Copyright ©2008 by Kendall Hunt Publishing Company. Reprinted by permission

quotients with their actual values as these are written right by the partial dividends (see Fig. 2.6). SFAW focuses on division with divisors of 0 to 12 in grade 3. Division as computation begins in grade 4, in which students do division by using the standard algorithm. In grade 5, students do division with larger numbers. SFAW concentrates on multiplication and division in grade 5, in which there are no lessons on addition and subtraction with whole numbers.

2.4.5  Whole Number Operations in the Five Programs There are some significant differences and distinct features in the scope and sequence of whole number and operations in the five curriculum programs. EM Uses Alternative Algorithms Extensively EM uses alternative algorithms extensively and does not introduce the standard algorithms in grades 3–5, as shown previously. In fact, some of the alternative algorithms (e.g., the column-addition algorithm for addition, the trade-first algorithm for subtraction, the partial-products method for multiplication, and the partial- quotients method for division) can help students make sense of the standard algorithms for the four operations. But, the program focuses on using alternative algorithms. MTB also uses alternative algorithms but introduces the standard algorithms, and the alternative algorithms eventually support students’ grasp of the standard algorithms in the program. I NV Places a Great Emphasis on Student Strategies and Mathematical Relationships in Whole Number Computation INV’s emphasis in whole number operations is on the meaning and relationships of the operations, and representing, describing, and analyzing various strategies of each operation, which encourages students’ own strategies and thinking. In INV, the

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standard algorithms are introduced for addition (grade 4), subtraction (grade 5), and multiplication (grade 5). Figure 2.7 illustrates how the standard algorithm for subtraction is introduced in INV. Although the standard algorithms are introduced, students are still encouraged to use various strategies to solve addition, subtraction, multiplication, and division problems, as shown in Figs. 2.4 and 2.8. In addition, INV encourages students to use the inverse relationship, especially between multiplication and division, and lessons on multiplication are combined with lessons on division. INV also uses conceptual approaches to computations, for example, equivalence in multiplication (e.g., 2 × 9 = 6 × 3, Fig. 2.9) and equivalence in division (e.g., 120÷20 = 60÷10, Fig. 2.10). In fact, the multiplication example illustrates the associative property of multiplication, i.e., 2 × 9 = 2 × (3 × 3) = (2 × 3) × 3 = 6 × 3. According to the curriculum, one number is tripled, the other is multiplied by a third, and their product is still the same as the product of the original numbers, i.e., 2 × 9 = 2 × (3 × 1/3) × 9 = (2 × 3) × (1/3 × 9) = 6 × 3. The division example in Fig. 2.10 illustrates that if both the dividend and the divisor are halved, then the answer is still same as the original division problem, i.e., 120÷20  =  (120÷2)÷(20÷2)  =  60÷10 (the same ratio). The examples indicate that INV uses whole number operations as a context to explore mathematical relationships, beyond computation, although students are not expected to discuss these ideas in a formal way.

674

600 + 70 + 4

600 + 60 + 14

674

– 328

– (300 + 20 + 8)

– (300 + 20 + 8)

– 328

300 + 40 + 6 = 346 346 Fig. 2.7  Subtraction standard algorithm introduced in INV. Excerpt from INV GR5 3.2.4, p. 77 illustrating the subtraction U.S. standard algorithm. From Investigations 2008 Curriculum Unit (Grade 5, Unit 3) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

35 × 28 35 × 30 = 1,050 I multiplied 35 × 10 and added 350 three times. 35 × 2 = 70 It’s not 30, it’s only 28, so I had to find two 35s. 1,050 – 70 = 980 Fig. 2.8  An example of student strategies for multiplication in INV (GR 5 1.2.1). Excerpt from INV GR5 1.2.1, p.  73 illustrating a sample multiplication strategy. From Investigations 2008 Curriculum Unit (Grade 5, Unit 1) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

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Fig. 2.9  Equivalent multiplication problems (6 × 9 = 3 × 18) shown in two different representations (INV GR5 7.1.2). Excerpt from INV GR5 7.1.2, pp. 35–36 illustrating equivalent multiplication problems. From Investigations 2008 Curriculum Unit (Grade 5, Unit 7) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

Fig. 2.10 Equivalent division problems (120÷12 = 60÷6) shown in arrays (INV GR5 7.1.4) excerpt from INV GR5 7.1.4, p. 42 illustrating equivalent division problems. From Investigations 2008 Curriculum Unit (Grade 5, Unit 7) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

 IF Packs Operations in a Few Lessons, Focusing on Using Procedures M and Properties MIF lessons on operations are very packed with small chunks from introduction to completion, focusing on using procedures and properties (e.g., associative property of multiplication). For example, MIF includes only two consecutive lessons for five days of teaching of whole number multiplication in grade 4, in which students are expected to multiply multidigit numbers by 1-digit numbers, then 2-digit or 3-digit numbers by tens and hundreds, then 2-digit by 2-digit numbers, and then finally 3-digit by 2-digit numbers. MIF also focuses on particular operations using the

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standard algorithms in particular grade(s). Multidigit addition and subtraction are in grade 3; multidigit multiplication and division are in grades 4 and 5.  M, INV, and MTB Spread Whole Number Computation throughout E the Year EM spreads whole numbers and operations in almost every chapter; even some chapters on geometry include lessons on operations explicitly. This placement is not a part of ongoing practice routine but rather operations are the main focus of a couple of lessons in the geometry chapters. INV also spends a great deal of time on operations throughout the year. For example, the program covers multiplication and division in three out of nine units in grade 4. MTB also includes extensive Daily Practice Problems (DPPs) with operations throughout the year (see more about DPPs in Sec. 5.2, Chap. 4, and Appendix D). In addition, the three programs revisit the four operations each year in grades 3–5.  FAW Progresses Lessons in Small Chunks, Focusing on Specific Aspects S of Computation in Each Lesson SFAW tends to move in small chunks from one lesson to the next deliberately to build students’ computation ability using the standard algorithms. That is, each lesson focuses on one specific aspect of computations. For example, one lesson in grade 3 centers on subtracting a number from a subtrahend with 0 in the tens place to address how to “borrow” from the 100  s place when the tens place is 0 (see Appendix E). Targeting the procedures on a specific number of digits, SFAW does not include variations in one day of teaching, and the subsequent lesson targets the procedure that is a little more complicated. Calculators Are Used as a Means to Teach Whole Number and Operations EM and MTB use calculators as a means to teach whole number and operations. For example, EM includes a place value puzzle (GR3 1.8) that requires students to explore place value by using a calculator: “Enter 35 [any two or three digit number] into your calculator. Try to change the display to 85 [a number that is 100 s larger/ smaller or 10s larger or smaller] by adding or subtracting a number” (p. 51). Students in MTB use a calculator to divide larger numbers and money, and develop strategies for whole number remainders (GR5 9.5). SFAW also allows students to use a calculator as one method to do operations, but its use is only minimal, just for calculation. The other two programs do not use calculators in the lessons of whole number operations.

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EM and MTB Place Content in Earlier Grades EM and MTB tend to introduce the content in earlier grades than common placement. For example, both EM and MTB introduce numbers up to trillions, and EM starts multidigit multiplication in grade 3. These programs also include other advanced content, such as scientific notations, and four quadrants of coordinate grids in grade 5. The MTB authors’ rationale for such content placement is that students can learn more when they are given challenging content (see Chap. 4).

2.4.6  Fraction and Decimal Concepts EM, INV, and MTB include various aspects of fraction concept to support students’ understanding of the concept, whereas MIF and SFAW have limited content on fractions (see Table 2.10). The programs encourage students to understand the meaning of fractions and find fractional parts of different wholes (e.g., a set/group, an object, a region/area) in grades 3 and/or 4. Comparing fractions using a benchmark and/or comparing fractions with a benchmark (e.g., ½) appear in grades 3 or 4 in all programs but SFAW. Mixed numbers and improper fractions are placed in grades 3 and/or 4 in all five programs. Relationships between fractions and percents are placed in grade 4 or 5 in all programs but SFAW. Fraction as division is placed in grade 4 or 5 in all programs but MIF. All programs start covering equivalent fractions in grade 3. The relationships between fractions and decimals are placed in all grades 3–5 in INV and MTB, and in grades 4–5 in EM and MIF. SFAW does not have lessons on fractions and decimal equivalents. Decimals are introduced up to thousandths by the end of grade 5 in all programs, and yet the placement of tenths and hundredths varies depending on the program (see Table 2.11). MIF does not have any lesson on decimals in grade 3, and EM introduces decimals to tenths, hundredths, and thousandths all in grade 3. Overall, other than place value, rounding decimals, and ordering and comparing decimals, Table 2.10  Fraction concepts placed in the five programs EM INV MIF MTB SFAW

Grade 3 Frprt, Ben, Equi, Mix Frprt, Equi, Mix, Deci Frprt, Ben, Equi Frprt, Ben, Equi, Deci, Frprt, Equi

Grade 4 Frprt Prct, Div, Equi, Deci Frprt, Ben, Mix, Deci, Mix, Deci Frprt, Ben, Equi, Deci Frprt, Equi, Mix

Grade 5 Prct, Equi, Deci Prct, Div, Deci Prct, Deci Ben, Prct, Div, Equi, Mix, Deci Div

Frprt stands for fractional parts of different wholes, Ben comparing fractions with a benchmark, Mix mixed numbers, Prct relationship between fractions and percents, Div fraction as division, Equi equivalent fractions, Deci relationship with decimals

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Table 2.11  Decimal content in the five programs EM

INV

Grade 3 Decimals to tenths, hundredths, and thousandths Decimal fractions (0.50, 0.25) as equal parts of a whole (an object, an area, a set of objects)

MIF

MTB

Decimals up to hundredths; compare and order tenths; translate between base-ten pieces, money, and decimals SFAW Decimals up to hundredths

Grade 4 Decimals to thousandths; compare and order decimals Decimals to hundredths; compare decimals

Decimals up to tenths and hundredths; compare and order decimals; round decimals to the nearest whole number or tenth Decimals up to hundredths; compare and order decimals; count by tenths and hundredths

Compare and order decimals

Grade 5 Round decimals

Decimals to thousandths; compare decimals to 0, ½, and 1; order decimals; repeating decimals Decimals up to tenths and hundredths; compare and order decimals; round decimals to the nearest whole number or tenth Decimals up to thousandths; translate between different representations of decimals; compare and order decimals; round decimals; Repeating decimals Decimals up to thousandths

the concept of decimals gets less attention than the concept of fractions in the programs and is often explored in relation to fractions and/or percents in all but SFAW. Repeating decimals are introduced in INV and MTB, as decimal equivalents to fractions, such as 1/3. INV provides lessons on decimals (0.50, 0.25) as equal parts of a whole (an object, an area, or a set of objects) in grade 3, which is unique in the five curriculum programs.

2.4.7  Operations with Fractions and Decimals MIF and SFAW start addition and subtraction with fractions in grade 3 and other programs in later grades. Multiplication and division with fractions are introduced in grade 5 in EM, MIF, and SFAW. MTB does only multiplication with fractions in grade 5. INV does not expect students to do multiplication or division with fractions (see Table 2.12). EM starts lessons on addition and subtraction with fractions (using representations such as pattern blocks and clock face) in grade 4 and introduces using common denominators to do the two operations in grade 5. MIF starts adding and subtracting like fractions in grade 3, and moves to subtracting fractions from whole numbers in grade 4, and then to adding and subtracting unlike fractions, and adding and subtracting mixed numbers in grade 5. SFAW also moves from addition and subtraction of like fractions (grade 3) to unlike fractions using common denominators informally (grade 4) and formally (grade 5).

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Table 2.12  Operations with fractions and decimals in the five programs Curriculum EM INV MIF MTB SFAW

Operations with fractions Grade 3 Grade 4 +− + +− − + +− +−

Grade 5 + − × ÷ +− + − × ÷ + − × + − × ÷

Operations with decimals Grade 3 Grade 4 + − x ÷ +− +− +−

+−

Grade 5 + − x ÷ +− ×÷ + − × + − × ÷

+ addition; − subtraction; × multiplication; ÷ division

In contrast, INV and MTB approach these operations quite differently from the other programs. Both programs do not use explicitly the common denominator method; instead, they use representations and manipulatives to find sums and differences or combine fractions to make up ½ or 1. In these two programs, addition and subtraction with fractions are explored informally by breaking a fraction into smaller fractions or combining smaller fractions to form a larger number and with reasoning and representations. INV does not have any lessons on multiplication and division with fractions. MTB has only multiplication with fractions, specifically fractions by whole numbers and fractions by a fraction by using informal strategies with a diagram, pattern blocks, and paper folding. The three other programs include some varied lessons on multiplication and division with fractions in grade 5. EM has lessons on multiplying fractions (and mixed numbers) by whole numbers and dividing fractions using a common denominator method (e.g., 4 ÷ 4/5  =  20/5 ÷ 4/5  =  20 ÷ 4  =  5; 3 1/3 ÷ 5/6 = 10/3 ÷ 5/6 = 20/6 ÷ 5/6 = 20 ÷ 5 = 4). MIF covers multiplying proper and improper fractions with fractions, and mixed numbers by whole numbers, and dividing fractions by whole numbers. SFAW addresses multiplication with fractions (and mixed numbers) and division of whole numbers by a fraction. Adding and subtracting decimals first appear most often in grade 4, but are first introduced in grade 5 in MTB and in grade 3 in SFAW (see Table 2.12). Both INV (grades 4–5) and MTB (grade 5) ask students to use a 10 × 10 grid to add and subtract decimals. The placement of multiplication and division with decimals varies greatly in the five programs. EM requires students to multiply and divide decimals by whole numbers by using the partial-products and partial-quotients methods in grade 4, and then to do the two operations with decimals in grade 5. INV does not have lessons on multiplication and division with decimals. MIF and SFAW introduce multiplication and division with decimals in grade 5. As with addition and subtraction with decimals, MTB asks students to use a 10 × 10 grid first and then a paper-and-pencil method to multiply decimals in grade 5; no lessons are included on division with decimals in MTB. Examining individual lessons on fractions and decimals reveals that MTB, INV, and EM use informal strategies, and visual and physical representations (e.g., pattern blocks, clock face, 10 × 10 grids, paper folding) to varying degrees. MIF and SFAW also use visual models (diagrams), but they are used to illustrate paper-and-­ pencil procedures mostly.

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2.5  The Nature of Mathematical Work The nature of mathematical work in the five programs was examined in terms of cognitive demand of lesson activities and the characteristics of ongoing practice.

2.5.1  Cognitive Demand of Lesson Activities The main activities and tasks of the lesson segments from 30 lessons of each curriculum program in the content strand of number and operations, and algebra were coded in terms of the level of cognitive demand by using the analytic framework of cognitive demand of lesson activities (see Table 2.13). Activities and tasks categorized as memory, procedures without connections, and procedures with superficial connections are in low levels, and those categorized as procedures with connections and doing mathematics are in high levels. EM, INV, and MTB have 80–100% of high-level instructional activities. INV includes no low-level instructional activities. EM includes instructional activities at the low level, such as the rules for the order of operations, 50-facts tests, and the lattice method for multiplication. For procedural fluency and practice, EM and INV use games extensively. Games in EM range from memory to doing math. INV usually includes instructional segments called workshops with 2–4 different activities at the same time so that students can choose one that they like to do for practice and do another if time permits, which are mostly procedure with extended connections. In contrast, SFAW includes the lowest percent of high-level activities and tasks (8.3%). In fact, SFAW does not have any activities in the level of procedures with extended connections among the 30 lessons randomly selected. Overall, SFAW includes mostly low-level activities (91.7%) with 41.7% of procedure with superficial connections and 50% of procedures without connections. MIF includes all five levels of instructional activities, 66.6% of low level and 33.4% of high level. The largest portion of instructional activities in MIF is a procedure with superficial connections (36.1%). As indicated in the analytics framework, Table 2.13  Cognitive demand of main instructional activities in the five programs (%) Levels of cognitive demand Procedures without Memory connections EM 3.4 5.1 INV – – MIF 4.6 25.9 MTB – 3.3 SFAW – 50.0

Procedures with superficial connections – – 36.1 1.7 41.7

Procedures with extended connections 74.6 40.3 27.8 71.7 –

Doing mathematics 16.0 59.7 5.6 23.3 8.3

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activities in this category target mastery of procedures with connections for scaffolding but without the ultimate goal of understanding and meaning. MIF often starts with meaningful contexts but soon drops the attention to procedures themselves. For example, the lesson on rounding numbers in grade 3 uses a number line but quickly moves to the rule, “If it is 1, 2, 3 or 4, then the number is rounded down to the hundred that is less. If the digit in the tens place is 5, 6, 7, 8 or 9, then the number is rounded up to the hundred that is greater” (GR3 2.4, p.  54). Another example is in a lesson on division: “Remind the students that the product and the dividend will be always greater than the divisor” (GR3 6.7, p. 179). One third grade lesson uses number bonds to show the part-whole relationship of a number (e.g., 100 has bonds of 98 and 2), which is helpful for students to make sense of the procedures, such as 98 + 95 = 100 + 95–2 = 193, but the focus is given to “commit to memory” eventually. Tasks in the category of procedure with extended connections in MIF have usually a specific structure to follow even in practice problems; there is no other way to answer and often with many blanks to fill in (see Fig. 2.11). Strategies are always Fig. 2.11  Illustration of a Procedure with Extended Connections task in MIF. This illustration was created by the authors based on the image that appears in MIF GR3 teacher’s guide (MIF GR3 6.2, p.155)

6×6=? Start with 5 groups of 6 1 2 3 4 5 6 1

1 2 3 4 5 6 1

2

2

3 4

3 4

5

5 6

5 × 6 = 30

6 × 6 = 5 groups of 6 + 1 group of 6 = 30 + 6 = 36

8×6=? Start with 10 groups of 6

1 2 3 4 5 6

1 2 3 4 5 6 1

1

2

2

3 4

3 4

5 6

5 6

7

7

8 9

8

10

10 × 6 = 60

8 × 6 = 10 groups of 6 – 2 groups of 6 = 60 – 12 = 48

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given upfront, and problems are often presented to make students use a particular way of finding the answer as shown in Fig. 2.11, although there are multiple ways to use arrays to figure out the products of the given multiplication problems. In examining the potential for teacher learning and learning demands on teachers, Stein and Kim (2009) analyzed cognitive demand of tasks from two curriculum programs in grades 1–5: Everyday Mathematics (EM) and Investigations in Number, Data, and Space (INV), both of which were the first editions. They randomly selected one lesson from each chapter in EM (a total of 57 lessons) and one session (lesson) from each unit in INV (a total of 44 lessons). They found 91% and 100% of the tasks analyzed were categorized as high-level tasks in EM and INV, respectively. This result is quite similar to our results, although we focused only on the content strands of number and operations, and algebra, using different editions than those Stein and Kim analyzed. An additional difference is that Stein and Kim coded the cognitive demand of each lesson  by using the original  task analysis guide, whereas we looked at each segment of lessons by using the modified framework (see Table 2.1).

2.5.2  The Nature of Ongoing Practice All of the curriculum programs, except for MIF, included various routines and ongoing practice problems other than practice of the content of the lesson. Such daily practice was routinized, usually distributed throughout the year, as the curriculum authors explained in Chap. 4. In this section, we describe the nature of daily practice included in individual lessons. The curriculum programs had daily practice incorporated into lesson guides and further assigned as homework. Table 2.14 summarizes the five curriculum programs’ daily practice. Table 2.14  Daily practice in the five programs Curriculum Component EM Math Boxes; Mental Math and Reflexes INV Ten-minute Math; Daily Practice MIF MTB

SFAW

Purpose Cumulative review or assessment of concepts and skills (Math Boxes); mental math (Mental Math and Reflexes) Mental strategies and problem solving (Ten-minute Math); ongoing review of concepts and skills (Daily Practice) None No ongoing practice other than practice of the content of the lesson Daily Practice Problems Quick review of a topic or focused practice on a specific (DPPs) skill in short items (bits), use of concepts in a new context (tasks), or extension of concepts in a challenging new situation (challenges) Problem-solving skills and strategies (Problem of the Problem of the Day; Day); test-taking skills (Spiral Review and Test Prep; Spiral Review and Test Prep; Mixed Review and Mixed Review and Test Prep) Test Prep

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EM lessons are usually composed of three parts: teaching the lesson, ongoing learning and practice, and optional activities for individualizing. The second section is devoted to both practice for the content of the lesson and the ongoing practice of concepts and skills. The component called Math Boxes is designed for distributed practice. The problems are mostly fill-in-the-blank or short-answer types. EM lessons also include quick mental math activities called Mental Math and Reflexes at the start, which provide some suggestions for the teacher to choose from (three levels of difficulty). Students are encouraged to flexibly use their knowledge of numbers and operations in this routine (e.g., 16 + 10 + 14 + 23; see other examples in Appendix B). In fact, the fifth principle of the EM program emphasizes the role of “practical routines to help build the arithmetic skills and quick responses” needed for problem solving (see Chap. 4). EM’s approach to ongoing practice supports this principle. INV also includes the section of Daily Practice at the end of each lesson plan, which usually provides 2–3 open-ended problems for ongoing review. Ten-minute Math activities, summarized in the scope and sequence chart of the Implementation Guide in each grade, are designed for ongoing skill building, practice, and review, without a worksheet, often targeting mental strategies and problem solving. In each grade, 2–4 units target a particular topic, such as closest estimate, practicing place value, and quick survey. For example, quick survey activities in three units in grade 4 require students to describe features of data and interpret a set of data. Both Daily Practice and Ten-minute Math require additional time for practice outside the 60-min lesson. (See examples in Appendix C for Ten-minute Math.) As presented in Table 2.14, MTB lessons have Daily Practice Problems (DPPs) including different levels of problems: bits, tasks, and challenges. Some of these are short-answer questions; others are open ended. (See examples in Appendix D.) DPP challenges are meant to develop student thinking and problem-solving skills, and they usually take more than 15  min to complete according to the Unit Resource Guides. Most lessons have 2–4 DPPs (one bit and one task, two bits and two tasks, etc.) and some have all three types of problems. DPPs can be used in class for practice and review or for homework. Like INV, MTB includes DPPs in the scope and sequence chart to keep track throughout the year. In Chap. 4, the MTB authors explain that the DPPs are designed to support both the learning of new content as well as distributed practice. SFAW lessons have three kinds of ongoing review and practice (see Table 2.14). Problem of the Day provides one problem with context but usually producing a simple answer. Spiral Review and Test Prep includes a set of problems (about half of them multiple-choice items, and the other half divided into fill-in-the-blank and short-answer types). Problem of the Day and Spiral Review and Test Prep are outside of individual lessons, provided as suggestions for spiral review. Mixed Review and Test Prep, a part of practice in an individual lesson, provides a small set of problems, mostly fill-in-the-blank and multiple-choice types requiring quick responses. Whereas Problem of the day is to reinforce previous mathematics content and enhance problem-solving skills and strategies, the other two are for test-taking skills. (See examples in Appendix E.)

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The programs provide resources for students to continue to improve their skills and fluency in computation on a daily basis. Putnam (2003) described the importance and role of computational skills in students’ learning of mathematics and considered computational skills as one element to assess in order to evaluate the effectiveness of the curriculum. By including ongoing practice throughout, the four programs certainly acknowledge this viewpoint. However, ongoing practice in SFAW remains in the skill level mostly and focuses on test-taking skills. EM also focuses on the mastery of arithmetic skills along with conceptual foundation. The other two programs have a more diverse approach to ongoing practice, moving beyond computation skills and targeting problem-solving strategies and more.

2.6  Visual and Physical Representations Representations play a versatile role in mathematics teaching and learning. Representations support students’ understanding of concepts presented; representations are tools to communicate mathematical ideas; representations are tools to solve problems. In this chapter, we examined visual and physical representations used to support student learning of mathematics in each lesson segment from 30 lessons randomly selected in each curriculum program. Table  2.15 indicates percentages of lesson segments with no, one, and multiple visual/physical representations. The last column of the table shows percentages of representations with and without connections among the multiple representations used in the lesson segments. Overall, about 30 to 47% of the instructional segments analyzed do not use any visual/physical representations, and about 34 to 47% use one representation. This indicates that the instructional activities analyzed mostly use no or one visual/physical representation. INV uses multiple visual/physical representations the most (about 36%) among the programs, 87% of which are with explicit connections among the representations. INV sometimes uses three representations along with connections among them. For example, one of the lesson segments focusing on place value in grade 3 uses (a) strips of 10 and single stickers, (b) a 100 chart, and (c) cubes in towers of 10, which are explicitly related from one representation to another to support students’ thinking about tens and ones. Later in the lesson, the Table 2.15  Patterns in the use of visual/physical representations in instructional segments of the five programs Percent of segments with no representations EM 36.4 INV 30.0 MIF 43.1 MTB 31.6 SFAW 46.7

Percent of segments with multiple representations (with; without Percent of segments with one representation connections) 45.5 16.4 (66.7; 33.3) 34.1 35.9 (87.0; 13.0) 40.2 15.7 (68.8; 31.3) 47.4 21.1 (83.3; 16.7) 46.7 6.7 (0.0; 100.0)

2  Examining the Mathematical Emphasis in Five Curriculum Programs Picture

Strips of 10 Singles 4 strips

# Stickers

6 singles 46 stickers

57

Equation 40 + 6 = 46

Fig. 2.12  Multiple representations with connections in INV. Excerpt from INV GR3 1.1.1, p. 29 illustrating equivalent division problems. From Investigations 2008 Curriculum Unit (Grade 3, Unit 1) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

strips and stickers (cubes in towers of 10 and singles) are recorded as a picture of vertical lines and single dots along with equations (see Fig. 2.12). SFAW includes the smallest portion of multiple representations, all of which were coded as without connections among the representations. Representations included in the 30 lessons analyzed from each program are listed in Table  2.16, which helps show a range of visual/physical representations used in the five programs.

2.6.1  Common Visual/Physical Representations Representations commonly used across the programs are base-ten blocks and place value charts, number lines, and arrays (see Table 2.16). MTB uses base-ten blocks and place value charts, especially with drawings of 100  s, 10s, and 1  s, more ­extensively than any other programs analyzed (see Figs. 2.1 and 2.13). INV uses number lines extensively to represent addition and subtraction problems with various strategies. Arrays (see Figs. 2.8 and 2.9 for example) are used to represent multiplication problems mostly, but division problems as well in INV. In addition, the use of various contexts is evident in the lessons from all five programs. These involve a range of visual and physical representations depending on the context, such as thermometers, coordinate graphs, maps, a globe and measuring tapes, live plants, and magic squares.

2.6.2  Unique Visual/Physical Representations and Variations Each program has its specific representations commonly used across lessons, or in a few particular lessons. EM uses Fact Triangles, which include both addition and subtraction facts or multiplication and division facts on a single triangular flash card (see Fig. 2.14 for example). EM also uses a representation similar to a 100 chart, called Number Grid, which includes 0 right above 10 and additional tens beyond

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Table 2.16  Representations included in the lessons analyzed from the five programs EM Number grid, filled Number grid with blanks Fact triangle Addition table Function machine Mult/div diagram Counters Calculators Base-ten blocks Number line Ratio table Multiplication table Glove & measuring tape Arrays Fraction models Pan balances Line graphs Factor trees Factor rainbows Situation diagrams

INV Strips of 10 & single stickers 100 chart 200 chart 301–600 chart 1000 chart Cubes in towers of 10 Shorthand representation of 10s, and 1 s Thermometer Temperature chart Coordinate graphs Number line Arrays Factor rainbows 10,000 chart Line graphs Student’s choice

MIF Number bonds Number line Base-ten blocks Place value chart with place value chips Arrays Pictures of equal groups Counters Factor “L” Bar model Dot and circle models Pan balances and masses Student’s choice

MTB Magic Squares Spinners with numbers & game boards Base-10 blocks Place value chart Bar graphs Line graphs Triangle fact cards Number cards Counters Factor trees Area/array models Calculators Tables with number patterns Function machines Shorthand representation of 100 s, 10s, and 1 s Grid paper Number charts (e.g., 40 chart) Sieve of Eratosthenes Square titles Coordinate graphs Student’s choice

SFAW Base-ten blocks Place value chart Bar models Number lines Clocks Arrays Pictographs Bar graphs Patterns with shapes Pictures of equal groups Factor trees Ratio Tables A picture of a balance with squares

Fig. 2.13  Base-ten blocks shorthand for subtraction in MTB (GR5 2.3, p. 76). Excerpt from Math Trailblazers, Grade 5, Unit 2 Resource Guide by TIMS.  Copyright ©2008 by Kendall Hunt Publishing Company. Reprinted by permission

100. Like Number Grids in EM, INV uses variations of a 100 chart, such as a 200 chart and a 10,000 chart for exploration of large numbers. MTB also uses variations of a 100 chart for explorations of factors of a number (e.g., numbers from 1 to 40 on a 4 × 10 grid to determine the factors of 40), and Triangle Fact Cards similar to fact triangles in EM.

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Fig. 2.14  Addition/subtraction fact triangles in EM. Excerpt from Everyday Mathematics, Grade 3, Teacher’s Lesson Guide, Volume 1, p. 101. (University of Chicago School Mathematics Project). Copyright 2007 by McGraw Hill. All rights reserved. Reprinted by permission of the publisher Ten Thousands

Thousands

Hundreds

Tens

Ones

Ten Thousands

Thousands

Hundreds

Tens

Ones

10,000 10 Thousands = 1 Ten Thousand

Fig. 2.15  Illustration of using Place Value Chips in MIF.  This illustration was created by the authors based on the image that appears in MIF GR4 teacher’s guide (MIF GR4 1.1, p. 5)

Representations that are unique but commonly used in MIF include (1) number bonds to show the parts and the whole together; (2) Place Value Chips along with a place value chart, which are chips with different colors for different places (see Fig. 2.15); and (3) bar models, which can show part–whole relationships like number bonds, but can be used in more complex situations than those for number bonds. SFAW also uses bar models, but MIF incorporates them more extensively; EM uses representations to show part–part–whole relationships, called situational diagrams (e.g., part–part–whole, change to more, change to less, and comparison in GR5 2.4), which is similar to bar models in MIF.  Although place value chips and base-ten blocks target the same concept (place value), it seems that place value chips are not as clear in showing the distinctions between different places and size of numbers as base-ten blocks because chips in different places have the same size. As shown in Table 2.15, MTB has the longest list of representations used in the 30 lessons analyzed. MTB uses a range of different representations along with commonly used ones across the five programs. In particular, MTB uses data tables with number patterns extensively to find or apply a rule (i.e., number relationship), using letter notations for variables and coordinate graphs in all grades 3–5.

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2.6.3  Notable Approaches to Using Visual/ Physical Representations It is notable that INV, MIF, and MTB have lessons with no specific visual/physical representations that students are required to use. Instead, in these lessons, students are asked to choose their own representation to solve problems or for practice. For example, in one lesson segment on division with whole numbers (3-digit numbers by 2-digit numbers) in grade 5 in INV, students are asked to draw a representation of the problem and its solution—for example, cubes, an array, or groups—if they need to do so. INV also often asks students to generate story contexts that match a graph or an operation in a way to help them make sense of the concept that they learn. EM also asks students to create and solve story problems. It is also notable that representations other than student explorations are used in two programs: MIF and SFAW.  Among the 30 lessons analyzed, MIF has one instructional activity/segment with representations for teacher use only. In SFAW, 21.7% of the instructional activities analyzed have representations for teacher use only or for illustration purposes. This means that the representations are not intended for student exploration of tasks; they are for teacher demonstration or illustration. One example of such use in a lesson is a picture of a balance with squares on top of each side in two different colors to illustrate a problem (“Mis had 5 CDs and bought 2 more. Juan had 3 CDs and bought 4 more. You could write 5 + 2 = 3 + 4 to show that each has the same number of CDs.” GR5 12.1, P. 696). Then, the lesson asks students to tell what they can “do to both sides of an equation so that the sides stay equal.” A teacher can choose to use a real balance or a picture to discuss this with students, but there is no indication in the lesson guide that balances are used for students’ exploration. Apparently, the picture of a balance is only for an illustration of the context. The other three programs had no such use of representations in any of the instructional activities analyzed.

2.7  Discussion: Implications for Teachers and Teaching A major takeaway of our comparative analysis of the mathematical emphasis in the five programs is that variation is the norm. We found substantial differences in what mathematics was included and how it was sequenced, developed, and represented, even across the three Standards-based programs (i.e., EM, INV, and MTB). Important findings in this chapter include: 1. There are substantial variations in the scope and sequence of numbers and operations. 2. Some programs place emphasis on strategies and relationships. 3. Many tasks start with connections, but some connections are more superficial than others.

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4 . Ongoing practice is not just for mastery of skills. 5. The kind of representation matters, and yet how it is used needs more attention. We discuss implications for teachers and teaching for each finding below.

2.7.1  T  here Are Substantial Variations in the Scope and Sequence The five programs had relatively common targets for operations with whole numbers by the end of grade 5, but we found considerable variation in the sequence of operations with whole numbers, fractions, and decimals. In fact, the variation in operations with fractions and decimals was more extensive than whole number operations. The programs analyzed were published before the Common Core State Standards (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010), which essentially replaced 50 different state standards in the USA. In other countries, there may be a possibility of significant differences across programs as well. It seems that the Common Core can provide a holistic picture of content and sequence, and yet the document does not specify the sequence within grade levels. Nevertheless, the Common Core provides only the final goal by the end of each grade; how to achieve the goal within grades is still curriculum developers’ decision to make. Curriculum developers made a great effort to determine the sequence (content placement) in their own examinations and in consultation with existing research (see Chap. 4). Confrey et  al. (2014) made a series of efforts to delineate and refine learning trajectories using the Common Core State Standards in the United States, developed materials for teaching and assessment, and implemented and revised the materials. Given that learning trajectories are not uniform, linear, or fixed, Confrey et  al. argued that coherence requires “careful curricular, instructional, and assessment design, and subsequent interpretation of implementation in relation to achieved outcomes” (p. 731). What does this all imply for teachers and teaching? Curriculum developers decide the detailed sequence of the mathematics topics in different grain sizes and there are variations in the sequences in nature. In this circumstance, when using a curriculum program to teach mathematics, teachers need to make sense of the sequence provided, i.e., how concepts and ideas are built on the previous content, regardless of the diversity of the sequence. They need to support students to develop the concepts and ideas in the sequence laid out. When they find a gap or leap in the sequence or determine that a better organization of the sequence is needed, they can fill in a stepping stone necessary for students to develop the concepts/ideas properly. Also, curriculum developers need to help teachers make sense of the scope and sequence by making the rationale clear and transparent to teachers and providing sufficient explanations about the mathematical flow and connections in the sequence. Analyzing teachers using three different curriculum programs, Kim (2018) found

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that the teachers’ sequencing decisions often hindered their students’ learning of the target mathematics of the lessons. The teachers’ rationale for moving away from the sequence laid out in the programs was largely students’ readiness/need and the teachers’ previous experiences, although many of those decisions were not productive. Curriculum programs target broader audiences and place parameters with users (teachers) in mind; it is the teachers who make decisions for instruction in the end. It is critical to support teachers in organizing instruction in a reasonable route by making the sequence clear and explicit with rationale and providing professional development on sequencing lesson activities and content.

2.7.2  S  ome Programs Place Emphasis on Strategies and Relationships INV and MTB placed emphasis on student strategies, mathematical relationships, connections among representations used, and informal strategies with fractions and decimals in grades 3–5. In INV, the standard algorithms were treated as one option to do operations and various strategies were discussed and compared, along with mathematical practice of reasoning, thinking, and making connections. EM and MTB’s alternative algorithms attend to the meaning of the procedures with extended connections. In contrast, MIF and SFAW place more emphasis on the development of arithmetic skills, focusing on procedures and algorithms from early on. The Common Core State Standards in the USA includes a range of strategies in grade-­ level expectations for number and operations. Algorithms are useful tools as general procedures, and yet students need to know various strategies to solve problems and be able to choose one that is appropriate for a given problem (National Research Council, 2001). Curriculum programs including student strategies provide support features to help teachers make sense of the strategies and the mathematics in them, and how to encourage students to generate and use such strategies. Teachers need to see the importance of various strategies in order to support not only students’ computational fluency, but also exploration of mathematical relationships through numbers and operations.

2.7.3  M  any Tasks Start with Connections, But Some Connections Are More Superficial Than Others Since the focused content strand for our analysis was number and operations, there were a number of different procedures contained in the lessons analyzed. A significant portion of tasks and problems were categorized as procedures with superficial connections in MIF and SFAW. These tasks had a potential for high-level demand, although the target goal was more concerned with the mastery of procedures.

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Curriculum authors need to clarify elements in the tasks toward high-level demand. Also, teachers need to see the potential of those tasks and enact them in a way that requires students to engage in the mathematics of the tasks significantly with high demand, which demands careful teacher preparation and professional development toward this critical evaluation of tasks and problems.

2.7.4  Ongoing Practice Is Not Just for Mastery of Skills Whereas MIF did not have any distributed practice, the other four programs provided ongoing review and practice of concepts and skills throughout the year. Some were geared toward skills mostly; some have concepts and further extension. For example, SFAW focused on skill practice and test-taking strategies; INV attended to mathematical practice of reasoning, thinking, explaining, justifying mathematical ideas and relationships in ongoing practice. Researchers recommended that practice problems/tasks need to be beyond basic skills (Barzel et  al., 2013; Wittmann, E. C. H., 2001) and yet the view of practice as limited to a mastery of skills is still prevalent. As the MTB authors explained in Chap. 4, the purpose of the ongoing review/practice problems needs to be clear in order for teachers to use the problems to support students’ learning of multifaceted mathematics, not just skills and procedures.

2.7.5  T  he Kind of Representation Matters, and Yet How It Is Used Needs More Attention Although we observed common representations such as base-ten blocks and number lines, there were variations in representations used across the five programs. Moreover, some programs had explicit connections between those representations when the lessons used more than one representation and offered students a choice of which representation to use. Such representation use supports students’ representational fluency (Lesh & Jawojewski, 2007). Some programs included representations mainly for illustration purposes or teacher use only. In these cases, representations were not for students to use to explore mathematical ideas and solve problems, which limits students’ representational fluency and also determines the kind of learning that the lessons provoke. Such representation use indicates a low level of cognitive demand (Stein et al., 2000). It is certain that the kind of representation matters, and yet how it is used needs more attention, as the curriculum authors explained in Chap. 4. Certain representations are chosen for particular ideas; some are used repeatedly and others are fairly confined to one or a few ideas. Teachers and students need to learn how to make use of each representation for particular and general ideas. Both curriculum authors and teachers need to think about how to help students to use representations fluently in their exploration of mathematics.

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References Barzel, B., Leuders, T., Prediger, S., & Hußmann, S. (2013). Designing tasks for engaging students in active knowledge organization. In A. Watson, M. Ohtani, J. Ainley, J. B. Frant, M. Doorman, C. Kieran, A. Leung, C. Margolinas, P. Sullivan, D. Thompson, & Y. Yang (Eds.), Proceedings of study conference–ICMI study 22 on task design (pp. 285–294). Oxford: Oxford Press. Boston, M. D., & Candela, A. G. (2018). The instructional quality assessment as a tool for reflecting on instructional practice. ZDM Mathematics Education, 50, 427–444. Charles, R. I., Crown, W., Fennell, F., et al. (2008). Scott Foresman–Addison Wesley Mathematics. Glenview, IL: Pearson. Clements, D.  H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89. Confrey, J., Maloney, A. P., & Corley, A. K. (2014). Learning trajectories: A framework for connecting standards with curriculum. ZDM Mathematics Education, 46, 719–733. Corcoran, T., Mosher, F. A., & Rogat, A. (2009). Learning progressions in science: An evidence-­ based approach to reform. Research Report No. 63. Madison, WI: Consortium for Policy Research in Education. Doyle, W. (1983). Academic work. Review of Education Research, 5, 159–199. Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23(2), 167–180. Doyle, W., & Carter, L. (1984). Academic tasks in classroom. Curriculum Inquiry, 14, 129–149. Gelman, R., & Brenneman, K. (2004). Science learning pathways for young children. Early Childhood Research Quarterly, 19, 150–158. Hiebert, J. (Ed.). (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale NJ: Lawrence Erlbaum. Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational Research Journal, 30(2), 393–425. Kim, O. K. (2018). Teacher decisions on lesson sequence and their impact on opportunities for students to learn. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Research on mathematics textbooks and teachers’ resources (pp. 315–339). Berlin: Springer. Lesh, R., & Jawojewski, J. (2007). Problem solving and modeling. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 262–804). Charlotte, NC: Information Age. Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards mathematics. Washington, DC: Author. National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. Otten, S., & Soria, V. M. (2014). Relationships between students’ learning and their participation during enactment of middle school algebra tasks. ZDM Mathematics Education, 46, 815–827. Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teachers’ work and interactions: A collective perspective on resources, their use and transformation. ZDM, 45(7), 929–943. Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: A way to understand teaching and learning cultures. ZDM, 33(5), 1–20. Putnam, R. (2003). Commentary on four elementary mathematics curricula. In S.  Senk & D. Thompson (Eds.), Standards-based school mathematics curricula: What are they? What do students learn? (pp. 161–178). Mahwah, NJ: Erlbaum. Remillard, J. T., Harris, B., & Agodini, R. (2014). The influence of curriculum material design on opportunities for student learning. ZDM Mathematics Education, 46, 375–349.

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Remillard, J.  T., & Kim, O.  K. (2017). Knowledge of curriculum embedded mathematics: Exploring a critical domain of teaching. Educational Studies in Mathematics, 96(1), 65–81. Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13. Simon, M.  A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145. Sleep, L. (2009). Teaching to the mathematical point: Knowing and using mathematics in teaching. Unpublished doctoral dissertation. University of Michigan, USA. 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. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344–350. Son, J.-W., & Kim, O. K. (2015). Teachers’ selection and enactment of mathematical problems from textbooks. Mathematics Education Research Journal, 27(4), 491–518. Stein, M. K., Grover, B. W., & Henningsen, M. A. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classroom. American Educational Research Journal, 33(2), 455–488. Stein, M. K., & Kim, G. (2009). The role of mathematics curriculum materials in large-scale urban reform: An analysis of demands and opportunities for teacher learning. In J.  T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 37–55). New York: Routledge. 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. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-­ based mathematics instruction: A casebook for professional development. Reston, VA: National Council of Teachers of Mathematics. Stylianou, D. A. (2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: Towards an organizing scheme. Educational Studies in Mathematics, 76(3), 265–280. Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147–156. TIMS Project. (2008). Math trailblazers unit resource guide grade 5, unit 4 division and data. Dubuque, IA: Kendall/Hunt Publishing Company. TERC. (2008). Investigations in Number, Data, and Space (2nd edition). Glenview, IL: Pearson Education Inc. TIMS Project (2008). Math Trailblazers (3rd Edition). Dubuque, IA: Kendall/Hunt Publishing Company. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill. Wittmann, E. C. H. (1995). Mathematics education as a design science. Educational Studies in Mathematics, 29, 355–374. Wittmann, E. C. H. (2001). Developing mathematics education in a systemic process. Educational Studies in Mathematics, 48, 1–20.

Chapter 3

Examining the Pedagogical Approaches in Five Curriculum Programs Janine T. Remillard, Ok-Kyeong Kim, and Rowan Machalow

Abstract  This chapter explores the pedagogical approach incorporated into each of the five curriculum programs we analyzed (The five programs are Everyday Mathematics (EM), Investigations in Number, Data, and Space (INV), Math in Focus (MIF), Math Trailblazers (MTB), and Scott Foresman–Addison Wesley Mathematics (SFAW). See Chap. 1 for more details about the programs.). Specifically, we consider explicit and implicit messages about how students should interact with mathematics, one another, the teacher, and the textbook around these mathematical ideas. The analysis considers the role the teacher is expected to play and the corresponding roles the textbooks and students play in shaping these interactions. Following Hiebert and colleagues’  notion that teaching is an interactive system, involving a constellation of interrelated features, we present the approach of each program in a holistic profile, which overviews five dimensions: participant structures, nature of student work, role of the teacher, sources of knowledge, and cognitive demand of tasks. We also consider ways that these features bolster or moderate one another. Using a dialogic-to-direct instruction continuum characterized by Munter and colleagues, we highlight several key differences of the pedagogical approaches across the five programs. We also discuss ways that a single continuum is insufficient for characterizing the way pedagogical approach is represented in teacher’s guides. Keywords  Curriculum analysis · Mathematics curriculum materials · Curriculum materials pedagogy · Teacher’s role · Dialogic instruction · Source of knowledge · Cognitive demand · Participant structure · Teacher’s guide · Role of teacher · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman-Addison Wesley Mathematics

J. T. Remillard (*) · R. Machalow Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] O.-K. Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_3

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3.1  Introduction Examinations of pedagogical approaches and underlying pedagogical visions in mathematics instruction have long been a focus on analysis in mathematics education research (Franke, Kazemi, & Battey, 2007: Hiebert et al., 1997; Munter, Stein, & Smith, 2015). Our focus in this analysis is how these intentions are represented and communicated to teachers through guidance in curriculum materials. The analysis in this chapter builds on the previous chapter, which looked across the five programs to elaborate different ways of organizing, sequencing, and representing mathematical concepts and different views on the nature of mathematical work students should be expected to do. By focusing on the pedagogical approach, we inquire into how and by whom students’ learning experiences are structured, particularly by the teacher. In this analysis, we do not consider whether these intended interactions or roles are reasonable or how well they are supported by the materials provided. The chapters in Pt. II of this volume offer analyses of how and whether the curriculum programs communicated with teachers to support them in enacting the roles set out for them. In order to situate our analysis, we begin by defining the construct of pedagogical approach, drawing on frameworks offered by other scholars. We then review how pedagogical recommendations became part of curriculum materials. We also review existing analyses of pedagogical approach in curriculum materials and in classrooms more generally. We then present our analytical framework and methods for analysis before presenting our findings, looking at each program holistically. We conclude by considering how the analysis of pedagogical approach informs our understanding of curriculum materials.

3.2  Pedagogical Approach in Curriculum Materials? We use the term pedagogical approach to refer to expectations and underlying assumptions in curriculum materials, and in the teacher’s guide in particular, intended to shape the learning opportunities available to students during mathematics instruction (Century, Rudnick, & Freeman, 2010; Heck, Chval, Weiss, & Ziebarth, 2012; Hiebert et al., 2005). In particular, we focus on messages about (a) how students should be supported to engage with mathematical ideas and (b) how this engagement is facilitated by the teacher and the textbook. These assumptions and expectations are communicated through both explicit and implicit messages in curriculum materials. In many cases, these messages are intertwined with the mathematical emphasis. For example, tasks categorized as doing mathematics, according to our modified version of Smith and Stein’s (1998) cognitive demand framework, engage students in developing strategies to solve nonroutine problems. (See Chap. 2.) Tasks of this sort assume an approach to teaching that allows students the time, opportunity, and support to do this type of mathematical work. We note, however,

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that discussions of efforts to shift approaches to teaching mathematics often attend to mathematical emphasis more directly than the underlying pedagogical approaches. Our aim in this chapter is to focus in detail on how different curriculum authors conceptualize mathematics pedagogy as a component of the materials they designed. In the following section, we discuss the role pedagogical approach has played in mathematics curriculum materials in the USA and the ways it has been examined in other curriculum analyses. We then draw on research on teaching practices and instruction to offer an analytical framework for our analysis.

3.2.1  T  he Emergence of Pedagogical Approach in Mathematics Curriculum Materials Both attention to and analysis of pedagogical approaches in mathematics curriculum materials are relatively recent phenomena. Around the world, governing entities identify content and curriculum objectives or attainment targets that must be taught in mathematics (Fan, Zhu, & Miao, 2013; Houang & Schmidt, 2008; Jones & Fujita, 2013; Remillard & Heck, 2014). In the USA these objectives have come to be known as Standards. It is less common for such documents to specify the pedagogical approach used when teaching toward these goals. In many contexts, such decisions are left to teachers or other local decision makers (schools, local school districts, etc.). This is not to say that such documents are free of pedagogical implications, but they are most often implicit, rather than explicit. Pepin, Gueudet, and Trouche (2013), for example, analyzed national curriculum documents from France and Norway and, attending to discourse patterns in the language used, found more references to egalitarian notions in the Norwegian documents and more references to individualism in the French documents. These differences reflect the educational traditions in each country, and also have connections to implied pedagogical practices, although they are not stated as such. Historically, mathematics teacher’s guides, as a genre, have been limited in offering pedagogical guidance to teachers, focusing primarily on the content of the student textbook (Ball & Cohen, 1996; Davis & Krajcik, 2005). In fact, the majority of research on textbooks and curriculum materials around the world has focused on the student textbooks (Fan et  al., 2013; Osterholm & Bergqvist, 2013), with limited attention to recommendations in the teacher’s guide. In the USA, the content and presumed role of the teacher’s guide underwent substantial change after the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) and subsequent teaching standards (NCTM, 1991) were published and began to influence both state- and district-level curriculum frameworks, classroom practices, and curriculum development. As described in Chap. 1, the National Science Foundation, funded a number of Instructional Materials Development (IMD) projects in 1991 (Senk & Thompson, 2003), which included explicit attention to the nature of learning opportunities envisioned for

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students and the role that the teacher plays in shaping them. (See Chap. 4 for an explication of developers’ visions for classroom learning experiences of the three Standards-based programs.) Some argue that the NCTM Standards influenced mathematics education around the world (e.g., Boesen et al., 2014). The Standards-based curriculum materials, which began to be adopted by school districts across the USA in the mid-1990s, ushered into the textbook market a shift in the genre of the teacher’s guide. In addition to offering substantially different approaches to mathematics content, these materials directly addressed the teachers’ and students’ roles during instruction, giving greater attention to the nature of classroom interactions and the teacher’s role in them, and in many cases decreasing the prominence of the student textbook pages as a vehicle for learning. In this way, the mathematics teacher’s guide became a messenger of mathematics content and related pedagogical practices. Typical textbooks and teacher’s guides prior to this new genre of materials, most of which were commercially developed and marketed (Stein, Remillard, & Smith, 2007), offered limited pedagogical guidance to teachers.1 Despite the lack of attention to pedagogy, we argue that such teacher’s guides embody an implicit view of the mathematics learning, classroom instruction, and the teacher’s role. This embodiment is evident in our analysis of the SFAW teacher’s guides. The inclusion of explicit attention to pedagogical approaches and strategies in teachers’ guides in the 1990s, inspired by the influence of the NCTM Standards, had a transformative effect on the U.S. curriculum market. Currently, both curriculum materials developed by members of the mathematics or mathematics education community, along with those developed by commercial publishers, include some level of attention to intended pedagogical practices and classroom structures. Even though the Common Core State Standards, the curriculum policy governing much of the curriculum development in the USA since 2010 (see Remillard & Reinke, 2017), do not promote a particular pedagogical approach, teacher’s guides published today continue to devote substantial attention and real estate on the page to pedagogical guidance.

3.2.2  Analysis of Pedagogical Approach in Curriculum Materials Given the relatively recent emergence of pedagogical approach in mathematics curriculum materials, it should not be surprising that this construct as a potential characteristic of textbook or teacher’s guides has received minimal attention in research. This absence is strikingly evident in a 2013 special issue of The International Journal on Mathematics Education (ZDM), entitled Textbook Research in

 Some less typical curriculum resources existed as exceptions to this rule, but they were not part of the mainstream curriculum market. 1

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Mathematics Education. In a reflective article on mathematics textbook development and research, Howson (2013) discusses the attributes of textbooks “that a reviewer of a textbook” might consider. He offers 12, which range from “mathematical coherence” to “gender, racial and other social balance” to “the physical attractiveness of the text” (pp. 653–4). Pedagogical approach or emphasis is not included in this list, although the final item on the list is provisions that “go beyond answer books and balance the twin demands of developing the teachers’ mathematical understanding and assisting the management of the lesson.” We see this attribute as akin to educative features described by Davis and Krajcik (2005), but distinct from pedagogical recommendations. In the same special issue, Fan et al. (2013) offer a survey of research on mathematics textbooks, beginning in 1950. The section characterizing different approaches to curriculum analysis and comparisons includes a subsection on cognition and pedagogy. For the most part, the 11 examples cited in this section examine features of mathematics problems that place particular types of demands on students, including higher order thinking, problem solving, and the presence of nonroutine tasks. Pedagogy and instructional approach are mentioned in passing, but appear to be limited to how the textbook presents content to students. Vincent and Stacey (2008), for example, used the term “shallow teaching syndrome” to refer to the types of problems, the amount of repetition, and other features of the mathematical task in textbooks. Pepin and Haggarty’s (2001) comparative analysis of English, French, and German mathematics textbooks also treats pedagogy as an element of the mathematics and the guidance written into the student text. They use the term “pedagogical intentions of textbook” to refer to “ways in which the learner is helped (or not) within the content of the text, the methods included in the text; and by the rhetorical voice of the text” (p. 4). Given the interest in shaping pedagogical practices in the USA through curriculum materials and teacher’s guides (Senk & Thompson, 2003), it is to be expected that the majority of explicit attention to teaching approaches in curriculum materials also comes from U.S. researchers. Many of these efforts were guided by a desire to develop analytical methods for assessing how well explicit or implicit pedagogical intentions in the written materials are upheld during instruction. Recognizing that, increasingly, new curriculum materials included both mathematical and pedagogical guidance, researchers interested in assessing the curriculum fidelity developed frameworks that explicitly examined pedagogical recommendations and expectations. Century et al.’s (2010) framework for measuring fidelity of implementation identifies “critical components” of the given curriculum program under both structural and instructional dimensions. Under the heading Pedagogical Components, which falls within the instructional dimension category, they include: (a) facilitating student engagement with others; (b) facilitating student engagement with content; (c) facilitating student role as learner; and (d) pedagogical strategies, which include teachers’ use of materials and tools, differentiation, and assessment (p. 206). Heck et al. (2012) also distinguished between mathematical and pedagogical elements of curriculum materials in the tool they developed. They use the term “mathematical storyline” to characterize how the content is covered, how it is organized, and the

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specific mathematical learning goals. They use the term “pedagogical storyline” to refer to the “means of engaging students with those topics and goals” (p.  69). Prominent pedagogical features include how and with whom students are expected to interact during the lessons, the amount of time spent on different types of activities, the source and process of new learning (e.g., teacher demonstrations, examples in the text, student exploration, and sense making), and the way new concepts are sequenced. Some frameworks for analyzing mathematics curriculum materials or tasks in the written curriculum do not distinguish between mathematical and pedagogical elements, but instead make tacit reference to pedagogical expectations when characterizing the nature of mathematical tasks. Stein, Grover, and Henningsen’s (1996) framework for assessing cognitive demand characterizes mathematical tasks as either low or high demand. Some of their criteria include implicit reference to what classroom interactions teachers might engage in. For example, procedures without connections tasks, one type of low-demand task, require no explanations or explanations that focus solely on describing the procedural steps used, while in higher demand tasks, such as procedures with connections, students must engage with conceptual ideas that underlie the procedures in order to complete the task. Other criteria reference more mathematical components of tasks, including the type of mathematical knowledge or thinking emphasized.

3.2.3  A  nalysis of Pedagogical Approach Outside of Curriculum Materials Even though it has not been a strong focus of curriculum analysis, pedagogical approach has been a focus of analysis of mathematics instruction in general and informs our analysis of the pedagogical approaches in the curriculum programs. In the USA, there is a history of defining two major instructional models that each includes a constellation of interrelated factors. Munter et  al. (2015) characterize these two models as direct and dialogic, acknowledging that each might be considered as an extreme point on a continuum along which most teaching falls and that many have argued for models that avoid such dichotomies (e.g., National Research Council, 2001). In the direct instructional model: Pedagogy consists of [the teacher] describing an objective, articulating motivating reasons for achieving the objective and connections to previous topics; presenting requisite concepts (if they have not been presented previously); demonstrating how to complete the target problem type; and providing scaffolded phases of guided and independent practice, accompanied by corrective feedback. (p. 7)

In contrast, the dialogic model requires that “Students must have opportunities to (a) wrestle with big ideas, without teachers interfering prematurely, (b) put forth claims and justify them as well as listening to and critiquing claims of others, and (c) engage in carefully designed, deliberate practice” (p.  8). Munter et  al. also

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include the following components of the teacher’s role in dialogic approaches: “orchestrate discussions that make mathematical ideas available to all students and steer collective understandings toward the mathematical goal of the lesson” and “sequence classroom activities in a way that consistently positions students as autonomous learners and users of mathematics” (p. 8). Although defining these two pedagogical approaches provides insight into many debates about mathematics education in the USA over the past 50 years, these two systems are not monolithic. Munter et al. (2015) identified nine key areas that are important to both models, but taken up in very distinct ways. Among these areas are the following, which are particularly pertinent for the analysis in this chapter: 1 . The importance of and role of talk. 2. The importance of and role of group work. 3. The nature and ordering of mathematical instructional tasks. 4. The nature, timing, source, and purpose of feedback from the teacher. 5. The emphasis on creativity, which includes students’ authoring their own understandings and approaches and mathematizing subject matter from reality (pp. 9–11). Based on the analysis of video-recorded lessons from seven different countries, collected through the Third International Mathematics and Science Study (TIMSS), Hiebert et al. (2005) argue for characterizing classroom teaching as a system with a constellation of interrelated features, rather than studying individual features in isolation. By studying a number of features in concert, it is possible to understand how they reinforce each other and build together to create normative models of teaching and learning. Hiebert et al. identified 75 features, grouped into three categories: (a) structure and organization of daily lessons, (b) nature of math presented, and (c) the way that math was worked on during the lesson. They point out that the interactive nature of these systems often means that attempts to change one or more features without changing others that they are dependent on is often ineffective. Viewing components of instruction as a system of interrelated factors makes it possible to consider how curriculum programs might choose to emphasize a set of factors that are more similar to dialogic or direct instructional models or a combination of both (Hiebert et al. 2005; Munter et al. 2015). Our analysis is designed to bring a more nuanced understanding to ways that each of these models might be realized through different curriculum programs. Our particular interest in analyzing the pedagogical approach of the curriculum programs is related to our overall focus on understanding what it means for teachers to use curriculum materials. In the following section, we describe our analytical framework for undertaking this analysis. Our analysis seeks to answer two central questions about the five curriculum programs we analyzed: 1. What types of mathematical work are students expected to engage in during mathematics instruction? 2. What roles are the teacher, the student textbook, and other students expected to play in supporting this type of work?

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3.3  A  nalytical Framework: Surfacing Pedagogical Approach in Teacher’s Guides Acknowledging that a teacher’s guide or any designed curriculum plan cannot dictate pedagogical practices, our analysis considered explicit and implicit design elements and messages in the program that signal pedagogical intentions. In specifying a framework, we were conscious of Hiebert et al.’s (2005) assertion that teaching must be understood as an interactive system and not a list of isolated items. We drew on existing frameworks for analyzing teaching practices and curriculum materials (Century et al. 2010; Munter et al. 2015) to identify critical elements of the system that together support opportunities for student learning. (See Chap. 1 for elaboration of this construct.) Specifically, we considered the following categories (a) the overarching perspective on mathematics learning explicitly promoted in curriculum documents, (b) the types of opportunities to learn created within the daily lessons, (c) how mathematics learning is conceptualized, and (d) the role the teacher plays in supporting students’ learning opportunities. The first two categories of the framework were descriptive in nature; the second two were more analytical than descriptive, in that they involved coding suggested activities for students and teachers during lessons in order to surface guiding perspectives and views on ideal student learning, as well as the specified or implied role the teacher should play. Importantly, the analysis in this chapter was fully based on curriculum documents and did not take into account how program developers described their intentions, as represented in Chap. 4.

3.3.1  Explicit Perspective on Mathematics Learning Our analysis began with an examination of how the curriculum documents represented the overarching view on mathematics learning or priorities of the program. When explicitly stated, this information was often in supplementary documents, such as implementation or assessment guides, or in the introduction to the teacher’s guide.

3.3.2  Lesson and Participant Structure This category describes the types of structures and tasks designed into daily lessons that provide opportunities for teachers and students to interact with mathematical concepts and one another. As we discuss in the methods section, we focus our analysis of the pedagogical approach on the part of the lessons we identified as the main

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body of the lesson, that is the portions of the lesson description intended as the primary focus of the lesson and with which students and teachers would spend the majority of time during a mathematics lesson. In contrast to Stein et al. (1996), who focused their analysis on the two tasks intended to take the most time during the lesson, we analyzed almost all segments identified as part of the standard lesson, including any warm-up activity, instructional components, and student work. Each program we analyzed included components that were not intended to be part of the daily lesson, including opportunities for repeated practice, review, or enrichment. Other components were clearly described as or appeared to be optional, allowing the teacher to determine whether to use or omit them in order to customize or extend the lesson. We referred to these components as beyond the main body of the lesson. Our analysis suggested that the use or omission of these components did not significantly alter the core focus of the lesson or the participant structures included. Within the main body of the lesson, we examined the lesson structure and the major participant structures, as detailed below. The lesson structure dimension detailed the way lessons were typically organized, including the types of activities routinely included in lessons and how they are sequenced and connected to one another. It also addressed the extent to which these activities varied or were stable across multiple lessons. When the anticipated time-frame of different lesson components was identified, which is not always the case, we include these recommendations as well. Major participant structures characterized how students were expected to participate in the activities of the lesson, including the variation and the frequency with which they were expected to work in a whole-class, teacher-led session, with partners or small groups, or individually. While some authors made explicit recommendations for participant structures, others left these decisions to the teacher. The primary focus of our analysis of pedagogical emphasis was the main body of the lessons. That said, we also provided a brief description of consistent peripheral topics and activities that we categorized as beyond the main body. Including these components in the description helped to provide a full picture of the program, especially when peripheral activities differed considerably from the main body of the lesson. The three dimensions of this category of the framework are summarized in Table 3.1.

Table 3.1  Lesson and participant structure dimensions and indicators Dimension Lesson structure Major participant structures Peripheral topics and activities

Descriptive indicators Typical lesson structure, routine activities, time frames. How students were expected to participate: whole class, partners or small groups, individually. Description of activities included in daily lesson descriptions, but not central to them: differentiation options, extension activities.

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3.3.3  Nature of Student Learning and Work This element considered how student learning was conceptualized in the lessons, based on the types of opportunities students had to engage with mathematics concepts and do mathematical work. We used the following interrelated dimensions to identify indicators of possible positions along several dialogic and direct continua: (a) types of mathematical work expected of students; (b) the cognitive demand of the main instructional activities, and (c) sources of mathematical knowledge and knowing. See Table 3.2 for indicators for each dimension and the descriptions used to define codes. Table 3.2  Nature of student learning and work dimensions and indicators Analytical Dimension indicators Types of mathematical Answer, no work expected of students explanation Describe steps

Description No explanations are required for the entire lesson section. Students are asked to list a set of steps they followed. Show work Students are asked to show the work they did to arrive at an answer, using numerals or representations. Notice, describe Students are asked to identify (and sometimes patterns explain) patterns. Generate, explain, Students are asked to create and explain their justify answers own strategy, solution, or understandings. Memorization Tasks require recall of facts, rules, and Cognitive demand definitions. instructional activities (See Chap. 2 for more Procedures without Tasks focus on mastery of procedures without detail.) connections meaning. Tasks focus on mastery of procedures using Procedures with initial or limited connections to meaning as superficial possible scaffolds. connections Tasks emphasize learning procedures in Procedures with relation to underlying mathematical meaning. extended connections Doing mathematics Tasks aimed at developing and using strategies for solving unstructured problems in order to learn and use mathematical concepts and reason about them. Sources of knowledge and The teacher Teacher imparts knowledge and determines knowing correctness. Students Students generate knowledge and determine correctness through reasoning. The student Knowledge and correct answers are textbook determined by the textbook. Prior sources Knowledge is commonly shared, based on prior learning.

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The first dimension, types of mathematical work expected of students, refers to the types of thinking students were expected to do and the artifacts they were expected to produce, verbally, collaboratively, in writing, or through other forms or media. On a continuum from most direct to most dialogic, the indicators were (a) provide answers with no explanation, (b) describe steps of a procedure, (c) show work, (d) notice, describe patterns, and (e) generate, explain, and justify answers. In addition to being more aligned with dialogic models of instruction, the latter three expectations are more likely to be associated with higher cognitive demand tasks. Expecting students to communicate mathematical ideas in writing, with spoken words or other representational forms, provides opportunities for them to make sense of concepts, formalize strategies, and use mathematical terminology to increase the level of cognitive demand of a task. As mentioned earlier, the concept of cognitive demand, introduced by Stein et al. (1996) and taken up and modified by a number of scholars to characterize the levels of thinking mathematical tasks require of learners (e.g., Boston & Candela, 2018; Otten & Soria, 2014; Smith & Stein, 1998) has implications for our analysis of both the mathematical and pedagogical aspects of curriculum materials. The levels of cognitive demand, referenced briefly in Table 3.2, are described in detail in Chap. 2 and are also relevant for characterizing the nature of mathematical work expected of students. For this reason, we consider the extent to which the main tasks in the lessons asked students to engage in (a) memorization, (b) procedures without connections, (c) procedures with superficial connections, (d) procedures with extended connections, or (e) doing mathematics. The final dimension of the nature of student learning and work was the sources of mathematical knowledge and knowing. Given the interest in positioning students as autonomous thinkers and learners within the dialogic approach, in contrast to the emphasis on modeling procedures and providing scaffolded opportunities for repeated practice within the direct approach, we needed our framework to take into account differences in how knowledge is generated and imparted in the curriculum materials. This dimension captures ideas about learning, since two possible indicators were the teacher and the student. It also surfaces assumptions about where the mathematical authority lies, since a third indicator is the student textbook. While positioning the student as the source of knowledge falls under the dialogic instructional model, locating the teacher or the textbook as sources of knowledge are both more aligned with a direct approach. A key difference in these latter two indicators is that they frame learning and the teacher’s role differently. We also included prior sources as a fourth indicator in this dimension because it was not uncommon for the mathematical knowledge central to the lesson to be commonly shared, based on prior learning.

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3.3.4  Teacher’s Role The final category in the framework, composed of a single dimension, was the teacher’s role in supporting student learning opportunities during the lesson. Throughout each lesson, teachers were guided, sometimes explicitly, other times implicitly, to take up different instructional roles. This category was informed by Century et al.’s (2010) pedagogical components, which consider how the teacher facilitates students’ engagement with others, the content, their own learning, and the available materials (p. 206). The indicators used in this category emerged from our analysis of the teacher’s guides themselves. In keeping with Hiebert et al.’s (2005) notion that teaching is an interactive system, these roles are related to dimensions described above, including how learning is conceptualized and the sources of knowledge. Further, while they tend to fall along a continuum from direct to dialogic instructional models, they also highlight some differences in where the mathematical authority is located. The indicators of the teacher’s role that we identified within the teacher’s guides are: Telling and showing, Telling and showing through the student text, Guiding, Facilitating, Orchestrating, and Stepping back. (See Table 3.3.) Telling and showing is the role most direct in nature. It includes the teacher modeling, explaining, or walking students through steps and procedures. Telling and showing through the student text has many similarities to the previous indicator, in that the teacher’s role is to provide information to students, but it also positions the Table 3.3  The teacher’s role indicators and descriptions Indicator Telling and showing Telling, showing through the student text Guiding

Facilitating

Orchestrating

Stepping back

Description The teacher demonstrates, explains, or walks students through a procedure. (The teacher is the primary source of knowledge.) The text shows, explains, or walks students through a concept or procedure. The teacher reiterates the information in the text and ensures that students understand what is written. (The text is the primary source of knowledge.) The teacher shapes classroom interactions through a series of scaffolded questions designed to prompt student thinking or expected answers. (The teacher is the primary source of knowledge, often in conjunction with the text.) The teacher encourages student exploration and meaning making by prompting and supporting them to make connections or overcome difficulties, as they work on tasks. (Students are the primary source of knowledge.) The teacher creates a classroom environment in which students model, explain, or discuss strategies as a community. The teacher participates in the discussion less than the students do. (Students, as a class, are the primary source of knowledge and determine correctness.) The teacher steps back while students work on tasks independently or in small groups, with the possibility of monitoring.

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text as the primary source of knowledge. The teacher ensures students understand this information. The guiding role positions the teacher in a somewhat more dialogic role, as it involves shaping classroom interactions and prompting students to share their thinking and solutions, but the teacher, or the text, remains the primary source of knowledge and the teacher’s task is to guide students to the correct understanding. The facilitating role is more dialogic in nature in that it positions students as the primary source of knowledge. The teacher fosters student exploration and meaning making by encouraging them to make connections or overcome difficulties, as they work on tasks, often in small groups. The orchestrating role is also dialogic in nature. The teacher’s task is to create a classroom environment in which students model, explain, or discuss strategies as a community. The teacher plays an important shaping role, but participates in the discussion less than the students do. As in many models of dialogic instruction, some teacher’s guides suggest facilitating and guiding go hand in hand. A lesson segment might begin with students working on a task in small groups and move into a whole-class discussion, which draws on this work and in which the teacher plays an orchestrating role. The final role, stepping back, places the teacher in a less active role than the others described above. In this role, the teacher steps back while students work on tasks independently or in small groups. This role was most often implicit and included lesson segments that specified what the students should be doing, but did not always indicate a role for the teacher. If the teacher’s guide indicated that the teacher was expected to assess or support student work, we generally labeled it as facilitating.

3.4  Methods Following the analytical framework, described above, we analyzed the five curriculum programs to characterize the intended pedagogical approach of each. Our analysis focused primarily on the 30 lessons selected from each program. As described in Chap. 1, we randomly selected 10 lessons from each of grades 3–5 in the Numbers and Operation strands. We also reviewed supplementary documents, such as implementation or assessment guides, as well as the introductory sections in the teacher’s guides to uncover any explicit statements authors made about intended pedagogy, opportunities for student learning, or other priorities.

3.4.1  Coding Using the descriptions of the indicators detailed in the previous section, we developed codes for the dimensions in each coding category, with the exception of three dimensions (authors’ explicitly stated perspectives, lesson structure, and peripheral topics and activities), which we characterized descriptively through reviewing mul-

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tiple lessons and other documents. Using the analytic codes, we coded all 150 lessons, focusing on the core components of each lesson, or what we referred to as the main body of the lesson. The three descriptive categories allowed us to contextualize the core lesson features in a broader set of components, some of which were routine, but not incorporated into the lesson, such as daily fluency practice or homework, others of which were quite peripheral or optional, such as extension activities. Our analysis includes descriptions of these components to provide a greater sense of the context in which the main body of the lesson fits, but we did not code them as part of our analysis of pedagogical approach. When a single lesson was intended to be taught over several days, we coded all of the days. Our coding unit of analysis was what we refer to as the lesson segment, a continuous chunk of a lesson, as specified in the teacher’s guide, which generally addressed the same mathematical topic. In cases where students were presented with different types of problems or tasks, but all addressed the same mathematical topic, we treated these as a single lesson segment. Some continuous lesson segments had blended participant structures, for example, the teacher was guided to move between a whole-class structure and small group work on the same mathematical topic. We treated these as a single lesson segment, but double coded them. Similarly, we treated “mixed review” of several unrelated problem types and topics as a single lesson segment with a review theme. When coding, we allowed for multiple codes in the same dimension to be applied, when appropriate. When coding for cognitive demand, we identified the main instructional task or tasks in each segment and coded it or them. These were the tasks that students spent the most time on during the lesson segment. We did not assess the cognitive demand of tasks included in a segment in a more peripheral way, such as warm-ups (e.g., 5-min warm-up in MIF), optional activities, and other routines for practice. In a few cases, two different tasks were central to a segment and each received a cognitive demand code. See Chap. 2 for more detail on the cognitive demand codes.

3.4.2  Analysis For each curriculum program, we compiled the codes for each coding category and dimension into a single count. In order to make comparisons across the five programs, we used percentages to determine the proportion of lesson segments in which each indicator was prominent. Since we occasionally gave more than one task in a segment a cognitive demand code, we determined the proportion of all tasks at each level of cognitive demand. We then used these percentages, along with the descriptive analysis, to characterize the pedagogical approach of each program. In the following sections, we present our findings, along with illustrative examples from each program.

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3.5  Pedagogical Emphasis: Comparison of Five Programs In this section, we characterize the pedagogical approach of each curriculum program, combining descriptive details and the results of our analytical coding. For the purpose of making comparisons, we have sequenced our presentation of the five programs from the one we assess to have the most dialogic instructional approach to least, using the following abbreviations to refer to them: Investigations in Number, Data, and Space (INV), Math Trailblazers (MTB), Everyday Mathematics (EM), Math in Focus (MIF), and Scott Foresman–Addison Wesley Mathematics (SFAW). Each description includes the following sections: (a) the lesson and participant structures, (b) the nature of student learning and work during the lessons, and (c) the teacher’s role in supporting this work. Illustrative examples are interwoven into the descriptions.

3.5.1  I nvestigations: Students as Agents of Their Own Learning One of the three guiding principles of INV, stated in the Implementation guide and in the introductory portion of each unit, is the assertion that students (and teachers) are “agents of their own learning.” They come to school with ideas about the major topics of the school mathematics curriculum, numbers, shapes, patterns, etc. The authors further argue that: If given the opportunity to learn in an environment that stresses making sense of mathematics, students build on the ideas they already have and learn about new mathematics they have never encountered. Students learn that they are capable of having mathematical ideas, applying what they know to new situations, and thinking and reasoning about unfamiliar problems. (p. 1).

Through our analysis of INV lessons, we found this guiding principle to be evident in the way lessons and student participation were structured and the types of mathematical work expected of students. Our analysis of the teacher’s role and the role of the text illustrate how student mathematical agency is developed. Lesson and Participation Structure Lessons in INV are intended to take one hour and are typically composed of three or four segments. There are five types of segments used throughout the INV program, but the types of segments used and in which sequence varies and seems to be influenced by the particular goals of the lesson. Most lessons begin with a whole-­ class activity, during which the teacher introduces a task, often emphasizing key mathematical ideas, and then provides time for students to work on a related task in pairs. All lessons conclude with a session follow-up, titled daily practice task and

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homework and include recommendations for practice and workbook pages students might be assigned. One or two additional segments fall between the first and last. Most frequently, the second segment of each lesson is a discussion, a teacher-­ facilitated, whole-class conversation around a particular mathematical concept or problem, in which students share strategies and understandings. In many lessons, the second or third segment takes the form of a second activity, involving whole-­ group, paired, or individual work on specified tasks. For 30–40% of the lessons focusing on Number and Operations, the second or third activity is called a Math Workshop. During Math Workshops, students progress through 2–3 different tasks or games in pairs or individually. In addition to providing students opportunities to review and practice concepts they have learned, this format is intended to provide time for the teacher to observe, assess, and provide additional support to students as needed. Frequently, the same set of Workshop tasks are used over a 2- to 3-day period. After about 6 or 7 lessons, grouped together as an investigation, one lesson segment takes the form of an assessment activity. These are individual assessment tasks, which focus on one or more benchmarks of the unit. The final lesson in each unit takes the form of an end-of-unit assessment, which measures students’ work on all unit benchmarks. Descriptions of the segment types and the frequency with which they occurred in the Number and Operations strands in grades 3–5 can be found in Table 3.4. In grades 3–5, each lesson includes a Ten-Minute Math activity, listed on the overview page of the teacher’s guide (see Appendix C). These activities are designed to foster fluency and review of recurring concepts. According to the implementation guide, these activities should be inserted into brief chunks of time throughout the day, such as just before or during the transition, or during a morning meeting, but not during the math lesson. INV lessons conclude with a section called Daily Practice, which includes recommendations and resources for reinforcing the content of the lesson and providing homework. Because they were not incorporated into the main body of the lesson, we treated these activities as peripheral and did not Table 3.4  Primary segments in INV lessons and frequency of occurrence in grades 3–5 lessons Segment type Activity Discussion

Workshop Assessment Daily Practice and Homeworkb

Description Combination of whole-class and partner work on a specific task. Teacher facilitated, whole-class conversation focused on specified math focus points; students share strategies and explain thinking. Students work individually or in pairs on several different tasks or games, distributed over several lessons. Formative assessment tasks, focused on unit benchmarks. Indicates one page from Student Activity Book for practice during the lesson and another for homework.

Frequencya 93% 75%

40% 15% 100%

Percentage of lessons containing at least 1 of each type of segment. A number of lessons had more than one activity b The Daily Practice falls outside of the 60-min allotted for each lesson a

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consider them in our analysis of the pedagogical approach, which follows. See Sect. 3.2 for more detail on this analytical decision. Throughout the INV lessons, students are expected to spend a great deal of time either working in pairs on tasks or in whole-class discussions, during which students are asked to explain their thinking or share the strategies they used on the tasks. As shown in the curriculum program profile (Fig.  3.1), 51% of the lesson segments analyzed recommend a whole-class structure; 25% recommended students work in pairs and 25% of the segments were designed for students to work independently.

Participant Structure

Independent 25%

Small group 25%

Teacher’s Role Tell/show: text 0%

Whole class 51%

Step back 33%

Orchestrate 40%

Facilitate 17%

Tell/show 3%

Guide 6%

Nature of Student Learning and Work 67%

60%

57%

40%

Cognitive Demand

25%

Generate/explain/justify

Show work

Describe steps

Answer only

See/describe patterns

6%

1% Prior lessons

0% Student

3%

Student text

12%

Teacher

DM

0% PWEC

0%

PWSC

Mem

0%

PWOC

30%

Sources of Knoweldge Types of Mathematical Work

Fig. 3.1  Curriculum program profile for INV

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Nature of Student Learning and Work We categorized INV as fully dialogic in nature (Munter et al. 2015), based on the teacher’s and students’ roles, the source of knowledge, and the types of mathematical work expected of students. As the profile in Fig. 3.1 shows, in 57% of the lesson segments, students were expected to solve problems using strategies they developed and to explain and justify their answers and approaches. In 25% of the segments, they were expected to show the work they did to arrive at an answer using numerals or representations. Noticing patterns (6%), describing steps (1%), and providing answers with minimal or no explanations (12%) were much less prominent across all INV lessons. This characterization of students’ mathematical work aligns with the analysis of cognitive demand of the main tasks in the lesson, discussed in Chap. 3. The majority of INV tasks (60%) were categorized as doing mathematics and 40% as procedures with extended connections. In a similar vein, in 67% of the lesson segments coded, students were framed as the source of knowledge. In those segments, students are intended to be engaged in generating strategies and understandings, compared to 3% of the segments in which the teacher is guided to provide students with information. Instances in which information is explicitly given by the teacher typically occur after students have engaged in exploratory work, as illustrated in the following example. In a grade 4 lesson on arrays and multiples, for instance, students are asked to figure out the different ways 12 objects can be arranged in a rectangular array. Once the class has generated all possible arrays, labeling each with its dimensions (e.g., 3 × 4), the teacher is guided to explain: “Each of the dimensions on this list is a factor of 12. That means 12 can be divided by each of these numbers with no leftovers.” The teacher is then guided to reiterate the connection by asking, “What are all of the factors of 12?” (GR4 1.1.2, p. 33). We did not find any instances in which concepts or strategies were introduced or presented directly in the student text. The remaining 30% of segments analyzed involved students reviewing and applying concepts and skills learned in previous lessons. The Teacher’s Role: Supporting Student Agency Several characteristics of the INV lessons we analyzed appear to be designed to support students in engaging in the types of strategy-generating work described above. In particular, our classifications of the predominant instructional approach outlined for the teacher surfaced the extent to which teachers were intended to facilitate students’ work as problem solvers and sense makers in the lessons. The most commonly prescribed instructional approach, for 40% of the segments, was orchestrating. In these segments, the teacher was expected to lead a whole-class discussion during which students model and explain their strategies or understandings. In INV, the teacher’s role of orchestrating usually followed a segment in which the teacher was guided to play a facilitating role (occurring in 17% of segments) by supporting

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individuals and groups of students to make sense of and work on tasks. In 33% of segments, the teacher was given the role of stepping back while students worked or played games, designed to provide fluency practice and encourage strategy development, which involved supporting students individually or in small groups to solve problems or make sense of concepts. We coded the teacher’s role as guiding for 6% of the segments and telling or showing for 3%. (See Fig. 3.1.) Example Lesson The following excerpt from the grade 3 lesson, entitled, Adding More than Two Numbers, illustrates many of the patterns described above. The lesson begins with a whole-class activity intended to take 15 min and introduce addition involving multiple addends. The teacher is guided to write the following problem on the board: 186 + 92 + 84 The following appears in blue font, indicating exact words the teacher might use: “Let’s estimate this first before we solve it. How many hundreds do you think the answer will have?” The teacher’s guide further directs the teacher in facilitating student generation and explanation of strategies: Students’ responses are likely to range from 100 (because they see only 1 hundred) to 400. As students give their estimates, ask them to explain their reasoning. Listen for explanations that include thinking of 92 and 84 as each being close to 100, and 186 as being close to 200, making the sum about 400 altogether. Some students may recognize that because each of the numbers is less than the nearest hundred, the sum will be less than 400, so a number in the 300s is a good estimate. (INV GR3 8.2.4, p. 86)

Next, the guide suggests the teacher ask students to solve the problem to get an exact answer, possibly with a partner, keeping track of their thinking on paper and then have several students share their solutions. During the sharing, the guide recommends the teacher focus on “how students broke the numbers apart and recombined them.” It even provides illustrative examples of ways the teacher could press students to explain their thinking further, such as: Denzel, when you added 186 plus 14 to get 200, how were you thinking about solving the problem?

The remaining 45 min of the lesson are devoted to a Math Workshop, during which students estimate and then solve 3-addend addition problems, practice solving addition problems with 3-digit addends, and play a game in which they generate and solve 2- and 3-digit addition problems. In all cases, students are not told what strategy to use and are expected to show their solutions. The teacher’s role during the workshop is to step back and assess student understanding. This lesson is typical of the INV lessons we analyzed in the emphasis on making sense of mathematical concepts, in this case number and operations, and the expectation that students will generate and explain their own strategies based on their under-

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standing. As shown in Table 3.4, the workshop format occurs periodically throughout most INV units, depending on the topic and grade level, but less frequently than other structures, including paired exploratory work or whole-class discussions. Another lesson in the same unit, entitled Strategies for Subtraction, begins with a 40-min activity, during which students work individually on a set of 3-digit subtraction problems, half embedded in story contexts, half presented numerically, while the teacher observes students to assess their understanding. Then, the teacher brings the class together for a 20-min discussion, during which students share their subtraction strategies, first with partners and then with the whole class. During the whole-class discussion, the teacher is encouraged to represent students’ strategies on chart paper, which can be posted in the classroom, “where students can refer to them as they continue to practice subtraction” (INV GR3 8.2.5, p. 136).

3.5.2  Math Trailblazers: An Integrated Approach Three commitments for mathematics learning are highlighted in the MTB materials: First, mathematics instruction should be grounded in real-world experiences that students can relate to; second, mathematics instruction should challenge students intellectually by introducing them to ideas and ways of thinking beyond those common to most elementary curricula; and third, mathematics instruction should embrace and balance the development of advanced mathematical thinking along with learning of procedural skills common to traditional mathematics curriculum programs in the USA. These ideas are reflected in the following excerpt from the Teacher Implementation Guide: The primary goal of Math Trailblazers has been to create an educational experience that results in children who are flexible mathematical thinkers, who see the connections between the mathematics they learn in school and the thinking they do in everyday life, and who enjoy mathematics. We believe that children can succeed in mathematics and that if more is expected of them, more will be achieved. This curriculum incorporates the best of traditional mathematics, while widening the horizons of students mathematically. (TIMS Project 2008, p. v)

Our analysis of the lesson structure, the nature of the learning opportunities for students, and the teachers’ role illustrates how these commitments are incorporated into the designed curriculum. Lesson and Participation Structures The MTB developers anticipate that 60 min will be devoted to mathematics instruction each day. Within those expectations, the typical MTB lesson was designed to last more than one class session. As Table  3.5 shows, just under half of the 30 ­lessons analyzed were designed to take 2 days and only seven lessons were designed to be 1 or 1–2 days.

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Table 3.5  The number of days devoted to the MTB lessons analyzed Estimated class sessions 1 or 1–2 days 2 days 2–3 days

Grade 3 4 4 2

Grade 4 1 5 4

Grade 5 2 5 3

Total 7 14 9

Fig. 3.2  The opening vignette in MTB GR 4.3.1 presents 30 basketballs to be donated to five schools. Excerpt from Math Trailblazers, Grade 4, Unit 3, Lesson 1 Resource Guide 3 by TIMS. Copyright ©2008 by Kendall Hunt Publishing Company. Reprinted by permission

The structure of the lesson, laid out in the teacher’s guide, closely mirrors the contents in the student book. The portion in the guide labeled Teaching the Activity is typically divided into two or three parts. Each part is labeled with a content-­ specific heading, such as, Using Fact Families or Multiplying with 0 and 1. Each lesson is designed to be used in conjunction with the student book. That said, it is evident from the guidance provided that the majority of lesson segments assume a whole-class format (56%), with pair or small-group work periodically interspersed. The rest of the segments were devoted to either small group or pair format (12%) or independent work (32%), however, the participant structure is not always specified and the notes to the teacher suggest that the entire class moves through much of the lesson together, fluctuating somewhat fluidly between whole-group, pairs, and independent structures. The typical MTB lesson begins by guiding the teacher to read or have students read a vignette from the student book, which is used to launch the lesson (see Appendix D). A grade 3 lesson, called Number Sense with Dollars and Sense (GR3 2.8), shows a picture of a farmers’ market with prices of various fruits and vegetables labeled. A grade 4 lesson, Multiplying and Dividing with 5 s and 10s (GR4 3.1), describes a situation in which 30 basketballs are being donated to five different schools, and includes two illustrations: first, a group of 30 basketballs and second, five groups of 6 basketballs each (Fig. 3.2). The student book then leads the students through discussion or questions that involve making sense of the situation mathematically.

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The student book in the grade 4 lesson, for example, introduces the “division sentence” 30 ÷ 5 = 6, along with the terms quotient, dividend, and divisor. The book also extends the story to demonstrate the relationship between 6 + 6 + 6 + 6 + 6, 5 × 6, and 30 ÷ 6. Students are then asked to discuss a similar situation, both in terms of how many balls each school will receive and how the different number sentences represent the relationships in the story. From the opening vignette, MTB lessons typically proceed by introducing new, but related situations or extending the work into more symbolic mathematical territory. Lessons designed to take two or three days include several different activities. In addition to discussing and responding to questions in the student book, students gather and represent data, use concrete materials to model numeric relationships, and play games to develop fluency and strategies. The grade 4 lesson described above, which begins with grouping basketballs, is intended to take two class sessions. After the vignette, the text introduces additional situations involving division and asks students to describe key components of the situation and use number sentences to represent it, emphasizing the relationship between multiplication and division and how these relationships are expressed numerically. Parts 2 and 3 of the lesson continue the focus on multiplicative relationships, but with different contexts, including multiplying large numbers by 1 or 0 and practicing multiplication facts using Triangle Flash Cards. The teacher is encouraged to have students work in pairs to complete all of these activities. Although they were not integrated into the daily lessons, MTB includes a set of Daily Practice Problems (DPP), which are distributed across almost every lesson over the year to review previously taught content and develop fact fluency. These problems are not incorporated into the daily lessons; however, they are briefly referred to at the end of each lesson in the teacher’s guide with a statement of how they might be used, such as practice with Math Facts and for Homework and Practice. Some of the lessons also include a Journal Prompt, listed in the teacher’s guide, but not the student book, providing the option of responding to one of the questions on the student page in a journal. We classified these activities as beyond the main body of the lesson and did not include them in our analysis of the pedagogical approach. Nature of Student Learning and Work Similar to INV, we categorized the approach to supporting learning in MTB as primarily dialogic, particularly when considering the nature of student work and implicit views of student learning. In 46% of the segments coded, students were asked to generate strategies, explain, and/or justify their answers or approaches, as illustrated by examples in the previous paragraph. In 19% of the segments, students were expected to show their work on problems or other tasks. The lesson guide frequently reminds the teacher that students should be encouraged to use strategies that make sense to them and share their strategies with others, as illustrated in the excerpt from a grade 3 lesson guide:

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Keep in mind, however, that the goal of this activity is to engage students in using their own strategies to add and subtract numbers rather than to teach them formal procedures. Give students ample opportunities to share their strategies for solving problems with the class. (GR3 6.1, p. 29)

In 16% of the segments, students were expected to observe, notice, and describe patterns, often in data they generated during the lesson or phenomena introduced through the vignettes that begin each lesson. Twelve percent of the segments asked for students to provide an answer without any explanation and 7% of the segments asked students to describe the steps they used to solve a problem or complete a computational task. This pattern was also reflected in the analysis of the cognitive demand of the main instructional tasks in the lessons, described in Chap. 3. We coded 23% of the tasks as doing mathematics and 72% as focusing on procedures with extended connections. Only 2% of the main tasks focused on procedures with superficial connections and 3% involved procedures without connections to underlying concepts. None of the main tasks focused on memorization. Students were treated as the primary source of knowledge in 60% of the segments. In grade 3, lesson 2.8, the teacher is guided to “Encourage students to develop their own strategies,” and, later, “Let the students teach one another through discussion” (p.  108). In 15% of the segments, the student book was the primary source of knowledge or information, as is illustrated by the number sentences and terms offered by the student book in the lesson described above (GR4 3.1). The teacher was the primary source in 12% of the segments. For example, in the same lesson, the teacher is guided to write four related number sentences on the board and “Tell students that” they “make a fact family” (GR4 3.1, p. 29). The other 13% of segments involved the use of concepts and facts that had been previously learned and were treated as shared knowledge, such as, “Remind students that the two multiplication sentences are turn-around facts (5 × 6 and 6 × 5),” from the same lesson. See Fig. 3.3 for MTB curriculum profile. The Teacher’s Role: Guiding Students to Make Sense It was in our analysis of the teacher’s role that we saw great variation in how the teacher was positioned in relation to student learning. Similar to INV, MTB conceptualizes the teacher as a facilitator and guide in students’ processes of making sense of phenomena and developing mathematical approaches and understandings. At the same time, the teacher’s role was often more leading. In 27% of the segments, the teacher’s role was to orchestrate whole-class discussion and sense making. In 18% of the segments, the teacher was a facilitator, supporting individuals and groups of students to develop strategies and work through problems. In 28% of the segments, the teacher’s role was to guide, carefully leading students down a developmental path from concrete examples to more generalized or symbolic representations. To a large extent, the sequence of questions laid out on the student page formulates this

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Teacher’s Role Tell/show 0% Step back 26%

Independent 32% Whole class 56%

Tell/show: text 6%

Small group 12%

Orchestrate 27% Guide 28%

Facilitate 18%

Nature of Student Learning and Work 72% 60% 46%

Cognitive Demand

Generate/explain/justify

See/describe patterns

19% 16%

Show work

7% Describe steps

Answer only

Prior lessons

13% 12% Student

Student text

12% 15% Teacher

DM

2% PWEC

3%

PWSC

Mem

0%

PWOC

23%

Sources of Knoweldge Types of Mathematical Work

Fig. 3.3  Curriculum program profile for MTB

pathway and the teacher’s role, as characterized in the lesson guide, is to guide students along it. This guiding role is illustrated well by lesson GR4 3.1, described above. The teacher’s guide summarizes the basketball vignette and the multiplication and division number sentences used to represent the relationship, noting that “Students should see that division is the opposite (inverse) operation of multiplication” (p. 28). The student text then presents a different situation, 30 soccer balls and each school receiving a crate of six balls. The questions in the student text ask them to compare the two situations and write a number sentence to describe the new situation. The teacher’s guide provides additional questions the teacher might ask to guide the

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students’ work on the questions in their text, such as “What information is given in the problem?” and “What does the answer, the quotient, tell you?” As the student text proceeds through additional questions, the teacher’s guide suggests how the teacher should support students’ work by explaining the purpose of some of the questions, suggesting that students work in pairs on certain questions, or indicating that the teacher should remind students of something when they get to a particular question. We did not find any segments in which the teacher’s primary role was to present information or show steps, although we found that in 6% of the segments, like the example above, the teacher presented information directly, using material included in the student text. In 28% of the segments, the teacher’s role was to step back, allowing students to work on tasks, gather data, or play games.

3.5.3  E  veryday Mathematics: Combining Meaning and Repeated Practice The EM materials assert that the development of the program was informed by research findings that suggest students in the USA “can, and must, learn more mathematics than has been expected from them in the past” (EM teacher’s guide). Informed by approaches used to teach mathematics in highly successful countries, the authors set out to design a program that builds a deep understanding of concepts over time, along with solid mastery of skills. EM is known for its spiral approach to sequencing mathematics content, in which topics are touched on multiple times across the year, allowing students to deepen their understanding with repeated exposure and practice. Our analysis of the EM lessons from a pedagogical perspective illustrates the ways that these commitments are combined and integrated into daily lessons. Lesson and Participation Structures Daily lessons in EM are intended to take 60 min and are comprised of several consistent elements. Each lesson begins with three recurring activities, listed under the heading Getting Started in the teacher’s guide (see Appendix B). The first, Mental Math and Reflexes, includes a small set of exercises that students are expected to complete mentally; the second, Math Message, provides a question or problem the teacher writes or projects on the board, so that students can work on it prior to the formal beginning of the lesson; the third, Study Link Follow-Up, offers a way to have the class follow-up on the previous day’s homework. Although these activities are included in each EM lesson guide, they are not included in the lesson overview or as part of the daily lesson. The daily lessons are composed of two types of activities: First, several activities, described under the heading, Teaching the Lesson,

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engage the students in the primary mathematical focus of the lesson. Then, the lesson moves into the Ongoing Learning and Practice section, which provides “essential review and practice for maintaining skills” (teacher’s guide). During this portion of the lesson, students review previously taught material by playing games or completing a student page from the Math Journal, or student textbook. This segment of the lesson consistently includes a student page called Math Boxes. A hallmark of the EM program, the Math Boxes page is composed of four to six mixed review problems that give students practice with different material previously taught. As with INV and MTB, games are also used consistently throughout EM to incorporate practice, fluency, and strategy development. Each lesson in the EM teacher’s guide also includes a section entitled Differentiation Options. This section includes suggestions and resources to provide students with enrichment and extra practice. We classified these sections, along with those listed under the heading Getting Started, as beyond the main body of the lesson and did not include them in our analysis of the pedagogical approach. Although specific times are not provided in the teacher’s guide, the activities described under the heading Teaching the Lesson are intended to take a significant portion of the lesson. It is during these activities that new concepts are introduced and developed. Each begins with a whole-class discussion, described under the heading Math Message Follow-Up, during which the teacher is guided to ask students to share strategies used to complete the Math Message task and then develop the concept further. Following the opening discussion, this segment of the lessons typically includes two or three additional tasks, which generally develop the concept further, while engaging students in whole-class, partner, or independent activities. The following example from grade 4 lesson 3.10 illustrates the relationship between the Math Message, the Math Message Follow-Up, and the remaining segments of the lesson. The Math Message is as follows: Tell whether each number sentence is true or false. 28 – 6 + 9 = 31 28 – 6 + 9 = 13 Be ready to defend your answer. (p. 209) During the Math Message Follow-Up, the teacher leads the class in discussing how each number sentence can be true and then introduces and demonstrates the use of parentheses to “show which operation to do first” (p. 210). The next two lesson segments build on this idea. First, the teacher writes several number sentences containing parentheses on the board and asks students to indicate T or F. Then the students work with partners on a journal page involving using parentheses when computing answers and adding parentheses to number sentences to make them accurate. The EM teacher’s guide specifies the intended participant structure for each lesson activity. As the above description illustrates, students spend a great deal of time working as a whole class (45% of the segments analyzed) or in pairs or small groups (25%). Thirty percent of the lesson segments analyzed were intended for independent work.

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Nature of Student Learning and Work Lessons in EM represent a mixture of dialogic and direct approaches. The lessons were designed to engage students in exploring mathematical ideas and relationships, but also place a strong emphasis on repeated practice and student production of answers. For example, students were asked to provide answers with minimal or no explanation in 59% of the segments analyzed. This large proportion of tasks requiring minimal explanations is explained in part by the fact that mixed practice problems were included in every lesson. Students were expected to describe the steps they used in 20% of the segments and show their work and thinking in 2% of the segments. They were asked to generate, explain, or justify their answers or strategies in 14% of the segments and notice or describe patterns in 5% of the segments. Our analysis of the cognitive demand of the major tasks in each lesson segment highlights that emphasis in EM is on concepts and underlying meanings. The largest proportion of tasks focused on procedures with extended connections to concepts (75%). We coded 16% of the tasks as doing mathematics, including a number of games that require students to develop and use strategies. The remainder of the tasks were coded as low demand, 5% being procedures without connections and 4% being memorization. Our analysis of the implied sources of knowledge during the lessons suggested that the teacher and the students shared some intellectual authority. In 30% of the segments, the teacher was the source of information or responsible for determining correctness. In 24% of segments, the students were the source. The student text was the source in 6% of the segments. And in 39%, the knowledge came from previously learned material, usually in the form of review tasks. The way these sources of knowledge come into play during a lesson is illustrated by grade 4 lesson 3.10, briefly introduced previously. The students explore two number sentences and are encouraged to consider how both can be true. The teacher then introduces the use of parentheses and shows how they can be used to show which operation to complete first. The students then practice by evaluating number sentences containing parentheses to determine whether they are true or false. The main task of this segment was coded as procedures with extended connections, but much of the follow-up work the students are asked to do involves providing answers or showing their work. See Fig. 3.4 for curriculum profile. The Teacher’s Role: Multiple Approaches to Supporting Student Learning In keeping with the mixture of dialogic and direct approaches described above, the teacher’s role also varied from telling and showing (18%), to guiding (20%), to facilitating (15%). Only 2% of the segments we coded led the teacher to orchestrate a whole-class discussion in which students generated strategies or explanations for them. In 43% of the segments, the teacher’s role was to step back, while students worked in groups or individually. Many of these instances occurred during the Ongoing Learning and Practice portion of the lesson, when students were expected

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Participant Structure

Teacher’s Role Orchestrate 2%

Independent 30%

Guide 20% Whole class 45%

Step back 43%

Small group 25%

Facilitate 15% Tell/show 18%

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Sources of Knoweldge Types of Mathematical Work

Fig. 3.4  Curriculum program profile for EM

to complete journal pages associated with the lesson, work on Math Boxes pages, or play games as a form of practice. In only 2% of the segments, students are shown or provided information through the student textbook. The teacher’s role during the lesson segment described above (GR4 3.10) was coded as telling or showing because the guide led the teacher to introduce parentheses as a mathematical convention and lead the students through exercise during which they used them to communicate and interpret the meaning of expressions. In a grade 3 lesson on Finding Differences, the teacher is led to support students as they use a number grid to find differences between 2-digit numbers. The prompt in

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the teacher’s guide reads: “Use your number grid. Start at 30. How many spaces is it to 57?” (GR3 1.8, p. 52). The teacher is then encouraged to give students a minute to find the difference before asking for volunteers to “describe what they did.” The teacher then introduces the term “difference” and its meaning and leads the students through several more problems as a whole class, before asking them to complete a journal page of related problems. This segment, illustrates the guiding role, commonly recommended in the EM teacher’s guide. We coded the mathematical task as procedures with extended connections and both the teacher and the student as sources of knowledge.

3.5.4  Math in Focus: Scaffolding Student Learning The MIF teacher’s guide and related documents present the program as “drawing on success in Singapore,” referring to the nation’s success on international comparisons, such as TIMSS. The description of the program goals appears to be similar to those embraced by EM, including attention to both concept and skill development and digging more deeply into fewer topics in a single year. The material also emphasized a strong focus on the use of models and a developmental progression that moved from concrete to pictorial to abstract. As discussed in Chap. 2, MIF employs number models or representations common to the Singapore approach, such as bar models and number bonds. Examining MIF from a pedagogical perspective uncovered some differences that distinguish it from the three programs discussed previously. The authors highlight a scaffolded instructional approach, which begins with concepts being “presented in a straightforward visual format, with specific and structured learning tasks,” followed by guided and then independent practice (MIF teacher’s guide). In the USA, particularly in reading instruction, this approach is often referred to as a gradual release of responsibility model (Duke & David Pearson, 2002) and is aligned with the direct instructional model described by Munter et al. (2015). Lesson and Participation Structures Many of the MIF lessons in the teacher’s guide were designed to take more than one session. Table 3.6 shows the number of days given for the 30 lessons we analyzed. Each lesson begins with a 5-Minute Warm Up activity, which typically involves some sort of mental fluency activity, intended to prepare the students for the coming lesson. The 5-Minute Warm Up for grade 4, lesson 2.2 states: “Have students recite the multiplication tables through 10 × 10 with a partner. This activity prepares them for easy recall of multiplication and division facts to find factors of numbers” (p. 44). Other 5-Minute Warm Up activities involve using counters to show groups (grade 3), placing numbers on an open number line (grade 4), or translating a story problem to an algebraic expression (grade 5).

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Table 3.6  Number of days allotted for the MIF lessons analyzed Estimated class sessions 1 day 2 days 3 days

Grade 3 8 2

Grade 4 1 6 3

Grade 5 4 4 2

Total 13 12 5

After the warm-up, the MIF lessons are primarily teacher led and closely aligned to the content of the student textbook. In fact, a statement of the lesson objective can be found on the first page of the lesson in the student textbook, just as it is written in the teacher’s guide. Following a gradual release of responsibility model (Duke & David Pearson, 2002), the instruction begins with a segment of direct instruction, during which the teacher walks students through examples and exercises on the student page. These segments are labeled Learn in the student text and Teach in the teacher’s guide. Next, the students engage in Guided Practice, labeled as such in both the teacher’s guide and student text and composed of a set of exercises that involve and extend the concepts just introduced. Finally, the text includes a page labeled Let’s Practice, which includes additional opportunities for students to practice the strategy or skill taught during the lesson. It is often the case that additional Learn and Guided Practice segments are included before the final Let’s Practice segment. In grade 4, lesson 2.2, after the warm-up, described above, the teacher is guided to introduce the concept of factors by explaining defining characteristics, including “that the factor of a number is a whole number and when a given number is divided by its factor, it does not leave any remainder” (p. 44). The teacher shows students how to find factors using division. During the Guided Practice portion of the lesson, the students work in pairs to find the factors of 32 and 24. A second Learn segment follows, during which the teacher is guided to use the student text which shows students how to use the long division algorithm to determine if one number is a factor of another number. After another brief segment of Guided Practice, the teacher takes the class through two more Learn and Guided Practice segments, before having students complete the Let’s Practice pages. Most of the lessons we analyzed also included one or two activities that did not follow the gradual release model. All of these activities were related to the concepts being taught in the lesson, but engaged students in different types of exploration or reflective tasks, including working with materials, playing games, solving problems, or writing explanations of mathematical concepts. These activities typically occurred during the guided practice portion of the lesson. The most common of these was labeled, Hands on Activity, and involved working with models or diagrams that represented concepts in the lesson. The Hands on Activity in grade 4, lesson 2.2 is titled Find Prime Numbers to 50 and introduces the “Sieve of Eratosthenes” to eliminate composite numbers and identify primes. Students work in pairs to complete the activity.

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Table 3.7 shows the distribution of the different types of activities across the 30 lessons analyzed. All but 20% of the lessons included at least one of these activities and several included two. In keeping with our approach to focusing our analysis on the main body of the lesson, we did not include the 5-Minute Warm Ups or differentiation options in our analysis of the daily lessons. We did incorporate the practice pages and Exploration activities in our analysis of the pedagogical approach. The majority of lesson segments in MIF (49%), including the direct instruction component of each lesson, were intended to occur in a whole-class structure. We coded the guided practice and practice components of the lessons as independent, unless otherwise specified. These comprised 33% of the lesson segments analyzed. In some cases, the teacher’s guide recommended that students complete some of the guided practice problems with a partner. Another 18% of lesson segments indicated students should work in pairs or small groups. Nature of Student Learning and Work Lessons in MIF consistently followed a direct instructional approach. Student exploration was not fully absent, but was heavily guided by the teacher or the text. In 33% of the segments analyzed, the teacher was the primary source of knowledge and the textbook was the primary source in 31% of the segments. The student was the source of knowledge in 11% of the segments. In 25%, the knowledge came from previously learned material. Figure 3.5 helps to illustrate the central roles the teacher and the student text, together, played as sources of knowledge. The guidance under the heading “Teach” is replicated from the teacher’s guide and corresponds with page 86 in the student textbook. The student textbook is partially replicated on the left. (See Remillard 2018 for full reprint.) The implication is that students follow Table 3.7  Distribution of exploration activities across MIF lessons by grade level Hands on activity—Students work with visual or concrete models. Game—Students play game that allows for practice with concept. Critical thinking skills—Students solve nonroutine problems. Math journal—Students write problems or explanations of concepts. Let’s explore—Students work with a partner to explore and observe patterns. None

Grade 3 Grade 4 Grade 5 Totala 4 4 5 13 4

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Learn

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Multiply by Tens

Learn

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Kevin had 4 bags of apples. Each bag contains 10 apples. How many apples does Kevin pack altogether? 4 x 10 = 4 x 1 ten = 4 tens = 40 Kevin packs 40 apples. Rafael buys 3 packages of crayons. Each package contains 20 crayons. How many crayons does Rafael buy? 3 x 20 = 3 x 2 tens = 6 tens = 60 Rafael buys 60 crayons.

Teach

Multiply by Tens (page 86)

Students learn to multiply by 2-digit numbers in the form of tens. • Help students recall the strategy of multiplying a number by tens by working through the examples in the student book. • In the first example, express 10 as 1 ten. So 4 x 1 ten = 40. • Using the strategy, work through the second example with students. • First, express 20 as 2 tens. So 3 x 20 = 3 x 2 tens = 6 tens = 60. • For students who cannot visualize multiplying by tens, use a place-value chart to show the connection.

Student Book A p. 86

Fig. 3.5  Illustration of the relationship between the student text and teacher’s guide in MIF. This illustration was created by the authors based on images that appears in MIF GR4 teacher’s guide (MIF GR4.3.2, p. 86)

along on their page, while the teacher provides related instruction to the whole class. Following the direct teaching guidance in the teacher’s guide, shown in Fig. 3.5, there is a teacher note with the heading: Best Practices, which illustrates the somewhat superficial references to student-generated knowledge and partner work. The note states, “After teaching each Learn section, have students work in pairs to write a question each, based on what they have learned” (MIF GR4.3.2, p. 86). Students should then answer each other’s questions. In keeping with this trend, our analysis of the types of mathematical work expected of students indicated that students’ primary role was to provide answers with no explanation in 67% of the segments. In 22% of the segments, students were expected to show or describe the steps they used to solve a problem. Students were expected to notice or describe patterns in 8% of the segments and justify or explain their answers in only 3% of segments. These patterns were supported by the cognitive demand analysis presented in Chap. 3. Our analysis of the main tasks of each segment indicated that 36% emphasized procedures with superficial connections. These tasks tended to use conceptual models to introduce concepts, but moved somewhat quickly to abstract notations without maintaining a solid connection to underlying meanings. See Fig. 3.6 for the curriculum profile for MIF.

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Fig. 3.6  Curriculum program profile for MIF

The Teacher’s Role: Scaffolding Students’ Learning Through the Text Figure 3.5 also illustrates the teacher’s role in MIF, which is both directive and text based. All of the lessons are designed to be used along with the student textbook. In fact, 34% of the segments involved the teacher telling or showing by use of the student text. In 45% of segments, the teacher steps back while students work on the textbook pages. In 21% of the segments, the teacher plays a guiding role, as illustrated by the Hands on Activity described above.

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3.5.5  Scott Foresman–Addison Wesley: A Mathematics Textbook Approach Scott Foresman–Addison Wesley Mathematics, originally developed by publisher Scott-Foresman, before it merged with Addison–Wesley, represents a typical publisher developed program (Stein et  al., 2007) available in the USA prior to the release of the CCSSM. We selected SFAW to illustrate the mainstream approach to elementary curriculum materials that dominated the U.S. textbook market for decades, including the 1990s and 2000s, when the other four programs analyzed were being developed. As discussed at the beginning of this chapter, attending to teaching practices was a prominent feature of mathematics textbooks prior to the 1990s. And, as we discuss in the following section, the teacher’s role remains in the background. SFAW offers a textbook-centric approach to mathematics instruction to which elements of differentiated instruction and problem solving have been added, somewhat around the edges. Lesson and Participation Structures SFAW lessons are highly consistent in their design. The teacher’s guide includes two pre-lesson activities, which can be found on a page prior to the lesson description. (See Appendix E.) Both appear under the heading, “Getting Started.” The first includes two “spiral” review tasks: a problem of the day and a worksheet, containing review problems from previous lessons. The section activity is labeled “Investigating the Concept” and is a 10- to 15-min activity around the topic of the lesson. Many of these activities involve concrete or visual representations and students working in pairs or the whole class to answer questions posed by the teacher. The bulk of the lesson is oriented around the pages in the student textbook. It begins with a short warm-up, during which the teacher is guided to “activate” students’ prior knowledge by reviewing a previously taught concept related to the lesson topic. A grade 5 lesson entitled Place Value Through Thousandths, for example, suggests the teacher “review the place and value for digits in whole numbers.” The next section entitled Teach, corresponds with the first page of the student textbook, entitled Learn (the first two pages for grade 5), which introduces the students to the main concepts of the lesson. Two questions at the bottom of the page, allow the teacher to check students’ understanding. For the remainder of the lesson, students complete practice problems in the textbook independently. The margin of the teacher’s guide includes one or two suggestions for checking student understanding as they work. Each includes an “If” followed by an error a student might make, such as misread decimals using the wrong place value, and a “Then,” which suggests how the teacher might respond to the error (see Chaps. 6 and 7 also). In addition to practice problems, each lesson includes several problems under the headings, reasoning, and problem solving, test-taking practice, and extensions.

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The final activity in the lesson, in the teacher’s guide, but not on the student page, falls under the heading Assess in the teacher’s guide and offers a Journal Idea. These tasks engage students in using their concepts taught in the lesson, often through writing. The Journal Idea accompanying the grade 5 lesson on Place Value Through Thousandths states: “Remind students that a meter stick shows 100 centimeters. Have them explain how a meter stick can be used to represent hundredths and thousandths” (GR5 1.3, p. 11). In keeping with our approach to focusing our analysis on the main body of the lesson, we did not include the Getting Started suggestions or any differentiation options in our analysis of the daily lessons. We focused our analysis on the instruction surrounding the student textbook pages represented as the core component of the lessons. As shown in the curriculum profile for SFAW (Fig. 3.7), the two dominant participant structures in SFAW are the whole-class format (50% of the segments coded) and independent work (47%). Only 3% of the segments explicitly suggest a small group or paired activities. Nature of Student Learning and Work Similar to MIF, the lessons in SFAW are primarily direct in their instructional approach and are oriented around the student textbook. Students have some, but minimal, opportunities to engage in dialogic practices. SFAW differs from MIF in that the textbook appears to provide the majority of instruction and the teacher’s role appears minimal. In the grade 5 lesson referenced above, the images and text on the student page show “different ways to represent 1.5.16. The images include a grid sectioned into hundredths, a number line, and a place-value chart. The number is shown in expanded form, standard form, and written in words. In contrast to this detail, the guidance under the heading Teach in the teacher’s guide states: Using place-value blocks or girds for decimals requires defining what ‘1 whole’ means. If a 10-by-10 square is 1 whole, then 1 row of 10 small squares is 0.1 and 1 square is 0.01” (GR5 1.3, p. 8). As is illustrated in the example above, we found the student textbook to be the primary source of knowledge in 41% of the segments, compared to the teacher, who was the primary source in 19% of the segments. In 38% of the segments, the ­knowledge came from previously learned material. The students were treated as the source of knowledge in only 3% of the segments. Similarly, our analysis of the types of mathematical work expected of students revealed that they were expected to show and describe the steps of procedures in 44% of segments and provide answers with no explanations in 39% of segments. In 10% of the segments, students were expected to generate, explain, or justify their answers and in 7%, they were expected to notice or describe patterns. Opportunities to explain or justify answers tend to occur after a concept is introduced on the student page and more frequently in grade 5 than grades 3 or 4. A

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Teacher’s Role Guide 2%

Tell/show 0% Independent 47%

Whole class 50%

Facilitate 2%

Step back 48%

Small group 3%

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Orchestrate 0% Nature of Student Learning and Work 50% 42%

41%

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Sources of Knoweldge Types of Mathematical Work

Fig. 3.7  Curriculum program profile for SFAW

question at the bottom of the student page described above notes that there are two 1 s in 1.516 and asks, “Does each 1 have the same value? Explain.” As shown in Fig. 3.7, the majority of tasks in the SFAW lesson were coded as procedures without connections to meaning (50%) or procedures with superficial connections (42%). We did not find any instances of procedures with extended connections to meaning, since the type of visual scaffolding used to introduce a concept was typically removed by the second page of the lesson. We did find a relatively small number of tasks focus on doing mathematics (8%).

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The Teacher’s Role: Deferring to the Textbook As mentioned above, the SFAW teacher’s guide provides minimal guidance for the teacher and includes much of the mathematical instruction on the student page. Our analysis suggested that the teacher had two primary roles: telling and showing by way of the student textbook (48%) and stepping back while students practiced (48%). Teachers were expected to play guiding roles in 2% of the segments and facilitating roles in another 2%. We did not find any instances in which teachers are expected to directly present or demonstrate material without the use of the student textbook.

3.6  Looking Comparatively at Pedagogical Approach We conclude our analysis of the pedagogical approaches of five curriculum programs by considering what we have learned about the programs and about examining the intended pedagogical approach in curriculum materials. We also touch on what is more challenging to learn from such an analysis and a possible limitation of the analytical decisions we made. Our analysis was heavily influenced by Munter et  al.’s (2015) characterization of direct and dialogic instructional models. These two models capture a number of key differences that were evident in our findings and our analysis may offer additional ways to understand these models, particularly from the perspective of curriculum materials. At the same time, our analysis suggests that a single continuum is insufficient for understanding how pedagogical approaches can be identified in the design of curriculum materials. We suggest that considering a constellation of features, as Hiebert et al. (2005) propose, surfaces more degrees of variation and uncovers aspects of pedagogical design likely to matter for students’ experiences as learners.

3.6.1  Considering a Dialogic Versus Direct Lens In the previous section, we sequenced our description of the curriculum programs from what we determined to be the most dialogic to direct, based on Munter et al.’s (2015) characterizations. Looking at the programs in this sequence illustrates the continuum itself and surfaces some important differences in what it means for a program to adopt a dialogic or direct approach. For instance, we identify both INV and MTB as dialogic in approach, considering their tendency to position students as the primary source of knowledge, expect them to explain and justify answers, and their consistent use of high cognitive demand tasks. At the same time, several features of these two programs represent differences within the dialogic end of the continuum. INV uses more doing mathematics tasks and positions the teacher as orchestrator of whole-class discussion and sense making, whereas MTB uses more

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procedures with extended connections tasks and leads the teacher to play several roles, including guiding, which is more teacher directed, along with facilitating and orchestrating. We identified MIF and SFAW as both relying on primarily direct approaches. The teacher or the text is the primary source of knowledge and the teacher’s primary roles are to instruct students through showing or telling, often in conjunction with the textbook, and stepping back while students practice. Importantly, two characteristics of MIF align it more closely with direct models of instruction characterized by Munter and colleagues, which emphasize rigorous and well-sequenced content and a highly active role for the teacher. The majority of MIF tasks, for example, involve procedures with superficial or extended connections, whereas SFAW has very few high demand tasks. Moreover, MIF positions the teacher as a primary guide for students, whereas SFAW gives the teacher a minimal role. We view EM as occupying the middle ground along this continuum, adopting somewhat of a blended instructional approach, attending to both dialogic and direct features. The program places a strong emphasis on high cognitive demand tasks, but often positions the teacher as the source of knowledge and in a guiding or telling role. The consistent emphasis on review of previously learned material through the program means that students are frequently expected to provide answers with no explanation.

3.6.2  From a Constellation of Features Lens While the direct to dialogic continuum allows us to consider underlying assumptions about learning mathematics and the associated role of the teacher along a single continuum, our analysis suggests that a single continuum may be insufficient for understanding pedagogical approach in curriculum materials. The analytical framework we used was intended to capture a constellation of features that contributed to shaping the pedagogical approach, including lesson and participant ­structures, the nature of student work, and the teacher’s role. Comparing the five programs with respect to these factors allowed as to surface some more nuanced variations across the programs. When considering the sources of knowledge dimension, for instance, recall that both INV and MTB tended to position students as the primary source of knowledge, locating in them considerable authority for determining correctness. When combining this analysis with the primary teacher’s role, we noticed that INV’s tendency to assign the teacher an orchestrating role locates this authority with the classroom community of students, whereas, MTB’s emphasis on facilitating and guiding locates the authority with individual and small groups of students. When comparing the sources of knowledge among the most directive programs, we found that the authority rests primarily with the teacher in MIF and with the textbook in SFAW. Looking closely at several dimensions also helps to illustrate EM’s blended approach. Despite our finding that the main tasks in EM tended to be high demand,

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our analysis of the type of work expected of students across the lesson segments revealed more emphasis on providing answers than on explaining and justifying. Further, our analysis of the teacher’s role revealed comparable attention to telling and showing as guiding and facilitating and minimal attention to orchestrating. In other words, the roles ascribed to teachers and students in the teacher’s guide seem likely to moderate how high cognitive demand tasks play out.

3.6.3  Remaining Curiosities and Limitations Two coding categories were somewhat challenging to interpret in terms of their broader meanings, based on available evidence. First, in our analysis of the teacher’s role, we found that stepping back played a substantial role across all five programs. Between 25% (MTB) and 48% (SFAW) of segments were coded this way. In some sense, it is not surprising that all curriculum programs anticipate times that teachers allow students to work independently or in groups, either completing practice pages or playing games intended to develop strategies and fluency. In most of the programs, it was often challenging to glean what the teacher was intended to be doing during these periods. Second, for all but MTB, previously taught content was a substantial source of knowledge. In some cases, these previously taught concepts were being further developed, in other cases they were being reviewed and practiced (see Chap. 2). Most frameworks for examining pedagogical approach tend to focus on how new concepts are introduced to students and have much less to say about the role of review and practice. We made the decision to include all segments of the main body of the lesson in our analysis of pedagogical approach, because of their likely impact on students’ experiences and opportunities to learn. Depending on the amount of attention given to review in each lesson, these segments influenced our overall analysis of pedagogical approach. Had we opted to focus our analysis on the introduction of new concepts and skills alone, our results would have been different, especially for EM.  Our analytical approach sought to characterize the nature of learning and teaching throughout the full lesson.

References Ball, D.  L., & Cohen, D.  K. (1996). Reform by the book: What is—or might be—the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–8. Boesen, J., Helenius, O., Bergqvist, E., Bergqvist, T., Lithner, J., Palm, T., & Palmberg, B. (2014). Developing mathematical competence: From the intended to the enacted curriculum. The Journal of Mathematical Behavior, 33, 72–87. Boston, M. D., & Candela, A. G. (2018). The instructional quality assessment as a tool for reflecting on instructional practice. ZDM Mathematics Education, 50, 427–444.

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Century, J., Rudnick, M., & Freeman, C. (2010). A framework for measuring fidelity of implementation: A foundation for shared language and accumulation of knowledge. American Journal of Evaluation, 31(2), 199–121. Charles, R. I., Crown, W., Fennell, F., et al. (2008). Scott Foresman–Addison Wesley Mathematics. Glenview, IL: Pearson. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Duke, N. K., & David Pearson, P. (2002). Effective practices for developing reading comprehension (pp.  205–242). In: What research has to say about reading instruction (3rd ed.). International Reading Association. Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: development status and directions. ZDM The International Journal on Mathematics Education, 45(5), 633–646. Franke, M. L., Kazemi, E., & Battey, D. (2007). Mathematics teaching and classroom practice. In F.  K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 225–256). Charlotte, NC: Information Age Publishing. Heck, D. J., Chval, K. B., Weiss, I. R., & Ziebarth, S. W. (2012). Developing measures of fidelity of implementation for mathematics curriculum materials enactment. In D. J. Heck, K. B. Chval, I. R. Weiss, & S. W. Ziebarth (Eds.), Approaches to studying the enacted mathematics curriculum (pp. 67–87). Charlotte, NC: Information Age Publishing. Hiebert, J., Stigler, J. W., Jacobs, J. K., Garnier, H., Smith, M. S., Hollingsworth, H., Manaster, A., Wearne, D., & Gallimore, R. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 Video Study. Educational Evaluation and Policy Analysis, 27(2), 111–132. Hiebert, J., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, ME: Heinemann. Houang, R. T., & Schmidt, W. H. (2008). TIMSS international curriculum analysis and measuring educational opportunities. 3rd IEA International research conference TIMSS, 1–18. Retrieved from http://www.iea.nl/fileadmin/user_upload/IRC/IRC_2008/Papers/IRC2008_Houang_ Schmidt.pdf Howson, G. (2013). The development of mathematics textbooks: Historical reflections from a personal perspective. ZDM The International Journal on Mathematics Education, 45(5), 647–658. Jones, K., & Fujita, T. (2013). Interpretations of National Curricula: The case of geometry in textbooks from England and Japan. ZDM The International Journal on Mathematics Education, 45, 671–683. Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. Munter, C., Stein, M. K., & Smith, M. S. (2015). Dialogic and direct instruction: Two distinct models of mathematics instruction and the debate(s) surrounding them. Teachers College Record, 117(11), 1–32. National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. NCTM. (1991). The professional standards for teaching mathematics. Reston, VA: Author. Osterholm, M., & Bergqvist, E. (2013). What is so special about mathematical texts? Analyses of common claims in research literature and of properties of textbooks. ZDM The International Journal on Mathematics Education, 45, 751–763. Otten, S., & Soria, V.  M. (2014). Relationships between students’ learning and their participation during enactment of middle school algebra tasks. ZDM The International Journal on Mathematics Education, 46, 815–827. Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teachers’ work and interactions: A collective perspective on resources, their use and transformation. ZDM The International Journal on Mathematics Education, 45(7), 929–943.

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Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French, and German classrooms: A way to understand teaching and learning cultures. The International Journal on Mathematics Education (ZDM), 33(5), 158–175. Remillard, J.  T., & Heck, D.  J. (2014). Conceptualizing the curriculum enactment process in mathematics education. ZDM The International Journal on Mathematics Education, 46(5), 705–718. Remillard, J.  T., & Reinke, L.  T. (2017). Mathematics curriculum in the United States: New challenges and opportunities. In D.  R. Thompson, M.  A. Huntly, & C.  Suurtamm (Eds.), International perspectives on mathematics curriculum (pp.  131–162). Greenwich, CT: Information Age Publishing. Remillard, J. T. (2018). Examining Teachers’ Interactions with Curriculum Resource to Uncover Pedagogical Design Capacity. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Recent Advances in Research on Mathematics Teachers’ Textbooks and Resources (pp. 69–88). New York: Springer. Senk, S. L., & Thompson, D. R. (2003). Standards-based school mathematics curricula: What are they? What do students learn? Mahwah, NJ: Lawrence Erlbaum Associates. Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344–350. Stein, M. K., Grover, B. W., & Henningsen, M. A. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classroom. American Educational Research Journal, 33(2), 455–488. Stein, M. K., Remillard, J. T., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–369). Greenwich, CT: Information Age Publishing. TIMS Project. (2008). Math trailblazers teacher implementation guide, grades 3, 4, 5. Dubuque, IA: Kendall/Hunt Publishing Company. TERC. (2008). Investigations in Number, Data, and Space (2nd edition). Glenview, IL: Pearson Education Inc. TIMS Project (2008). Math Trailblazers (3rd Edition). Dubuque, IA: Kendall/Hunt Publishing Company. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill. Vincent, J., & Stacey, K. (2008). Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS Video Study criteria to Australian eighth-grade mathematics textbooks. Mathematics Education Research Journal, 20(1), 82–107.

Chapter 4

Authors Retrospective Reflections on Designing Opportunities for Student Learning Ok-Kyeong Kim and Janine T. Remillard

Abstract  This volume presents our analysis of five elementary mathematics curriculum programs from three perspectives that have implications for teachers’ use of them: mathematical emphasis, pedagogical approaches, and how authors communicate with teachers. In this chapter, we offer perspectives on these design features from the vantage point of three teams of curriculum authors. Drawing on extensive conversations with the author teams, we describe their reflections on several major design decisions, including mathematical goals and sequencing, pedagogical approach, and communicating intentions to teachers. Our aim is to highlight the relationship between curriculum authors’ intention, their subsequent design decisions, and the resulting curriculum resource. By using the curriculum authors’ voices, we attempt to highlight their intentions, positions, and reflections. We also describe the ongoing process of curriculum development and the evolving nature of curriculum, along with challenges the curriculum authors faced. In doing so, we seek to juxtapose our analysis of the designed curriculum materials with the authors’ intentions. Keywords  Curriculum analysis · Mathematics curriculum materials · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman–Addison Wesley Mathematics · Curriculum development · Curriculum author · Communication with teachers · Authors’ intentions · Design decisions

O.-K. Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA e-mail: [email protected] J. T. Remillard (*) Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_4

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4.1  Introduction This volume presents a range of analyses of curriculum resources focusing on opportunities for student learning and the potential of curriculum resources, especially from the users’ (teachers’) perspective. In these analyses, we attend to the nature of the intended opportunities to learn mathematics designed into the curriculum programs and how the authors communicate their intentions with teachers. In this chapter, we present the perspectives of the authors from three of the five curriculum programs—their goals, decisions, rationales, efforts, and reflections. Although they all intended to develop curriculum resources to support students’ understanding of and success in mathematics, the authors made a number of very different decisions in developing their programs. This chapter highlights the types of design decisions curriculum authors make, questions they ask, challenges they face, and what the user (the teacher) encounters in the curriculum resources. This chapter also illustrates how similar goals and intents play out in different ways and illuminates how the constraints of the curriculum design process can lead to potential gaps between author intention and the actual product.

4.1.1  The Author Interviews We were able to schedule group interviews with curriculum authors of three of the programs analyzed1 (see Chap. 1 for details about the programs): three authors from Everyday Mathematics (EM), two from Investigations in Number, Data, and Space (INV), and two from Math Trailblazers (MTB). The three programs were developed through funding from the National Science Foundation for instructional materials, originally awarded in 1991, to support the changes called for in the National Council of Teachers of Mathematics (NCTM) Standards (Senk & Thompson, 2003). We spoke with Andy Isaacs, Debbie Leslie, and another author of Everyday Mathematics. Andy worked on the second edition of the curriculum and was the Director of the third and fourth editions; Debbie also worked on the second, third, and fourth editions, and was a grade-level leader for the third and fourth editions. The other author we spoke with also had development and leadership roles in these editions. We also spoke to Catherine R. Kelso, who worked on all four editions of Math Trailblazers and Jennifer Mundt Leimberer, who led the fourth Edition revisions of the curriculum. Finally, we spoke with Susan Jo Russell and Karen Economopoulos, who worked on all three editions of the Investigations in Number, Data and Space curriculum and had leadership roles on the design team. In this chapter, we use the term “team” to refer to the curriculum

 We made efforts to contact all curriculum authors, but our requests were not returned from the authors of Math in Focus (MIF) and Scott Foresman–Addison Wesley Mathematics (SFAW). 1

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authors we spoke with in the interviews. The interviews were conducted while the analyses in this volume were being finalized. We developed common interview questions for all three programs. In particular, we created questions that helped us dig into various design decisions (e.g., mathematical emphasis, sequencing, and pedagogical approaches) each curriculum team made and the thinking behind them. We also generated questions general to each program overall and those particular to the specific edition that we analyzed in this volume, and had questions about anticipated challenges in using the programs and teacher feedback. In addition, we included some questions specific to each curriculum team regarding some findings in the analysis of the program. For example, we asked the EM team to explain the difference in objective, key activities, and key concepts and skills listed in each lesson. One week prior to each interview, we sent to the curriculum authors the interview questions (see Appendix A) and a sample lesson of our choice to facilitate the conversations with the curriculum teams. Each curriculum team also sent us sample lessons and additional documents to highlight what they intended in various decisions they made and their rationale. Each interview lasted one and a half to two hours. Although we had the common set of questions prepared, the three interviews had slightly different flows depending on the authors’ comments and our inquiry in specific areas. Overall, we were able to ask the questions prepared and the authors provided additional clarifications and elaborations. Through the interviews, the curriculum authors reflected on their own work and articulated their decision-making. Each interview was videotaped and transcribed. Each interview transcript was read multiple times while the main ideas and key points were identified and refined. Then, for each theme, such as overall program goals, the three programs were compared to tease out common aspects and distinct characteristics. We first attended to the authors’ design decisions on various aspects of curriculum development, such as sequencing the mathematics content and communicating to teachers. Then, we conducted another layer of analysis to find additional themes surfacing in developing a curriculum program. Ten years after the publication of the curriculum programs (the editions we analyzed in this volume), the curriculum authors talked about those particular editions, but they also explained how they made decisions in general. Although the curriculum authors largely described curriculum programs from a particular period in time, our intention is to focus on design decisions, curriculum components, the relationship between author intention and design decisions, and what that ends up looking like in a designed resource in general. Each interview captured different aspects of processes and different challenges, and as a whole the three interviews gave a clearer picture of what it is like to develop a curriculum program and what is involved in it. By using the curriculum authors’ voices, we attempt to highlight their intentions, positions, and reflections from the perspective of teacher use in this chapter. We see this chapter as juxtaposing our analyses of the results of the design process with the authors’ actual voices.

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4.2  The Curriculum Authors’ Design Decisions The results of our analyses of the five programs prompted us to inquire about the curriculum authors’ overall design decisions, such as program goals and visions, and ways to communicate mathematical goals and pedagogical approaches. We summarize the curriculum authors’ comments about overarching program goals, ways to communicate mathematical goals, decisions on sequencing, and pedagogical approach and communication to teachers. We focus on these aspects of design decisions because these are related to our analyses in this volume.

4.2.1  Overarching Program Goals and Principles We first describe the three curriculum programs’ overarching goals and principles the curriculum authors had, along with what they valued and what they believed. We were interested in understanding the authors’ major goals in developing their programs, which reveal what the authors intended and pursued. Although the three programs had some common values and assumptions largely aligned with the National Council of Teachers of Mathematics (NCTM) Standards (e.g., NCTM, 2000), they each had their own specific goals and distinct approaches to pursuing the goals. The authors used different terminologies to highlight what they pursued. The curriculum authors also described their core goals in developing the programs although not always directly presented in the written materials. When sharing their goals and principles with us, all authors pointed to or read from their curriculum documents, such as implementation guides. Goals and Principles of Everyday Mathematics The EM team shared the principles of the EM curriculum as follows: • The curriculum should begin with children’s everyday experience and should work to connect that experience with the discipline of mathematics. • Children begin school with a great deal of knowledge and intuition on which to build. • Excellent instruction is important. • Reforms must take account of the working lives of teachers. • The curriculum should include practical routines to help build the arithmetic skills and quick responses that are so essential in a problem-rich environment. The EM team believed that children can think mathematically and learn conceptual basis for many mathematical skills and processes early on: “Those [e.g., patterns in number] are things kids can handle really intellectually, even if they’re not doing worksheets about them, in fact. I think it’s partly just the concepts you see

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kids encountering at different points.” The EM team believed that children have their own knowledge of mathematics and that instruction needs to build on that to support their learning of mathematics with challenging content. In the EM curriculum, arithmetic skills were routinely practiced for solving more complex problems eventually. The EM team also valued the role of teachers in enacting the curriculum. It was the idea that they [the initial developers of EM] really believed that kids came to school with more mathematics knowledge than we were giving them credit for and we needed to really enhance that, and challenge them and build upon what they came with that already. And part of the program also I think was that they really respected the teacher as a professional and built that into the program. (EM team)

When developing the program, the EM team tried to convey the “messages of [mathematics being] interesting, useful, enjoyable, [and] understandable, and [mathematics] had to be for everyone.” Ultimately, the EM team wanted to design “a program that kids and teachers get to do, not that they have to do.” We’ve tried to convey that math is interesting. That math is understandable, that math is useful and that math is enjoyable. And that was one of the things that I think we tried to do in the third edition was make sure that people realize that those were things that were for everyone. And not just for some kids. (EM team)

Goals and Principles of Investigations in Number, Data, and Space The INV team elaborated its goals and guiding principles presented in the Implementation Guides in all grades. According to the Implementation Guides, the INV curriculum (second ed.) was designed to: • Support students to make sense of mathematics and learn that they can be mathematical thinkers • Focus on computational fluency with whole numbers as a major goal of the elementary grades • Provide substantive work in important areas of mathematics—rational numbers, geometry, measurement, data, and early algebra—and connections among them • Emphasize reasoning about mathematical ideas • Communicate the mathematics content and pedagogy to teachers • Engage the range of learners in understanding mathematics (TERC, 2008a, 2008b, 2008c, p. 1) According to the INV team, their “vision has to do with student ideas being central and math is about making sense and reasoning.” Three guiding principles were underlying the six goals above: (1) students have mathematical ideas; (2) teachers are engaged in ongoing learning about mathematics content, pedagogy, and student learning; and (3) teachers collaborate with the students and curriculum materials to create the curriculum as enacted in the classroom. Like the EM team, the INV team believed that “all students can do significant mathematical work.” The INV team saw teachers as their primary audience for their materials. They wanted to design a program that was “as much to be a dialogue with teachers as to

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be a core of content for students.” In fact, the INV team viewed the curriculum as “a vehicle for professional development as a primary goal” and wanted it to be used for professional development in various contexts, such as individual, school, and district settings. We know that teachers individually mostly do not read everything that’s there. They certainly don’t read it the first year and we don’t even expect that, but the material is there for teachers to go back to as they become more and more familiar with the curriculum. The materials are also there for use in professional development structures that are in place at a school, or at a district, so that they don’t have to create from scratch, professional development around the curriculum. There’s material already here that teachers can use together to think about the mathematics and about student thinking. So we don’t necessarily expect that every teacher is going to read all that material but that the material is there to be used in a lot of different ways. We use it in professional development with schools and districts and we encourage schools to use it in collective ways. (INV team, italics added)

They also explained that the second goal—computational fluency—was added in the second edition as a response to criticisms associated with the “Math Wars” (Schoenfeld, 2004) at that time to “help teachers be able to be articulate about how computational fluency is a focus in these grades.” Goals and Principles of Math Trailblazers When describing their major goals, the MTB team also referred to its Implementation Guide for each grade. The primary goal of the MTB program is “to create an educational experience that results in children who are flexible mathematical thinkers, who see the connections between the mathematics they learn in school and the thinking they do in everyday life, and who enjoy mathematics” (TIMS project, 2008a, 2008b, 2008c, p. v). The MTB program is based on the premise that “children can succeed in mathematics and that if more is expected of them, more will be achieved” (TIMS project, 2008a, 2008b, 2008c, p. v), which specifically indicates that children can learn more challenging mathematics than covered in traditional mathematics curricula. The MTB curriculum was organized based on goals and approaches in the NCTM Standards (2000). Not only the content standards, but also five process standards were emphasized throughout the program: problem-solving, reasoning and proof, communication, connections, and representation. The MTB curriculum was unusual in that it incorporated investigative labs and literature extensively to meet the primary goal, that is, to provide educational experiences for students to think flexibly, make connections, and enjoy mathematics. MTB lessons often started with a context from children’s literature (e.g., 500 Hats, a book by Dr. Seuss) to explore mathematical concepts (see Chaps. 3 and 9). The MTB team said, “Almost every grade level has some kind of investigative open-­ ended task right at the start of the school year to lay the groundwork for the practices, discourse, and social norms.” The MTB team gave examples of lessons to illustrate how the context and the labs provided opportunities for students to learn. In one lesson students interact with and describe a doubling pattern presented in the

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context of an old folk tale. They then extend and generalize that pattern using various representations such as tables and symbols. In a later lesson, students investigate the relationship between the bounce height and the drop height of various balls and represent that relationship with multiplicative language (e.g., doubling, halving), tables, graphs, and symbols and use those representations to solve problems. Comparison of the Three Programs’ Goals and Principles Although the curriculum authors used slightly different language, they aimed to design curriculum programs that provided all students with opportunities to learn mathematics based on what they already knew and make connections among contexts, representations, and mathematics. Providing a good context for students to learn was supported by all three teams. Table 4.1 highlights the common goals and principles that the curriculum authors mentioned. Although not all were explicitly mentioned by the authors, most of the items seem true for all three programs. One difference among the programs is the belief that students can learn challenging mathematics. The EM and MTB programs provided challenging mathematics content for students to learn in earlier grades than usual (e.g., exponential growth in MTB) whereas the INV program focused on reasoning and proof early on in various areas of mathematics including data and early algebra. It seems that the INV program emphasized mathematical practice such as reasoning and thinking early on and the other two programs covered more advanced mathematics in earlier grades (see examples in Chap. 2). The curriculum authors made their own critical design decisions to pursue their goals and vision, and in turn, the goals and principles influenced their design decisions, such as sequencing and support features for teachers. Such different decisions made the curriculum programs distinct from each other.

Table 4.1  Common goals and principles EM Math is for everyone

INV Designed for the range of learners

Math is enjoyable

Children can be challenged (they have much knowledge and intuition) Routines for arithmetic skills Connections of everyday experience with mathematics Importance of teacher role

Mathematical thinkers (reasoning) Important mathematics including rational numbers, data, and early algebra Computational fluency Connections among different areas of mathematics Importance of teacher role

MTB For all students Math is enjoyable Flexible thinkers Children can learn challenging math and they need to be challenged Connections (and other process standards by NCTM)

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4.2.2  Communicating Mathematical Goals to the Teacher We were interested in the mathematical emphasis of each program and how each curriculum team communicated their mathematical goals to teachers. We analyzed these aspects and present our analyses in Chaps. 2, 6, and 8 in this volume. We asked the curriculum authors to talk about their intentions and decisions regarding these aspects. All three teams mentioned that assessments were very important in communicating their mathematical goals, and elaborated on how their goals were organized and tied to various forms of assessments along with specific examples. Structure of Goals in the Three Programs The three programs used different terms and structures to indicate the mathematical goals of lessons (see Table 4.2). The curriculum authors acknowledged, “Teachers interpret what was written in the curriculum, which is not necessarily the same as authors’ intention.” Therefore, they made efforts to convey the mathematical goals and expectations to teachers in an organized way. Listing the core mathematics content for each lesson was to support teachers to understand what they need to attend to in each lesson. The EM team used the term, key concepts and skills, to indicate the mathematics important in the lesson. The key concepts and skills were a way to help the teacher see, ‘[There are] so many little things going on that … might not be immediately obvious to you, but this is where these things … come back and kids are touching on other things’ and we just highlighted them through that feature. (EM team)

The INV team used the term, Math Focus Points, to make the core mathematics of the lessons clear: “We were really trying to help teachers focus on what was the core of the math rather than maybe some other peripheral things that weren’t the core of the mathematics.” The INV program also included the Math Focus Point in the discussion portion of the lesson in order to guide teachers to what key idea(s) they need to focus on during the discussion since the INV team noticed that teachers sometimes had difficulty orchestrating discussions: “the reason we handled discussions Table 4.2  Terms and structures to indicate mathematical goals EM Key concepts and skills: Identified for each lesson and linked to grade-level goals Grade-level goals: Organized by content strand, articulated across grade levels, and listed along with key concepts and skills in each lesson

INV Math focus points: Identified for each unit (across lessons) and listed for each lesson Math emphases: Organized by content strand in the scope and sequence chart, and in the math essay in each unit; covered in one or more units, and subdivided into Math focus points

MTB Key content: Identified for each lesson Specific standards/expectations (no specific term indicated): Listed for each unit in the scope and sequence chart to indicate the development of each of the five content standards/strands

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differently in the 2nd edition is because teachers had a harder time facilitating a discussion, and seeing how the discussion supported the math of the lesson. Giving a focus to the discussion seemed more necessary than for the other activities.” This feature of indicating the core mathematics in discussion was unique in the INV program. The MTB team provided the important content of each lesson (key content) in the Unit Resource Guides (i.e., teachers’ guides for daily lessons) and specific expectations/goals to meet in each unit (e.g., maintain fluency with the subtraction facts) in the Implementation Guides. Often, key content in each lesson is an extensive list of mathematical expectations (see Chap. 6). The goals and expectations in each unit are organized by NCTM’s (2000) five content and five process standards. Assessments Tied to Mathematical Goals All the curriculum authors mentioned that assessments (ongoing and periodic) were important tools to communicate their program goals or grade-specific goals with teachers. These assessments were embedded in individual lessons so that teachers could check on students’ progress in meeting the goals on a daily basis. The MTB team indicated, “There’s no end-of-unit tests. We were trying to get away from the teach and test cycle.” All three programs used detailed rubrics or descriptions with sample student work to help teachers assess where individual students were, which were intended to inform teachers what it is like when a student had a good understanding of a concept or a procedure. These rubrics and descriptions of student work were tied together with lesson goals, target concepts and skills, and expected understandings at particular times in the grade. Although EM had end-of-unit assessments, a major emphasis was placed on ongoing assessment opportunities highlighted in each lesson. Teachers using the EM program were expected to recognize student achievement by using specific problems (indicated with a pink star) in each lesson. These lesson-specific assessment opportunities were summarized in charts by unit in grade-level assessment handbooks (University of Chicago School Mathematics Project, 2008a, 2008b, 2008c). Each chart includes the EM grade-level goals that the highlighted problems assessed. In addition, these assessment handbooks provided information about the end-of-unit “Progress check” lessons, including pointers to tables specifying which grade-level goals each progress-check item assessed, as well as rubrics for the “open-response” problem on the assessment. The EM assessment handbooks also included cross-grade grade-level goals charts that showed the mathematical trajectory of each concept or skill across all grades of the program. Although not providing specific rubrics for teachers to use, the INV authors tried to help teachers be clear about the mathematical foci in lessons and how they were tied to assessments and grade/unit-level benchmarks. In doing so, the INV authors included a detailed list of questions and points of concern in the section of “Ongoing Assessment” in each lesson. Moreover, the INV team provided extensive notes (called Teacher Notes) about student thinking.

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O.-K. Kim and J. T. Remillard The addition of benchmarks in the 2nd edition helped with this [goal]. In the first edition, we had assessments, but it wasn’t always clear what kids were aiming for in a unit. I think that connecting the benchmarks and the assessments and the analysis of the assessments that we have in the Teacher Notes with the sample student work and using examples of student work to show whether students were meeting the benchmarks [at the unit level in each grade], partially meeting the benchmarks, not meeting the benchmarks, was also something that was helpful. (INV team)

Similar to the EM and INV programs, in the MTB curriculum the goals and objectives were articulated by the key content and assessments in each unit (including assessment indicators in each unit and assessment lessons). The MTB team mentioned that assessment indicators “are more unit-based and these are what we want kids to get by the end of the unit, and the various assessment pieces we picked out.” The team further explained, I think we were more clearly trying to tie the lessons and the unit and the big ideas to the assessment and to have the assessments indicate for the students what they knew and could be able to do and if they couldn’t, then what to do about it. (MTB team)

In the fourth (most recent) edition, the MTB team tried to help teachers use assessments as intended in the curriculum by making “the embedded assessment more explicit, tied to the expectations, and then tied to what you do if [students don’t understand].” Also, the team wanted to support teachers’ thinking about what assessing students’ learning of mathematics means. We want them to get better thinking about what it looks like for a student to really have a robust understanding of this idea. What do those connections look like, what do those behaviors look like? And one of those behaviors is, for example, kids can translate between multiple representations. (MTB team)

All three teams tried to communicate mathematical goals to teachers by listing key mathematics content and mathematical foci along with grade-level goals and benchmarks, which were tied to various forms of assessments and explanations about student thinking and work meeting the goals. The EM and MTB programs provided specific rubrics and assessment indicators; the INV program provided points of ongoing assessment and student work with commentaries. The alignment between the goals and the assessments was critical in all of the three programs.

4.2.3  Sequencing the Mathematics Content The analysis of the curriculum programs’ sequence in number and operations (see Chap. 2) prompted us to think about how curriculum authors made decisions regarding sequencing the mathematics content in the programs. We found that the curriculum developers sequenced the mathematics content in very different ways (see Chap. 2 for more details). We asked the authors to explain how they sequenced the major topics, including distribution of practice and coherence in students’ learning progression. The curriculum authors explained what they based their decisions on

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when sequencing topics and lessons, how they made these decisions, and factors that influenced them. Each of the three programs provided a scope and sequence chart detailed in each grade as resources for teachers. Developing the Scope and Sequence Chart The three programs had a similar approach to determining the scope and sequence although they opted for somewhat different content placements within and across grades. Determining the sequence within and across grades, the three curriculum teams considered both the new content for learning and ongoing practice, and both conceptual and skill content. The EM team’s explanations revealed a complex process to determine a route or journey of mathematical ideas and concepts in lessons within and across grades. The EM team described the process they went through in great detail—the process similar to the other two programs’. The team had to think about the sequence across grades first and then the sequence in each grade considering both new content and practice components. We did start with the big cross-grade thing. Obviously that was a big deal for us. … it went to grade level team to essentially map out and we thought of it in a few ways. We thought of focus activities. … how [new content] would progress over the course of the year. And then, how we would distribute practice content using the features of EM over the course of the year. And so then we had kind of an equally big but more detailed chart for a particular grade level that mapped that out before we began writing. (EM team, italics added)

The EM team also explained what it was like to organize the sequence in each grade and make sure the flows/progression of the mathematics concepts and skills students were supposed to learn and practice. So, each of us had a chart that listed like all the lessons on the left-hand side and of the year-­end goals across the top broken into the smaller part of those year-end goals. And then what we did, I mean these charts were as big as an entire wall in an office and for every lesson we would fill in exactly what was happening in terms of the practice, the direct instruction, the part three optional activities. [We wanted to] make sure that you can clearly say that I am following this concept throughout the curriculum and you could see waves… where [the concepts] start where they would end. (EM team) … before we wrote any lessons or revised any lessons or moved any lessons or anything, we wanted it to be really clear internally about how, how these things would progress. And so these topics, some of the rows are a little bit more sort of conceptual, conceptual is by definition. Others are more skills like fluency with addition facts… you’ll see both. … So each person working on a grade level had a very clear map of exactly what they were responsible for at that grade level. And we were trying very hard to make sure there were no hole [in the sequence]. (EM team, italics added)

The EM team’s comments illustrate what it was like to determine the sequence of the mathematics content within and across grades. It involved a complicated process of checking and double-checking the placement of the content and the flow of student learning from one lesson to the next.

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Reasoning Behind the Scope and Sequence How did the curriculum authors determine whether there was or was not a hole in the sequence? How did they know the sequence would work? They determined their specific sequence based on a combination of various rationales, such as their own knowledge, research-based data, and the NCTM Standards. The INV team gave detailed explanations about their decision on sequencing, and we mostly  use the INV team’s rationale to describe what curriculum authors base decisions about sequencing on in general in this section. Rationale 1: Authors’ Knowledge About Mathematics Content The INV team commented that they thought about big ideas in each grade and how those ideas developed along with specific concepts and skills, based on their knowledge of the mathematics content. This is similar to the other two teams’ rationale for sequencing. We made decisions about content and about how it would be sequenced within a grade and across grades based on our knowledge of the math content of how these concepts and skills and ideas developed over time. We considered, in each content area what were the big ideas, that’s where we typically started, for this age group of five to 12-year-olds. What are the big ideas in each content area [e.g., “composing numbers up to 20 with two or more addends” in Number and Operations in Grade 1]. And then we thought about what concepts and skills were foundational to each of those big ideas. So we kind of started I’d say at the macro, and then kept refining our work. (INV team, italics added)

Rationale 2: Authors’ Positions About Students’ Cognitive Development The INV team also mentioned that they had to consider the notion of continuum and cognitive development of students, because the sequence of the mathematical ideas should fit and support students’ progressions in learning. While we were initially focusing on K-5, we also felt like teachers had to know a little bit about where these ideas came from and also where they connect to in middle school. …and similarly, within any grade level, how do ideas develop within a grade level but what comes before that and what comes after that mattered a lot. So that was part of our laying out of the continuum in order to be able to see how mathematics would develop over time. It’s whatever the time is, over the course of a unit, over the course of a year, over the course of a grade band, of Elementary School. … when we were thinking about, making these decisions about sequencing content, we were also thinking about the cognitive development pieces, you know, …What do we know about 7-year-olds and how they learn? How they think and how they act… (INV team, italics added)

The INV team’s position regarding cognitive development, however, was different from the other two teams. The EM team wanted to introduce advanced topics at the level of exposure and sense making in earlier grades. Similarly, the MTB program placed many advanced contents, such as exponential growths (doubling) with exponents and scientific notations in fifth grade, in earlier grades than usual. It is clear that this sequencing decision is based on the EM team’s and MTB team’s positions as explained in their goals and principles earlier.

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Rationale 3: Field-Based Data and Research The INV team also mentioned that their decisions on the scope and sequence were based on field-based data they gathered in the classrooms: “this has always been very classroom-based work and what we’re basing a lot of these decisions on is what we see students need from working across multiple classrooms.” For example, the INV curriculum had extensive work of addition and subtraction placed in grades 3–5 compared to other programs (see Chap. 2) and that decision was based on the curriculum authors’ findings and beliefs from their fieldwork. We’ve always believed from our work with kids and continue to believe that students are not finished with addition and subtraction at the end of second grade, that there is, there is not this automatic transfer to larger numbers, and an automatic generalization of the base-10 number system, and that students really still need a lot of work on addition and subtraction in third grade, which was reflected in this third grade sequence of the second edition. We’ve had to make some compromises with that in the third edition because of the Common Core. Nevertheless, we still have not given up on substantive work on addition and subtraction in third grade. And even into fourth grade, in fact. (INV team)

The EM and MTB teams also mentioned that research was one of the sources to determine their sequence. The MTB team said, “When you think about the decisions we made and why we made them, the mathematics education field came a long way since 1989 or 1992 or whenever we started right. So we had input from other researchers.” Although the curriculum teams used research to sequence mathematics content, the resulting sequences were rather different (see Chap. 2). Rationale 4: Curriculum Documents As mentioned above, the MTB team mentioned that they used the NCTM Standards to place content goals within and across grades, and then made sure that different contexts embedded the target mathematics content: “From the very beginning, we were aligned with the NCTM Standards. That was our idea. And so, we thought about the mathematics across the grade levels based on the NCTM Standards.” According to the three curriculum teams, the Common Core State Standards (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010) adopted by most states in the USA also influenced the curriculum teams’ sequencing decisions in later versions. Sequencing Decisions with Different Design Features The three curriculum programs had different design features that played a role in the authors’ sequencing decisions. Such features include a spiral approach, a unit approach, and specific aspects of lessons or routines, such as Daily Practice Problems (DPPs) in the MTB program. Below we describe those features that the authors mentioned during the interviews.

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Feature 1: Spiral, Unit, or Unit–Spiral Approach The EM team used a spiral approach to sequence the mathematics content across and within grades, and the other two teams were in different positions. The ­curriculum authors’ decisions on sequencing were largely based on their position regarding this notion of spiral. Using a spiral approach, the EM programs’ grade-level goals were carefully developed within a grade and also articulated across all grades in the program. “Everyday Math valued exposure to concepts and skills, and then mastery at a later time.” Sequencing in the EM program involved a careful layout of these goals within a single grade and across grades. The EM team commented that the grade-level goal charts were important to help teachers see concepts and skills in individual lessons in the big picture: “I think those [charts] were attempts to help teachers sort of see the big picture and see what their place was in the big picture.” Although the MTB curriculum used unit-based lessons like the INV curriculum (see Chap. 3), the MTB team mentioned that they used the notion of spiral curriculum to help students move through different phases of development to fluency. Their notion of spiral is that revisiting the content to develop it rather than repeating so that students “get a deeper and deeper and deeper notion of those ideas across the years as well as across the grades.” The MTB team added, “the idea is for kids that come to it and then deepen their understanding the next time it comes back and to increase their knowledge.” The MTB team described how they used this notion of spiral curriculum to sequence the content of operations in unit-based lessons. …like whole number operation, that’s one example. You’ll notice there’ll be several units within a grade level that deal with the development of those concepts. It’s not just we’re going to do addition as the second unit in the year and then we don’t touch it again…. So there’s several places across the year where [there are] several units. So earlier units will focus in on the of invention of strategies by putting kids in context where they would have to multiply multi digit and then later on in the year there are units and lessons that focus in on developing those algorithms and different representations and then later on in the year as we move down there is stuff that’s focusing on fluency (flexible, appropriate, accurate, efficient). (MTB team)

The INV curriculum, organized into units, had a style similar to the MTB curriculum’s in some content areas. The INV team made “the choice to structure the math content around three to four weeks, say a focused study around a particular content area and, kind of, to return to that, those ideas over time.” The team explained how the content revisited across units in the INV program. …it’s showing some content areas, for example, in the number and operation strand. There’s three or four units that are placed across the year where students are returning to some ideas or building on those ideas and then focusing on new ones. Kind of stepped it up a little bit. But even in non-number units, for example a geometry unit, the content often supports the work that they had just done with multiplication and division as they are moving into thinking about finding area. (INV team)

Feature 2: Aspect of Lessons—Distributed Practice All three teams described that their programs had practice problems distributed throughout the year with a particular trajectory intended (see Chap. 2). In particular,

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the MTB team had Daily Practice Problems (DPPs) in every lesson, which prompted students to get ready for mathematical exploration and practice skills needed. The MTB team also emphasized further that the DPPs were not just practice but ­groundwork for formalizing the ideas and strategies, in the context of multiplication in Grade 3: DPPs [daily practice problems] and the Home Practice are really interleaved or distributed practice. The math facts trajectory lives in the DPPs and the Home Practice as well. Grade 3 has a series of lessons that really goes deep on modelling and understanding and visualizing multiplication or understand the context of multiplication, but then the trajectory after that, the practice trajectory, lives in the DPPs and the Home Practice and is broken out across units but then you’ll have another unit a little later, which formalize. “We did of this to encourage invention [strategies]. From Unit 3 students are inventing and then in the DPPs for Units 4, 5, 6, 7. Now, in unit 8 students are going to figure out what students really have and formalize what they have. We will organize the thoughts that you have, look at all the different strategies and compare Janine’s strategies to Jennifer’s strategy and give it a name.” That happens across the curriculum. (MTB team, italics added)

The MTB team also emphasized that without the ongoing DPPs, students would not be ready to learn the content they were supposed to in later units because DPPs were provided in a careful sequence. …the formalization doesn’t necessarily happen in the DPPs, but without the DPPs I’m not sure that kids would be continuing to invent strategies to multiply. They wouldn’t be drawing pictures and trying to play with this a little bit and trying to find some other strategies. So, when you get to unit 8 you have some things to talk about. If the DPPs weren’t done the class would not have a lot to talk about because the kids probably aren’t doing much multiplication. (MTB team)

Feature 3: Aspect of Lessons—Investigative Labs in MTB Another aspect of lessons that influenced the MTB authors’ decision on sequencing was investigative labs. Incorporating labs in the curriculum, the MTB team had to think about how to use the labs to sequence lessons and the mathematics content. This placed an additional challenge to the MTB team. …this is the science piece and how the science piece is embedded … Through the third edition we wanted eight labs. … For each lab, we were trying to develop a mathematical concept as well as something about science. … And often it was just the science process and then link theory of math. Volume and time were big ideas that got it every year as I pointed out before. But then within those there were mathematics to test. So, for example, when the kids did marshmallows versus containers they were learning about volume because they were filling graduated cylinders of different sizes. You know a tall skinny container with a big fat container with marshmallows. But the other big idea was counting and grouping with counting. And so, the mathematics was organized in that way so that all of the labs, and then by the time again when you got a volume you did mass and volume and density and that was a big idea. You hit, ratio and proportional reasoning got hit at that point. So that was a big way that we developed and how we got to the mathematics within a context within a way that kids could do it concretely and as well as symbolically. (MTB team)

To summarize, it is clear that the three curriculum teams made great efforts to place mathematics contents deliberately in lessons within and across grades. The decisions on sequencing were based on various resources (e.g., state standards, NCTM

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Standards, and research) and authors’ knowledge and beliefs. The curriculum teams also had to work with the formats and styles of the program they chose (such as the MTB using investigative labs) in determining the learning trajectory for students. The diversity in the design features, and authors’ knowledge and beliefs, and different resources drawn on, such as research and policy documents, led the curriculum programs to present their own mathematics journey of student learning.

4.2.4  P  edagogical Approach and Communicating It to the Teacher As we allude to in the section on goals and principles, the curriculum teams had specific ideas about the types of learning opportunities they wanted students to have and the role the teacher should play in facilitating those opportunities. In Chap. 3, we elaborate what teachers can encounter in the written materials and how they can interpret the ideas presented in the materials. During the interviews, we wanted to capture the curriculum authors’ pedagogical intention along with how they attempted to communicate the ideas to teachers. In this section, we describe the three programs’ pedagogical approach and efforts to communicate it to teachers. We describe the vision for classrooms that the authors imagined and then discuss key design features the authors included to make this vision happen. The Vision for Classrooms As described earlier, the three curriculum teams had shared visions and goals. They also had similar pedagogical ideas about the types of learning opportunities they hoped to create. For example, they all imagined classrooms where students’ ideas were valued and  teachers facilitated discussions of mathematical ideas and supported students to reason, solve problems, and make connections. The EM team said, “We were definitely envisioning classrooms where there was a lot of discussion about math, where there [were] a lot of kids doing the work of the math and not necessarily just receiving the information from the teacher.” The INV team also had a similar vision: “the vision has to do with student ideas being central and that math is about making sense and reasoning. The discussion is about students building on each other’s ideas and the teachers have a key role here.” The MTB team also commented, (1) Kids learn math by doing the work of math: making decisions, inventing, listening, and talking. Creating a meaningful context to foster socialization, develop visualizations that connected to context to develop meaning was a goal. (2) Kids need time to invent, then see other invention, develop reasoning strategies supported by visuals of quantity and number, then use those strategies to develop flexible, appropriate, and efficient strategies. (MTB team)

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All three teams envisioned classrooms where students are actively engaged with the mathematics that they learn by solving problems, making connections, discussing their ideas, and reasoning about mathematical ideas and strategies. Without t­ eachers’ key role, their vision would be just an imagination. Therefore, the curriculum authors developed and used various features of their programs to communicate their vision to teachers. Design Features Highlighting the Vision Besides implementation guides directly explaining the vision of the curriculum programs, there was a range of different features the curriculum authors mentioned that embedded their vision for classrooms and conveyed the pedagogical approach they emphasized (see Table  4.3). In this section, we highlight three specific examples (problems, assessments, and games) to illustrate how the authors used different design features to deliver their vision for student learning and pedagogical approach to teachers. Design Feature 1: Problems All curriculum authors emphasized the importance of problems and how to use those problems in the classroom. The EM team commented, we had to be pretty explicit… that we gave them a problem that they don’t know how to solve at first… so you definitely see that like the Math Message is something that is not applying a procedure they already know. It’s kind of providing access to a problem. We’re gonna (sic) explore with hands-on stuff and talking today and then we’ll practice. (EM team) Table 4.3 Three programs’ design features communicating their vision and pedagogical approaches EM • Teacher manual (teacher’s guides, lesson guides) • Assessment handbook (assessment notes) • Differentiation handbook • “Unit front matter” (unit introduction) • Key concepts and skills • Student materials • Sample answers and possible strategies • Assessment lessons (open-ended, big question) • Games

INV • Implementation guide (collaborating with the authors, etc.) • Dialogue boxes with commentaries • Sample student thinking/work and representations • Teacher notes • Representations and contexts • Images (pictures of student work, student worksheets, etc.) • Sidebar notes (“highlight to help the teacher in the moment of what am I doing.”) • Scope and sequence charts • Benchmarks and math focus points • Student work and assessment commentaries

MTB • Challenging problems and high-interest contexts • Learning progression (sequence) • DPPs (distributed practice) • Unit resource guides (teacher’s guides) • Teacher implementation guide • Key content (goals and objectives) • Assessments and assessment indicators • Multidimensional rubrics • Representations • Possible solutions, sample responses, and patterns • Content notes • Unit overviews

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According to the MTB team, the curriculum provided specific directions for teachers along with teacher questions and anticipated student thinking. The MTB team claimed that the way the DPP was structured called for a shift in teaching practice toward their vision. So, the first thing to do was to come up with a really good problem, or series of problems. But the next thing to do was to communicate to teachers that you don’t tell the kids how to solve the problem and they don’t need to know their facts of 7 minus 5 to solve a word problem that includes seven subtract 5 and all of that. Now that’s not as much of a problem. But when teachers were transitioning from traditional curricula to MTB curricula they said you can’t ask students to solve those kinds of problems. Those kinds of problems because they can’t solve this problem [because] they don’t want seven minus five. So, we had to say, “Do these problems and then ask the kids how they work them out. Give them the tools, give them the stories, the tools, and then ask them for their various ways to solve the problem” and then say kids may do this, this, this, or this. The first edition that was a little tricky because we hadn’t, except for the field tests, we didn’t have that many ways that kids, you know, we didn’t see it all. But in later editions we were able to say you know kids may do this, kids may do this, kids may do that. (MTB team, italics added)

Design Feature 2: Assessment According to the curriculum authors, assessments were closely related to their target mathematical goals. Assessments also deliver a message about what it means to assess students’ achievement and progress. The EM team commented, One of the things in Everyday Math three [EM third edition], we talked about two different kinds of assessment notes within lesson, and the idea that teachers could recognize student achievement where we would look at student work and evaluated and we would give an expected level of adequate progress. Because we want teachers to understand that assessment came in a variety of ways and you want to know what the kids get, have many different opportunities to show you what it is they know how to do. So that was one thing. Um, another flavor of assessment was on informing instruction where we would get at common misconception as a way to highlight those for teachers. (EM team, italics added)

Design Feature 3: Games in EM The curriculum authors indicated that a range of different features embedded their pedagogical ideas. One of those mentioned by the EM team was games, which delivered a new message about assessing students’ fluency and meaningful practice. …we try to develop record sheets for every game so that there was a tangible something, that a child play the game and there was a record of them practicing. And we also made a big effort in the practice portion of the lesson, part 2 that games were revisited on a regular basis. Um, so that teachers would see, “hey bring these back again.” (EM team)

There were many other design features highlighting the authors’ vision, such as Dialogue Boxes and Teacher Notes in INV. In the remainder of the section, based on the authors’ comments we describe particular efforts they made to communicate their vision for classrooms and pedagogical approaches to teachers, along with specific features used.

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Communicating Pedagogical Approaches to Teachers Pedagogical approaches are presented in the resources for instruction, also intended for teacher learning, called educative features (Davis & Krajcik, 2005; Davis, Palincsar, Arias, Bismack, Marulis, & Iwashyna, 2014). Through field-tests and professional development, conversations, and exchanges with teachers, the curriculum developers received feedback from teachers. The authors used the feedback to revise their resources to convey their ideas more clearly. There were various forms of communicating the pedagogical approaches: icons and images, expository texts and narratives, implementation guides, context and representations, assessments (assessment indicators and assessment items and rubrics), etc. Using the different features, the curriculum authors made efforts to communicate their ideas about teaching. Using the case of the INV team, we illustrate how the curriculum authors attempted to speak with teachers. The INV team provided information about student thinking in great detail in the program using various features. We also anticipated that teachers would need more support around looking at and understanding the work that their students would be producing. We wanted to help teachers focus on a student’s solution or their strategy for solving a problem. … And so, we try to provide more support for the teachers (a) by showing more examples of work. Just having the work there helps teachers get a sense that this is going to look different. In terms of the Assessment Teacher Notes in the back of the book, that went with the assessments, not only showing examples of anchor papers but, these were all related to the benchmarks, so showing and talking about pieces of work that met the benchmark and why, and pieces of work that partially met a benchmark and why, and a piece of the work that did not meet a benchmark and why, and oftentimes how you might help, how you might intervene, to support [a student.] (INV team)

The INV curriculum also included resources about reasoning and proof in relation to early algebra, hoping to support teachers to see what it is like for elementary students to reason and prove and encourage students to engage in such mathematical practices. the idea of what does it mean to prove something in the elementary grades. … that was these essays on reasoning and proof, that are actually Teacher Notes in the back of the number units and the function units. We’re really, we’re trying to say here’s what proof looks like for a mathematician and here is what proof looks like in the elementary grades. Those were sort of, those were certainly new. We thought would be new ideas for teachers. (INV team)

All three programs placed various support for teachers in different locations in different lengths. For example, the INV curriculum includes guidance to teacher actions and information about students’ thinking, which were in the usual main body of the lesson. Then, Teachers Notes were devoted to explain both mathematics content and pedagogy in the sidebar (shorter ones) in the lesson and in the back of the unit (longer ones). How did we communicate these decisions to teachers? So, we tried to not only structure the curriculum, you know, how the curriculum was unveiled over time, in a very deliberate and sequential and coherent way for teachers, so what are they actually doing with kids? That was one way we communicated it to teachers. We also offered teachers support and we’ve

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already talked about these structures in the form of Teacher Notes - so, information about, both about the content that they were thinking about, and also how children of their age group learned that content or understood or even misunderstood that content. And also the sidebar notes which I think we mentioned. Some of this support is not only in different places but it’s a different grain size. So, a Teacher Note is a much longer piece of writing about a certain topic. And a sidebar note is right at point of use, right when you’re on top of that lesson. (INV team, italics added)

Next, we describe the three curriculum teams’ ways to talk to teachers about how to organize classroom discussions using different design features. Three Different Approaches to Supporting Discussion All three curriculum teams acknowledged that teachers need support for organizing classroom discussions. The EM team commented, “Certainly [if] you don’t have the depth of knowledge, it’s going to be harder to run a discussion about it.” The INV team further commented on the importance of discussion and the importance of the teacher’s role in it, requiring teachers to be more than facilitators. The discussion is about students building on each other’s ideas and the teachers have a key role here. I think this is one of the things we still hear from teachers, that we tried to work with, which is teachers is feeling like if they’re only a facilitator, then they don’t have an agenda for what’s going to happen. And so it’s helping teachers see that they have a key role; that they’re not just kind of the master or mistress of ceremonies; that they have agendas about content and that they listen and question students with these agendas in mind. (INV team, italics added)

Although the discussion was an important pedagogical emphasis by all three programs, they took different approaches to address this important component in instruction. The MTB team shared that they struggled with how to guide teachers for classroom discussion and decided to provide questions to ask, although their later version (fourth edition) included short sample dialogues in the margin (sidebar) of the main lesson flow, along with open-ended teacher questions and possible responses. We grappled with whether to script lessons and give teachers a series of questions and then possible answers. And that doesn’t work so well if the kids can go on a zillion directions, so except on rare occasions we chose not to do that. But we did want to give them good questions to ask and then some possible responses. (MTB team, italics added)

The EM team was also not sure about whether a “pretend conversation” would be beneficial to teachers “because the flow may differ than what is written, depending on actual student responses.” The EM team took a bit more indirect approach to supporting teachers to have a classroom discussion. One EM author pointed out that sample answers and strategies were provided to help teachers to have a discussion about important mathematical ideas in EM lessons. So sample answers are kind of, they may be a little more subtle, like they’re not labeled in a box … But we think a lot about the sample answers. I would say, perhaps certainly more than probably the people reading them, but we hope they sort of, those give specific

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­ essages about what might come up. And also more general messages about the kind of m conversation we hope kids, they might have in the classroom. (EM team, italics added)

The EM team mentioned that they wanted to support teachers to have discussions, but they felt their approach may not be sufficient: “we may not have talked about that as much, at least not as explicitly.” Despite that, what would be good ways to do so was a question the EM team was still actively considering and exploring. Is it better to have the kind of pretend conversations? Would that be a better approach? So I would say that we haven’t, as long as I’ve worked on them, we haven’t completely shifted our style and our voice from a few sample answers. We’ve used more open-ended questions. But again, I feel like even now this is something we’re thinking about, is this coming across right? Are people having more success with a different type of questioning pattern? … So I feel like we maybe are becoming an, coming to a place where we’re feeling like possibly trying something a little bit different and seeing if we get some different effects. I think partly that’s something, we’re taking away from looking at how people say, use the Eureka math style of, this sort of pretend conversation. (EM team, italics added)

The INV curriculum provided more extensive and explicit support for teachers to conduct a classroom discussion. First, the curriculum included a lesson component called “discussion” in almost every lesson, and the component had its mathematical focus clearly indicated as described earlier. we saw a lot of discussions where that sort of sharing strategy discussions, where each kid said a thing but there were no connections being made among them. And so, we realized that this whole mathematics discussion was really difficult for teachers and so trying to help teachers focus more on “what’s the heart of this discussion?” “what’s its purpose?” was something we really worked on in both the second and third editions. (INV team, italics added)

Next, the curriculum included a set of teacher questions and sample responses and representations used taken from real students in the field-tests. One other kind of support in the INV curriculum was Dialogue Boxes along with commentaries, which illustrated what a discussion on the given topic and focus might look like. We always had Dialogue Boxes in the first edition. But in the 2nd edition we added an intro or orientation for the reader and then at the end we added a reflection about what’s important about this conversation. So, it wasn’t just about classroom dialogue anymore. We really wanted to make sure we highlighted the reason why we put this topic [in the Dialogue Box]. … this is having an introduction and some commentary about what you might pay attention to here. (INV team, italics added)

The curriculum also included a section on discussing mathematical ideas in the Implementation Guide in each grade, containing how to focus whole discussions, and students building in each other’s ideas, etc. (TERC, 2008a, 2008b, 2008c). Despite similar pedagogical emphasis, the three programs used different styles and formats to communicate it, as seen in the example of supporting teachers to organize discussions above. The EM team provided sample answers and strategies; the MTB team listed questions to ask and some possible responses; the INV highlighted the content focus of discussion and provided sample dialogues with commentaries and specific notes about having a discussion in various locations (in the lesson, back of the unit, and implementation guides).

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4.3  The Nature of the Work of Curriculum Development The interviews with the curriculum authors illuminated that curriculum development is a dialectical process and a curriculum program is an ongoing, evolving product. Therefore, the editions we analyzed in this volume were only temporally finished products; they evolved from earlier editions and further revisions were anticipated, based on responses from teachers and students, research, and changes in the policy environment. The ongoing efforts to align the authors’ aims and ideas with the curriculum product generated a cycle of design, test, and revision. This evolving nature of curriculum programs was a prominent theme in the conversations with the authors. This nature was largely due to the authors’ aims to help teachers better understand what was behind the design. In fact, both the curriculum programs and the teachers using them were in the process of evolving through their interactive relationships. The INV team commented, we use a variety of resources for teachers and they say everything doesn’t speak to everyone. But we really feel like the curriculum can grow with the teacher when he or she is ready to dive more into some aspect of mathematics. The information is there for them in the second year or the third year. We have teachers so often say, ‘Oh my gosh I never noticed. I’ve taught this unit three times but I never noticed X’ but I think it’s because now they, for whatever reason, they’re freed up to pay attention to X in a way that they might not have been in previous ways through. (INV team, italics added)

In fact, the three programs made the field rethink about the notion of curriculum. The INV team mentioned that they were thinking about “helping teachers to broaden their notions or ideas about what curriculum could be” when they started developing the INV curriculum. Since these curriculum programs came out, there have been significant changes in formats and support features (educative features) of curriculum programs in general. In this section, we illustrate the evolving nature of curriculum development along with examples of change or improvement made in some elements or features of the programs in response to teachers, such as mathematical goals, assessments and follow-up actions, support for flexible implementation, and student resources/texts. Evolving Nature 1: Making Goals Clearer All three curriculum programs in this chapter were largely based on the goals and vision presented in the NCTM Standards (e.g., NCTM, 1989, 1991, 2000). The curriculum authors found that they needed to make elements of their programs, such as goals, assessments, and how to use certain elements (assessments, problems, representations) clearer to support teachers to use the materials productively. In the process of making things clearer and revising/improving elements of the recourses, the curriculum programs were evolving and growing. Mathematical goals for students within and across lessons/grades were one of the important elements the authors had to present clearly to teachers. The following comment by the INV team was about their efforts and struggles in making mathematical goals clear to teachers. One particular thing that I remember stood out to me was when we thought the math goals of a lesson were really clear, they weren’t necessarily clear to teachers and so they saw they

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weren’t seeing the way certain math ideas built. And so in the second edition and then even more in the third edition we really worked on these Math Focus Points and the large math ideas in each unit and how those were tied to the benchmarks for students. I remember that even moving from the second edition to the third edition that we still thought those Math Focus Points were not always the same grain size, the right grain size or clear enough in the way they progressed through a unit or through a grade and we really worked hard on them yet again… (INV team, italics added)

The MTB team explained that they tried to support teachers to teach toward student expectations about key mathematics content, by providing a list of student expectations for each unit and organizing each unit around them. In our own research, setting and supporting rigorous student expectations was one of the biggest challenges for teachers. Each unit was, is now, organized around a set of expectations. These expectations grow and morph across the year and across multiple years to assist teachers listening and responding to students. (MTB team)

Evolving Nature 2: Improving Other Elements to Achieve Goals All three curriculum teams mentioned a few times instances where teachers had misinterpretations or misunderstandings about their intentions. Regarding the issue of clarity and explicitness, they all realized that their intention was not always transparent. To avoid teachers’ confusion, misunderstanding, or misinterpretation, the authors had to make support elements clear and transparent. Not only did the curriculum authors try hard to make their goals clear, they attended to other important elements, which also needed improvement to achieve their mathematical goals. The MTB team explained their efforts to make assessments clearer to teachers. We found in this [third] edition that the [assessment] descriptors weren’t enough to really help teachers make decisions. Right? So assessments have to give teachers enough information about where they’re at and understand what they’re seeing, because they might not be sure what they’re looking at. …So in this [third] edition, we struggled with that as we struggled with the teachers because they just didn’t have enough description. They didn’t know what decisions to make next. They didn’t really have good ways of collecting information. … That’s a pretty significant shift in the fourth edition. And now that there’s a shift to standards-based report cards. It is really more on performance descriptors and that kind of thing. Trailblazers is set up to feed right into that kind of conversation. (MTB team)

The INV team commented that they tried to improve specific differentiation (intervention and extension) that could be offered in response to students’ thinking evident in ongoing assessment during instruction. …this [differentiation] was something that teachers also were asking for more support around, you know, how do I meet the needs of a range of learners in my classroom? And so, this is one of the features where we called out, as part of the ongoing assessment section for teacher, we called out the differentiation support with intervention and extension suggestions and also in some sessions there are English language learner suggestions. (INV team)

Evolving Nature 3: Responding to Teachers Based on anticipated points of support needed and teacher difficulty, the curriculum authors developed support materials to help teachers make sense of specific goals or broad approaches and do actions accordingly for students’ learning. Then, they

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revised the support materials based on the feedback from teachers and the results of field-tests. The MTB team’s work on assessment and the INV team’s attention to differentiation above are examples of the authors’ responding to teachers. Also, the EM team commented that they had to consider how teachers read and perceived their curriculum elements and make them accessible to teachers. As we always said in the first and second edition, the differentiation is there, it’s already there, the assessment is already there. And I think the early adoption teachers were at that level where they could … run with the materials that are there, I think by the time we came to the third edition and it was more widely used, we realized that yes it was there, but for some teachers we were going to need to pull it out a little bit more and we were going to need to highlight it more so that they would be able to use it more easily and they would understand where these things actually already existed in the program. (EM team, italics added)

Teachers are in need of various support for instructional decisions. The comment by the EM team below indicates that the authors tried to respond to teachers’ need of implementing the curriculum flexibly in their own situations. I think there was the sense that the people had a hard time thinking about implementing EM using a variety of classroom structures. They wanted to do more small-group work or they wanted to run math in a more of a workshop format or something like that. I think they had a hard time translating from the way our lessons written up. Some teachers did do that. They needed some permission and then some specific suggestions. I feel like once we were able to give them that for some people that was actually the hurdle they were getting by. They wanted to do it a little bit differently and didn’t think it was allowed. So, helping them sort of see how it could be more flexibly implemented cause that’s not really explicit in third edition so much. (EM team)

Evolving Nature 4: Developing a Useful Element—MTB’s Student Text Initially, the MTB team did not plan to have a student text, but wanted to have a resource book for students. Yet, they ended up developing a student text. The team provided their rationale behind this decision—advantages of having a student text. so we were going to have a core book or something similar to what Everyday Mathematics has like the student reference guide, which is not a textbook but you know just had the mathematics in it. And we tried that in the first edition in the field test and it just didn’t work. It was more helpful to have the mathematics represented in the student book in the lessons and then to have the questions there for the students to work through. … You have the kids read the thing, read the context, you know, work on a table and then they can work through in a group a series of questions that helps them develop the mathematics without having to stop and start and have the teacher ask all those questions. (MTB team, italics added)

The MTB team also mentioned that the student text made teachers’ implementation of the program more thorough and flexible, and made their design rationale more transparent. Well [the student text] also makes a little more flexible for implementation. You could essentially keep the book closed and do some of that stuff with the whole class, if that’s what you decided to do, but is it also breaking up into smaller groups and it’s all there. Yeah I think when we shifted to fourth edition we even found that, for example, there were lessons that weren’t in the student guide that were only in the teacher’s guide. Well those ­lessons were skipped. We found that the student guide became a tool to help us make the design rationale transparent and obvious. (MTB team)

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The MTB team thought that their student text signaled what was important in the lesson and provided important information, such as misconceptions, representations, and strategies: “if there is a misconception or representation or a particular strategy we really want to make sure kids experience or are touched by students, it’s in the student guide. So, you can’t miss it.” The team also mentioned that the student text eliminated the need to photocopy. They saw many more advantages than downsides of having the student text. They even developed animated student pages using digital resources in the recent edition. The entire process of developing student text is another example of an evolving curriculum. The authors’ voices conveyed their efforts and struggles in designing and improving their curriculum programs. The authors sometimes had to work with a certain policy, criticism, or dilemma. Some challenges the curriculum authors faced were curriculum specific. The style and structure of the curriculum itself generated certain challenges. For example, having multiday lessons, the MTB team thought carefully about how to chunk the lessons into parts, and instead of giving a specific direction, the team decided to encourage teachers to break a multiday lesson into parts in a way that worked in their classrooms. The EM team, having a spiral approach and grade-specific goals, tried hard to make them clear for classroom implementation. With a unit-based organization, the INV team put efforts to make sure the learning progression within and across grades in each topic coherent. All of these different efforts were to support teaching and learning to be more productive. In the next section, we describe some common challenges the authors faced.

4.4  Challenges of Developing Curriculum Resources In the cycle of revision, the curriculum authors developed and improved features to support teachers based on research and feedback from teachers. Along with the mathematics content for instruction, the authors continued to think about, “What is something that is actually useful for teachers?” All three programs had extensive materials for teachers, some of which were student pages and assessments for photocopy and others were educative features supporting teachers in terms of content and pedagogy. The materials for teachers built a large resource pool, containing a range of different aspects related to teaching mathematics. The guides for everyday teaching were especially closely related to student actions and teacher role in instruction (see Pt. 3 of this volume). The EM team mentioned, “the design of having a pretty big teacher’s manual [teacher’s guides for everyday lessons] is the heart of the program.” As indicated previously, however, developing and refining the curriculum programs with support features was a challenging task. Teachers were engaged in different types of practices and worked in different situations. Communicating something that went against typical practice generated a number of challenges. Although the curriculum teams took a range of different approaches to supporting teachers to teach mathematics and making these support features presented clearly

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to teachers, they faced some similar challenges in the development and revision process. In this section, we explain those common challenges surfaced during the interviews: making the design rationale clear, determining where to provide educative features, providing the “right” amount of resources, working with tension between authors’ and teachers’ perspectives, and working with standards, policies, and outside pressures. Challenge 1: Making the Design Rationale Clear The curriculum authors expressed several times that they tried to make their intention clear and transparent to teachers as they saw teachers were struggling. How did the curriculum authors make things clear and transparent to teachers? Apparently, this was one of the biggest challenges they faced. The INV team said, “…part of the feedback overall made it clear that things that we thought were obvious in the curriculum were not necessarily obvious to teachers.” There was a range of different elements that needed attention for refinement and revision for clarification. For example, making mathematical goals clear was not a trivial task. The comment made by the INV team earlier indicated that the team continued to work hard to make their mathematical goals clearer to teachers. The team indicated that presenting the goals involved language understandable to teachers and yet mathematically accurate and that also had to be in a coherent manner. … we still felt [Math Focus Points] needed more work, grain size was sometimes an issue; not in every single one but sometimes; making sure what we were saying in words was mathematically correct; we want this wording to really be understandable to teachers and accurate. (INV team)

Challenge 2: Determining Where to Provide Educative Features The curriculum authors learned what elements were not clearly communicated to teachers based on feedback from teachers and research. In the process of development and refinement, they had to decide where they needed to provide the critical information (e.g., right at the lesson or in the unit level) and in what ways to make things explicit in order to better communicate their ideas to teachers. The MTB team indicated that they revised their current edition to have design rational more transparent at the lesson and unit level rather than the grade-level Teacher Implementation Guide. Teachers could see the related information right there as they prepared a lesson. The MTB team commented, they’re [content expectations are] more clearly tied to lessons and [this helps teachers in terms of] how they would be assessed and if [students don’t meet the expectations], what you can do about it. So that was a big idea that came out of that research and revision study about what we needed to improve about the second and third edition. (MTB team)

In contrast, the INV team moved text-heavy resources from individual lessons to the back of each unit so that teachers could look it up when needed. How much information needs to be provided in the lesson or unit level and what could be explained in the resources for overall guidance, such as handbook or implementation guide (by grade or grade band)? What and how much detail in individual lessons? Such decisions to make were related to another challenge.

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Challenge 3: Providing the “Right” Amount of Resources All three curriculum programs included extensive resources to support teachers’ learning of new content (e.g., early algebra in INV and advanced content in EM and MTB) and pedagogical approach. The comment by the MTB team below indicated the challenge that the curriculum authors had in supporting teachers with the “right” amount of information. The publishers did not like to include lengthy information for teachers, and teachers might not read it. How much information for teachers is “sufficient” for all? Using fewer words but providing sufficient support is a challenging task. It’s hard to teach an open-ended lesson where you want the mathematics to get some place unless you understand that mathematics deeply. So those were a lot of the things that we grappled with as we tried to put it together, and had a huge fight with the publisher because they didn’t want us to use so many words. And this is a problem, it’s a tension, because if you put too many words on there they don’t even start. Yeah. So, it took us a while to try to come to grips with the fact that we’d better tighten it up or they weren’t going to read it at all. Yeah. And so, this was always a tension that we grappled with all the way through. (MTB team, italics added)

In fact, it seemed that the text-heavy materials created both comfort and discomfort to teachers. Whereas teachers could find the information in the materials when needed, heavy texts overwhelmed them as well, which could cause them to miss or even not read important information and guidance. Challenge 4: Working with Tension Between Authors’ Intention and Teachers’ Perception and Need Certainly, various features of curriculum materials support teachers’ continued learning on the content they teach and the methods to teach mathematics, and yet teachers determine what to read based on what they think they need to know more. The curriculum authors worked with the tension between what the curriculum authors thought teachers needed and what teachers thought they needed, and the tension between the kinds and the amount of resources the authors wanted to provide and what teachers could attend to. …when we first started on the first edition all of us was how naive we were… We were newbies to say the least and I think one of the things that we were naive about was, again there’s a tension between how much you, what you should tell a teacher is going on in that lesson and what they can attend to…. I look back at what’s in the second and third editions. I can see why the teachers were frustrated, because there was a really good, we would say you can in some things we would just say you can use the DPP to assess and we didn’t and you had to go dig back maybe to the beginning of the unit, at the unit level things to see what it could assess. Well no wonder they were, you know, they were frustrated about that. And while we had high minded ideas about embedded assessment, we didn’t really, I would argue, do that great a job helping teachers track what the kids knew through that embedded assessment. (MTB team, italics added)

Challenge 5: Working with Standards, Policies, and Pressure Another challenge all three teams faced throughout different editions they had developed was the difficulty working with the outside forces or policies, such as NCTM Standards (e.g., NCTM, 2000), 50 State Standards, the Common Core State

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Standards (National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010), and the pressure from the field (e.g., the INV team’s emphasis on computational fluency), especially when the approach and basic tenets were different from the curriculum authors’. For example, all three programs had to incorporate the Common Core State Standards in their programs since 2010 because the Standards were adopted by most states in the USA.  The curriculum teams had to compromise what they believed in terms of scope and sequence, which created additional challenges in their revisions. The MTB team also mentioned that they had to use important content as a context to support students’ learning of others because those were not required in K-5 in the Common Core. For example, Probability is another one of those that comes up as kids get older because it happens outside of the K-5 standards. But there are problems that really are probability problems or probability contexts, but we can get kids to engage in them anyway even though there is no probability and that standard, we really don’t care about the probability part of it. We care about the comparative representation ratio and fractions part of it, and that’s all that matters. (MTB team, italics added)

Despite all of the authors’ efforts to overcome the challenges to design the curriculum programs, communicating the authors’ intentions, and goals to teachers clearly was constrained by mostly written formats. The EM team mentioned, “even if a teacher read every word of our books, it’s unclear whether they would get all of it.” This indicated that professional development was critical in using the programs productively. A few questions arose regarding different approaches taken by the curriculum teams. How would teachers perceive these different approaches, especially presented in the written format? How would they feel about the support (guidance and educative features) provided? What would they think is useful in teaching? The curriculum authors mentioned a few times that they used research findings in the development and revision process. In this volume, we present our analyses of the programs from the perspective of teacher use and teaching. We hope that our analyses provide a new point of view to the research on curriculum, which supports both the design and the use of curriculum products.

References Charles, R. I., Crown, W., Fennell, F., et al. (2008). Scott Foresman–Addison Wesley Mathematics. Glenview, IL: Pearson. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Davis, E.  A., Palincsar, A.  S., Arias, A.  M., Bismack, A.  S., Marulis, L.  M., & Iwashyna, A. K. (2014). Designing educative curriculum materials: A theoretically and empirically driven process. Harvard Educational Review, 84(1), 24–52. Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

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National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards mathematics. Washington, DC: Author. Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253–286. Senk, S. L., & Thompson, D. R. (Eds.). (2003). Standards based school mathematics curricula: What are they? What do students learn? Mahwah, NJ: Erlbaum. TERC. (2008). Investigations in Number, Data, and Space (2nd edition). Glenview, IL: Pearson Education Inc. TERC. (2008a). Investigations in number, data, and space: Implementing investigations in grade 3 (2nd ed.). Glenview, IL: Pearson Education. TERC. (2008b). Investigations in number, data, and space: Implementing investigations in grade 4 (2nd ed.). Glenview, IL: Pearson Education. TERC. (2008c). Investigations in number, data, and space: Implementing investigations in grade 5 (2nd ed.). Glenview, IL: Pearson Education. TIMS Project (2008). Math Trailblazers (3rd Edition). Dubuque, IA: Kendall/Hunt Publishing Company. TIMS Project. (2008a). Math trailblazers teacher implementation guide grade 3. Dubuque, IA: Kendall/Hunt. TIMS Project. (2008b). Math trailblazers teacher implementation guide grade 4. Dubuque, IA: Kendall/Hunt. TIMS Project. (2008c). Math trailblazers teacher implementation guide grade 5. Dubuque, IA: Kendall/Hunt. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill. University of Chicago School Mathematics Project. (2008a). Third grade everyday mathematics assessment handbook. Chicago, IL: McGraw-Hill. University of Chicago School Mathematics Project. (2008b). Fourth grade everyday mathematics assessment handbook. Chicago, IL: McGraw-Hill. University of Chicago School Mathematics Project. (2008c). Fifth grade everyday mathematics assessment handbook. Chicago, IL: McGraw-Hill.

Part II

How Curriculum Authors Communicate with Teachers

Chapter 5

Beyond the Script: How Curriculum Authors Communicate with Teachers as Curriculum Enactors Janine T. Remillard and Ok-Kyeong Kim

Abstract This chapter introduces our approach to analyzing how curriculum authors communicated with teachers. Built on recommendations by Ball and Cohen (Educational researcher 25:6–14, 1996), Davis and Krajcik (Educational Researcher 34:3–14, 2005), and Remillard (Curriculum Inquiry 29:315–342, 1999, Elementary School Journal 100:331–350, 2000), we explored whether and the extent to which curriculum authors provided guidance intended to support teachers in their roles as curriculum enactors. The chapter reports on the coding framework used by the ICUBiT team to categorize different approaches of communicating with teachers in mathematics lesson guides and presents findings from quantitative analysis of these data. We found differences in the quantity of communication and the authors’ tendencies to direct teachers’ actions versus communicate to them about mathematics, student thinking, or design rationale (which we considered potentially educative). We also found that one program tended to combine directive and educative communication much more extensively than others. When looking across findings from previous chapters, we found alignment between mathematical emphasis, pedagogical approach, and approach to communicating with the teacher in the lesson guide. The findings in this chapter set up Chaps. 6, 7, and 8, which offer an in-­ depth analysis of communication within each type of support for teachers.

The findings in this chapter reflect the work of the entire ICUBiT team, all of whom were fully involved in conceptualizing coding categories and undertaking coding of the 75 lesson guides. These team members include: Napthalin Atanga, Shari McCarty, Luke Reinke, Dustin Smith, Joshua Taton, and Hendrik Van Steenbrugge. J. T. Remillard (*) Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] O.-K. Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_5

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Keywords  Curriculum analysis · Mathematics curriculum materials · Educative curricula · Communication with teachers · Mathematical explanations · Student thinking · Design transparency · Design rationale · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman–Addison Wesley Mathematics · Teacher’s guide

5.1  Introduction to Part II In Chaps. 2 and 3, we compared the content of the five elementary curriculum programs1 with respect to mathematical treatment and emphasis and pedagogical approach. Our findings revealed substantial variation in how the mathematical content was sequenced and represented, the types of mathematical work students were expected to do, and the role the teacher should play in supporting student learning, even across the programs whose authors shared similar perspectives on mathematics and student learning, as described in Chap. 4. This variation is likely to contribute to qualitatively different opportunities to learn mathematics. It is also likely to place different types of demands on teachers, who are key players in leveraging the contents of designed curriculum materials to enact and support these mathematics learning opportunities for students. The analysis in Pt. II of this volume examines whether and how the teacher’s guides were designed to support teachers as curriculum enactors. We examine how curriculum authors communicated with teachers and what they communicated about. This part is composed of four chapters. We conducted a detailed analysis of all sentences and phrases in the guides for 75 lessons (half of those analyzed for Chaps. 2 and 3), 15 from each program. This chapter presents the conceptual framework that guided the analysis and the methodology used to examine how curriculum authors communicate to teachers. We also discuss general patterns found in types of guidance across all coding categories and programs. The three chapters that follow provide more focused analyses of three key types of support analyzed: (a) explaining the mathematics, (b) supporting teachers to anticipate student thinking, and (c) making design decisions transparent. In these chapters, we use quantitative and qualitative analyses to look at the ways that program authors employ each type of support and we consider the potential consequences for teachers of these design decisions.

 The five programs are Everyday Mathematics (EM), Investigations in Number, Data, and Space (INV), Math in Focus (MIF), Math Trailblazers (MTB), and Scott Foresman–Addison Wesley Mathematics (SFAW). See Chap. 1 for more details about the programs. 1

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5.2  A  n Approach to Examining How Curriculum Authors Communicate with Teachers In an article entitled Reform by the book: What is—or might be—the role of curriculum materials in teacher learning and instructional reform? Ball and Cohen (1996) question the way that textbooks and curriculum materials are used as a lever of curricular and pedagogical reform. They speculate that “curriculum materials could contribute to professional practices if they were created with closer attention to processes of curriculum enactment” (p. 7). They use the term curriculum enactment to refer to the work teachers do in the moment, which includes listening to and interpreting students’ responses, deciding how to respond to individual students while engaging the entire class and moving them toward specified learning goals. Noting that conventional teacher’s guides tended to direct teachers’ instructional actions, Remillard (1999, 2000) argued that curriculum authors might give greater attention to communicating directly with teachers about the aims and ideas in the materials. Davis and Krajcik (2005) build on these ideas, using the term “educative” to refer to curriculum materials designed to speak directly to teachers and support them in planning and enacting curriculum. This chapter, and the three that follow, present findings from our comparative analysis of these communicative supports. Our analysis was guided by the following questions: 1. What types of communicative supports do curriculum authors provide teachers to aid them in planning and enacting the curriculum? 2. How do these supports differ across five curriculum programs with respect to their quantity, what they communicate about, and how they communicate with teachers? 3. What do these differences suggest about the way each program might contribute to teachers’ use of them? Research on these questions is critical to understanding the potential role that curriculum materials might play in supporting teachers to deepen and enrich their mathematics instruction. Because they are used by teachers across the USA, and many other parts of the world, curriculum materials have been seen as a primary vehicle for instructional change in classrooms. The need for these materials to support teachers is especially strong in schools that are under resourced, primarily in low-income communities, where other types of professional supports are scarce.

5.2.1  Conceptual and Empirical Background In this section, we describe frameworks in the literature that have guided our analysis. We also review the literature on the relationship between communication in curriculum materials and curriculum enactment. We then present the analytical framework that guided our analysis.

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How Curriculum Authors Communicate with Teachers Over the last two decades of curriculum development and research on teachers’ use of curriculum materials, increasing attention has been paid to how and what curriculum designers communicate to teachers. The idea that these resources might exert their influence on teachers not simply by scripting instruction, but by providing guidance that is educative and that supports teachers in enacting curriculum (Ball & Cohen, 1996; Davis & Krajcik, 2005; Schneider & Krajcik, 2002) has garnered interest among researchers and curriculum developers. Curriculum materials have typically organized and provided material to present to students during instruction. Remillard (1999) refers to this approach as speaking through the teacher, that is, providing guidance by directing teacher actions. Remillard suggests that curriculum materials seeking to support enactment need to speak to the teacher about the rationale behind the design. This position is shared by several scholars of teaching and assumes that even when following curriculum guides, teachers play a fundamental role in enacting curriculum in the classroom with their students (Ball & Cohen, 1996; Ben-Peretz, 1990; Snyder, Bolin, & Zumwalt, 1992; Stein, Grover, & Henningsen, 1996). Positing that the design of curriculum materials also matters for their use, Remillard (2005, 2019) argues for adopting a participatory perspective when studying curriculum design and enactment. In short, “examining teachers’ work with curriculum resources requires taking into account how teachers interact with resources, as well as how these interactions are mediated by both the teacher and the materials” (Remillard, 2019, p. 176). Davis and Krajcik propose a set of organizing categories for how curriculum authors might offer educative supports in curriculum materials. They could potentially (a) help teachers attend to student thinking, (b) provide subject-specific content support, (c) help teachers connect ideas within a given discipline, (d) communicate curriculum designers’ rationale for pedagogical choices, and (e) foster teachers’ ability to effectively mobilize curricular materials within a specific classroom context. Our examination of five elementary curriculum programs suggests that most materials address some of these design elements, but that significant variation exists in both quantity and approach used. As we discuss in the following review of literature, a developing body of evidence suggests that what and how curriculum authors communicate to teachers matters for their teaching. Does Communication in Curriculum Materials Matter for Teaching? Growing evidence suggests that under particular conditions, curriculum programs do influence teacher decisions, classroom practice, and even student learning (Agodini et al., 2013; Tarr et al., 2008). Researchers and curriculum developers promoting the inclusion of educative features in curriculum materials hypothesize that when curriculum authors communicate directly with teachers, they are more likely to use the supports provided (Davis et al., 2014).

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Researchers in the USA have begun to explore teachers’ use of educative supports in teacher’s guides and their impact on their teaching, producing evidence in both elementary science and mathematics, that educative supports can influence instruction, when teachers attend to them. Schneider and Krajcik (2002) describe how science teachers who read the educational supports in their curriculum materials were able to effectively enact particular strategies in ways that supported students learning. They also found evidence that supports describing how students make sense of particular scientific representations aided teachers in responding to students’ thinking during lessons. In a study aimed specifically at examining the influence of supports on instruction, Arias, Bismack, Davis, and Palincsar (2016) traced how the language and teaching moves suggested by educative components in the curriculum materials appeared during teachers’ enacted science lessons. In a quasi-experimental study, Beyer and Davis (2009) tested different approaches to communicating with teachers. They found that pre-service teachers who read supports that provided expository descriptions of science concepts when preparing for a lesson were able to identify principles of practice they could apply to other lessons. Those who read narrative supports gleaned information that was specific to the particular lesson, but less generalizable. Others analyzing the guidance in teacher’s guides and the corresponding instruction have found important relationships between what and how curriculum authors communicate and lesson enactment. Research on teachers’ use of educative supports in mathematics education have found that the content and format of the educative supports matter for how they are taken up by teachers. Grant, Kline, Crumbaugh, Kim, and Cengiz (2009) studied teachers’ use of supports related to facilitating classroom discussions in the INV curriculum. They found that the supports in the teacher’s guide were more effective at helping teachers elicit student thinking about a specific concept than they were at helping students extend their thinking to general understandings. An analysis of the communication in the teacher’s guide led Grant et al. to conclude that they “conveyed a somewhat general image of the role of the teacher as questioner and facilitator of student discussion” (p. 113). Stein’s and Kaufman’s (2010) study of teachers’ use of educative supports in INV and EM examined how curriculum authors helped teachers attend to the central mathematical point of the lesson. They found that when teacher’s guides identified the “big mathematical ideas” of a lesson, teachers who attended to this information were more likely to enact lessons that matched the goals of the program and that were rated as higher quality using several measures. Remillard, Reinke, and Kapoor (2019) analyzed how four of the programs included in the ICUBiT Study (EM, INV, MIF, and MTB) communicated the mathematical goals of a lesson and how and whether they orient the activities in the lesson toward those goals. Analysis of lessons and corresponding teachers’ enactment of these lessons indicated a correlation between the depth to which the goals were communicated and the extent to which teachers steered the instruction toward the goals. These findings lay the groundwork for the analysis in this chapter and the three chapters that follow. We examine the different approaches to communicating with teachers used in the five curriculum programs and consider their implications for supporting teachers. Our analysis focused on the teacher’s guide and specifically the guidance for the daily lessons.

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Teacher’s Guides as Tools of Communication for Teachers In Chap. 1, we argue that full consideration of the potential opportunities to learn mathematics in curriculum materials must take into account the role of the teacher and how the materials support the teacher. The teacher’s guide, written for teachers, is typically intended to support them in shaping lessons and using the student-facing resources in a curriculum program. These guides vary in the types of guidance they provide and the forms of communication they use. The teacher’s guide, and specifically the guidance provided for the daily lessons (or lesson guides) were the primary focus of our analysis in our effort to understand how curriculum authors communicate with and support teachers. Because they are intended for daily use and offer guidance on shaping each mathematics lesson, these guides offer the best representation of the designers’ intended curriculum.

5.2.2  Analytical Framework The conceptual categories guiding our analysis draw on Ball and Cohen’s (1996) and Davis and Krajcik’s (2005) concepts of supporting teacher enactment and educative materials. We identified five types of guidance that curriculum authors provide teachers (Table  5.1). The first two, providing referential information and directing teachers’ and students’ actions, include basic information about the planned curriculum. In contrast, the next three categories provide examples of ways curriculum authors might communicate to teachers about aims and ideas underlying

Table 5.1  Primary codes, specifying types of guidance Code Title 0 Providing referential information 1

2

3

4

Include statements that: Provide information for the teacher about the lesson without simultaneously accomplishing aims specified in categories 1–4. Includes lists of material or vocabulary, references to standards, other pages or resources, and answers to questions asked of students. Directing teachers’ Indicate what teachers and students should do or say during or in preparation for the lesson. or students’ actions Providing design Explain or clarify to teachers the curriculum designers’ decisions transparency about goals, lesson structure, what the lesson is intended to accomplish, or how the lesson is related to other lessons. Attending to Address aspects of student understanding, including what students students’ thinking should know or be able to do, ways they may respond, difficulties they may encounter, strategies they may use, or schemas they may possess or develop. Describe mathematical relationships, ideas, definitions, or properties Attending to to the teacher; specify the mathematical importance of a particular mathematical concept or idea. ideas

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the curriculum design (Remillard 1999). These last three categories potentially fit the classification of educative supports, in that they provide information and guidance that would support the teacher in enacting the lesson. We note here that whether such guidance is truly educative is an empirical question we seek to explore in Chaps. 6, 7 and 8.

5.3  Methods Analysis for this chapter and the three that follow involved two stages of coding and analysis. Using the coding categories detailed in Table  5.1, we assigned primary codes to communication to the teacher in 15 lessons per curriculum program. Findings from the analysis of the primary coding are detailed in this chapter. Chapters 6, 7, and 8 present findings from our secondary coding, which involved a more in-depth analysis of several types of communication found in the teacher’s guides. We detail our coding and analytical process for the primary stage of analysis below.

5.3.1  Curriculum Programs and Sampling As described in Chaps. 1 and 4, we analyzed a set of lessons randomly selected from teacher’s guides of five curriculum programs: Everyday Mathematics (EM) (University of Chicago School Mathematics Project 2008), Investigations in Number, Data, and Space (INV) (The Education Research Collaborative at TERC 2008), Math in Focus (MIF) (Marshall Cavendish International 2010), Math Trailblazers (MTB) (TIMS Project University of Illinois at Chicago 2008), and Scott Foresman–Addison Wesley Mathematics (SFAW). These programs are described in Chap. 1. Sample Selection For this analysis, we used 75 of the lessons analyzed for Chaps. 2 and 3. They were drawn from the numbers, operations, and algebra (NOA) domains of the five curriculum programs. Five randomly selected lessons per grade (grades 3–5) for a total of 15 lessons per program. Our preliminary analysis of a subset of these lessons revealed substantial internal consistency in their structure, format, and modes of communicating with the teacher (Remillard, 2013). Because we were primarily interested in how curriculum authors communicated with teachers, we focused on the guidance intended for the teacher and not the student text, even if the student text was pictured in the guide.

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Unit of Analysis For the primary coding, we treated the sentence (or phrase or image)2 as the unit of analysis. This decision was informed by preliminary examinations of the lesson guides and/or interest in characterizing how and how much curriculum authors communicated with teachers. As is evident in the INV excerpt in Fig. 5.1, lesson guides provide a great deal of information that often intertwines directive s­ uggestions with explanations about the design, insights about how students might respond, and information about the mathematics. Coding at the sentence level allowed us to look at the smallest complete unit in the guides and take account of the specific focus of each. It also seemed to be the best way to assess the extent to which curriculum

Fig. 5.1  Excerpt from INV GR3 1.1.2, p. 41 illustrating sentences assigned to each primary code. From Investigations 2008 Curriculum Unit (Grade 3, Unit 1) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved  The complete sentence was our primary unit of analysis, however, when phrases or images were used to communicate information to the teacher, we treated each as a single unit. In reporting our findings, we use sentence to refer to all of these units. 2

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authors communicated about particular ideas or in particular ways, and how they represent these values quantitatively. Using this approach does not presume that any sentence in a teacher’s guide is intended to be read in isolation of others.

5.3.2  Primary Coding and Analysis Six research team members coded each sentence or phrase in each lesson guide, using the coding scheme detailed in Table  5.1, using ATLAS.ti. These primary codes indexed each sentence or phrase, according to what it communicated to the teacher about or the kind of guidance it provided. We allowed for double coding in a few specific instances. In cases where the sentence both directed the teacher’s actions (code 1) and, concurrently and with equal emphasis, provided educative guidance about design rationale (code 2), student thinking (code 3), or mathematics (code 4), we allowed for double coding. We refer to these sentences, for coding purposes, as hybrids (1/2, 1/3, 1/4, or 1/3/4). We also allowed for double coding with codes 3 and 4 when appropriate, as combining mathematical explanations with information about student thinking or strategies was common in several programs. See Table 5.2 for examples of possible double codes. In addition, we used structural codes to indicate where in the lesson guide each sentence was located: (a) Introductory and orienting material, (b) main body of the lesson, and (c) beyond the main body of the lesson. Introductory material refers to the portions intended to orient the teacher to the lesson’s goals, objectives, materiTable 5.2  Examples of possible double codes Code nos. Coding categories 1/2 Directing Action (DA) + Design Transparency 1/3

DA + Attending to Students’ Thinking

1/4

DA + Attending to Mathematical Ideas

1/3/4

DA + Attending to Students’ Thinking + Mathematical Ideas

3/4

Attending to Students’ Thinking + Mathematical Ideas

Example To support English language learners, write prime numbers and composite numbers along with the definition… (EM) Use journal page 169, Problem 10 to assess children’s progress toward using relationships between units of time to solve number stories. (EM) Guide students to simplify algebraic expressions based on the concept that: a + a + a + … + a(n terms) = n × a. (MIF) Listen for explanations that focus on how subtracting exactly 10 or a multiple of 10 affects the number of tens (represented by the tens digit in the difference), while the number of ones (represented by the ones digit in the difference) is the same as the ones digit in the original amount. (INV) Another student may refine the estimate by computing 8 × 500 = 4000 and choosing a number between 4000 and 4800 such as 4500. (MTB)

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als, vocabulary, activities, practice problems, homework, and assessment resources. The main body of the lesson refers to material intended for the primary focus of the lesson. It typically included guidance for the lesson as well as teaching notes associated with this guidance. We coded material as beyond the main body of the lesson if it was material teachers had the option to use or omit in order to customize or extend the lesson. Use or omission of this material did not significantly alter the main objectives of the lesson. The intent of the structural codes was to provide a mechanism for sorting between guidance designed to support the primary instructional activities of the lesson, guidance intended to provide orienting information, and guidance that was not core to the lesson. Rater Agreement All team members who coded were also involved in conceptualizing and defining the codes, which were documented in a project coding manual. Still, achieving the desired agreement proved to be a challenge. All team members were experienced educators and were inclined to interpret statements in different teacher’s guides through their own lenses. We addressed this challenge by (a) refining and further specifying our coding guidelines and holding a week of extensive training and practice, and (b) conducting all coding within a relatively short window of time (1–2 weeks), to ensure that coding rules were fresh in all coders’ minds. All coders first coded the same five lessons, one from each program, to assess intercoder reliability, using Krippendorff’s alpha (Krippendorff, 2004, 2011). We first assessed reliability differentiating between codes 0, 1, and 2–4, since these clusters involved between 700 and 800 total sentences. The alphas were all over 0.8, as shown in Table 5.3. We then assessed the reliability of each 2–4 code (those that communicated to teachers) individually, which involved coding between 300 and 700 total sentences per category. These alphas were between 0.66 and 0.76, which is considered acceptable for conservative indices, such as Krippendorff’s alpha (Lombard, Snyder-Duch, & Bracken, 2004). Satisfied with the level of agreement, we p­ roceeded with individual coding. Each coder coded between 10 and 13 lessons independently.

Table 5.3 Coder interrater reliability calculations

Code 0 1 2–4 2 3 4

Category or cluster Referential information Directing action Communicating to teachers Design transparency Students’ thinking Mathematical ideas

Krippendorff’s alpha 0.85 0.82 0.81 0.76 0.66 0.72

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Analysis We exported a summary of coded sentences and the associated codes into an Excel spreadsheet and compiled counts and percentages for each coding category (type of communication) by curriculum program. As detailed in Sect. 5.4, we used different data representations to answer our different research questions.

5.4  Results of Primary Coding In this section, we present quantitative findings from the primary analysis of how curriculum authors communicate with teachers in lesson guides. By compiling our data by type of communication and curriculum program, we offer a picture of the quantity and location of different types of communicative supports, what they communicate about, and how they communicate with teachers. In the sections that follow, we use tabular data and descriptive statistics to present our findings and make comparisons across the curriculum programs.

5.4.1  Quantity and Location of Communication In order to get a general picture of how communicative supports for teachers were used across the five programs (in the lesson guides for 15 lessons per program analyzed), we calculated the mean number of sentences (or phrases) per lesson. We also examined how these sentences were distributed across the three structural components of the lesson guides, introductory or orienting content, main body of the lesson, and content beyond the main body of the lesson. As discussed in Chap. 3, the amount and prominence of material included in the introductory and the content beyond the main body of the lesson varied across the five programs. Our primary analytical interest was in the content considered in the main body of the lesson, since this is the material teachers were likely to spend the most time with when preparing and teaching the lessons. Table 5.4 shows the mean number of sentences and phrases per lesson and differences in the distribution of sentences across the three structural components within each program. Several patterns stand out. First, the difference in quantity of communication to the teacher is striking. By treating the sentence or phrase as the unit of analysis, we are able to see that EM and MTB provided substantially more communication to the teacher than the other three programs. These patterns change, however, when looking at the proportion of sentences found in each section of the lesson guide. The column indicating the number and proportion of sentences per lesson devoted to the Main Body of the lesson shows that EM, INV, MIF, and MTB included substantially more sentences in this part of the lesson guide (ranging from 88.6 to 127 sentences, compared to 28.5 in SFAW). Considering these numbers as a

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Table 5.4  Distribution of sentences within each structural component across five curriculum programs

EM

Mean sentences per lesson 173.4

INV

109.4

MIF

113.8

MTB

152.5

SFAW 103.9

Number (percent) of sentences per lesson categorized as: Introductory/ Main Beyond the main orienting body body 20.9 (12%) Mean 25.5 (15%) 127 8–30 Range 21–31 (73%) 92–158 7 (6%) Mean 13.8 (13%) 88.6 3–13 Range 8–18 (81%) 36–52 8.3 (7%) Mean 7.9 (7%) 97.6 3–22 Range 6–11 (86%) 53–148 7.9 (5%) Mean 42.6 (28%) 102 0–38 Range 30–70 (67%) 43–262 68 (66%) Mean 7.4 (7%) 28.5 48–79 Range 4–11 (27%) 17–50

proportion of the total number of sentences per lesson indicates that these four programs were fairly comparable in the relative amount of guidance devoted to the actual lesson, ranging from 67 to 86%. SFAW, in contrast, devoted only 27% of its communication to the main lesson, focusing much more on additional and supplementary material (66%). The amount of guidance provided with the main part of the lesson might be interpreted as the presumed amount of support teachers are likely to need to enact the lesson. Following this assumption, it appears that the authors of SFAW viewed teachers as needing minimal guidance. One explanation for this approach can be found in the pedagogical approach analyzed in Chap. 3. SFAW positioned the learning as occurring primarily between the student and the textbook, which suggests a diminished role for the teacher. The large proportion of support placed prior to the main lesson in the SFAW lesson guide included a full-page spread of optional activities before the guidance on teaching the lesson (see Appendix E). These options include sections called Spiral Review and Investigating the Concept, as well as activities aimed at different levels of students (English language learners, those needing additional support, or extension). Although it was not entirely clear whether some of these activities were intended to be part of the main body of the lesson, they were not incorporated into the lesson description, which was organized under four headings: “1 Warm Up, 2 Teach, 3 Practice, 4 Assess.” Another difference that stands out in Table 5.4 is that MTB devotes almost 30% of its communication to the orienting elements of the lesson guide. Each MTB lesson guide includes at least two pages of orienting guidance, including a lesson overview, lists of key content and vocabulary, and required materials. The main points of the lesson are also summarized in a page entitled At a Glance. EM and INV devote

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a single page to orienting guidance. In contrast, MIF and SFAW both devote the margin of one page to this type of guidance. (See Appendices B–E for sample lesson guides from four of the programs.) Much of the information in the orienting sections of lesson guides would be coded as providing design transparency, as it explains designers’ decisions about goals, lesson structure, intent, and connections across lessons. MTB authors, along with INV and EM (all three Standards-based programs), placed much more stock in providing such orienting information for teachers than MIF and SFAW. (See Chap. 8 for more details on these types of supports.)

5.4.2  What Curriculum Authors Communicate About Given our interest in what curriculum authors communicate to teachers about, we coded each sentence in the lesson guides for the type of guidance it provides the teacher, using the codes described in Table 5.1. This process allowed us to differentiate, for the most part, statements aimed at directing teacher action from those communicating to teachers about student thinking, mathematics, or various design decisions. As detailed in Sect. 5.3.2, we found that it was not uncommon for some programs to combine different types of guidance in a single sentence and, as a result, both directing teacher action and offering an explanation about the mathematics or supporting teachers to anticipate student thinking. We referred to sentences in which authors embedded information about mathematics, student thinking, or design transparency in statements that directed teacher or student actions as hybrid sentences. (See Table 5.2 for examples.) We also found some cases in which authors provided guidance about student thinking and the mathematics in the same sentence, which we double coded, but did not refer to as hybrids. Our findings from this analysis are compiled in Table 5.5. Each column provides the mean and range percentages for one type of guidance, as specified by a coding category. The percentages in columns 2–5 (Directing Action through Design Transparency) are inclusive of all multiply coded sentences. For example, a sentence coded as both directing action and anticipating student thinking is represented in both columns 2 and 4. The data in Table 5.5 suggest a number of commonalities in the types of communication, despite moderate differences. For all programs, the most prominent type of communication was directing teacher or student action, although in MTB providing design transparency was equally prominent (30% of all sentences). This finding is not surprising, given that guiding teachers to enact lessons is the central purpose of lesson guides. Still, we see some variation in this category, with MIF and SFAW devoting just over half of their communication to directing actions and EM and INV devoting between 40% and 48%, respectively. In contrast, MTB devoted relatively less attention to directing action (30%). For EM, INV, MIF, and SFAW, providing design transparency was the second most common type of communication. In general, explaining the mathematics and anticipating student thinking

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Table 5.5  Variations in total guidance for teachers across five curriculum programs

EM

Mean sentences per lesson 173.4

INV

109.4

MIF

113.8

MTB

152.5

SFAW 103.9

Meanb Rangec Mean Range Mean Range Mean Range Mean Range

Percent of total numbera of sentences devoted to… Attending Attending Providing to student referential Directing to math thinking ideas information action 28.1 40.0 14.9 14.0 20–34 35–49 2–28 5–37 15.0 48.2 8.8 19.1 6–24 23–67 2–21 8–34 22.0 53.0 23.5 19.0 15–28 41–68 8–42 4–34 23.3 29.9 20.1 17.2 15–36 5–48 2–38 2–29 21.1 51.0 9.0 14.4 15–28 44–57 1–22 6–24

Design transparency 23.0 17–32 26.6 14–43 22.8 17–32 30.6 12–55 17.2 14–22

The percentage in each coding category is inclusive of all double-coded sentences. Thus, each row does not sum to 100% b The mean in each coding category refers to the mean percentage for the 15 lessons analyzed for each program c The range shows the highest and lowest percentage coded in each coding category for a single lesson, rounded to whole number values a

received less attention than the other categories for all but MIF, which was fairly balanced in its focus on these three types of communication. As we explore in Sect. 5.4.3, this balance is attributable to the tendency for MIF authors to combine messages about these topics with directing teacher or student actions. As noted earlier, the mean and range in each column are inclusive of all double-­ coded sentences. That means hybrid sentences are represented in both the column for Directing Action and one or more of the columns for codes 2–4. Thus, it is difficult to tell from this analysis how curriculum authors communicated with teachers about these different topics. In the following section, we remove the hybrid sentences (those that directed teacher action while also communicating to teachers about student thinking, the mathematics, or various design decisions), in order to explore how authors communicated with teachers.

5.4.3  How Curriculum Authors Communicate with Teachers In order to consider the extent to which curriculum authors communicated to teachers as opposed to through them, we represented the hybrid sentences (those that combined speaking through and to the teacher) in a separate column in a new table. In Table 5.6, the first four columns display the mean and range percentages of sentences per lesson devoted to only each coding category, with the exception of the two columns on the right. The column second from the far-right shows the percentage of sentences coded as communicating about mathematics and student thinking. The percentage in the far-right, shaded column shows the percentage of hybrid sentences in

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Table 5.6  Variations in focused types of guidance for teachers across five curriculum programs

EM

Mean Ranged INV Mean Range MIF Mean Range MTB Mean Range SFAW Mean Range c

Percent of total numbera of sentences devoted ONLY to… Math and Attending to student Design Directing Attending to student transparency thinking action math ideas thinking 27.6 4.1 4.6 19.1 4.1 14–35 0–19 1–14 14–25 0–19 36.1 2.0 10.0 20.6 4.2 18–55 0–8 5–22 10–41 0–15 28.1 0.8 2.4 19.3 2.6 19–43 0–9 0–4 15–28 0–10 20.1 7.2 4.3 26.8 8.5 5–34 2–18 0–9 12–47 1–21 42.8 2.1 6.9 16.4 2.4 31–52 1–9 2–16 12–22 0–8

Hybridsb 12.4 12.1 24.8 9.8 8.3

The percentage in each coding category refers to sentences coded as ONLY in that category. With the exception of Math and Student Thinking, all double codes have been excluded b Sentences that combined speaking through and to the teacher c The mean in each coding category refers to the mean percentage for the 15 lessons analyzed for each program d The range shows the highest and lowest percentage coded in each coding category for a single lesson, rounded to whole number values a

the data, which were removed from the other column in Table 5.5 because they combine speaking through and to the teacher. The percentages in the Directing Action column, for example, refer to sentences written to direct teachers’ or students’ actions, without also communicating about mathematics, student thinking, or design transparency. Sentences that provide a script (e.g., Did anyone start by adding 40 + 30?) or suggested action, like “Call on a student who used this strategy” (Fig. 5.1) would be classified in this way. In columns 2–5, the percentages refer to sentences in which curriculum authors were communicating directly to teachers about the specified topics, without also directing actions. For example, “Students are likely to have solved the problem with different strategies, including adding by place or adding one number in parts” (Fig. 5.1) would have been classified as anticipating student thinking. Because all hybrid sentences have been removed from the other categories in Table 5.6, we can see patterns in how authors communicated with teachers. Only 20% of the sentences in a typical MTB lesson direct teacher or student action exclusively, whereas almost 43% of the sentences in SFAW do so. When summing the percentages in columns 2–5, the four categories for communicating to teachers, MTB stands out as devoting the most relative attention to this type of communication (46.8% per lesson); MIF, on the other hand, communicated directly to teachers the least (25.2% per lesson). The others fall in between INV (36.8%), EM (31.9%), and SFAW (27.8%). When comparing these percentages to those in the directing action column, we see that each of the Standards-based programs communicated to the teacher as much as or more than through the teacher. MIF’s approach of communicating with teachers is also evident in the far-right, hybrid column, which shows that almost 25% of the sentences in an MIF lesson combine directing action with other information.

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Removing the hybrid sentences from columns 2 to 5 of Table 5.5 also reveals how much or how little curriculum authors communicated directly to teachers. In general, the percentages under both attending to mathematical ideas and attending to student thinking are quite low, suggesting that overall curriculum authors tend to not communicate with teachers directly about these topics. The differences in the values in each cell between Tables 5.5 and 5.6 show just how much of the communication about mathematics or students tends to be embedded in directive guidance. There are several distinctions worth noting. INV is higher than the others in the percentage of sentences devoted only to attending to student thinking per lesson (10% compared to 6.9% or less). Throughout the lesson descriptions, INV authors consistently provide example strategies and explanations students are likely to produce. (Chap. 7 provides a more detailed analysis of how the different programs communicate about student thinking.) MTB is higher than the others in the percentage of sentences devoted to explaining mathematics (7.2% compared to 4.1% or less) and those that combine student thinking and mathematical explanations (8.5% compared to 4.2% or less). (See Chap. 6 for a more detailed analysis of the different ways curriculum authors communicate with teachers about mathematics.) It is also noteworthy that the percentages in the column for design transparency decrease only moderately in Table 5.6, when compared to 5.5, suggesting that all program authors were more likely to communicate to teachers directly about elements of the design of the lesson than the other topics. One reason for this pattern is that sentences coded under design transparency included statements of learning goals and participant structures, features common to each lesson. (See Chap. 8 for a detailed analysis of approaches to design transparency in the five programs.)

5.5  Discussion and Next Steps The analysis presented above offers a high-level, quantitative analysis of what, how, and how much the curriculum authors communicated with teachers to aid them in planning and enacting the designed curriculum. Several differences emerged in the quantity and location of communication, what authors communicate about, and the extent to which they communicate through and to the teacher. In this section, we discuss several key differences, making connections back to the analysis in Pt. I, and consider implications for teachers using these curriculum materials.

5.5.1  Understanding Different Approaches to Communication As discussed in Sect. 5.2, the questions underlying the analysis in this chapter are grounded in a participatory perspective on curriculum material design and use (Remillard, 2005). This perspective assumes that the particular design of curriculum

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materials can mediate teachers’ use of them. Building on this perspective, researchers have brought attention to how curriculum authors communicate with teachers (Ball & Cohen, 1996; Beyer & Davis, 2009; Grant et al., 2009; Remillard, 1999) and what they communicate about (Remillard et al., 2019; Stein & Kaufman, 2010), finding that particular design features and educative supports can mediate teachers’ use. Looking across the different approaches to communication found in the five programs, three primary modes of communication surface. First, as discussed in Sect. 5.4.1, SFAW stands out in its tendency to communicate through the student textbook, which is visually dominant in the lesson guides (Appendix E). Communication in the lesson guide is minimal and focused primarily on directing teacher and student actions, particularly as they relate to the student text. Second, as discussed in Sect. 5.4.3, MIF provides a substantial amount of guidance intended for teachers, communicating primarily through directing action, but incorporating mathematical guidance into these directives. Perhaps the authors considered elementary teachers and their students to be the audience of this guidance, rather than just teachers. The image of the student text is also dominant in the MIF lesson guides. Third, the three Standards-based programs, while differing in important ways, communicate to the teacher as much as through the teacher and attend to all three categories of educative supports. These different modes of communication likely reflect different assumptions or philosophies about the role of curriculum materials in teaching. They may also reflect differences in the mathematical and pedagogical approaches of the program and the related demands these approaches place on the teacher. In a comparative analysis of EM and INV,3 Stein and Kim (2009) contrast the cognitive demand of the two programs and the types of educative supports provided. They note that the consistent tendency to include high demand tasks in INV, especially those involving “doing mathematics,” was matched by a greater frequency of educative supports in the teacher’s guide. EM, in contrast, offered fewer high demand tasks and those tended to involve procedures with connections. EM also provided relatively fewer educative supports for teachers. Stein and Kim argue that it was reasonable for the program placing greater demands on teachers both mathematically and pedagogically to provide more educative supports. By considering key elements of the analyses in Chaps. 2 and 3, we can consider whether there is alignment between the different modes of communication and the demands they place on teachers. For this analysis, we looked at levels of cognitive demand to consider the mathematical demands of the program and the dominant teacher role and primary source of knowledge to consider the pedagogical approach. These are summarized in Table 5.7. Cognitive demand levels are represented by the percentage of tasks coded as doing mathematics (most demanding for teachers and students) and procedures with connections (also high cognitive demand). To summarize the mode of communication from this chapter, we included the mean num-

 Stein and Kim (2009) analyzed earlier editions of EM and INV than those analyzed in this volume.

3

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ber of sentences per lesson and the percent of those sentences devoted to communication to the teacher. Our intent was to represent the amount of guidance and how the authors communicated. Like Stein and Kim (2009), we see some alignment between the mathematical and the pedagogical demands of the programs and the quantity and nature of supports. SFAW is the least demanding and positions the student textbook as the primary source of knowledge. The teachers’ role is to support students as they work through the exercises in the textbook. It is, then, not surprising that the SFAW guides provided minimal supports for teachers and communicated primarily through the student page. MIF had a moderate proportion of demanding tasks but positioned the teacher and the student textbook as key sources of knowledge. The teachers’ primary role is to tell and show students, along with the textbook. In support of this role, MIF authors provided more guidance to teachers, but did so primarily by directing action. INV, MTB, and EM all offered high demand tasks, although INV stands out as offering more that involves doing mathematics than the other two. EM positions the teacher as a guide and the primary source of knowledge. The student text plays only a supporting role in EM lessons. Given the strong role the teacher plays, it is unsurprising that EM authors offer the greatest amount of guidance per lesson, as measured by the total number of sentences. That said, compared to INV and MTB, they provide less communication to the teacher. Both INV and MTB position the teacher primarily as a guide or orchestrator of student work during lessons and the students as the primary source of knowledge and strategies. Both these programs also communicate to teachers relatively more than the other programs. Following Ball and Cohen’s (1996) call, these authors may be seeking to support teachers in the process of enactment by communicating to them about the mathematics, student thinking, and the design of the program. Presumably, the more Table 5.7  Correlating mathematical demands, pedagogical approach, and modes of communication across the five programs

INV MTB

High cognitive demand, Chap. 2 Percent procedures Percent doing with mathematics connections 59.7 40.3 23.3 71.7

EM MIF

16.0 5.6

74.6 27.8

8.3

0.0

SFAW

Pedagogical approach, Chap. 3

Dominant teacher rolea Orchestrate Guide, orchestrate Guide Tell/show via text Tell/show via text

Primary source of knowledge Students Students

Communicating in lessons, Chap. 5 Mean no. Percent of sentences that of sentence communicate to per lesson teacher 109.4 36.8 152.5 46.8

Teacher 173.4 Teacher, 113.8 student text Student text 103.9

31.9 25.2 27.8

Step back, which was often recommended when students were expected to work on review or practice tasks has been removed from this column. It played a significant role in all five programs, but was the dominant role for EM and MIF

a

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teachers understand about the intent of the lesson or tasks within it and are able to interpret students’ responses, the more equipped they will be to enact their roles as guides or orchestrators of student thinking.

5.5.2  Need for Further Analysis The quantitative analysis discussed in this chapter points to important differences in how curriculum authors communicate with teachers and reveal reasonable correlations between these approaches and the mathematical and pedagogical designs of the programs. What is not revealed in this analysis, but was abundantly apparent when we examined and coded lessons from the five programs, is that each set of authors used very different ways of communicating with teachers within each coding category. In order to look more deeply and comparatively at these communicative approaches, we undertook a secondary analysis of each set of sentences in the following coding categories: (a) mathematical ideas, (b) student thinking, and (c) design transparency. Our findings are presented in the following chapters. Chapter 6 examines approaches to discussing mathematical ideas and strategies; Chap. 7 examines approaches to supporting teachers to notice and respond to student thinking; Chap. 8 examines whether and how design transparency is communicated.

References Agodini, R., Harris, B., Seftor, N., Remillard, J.T., & Thomas, M. (2013). After two years, three elementary math curricula outperform a fourth. (NCEE 2013-4019). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Arias, A.  M., Bismack, A.  S., Davis, E.  A., & Palincsar, A.  S. (2016). Interacting with a suite of educative features: Elementary science teachers’ use of educative curriculum materials. Journal of Research in Science Teaching, 53(3), 422–449. Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is—Or might be—The role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–14. Ben-Peretz, M. (1990). The teacher-curriculum encounter: Freeing teachers from the tyranny of texts. Albany, NY: State University of New York Press. Beyer, C. J., & Davis, E. A. (2009). Using educative curriculum materials to support pre-service elementary teachers’ curricular planning: A comparison between two different forms of support. Curriculum Inquiry, 39(5), 679–703. Charles, R. I., Crown, W., Fennell, F., et al. (2008). Scott Foresman–Addison Wesley Mathematics. Glenview, IL: Pearson. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Davis, E.  A., Palincsar, A.  S., Arias, A.  M., Bismack, A.  S., Marulis, L.  M., & Iwashyna, A. K. (2014). Designing educative curriculum materials: A theoretically and empirically driven process. Harvard Educational Review, 84(1), 24–52.

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Grant, T. J., Kline, K., Crumbaugh, C., Kim, O. K., & Cengiz, N. (2009). How can curriculum materials support teachers in pursuing student thinking during whole-group discussions? In J.  T. Remillard, B.  A. Herbel-Eisenmann, & G.  M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 103–117). New York: Routledge. Krippendorff, K. (2004). Content analysis: An introduction to its methodology. Thousand Oaks, CA: Sage. Krippendorff, K. (2011). Computing Krippendorff’s alpha-reliability. Philadelphia: Annenberg School for Communication Departmental Papers. Retrieved July 6, 2011, from http://repository.upenn.edu/cgi/viewcontent.cgi?article=1043&context=asc_papers. Lombard, M., Snyder-Duch, J., & Bracken, C. C. (2004). Practical resources for assessing and reporting intercoder reliability in content analysis research projects. Retrieved September 29, 2019, from http://www.temple.edu/sct/mmc/reliability/ Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers’ curriculum development. Curriculum Inquiry, 29(3), 315–342. Remillard, J. T. (2000). Can curriculum materials support teachers’ learning? Elementary School Journal, 100(4), 331–350. Remillard, J. T. (2005). Examining key concepts of research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Remillard, J. T. (2013, May). Beyond the script: Educative reatures of five mathematics curricula and how teachers use them. Paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA. Remillard, J. T. (2019). Teachers’ use of mathematics resources: A look across cultural boundaries. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), The ‘Resource’ approach to mathematics education. New York: Springer. Remillard, J. T., Reinke, L. T., & Kapoor, R. (2019). What is the point? Examining how curriculum materials articulate mathematical goals and how teachers steer instruction. International Journal of Educational Research, 93, 101–117. Schneider, R. M., & Krajcik, J. S. (2002). Supporting science teacher learning: The role of educative curriculum materials. Journal of Science Teacher Education, 13(3), 221–245. Snyder, J., Bolin, F., & Zumwalt, K. (1992). Curriculum implementation. In P. W. Jackson (Ed.), Handbook of research on curriculum (pp. 402–435). New York: Macmillan. Stein, M. K., Grover, B. W., & Henningsen, M. A. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classroom. American Educational Research Journal, 33(2), 455–488. Stein, M. K., & Kaufman, J. H. (2010). Selecting and supporting the use of mathematics curricula at scale. American Educational Research Journal, 47(3), 663–693. Stein, M. K., & Kim, G. (2009). The role of mathematics curriculum materials in large-scale urban reform: An analysis of demands and opportunities for teacher learning. In J.  T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 37–55). New York: Routledge. Tarr, J. E., Reys, R. E., Reys, B. J., Chávez, Ó., Shih, J., & Osterlind, S. J. (2008). The impact of middle-grades mathematics curricula and the classroom learning environment on student achievement. Journal for Research in Mathematics Education, 39(3), 247–280. TERC. (2008). Investigations in Number, Data, and Space (2nd edition). Glenview, IL: Pearson Education Inc. TIMS Project (2008). Math Trailblazers (3rd Edition). Dubuque, IA: Kendall/Hunt Publishing Company. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill.

Chapter 6

Examining Communication About Mathematics in Elementary Curriculum Materials Ok-Kyeong Kim and Janine T. Remillard

Abstract  In this chapter, we examine the types of mathematical support offered to teachers and the approaches used to communicate about mathematics to teachers in five elementary mathematics programs. The results show that the location, extent, and type of mathematical support vary across the programs. All five programs, however, (1) explain mathematics to teachers directly, (2) communicate mathematics by illustrating anticipated student strategies and thinking, (3) communicate mathematical ideas by embedding them in teacher/student actions, and (4) communicate strategies and thinking by embedding them in teacher/student actions. The programs also provide mathematical goals and vocabulary of lessons and highlight the mathematics in the titles of lessons. In addition, some programs use the headings in the lesson guide to highlight mathematical ideas. The five programs communicated various types of mathematics to teachers: (1) descriptions of procedures and steps, (2) explanations of strategies, (3) explanations of concepts, definitions, and conventions, (4) explanations of representations, and (5) explanations of connections, relationships, and applications. Based on the findings of the study, we discuss the type of mathematics teachers need to know to teach these programs and ways in which mathematics can be better communicated to teachers. Keywords  Curriculum analysis · Mathematics curriculum materials · Educative curricula · Teacher’s guide · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman-Addison Wesley Mathematics · Mathematical explanations · Mathematical support

O.-K. Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA e-mail: [email protected] J. T. Remillard (*) Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_6

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6.1  Introduction Earlier in this volume, we described mathematical emphasis in the five programs as presented in the content and sequence, cognitive demand, routine practice problems, and visual and physical representations. We noticed that there were great variations in the mathematical emphasis of the programs, even in the three Standards-based programs (see Chap. 2). The interviews with curriculum authors revealed that although all were aligned with the NCTM Standards, they made different decisions on what to explain to teachers and how to communicate their ideas to them (see Chap. 4). Curriculum authors’ design decisions produced different outcomes and products. This chapter examines the types of mathematics that the five elementary curriculum programs communicated to teachers and the ways in which mathematical support was provided in these programs. As seen in the notion of educative curriculum materials (Davis & Krajcik, 2005; Davis et al., 2014), mathematical explanations provided in curriculum materials are to facilitate teachers’ own understanding of the content they teach, which supports teachers’ instructional design and enactment. In other words, providing mathematical support eventually aims to increase teacher capacity to use the curriculum resources to design instruction. In designing and refining educative features of elementary science curriculum materials, Davis et al. (2014) included explanations of science content focusing on connections of concepts in lessons and units. They also attended to content support as background knowledge for teachers. We placed these [content boxes] next to the lesson’s background sections or the procedural steps and signaled particularly important content to highlight for children, through discussion, additional information that might be helpful as background for the teacher, or even content that was missing from the base curricular materials that seemed important for the teacher to be able to foster a robust understanding among the students (p. 37).

We heard similar explanations from the curriculum authors in Chap. 4. They considered different formats, locations, lengths, and types of mathematical support for teachers to enact lessons as intended. As the curriculum programs evolved, the authors took some different approaches. We examine the authors’ approaches to providing mathematical support in the editions of the programs we analyzed. Investigating mathematical support given in the curriculum programs helps understand the teacher–curriculum relationship (e.g., Remillard, 2005) in terms of the content communicated between them. As teachers decide what to read and use in the curriculum, the types of mathematical support and ways in which it is communicated will greatly influence their use and implementation of it. Building upon the current research, we describe how mathematical support is provided in the five curriculum programs: the location and extent of mathematical support, and the ways in which the mathematics is communicated to teachers. Then, we attend to the types of mathematics communicated in individual lessons. Findings from these areas will stimulate discussion of what mathematics teachers are expected to know in order to teach mathematics using these programs, and in what ways mathematics can be better communicated.

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6.2  Theoretical Background This chapter explores ways in which the curriculum attempts to realize better enactment of the lesson, by communicating with the teacher, and increasing teachers’ knowledge about the content they teach. Therefore, this chapter draws on literature discussing teacher knowledge for teaching mathematics and content support for teachers’ design and enactment of the lesson regarding the mathematical focus.

6.2.1  Teachers’ Content Knowledge for Teaching Teacher knowledge influences teaching practice, the mathematical quality of instruction, and eventually student learning (Baumert et al., 2010; Hill, Rowan, & Ball, 2005; Kelcey, Hill, & Chin, 2019). Shulman (1986) described three types of content knowledge needed for teaching: subject matter knowledge, pedagogical content knowledge, and curricular knowledge. One of the main points that Shulman highlighted is the importance of teachers’ understanding of the content from the perspective of teaching. Building on Shulman’s work, Ball and her colleagues (e.g., Ball, Thames, & Phelps, 2008) elaborated on what they called mathematical knowledge for teaching (MKT)—the type of knowledge specifically needed for teaching mathematics. Unpacking the construct of MKT, they further detailed subject matter knowledge (common content knowledge, specialized content knowledge, and horizon content knowledge) and pedagogical content knowledge (knowledge of content and students, knowledge of content and teaching, and knowledge of content and curriculum). Ball and her colleagues’ conceptualization of MKT using such subdomains also emphasizes that teacher knowledge of the content is deeply rooted in the context of teaching and is related to various elements involved in this context, including students and curriculum. The mathematics explained in curriculum programs can cover all of the various subdomains of MKT. The curriculum can provide information about the content at a general level (i.e., common content knowledge) as well as information specific to the teaching context in a particular grade level (i.e., specialized content knowledge). It can also elaborate how students’ learning of the concept addressed in the current lesson would be related to their learning of the concept in the future (i.e., horizon content knowledge, and knowledge of content and curriculum). Mathematical explanations can be about anticipated student thinking and reasoning about a given problem (i.e., knowledge of content and student). Explanations can also be about how the content can help teachers make decisions about what to do in teaching a certain concept (i.e., knowledge of content and teaching). Mathematical foundations are necessary to make proper instructional decisions in planning and enactment of lessons (Rowland, 2013; Rowland, Huckstep, & Thwaites, 2005; Shulman, 1987). According to Shulman (1987), and Rowland and his colleagues, teachers use their mathematical knowledge to transform the content

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presented in curriculum materials for students’ learning and make in-the-moment decisions during instruction. In fact, teachers’ content knowledge is the basis of other domains of teacher knowledge (e.g., Koponen, Asilainen, Viholainen, & Hirvonen, 2019; Remillard & Kim, 2017) needed for appropriate instructional decisions. Teachers need to have a profound understanding of the mathematics that they teach (Ma, 1999).

6.2.2  Content Support for Teachers Curriculum materials can play a significant role in supporting teachers with the content that they teach (e.g., Davis & Krajcik, 2005). Morris, Hiebert, and Spitzer’s (2009) study illuminated a relationship between mathematical knowledge for teaching and curriculum. They studied preservice elementary teachers’ articulation of the mathematical learning goals of a lesson and found that these preservice teachers were able to identify goals and subgoals for lessons, but struggled in developing a teaching plan addressing them. Explicit explanations of subgoals and concepts related to the goals in the curriculum are beneficial, as Morris et al. (2009) found that unpacking subgoals for an activity did not come naturally. Brown, Pitvorec, Ditto, and Kelso (2009) also highlighted that it is not trivial to teach to the curriculum designer’s intention, especially to meet the mathematical foci of the lesson. Kim (2018, 2019) found that understanding the mathematical goals of lessons matters, but teaching toward the goals is even more challenging for both in-service and preservice teachers. Articulating the work of teaching mathematics, Sleep (2012) described the importance of articulating the mathematical point and organizing the lesson around the mathematical point. She also identified seven central tasks in this work of teaching, some of which are spending instructional time on the intended mathematics, developing and maintaining a mathematical storyline, opening up and emphasizing key mathematical ideas, and keeping a focus on the meaning of mathematical concepts. Examining mathematical support provided by various curriculum programs is important in that curriculum materials can explicitly provide information pertinent to these identified critical tasks and teachers can be better prepared for teaching to the mathematical points. According to Davis et al. (2014), educative features refer to “text and graphics that can be incorporated into curriculum materials with the intention of supporting teacher learning” (p. 25). The purpose of those educative features in general, and content support (e.g., mathematical support) in particular, is to help teachers recognize the essence of the support feature and use that in their own teaching (Davis et  al., 2014). The former is for teachers’ own learning whereas the latter is for ­students’ learning. So, the educative features target both teachers’ own learning and students’ learning. Attending to the knowledge base teachers need to have in order to use curriculum materials effectively, Davis and her colleagues (e.g., Davis & Krajcik, 2005; Davis et al., 2014) examined ways in which curriculum materials can play a role in teacher

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learning. Teacher learning is situated in teaching contexts, which influences instructional strategies. Content and pedagogical challenges teachers may face in using the curriculum materials need to be addressed in educative features. Davis and Krajcik (2005) proposed heuristics for designing educative curriculum materials, attending to the areas that support teacher learning in science. One of those areas focused on subject matter knowledge: not only facts and concepts, but also disciplinary practices (e.g., developing and evaluating mathematical arguments and communicating mathematical thinking coherently) within the subject. In Chap. 4, it is clear that the elementary mathematics curriculum authors made efforts to support teachers in the content they teach, especially with the content where teachers may have a misunderstanding (e.g., there is no place value content without base-ten blocks) and/or the content that is new to them (e.g., early algebra). The authors also mentioned that they considered different locations of the content support (e.g., unit or lesson overview and side note or margin of lesson descriptions) and lengths of the explanations to increase teacher access to it and its readability. As Collopy (2003) argued, the curriculum authors intended their curriculum materials as a professional development tool for both pedagogy and content. Mathematical support in curriculum materials, however, has rarely been investigated. Chapter 5 presents overall results of educative features attending to mathematical ideas, which indicate that there is variation across the five programs in the extent and forms (direct or indirect; to the teacher vs. through the teacher) of mathematical support for teachers. We further analyze the mathematical support provided in the programs by using our analytic framework that we explain below.

6.3  A  pproaches to Communicating Mathematics to Teachers and Analytic Framework Investigating teaching and learning of mathematics attends to the interaction among teacher, students, and mathematics, which refers to the instructional triangle (National Research Council, 2001; Nipper & Sztajn, 2008). This instructional triangle cites contexts and environments as important in the interaction among the three elements. Rezat and Sträßer’s (2012) tetrahedron model of the didactical situation includes artifacts as another important element in the interaction in teaching and learning of mathematics, which presents a more dynamic relationship in the 3D model shown below.1 One of the critical artifacts is the curriculum materials. Then, the four elements of the tetrahedron model are teacher, students, mathematics, and curriculum materials, which helps us think about the role of curriculum in supporting teachers’ and students’ interaction with mathematics. We use this tetrahedron to guide our analysis (see Fig. 6.1).

 Rezat and Sträßer’s (2012) socio-didactical tetrahedron model includes socio-cultural contexts, such as family, institution, and conventions. We only attend to the part in the classroom level. 1

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In particular, the tetrahedron model helps consider different ways curriculum materials can provide mathematical support to teachers and the different sources teachers learn mathematics from in the curriculum materials. Mathematics can be explained directly to teachers in the curriculum materials, which is represented by code 4 in our coding scheme.2 Such mathematical support targets the content teachers need to know as background knowledge. Mathematical support can also be provided in the form of explaining student thinking and strategies (code 3/4). As described in Chap. 5, curriculum authors can also embed mathematical support for teachers in statements that direct teacher and student actions (codes 1/4 and 1/3/4). These are different choices available that curriculum authors can make when providing content support for teachers. Combining mathematical support with directing actions or explaining student thinking is an indirect approach to communicating mathematics to teachers, which also serve different purposes at the same time (see Chap. 5). Based on the author interviews (Chap. 4), the results from Chap. 5, and the literature we draw on, in this chapter we focus on three main areas in our analysis of mathematical support provided in the five programs: (1) the location and extent of mathematical support within and outside individual lessons, (2) the ways in which mathematics is communicated in individual lessons, and (3) the types of mathematics communicated in individual lessons (see Table 6.1). First, identifying the location and extent of mathematical explanations is important in order to understand the comparative amount of support given and its relative accessibility across programs. We examin both day-to-day lessons and other special sections of the programs for this focus. Second, the ways in which individual lessons communicate the mathematics are explored to seek the common and distinct approaches the programs take. This includes examining the modes of mathematical support (mathematics directly explained, mathematics embedded in student thinking, or mathematics or strategies embedded in teacher/student actions) and various components of lessons, such as titles and content goals of lessons, that may contain mathematical support for teachCurriculum

Fig. 6.1 Instructional Tetrahedron Modified from Rezat and Sträßer’s (2012)

Teacher Students

Mathematics  In our coding scheme, Code 1 was given to sentences directing teachers’ or students’ actions; Code 2 to sentences for design rationale and transparency; Code 3 to sentences attending to student thinking; Code 4 to sentences attending to mathematical ideas explained directly to teachers. Multiple codes, such as 3/4, 1/4 and 1/3/4, were given to sentences attending to more than one area at the same time (hybrid statements). See Chap. 5. 2

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Table 6.1  Analytic framework for mathematical support for teachers Location and extent of mathematical support

Ways in which mathematics is communicated in individual lessons

Types of mathematics explained

Location and extent within individual lessons  Introductory part  Main body (embedded in or separated from the main flow, i.e., main flow versus side note)  Optional part Location and extent outside individual lessons  Unit/chapter/lesson overview  Implementation guide or other resources for overall mathematics content Mathematical statements (modes of mathematical support)  Mathematics directly explained to teachers (for teachers’ own knowledge as well)  Mathematics embedded in anticipated student thinking  Mathematics embedded in teacher/student actions  Strategies and thinking embedded in teacher/student actions Components of lessons  Titles of lessons  Lesson goals (content goals)  Vocabulary list  Headings used in the lesson guide Categories emerged from the data (see Sect. 6.4.2.)

ers. The third focus, examining the types of mathematics communicated at the lesson level, helps explore the types of mathematical support for lesson enactment and/ or for teachers’ own learning. Given what is offered in the curriculum, we can discuss what mathematics teachers need to know to teach these programs, which is critical for teacher education and curriculum development. In this chapter, examining mathematics communicated in the five curriculum programs, we attend to the types of mathematical support offered in different curriculum programs and the approaches used to communicate mathematics to teachers. This will help us account for how such support can help teachers focus on and teach toward the mathematical goal of the lesson. Details of the methods are described in the next section.

6.4  Methods As described in Chap. 5, this study used a total of 75 lessons (15 lessons in grades 3–5 per program, 5 lessons per grade) randomly selected in the content strands of number and operations and algebra from the five elementary mathematics curriculum programs: Everyday Mathematics (EM) (University of Chicago School Mathematics Project, 2008), Investigations in Number, Data, and Space (INV) (The Education Research Collaborative at TERC, 2008), Math in Focus (MIF) (Marshall Cavendish International, 2010), Math Trailblazers (MTB) (TIMS Project University of Illinois at

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Chicago, 2008), and Scott Foresman–Addison Wesley Mathematics (SFAW) (Charles et  al., 2008).3 In addition, we examined places explaining mathematics other than those individual lessons, such as unit/chapter overviews and implementation guides. Below we describe data we examined and methods of analysis.

6.4.1  Compilation of the Data Set Once all the codes of individual sentences, phrases, and images (hereafter sentences) in 75 lessons of the five programs were obtained as described in Chap. 5, we further analyzed the mathematics communicated to teachers in those lessons. In doing so, as described above we examined not only code 4 (mathematical explanation directly to teachers), but also code 3/4 (explanation of student thinking with mathematics), because mathematical concepts and ideas were addressed often when student thinking and strategies were explained.4 We also examined codes 1/4 (mathematics embedded in teacher/student actions), and 1/3/4 (teacher/student action with student thinking and mathematics) because teacher/student actions sometimes included mathematical support. Keeping these codes while removing all others in the software (Atlas.ti) we used to code individual sentences, we compiled a data set (a total of 1748 sentences) representing the specific foci described previously. In addition, we collected various resource materials from each curriculum program, such as implementation guides, to examine mathematical support beyond the individual lessons.

6.4.2  Data Analysis Data analysis had three phases: (1) examining the location and extent of mathematical support within and outside the 75 lessons from the five programs, (2) examining ways in which mathematics is communicated within individual lessons, and (3) examining types of mathematical support in the lesson guidance for the 75 lessons. Examining the Location and Extent In terms of location, we examined whether mathematical explanations were provided in the overview of the lesson, the main body of the lesson, or beyond the main body of the lesson. If in the main body, we further examined the lesson to determine

 See Chap. 1 for details about the programs.  Chapter 7 presents the analysis of curricular support for teachers’ noticing of students’ mathematical thinking. 3 4

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whether explanations appeared in the main flow of the guidance for the lesson or in a place separated from the main flow (e.g., in a separate box or in the margin/column distinguished from the main guidance for the lesson activities). The rationale for the analysis of where mathematical support appeared was that these different locations might influence teachers’ decisions about what to read and use for teaching. The extent of mathematical support was determined and compared in terms of the number of sentences addressing mathematics in individual lessons (by location and total), across grades, and across programs. To describe the extent and ways in which mathematics was explained, we also searched places other than the 75 lessons analyzed, such as the overview of a chapter/ unit, the overview of a grade, and mathematics tutorial materials for teachers in a distinct volume. The number of pages, topics addressed, and overall approaches to explaining mathematics were summarized in each program and compared across programs. Examining Ways of Explaining Mathematics in Individual Lessons To examine ways in which the five curriculum programs communicate mathematics to teachers, we first compared the numbers of sentences across codes 4, 3/4, 1/4, and 1/3/4, because these codes themselves are different ways in which mathematics is communicated to teachers. In doing so, we created summary tables per grade and across lessons within each program in relation to locations (e.g., introductory part, main body, and optional part of lesson) of mathematical support and codes (4, 3/4, 1/4, and 1/3/4), in order to compare three grades in each program, five programs in each grade, and overall approaches to explaining mathematics in the five programs. Also, objectives (mathematical goals) of the lessons and lists of vocabulary (i.e., mathematical terms used in the lessons) provided in the programs were examined to see the extent to which mathematics was highlighted in them. In fact, objectives were coded as 2 (i.e., providing rationale and transparency5) in our overall analysis since they guided teachers to the mathematical focus of the lessons (i.e., what the lesson is about—design rationale), whereas vocabulary lists were coded as 0 because they provided referential information, rather than explaining any mathematics or design rationale explicitly (see Chap. 5). Despite that, objectives and vocabulary lists were included in our analysis of mathematical support because they summarize the mathematics content of the lesson in a succinct way. We also examined lesson titles and headings used in the guidance of the lesson to see whether lesson titles in each program expressed mathematics, whether each program used headings in the lesson to indicate the mathematics, and, if so, to what extent and in what ways the titles and headings addressed the mathematics of the lesson. We listed all the titles of the lessons from each program and determined whether each title provided or indicated a mathematical point in it. We also listed all the headings in the main body of the lesson that were not routinely provided. Since

 See Chap. 8 for the analysis of design rationale and transparency.

5

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headings that were routinely provided, such as “Teaching the Activity” or “Guided Practice,” were not specific to the content of the lesson, we focused on only nonroutine headings provided, depending on the content of individual lessons. The analysis of lesson titles and headings was conducted beyond the 75 lessons that were included in the overall curriculum analysis; in order to see if the pattern was consistent in each program, several additional random lessons across grades had to be examined. Examining Types of Mathematical Support To examine the types of mathematical support in the guidance of lessons, we used the 1748 sentences from the 75 lessons and developed a set of categories that were meaningfully distinct (see Table 6.2). First, by reviewing sentences from our data set, we created a list of preliminary categories for various types of mathematics explained.6 Then, we refined the list by testing the initial categories with additional sample sentences chosen from each curriculum program. After examining sample sentences with mathematical support from various programs, we determined five categories (see Table 6.2). Two researchers of the Improving Curriculum Use for Better Teaching (ICUBiT) project7 individually coded about one-third of the data set and they assigned the same categories to about 82% of the total sentences tested out. We discussed those with conflicts to reach an agreement on a particular category. Then, the rest of the sentences were coded by one of the two researchers, using Atlas.ti, the same software used for primary coding—0, 1, 2, 3, 4, etc. Then, summary tables were created to search for patterns within and across grades, and within and across programs. The five programs vary in the way they provide mathematical support for teachers. They exhibit differences in the location and extent of mathematics explained, types of mathematical explanations and emphases in them, and mathematical support across lessons and grades, although some similar approaches across programs are found. Below we describe findings of the study in detail. Table 6.2  Categories for types of mathematical explanations Code Types of explanation E1 Descriptions solely about procedures and steps (what to do and how to do it) E2 Explanations of procedures and strategy (why), including general approaches as well as specific steps E3 Explanations about concepts, definitions, conventions, and mathematical information; usually general descriptions E4 Explanations about representations and tools (how to use them, what ideas they represent and in what ways). Figures are usually in this category E5 Explanations about mathematical connections, relationships, and applications (e.g., discussing a story context of a given problem)

 Dustin Smith contributed this first stage of categorization.  See Chap. 1 for more about the project.

6 7

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6.5  Extent and Location of Mathematical Explanations The extent and location of mathematical support were examined first in individual lessons and then in resources beyond individual lessons, in order to see the comparative amount of support given and its relative accessibility across programs. The extent and different locations might influence teachers’ decisions about what to read and use for teaching. Our analysis revealed an overall tendency of the programs and differences among them in the amount and locations of mathematical support.

6.5.1  E  xtent and Location of Mathematical Explanations in Individual Lessons The extent to which mathematics is communicated varies across the five programs. We analyzed this first by counting the total number of sentences in individual lessons. Table 6.3 summarizes the number of sentences that communicate mathematics in the 15 individual lessons analyzed per curriculum program. On average, MTB lessons included the most sentences communicating mathematics and SFAW lessons provided the least. (Note that SFAW includes the least number of sentences overall, as shown in Chap. 5). INV lessons do not provide as many sentences containing mathematical support in individual lessons as in other Standards-based programs like EM and MTB, even though INV includes extensive “Teacher Notes” in the back of each unit that is devoted to explaining the content of the unit from the perspective of teaching (more details to follow in Sect. 6.5.2). Also, there is great variation within each program; some lessons explain mathematics extensively, whereas some lessons have a minimal number of sentences about mathematics. Although not explicit, this phenomenon depends on the content of the lesson. For example, lessons on factors, prime and composite numbers, simplifying algebraic expressions (MIF), representing numbers and operations in the base-ten number system, patterns with square numbers, function machines (MTB), factor trees, modeling contexts in equations and inequalities (EM), representing constant changes and algebraic relationships (INV), and prime and composite numbers (SFAW) include more sentences than others in each program. The complexity of the content and/or the introduction to the content seem to require more extensive elaborations. In contrast, the lesson with the least number of sentences in SFAW expects students to “tell in words what is known and what needs to be determined in given word problems,” which does not seem to require a lot of support for teachTable 6.3  Number of sentences explaining mathematics in individual lessons Curriculum Range Median (mean)

EM 3–55 24 (26.5)

INV 1–33 16 (15.3)

MIF 8–63 28 (27.7)

MTB 2–98 35 (37.4)

SFAW 1–25 8 (9.5)

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ers in terms of the content addressed in this lesson. Although not extensively examined, it is possible that whether building on previous lessons or introducing a new concept also influences the extent to which mathematics is explained in individual lessons. The extent of communicating mathematics in individual lessons was examined in relation to the locations where mathematical support was given. The overall analysis of the lessons distinguished three parts of the lesson: lesson introduction, main body (core lesson activities), and beyond the main body of the lesson (optional portion of the lesson).8 As indicated in Chap. 5, the intention of attending to the location was to determine whether mathematical support was for the primary instructional activities of the lesson, as opposed to orienting information and optional activities. Table 6.4 summarizes the number of sentences containing mathematical support in these three locations by program in each of the four categories. This table also provides a summary of data distribution (percentage in parentheses) in all five programs by each category. Most sentences provided beyond the main body of the lesson are from SFAW (44.3% of total sentences beyond the main body from all programs), which is, in fact, 37.8% of total sentences with mathematical support in SFAW lessons. That is driven by the fact that all SFAW lessons offer optional hands-on activities. EM lessons also have some mathematical support beyond the main body as they tend to offer optional, extension activities/tasks. Note that in Chap. 5, considering all types of categories, not just mathematical support, it is also the case that SFAW and EM provide a lot of support outside the main body of the lesson (see Table 5.4  in Chap. 5). Overall, the curriculum programs, except for SFAW, rarely provide mathematical support for teachers in the lesson introduction. In SFAW, 17.5% (25 out of 143) of the total sentences explaining mathematics are in the introductory part of lesson. A majority (91.5%) of the sentences containing mathematical support across programs are provided in the main body of lesson (EM 91.7%, INV 98.3%, MIF 95.7%, MTB 97.3%, and SFAW 44.8%). The data in the fourth column Main Body in Table 6.4 are summarized by program in Fig. 6.2. Mathematical support in the main body of the lesson is given mostly in the form of student thinking and strategies (code 3/4, 35%) and student thinking embedded in teacher/student actions (code 1/3/4, 31%). Less common is mathematics directly explained to teachers (code 4, 18%) and mathematical support embedded in teacher/student actions (code 1/4, 16%). Overall, the programs except for MIF provide mathematical support mostly in the form of student thinking and reasoning in the main body of lesson (code 3/4). In fact, most mathematical support in the main body of MIF lessons is given in the form of directing actions (i.e., codes 1/4 and 1/3/4), 90% in total. In our analysis, we further considered two different parts within the main body of the lesson: the main flow of the lesson and side notes (those in the margin or in a separate box—see Figs. 6.3 and 6.4, for example). The latter may not get as much

 See Chap. 5 about the three parts.

8

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Table 6.4  Location and extent of mathematical support in individual lessons (Number of sentences) Category (code) Program Mathematical explanation directly to EM teachers (4) INV MIF MTB SFAW Total

Introduction 0 0 0 1 22 23 (6.7%)

Mathematics embedded in student thinking (3/4)

EM INV MIF MTB SFAW Total

0 0 0 0 1 1 (0.2%)

Mathematics embedded in teacher/ student actions (1/4)

EM INV MIF MTB SFAW Total

0 0 0 1 2 3 (1.0%)

EM Student strategies and thinking embedded in teacher/student actions INV (1/3/4) MIF MTB SFAW Total

0 0 0 0 0 0 (0.0%)

All

27 (1.6%)

Grand Total

Main body 87 36 13 151 2 289 (83.5%) 118 113 26 277 26 560 (92.4%) 65 24 125 32 12 258 (87.8%) 95 53 234 86 24 492 (98.0%) 1599 (91.5%)

Beyond 19 1 1 3 10 34 (9.8%) 1 2 15 9 18 45 (7.4%) 9 1 0 1 22 33 (11.2%) 4 0 2 0 4 10 (2.0%) 122 (7.0%)

Total 106 37 14 155 34 346 (100%) 119 115 41 286 45 606 (100%) 74 25 125 34 36 294 (100%) 99 53 236 86 28 502 (100%) 1748 (100%)

The percentages in parentheses indicate the distribution of mathematical support across three locations in each category. The total number of sentences communicating mathematics in each program: EM = 398, INV = 230, MIF =416, MTB = 561, SFAW = 143 EM (N=365) INV (N=226) MIF (N=398) 3 MTB (N=546) SFAW (N=64) 3 All (N=1599)

24

32

16

18

50

7

31 41 18

50

6

19

37

35 20% code 4

23

59

28

0%

26 11

16 40%

code 3/4

60% code 1/4

16 31

80% code 1/3/4

Fig. 6.2  Distribution of four ways in the Main Body of the lesson in each program (%)

100%

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Fig. 6.3  Mathematical explanations in the margin (INV GR3 1.1.2). From Investigations 2008 Curriculum Unit (Grade 3, Unit 1) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

Fig. 6.4  Mathematical explanations in a separate box (MTB GR4 15.4, p. 76). Excerpt from Math Trailblazers, Grade 4, Unit 15 Resource Guide by TIMS.  Copyright ©2008 by Kendall Hunt Publishing Company. Reprinted by permission

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teacher attention as the former, because teachers tend to focus on core directions for the lesson that appears in the main flow of the lesson guide. Table 6.5 presents the distribution of mathematical explanations in the main flow and side notes (i.e., separate boxes or margins/columns) in the main body of lesson in each program. Overall, there is a strong tendency in the five programs that mathematical support is provided for teachers in the main flow of lesson. Across the five programs about 80% of the mathematical explanations were in the main flow. MIF has the highest portion (87.4%) and INV has the lowest (65.0%). In contrast, the portion of mathematical explanations in the side notes is as high as 35% (INV) and as low as 12.6% (MIF). Figure 6.3 above shows an example of INV providing mathematical explanations in the side notes.

6.5.2  E  xtent and Location of Mathematical Explanations Beyond Individual Lessons All five programs provide explanations about the mathematics of lessons in the overview of the unit/chapter. However, this varies across the programs: MIF provides a couple of paragraphs, EM, MTB, and SFAW give 1–2 pages of explanation, and INV provides 4–6 pages. Both EM and MTB have an extensive number of tutorial pages in one designated place (i.e., a volume or chapter) for teachers to address mathematics outside individual lessons. MTB provides the Implementation Guide for each grade, which includes a chapter called TIMS Tutors with about 140 pages of elaboration on the mathematics topics covered in the grade. Whereas each Unit Resource Guide (i.e., the teacher’s guide for each unit) includes brief explanations about the mathematics students learn in the unit, these TIMS tutors chapters provide in-depth, extensive explanations of the mathematical concepts and ideas behind the curriculum.

Table 6.5 Mathematical support in main flow vs. side notes in the main body of lesson (%)

Program EM INV MIF MTB SFAW All

Main flow 83.8 65.0 87.4 79.1 75.0 80.1

Side note 16.2 35.0 12.6 20.9 25.0 19.9

The total number of sentences in the main body of lesson: EM  =  365, INV  =  226, MIF  =  398, MTB  =  546, SFAW = 64, All = 1599

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EM has a volume called Teacher’s Reference Manual (one for grades 1–3 and one for grades 4–6), which explains the mathematical background of the curriculum on hundreds of pages (250–350), including a management guide (pedagogical approaches to content), ten mathematical topics (e.g., number and operations, measurement), and glossary. When providing a mathematical background (1–6 pages) in each chapter overview, the Teacher’s Reference Manual is referred to for more detail. In contrast, even though INV has a smaller number of sentences explaining mathematics in individual lessons than other programs (see Table 6.3), there is more extensive mathematical support spread out in the curriculum. First, INV has a separate volume (the Implementing Investigations guide) explaining the mathematics of each grade with an emphasis on the teaching of the content. The Implementing Investigations guide for each grade devotes about 7 pages to explaining the content itself by strand; in addition, it includes a collection of extensive notes for professional development called Teacher Notes. Each of these notes elaborates key ideas addressed in the curriculum from the perspective of teaching, with a specifically focused topic (e.g., Foundations of Algebra in the Elementary Grades) on 2–6 pages (a total of 25–30 pages in each grade). Besides these extensive Teacher Notes in the Implementing Investigations guide, INV includes about 10 Teacher Notes that are specific to the unit and about five Dialogue Boxes (i.e., classroom vignettes discussing the content of the unit) in the back of each unit (a total of about 30 pages per unit). These Teacher Notes and Dialogue Boxes are indicated with numbers in the main flow of the lesson so that teachers can refer to these notes in the back of the unit (see the number 5 inside a small circle and the note in the margin in Fig. 6.3, for example). Since each grade has nine units, there are about 250–300 pages of resources that explain the mathematics content of the grade, which is almost equivalent to EM’s single volume of Teacher’s Reference Manual. Whereas EM’s single volume of Teacher’s Reference Manual is for three grades, INV provides mathematical support distributed into nine units for each grade. Overall, INV provided the most mathematical support outside individual lessons, followed by MTB and EM next. INV scattered mathematical support throughout individual units, whereas EM and MTB provided it in one single place. Another difference was that INV explained mathematics in relation to teaching more so than EM and MTB. MIF and SFAW do not have extensive tutorial pages other than the overview of each chapter.

6.6  H  ow Mathematics is Communicated in Individual Lessons The five programs’ explanations about the mathematics in individual lessons vary, as shown in Table 6.4. Even though the five programs communicate mathematics in different ways, they also have some common approaches to explaining the mathematics

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to teachers: (1) explaining mathematics to teachers directly, (2) communicating mathematical ideas by embedding them in teacher/student actions, (3) communicating mathematics by illustrating anticipated student strategies and thinking, and (4) communicating student strategies by embedding them in teacher/student actions. The programs also provide mathematical goals and vocabulary of lessons, and highlight the mathematics in the titles of lessons. In addition, some programs use the headings in the lesson guide to highlight mathematical ideas. Below, common ways of communicating as used within individual lessons are elaborated. Also, distinct characteristics of each program are addressed, as each program exhibits different styles of providing teachers with information about the mathematics they are to teach.

6.6.1  Mathematical Explanations in Individual Lessons In this section, the first four ways of communicating with teachers as listed above are explained. Overall, the programs use all four ways of communicating mathematics (background information, mathematical ideas, and student strategies) to teachers, with some variation. Figure 6.4 summarizes mathematical support in individual lessons in terms of the four categories by program, regardless of the location of the support (i.e., introduction, main body, or optional activities of the lesson). Note that Fig. 6.2 presents the data only from the main body of the lesson. EM, MTB, and SFAW provide mathematical explanations directly to the teacher (code 4) more than INV and MIF. Except for MIF, the four programs tend to communicate mathematics by embedding it in student thinking (code 3/4); they have the highest portion of their mathematical support in this way, especially INV and MTB at about 50%. MIF and SFAW tend to have mathematical ideas embedded in teacher/student actions (code 1/4), at 30% and 25%, respectively. In fact, the last two parts of each bar in Fig. 6.5 indicate mathematical support embedded in teacher/student actions, whether it is about mathematical ideas (1/4) or student thinking/strategies (1/3/4). MIF is unique in that about 87% of mathematical support is conveyed in teacher/student actions (codes 1/4 and 1/3/4). Mathematical support is combined with teacher/student actions in those cases. In contrast, MTB provides mathematical support directly to teachers, rather than embedding it in teacher/student actions (code 4 and 3/4, 79% EM (N=398) INV (N=230) MIF (N=416) 3 MTB (N=561) SFAW (N=143) All (N=1748) 0%

27

30 50

16 10

18

30

28 24 20

25 23

11 57 51

31 35 20%

code 4

17

40% code 3/4

6

15 20

25 60% code 1/4

28 80%

100%

code 1/3/4

Fig. 6.5  Distribution of four ways to communicate mathematics in each program (%)

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in total). Below each of the four ways is explained with specific examples from each program. In general, mathematical support is often given when explaining anticipated or expected student thinking and strategies (codes 3/4 and 1/3/4). Mathematics Explained Directly to Teachers As described earlier, the programs have designated places (e.g., an overview of the unit, or a chapter or entire resource book on math topics) to elaborate on the mathematics covered in a particular unit, a particular grade, and across grades. The mathematics addressed in these resources directly targets teachers’ understanding of the content that they teach. Mathematical support for teachers is also housed within individual lessons. This support includes conventions, procedures, definitions, and mathematical statements that should be covered in instruction, and sometimes go beyond the content that students are expected to learn. For example, in an MTB lesson on function machines in grade 4, definitions of a function are explained from various approaches, which include those that students need to learn in the lessons as well as those at an advanced level (see Fig. 6.4). Mathematical explanations provided directly to the teacher are sometimes in side notes, such as boxes (as shown in Fig. 6.4) and in the margins (as shown in Fig. 6.3) of the lesson. Often, these are intended to educate teachers, rather than explain the mathematics that students are expected to learn. In contrast, mathematics communicated in the main flow of the lesson mostly elaborates on what students need to learn (which, of course, teachers have to understand as well). Examples of mathematics directly explained to teachers are presented from each program in Table 6.6. The examples from INV and MTB in Table 6.6 are for teachers’ own knowledge; the examples from the other programs are for both teachers’ and students’ learning toward the goal of the lesson. EM (27%) and MTB (28%) have the highest portion of sentences explaining mathematics directly to teachers (see Fig. 6.5). MTB often has extensive explanations enclosed in a separate box as in Fig.  6.4. Below is another Content Note explaining mathematics directly to teachers. Using the graph to find data points that lie between those in your data table is called interpolation. Inter means “between” or “among.” Since we have data for four pitchers and eight pitchers, using the graph to answer Question 6 (five pitchers) is interpolating. Using the Table 6.6  Mathematics directly explained to teachers (code 4) Program Example EM The lattice method is a very efficient algorithm, no matter how many digits are in the factors INV When students record the number of spaces they have moved in Capture from 300 to 600, they are actually recording the absolute value of their move MIF In the second example, there are two algebraic expressions MTB Line math puzzles are found in many shapes while magic squares are always squares SFAW When a number is divisible by another number, there is no remainder. Divisibility rules can help in solving some problems mentally

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graph to make predictions about data points that lie outside those in your data table is called extrapolation. Extra means “outside” or “beyond.” Since 72 lemons lie beyond the largest number of lemons reported in the data table, using the graph to answer question 8 is extrapolating. (MTB GR3 7.1, p. 33)

Mathematics Communicated in Anticipated Student Thinking As noted earlier, sentences explaining anticipated student thinking and strategies highlight a lot of the mathematics that teachers need to know in order to teach the lesson, which support teachers to be prepared for attending to student thinking (see Chap. 7 for detailed analysis of this type of support). These types of mathematical supports include sample student responses to a given problem, and thinking and reasoning required for a problem/concept/idea, which provide teachers with the boundary and details of the content in the grade they teach. In this way of mathematical support, as explained in Chap. 5, the information is directly communicated to teachers so that they can be prepared to teach the lesson. This is also the case for the category above (i.e., mathematics explained directly to teachers). Table 6.7 presents examples of mathematics embedded in anticipated student thinking, which show a range of student thinking and strategies anticipated. Some are based on procedures, facts, properties, or definitions as in examples from SFAW and MIF; some address procedures and reasoning/meaning for the procedures as in Table 6.7  Mathematics embedded in anticipated student thinking (code 3/4) Program Example EM  •  A possible response: … Then I moved down one more row to 50—that is 10 more spots.  •  So I can add 500 eight times: 500 + 500 + 500 + 500 + 500 + 500 + 500 + 500 = 4000.  •  Possible strategies [for 6 × 45]: Some children might reason that since 45 is 3 times as much as 15, 6 [45s] are 3 times as much as 90, or 270. INV  •  [Finding 3218 in the 10,000 chart] Another may visualize the 3300 chart as they would any 100 chart and know that it would be in the 18th place or in the eights column on the chart.  •  All the strategies in this discussion involve breaking apart one or both addends and combining the parts in a different order. MIF  •  [Finding the sum of 99 and 99 mentally] 99 is 100 minus 1; 100 + 100 − 2 = 198.  •  Students will find that the 10,000 digits are the same for all four numbers.  •  Or, they can write 120 as 100 + 20 and use the Distributive Property. MTB  •  A more sophisticated strategy is to note that there are 4 rows of 8 cm2 each (or 8 columns of 4 cm2 each) and they can multiply or skip count to get 32 cm2.  •  Students might say that Solution 3 is like Solution 1 except that the twos and sixes are interchanged or that Solution 4 is like Solution 1 except that the first and last columns are swapped. Each row and each column has a two, a four, and a six. SFAW  •  Students need to multiply by 3 to change it to feet.  •  To be in the same fact family, the factors need to be the same.  •  They need to find only one factor that is not 1 or the number itself.

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EM and INV examples; some are comparing solutions or strategies as in INV and MTB examples. These are typical patterns in the programs. The programs often use some specific forms to address mathematics in anticipated student thinking. For example, EM provides sample student responses in red print; INV includes an actual student utterance in quotation marks with a picture of the student; SFAW provides desired student responses in parentheses. All of these indicate a visual signal to teachers that the provided resources highlight the mathematics through student reasoning and thinking, which teachers need to look for during instruction. As mentioned previously, MIF rarely communicates mathematics by embedding it in student thinking and strategies—only about 10% of its mathematical support is embedded in anticipated student thinking. The other four programs provide mathematical support in this manner at a much higher rate than MIF, ranging from 30% (EM) to 50% (INV) in Fig. 6.5. In fact, communicating mathematics in the form of student thinking and strategies is the most popular across the programs among the four different ways (35%). Mathematics Embedded in Teacher/Student Actions The curriculum programs include mathematical support embedded in particular teacher/student actions, which explicitly highlights the mathematics students need to learn in instruction. Such mathematical support, while targeting students’ understanding of mathematics, promotes teachers’ learning of the content that they teach. This way of communicating mathematics is different from the two ways described above in that mathematical ideas to students through the teacher are combined with particular teacher/student actions, as explained in Chap. 5. As indicated in the verbs in bold in the examples in Table 6.8, the lesson guides suggest that teachers do certain actions (e.g., explain, show, remind, point out, tell, and suggest) regarding the mathematics addressed. As shown in the examples, some teacher actions, such as tell, point out, show and explain, are directly delivering the mathematical ideas to students; others, such as discuss, guide, and generalize, seem to be more complex actions teachers have to take regarding the mathematics students are to learn. MIF includes extensive mathematical support embedded in teacher/student actions, as shown in Fig.  6.5 (30%); MTB provides the least mathematical support in this form (6%). Strategies and Thinking Embedded in Teacher/Student Actions Some of the mathematical explanations embedded in teacher/student actions are specifically about student strategies and thinking as opposed to mathematical ideas and concepts. We distinguish the former (1/3/4) from the latter (code 1/4) because the former provides information about anticipated and expected students’ thinking, which also supports teachers’ learning of the content and helps them prepare to

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Table 6.8  Mathematics embedded in teacher/student actions (code 1/4) Program Example EM  •  Tell the class that the number of spaces from 30 to 57 is called the difference between 57 and 30.  •  Remind students that the numbers in the shaded row and shaded column in the table are called factors, and the rest of the numbers are called products.  •  Tell the class that this last strategy makes use of the place value of each digit.  •  Include the following points [in the whole group]  – A factor can be any type of number. However, factor pair, factor string, a composite number, and prime number refer only to whole numbers. INV  •  As students explain how they use the 10th multiple, 20th multiple, 30th multiple, and so on, point out that these are called landmark multiples.  •  Point out that 3 and 4 are both factors of 12 and that you can turn one array to look exactly like the other.  •  Have a brief discussion—no more than 5 min—about how these different notations mean either how many groups of 12 are in 374 or, if 374 is divided into 12 groups, how many are in each group. MIF  •  Re-emphasize the convention that the first factor refers to the number of groups and the second factor as the number of items in each group.  •  Read the question aloud and explain how the model represents the 5 equal groups, with 12 pencils in each group.  •  Show students how division is used to determine whether a number is a factor of another number.  •  Explain that consecutive whole numbers are numbers in running order in increasing steps of 1.  •  Guide students to simplify algebraic expressions based on the concept that: a + a + a + … + a(n terms) = n × a. MTB  •  Explain that we often cannot or do not need to find exact answers—and estimate is good enough.  •  Describe how each side of the square meter measures 1000 mm.  •  Illustrate this by placing ten skinnies [tens pieces in base-ten blocks] side-by-­ side to show they are exactly the same size as a flat [a hundred piece in base-ten blocks].  •  Generalize that the area of a rectangle is the length times the width. SFAW  •  Suggest that students think of the inverse operation as going in reverse.  •  Discuss why division is used to solve the problem, pointing out that the number of cartons in each group will be the same.  •  Emphasize that in any factor tree (a) the product of the factors in any row must equal the original number and (b) the last row contains only prime numbers. Bold added for emphasis

notice students’ thinking and respond to it (see Chap. 7 for detailed analysis of this type of support). Across all programs, this way of communicating mathematics to teachers is the second most popular approach (28%); MIF has the highest portion (57%) of this type of mathematical support within the program and across the programs and MTB the least (15%) as shown in Fig. 6.5. Table 6.9 presents some examples of this category by curriculum program. The verbal words or phrases in bold in the examples in Table 6.9 indicate that teachers are guided to attend to specific student thinking, work, strategies, or explanations in relation to the mathematics of the lesson. Note that many verbs in the examples

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Table 6.9  Strategies and thinking embedded in teacher/student actions (code 1/3/4) Program Example EM   •  Watch for students who think that certain numbers, such as the total length of the bookshelves in the Library of Congress (535 miles), are counts rather than measurements.   •  When students are explaining the steps in the partial-products algorithm, watch for those who say “6 ∗ 8”: the 8 in the problem to the right, for example, is in the hundreds place and has a value of 800, not 8. INV   •  Make sure that students understand the difference between “up” and “down” as directions on the [100] chart and the use of the phrases “numbers going up” to mean that the numbers are increasing and “numbers going down” to mean that the numbers are decreasing.   •  Listen for explanations that focus on how subtracting exactly 10 or a multiple of 10 affects the number of tens (represented by the tens digit in the difference), while the number of ones (represented by the ones digit in the difference) is the same as the ones digit in the original amount. MIF   •  Explain the following steps of the strategy: … Think of adding 95 as adding 100 and subtracting 5.   •  Ensure that students realize that adding 95 is the same as adding 100 and subtracting 5, and adding 99 is the same as adding 100 and subtracting 1.   •  Point out that in this case, since 10 × 6 = 60 is the easier, known fact, you have to subtract a group of 6 from the 10 groups of 6 to find 9 groups of 6: 9 × 6= (10 groups of 6) − (1 group of 6)—Distributive Property (Subtraction). MTB   •  Point out that when they add the 6 bits with the 7 bits, they can regroup and think, “That’s 1 skinny and 3 bits”.   •  Encourage them to measure each side to the nearest tenth of a centimeter and then multiply length by width on their calculators—5.7 cm × 2.8 cm = 15.96 cm2 or 16 cm2 to the nearest sq. cm. SFAW   •  If students have trouble identifying the correct operation, let them simplify the problem temporarily by substituting a specific number in place of the variable to decide on the operation needed.   •  Suggest that students think of the inverse operation as going in reverse. Bold added for emphasis

from Tables 6.8 and 6.9 are different, as those in Table  6.9 gear toward specific thinking and strategies to support students’ learning of the mathematics of the lesson. Some of the mathematical support in this form includes suggested follow-up teacher actions as well. This occurs often in SFAW as shown in the first example of SFAW in Table 6.9. SFAW has a structured format of If/Then in a separate box to provide such support. Other programs also offer teacher action for certain student misconception or misunderstanding, although not structured like SFAW.  In fact, teacher actions suggested in SFAW are mostly procedural as in the example in Table 6.9. INV often embeds mathematical support in teacher actions, especially in a series of teacher prompts, desired student thinking and the mathematics teachers need to know. Your arithmetic expression of the perimeter for 100 rows is (2 × 100) + 20. What is the 100 in your expression? What is the 20? If you picture the rectangle of 10 tiles across with

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100 rows, can you explain to me why this expression works? How is it the same or different from the expression you used to figure out the perimeter for 100 rows of the 5-tile rectangle? (INV GR5 8.2.4)

The example above illustrates teacher comments and questions (as indicated in blue font in the lesson guide) to support students’ thinking about the problem in the context of perimeter. The suggested teacher prompts go beyond the steps and the answer, focusing on the meaning of the steps, and connections between the problem and the solution that are desired for student learning. Such prompts also support teachers’ in-depth thinking about what students are expected to learn.

6.6.2  Components of Lessons Various structural components of lessons can be used to communicate mathematics to teachers. Below, we describe the extent to which mathematics is communicated to teachers through lesson goals, vocabulary lists, titles of lessons, and nonroutine headings. Lesson Goals and Vocabulary List In all five programs, one or multiple mathematical goals are provided in every lesson. These are called objectives (MIF and SFAW), objectives, and key concepts and skills (EM), math focus points (INV), or key content (MTB). Note that in our overall analysis, lesson goals are coded as 2—transparency and design rationale (see Chap. 5 for more details). Content goals, however, can provide significant mathematical support for teachers. For this reason, we examined the extent to which these goals play a role of communicating about the mathematics of the lesson to teachers. MIF and SFAW include very brief objectives, usually one in each lesson, whereas INV and MTB tend to have a detailed list of content goals. Examples of lesson goals (all from grade 3) are presented in Table 6.10. These lesson goals highlight the mathematics of each lesson, including mathematical concepts, ideas, strategies, and even models. See Chap. 8 for detailed analysis of communication of mathematical goals to teachers. EM provides objectives for teachers, that is, the objective that teachers have to meet in the lesson as well as key concepts and skills for students to learn (see Table  6.10 for example), whereas the other four programs directly address what students are expected to accomplish in the lesson. In Table 6.10, it is notable that some goals are written in a way to reveal those mathematical ideas, although more clarification is needed. For example, in the EM lesson, students “identify number-­ grid patterns and use them to find differences between pairs of numbers.” In the INV lesson, students solve “addition problems by using strategies that involve breaking numbers apart by place or adding one number in parts.” In the MTB lesson, students translate “between representations of addition (base-ten pieces and symbols).”

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Table 6.10  Sample lesson goals and vocabulary list Program Title of lesson EM Finding differences

Vocabulary Difference

INV

Equation Tens place Ones place Difference

MIF MTB

SFAW

Lesson goal To guide children as they identify number-grid patterns and use them to find differences between pairs of numbers   •  Compare whole numbers   •  Find differences between pairs of numbers   •  Use a number grid to solve problems   •  Recognizing and representing the place value Adding and of each digit in 2- and 3-digit numbers subtracting 2-digit   •  Adding and subtracting multiples of 10 numbers   •  Solving addition problems with 2-digit numbers by using strategies that involve breaking numbers apart by place or adding one number in parts More mental Use different strategies to add 2-digit numbers close addition to 100 mentally Base-ten addition   •  Understanding place value   •  Solving addition problems and explaining mathematical reasoning   •  Representing addition problems using base-ten pieces   •  Adding multidigit numbers using manipulatives and drawings   •  Translating between representations of addition (base-ten pieces and symbols) Subtracting Subtract two-digit numbers using paper-and-pencil two-digit numbers methods

None Regrouping

None

Although it is not clear whether it was intended by the curriculum authors, these goals support teachers to think about the mathematical ideas embedded in the lesson goals. It is unique that INV also provides math focus points (i.e., objectives) for whole-­ class discussion during the lesson. A typical INV lesson has 2–3 main activities, one of which is often devoted to classroom discussion. The math focus point(s) that should be addressed during the discussion (some of those that are listed as math focus points of the lesson) are always indicated at the beginning of the guidance for the discussion. For example, the guide for the discussion on “Representing adding and subtracting 10s” in the lesson of “Adding and Subtracting 2-Digit Numbers” in INV lists the first two of the three math focus points in Table 6.10. The INV design team described in Chap. 4 that they provided the math focus point(s) of each discussion to guide teachers to organize a discussion around specific mathematical ideas. All five programs include a list of vocabulary along with lesson goals, even though some lessons do not have any vocabulary (see Table  6.10, for example). (Note that in our overall analysis, a list of vocabulary was coded as a 0—referential information. See Chap. 5 for more detail.) The extent to which vocabulary is provided depends on the lesson: some have a lengthy list and some have none. Overall,

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EM, INV, and MTB include more vocabulary per lesson than SFAW and MIF do. Vocabulary lists offer mathematical terminology that students are expected to understand and use, which also suggests that teachers understand the meaning of such terms, introduce them accurately, and reinforce students’ use of correct terms. Titles of Lessons and Headings in the Main Body Titles of lessons and nonroutine headings in individual lessons also provide mathematical support for teachers. As shown in the examples of lesson titles in Table 6.10, all five programs provide mathematical focus of the lessons in their titles, except for a few rare cases that usually capture the context of the lesson (e.g., A Visit to Washington, D.C. in EM) rather than the mathematics. The lessons we analyzed have titles such as Prime and Composite Number (EM), Representing a Division Problem (INV), Rounding Numbers to Estimate (MIF), Subtraction Facts Strategies (MTB), and Relating Multiplication and Division (SFAW). Such titles of lessons, along with objectives and vocabulary lists, provide teachers with an image of what the lesson is about. Unlike titles of lessons, nonroutine headings in the main body of the lesson demonstrate great variation across the five programs. SFAW does not provide any nonroutine headings in the main body of the lesson; usually, there are one or two headings only on student pages that are included in the teacher’s guide, in the form of a question regarding mathematical ideas addressed in the lesson (e.g., How are multiplication and division related?). MTB does not provide nonroutine headings in the main body in general, although there were a few special cases. EM, INV, and MIF contain nonroutine headings on a regular basis that are specific to the content of the lesson. These headings are listed with each student activity or task, which indicate the mathematical focus of the activity or task. MIF tends to include headings in great detail and extensively. For example, a lesson that is intended for 1-day instruction (most MIF lessons are for multiple days) includes headings as follows: • • • • • • •

Algebraic Expressions Can Be Simplified Like Terms Can Be Added A Variable Subtracted from Itself Results in Zero Like Terms Can Be Subtracted Use Order of Operations to Simplify Algebraic Expressions Collect Like Terms to Simplify Algebraic Expressions Make Closed Figures Using Craft Sticks to Practice Writing Algebraic Expression in Context (MIF GR5 5.2 Simplifying Algebraic Expression)

These headings are explicit to the content of the lesson, specifically summarizing the ideas, concepts, and procedures teachers need to address in the instruction. Skimming through such headings gives more ideas about the mathematics of the lesson beyond the title, vocabulary, and objectives of the lesson.

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6.7  Types of Mathematics Communicated to Teachers As explained earlier, all of the 1748 sentences compiled from the five programs were coded in terms of the types of mathematics addressed in each of them. The five categories we developed to examine types of mathematical explanations are descriptions of procedures and steps (E1); explanations of strategies (E2); explanations of concepts, definitions, and conventions (E3); explanations of representations (E4); and explanations of connections, relationships, and applications (E5). Table  6.11 provides examples of these different types of mathematical explanations in the five programs. Attending to these different aspects seems significant for mathematical proficiency (National Research Council, 2001). The examples from MIF and SFAW in the category of procedures and steps (E1) and concepts, definitions, and conventions (E3) in Table 6.11 show a typical type of mathematics communicated in these two programs. The two examples of strategies (E2) illustrate how procedures are communicated with meaning in the problem context, which is common not only in the Standards-based problems, but also in the other two programs. The examples of representations (E4) and relationships, connections, and applications (E5) exhibit diverse explanations in these two categories from the three Standards-based programs: how mathematics can be made sense of with the representations used, and what kind of relationships, connections, and applications are attended to. MTB attends to patterns and relationships much more than the other programs and this tendency is shown in the examples of E5. The example of representations (E4) from INV illustrates one approach of the program to providing mathematical support. As explained earlier, suggested teacher prompts and questions guide teachers (rather than directly explaining to teachers) regarding desired student thinking and the mathematics teachers also need to know. In such cases, teachers are expected to build a complete picture by thinking about and doing the related problem, and making sense of the mathematics in it. Figure 6.6 summarizes the types of mathematics addressed in the five programs in terms of the percentage of the corresponding category. Although it is noticeable that the five programs have different emphases in explaining mathematics, overall, E2 strategies (24%) and E3 concepts, definitions, and conventions (27%) are the two most popular categories, and E1 (procedures and steps) appears the least (12%). In general, the curriculum programs devote a large portion of their mathematical support to address strategies for solving problems (E2), INV the most (29%) and SFAW the least (18%). Also, the programs pay great attention to concepts, definitions, and conventions overall; MIF (35%) and SFAW (37%) attend to this type of mathematical support more than the other four types, whereas INV includes such support the least (13%) in the five programs. INV and MTB rarely include a sentence describing procedures or steps only (less than 0.5% and 2%, respectively), whereas both EM and MIF allot over 20% of their mathematical explanations for this purpose. EM pays significant attention to every category, unlike the other programs. INV attends to strategies (E2), representations (E4), and connections and applications (E5). (See more examples of INV in Chap. 2.) MIF communicates procedures (E1),

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Table 6.11  Examples of types of mathematics communicated in individual lessons Types of mathematics E1. Procedures and steps

E2. Strategies

E3. Concepts, definitions, and conventions

E4. Representations and tools

E5. Relationships, connections and applications

Examples Go through the following number bond strategy and get students to commit to memory:  •  Add 99 is to add 100 minus 1  •  Add 98 is to add 100 minus 2  •  Add 97 is to add 100 minus 3  •  Add 96 is to add 100 minus 4. (MIF) If students have trouble finding the number of marbles, have them skip count by 1000 to reach 10,000. (SFAW) Children share solution strategies as you record them on the board. Possible strategies:  •  Each lap is 500 meters. So I can add 500 eight times: 500 + 500 + 500 + 500 + 500 + 500 + 500 + 500 = 4000. If someone suggests this strategy, it can be used as a lead-in to the next strategy  •  Group the 500s in pairs to show that the sum of each pair is 1000. Then the sum of each pair of 1000s is 2000; and the sum of each pair of 2000s is 4000  •  Repeated doubling: 2 laps = 1000 m, so 4 laps = 2000 m, and 8 laps = 4000 m … (EM) For example, one common strategy [for solving 26 × 368] is to think of twenty-six as 10 + 10 + 6, twenty-six 368’s is 10 × 368 + 10 × 368 + 6 × 368. (MTB) When we skip count by a certain number, we are finding multiples of that number. (INV) Point out that [in 24 × 300] 24 × 3 × 100 = 24 × 100 × 3 according to the Associative Property of Multiplication. (MIF) Place value helps you understand a number by organizing it into places and periods. The comma helps organize the standard form to make it easier to read and write. (SFAW) Remind them when they use the number grid, they can just move down one row for each 10. (EM) Students complete stories to match temperature graphs and consider what elements in the stories correspond to features of the graphs …  •  Do students use features of the stories to identify the shape of the corresponding graph? (INV) They may see that each difference in the “Subtract 10” column is the same as the ones digit of the first number in the sentence. They may also notice that each difference in the “Subtract 9” column is one more than the ones digit of the first number in the sentence. (MTB) Story contexts can be particularly useful in helping students understand the meaning of remainders in division problems. …For example, they might think of 374 ÷ 12 as packing 374 balls into boxes that each hold 12 balls. What does the solution 31 R2 mean in this situation? The 31 represents 31 full boxes with 12 balls in each box. The 2 represents 2 additional balls that are left to pack. (INV) All methods are based on the idea of breaking up the multiplication problem into groups and computing the groups individually, although this is not always so obvious. (MTB)

Bold added for emphasis

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O.-K. Kim and J. T. Remillard EM (N=398) 22 INV (N=230) 0 29 MIF (N=416) 23 MTB (N=561) 2 23 SFAW (N=143) 6 18 All (N=1748) 12 0% E1 Steps

10%

23 13 26 28

22 29

22 35 23

37 24 20%

E2 Strategies

16 27

30%

40%

E3 Concepts

50%

11 29

19 60%

70%

E4 Representations

80%

8 24 23 18

5

90%

100%

E5 Connections

Fig. 6.6  Types of mathematics communicated in individual lessons (%)

strategies (E2), and conventions (E3) much more than representations (E4) and connections and application (E5). In fact, MIF includes a limited number of figures to illustrate mathematical ideas, whereas the other four programs include significantly more. These figures are usually provided to explain a mathematical idea using a representation. Moreover, a very small portion of mathematical support in MIF is about connections and applications. MTB includes a significant portion of each category except for procedures alone (E1). These patterns in the types of mathematical support are aligned, to some extent, with the mathematical treatment and emphasis of the programs in Chap. 2. For example, INV’s emphasis in number and operations is on student strategies and mathematical relationships, and through this emphasis it supports students’ representational fluency. Such mathematical emphasis is communicated to teachers in the lesson guidance in INV.  This is also the case with MTB.  In contrast, MIF’s emphasis is on procedures, algorithms, and underlying properties; the program’s mathematical support for teachers is focused on procedures, strategies, definitions, and conventions. EM’s emphasis on strategies and alternative algorithms as well as ongoing skill practice is also reflected in the types of mathematics communicated to teachers, seen in its significant attention to all five types.

6.8  Discussion The results of this study reveal some common and unique approaches to providing mathematical support in the five elementary mathematics curriculum programs. The results are used to infer the types of mathematical knowledge curriculum authors think teachers need to have in order to teach the subject using their programs, and to discuss issues requiring further research in terms of ways in which a curriculum can better communicate with teachers about the mathematics of its lessons.

6.8.1  Mathematics Teachers Need to Know All five programs pay significant attention to explaining strategies for problem solving, beyond mere procedures and steps. In particular, EM, INV, and MTB often explain multiple strategies for a given problem. Overall, simply providing a set

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procedure without attending to plausible support for it is less common than providing strategies and reasoning. Curriculum authors’ communication of procedures and strategies suggests that teachers need to know not only the procedure and steps to solve a problem, but also the reasoning behind the procedures and diverse approaches to a problem. MIF describes procedural steps in a very organized way (a set of steps listed one step after another in a bulleted format) throughout the lessons analyzed. In contrast, INV focuses mostly on explanations of strategies and reasoning with minimal descriptions of steps per se. The programs often use representations and visual models to explain the mathematics conveyed in the strategies. Representations are important tools to explore; they present mathematics and are fundamental in teaching and learning of school mathematics (Lesh & Jawojewski, 2007; National Council of Teachers of Mathematics, 2000). A portion of mathematical support is allotted to explain how representations can be used to address the mathematical ideas and concepts that students need to learn. The inclusion of this type of support suggests that curriculum designers expect teachers to be able to use such representations flexibly to address the mathematics of the lesson and promote student thinking about mathematical concepts and strategies. Mathematical relationships, connections, and applications are commonly explained across the programs, which indicates that teachers are expected to know the content they teach in a connected way rather than as isolated pieces. One common aspect of such mathematical explanations is about how mathematical concepts and ideas in a particular lesson are related to those in previous and future lessons/ units/grades. This suggests that teachers are expected to understand the development of a concept within and across grades in these programs, as described in Shulman’s (1986) notion of lateral and vertical curricular knowledge and Sleep’s (2009) elaboration of knowledge of content and curriculum along with a mathematical storyline. As noted in Chap. 2, it seems that EM, MTB, and MIF expect quite an extensive teacher content knowledge, as these programs place advanced content in earlier grades (e.g., EM—adding and subtracting positive and negative numbers in grade 4, and square roots in grade 5; MIF—operations with fractions in grade 3, and simplifying algebraic expressions in grade 5; MTB—function machine in grade 4, and prime numbers and exponents in grade 3). This means that teachers need to have knowledge of advanced content and its forms that are most appropriate for ­elementary students. EM and MTB provide more extensive information about the advanced content within and outside lessons (Teacher’s Lesson Guides and Teacher’s Reference Manual in EM, and Unit Resource Guides and TIMS Tutors in MTB) than MIF does. It seems that the designers of MIF assume teachers already know much of the content that is addressed in its lessons. This seems to be the case with SFAW as well, in that it provides mathematical support least frequently among the five programs. In contrast, INV organizes mathematical support features to explain mathematics of the lessons from the perspective of teaching more extensively than the other four programs. For example, INV provides a professional development section in the

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back of each unit that is composed of a collection of Teacher Notes and Dialogue Boxes. Each of the Teacher Notes addresses a particular mathematical concept in-­ depth explicitly for teaching; Dialogue Boxes illustrate examples of classroom discussions on certain concepts and strategies. This reflects the consensus in our field that simply knowing more mathematics is not sufficient for teaching mathematics, and that instead, teachers need to understand the content from a pedagogical perspective (e.g., Ball et  al., 2008; Remillard & Kim, 2017; Rowland et  al., 2005; Shulman, 1987). Overall, the types of mathematics communicated to teachers in the five programs suggest that the programs attempt to provide mathematical support that reflects Ma’s (1999) notion of a “profound understanding of fundamental mathematics” required to teach mathematics, and Skemp’s (1987) notion of relational understanding as opposed to instrumental understanding. The programs include various mathematical supports beyond procedures, providing procedures with mathematical ideas and concepts, making connections among strategies, explaining mathematical ideas using visual models and tools, and relating mathematical topics. However, the extent of such support varies across the programs, and it seems that the designers of the programs have different expectations of the teacher’s knowledge necessary for using their programs.

6.8.2  Seeking Better Communication About Mathematics The five programs exhibit different styles of communicating mathematics to teachers. Examining these different ways of communicating mathematics raises several questions about supporting teachers’ design and enactment of lessons that require further investigation and debate. In this final section, we summarize some findings and recommendations along with ideas for future research that is potentially beneficial to understand and improve the communication about mathematics between curriculum authors and teachers enabling better teaching. The five programs convey mathematical information and explanations in various forms: direct explanations to teachers, mathematical support given in the context of student thinking and reasoning, and mathematical explanations and student strategies/thinking embedded in directions for teacher or student actions. Sometimes they communicate about the mathematics to teachers, but more often tend to do so through teachers. Using these various forms and styles of communication seems reasonable, given the complexity of the mathematical knowledge necessary for teaching. As discussed in Chap. 5 and in this chapter, each program tends to favor a particular form of communication. MIF, for example, tends to incorporate mathematical explanations into directions for teacher/student actions; MTB, in contrast, explicitly describes mathematical ideas, student strategies, and follow-up actions. Comparing the approaches to communication about mathematics across the five programs illuminates the different options available and might lead curriculum authors to be more conscious about the way they explain mathematics within indi-

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vidual lessons to teachers. That said, research on how teachers perceive the information provided in these different forms and styles can lead to improvements in how curriculum materials communicate with teachers about mathematics. The five programs also include mathematical support in various locations. They all have a unit/chapter overview that describes the mathematics of the unit/chapter and explain the mathematics within individual lessons as a way to guide teachers to enact it, even though the extent and types of mathematics communicated in these locations vary across programs. Beyond the unit overview and the explanations in individual lessons, EM, MTB, and INV designate a specific place to elaborate on the mathematics in their program—a separate volume, a large portion of the implementation guide, and a collection of extensive professional development notes in the back of each unit, respectively. These are in-depth, informative, and educative resources for teachers regarding the content they teach. Questions that arise are: How easy is it to access the resources when needed? How do teachers know where to look for content help, and when do they need to? As shown in Fig. 6.3, INV gives a signal with a number in a circle to teachers regarding a related resource provided either in the margin or in a place outside the lesson. This way of providing mathematical explanations may resolve the issue with the second question, yet still it is not known how likely it is that teachers refer to the materials outside the main flow of the lesson (i.e., mathematical support in the side note and outside the lesson) as opposed to those within the main flow. When considering mathematical support within individual lessons, there are also different places in which such support is provided. For example, SFAW tends to provide mathematical explanations in the lesson introduction, whereas MTB and INV often use a separate location (separate boxes or margins, as seen in Figs. 6.3 and 6.4) in the main body of the lesson. In general, we do not know much about which parts teachers read and where teachers look for information. Systematic research is needed to investigate the accessibility of mathematical support in different locations within and outside of individual lessons. One form of communication that may be underutilized by curriculum authors is using various elements of the lesson that capture the mathematics in a rather succinct way—the title of the lesson, lesson goals, vocabulary list, and headings in the lesson guide. Curriculum authors need to consider ways in which these elements can be provided to help teachers articulate the mathematical point of the lesson and the pathway of its core ideas. All of the five programs use titles and goals of lessons to highlight the mathematics embedded in them. EM, INV, and MIF also provide headings in the lesson that summarize the mathematical foci of its activities/tasks. It is unique that INV lists mathematical goals at the beginning of each discussion. Building on INV’s approach, curriculum authors can explicitly indicate goals and mathematical foci of individual activities within the lesson, in addition to the overarching goals of the lesson. Chapter 8 describes lesson goals in terms of design rationale and transparency. When teachers clearly understand the mathematical ideas that should be highlighted in an activity, it is likely that they focus on them during the activity (Remillard, Reinke, & Kapoor, 2019). Further investigation on

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the impact of different elements of the lesson on teachers’ articulation of and teaching toward the goals of individual activities is needed. Although the programs attend to various types of mathematics overall, the types of mathematical support (e.g., procedures, strategies, concepts, representations, and connections) may need to be more balanced among them in some programs. Balancing different types of mathematical support does not mean that there should be an even distribution among the different types of mathematical support. Some categories may need more attention than they are currently given. For example, MIF needs to explain more about representations and mathematical ideas, and connections; 8% and 5% of explanations are dedicated to each of the two categories, respectively. Kim (2018) found a teacher enacting MIF lessons did not use representations useful for students’ understanding of fraction procedures; those lessons included the representations extensively but did not explain them at all. Explanations about the representations for the fraction procedures might have supported the teacher in using them to teach the lessons productively. One area of possible research could focus on the relationship between mathematical support and lesson enactment, especially, the types of mathematical support provided and their impact on lesson enactment. The analysis of mathematical support in this chapter reveals a wide variety of options curriculum authors have in communicating about mathematics with teachers in order to assist their learning of the content and instructional design decisions. It is important to examine the types of mathematical support and the ways in which such support is provided in the curriculum, which provides insight into how such mathematical support can be provided more effectively. As discussed above, understanding the impact of such mathematical support and forms of communication requires further research, especially, how teachers perceive and use mathematical support provided in different forms.

References 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. Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., Klusmann, U., Krauss, S., Neubrand, M., & Tsai, Y. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133–180. Brown, S.  A., Pitvorec, K., Ditto, C., & Kelso, C.  R. (2009). Reconceiving fidelity of implementation: An investigation of elementary whole-number lessons. Journal for Research in Mathematics Education, 40(4), 363–395. Charles, R. I., Crown, W., Fennell, F., Caldwell, J. H., Cavanagh, M., Chancellor, D., et al. (2008). Scott Foresman–Addison Wesley mathematics. Glenview, IL: Pearson. Collopy, R. (2003). Curriculum materials as a professional development tool: How a mathematics textbook affected two teachers’ learning. Elementary School Journal, 103(3), 287–311. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14.

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Davis, E.  A., Palincsar, A.  S., Arias, A.  M., Bismack, A.  S., Marulis, L.  M., & Iwashyna, A. K. (2014). Designing educative curriculum materials: A theoretically and empirically driven process. Harvard Educational Review, 84(1), 24–52. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406. Kelcey, B., Hill, H. C., & Chin, M. (2019). Teacher mathematical knowledge, instructional quality, and student outcomes: A multilevel quantile mediation analysis. School Effectiveness and School Improvement, 30(4), 398–431. https://doi.org/10.1080/09243453.2019.1570944. Kim, O. K. (2018). Teacher decisions on lesson sequence and their impact on opportunities for students to learn. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Research on mathematics textbooks and teachers’ resources: Advances and issues (pp.  315–339). New  York: Springer. Kim, O.  K. (2019). Preservice teachers’ learning to use existing resources productively. In Proceedings of the third international conference on mathematics textbook research and development (ICMT3). Paderborn. Koponen, M., Asilainen, M. A., Viholainen, A., & Hirvonen, P. E. (2019). Using network analysis methods to investigate how future teachers conceptualize the links between the domains of teacher knowledge. Teaching and Teacher Education, 79, 137–152. Lesh, R., & Jawojewski, J. (2007). Problem solving and modeling. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 262–804). Charlotte, NC: Information Age Publishing. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum. Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. Morris, A. K., Hiebert, J., & Spitzer, S. M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: What can preservice teachers learn? Journal for Research in Mathematics Education, 40(5), 491–529. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. Nipper, K., & Sztajn, P. (2008). Expanding the instructional triangle: Conceptualizing mathematics teacher development. Journal of Mathematics Teacher Education, 11, 333–341. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Remillard, J.  T., & Kim, O.  K. (2017). Knowledge of curriculum embedded mathematics: Exploring a critical domain of teaching. Educational Studies in Mathematics, 96(1), 65–81. Remillard, J. T., Reinke, L. T., & Kapoor, R. (2019). What is the point? Examining how curriculum materials articulate mathematical goals and how teachers steer instruction. International Journal of Educational Research, 93, 101–117. Rezat, S., & Sträßer, R. (2012). From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation. ZDM—The International Journal on Mathematics Education, 44(5), 641–651. Rowland, T. (2013). The knowledge quartet: The genesis and application of a framework for analyzing mathematics teaching and deepening teachers’ mathematics knowledge. Journal of Education, 1(3), 15–43. Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255–281. Shulman, L.  S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

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Shulman, L.  S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–21. Skemp, R. (1987). The psychology of learning mathematics. Hillsdale, NJ: Lawrence Erlbaum. Sleep, L. (2009). Teaching to the mathematical point: Knowing and using mathematics in teaching (Unpublished doctoral dissertation). Ann Arbor, MI: University of Michigan. 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. TERC. (2008). Investigations in number, data, and space (2nd ed.). Glenview, IL: Pearson. TIMS Project University of Illinois at Chicago. (2008). Math trailblazers (3rd ed.). Dubuque, IA: Kendall/Hunt. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill.

Chapter 7

How Curriculum Materials Support Teachers’ Noticing of Student Thinking Rowan Machalow, Janine T. Remillard, Hendrik Van Steenbrugge, and Ok-Kyeong Kim

Abstract  Professional noticing of children’s mathematical thinking (PNCMT) is the ability for teachers to attend to students’ mathematical thinking, analyze it within a larger framework, and use that analysis to respond differentially to students, based on the strategies that they choose or mis/understandings that they demonstrate. This chapter analyzes whether and how teacher’s guides in five elementary mathematics curriculum programs offer support to teachers related to noticing student thinking. As reported in Chap. 5, we found that all five curriculum programs provided some supports for attending to and analyzing students’ thinking and work. Our analysis in this chapter reveals that only two of the programs frequently modeled the explicit connections to conceptual understanding necessary for PNCMT. Two other programs frequently modeled evaluating students based upon correctness or general characteristics, without considering the strategies they used. The final program used a combination of the two approaches. As teachers do not tend to use PNCMT without focused education and support, we hypothesize that curriculum programs that provide models for PNCMT are more likely to support teachers in learning to attend to student thinking. Our findings provide examples of ways that curriculum authors might support teachers in developing PNCMT.

R. Machalow · J. T. Remillard (*) Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected]; [email protected] H. Van Steenbrugge Stockholm University, Stockholm, Sweden e-mail: [email protected] O.-K. Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_7

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Keywords  Curriculum analysis · Mathematics curriculum materials · Educative curricula · Student thinking · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman-Addison Wesley Mathematics · Teacher’s guide · Noticing student thinking · Teacher noticing

7.1  Introduction How might mathematics curriculum materials support teachers to anticipate and respond to student thinking? While it is common for teachers to attend primarily to whether students’ answers are correct, teachers who can analyze students’ written and oral work and place students’ strategies within a larger framework of mathematics concept development can use that information to guide student learning (Choppin, 2011; Jacobs, Lamb, Philipp, & Schappelle, 2011). This chapter examines whether and how features in the five curriculum programs1 have the potential to support teachers in developing skills in attending to student thinking. Both Ball and Cohen (1996) and Davis and Krajcik (2005) emphasize these skills in their conceptualizations of how curriculum materials might be designed to be educative for teachers. Ball and Cohen propose that: Teachers’ guides could help teachers to learn how to listen to and interpret what students say, and to anticipate what learners may think about or do in response to instructional activities… [by providing] examples of a range of student work in the context of the material at hand, and comment on the meaning of the work, instead of simply stating lamely that ‘answers will vary.’ (p. 7)

Building on this proposal, Davis and Krajcik suggest that educative features should help teachers “learn how to anticipate and interpret what learners may think about or do in response to instructional activities, …[d]escribing why students might hold particular ideas and giving suggestions for how to deal with those ideas” (p. 5). As described in Chap. 5, attending to student thinking was one of the codes we applied to the set of 75 lessons in order to examine what and how curriculum authors communicated about student thinking through teacher’s guides. We noted all places where teacher’s guides communicated with the teacher about what students should know or be able to do, ways they may respond, difficulties they may encounter, strategies they may use, or schemas they may possess or develop. By looking at the quantitative results of our coded lessons, in Chap. 5, we uncovered how much each program communicated with teachers about student thinking (along with how much  The five programs are Everyday Mathematics (EM), Investigations in Number, Data, and Space (INV), Math in Focus (MIF), Math Trailblazers (MTB), and Scott Foresman–Addison Wesley Mathematics (SFAW). See Chap. 1 for details about the programs. 1

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they communicated about the mathematics, the design of the lessons, or about what teachers needed to do). We found that all programs communicated with teachers about various aspects of student thinking, but they did so in substantially different ways. INV, for example, tended to communicate to teachers directly about student thinking, whereas MIF almost always combined information about student thinking with directive guidance. Through qualitative analysis of the 75 lessons, we were able to explore these differences in more detail. We present the findings of this analysis in this chapter. Our analysis was guided by the literature on teacher noticing, which explores how teachers attend to, interpret, and respond to students’ written and verbal work as a tool for understanding their thinking (Ebby, 2015; Jacobs, Lamb, & Philipp, 2010; van Es & Sherin, 2002, 2008). Ideally, teachers would engage in what Jacobs et  al. (2010) call professional noticing of children’s mathematical thinking (PNCMT), which puts awareness and analysis of mathematical strategies at the center of teachers’ noticing. A number of studies, however, demonstrate that without targeted professional development, teachers are more likely to evaluate students based upon behavior, correctness (getting a right or wrong answer), or overall performance (being “below basic” or “advanced”) than attending to their strategies and mathematical ideas, or that teachers’ awareness of students’ strategies extended only to describing them rather than analyzing them or making decisions based upon them (Goldsmith & Seago, 2011; Jacobs, Franke, Carpenter, Levi, & Battey, 2007; Jacobs et al., 2011; van Es & Sherin, 2002, 2008). In addition, Ebby (2015) describes how teachers frequently evaluate student work by drawing on working knowledge of students (work habits, learning profiles such as ELL or special education, performance level, home life, and emotions about math). While this knowledge was used by some teachers to enhance their professional noticing of children’s mathematical thinking (e.g., interpreting the same error as a hasty mistake by one student and an area of conceptual confusion for another), all too often the working knowledge of students replaced attention to strategies, causing teachers to dismiss a student’s actual work in favor of prior assumptions about the student (Ebby, 2015; Ebby & Sam, 2015). We hypothesized that teacher’s guides may have the potential to support teachers in developing professional noticing of students’ mathematical thinking through modeling the component practices. On the flip side, teacher’s guides might instead reinforce common habits of focusing on overall performance, working knowledge of students, or a correctness-based analysis of students’ strategies if they utilize these approaches. To this end, our analysis explored how curriculum authors communicated with teachers about student thinking and whether these approaches were likely to support teachers in engaging in PNCMT. Our analysis was guided by the following research questions: 1. What is the range and variation of ways curriculum authors communicate with the teacher about student thinking, strategies, and solutions in mathematics lesson guides?

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2. To what extent and in what ways do lesson guides support teachers in noticing children’s mathematical thinking, as opposed to attending to correctness or working knowledge of students?

7.2  Conceptualizing Student Thinking in Teacher’s Guides Our analysis of whether and how teacher’s guides support teachers in noticing, engaging with, and supporting student thinking is informed by existing research on mathematics teaching that supports student growth in deep mathematical understanding. Three prominent factors stand out as being critical to teachers and teaching from this work: (a) attending to and supporting student thinking, (b) creating a climate that emphasizes strategies over answers and that allows students to choose their own strategies, and (c) development of mathematical and pedagogical knowledge that supports these teaching practices. These comprise our conceptual framework, as described below.

7.2.1  Teacher Noticing Studies of teacher noticing address the ways in which teachers selectively observe, make sense of, and respond to the events in their classrooms (Goldsmith & Seago, 2011; Jacobs et al., 2010). This research stems from expert–novice studies, which suggest that experts in a field are able to rapidly take in a specific situation and identify patterns linked to larger concepts that can guide an effective response, whereas novices are likely to give all components of a situation equal attention and see each situation as unique and disconnected from larger patterns resulting in less successful and purposeful responses (Jacobs et al., 2010; van Es & Sherin, 2002). Studies suggest that teachers’ expertise in noticing does not simply develop over time and exposure, but rather through guidance and practice related to directing attention and actively developing frameworks (Goldsmith & Seago, 2011; van Es & Sherin, 2008). Further, what teachers notice is likely to reflect their experiences, philosophies, cultural backgrounds, beliefs, and learning experiences, creating “expectancies” around where they focus their attention (Jacobs et al., 2010; van Es & Sherin, 2008). While teacher noticing studies may apply to any topic worthy of a teacher’s attention, such as cultural funds of knowledge, supporting English language learners, or using a teacher’s guide effectively (Choppin, 2011; Drake, Land, & Tyminski, 2014; Roth McDuffie et al., 2014; Schack, Fisher, & Wilhelm, 2017; Stoehr et al., 2016), this chapter focuses on what Jacobs et al. (2010) call professional noticing of children’s mathematical thinking (PNCMT). There are two major frameworks used for analyzing PNCMT. Jacobs et al. (2010) offer a framework that addresses teachers’ expertise in three interrelated skills: (a) attending to children’s strategies, (b) interpreting children’s understandings, and (c)

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deciding how to respond on the basis of children’s understanding. van Es and Sherin (2002, 2008) offer a similar and compatible Learning to Notice Framework, which focuses on the attending and interpreting (but not responding) aspects of noticing: (a) identifying what is important in a teaching situation, (b) using what one knows about the context to reason about a situation, and (c) making connections between specific events and broader principles of teaching and learning. Jacobs et al. (2010) note that during in-the-moment decision-making, all three steps of attending, interpreting, and responding happen almost instantly and in the background as an integrated teaching move. Professional development efforts to make these aspects of teacher noticing more visible to teachers and allow them to build their expertise have engaged them analyzing student work (written solutions to open-ended problems) or videos of classroom teaching as a way to develop expert constructs for understanding and responding to students’ ideas (Fennema et  al., 1996; Goldsmith & Seago, 2011; Jacobs et al., 2007; van Es & Sherin, 2002, 2008). Researchers have found that over the course of a year or two, teachers in these programs shifted from attending mostly to teachers’ actions, classroom climate, and student behavior (for video lessons) and general pedagogy or intended answers, to attending mostly to students’ strategies and thinking. They also shifted to predominantly interpreting (making sense of actions in a mathematically conceptual context), rather than describing steps without analysis or evaluating based upon correctness (good/bad or wrong/right). As teachers increased their attention to students’ strategies, their teaching responses were increasingly based upon students’ mathematical understandings (Fennema et al., 1996; Goldsmith & Seago, 2011; Hiebert et al., 1996; Jacobs et al., 2007, 2011; van Es & Sherin, 2002, 2008). Most studies of teacher noticing in mathematics have bifurcated the topic of teachers’ noticing into mathematical and non-mathematical, with little attention paid to the latter. Ebby’s (2015) analysis of the range of topics that teachers attend to when analyzing student work attends to both. She found that most teachers focused on either evaluating or describing the work, while a smaller group was able to articulate the concepts underlying a strategy, identify a specific misunderstanding, or place the strategy within a developmental framework. She found that instead of or in addition to focusing on strategies, teachers frequently drew on what she calls working knowledge of students, including work habits, previous work, performance level (e.g., advanced, struggling), learning profile (ELL, special education), home life, dispositions (emotions surrounding math), and classroom behavior. Teachers proposed instructional responses that generally mirrored their analyses, with most focused on non- mathematical responses (having students work in small groups, reminding them to write neatly, etc.) or procedural responses (more practice, general review of the procedure), with a smaller portion of the responses devoted to helping students understand the concepts behind a procedure or helping students transition to more sophisticated strategies. Working knowledge of students should not be seen as an irrelevant type of noticing, as instructional responses such as building students’ confidence or managing behaviors are also necessary for student learning (Ebby, 2015; Jacobs et al., 2011).

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At the same time, Goldsmith and Seago (2011) found that when analyzing their own students’ work, teachers paid less attention to mathematical strategies than they did when analyzing similar work from unknown students. Instead, they relied on memories of the lesson or students’ identities rather than attending to the strategies written on students’ papers, causing them to engage in only superficial analysis based on descriptions of the lesson or vague evaluations. This finding suggests that working knowledge of students has the potential to divert teachers’ attention from students’ strategies. Ebby and Sam (2015) suggest that working knowledge of students could be used as an entry point for teachers into the study of student work, but emphasize that it should complement, rather than replace, attention to students’ strategies.

7.2.2  Supporting Multiple Strategies Teachers’ ability to analyze students’ strategies is particularly important when students are engaged in generating and defending their own strategies as well as critiquing the strategies of others, as recommended by the National Council of Teachers of Mathematics (NCTM, 2000) and the Common Core State Standards for Mathematics (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010). Although American teachers have historically taught basic operations using a single, standard algorithm through rote practice (Henningsen & Stein, 1997; Stigler & Hiebert, 1999), it is well established that students can develop strong conceptual understandings when teachers support them in inventing their own procedures first (Carroll & Porter, 1997; Hiebert et al., 1996; Kamii & Dominick, 1998) or guide them through developing or using multiple strategies and representations from concrete to abstract (Daro, Mosher, & Corcoran, 2011; Heritage, 2008). To support this development, teachers must be able to identify salient mathematical ideas in students’ responses, support students in identifying mathematical misconceptions or errors in incorrect responses, make connections between students’ ideas and conventions used by the larger mathematical community, and summarize students’ ideas in ways that make them accessible to the whole class (Boerst, Sleep, Ball, & Bass, 2011; Chazan & Ball, 1999; Sleep, 2012; Stein, Engle, Smith, & Hughes, 2008). Based upon this research, we propose that curricular supports aimed at helping teachers guide students to develop strategies, make connections between multiple strategies, or supporting teachers in understanding student-generated or student-­ chosen strategies would support teachers in developing these practices. In contrast, when curricular supports focus on accurate replication of a teacher-taught strategy, or when the teacher-taught strategy is shown as the only acceptable solution to a task, this may reduce the opportunity for teachers to learn about alternative algorithms or student-generated algorithms. In the middle ground, teachers might use a rote approach to teach multiple strategies to students. Thus, the use of multiple strategies is necessary but not sufficient to promote PNCMT.

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7.2.3  Building Foundational Knowledge In addition to encouraging teachers to analyze student work, teacher’s guides may also include supports that provide background on mathematical principles, pedagogical strategies, and common student misconceptions (Ball & Cohen, 1996; Davis & Krajcik, 2005). When teachers have a greater foundational knowledge of mathematical and pedagogical principles they may be better able to anticipate students’ responses and areas of confusion, and identify when and how to apply efficient models, activities, explanations, and alternative explanations in response to students’ work (Ball, Thames, & Phelps, 2008; Hill, Ball, & Schilling, 2008). Thus, our analysis considered how and the extent to which teacher’s guides provided foundational guidelines and knowledge that might help them understand their students’ thinking.

7.3  Analytical Framework Guided by the PNCMT literature, our analysis considered whether guidance in the materials might help teachers to (a) decide what to focus on (attend), (b) make sense of it (analyze), and (c) adapt their teaching based upon that analysis (respond). The analysis in this chapter focuses predominantly on the attending and analyzing steps. Throughout this analysis, we considered student work to include both written and verbal responses (anticipated comments during class discussions, answers to teachers’ questions, etc.). In addition, we noticed that while the majority of tasks required students to solve a mathematics problem, some asked students to provide strategies, definitions, examples, or explanations. All of these anticipated student responses, we believe, provide evidence of student thinking that the teacher has the opportunity to notice and analyze. We used the teacher noticing literature to develop our analytical framework for considering how teacher’s guides may support (or not support) teacher noticing, shown in Fig. 7.1. We recognize that although the goal of PNCMT is to analyze students’ strategies with attention to mathematical concepts, in practice teacher’s guides often suggest that teachers analyze student work based on correctness or suggest differentiation or remediation steps without first attending to students’ responses. To explore how teacher’s guides communicate with teachers about noticing, we classified these noticing supports by where their attention is focused: attending, analyzing, or providing foundational guidelines for those steps (top row of Fig. 7.1). We also addressed the rigor level of each noticing support using a scale modified from those used in the teacher noticing literature (bottom row). The coding categories within each type of noticing emerged primarily from our initial analysis. Under the first step of PNCMT, attending to or anticipating students’ thinking, curriculum authors tended to provide samples or descriptions of possible student responses to specific tasks or questions. Possible student responses

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Noticing Supports

Attending to and anticipating student thinking • Possible student responses

Analyzing and evaluating students and their thinking • Strategy-specific evaluations • Evaluations based on correctness/ unspecified difficulties • No/implicit evaluation • Working knowledge of students

Foundational guidelines that support noticing

Mathematical Quality of Noticing Supports Rigor level • Explicitly conceptual • Potential conceptual • Procedural (based on rote repetition & memorization)

Supporting multiple strategies

Fig. 7.1  Analytical framework of how teacher’s guides communicate about student thinking

can provide a window into understanding students’ thinking; the strategies that students choose, the explanations they provide, and the errors that they make can give teachers insight into what specific concepts students understand and where they hold misconceptions. While it is possible that teachers interpret these possible student responses as the “right answers” that require no further thought, we propose that curriculum authors include possible student responses to draw teachers’ attention to students’ work. The second step of PNCMT involves analyzing and evaluating students and/or their work. Some noticing supports from the teacher’s guides suggest analyzing the responses that students give in order to understand students’ thinking or plan a response, a necessary component of noticing children’s mathematical thinking. However, many noticing supports suggest evaluating students’ work based on overall correctness or unspecified difficulties without attending to the strategies used or the misconceptions that students might hold. And just as teachers may evaluate students without attending to their work and instead base decisions on working knowledge of students (Ebby, 2015; Jacobs et al., 2010; van Es & Sherin, 2008), teacher’s guides may direct teachers to choose teaching responses based on pre-­ assigned groups or labels without first assessing students’ abilities on that day’s lesson. Our analysis explored how teacher’s guides suggest that teachers evaluate

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their students’ needs based on these foci. While the two types of noticing supports above address individual tasks, the third category, foundational guidelines to support noticing, is broader. Foundational guidelines may place the work that students are doing within a larger context, explain why or how a strategy or resource can be used, suggest adaptations that can be used across many lessons, set expectations for student’s capabilities based on their grade or point in the year, and so on. As with the previous noticing support, these foundational guidelines may suggest that teachers attend to students’ strategies or evaluate students based on general difficulty. Because the quality and focus of noticing supports can vary so dramatically, we determined that it was important to assess their quality. We use a modified version of the scale that Ebby (2015) used to categorize teacher’s noticing abilities by level of conceptual rigor. Teacher’s guides may lead teachers to analyze student work at three rigor levels: procedural (based on correctness, description, or completion of rote procedures), potentially conceptual (making reference to concepts without fully developing or elucidating them), and explicitly conceptual (providing conceptual explanations of students’ strategies or misconceptions). Of these three rigor levels, only explicitly conceptual analyses meet the criteria for PNCMT from Jacobs et al. (2010), leaving a wide range of opportunities for curricular supports to direct teachers toward other types of noticing that either avoid mathematics altogether or fail to lead to conceptual understanding. Similarly, just as teachers may successfully guide students in exploring multiple strategies or promote only rote repetition of a single, teacher-taught strategy, curricular supports may direct teachers down one of these paths. However, the simple presence of a multiple strategies or a guided discussion does not necessarily imply that the feature addresses conceptual understanding; students may be asked to solve problems using two teacher-taught strategies without comprehension. Thus, the presence of multiple strategies within a noticing support is an important component, but should still be considered in light of its level of rigor.

7.4  Methods As described in Chaps. 1 and 5, we analyzed a set of lessons randomly selected from teacher’s guides of five curriculum programs: Everyday Mathematics (EM) (University of Chicago School Mathematics Project, 2008), Investigations in Number, Data, and Space (INV) (TERC, 2008), Math in Focus (MIF) (Marshall Cavendish International, 2010), Math Trailblazers (MTB) (TIMS Project, 2008), and Scott Foresman–Addison Wesley Mathematics (SFAW) (Charles et al., 2008). We used the same lessons that were used in Chaps. 5, 6, and 8 of this monograph: 5 randomly selected lessons per grade (grades 3–5) for a total of 15 lessons per program and 75 lessons altogether. Because we were primarily interested in how curriculum authors communicated with teachers about student thinking, we focused our analysis on content in the lesson guide, but not in the student text, even if the student text was pictured in the guide. We did not analyze potential student answers

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written within the inset student text for consistency with other analyses in this volume. As all five curriculum programs had extensive instructions in the teachers’ guides which recapitulated the content of the student text, we felt that we were not missing any information from the student text that might significantly change our results.

7.4.1  Unit of Analysis We began with the sentences or phrases coded as attending to student thinking in the primary coding, described in Chap. 5 (n = 1675). We found, however, that curricular supports pertaining to understanding student thinking often occurred in discrete multi-sentence chunks, sometimes delimited by visual signifiers such as icons, boxes, color changes, font changes, or formatted headings. Because these curricular supports had a clear beginning and end (either delimited or easy to identify by the researchers as a complete example or explanation), we determined that analyzing these chunks holistically would best help us understand the intention of the curriculum authors and their likely use by teachers. This decision to work with delimited units sometimes extended the focus of our analysis beyond specific references to student thinking to include references to how student work and thinking might be prompted or how teachers might respond. Further, we determined that using these discrete chunks as a unit of analysis was appropriate given our guiding framework. As discussed earlier, the teacher noticing literature considers attending to, analyzing, and responding to student thinking as an integrated teaching move. By focusing on cohesive chunks, we were able to count the number of unique instances where the guides referred teachers to attend to student thinking or missed the opportunity by focusing teachers’ attention on correctness or working knowledge of students.

7.4.2  Coding We coded the chunked segments in several waves using a combination of emergent and theoretically guided approaches. We derived codes from our conceptual framework based on PNCMT and Working Knowledge of Students. Through preliminary coding, we clarified and revised the codes to address emergent patterns. The resulting codes are shown in Table 7.1. Using Atlas.ti, we assigned all segments a code from two sets: a type of noticing support and a rigor level indicating the quality of the noticing support. Segments received additional codes if (a) multiple strategies were present or (b) working knowledge of students was indicated.

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Table 7.1  Code descriptions and examples Noticing supports Anticipating/attending to student’s work Possible student Possible student responses to responses tasks, either given in the student font or within the lesson text

“27 is divisible by 9 because the result is 3—a whole number. 27 is not divisible by 5 because the result is 5.4—not a whole number.” (EM GR5 1.5)

Analyzing and evaluating students and student work Strategy-specific Instructions to the teacher for evaluations evaluating students’ responses based on the strategy the student uses, including common errors

“Ongoing Assessment: Observing Students at Work Can students solve the problem by using relationships on the multiple tower? Do they use the solutions to one problem to help them solve others?” (INV GR5 1.3.2) “If students have trouble getting started, Evaluations based Instructions to the teacher for have them first add to find the total time evaluating students’ responses on correctness or for the two 50-meter races.” (SFAW based on students’ general unspecified GR5 1.14) (Emphasis added) correctness or difficulty difficulty “Some students might find it easier to No/implicit Suggestions for adapting the evaluation lesson that are given without first work on a full sheet of dot paper for Problem 2.” (EM GR5 1.2) evaluating students’ work. (Includes working knowledge of students and adaptations where no criteria are given.) See examples in each category below Co-codes used when a header, Working icon, or standard phrase (e.g., knowledge of “For students who are learning students English…”) indicates that working knowledge of students should be used • Targeted group Statement guiding evaluation or “Differentiated Instruction: English Language Learners Before comparing adaptations based on labels assigned to students (e.g.,special algebraic expressions on pages 226 and 227, have students use , and = to education or English language compare numerical expressions.” (MIF learners GR5 5.3) “DIFFERENTIATION: • Performance Evaluations based on students’ level generally being below level, on Intervention For those students who need to use tools to solve these level, or advanced problems, provide connecting cubes in towers of 10 or grid paper for drawing arrays.” (INV GR4 3.3.3) “Early Finishers Challenge students to • Finish early Evaluations based on students completing an assignment faster find the numbers from 10 to 900,000 that have the same digit in every place.” than their peers (SFAW GR3 1.5) (continued)

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Table 7.1 (continued) “Math and Art Show Three Methods • Each group creates a poster that shows three addition or subtraction problems solved by three different methods: mental math, paper and pencil, and calculator.” (SFAW GR4 2.8) “Convention dictates that we begin Background information on Foundational [multi-digit subtraction] at the right. mathematical approaches, guidelines that common mistakes, expectations However, it is intuitive to begin on the support noticing left. No harm is done if students do so about what students should be though it may involve some extra able to do, or how teachers can trading.” (MTB GR5 2.3) support learning Mathematical quality of noticing supports (Co-coded with all noticing supports) Rigor levels “It is expected that most fifth graders are Explicitly Noticing supports that provide able to solve division problems without conceptual explicit explanations of drawing a representation first. However, underlying mathematical in this session students are asked to use concepts a story context and representation…. Students may use these tools when they encounter a problem they are not sure how to solve [or] to help them keep track of their partial solutions” (Co-coded with Foundational Guidelines) (INV GR5 7.3.1) “For Advanced Learners: Use concrete Potentially Noticing supports that mention materials to show groups of sixes. Count conceptual conceptual understandings in sixes: 6, 12, 18, 24, 30, 36, 42, 48, 54, without fully developing or 60. Students can look for a number elucidating them pattern in the ones digits of the multiples of 6.” (Co-coded with No/ Implicit Evaluation and Performance Level) (MIF GR3 6.2) “Ask: Can someone explain what to do Procedural Noticing supports focused on if a diagonal sum [in lattice practicing rote procedures, memorizing definitions, arriving multiplication] is a 2-digit number? Write the ones digit of the sum, and then at correct answers without attending to strategies, classroom add the tens digit to the sum in the next diagonal.” (Co-coded with Possible management, or other non-­ Student Response) (EM GR3 9.9) mathematical issues Supporting multiple strategies Multiple strategies This code identifies the presence “Students may suggest several ways of of multiple strategies or solutions estimating. One way is to say $76 is close to $75. Another person may find 80 to be a convenient number. Your students may have other suggestions.” (Co-coded with Possible Student Response and Potentially Conceptual) (MTB GR5 2.6) • Learning style/ Evaluations based on students interest learning style (kinesthetic, etc.), interests in other academic subjects (music, reading, etc.), or further interests in math

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7.4.3  Analysis After each segment was coded, we explored each noticing support using the mathematical quality co-codes, rigor level, and support of multiple strategies. This allowed us to both quantify how each teacher’s guide supported teachers in attending to children’s mathematical thinking, and to consider the qualities of statements that were more and less helpful in developing PNCMT. In some cases, potentially conceptual or procedural noticing supports were coded as such because the underlying concepts had been described and coded elsewhere in the lesson. This decision allowed us to answer questions such as, how do authors use possible student answers (or evaluations or foundational guidelines) to support teachers in noticing children’s mathematical thinking? We were not aiming to assess the mathematical quality of the curriculum overall. (See Chap. 2 for a comparative analysis of the treatment of mathematics across the five programs and Chap. 6 for an analysis of how the mathematics is communicated to teachers in the teacher’s guides.) In addition, we analyzed one noticing support type, No/Implicit Evaluation, using the prevalence of evaluations based on working knowledge of students.

7.5  Results In this section, we consider patterns from our analytical coding to report findings on the extent to which the five curriculum programs supported teachers’ noticing of student thinking. We discuss each type of noticing support by first showing its prevalence at each rigor level for each of the five programs, except for the section on no/ implicit evaluations, which focuses on working knowledge of students. We then provide illustrative examples of each noticing support at each of the rigor levels and note interesting distinctions between programs. We discuss the noticing supports following the order in which the PNCMT steps are described: First, attending to or anticipating students’ work by evaluating possible student responses; then, analyzing or evaluating students’ work; and finally, foundational guidelines that support noticing. The final section provides a holistic overview of how each of these pieces fits together within each of the teacher’s guides, and the potential that they offer to teachers in developing PNCMT.

7.5.1  A  nticipating/Attending to Student Work: Possible Student Responses The first step of PNCMT is anticipating and attending to students’ work. This type of work relies on having a sense of the typical or expected student responses to a task, including solutions, strategies, explanations, definitions, incorrect responses

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based on misconceptions, and so on. We coded examples of student work or descriptions of student work provided in the teacher’s guide to the lesson, but not on inset student pages. We coded only possible student responses that held at least the potential for addressing student thinking by showing the approach used (in representations, steps, equations, or words), but not simple numerical answers such as 6 or 2/3. Possible student responses ranged in the level of detail, incorporation of underlying concepts, and opportunity for students to choose between strategies, which we addressed with the rigor level and multiple strategies co-codes. Note that the rigor level co-codes did not address whether the task was explicitly conceptual, potentially conceptual, or procedural for the student. Instead, we evaluated the degree to which the answer supported teachers in developing PNCTM. For example, asking students to write equations to describe a story problem with attention to tens and ones is explicitly conceptual for the student. The following possible student response, however, is only potentially conceptual for the teacher because it alludes to underlying concepts without making them explicit: “Students are likely to suggest 46 + 32 = _____ or 46 + 30 + 2 = ____” (INV GR3 1.1.2). Our analysis was guided by the belief that explicitly conceptual student responses model for teachers’ attention to strategies and concepts that would support PNCMT, while potentially conceptual responses might support this orientation if used in conjunction with other explicitly conceptual curricular supports. Not only do explicitly and potentially conceptual responses demonstrate for teachers what student work might look like, but they can also play an educative role in clarifying mathematical concepts and modeling the type of thinking teachers should elicit in the classroom. Similarly, the presence of multiple strategies at these levels may support teachers in making connections between strategies or supporting students in developing their own strategies or choosing strategies. Procedural responses, however, are unlikely to support this orientation and may reinforce for teachers that mathematics instruction should focus on procedural steps or correctness. In general, when multiple procedural strategies were included, both the student task and the potential responses showed two strategies previously introduced by the teacher without attention to the relationships between them, consideration of when either might be appropriate, or attention to the concepts underlying them. Figure 7.2 shows how possible student responses in each of the five programs varied by levels of rigor and the presence of multiple strategies. The graph on the MTB INV EM MIF SFAW

Explicitly Conceptual Potentially Conceptual 0

20

40

60

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Procedural/ Rote

MTB INV EM MIF SFAW 0% 20% 40% 60%

Fig. 7.2  Rigor level of possible student responses: Total number at each rigor level (left) and percent of those responses with multiple strategies/solutions (right)

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left represents the total noticing support counts by rigor level from five sample lessons per grade from grades 3–5 (15 total lessons). Note that because the total number of noticing supports varied widely between teachers’ guides, we chose to use stacked bar graphs based upon the total number, rather than percentage, which could be misleading. For example, MTB and MIF have roughly the same percentage of explicitly conceptual possible student responses (~50%), however, MIF has only 8 total (averaging 0.5 per lesson), while MTB has nearly 40 (averaging 2.6 per lesson) thereby offering many more opportunities to build PNCMT. The graph on the right, which shows the noticing supports that use multiple strategies or solutions, shows them as a percentage of the total shown in the graph to the left. We use this format for graphs throughout this chapter, unless otherwise indicated. Explicitly conceptual possible student responses not only provide a sample strategy or solution, but make connections to underlying concepts. Responses in MTB and INV were frequently explicitly conceptual and showcased multiple possible approaches about half of the time. For example, the possible student response at the left of Fig. 7.3 explains the concepts behind using number decomposition as a multiplication strategy, while the potential student answers at right make conceptual connections between multiplication and addition. While EM, MIF, and SFAW had few answers with multiple strategies/solutions, when they did they were almost entirely at the explicitly conceptual level. For example, the excerpt in Fig. 7.4 from SFAW GR4 5.1 not only encourages students to solve the problem three ways, but also makes multiplication based on place value clear in a way that is atypical for this program. There are a great many strategies students can use. For example, one common strategy is to think of twenty-six as 10 + 10 + 6, twenty-six 368’s is 10 × 368 + 10 × 368 + 6 × 368. So, the result is: 10 × 368 = 3680 10 × 368 = 3680 6 × 368 = 2208 since 3680 + 3680 + 2208 = 9568 then 26 × 368 = 9568

Students might say: “Multiply 100 by 3 and then add 4.”

“It’s 4 + 3 + 3 + 3 + 3 + 3 . . . and do that 100 times.”

Fig. 7.3  Two examples of explicitly conceptual possible student responses with multiple solutions from MTB GR5 9.3 (left) and INV GR5 U8 1.3 (right). Left excerpt from MathTrailblazers, Grade 5, Unit 9, Lesson 3 Resource Guide by TIMS.  Copyright ©2008 by Kendall Hunt Publishing Company. Reprinted by permission. Right image from Investigations 2008 Curriculum Unit (Grade 5, Unit 8) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved • What are two ways to use repeated addition to find the product? (Add 3 groups of 40 or add 40 groups of 3.) • How can you find the product by using place value? (40 = 4 tens, so 3 × 40 = 3 × 4 tens = 12 tens = 120)

Fig. 7.4  Explicitly conceptual possible student response with multiple solutions from SFAW GR4 5.1. From Scott Foresman Addison Wesley Mathematics: Grade 4, Volume 2 (Teachers Edition) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

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All of the teacher’s guides except MIF included a relatively large number of potentially conceptual answers. These sample answers might be thought of as missed opportunities, where underlying concepts could be made clear with only a little more explanation, but those last few steps are not taken (though they may appear in other parts of the lesson). For example, the possible student response in Fig. 7.5 could be used to discuss properties of divisibility, evens and odds, and number sense, but instead the answers are given without commentary or suggestions for discussion, and the teacher could easily interpret them as rules to memorize. At the opposite end of the spectrum, some possible student responses take on a procedural interpretation of mathematics teaching by focusing on rote procedures, complete sentences, or correct answers without showing work, (although, again, concepts may be developed elsewhere in the lesson). For example, this possible student response (typical for EM and SFAW) focuses on getting the correct answer without attending to the strategy used: “Mrs. Snow was 25 years old when Kevin was born” (EM GR5 2.4). SFAW, EM, and MIF often use possible student responses that focus on repeating memorized strategy steps, such as: “To check if students understand the problem, ask: What are the three parts you need to include in your answer? (The strategy I used, how I solved the problem, and how I checked my answer.)” (SFAW GR3 1.14). Similarly, Fig. 7.6 from MIF ignores the context of the story problem it is based on, demonstrates rote application of the standard algorithm as the only solution strategy, and is also missing subtraction symbols that might support student understanding. In general, these procedural student responses rarely involved multiple strategies/solutions, though in several cases teacher’s guides would simply state the name of strategies without demonstrating them. Looking across the curriculum programs reveals the range of types of student responses offered to help teachers attend to student thinking. This range raises questions about the potential power that sample student responses might have in shap-

Ask: How can you know that a number is divisible by 2 without actually doing the division? Numbers that end in 0, 2, 4, 6, or 8 are divisible by 2. Can you tell whether a number is divisible by 10 without dividing? Yes; numbers that end in 0 are divisible by 10. Fig. 7.5  A potentially conceptual answer from EM GR5 1.5. Excerpt from Everyday Mathematics, Grade 5, Unit 1–5 Resource Guide by the University of Chicago School Mathematics Project. Copyright © 2007 by McGraw Hill Publishing Company. Reprinted by permission

Fig. 7.6  Example of a procedural answer found in MIF. This image was created by the authors based on an image found in MIF GR4 3.3, p. 97

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ing, not only the type of work that teachers expect, but also how they might make meaning of that work. It is possible that the types of student responses provided in the teacher’s guide will lead teachers to encourage particular types of student work, from rote replication to creative problem solving.

7.5.2  Analyzing and Evaluating Students and Student Work This section explores how and when the lesson guides encourage teachers to analyze student work based on strategies, as opposed to making evaluations based upon correctness or working knowledge of students. In the majority of cases, the guides suggested that teachers analyze or evaluate students or their work in order to provide remediation and adaptations. For example, the first sentence of this excerpt provides an analytical lens for evaluating students’ work and the second provides an adaptation to the lesson: “Watch for children who have difficulty saying or writing the correct fact family. Remind children that all the number sentences in a fact family use the same numbers.” (EM GR3 1.8). For this section, we focused on the first and second steps of PNCMT, noticing and evaluation/analysis (“Watch for students…”), but not the third step, deciding how to respond based upon the analysis (“Remind them that…”), although we generally provide both in our examples for context. We chose to focus on the noticing and evaluation steps because these are at the heart of the PNCMT practice and without going through these steps any responses are likely to respond to superficial evaluations. In addition, while we found many examples of evaluating whether to use a single teacher response, we found only a handful of examples (all in INV) where there were several different responses presented for teachers to choose among based on a students’ thinking. The PNCTM literature (Jacobs et al., 2010; Thomas et al., 2015) suggests that teachers often need support in making sense of strategies, partial conceptions, or common errors at an analytical level. The same literature, however, suggests that even when teachers attend to students’ strategies, they may be evaluating them only for the correct replication of a rote procedure. For this reason, we first identified evaluations based upon students’ strategies and then co-coded them by rigor level. In addition, we found that, like teachers in the PNCMT literature, teacher’s guides often encourage teachers to evaluate students’ work based on correctness of the responses and unspecified difficulties rather than attending to strategies. For example, the evaluation “If students have difficulty changing between units of time…” (SFAW GR4 4.2) does not specify what difficulties or misconceptions students might have or how teachers’ responses might vary based upon those specific difficulties. Similarly, teacher’s guides often privilege working knowledge of students (e.g., ELL, SPED, advanced designations) over their work. When teachers are advised to evaluate students without attending to their work, we term this “no/ implicit evaluation” because the teacher is often instructed to give the same ­adaptation or task to all students with the same designation without first assessing their comprehension or needs relating to the lesson. The frequency of each of these

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No/Implicit Eval. (inc. Working Know. of Stu.)

MTB INV EM MIF SFAW

Correctness/Unspecified Difficulty-Based Eval. Strategy-Specific Evaluation 0

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40

60

80

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Fig. 7.7  Total counts of noticing supports that evaluate students or their work in each program, by type of evaluation

approaches to analyzing and evaluating students and their work in each of the five programs is shown in Fig. 7.7. We discuss each approach in more detail in the following sections. Strategy-Specific Evaluations of Student Work Strategy-specific evaluations focus on the students’ mathematical solution, strategy, misconception, or error. This type of analysis was used in about half of the MTB evaluations and two-thirds of INV evaluations, but made up a relatively small proportion in SFAW, EM, and MIF evaluations. While strategy-specific evaluations have the potential to address PNCMT, this is not a guarantee. To better understand the focus and format of strategy-specific evaluations, we further co-coded them with rigor levels and indicated the percent of them that referenced multiple strategies, as shown in Fig. 7.8. Explicitly conceptual strategy-specific evaluations attended to both the strategy that students were using and addressed the underlying concepts, relationships between strategies, or conceptual misconceptions that students might hold, modeling the level of analysis necessary for professional noticing of children’s mathematical thinking. Out of all 390 evaluations across the 75 lessons, only 33 (8.4%) fell in this category, of which 21 appeared in INV. In INV, many of the explicitly conceptual mathematical evaluations appeared in the “Ongoing Assessment” educative feature. This section modeled the type of questions that teachers should ask themselves for analyzing student work, such as, “How do students determine the number of stickers for each person in problems 1, 2, and 7? Do they count by 10s? Do they see three 10s as 30, four 10s as 40, and so on? How do they determine the total number of stickers in the first two problems? Do they add 30 and 30 mentally? Do they count up by 10s from the first number?” (INV GR3 1.1.1). These evaluations are not matched to specific instructional responses, but seem to be designed only to raise teachers’ awareness of students’ strategies. EM modeled explicitly conceptual mathematical evaluations through a similar educative feature called “Ongoing Assessment: Informing Instruction” which described common misconceptions, though they did not appear in every lesson. For example, “When students are

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SFAW 0

10

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Procedural/ 40 Rote

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MTB INV EM

MIF SFAW 0%

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Fig. 7.8  Number of strategy-specific evaluations by rigor level (left) and percent of those evaluations that use multiple strategies (right)

explaining the steps in the partial-products algorithm, watch for those who say ‘6 ∗ 8’: the 8 in the problem to the right, for example, is in the hundreds place and has a value of 800, not 8. Encourage students to think and say ‘6 [800s]’ or ‘6 ∗ 800.’” (EM GR4 5.5). Noticeably, while these features appeared in both programs, they were used in the main body of the lesson for INV, potentially increasing their chance of being used, but in the sidebars of EM, thereby possibly being interpreted by the teacher as optional or less important. In MTB, these features often occurred as part of the main lesson text and were used to help teachers decide when to continue reinforcing a concept and when to move on. For example, in a lesson where students transition from addition using base-10 boards to addition using the standard algorithm, the section on the standard algorithm opens by explaining: The amount of time you spend on this depends on students’ experiences. Be cautious, however, of those who learned the algorithm by rote and do not understand what is behind it. As an informal assessment, ask: • Explain how using the algorithm and working with base-ten pieces are similar. • Explain how you use marks to show regrouping. (MTB GR3 6.3)

Finally, INV sometimes used explicitly conceptual mathematical evaluations to describe students’ strategies as an alternative to providing a possible student response when the task was too broad to have specific possible responses. For example, in an INV lesson on memorizing multiplication facts, a “Teaching Note” explained: Some students begin to visualize the multiplication combinations as arrays, and this can help them find and remember the product of combinations they do not yet know fluently. For example, a student visualizing 8 × 9 may realize that it is one row of 9 less than 9 × 9; or a student might visualize 8 × 6 as two groups of 8 × 3. (INV GR4 1.2.2)

Potentially conceptual evaluations attended to strategies and referenced underlying understandings, but did not take the last steps toward making those understandings clear (though they may be clarified elsewhere in the lesson). These features were common in MTB and INV, where they predominantly occurred in the main body of the lesson. For example, these instructions to the teacher drew attention to assessing students’ understanding or strategies, but did not provide conceptual details: “Though the

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placement of carry marks can vary, it is important to assess whether students understand the reasons for carrying and the value of the carry marks” (MTB GR3 6.3), and “As students are working, circulate and observe whether they are using the multiple tower to answer the questions” (INV GR5 1.3.2). Some of the INV features at this level were Ongoing Assessments which simply provided less clarity around concepts than in other lessons, such as this one: “Can students multiply by a multiple of 10 accurately? • Can students show their thinking with equations? Do they illustrate their thinking with visual representations (arrays, pictures, or diagrams)?” (INV GR4 3.3.3) while others appeared as adaptations based on students’ readiness, such as, Some students will understand the result of adding and subtracting l0s without using place value materials. Challenge these students by asking them to add larger multiples of 10. In Problem 3, you figured out that Jasmine now has 92 stickers. How many stickers will she have if she buys five more strips of 10? (INV GR3 1.1.1)

That is, while these features in MTB and INV might have lacked the conceptual clarity of explicitly-conceptual strategy-specific evaluations, they still encouraged teachers to attend to strategies throughout the lesson. In MIF, EM, and SFAW, however, potentially conceptual strategy-specific evaluations occurred rarely and entirely in sidebars. In EM and SFAW, these features occurred alongside lessons that were otherwise mostly procedural, giving teachers the option of attending to students’ strategies to some degree, without making it central. For example, Students may have difficulty with expressions involving multiplication when no operation sign is used. When expressions such as 3n are written on the board, write 3 × n at the side with an arrow pointing to 3n to remind them of the meaning of the expression. (SFAW GR5 2.13) Watch for children who count by 10s one space at a time in the Mental Math and Reflexes. Remind them that when they use the number grid, they can just move down one row for each 10. (EM GR3 1.8)

In MIF, on the other hand, the teacher’s demonstrations in the lesson often attended to strategies at a higher conceptual level than the sidebar evaluations, which mostly reinforced procedures, such as in this feature: “Common Error: For Exercise 7 help students align the numbers by place value, rather than comparing the first digit in each number” (MIF GR4 1.2). Strategy-specific evaluations that attended to procedural steps were most common in SFAW, EM, and MIF. Some were placed in the margin of the page and focused on identifying procedural errors and giving students techniques for completing them more easily, such as the example from SFAW in Fig. 7.9. The majority of these evaluations occurred as assessments, intended to determine whether students were making adequate progress (in SFAW) or demonstrate how to score open-­ended questions (in EM). For example, the Test Taking Practice feature from SFAW, shown in Fig. 7.10, is representative of those that occurred in about half of the SFAW sessions, and attended only to corrections and completion, rather than conceptual understanding. The range of ways that curriculum programs suggest strategy-specific analyses of student work provides insight into the types of noticing teachers are expected to

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Fig. 7.9 Strategy-specific, procedural evaluation from SFAW GR3 1.5. From Scott Foresman Addison Wesley Mathematics: Grade 3, Volume 1 (Teachers Edition) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

Fig. 7.10  Strategy-specific, procedural evaluation from SFAW GR5 3.11. From Scott Foresman Addison Wesley Mathematics: Grade 5, Volume 1 (Teachers Edition) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

do. While INV and MTB provide models of how teacher’s guides could support teachers in developing PNCMT, the reliance on procedural analyses in EM, MIF, and SFAW suggests that even when multiple strategies are supported for students, teachers should focus on the correctness of the required student algorithms, rather than analyzing students’ choice of algorithms to understand students’ thinking. Evaluations Based on Correctness/Unspecified Difficulty As shown previously in Fig.  7.7, evaluations based upon correctness (correct answers or ability to correctly implement a procedure without errors) and general difficulties with a task made up around 40–60% of SFAW, MIF, INV, and MTB. As

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many of the PNCMT studies showed, this type of analysis was extremely common among teachers unless they were guided to analyze student strategies (Ebby, 2015; van Es & Sherin, 2002). As is evident in Chap. 4, curriculum authors typically face a tension between giving teachers so much detail that lessons become overprescribed and trusting in teachers’ judgment at the risk of poorly implemented lessons. By choosing to provide less detail and suggest evaluations based upon correctness or general difficulty, curriculum authors may support teachers in this type of surface-level thinking. In general, evaluations based on correctness or unspecified difficulty looked very similar across the five curriculum programs and took the form of short phrases leading into an instructional response. In each of the examples that follow, it is unclear what difficulties students might be having or how difficulties might be analyzed or evaluated: “Watch for children who are having difficulty with Problems 7 through 9” (EM GR3 7.6). “Some students may benefit from spending more time working with the first set of Change Cards and their 1,000 books before moving on to working with the new cards and the 10,000 chart” (INV GR4 5.3.3). “If students are uncertain, sketch the rectangles for them” (MTB GR5 4.1). MIF contained a feature in each lesson called On Your Own: Differentiation Options, which at least nominally addresses students’ correctness on the day’s lesson and is therefore categorized here, though the use of targeted groups could also incline teachers to use preset groups without looking at students’ work. For example, “Depending on students’ success with the Workbook pages, use these materials as needed. Struggling: Reteach 5A, pp.  63–70 On Level: Extra Practice 5A, pp. 31–32” (MIF GR5 2.6). This use of targeted groups is further discussed in the next section. Every EM lesson contained a feature called Ongoing Assessment: Recognizing Student Achievement, which was often based on correctness. For example, “Use journal page 169, Problem 10 to assess children's progress toward using relationships between units of time to solve number stories. Children are making adequate progress if they complete Problem 10a successfully. Some children may complete Problems 10b and 10c successfully” (EM GR3 7.6). Some of these features are written in a way that seems to address students’ strategies, though the overall use of the feature suggests that the teacher should be evaluating student work based upon correct completion of the assigned procedure (e.g., using the partial products algorithm correctly) rather than an analysis of student thinking. For example, “Use journal page 118, Problems 1 and 2 to assess students’ ability to use the partial-products algorithm to multiply a 1-digit number by a 2-digit number. Students are making adequate progress if they can correctly calculate and then add the partial products. Some students may be able to solve Problems 3–6, which involve the multiplication of a 1-digit number by a 3- or 4-digit number” (EM GR5 1.5). These examples illustrate the ways that focusing on correctness or not specifying areas that might cause students difficulty could compound the challenge of supporting teachers’ PNCMT. In addition to communicating an emphasis on correctness, these responses do not provide teachers with insights that might help them analyze student strategies or errors.

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No/Implicit Evaluations and Working Knowledge of Students As shown in Fig. 7.7, in about 60% of EM and SFAW evaluations and 30% of MIF evaluations, teachers were guided to provide an adaptation to the lesson for a subgroup of students without attending to their strategies or even general performance on the lesson’s content. Instead, these evaluations privileged working knowledge of students, using common classifications (performance level or targeted groups), with some attention to students who finished early or who might want to pursue a particular interest or benefit from a different style of learning (kinesthetic, etc.), as shown in Fig. 7.11. Even though assigning work or additional support to students based on these characteristics no doubt makes teachers’ decision-making more efficient, the PNCMT literature suggests that this focus can decrease teachers’ attention to students’ strategies (Ebby & Sam, 2015; Goldsmith & Seago, 2011). SFAW, MIF, and EM use headers or icons with the name of a student group to replace the analysis of student work. These highly visual features are usually in the sidebars or come before or after the lesson, which may further suggest to teachers that differentiation should be pre-planned based upon working knowledge of students rather than implemented in response to students’ work. For example, Fig. 7.12 uses the “ELL Support” icon to designate that students should be evaluated for participation based upon their targeted groups (English language learners), rather than on their overall performance or the strategies they used. This can imply that all ELLs are struggling with identifying patterns on the number grid and would benefit from the same intervention. Similarly, each SFAW lesson concluded by assigning students to solve problems based upon their performance level, as shown in Fig. 7.13. While teachers might assess students’ levels based upon their performance on that day’s lesson, the lack of instructions in this area instead suggests that teachers should assign problems based upon students’ general level of performance. The assignment for Early Finishers also suggests that all students who finish early are ready for more advanced work, rather than considering that some faster students may struggle with concepts while some slower students may be ready for a challenge. Evaluations made implicitly or based on working knowledge of students may suggest to teachers that there is no value in even taking the first step in PNCMT, attending to student work, to recognize that it has the potential to provide insights

MTB INV EM MIF SFAW 0 10 20 Performance Level

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Fig. 7.11  Number of evaluations based on predetermined classifications of students

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Fig. 7.12  An implicit evaluation based on students’ prior designation as ELLs (EM GR3 1.8). Excerpt from Everyday Mathematics, Grade 3, Unit 1–8 Resource Guide by the University of Chicago School Mathematics Project. Copyright © 2007 by McGraw Hill Publishing Company. Reprinted by permission

Fig. 7.13  Implicit evaluation based on performance level and finishing early from SFAW GR5 1.3. From Scott Foresman Addison Wesley Mathematics: Grade 5, Volume 1 (Teachers Edition) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

into student thinking. The next section, however, provides an alternative that refocuses attention on student work. Strategy-Based Evaluations Supported by Working Knowledge of Students In MTB and INV, performance levels and targeted groups were rarely used to help determine students’ needs. When they were used, the group seemed to indicate a first level of attention that should be followed up with a more detailed evaluation based upon assessments of students’ thinking. While INV lessons also use visual icons and headers based on working knowledge of students, teachers are encouraged to make at least a superficial evaluation of students’ needs even when working with a specific group. For example, the INV intervention in Fig. 7.14 suggests that,

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Fig. 7.14  Strategy-based evaluation that begins with a tag based on working knowledge of students and then further differentiates. From INV GR3 1.1.2. From Investigations 2008 Curriculum Unit (Grade 3, Unit 1) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

rather than targeting students who generally struggle with math, the teacher should identify the students who are struggling with the lesson topic and make further assessments within the intervention. Similarly, in INV it is not enough that students finish early, but there is also a check for conceptual understanding before moving on: “If you have time and all students seem to be understanding how such an expression as (100 × 3) + 4 represents the situation, ask students to…” (INV GR5 U8 1.3). By taking this approach, curriculum authors can support teachers in using working knowledge of students to support PNCMT rather than replacing it.

7.5.3  Foundational Guidelines that Support Noticing Foundational guidelines that support noticing not only provide teachers with information about students’ thinking, but set it within a larger pedagogical context. These guidelines were usually rather lengthy, found in marginal sidebars, and often addressed a general strategy or class of problems, rather than a single task. They often provided a rationale for the strategies and concepts themselves, or for students using them to learn, thereby giving teachers some background information and purpose for engaging with PNCMT.  For example, one foundational guideline opens with, “Story contexts can be particularly useful in helping students understand the meaning of remainders in division problems” (INV GR5 7.3.1) and then provides several story problems that illustrate how remainders can be used. They may also suggest ways that teachers can support students in developing skills or concepts through outlining approaches that students can use. For example, “NOTE: Devise a shorthand notation to record children’s solution strategies… For example, 8 [500s] represents 8 five hundreds. Children will quickly adapt to whatever notation you adopt, as long as it is used consistently” (EM GR3 7.6). Foundational guidelines like these provide information on the types of student responses that teachers should

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Explicitly Conceptual Potentially Conceptual 0

5

10

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Procedural/ Rote

MTB INV EM MIF SFAW 0%

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Fig. 7.15  Rigor levels of foundational guidelines that support noticing (left) and percent of those guidelines that used multiple strategies/solutions (right)

anticipate. In comparison to other types of noticing supports, foundational guidelines were more likely than other noticing supports to be explicitly or potentially conceptual and to use multiple strategies, as shown in Fig. 7.15. Across the teacher’s guides, the vast majority of foundational guidelines that support noticing addressed the value of having students learn non-standard algorithms or use visual representations/manipulatives before learning the standard American algorithms or not equating fact/procedural fluency with comprehension. As many teachers did not learn these alternative approaches as students and were exposed to them for the first time after the publication of the NCTM Standards (1989, 2000), this may indicate that the curriculum developers expected teachers to need additional pedagogical support in these areas. In particular, the majority of foundational guidelines focused on the core operations (+, −, ×, and ÷) where teachers might be learning new algorithms alongside their students, as well as having students share their strategies, develop their own algorithms, or use visual representations—all teaching strategies that were largely introduced to teachers through the NCTM Standards (1989, 2000). The following foundational guideline is a good example of the philosophy that the NCTM Standards urged teachers to adopt: NOTE Some children may discover that one way to find products such as 9 × 500 is to multiply the nonzero digits (9 × 5 = 45), count how many zeros are in the multiple (two), and attach the zeros to the product to get 4,500. If a child mentions this discovery to the class, support the other children by explaining and writing the strategy as (9 × 5) × 100, not as just counting zeros and attaching them to the end. The authors suggest that you not introduce this strategy to your children; using this shortcut prematurely may discourage children from thinking the problem through and often leads to errors when multiplying decimals. For example, 0.2 × 0.30 is not equal to 0.60. (EM GR3 7.6)

Several foundational guidelines that support noticing also reminded teachers to allow students to build up a concept over time, rather than expecting immediate success. For example, an MTB lesson on the standard algorithm for addition recommends: Use your professional judgment to determine the amount of practice necessary in class and at home. It is best to let the algorithm sink in slowly. Assigning a few problems a day over a long period of time is better than assigning many at once and expecting immediate fluency. (MTB GR3 6.3)

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In contrast to most foundational guidelines that support noticing, which were explicitly or potentially conceptual, SFAW included a few that focused on rote procedures, such as: Students may initially find the division algorithm confusing, because the process moves from left to right through the place values, while the processes for other operations move from right to left. The two previous lessons focusing on dividing tens and then ones can help students prepare for this concept. (SFAW GR3 11.14)

Other teacher’s guides contained quotes coded as foundational guidelines at the procedural level primarily because they did not fit into other classifications, such as notes about what students might like or dislike. For example, an MTB lesson on the lattice method notes that “Many students enjoy using this method” (MTB GR5 9.3); while this may help teachers anticipate students’ responses, it focuses on students’ feelings rather than their conceptual understanding. Foundational guidelines are the location where teacher’s guides most directly addressed teachers in setting expectations for anticipating, analyzing, or evaluating student work, along with the pedagogical intentions behind those expectations. The large number and conceptual focus of these noticing supports in MTB, INV, and EM suggest that curriculum authors placed value in having teachers understand the concepts and philosophy of the mathematics they were teaching. These understandings are also likely to increase teachers’ capacity to develop PNCMT.

7.5.4  Teacher’s Guide Profiles In this section, we look across the different types of supports described in the previous section to present holistic profiles of how each program communicates about student thinking and consider how and when they support teachers in developing professional noticing of children’s mathematical thinking. Figures  7.16 and 7.17 summarize the findings of the sections above with an overall look at the types of curricular supports (Fig. 7.16), levels of rigor, and prevalence of multiple strategies/ solutions (Fig. 7.17) found in each teacher’s guide. Using these data, we present overviews of each program that address how they communicate with teachers about how they expect teachers to engage with students’ responses. In general, the mathematical quality of the noticing supports in each program mirrors the larger trends described in Chaps. 2 and 3. INV and MTB both reflect Standards-based principles in which students invent and choose their own algorithms. To effectively teach students in this situation, teachers must develop the skills of PNCMT, and both teacher’s guides offer a number of educative features to support this development although they go about it differently. INV is characterized by rich examples of student strategies and thoughtful evaluations of student work, often at the conceptual or potentially conceptual level and often with multiple solutions presented as alternatives. MTB, on the other hand, gives teachers very little explicit direction on how to evaluate student work or even clear distinctions between

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Fig. 7.17  Percent of noticing supports at each level of rigor (left) and percent of those noticing supports using multiple strategies/solutions (right)

whether the teacher or students should be presenting strategies in a discussion. Instead, the lessons focused on core mathematical ideas that should be developed, flowing casually between teachers and students building on each other’s ideas, largely at the explicitly conceptual level. Both INV and MTB frequently provided multiple strategies/solutions, often at high levels of rigor, reflecting their orientation toward having students choose, invent, or co-create solutions to problems which are then discussed as a class under the guidance of a teacher who must be familiar with common approaches students might take. Even though their structures differed substantially, both teacher’s guides provide strong examples of supports that can be used to help teachers develop PNCMT. Although EM was designed to reflect the vision of the NCTM Standards, it adopted a number of approaches consistent with conventional teacher’s guides in the USA. In Chap. 3, we described its pedagogical approach as occupying a middle between dialogic or conceptual instructional models and direct or conventional approaches (Munter, Stein, & Smith, 2015). Its mode of communicating with teachers about students’ thinking takes a similarly blended approach. EM uses a range of types of noticing supports and has more examples of multiple strategies/solutions than both SFAW and MIF, but proportionally less than INV and MTB. Further, the student response options are largely based on tasks where students must complete or may choose between two rote strategies. Compared to INV and MTB, EM makes a large number of evaluations without attention to students’ work, and its possible

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student responses and analyses/evaluations of students’ work are largely procedural or rote. It is worth noting that EM routinely includes foundational guidelines that present Standards-aligned approaches that support PNCMT as an optional and briefly explained process. These approaches, however, are not actualized throughout the lesson, leaving teachers with a conceptual overview of PNCMT and other Standards-based approaches, but few examples of how to enact them in the classroom. MIF, which was influenced by the national curriculum in Singapore, largely communicates with teachers about student thinking with models on the student pages, and provides relatively few possible student responses and examples of multiple strategies/solutions, as students are expected to replicate the approaches introduced by the teacher and shown in the textbook. Following this direct instructional approach (Munter et  al., 2015), there are frequent evaluations of student work, although these are typically based on correctly replicating a procedure. What provides MIF with a unique flavor is that around half of its teacher supports are at the potentially or explicitly conceptual levels. Thus, teachers are expected to notice when students have demonstrated understanding of specific conceptual outcomes, even though the demonstration comes through the use of rote strategies. SFAW follows a traditional U.S. instructional model in which the textbook presents a strategy and students replicate it. Its approach to communicating with teachers about student thinking tends to be somewhat formulaic and procedural: teachers are guided to make implicit evaluations based on working knowledge of students twice as often as they are guided to evaluate students’ work, and only a tiny fraction of teacher supports are either explicitly conceptual or address multiple strategies or solutions. Rather than providing supports that encourage teachers to develop PNCMT, SFAW seems to guide teachers toward seeing math as a series of rote procedures, so that evaluations are based upon working knowledge of students, completion, and correctness rather than students’ strategies.

7.6  Discussion We draw on the findings above and the PNCMT literature to consider the extent to which the five teacher’s guides have the potential to support teachers in noticing, engaging with, and supporting student thinking. The PNCMT literature suggests that teachers largely evaluate students and make mathematical teaching decisions based on working knowledge of students or answer correctness, but that with ­professional development teachers can shift their focus to analyze students’ strategies and build instruction based upon that analysis (Ebby, 2015; Jacobs et al., 2007, 2010; van Es & Sherin, 2002, 2008). Our analysis suggests that, rather than leading teachers toward analyzing students’ strategies, most of the teacher’s guides we analyzed often suggested evaluating students’ work without attention to their strategies or based only on procedural fluency or answer correctness. Implementation of PNCMT (analyzing students’ strategies to understand their thinking and make deci-

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sions) was extremely rare in U.S. curriculum programs. It occurred only 12 total times across the 60 lessons that we analyzed for EM, MIF, MTB, and SFAW, giving teachers limited opportunities to learn the skill. INV stood alone in offering a strong model of how teacher’s guides might encourage teachers to engage in the analyze/ evaluate step of PNCMT with 21 examples over 15 lessons. In general, possible student responses were more successful than analyses/evaluations at modeling what students’ strategies might look like at potentially or explicitly conceptual levels. INV and MTB, in particular, provided strong examples of conceptually driven possible student responses, which have the potential to educate teachers in the first step of PNCTM, attending to student strategies. The inclusion of these higher level possible student responses in the main body of the MTB and INV teacher’s guides (rather than optional notes in the sidebars) may also indicate the importance of possible student responses to the textbook authors and to teachers. At the other end of the spectrum, EM, MIF, and SFAW provided substantially more possible student responses that focused on correctly completing rote procedures, which may send a message to teachers that they should evaluate their students based on correctly replicating teacher-taught algorithms rather than students’ strategies or thinking. Although each of these teacher’s guides had some examples of explicitly conceptual sample answers, their low prevalence might indicate to teachers that conceptually driven answers are only appropriate for some types of problems, topics, or students. In addition, as the conceptual answers often appeared in sidebars or optional problems, this might suggest to teachers that attending to students’ strategies is optional rather than central to mathematics teaching.

References Ball, D.  L., & Cohen, D.  K. (1996). Reform by the book: What is—or might be—the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–14. 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. Boerst, T. A., Sleep, L., Ball, D. L., & Bass, H. (2011). Preparing teachers to lead mathematics discussions. Teachers College Record, 113(12), 2844–2877. Carroll, W.  M., & Porter, D. (1997). Invented strategies can develop meaningful mathematical procedures. Teaching Children Mathematics, 3(7), 370–374. Charles, R. I., Crown, W., Fennell, F., et al. (2008). Scott Foresman–Addison Wesley Mathematics. Glenview, IL: Pearson. Chazan, D., & Ball, D. L. (1999). Beyond being told not to tell. For the Learning of Mathematics, 19(2), 2–10. Choppin, J. (2011). The impact of professional noticing on teachers’ adaptations of challenging tasks. Mathematical Thinking and Learning, 13, 175–197. Daro, P., Mosher, F. A., & Corcoran, T. B. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. CPRE Research Reports. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14.

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Drake, C., Land, T. J., & Tyminski, A. M. (2014). Using educative curriculum materials to support the development of prospective teachers’ knowledge. Educational Researcher, 43(3), 154–162. Ebby, C.  B. (2015). How do teachers make sense of student work for instruction? In National Council of Teachers of Mathematics Research Conference. NCTM. Ebby, C. B., & Sam, C. (2015). Understanding how math teachers make sense of student work for instruction. In Annual Meeting of the American Educational Research Association. Chicago. Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal of Research in Mathematics Education, 27(4), 403–434. Goldsmith, L.  T., & Seago, N. (2011). Using classroom artifacts to focus teachers’ noticing: Affordances and opportunities. In V. R. Jacobs, M. Sherin, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 169–187). London: Routledge. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-­ based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549. Heritage, M. (2008). Learning progressions: Supporting instruction and formative assessment. Washington, DC: Council of Chief State School Officers. Hiebert, J., Carpenter, T.  P., Fennema, E., Fuson, K., Human, P., Murray, H., et  al. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(4), 12–21. Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400. Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38(3), 258–288. Retrieved from http://homepages.math.uic. edu/~martinez/PD-EarlyAlgebra.pdf. Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. Jacobs, V. R., Lamb, L. L. C., Philipp, R. A., & Schappelle, B. P. (2011). Deciding how to respond on the basis of children’s understandings. In M. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 97–116). London: Routledge. Kamii, C., & Dominick, A. (1998). The harmful effects of algorithms in grades 1-4. In Yearbook (National Council of Teachers of Mathematics) (pp. 130–140). Reston, VA: National Council of Teachers of Mathematics. Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. Munter, C., Stein, M. K., & Smith, M. S. (2015). Dialogic and direct instruction: Two distinct models of mathematics instruction and the debate(s) surrounding them. Teachers College Record, 117(11), 1–32. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.C. Roth McDuffie, A., Foote, M. Q., Bolson, C., Turner, E. E., Aguirre, J. M., Bartell, T. G., et al. (2014). Using video analysis to support prospective K-8 teachers’ noticing of students’ multiple mathematical knowledge bases. Journal of Mathematics Teacher Education, 17(3), 245–270. https://doi.org/10.1007/s10857-013-9257-0. Schack, E. O., Fisher, M. H., & Wilhelm, J. A. (Eds.). (2017). Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks. Cham: Springer International Publishing.

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Sleep, L. (2012). The work of steering instruction toward the mathematical point: A decomposition of teaching practice. American Educational Research Journal, 49, 935–970. 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, 313–340. Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: Free Press. Stoehr, K.  J., Bartell, T., Mcduffie, A.  R., Witters, A., Drake, C., Sugimoto, A.  T., & Foote, M. Q. (2016). Early career elementary mathematics teachers’ noticing related to language and language learners. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Proceedings of the 38th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 347–355). Tucson, AZ: The University of Arizona. TERC. (2008). Investigations in Number, Data, and Space (2nd edition). Glenview, IL: Pearson Education Inc. TIMS Project. (2008). Math Trailblazers (3rd Edition). Dubuque, IA: Kendall/Hunt Publishing Company. Thomas, J.  N., Eisenhardt, S., Fisher, M.  H., Schack, E.  O., Tassell, J., & Yoder, M. (2015). Professional noticing: Developing responsive mathematics teaching. Teaching Children Mathematics, 21(5), 294–303. van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions. Journal of Technology and Teacher Education, 10, 571–596. van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education, 24, 244–276. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill.

Chapter 8

Examining Design Transparency in Elementary Mathematics Curriculum Materials Luke T. Reinke, Janine T. Remillard, and Ok-Kyeong Kim

Abstract  To support teachers in understanding the design of their curriculum program and effectively transforming the written lesson as they enact it with their students, some curriculum authors have taken up recommendations to add transparency to their curriculum materials. Our analysis characterizes the different ways the authors of five elementary mathematics curriculum programs make design decisions transparent to teachers and the frequency, regularity, and depth of this support. We found that authors provided rationale for the design of the curriculum in varied and limited ways. While four of the five programs regularly communicated the purpose of instructional activities, they seldom elaborated why these particular activities or actions were appropriate or effective for serving these purposes. The authors of three of the programs provided storyline support aimed at helping teachers understand how the curriculum developed across time, most often making explicit connections across lessons and units. The depth of these connections, however, varied considerably, and authors rarely went so far as to explain how teachers might leverage these connections during instruction. Through analysis and illustrative examples, the chapter illuminates the possibilities for future design and research.

Napthalin Atanga provided valuable insights during the analysis of the data and contributed to an earlier version of the paper presented at the Annual Meeting of the American Educational Research Association (Reinke & Atanga, 2013). L. T. Reinke University of North Carolina at Charlotte, Charlotte, NC, USA e-mail: [email protected] J. T. Remillard (*) Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] O.-K. Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_8

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Keywords  Curriculum analysis · Mathematics curriculum materials · Educative curricula · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman–Addison Wesley Mathematics · Teacher’s guide · Design transparency · Design rationale · Mathematical storyline · Curriculum sequence · Mathematical goals

8.1  Introduction Curriculum authors face an abundance of design decisions when developing curriculum programs. They decide what mathematical topics will be addressed and in what order. After deciding upon an overarching pedagogical approach, they design individual lessons, each the result of countless more decisions. Historically, the rationale behind these decisions has not been visible to teachers (Ball & Cohen, 1996). The absence of such rationale would not be concerning if curriculum materials represented a set of instructions that could simply be followed, like those contained in a recipe or describing the rules of a board game. But teaching is complex work, and in planning and enacting lessons, teachers too face an abundance of decisions. They select and adapt tasks to align with state or local standards or learning objectives. They develop a vision for transforming these tasks from the written page into an enacted lesson with their students (Remillard, 1999). Finally, once the lesson begins, they respond to students’ ideas, questions, and difficulties, keeping an “ear to the ground, listening to … students” and “eyes … focused on the mathematical horizon” (Ball, 1993, p. 376). When teacher’s guides provide only a set of instructions or recommendations, without providing insight into the rationale behind these recommendations, teachers are left to make their own decisions, and are, as Stein and Kim (2009) put it, held “hostage to a set of actions without the knowledge needed to select and adapt tasks” (p. 44). Some curriculum authors, however, have taken up recommendations to communicate directly to teachers about the design decisions, in an effort to add transparency to their curriculum materials. Stein and Kim (2009) define transparency as: …visibility of the curriculum developers’ rationales for specific instructional tasks or particular learning pathways found in the base curriculum materials. Transparent materials contain explanations for why a particular task or route through a teaching-and-learning territory was selected, including how that task or route might lead to students’ understanding of worthwhile mathematical processes and ideas. (p. 47)

When teacher’s guides include discussion of the affordances and limitations of the activities and representations described therein, teachers are better equipped to flexibly enact the curriculum in consideration of the conditions in the classroom (Ball & Cohen, 1996; Davis & Krajcik, 2005; Stein & Kim, 2009).

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In this chapter, we examine how authors of five elementary mathematics curriculum programs1 support teachers by making their design decisions transparent in their teacher’s guides. As described in Chap. 5, providing design transparency was one of the codes we applied to the set of 75 lessons in order to examine what and how curriculum authors communicated to teachers in the lesson guides. We noted all places that the guides explained or clarified the curriculum designers’ decisions about goals, lesson structure, what the lesson is intended to accomplish, or how the lesson is related to other lessons. When looking at our data quantitatively, we found modest variation in the proportion of statements devoted to design transparency, ranging from an average of 31% (MTB) to 17% (SFAW). Unlike the statements that explained the mathematics or supported teachers to anticipate student thinking, the majority of statements coded as communicating about design transparency were not also coded as directing teachers’ actions. In other words, most coded statements focused purely on communicating features of the design. (See Chap. 5 for more detail.) The quantitative analysis in Chap. 5 offered an overall picture of how much each program communicated about aspects of the design. It did not provide any detail about what these types of supports looked like in each curriculum program or how they differed across programs. The analysis discussed in this chapter hones in on the statements coded as providing design transparency and uses a combination of qualitative and quantitative approaches to examine the following questions: 1. What are the different types of support elementary mathematics curriculum authors provide to make design decisions transparent to teachers? 2. What is the frequency, regularity, and depth with which this support is provided?

8.2  Conceptualizing Design Transparency in Teacher’s Guides Our analysis of whether and how teacher’s guides make curriculum design decisions transparent to teachers is informed by existing research on mathematical knowledge for teaching (Ball, Thames, & Phelps, 2008) and the design and use of mathematics curriculum materials (Ball & Cohen, 1996; Davis & Krajcik, 2005; Remillard, 2000). Two categories of design transparency are identified as potentially meaningful for teachers: communicating rationale for design decisions and attending to the development of curriculum across time. These two categories comprise our conceptual framework for design transparency described below and are further elaborated through our analysis of the teacher’s guides.

 The five programs are Everyday Mathematics (EM), Investigations in Number, Data, and Space (INV), Math in Focus (MIF), Math Trailblazers (MTB), and Scott Foresman–Addison Wesley Mathematics (SFAW). See Chap. 1 for details about the programs. 1

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8.2.1  Mathematical Knowledge for Teaching Studies of teacher knowledge have examined the various types of knowledge that teachers draw on as they perform their work. Shulman (1986) proposed three categories of teacher knowledge: (a) subject matter content knowledge, (b) pedagogical content knowledge, and (c) curriculum knowledge. The latter involves knowledge of the “variety of instructional materials available … and the set of characteristics that serve as both the indications and contraindications for the use of particular curriculum or program materials in particular circumstances” (p. 10). Curriculum knowledge also includes horizontal curriculum knowledge, or what students are studying concurrently in other subjects, and vertical curriculum knowledge: “familiarity with the topics and issues that have been and will be taught in the same subject area during the preceding and later years in school, and the materials that embody them” (p. 10). Specific to the field of mathematics, Ball and colleagues (e.g., Ball et al., 2008) have worked to unpack and refine Shuman’s constructs into two broad categories: subject matter knowledge and pedagogical content knowledge. They identify three constructs within subject matter knowledge: common content knowledge, specialized content knowledge that is specific to teaching, and horizon content knowledge, which includes knowledge of “how mathematical topics are related over the span of mathematics included in the curriculum” (p. 403). Pedagogical content knowledge is composed of three categories. Knowledge of content and students involves understanding how students typically think about mathematical ideas and anticipating how they will respond to particular tasks. Knowledge of content and teaching includes mathematical knowledge related to instructional design including how to sequence examples and the affordances of possible representations and approaches. Finally, knowledge of content and curriculum contains the aspects identified by Shulman (1986) as curriculum knowledge. These descriptions of the knowledge needed for teaching mathematics suggest two ways curriculum authors can support teachers’ development by making their design decisions more transparent: (a) by providing rationale for specific design decisions and (b) by helping teachers understand the sequencing and connections within and across lessons and units. By communicating the rationale behind decisions influencing the design of a given activity or lesson, curriculum authors support teachers in developing their understanding of the affordances and limitations of particular mathematical representations and pedagogical approaches (knowledge of content and teaching) as well as their understanding of the types of instructional tasks that can be used to develop these representations and enact these pedagogical approaches (knowledge of content and curriculum). When they explain the rationale behind the sequencing of mathematical content and instructional tasks and point out connections within and across lessons and units, authors support teachers in developing a vision for how mathematical topics are related (common content knowledge and horizon content knowledge) and how student learning is mediated by the order in which these topics are presented (knowledge of content and teaching). As t­ eachers

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develop this knowledge, they become better equipped to effectively enact curriculum materials in their particular instructional setting. In the next two sections, we review existing literature related to these two categories of design transparency, in order to develop an analytical framework for characterizing design transparency in the materials we sampled.

8.2.2  Communicating Rationale In their recommendations for educative curricula, Ball and Cohen (1996) and Davis and Krajcik (2005) propose that providing rationale support equips teachers to effectively select and adapt particular instructional tasks during planning and, with their students, co-construct a lesson that accomplishes the intended goals. Remillard (2000) provides an illustrative example of how this type of support might be helpful for teachers. She describes her work with a fourth grade teacher who tended to invent her own tasks, rather than use the tasks suggested by the developers, and hypothesizes that, if the designers had included commentary aimed at supporting the teacher in understanding the mathematical goals underlying a task and the reasons supporting the design of the task, the teacher might have been more inclined to use the task and learn from that experience. If, after reading this commentary, the teacher ultimately opted to not use a particular task as written, the teacher would have been supported in adapting the task in such a way that preserved the essential characteristics intended by the developers. Remillard’s (2000) example suggests that by providing rationale for their design decisions, curriculum authors can support teachers in the work of steering their instruction toward the mathematical point of the lesson (Sleep, 2012). If teachers understand the mathematical objectives and the relationship between these objectives and the recommended activities, they are then equipped to make planning and in-the-moment decisions to achieve those objectives. Through our work on the ICUBiT project,2 we have identified a number of ways curriculum authors can help teachers to understand and achieve the desired objectives. In addition to articulating the learning goals, authors can provide rationale for these goals, unpack the goals (Morris, Hiebert, & Spitzer, 2009), and orient teachers to the ways in which the various instructional activities contribute to the opportunity for students to achieve those goals (Kim, 2018; Remillard, 2018). By analyzing the extent to which curriculum authors provide these four forms of support around the learning goals and teachers’ enactments of the corresponding lessons, Remillard, Reinke, and Kapoor (2019) found a positive correlation between the depth of the steering support provided for each goal and the extent to which teachers steered toward these goals during enactment.

 See Chap. 1 for details about the project.

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Rationale support also helps teachers connect their use of instructional materials to disciplinary or pedagogical principles (Davis & Krajcik, 2005). For instance, Davis et al. (2014) demonstrated how developers can help teachers connect their practice to generalizable theoretical constructs: in their case, key scientific practices. In their educative supplements, the designers (who were also the researchers) introduced the scientific practices in depth outside of the lesson flow. Then, in the individual lessons, they included what they called “why-and-how” support, where they reminded teachers of reasons why the practice was important and recommend particular actions to take within the lesson in order to develop students’ understanding of that practice. Arias, Bismack, Davis, and Palincsar (2016) found evidence of the scientific practices, and sometimes even the rationales provided for teaching these practices, in the words and actions of the teachers as they enacted the lessons with students. Empirical studies into how teachers take up and learn from rationale support, like the one described by Arias et al. (2016), are limited. This is not surprising given the absence of design transparency in many curriculum materials, particularly in the USA. Comparative studies of mathematic textbooks indicate that rationale support in U.S. curriculum materials is less consistent than countries like Japan (Watanabe, 2001) and China (Li, 2004); texts from Japan and China provide some insight into what types of rationale support might be provided. Watanabe found that a Japanese text consistently contained statements of rationale for problem settings and for individual steps of the lesson, and Li found rationale support in Chinese texts in the introduction to the book, chapter notes, in addition to extensive explanation of design rationale at the lesson level. For instance, in explaining a lesson focused on multiplying two-digit numbers, the authors of the Chinese texts write: The text first arranges two review problems. The first one is to review multiplying by a 1-digit number and multiplying by a multiple of ten. Through reviewing multiplying by a 1-digit multiplier, the review helps students recall when multiplying by a 1-digit number one needs to use the multiplier to multiply every digit of the multiplicand. This will prepare students to learn the order when multiplying by a 2-digit multiplier. (from PEP Math, Grade 3, vol. 1 (Elementary Mathematics Department, 1997) as translated by Li (2004, p. 146))

The authors explain the organization and purpose of the problems in the lesson, situating them as prerequisites for mathematics to be learned in the future. Watanabe and Li found that traditional U.S. elementary mathematics texts, designed by textbook publishers, tended not to contain rationale support, while texts aligned with NCTM Standards that were designed at academic centers with support from the National Science Foundation (NSF) contained varying degrees of rationale support. Indeed, considerable variation exists across the Standards-based programs. Stein and Kim (2009) analyzed two Standards-based curriculum programs, Everyday Mathematics (EM) and Investigations in Number, Data, and Space (INV) and found that, in their sample, 80% of the lessons from INV provided at least one statement of rationale behind the activities and mathematics included therein, compared to only 21% of EM lessons.

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To better understand the different ways authors can make their materials transparent by explaining the rationale behind design decisions, Males (2011) further delineated categories of rationale support and identified 12 categories of rationale support present in middle school mathematics textbooks: rationale for (a) particular experiences, (b) activity sequences, (c) representations, (d) tools, (e) questions, (f) approaches to justification, reasoning and proof, (g) terminology, (h) participation structures, (i) curriculum philosophy, (j) curriculum features, (k) curriculum storyline, or (1) particular goals. Analyzing three Standards-based texts and one publisher designed text, she found that only 6% of the support provided in middle school math curricula was rationale guidance, and noted how minimal this was compared to 20% found in science curriculum (Beyer, Delgado, Davis, & Krajcik, 2009). She also found that across the curriculum programs, the most frequently included categories of rationale support were rationale guidance for experiences, representations, tools, questions, philosophy, and storyline. From these studies of the rationale support provided to teachers, we identified four primary dimensions through which authors can support teachers by providing rationale: (1) providing rationale for overarching curriculum design approaches; (2) connecting to disciplinary or pedagogical philosophy principles; (3) identifying, unpacking, and providing rationale behind the mathematical goals of particular lessons or sets of lessons; and (4) explaining the design of particular tasks or activities, including the benefits of particular representations, tools, teacher moves, and/or participation structures.

8.2.3  Attending to the Development of Curriculum Over Time In addition to providing support in the form of rationale for the design of the curriculum, authors can also support teachers in attending to the development of mathematical ideas over time and make connections across time (Ball & Cohen, 1996; Davis & Krajcik, 2005; Remillard, 2000). In order to effectively communicate a mathematical storyline, teachers must understand how the curriculum develops over activities in a lesson or unit (Remillard, 2000), across units, within a year (Ball & Cohen, 1996; Davis & Krajcik, 2005), and across grades (Shulman, 1986). Support across all of these dimensions has the potential to develop teachers’ knowledge of teaching and curriculum (Ball et al., 2008) as well as their horizon content knowledge. Understanding how mathematical ideas develop in a specific curriculum is one component of what Drake and Sherin (2008) call curriculum vision, or a vision of “where the curriculum was going mathematically” (p. 325). We know very little about how mathematics teacher’s guides provide support for developing teachers’ understanding of how the curriculum develops over time. Although Males (2011) coded middle school mathematics teacher’s guides for guidance addressing the “curricular storyline,” meaning connections between “previous and future mathematics content, including within and across disciplines, courses,

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units, sections and lessons” (p. 44), the only finding reported is that rationale support for storyline was one of the more frequent types of rationale support. No research could be found describing the extent to which elementary mathematics textbooks in the USA provide storyline support.

8.3  Analytical Framework The analysis described in this chapter is based on an assumption that, by making the design of curriculum materials transparent, authors can support teachers in developing multiple knowledge domains. This knowledge, in turn, equips teachers to effectively use and adapt the materials to respond to their specific context and students in order to achieve the intended learning goals. By synthesizing the literature on teacher knowledge and the design and use of educative curriculum materials, we developed an analytical framework for characterizing the design transparency of curriculum materials, shown in Fig. 8.1. We have identified two primary categories of design transparency: rationale support, which explains the design of particular lessons and the activities within these lessons; and storyline support, which is aimed at helping teachers to enact instruction that is coherent within and across lessons, units, and grades. To address the significant gaps in our understanding of how authors of elementary mathematics textbooks used in the USA support teachers by providing design transparency, we used this analytical framework to conduct a systematic analysis.

Design Transparency Rationale Support • Identifying, unpacking and providing rationale for the mathematical goals • Connecting to disciplinary or pedagogical philosophy principles • Explaining design of the task • Explaining the benefits of particular representations, teacher moves, participation structures

Storyline Support • Explaining the sequencing and connections across tasks in a lesson • Explaining the sequencing and connections within and across units • Explaining the sequencing and connections within and across grades

Fig. 8.1  Analytical framework of aspects of curriculum design transparency hypothesized to support teacher learning and decision-making

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8.4  Methods As described in Chap. 5, we analyzed a set of lessons randomly selected from teacher’s guides of five curriculum programs: Everyday Mathematics (EM) (University of Chicago School Mathematics Project, 2008), Investigations in Number, Data, and Space (INV) (TERC 2008), Math in Focus (MIF) (Marshall Cavendish International 2010), Math Trailblazers (MTB) (TIMS Project, 2008), and Scott Foresman–Addison Wesley Mathematics (SFAW) (Charles et al., 2008). We used the same lessons that were used in Chaps. 5, 6, and 7 of this volume: 5 randomly selected lessons per grade (grades 3–5) for a total of 15 lessons per program and 75 lessons in total. Because we were primarily interested in how curriculum authors made design elements and decisions transparent to teachers, we focused our analysis on content in the guidance provided for each lesson, which we call the lesson guides. We did not analyze the student text, even if the student text was pictured in the guide. Our analysis focused only on the lesson guides, as opposed to the information provided in other parts of the teacher’s guide. This approach does not provide a full picture of the support provided by the curriculum authors. But, in light of evidence suggesting that teachers may not pay close attention to support that does not address a particular lesson (Beyer & Davis, 2009; Schneider & Krajcik, 2002), our approach focuses the analysis on the portion of the teacher’s guides that teachers are most likely to interact with on a daily basis.

8.4.1  Unit of Analysis and Coding Our data set was composed of all individual sentences and images from the set of 75 lesson guides coded as Design Transparency (n = 2284). These were sentences that we inferred in the coding process were intended “to help teachers understand the purpose or rationale behind the lesson and to make sound decisions while teaching the lesson that are aligned with the intent of the curriculum designers” as well as “explain the goals, topic selection, or organization of lessons in the curriculum” (ICUBiT primary coding manual). As with the primary coding described in Chap. 5, our unit of analysis was the sentence as it appeared in the teacher’s guide. To further investigate the different types of rationale and storyline support provided by the curriculum authors, two coders began to examine and assign each sentence a secondary code, using Atlas.ti. These codes corresponded to the aspects of design transparency identified in Fig. 8.1. When a clear fit to one of these categories was not evident, we identified a new, emergent category. The resulting 14 categories were clarified and revised through discussions of the 2 coders as they coded sentences from 25 lessons. The lead author then used this stabilized coding system to code the sentences from the remaining 50 lessons. After sentences had been coded, the lead author validated each category by checking to make sure each sentence fit the code description.

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Table 8.1  Design transparency codes Codes Category: rationale Objectives Rationale for lesson objectives Purpose of an activity Purpose of an action Rationale for universal design decisions Category: connections Connection between activities in one lesson Connections to other lessons within a unit Connections to other lessons within a particular grade Connections to material in other grades Category: other transparency Explain pedagogical approach Time Participation structure Explain curriculum specific materials Explaining the use of a supplemental curriculum component

Definition Identifies lesson objectives Explains the significance of the objectives for students’ futures, mathematical or otherwise Explains the purpose for a given activity Explains the purpose for a specific action Provides rationale for particular design decisions that span across lessons Explicitly indicates connections between separate activities or segments within a lesson Explicitly identifies connections between lessons in the unit Explicitly identifies connections between lessons in a different unit but the same grade Explicitly identifies connections between lessons or units in different grades Explains the perspectives of the authors related to a particular aspect of instruction without providing rationale Suggests time to complete an activity Indicates group size: Individual, partner, small group, and whole class Explains supplemental materials that are specific to some curriculum materials (e.g., Fact Triangles for EM) Explains the role of particular components. For example, for assessment or to organize student data

The final 14 codes are defined in Table 8.1. For the purposes of organization, the secondary codes are consolidated into three broad categories describing their relationship with the constructs identified from the literature: statements through which designers communicate purpose or rationale for the curriculum design, statements that describe authors’ insights into the storyline or connections between various parts of the curriculum, and other statements that make the design of the curriculum transparent but do not fall into the other two categories.

8.4.2  Analysis We began the analysis by determining the frequency with which the designers included each type of rationale support and the percentage of lessons that contained each type of support across the data set. We then examined the instances within each

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category qualitatively to characterize the depth of the support provided and make comparisons among the five programs. In this step of the analysis, we expanded our focus to include sentences surrounding the particular coded sentence, in order to understand the sentences in the context that they appeared in the lesson guides.

8.5  Results In this section, we describe the different ways that curriculum developers make the design of the curriculum transparent to teachers. We begin by examining the support aimed at helping teachers understand the purpose and rationale of the various recommendations in the teacher’s guides. After describing the frequency and distribution of codes, we then characterize the variation across the sentences containing the same code in terms of the depth of support. We then examine the various ways curriculum authors provide support for understanding the mathematical storyline within and across lessons, units, and grades. Finally, we discuss other forms of design transparency that emerged through our analysis.

8.5.1  Communicating Rationale We began by characterizing the  frequency and regularity with which the lesson guides provided each of the five different categories of rationale support: (a) identifying the mathematical objectives or goals for a lesson, (b) providing rationale for those objectives, (c) explaining the purpose of an instructional activity or a particular aspect of that activity (representations or tools used), (d) explaining the purpose of a recommended teacher or student action, and (e) providing rationale for a universal design decision. Figure  8.2 provides the raw frequency for each category, meaning the number of sentences that contained that type of support over the sample of 75 lessons. Several patterns stand out. The three most commonly used types of support across all five programs were: describing the purpose of an activity, identifying lesson objectives, and explaining the purpose of specific actions. Describing the purpose of an activity was most frequently used in all except SFAW. Specific rationales for universal design decisions or for lesson objectives were provided much less frequently. To identify whether the sentences containing these types of support were clustered into only a few lessons or spread throughout the 15 lesson samples of each program, we calculated the percentage of lessons that feature at least one sentence containing support aligned to the five rationale categories (Fig. 8.3). Each of the lessons analyzed stated at least one mathematical objective, however, rationale for these objectives was rarely provided, or never in the case of MIF and INV. Almost all lessons across the dataset contained at least one sentence explaining the purpose

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EM n=296

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Fig. 8.3  Percentage of lessons from each program that contained each type of rationale support

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of a given activity, and this form of support received the most attention from curriculum authors. About half of the INV and SFAW lessons contained an explanation of the purpose for specific teacher or student actions; the percentage of lessons containing this support was higher in the other three curricula; MIF stands out with this type of support in over 85% of lessons. Finally, a majority of lessons did not contain any rationale for more universal design decisions with the distribution ranging from about a third of lessons (in EM and INV) to none of the lessons (in MIF). In the following sections, we describe how each type of support was provided across the five programs. Lesson Objectives and Goals Each of the lessons analyzed provided some statement or set of statements that described the skill or concept students should be learning. These goals were always provided at the beginning of the lesson. The programs varied, however, in the terms used to identify these objectives. For example, SFAW and MIF identified Lesson Objectives, MTB used the term Key Content, and INV listed the Math Focus Points for each lesson. EM, as described below, offered two types of objectives or goals for each lesson. Each EM lesson guide began with an Objective. These objectives were not phrased as student performance outcomes, rather, they explained the purpose of the lesson from a curriculum design perspective. For example, the objective from EM GR4 2.1, was “To review fact families and number families; and to review the inverse relationship between addition and subtraction” (p.  100). These objectives frequently began with the words “To review…” or “To introduce…” Following these objectives, the authors list Key Concepts and Skills, which identified student performance goals, as in the case of EM GR5 12.1: “Identify the prime factorization for a number” (p. 914). EM was designed as a spiraling curriculum, meaning that particular skills and concepts developed over time and were touched on periodically. A given concept may be introduced in one lesson, elaborated on in another lesson that may or may not directly follow, and reinforced throughout a number of subsequent lessons. The “Objective” appears to be written to help teachers understand the role the lesson plays in this sequence. Notably, none of the five curriculum programs consistently included with their objectives an indication of the level of mastery students should be expected to attain by the end of the particular lesson; at least this information was not provided in the lesson guide itself. The authors of EM, for instance, could hypothetically note that students should not be expected to demonstrate fluency after a lesson in which a particular idea is introduced. Authors of curriculum that are not intended to spiral could point out instances where particular concepts or skills will be reinforced in later lessons or necessary as a prerequisite before introducing a new idea.

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Rationale for Chosen Objectives and Goals Although each teacher’s guide provided objectives for each lesson analyzed, authors rarely (or never in the case of MIF and INV) provided explanations for why these mathematical objectives or goals are important for students to learn, either in service of future mathematical development or life beyond mathematics. The few instances that we observed provide the opportunity to consider the value such support could add. An example is found in EM GR5 2.4: You can expect that many students will be able to solve simple addition and subtraction number stories without first having to write an open number sentence. The main purpose of this lesson is to introduce the use of open number sentences so students can use them to solve more complex problems. (p. 100, italics added to identify the sentence coded as “rationale for objectives”)

Here the developers explained why the lesson focuses on introducing number sentences by describing the future utility. This statement provides a guide for teachers who otherwise may be tempted to skip the lesson, because they anticipate that their students could solve the number stories without writing open number sentences or to those who would teach the lesson but de-emphasize or omit the writing of number sentences. Another example from SFAW attempts to explain the usefulness of the objective outside mathematics: Learning and applying problem solving skills need not be limited to mathematics. The basic principles-understand the problem, make a plan and implement it, and look back and check-­ can be applied to a wide range of ‘problems’ that arise in the classroom. (SFAW GR3 1.14, p. 42A)

Statements like these that provide rationale for the lesson objectives have the potential to support the teacher in motivating students and answer questions students commonly ask, such as “why do we learn this?” “what good is this?” and “where am I going to use this?” In this sample of curriculum materials, questions such as these are largely left for the teachers to navigate on their own. Purpose of an Activity The most prevalent form of rationale support across the data set (in terms of frequency) was descriptions of the purpose of instructional activities included in the lessons (Fig.  8.2). We found instances of this type of explanation in 87% of the SFAW lessons and 100% of the EM, INV, MIF, and MTB lessons (Fig. 8.3). That said, looking across the sentences in this category, we found that curriculum authors described the purpose for particular activities in a number of different ways. One way was through statements summarizing the mathematical action students perform in the activity; these statements identify the objective for the particular activity, which in some cases could clarify which of multiple lesson objectives were addressed by a particular activity. For example, anytime the authors of INV ­identified

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Fig. 8.4  An illustration of INV authors clarifying the purpose of a particular discussion from INV. From Investigations 2008 Curriculum Unit (Grade 3, Unit 1, p. 37) © 2008 by Pearson K12 Learning LLC, or its affiliates. Used by permission. All Rights Reserved

that a particular portion of a lesson would be devoted to a discussion, the authors noted which of the Math Focus Points for the lesson should be targeted in that particular discussion. This approach is illustrated in the sentence in Fig. 8.4, from INV GR3 1.1.2. At the outset of the support for the discussion, the authors identified the Math Focus Points for Discussion are “Recognizing and representing the place value of each digit in 2- and 3-digit numbers” and “Adding and subtracting multiples of 10,” which match two of the three objectives for the overall lesson. By identifying these focal points, teachers are supported in steering the discussion toward the intended ideas. In EM and MTB, statements explaining the objective of particular activities tended to appear in the lesson overview, a one- or two-page summary of the lesson. EM developers also included a one-sentence description that explained the purpose of each of the supplementary activities, which were provided to aid the teacher in differentiating for those who are less mathematically prepared for the lesson (Readiness), those who would benefit from enrichment (Enrichment), and those who are English Language Learners (ELL Support): for example, under the Readiness heading from a third grade lesson in EM: “To explore patterns in skip counting, use Math Masters, p. 22, and have children skip count by 2s and 5s” (EM GR3 1.8). In MIF, most of the activities in the main body of the lesson included a one- or two-sentence description identifying the objective of the activity. For instance, in a Grade 3 lesson shown in Fig. 8.5, with the objective “Round numbers to estimate sums and differences,” the first sentence under the activity titled “Round a 4-Digit number to the Nearest Hundred” states that “Number lines are used to estimate the value of a number to the nearest hundred” (p. 54). The frequencies for this type of support were greatest in MIF and MTB because the authors routinely went beyond stating the purpose of a particular instructional segment or activity and identified the purpose of specific exercises or groups of exercises within the segment or activity. For example, from the MIF lesson described above, the designers identified the purpose of exercise 5 in the Guided Practice seg-

242 Fig. 8.5  An illustration of the way MIF authors clarified the purpose of a particular lesson segment. This illustration was created by the authors based on an image that appears in MIF GR3 2.4, p. 54

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Round a 4-Digit Number to the Nearest Hundred Number lines are used to estimate the value of a number to the nearest hundred. · Draw the number line in the Student Book on the board, showing 2,300, 2,350, and 2,400. · Ask volunteers to locate 2,329 and 2,382 on the number line. Use additional markings at intervals of 10, if necessary.

ment: “This exercise provides practice with rounding 4-digit numbers to the nearest hundred,” before explaining that exercises 6–9 “reinforce students’ ability to round 4-digit numbers to the nearest hundred and mark each number on the number line” (MIF GR3 2.4, p. 55). This type of specific purpose statement can support teachers as they make decisions about which exercises to assign, especially given limitations of time. Looking across the statements coded as communicating the purpose of an activity, we found that some included an additional layer of rationale: a description of the role the activity played in the learning process. For example, in the first activity in EM GR3 7.6, the authors explain that “Children develop strategies for multiplying and dividing using 1-digit numbers and multiples of 10, 100 and 1000” (p. 606). Then in the second activity, labeled Ongoing Learning & Practice, “Children practice basic multiplication facts by playing Beat the Calculator” (p. 606). By using the verbs “develop” and “practice,” the designers appeared to be communicating the role of the two activities within the pedagogical storyline. As shown in Fig. 8.2, SFAW was unique among the programs analyzed in that it contained very little support identifying the purpose of particular activities. In fact, of the excepts coded in SFAW as communicating the purpose for activities, all but one were simply lists of mathematical topics covered in the section titled Spiral Review as in SFAW GR 5 11.7, “Topics reviewed: Scale drawings, Problem-Solving Strategy: Make a Table” (664A). We also found that the lesson guides rarely supported teachers in communicating the purpose or rationale of a given activity to students; this occurred in only four instances in the entire data set (two instances from EM, one instance from INV and one instance from MTB). An example is provided in EM GR4 3.2: “Tell the class they will use Fact Triangles to practice multiplication facts and another tool, the Multiplication/Division Facts Table, to discuss the terms factor, product and multiple.” Support like this suggests to teachers that students will benefit from a clear articulation of the goals of a given activity. Finally, across all instances of this type of support, the following quotation stands out as an exemplar of the potential, because it provides deep insight into the design of an activity: It is expected that most fifth graders are able to solve division problems without drawing a representation first. However, in this session students are asked to use a story context and representation as a reminder that these are useful tools. Students may use these tools when

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they encounter a problem they are not sure how to solve, to help them keep track of their partial solutions, and to help them know when they are done solving the problem. (INV GR 5 7.3.1, p. 69, italics added)

This statement, explaining why the activity includes a context and a visual representation, could help a teacher explain to students why they are being asked to perform these steps and would also provide a learning opportunity for a teacher who might want to omit or adapt the task. Overall, in four of the five curriculum programs, the authors consistently provided rationale for activities by describing the mathematical goal the activity served. These explanations of purpose for recommended activities could assist teachers as they select from, enact, and possibly supplement the various offerings present in the lesson guides. It is worth noting, however, that we rarely observed statements explaining why an activity was designed in a particular way or explicit support for communicating the purpose of an activity to students. Purpose of an Action We also observed support designed to clarify why a particular action, often a pedagogical move on the part of the teacher, might be valuable. These instances were modest in quantity, never averaging more than two instances per lesson in any of the programs, but they were fairly well distributed across lessons, showing up in almost two-thirds of those sampled. As shown in Fig. 8.3, this type of support occurred in about half of the lessons in EM, INV, and SFAW and slightly more consistently in MIF and MTB. In EM, we observed only 17 sentences in 9 lessons, and the rationale provided in 6 of these 17 was simply “To support English Language learners…” Similarly, this form of support was seldom provided in INV, with 20 instances spread over 7 lessons. We found only nine sentences in the SFAW lessons that provided purposes for recommended actions. The low frequency of this type of support in the SFAW lessons is not surprising, given that there are far fewer sentences that direct teacher actions in this curriculum program. The teacher’s guides of the other two programs contained more examples of this type of support, although the frequencies are not drastically different: MTB contained 25 instances over 11 lessons, and MIF was most evenly distributed with 26 examples, with at least one example occurring in 13 of the 15 lessons. Eight of these examples were found under a section titled Best Practices, through which the designers recommended a pedagogical move and provided the purpose behind it. The depth of the justifications provided across this type of support varied considerably. On the most surface level, sentences were coded as containing “Purpose of an action” when they contained a pragmatic explanation of a particular action, such as “For more durable flash cards, copy the Triangle Flash Cards onto card stock or laminate them” (MTB GR3 3.1). We applied this code even to sentences that simply identify which students a particular pedagogical action is intended to help: For example, “To support English language learners, write prime numbers and composite numbers along with the definition…” (EM GR5 12.1).

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In some cases, though, the statements coded as “Purpose of an action” provide a deeper level of rationale. Some statements explained the potential impact of a teacher move on student learning, for instance, “Once a child has shared a strategy, encourage other children to try it. This validates the child who suggested the strategy and offers the rest of the children the opportunity to expand their repertoire of strategies” (EM GR3 9.2, italics added). Other instances clarify the role of the move in the teaching process. One example is from a third grade lesson from SFAW; the authors state “to check if students understand the problem, ask: what are the three parts you need to include in your answer?” (SFAW GR3 1.14, p. 43). Both of these examples provide valuable information to the teacher as she decides whether to follow the suggestion of the authors or not. Of particular theoretical significance are those examples that provide glimpses into generalizable practices, or practices that are not specific to a given lesson, because these messages may contribute to the development of teachers’ knowledge differently than those that are lesson specific. The practice described in the EM example in the preceding paragraph could apply to any lesson, and the SFAW example reminds teachers of the importance of checking for student understanding. Other examples signal the generalizability of the practice by using wording that transcends a given lesson, as is the case in the sentence in Fig. 8.6 from MIF, labeled as part of a series of suggested Best Practices. Rationale for Universal Design Decisions Sentences coded as rationale for universal design decisions provided reasons for particular aspects of the design that were focused on more overarching aspects or features of the program, rather than a particular lesson, activity, or action. A number of the instances explained the affordances of particular representations used in the curriculum programs. For instance, the authors of MTB provide rationale for the use of “fact families” and EM explains the benefits of “Fact Triangles” in the following sentences: Fact families are introduced so students can use multiplication facts to learn related division facts. (MTB GR4 3.1) As an alternative to traditional flash cards, Fact Triangles are ideal for children who need practice with basic facts. Using the cards reminds children that one way to do subtraction is to ask, “How much do I add?” (EM GR3 2.1)

Best Practices

You may want to have student pairs share their answers with the class. Student-to-student communication is often helpful in identifying misconceptions.

Fig. 8.6  A general pedagogical recommendation about student-to-student communication, labeled as a Best Practice from MIF GR4. This illustration was created by the authors based on an image that appears in MIF GR4 1.2, p. 16

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The EM example is particularly notable as an instance where the authors explain the strengths of the chosen approach that go beyond what traditional flash cards offer. This type of support generalizes to multiple lessons and has the potential to foster teachers’ awareness of the affordances of particular representations. Other commentary was even more global, describing the logic behind higher level design principles. The INV authors, for instance, describe the use of models to represent operations: Using models to represent the actions of the operations [for example, removing or crossing out for subtraction] not only supports computation but also provides tools for reasoning about the generalizations that underlie the computation. (INV GR3 1.1.2)

This commentary could support teachers who would otherwise have considered omitting or deemphasizing the use of models in their instruction of the importance of these models. Summarizing Rationale Support Overall, the lesson guides in this sample regularly provided explanations of the purpose of a lesson by communicating the lesson objectives, but they rarely provided rationale for why these objectives are important. Similarly, the guides contain explanations of the purpose of activities, but rarely provided rationale for the ways in which particular activities were designed. Finally, curriculum authors occasionally provided support for the teacher and student actions recommended in the lesson guides. These patterns suggest efforts to make aspects of the design of the lesson transparent to teachers but more limited attempts to make the decisions and rationales behind these decisions transparent. Furthermore, across all of the statements coded under the rationale category, we found very few references to research or other professional literature. SFAW was the only program to provide research citations as evidence for particular design decisions, which occurred twice in our sample. It is important to point out that this particular analysis focused solely on the lessons themselves and did not include material elsewhere in the teacher’s guide or in supplemental documents. We are aware that authors did include some explanation of design principles that span lessons, units, or the entire program in other locations outside the lesson guides. That said, if teachers read only the lesson level support, they are provided little support for understanding why lessons and activities were designed the way they were.

8.5.2  Storyline Support In this section, we describe the ways curriculum authors support teachers in understanding how the designed curriculum develops a mathematical storyline (Heck, Chval, Weiss, & Ziebarth, 2012) over time by identifying connections between vari-

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ous points in the curricular story within and across lessons, units, and grades. Figures 8.7 and 8.8 provide the frequency and consistency with which the lesson guides made these different types of connections. Overall, we found substantial variation in frequency and consistency of storyline support across the five programs and types of connections made. The figures show that lessons from MTB, INV, and EM (the three NCTM-aligned programs in the sample) contained significantly more storyline supports overall when compared to SFAW and MIF. INV and EM authors frequently identified connections across lessons within a given unit, and MTB and EM authors frequently identified connections across units within a given grade. We discuss the treatment of each type of connection in the following sections.

EM n=52

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Connections Within Lessons Curriculum authors can help teachers understand how the designed curriculum unfolds over a very short time period by describing the connections between activities with in particular lessons. In our sample, we observed few instances where developers explicitly made connections between two different activities in the same lesson. These connections occurred in roughly one-third of the lessons and of those lessons, roughly once per lesson. The frequency of this type of support was highest in INV and MTB. As described in Chap. 3, INV lessons were often composed of multiple activities and discussions; the sentences coded as connections within lessons support the teacher in understanding the relationship between the activities, and in some cases, help the teacher communicate that information to students. An example of the latter type of support comes from INV GR5 1.3.2: We’re going to count again by 21s, but now when you say a number, I will write it on this strip of paper, which we’re going to call a multiple tower. Later we’ll use this list of multiples to help solve multiplication and division problems. (p. 122)

In both MIF and MTB, where developers typically provide commentary at the level of individual questions or exercises, they sometimes explain how these questions or exercises are linked. For example, “Students further investigate, in Question 6, the relationship between square numbers and odd numbers that appears in the patterns of Questions 3–5” (MTB GR5 11.3, p. 49). This sort of support is aimed at helping teachers understand the design and sequence of the activity, and could support teachers in understanding the purpose of particular questions that they would otherwise omit. Similarly, if a teacher were adapting the task, she could do so in a way that preserved the development of the idea as intended by the designers. The infrequent use of within lesson mathematical connections might be a result of our coding decisions. We only coded sentences where these communications were explicit. This approach omitted connections in headings. As a result, a common feature in EM, signaled by a heading, was not coded, even though it referenced another activity within the lesson. Each Teaching the Lesson segment opens with an activity labeled Math Message Follow-Up; this section implicitly links students’ work on the Math Message (a warm-up activity meant to take place prior to the mathematics lesson) to the main activities of the lesson. It is also possible that authors believed many connections between lesson activities would be obvious to teachers and chose to not make explicit connections between them. For instance, when a lesson featured two activities, such as Addition with Base-Ten Pieces and Subtraction with Base-Ten Pieces, the notion that these two activities are linked by the use of the same representation may  have been assumed to be obvious. Connections across lessons and units within a single grade and across grades are likely to be less obvious to teacher. We discuss how these connections were made in the following sections.

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Connections to Other Lessons Within a Unit The authors made explicit connections to mathematics in other lessons within a unit far more frequently than within lessons in our sample. We found the highest frequency of within-unit support in the INV guides where we observed 32 instances across 80% of the lessons analyzed. This frequency might be expected given that each INV unit is organized into Investigations, and each of these Investigations is comprised of several sessions (or lessons). The authors consistently made connections between these sessions by identifying when particular instructional activities carry over from one session to another. For example, In this Math Workshop, students play the new game Capture from 300 to 600. They also continue working on the paper clip problems from the previous session. Students will discuss Problem 6 at the beginning of the next session. (INV GR3 8.1.2)

The authors of INV also provided support for helping students see connections between lessons within a unit. In an example from Grade 4, the authors provide a sample script of what the teacher might say to students: Today you will compare the ways two of your plants have been growing by putting the growth of both plants on the same piece of grid paper, using the same set of axes—just as we did with the Penny Jar Comparisons. (GR4 9.3.2, p. 115)

The lessons from EM contained the next highest frequency of explicit reference to other lessons within the unit, compared with the other curriculum programs: 23 instances and at least one instance in each lesson. Fourteen of these occurrences came in the mixed review section, where references were made to other lessons with “paired” content: “Math Boxes in this lesson are paired with Math Boxes in Lesson 2-3” GR 4.2.1, p. 87.” Statements like these help teachers to understand the spiraling nature of the EM curriculum; however, it is unclear from the lesson support how this information could be used to guide instruction. One feature that is notable from EM is a repeated subheading called Link to the Future, which provides transparency about how the curriculum develops over time. The following is an illustrative example of both storyline and rationale transparency: Because children’s mental math procedures often anticipate more formal algorithms, it is important to provide them with opportunities to devise their own strategies before presenting formal algorithms. Formal multiplication algorithms will be presented in Lessons 9-4, 9-9, 9-11, and 9-12. (EM GR 3.9.2, p. 71, italics added)

The first sentence provides rationale for the design of the curriculum, in particular, the inclusion of tasks aimed at prompting students to devise their own strategies prior to the introduction of conventional algorithms in the instructional sequence. The second sentence, which was coded as within unit storyline support provides transparency as to where in the program the instructional sequence is continued. This type of support has the potential to help teachers anticipate where the program is going mathematically and is unique in that the designers chose to explicitly point out the type of support through the heading Link to the Future.

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Fig. 8.9  An example of connections across units from MTB (GR3 6.3, p. 41). Excerpt from Math Trailblazers, Grade 3, Unit 6 Resource Guide by TIMS.  Copyright © 2008 by Kendall Hunt Publishing Company. Reprinted by permission

Connections to Other Units Curriculum authors sometimes included explicit connections to other units, although this practice varied from program to program. In some cases, the approach taken overlapped with connections made across lessons in a unit. By explicitly identifying connections between units in the program, authors can help teachers see how mathematical ideas are connected and build on one another over time. In the lessons sampled, MTB and EM consistently contain connections to other units, in almost every lesson sampled (Fig. 8.9). MTB lessons contained this type of support 39 times over 13 lessons. Like the Link to the Future component in EM described above, the MTB guide contained a Curriculum Sequence note in 8 of the 15 lessons in our sample. This component identified prior related experiences and future opportunities that reinforce the focal concept. For example, Fig. 8.9 shows a sentence from MTB GR3 6.3, which introduces a standard algorithm for adding two-digit numbers. In the sentence, the authors explain that the lesson builds from the work students did with base-ten pieces in a previous unit and will be reinforced through practice in later units. This support helps teachers understand how the mathematical ideas are developed and reinforced over time. It is notable that  this  Curriculum Sequence note occurred in the introduction to the lesson, and support for how to use this information to inform instruction is not provided. The EM guides contained the next highest frequency of support that made connections across units. Fourteen of the 16 instances occurred in the Mixed Review section. In these instances, authors used form language to indicate that a particular problem was previewing the content in the next unit, for instance, “The skill in Problem 6 previews unit 2 content” (EM GR3 1.8, p. 54). Interestingly, we did not find many instances in which teachers were supported to help their students identify mathematical connections across units. An exemplar occurred in MTB GR4 15.1, in which the authors state: “To answer Question 14, stu-

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dents can compare their plant growth graphs with some of the other graphs they drew this year: Bouncing Ball (Unit 5 Lesson 4), the rice on a checkerboard graph (Unit 6 Lesson 2), and Downhill Racer (Unit 10 Lesson 4)” (p. 30). It is notable that this type of support was observed in only two sentences in our sample, the example from MTB above and another from INV, which advised the teacher to “Remind students of the work they did with division in Unit 1, Number Puzzles and Multiple Towers, and other division practice they have had throughout the year” (INV GR5 7.3.1, p. 69). Connections to Other Grades Curriculum authors can also provide support aimed at helping teachers identify connections to mathematics students may learn in other grades. Across the data set, roughly one-third of the lessons (23/75) contained links to content in other grades. As shown in Figs. 8.8 and 8.9, MTB lessons include 15 instances over 6 of the lessons and the authors of EM made these connections 10 times over 9 of the 15 lessons sampled. These connections were even more rare in INV, MIF, and SFAW. In MTB, nine of the connections to material in other grades were found in the introduction of the lesson within the Curriculum Sequence section described above, which describes prior and future experiences. An example from a grade five guide explains the previous experiences students who have used the curriculum should have: Students using Math Trailblazers had many experiences with area in Grades K-4. In third grade, they found the area of figures with straight sides and estimated the area of figures with curved sides by counting square centimeters (see Grade 3 Unit 5). In fourth grade, students found the area and perimeter of shapes built with square-inch tiles (see Grade 4 Unit 2). (MTB GR5 4.1, p. 33)

As in most of the cases in the sample, the teacher is not provided guidance on how to incorporate this information into instruction. Since connections are simply stated, a great deal of design capacity is still expected of teachers. Teachers still have to understand the connections and determine their significance for a given lesson. An exception to this trend comes from EM: For Problem 4, have students provide additional reference points by locating the years halfway between the given years. Encourage students to think in terms of the Frames-and-­ Arrows problems they have solved since First Grade Everyday Mathematics to find the additional dates. (EM GR4 2.1, p. 86, italics added)

Statements like these cue teachers to remind students or prior experiences but leave the specifics of what to say to these students (and those who may not have experienced EM in the earlier grade) up to the teachers’ discretion. Summarizing Storyline Support Support for helping teachers see connections across activities, lessons, units, and grades varied considerably across the curriculum programs. Explicit connections between activities in a single lesson were rare. The INV guides contained the most

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statements explicitly identifying connections across lessons (or sessions as they are called in INV), and this makes sense given that multiple sessions make up a single investigation. Connections across units and grades were most consistent in MTB, likely because of the repeated section called Curriculum Sequence. Looking across each of these types of storyline support, seldom did these statements provide suggestions to the teacher for how these connections might be useful or leveraged when planning or during instruction.

8.5.3  Other Design Transparency Not all of the supports that provide transparency about the design of the curriculum include a statement of purpose, rationale, or provide storyline support. In this section, we describe the other categories of transparency support organized into two categories: (a) explanations of curriculum-specific elements and (b) recommended instructional structures. These categories of transparency are aimed at providing teachers with insight into the authors’ intentions and perspectives, as well as general information about key design components. Explanations of Curriculum-Specific Elements Our coding process surfaced three types of curriculum-specific elements that authors periodically explained to teachers: pedagogical positions, curriculum-­ specific materials, and use of supplemental components of the program. The fre-

EM

INV

MIF

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Identify a pedagogical position Explain curriculum specific materials Explain use of supplemental component 0

5

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Fig. 8.10  Number of sentences that contained explanations of curriculum-specific elements per program

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quency of each type of curriculum-specific element is displayed in Fig. 8.10. As can be seen in the figure, statements of these three types were very infrequent, typically accounting for less than 10 sentences across the 15 lessons sampled from each curriculum, with a few exceptions: the identification of pedagogical positions and use of supplemental components in MTB and explanations of curriculum-specific materials in EM. Statements about the pedagogical position of the program provided the authors’ perspective about particular decisions without providing any rationale. Although these statements did not fit into the other categories, they still provide some insight into the authors’ intentions. The following instance from EM provides an example: Children should not be required to learn this [the lattice] method [of multiplication], but they should try it. If they like it, they should not be discouraged from using it (EM GR3 9.9, p. 762). This statement explains the designers’ perspective on the role lattice multiplication plays in the program, as something to expose students to but not foundational to future learning, but the guide provides no rationale for this perspective. An example from MTB provides similar commentary: The Many Ways to Multiply section in the Student Guide points out that there are many different correct methods for multiplying numbers using pencil and paper. All methods are based on the idea of breaking up the multiplication problem into groups and computing the groups individually, although this is not always so obvious. You may find that some students will come into your class having learned a different algorithm. If they have a method that works, that’s fine. They should continue using their method if it is efficient. (MTB GR5 9.3, p. 68, italics added)

These statements are potentially valuable to teachers, because they can help teachers understand the intentions behind methods that are introduced in the curriculum programs. A second type of explanation we found involved authors explaining the use or purpose of materials specific to the program. We found only 14 occurrences of this type of explanation. Ten of these were contained in EM, three in MTB, and one in INV. The following example, from EM, explains the use of fact triangles: Fact Triangles were used in Second Grade Everyday Mathematics to help children memorize addition and subtraction facts. To generate addition facts, cover the sum (marked with a dot). To generate subtraction facts, cover either of the other numbers. (EM GR3 2.1, p. 100)

A similar statement in MTB explained the triangular flash cards specific to that program. The remainder of statements referring to curriculum-specific elements referred to supplemental materials in the program and explained their use. Twenty-five of these statements came from MTB and many of these were routine statements that were repeated in every lesson. For instance, 12 of them took the form of a repeated sentence in MTB: “Classrooms whose pacing differs significantly from the suggested pacing of the units should use the Math Facts Calendar in Section 4 of the Facts Resource Guide to ensure students receive the complete math facts program” (MTB GR4 3.1, p. 26). These statements could help the teacher understand the curriculum support provided for differentiated pacing and enhance the transparency of

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the program. Another five statements in MTB recommended that teachers “Use the Observational Assessment Record to” note student data related to a particular skill. An example from EM recommends that teachers consult a supplemental resource for more information about classroom routines: “See the Teacher’s Reference Manual for ideas on establishing a game routine in your classroom” (EM GR3 2.1, p. 103). Overall, these statements support the teacher in effectively navigating the program resources, although most took the form of boilerplate text. Recommended Instructional Structures It is perhaps not surprising that each of the five curricula provides some sort of transparency regarding general instructional structures. The most common of these were timeframes and participant structures. These statements were coded as providing transparency because they made clear how the authors envisioned the curriculum would be enacted, often without making statements that explicitly directed the teachers’ actions. All program authors provided some guidance on the amount of time to be allotted for different aspects of the lessons, but they focused on substantially different timeframes. In MTB lesson guides, the designers identified the number of days they envisioned each lesson should take by providing an Estimated Class Sessions note in the lesson overview and At a Glance summary. As with the other curricula, the recommended length of a class session was not provided in the lesson guides, but was made known in other portions of the teacher’s guide. These authors did not provide suggestions for the duration of the various activities within the lesson, leaving teachers to decide appropriate points to break up lessons. MIF also provided indicators of how many days the designers imagined each lesson should take and where daily lesson breaks should occur. In EM, each lesson was assumed to take one class session, and the developers identified the recommended time only for the differentiation options, not the activities within the main body of the lesson. SFAW recommended a duration for the Differentiation options and the Investigating the Concept portion prior to the main lesson. Only INV provided guidance on how long designers anticipated each of the activities within the main body of the lesson should take. Each of the curriculum programs communicated the authors’ vision for the participant structures in the lesson, referring to whether particular activities should be done in a whole class, small group, partner, or individual format. EM, INV, MIF, and SFAW used specific, recurring icons to signal the intended participation structure for each activity. MTB was unique in that the participation structure recommendations were embedded, and perhaps buried when compared to the other programs, in the main text, for instance: “Have student pairs complete Questions 16–17 before discussing them together as a class” (MTB GR4 3.1). As is illustrated in this sentence, MTB tended to recommend frequent transitions between whole group and individual or paired structures during the lesson. (Chapter 3 provides more detail on the participant structures in all five programs.)

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8.6  Discussion Researchers who see the relationship between teachers and curriculum materials as dynamic and collaborative argue that, rather than simply speaking “through teachers,” curriculum designers need to speak “to teachers” (Remillard, 2000, 2005); in this analysis, we examined support that explicitly spoke to teachers by making aspects of the designed curriculum and the rationale for these design choices transparent. From this sample, it is evident that designers of elementary curriculum materials have heeded the recommendation to speak to teachers, but in limited and varied ways. All but one of the programs (SFAW) provide a rationale for activities at least once in every lesson, often by clarifying the mathematical goals underlying activities, as recommended by Remillard (2005). This type of transparency aligns with Males’s (2011) finding that rationale for particular activities (or experiences) was one of the most prevalent forms of rationale support. By including this commentary, the authors offer help to teachers as they select from, enact, and possibly modify and supplement the various offerings found in the guides. However, both Ball and Cohen (1996) and Davis and Krajcik (2005) advocate for a deeper level of communication of rationale. In 1996, Ball and Cohen noted that, at that time, curriculum authors rarely discussed “the strengths and weaknesses of particular designs” (p. 7). This observation remains true for this sample; even as these programs explained the intended purpose of the activities and actions, we saw only isolated instances of support that described why these particular activities or actions were appropriate or effective for serving these purposes, and curriculum authors almost never mentioned alternatives that were deliberately omitted from the designs. Deeper levels of rationale are necessary for teachers to develop what Ball et al. (2008) describe as knowledge of content and teaching, including “the instructional advantages and disadvantages of representations used to teach a specific idea and identify what different methods and procedures afford instructionally” (p. 401). Davis and Krajcik (2005) note that curricular materials could help teachers “make connections between theory and practice” (p. 5), and both Ball and Cohen (1996) and Davis and Krajcik (2005) recommend that curriculum materials help teachers understand the way the curriculum develops over time, but from this sample, it appears that developers of elementary mathematics curriculum programs have not universally adopted these recommendations. None of the curricula made references to research literature, and only MTB consistently provided significant commentary on how the mathematics unfolds across units and grades. Furthermore, in her work identifying practices that teachers engage in to steer instruction toward the mathematical point of lessons, Sleep (2012) highlights the importance of helping students see and understand the mathematical storyline of lessons and the connections to prior and future work. The absence of support for communicating this storyline to students in the lesson guides places significant demands on teachers in emphasizing these connections to their students.

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Our analysis was limited in that we only analyzed the lesson guides. It is certainly possible that this sort of support is found elsewhere in the curriculum materials, but there is evidence that suggests that teachers may not pay close attention to support that does not address a particular lesson (Beyer & Davis, 2009; Schneider & Krajcik, 2002). Teachers who choose to only read the lesson-specific support in the teacher’s guides analyzed in this study would not be consistently supported in making connections between theory and practice. More research is necessary to understand the relationship between the location of various types of support, especially that which pertains to the program as a whole or spans across lessons or units, and the ways the support is taken up by teachers. Our analysis of 75 lesson guides sampled from five different curriculum programs characterized the extent to which curriculum authors speak to teachers about their program and the decisions that went into its creation. By finding that this type of support was limited and providing examples of its varied nature, our study encourages curriculum authors to carefully consider rationale and storyline support so that teachers can make educated and productive adaptations as they implement the curriculum programs in their local contexts.

References Arias, A.  M., Bismack, A.  S., Davis, E.  A., & Palincsar, A.  S. (2016). Interacting with a suite of educative features: Elementary science teachers’ use of educative curriculum materials. Journal of Research in Science Teaching, 53(3), 422–449. Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373–397. Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is—Or might be—The role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–14. 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. Beyer, C. J., & Davis, E. A. (2009). Using educative curriculum materials to support preservice elementary teachers’ curricular planning: A comparison between two different forms of support. Curriculum Inquiry, 39(5), 679–703. Beyer, C. J., Delgado, C., Davis, E. A., & Krajcik, J. (2009). Investigating teacher learning supports in high school biology curricular programs to inform the design of educative curriculum materials. Journal of Research in Science Teaching, 46(9), 977–998. Charles, R. I., Crown, W., Fennell, F., et al. (2008). Scott Foresman–Addison Wesley Mathematics. Glenview, IL: Pearson. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Davis, E. A., Palincsar, A. S., Arias, A. M., Bismack, A. S., Marulis, L., & Iwashyna, S. (2014). Designing educative curriculum materials: A theoretically and empirically driven process. Harvard Educational Review, 84(1), 24–52. Drake, C., & Sherin, M. G. (2008). Developing curriculum vision and trust. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 321–337). New York: Routledge.

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Elementary Mathematics Department. (1997). PEP math, grade three (Vol. 1). Beijing: People’s Education Press. Heck, D. J., Chval, K. B., Weiss, I. R., & Ziebarth, S. W. (2012). Developing measures of fidelity of implementation for mathematics curriculum materials enactment. In D. J. Heck, K. B. Chval, I. R. Weiss, & S. W. Ziebarth (Eds.), Approaches to studying the enacted mathematics curriculum (pp. 67–87). Charlotte, NC: Information Age Publishing. Kim, O. K. (2018). Teacher decisions on lesson sequence and their impact on opportunities for students to learn. In L. Fan & L. Trouche (Eds.), Recent advances in research on mathematics textbooks and teachers’ resources (pp. 315–339). New York: Springer. Li, J. (2004). A comparative study of United States and Chinese elementary mathematics textbook teacher guides. Unpublished doctoral dissertation, University of Chicago, Chicago, IL. Males, L.  M. (2011). Educative supports for teachers in middle school mathematics curriculum materials: What is offered and how is it expressed? Unpublished doctoral dissertation, Michigan State University, East Lansing, MI. Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. Morris, A. K., Hiebert, J., & Spitzer, S. M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: What can preservice teachers learn? Journal for Research in Mathematics Education, 40, 491–529. Reinke, L. T., & Atanga, N. A. (2013, April 27–May 1). An analysis of authors’ communication of transparency and rationale for design in five elementary mathematics curriculum guides. Paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA. Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers’ curriculum development. Curriculum Inquiry, 29(3), 315–342. Remillard, J. T. (2000). Can curriculum materials support teachers’ learning? Two fourth-grade teachers’ use of a new mathematics text. The Elementary School Journal, 100(4), 331–350. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Remillard, J.  T. (2018). Examining teachers’ interactions with curriculum resource to uncover pedagogical design capacity. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Recent advances in research on mathematics teachers’ textbooks and resources (pp. 69–88). New York: Springer. Remillard, J. T., Reinke, L. T., & Kapoor, R. (2019). What is the point? Examining how curriculum materials articulate mathematical goals and how teachers steer instruction. International Journal of Educational Research, 93, 101–117. Schneider, R. M., & Krajcik, J. S. (2002). Supporting science teacher learning: The role of educative curriculum materials. Journal of Science Teacher Education, 13(3), 221–245. Shulman, L.  S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. 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. Stein, M.  K., & Kim, G. (2009). The role of mathematics curriculum materials in large-scale urban reform. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (p.  37). ­ New York: Routledge. TERC. (2008). Investigations in Number, Data, and Space (2nd edition). Glenview, IL: Pearson Education Inc. TIMS Project (2008). Math Trailblazers (3rd Edition). Dubuque, IA: Kendall/Hunt Publishing Company. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill. Watanabe, T. (2001). Content and organization of teacher’s manuals: An analysis of Japanese elementary mathematics teacher’s manuals. School Science and Mathematics, 101(4), 194–205.

Part III

Synthesis and Commentary

Chapter 9

Complexity of Curriculum Materials as Designed Artifacts: Implications and Future Directions Janine T. Remillard and Ok-Kyeong Kim

Abstract  This chapter synthesizes and builds on findings from the previous chapters to consider what we have learned about mathematics curriculum materials as tools for teachers. Looking back at our analysis in Chaps. 2, 3, 4, 5, 6, 7, and 8, we briefly summarize our overall key findings and consider how these findings were influenced by key methodological decisions. We then discuss the complexity of curriculum materials as designed artifacts and the challenges associated with analyzing them. Using Hiebert and colleagues’ notion of a constellation of features that are needed to characterize classroom teaching, we argue that examining curriculum materials also involves considering a cluster of features. Using existing literature and our analysis in this volume, we also conceptualize curriculum materials as artifacts of design decisions comprised of multiple layers. We distinguish two main components, i.e., objectively given structures (what teachers see physically) and authors’ ideas and values, and discuss the relationship between them from curriculum authors’, the researchers’, and teachers’ perspectives. Finally, we provide some implications for teachers, curriculum designers, and researchers based on our findings and reflections on our work. Keywords  Curriculum analysis · Mathematics curriculum materials · Teacher’s guide · Designed artifact · Objectively given structure · Constellation of features · Design decisions · Everyday Mathematics · Investigations in Number, Data, and Space · Math in Focus · Math Trailblazers · Scott Foresman–Addison Wesley Mathematics · Author intentions · Curriculum interpretations · Teacher interpretations

J. T. Remillard (*) Graduate School of Education, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] O.-K. Kim Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_9

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9.1  Introduction This volume has two central aims, to introduce a framework for analyzing mathematics curriculum materials from the perspective of teachers as the primary users of these resources and to use this framework to analyze five elementary curriculum programs. The framework we introduce includes three lenses: (1) the mathematical emphasis of curriculum programs; (2) the pedagogical approach, and (3) modes of communicating with teachers. With the last lens, we attend to mathematical support for teachers, supports for noticing and responding to student thinking, and design rationale and transparency. The findings from the analysis presented in Chaps. 2, 3, 4, 5, 6, 7, and 8 provide insight into particular design features of each of the programs analyzed. More importantly, however, the analysis sheds light on the characteristics of curriculum materials more generally, highlighting challenges associated with the design and use of curriculum materials. In this chapter, we explore some of the characteristics of curriculum materials that emerged through our analysis and discuss implications for teachers and teaching, curriculum design, and research.

9.2  Key Findings: Trends Within Variation The five curriculum programs analyzed varied considerably across a wide range of dimensions. In fact, we see the range and types of variation that surfaced through our analysis as one of our key findings. We were unable to array the programs along a single continuum or fit all into an overarching classification system. Instead, we found multiple continua and groupings appropriate for classifying and characterizing features of the five programs. Within the lens of treatment of mathematics, we found variation in how mathematical topics were sequenced, the types of mathematical thinking expected, the role of representations, and the purpose of practice. Within the lens of pedagogical approach, we found differences in how lessons were structured, the type of mathematical work students were expected to do, the sources of mathematical knowledge that were valued, and the role the teacher was expected to play in supporting student learning. We also found notable variation in how curriculum authors communicated with teachers, including how much and where, what they communicated about, and the approach to communication. In particular, we found significant differences in the ways the programs guided teachers to consider and respond to student thinking and explained elements of the curriculum design. These differences hint at a range of views among curriculum authors about how teachers might be supported to enact the lessons and different options for offering that support. Our analysis also surfaced some consistent trends that are worthy of note. First, across a number of our analytical lenses, a common clustering often emerged. INV and MTB consistently fell toward one end of many continua, placing greater emphasis on conceptually oriented mathematical tasks and student-generated strategies

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and positioning the teacher as an orchestrator of student sense making, while addressing these commitments to differing degrees and in their own ways. Despite considerable variation between them, MIF and SFAW typically fell toward the other end of the same continua, placing more emphasis on mathematical procedures and positioning the teacher and the textbook as major sources of knowledge. EM fell at various positions in the middle of these continua, at times closer to INV and MTB, other times closer to the other two, and often represented a blended position. This common clustering across several analytical dimensions leads us to the second trend: We found discernable alignment within the curriculum programs among the mathematical emphasis, pedagogical approach, and mode of communicating with and supporting teachers as curriculum enactors. This alignment is illustrated in Chap. 5, which demonstrates how the approach to communicating with the teacher reflected the mathematical and pedagogical demands of the program and supported the intended role of the teacher and the relationship between the teacher and the student textbook. Another instance of alignment is also illustrated in Chap. 7, which considers whether and how the five programs offer supports for noticing and responding to student thinking. The two programs that consistently made explicit connections between student thinking and conceptual understanding, also placed more emphasis on both in general. The two programs that tended to guide the teacher to evaluate student work based on correctness placed more emphasis on procedural knowledge and positioned the teacher or textbook as the primary source of knowledge. In other words, we found within-program consistency across multiple analytical lenses, including how authors communicated with, and provided guidance for, the teacher. Despite the finding that programs tended to provide guidance that aligned with the mathematical and pedagogical demands placed on the teacher, our analysis also revealed considerable gaps in supports intended to help them enact the designed curriculum. Overall, we found a tendency for curriculum authors to communicate through directing action as opposed to communicating to teachers. In particular, MIF provided minimal direct explication of the underlying mathematical ideas (Chap. 6). Because mathematical explanations, when they were included, tended to be embedded in the instructional steps, teachers were left to extract the important mathematical ideas on their own. Our analysis of whether and how design transparency was communicated (Chap. 8) revealed limited discussion of the rationales for particular design decisions. Similarly, authors provided some explication of the mathematical storyline across lessons or units, but these connections were often brief and superficial and rarely explained how teachers could leverage connections during instruction. In the remainder of this chapter, we discuss what these findings contribute to understanding about what designed curriculum materials are, their critical features, and implications for teachers, curriculum developers, and future research. First, however, we review and reflect on the methodological decisions we made and how they shaped our findings.

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9.3  Consequences of Methodological Decisions Like all research, the findings summarized above must be understood in relation to the methodological choices we made in undertaking the analysis. In Chap. 1, we outlined several levels of methodological decisions made in the process of curriculum analysis. The consequential decisions that affected our analysis occurred at multiple levels, including overall purpose and focus, conceptual and analytical frameworks employed, sampling, and basic analytical techniques. These decisions guided which components of the curriculum programs we examined and how we examined them, having a determining effect on what we were able to uncover. In contrast to many of the curriculum analyses discussed in Chap. 1, we identified multiple foci that we considered in relation to one another, seeing them as jointly constituting potential opportunities for learning. These included mathematical treatment and emphasis, pedagogical approach, and communication with the teacher. The treatment of mathematics is a common focus of curriculum analyses. Our interest in considering all three foci from the perspective of the teacher responsible for enacting the designed curriculum, led us to look more deeply at aspects of the pedagogical approach and communication with the teacher infrequently considered in curriculum analyses. These foci required us to direct our primary attention to the teacher’s guide and the daily lessons, which are usually inclusive of the student-­ facing material. In contrast, many curriculum analyses place primary emphasis on the student text. Once the focus of our analysis was established, how we conceptualized the dimensions within each lens further influenced what we paid attention to. Mathematical emphasis and pedagogical approach, two of our analytical lenses, can be understood and examined in multiple ways. Drawing on research literature, we identified particular dimensions to examine within each focus, and, by necessity, excluded others. In each case, we sought to consider each lens broadly, using multiple dimensions. Doing so allowed us to create a complex picture of each program. At the same time, we are aware that other researchers might favor different dimensions. Table 9.1 provides a summary of the three lenses and dimensions within each. Not represented in the table, but detailed in the previous chapters, are Table 9.1  Summary of lenses and dimensions used for the analysis of five curriculum programs Mathematical emphasis •  Scope and sequence •  Cognitive demand of tasks •  Treatment of ongoing practice •  Use of representations

Pedagogical approach •  Lesson and participant structure •  Nature of student learning and work –  Types of mathematical work –  Cognitive demand of tasks –  Source of knowledge, knowing •  Teacher’s role

Communication with teacher •  Quantity of communication •  Topic (purpose) of communication – Mathematics –  Student thinking –  Design transparency •  Mode of communication •  Location of communication

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the indicators we used to mark variation within each dimension. In some cases, like cognitive demand of tasks, these indicators were based on existing classification schemes in the literature (e.g., Stein, Grover, & Henningsen, 1996). In other cases, we developed our own classification system by integrating across the literature or through emergent types of variation. The source of knowledge dimension, under pedagogical approach, for example, was originally important to us in connection with Munter, Stein, and Smith’s (2015) distinction between direct and dialogic instructional models. The direct instructional model positions the teacher as the primary source of knowledge, whereas the dialogic model places a high value on student generation of knowledge. Through examining the explicit and implicit sources of knowledge in the daily lessons, we determined the need for additional indicators, including the student textbook and prior sources. This decision allowed us to make distinctions between SFAW and MIF that were informative for our analysis. We are aware that choices such as these may also mask other distinctions. Finally, in order to undertake the analysis, we made a number of decisions related to sampling and coding that impacted our findings. To narrow the focus of our analysis somewhat, we selected the content domains of Number and Operations and Algebra, inclusive of fractions and decimals, in grades 3–5. Doing so somewhat reduced the possible number of lessons that we needed to sample from, although, these content domains comprised 60 and 70% of the lessons in each program. Still, we are cautious about extending our claims to other content and especially to other grade levels. We also decided to randomly sample 30 lessons per program, 10 from each grade level for much of the analysis. Having perused many additional lessons in each program, we noted a great deal of consistency across lessons within each program. Still, we continue to wonder if a larger sample or intentionally selected lessons might have influenced our findings. Further, because the curriculum programs contain a great deal of resources and options, some less critical to the core lesson, we opted to focus our analysis primarily on what we called the main body of the lesson. This decision allowed us to give more importance to what we assessed to be the principal part of the daily lesson than to supplementary and optional material. At the same time, determining what was core and what was optional was not always straightforward for each program. Finally, when coding lessons, determining the unit of analysis presented another challenge, especially when we intended to quantify the results. We ended up making different determinations about unit of analysis based on the focus and dimension and the questions we were asking. For instance, in Chap. 5, we explain our decision to code communication with the teacher at the sentence level, in order to compare the quantity of different types of communication. In other chapters, we used lesson segments, tasks, or complete teacher notes to compare what the authors communicate to teachers, aspects of the mathematical, or pedagogical approach. We now turn to a discussion of what our particular findings, given our methodological approach, reveal about mathematics curriculum materials as designed artifacts and the work teachers must do to use them.

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9.4  Curriculum Materials as Complex and Layered Undertaking this analysis has left us struck by the depth and complexity of curriculum programs, characteristics that may be underappreciated by many educators, policy makers, and researchers. Our analytical framework focused on three major lenses, mathematical emphasis, pedagogical approach, and communication with the teacher. Each lens marks a type of guidance included in curriculum materials. Within each of these lenses, we identified a number of different dimensions that contributed to a full representation of the lens (see Table 9.1). The number of dimensions we identified and the variation within each, as described in Sect. 9.2, illuminates the way that mathematics curriculum materials are composed of layers of design elements that must be considered together. We see these layers of design elements as reminiscent of Hiebert et al.’s (2005) notion of a constellation of interrelated features that typically characterizes a particular classroom system in both the number of elements and their interrelationships. In their analysis of lessons from 7 different countries, they identified 75 different features that might be considered to characterize one system. Our analysis focused on three lenses and at least 17 dimensions and subdimensions across them. A review of the analytical chapters in this volume suggests a number of additional features and characteristics within each lens that can be uncovered and compared. Like classroom teaching, curriculum programs cannot be characterized simply by a single or a small number of components or features. Our analysis mirrors Hiebert and colleague’s findings about the relationship between features as well. When looking at teaching as a system, they were able to identify a “constellation of features” that worked together to reinforce overarching characteristics of the system in both predictable and unpredictable ways. In our summary of findings, we noted an unsurprising pattern of internal alignment within each curriculum program across the different dimensions we analyzed; characteristics of the mathematical emphasis aligned with the pedagogical approach and appeared to be supported by the communicative support provided to teachers. We also found several ways dimensions of the designs of curriculum materials worked together in more complicated ways, often moderating the overall nature of learning opportunities. This tendency was especially pronounced in EM, when compared to the other two Standards-based programs, INV and MTB (see Table 9.2). The majority of focal tasks in all three programs were categorized as high cognitive demand; they either connected procedures to underlying meaning or required students to solve unstructured problems or develop and use strategies. When looking across other dimensions (such as those shown in Table 9.2), we see clusters of features that work together to shape the opportunities to learn differently. We use Hiebert et al.’s (2005) concept of constellation of features to describe two such clusterings. The first constellation includes the focus of ongoing practice and the nature of student work and illustrates how these two features work together to shift or reinforce the overall nature of student learning opportunities. All three programs included routines for ongoing practice across the curriculum, but the focus of the

9  Complexity of Curriculum Materials as Designed Artifacts: Implications and Future… 265 Table 9.2  Dimensions and characteristics that worked together to shape opportunities to learn in Standards-based programs

EM

High cognitive demanda Percent procedures with extended Percent doing connections mathematics 74.6 16.0

INV

40.3

59.7

MTB 71.7

23.3

Treatment of mathematics Focus of ongoing practice Short answer, automaticity Open-ended, mathematical relationships Combines short answer, application, extension tasks

Computational procedure development Alternative algorithms St.-generated strategies; relationships St.-generated strategies; alternative → standard algorithms

Pedagogical approach Type student work Answer only Generate, explain, justify Generate, explain, justify

Primary teacher’s role Guide, step back Orchestrate, step back Guide, orchestrate, step back

For the purposes of this discussion, we include the percentages of tasks that were categorized in the two highest cognitive demand categories

a

practice differs. INV’s Ten-Minute Math activities and Daily Practice assignments were designed to engage students in open-ended or strategy-oriented work to reinforce previously taught concepts and develop fluency. MTB’s Daily Practice Problems offer a combination of a few short-answer review problems and application and extension tasks. In both cases, the approach to daily practice is consonant with the majority of tasks throughout the program. These approaches incorporate ongoing practice while reinforcing the predominant type of student work, which we classified as generating, explaining, and justifying. Ongoing practice in EM is focused on practice of previously taught skills (using Math Boxes) and developing speed and automaticity with computational skills through Mental Math and Reflexes exercises that begin every lesson. These ongoing practice formats require short answers or quick responses and rarely require explanation. As a result, students spend a considerable amount of time during each lesson providing answers and only about 14% of lesson time, on average, generating strategies or explaining or justifying their answers (see Chap. 3). In essence, the emphasis of the ongoing practice in EM diminishes the potential for student work more commonly associated with high demand tasks. A second cluster of features that impacts opportunities to learn includes the way computational procedures are developed, cognitive demand of main tasks, and the teacher’s role. As detailed in Chap. 2, but referenced in Table 9.2, EM, INV, and MTB take somewhat different approaches to developing computational procedures. EM uses alternative algorithms to make the computational process visible to students. They are introduced to students by the teacher and textbook and used throughout the program, without the intention of transitioning to the standard U.S. algorithms. These alternative algorithms and the way they are handled in EM lessons are prototypical examples of procedures with connections tasks. Students are introduced to

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and practice an algorithmic procedure, but the instruction surrounding the procedure and the procedure itself make connections to place value and the meaning of the operation explicit. The analysis of the teacher’s role in Chap. 3 indicated that guiding students and stepping back while students practiced were the primary roles assigned to the teacher. These factors worked together to shape learning opportunities that tended to focus on procedural steps specified by the teacher (and student text), despite the emphasis on mathematical meaning. In contrast, MTB and INV begin computation work with student-generated strategies. MTB is quite similar to EM in the percentage of tasks coded as procedures with connections versus doing mathematics, but other factors place more emphasis on student sense making and production of knowledge. The predominant teacher roles are to both guide and orchestrate, as well as step back. In particular, teachers are encouraged to take an orchestrating role when students generate strategies for computation. MTB also introduces and encourages the use of multiple models for computation, including alternative algorithms, concrete models (such as base-ten blocks) for computational algorithms, using base-ten blocks, and standard algorithms. Students are encouraged to use approaches that make sense to them. As we have discussed previously, the INV program had the most tasks classified as doing mathematics, which involves working on nonroutine problems. In keeping with this emphasis, orchestrating is the predominant teacher’s role. In the case of algorithms, students are supported to generate strategies based on their understanding of number relationships and operations. INV introduced standard U.S. algorithms as one option to solve computational problems, but students are primarily encouraged to use their own strategies. These characteristics in both MTB and INV work together to create learning opportunities in which students have autonomy to decide how they will solve problems. The differences between the three Standards-based programs detailed above align with our findings from the interviews with authors, summarized in Chap. 4. All three author teams expressed a commitment to deepening students’ computational fluency, but they made different decisions about how to incorporate this commitment into the designed curriculum, based in part on other goals for student learning. Despite similarities in the authors’ commitments and visions, the design decisions they made resulted in substantial differences in the resulting curriculum artifacts. We discuss this phenomenon in the following section.

9.5  Curriculum Materials as Artifacts of Design Decisions Considering our findings from the curriculum analysis alongside what we learned from the curriculum authors about their design decisions and processes (summarized in Chap. 4) illustrates an important characteristic of curriculum materials— that they are artifacts of designers’ values and design decisions. Our conversations with authors of three of the programs uncovered a number of common views about mathematics teaching and learning and, at the same time, a range of possible

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approaches they could take in the pursuit of their visions when designing the curriculum artifacts. At various points in the development process, authors made different design decisions, which led to different outcomes. In this section, we use findings from our analysis to explore the concept of curriculum materials as artifacts of authors’ design decisions. First, we develop a conceptual model to characterize the components of designed curriculum artifacts and their relationship to the authors’ intentions. We then use examples from the programs we analyzed to explore the relationships between these designed artifacts and curriculum authors’ aims and underlying values. We conclude the section with a discussion of implications for teachers using these artifacts.

9.5.1  C  onceptual Model of Curriculum Materials as Artifacts of Design The previous description of what the written materials represent with respect to authors’ intentions and decisions illustrates the way that curriculum materials are a particular genre of communication (Remillard, 2005, 2012). As such, they have their own purpose (to represent and guide instructional decision making) and a number of conventions and “physical, visual, and substantive forms” to use when communicating their intent (Remillard, 2012, p. 108). Our model takes into account both the purpose of communication and the forms of communication. Brown, Pitvorec, Ditto, and Kelso’s (2009) distinction between curriculum authors’ ideas and written lessons, discussed in Chap. 1, helps to consider the purpose of designed curriculum artifacts, particularly in relation to authors’ intentions. Written lessons are based on authors’ ideas about the content to be learned, coupled with ideas about activities and tasks that will engage students in the desired learning. Further, they are representations of authors’ ideas, not the ideas themselves. When designing materials, based on these ideas, curriculum authors deploy various forms of representation and communication, which we think of as design features, to make their intended ideas available to teachers and students. (As should be evident from our analysis, we have a particular interest in examining how intended ideas, both about content to be learned and activities to support this learning, are made available to the teacher.) The resulting curriculum artifacts and the various design features that comprise them send explicit and implicit messages about the authors’ underlying values and priorities. We use Otte’s (1986) characterization of textbooks to help us describe the difference between the form or design features and their relationship to underlying values and priorities. He used the term “objectively given structure of information” to refer to the physical (or visual) form a textbook takes. We think of this aspect of curriculum artifacts as what teachers (and students) see and interact with. The two columns in Fig.  9.1 represent objectively given structures and authors’ ideas and values underlying them. The authors’ ideas and values are typically not directly visible to

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Objectively Given Structures

Authors Ideas and Values

Forms (How substance is communicated)

• How best to communicate ideas to teachers

Focus (What is communicated)

• Mathematical-instructional objects Tasks, problems, exercises, examples Representations • Curricular sequences • Instructional guidance • Supports for teachers’ enactment mathematical support support for noticing student thinking, Design rationale and transparency • Organization and arrangement of the page Page layout and sectioning Placement of items on the page • Visual images • Amount/extent of detail • Voice Quantity, focus Mode of communication

• What mathematics is valued • What it means to know mathematics • The mathematical learning point • What students need to do to learn mathematics • How learning should be supported • What support teachers need

Fig. 9.1  Curriculum materials as artifacts

teachers. Our analysis suggests that objectively given structures have two key elements that need to be considered: the focus of the message, or what the authors are communicating about, and the forms of communication, or how the authors use features of the medium to communicate. These two elements are shown in the two rows in Fig. 9.1. The items in the focus row, under objectively given structures in Fig. 9.1, include key design components that we considered in our analysis. There are many ways to categorize and name the layers of components or design features in curriculum materials; we have opted for four broad categories that align with our analytical framework. In the right-hand column, we list the types of authors’ ideas and values surfaced by our analysis of these components. Mathematical-instructional objects are “mathematical ideas packaged for the purpose of instruction,” including tasks, activities, exercises, and representations (Remillard, 2018). These objects are designed for students to engage with and are intended to facilitate their learning of specific mathematics. These objects engage students with mathematical ideas, but they are not the ideas themselves (Sleep, 2009, 2012). Curricular sequences refer to how the mathematical content is organized for learning over a variety of different timeframes, within a lesson, across lessons, or across grades. Both mathematical-­ instructional objects and curricular sequences were a focus of our analysis in Chap. 2. They reflect the authors’ ideas about what is important mathematically and what it means to know and learn it, and they are aimed at addressing particular mathematical learning points. Instructional guidance refers to various types of information intended to guide the teacher’s enactment of the lesson, including scripts and other types of suggestions regarding the teacher’s role during lessons. Our analysis of instructional guidance in the five programs and what it suggests about authors’ ideas about what students should do to learn mathematics and how learning should be supported by the teacher or other learning sources can be found in Chap. 3. Support

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for teachers’ enactment refers to the type of guidance that Davis and Krajcik (2005) might consider educative, in that it provides communicative support likely to help teachers make decisions and navigate student responses during lessons. As we discuss in Chaps. 5, 6, 7, and 8, this type of support provides insight into authors’ ideas about the types of support teachers might need to enact the designed curriculum plans, which sometimes include more explicit articulations of underlying ideas. The items in the forms row in Fig. 9.1 include a set of communicative tools curriculum developers have available within the genre of curriculum materials and specifically teacher’s guides. This idea is informed by our view of curriculum materials as multimodal. Bezemer and Kress (2008, 2016) note that contemporary textbooks combine writing, image, page layout, and other modes of communication to shape the user’s engagement and the learning experience. Teacher’s guides are similar, in that they use all of these communicative resources, although certain forms and styles are particular to the genre of teacher’s guides, including images of student pages, sectioning the page into a main column and a margin, colored font to indicate what teachers should say or student answers, and using boxes, icons, and headings to signal certain types of information. Some of the tools or forms curriculum developers have at their disposal are organization and arrangement of the page, visual images, the amount and extent of communication, and voice, including quantity and mode of communication. These forms, which provide insight into authors’ ideas about desired or appropriate ways to communicate with teachers, are discussed in Chaps. 5, 6, 7, and 8, they can also be seen in the sample lesson guides in Appendixes B–E.

9.5.2  Interpretation of Designed Curriculum Artifacts The model presented in the previous section illustrates the complexity of curriculum materials as designed artifacts, both in terms of their design and their relationship to authors’ or designers’ ideas, values, and priorities. Considering both the focus and forms of designed materials allows us to see that any component of a designed artifact (objectively given structures) represents authors’ efforts to communicate about mathematics, learning, and teaching in forms that will be accessible and useful to teachers. This design process is represented by the top arrow in Fig.  9.2. These structures are what teachers see and interact with and, in the process, make their own determinations about the authors’ ideas, values, and priorities. This interpretive work, from objectively given structures to authors’ ideas, is represented by the return arrow in Fig. 9.2. Importantly, the ideas that result from teachers’ i­ nterpretation Fig. 9.2 Relationship between authors’ ideas and values, and objectively given structures

Authors’ ideas and values

Design

Interpretation

Objectively given structures

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might differ from the authors’ original ideas. The bottom arrow also represents the work of our analysis of the five curriculum programs. We scrutinized these structures from different perspectives and in great detail in order to identify our own views of authors’ ideas. We also spoke with several curriculum authors in order to understand the decisions behind their designs. Because we were able to juxtapose our findings with the words of some of the curriculum authors, we also uncovered underlying ideas and values that authors had not made explicit to us during the interviews. We illustrate the relationship between the authors’ ideas and values, the objectively given structures, and potential interpretations of these structures using examples from the three Standards-based curriculum programs. In doing so, we describe some specific characteristics of each program, design features highlighting the characteristics, curriculum authors’ comments about them, and our interpretations of authors’ values and ideas based on our analysis. EM: Fluency Through Mental Math and Reflexes In the previous section, we summarized EM’s approach to ongoing practice and automaticity, discussed in Chap. 2. Each lesson begins with a brief segment, called Mental Math and Reflexes, comprised of a small set of exercises, involving computation or number fluency. Many of these exercises involve more than one step, thinking flexibly, or are set in contexts, and none can be completed by simply producing memorized facts. The stopwatch icon consistently accompanies these exercises. Periodically, the guide recommends students show their answers by holding up personal slates. Correct answers are provided in the teacher’s guide in red font. The guide sometimes suggests students should explain their solution strategies after each problem, as in the example in Fig. 9.3. The shaded circles to the left of each question indicate difficulty from easiest to most difficult. How might this routine component of EM be interpreted by teachers? The daily occurrence of the Mental Math and Reflexes task and its characteristics are likely to signal two different ideas and values. First, the importance of speed and automaticity is evident in the regularity of this type of task, the indication that it should happen quickly indicated by the stopwatch, and even the use of the word “reflexes” in the title. Second, the nature of these fluency tasks, that they involve much more than straightforward computation or recall of facts, but instead require some level of understanding of number relationships or other concepts in order to arrive at an answer, suggests that the authors value reasoning and application as well as procedures. Our conversation with the EM author team confirmed a dual commitment to “arithmetic skills and quick responses” essential to problem solving and conceptual development. As we illustrated throughout this volume, combining attention to computational fluency and sense making was a characteristic theme of the EM program. We also point out that this dual commitment frequently positioned the EM program in the middle of various continua we could use to characterize both

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Fig. 9.3  Example mental math and reflexes from EM.  Excerpt from Everyday Mathematics, Grade 3, Teacher’s Lesson Guide, Volume 1, p. 607. (University of Chicago School Mathematics Project). Copyright 2007 by McGraw Hill. All rights reserved. Reprinted by permission of the publisher

­ athematical and pedagogical approaches. We posit that understanding the imporm tance of both these commitments and how they might work together to support student learning is critical to making sense of the design decisions made by EM authors and might be the greatest interpretive challenge for teachers using the program. MTB: Context-Driven Lessons Almost every MTB lesson begins with a vignette, drawn from fictitious situations or literature, to launch the lesson (explained in Chap. 3). These situations provide an everyday or relatable context about which students gather and represent data, model the data with graphs, tables, and symbols, or solve problems. The lesson in sample lesson guides in Appendix D used the following context: Help wanted: Your creative talents are needed to help make costumes for the school play “Michael and the Land of Many Colors.” The students want to cover the front of the costumes with a special, colorful material. They need your help in figuring out how much of this material they will need. To do this, your group will need one coat. (MTB, GR3 6.2, p. 36)

This introduction is followed by a set of instructions in the student text, leading them to trace the outline of the coat and use base-ten blocks (i.e., 100 s, 10 s, and 1 s blocks) to find the area of the coat (See Fig. 9.4 for an illustration included in the lesson guide). A note in the teacher’s guide suggests the teacher “encourage students to use their knowledge of symmetry to make the measuring faster” and states that “students should realize that measuring the right or left half is sufficient because the coat is symmetrical around the zipper” (p. 37). The teacher’s guide notes that students should report their answer in square centimeters and indicates that in order to do so, they will need to “interpret that, for example, 22 flats [100 s], 19 skinnies

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Fig. 9.4  Making outlines around the base-ten blocks (MTB GR3 6.2, p. 37). Excerpt from Math Trailblazers, Grade 3, Unit Resource Guide by TIMS. Copyright © 2008 by Kendall Hunt Publishing Company. Reprinted by permission

[10 s], and 54 bits [1 s] are the same as 2200 bits, 190 bits, and 54 bits, or 2444” (p. 37). Later in the lesson, students compare and order the areas of all the coats they found (all of which involve 4-digit numbers). Teachers using MTB lessons like this one are likely to notice that this fictional situation, finding the area of a coat (an irregular, but symmetric shape) to make costumes for a school play, is used to generate opportunities for students to relate different mathematical ideas, such as finding area, using symmetry to do so efficiently, and working with place value to compose and compare large numbers. In fact, the lesson lists five mathematical topics under Key Content, including measuring area, understanding place value, and comparing and ordering four-digit numbers. The context is not just included to introduce the lesson, but is incorporated throughout. Although teachers are likely to see the relevance of the context in the lesson, they may also have difficulty navigating multiple mathematical ideas integrated into the lesson, especially since many of the concepts are intended to emerge naturally as students’ work. Teachers might be unsure which mathematical ideas to highlight at different instructional moments and how they are all connected to support students’ learning. Further, a number of mathematical decisions are left to the teacher. The teacher’s guide indicates, for example, that students should recognize that the symmetry of the jacket can help them, but does not say how students might be guided to use this information. The role of contexts as drivers of mathematical work in MTB was mentioned in our conversation with the authors. They emphasized the importance of students seeing “the connections between the mathematics they learn in school and the thinking they do in everyday life.” Initially designed to integrate mathematics and science, many of the contexts involve collecting and working with some sort of data. The example above illustrates how a context was used to generate large numbers with base-ten material that students needed to interpret. These contexts routinely integrate different mathematical concepts, using them opportunistically to encourage students to make connections across concepts. The concept of symmetry, for example, is raised briefly and naturally to make finding the area of the coat more efficient and, in so doing, require students to combine (by addition or multiplication) two 4-digit numbers. The way the fictitious situation and multiple mathematics topics are used in this lesson is characteristic of the way MTB builds lessons around contexts, both realistic and fictitious (Freudenthal, 1973; Treffers, 1987).

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INV: Supporting the Teachers’ Role in Discussion Discussion was a prominent feature in all three Standards-based programs, but the nature of these discussions and the support for facilitating them in the INV teacher’s guide stands out as a unique design feature. Seventy-five percent of INV lessons on Number and Operations included a discussion as a distinct lesson segment (see Chap. 3). During these discussions, students offer and are exposed to different solution strategies or understandings of concepts and the teacher supports them in connecting their understandings to key mathematical ideas. Each discussion segment in the teacher’s guide included a number of components written expressly for the teacher: (1) Math Focus Point(s) listed at the beginning of the discussion segment; (2) explicit guidance for how to initiate the discussion, including specific teacher questions and prompts in blue font; (3) anticipated student strategies and images of student work; (4) Teaching Notes highlighting important points to be aware of (INV authors referred to these as sidebar notes); (5) Teacher Notes and Dialogue Boxes in the back of each unit, along with a reference note in the lesson guide, providing more detailed guidance about the mathematical ideas and how the discussion might go; and (6) a section on classroom discussion in the grade-level Implementation Guides. As a set, these features provide instructional guidance and communicative support to prepare teachers to guide the classroom discussions. The excerpted lesson in Appendix C on using arrays to model multiplication includes a discussion segment (INV GR4 1.2). The Math Focus Point of the discussion is: “Developing strategies for multiplying that involve breaking apart numbers.” This 15-min discussion comes after a 45-min activity during which students use two arrays to make a larger array (e.g., 4 × 4 and 4 × 5 arrays are used to make a 4 × 9 array). During the discussion, the teacher asks students to share different ways to break an 8  ×  9 array into two smaller arrays. The pages that follow the introduction of the discussion include three examples of potential student responses, guidance for how the teacher might respond and what to look for in students’ responses, and ways to represent students’ ideas using symbols, such as: 8 × 9 = (8 × 5) + (8 × 4) 72 = 40 + 32 The teacher is guided to “Emphasize the idea that multiplication problems with larger numbers can always be broken down into smaller parts that make them easier to solve” (p. 41). The guide further explains that the arrays can help students keep track of the factors in a multiplication problem, especially when working with larger numbers. The Teaching Note highlights some of the mathematical backgrounds of the discussion (i.e., the distributive property of multiplication). Discussions in other lessons include potential student strategies and thinking, along with photos of children (as illustrated in Chap. 7). Together, these components communicate a number of messages about the role of discussion in the INV designed curriculum. Their frequency and their placement after exploration or problem-solving tasks suggest they play an important role in supporting student learning by consolidating concepts and strategies. The fact that

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they make students’ strategies and ideas central indicates the authors’ commitment to positioning students as key sources of knowledge (described in Chap. 3). The extensive structuring of the discussions and specifying of the understandings they should generate provides evidence that the authors intended these discussions to go beyond students simply sharing their work via show and tell (Stein, Engle, Smith, & Hughes, 2008). The broad collection of guidance and supports in the teacher’s guide further reveals the critical role the teacher should play in orchestrating classroom discussions. For teachers, the quantity of guidance for discussions found in the INV teacher’s guide could be seen as both scaffolding and overwhelming. The specific prompts and sample student answers can help them launch discussions and anticipate student responses. At the same time, the amount of guidance and its directive character might reduce teachers’ sense of agency and decision-making, especially if their students do not respond in the same way as the children in the book. Further, the complexity of different forms with which this guidance is provided, including notes in the margins and at the back of the teacher’s guide, may lead to teachers overlooking or dismissing some of the guidance provided. Our analysis of the level of support for discussions in the teacher’s guides not only revealed the INV authors’ views about what discussions should look like and accomplish, but also provided evidence of their views about the type of support teachers might need to enact them (see Chaps. 5 and 7). Our conversation with the INV author team, reported in Chap. 4, illuminated the authors’ decision-making process. During field testing, they reported observing classroom discussions that simply involved sharing strategies and minimal mathematical connections being made. They concluded that mathematics discussions were “really difficult for teachers.” When designing the second edition of the program, they designed additional supports. In particular, they tried “to help teachers focus more on ‘what’s the heart of this discussion?’ ‘What’s its purpose?’” (Chap. 4). They set out to provide a clear mathematical focus to clarify the purpose of the discussion, sample student responses to help teachers anticipate student thinking, notes about important mathematics ideas, sample dialogues with comments about what teachers needed to pay attention to during the discussion, and specific guidance about orchestrating a discussion. While some might interpret this extensive guidance as scripting, the INV authors viewed it as part of their mission to engage teachers in “ongoing learning about mathematics content, pedagogy and student learning” (e.g., TERC, 2008). Cross-Cutting Points We intend these examples to illustrate several related points about curriculum materials, and specifically teacher’s guides, as artifacts of design that are relevant for considering teachers’ use of them. First, when using teacher’s guides, teachers encounter and interact with the visible outcomes of the design process, what Otte (1986) referred to as the objectively given structures in materials. These structures, which might include a task, a list of content included, or a teacher’s note, are

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d­ eveloped to communicate authors’ ideas on two levels: intended mathematical learning goals for students and how those goals might be pursued through different types of instructional activities and teacher guidance. Second, teacher’s guides include layers of features and forms, including mathematical concepts, tasks, representations intertwined with pedagogical guidance and explanations that teachers must navigate through in order to glean and interpret the authors’ ideas. Third, not all authors’ ideas and values are equally accessible to teachers. Two characteristics of the genre of teacher’s guides mediate teachers’ access to the ideas underlying the designs. They tend to communicate with teachers through directing action and student activity and they often do so succinctly with minimal transparency. Authors’ ideas are encoded1 in these design features and are often spread across clusters of features, but are not always made explicit. As we discuss in the following section, these characteristics have implications for teachers using curriculum materials and the curriculum–teacher relationship.

9.6  Implications for the Curriculum–Teacher Relationship Curriculum materials are complex and multilayered artifacts that use multimodal forms to communicate with teachers. They reflect a tremendous amount of deliberate design and coordination on the part of curriculum authors, a point that was readily apparent to us during our author interviews. Despite these efforts and regardless of the amount of detail and specification included in these designed artifacts, curriculum authors are dependent on teachers to interpret and bring their ideas and visions to life in classrooms. For this reason, Bruner’s (1977) assertion that curriculum materials must “change, move, perturb, inform teachers” in order to have an effect on students (p. xv) seems highly apropos; it also underlines a critical challenge that curriculum authors must grapple with. In this final section of this chapter and this book, we discuss several implications related to what we think of as the interdependence between curriculum authors and teachers.

9.6.1  Implications for Teachers: Developing Design Capacity We begin with the idea that curriculum authors are dependent on teachers to bring their ideas to life in classrooms. Our deepened understanding of the complexity of curriculum materials underlines the demands that this expectation places on  Elsewhere, we have used “embedded” to refer to the way mathematical ideas are located within mathematical-instructional objects (Remillard & Kim, 2017). Our use of the term “encoded” here is influenced by Jill Adler’s (2019) assertion that resources are encoded with mathematical, social, and cultural meanings. And, importantly, while produced with mathematical intentions, these intentions were not always “shining through” them. 1

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t­eachers and hints at the expertise required. As shown in Fig.  9.2, curriculum materials contain multiple layers of components, represented through different modes and forms. The teacher’s role is to read, interpret, and leverage the ideas presented in curriculum materials for use in a particular classroom. Matthew Brown (2009) suggests that using curriculum materials productively involves perceiving and mobilizing affordances in the resources to “craft instructional episodes” (p.  29). Perceiving involves reading, interpreting, and evaluating the contents of a lesson in order to consider what to use and how to use it (mobilizing). While teacher’s guides are written for the generic classroom, teachers perceive the contents of these guides through the lens of their own classroom and students. An important aspect of interpreting involves determining the mathematical point or points of the lesson and evaluating and orienting instructional activities with respect to the point. Sleep (2012) used the term mathematical purposing to describe this type of work. Because teacher’s guides tend to communicate with teachers through directing teacher and student action, teachers often need to discern the mathematical point encoded in these directions. Teachers also might perceive the learning sequences built into the design of a lesson, or a set of lessons, and assess their appropriateness for specific students. Based on their assessments, teachers mobilize the affordances by making determinations about which components to use and how. Brown (2009) used the term pedagogical design capacity (PDC) to characterize the teacher’s ability to perceive the affordances in instructional resources and leverage them to create “deliberate, productive designs that help accomplish their instructional goals” (p.  29). Others have added to this construct, both conceptually and empirically. Leshota and Adler (2018), for example, begin with the context of instruction and emphasize the “teacher’s capacity to perceive needs and opportunities in their classrooms,” as well as “the teacher’s capacity for opening up opportunities for mediation with the available personal and external resources” (p.  93). Coming from a South African context where instructional resources can be limited, Leshota and Adler examined the use of a variety of resources, including the teacher herself. This perspective allows for extending mobilization to include inserting instructional elements into lessons from the prescribed textbook from other sources. They determined PDC according to the quality of teachers’ insertions and omissions in relation to the instructional sequence and in light of an understanding of the local context. In another example, Kim (2019) analyzed teachers’ use of curriculum resources and existing literature (e.g., Brown, 2009; Sleep, 2012) to elaborate tasks that teachers engage in when using curriculum resources productively to plan and enact mathematics instruction. These tasks include (1) identifying the mathematical points of individual lessons (within and across), (2) steering instruction toward the mathematical points, (3) recognizing affordances and constraints of the resources, (4) using the affordances of the resources, and (5) filling in the gaps and holes in the resources. All of these tasks require teachers’ careful examination of the resources they use and subsequent decision-making to design instruction, which includes determining what to use and how to use it. Kim also found that when teachers’ interpretation of the resources was not aligned with the mathematical point as

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c­ ommunicated by the authors, they performed one or more tasks unproductively or in ways that did not support students’ mathematics learning. Pepin, Gueudet, and Trouche (2017) draw on curriculum research to conceptualize teacher design capacity, which they further refine through empirical analysis of groups of teachers engaged in different design projects. They propose three components essential to curriculum design: (1) an orientation, a goal, or point of reference for the design, including knowing students’ needs and where students need to go mathematically in the short and long term; (2) a set of firm but flexible design principles, which are informed by evidence; and (3) the particular type of reflection that allows teachers to anticipate relationships or potential outcomes from action (pp. 802–803). They see teachers’ use of curriculum resources (and other types of instructional resources) within the context of curriculum design as “a participatory two-way process of mutual adaptation” (p. 803). In Pepin et al.’s model, didactical flexibility, or the ability to adapt to new designs and adapt designs to new contexts, is a critical component of design capacity. They also found that the process of designing and using the curriculum to fit a particular context, especially in digital formats, led to growth in teacher capacity. In an earlier publication (Remillard & Kim, 2017), we introduced the concept of knowledge of curriculum embedded mathematics (KCEM), a primary element of PDC. KCEM refers to the mathematical knowledge teachers activate when reading and interpreting mathematical tasks, instructional designs, and representations in curriculum materials and specifies how it might be applied to fundamental elements of these designs. We propose four overlapping dimensions of designed curriculum materials that are encoded with mathematical meaning: (1) foundational mathematical ideas; (2) representations and connections across them; (3) relative problem complexity; and (4) mathematical sequences and learning pathways. Teachers activate KCEM in order to uncover the mathematical meanings in each of these curricular formats. Supporting the Development of Teachers’ Design Capacity We draw on these conceptions of teacher capacity in relation to the use of curriculum materials to consider ways teachers might be supported in this work. First, we argue that the work of using curriculum materials needs to be understood as a dynamic professional process, involving consideration of the needs and opportunities in one’s context or classroom, uncovering the mathematical and pedagogical visions in designed resources, and flexibly connecting the two. Here educators in the USA have much to learn from colleagues in Europe and Japan. The work undertaken by Gueudet and Trouche (2009) and Pepin, Gueudet, and Trouche (2013), which conceptualizes the documentation approach to didactics (DAD) offers a foundation for moving in this direction. DAD theorizes how teachers take up resources and imbue them with a goal-driven utilization scheme, transforming them into purposeful documents. The Japanese concept of Kyozaikenkyu, which is a crucial step in Japanese lesson study and translates to study of instructional material

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(Watanabe, Takahashi, & Yoshida, 2008) also provides a strong image for how teachers might engage with curriculum materials. Multiple and differing descriptions of this practice can be found in the literature (Melville, 2008), but a thread that runs throughout all of them is the appreciation of the complexity of the design of instructional materials and the assumption that uncovering its potential and considering its role in one’s teaching involves careful study and analysis. There is some evidence that teachers in the USA can benefit from engaging in deliberative and careful study of curriculum materials during the lesson study cycle (Shouffler, 2018; Watanabe et al., 2008). We believe that teachers need more opportunities to participate in this type of study and that by doing so it may lead to an increased understanding of curriculum resources and capacities to use them to design lessons and curriculum. Our second recommendation is an amendment to the first: The work of studying resources in relation to one’s context is best when it occurs with colleagues and is closely connected to teachers’ daily work. The value of collaborative inquiry around lesson design and analysis is well illustrated by research on lesson study (Fernandez & Yoshida, 2004). Through cycles of joint design, classroom observation, and refinement of a single lesson that is shared by all members of the group, teachers, and school leaders seek to improve their daily teaching practice (Melville, 2008). Further, research on teacher inquiry groups, communities of practice, and professional learning communities provides ample evidence of the generative power of collective inquiry to reduce isolation and build social and human capital through collaborative construction of knowledge (Grossman, Wineburg, & Woolworth, 2001; Little & Horn, 2007; van Es, 2009). Although lesson study is gaining popularity in the USA and elsewhere in the world, we also know that instituting such opportunities for collaborative inquiry and observation of colleagues’ classes can be challenging in some school systems (Winsløw, 2012). In the area of teachers’ work with curriculum documents, Gueudet and Trouche’s (2012) writing goes a long way toward conceptualizing the collective nature of teachers’ work in general and as it relates to documentation work. From the French perspective, teaching is seen as collaborating, whether it happens explicitly or implicitly. They point out that collective aspects of teachers’ work are “always present: each teacher necessarily has relationships with her colleagues, and further, teachers are related through their communication work” (p. 305). Building on Wenger, McDermott, and Snyder’s (2002) conceptualization, they offer the term “community documentational genesis for describing the process of gathering, creating and sharing resources to achieve the teaching goal of the community” (p. 309). By following the collective activities of two such communities, Gueudet and Trouche illustrate the process of “two-way mutual adaptation” described by Pepin et al. (2017). Both teachers’ roles and participatory practices changed as the documents evolved. From our perspective, the term genesis might refer to the community as well as the documentary work. We see this type of engagement with curriculum resources among teachers as holding great potential for developing teachers’ design capacity. We are also cognizant of the potential institutional and structural barriers that creating such opportunities for elementary teachers in the USA might face. Our

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third recommendation considers existing opportunities in the USA that might be leveraged to build teachers’ capacities to use curriculum resources. Teachers have opportunities to learn about strategies for mathematics teaching through initial teacher education and ongoing professional development programs. Examining curriculum materials as tools or considering one’s role in the curriculum–teacher partnership is seldom part of these types of learning opportunities. In fact, the view that good teachers do not use textbooks is promoted by many teacher education programs (Ball & Feiman-Nemser, 1988) and shared among practicing teachers (Lloyd, 1999; Remillard & Bryans, 2004; Remillard & Taton, 2015). This view is often subtly communicated through emphasis on lesson and unit design and introduction of rich tasks not situated in a curriculum framework. In a review of effective professional development practices, Desimone and Garet (2015) argue for strong connections to teachers’ daily work and the materials teachers are expected to use. We believe that teacher education and professional development provide rich opportunities for teachers to learn to read and interpret the complex layers of ideas and messages in curriculum materials and deliberate about how to mobilize these resources in the classroom. Our final recommendation underlies all of the previous suggestions; we call for a reframing of how the teacher–curriculum relationship is conceptualized and described, particularly for teachers. Remillard and Taton (2015) identify several myths that dominate educational discourse that would benefit from reframing. Two of these myths are particularly relevant to our discussion of the curriculum–teacher relationship. The myth that good teachers do not use prepared curriculum materials, but instead design their own, might be rewritten as “Good teachers partner with curriculum resources” (p. 52). Here teacher expertise is framed as “deliberative, purposeful, use of curriculum materials” (p.  52). A second myth, which takes the opposite position, assumes that adopting a well-designed curriculum program and getting it in the hands of teachers is the ultimate action and investment school leaders make to guide the curriculum. While high-quality curriculum materials that align with the philosophy and goals of the school are essential, there is great danger in ignoring the role teachers play in adapting the program for the local context. Incorporating a new program into an existing system should be treated as a constructive process, requiring “active engagement, new learning, and adjustments on the part of teachers and other professionals” (p.  56). We argue for reframing the myth from curriculum adoption to adaptation.

9.6.2  I mplications for Curriculum Authors: Supporting Teachers Design Capacity Brown (2009) and others (Remillard, 2018) note that PDC is not simply located in the teacher, but is distributed across teachers and curriculum resources. Thus, we need to also consider the role that the design of curriculum materials plays in the

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curriculum–teacher relationship. Components of curriculum materials place different types of demands on teachers and vary in the types of supports and guidance they provide to help teachers perceive and mobilize these resources in their teaching. As a result, an individual teacher’s design capacity reflects the contributions of both the teacher and the curriculum resources being used. This point brings us back to a proposition offered by Ball and Cohen (1996), cited elsewhere in this volume: “curriculum materials could contribute to professional practices if they were created with closer attention to processes of curriculum enactment” (p. 7). Our recommendations for curriculum authors consider their role in the curriculum–teacher relationship and offer several ways they might pay closer attention to the process of curriculum enactment. We begin by taking up the last recommendation in the previous section, but from the perspective of the curriculum authors. We believe it is critically important for authors to be conscious of the work teachers do to make sense of the intentions and ideas underlying the features of designed artifacts. As designers, authors need to anticipate how teachers are likely to use various features of their designs to perceive these ideas and mobilize them to design mathematics lessons. The three author teams we interviewed expressed a commitment to supporting teachers who were using their programs. They shared ideas about what teachers wanted in a mathematics program and acknowledged respect for teachers as professionals. Their explicit understanding of or stance toward teachers’ work in relation to using their materials was less clear across the board, although INV may have been an exception. The INV authors took the less typical approach of positioning teachers as learners and collaborators. In Chap. 4, we note that two of the three guiding principles of INV elaborated the teacher’s role: “Teachers are engaged in ongoing learning about mathematics content, pedagogy and student learning” and “Teachers collaborate with the students and curriculum materials to create the curriculum as enacted in the classroom” (TERC, 2008, p. 1). We do not suggest that the other program authors did not view teachers in this way, but an acknowledgment of this aspect of the teacher–curriculum relationship was not highlighted in the materials or the interviews.2 In the following discussion, we offer some recommendations for designing materials that anticipate teachers’ work using them and view teachers as learners. Our findings in Chaps. 5, 6, 7, and 8 revealed considerable gaps in support for teachers and a tendency to focus on directing action, with limited attention to explaining underlying ideas, anticipating student responses and challenges, and elaborating the rationale behind design decisions. In order to fulfill their role in the curriculum-teacher relationship, teacher’s guides need to contain greater transparency about the mathematical and pedagogical ideas and intentions underlying their designs. Having access to such information can contribute to the work of perceiving  Elsewhere (Brown et al., 2009), MTB authors, together with researchers, describe the process by which teachers interpret features of lesson guides to develop a model of authors’ intended lesson, clearly demonstrating an understanding of teachers’ interpretive work. We draw on this framework earlier in this chapter, but do not refer to it here because it was not made explicit in the material we analyzed or our interviews with the authors. 2

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and mobilizing the designs in materials by helping them develop a deeper understanding of authors’ ideas underlying lesson components (which are often signaled through directing action). When armed with an understanding of the underlying ideas, teachers can assess the appropriateness of the suggested tasks for their students and make modifications and decisions during the lesson that reflect these ideas. Without this type of understanding, teachers are simply left with directions to follow. Beyond greater transparency, curriculum authors might also provide features intended to directly support teachers in interpreting curriculum designs. In Chaps. 5, 6, 7, and 8, we draw on Davis and Krajcik’s (2005) ideas about designing curriculum materials that are educative for teachers to consider ways that authors support teachers. We use the term communicative supports to refer to features in materials designed to help teachers understand the designs and learn more about the mathematics, student learning, or intended teacher roles. These types of support include, but also go beyond, transparency. For instance, they might take the form of offering several possible student strategies for solving a problem or explaining how the mathematical tasks and representations in one lesson build on previous lessons. These types of supports not only have the potential to help teachers enact the particular lesson, but can position teachers to learn from doing so. For instance, in Chap. 7, we noted that some supports in the teacher’s guides were written to help teachers notice key aspects of children’s mathematical thinking, while others attended to superficial aspects of students’ work or previously determined ways of categorizing or positioning students (such as performance level). One critical form that many educative supports discussed in Chap. 5 take is communication to the teacher, rather than simply through the teacher. We recommend the creative use of educative supports as a way to help teachers learn from using the program. The programs we analyzed in this volume were print-centric. The digital components associated with the programs at the time of the study were static (i.e., PDF versions of the teacher’s guide). Interactive and dynamic digital components currently in use hold great potential for incorporating educative supports in ways that are easily accessible to the teacher. The potential use of scroll-over pop-up boxes with additional information, embedded videos of sample teacher practices or student explanations, and images of student work can offer an educative layer that teachers can access when planning lessons and return to afterward. We have observed many advances in the realm of dynamic digital curriculum programs for teachers and students, but have not seen strategic use of these formats to advance teacher design capacity.

9.6.3  Implications for Research and Innovation Within many of the chapters in this volume, we have offered specific areas where further research is needed to understand how teachers use particular curriculum features or their impact on classroom instruction. Here we focus on several recom-

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mendations for future work pertinent to the themes in this chapter—the complex and layered characteristics of curriculum materials as designed artifacts and the need to consider the work that teachers do when using them. As noted earlier, the framework guiding our analysis considered several different lenses and multiple dimensions within these lenses. This framework and our findings can inform future curriculum analysis and curriculum development. Our broad focus and the many dimensions we considered allowed us to characterize the complexity of designed curriculum materials. In Chap. 1, we categorized different foci adopted by those who analyze curriculum materials. Our approach fits into the comprehensive focus, which takes into account many aspects of materials in relation to one another and seeks to offer a holistic perspective. Few curriculum analyses have taken a comprehensive approach. Two of our three lenses (pedagogical approach and communication with the teacher) have infrequently been the focus of curriculum analyses. Further, within each lens, we examined multiple dimensions, uncovering nuanced components of designed curriculum materials that comprise the artifacts teachers engage with. We believe the comprehensive nature of our analytical framework could be considered somewhat of a road map for future researchers considering curriculum analysis. We are not arguing for a single approach, but the comprehensive framework contributes to mapping out the terrain of lenses and dimensions of significance. Future researchers are likely to identify additional lenses and dimensions that are also of importance, further adding to the map. Additionally, the discussion of the methodological decisions we made and the questions they raised can inform future researchers undertaking similar work. In her commentary (Chap. 10), Kirsti Hemmi discusses the value that the detailed descriptions of methodological decisions provided in this volume could have for other researchers. She notes that within the field “There exists no manual for the work [curriculum analysis].” Our analytical approach and our findings can inform curriculum authors as well. Our analytical framework examines curriculum programs through lenses they may not have previously considered, placing particular attention on how authors’ intentions are made accessible to teachers. Our findings also reveal how various design decisions play out in relation to others and within different overarching approaches. We hope that these insights raise new questions for curriculum developers about how designs they create might be interpreted by teachers and encourage them to explore new approaches. We conclude with a final recommendation about research on teachers’ use of curriculum materials. In the previous sections, we argued for work aimed at strengthening the curriculum–teacher relationship and teachers’ capacity to use curriculum materials productively. The field is in need of research that contributes to ­understanding of this relationship and how to support teachers in developing robust and flexible ways of interacting with curriculum materials to meet their instructional goals. As curriculum developers experiment with new approaches to designing materials for teachers, using print or digital formats, these efforts need to be accompanied by research on how teachers read, interpret and use them. This research might also be extended to how teachers navigate and make use of the increasingly

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wide range of online and Open Educational Resources currently available to teachers, discussed in Chap. 1. Given the complex nature of curriculum resources and the regularity with which teachers make use of them, we see this research as essential to understanding and supporting a core aspect of the work of teaching.

References Adler, J. (2019, September). Revisiting resources as a them in mathematics (teacher) education. Plenary address given at the Third International Conference on Mathematics Textbooks, Paderborn. Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is—Or might be—The role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–8. Ball, D. L., & Feiman-Nemser, S. (1988). Using textbooks and teachers’ guides: A dilemma for beginning teachers and teacher educators. Curriculum Inquiry, 18(4), 401–423. Bezemer, J., & Kress, G. (2008). Writing in multimodal texts—A social semiotic account of designs for learning. Written Communication, 25(2), 166–195. Bezemer, J., & Kress, G. (2016). Multimodality, learning and communication: A social semiotic frame. London: Routledge. Brown, M.  W. (2009). Toward a theory of curriculum design and use: Understanding the teacher-tool relationship. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 17–36). New York: Routledge. Brown, S.  A., Pitvorec, K., Ditto, C., & Kelso, C.  R. (2009). Reconceiving fidelity of implementation: An investigation of elementary whole-number lessons. Journal for Research in Mathematics Education, 40(4), 363–395. Bruner, J. (1977). The process of education. Cambridge, MA: Harvard University Press. Charles, R. I., Crown, W., Fennell, F., et al. (2008). Scott Foresman–Addison Wesley Mathematics. Glenview, IL: Pearson. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Desimone, L. M., & Garet, M. S. (2015). Best practices in teacher’s professional development in the United States. Psychology, Society, & Education, 7(3), 252–263. Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Mahwah, NJ: Lawrence Erlbaum Associates. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel Publishing. Grossman, P., Wineburg, S., & Woolworth, S. (2001). Toward a theory of teacher community. Teachers College Record, 103, 942–1012. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71(3), 199–218. Gueudet, G., & Trouche, L. (2012). Communities, documents and professional geneses: Interrelated stories. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources (pp. 305–322). New York: Springer. Hiebert, J., Stigler, J. W., Jacobs, J. K., Garnier, H., Smith, M. S., Hollingsworth, H., et al. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 Video Study. Educational Evaluation and Policy Analysis, 27(2), 111–132. Kim, O.  K. (2019). Teacher capacity for productive use of existing resources. In L.  Trouche, G. Gueudet, & B. Pepin (Eds.), The ‘resource’ approach to mathematics education. New York: Springer.

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Leshota, M., & Adler, J. (2018). Disaggregating a mathematics teacher’s pedagogical design capacity. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Recent advances in research on mathematics teachers’ textbooks and resources (pp. 89–118). New York: Springer. Little, J. W., & Horn, I. S. (2007). ‘Normalizing’ problems of practice: Converting routine conversation into a resource for learning in professional communities. In L. Stoll & K. S. Louis (Eds.), Professional learning communities: Divergence, detail and difficulties (pp.  79–92). Maidenhead: Open University Press. Lloyd, G. M. (1999). Two teachers’ conceptions of a reform-oriented curriculum: Implications for mathematics teacher development. Journal of Mathematics Teacher Education, 2(3), 227–252. Marshall Cavendish International. (2010). Math in focus: The Singapore approach by Marshall Cavendish. Boston: Houghton Mifflin Harcourt. Melville, M. D. (2008). Kyozaikenkyu: An in-depth look into Japanese educators’ daily planning practices. Thesis, Brigham Young University. Munter, C., Stein, M. K., & Smith, M. S. (2015). Dialogic and direct instruction: Two distinct models of mathematics instruction and the debate(s) surrounding them. Teachers College Record, 117(11), 1–32. Otte, M. (1986). What is a text? In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on math education (pp. 173–202). Dordrecht: Kluwer. Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teachers’ work and interactions: A collective perspective on resources, their use and transformations. ZDM – The International Journal on Mathematics Education, 45(7), 929–944. Pepin, B., Gueudet, G., & Trouche, L. (2017). Refining teacher design capacity: Mathematics teachers’ interactions with digital curriculum resources. ZDM – Mathematics Education, 49(5), 799–812. Retrieved from http://rdcu.be/tmXb. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Remillard, J. T. (2012). Modes of engagement: Understanding teachers’ transactions with mathematics curriculum resources. In G.  Gueudet, B.  Pepin, & L.  Trouche (Eds.), From text to ‘lived’ resources (pp. 105–122). New York: Springer. Remillard, J.  T. (2018). Examining teachers’ interactions with curriculum resource to uncover pedagogical design capacity. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Recent advances in research on mathematics teachers’ textbooks and resources (pp. 69–88). New York: Springer. Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientations toward mathematics curriculum materials: Implications for teacher learning. Journal of Research in Mathematics Education, 35(5), 352–388. Remillard, J.  T., & Kim, O.  K. (2017). Knowledge of curriculum embedded mathematics: Exploring a critical domain of teaching. Educational Studies in Mathematics, 96(1), 65–81. Remillard, J. T., & Taton, J. (2015). Rewriting myths about curriculum materials and teaching to new standards. In J. A. Supovitz & J. Spillane (Eds.), Challenging standards: Navigating conflict and building capacity in the era of the common core (pp. 49–58). Lanham, MD: Rowman & Littlefield. Shouffler, J. (2018). Teacher learning within United States lesson study: A study of a middle school mathematics lesson study team. Dissertation, University of Pennsylvania. Sleep, L. (2009). Teaching to the mathematical point: Knowing and using mathematics in teaching. Unpublished doctoral dissertation, University of Michigan. 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. 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.

9  Complexity of Curriculum Materials as Designed Artifacts: Implications and Future… 285 Stein, M. K., Grover, B. W., & Henningsen, M. A. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classroom. American Educational Research Journal, 33(2), 455–488. TERC. (2008). Investigations in Number, Data, and Space (2nd edition). Glenview, IL: Pearson Education Inc. TIMS Project (2008). Math Trailblazers (3rd Edition). Dubuque, IA: Kendall/Hunt Publishing Company. Treffers, A. (1987). Three dimensions: A model of goal and theory description in mathematics instruction—The Wiskobas project. Dordrecht: Reidel Publishing. University of Chicago School Mathematics Project. (2008). Everyday Mathematics (3rd Edition). Chicago, IL: McGraw-Hill. van Es, E. A. (2009). Participants’ roles in the context of a video club. The Journal of the Learning Sciences, 18, 100–137. Watanabe, T., Takahashi, A., & Yoshida, M. (2008). Kyozaikenkyu: A critical step for conducting effective lesson study and beyond. In F. Arbaugh & P. M. Taylor (Eds.), Inquiry into mathematics teacher education (pp.  139–142). San Diego, CA: Association of Mathematics Teacher Educators. Wenger, E., McDermott, R. A., & Snyder, W. (2002). Cultivating communities of practice: A guide to managing knowledge. Boston, MA: Harvard Business School Press. Winsløw, C. (2012). A comparative perspective on teacher collaboration: The cases of lesson study in Japan and of multidisciplinary teaching in Denmark. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources (pp. 291–304). New York: Springer.

Chapter 10

Commentary Kirsti Hemmi

Abstract  I have enjoyed reading the chapters of this book where each of them alone, and together with the others, offers a unique perspective on the potential of the five chosen curriculum programs to communicate with teachers, and hence, indirectly influence classroom practices and student learning. The contribution of the book ranges from textbook authors’ intentions, to the character of mathematics that the students and teachers are asked to meet, to numerous pedagogical aspects and communication with teachers. In this commentary chapter, I first briefly describe my perspective. I then proceed to reflect on some specific issues that I find interesting when reading the chapters with an outsider’s eyes. Then, I discuss the visions of curriculum authors and researchers in relation to classroom reality in different contexts. I conclude by discussing the main contributions of the book to the research field, including some future visions and challenges. Keywords  Curriculum analysis · Mathematics curriculum materials · Commentary · Finnish perspective · Finnish textbooks · Swedish textbooks · Globalization of education · Standards-based curricula · Teacher’s guides · Curriculum digitization

10.1  My Perspective Despite the globalization of education and the international evaluations that bring people working with education closer to each other, there are still huge differences between the educational cultures in different countries. These differences might concern the expectations of what children need to learn, teacher education, the view of mathematics and pedagogy, as well as factors outside the school system, such as the roles of parents. Obviously, the character of curriculum materials, their role in

K. Hemmi (*) Åbo Akademi University, Vaasa, Finland e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0_10

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education and teachers’ interaction with them differ accordingly. I read and ­interpret the studies in this book from a background as a researcher and a teacher educator in both Finland and Sweden, two neighboring Nordic countries with many similarities in their educational systems, but clear differences in classroom cultures and curriculum materials (Hemmi, Krzywacki, & Liljekvist, 2018). The compulsory schools (Grades 1–9) in both countries are inclusive and do not track students. Moreover, a long-lasting contact with one class teacher is appreciated and it is usual that teachers follow their class at least 2 years, sometimes even 6 years (from Grade 1 to 6). As opposed to the situation in the USA, where teachers have to follow a certain curriculum program, the national curriculums in Finland and Sweden leave a lot of space for teachers’ own decisions. Teachers in both countries can quite freely choose to use the material they like or choose not to use any specific “program.” This autonomy has, especially in Sweden, been desirable and viewed as a characteristic of a good teacher in the view of national school agencies and teacher educators since the 1990s. According to this view, an ideal teacher does not follow a certain program but creatively plans their own ways of designing and enacting mathematics lessons suitable for a specific class and individual students (compare with American teacher education in the 1980s in Ball and Feiman-Nemser (1988)). Yet, curriculum materials that are commercially produced play an important role in mathematics classroom practices in both countries, and should, in my opinion, be of utmost interest among policymakers, researchers, teacher educators, and school agencies in our countries. In contrast to the situation in Sweden, there is a long tradition in Finland of producing profound teacher’s guides separately from textbooks, written by teams of teachers and teacher educators, with a clear focus on designing lessons and an eye to mathematical storylines (Hemmi, Krzywacki, & Koljonen, 2018). In Sweden, textbooks have been the major source of knowledge in individualized instruction where students work with their textbooks at their own pace (e.g., Bråting, Madej, & Hemmi, 2019). Hence, in Sweden, designing specific lessons has not been a focus in the materials, and teacher’s guides have played a minor role in teachers’ work (Hemmi, Krzywacki, & Liljekvist, 2018). The globalization of education, including the international evaluations, has led to growing interest in specific features of other countries’ mathematics classrooms, especially of the countries where student achievement is comparatively high. Relative to the situation in Finland, Swedish students’ results in international comparisons of mathematics have been poor, and a Finnish curriculum program is gaining popularity in Sweden. Recently, in line with efforts in the USA (MIF, one of the programs analyzed in this book), a textbook series in mathematics following Singapore mathematics has also been launched in the Swedish market, as well. When reading the studies in this book, I make sense of them with the knowledge of international research on curriculum materials, knowledge of NCTM Standards, and their connections to recent educational trends in our countries. I also relate the results to the situations in the Nordic countries with which I am familiar.

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10.2  A  Reflection on the Studies Seen with an Outsider’s Eyes The authors investigate the five curriculum programs they have chosen from different perspectives and manage to offer an exceptionally nuanced picture of each. Every analysis reveals unexpected differences and similarities between the programs. In Chap. 4, concerning the curriculum authors’ intentions, the concept of “design features” is presented. The concept serves well for sorting out general features of curriculum programs such as sequencing of the content, the structure and positioning of specific activities, and information for teachers across different parts of the materials, all of which are important to take into account when analyzing curriculum materials. Herein this, we get a glint of the variety of ways in which curriculum programs may differ from each other. The same visions are realized using different design features leading to products that may communicate differently to teachers the messages based on NCTM Standards. In Chap. 2, the authors deeply explore the differences between the materials by analyzing the mathematical content in chosen areas. The selected mathematical content also reflects the pedagogical views of the authors. While all three Standards-­ based programs include activities at a high cognitive level, such as “procedures with connections” and “doing mathematics,” INV stands out by totally excluding activities of lower cognitive demand. Globalization of education has resulted in an enhanced exchange of ideas, and it is possible to identify similar pedagogical trends in different parts of the world across the past several decades. The NCTM Standards are based on international research, and therefore similar goals permeate, to some extent, the national curriculums in Finland and Sweden as well. These curriculums base mathematics teaching on students’ thinking, investigating, and inventing their own strategies instead of learning standard algorithms. Students should also discuss and compare various strategies, make everyday connections, and use different representations. Teachers’ roles in mathematics classrooms should be supporting student agency, rather than transmitting knowledge. At least in Sweden, similar ideas of good teaching can also be found within teacher education discourses (Hemmi & Ryve, 2015). However, looking closer to the educational practices, classroom cultures, curriculum materials, and their use by teachers, we find various interpretations of the same goals. These interpretations differ substantially, even between the actors in Finland and Sweden, despite the geographical and cultural closeness as shown below. Chapter 3 reveals interesting differences concerning pedagogical approaches of the five programs in relation to the participant structure, a teacher’s role, and the nature of student learning and work. All five programs show a participant structure where whole-class activities take about half of the lesson time. Further, we learn that the three programs following the NCTM Standards emphasize students as the source of knowledge in proportions of the lesson time ranging from 24% in EM to at least 60% in MTB and INV. This takes the form of different activities, such as

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orchestrating whole-class discussions and facilitating students’ individual or group work. Orchestrating whole-class discussions takes a major part of the potential lessons in INV (40%) and MTB (27%), but only 2% in EM. In Chap. 7, we learn more about the ways in which the different materials guide a teacher in anticipating, attending to, and evaluating students’ input. This is important when encouraging students to invent their own methods and building their collective mathematics knowledge. It is worth noting that the two programs that stress whole-class discussions (INV and MTB) also frequently provide examples of multiple strategies and possible student thinking, as well as advice at a high level of rigor on how a teacher can use student strategies in mathematically productive whole-class discussions. This kind of support in attending to student thinking is absent from curriculum materials in Finland and Sweden. I have not seen teacher’s guides in our countries that describe in such detail the flow of the expected classroom discussion based on potential student thinking, as for example INV (see Fig. 6.2). In fact, such detail is unthinkable in our contexts, considering the ideal role of the teacher requires continuously adapting her/his actions in response to what happens in the classroom and not leaning too much on the curriculum materials. For example, Finnish materials are created more like mind-maps circulating the mathematics that is the focus of the lesson, and teachers choose which elements to teach, and in what order they enact their lessons (Hemmi, Krzywacki & Koljonen 2018; Hemmi et al., 2018a). This is also an explanation for why, in an analysis of Swedish and Finnish curriculum materials, the parallel structural categories referring to the place of the lesson flow could not be meaningful as their teachers’ guides do not usually have instructions about how to proceed in a lesson. In my opinion, however, the kind of support offered in the three Standards-based programs would be valuable in our countries as well, since both the Finnish and Swedish curriculums expect teachers to lead discussions on different solutions to mathematical problems. Teachers might not be used to leading open discussions, and they may not be able to anticipate students’ ideas and connect them to rigorous mathematical ideas. In Sweden, the idea of basing mathematics on student thinking has been conceptualized in a very different manner than in the USA, namely in striving to create a starting point for discussions from students’ everyday experiences, and their spontaneous questions. The problem has been that in these discussions, the mathematical storyline, as well as doing mathematics and making mathematical connections, may not be realized (Hemmi & Ryve, 2015). Chapter 6 focuses on how mathematics is communicated to the teachers. In line with the authors, we also used Davis and Krajcik’s (2005) ideas of educative features in our analyses of Finnish teacher’s guides (Hemmi, Krzywacki, & Koljonen, 2018). Yet, we interpreted mathematical knowledge provided by the teacher’s guides as a kind of background knowledge about the subject beyond the knowledge their students need to learn. Hence, we did not find much of such information in the guides. Moreover, in line with the design of MIF, it is assumed that Finnish class teachers have good knowledge of mathematics already when they enter the university and therefore some of the results presented in Chap. 6 seem trivial in light of this background. However, when reading the results, I have started to ponder

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whether this kind of finer-grained analysis could be useful in Swedish curriculum materials, including the imported materials from Finland and Singapore, as there are many class teachers who have a poor educational background in mathematics in Sweden. Sleep (2012) stresses the importance of articulating the mathematical point and keeping the focus of lessons on the mathematical storyline, which is certainly realized in the Finnish materials but not in the traditional Swedish materials. Some of them do not even use mathematical titles for chapters. Swedish teachers who started to use the Finnish material appreciated in particular the consequent use of mathematical concepts and making connections between them as experienced in the Finnish teacher’s guides (Hemmi, Krzywacki, & Liljekvist, 2018). The method of analysis in Chap. 6 would certainly reveal interesting differences between the traditional Swedish materials and the imported materials that have important consequences for teacher learning from curriculum materials. Comparing the five materials with Finnish and Swedish materials, there are similarities between the MIF and the Finnish materials, in that they both reflect a responsibility of the teacher to scaffold student learning. There is no place for students’ inventions, as the teacher is the source of knowledge. Yet, in Finnish guides, game activities, mental calculations, and concrete materials play an important role. Comparatively, SFAW is the material that I would characterize as closest to traditional Swedish materials because the main source of knowledge is the student textbook. Chapter 8 investigates how the materials make design decisions transparent to teachers. This question is interesting as textbook productions often take for granted their own traditions, and sometimes the rationale is invisible even for the authors. The production of Standards-based materials has obviously increased a focus on design transparency, something that we can see in the results in Chap. 8. As the authors point out, this kind of information is more likely to be found in program, unit or chapter level support. The rationale for why only lessons were analyzed was that it was assumed teachers are more likely to read them than the information elsewhere. In my experience, at least Swedish teachers study new materials trying to find a rationale behind the chosen design features and other decisions (Hemmi, Krzywacki, & Liljekvist, 2018), especially when they make their long-term plans in the beginning of a term. It is possible that teachers who are used to following a program lesson-by-lesson do not think about a rationale. If a teacher uses materials that follow a certain familiar tradition, it is possible that the teacher takes for granted the order and the sequencing of the content as well as the representations and the pedagogical approach used in the material. We have noticed that in the Finnish context, one can find the same activities, features, and qualities that Finnish teacher educators consider characteristics of good mathematics teaching (Hemmi & Ryve, 2015) in all four of the programs covering nearly 90% of the market (Hemmi, Krzywacki, & Koljonen, 2018). There is very little design transparency found in these programs, but we assume that Finnish teachers are already familiar with similar ideas from their teacher education and from being part of a certain educational culture. This is not the case with Swedish teachers who worked with the Finnish materials (Hemmi, Krzywacki, & Liljekvist, 2018), and I think that design transparency is especially

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important when importing materials from another context or introducing new ideas for mathematics classrooms. Of course, it is important for all actors such as researchers, national curriculum authors, and textbook authors to reflect on rationale behind different recommendations and decisions. A part of design transparency concerns knowledge of curriculum. I find it important for both teachers and students to acknowledge when they can make connections between tasks and key ideas across units and grade levels, as seen in MTB and EM. These connections might be even more important for teachers who teach only one grade level. Rather than a mathematical storyline, I would like to think of the progression in mathematics in terms of a network of various threads that are connected to each other in more or less complex manners, both horizontally and vertically (Bernstein, 2000). Materials could help both teachers and students make various connections between the threads and their work in them. Limiting the study on design transparency in Chap. 8 to lesson pages allows us to find the ways in which a teacher could learn in close connection with classroom practice, as opposed to long-term planning situations where the actual enactment of instruction is still imaginary. The ideal, I think, would be both deep reflection of the design rationale and the big ideas behind it at the program, unit, and Chap. level, and references to them on the lesson level. In that way, the abstract ideas would become more concrete, while learning by doing could reinforce the understanding of the big ideas and the important decisions behind the mathematical contents and pedagogical approaches, as well as other design features at a more general level. Finally, in Chap. 8, the authors reflect on the possible affordances that digitalization could offer for providing teachers design transparency. Certainly, it is easier to create a map of a network and help teachers and students to connect separate activities to different parts of the network with the help of digital techniques. In addition, hyperlinks to deeper descriptions in other lesson pages allow teachers to easily refresh their memory concerning what they have learned when planning a longer sequence of instruction. Further, one figure could speak to more than several pages of text. At least in some old Finnish teachers’ guides, the progression is illustrated with a kind of network map where a teacher can see the optional development over several school years on the first pages. With the help of digital techniques, communication of these kinds of figures could be made even more effective.

10.3  Who Knows Best? Next, I reflect on some issues connected to the different pedagogical philosophies behind the approaches described in Chap. 3. An important question for research is how to take into account different social factors surrounding the context of teaching and learning in the production of curriculum materials (cf. Jaworski, 2009). For example, Doerr and Chandler-Olcott (2009) contemplate the literacy demands of one curriculum program following the NCTM Standards and the difficulties experienced by teachers working with second language learners, students who struggle

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with academic literacy, and students who struggle with mathematics. The working manners advocated in curriculum may not always provide enough support for working with individuals with special needs. This is especially important in Finland and Sweden where the classes are often heterogeneous. My concern is also what happens with students who have difficulties with concentration in classrooms where the knowledge is to be constructed in whole-class discussions. What happens with children who do not manage to invent any methods? Is their role to listen and try to make sense of others’ presentations and discussions which may not always be clear and logical, and may therefore be difficult to follow? Thinking about the pedagogical theories, for these individuals, what is the difference between this practice and listening to teachers’ explanations and demonstrations, except that they could be clearer? The same pedagogical approaches may not suit all students. For this reason, flexibility of use is of utmost interest in the development of curriculum materials (cf. Brown, 2009). Rather than directing teachers’ actions, it might be better to offer well-founded ideas for each topic with which a teacher could compose her/his own music according to the needs of students. An important question that Pimm (2009) raises is “who knows best” what is to be dealt with in mathematics classrooms, and what are the best design features in enacting instruction. What is possible to realize in different contexts? The textbook authors in Chap. 4 state that they attempt to follow up on teachers’ experiences and use of the material and continuously develop it. One way to do this could be to use teachers’ social media. For example, the company publishing the Finnish curriculum materials in Sweden has created a Facebook site where teachers can pose questions and discuss, with both each other and the authors, issues connected to teaching, learning, and the materials. There are now over 10,000 members, which is a lot in a country with only about 10 million citizens. This is in line with the idea of seeing teachers not only as consumers but as co-producers of materials (cf. Gueudet & Trouche, 2009). This could also be the first step in gathering students’ voices through their teachers in order to develop the materials accordingly. A key concern is also the place of educational research in the production and use of curriculum materials. Yet, there was very little reference to research in the lesson pages of the materials as shown in Chap. 8.

10.4  Contribution and Future Visions We know from research on textbooks and teacher’s guides that they have a potential to affect the quality of instruction in various ways that are not easy to predict (Stein, Remillard, & Smith, 2007). Each chapter of the book contains several important results that shed light on this complex relation from different points of view and provides a thoughtful and meticulous picture of important characteristics of each program and their potential to offer possibilities for both student and teacher learning. As the authors point out, pedagogical aspects in curriculum materials have not gained attention in the field of curriculum studies. Not only do we need knowledge

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about the contents, but also the ways in which they are dealt with connected to the most basic questions within mathematics education concerning the character of the subject and its effective instruction. The studies presented in the book make a substantial contribution by both developing relevant methods for this purpose and presenting well-founded results. There is significant work behind every study of the book offering a new viewpoint for the cases. From my own experience, I know that there are several difficult decisions to be made when analyzing this kind of material. There exists no manual for the work. The list of methodological challenges stated in earlier research (Chap. 1) only offers a general frame to begin with. Within every challenge, there are several important decisions to be taken. Even the question of when it is meaningful to quantify, and in that case in what way, is difficult to solve. Programs are differently structured and the amount of text varies between different parts of the materials. Besides, sometimes a figure can address several pages. All of these differences have to be taken into account in a relevant manner. In my opinion, the authors have succeeded in an impressive way. The challenges have dealt with meticulous considerations and the authors point out clearly how certain compromises in their work should be taken into account when interpreting the results. The authors have also succeeded in their search for relevant theoretical stances for analyses by creatively utilizing theories developed within classroom studies and studies concerning teacher knowledge and operationalizing these theories for analyses of different parts of the curriculum materials in a purposeful manner. As an example, by viewing components of instruction as a system of interrelated factors (Hiebert et al., 2005) the authors avoid dichotomizing the scene when characterizing the pedagogical approaches of the five different programs. In addition, the procedures and the decisions made in connection with different analyses are described in a transparent way, allowing other researchers to conduct similar studies in other contexts by applying, modifying, and developing the methods. To conclude, careful and detailed studies on curriculum materials, including the pedagogical approaches, are urgent when the digitalization rapidly reshapes classroom practices. Various convenient tools for teachers, cognitive assistant tools, and tutorial systems for students are often based on ideas in written curriculum materials. Very soon, social robots with artificial intelligence will be common as assistants in mathematics classrooms. If all these inventions are to work in an ideal manner in different contexts and individuals, we have to be aware of the mathematical contents and the pedagogical approaches in materials that these developments will be based on. Therefore, it is important to consider all the factors presented in this book when developing digital resources for different purposes and contexts. With deep knowledge of the resources, including their pedagogical and mathematical visions, the development of digital instructional support will avoid the deficiencies of existing materials, but also identify what exactly could be the added value of the new techniques.

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References Ball, D. L., & Feiman-Nemser, S. (1988). Using textbooks and teachers’ guides: A dilemma for beginning teachers and teacher educators. Curriculum Inquiry, 18(4), 401–423. Bernstein, B. (2000). Pedagogy, symbolic control and identity: Theory, research, critique. Lanham, MD: Rowman & Littlefield Publishers. Bråting, K., Madej, L., & Hemmi, K. (2019). Development of algebraic thinking: Opportunities offered by the Swedish curriculum and elementary mathematics textbooks. Nordisk matematikkdidaktikk, NOMAD: [Nordic Studies in Mathematics Education], 24(1), 27–49. Brown, M. W. (2009). The teacher-tool relationship: Theorizing the design and use of curriculum materials. In J. T. Remillard, B. Herbel-Eisenmann, & G. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 17–36). New York: Routledge. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Doerr, H. M., & Chandler-Olcott, K. (2009). Negotiating the literacy demands of Standards-based curriculum materials. In J.T. Remillard, B. Herbel-Eisenmann, & G. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 283–301). New York: Routledge. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers. Educational Studies in Mathematics, 71(3), 199–218. Hemmi, K., Krzywacki, H., & Koljonen, T. (2018). Investigating Finnish teacher guides as a resource for mathematics teaching. Scandinavian Journal of Educational Research, 62(6), 911–928. https://doi.org/10.1080/00313831.2017.1307278. Hemmi, K., Krzywacki, H., & Liljekvist, Y. (2018a). Challenging traditional classroom practices: Swedish teachers’ interplay with Finnish curriculum materials. Journal of Curriculum Studies, 1–20. https://doi.org/10.1080/00220272.2018.1479449. Hemmi, K., & Ryve, A. (2015). Effective mathematics teaching in Finnish and Swedish teacher education discourses. Journal of Mathematics Teacher Education, 18(6), 501–521. Hiebert, J., Stigler, J. W., Jacobs, J. K., Garnier, H., Smith, M. S., Hollingsworth, H., et al. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 Video Study. Educational Evaluation and Policy Analysis, 27(2), 111–132. Jaworski, B. (2009). Development of teaching through research into teachers’ use of curriculum materials and relationships between teachers and curriculum. In J.  Remillard, B.  Herbel-­ Eisenmann, & G. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 338–346). New York: Routledge. Pimm, D. (2009). Who knows best? Tales of ordination, subordination and insubordination. In J.  Remillard, B.  Herbel-Eisenmann, & G.  Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp.  190–196). New  York: Routledge. 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. Stein, M. K., Remillard, J.T, & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics 1(1) (pp. 319–370). Gweenwich, CT: IAP.

List of Appendices

A—Interview Questions for Curriculum Authors Appendices B–E contain 3 pages from a single lesson found in four of the five curriculum program teacher’s guides. These samples are included to provide a visual image of the content and layout of the different guides. The particular pages were selected to illustrate points made in the analysis chapters. We have not included samples from MIF because the publishers did not respond to our requests for permission to reprint. Selected photographs have been covered in SFAW at the publisher’s request. B—Sample pages from EM Lesson GR3 7-6 C—Sample pages from INV Lesson GR4 U3 1-2 D—Sample pages from MTB Lesson GR3 6-2 E—Sample pages from SFAW Lesson GR3 3-10

© Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0

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Appendix A: Interview Questions for Curriculum Authors Introduction: In our study we analyzed [curriculum-edition], published in 2008. We recognize that you have done a great deal of work on [curriculum name] since working on this edition, so it may be challenging to remember the particulars. First, we will ask some questions about your development of [curriculum name] in general and then will ask questions about specific aspects of the design. 1. What were your major goals in developing [curriculum name]? And [curriculum-edition]? a. What are some of the design features of [curriculum name] that reflect these goals? 2. In our study, we analyzed [curriculum-edition]. What does the edition represent in terms of how [curriculum name] evolved? The following questions allow us to dig into various design decisions your team made and the thinking behind them. Please share your general responses and any responses that are particular to [curriculum-edition]. 3. First, we would like to understand the mathematical emphasis of [curriculum name]. What do you expect a student will be able to do and/or understand after learning mathematics through using [curriculum name]? a. What are some of the features of the program that illustrate this emphasis? b. How, if at all, did you communicate this emphasis to teachers? 4. How did your development team make decisions about sequencing content within and across grades? Especially, in number and operations? c. How are these decisions evident in the structure of the program? d. How, if at all, did you communicate these decisions to teachers? 5. What is the vision your team had for teachers’ pedagogical practices when using [curriculum name]? e. How is this vision evident in the program? f. How, if at all, did you communicate this vision to teachers? 6. Some curriculum authors include different types of information for teachers (such as about the mathematics or common student misconceptions), which have come to be called educative features. How did your curriculum design team think about providing these types of information? Questions that are curriculum specific: 7. Each of the lesson overviews in EM3 include statements of the lesson objectives, the key activities, and the key concepts and skills. Can you describe the difference in these categories?

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8. Each of the lesson overviews in Investigations includes Math Focus Points. How did you determine them for each lesson? We also find that often the same math focus points appear in multiple lessons. Can you describe the rationale for this? 9. We noticed that Math Trailblazers, more than other programs, includes lessons intended for two or more days; Tell us about the decision to structure lessons in this way. How did you think about how teachers might break up the sessions? How, if at all, did you communicate these decisions to teachers? 10. Another noticeable feature of Math Trailblazers is the Daily Practice Problems Section. What were you trying to accomplish with this daily set of tasks? 11. Each of the lessons begins with a Lesson Overview, which is a brief description of the lesson, followed by a list of Key Content. Can you describe how you think about these two things? How did you decide what would be on the Key Content list? 12. How were you hoping teachers were using this information [those from #7 to #11] when planning or teaching? 13. We’d like to look at a specific lesson to get a sense of how you were thinking about designing the lesson guides for teachers’ use (see the attached lesson for example). Using the attached lesson or any lesson you pick, would you help us understand or see examples of your responses to the above questions? General questions about [curriculum name] in context: 14. What components of the program did you anticipate teachers would find challenging to use? 15. In your experience, what did teachers find challenging to use? a. Are there changes you made to respond to these challenges? 16. In what ways was [curriculum name] distinct from the curriculum programs available when you first developed it? 17. In what ways was [curriculum name] distinct from other NSF funded programs?

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Appendix B: Sample pages from EM Lesson GR3 7-6

Objectives

To guide children as they multiply 1-digit numbers by multiples of 10, 100, and 1,000 and divide such multiples by 1-digit numbers.

1

materials

Teaching the Lesson

Key Activities

Math Journal 2, p. 169

Children develop strategies for multiplying and dividing using 1-digit numbers and multiples of 10, 100, and 1,000.

Home Link 7 5 slate

Key Concepts and Skills • • • • •

Recognize multiples of 10. [Number and Numeration Goal 3] Use multiplication facts to solve problems. [Operations and Computation Goal 3] Use multiplication facts to solve division problems. [Operations and Computation Goal 3] Share solution strategies for solving number stories. [Operations and Computation Goal 4] Use relationships between units of time to solve number stories. [Measurement and Reference Goal 3]

Key Vocabulary extended facts

Ongoing Assessment: Informing Instruction See pages 608 and 610. Ongoing Assessment: Recognizing Student Achievement Use journal page 169. [Measurement and Reference Frames Goal 3]

2

Ongoing Learning & Practice

materials

Children practice basic multiplication facts by playing Beat the Calculator.

Math Journal 2, p. 170

Children practice and maintain skills through Math Boxes and Home Link activities.

Student Reference Book, p. 279 Home Link Master (Math Masters, p. 220) number cards 1–10 (4 of each) calculator

3

materials

Differentiation Options

READINESS

Children use calculators and base-10 blocks to find patterns in counting.

EXTRA PRACTICE

Children practice counting with 10s, 100s, and 1,000s.

Teaching Master (Math Masters, p. 221) Minute Math ®+, pp. 5, 7, and 14 base-10 blocks; calculator

Technology Assessment Management System Journal page 169 See the iTLG.

606

Unit 7 Multiplication and Division

List of Appendices

301

Getting Started Mental Math and Reflexes

Math Message

Pose the following problems.

The distance around a racetrack is 500 meters. How far does a racer travel in 8 laps? Record the answer on your slate. 4,000 meters

Children record their answers on their slates. Have them share solution strategies after each problem. One side of a square is 10 centimeters long. What is the perimeter of the square? 40 cm The sides of an equilateral triangle are 5 meters long. What is the perimeter of the triangle? 15 m (You may want to review the meaning of equilateral.) The longer side of a rectangle measures 8 inches. The shorter side is half as long. What is the perimeter of the rectangle? 24 inches

Home Link 7 5 Follow-Up Have children share a few ways to get 15 points in basketball.

1 Teaching the Lesson Math Message Follow-Up

WHOLE-CLASS DISCUSSION

Children share solution strategies as you record them on the board. Possible strategies: Each lap is 500 meters. So I can add 500 eight times: 500 500 500 500 500 500 500 4,000. 500 If someone suggests this strategy, it can be used as a lead-in to the next strategy.

NOTE Devise a shorthand notation to record children’s solution strategies. In Everyday Mathematics, brackets are used to separate pairs of quantities. For example, 8 [500s] represents 8 five hundreds. Children will quickly adapt to whatever notation you adopt, as long as it is used consistently.

Group the 500s in pairs to show that the sum of each pair is 1,000. Then the sum of each pair of 1,000s is 2,000; and the sum of each pair of 2,000s is 4,000. Repeated doubling: 2 laps 1,000 meters, so 4,000 meters. 4 laps 2,000 meters, and 8 laps 500 5 hundreds, so 8 [500s] are 8 times as many hundreds, or 40 hundreds, or 4,000. 8 [5s] are 40, so 8 [50s] are 10 times as much as 40, or 400, and 8 [500s] are 10 times as much as 400, or 4,000. There are a number of ways to solve problems like 8 [500s]. Children should use whatever method makes sense to them.

Lesson 7 6

607

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List of Appendices

Student Page Write the number of 3s in each number.

10 100 1,000

1. How many 3s in 30? 2. How many 3s in 300? 3. How many 3s in 3,000?

Solve each

,

5. How many 3s in 120? 6. How many 3s in 1,200?

2,000

7.

,

60

300

2

4

240

3

5

300

1,200 1,500

,

500

1,000

3

1,500

3,000

4

,

5

9.

800 1,000 8,000 32,000 40,000 200

6 3,000

Watch for children who are having difficulty with Problems 7 through 9. Consider drawing the table for the example on the board. Instead of recording the products in the table, write the multiplication expression for each product (for example, 2 × 300, 2 × 2,000). Carry out each multiplication problem and write it in the table. Help children find the missing factors for Problem 9 and continue as above.

A 30-minute television program has two 60-second commercials at the beginning, two 60-second commercials at the end, and four 30-second commercials in the middle.

6 minutes (unit) 24 minutes (unit) 6 = 24

a. How many minutes of commercials are there?

c. Number model:

Sample answer:

30

(Math Journal 2, p. 169)

Ongoing Assessment: Informing Instruction

6,000

10. Solve the number story.

b. How many minutes is the actual program?

PARTNER ACTIVITY

Children complete journal page 169 on their own or with a partner. Note that two of the factors are missing in Problem 9. Thus, multiplication and division are called for to complete the puzzle.

puzzle. Fill in the blanks.

300

,

8.

4 40 400

4. How many 3s in 12?

Solving Extended Multiplication and Division Facts

Math Journal 2, p. 169

Ongoing Assessment: Recognizing Student Achievement

Journal Page 169 Problem 10

Use journal page 169, Problem 10 to assess children’s progress toward using relationships between units of time to solve number stories. Children are making adequate progress if they complete Problem 10a successfully. Some children may complete Problems 10b and 10c successfully. [Measurement and Reference Frames Goal 3]

2 Ongoing Learning & Practice Student Page

Playing Beat the Calculator (Multiplication)

LESSON

7 6

1.

Solve.

2.

6

6

36

7

7

49

7

64 9

9

100

10

10

8

2 7 8

28

81

8

Fill in the missing whole number factors. Sample answers:

3. Add parentheses to complete the

32

6 9

(46 (4

(10

2)

23) 2)

6 in

Complete.

4 8

20

10

50

25

8 10

2

out

16

10

13 6

Children practice basic facts by playing Beat the Calculator (Multiplication). Detailed instructions are on Student Reference Book, page 279.

48

number models. 30

8

Answers vary.

5.

6.

Solve. 6

10

6

30

60 180

50

6

300

420 6

90

70

6

540

Color

1 2

How many fourths are shaded? fourths

Math Journal 2, p. 170

610

Adjusting the Activity Play a variation to continue practice on a specific set of facts. Have children keep one of the two numbers the same for each round so it is always a factor (e.g., always keep 6 as a factor, and draw one other card each turn so the child is always multiplying 6 × n).

of the circle.

Sample answer:

2

(Student Reference Book, p. 279)

14

4 4 8

54 4.

SMALL-GROUP ACTIVITY

Unit 7 Multiplication and Division

A U D I T O R Y

¨

K I N E S T H E T I C

¨

T A C T I L E

¨

V I S U A L

List of Appendices

Appendix C: Sample pages from INV Lesson GR4 U3 1-2

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List of Appendices

List of Appendices

305

306

List of Appendices

Appendix D: Sample pages from MTB Lesson GR3 6-2

Estimated Class Sessions

• • • • •

List of Appendices

307

308

List of Appendices

List of Appendices

Appendix E: Sample pages from SFAW Lesson GR3 3-10

309

310

List of Appendices

List of Appendices

311

Author Index

A Adler, J., 276 Agodini, R., 8, 12, 30, 144 Arias, A.M., 145, 162, 164, 232 Asilainen, M.A., 164 Atanga, N.A., 2 B Ball, D.L., 11, 21, 69, 143, 144, 146, 157, 158, 163, 190, 196, 200, 201, 228–231, 233, 254, 279, 280, 288 Barzel, B., 37, 63 Bass, H., 200 Battey, D., 68, 197 Baumert, J., 163 Ben-Peretz, M., 9, 144 Bergqvist, E., 69 Bernstein, B., 292 Beyer, C.J., 145, 157, 233, 235, 255 Bezemer, J., 269 Bianchi, L.J., 2 Bishop, A.J., 14 Bismack, A.S., 145, 162, 164, 232 Boerst, T.A., 200 Boesen, J., 70 Bolin, F., 144 Boston, M.D., 35, 77 Bracken, C.C., 150 Bråting, K., 288 Brenneman, K., 34 Brown, M.W., 9, 10, 276, 279, 293 Brown, S.A., 2, 9, 164, 267 Bruner, J., 2, 275 Bryans, M.B., 279

C Cai, J., 2, 12, 17, 20 Candela, A.G., 35, 77 Carpenter, T.P., 197 Carroll, W.M., 200 Carter, L., 31 Cengiz, N., 145 Century, J., 15, 16, 68, 71, 74, 78 Chandler-Olcott, K., 292 Charles, R.I., 168 Chávez, Ó., 2 Chazan, D., 200 Chin, M., 163 Choppin, J., 2, 3, 196, 198 Chval, K.B., 2, 68, 245 Cirillo, M., 2, 12, 17, 20 Clements, D.H., 33, 34 Cogan, L., 6 Cohen, D.K., 11, 21, 69, 143, 144, 146, 157, 158, 196, 201, 228, 229, 231, 233, 254, 279, 280 Collopy, R., 2, 165 Confrey, J., 32, 34 Corcoran, T., 33, 200 Corley, A.K., 34 Crumbaugh, C., 145 D Daro, P., 200 Davis, E.A., 11, 12, 21, 69, 71, 143–146, 157, 162, 164, 196, 201, 228, 229, 231–233, 235, 254, 255, 269, 281, 290 Delgado, C., 233 Desimone, L.M., 279

© Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0

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Author Index

314 Ditto, C., 2, 9, 164, 267 Doerr, H.M., 292 Dominick, A., 200 Dowling, P., 14, 16 Doyle, W., 31–33 Drake, C., 198 Duke, N.K., 95, 96

Houang, R.T., 2, 6, 69 Howson, G., 71 Hußmann, S., 37 Huckstep, P., 163 Hughes, E.K., 200, 274 I Iwashyna, A.K., 162, 164

E Ebby, C.B., 197, 199, 200, 202, 203, 216, 217, 223 Edgington, C., 32 Engle, R.A., 200, 274 F Fan, L., 3, 12, 16, 17, 69, 71 Feiman-Nemser, S., 279, 288 Fennema, E., 199 Fernandez, C., 278 Fisher, M.H., 198 Floden, R., 8 Franke, M.L., 68, 197 Freeman, C., 15, 68 Freudenthal, H., 272 Fujita, T., 15, 16, 69 G Garet, M.S., 279 Gelman, R., 34 Goldsmith, L.T., 197–200, 217 Grant, T.J., 145, 157 Grossman, P., 278 Grouws, D.A., 8, 11 Grover, B.W., 8, 32, 72, 144, 263 Gueudet, G., 15, 30, 69, 277, 278, 293 H Haggarty, L., 2, 14, 30, 71 Harris, B., 8, 12, 30 Heck, D.J., 68, 69, 71, 245 Hemmi, K., 288–291 Henningsen, M.A., 8, 32, 72, 144, 200, 263 Herbel-Eisenmann, B.A., 13, 14 Heritage, M., 200 Hiebert, J., 8, 11, 31, 32, 68, 73, 74, 78, 103, 164, 199, 200, 231, 264, 294 Hill, H.C., 163, 201 Hirvonen, P.E., 164 Horn, I.S., 278

J Jacobs, V.R., 21, 196–200, 202, 203, 211, 223 Jawojewski, J., 33, 63, 189 Jaworski, B., 292 Jones, K., 15, 16, 69 K Kamii, C., 200 Kapoor, R., 10, 145, 191, 231 Kaufman, J.H., 7, 145, 157 Kazemi, E., 68 Kelcey, B., 163 Kelso, C.R., 2, 9, 164, 267 Kim, G., 2, 13, 15, 16, 18, 32, 54, 157, 158, 228, 232 Kim, O.K., 2, 10, 32, 33, 61, 145, 164, 190, 192, 231, 232, 276, 277 Kline, K., 145 Koljonen, T., 288, 290, 291 Koponen, M., 164 Koyama, M., 14 Krajcik, J.S., 11, 12, 21, 69, 71, 143–146, 162, 164, 196, 201, 228, 229, 231–233, 235, 254, 255, 269, 281, 290 Kress, G., 269 Krippendorff, K., 150 Krzywacki, H., 288, 290, 291 L Lamb, L.L.C., 21, 196, 197 Land, T.J., 198 Lesh, R., 33, 63, 189 Leshota, M., 276 Leuders, T., 37 Levi, L., 197 Li, J., 232 Liljekvist, Y., 288, 291 Little, J.W., 278 Lloyd, G.M., 279 Lombard, M., 150

Author Index M Ma, L., 164, 190 Madej, L., 288 Males, L.M., 233, 254 Maloney, A.P., 34 Malzahn, K.A., 18, 19 Marulis, L.M., 162, 164 McCallum, W., 6 McDermott, R.A., 278 Melville, M.D., 278 Mesa, V., 13, 14, 16 Miao, Z., 3, 69 Morris, A.K., 164, 231 Mosher, F.A., 33, 200 Munter, C., 8, 11, 68, 72–74, 84, 95, 103, 104, 222, 223, 263 N Nipper, K., 165 O Osterholm, M., 69 Otte, M., 267, 274 Otten, S., 35, 77 P Palincsar, A.S., 145, 162, 164, 232 Parker, K., 14 Pearson, P.D., 95, 96 Pepin, B., 2, 14, 15, 30, 69, 71, 277, 278 Phelps, G., 163, 201, 229 Philipp, R.A., 21, 196, 197 Pimm, D., 293 Pitvorec, K., 2, 9, 164, 267 Polikoff, M.S., 6 Porter, D., 200 Prediger, S., 37 Putnam, R., 56 R Reinke, L.T., 6, 7, 10, 70, 145, 191, 231 Remillard, J.T., 2, 3, 6–8, 10–12, 21, 30, 33, 69, 70, 143–145, 147, 156, 157, 162, 164, 190, 191, 228, 229, 231, 233, 254, 267, 268, 277, 279, 293 Reys, B.J., 2, 14 Reys, R.E., 14 Rezat, S., 165, 166 Rogat, A., 33 Roth McDuffie, A., 198

315 Rowan, B., 163 Rowland, T., 163 Rudnick, M., 15, 68 Ryve, A., 289–291 S Sam, C., 197, 200, 217 Sarama, J., 33, 34 Schack, E.O., 198 Schappelle, B.P., 196 Schilling, S.G., 201 Schmidt, W.H., 2, 6, 69 Schneider, R.M., 144, 145, 235, 255 Schoenfeld, A.H, 5, 11, 114 Seago, N., 197–200, 217 Seah, W.T., 14 Seftor, N., 12 Senk, S.L., 5, 11, 15, 16, 69, 71 Sfard, A., 31 Sherin, M.G., 21, 197–199, 202, 216, 223 Shouffler, J., 278 Shulman, L.S., 163, 189, 230, 233 Silver, E.A., 32 Simon, M.A., 33, 34 Skemp, R., 190 Sleep, L., 3, 10, 33, 164, 189, 200, 231, 254, 268, 276 Smith, M.S., 2, 8, 32, 35, 68, 70, 77, 200, 222, 263, 274, 293 Snyder, J., 144 Snyder, W., 278 Snyder-Duch, J., 150 Son, J.-W., 15, 16, 32 Soria, V.M., 35, 77 Spitzer, S.M., 164, 231 Stacey, K., 14, 71 Stein, M.K., 2, 8, 11–13, 15, 16, 18, 32, 35, 54, 68, 70, 72, 75, 77, 100, 144, 145, 157, 158, 200, 222, 228, 232, 263, 274, 293 Stigler, J.W., 200 Stoehr, K.J., 198 Sträßer, R., 165, 166 Stylianides, A.J., 15, 16 Stylianou, D.A., 33 Sztajn, P., 32, 34, 165 T Takahashi, A., 278 Tarr, J.E., 2, 144 Taton, J., 279 Thames, M.H., 163, 201, 229

Author Index

316 Thomas, J.N., 211 Thomas, M., 12 Thompson, D.R., 5, 11, 69, 71 Thwaites, A., 163 Treffers, A., 272 Trouche, L., 15, 30, 69, 277, 278, 293 Tyminski, A.M., 198 V Valverde, G.A., 2, 13, 14 van Es, E.A., 21, 197–199, 202, 216, 223, 278 Viholainen, A., 164 Vincent, J., 14, 71 W Watanabe, T., 232, 278 Wearne, D., 31, 32

Weiss, I.R., 68, 245 Wenger, E., 278 Wilhelm, J.A., 198 Wilson, P.H., 32 Wineburg, S., 278 Winsløw, C., 278 Wittmann, E.C.H., 9, 37, 63 Wolfe, R.G., 2 Woolworth, S., 278 Y Yoshida, M., 278 Z Zhu, Y., 3, 69 Ziebarth, S.W., 68, 245 Zumwalt, K., 144

Subject Index

A Anticipated student thinking, 163, 179, 180 Assessment activity, 82 Authors’ intentions, 116, 135, 267, 282 C Cognitive demand, 11, 21, 30–32, 34–38, 52–54, 63, 68, 72, 76, 77, 80, 84, 89, 93, 98, 103–105, 157, 158, 162, 262–265, 289 Common Core State Standards for Mathematics (CCSSM), 6, 7, 62, 70, 121, 135–136 Communication with teachers assessments tied to mathematical goals, 117, 118 curriculum enactment, 143 curriculum materials, 144–145 design, 144 assessments, 126 games, 126 implementation guides, 125 pedagogical approaches, 127, 128 problems, 125, 126 vision and pedagogical approaches, 125 guidance, curriculum authors, 146 learning opportunities, 124 mathematical emphasis, 116 mathematical ideas, 128 mathematics content (see Mathematics content) organizing classroom discussions, 128 pedagogical emphasis, 128, 129 structure of goals, 116, 117

teacher’s guide, 146 teachers’ use, curriculum materials, 144 vision for classrooms, 124, 125 Constellation of features, 264 Content knowledge, 163, 164, 189 Content support, 162, 164–166 Correctness/unspecified difficulty, 215, 216 Critical components, 71 Curricular sequences, 268 Curriculum analysis, 147, 151, 168, 170, 171 clarifying terms, 3, 4 components/features, 12 curriculum program selection (see Curriculum program selection) design components, 268 instructional resources, 2 learning opportunities students, 2 level of support, 274 mathematical emphasis and pedagogical approach, 262 mathematics textbooks, 12 methodological decisions, 20 methodological issues, 14, 15, 17 primary coding, 235 purpose of analysis, 13, 15 rationale support, 236 teacher learning, 2 teachers, 2, 3 types of opportunities, 2 Curriculum and Evaluation Standards for School Mathematics, 4–5, 11 Curriculum author challenges content and pedagogy, 133 design rationale clear, 134

© Springer Nature Switzerland AG 2020 J. T. Remillard, O.-K. Kim, Elementary Mathematics Curriculum Materials, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-38588-0

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Subject Index

318 Curriculum author (cont.) educative features, 134 “right” amount of information, 135 standards, policies and pressure, 135, 136 tension, 135 Curriculum development elements to achieve goals, 131 making goals clearer, 130, 131 policy environment, 130 responding to teachers, 131, 132 Curriculum digitization, 292, 294 Curriculum enactment, 143 Curriculum interpretations, 269–270 Curriculum materials, 3, 288, 294 as artifacts, 268 as artifacts of design, 267–269 as complex and layered, 264–266 and curriculum–teacher relationship (see Curriculum–teacher relationship) teachers’ interpretation, 269 Curriculum programs, 3, 147, 154 Curriculum sequence, 249–251 Curriculum-specific elements, 251, 252 Curriculum–teacher relationship, 280 D Daily Practice, 82 Daily Practice Problems (DPPs), 48, 55, 88, 123 Design decisions, 110, 111, 115, 261, 266, 282 implementation guides, 112 See also Curriculum materials Designed artifacts, 263, 267, 269, 275, 280, 282 Designed teaching units, 9 Design features, 289 Design rationale, 149 Design transparency, 146, 149, 150, 153–156, 159, 292 analysis and coding, 235, 236 curriculum-specific elements, 251–253 instructional structures, 253 rationale support, 231–233, 237 Digital capabilities, 3 Digital instructional resources, 7 Directing actions, 153 E Educative curricula, 143, 144, 162, 165, 208 Educative features, 127, 164, 165

Educative supports, 145, 147, 157, 232 attending to student thinking, 204 EngageNY, 7 Equivalent multiplication problems, 47 Everyday Mathematics (EM), 19, 54, 110, 112, 113, 147, 151–155, 157, 158, 167, 171, 172, 175, 176, 178–184, 186, 210, 218, 232, 235, 239, 241–245, 247–250, 252, 253, 265, 270–271 lesson and participation structures, 91, 92 student learning and work, 93 teacher’s role, 93, 95 uses alternative algorithms extensively, 45 Expression of educational potential, 9 F Fact Triangles, 57 Fidelity of implementation, 13 Finnish perspective, 287, 288 Foundational guidelines, 219 Fractions and decimals, 49–51 G Globalization of education, 287–289 Guidance for teachers, 155 Guided Practice, 96 H History of U.S. mathematics curriculum materials, 4 I Implementation guides, 37 Improving Curriculum Use for Better Teaching (ICUBiT) project, 4, 170 Instructional Materials Development (IMD) program, 5, 69 Instructional resources, 3 Investigations in Number, Data, and Space (INV), 19, 54, 110, 113, 114, 145, 147, 148, 151–158, 167, 171, 175, 176, 179–184, 186, 189, 191, 203, 209, 212–215, 218, 221, 222, 224, 232, 235, 237, 239–243, 245–248, 250, 252, 273 instructional approach, 81, 84 lesson and participation structures, 81–83

Subject Index student learning and work, 84 teacher’s role, 84–85 K Knowledge of curriculum embedded mathematics (KCEM), 277 L Lattice method, 42 Learn and Guided Practice segments, 96 Lesson goals, 183, 184 Lesson segment, 35, 80 Lesson-specific assessment opportunities, 117 Lesson structure dimension, 75 M Major participant structures, 75 Mathematical emphasis decimal concepts, 49–51 fraction concepts, 49–51 instructional activities, 30 learning pathways, 33, 34 kind of representation, 63 mathematics knowledge, 31 methods cognitive demand, 34–36 representation, 35, 36 scope and sequence, 37, 38 nature of mathematical work (see Nature of mathematical work) procedures with superficial connections, 62 representation, 32, 33 scope and sequence, 61, 62 skills, 63 strategies and relationships, 62 visual/physical representations, 56, 57, 59, 60 whole numbers (see Whole numbers) Mathematical explanations, 149, 156 boxes and in margins of lesson, 178 in individual lessons, 171 main flow and side notes, 175 student thinking and reasoning, 163 types, 170 Mathematical foundations, 163 Mathematical goals, 231, 233, 243, 254 Mathematical ideas, 146, 149, 150, 156, 159 Mathematical-instructional objects, 268 Mathematical intentions, 30

319 Mathematical knowledge for teaching (MKT), 163 common content knowledge, 230 horizon content knowledge, 163, 230, 233 knowledge of content and curriculum, 163 knowledge of content and students, 163, 230 knowledge of content and teaching, 163, 230, 254 specialized content knowledge, 163, 230 Mathematical practices, 30 Mathematical storyline, 71, 233, 237, 245, 254 Mathematical support, 171–173 anticipated student thinking, 180 content support, 164 direct explanations to teachers, 190 follow-up teacher actions, 182 lesson enactment, 167 lessons and nonroutine headings, 185 main body of the lesson, 172 main flow of lesson, 175 main flow vs. side notes, 175 objectives and vocabulary lists, 169 sample student responses, 179 student thinking and strategies, 166 for teachers, 178, 188 teacher/student actions, 180, 181 Mathematical support framework, for teachers, 167 Mathematical treatment and emphasis, 11 Mathematics content scope and sequence, 119–121 sequencing decisions distributed practice, 122, 123 spiral approach, 122 students’ learning progression, 118 Mathematics curriculum materials curriculum development and research, 144 as designed artifacts, 263, 267, 269, 275, 280, 282 design transparency (see Design transparency) educative features, 162, 164 mathematical knowledge, 163 mathematical support, 164 professional practices, 143 role, 143 student thinking, 196 support enactment, 144 in supporting teachers, 164 teacher learning, 165 in teaching, 157 tetrahedron model, 166

Subject Index

320 Math in Focus (MIF), 19, 147, 151–155, 157, 158, 167, 172, 175, 177, 179–182, 185, 186, 188–190, 192, 203, 209, 210, 212, 214–217, 222, 223, 235, 239–241, 243, 244, 246, 247, 250, 253, 261, 263 instructional approach, 95 lesson and participation structures, 95–97 student learning and work, 97, 98 teacher’s role, 99 Math Trailblazers (MTB), 19, 45, 58, 110, 114, 115, 123, 124, 147, 151–156, 158, 167, 175–186, 188–190, 203, 235, 241, 243, 244, 246, 247, 249–253, 271–272 learning opportunities, 86 lesson and participation structures, 86–88 mathematics instruction, 86 mathematics learning, 86 nature of student learning and work, 88, 89 teacher’s role, 89, 91 Math Workshop, 82, 85 Mental Math and Reflexes, 55, 265, 270, 271 N National Council of Teachers of Mathematics (NCTM) Standards, 4, 31, 32, 34, 112 National Science Foundation (NSF), 4, 5, 9, 18, 19, 69, 110, 232 National Survey of Science and Mathematics Education, 19 Nature of mathematical work cognitive demand of lesson activities, 52–54 ongoing practice, 54–56 NCTM Curriculum Standards, 9 No Child Left Behind (NCLB) Act, 5 Noticing student thinking, 196, 198, 204, 207, 221, 223 Noticing supports, 219–221 Number grid, 57 O Objectively given structure, 267–270, 274 Ongoing assessment, 117 Open Educational Resources (OER), 7 Opportunity to learn (OTL) analytical framework, 10–12 core assumptions, 8–10 curriculum programs, 8

mathematics instruction, 8 student learning, 8 P Partial-products method, 42 Partial quotients method, 44, 45 Pedagogical approach, 11, 12 analytical framework, 104 classroom community, 104 explicit and implicit messages, 68 dialogic vs. direct lens, 103, 104 direct and dialogic instructional models, 103 learning opportunities, 68 lesson and participation structure, 81–83 mathematical work students, 68 mathematics education research, 68 methods analysis, 80 analytical framework, 79 coding, 79, 80 nature of student learning and work, 84 supporting student agency, 84, 85 teacher’s guide categories, 74 lesson and participant structure, 74, 75 mathematics learning, 74 student learning, 74 student learning and work, 76, 77 teacher’s role, 78, 79 Pedagogical components, 71 Pedagogical content knowledge, 230 Pedagogical design capacity (PDC), 276, 277, 279 Pedagogical storyline, 72 Professional noticing of children’s mathematical thinking (PNCMT) anticipating and attending to students’ work, 207–211 coding, 201, 204–206 goal, 201 guidance, in materials, 201 interrelated skills, 198 noticing and evaluation/analysis, 211 professional development, 197 Professional Standards for Teaching Mathematics, 11 R Rater agreement, 150 Rationale support, 231–234, 236, 237, 240, 245, 254

Subject Index S Scott Foresman–Addison Wesley Mathematics (SFAW), 20, 147, 152–155, 157, 158, 168, 171, 172, 177–182, 184–186, 189, 191, 203, 209, 210, 212, 214, 215, 217, 221–224, 235, 237, 239, 240, 242–246, 253, 254, 261, 263 lesson and participation structures, 100, 101 student learning and work, 101, 102 teacher’s role, 103 textbook-centric approach, 100 Sequencing decisions, 62 Shallow teaching syndrome, 71 Situational diagrams, 59 Social relations, 16 Standards-based materials, 6, 291 Standards-based programs, 5, 16, 153, 155, 157, 171, 186, 232, 290 Static representations, 9 Story contexts, 219 Strategy-based evaluations, 218, 219 Strategy-specific evaluations, 212–215 Student thinking, 142, 144, 145, 149, 153–156, 158, 159, 179 attending to student thinking, 196 teacher’s guides foundational knowledge, 201 multiple strategies, use, 200 teacher noticing, 198–200 Subtraction standard algorithm, 46 Swedish textbooks, 288 T Teacher- and student-facing material, 9 Teacher knowledge, 230 curriculum knowledge, 230 horizontal curriculum knowledge, 230 pedagogical content knowledge, 230 subject matter content knowledge, 230

321 vertical curriculum knowledge, 230 working knowledge of students, 217 Teacher learning, 164, 165 Teacher noticing, 197–201, 204 Teacher’s guide, 3, 10, 20, 142, 143, 145–147, 149, 150, 157, 175, 185, 223 affordances and limitations, 228 communicative resources, 269 curriculum programs, 235 curriculum-teacher relationship, 280 design transparency (see Design transparency) instructions/recommendations, 228 level of support, 274 Teacher’s perspective, 16 Teacher’s interpretation, 269, 276 Teaching Integrated Mathematics and Science Project (TIMS), 19 Ten-Minute Math activity, 82 Tetrahedron model, 165, 166 Third International Mathematics and Science Study (TIMSS), 5, 73 Treatment of learning, 16 U Unit Resource Guides, 55, 117 V Vocabulary list, 184, 185 W Whole numbers addition and subtraction, 38–41 calculators, 48 division, 43, 45 mathematical relationships, 45, 46 multiplication, 41, 42 student strategies, 45, 46