Taking Action : Implementing Effective Mathematics Teaching Practices in Grades 6-8 [1 ed.] 9780873539753

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Taking Action Implementing Effective Mathematics Teaching Practices

Margaret S. Smith Series Editor

Michael D. Steele Mary Lynn Raith

Grades 6-8

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Taking Action: Implementing Effective Mathematics Teaching Practices

in Grades 6–8 Margaret S. Smith University of Pittsburgh Series Editor

Michael D. Steele University of Wisconsin–Milwaukee Mary Lynn Raith Pittsburgh Public Schools (Retired)

more www.nctm.org/more4u Access code: TAP15200

more Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Copyright © 2017 by The National Council of Teachers of Mathematics, Inc. 1906 Association Drive, Reston, VA 20191-1502 (703) 620-9840; (800) 235-7566; www.nctm.org All rights reserved Library of Congress Cataloging-in-Publication Data Names: Smith, Margaret Schwan. | Steele, Michael D. (Michael David) | Raith, Mary Lynn. Title: Taking action : implementing effective mathematics teaching practices in grades 6–8 / Margaret S. Smith, University of Pittsburgh, Michael D. Steele, University of Wisconsin– Milwaukee, Mary Lynn Raith, Pittsburgh Public Schools. Description: Reston, VA : The National Council of Teachers of Mathematics, [2017] | Includes bibliographical references. Identifiers: LCCN 2016043810 (print) | LCCN 2016044177 (ebook) | ISBN9780873539753 (pbk.) | ISBN 9780873539982 (ebook) Subjects: LCSH: Mathematics teachers--Training of. | Mathematics--Study and teaching (Middle school) Classification: LCC QA10.5 .S65 2017 (print) | LCC QA10.5 (ebook) | DDC510.71/2--dc23 LC record available at https://lccn.loc.gov/2016043810 The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for each and every student through vision, leadership, professional development, and research. When forms, problems, or sample documents are included or are made available on NCTM’s website, their use is authorized for educational purposes by educators and noncommercial or nonprofit entities that have purchased this book. Except for that use, permission to photocopy or use material electronically from Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 6–8 must be obtained from www .copyright.com or by contacting Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. Permission does not automatically extend to any items identified as reprinted by permission of other publishers or copyright holders. Such items must be excluded unless separate permissions are obtained. It is the responsibility of the user to identify such materials and obtain the permissions. The publications of the National Council of Teachers of Mathematics present a variety of viewpoints. The views expressed or implied in this publication, unless otherwise noted, should not be interpreted as official positions of the Council. Printed in the United States of America Credits and permissions are listed on page 233, an extension of this copyright page.

Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Chapter 1 Setting the Stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 Establish Mathematics Goals to Focus Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 3 Implement Tasks That Promote Reasoning and Problem Solving. . . . . . . . . . . . . . . . . . . . . . 29 Chapter 4 Build Procedural Fluency from Conceptual Understanding. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Chapter 5 Pose Purposeful Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Chapter 6 Use and Connect Mathematical Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter 7 Facilitate Meaningful Mathematics Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

    Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Chapter 8 Elicit and Use Evidence of Student Thinking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Chapter 9 Support Productive Struggle in Learning Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Chapter 10 Pulling It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Appendix A Lesson Plan for the Hexagon Task. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Appendix B A Lesson Planning Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Accompanying Materials at More4U ATL 2.3 ATL 2.3

Kelly Polosky Triangle Video Clip Kelly Polosky Triangle Transcript

ATL 5.2 ATL 5.2

Elizabeth Brovey Calling Plans Video Clip Elizabeth Brove0ing Plans Transcript

ATL 7.1 ATL 7.1

Peter Dubno Counting Cubes Video Clip Peter Dubno Counting Cubes Transcript

ATL 8.3 ATL 8.3

Elizabeth Brovey Storage Tanks Video Clip Elizabeth Brovey Storage Tanks Transcript

ATL 9.1 ATL 9.1

Patricia Rossman Hexagon Video Clip Patricia Rossman Hexagon Transcript

iv   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Preface In April 2014, the National Council of Teachers of Mathematics published Principles to Actions: Ensuring Mathematical Success for All. The purpose of that book is to provide support to teachers, schools, and districts in creating learning environments that support the mathematics learning of each and every student. Principles to Actions articulates a set of six guiding principles for school mathematics —  Teaching and Learning, Access and Equity, Curriculum, Tools and Technology, Assessment, and Professionalism. These principles describe a “system of essential elements of excellent mathematics programs” (NCTM 2014, p. 59). The overarching message of Principles to Actions is that “effective teaching is the nonnegotiable core that ensures that all students learn mathematics at high levels and that such teaching requires a range of actions at the state or provincial, district, school, and classroom levels” (p. 4). The eight “effective mathematics teaching practices” delineated in the “Teaching and Learning Principle” (see chapter 1 of this book) are intended to guide and focus the teaching of mathematics across grade levels and content areas. Decades of empirical research in mathematics classrooms support these teaching practices. Following the publication of Principles to Actions, NCTM president Diane Briars appointed a working group to develop the Principles to Actions Professional Learning Toolkit (http:// www.nctm.org/ptatoolkit/) to support teacher learning of the eight effective mathematics teaching practices. The professional development resources in the Toolkit consist of gradeband modules that engage teachers in analyzing artifacts of teaching (e.g., mathematical tasks, narrative and video cases, student work samples). The Toolkit modules use a “practice-based” approach to professional development, in which materials taken from real classrooms give teachers opportunities to explore, critique, and examine new practices (Ball and Cohen 1999; Smith 2001). The Toolkit represents a collaborative effort between the National Council of Teachers of Mathematics and the Institute for Learning (IFL) at the University of Pittsburgh. The Institute     Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

for Learning (IFL) is an outreach of the University of Pittsburgh’s Learning Research and Development Center (LRDC) and has worked to improve teaching and learning in large urban school districts for more than twenty years. Through this partnership, the IFL made available to the working group a library of classroom videos featuring teachers engaged in ambitious teaching. These videos, a key component of many of the modules in the Toolkit, offer positive narratives of ambitious teaching in urban classrooms. The Taking Action series includes three grade-band books: grades K–5, grades 6–8, and grades 9–12. These books draw on the toolkit modules but go far beyond the modules in several important ways. Each book presents a coherent set of professional learning experiences, with the specific goal of fostering teachers’ development of the effective mathematics teaching practices. The authors intentionally sequenced the chapters to scaffold teachers’ exploration of the eight teaching practices using practice-based materials, including additional tasks, instructional episodes, and student work to extend the range of mathematical content and instructional practices featured in each book, thus providing a richer set of experiences to bring the practices to life. Although each Toolkit module affords an opportunity to investigate an effective teaching practice, the books provide materials for extended learning experiences around an individual teaching practice and across the set of eight effective practices as a whole. The books also give connections to resources in research and equity. In fact, a central element of the book is the attention to issues of equity, access, and identity, with each chapter identifying how the focal effective teaching practice supports equitable mathematics teaching and learning. Each chapter features key ideas and literature surrounding ambitious and equitable mathematics instruction to support the focal practice and provides pathways for teachers’ further investigation. We hope this book will become a valuable resource to classroom teachers and those who support them in strengthening mathematics teaching and learning. Margaret Smith, Series Editor Melissa Boston DeAnn Huinker

vi   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Acknowledgments The activities in this book are drawn in part from Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the team that includes Margaret Smith (Chair) and Victoria Bill (cochair), Melissa Boston, Fredrick Dillon, Amy Hillen, DeAnn Huinker, Stephen Miller, Mary Lynn Raith, and Michael Steele. This project is a partnership between the National Council of Teachers of Mathematics and the Institute for Learning at the University of Pittsburgh. The Toolkit can be accessed at http://www.nctm.org/PtAToolkit/ The video clips used in the Toolkit and in this book were taken from the video archive of the Institute for Learning at the University of Pittsburgh. The teachers featured in the videos allowed us to film their teaching in an effort to open a dialogue about teaching and learning with others who are working to improve their instruction. We thank them for their bravery in sharing their practice with us so that others can learn from their efforts.

    Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

CHAPTER 1

Setting the Stage Imagine walking into a middle school classroom where students are working on a statistics unit in which they are investigating patterns of association between two quantities. While students enter the classroom, the teacher gives each student a sheet of paper that contains the shoeprint shown in figure 1.1. The teacher explains that when investigators find shoeprints at the scene of a crime, forensic scientists can use the prints to identify suspects. She asks students to consider how a footprint could help someone solve a crime. After a brief discussion, students conclude that a shoeprint can indicate the type of shoe that a suspect wore, as well as the size of the suspect.

Fig. 1.1. Shoeprint distributed to students (From Pixabay)

The teacher explains that the students are going to investigate the relationship between shoe size and height so that they can determine the height of the suspect. While students work in pairs, measuring each other’s height and shoe length, the

    Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

teacher monitors the activity and asks and answers questions as needed to support students’ efforts. When pairs finish measuring, they add their data (red dots for girls and green dots for boys) to a large graph   —   with the x-axis labeled as shoe length and the y-axis labeled as height  —  posted in the front of the room. When all the students have added their data points to the graph, the teacher asks students to talk with their partners about the patterns that they notice. After a few minutes, the students share their observations, which the teacher records: for example, no two people have the same shoe size and height, most girls have smaller feet and are shorter than the boys, tall people have bigger feet than short people, the data go up from left to right, and the data are kind of linear. The teacher tells students that their next step is to find a line that models these data  —  a line of best fit. She directs students to a Web-based applet, where they plot the class data in two-pair teams, guess at a line of best fit, and check their guesses. (An applet that supports this investigation is at http://illuminations.nctm .org/Activity.aspx?id54186.) The class concludes with a lively whole-group discussion, during which teams share their findings regarding the line of best fit, discuss the meaning of the slope and y-intercept in context, and consider how confident they are that the equation will be a good predictor of a person’s height based on a shoeprint. In the final five minutes of class, students complete an exit ticket in which they indicate how tall they think the suspect is and present their reasons. The authors have adapted the preceding lesson from NCTM Illuminations. http://illuminations.nctm.org/Lesson .aspx?id52838

A Vision for Students as Mathematics Learners and Doers The lesson in the opening scenario exemplifies the vision of school mathematics for which the National Council of Teachers of Mathematics (NCTM) has been advocating in a series of policy documents for more than 25 years (1989, 2000, 2006, 2009). In this vision, as in the scenario, students are active learners, constructing their knowledge of mathematics through exploration, discussion, and reflection. The tasks in which students engage are both challenging and interesting, and students cannot quickly complete them by applying a known rule or procedure. Students must reason about and make sense of a situation and persevere when a pathway is not immediately evident. Students use a range of tools to support their thinking and collaborate with their peers to test and refine their ideas. A whole-class discussion provides a forum for students to share ideas and clarify understandings, develop convincing arguments, and learn to see things from the perspectives of other students.

2   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

In the shoeprint scenario, students faced a real-world problem, and they needed to collect and analyze data to solve it. All students could enter the problem by measuring, recording data on the graph, and making observations. Students’ observations  —  including that “tall people have bigger feet than short people,” that data were “kind of linear” and “went up from left to right” — furnish evidence that students were attending to salient features of the graph and the relationship between the quantities. These observations then positioned the students to focus more narrowly on finding a line that modeled the data. Through the use of the applet, students were able to guess at the line of best fit and then check their guesses. During the discussion, students reported on their work, but the teacher also pressed them to consider what the equation meant in context. When the issue of how confident they should be about their equation came up, the teacher could then introduce and discuss the meaning of the correlation coefficient (which the applet generated). During the closing minutes of the lesson, the teacher asked students to determine how tall the suspect must be. This information could give the teacher insight into the extent to which students recognized the utility of the model to make predictions beyond the data set. The vision for student learning that NCTM advocates and that our opening scenario represents has gained growing support over the previous decade while states and provinces have put into place world-class standards (e.g., National Governors Association Center for Best Practices and Council of Chief State School Officers [NGA Center and CCSSO] 2010). These standards focus on developing conceptual understanding of key mathematical ideas, flexible use of procedures, and the ability to engage in a set of mathematical practices that include reasoning, problem solving, and communicating mathematically.

A Vision for Teachers as Facilitators of Student Learning Meeting the demands of world-class standards for student learning requires that teachers engage in ambitious teaching. Ambitious teaching stands in sharp contrast to the welldocumented routine found in many classrooms. That routine consists of homework review, teacher lecture, and demonstration, followed by individual practice (e.g., Hiebert et al. 2003). This routine has been translated into the gradual release model: I do (the teacher tells students what to do); we do (the teacher practices with students); and you do (the students practice on their own) (Santos 2011). In instruction that uses this approach, the focus is on learning and practicing procedures with limited connection with meaning. Students have few opportunities to reason and solve problems. Although they may learn the procedure as intended, they often do not understand why it works and apply the procedure in situations in which it is not appropriate. According to Martin (2009, p. 165), “mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results.” That is, at times students

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get answers that make no sense; however, they have no idea how to judge correctness because they are mindlessly applying a procedure that they really do not understand. In ambitious teaching, the teacher engages students in challenging tasks and then observes and listens while they work so that he or she can offer an appropriate level of support to diverse learners. The goal is to ensure that each and every student succeeds in doing highquality academic work rather than merely executing procedures with speed and accuracy. In the opening scenario, a teacher is engaging students in meaningful mathematics learning. She has selected an authentic task for students to work on, provided resources to support their work (e.g., a method for measuring and recording data, an applet for investigating line of best fit, partners with whom to exchange ideas), monitored students while they worked, gave support as needed, and orchestrated a discussion in which students’ contributions were essential. However, what we do not see in this brief scenario is exactly how the teacher is eliciting thinking and responding to students so that she supports every student in his or her learning. According to Lampert and her colleagues (Lampert et al. 2010, p. 130): deliberately responsive and discipline-connected instruction greatly complicates the intellectual and social load of the interactions in which teachers need to engage, making ambitious teaching particularly challenging. This book intends to support teachers in meeting the challenge of ambitious teaching by describing and illustrating a set of teaching practices that facilitate the type of “responsive and discipline-connected instruction” that is at the heart of ambitious teaching (Lampert et al. 2010, p. 130).

Support for Ambitious Teaching Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014) gives guidance on what will make ambitious teaching, as well as rigorous content standards that it targets, a reality in classrooms, schools, and districts to support mathematical success for each and every student. At the heart of this book is a set of eight teaching practices that provide a framework for strengthening the teaching and learning of mathematics (see fig. 1.2). These teaching practices describe intentional and purposeful actions that teachers take to support the engagement and learning of each and every student. These practices, based on knowledge of mathematics teaching and learning accumulated over more than two decades, represent “a core set of high-leverage practices and essential teaching skills necessary to promote deep learning of mathematics” (NCTM 2014, p. 9). Subsequent chapters of this book examine each of these teaching practices in more depth through illustrations and discussions.

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Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions. Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies. Use and connect mathematical representations. Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving. Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments. Pose purposeful questions. Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships. Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems. Support productive struggle in learning mathematics. Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships. Elicit and use evidence of student thinking. Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

Fig. 1.2. Eight Effective Mathematics Teaching Practices (NCTM 2014, p. 10)

Ambitious mathematics teaching must be equitable. Driscoll and his colleagues (Driscoll, Nikula, and DePiper 2016, pp. ix–x) acknowledge that defining equity can be elusive but argue that equity is really about fairness in terms of access “providing each learner with alternative ways to achieve, no matter the obstacles they face” and potential, “as in potential shown by students to do challenging mathematical reasoning and problem solving.” Hence, teachers need to pay attention to the instructional opportunities that students receive, particularly

Setting the Stage   5 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

to historically underserved or marginalized youth (i.e., students who are black, Latina/ Latino, American Indian, low income) (Gutierrez 2013, p. 7). Every student must participate substantially in all phases of a mathematics lesson (e.g., individual work, small-group work, whole-class discussion), although not necessarily in the same ways ( Jackson and Cobb 2010). Toward this end, throughout this book, we relate the eight effective teaching practices to specific equity-based practices that strengthen mathematical learning and cultivate positive student mathematical identities (Aguirre, Mayfield-Ingram, and Martin 2013). Figure 1.3 lists five equity-based instructional practices along with brief descriptions of them. Go deep with mathematics. Develop students’ conceptual understanding, procedural fluency, and problem solving and reasoning. Leverage multiple mathematical competencies. Use students’ different mathematical strengths as a resource for learning. Affirm mathematics learners’ identities. Promote student participation and value different ways of contributing. Challenge spaces of marginality. Embrace student competencies, value multiple mathematical contributions, and position students as sources of expertise. Draw on multiple resources of knowledge (mathematics, language, culture, family). Tap students’ knowledge and experiences as resources for mathematics learning.

Fig. 1.3. The Five Equity-Based Teaching Mathematics Practices (Adapted from Aguirre et al. 2013, p. 43)

Central to ambitious teaching — and at the core of the five equity-based practices — is helping each student develop an identity as a doer of mathematics. Aguirre and her colleagues (Aguirre et al. 2013, p. 14) define mathematical identities as follows: the dispositions and deeply held beliefs that students develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics in powerful ways across the contexts of their lives. Many middle school students believe that they are not good at mathematics and approach mathematics with fear and lack of confidence. Their identity, developed through their previous years of schooling, can affect their school and career choices. Allen and Schnell (2016, p. 398) argue that “middle school mathematics teachers have a unique opportunity to steer their

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students’ mathematical development in a more positive direction.” The effective teaching practices that this book discusses and illustrates attempt to help in this regard.

Contents of This Book This book is written primarily for teachers and teacher educators who are committed to ambitious teaching practice that provides their students with increased opportunities to experience mathematics as meaningful, challenging, and worthwhile. It is likely, however, that any education professionals working with teachers would benefit from the illustrations and discussions of the effective teaching practices. Educators can use this book in several different ways. Teachers can read through the book on their own, stop to engage in the activities as suggested, or try activities in their own classrooms. Alternatively, and perhaps more powerfully, teachers can work their way through the book with colleagues in professional learning communities, department meetings, or when time permits. We believe that exchanging ideas with one’s peers adds considerable value. Teacher educators or professional developers can use this book in college or university education courses for practicing or preservice teachers or in professional development workshops during the summer or school year. The book might be a good choice for a book study for any group of mathematics teachers interesting in improving their instructional practices. This book supplies a rationale for and discussion of each of the eight effective teaching practices, and we connect them with the equity-based teaching practices when appropriate. We give examples and activities intended to help teachers of students in the middle grades develop their understanding of each practice, how they can enact it in the classroom, and how it can promote equity. Toward this end, we invite the reader to actively engage in two types of activities that the book presents throughout: Analyzing Teaching and Learning (ATL) and Taking Action in Your Classroom. ATL activities invite the reader to actively engage with specific artifacts of classroom practice (e.g., mathematics tasks, narrative cases of classroom instruction, video clips in More4U, student work samples). Taking Action in Your Classroom activities offer specific suggestions that indicate how a teacher can begin to explore specific teaching practices in her or his classroom. The Analyzing Teaching and Learning activities come, in part, from activities found in the Principles to Actions Professional Learning Toolkit (http://www.nctm.org/PtAToolkit/). We have added additional activities, beyond what the Toolkit includes, to facilitate a more extensive investigation of each of the eight effective mathematics teaching practices. The video clips featured in the Analyzing Teaching and Learning activities show teachers who were endeavoring to engage in ambitious instruction in their urban classrooms and students who are persevering in solving mathematical tasks that require reasoning and problem solving. The videos, made available by the Institute for Learning at the University of Pittsburgh, Setting the Stage   7 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

give images of aspects of effective teaching. As such, they are examples to analyze rather than models to copy. To access the video clips, visit NCTM’s More4U website at nctm.org/more4u and enter the access code on the title page of this book. While you read this book and engage with both types of activities, we encourage you to keep a journal or notebook in which you record your responses to questions that the book poses, in addition to noting issues and new ideas that emerge. These written records can serve as a basis for your own personal reflections, informal conversations with other teachers, and for planned discussions with colleagues. Each of the next eight chapters focuses explicitly on one of the eight effective teaching practices. We have arranged the chapters in an order that makes it possible to highlight the interrelationships among the effective teaching practices. (Note that this order differs from the one shown in figure 1.2 and in Principles to Actions [NCTM 2014].) Chapter 2: Establish Mathematics Goals to Focus Learning Chapter 3: Implement Tasks That Promote Reasoning and Problem Solving Chapter 4: Build Procedural Fluency from Conceptual Understanding Chapter 5: Pose Purposeful Questions Chapter 6: Use and Connect Mathematical Representations Chapter 7: Facilitate Meaningful Mathematical Discourse Chapter 8: Elicit and Use Evidence of Student Thinking Chapter 9: Support Productive Struggle in Learning Mathematics Each of these chapters follows a similar structure. We begin chapters by asking the reader to engage in an Analyzing Teaching and Learning activity that sets the stage for a discussion of the focal teaching practice. We then relate the opening activity to the focal teaching practice and highlight the key features of the teaching practice for teachers and students. Each chapter also highlights key research findings that relate to the focal teaching practice, describes how the focal teaching practice supports access and equity for all students, and includes additional ATL activities and related analysis as needed to provide sufficient grounding in the focal teaching practice. Each chapter concludes with a summary of the key points and a Taking Action in Your Classroom activity that encourages the reader to purposefully relate the teaching practice that the chapter examines to her or his own classroom instruction. Although we present each of the effective teaching practices in a separate chapter, within each chapter we highlight other effective teaching practices that support the focal practice. In the final chapter of the book (Chapter 10 — Pulling It All Together), we consider how the set of eight effective teaching practices relate to one another and how they work in concert to support students’ learning. In chapter 10, we also consider the importance of thoughtful and thorough planning in advance of a lesson and evidence-based reflection following a lesson as

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critical components of the teaching cycle that are necessary for successful use of the effective teaching practices.

An Exploration of Teaching and Learning We close the chapter with the first Analyzing Teaching and Learning activity, which takes readers into the classroom of Patrick Donnelly, where seventh-grade students are exploring proportional relationships. The case presents an excerpt from his classroom in which he and his students are discussing and analyzing the various strategies that students used to solve the Candy Jar task. When chapters 2 through 9 introduce new teaching practices, we relate the new practice to some aspect of “The Case of Patrick Donnelly.” In so doing, we are using the case as a touchstone to which we can relate the new learning in each chapter. Hence, the case provides a unifying thread that brings coherence to the book and makes salient the synergy of the effective teaching practices (i.e., the combined effect of the practices is greater than the impact of any individual practice).

Analyzing Teaching and Learning 1.1 Investigating Teaching and Learning in a Seventh-Grade Classroom As you read “The Case of Patrick Donnelly,” consider the following questions and record your observations in your journal or notebook so that you can revisit them when we refer to this case in subsequent chapters: • What does Patrick Donnelly do during the lesson to support his students’ engagement in and learning of mathematics? • What aspects of Patrick Donnelly’s teaching are similar to or different from what you do? • Which practices would you want to incorporate into your own teaching practices?

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2

Exploring Proportional Relationships: The Case of Patrick Donnelly

3 4 5 6 7

Patrick Donnelly wanted his students to understand that quantities that are in a proportional (multiplicative) relationship grow at a constant rate and that students can use three key strategies to solve problems of this type: scaling up, a scale factor, and a unit rate. He selected the Candy Jar task for the lesson because it aligned with his goals, was cognitively challenging, and had multiple entry points.

8

The Candy Jar Task

1

9 10 11 12

13 14 15 16 17 18 19 20 21 22 23 24 25 26

A candy jar contains 5 Jolly Ranchers (JRs) and 13 jawbreakers (JBs). Suppose you had a new candy jar with the same ratio of Jolly Ranchers to jawbreakers, but it contained 100 Jolly Ranchers. How many jawbreakers would you have? Explain how you know.

While students began working with their partners on the task, Mr. Donnelly walked around the room, stopping at different groups to listen in on their conversations and to ask questions as needed (e.g., How did you get that? How do you know that the new ratio is equivalent to the initial ratio?). When students struggled to figure out what to do, he encouraged them to look at the work that they had done the previous day, which included producing a table of ratios equivalent to 5 JRs : 13 JBs and a unit rate of 1 JR to 2.6 JBs. He also encouraged students to consider how much larger the new candy jar must be when compared to the original jar. As he made his way around the room, Mr. Donnelly also noted the strategies that students were using (see fig. 1.4) so that he could decide which groups he wanted to ask to present their work. After visiting each group, he decided that he would ask groups 4, 5, and 2 to share their approaches (in that order), because each of those groups used one of the strategies that he was targeting and the sequencing reflected the sophistication and frequency of strategies.

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27 28 29 30 31 32 33 34 35 36 37 38

During the discussion, he asked the presenters (one student from each of the targeted groups) to explain what their group did and why, and he invited other students to consider whether the approach made sense and to ask questions. He made a point of labeling each of the three strategies, asking students which strategy was most efficient in solving this particular task and asking them questions that helped them make connections between the different strategies and with the key ideas that he was targeting. Specifically, he wanted students to see that that the scale factor that group 5 identified was the same as the number of entries in the table that group 4 created (or the number of small candy jars that would make the new candy jar) and that the unit rate that group 2 identified was the factor that connected the JRs and JBs in each row of the table. The following is an excerpt from the discussion that took place around the unit-rate solution that Jerry from group 2 presented.

39 40

Jerry:

41

Mr. D.:

42

Jerry:

43

Mr. D.:

44

Danielle: How did you know to do 13 divided by 5?

We figured that there was 1 JR for 2.6 JBs, so that a jar with 100 JRs would have 260 JBs. So we got the same thing as the other groups. Can you tell us how you figured out that there was 1 JR for 2.6 JBs? We divided 13 by 5.

Does anyone have any questions for Jerry? [Pause] Danielle? See, we wanted to find out the number of JBs for 1 JR; so if we wanted JRs to be 1, we needed to divide it by 5. So now we needed to do the same thing to the JBs.

45 46 47

Jerry:

48

Danielle: How did you then get 260 JBs?

49 50

Jerry:

Well, once we had 1 JR to 2.6 JBs, it was easy to see that we needed to multiply each type of candy by 100 so we could get 100 JRs.

So Jerry’s group multiplied by 100, but Danielle and her group (group 5) multiplied by 20. Can they both be right? Amanda?

51 52

Mr. D.:

53 54 55 56

Amanda: Yes. Jerry’s group multiplied 1 and 2.6 by 100, and Danielle and her group multiplied 5 and 13 by 20. Jerry’s group multiplied by a number 5 times bigger than Danielle’s group because their ratio was one-fifth the size of the ratio Danielle’s group used, so it is the same thing.

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Do others agree with what Danielle is saying? [Students nod their heads and give Danielle a thumbs-up.] What is important here is that both groups kept the ratio constant by multiplying both the JRs and JBs by the same amount. We call what Jerry and his group found the unit rate. A unit rate describes how many units of one quantity (in this case, JBs) correspond to one unit of another quantity (in this case, JRs). [Mr. Donnelly writes this definition on the board.]

57 58 59 60 61 62 63

Mr. D.:

64 65

Mr. D.:

66 67

Mr. D.:

68 69

Kamiko: We noticed that if we looked at any row in our table that the number of JBs in the row was always 2.6 times the number of JRs in the same row.

I am wondering if we can relate the unit rate to the table that group 4 shared. Take two minutes, and talk to your partner about this. [Two minutes pass.] Kamiko and Jerilyn [from group 4], can you tell us what you were talking about?

Yeah, we saw that too, so it looks like any number of JRs times 2.6 will give you the number of JBs.

70 71

Mike:

72

Mr. D.:

73

Mike:

74 75 76

Mr. D.:

77

[After two minutes, the discussion continues.]

78 79 80 81 82 83 84

Toward the end of the lesson, Mr. Donnelly placed the solution that group 1 produced on the document camera and asked students to decide whether this approach was a viable one for solving the task and to justify their answer. He told them that they would have five minutes to write a response that he would collect while they exited the room. He thought that this exercise would give him some insight as to whether individual students were coming to understand that ratios needed to grow at a constant rate that was multiplicative rather than additive.

What if we were looking for the number of JBs in a jar that had 1000 JRs? Well, the new jar would be 1000 times bigger, so you multiply by 1000.

Take two minutes, and see if you and your partner can write a rule that we could use to find the number of JBs in a candy jar no matter how many JRs are in it.

Margaret Smith of the University of Pittsburgh drew on two sources in writing this vignette: NCTM (2014) and Smith et al. (2005). The task is adapted and reprinted by permission of the Publisher. From Margaret Schwan Smith, Edward A. Silver, and Mary Kay Stein. Improving Instruction in Rational Numbers and Proportionality: Using Cases to Transform Mathematics Teaching and Learning (volume 1) New York: Teachers College Press. Copyright 2005 by Teachers College, Columbia University. All rights reserved.

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Group 1, First Solution) (incorrect additive) 100 JRs is 95 more than the 5 we started with. So we will need 95 more JBs than the 13 I started with.   5 JRs 1 95 JRs 5 100 JRs 13 JBs 1 95 JBs 5 108 JBs

You had to multiply the five JRs by 20 to get 100, so you would also have to multiply the 13 JBs by 20 to get 260. (320)   5 JRs  ➞  100 JRs 13 JBs  ➞  260 JBs (320)

Group 2 (unit rate) Since the ratio is 5 JRs for 13 JBs, we divided 13 by 5 and got 2.6. So that would mean that for every 1 JR, there are 2.6 JBs. So then you just multiply 2.6 by 100. (3100)   1 JR   ➞  100 JRs 2.6 JBs  ➞  260 JBs (3100)

Group 1, Second Solution; Group 4; and Group 7 (scaling up)

Groups 3 and 5 (scale factor)

JR

JB

JR

JB

5

13

55

143

10

26

60

156

15

39

65

169

20

52

70

182

25

65

75

195

30

78

80

208

35

91

85

221

40

104

90

234

45

117

95

247

50

130

100

260

Group 6 (scaling up) JRs

5

10

20

40

80

100

JBs

13

26

52

104

208

260

We started by doubling both the number of JRs and JBs. But then when we got to 80 JRs, we didn’t want to double it any more because we wanted to end up at 100 JRs and doubling 80 would give me too many. So we noticed that if we added 20 JRs: 52 JBs and 80 JRs: 208 JBs, we would get 100 JRs:260 JBs.

Group 8 (scaling up) We drew 100 JRs in groups of 5. Then we put 13 JBs with each group of 5 JRs. We then counted the number of JBs and found we had used 260 of them.

Fig. 1.4. The work of Patrick Donnelly’s students

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Moving Forward Mr. Donnelly’s instruction has many noteworthy aspects, and the vignette gives examples of his use of the effective teaching practices. However, we are not going to analyze this case here. Rather, as you work your way through Chapters 2 through 9, you will revisit the case of Mr. Donnelly and consider the extent to which he engaged in the focal practice and the impact that it appeared to have on student learning and engagement. While you progress through the chapters, you may want to return to the observations that you made during your initial reading of the case and consider the extent to which you are viewing aspects of the case differently. As you read the chapters that follow, we encourage you to continue to reflect on your own instruction and how the effective mathematics teaching practices can help you improve your teaching practice. The Taking Action in Your Classroom activity at the end of each chapter is intended to support you in this process. Cultivating a habit of systematic and deliberate reflection may hold the key to improving one’s teaching as well as sustaining lifelong professional development.

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CHAPTER 2

Establish Mathematics Goals to Focus Learning The Analyzing Teaching and Learning (ATL) activities in this chapter engage you in exploring the effective teaching practice, establish mathematics goals to focus learning. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 12): Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions. Establishing clear and specific goals for student learning in a lesson sets the stage for everything else that a teacher does, from selecting a task on which students will work, to identifying questions that will help determine what students already know related to the goal and move them toward it, to determining how to organize a whole-class discussion so that the work of students can be used as a basis for achieving lesson goals. The goals should serve as a beacon that provides guidance and direction during a lesson. As such, goals need to identify what students will come to understand about mathematics during the lesson rather than focusing on what students will do. In this chapter, you will —  • explore and compare different goal statements created for a lesson on proportional relationships;

• consider the ways in which lesson goals can support teaching and learning by connecting goals with specific teaching moves in both narrative and video cases;

• review research findings that relate to the importance of establishing mathematics goals to focus learning; and

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• analyze the relationships among your classroom goals, your teaching practices, and possible student learning outcomes. For each Analyzing Teaching and Learning activity, note your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, establish mathematics goals to focus learning. If possible, share and discuss your responses and ideas with colleagues. After you have written or shared your ideas, read the analysis, in which we offer ideas relating the ATL activity to the focal teaching practice.

Exploring Lesson Goals We begin the chapter by asking you to engage in ATL 2.1. In this activity, you compare three different goal statements that could be written for a lesson on proportional relationships.

Analyzing Teaching and Learning 2.1 Comparing Goal Statements Review goal statements A, B, and C (shown below), written for a lesson on proportional relationships, and consider the following: • How are they the same, and how are they different? • How might the differences matter? Goal A: Students will learn the procedure (cross multiplication) for finding the missing value in a proportional situation. Goal B: Students will be able to (SWBAT) use cross multiplication to find the missing value in problems in which the quantities being compared are in a proportional relationship. Goal C: Students will recognize that a proportion consists of two ratios that a ax are equivalent to each other (e.g., b5bx ) and that missing values in the proportion can be found by determining the scale factor x that relates the two ratios or by determining the unit rate — the relationship (multiplicative) between a and b and recognizing that ax and bx must have the same relationship as a and b.

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Analysis of ATL 2.1: Comparing Goal Statements The three goal statements in ATL 2.1 are similar in a number of ways. All three goal statements relate to proportional reasoning and the ability to determine missing values in a proportional situation. The goals are all mathematical in nature, and meeting those goals would equip students to solve certain types of proportional reasoning problems. A number of important differences exist in the sets of goals. Goals A and B focus on a single specific strategy (the procedure for cross multiplication). Both goals are fairly narrow and do not require students to understand the reasons why the procedures work, to connect those procedures with the concept of proportionality or proportional reasoning, or to use multiple methods to solve proportional reasoning tasks. Goal C focuses on two different strategies — scale factor and unit rate — and links these strategies to the underlying relationships of scale that exist in proportional situations. Students who achieve goal C might be more likely to be able to solve a wider range of proportional reasoning problems than students who achieve goals A or B. Differences also exist in what would count as evidence of meeting the goals. One might determine evidence of meeting goals A and B by looking at students’ written computational work, whereas evidence of meeting goal C would likely require students to explain their thinking verbally or in writing. Goal C represents a strong balance between the important conceptual underpinnings of a proportional relationship (two ratios that are equivalent to each other) and two possible methods for describing the relationship between the ratios. The goal notably does not describe the notion of a missing-value problem; instead, it describes the ways in which a scale factor relates two equivalent ratios. This fact can be used to determine missing values in a proportional relationship and has implications that extend far beyond a straightforward missing-value problem. If students understand how to use a scale-factor strategy, they are more likely to be able to generate new equivalent ratios more flexibly and describe the overall situation (e.g., for every three additional balloons that we want to buy, we will spend $2 more). The second aspect of the goal is something that educators can assess through examining computational steps, whereas the first aspect of the goal is likely to require students to explain their thinking verbally or in writing. One way to think about differences among goals is the distinction between a performance goal and a learning goal. A performance goal describes a specific written or spoken performance that students should demonstrate as a result of the lesson. For example, goals that educators frame with the opening phrase, “Students Will Be Able to” (often abbreviated SWBAT) tend to be performance goals. The mathematics that a teacher is looking for is highly visible but is also limited to demonstrating a process that the goal lists, as evidenced in goals A and B. A learning goal, in contrast, focuses on what students can learn as a result of engaging in a lesson. Consider goal C — this goal describes a particular set of understandings related to equivalent ratios and connects them with two possible methods for using equivalent ratios in

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understanding a proportional relationship. The ways in which students might write about or talk about this goal can vary. The goal itself describes the nature of the learning rather than the nature of the performance. Do the differences matter? In comparing the three goals in ATL 2.1, the differences do matter. Two of the goals specify what students will do, whereas goal C specifies what students will learn from engaging in the mathematics lesson. The differences between goals A, B, and C set different expectations for the tasks that teachers might use to structure the lesson, how students will work toward the goals, and what teachers and students will accept as evidence of meeting the goal. Another type of goal would be the statement of a standard from a state or national standards document, such as the one shown in figure 2.1. Teachers may often be required to state a standard in a lesson plan or on the board. Standards, however, do not always clearly specify the nature of learning; and they may be broader than a goal for a single lesson. For example, the standard in figure 2.1 contains a great deal of information about a number of ways that students might represent, describe, and identify proportional relationships. The standard describes a target that may be more appropriate at the end of a series of connected lessons or a unit. Goal C in ATL 2.1 reflects some aspects of the standard but provides more specific direction for a teacher with respect to what student learning might look like at the lesson level.

7.RP.A.2 Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships c. Represent proportional relationships by equations

Fig. 2.1. A seventh-grade proportional reasoning standard from the Common Core State Standards (CCSS) for Mathematics (NGA Center and CCSSO 2010)

A strong mathematics goal should be more focused than a standard. Also, if the ultimate goal is for students to learn the conceptual underpinnings behind a particular procedure or set of procedures, the goal should not be so narrow that it specifies a single algorithmic approach. A strong mathematics goal should connect with bigger mathematical ideas and provide opportunities to build lessons that help students understand how their learning connects with prior knowledge and why the new mathematical learning is important. The goal should also help students and teachers monitor progress toward meeting the goal. When setting goals for student learning, teachers might also consider the effective teaching practices of promoting productive struggle and eliciting and using evidence of student thinking. 18   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

The goals that a teacher sets and the tasks that a teacher chooses to support students in meeting those goals should give students opportunities to struggle productively. If teachers wish for students to develop a belief that they are capable of engaging in worthwhile mathematics and to develop a productive disposition toward mathematics as a sense-making endeavor, how do a teacher’s goals and instructional decisions support this work? Eliciting and using evidence of student thinking is essential in determining whether a teacher’s goals are being met. With the goal in mind, teachers should think about the strategies that they will use to collect evidence of student thinking and how this evidence informs teachers about the extent to which students are meeting the lesson’s goals.

Exploring Lesson Goals in “The Case of Patrick Donnelly” In ATL 2.2, we return to “The Case of Patrick Donnelly,” which was introduced in chapter 1, and consider the goals that he set and the ways in which the goals help focus the lesson.

Analyzing Teaching and Learning 2.2 Connecting Patrick Donnelly’s Goal and Teacher Moves Revisit the case of Patrick Donnelly presented in chapter 1. • What were Mr. Donnelly’s goals for the lesson that the case featured? • Identify the teacher moves that help focus learning toward the goal and the student talk and work that represent progress toward the goal.

Analysis of ATL 2.2: Connecting Patrick Donnelly’s Goal and Teacher Moves Lines 3–6 of the case state Mr. Donnelly’s mathematical goal: He “wanted his students to understand that quantities that are in a proportional (multiplicative) relationship grow at a constant rate and that students can use three key strategies to solve problems of this type: scaling up, a scale factor, and a unit rate.” This clear goal includes aspects of conceptual understanding of proportional relationships and identifies the range of strategies that Mr. Donnelly intended for his students to understand as useful in proportional situations. This goal is grade-level-appropriate and connects with rigorous mathematical standards like CCSS mathematics standard 7.RP.A.2 (see fig. 2.1).

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Mr. Donnelly made several teaching moves that focused the learning related to his goal. First, he chose a lesson and a task that built on students’ prior knowledge. Students had already built tables of ratios equivalent to the 5 Jolly Ranchers: 13 jawbreakers ratio that was introduced in the problem on the previous day. Students also understood the concept of a unit rate and had ways to determine the unit rate (lines 18–19). These understandings laid the groundwork for further exploration of the proportional relationships and for Mr. Donnelly’s students to use and connect the three key strategies identified in his goal. Mr. Donnelly moved students toward this goal in the choices that he made to elicit student thinking. He set up a contrast between two groups (groups 2 and 5) that used different scale factors (100 and 20) to find the same answer (lines 51–52 done) and asked the class to determine whether both groups could be right. This move has the explicit effect of asking students to compare the scale-factor strategy (group 5) with a unit-rate strategy (group 2). Later in the discussion, Mr. Donnelly asked students to connect the unit-rate strategy with the table that group 4 produced, representing the third strategy, scaling up (lines 64–65). Mr. Donnelly’s exit slip (lines 78–82) asked students to identify whether an incorrect additive strategy was viable, thereby giving students opportunities to convey their understanding of the ratio relationships in a proportion and to use any of the three strategies to justify why the additive strategy was not correct. Students’ work and thinking showed that Mr. Donnelly’s class was making significant progress toward his goals. Of the eight groups, seven produced a viable solution using one of the three strategies for working with proportional relationships. Mr. Donnelly labeled the three targeted strategies (lines 29–30, 60) so that students could reference them while they discussed their solutions. This labeling also marked the strategies as important for students to know and understand. Students asked questions about the specific strategies that other groups used (Danielle, lines 44, 48) and made connections among the strategies (Amanda, lines 53–56). They extended one strategy to a new ratio (Mike, line 73); and Mr. Donnelly asked them to consider a fourth strategy, an incorrect one, in comparison for their end-of-lesson exit-ticket assessment (lines 78–82). Mr. Donnelly hoped that student work on the exit ticket would give him additional insight into students’ progress toward the goal, in particular, understanding the multiplicative nature of proportional relationships. In “The Case of Patrick Donnelly,” we see important teacher and student actions that are guided and focused by the establishment of a meaningful mathematical goal. Mr. Donnelly’s goal was not just something that he wrote in his lesson plan and posted on the board for students — rather, it guided his instructional decision making before and during the lesson. The mathematical task that he chose aligned well with his mathematical goal; and the ways in which he interacted with small groups, elicited student thinking, and fostered meaningful mathematical discourse all connected directly with the goal. This effort paid off in that a significant amount of students’ talk and work helped Mr. Donnelly measure student progress toward that goal, both informally and formally. 20   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Considering How Lesson Goals Support Teaching and Learning In Analyzing Teaching and Learning activity 2.3, we go into the classroom of Kelly Polosky, where her fifth-grade students were trying to create a formula for finding the area of a right triangle (see below). Ms. Polosky gave the students cardboard triangles and other tools such as scissors and grid paper to support their work on the task. The students had just begun a geometry unit. Before the lesson featured in the video clip, the students sorted polygons and nonpolygons, and they identified the characteristics of polygons. They also found the formulas for the areas of rectangles and squares. The lesson featured in the video clip is the first in a series of lessons that help students develop an understanding of the formula for finding the area of a triangle. As a result of engaging in the featured lesson, Ms. Polosky wanted her students to understand that —    1. The area of a triangle is 1/2 of its length times its width.

  2. The relationship between area, length, and width of a triangle can be generalized to a formula.

  3. There are several equivalent ways of writing the formula for the area of a triangle, and each can be related to a physical model.

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The Triangle Task

Fig. 2.2a

Fig. 2.2b

Using the triangles shown above and the grid paper, construct a formula for the area of a right triangle. You may cut your triangles out and manipulate them in any way that might help you make your argument. After you explore with your particular triangle, make a general mathematical argument using words, symbols, and diagrams about the formula for the area of any right triangle. Adapted from University of Chicago School Mathematics Project (2007).

Analyzing Teaching and Learning 2.3 more Connecting Goals and Instructional Decisions Watch the video clip and download the transcript of the discussion of the triangle task in Kelly Polosky’s classroom. As you watch, make notes about each of the following —  • ways in which the goals that Ms. Polosky has established for the lesson are evident in the discussion; • what Ms. Polosky did to keep students focused on the main points of the lesson; • specific instances in which Ms. Polosky makes an instructional decision that relates directly to her goal; and • what students say or do as a result of that move. You can access and download the videos and their transcripts by visiting NCTM’s More4U website (nctm.org/more4u). The access code can be found on the title page of this book.

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Analysis of ATL 2.3: Connecting Goals and Instructional Decisions In this clip, we see how Ms. Polosky’s instructional decisions reflected the three components of her instructional goals. The specificity of her goals allowed us to make strong connections among her goals, her teaching moves, and what students did and said in response to those teaching moves. First, Ms. Polosky framed the discussion by asking students to share possible formulas for finding the area of the two given triangles (lines 3–5, 20–21 in the transcript). The result was that students shared two different formulas (lines 6–7, 24). She probed students with respect to how they derived their formulas (l × w  2 and 1/2 lw) and how to apply them to the triangle examples that the class was using (lines 8, 13–14, 17), which continually brought into the discussion the two components that determine the area of a triangle — length and width (lines 9–12, 15–16, 18–19). She pressed students to determine whether the formulas are equivalent (lines 25, 27–28); and in their justifications (lines 32–35, 37, 39, 41, 43, 54–55, 61–64), students used the specific examples and moved toward a generalization (lines 66–68, 70–72, 74–78). This conversation ended with Ms. Polosky asking whether students could use the formula for any right triangle — a very important aspect of her first goal, that is, for students to understand that the area of a triangle is one-half of its length times its width. Ms. Polosky was more present in the conversation at the end, while she synthesized students’ contributions and worked with students to connect the formula with the physical model, thereby connecting all three goals. A number of other noticeable features of this discussion are hallmarks of ambitious teaching practice. First, Ms. Polosky made sure to record the formulas publicly (lines 44–45), which allowed students access and allowed them to build their arguments on the elements of the formula (lines 57–59, 70–72). Ms. Polosky persisted in soliciting students’ contributions and often moved from one to the next without telling students whether they were correct or incorrect, thereby allowing for multiple student voices to contribute to the conversation and for students to build their arguments organically, agreeing and disagreeing with one another and aggregating mathematical evidence. Even when statements were ambiguous or incorrect, such as in lines 57–59, in which a student appeared to be confusing a lowercase L with the number 1, Ms. Polosky allowed the mathematical justification to unfold organically. Some students focused on the specific values for length and width in the given problem, and others used more general language about the length and width; and she drew these two together in the end. She strategically shaped the conversation in lines 44–45 and 79–81, in which she asked questions that she designed to bring ideas together and to target aspects of her mathematical goals. Allowing for rich student discourse while choosing some key moments in which to intervene produced a meaningful discussion that was both mathematically substantive and dominated by students’ own thinking. Hence, the goals that Ms. Polosky set for the lesson gave her a clear target for focusing student learning. Although such a lesson could have turned into a focus on the procedural aspects of finding the area of a right triangle, she consistently pushed

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for conceptual understanding, which was at the heart of what she intended to accomplish with the lesson. In summary, goals drive the teaching decisions that relate to many of the other effective mathematics teaching practices, as we saw in the examples of the classrooms of Patrick Donnelly and Kelly Polosky. The questions that teachers ask and the nature of the mathematics discourse that teachers seek to promote should help teachers advance students toward the mathematical goals for the lesson. Questions should help students move along their broader mathematical trajectories as well. Goals and the broader learning trajectories should also attend to the issue of developing procedural fluency from conceptual understanding. When teachers introduce a new topic, early lessons should focus on goals that relate to understanding the mathematical ideas at a conceptual level and developing meaning for the related procedures that follow. Subsequent lessons might move more toward fluency with using and applying the procedures, embedded in connections back to the underlying conceptual understandings.

Establishing Goals to Focus Learning: What Research Has to Say Establishing mathematical goals has important effects on the teaching and learning of mathematics in a wide variety of ways. Goals are critical so that teachers can identify their anticipated student learning outcomes for a class; and as shown in the example of Mrs. Polosky’s class (ATL 2.2), to make key strategic decisions during a lesson. Hiebert and colleagues note the myriad ways in which clear and explicit learning goals influence teacher decision making in a lesson: Without explicit learning goals, it is difficult to know what counts as evidence of students’ learning, how students’ learning can be linked to particular instructional activities, and how to revise instruction to facilitate students’ learning more effectively. Formulating clear, explicit learning goals sets the stage for everything else (Hiebert, Morris, Berk, and Jensen 2007, p. 51). Beyond just a single instructional episode, goals can and should build from lesson to lesson in ways that trace a coherent, meaningful, and rigorous trajectory through important mathematical ideas. Goals should be situated in larger mathematics learning trajectories or progressions (Daro, Mosher, and Corcoran 2011), such as the work of the Common Core Standards Writing Team (see http://math.arizona.edu/~ime/progressions/) and the Turn on CC Math project (https://turnonccmath.net). Goals should relate to key thematic ideas in middle grades mathematics, such as proportional reasoning, the rational number system, functions, transformations, and statistical thinking (NCTM 2006; NGA and CCSSO 2010). A strong set of goals that connect with big mathematical ideas will support students 24   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

in seeing mathematics as a meaningful, well-connected body of ideas. A number of sources exist for situating the content of learning goals in a broader set of ideas, including essential understandings (e.g., Sinclair et al. 2012); learning progressions and trajectories (e.g., Clements and Sarama 2004), big ideas (Charles 2005), and key developmental understandings (Simon, Placa, and Avitzur 2016). When a teacher faces a decision point in a lesson (e.g., do I pursue this mathematical idea, this student contribution, or this solution path) considering the mathematical goals can help teachers make sound and thoughtful decisions. Having a sound understanding of the goals and the multiple pathways that students can (and cannot) take to reach them is a hallmark of ambitious mathematics teaching (Leinhardt and Steele 2005). The decisions that teachers make about which student contributions to incorporate into the lesson and which contributions to avoid exploring in depth can help shape the trajectory that individual students and the class take toward the mathematical goals. Goals are also important to support students in their mathematical work. Many studies of instructional practice have suggested that self-monitoring of learning is an important factor in student achievement (e.g., Ames and Archer 1988; Engle and Conant 2002; Fuchs et al. 2003; Henningsen and Stein 1997). Sharing instructional goals with students in some form facilitates this self-monitoring of learning. Although teachers do not have to explicitly post written goals in classrooms, students must have a very clear sense of what their learning target is and whether they are making progress toward that target while the mathematics lesson unfolds. Specifically, these goals can motivate student learning when students perceive the goals as challenging but attainable (Marzano 2003; McTighe and Wiggins 2013). Throughout instruction, teachers need to find meaningful ways to refer to goals to focus student learning and help support students in monitoring their progress (Clarke, Timperley, and Hattie 2004; Zimmerman 2001).

Promoting Equity by Establishing Goals to Focus Learning An equitable mathematics program gives all students access to important mathematical ideas, with a focus on both promoting conceptual understanding of mathematical ideas and developing fluency with mathematical procedures. In considering how to promote an equitable classroom, the teacher must consider the sets of goals that he or she has for students in a set of lessons, in a unit, and in a year. What is the balance between performance goals and learning goals? Do some students have access to learning goals more regularly than other students? Productive disposition in mathematics is one of the five strands of mathematical proficiency identified in Adding It Up (NRC 2001), and the teacher’s goals have a profound impact on students’ development of productive disposition. A productive disposition is one in which students see mathematics as a sensible and useful endeavor and one in which they

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are capable of engaging. The goals that a teacher chooses influence the development of a productive disposition in the mathematics classroom (Gresalfi and Cobb 2006; Gresalfi 2009), and an overemphasis on performance goals has negative effects on student motivation (Ryan et al. 1998). Students must be able to see the goals of the classroom as achievable, as connecting mathematical ideas in sensible ways, and as being able to relate to their lives outside the classroom. The teacher’s goals for a lesson frame the tasks that teachers choose and therefore students’ opportunities to learn. Consider the ways in which the goals of both Mr. Donnelly and Ms. Polosky framed students’ learning opportunities. If these teachers had selected more performance-based goals, would they have chosen the same mathematical tasks? Would the nature and type of student thinking evident in their classes have been different? In what ways? As mentioned previously, strong mathematical goals should connect with a mathematics learning trajectory that traces in broad strokes the mathematics content that we want students to learn. An important consideration for a mathematics program, particularly in the middle grades, is whether all students have access to the same trajectories and goals. Acceleration and tracking of students can limit students’ possible pathways through mathematics and can give some students access to mathematics that other students are denied (Oakes 1990). By analyzing goals at the lesson, unit, and course level, mathematics departments can determine the extent to which students in a particular grade level have access to an equitable mathematics curriculum. The extent to which goals are visible to students is also important. Although students do not need to see goals posted verbatim in classrooms every day, students need to emerge from their mathematical work in the classroom with an understanding of what the learning goals were and the extent to which they met those goals. This can be accomplished through the task, the norms and expectations for classroom work, the discourse with students, and by posting learning goals and reflecting on them. It is also critical within both lessons and units to give students opportunities to reflect on the extent to which they achieved the mathematical goals. Students who enter middle school with strong mathematical thinking skills are more likely to be able to abstract the essential points of a lesson. Students who struggle to determine how mathematical ideas fit together may need time and support to reflect on their learning and to draw connections among mathematical ideas. Returning to goals at the close of instruction and encouraging students to reflect explicitly on those goals can help students experience mathematics as a connected, meaningful whole.

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Key Messages • Goals should be more than just a learning standard or the description of a procedure and should include information about the conceptual understandings that students will gain from the lesson. • Establishing mathematical goals should include identifying how to assess the goal and how students can determine whether they are making progress toward the goal. • Mathematical goals should connect with broader learning trajectories at the unit and the course or grade level.

• Goals provide teachers with support in making instructional decisions about the tasks to use for a lesson and making good choices about students’ contributions to pursue and probe.

Taking Action in Your Classroom: Making Learning Goals Explicit In Taking Action in Your Classroom, we invite you to explore how you can make your goals for student learning more explicit and how this explicitness might enhance your teaching and student learning.

Taking Action in Your Classroom Making Learning Goals Explicit Consider a lesson that you have recently taught in which the learning goal was not explicit. • Rewrite the learning goal so that the mathematical idea that you wanted students to learn is explicit. • How might your more explicit goal statement guide your decision making before and during the lesson?

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CHAPTER 3

Implement Tasks That Promote Reasoning and Problem Solving The Analyzing Teaching and Learning (ATL) activities in this chapter engage you in exploring the effective teaching practice, implement tasks that promote reasoning and problem solving. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 17): Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies. Tasks that promote reasoning and problem solving require students to engage in thinking and sense making. Students need to explore a situation, determine what they know that relates to the problem at hand, and decide on an appropriate strategy to employ or a particular pathway to follow. Students can often solve such tasks in different ways and by using different representations. In this chapter, you will —  • solve and compare mathematical tasks;

• analyze two narrative cases and consider the factors that affect task implementation and student learning;

• review key research findings related to the mathematical tasks; and • reflect on task selection and use in your own classroom.

For each ATL activity, note your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, implement tasks that promote reasoning and problem solving. If possible, share and discuss your responses and ideas

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with colleagues. After you have written or shared your ideas, read the analysis of the ATL activity, in which we offer ideas relating the activity to the focal teaching practice.

Comparing Mathematical Tasks In Analyzing Teaching and Learning activity 3.1, you solve and compare two mathematical tasks that focus on proportional relationships. While you solve each task, consider the strategies to which you are drawn and the extent to which the strategies that you use are ones that make sense to you and that you can explain mathematically.

Analyzing Teaching and Learning 3.1 Comparing Two Tasks Involving Proportional Relationships   1. Solve the Candy Jar task and the Finding the Missing Value task. Consider the strategy that you use to solve each task and the features of the problem that led you to use it.   2. Consider the ways in which the two tasks are the same and the ways in which they are different. Which one is more likely to promote reasoning and problem solving? Why?

The Candy Jar Task

The Finding the Missing Value Task

A candy jar contains 5 Jolly Ranchers and 13 jawbreakers. Suppose that you had a new candy jar with the same ratio of Jolly Ranchers to jawbreakers, but it contains 100 Jolly Ranchers.

Find the value of the unknown in each of the proportions shown below.

How many jawbreakers would you have? Explain how you know.

5    y  5 2 10   a     7 5 24 8 n    3   5 8 12 30   b 5 6 7

Adapted from Smith et al. 2005.

  5     3 5 20 d 3    4 5 x 28

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Analysis of ATL 3.1: Comparing Two Tasks Involving Proportional Relationships A seventh-grade unit on ratio and proportion could include both these tasks. At a basic level, each task involves finding a missing value in a proportional relationship, and students can find these missing values by using cross multiplication. However, a deeper analysis of the tasks suggests that they are in fact different in the level and kind of thinking in which students might engage while solving them. The Candy Jar task does not state or imply any solution strategy. Hence, students must expend effort to find a strategy for solving the task and justify why their approach makes sense. “The Case of Patrick Donnelly,” presented in chapter 1, gives the range of strategies and explanations that students provided. Students solved the task by scaling up the original ratio of 5 Jolly Ranchers to 13 jawbreakers until they found the new ratio of 100 Jolly Ranchers and 260 jawbreakers (groups 1, 4, 6, 7, 8), identifying the scale factor that indicates that the new jar is 20 times bigger than the original jar (groups 3 and 5); and finding the unit rate that shows that for every 1 Jolly Rancher, there are 2.6 jawbreakers (group 2). In addition to being open to the use of different strategies, the Candy Jar task also has multiple entry points. Students can build candy jars by using manipulatives to represent each candy, by drawing pictures of candy jars (group 8), or by making a table that shows repeated additions of 5 Jolly Ranchers and 13 jawbreakers (groups 1, 4, and 7). All these approaches give students access to the task. That is, when the teacher presents the task, all students are able to do something at their current level of understanding that will help them make progress toward the mathematical goals of the lesson. Although students could solve the Candy Jar task by using cross multiplication, this task would be used most productively before the students know the procedure, as in Mr. Donnelly’s classroom. (The following chapter more fully explores conceptual understanding as a basis on which to build procedural fluency.) By contrast, the Finding the Missing Value task requires limited effort. The way in which the task is set up, as well as the directions given, imply that students know a procedure for solving such problems, probably cross multiplication (i.e., setting the product of the means and the product of the extremes equal and solving algebraically for the unknown). However, if students do not know a procedure for solving the task, they are unlikely to have any entry into the problem. The goal of the task is to find the correct numeric answer, as evidenced by the absence of a context or the requirement to explain why the procedure or the answers make sense. Although such tasks serve a function in the curriculum (that is, they provide an opportunity for students to practice procedures that they have learned and to gain fluency), they do not require students to engage in thinking, reasoning, and problem solving. Further, the use of such tasks without first laying a foundation of conceptual understanding can lead to students applying rules with limited or no understanding of why they work or when they should apply the rules.

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The Task Analysis Guide (Smith and Stein 1998), shown in figure 3.1, gives criteria for determining the level of thinking or cognitive demand required for solving a task. Using these criteria, the Candy Jar task would be categorized as having high cognitive demands, whereas the Finding the Missing Value task would be classified as having low cognitive demands. Highdemand tasks such as the Candy Jar task have the potential to engage students in reasoning and problem solving. Specifically, the Candy Jar task would be in the doing mathematics category (the last category in fig. 3.1) because the task does not prescribe what students need to do or how they should do it and students must expend considerable effort to determine a course of action.

Task Analysis Guide Lower-level demands: Memorization • Involve either reproducing previously learned facts, rules, formulas, or definitions or committing facts, rules, formulas or definitions to memory. • Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure. • Are not ambiguous. Such tasks involve the exact reproduction of previously seen material, and what is to be reproduced is clearly and directly stated. • Have no connection to the concepts or meaning that underlie the facts, rules, formulas, or definitions being learned or reproduced. Lower-level demands: Procedures Without Connections • Are algorithmic. Use of the procedure either is specifically called for or is evident from prior instruction, experience, or placement of the task. • Require limited cognitive demand for successful completion. Little ambiguity exists about what needs to be done and how to do it. • Have no connection to the concepts or meaning that underlie the procedure being used. • Are focused on producing correct answers instead of on developing mathematical understanding. • Require no explanations or explanations that focus solely on describing the procedure that was used. Higher-level demands: Procedures With Connections • Focus students’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas. • Suggest explicitly or implicitly pathways to follow that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts. continued on next page

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• Usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols, and problem situations. Making connections among multiple representations helps develop meaning. • Require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding. Higher-level demands: Doing Mathematics • Require complex and nonalgorithmic thinking — a predictable, well-rehearsed approach or pathway is not explicitly suggested by the task, task instructions, or a worked-out example. • Require students to explore and understand the nature of mathematical concepts, processes, or relationships. • Demand self-monitoring or self-regulation of one’s own cognitive processes. • Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task. • Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions. • Require considerable cognitive effort and may involve some level of anxiety for the student because of the unpredictable nature of the solution process required. These characteristics are derived from the work of Doyle on academic tasks (1988) and Resnick on highlevel-thinking skills (1987), the Professional Standards for Teaching Mathematics (NCTM 1991), and the examination and categorization of hundreds of tasks used in QUASAR classrooms (Stein, Grover, and Henningsen 1996; Stein, Lane, and Silver 1996).

Fig. 3.1. A guide for examining the cognitive demand of mathematical tasks (From Smith and Stein 1998, p. 348)

Low-demand tasks, such as the Finding the Missing Value task focus on the memorized facts and rules and on applying known procedures. Specifically, the Finding the Missing Value task falls into the procedures without connections category (the second category in fig. 3.1) because of the lack of ambiguity regarding what students need to do and how to do it and the limited effort that students must expend. Although students need opportunities to engage in all types of tasks, research shows that high-level tasks are essential in order for students to develop the capacity to engage in thinking, reasoning, and problem solving (Stein and Lane 1996). In ATL 3.2, you will use the Task Analysis Guide to analyze four different tasks and decide the level and type of thinking in which one must engage to complete the tasks.

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Analyzing Teaching and Learning 3.2 Categorizing Four Tasks Based on the Level of Thinking Required   1. Use the Task Analysis Guide (shown in fig. 3.1) to decide which of the four categories (memorization, procedures without connections, procedures with connections, doing mathematics) best describes the level of thinking required for each of the four tasks (A–D) shown below.   2. Identify specific criteria from the Task Analysis Guide to support your categorization. A. The local nature club is carrying out a survey of the number of ducklings in each family of ducks in the lake. Here are the results of the survey: 4, 7, 6, 5, 8, 7, 5, 4, 10, 4, 9, 6, 5, 4, 4, 5, 9, 8, 4 How many ducks are in a typical family? Use tables, graphs, or arithmetic to justify your answer. Adapted from http://www.insidemathematics.org /assets/common-core-math-tasks/ducklings.pdf (Noyce Foundation, 2012)

B. The pairs of numbers in (a) 2 (c) represent the heights of stacks of cubes to be leveled off. On grid paper, sketch the front views of columns of cubes with these heights before and after they are leveled off, as shown in the example below of 9 and 5. Write a statement under the sketches that explains how your method of leveling off is related to finding the average of the two numbers.

(a) 14 and 8 (b) 16 and 7 (c) 7 and 12 Taken from Visual Mathematics Course I, Foreman and Bennett, 1995, Lesson 10, Follow-up Student Activity 10.1, #1, p. 121.

C. Justin scored the following grades on his math test:

D. What is the formula for finding the mean of a set of data?

95, 71, 89, 100, 82. What was his mean grade?

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Analysis of ATL 3.2: Categorizing Four Tasks on the Basis of the Level of Thinking Required A middle grades unit on statistics and probability might use all four of the tasks. The tasks, however, differ in cognitive demand (i.e., the level of thinking required to solve the task). Task A is a high-level doing mathematics task because the task does not suggest or imply a pathway and the students must decide what typical means and justify why their solution fits those criteria. Students can initially organize the data by using a table, a graph, or an ordered listing. They can complete the task by finding the mean, median, or mode. Regardless of the measure of center or type of representation that students use, they need to create an argument that supports their choices. Task B is a procedures with connections task. Although the task tells students exactly what to do — sketch the front views of columns of cubes before and after leveling them off — it then asks them to write a statement that explains how the method of leveling off relates to finding the average of the two numbers. The task specifies what to do and how, limiting the approaches that students can use to solve this task. However, it is the connection to meaning — linking the procedure that students followed in leveling off the columns to the meaning of average — that makes this task a high-level one. Tasks in this category help students create meaning for procedures and develop an understanding of why things work. Task C is a procedures without connections task. The task tells students to find the mean of a set of numbers. The use of the word mean suggests that students know what it is and how to find it. Although the task is set in a real-world context, it requires a straightforward application of a procedure (add up the test scores and divide by the number of scores) and requires no explanation, interpretation, or connection to meaning. Task D is a memorization task because it requires repeating a learned rule. Neither of these low-level tasks (tasks C and D) allow for entry into the task unless students already know the specific rule or fact. Hence high-demand tasks (i.e., procedures with connections and doing mathematics) are the only types of tasks that have potential to engage students in reasoning and problem solving. As Boston and Wilhelm (2015, p. 24) note —  In general, the potential of the task sets the ceiling for implementation —  that is a task almost never increases in cognitive demand during implementation. This finding, robust in its consistency across several studies, suggests that high-level instructional tasks are a necessary condition for ambitious mathematics instruction. However, the use of high-level tasks does not guarantee that the potential will be realized during instruction.

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Considering How Task Implementation Affects Learning Opportunities Although cognitively demanding tasks can engage students in high-level mathematical thinking and reasoning, research has shown that teachers often implement high-level tasks in ways that lower the demands of the task (e.g., Stein, Grover, and Henningsen 1996; Stigler and Hiebert 2004). When this occurs, students’ opportunities to engage in thinking, reasoning, and problem solving diminish; and they are left to carry out an identified procedure with no connection to meaning or understanding. Research has identified a set of factors (shown in fig. 3.2) that are associated with the maintenance and decline of high-level tasks (Henningsen and Stein 1997). Specifically, in classrooms in which tasks decline during implementation, a subset of the factors listed on the left of figure 3.2 are often at play; in classrooms where the level of the task is maintained, a subset of the factors listed on the right of figure 3.2 are often at play. In ATL 3.3, you will analyze the implementation of the Candy Jar task in two different seventh-grade classrooms and determine whether the demands of the task were maintained and the factors that account for the outcome.

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Factors Associated with Decline

  1. Problematic aspects of the task become routinized (e.g., students press teacher to reduce task complexity by specifying explicit procedures or steps to perform; teacher “takes over” difficult pieces of the task and performs them for the students or tells them how to do it).   2. Teacher shifts emphasis from meaning, concepts, or understanding to correctness or completeness of the answer.   3. Not enough time is provided for students to wrestle with the demanding aspects of the task or too much time is provided and students flounder or drift off task.   4. Classroom management problems prevent sustained engagement.   5. Task is inappropriate for the group of students (e.g., lack of interest, lack of motivation, lack of prior knowledge needed to perform, task expectations not clear enough to put students in the right cognitive space, etc.).

Factors Associated with Maintenance   1. Scaffolding (i.e., task is simplified so student can solve it; complexity is maintained, but greater resources are made available). Could occur during whole-class discussion, presentations, or during group or pair work.   2. Students are provided with the means of monitoring their own progress (e.g., rubrics are discussed and used to judge performance; means for testing conjectures are made explicit and used).   3. The teacher or capable students model high-level performance.   4. Sustained press for justifications, explanations, meaning through teacher questioning, comments, feedback.   5. Tasks are selected that build on students’ prior knowledge.   6. Teacher draws frequent conceptual connections.   7. Sufficient time to explore (not too little, not too much).

  6. Students not held accountable for high-level products or processes (e.g., although asked to explain their thinking, unclear or incorrect student explanations are accepted; students were given the impression that their work would not “count” (i.e., be used to determine grades).

Fig. 3.2. Factors of maintenance and decline of high-level tasks (Adapted from Stein et al. 2009, p. 16)

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Analyzing Teaching and Learning 3.3 Comparing Instruction in Two Classrooms Revisit “The Case of Patrick Donnelly,” introduced in chapter 1, and read “The Case of Sandra Pascal,” which follows.   1. How are the two classes the same, and how do they differ?   2. What factors (from fig. 3.2) might help account for the differences?   3. Do you believe that the differences matter? Why or why not?

1

The Case of Sandra Pascal

2 3 4 5 6 7 8 9

Ms. Pascal began the lesson by showing her students a small candy jar that she had filled with 5 jawbreakers ( JBs) and 13 Jolly Ranchers ( JRs). She explained to students that they would be working on a task (shown in fig. 3.3) that required them to make new candy jars that had the same ratio of the two types of candy as the initial jar (the one that she was holding) but that had greater amounts of each candy. Their job, she explained, was to determine the exact number of each type of candy in the new jar. She told students that she had placed bags of red and green tiles on their desks that they could use to represent the candies.

10

The Candy Jar Tasks

11 12 13

A candy jar contains 5 Jolly Ranchers (JRs) and 13 jawbreakers (JBs). Solve each of the following problems and explain your thinking:

14 15 16 17

  1. Suppose that you have a new candy jar with the same ratio of Jolly Ranchers to jawbreakers that Ms. Pascal had but it contains 100 Jolly Ranchers. How many jawbreakers would you have? Explain how you know.

18 19 20 21

  2. Suppose that you have a candy jar with the same ratio of Jolly Ranchers to jawbreakers as in the previous problem, but it contains 720 candies. How many of each kind of candy would you have? Adapted from Smith et al. (2005).

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22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

While students worked with their partners on the first candy jar problem, Ms. Pascal walked around the room and stopped at different groups to listen in on their conversations. The first group that she visited was happy to see her. They were not sure what to do and pleaded for her to help them. She asked the group what the first question was asking them to find. Alexis said that they needed to find the number of JBs in the new jar but they were not sure how to do it. Ms. Pascal then asked the students if they could use the red ( JB) and green ( JR) tiles that she had supplied to help them determine how the candy jar grew. When even this suggestion seemed to puzzle students, she told them to count out 5 JRs and 13 JBs and then to keep adding 5 green tiles ( JRs) and 13 red tiles ( JBs) to the pile to see what bigger jars with the same ratio would look like. When she moved on to other groups, she noticed that most groups had amassed a pile of JRs and JBs but were unable to describe what the piles meant other than “bigger jars.” Students seemed to have lost track of the goal — determining the number of JBs that they would need in a jar with 100 JRs. At this point, Ms. Pascal decided to call the class together to see whether she could give them a bit more direction without visiting each group individually. Students clearly were not making progress on the task, and she did not want them to waste any more time counting tiles. She reminded students of the table that they had created the previous day (see fig. 3.3), in which they had generated ratios equivalent to 5 JRs to 13 JBs and projected it on the interactive whiteboard. She suggested that they might want to consider extending the table to find out how many JBs they would have when the number of JRs was 100. She told them that if they were using the colored tiles, they could use the table to keep track of how many of each type of candy they had after adding an additional 5 JRs and 13 JBs. To make sure that students understood how the table would continue to grow, she asked the class how many JRs and JBs would be in the next biggest jar. Although the question seemed to confuse several students, a few hands shot up. Ms. Pascal called on Alicia, who tentatively said “25 and 65?” Ms. Pascal asked if anyone agreed with Alicia. Dominic said, “Yep; she is right. You just add 20 1 5 and 52 1 13.” Ms. Pascal then told students to continue working on the problem.

JR

5

10

15

20

JB

13

26

39

52

Fig. 3.3. The table of equivalent ratios that the class generated the previous day

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51 52 53 54 55 56 57 58 59 60 61 62 63 64

Students returned to their partner work with renewed energy. Ms. Pascal continued to monitor their progress and noted that students were easily able to extend the table to arrive at the correct answer. When she asked the groups how many JBs would be in a bag that had 100 JRs, they looked at the table and answered that it would have 260 JBs. When she asked one group why the answer was 260, they responded, “We kept adding 5 JBs and 13 JRs until we ended up with 100 JBs. Then we knew we had the right number of JRs.” After about 10 minutes, she brought the class together. She asked Ellen and Chuck to share their table with the class. Ellen and Chuck explained that they had continued the table by adding 5 JRs to the first row and 13 JBs to the second row until they reached 100 JRs and 260 JBs (see fig. 3.4). Ms. Pascal asked the class how many students had created a similar table. Nearly every hand in the class went up. Ms. Pascal thanked the group and reminded the class that tables could be helpful in solving all types of problems.

JR

5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

JB 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260

Fig. 3.4. Ellen and Chuck’s completed table 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

Ms. Pascal then told the class that they could start work on the second part of the task. She gave them a few minutes before she began checking in with the groups. She noticed that the groups were extending the table by continuing to add 5 and 13. When she asked a group for an explanation, the students responded, “We plan to keep going until one of the numbers is 720.” Because Ms. Pascal recognized that this strategy was not a productive one, she asked, “What are we looking for in this problem? Is it the same as the last problem? What does the 720 represent?” She hoped that these questions would help students see that 720 was the total number of candies and that they should add another row to their tables to represent the total. At that point, she noticed that the bell would ring in two minutes. She told students that their homework was to take the table that they had created for the first part of the problem, add a third column to represent the total number of JRs and JBs in each jar, add the number of Jolly Ranchers and jawbreakers in each row and place this number to the new column, and then consider how this process could help them answer the second part of the problem. She hoped that students would realize that they first had to find the jar that had 720 candies in it to know how many of each type of candy they had.

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Analysis of ATL 3.3: Comparing Instruction in Two Classrooms Although Mr. Donnelly and Ms. Pascal both used the Candy Jar task as the basis for instruction, the lessons were very different. Most notably, Mr. Donnelly supported students’ ability to work through the problem without taking over the thinking for them, whereas Ms. Pascal was the one who did most of thinking in her classroom. The factors associated with the maintenance of high-level tasks (identified on the right side of fig. 3.2) help account for the success of Mr. Donnelly’s lesson. He provided appropriate scaffolding for students (factor 1). When students struggled, he encouraged them to look at the work that they had done on the previous day (lines 16–19). He engaged students in a discussion of unit rate that gave the class access to this important idea and helped them define the relationship between JRs and JBs (lines 39–81). Mr. Donnelly had students model high-level performance (factor 3). He selected specific students to share their solutions during the discussion (line 23–26) to highlight the strategies that he had targeted (line 3). Mr. Donnelly pressed students to explain their thinking (factor 4). His questions helped students articulate their thinking (lines 13–16, 41, 66–67); to take on the reasoning of others (lines 51–52, 57); to extend their thinking (lines 64–65, 74–76); and to make sense of the incorrect additive solution (lines 78–80). Mr. Donnelly selected a task that built on students’ prior knowledge (factor 5). During the previous lesson, students had produced a table of ratios equivalent to 5 JRs : 13 JBs (lines 16–19) on which they were able to draw in making sense of the Candy Jar task. Mr. Donnelly made conceptual connections (factor 6). He asked questions that helped students make connections among the different strategies that students used, connections between each strategy and the problem context (lines 29–32), and connections with the key ideas in the lesson such as unit rate (lines 60–63). Finally — and perhaps most important — he had a clear goal for students’ learning (as discussed in chapter 2), he selected a task that aligned with this goal, and all his efforts throughout the lesson focused on helping students reach the goal that he had set. Although Ms. Pascal began the lesson with a high-level task, the level of demand declined during implementation as the result of her actions and interactions with students. The factors associated with the decline of high-level tasks (identified on the left side of fig. 3.2) help account for the lack of success of Ms. Pascal’s lesson. Ms. Pascal removed problematic aspects of the tasks (factor 1). When students in Ms. Pascal’s class struggled, she took over the thinking for them and told them exactly what to do (lines 29–31, 39–45). When students first indicated that they were struggling (lines 24–25), she responded by directing them to a particular strategy rather than ask questions that might assess their current thinking and identify ways to advance that thinking and support students’ productive struggle. Ms. Pascal focused on the correct answer (factor 2). The students all obtained the same correct answer and used the same strategy — the one that she had suggested (lines 51–53). Although she asked students questions (lines 55–57), the answers that students provided (and that Ms. Pascal accepted without further

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pressing) focused on using a procedure to obtain a numeric answer without any discussion of why the answer made sense. Ms. Pascal did not hold students accountable for high-level products (factor 6). She did not give students an opportunity to make a sensible choice regarding a strategy to use to solve the problem, and they were not in a position to evaluate the quality and utility of that strategy as compared with alternatives. As a result, students were not building a set of approaches that they could apply to a broad class of problems. Although creating a table may have met the immediate demand of helping students find a correct answer for the Candy Jar task, that strategy would not be efficient or even reasonable to use for larger numbers. Ms. Pascal’s instructional goal for this lesson was unclear, but if it extended beyond finding the number of JBs in the new jar, it does not appear that she achieved it. Without clarity regarding what she wanted students to learn about mathematics, she had no target toward which to move students. The differences between the two classes matter because students in the two classes had very different opportunities to learn mathematics and to see themselves as being capable mathematically. In Mr. Donnelly’s class, the students learned (or were in the process of learning) that quantities in a proportional relationship grow at a constant rate; what the unit rate is, how to find it, where it appears in the table, and how to use the unit-rate approach to find the number of JBs in any jar when given the number of JRs; and how different approaches connected with one another. In addition, students were learning to persevere in the face of struggle and find approaches that made sense to them. In this way, Mr. Donnelly was encouraging students to be authors and owners of knowledge. In Ms. Pascal’s class, students learned very little (if anything) about proportional relationships. Instead, Ms. Pascal’s students may have learned that when a problem is difficult, the teacher would tell them what to do; that students need the teacher to help them figure out problems; that one way exists to solve a problem and it is the teacher’s way, and that students do not need to understand why something works, they just need to know how to obtain the right answer. In classrooms such as Ms. Pascal’s, the teacher is the sole authority in the classroom and the students’ role is limited to following the approach that the teacher presents and providing answers. The unintended consequence of such experiences is that students do not see mathematics as something that they can explore and understand and do not see themselves as capable of figuring things out for themselves. As discussed in chapter 1, the type of instruction that took place in Mr. Donnelly’s class is ambitious teaching. That is, it is teaching that aims to support all students in understanding mathematical ideas, participating in the practices of the discipline, and solving high-level tasks that require reasoning and problem solving. When used in combination, the teaching practices support ambitious teaching, as you will see as you continue to explore the practices in subsequent chapters.

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Specifically, tasks that promote reasoning and problem solving should allow students to use different representations (e.g., the drawings, tables, and numerical solutions that Mr. Donnelly’s students produced) and to make connections among them. By helping students make connections among different representations (i.e., tables, arithmetic, content), Mr. Donnelly facilitated students’ development of a deeper and more flexible understanding of mathematics and a firm basis for developing conceptual understanding. Teachers must support students’ productive struggle on high-level tasks so that students feel ownership of their own learning and see themselves as capable mathematically. The teacher supports this struggle by posing purposeful questions that assess and advance thinking without taking over the challenging aspects of the task and by engaging students in meaningful discourse that elicits and uses students’ thinking by making connections among different approaches for solving a task and connections with the mathematics ideas that are central to the lesson. The examples of Ms. Pascal and Mr. Donnelly provide an interesting contrast in this regard. Mr. Donnelly supported productive struggle throughout the lesson and thereby gained insights into students’ thinking through small-group and whole-group interactions, whereas Ms. Pascal took over the thinking almost immediately and did not encourage her students to engage in productive struggle. As a result, what they learned about mathematics is not clear.

Considering How Sequences of Tasks Build Understanding Although lessons around high-level tasks, such as the one that took place in Mr. Donnelly’s classroom, can affect students’ disposition toward and learning of mathematics, learning important mathematical ideas rarely occurs within one lesson. As Hiebert and his colleagues have argued (Hiebert et al. 1997, p. 31), [the] teacher’s role in selecting tasks goes well beyond choosing good individual tasks, one after another. Teachers need to select sequences of tasks so that, over time, students’ experiences add up to something important. Teachers need to consider the residue left behind by sets of tasks, not just individual tasks. In ATL 3.4, you will analyze a sequence of four instructional tasks from an eighthgrade unit on rigid motion and consider how the tasks, as a set, can leave behind important residue — insights into the structure of mathematics and strategies or methods for solving problems (Hiebert et al. 1997).

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Analyzing Teaching and Learning 3.4 Exploring a Sequence of Tasks Review the four tasks from an eighth-grade unit on rigid motion shown in figure 3.5. Consider:   1. What does each task contribute to students’ understanding of rigid motion?   2. Use the Task Analysis Guide in figure 3.1 to decide how you would classify the cognitive demands of each of the tasks.   3. Why would you use the set of tasks rather than one individual task?

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Task 1: Reflect on This Engineers write computer programs that direct robots to move in specific directions and distances to complete complex tasks. Mathematicians also have developed ways to make figures move and change. 1. On a new piece of paper, draw the L and points D, E, and F. Then make an exact copy of line L and points D, E, and F on either patty paper or a transparency using a colored pencil or marker. Label the three points on your copy D’, E’, and F’. A. Use the copy to reflect D, E, and F across L. Tape the copy into its new position. B. Use your math tools to see what relationships you can find: a. among the original points D, E, and F and among the images D’, E’, and F’; and b. between the original points and their images. C. What preliminary conjectures can you make about the properties of reflection?

F

D

E

L

2. Now experiment with the reflection of a polygon across a line. A. On another piece of paper make an exact copy of quadrilateral GHIJ and the line of reflection L. Label the vertices G, H, I, and J. G H

J

I L

B. Next copy quadrilateral GHIJ and the line of reflection L on a transparency or patty paper, using a colored pencil. C. Use your copy to reflect quadrilateral GHIJ across the line of reflection L, and label its reflection quadrilateral G’H’I’J’. Tape the transparency or the patty paper to the original copy to show the reflection. D. Examine the relationships between features of the original figure and features of its image. What stays the same? What changes? E. Which of your preliminary conjectures still seem to be true? What new conjectures can you make? 3. Now it’s your turn. Draw a line of reflection and a polygon on a new piece of paper. Label its vertices. Try to think of a “case” that we have not tried — perhaps a different type of polygon or a different location or direction of the line of reflection. A. Use a transparency or patty paper to reflect your polygon across the line of reflection. Label its vertices. B. Examine the relationships between features of your original polygon and features of its image. What stays the same? What changes? C. Which of your conjectures still seem to be true? What new conjectures can you make? D. Use mathematical reasoning to justify why your conjectures make sense. Be prepared to share your reasoning with your group and the class. continued on next page

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Task 2: Slip and Slide 1. On a new piece of paper, draw the vector and ∆DEF. Make an exact copy of vector and ∆DEF on either patty paper or a transparency, using a colored pencil or marker. On your copy label the vertices of the triangle D’, E’, and F’. E

D V F

A. Slide your copy along vector to translate ∆DEF and create its image ∆D’E’F’. Tape the transparency or patty paper into its new position. B. Use your math tools to see what relationships you can find between and within the original triangle and its image. C. What preliminary conjectures can you make about the properties of translation? ⇀. 2. Now experiment with translation along vector w ⇀. A. On a new piece of paper, draw quadrilateral GHIJ and vector w G H

I

J

W

⇀ and quadrilateral GHIJ on a transparency or patty paper, using a colored B. Copy vector w pencil. Perform the translation along the vector, and label the translated quadrilateral G’H’I’J’. Tape the transparency into its new position.

C. Examine the relationships between and among the features of the original figure and the features of its image. What stays the same? What changes? D. Which of your conjectures still seem to be true? What new conjectures can you make? 3. Now it’s your turn to experiment with your own translation. Try to think of a “case” we have not yet tried. A. Draw a polygon and label its vertices. Draw a vector from one of the vertices. B. Use a transparency or patty paper to translate your polygon along the vector. Label the vertices of the translated polygon. C. Examine the relationships between features of your original polygon and features of its image. What stays the same? What changes? D. Which of your conjectures still seem to be true? What new conjectures can you make? E. Use mathematical reasoning to justify why your conjectures make sense. Be prepared to share your reasoning with your group and the class. continued on next page

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Task 3: Do the Twist M 1. Copy the center of rotation O, line segment OM, and points D and E onto patty paper or a transparency using a colored pencil or marker. A. Rotate the transparency or patty paper to show a counterclockwise rotation of 90° around O and tape the transparency or patty paper into place. After the rotation of points M, E, and D, label E the images M’, E’, and D’. B. Use your math tools to see what relationships you can O find between and among the original points and line segment and their images. C. What preliminary conjectures can you make about the properties of rotation. 2. Next experiment with the rotation of a polygon around a center of rotation. A. On a new piece of paper, draw quadrilateral GHIJ and the center of rotation O.

D

G H O

I J

B. Make an exact copy of quadrilateral GHIJ and the center of rotation O on a transparency or patty paper, using a colored pencil. C. Rotate quadrilateral GHIJ 180° counterclockwise around O, and label its image quadrilateral G’H’I’J’. Tape the transparency into place. Be prepared to explain how you determined that the degree of rotation is 180°. D. Examine the relationships between features of the original figure and features of its image. What stays the same? What changes? E. Which of your conjectures still seem to be true? What new conjectures can you make? 3. Now it’s your turn. Draw a center of rotation on a new piece of paper. Try to think of a “case” we have not yet tried. A. Draw a polygon and label its vertices. B. Select an angle of rotation and write it on your paper. Use a transparency or patty paper to rotate your polygon that number of degrees counterclockwise around the center of rotation. Label its vertices. Be prepared to justify how you know that your angle of rotation is correct. C. Examine the relationships between features of your original polygon and features of its image. What stays the same? What changes? D. Which of your conjectures still seem to be true? What new conjectures can you make? E. Use mathematical reasoning to justify why your conjectures make sense. Be prepared to share your reasoning with your group and the class. 4. Based on your explorations of reflection, translation, and rotation: A. What properties do these three transformations have in common? B. An original figure is congruent to its image when it is reflected, translated, or rotated. What do you think we mean when we say that two shapes are congruent? Extension: 5. Compare a reflection across a line, and a rotation of 180° around a point. A. Draw a vertical line of reflection L through point O on your work from Question #2 above. B. Reflect quadrilateral GHIJ across L. C. Compare the image produced by a rotation of 180° with the image produced by a reflection across a vertical line. What do they have in common? How do they differ? continued on next page

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Task 4: Robot Mouse Below is a diagram of a robot mouse in the original position M and the location of its final image M’. Assume that the robot is flat so that you can reflect it. Create different sequences of motions (translations, rotations, and reflections) that will move the robot mouse from position M to M’ on the field. To get started: • Make copies of the robot mouse on at least four different transparencies or pieces of patty paper. • Write sequences of steps to move the robot mouse from position M to position M’. Keep in mind that:  —  each step is a single transformation — either a reflection, translation, or rotation;  —  you will need to draw and label the vector, line of reflection, or center and angle of rotation for each step in your diagram; and  —  you will need to tape down a copy of the robot mouse to show the image created by each step.

M

M’

1. Describe and show a sequence of at least three steps that will move the robot mouse from position M to M’. 2. Describe and show a sequence of exactly two steps that will move the robot mouse from position M to M’. 3. Describe and show a sequence of steps that will move the robot mouse from position M to M’ that includes at least one reflection. Make a conjecture: A. Following a sequence of translations, reflections, and/or rotations, what will an original figure and its image always have in common? B. Explain why your conjecture makes sense.

Fig. 3.5. Tasks 1–4 in the rigid motion sequence Adapted from Institute for Learning 2015a. Lesson guides and student workbooks available at ifl.pitt.edu.

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Analysis of ATL 3.4: Exploring a Sequence of Tasks The four tasks shown in figure 3.5 can form the starting point for instruction in a unit on rigid motion. The first three tasks (see tasks 1, 2, and 3) help students develop an intuitive understanding of rigid motion. By exploring the relationships between an original image and a transformed image, students develop insights about reflections (Task 1 — Reflect on This), translations (Task 2 — Slip and Slide), and rotations (Task 3 — Do the Twist). On the basis of the criteria in the Task Analysis Guide (shown in fig. 3.1), the first three tasks are procedures with connections. Specifically, each task tells the student what to do: copy the given image (by using transparencies or patty paper), transform the image (reflect, translate, or rotate), explore the relationship between the given image and the transformed image by using mathematics tools (protractor and ruler), and justify why the conjectures make sense. However, students cannot mindlessly follow the procedures given. To complete the task, students must engage in exploring and analyzing the original and transformed images, determine what is the same and what is different, and articulate conjectures about the properties of each transformation. In addition, in task 3 (Do the Twist), students look across the first three tasks and identify the properties that the three types of transformations have in common and compare the reflection across a line and a translation of 180º around a point. From their explorations, students notice that rotations, translations, and reflections preserve segment length and angle measure and that transformed shapes are congruent to their original images. The final task in this sequence (Task 4 — Robot Mouse) asks students to describe a sequence of rigid motions that will move one shape onto another. This task allows students to use the knowledge gained in the first three tasks to create sequences of reflections, translations, and rotations, thereby solidifying their understanding of these concepts. On the basis of the criteria in the Task Analysis Guide (fig. 3.1), this task is a doing mathematics task. Although the task gives students the original image and students create a composition of a fixed number of transformations to arrive at the new image, multiple ways exist to achieve the desired outcome. At the end of the task, students generalize what the original figures and their transformed images will always have in common. Although each of these four tasks could be an interesting exploration for students, it is the sequence of the four tasks together that have potential to build students’ conceptual understanding of what transformations are and what they do. Hence, the residue (i.e., mathematical understanding) that remains (i.e., all transformations preserve size, area, angles, and line lengths) is a result of engagement in all four tasks. The ability to make generalizations about transformations requires opportunities to explore all three (reflections, translations, and rotations) individually and in combination.

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Implementing Tasks That Promote Reasoning and Problem Solving: What Research Has to Say So far this chapter has established that —  • tasks vary with respect to their cognitive demands, and low-level and high-level tasks provide different opportunities for student learning;

• high-level tasks are a necessary condition for ambitious instruction; 
• high-level tasks often decline during implementation in ways that reduce students’ opportunities to reason and problem solve; and • sequences of related tasks are necessary to build students’ understanding of key mathematical ideas.

A key question that emerges from our discussion is “What is the relationship between the tasks used as the basis for classroom instruction and student learning outcomes?” Research over the previous twenty-five years offers evidence that students who have ongoing opportunities to engage in high-level mathematical tasks show greater learning gains than students who spend most of their time engaged in procedural tasks (e.g., Hiebert and Wearne 1993; Stein and Lane 1996; Stigler and Hiebert 2004; and Boaler and Staples 2008). Two projects focused on the middle grades, the Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) project (Silver and Stein 1996) and the Trends in International Mathematics and Science Study (TIMSS) video study (Stigler and Hiebert 2004), provide additional insight on these phenomena. The QUASAR Project sought to improve student learning in urban middle schools that serve economically disadvantaged communities by focusing instruction on thinking, reasoning, and problem solving rather than on procedural fluency. Researchers collected and analyzed data on classroom instruction and student learning in four urban middle schools (grades 6, 7, and 8) over a four-year period. Students who performed best on project-developed measures of reasoning and problem solving were in classrooms in which teachers frequently set up and implemented tasks at high levels of cognitive demand; students who performed the worst on project-based measures were in classrooms where teachers consistently used low-level tasks (Stein and Lane, 1996). The results of the TIMSS video study (Stigler and Hiebert 2004) give additional evidence of the relationship between the cognitive demands of mathematical tasks and student achievement at the middle school level. In this study, a random sample of 100 eighthgrade mathematics classes from each of six countries that outperformed the United States on the 1995 TIMSS (Australia, the Czech Republic, Hong Kong, Japan, the Netherlands, Switzerland) and the United States were collected, videotaped, and analyzed during the 1999 school year. The study revealed that the six higher-achieving countries implemented a greater percentage of high-level (which that study called making connections) tasks in ways 50   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

that maintained the demands of the task. Although the higher-achieving countries (except Japan) did not use a greater percentage of high-level tasks than the United States, they more successfully avoided reducing these tasks into procedural exercises. The fundamental distinguishing feature between instruction in the United States and instruction in higherachieving countries is that students in U.S. classrooms “rarely spend time engaged in the serious study of mathematical concepts” (Stigler and Hiebert 2004, p. 16). The body of research on mathematics tasks establishes a clear link between the nature of classroom instruction (i.e., the type of task used and the manner in which teachers implemented it) and student learning outcomes. This finding emphasizes the need to give students ongoing opportunities to engage with cognitively challenging tasks.

Promoting Equity by Implementing Tasks That Promote Reasoning and Problem Solving Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014) identifies access and equity as an essential element of excellent mathematics programs. This principle makes salient that all students must “have access to a high-quality mathematics curriculum, effective teaching and learning, high expectations, and the support and resources needed to maximize their learning potential” (NCTM 2014, p. 59). This section considers how tasks that promote reasoning and problem solving can also provide access to all students and promote equity. Looking across the high-level tasks that this chapter has so far discussed indicates that these tasks share characteristics beyond their cognitive demands. These tasks can also be characterized as “low threshold, high ceiling tasks” (McClure 2011). Such tasks are ones that all students can enter and explore at some level but that have potential for engaging students in challenging mathematics. Doing mathematics tasks, such as the Candy Jar task (ATL 3.1), Duckling Survey task (ATL 3.2, task A) and Robot Mouse task (fig. 3.5, task 4), have no set pathways to follow and therefore invite students to select an approach that makes sense to them and that gives them entry to the task. Procedures with connections tasks, such as the Leveling Off task (ATL 3.2, task B), Reflect on This, Slip and Slide, and Do The Twist (fig. 3.5, tasks 1, 2, and 3) give a specific entry to the problem, but the guidance provided serves as a basis for exploring the mathematical question at hand. Engaging students in these high-level tasks affords them the opportunity to go deep with mathematics, one of the five equity-based mathematics teaching practices (Aguirre, Mayfield-Ingram, and Martin 2013). Making these high-level tasks accessible to all students (i.e., low threshold) requires launching the tasks so that students understand exactly what is expected of them and giving students appropriate resources that support their entry into the task and in ways that allow students to draw on multiple resources of knowledge (Aguirre, Mayfield-Ingram, and Martin 2013).

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Jackson et al. (2012) argue that launching a challenging task includes four aspects: discussing the key contextual features, discussing the key mathematical ideas, developing common language to describe the key features, and maintaining the cognitive demand. In a launch of the Candy Jar task that embodies these features, a teacher might do the following:   1. Access students’ prior knowledge regarding what a ratio is and what it means for ratios to be equivalent.   2. Show students a candy jar that contains 5 JRs and 13 JBs and ask students what they notice about the original jar (e.g., the jar has more JBs than JRs, the number of JBs is more than twice the number of JRs, 5 and 13 are odd numbers of candies) and what they might expect to find in the bigger jar (e.g., 100 JRs, more than 200 JBs, the same ratio as the first jar).   3. Avoid suggesting exactly what students should discover in their investigation or a specific procedure to use, unlike Ms. Pascal.

The launch phase of the lesson is important because research suggests that the way in which teachers set up or launch a task affects students’ ability to participate in whole-class discussions in meaningful ways ( Jackson et al. 2013). As discussed previously, students can complete the Candy Jar task by making a table and calculating additional entries by repeatedly adding 5 and 13, identifying the scale factor that indicates how much bigger the new jar is, or by finding the unit rate that shows the relationship between the two quantities being compared. In addition, making tiles or chips available in different colors affords some students a pathway to begin the problem that would otherwise be inaccessible to them. In general, physical materials can help students model abstract ideas, explore relationships, and communicate their thinking to others. The Candy Jar task, as well as other high-level tasks discussed in this chapter, has a high ceiling — the potential to engage students in challenging mathematics. As in “The Case of Patrick Donnelly,” the task can engage students in generalizing the relationship between JRs and JBs. Although Mr. Donnelly introduced the generalization during the discussion of the task, a teacher might include questions that press students to generalize when she or he first presents the task so that all students can extend their thinking before the whole-class discussion (see questions 2 and 3 in fig. 3.6).

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1. A candy jar contains 5 Jolly Ranchers (JRs) and 13 jawbreakers (JBs). Suppose that you had a new candy jar with the same ratio of Jolly Ranchers to jawbreakers, but it contained 100 Jolly Ranchers. How many jawbreakers would you have? Explain how you know. 2. Suppose that you have a huge candy jar that has the same ratio of Jolly Ranchers to jawbreakers, but it contained 1000 Jolly Ranchers. How many jawbreakers would you have? Explain how you know. 3. Write a rule that you can use to find the number of jawbreakers in any jar if you know the number of Jolly Ranchers.

Fig. 3.6. Questions that extend the Candy Jar task Adapted from Smith et al. 2005

Another characteristic that the high-level tasks that we have discussed share is that they are group-worthy (Lotan 2003). That is, the nonroutine nature of the high-level tasks gives students something to work on together, drawing on different strengths and resources of knowledge. Having students collaborate to solve these tasks generally results in a broader diversity of approaches and thinking than a student working alone would produce. Such tasks narrow the achievement gap by broadening the view of what counts as competence and who is seen as competent (Lotan 2003).

Key Messages • Tasks that promote reasoning and problem solving (i.e., high-level tasks) lead to the greatest learning gains for students when the demands of the task are maintained during implementation. • Sequences of related tasks rather than individual tasks are necessary to fully develop students’ understanding of mathematical ideas. • Tasks that promote reasoning and problem solving can give access to all students and promote equity by providing students with multiple entry points and ways to demonstrate competence.

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Taking Action in Your Classroom: Analyzing Task Selection and Implementation It is time to consider the implications of the ideas that this chapter has discussed for your own practice. You can begin this process by engaging in each of the Taking Action in Your Classroom activities described below.

Taking Action in Your Classroom Using the Task Analysis Guide to Examine Learning Opportunities Use the Task Analysis Guide (shown in fig. 3.1) to analyze the tasks that you have used in one of your classes during the previous few weeks. • To what extent did you provide your students an opportunity to reason and problem solve through the use of high-demand tasks? • Can you identify tasks in your textbook (or in other available resources) that require students to engage in reasoning and problem solving?

Using the Factors of Maintenance and Decline to Explain Lesson Outcomes Teach a lesson that you have planned based on a high-demand task. (You may want to make an audio or video recording of the lesson so that you have a record of what occurred.) Use the following questions to guide your reflection: • Was the implementation of your lesson more like that of Patrick Donnelly or more like that of Sandra Pascal? • What actions or interactions might account for the outcome of the lesson? • What factors (fig. 3.2) influenced the maintenance or decline of the task?

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CHAPTER 4

Build Procedural Fluency from Conceptual Understanding The Analyzing Teaching and Learning (ATL) activities in this chapter engage you in exploring the effective teaching practice, build procedural fluency from conceptual understanding. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 42): Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems. Supporting students in developing procedural fluency is an important mathematical goal for all teachers. Throughout their mathematical experiences, students should be able to select procedures that are appropriate for a mathematical situation, implement those procedures effectively and efficiently, and reflect on the result in meaningful ways. This procedural fluency, however, is fragile and meaningless without a sound conceptual understanding of the mathematics. Conceptual understanding and procedural fluency are essential and integrated components of mathematical proficiency. For students to make sound selections of procedures and carry them out effectively, teachers must support students in building a foundation of a conceptual understanding of mathematics on which rest a set of mathematical procedures. In this chapter, you will —  • establish the connection between procedural fluency and conceptual understanding;

• explore the ways in which you can sequence tasks to promote conceptual understanding first, then build procedural fluency;

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• review key research findings related to building procedural fluency from conceptual understanding; and

• reflect on ways that you can build procedural fluency from conceptual understanding in your own classroom. For each Analyzing Teaching and Learning activity, note your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, build procedural fluency from conceptual understanding. If possible, share and discuss your responses and ideas with colleagues. After you have written or shared your ideas, read the analysis, in which we offer ideas relating the ATL activity to the focal teaching practice.

The Relationship between Conceptual Understanding and Procedural Fluency Both conceptual understanding and procedural fluency are critical and connected components of students’ mathematical proficiency. A central question in this chapter is, how do procedural fluency and conceptual understanding relate to each other? Conceptual understanding must come first and must be the foundation on which to build procedural fluency. The activities in this chapter explore this complex relationship and provide the tools that will help you think about supporting students in developing conceptual understanding and building procedural fluency in your classroom. ATL 4.1 begins that exploration by focusing on mathematical tasks. This activity asks you to think about how exploring high-level tasks and developing different strategies can form the basis for developing procedural fluency.

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Analyzing Teaching and Learning 4.1 Exploring the Relationship between Conceptual Understanding and Procedural Fluency After students have had an opportunity to develop a set of strategies by solving a variety of contextual problems like the Candy Jar task, what might the students in Patrick Donnelly’s class be able to do when presented with the missing-value problem shown below? (You may want to return to the case in chapter 1 while you consider this question.) 5 127 = 13 x What do the three student solutions shown below indicate about what each student understands about proportional relationships? Solution 1

Solution 2

Solution 3

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Analysis of ATL 4.1: Exploring the Relationship between Conceptual Understanding and Procedural Fluency Students in Patrick Donnelly’s class used multiple mathematical representations and strategies when solving the Candy Jar task. These strategies all connected with the fundamental understanding of what it means to have a proportional relationship. Students explored the multiplicative nature of that relationship through a unit-rate approach, a scale-factor approach, and a scaling-up approach. Figure 4.1 summarizes these approaches.

Unit Rate (group 2) Since the ratio is 5 JRs for 13 JBs, we divided 13 by 5 and got 2.6. So that would mean that for every 1 JR, there are 2.6 JBs. So then you just multiply 2.6 by 100. (3100)   1 JR   ➞  100 JRs 2.6 JBs   ➞  260 JBs (3100)

Scale Factor (groups 3 and 5) You had to multiply the five JRs by 20 to get 100, so you also have to multiply the 13 JBs by 20 to get 260.

  5 JRs  13 JBs 

(320) ➞  100 JRs ➞  260 JBs (320)

Scaling Up (Group 6) JRs

5

10

20

40

80 100

JBs 13

26

52 104 208 260

We started by doubling both the number of JRs and JBs. But then when we got to 80 JRs, we didn’t want to double it any more because we wanted to end up at 100 JRs and doubling 80 would give me too many. So we noticed that if we added 20 JRs : 52 JBs and 80 JRs : 208 JBs, we would get 100 JRs: 260 JBs.

Fig. 4.1. Three approaches to solving the Candy Jar task

ATL 4.1 begins by asking what students might do with a more procedural problem after developing a set of strategies through solving several contextual problems similar to the Candy Jar task. One could imagine that those subsequent experiences would involve trying some of the same strategies, adding new strategies to the repertoire (such as scaling down), and testing the boundaries of those strategies to determine cases in which one strategy might be preferable to another. The “naked number” problem shown in the first part of ATL 4.1 represents the same sort of proportional relationship as the Candy Jar task. Many of the strategies used in the contextual problems are effective on this task as well. As modeled in solution 1, students can convert 5/13 to a unit rate (2.6). The unit rate represents a multiplicative relationship of 1 to 2.6. That same multiplicative relationship must hold for 127/x, and x must be equal to 127 3 2.6.

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A scaling-up approach might be cumbersome given the numbers, but it can give students important information to help them get started. Solution 2 illustrates a scaling-up approach, which then transitions into a scale-factor approach. The table in solution 2 shows the student scaling up the two quantities by 5 and 13 in each column until the values reach 20 and 52. From there, the student identifies a multiplier that takes 5 directly to 127, which is 25.4. Because the two quantities must increase by the same multiplicative rate, students would multiply 13 by 25.4 to arrive at an answer of 330.2. Solution 3 represents a more direct application of the scale-factor approach. The student appeared to identify a scale factor to multiply by 5 to reach 127 (i.e., how many times bigger 127 is than 5). The student then had to use that same scale factor to multiply 13, for a result of 330.2. The explanation in solution 3 does not indicate how the student determined the scale factor but uses the scale factor to justify why 330.2 is the correct answer. The approaches that these students used show considerable understanding about the nature of proportional relationships, their multiplicative relationships, and the ability to transfer what students learned in one context to another — one in which there is no context for sense making. Teachers might commonly associate this missing-value problem with a process of cross multiplication — a process that students may not have considered if they have worked with contextual problems before seeing this task. Cross multiplication has a high degree of efficiency for this particular type of problem, particularly as compared with some of the other approaches. This procedure, however, can often lead to misconceptions. Students frequently do not understand why they are applying the cross-multiplication procedure or why it is mathematically valid, they confuse cross multiplication with the procedure for multiplying two fractions (i.e., multiplying numerator by numerator and denominator by denominator), or they apply cross multiplication in inappropriate mathematical circumstances (such as in linear situations that are not proportional). All four approaches noted here — the three conceptually focused student solutions and cross multiplication — are generalizable, and students can broadly apply them across a number of contexts. The conceptually focused approaches may have a significant advantage over cross multiplication, because they may be more sensible and meaningful to students. After students have worked on contextual problems to develop their conceptual understanding of the nature of proportional relationships, a missing-value problem gives the teacher an opportunity to introduce cross multiplication and connect it with the other approaches in a way that builds meaning for the procedure (see Boston, Smith, and Hillen 2003 for more about developing meaning for cross multiplication). By sequencing contextual problems that develop conceptual understanding first, teachers can build a strong foundation for procedural fluency and make connections between the procedures and the meaning behind them. The next section considers a specific set of tasks that might follow the Triangle task from Ms. Polosky’s lesson featured in Chapter 2.

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The Role of Task Sequences Ms. Polosky, first encountered in Analyzing Teaching and Learning activity 2.2, wants her students to understand the formula for finding the area of any triangle. She began her instructional sequence by having students find the formula for determining the area of right triangles. (Before proceeding, readers may want to review the Triangle task video. You can access and download the video and transcript by visiting NCTM’s More4U website [nctm .org/more4u]. The access code can be found on the title page of this book.) ATL activity 4.2 returns to Ms. Polosky’s lesson on finding the area of a triangle. Here, you analyze a set of tasks and consider which might help students develop generalized procedures for finding the area of any triangle.

Analyzing Teaching and Learning 4.2 Considering How a Task Sequence Builds Fluency Ms. Polosky wants to build on the conceptual understanding that students are developing about the formula for finding the area of a right triangle and extend that understanding to a formula for finding the area of any triangle. She is trying to choose the task or tasks in which she should next engage her students. Consider tasks A–D shown below: • Which tasks might support Ms. Polosky’s students in advancing from a generalization for right triangles to a generalization for all triangles? • What aspects of the tasks might be challenging for her students to solve, given their current understanding regarding right triangles, and why? • In what sequence might you use some or all these tasks to follow up on Ms. Polosky’s initial lesson? continued on next page

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Task A

Task B   1. The City of Grandville is building a small park on some unused land at the corner of Delphi Road and Bridge Street. Use the diagram below to find the area of the park.

  1. Cut out copies of each triangle. Position each triangle on centimeter grid paper, and trace the triangle. (You can do this more than once if necessary.) a. Label the base and height of each triangle. b. Find the area of each triangle. Explain how you found the area, and show any calculations that you did.   2. Find a second way to place the triangle on the grid paper. a. Label the base and height of each triangle.

  2. The state of Idaho is a roughly triangular shape with a base of 380 miles and a height of 500 miles. Find the approximate area of Idaho.

a. Find the area of each triangle. Explain how you found the area, and show any calculations that you did.   3. Did changing the orientation of the triangle change the area? Explain.

Adapted from Connected Mathematics Project 2, Covering and Surrounding (Lappan et al. 2006).

continued on next page

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Task C

Task D

On a grid, draw a segment that is 6 cm long. Use the segment as a base for each triangle described in problem 1. Draw each triangle on a separate grid.   1. Using the 6 cm segment as a base —  a. sketch a right triangle with a height of 4 cm; b. sketch a different right triangle with a height of 4 cm; c. sketch an isosceles triangle with a height of 4 cm; d. sketch a scalene triangle with a height of 4 cm; e. Find the area of each triangle.   2. What do these four triangles have in common?   3. Two triangles each have a base of 5 cm and a height of 6 cm. Do they have the same area? Explain how you know.   4. Two triangles each have an area of 15 square cm. Do they have the same perimeter? Explain how you know.

Reprinted with permission from Illuminations, copyright 2008, by the National Council of Teachers of Mathematics. All rights reserved.

Adapted from Connected Mathematics Project 2, Covering and Surrounding (Lappan et al. 2006).

Analysis of ATL 4.2: Considering How a Task Sequence Builds Fluency The goal of the original Triangle task used by Ms. Polosky was to build an understanding of an area formula for a right triangle and to connect the formula with the physical characteristics of the triangle. In the video, students made connections between the geometric manipulations of the triangle and two different area formulas. One of these formulas involved replicating

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the triangle to create a rectangle, then taking half the area of the rectangle (see fig. 4.2a). The other involved slicing the triangle at the midpoint of the base perpendicular to the base and reorganizing the two pieces into a rectangle with a length of one-half the base of the original triangle and a width equivalent to the height of the original triangle (see fig. 4.2b). For students to move forward from this lesson, a teacher might want them to consider whether they can apply similar strategies to the areas of non-right triangles.

(a)

(b) Fig. 4.2. Two triangle manipulations used to justify the area of a right triangle

Of the four tasks presented in ATL 4.2, tasks A and C directly move students toward generalizing the formula developed in the lesson to any triangle. Task A is similar to the task that Ms. Polosky used in her lesson, but it does not explicitly address right triangles. The orientation of the triangles and the instructions to cut out the triangles leaves students with several options for orienting the base of the triangle on the grid paper. The second question in the task explicitly asks them to orient the triangle (and its base) a different way. This task would likely provide students with opportunities to connect their right-triangle area formulas implicitly with the area of the triangle and would move students a step further in their conceptual understanding by problematizing the notion that the base of a triangle is arbitrary. Task C has a slightly more direct connection to the previous lesson, because it begins with a right triangle and asks students to construct multiple right and non-right triangles. The construction of multiple triangles with the same base and height provides a strong connection to the concept that base and height are the two invariant quantities that contribute to the area. The lengths of the two nonbase sides of the triangle are arbitrary if the base and height remain the same. This concept builds meaningfully on students’ experiences in the initial lesson.

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This task does not raise the notion that any side of the triangle can serve as the base, as in task A. One could imagine, however, using task C and then task A to build toward both these important understandings. Tasks B and D involve concepts with which students should be fluent by the close of the unit, but they do not directly support students in thinking about right triangles. The framing of task D invites multiple procedures but may not give students enough scaffolding to enable them to move from the right triangle formula to a more general formula that is based on the understanding that Ms. Polosky’s students had at the close of the initial lesson. Students who learned one of the methods for finding the area of the right triangle in Ms. Polosky’s class could solve the first part of the task but might need additional support to solve the second part of the task with the acute triangle. The same observations hold true for task B. Although these two problems have a superficial context, they do not provide any support for Ms. Polosky’s students to enable them to extend their understanding of finding the area of a right triangle to finding the area of the acute triangle in the Idaho problem. Although the second problem includes the perpendicular height, it does not provide measurements that would allow students to break the larger triangle into two right triangles and find the area. If the Idaho problem had included measurements to help students find the area of two smaller triangles, it might offer a meaningful way to build students’ understanding from the right triangle formula to a more general formula for all triangles. A question may arise while you consider these tasks — if Ms. Polosky spent time helping students understand the conceptual basis for the right triangle formula, why not just tell students that this formula works for all triangles? As we noted in the discussion of cross multiplication, students often overgeneralize a procedure or formula and use it in inappropriate mathematical situations when they do not fully understand the conceptual underpinnings. In addition, asking students to develop the initial conjecture, justify that conjecture, and then broaden it to all triangles supports the development of students’ abilities to reason and prove, consistent with curricular recommendations (NCTM 2000, 2009; NGA Center and CCSO 2010).

The Role of Assessment ATL 4.3 considers task B in a different way. This activity helps connect the teaching practice build procedural fluency from conceptual understanding with the practice elicit and use evidence of student thinking, which chapter 8 investigates further.

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Analyzing Teaching and Learning 4.3 Connecting Conceptual and Procedural Knowledge with Assessment Consider task B. • If task B were the type of task that Ms. Polosky would want students to be procedurally fluent in solving as a part of an assessment, what aspects of conceptual understanding would help support that fluency? • How might a teacher foster those aspects of understanding?

Analysis of ATL 4.3: Connecting Conceptual and Procedural Knowledge with Assessment A teacher might want students to have several aspects of conceptual understanding so that they can fluently solve task B. Students should know that the base and height (or length and width) are the two quantities that determine the area of any triangle. They should understand that other variances, such as side length or angle measure, do not change area as long as the base and height remain the same. A general understanding of area as a two-dimensional covering of a shape helps students make sense of the task situation and the meaning of the answer. Understanding that the base and the height have a right-angle relationship and that students can choose any side as the base, as long as the height is clearly and correctly defined in relation to that base, may also help with this task, although no alternative ways exist for students to choose the base and arrive at an area measurement. Similarly, knowing that changing the orientation of a triangle does not change the base and height of the triangle is a conceptual understanding that teachers might hope that students would have, but it is not fundamental to the solution to this task. A teacher might foster conceptual understanding of the area of a triangle and collect evidence to assess whether students have these understandings in several ways. A teacher may first ensure that students have experience finding the area of a wide variety of triangles, with attention to varying angle types and orientation. A teacher might want to give students tasks that ask them to find the area of a triangle and then create a noncongruent triangle with the same area. To emphasize the notion that angles do not matter related to area as long as the base and height of a triangle are the same, the teacher might ask students to construct two triangles with the same angle measures and different areas. Across the lessons that comprise a unit on triangles, the teacher may also ask students to identify the base and the height on a wide variety of triangles and to find the area of the same triangle using two different base-height pairs. These sets of mathematical experiences are likely to give students a strong conceptual understanding for finding the area of a right triangle. This conceptual foundation makes it Build Procedural Fluency from Conceptual Understanding    65 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

likely that they will be able to use the procedure for finding the area of a triangle (A 5 1/2 bh) in sensible and appropriate ways. Strengthening the conceptual foundation first before moving to conceptual fluency is a key feature of ambitious mathematics instruction. The final activity of this chapter discusses mathematical procedures that are commonly taught in the middle grades and asks you to consider how they might provide students with opportunities to develop strong conceptual understandings that support knowledge of those procedures.

The Role of Conceptual Understanding in Minimizing Misconceptions ATL 4.4 identifies three common middle school topics that can be challenging for students to understand. If you were to teach these topics, in what sorts of experiences would you engage the students so that they develop a strong conceptual understanding?

Analyzing Teaching and Learning 4.4 Considering Conceptual Experiences That Support Troublesome Procedures Consider the three mathematical topics listed below. Each of these topics is commonly taught in the middle grades, and teachers want students to learn procedures associated with these topics. What types of experiences would you want to provide students to build their conceptual understanding before formalizing a procedure for each? • Solving systems of linear equations • Calculating mean absolute deviation (MAD) • Expanding factored binomials of the form (ax 1 c)(bx 1 d)

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Analysis of ATL 4.4: Conceptual Experiences for Troublesome Procedures One might engage students in a number of possible experiences related to each of the three mathematical topics to establish a sound conceptual foundation. This analysis offers some suggestions for each, with the recognition that many other possibilities exist that can serve students’ needs. The overarching principles in identifying these suggestions are that students should have experiences that support making meaning of the steps of the procedure when the class discusses the task and that making use of contexts and multiple mathematical representations can support the development of that meaning.

Solving Systems of Linear Equations

Often, students can become tangled in the procedural manipulations for solving systems of equations. Teachers should provide experiences that help motivate the importance of a solution of systems of equations both numerically and graphically and should connect the systems with a context to give meaning to the solution. Moreover, students should develop intuitive methods of identifying the solution of a system before experiencing the typical canon of procedures (e.g., substitution, elimination) to support the students in making good choices about which procedures to use. Tasks like the Two Storage Tanks task (see page 68) give a meaningful context for considering a system of linear equations. In this task, the intersection point and the solution to the system are meaningful quantities in the context of the problem. Students can solve the task by using a variety of representations (tabular, symbolic, graphical), and those solution paths require students to coordinate the meanings of the values of their variables, the meaning of the intersection point, and the rates of change of each of the linear equations. (Chapter 8 further discusses the Two Storage Tanks task.)

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The Two Storage Tanks Task Two large storage tanks, T and W, contain water. T starts losing water at the same time additional water starts flowing into W. The graph below shows the amount of water in each tank over a period of time. Assume that the rates of water loss and water gain continue as shown.

  1. When will the two tanks contain the same amount of water? Explain how you found your answer, and interpret your solution in terms of the problem.   2. If you have not already done so, write an equation for each storage tank that can be used to determine the amount of water in the tank at any given number of hours. • Explain what the different parts of each equation mean in terms of the problem. • Explain what the different parts of each equation mean in terms of the graph. Adapted from NAEP Released Items, 2003-8M10 #13. http://nces.ed.gov/NationsReportCard/nqt (National Center for Educational Statistics 2013).

A discussion of this task might focus on determining efficient solution strategies that could lead to establishing one or more procedures for solving a system of equations presented in symbolic form. After using either the graph or a table to determine the rate of change per hour, students might write the following equations for the water levels in each of the two tanks y 5 900 2 50x (Tank T) y 5 300 1 25x (Tank W)

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From that point, students might execute two possible general strategies. Students might use their knowledge of properties of equations to multiply the Tank W equation by 2 and then add the two equations — leading to the common elimination strategy. 2y 5 600 1 50x (Tank W equation multiplied by 2)   1    y 5 900 2 50x (Tank T)

3y 5 1500    y 5 500 (the two tanks will contain the same amount of water at 500 gallons) Substituting 500 for y into either equation results in an x value of 8 hours, answering the question. A second approach begins with recognizing that both the expressions on the right side of the equation are equal to y and that setting the two equal to each other gives the following result: 900 2 50x 5 300 1 25x. Students can connect to the idea that what they are doing in solving this equation is finding the differences in the starting points (900 2 300), which can be seen graphically, and the differences in the rates (250 and 25). In the resulting simplification of the equation, 600 5 75x, both the 600 and the 75 have meaning in this context. Students might also extend the graphs of the lines in the Two Storage Tanks task and use that information to determine the intersection. This is another generalizable strategy.

Mean Absolute Deviation

Mean absolute deviation (MAD) is an important introduction to understanding the variability of a distribution. The formula for finding the MAD is fairly straightforward — it is the sum of the absolute values of deviations of the individual data points from the mean. To build meaning for the MAD and motivate its use, teachers might consider using a task like the Temperature task with students.

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The City Temperature Task Consider the temperature data from seven cities shown in the figure. In what order would you classify the temperatures of the seven cities, from least variability to most variability? Justify your answer by explaining the process that you used to sort the cities.

Adapted from Engage NY Grade 6, Module 6, Lesson 9: “A Story of Ratios.” Eureka Math (2014).

A discussion of this task might focus on ways in which students could mathematically describe the differences between the plots, which all feature the same mean temperature but significantly different distributions of temperature observations. For example, students might make an argument that city E or city F is more variable because the gap between two temperature values is very wide. Students might debate whether city B or city G is more variable. Both appear to have temperatures in the same general range but with different clustering around the mean. Students might argue that city D has a great deal of range and may consider it more variable. This discussion motivates the need for a common algorithm that can help organize and order the plots from greatest variability to least variability. A teacher might ask students what 70   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

the mean temperature for each city is and ask them to calculate that value. The teacher might then ask students how they could characterize how far away from that average temperature each city is likely to be. That question might lead students to compare individual observations to the mean. The process of aggregating those differences is the algorithm for mean average deviation.

Expanding Binomial Expressions

Teachers often introduce the FOIL (first, outer, inner, last) procedure to students as a quick and efficient way to expand binomials (see fig. 4.5). It is also a procedure that students often implement in inappropriate ways. This FOIL procedure works for binomials but breaks down when students begin multiplying other polynomials (Karp, Bush, and Dougherty 2015). It also obscures the important mathematical ideas connected to the distributive property and the relationships between multiplication and area.

Fig. 4.3. First, inner, outer, last (FOIL)

Representations such as algebra tiles can allow teachers to make connections to distribution and to visually understand the multiplication of two variable terms to produce a squared term. Using a task like the Algebra Tiles task can motivate the procedure for multiplying binomials, as in figure 4.6. Students are familiar with area models for multiplication in the elementary grades, and the algebra tiles show the creation of an area model with a width of 3x 1 2 (with the long rectangles representing x and the squares representing 1) and a length of 2x 2 1 (with dark tiles indicating negative values). By filling in the area model with large squares (representing x by x, or x 2), long rectangles (representing x by 1), and small squares (representing 1 by 1), students can see the ways in which the two expressions are multiplied. They can also see each of the individual products that the FOIL procedure produces and why the procedure produces those products. Understanding the algebra tile representation gives

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students the conceptual tools that they need to move beyond a limited procedure like FOIL so that they can multiply a monomial times a binomial. With this solid foundation rather than just knowledge of the procedure, students can identify patterns through repeated calculations that will support them with more complex algebraic problems of similar types.

Use algebra titles to show these multiplications and make a sketch of your model. Write the product. 1. 2x(x 2 1) 2. (x 1 1)(x 1 2) 3. (x 2 1)(3x 1 3) 4. (x 2 3)(x 1 3) 5. (2x 1 2)(2x 2 2) 6. (x 1 3)(x 1 3)

(2x 2 1)

1

50

(3x 1 2) x 1 (2x) 5 0

(3x 1 2) (2x 2 1) 5 6x 2 2 3x 1 4x 2 2 5 6x 2 1 x 2 2

Fig. 4.4. Algebra Tiles task and a worked example of (3x 1 2)(2x 2 1) (Taken from Stein et al. 2009, p. 91)

In each of these three cases, an experience that focuses students on the meaning behind a procedure that the teacher has not yet introduced connects with previous mathematical experiences and ideas and allows the use of multiple representations that can help students develop sound conceptual understanding. After these experiences in developing conceptual understanding, procedures are more likely to take hold and be used fluently and appropriately by students, as summarized in the research in this area.

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Building Procedural Fluency from Conceptual Understanding: What Research Has to Say When learning mathematics focuses predominantly on remembering and applying mathematical procedures, students often struggle. If students have not had opportunities to develop an understanding of when and why to use a procedure, they often apply the procedure in ways that are inappropriate to the mathematical task at hand (Martin 2009). Conversely, when procedures connect to conceptual understanding, students are better able to retain the procedures and make more appropriate decisions in how they apply procedures to new mathematical situations (Fuson, Kalchman, and Bransford 2005). The balance between conceptual understanding and procedural fluency is delicate. In the middle school, much of the mathematical work involves generalizing arithmetic relationships and using properties in ways that build more powerful understandings, including proportional reasoning, function, and algebraic reasoning. Teachers must make sure not to rush students to procedural fluency and the abstraction that often comes with it in middle school without establishing and maintaining connections to mathematical meaning. Such a rush can damage students’ identities as mathematical knowers and doers and can promote increased mathematical anxiety (Ashcraft 2002; Ramirez et al. 2013). Instead, middle school work should focus on strengthening conceptual understandings. For example, students should enter middle school with fluency in multiplication. Middle grades teachers should make connections between multiplication and area — connections that were forged in the elementary grades — and extend that connection to multiplication with variable expressions. This understanding, which can build in part through the use of algebra tiles, supports students in understanding the meaning behind polynomial factoring and the distributive property beyond simply knowing how to execute the FOIL procedure. Although teachers often contrast procedural fluency and conceptual understanding in teaching practice, they are not mutually exclusive. Hiebert and Grouws (2007) synthesize the research outcomes that compared conceptually focused teaching with procedurally focused teaching. On the whole, students in conceptually focused classrooms developed both conceptual understandings and skill with procedures. In procedurally focused classrooms, students developed only procedural fluency. In other words, students can develop procedural fluency through conceptually focused experiences like those described in this chapter, not just through direct instruction. The order in which teachers sequence tasks is also important. “The Case of Monique Butler” (Stein et al. 2009) illustrates the dangers of teaching the procedure first without understanding and subsequently attempting to connect with conceptual understandings. Most commonly, students who learn the procedure first continue to apply the procedure without

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engaging in the challenging work of making meaning for the procedure. Monique Butler was a well-intentioned teacher — she wished for her students to learn the FOIL method so that they would be prepared for items on a standardized test. But when she returned the day after teaching them the procedure and asked them to demonstrate connections with visual area representations by using algebra tiles, students did not engage in the ways that she wanted them to engage. They had no motivation to use the visual representations because they could use the procedure that they had memorized to attempt to solve the tasks. Students also had no understanding behind the procedure, leading to overconfidence and errors. Developing conceptual understanding first with students is critical to supporting the effective and efficient fluency with procedures that is an important part of doing mathematics.

Promoting Equity by Building Procedural Fluency from Conceptual Understanding Building procedural fluency from conceptual understanding is critical to providing all students access to meaningful mathematics. Mathematics instruction that focuses solely on remembering and applying procedures favors students who are strong in memorization skills and may disadvantage students who are not. Moreover, because new procedures frequently rest on previously learned procedures, students who have struggled with mathematics and may not have strong procedural fluency at the start of instruction are more likely to be marginalized when instruction focuses solely on building new procedures. When instruction instead focuses on building conceptual understanding by using multiple mathematical representations and multiple solution paths, students have a wider range of options for entering a task and building mathematical meaning. If students subsequently build procedures on this strong foundation, they can always return to the conceptual understanding to confirm or regenerate the procedure if they do not remember it. For example, if a student in Ms. Polosky’s class does not remember the formula for the area of a right triangle, he or she can use one of the two demonstrated procedures to regenerate the formula and find the area of the triangle. Although all students should reach a mathematical understanding such that they do not have to regenerate common formulas, starting with a strong conceptual understanding gives students more mathematical flexibility and a more robust knowledge base. Two of the equity-based practices in mathematics classrooms that Aguirre, MayfieldIngram, and Martin (2013) identify have strong connections to building procedural fluency from conceptual understanding. When students begin exploring a new mathematical idea with conceptually focused instruction, this conceptual experience supports them in going deep with mathematics. Because students are more likely to misapply or forget procedures if they lack a conceptual grounding, going deep with the mathematics through a conceptual focus

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early in an instructional sequence helps all students gain access to important mathematical ideas. Moreover, the use of multiple entry points and representations for conceptual tasks makes it much more likely that each and every student will be able to start productively on a task and ultimately work his or her way toward procedural fluency. Building that fluency from conceptual understanding also connects to the equity-focused practice of leveraging multiple mathematical competencies. When students memorize a procedure without a conceptual backing, the students have only one avenue for successful use of that procedure — memorizing and reproducing it. When students have experiences like the ones described in Ms. Polosky’s class that focus on the meaning behind the procedures, they can leverage those experiences even if they do not immediately remember the formula or how to apply it, thereby affording broader access to students so that they can demonstrate their mathematical competence in diverse situations.

Key Messages • When students learn mathematical procedures without a sound conceptual underpinning, they often misapply the procedures or forget them.

• Instruction should begin with a focus on conceptual understanding, thereby allowing students to make mathematical meaning for the procedures that they will learn. • Conceptually focused instruction gives students both understanding and fluency with skills.

Taking Action in Your Classroom: Building Procedural Fluency from Conceptual Understanding Taking Action in Your Classroom invites you to explore ways in which you can build students’ conceptual understanding before introducing specific procedures for them to use.

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Taking Action in Your Classroom Building Procedural Fluency from Conceptual Understanding Think about a unit that you have recently taught, and analyze the tasks that you asked students to do. • What were your goals for conceptual understanding and for procedural fluency in the unit? • Which of the tasks that you provided students attended primarily to conceptual understanding, and which contributed primarily to procedural fluency? • Reflect on where these tasks were in the timeline of the unit. What changes, if any, might you make in teaching this unit in the future to strengthen conceptual understanding and procedural fluency?

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CHAPTER 5

Pose Purposeful Questions The Analyzing Teaching and Learning (ATL) activities in this chapter engage you in exploring the effective teaching practice, pose purposeful questions. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 35): Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships. Questions are the primary tool that teachers have to help them determine what students know and understand about mathematics. Specifically, purposeful questions should reveal students’ current understandings; encourage students to explain, elaborate, or clarify their thinking; and make the mathematics more visible and accessible for student examination and discussion. According to Weiss and Pasley (2004, p. 26), “Teachers’ questions are crucial in helping students make connections and learn important mathematics and science concepts.” In this chapter, you will —  • analyze a transcript from a whole-class discussion in which the teacher poses questions to support students’ investigation of two different solutions to a task;

• analyze a video in which a teacher is interacting with small groups during a lesson to assess what they currently understand about the task and to advance beyond their current understanding;

• write questions that teachers can use to assess and advance students’ understanding on a specific task based on anticipated solutions; • review key research findings related to purposeful questions; and • reflect on questions used in your own classroom.

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For each Analyzing Teaching and Learning activity, note your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, pose purposeful questions. If possible, share and discuss your responses and ideas with colleagues. After you have written or shared your ideas, read the analysis section, in which we offer ideas relating the ATL activity to the focal teaching practice.

Exploring the Nature and Purpose of Teacher Questions ATL 5.1 visits Marie Hanson’s class, where she is facilitating a discussion about the Candy Jar task that we first encountered in chapter 1. While students were working on the task, Ms. Hanson noticed that Jordan had used an incorrect additive strategy to solve the task. (Group 1 also used this strategy in Patrick Donnelly’s class — the case first explored in chapter 1.) That strategy is a common misconception in the domain of proportional reasoning. Ms. Hanson therefore decided to begin the classroom discussion with it because she wanted to make it public so that all students in the class could see why that approach would not work.

Analyzing Teaching and Learning 5.1 Exploring the Questions Asked during a Whole-Class Discussion Read the following transcript of what took place during Marie Hanson’s lesson while the class was discussing Jordan’s solution to the Candy Jar task. • What do you notice about the questions that the teacher asked? • What purpose did the questions appear to serve? • How did the teacher foster a discussion on multiplicative rather than additive strategies?

1

Discussion in Marie Hanson’s Classroom

2 3

( Jordan walks to the front of the class and places his work (shown below) on the document camera. He then explains how he arrived at his solution.)  5 1 95 5 100 13 1 95 5 108

4 5 6 7

Jordan:

Since I had to add 95 to get to 100 Jolly Ranchers, I did the same thing to the jawbreakers — I added 95, so the answer is 108.

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Does anyone have any questions for Jordan? Sarah, it seems like something isn’t sitting right with you. Do you have a question to ask Jordan?

8 9

Ms. H:

10 11 12

Sarah:

13

Jordan: You have to do the same thing to both numbers, so 108 has to be the answer.

14

Ms. H:

15 16 17 18

Jerry:

19

Jordan:

20 21 22 23 24 25

Jerry:

26



27



28



29



30 31

Ms. H:

32

Jerlyn:

33

Ms. H:

34 35 36

Jerlyn:

37

Ms. H:

38



39



40



41



Why are the jawbreakers and Jolly Ranchers almost even in the new jar? There had been a lot more jawbreakers than Jolly Ranchers in the first jar. Something isn’t right. Does everyone agree with Jordan?

I don’t agree, because the problem specifically said that the new candy jar had the same ratio of Jolly Ranchers to jawbreakers; and in Jordan’s new jar, the ratio was almost one to one. I don’t know why Jordan’s way is wrong, but I’m sure it is. (To Jerry) How did you do the problem?

(Comes to the front of the room and places his work at the document camera.) I used 1 Jolly Rancher to 2.6 jawbreakers. (The students had previously established that 5 JRs to 13 JBs was equivalent to 1 JR to 2.6 JBs.) If one Jolly Rancher turns into 100 Jolly Ranchers, it must have been multiplied by 100. And so the 2.6 jawbreakers also have to be multiplied by 100. I guess I did the same thing to both numbers, too. But I multiplied. You added.  1

2.6

(3100)

100

260

(3 100)

(To the class) What do you all think? Which jar has the same ratio of Jolly Ranchers to jawbreakers as our first jar? Jerry’s jar has exactly the same ratio as the first. How can you tell?

You can divide Jerry’s 260 jawbreakers and 100 Jolly Ranchers each by 20 and you get 13 jawbreakers and 5 Jolly Ranchers — that’s what we started out with. Jerlyn, is this what you are saying? (Mrs. H records Jerlyn’s explanation.) 100

260

(20)

(20)

5

13

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42

Jerlyn:

43

Owen:

44

Ms. H:

45 46 47 48

Owen:

49

Mrs. H:

50



51



52



53



54 55 56 57

Ms. H:

58 59 60

Jordan:

61 62 63 64 65

Ms. H:

Yes that is it!

I sort of did what Jerlyn did, but in the opposite direction. Owen, can you explain what you did?

I knew it was 260 jawbreakers because you had to multiply the 5 Jolly Ranchers by 20 to get 100, so you’d also have to multiply the 13 jawbreakers by 20 to get 260 — kind of like what Jerry did, but I started with 5 and 13, not 1 and 2.6 Is this what you did, Owen? (Mrs. H records Owen’s explanation.)  5 13

(320)

(320)

100 260

Jordan was right to point out that what we do to the Jolly Ranchers we also have to do to the jawbreakers. However, if we want to keep the same ratio, Jordan’s answer shows us that adding the same amount doesn’t work. Jordan, what do you think about your strategy now? Yeah, when I added the same amount, the jawbreakers and Jolly Ranchers didn’t get bigger at the same rate. The Jolly Ranchers sort of caught up with the jawbreakers.

That is a good way to think about it. As Jordan said, we need to keep increasing the jawbreakers and the Jolly Ranchers at the same rate, and that means that adding the same amount to each of them won’t work. But as we just saw with Jerry’s work and Owen’s work, multiplication does work because it keeps the ratio the same.

The authors have adapted this transcript from Smith et al. (2005), “The Case of Marie Hanson” (chapter 3).

Analysis of ATL 5.1: Exploring the Questions Asked during a Whole-Class Discussion The first thing you may notice in this brief discussion is that Ms. Hanson asked several questions, almost all of which were open-ended. Open-ended questions encourage students to explain, elaborate, or clarify their thinking or make the mathematics more visible and accessible for student examination and discussion. Such questions invite a variety of approaches and responses, making salient what students know so that a teacher can guide their thinking toward the goals of the lesson. Principles to Actions (NCTM 2014) identifies four types of questions as important in teaching mathematics; three of them (2, 3, and 4) are open-ended questions: 80   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

 1. Gathering information questions ask students to recall facts, definitions, or procedures.  2. Probing thinking questions ask students to explain, elaborate, or clarify their thinking.  3. Making the mathematics visible questions ask students to discuss mathematical structures and make connections among ideas and relationships.

 4. Encouraging reflection and justification questions ask students to reveal understanding of their reasoning and actions or make an argument for the validity of their work. At this point, with these question types in mind, you may want to quickly review the questions that Ms. Hanson asked and consider how you would categorize specific questions. Viewing the discussion through this framework of question types highlights the fact that Ms. Hanson asked different questions for different purposes. At several points, she asked probing thinking questions so that she could better understand what the students were thinking and the basis for their thinking. For example, after Jerlyn made a general statement supporting Jerry’s solution (line 32) Ms. Hanson asked her, “How can you tell?” (line 33). Jerlyn’s response to this question made clear the mathematical basis for her claim that the new ratio that Jerry had produced was equivalent to the original ratio. Ms. Hanson subsequently asked Owen to explain what he did (line 44). Owen’s initial explanation, “I sort of did what Jerlyn did, but in the opposite direction” was not clear. By asking him to explain, Ms. Hanson (and the other students in the class) heard a mathematically sound explanation that showed what Owen meant by “the opposite direction”  —  he meant that he had multiplied whereas Jerlyn had divided. (He actually used the inverse operation.) Ms. Hanson also asked questions to encourage reflection and justification. For example, after Jerry proposed his solution (lines 20–29), Ms. Hanson invited the class as a whole to reflect on Jerry’s new jar compared with Jordan’s new jar. She asked, “What do you all think? Which jar has the same ratio of Jolly Ranchers to jawbreakers as our first jar?” (lines 30–31), thereby challenging students to consider the two options and to argue for their point of view. In so doing, she was also positioning students to rely on their collective thinking to determine the mathematical correctness of the two solutions. Later in the lesson, Ms. Hanson returned to Jordan (line 56–57) to see what he thought about his initial solution in light of the subsequent discussion. This question caused Jordan to reflect on his earlier work, and he concluded, “The Jolly Ranchers sort of caught up with the jawbreakers.” Ms. Hanson selected Jordan as the first presenter because she wanted to make the mathematics underlying proportional relationships more visible. But rather than simply tell Jordan he was wrong, she asked students to consider the validity of Jordan’s approach (line 14) and later, to determine which of two candy jars ( Jordan’s or Jerry’s) had the same ratio as the initial jar (lines 30–31). Although, as previously noted, these questions encouraged reflection and justification, they also uncovered the multiplicative relationship between the Jolly Ranchers and the jawbreakers. In the end, Ms. Hanson brought the discussion surrounding the additive

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approach to a close by circling back to Jordan’s strategy (lines 54–57) and trying to form a classroom-wide consensus that using a multiplicative constant preserved the ratio. This particular excerpt does not show the teacher gathering information. Although such retrieval questions can be important in a lesson, they do not engage students in thinking and should not dominate the lesson. Ms. Hanson did ask students to verify whether what she had recorded matched the explanation that they had given (lines 37, 49). These questions did not engage students in thinking, but they did position students as authors of the mathematical ideas that they shared and communicated to them that she wanted to make sure that she had accurately captured the way that they had thought about the problem. Also noteworthy about this discussion was the way that Ms. Hanson asked students to take a position when she asked, “Does everyone agree with Jordan?” (line 14), and “Which jar ( Jordan’s or Jerry’s) has the same ratio of Jolly Ranchers to jawbreakers as our first jar?” (lines 30–31) Both questions set the poles of a debate, and students needed to decide which position to support. Chazan and Ball (1999, pp. 7–8) claim, “Disagreement — the awareness of the presence of alternative ideas — can be an important catalyst. . . . Hence, disagreement with others may cause students to re-evaluate and rethink their ideas.” By discussing competing solutions, the students in Ms. Hanson’s class were able to disprove some ideas ( Jordan’s) while agreeing on others ( Jerlyn’s and Owen’s ideas). Principles to Actions (NCTM 2014) also identifies two patterns of questioning: funneling and focusing (Herbel-Eisenmann and Breyfogle 2005). Funneling involves using a set of questions to lead students to a desired procedure or conclusion while giving limited attention to responses that differ from the desired path. Students do not have an opportunity to make connections or build their own understanding. Focusing involves the teacher’s honoring what the students are thinking by pressing students to communicate their thinking clearly and asking them to reflect on their thinking and the thinking of their classmates. Ms. Hanson’s pattern of questioning fits the focusing pattern, because she used questions that elicited and clarified students’ thinking and made the mathematics at the heart of the lesson visible. Her goal in asking these questions was not to direct the discourse toward a particular mathematical process for scaling up the proportional relationship but instead to present an open space for students to share valid multiplicative methods that focused on the comparison between those methods and the incorrect additive strategy. How does a teacher know what questions to ask? The next section argues that a teacher should carefully plan many of the questions that she or he will ask in advance of a lesson. The non-task-specific questions shown in figure 5.1 can be a good starting point for teachers who are attempting to foster an interactive classroom environment in which students explain and reflect on their understanding, in which students ask questions of both their peers and teachers, and in which the teacher encourages and supports discourse.

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Questions That Foster Discussion •

Helping students work together to make sense of mathematics What do others think about what ____ said? Do you agree? Disagree? Does anyone have the same answer but a different way to explain it? Can you convince the rest of us that that makes sense?

• Helping students to rely more on themselves to determine whether something is mathematically correct Why do you think that? Why is that true? How did you reach that conclusion? Does that make sense? Can you design a model to show that? •

Helping students to reason mathematically Does that always work? Is that true for all cases? Can you think of a counterexample? How could you prove that? What assumptions are you making?

•

Helping students learn to conjecture, invent, and solve problems What would happen if . . . ? What if not? Do you see a pattern? What are some possibilities here? Can you predict the next one? What about the last one? How did you think about the problem? What decision do you think he should make? What is alike and what is different about your method of solution and hers?

•

Helping students to connect mathematics, its ideas, and its applications How does this relate to . . . ? What ideas that we have learned before were useful in solving this problem? Have we ever solved a problem like this before? Can you give me an example of . . . ?

Fig. 5.1. General questions that foster discussion (Taken from NCTM 1991, pp. 3–4)

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Considering How Questions Support Student Learning ATL 5.2 takes you into the classroom of Elizabeth Brovey, where her eighth-grade prealgebra students are working on part 2 of the Calling Plans task (shown below). Mrs. Brovey’s class consists of 27 mainstreamed eighth-grade prealgebra students. Before the lesson featured in the video, students solved part 1 of the Calling Plans task; and Mrs. Brovey posted in the classroom the tables, graphs and equations that the students produced in response to that task.

The Calling Plans Task Part 1 Long-distance company A charges a base rate of $5 per month, plus 4 cents per minute that you are on the phone. Long-distance company B charges a base rate of only $2 per month, but it charges 10 cents per minute used. How much time per month would you have to talk on the phone before subscribing to company A would save you money? Part 2 Create a phone plan for a different company, company C, that costs the same as companies A and B at 50 minutes but that has a lower monthly fee than either of the plans. Adapted from Achieve, Inc. (2002).

Through their engagement in this task, Mrs. Brovey wanted her students to understand that —    1. the point of intersection is a solution to each equation (companies A, B, and C);   2. the rate of change (cost per minute) determines the steepness of the line; and

  3. if the y-intercept (monthly base rate) is lowered, then the rate of change (cost per minute) must increase for the new equation (company C) to intersect the other two (companies A and B) at the same point.

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Analyzing Teaching and Learning 5.2 more Assessing and Advancing Students’ Learning Watch the video clip and download the transcript of Elizabeth Brovey’s students working individually and in small groups. What is Mrs. Brovey trying to accomplish with the questions that she poses to her students? You can access and download the videos and their transcripts by visiting NCTM’s More4U website (nctm.org/more4u). The access code can be found on the title page of this book.

Analysis of ATL 5.2: Assessing and Advancing Students’ Learning According to NCTM (2000), asking questions that reveal students’ knowledge about mathematics allows teachers to design instruction that responds to and builds on that knowledge. Questions that reveal students’ knowledge are assessing questions. These questions allow the teacher to determine what the student knows and understands about key mathematical ideas, problem-solving strategies, or representations. They either probe thinking or gather information (question types 1 and 2 in fig. 5.1). Questions that build on students’ current knowledge are advancing questions. These questions move students beyond where they currently are toward the targeted goal of the lesson. They either make mathematics visible or cause students to reflect on and justify their thinking (question types 3 and 4 in fig, 5.1). Figure  5.2 shows the characteristics of assessing and advancing questions.

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Assessing Questions • Are based closely on the work that the student has produced • Clarify what the student has done and what the student understands about what she or he has done • Give the teacher information about what the student understands Teacher stays to hear the answer to the question.

Advancing Questions • Use what students have produced as a basis for making progress toward the target goal of the lesson • Move students beyond their current thinking by pressing students to extend what they know to a new situation • Press students to think about something that they are not currently thinking about Teacher walks away, leaving students to figure out how to proceed.

Fig. 5.2. Characteristics of assessing and advancing questions (Bill and Smith 2008)

In Mrs. Brovey’s interaction with her students, she asks both assessing questions (e.g., lines 2, 50, 77 in the transcript) and advancing questions (e.g., lines 35, 37–39, 81–84). For example, in Mrs. Brovey’s first assessing question, “Can you tell me what you’re doing?” (line 2), she asks the student to explain what he has done. Although the teacher can see what the student has written, she is not making any assumptions about what it means to the student. She is thereby communicating to the student that she cares about what he is thinking and what he understands and that he is capable of explaining and interpreting the mathematical ideas. Later in the lesson, Mrs. Brovey asks a student, “12 cents. And what does it stand for in terms of her calling plan?” (line 54). The answer to this question reveals the student’s understanding of the connection between the numbers in the equation and the constraints of the situation. One of Mrs. Brovey’s advancing questions is on line 81: “Why? I want you to think harder about where 50 cents comes in. Maybe if I asked you to think about how these three equations are connected and I’d like you to think, there’s another one, too. There’s another one. How are these three equations connected? I want you to try and figure that out.” Here she is pressing the student not only to come up with another equation that works but also to connect the equations and explain the role that 50 cents plays. Even though the students have done what the task asked them to do, the teacher is pressing them beyond the confines of the original task. The advancing question in this case serves as an extension question for a group that has finished the original assignment. In the video clip of Elizabeth Brovey, she stayed with a student or with the group when she asked assessing questions, yet moved on when she asked an advancing question. This distinction is critical because advancing students’ understanding is impossible unless the teacher 86   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

has determined what they currently understand. According to Mrs. Brovey, “If I leave them with that (advancing question), I’m letting them know that I believe that they can do that on their own and that I value the fact that they can do that on their own.” The advancing question allows students to engage in productive struggle while they grapple with a question that they cannot immediately answer. According to Cartier et al. (2013, p. 60) “by leaving students with this ‘parting shot’ the teacher gave them something to pursue and sent the message that they were capable of following through with the suggestion without her oversight.” How is it that a teacher such as Mrs. Brovey seems to be able to ask the right questions? Is she just really good at thinking in the moment? Perhaps, but for most teachers approaching a group, determining what the group is doing and quickly coming up with the right questions is challenging. It is more likely that before the lesson, she determined how students would approach the problem and how she would move them toward the goals of the lesson. As one experienced teacher commented: Coming up with good questions before the lesson helps me keep a highlevel task at a high level, instead of pushing kids toward a particular solution path and giving them an opportunity to practice procedures. When kids call me over and say they don’t know how to do something (which they often do), it helps if I have a ready-made response that gives them structure to keep working on the problem without doing it for them. This way all kids have a point of entry to the problem. (Smith, Bill, and Hughes 2008, p. 137) The teaching practice pose purposeful questions does not occur in isolation from the other effective teaching practices. Both teachers featured in this chapter (Ms. Hanson and Mrs. Brovey) engaged in many of the practices. They supported students’ productive struggle on the high-level tasks so that their students felt ownership of their own learning and saw themselves as capable mathematically. The teachers supported this struggle by posing questions that assessed and advanced thinking without taking over the challenging aspects of the task and engaged students in meaningful discourse that elicited students’ thinking and used it by making connections among different approaches for solving a task and to the mathematics ideas that were central to the lesson. How do teachers like Ms. Hanson and Mrs. Brovey prepare themselves to teach lessons that feature high-level tasks? Consider, for example, the chart shown in figure 5.3, which a teacher could create before a lesson featuring part 2 of the Calling Plans task. The figure lists anticipated solution strategies along with questions that the teacher could ask students producing each strategy. When armed with a set of questions that correlate with potential student solution strategies, the teacher is ready with questions to ask while she or he moves from group to group. By considering possible solutions and questions in advance of the lesson, not only is the teacher ready to respond to students who did what she or he expected, but she or he also has time to formulate and ask the right questions of students who used strategies that the teacher did not anticipate. Pose Purposeful Questions   87 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Anticipated Strategies Trial and error — started with a given monthly fee or started with a given cost per minute. Possible equations: y 5 0.11x 1 1.50 y 5 0.12x 1 1.00 y 5 0.13x 1 .50 y 5 0.14x

Assessing Questions

Advancing Questions

• How did you know that • If this is the equation of the fee would be $1.50 the company C plan, can (or $1, etc.)? describe the plan in words? • Why did you start with • How would the graph of 15¢ (or 14¢, etc.)? your plan compare to the • What do you mean by graphs for company A and “it didn’t work?” What company B? did you find? What did • Do any other plans work? you try next? • How did you find your equation?

Draw a sketch of a graph with plans A and B and many lines connecting points on the y-axis less than (0, 2) with (50, 7).

• How and why did you draw your sketch? Does the sketch give you enough information to help you determine a plan for company C? • Where do you see the monthly fee for company C in the graph? • How does the graph show that your plan for company C will work?

• How can you use your sketch to make a table of values or an equation? • How would you use your graph to find the details of the plan for company C? • How would you use your graph to find the equation for the company? • Are there other places on the y-axis where you could have chosen your point?

Pick a point that represents a starting value of less than $2 (e.g., 0, 1.00) and draw a line from that point to the point (50, 7). Then make a slope triangle to figure out the cost per minute (up 6 and over 50 would be 0.12).

• What made you construct this particular slope triangle? How does it relate to company C? • Which side of the triangle represents the cost per minute?

• Can you describe your plan for company C in words? • What does the graph for the three plans look like? • How do the three equations compare? • Are there other places on the y-axis where you could have chosen your point?

Fig. 5.3. Possible strategies and related questions

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Several beginning teachers with whom we have worked have noted the benefits of anticipating student strategies and designing possible assessing and advancing questions prior to teaching their lessons: First, I find that when I create assessing and advancing questions prior to a lesson, it forces me to really know the task I am giving to my students inside and out. It forces me to understand the conceptual learning the students should get out of the lesson, as well as the procedural skills, as well as any strategies or misconceptions students might use/have while completing the task . . . I create assessing and advancing questions because it helps me feel more confident going into a lesson. If the lesson is going to be a challenge for the students, I feel confident in my ability to help provide students with questions that probe their thinking and help them access the task, while at the same time maintaining the rigor . . . . (Kevin Olsen) Planning questions in advance allows for a more productive lesson overall. I am able to anticipate some of the difficulties students will have with the material and prepare questions that will help them overcome these difficulties. Also, if I’m able to plan and ask questions that cause students to extend their thinking, students will be more likely to understand and transfer the important concepts to future lessons. (Barbara Stevens)

Creating Questions That Assess and Advance Student Learning ATL 5.3 gives four responses to the Mixing Juice task (shown below). Imagine that the last time you taught this lesson your students produced those responses. Before teaching the lesson again, you decided to prepare questions that you can ask during the lesson to assess and advance student thinking so that you are better prepared to support students in making progress toward the mathematical goals of the lesson. Your goal for the lesson is for students to develop the ability to recognize proportional situations and to use ratios, fractions, decimals, rates, unit rates, and percentages to make comparisons among quantities. You selected this task for the lesson because it aligns with this goal, is cognitively challenging, and has multiple entry points and solution paths.

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The Mixing Juice Task Arvind and Mariah attend summer camp. Everyone at the camp helps with cooking and cleanup at meal times. One morning, Arvind and Mariah are in charge of making orange juice for all the campers. They plan to make the juice by mixing water and frozen orange juice concentrate. To find the mix that tastes best, they decide to test some recipes.

  1. Which recipe will make juice that is the most “orangey”? Explain.   2. Which recipe will make juice that is the least “orangey”? Explain. From Connected Math Project Grade 7: Comparing and Scaling. © 1997 by Michigan State University, G. Lappan, J. Fey, W. Fitzgerald, S. Friel, and E. Phillips. Used by permission of Pearson Education, Inc. All rights reserved.

Analyzing Teaching and Learning 5.3 Creating Assessing and Advancing Questions • Solve the Mixing Juice task shown above. • Review the four samples of student work shown below. For each response,

consider the following:  —  What do the students appear to know or understand about mathematics?  —  With your lesson goals in mind, what questions could you ask to assess and advance students’ thinking?

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Student Work for The Mixing Juice Task Solution A

Solution C

Solution B

Solution D

Fig. 5.4. Four solutions to the Mixing Juice task

Analysis of ATL 5.3: Creating Assessing and Advancing Questions Each of the groups A–D used a different strategy to solve the task. The students who produced solution A appeared to understand that to compare mixtures with different amounts of the same quantities (juice and concentrate), they needed a common basis for comparison. They created equivalent ratios by scaling down the given ratios to calculate unit rates of the amount of water needed for one can of concentrate. They then compared those rates by judging the most orangey mixture and the least orangey mixture by the most and least amount of water per can of concentrate. To organize their data, they created a table. The following are questions that the teacher could ask this group: Pose Purposeful Questions   91 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Assessing Questions

Advancing Questions

• Explain your table and how you determined the numbers in each column. • You rewrote the recipes so that they all use one cup of concentrate. How did you know that you could rewrite them in that way? • Why did you choose to make all the recipes use one cup of concentrate? • You noted that mix A has the fewest cans of water — why does that tell you that mix A is the most orangey?

• You found the amount of water needed if you had one cup of concentrate for each mix. How much concentrate would you need if each mix used one cup of water? Is mix A still the most orangey with this strategy? • Can you design a mix that is more orangey than mix A? Can you design a mix that is less orangey than mix B?

The students who produced solution B scaled up each recipe to make fifteen cups of juice and then multiplied each quantity by the scale factor to find the total amounts of concentrate and water in the larger mixture. They appear to understand that scaling up each ingredient by the scale factor and then comparing the fractions (cups of concentrate divided by total cups of juice) with common denominators of fifteen would determine the most orangey mixture and the least orangey mixture. Possible assessing and advancing questions that a teacher could ask this group include the following: Assessing Questions • Discuss the calculations that you did and why you did them. • What does the fifteen cups mean? Why did you decide to use fifteen cups? • For mix C, what does the 1.25 mean? (Similarly, what does the 1.875 for mix D mean?) • Which mix did you decide is the most orangey? The least orangey? How do you know?

Advancing Questions • You scaled the recipes up so that they all made fifteen cups of juice. Suppose that you only had enough concentrate to use one cup in each mix. How much water would you need for each mix? • Were there amounts other than fifteen cups that would have helped you solve the problem? • Can you design a mix that is more orangey than mix A? Can you design a mix that is less orangey than mix B?

The students who produced solution C exhibited a common misconception in proportional reasoning. The students only considered the mixture with the most cans of concentrate. They did not think about how the cans of water would affect the taste. They did not compare the two parts and so did not consider all the constraints of the problem. Possible assessing and advancing questions that a teacher could ask this group include the following:

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Assessing Questions • How did you decide that mix C is the most orangey and mix B is the least orangey? • How do the cups of water added to the concentrate affect the taste of the mixed juice?

Advancing Questions • How does the amount of water used in each recipe affect the strength of the juice? • What if mix E used four cups of concentrate and one cup of water — which would be stronger, mix C or mix E? Why? • Model each of the four mixes with number tiles or a diagram. Then make some observations about how the concentrate and the water compare to each other.

The students who produced solution D exhibited another common misconception in proportional reasoning — identifying the relationship between water and concentrate as additive rather than multiplicative. ( Jordan also identified the relationship as additive when solving the Candy Jar task in Marie Hanson’s class.) They subtracted the amounts of each and based their answer on the most and least differences instead of thinking of them as ratios. Possible assessing and advancing questions that a teacher could ask that group include the following: Assessing Questions • How did you decide that mix A was the most orangey and mix C was the least orangey? • Discuss your calculations. For example, for mix B you subtracted 1 from 4 and obtained a result of 3. What does the 4 mean? The 1? The 3?

Advancing Questions • What if mix E used one cup of concentrate and two cups of water — how would it compare with mix A? • Follow-up question if students say that mix A and mix E are equally orangey: Suppose that you want to use the recipe for mix A, but only one cup of concentrate is available. How much water should you use? Is that the same amount of water used in mix E? • Model each of the four mixes with number tiles or a diagram. Then make some observations about how the concentrate and the water compare to each other.

It is helpful to keep student work that illustrates typical strategies, unique strategies, and common misconceptions to assist in planning future lessons that focus on the same task or other proportional reasoning tasks. These student-work artifacts not only help teachers design potential questions for future lessons but also help in orchestrating the whole-class

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discussion. Although students in the class ideally generate the work that is shared in a wholeclass discussion, the teacher can introduce a particular model or strategy that students did not generate to ensure that the key ideas needed to accomplish the goals of the lesson are made public (Cartier et al. 2013). Tasks should allow students to use different representations and to make connections among representations (i.e., tables, arithmetic, equations, and context). All three tasks in this chapter — the Candy Jar task, the Calling Plans task, and the Mixing Juice task — invite the use of multiple representations and classroom discourse that highlight the similarities and differences among the strategies and representations, thereby fostering a deeper and more conceptual understanding of the mathematical ideas in the lesson.

Pose Purposeful Questions: What Research Has to Say As noted throughout this chapter, the questions that a teacher asks while students work on a task (individually or in small groups) and during a whole-class discussion can shape the mathematical content in which students engage during a lesson and the ways in which students engage with that content. Teachers’ questions can support high-level thinking, prompt for explanations and connections, and encourage students to delve more deeply into mathematics. Teachers’ questions can also elicit facts, procedures, or calculations. Teachers’ questions are a critical aspect of effective mathematics teaching and show strong links to gains in student achievement (e.g., Redfield and Rousseau 1981; Samson et al. 1987). For example, in a five-year longitudinal study of 700 students at three high schools, Boaler and Staples (2008) noted that when teachers used tasks with high cognitive demand, questioning was a key factor in teachers’ ability to maintain the level of demand during a lesson, contributing to the success of students at Railside High School (one of the three high schools in the study). According to the authors: . . . the support that teachers gave to students did not serve to reduce the cognitive demand of the work, even when students were showing signs of frustration. . . . At Railside, teachers were highly effective in interacting with students in ways that supported their continued thinking and engagement in the core mathematics of the problems. (2008, pp. 635–36) Unfortunately, the types of questions that teachers pose in many U.S. classrooms do not support reasoning and problem solving. For example, Stigler and Hiebert (1999) noted that teachers in the United States pose fewer high-level questions in middle school mathematics classrooms than teachers in other countries. In a study of more than 350 mathematics and science lessons in grades K–12, Weiss and Pasley (2004) concluded that questioning that encourages students to think deeply was rare and occurred in less than 20 percent of the observed classes. 94   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Many classrooms continue to follow the traditional questioning pattern of “initiateresponse-evaluate,” or I-R-E (Mehan 1979a). In I-R-E, the teacher initiates a question intended to elicit a specific answer, a student provides a response, and the teacher evaluates the response as correct or incorrect. The I-R-E questioning pattern leaves little room for students to express ideas or explain their thinking because the questions asked tend to be limited to gathering information. Incorporating the open-ended types of questions discussed in this chapter (e.g., probing thinking, making mathematics visible, encouraging explanations and justification) can help teachers move beyond the I-R-E pattern and create space for studentgenerated ideas, strategies, and representations in mathematics lessons. Through the previously described types of purposeful questioning, teachers send important messages to students that learning mathematics includes reasoning quantitatively and abstractly, constructing explanations and justifications, examining the reasoning of others, and making sense of mathematics. Teachers must therefore pose more higher-level questions that surface and make visible important mathematical structures and connections; that probe students’ mathematical understanding and reasoning; and that encourage justification, reflection, and generalizations. Purposeful questioning is a critical tool for effective teaching.

Promoting Equity by Posing Purposeful Questions Equitable teaching of mathematics requires the intentional use of teacher questioning to ensure that every student not only progresses in his or her learning of important and challenging mathematical ideas but also develops a strong mathematical identity (Aguirre, MayfieldIngram, and Martin 2013). Teacher questioning and positioning of students influences how students view themselves as members of the mathematics learning community in the classroom. Positioning includes the way that students are “entitled, expected, and obligated to interact with one another as they work on content together” (Gresalfi and Cobb 2006, p. 51). Are all students’ ideas and questions heard, valued, and pursued in the mathematics classroom? Who does the teacher call on to answer questions? What mathematical ideas does the class examine and discuss? Whose thinking does the teacher select for further inquiry, and whose thinking does the teacher disregard during small-group and whole-class discussions? Teachers make numerous decisions in their classrooms every day, and they need to be more aware of whom they are asking which types of questions, whom they are positioning as competent, and whose ideas are featured and privileged. The teacher needs to pay particular attention to including students from marginalized populations. Not only must teachers carefully consider what questions to ask their students about mathematics, they must also be conscious of which students have voice and authority in those mathematical conversations (Civil and Planas 2004; Heinz 2010). By eliciting and valuing the thinking of each student, teachers can work toward building identity and agency within students as individuals capable

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of learning and using mathematics to solve problems (Aguirre, Mayfield-Ingram, and Martin 2013; Schoenfeld 2014). Equitable opportunities for participation — as well as the sense of being capable, valued, and valuable in the mathematics classroom — can support students’ engagement and success in mathematics.

Key Messages • Questions should go beyond gathering information to probing thinking and requiring explanation and justification. • Questions that build on — but do not take over or funnel — student thinking advance student understanding. • Assessing questions allow the teacher to determine what the student knows and understands about key mathematical ideas, problem-solving strategies, or representations. Advancing questions move students toward the targeted goal of the lesson.

• Anticipating questions should be part of the lesson-planning process so that teachers can highlight possible strategies, representations, and misconceptions and support student learning and engagement without taking over the thinking for students.

Taking Action in Your Classroom: Analyzing Questions and Responses Taking Action in Your Classroom invites you to explore the questions that you asked during a lesson and to consider the insights that these questions give you about students’ understanding of mathematics.

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Taking Action in Your Classroom Analyzing Questions and Responses Teach a lesson using a high-level task (see criteria in fig. 3.1). Record the discussion.   1. Consider the extent to which the questions you asked —  • revealed students’ current understandings; • encouraged students to explain, elaborate, or clarify their thinking; and • made the mathematics more visible and accessible for student examination and discussion.   2. Consider the discussion from an equity perspective: • Did historically marginalized students have equal opportunities to answer questions of the type listed in question 1? • Did you ask some students only information-retrieval questions?

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CHAPTER 6

Use and Connect Mathematical Representations The Analyzing Teaching and Learning (ATL) activities in this chapter engage you in exploring the effective teaching practice, use and connect mathematical representations. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 24) —  Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving. What exactly does representations mean? As NCTM points out —  The term representation refers both to process and to product — in other words, to the act of capturing a mathematical concept or relationship in some form and to the form itself. . . . Moreover, the term applies to processes and products that are observable externally as well as to those that occur “internally,” in the minds of people doing mathematics. (NCTM 2000, p. 67). As shown in figure 6.1, representations refer to visual diagrams, symbolic notation, verbal descriptions, contextual situations, and physical models. The arrows that connect each pair of representations show the bidirectional nature of the relationship — one can start with either representational form and relate it to the other. Developing flexibility in using and moving between different representations increases students’ understanding of mathematical ideas.

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Visual (diagrams, graphs, and pictures)

Physical

Symbolic

(manipulatives and models)

(algebraic and numeric)

Contextual

Verbal

Fig. 6.1. Different representations and the connections between them (Adapted from NCTM 2014, p. 25)

In this chapter, you will —  • determine different representations that can be used to solve and make sense of a mathematical task;

• consider what different representations contribute to the development of students’ mathematical understanding;

• analyze an instructional episode in which the teacher uses different representations; • develop questions that can help students make connections among different representational forms; • review key research findings related to mathematical representations; and • reflect on representation use in your own classroom.

For each Analyzing Teaching and Learning activity, note your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, use and connect mathematical representations. If possible, share and discuss your responses

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and ideas with colleagues. After you have written or shared your ideas, read the analysis, in which we offer ideas relating the ATL activity to the focal teaching practice.

Exploring Different Representations Representations are essential in supporting students’ use and understanding of mathematical concepts and relationships, as well as in communicating mathematical thinking, strategies, solution paths, arguments, and reasoning to oneself and to others. Representations facilitate students’ recognition of the connections among mathematical concepts and students’ application of mathematics to real-world problem situations through modeling. When students experience and employ mathematical representations, they build a repertoire of tools that expand their ability to think mathematically. Representations — such as pictures, diagrams, graphical displays, tables, and symbolic expressions — have traditionally been part of the middle school curriculum; but each representation is often taught in isolation without connections to one another or to meaning. Analyzing Teaching and Learning activity 6.1 explores representations that students can use to solve a task. First, you solve the task yourself and consider the representation that you used and the extent to which the representation helped you make sense of the situation. Next, you analyze a set of solutions that middle school students produced and consider how the representations used could enhance students’ understanding of the mathematics.

Analyzing Teaching and Learning 6.1 Considering the Advantages of Different Representations   1. Solve the Cars and Motorcycles task shown on page 102. Think about the representation that you used and how it helped you make sense of the situation.   2. Review the samples of student work shown in figure 6.2. Consider the following: • What does each representation tell you about what the student understands about the situation? • How do the different representations connect with one another?

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The Cars and Motorcycles Task Diane looked out the window of her math classroom at the teachers’ parking lot and said, “There are 13 motorcycles and cars in the lot.” Steve looked out the window and said, “I see 42 wheels.” The teacher asked, “How many motorcycles and how many cars are in the parking lot?” Show all your work, and explain your thinking. This task was adapted from the QUASAR Cognitive Assessment Instrument (Lane 1993).

Student Work for the Cars and Motorcycles Task Student A

Student D

continued on next page

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Student B

Student E

Student C

Fig. 6.2. Five student solutions to the Cars and Motorcycles task

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Analysis of ATL 6.1: Considering the Mathematical Advantages of Different Representations Students can solve the Cars and Motorcycles task in many different ways — by using algebra to set up and solve a system of equations; by drawing pictures to represent the number of vehicles and wheels; by using a physical model to represent the number of vehicles and wheels; by creating a table and testing possible options; or by guessing and checking. In the solutions presented in ATL 6.1, student A used algebra to solve the task. That student appeared to understand how to use equations to represent the relationships between vehicles (C 1 M 5 13) and between the number of wheels (4C 1 2M 5 42). The student appeared to have rewritten the first equation as C 5 13 2 M, as evidenced by the substitution in the second equation 4(13 2 M) 1 2M 5 42; but the student did not show that step. The student then solved the equation and concluded that M 5 5. Although the student found that C 5 8, how the student came to that conclusion is not evident. Perhaps the student mentally determined that since the total number of vehicles was 13, the number of cars had to be 8. (The phrase appeared to understand is used rather than the word understood because inferring understanding from written work alone is not possible. For example, student A may have been following a pattern for solving such problems with limited understanding of elements of his or her work such as why he or she made the substitution of 13 2 M for C.) Student B used a numeric guess-and-check approach to solve the task. The student appeared to understand that the number of vehicles must total 13 and that the total number of wheels had to be 42 (with cars contributing 4 wheels each and motorcycles contributing 2 wheels each). The student tried three different combinations of vehicles until a pair yielded 42 wheels. After the first pair (6C and 7M), the combinations seem to be more than random guesses. When the first combination yielded 38 wheels, the student switched the number of cars and motorcycle (7C and 6M), which resulted in 40 wheels. The final combination (8C and 5M) shows an increase of 1 C and a decrease of 1 M from the previous combination. Student C also used a guess-and-check strategy, but instead of a series of arithmetic problems, that student used an organized table showing the number of motorcycles (column 1), the number of cars (column 3), and the total number of wheels (column 2). Beginning with the combination 1 and 12 (for motorcycles and cars, respectively), the table lists all combinations of cars and motorcycles that yield a total of 13 vehicles. The student then calculated the number of wheels until he or she found the correct combination of 5 and 8. The student did not continue the table. Student D created a physical model by using 42 bingo chips to represent the total number of wheels. The student initially paired the chips to represent the wheels of 21 motorcycles. The student then realized that 21 vehicles were too many, so she or he moved two chips from one row (representing the 2 wheels of a motorcycle) to another row. The four chips then represented one car. The student continued this process until there were 13 rows (vehicles).

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Student E drew a picture to model the situation and started with 13 vehicles. The student appeared to understand that each vehicle had at least 2 wheels, so he or she assigned 2 wheels to each vehicle and used a total of 26 wheels. He or she then distributed the remaining 16 wheels, two at a time, to arrive at a total of 8 cars and 5 motorcycles. Although all the approaches resulted in the correct solution to the problem, they represented different levels of sophistication, ranging from a basic method of guess and check (student B) to the more abstract algebraic approach (student A). Although the algebraic approach may be the intended target of instruction, the approaches used by students B, C, D, and E reveal a lot about their understanding of the situation and can ultimately link with the algebraic representation. The different representations allow all students to enter the problem and make sense of the relationships. According to Marshall, Superfine, and Canty (2010, p. 46), Students should have frequent opportunities to not only learn to use and work with representations in mathematics class but also to translate between and among representations. Sufficient opportunities help them see mathematics as a web of connected ideas. In supplying such opportunities, teachers should engage students in dialogue about representations and the relationships between them in order to help develop students’ representational competence, an important aspect of mathematical understanding. Hence, classroom discourse should make clear the mathematical contribution of each representation and the connections among these representations. The power of a whole-class discussion of the varied solutions lies in comparing the variety of solution strategies and examining the mathematical structures evident in each of the representations. Students should relate where each solution represents the two constraints, 13 vehicles and 42 wheels. The discussion might include students explaining the similarities and differences between solutions B and C, and between solutions D and E, as well as how each of the other solutions reflects the equations in solution A. For example, student D’s solution shows 13 vehicles (corresponding to the number of rows), 8 of which have 4 wheels (sets of four circles) and 5 of which have 2 wheels (sets of two circles). Student E’s solution also shows 13 vehicles, each represented by a circle, 8 that have 4 wheels and 5 that have 2 wheels. Each row in the table that student C created shows the sum of cars and motorcycles, and the number of wheels in the middle column represents the calculation 2M 1 4C. All three of these solutions relate directly to student A’s algebraic solutions by showing that the sum of cars and motorcycles (C 1 M) equals 13 and that cars each contribute 4 wheels, whereas motorcycles contribute 2 wheels (4C 1 2M 5 42). By relating the equation (symbolic–algebraic representation) to pictures (visual representation) and the table (symbolic–numeric representations), students can begin to understand how to

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construct equations in two variables and solve them by substitution, thereby building their algebraic repertoire. While students create, compare, and use various representations, they develop a facility with representations other than their own; and by listening, questioning, and making an effort to understand what their classmates are trying to communicate with their drawings or writings, they add to their own library of skills. Adding to their own library of skills is especially important when the representations involved are unconventional, as in solutions D and E. In discussing different solutions, the teacher should consider the order in which she or he has the students present their solutions. Beginning the discussion with a solution that all students can understand regardless of the approach that they used (e.g., solution C) and then moving to more abstract or sophisticated approaches builds the mathematical understanding of the entire class. (Chapter 7 includes further discussion of the importance of selecting particular strategies and sequencing them in a specific order.) The point is not for students to use different representations just because they are possible. What is important is that students are using and connecting representations as tools to solve problems and to build understanding of mathematical concepts and ideas. Students should understand that written representations of mathematical ideas are an essential part of learning and doing mathematics. The ways in which the teacher organizes the discussion about these representations and connects them should make the underlying mathematics visible. Encouraging students to represent their ideas in ways that make sense to them is important. Hence, different representations should —  • be introduced, discussed, and connected;

• focus students’ attention on the structure or essential features of mathematical ideas; and • support students’ ability to justify and explain their reasoning.

Exploring the Use of Representations during Instruction So how would a teacher use the different representations that students produce during an actual lesson? What purpose can these representations serve? To answer these questions, ATL 6.2 drops in on a discussion in Terry Daniel’s eighth-grade class, where students are exploring the handshake problem, a classic problem in mathematics that asks how many handshakes occur when n people shake hands with one another. Ms. Daniel’s two goals for her students’ learning in this lesson were —  • to enable students to recognize the relationship between people and handshakes and create a generalized solution to the problem; and

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• for students to understand that they can use different mathematical representations to make sense of and solve a task and to understand the relationships among different representations. Ms. Daniel selected a task that aligned well with her goals for this lesson, a critical factor in the potential for success in a lesson.

Analyzing Teaching and Learning Activity 6.2 Using Representations to Support Learning • Solve the Supreme Court Handshake problem shown below. • Read the excerpt from the discussion that took place in Terry Daniel’s classroom.  —  What different representations did the students in Ms. Daniel’s class produce?  —  What did Ms. Daniel do to help her students make connections among the different representations?  —  How did the use of these different representations support students’ learning?

Supreme Court Handshake When the nine justices of the Supreme Court meet each day, each justice shakes the hand of every other justice, to show harmony of aims, if not views.   1. If each justice shakes hands exactly once with each of the other justices, how many handshakes take place?   2. How can you determine the number of handshakes for a group too large to model? Reprinted with permission from Illuminations, copyright 2008, by the National Council of Teachers of Mathematics. All rights reserved. http://illuminations.nctm.org/Lesson.aspx?id=2112

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1

The Use of Representations in Terry Daniel’s Classroom

2 3 4 5 6 7 8 9 10 11 12

Terry Daniel presented the Supreme Court Handshake task to her seventh-grade students. She selected this task because it gave students an interesting context in which to create a function that described the relationship between two quantities and an opportunity to use and compare different representations of a function. After some initial discussion about handshakes and the Supreme Court, she told students to solve the first part of the task any way that they wanted. While the students began work on the task, she monitored the activities of each group, intervening as needed and keeping track of who did what. At one point, groups 2 and 6 asked her to help them act out the number of handshakes because they needed a ninth person. When the groups had all arrived at an answer, she called them together for a whole-group discussion. The following excerpt describes this discussion.

13 14

Ms. D:

15 16 17 18

Jada:

Jada (from group 2), could you come up and explain to us what your group did? You can just bring your work and place it on the document camera. Well, we decided to act it out, so we got Belinda’s group (group 6) and Ms. D to help us so we would have nine people. Then we just counted the number of handshakes and kept track of them in a table. When we were done, we just added them up and got thirty-six. Person

# of Handshakes

1

8

2

7

3

6

4

5

5

4

6

3

7

2

8

1

9

0

Could we reenact this for the rest of the class? Let’s have everyone from groups 2 and 6 come up and show us what you did. The rest of the class should keep track of the handshakes and see if what you did matches what Jada shared. [Students in groups 2 and 6, along with Ms. Daniel, model the handshakes.]

19 20 21 22 23

Ms. D:

24 25

Ms. D:

26

Students: Yes!

Do you agree with Jada that the total number of handshakes should be thirty-six?

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Can someone explain what seems to be going on here?

27

Ms. D:

28 29 30

Belinda: Well what we noticed was that the first person shakes the most number of hands and that every person after that shakes one less hand. By the time you get to the last person, there are no other hands left for them to shake.

31 32

Ms. D:

33 34 35

Devon:

Devon (group 3), I think your group came to a similar conclusion. Can you share what you did?

We sort of did the same thing, but instead of acting it out, we drew a picture with nine people and then made lines connecting them [places drawing on document camera]. Then we counted the number of lines we drew. 1 2

9

3

8

4

7 6

36

Tamika:

37 38

Ms. D:

39 40 41 42 43 44

Devon:

45

Ms. D:

46 47 48 49 50

Jamal:

51

Ms. D:

52 53

Tamika:

54 55

Ms. D:

5

I don’t get why you have all the lines. It is confusing.

Can you explain your picture to Tamika and the class in more detail to help them better understand your thought process?

Each line represents a handshake between two people. We started with person 1 and drew a line connecting to all the other people. This gave us eight lines. Then we went on to person 2, and this time we only have to make seven lines. So, as we kept going, we saw the number of handshakes that a person does decreases by one each time, just like Belinda said. So, our lines are really handshakes. Can someone repeat what Devon said in their own words?

Well, basically the picture is the same as what the groups acted out. Because each person only shakes hands with each other person once, the number of handshakes decreases. So, person 2 only has seven handshakes because they already shook hands with person 1. Person 3 only has six handshakes because he already shook hands with persons 1 and 2. Tamika, does that make sense to you now?

Yeah. I get it now. All I could see at first was a bunch of lines but now I see how they drew them and what they mean.

Michelle (group 5), I think your group did something similar, but you added up the numbers differently. Can you share that with us?

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56 57 58 59 60 61 62

Michelle: Well, we started doing a picture like group 3, but we didn’t finish it. Once we saw that each person after the first one had one less handshake, we realized it was going to be 8 1 7 1 6 1 5 1 4 1 3 1 2 1 1, so we stopped drawing the lines after the third person. But we didn’t want to just add them up, so we tried to find a shortcut. So, we paired the numbers together to make nines and then saw how many nines we had. So, we found that there were 4 nines and 4 3 9 5 36. Same as what everyone else got.

63

Derrick: Why didn’t you just add them up in the order that they came in?

64 65 66 67 68 69 70

Michelle: We could have; this just seemed easier. Ms. D:

Kenyon (group 4), your group also multiplied. Can you share with us how you approached the problem?

Kenyon: Well we figured the nine people would each shake eight hands because they don’t shake hands with themselves. 9 3 8 5 72. But everyone would be shaking hands twice, so you would take half of that. So it is thirty-six. Audra:

Where did seventy-two come from? I don’t get it.

Ricardo (group 1), I think the chart that your group made might help explain where the seventy-two came from. Could you come up and show us what you did?

71 72 73

Ms. D:

74 75 76 77 78 79 80 81 82

Ricardo: We made this chart that showed all nine judges. Each square in the chart shows a handshake between the judges. We started by showing all the possible handshakes and that turned out to be 9 3 9 5 81. But the judges can’t shake hands with themselves, so we subtracted those out. That is the diagonal that we shaded. That gave us seventy-two handshakes, which is what Kenyon’s group had. But then we realized that we double counted — like 2-3 and 3-2 are the same handshake — so we had to subtract each of the duplicates. We saw that we were left with thirty-six. So, it was the same thing as dividing by 2 like Kenyon did. 1 2 3 4 5 6 7 8 9

1 1-1 2-1 3-1 4-1 5-1 6-1 7-1 8-1 9-1

2 1-2 2-2 3-2 4-2 5-2 6-2 7-2 8-2 9-2

3 1-3 2-3 3-3 4-3 5-3 6-3 7-3 8-3 9-3

4 1-4 2-4 3-4 4-4 5-4 6-4 7-4 8-4 9-4

5 1-5 2-5 3-5 4-5 5-6 6-5 7-5 8-5 9-5

6 1-6 2-6 3-6 4-6 5-6 6-6 7-6 8-6 9-6

7 1-7 2-7 3-7 4-7 5-7 6-7 7-7 8-7 9-7

8 1-8 2-8 3-8 4-8 5-8 6-8 7-8 8-8 9-8

9 1-9 2-9 3-9 4-9 5-9 6-9 7-9 8-9 9-9

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But you did 9 3 9, and Kenyon did 9 3 8. How come?

83

Audra:

84 85

Ricardo: Well, we did 9 3 9 but then subtract 9 away, so that is really the same thing as 9 3 8. So, our chart helps shows why you need to divide by 2. What do you all think of the chart Ricardo’s group made?

86

Ms. D:

87 88

Jamie:

89 90

Charles: I like it, but I wouldn’t want to have to make one all the time — too much work! Looks like it would take a while.

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

Ms. D:

It kind of shows what we all did in just one place. You can see the 72 and the 36 and why you need to divide by 2.

You bring up a good point, Charles. Suppose we wanted to figure out the number of handshakes for a larger number of people, like twenty-five. What would you do? Take a few minutes to discuss this in your groups. I don’t want you to find the answer, but just think about how you could find an answer. What did you learn about doing handshakes with nine people that would help figure out the number of handshakes with twenty-five people?

After five minutes, Ms. Daniel called students back together to discuss what they had found. Students said that they thought that the answer would be the sum of the numbers from 1 to 24. They indicated that they would find this number by using group 5’s addition strategy of making pairs, which would give them 25 3 12 and by using group 4’s strategy, which would give them (25 3 24)/2. After the students had been armed with these ideas, Ms. Daniel told them that their job was to find the number of handshakes for any group of people and to write a generalization that always works. She planned to connect the strategies of group 5 and group 4 later in the lesson when the class discussed the generalization.

Analysis of ATL 6.2: Using Representations to Support Learning During the lesson, Ms. Daniel introduced, discussed, and connected a range of different representations; focused students’ attention on the structure or essential features of the mathematical ideas; and supported students’ ability to justify and explain their reasoning. Her discussion traced a mathematical storyline that began with a focus on the actions of the handshake and systematically built toward the relationship between the number of people shaking hands and the number of handshakes needed. Ms. Daniel began the discussion by focusing on “acting it out,” the concrete approach that groups 2 and 6 used (lines 13–14). Ms. Daniel asked the groups to explain and then act out their approach while the rest of the class kept track of the handshakes to see whether what they had done matched what group 2 and group 6 shared (lines 19–25). By then asking if someone in the class could explain “what seems to be going on here” (line 27), she engaged additional students in considering how the

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representation connected with their own work. This question also encourages students to move from considering the specifics of the handshakes to the general patterns evident while the students create and explain the representations. Group 3 used a geometric, pictorial, or network solution drawn so that each dot represented a person and each line segment represented a handshake between two people (lines 33–35). Again, Ms. Daniel asked a student (Devon) to explain the picture in more detail to help the class better understand the thought process of his group, and she asked another student ( Jamal) to revoice the explanation in his own words (lines 37–50). This process ensured that the students understood the connection between the concrete representation (“acting it out”) and the pictorial representation beyond seeing that the students’ answers were the same. Problem-solving strategies like drawing a picture or diagram and acting it out can be helpful for visualizing complex problems. According to Tripathi (2008, p. 441): . . . middle school is an appropriate time to initiate students into visual forms of representing mathematical ideas and the notion that a pictorial representation can represent objects and relationships between the objects. Such representations serve as a bridge between concrete objects that students may use to model concepts at earlier stages of understanding a concept and the symbolic or verbal forms that they may use later to refer to the concept. Next, Ms. Daniel asked the students in group 5 (lines 54–62) to share their solution, pointing out that the group initially did something similar to what group 3 had done. Michelle indicated that they had noticed the pattern and stopped after the third person, so that instead of continuing the picture, they just added the numbers by making sets of 9 and multiplying by the number of sets (4 3 9) instead of simply adding the numbers 1 through 8. This move represents an important step between considering the discrete set of handshakes to a relationship between the people and handshakes. Because group 4 also multiplied, Ms. Daniel then asked the students in that group to describe how they approached the problem (lines 65–69). Those students explained that they multiplied 9 3 8 to obtain 72 but realized that everyone would be shaking hands twice, so they needed half that amount, resulting in an answer of 36. Groups 5 and 4 both ultimately used numerical representations of the situation. The next representation that Ms. Daniel decided to highlight was the chart that group 1 designed (lines 71–82). The chart helped students see why (9 3 8)/2 (the explanation that group 4 gave) made sense. After the students worked on the number of handshakes for 25 people, Ms. Daniel called the students back together to discuss their strategies. The students said that they thought the number of handshakes would be the sum of the numbers from 1 to 24. They indicated that they could find this number by using group 5’s addition strategy of making pairs, which would give

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them 25 3 12, and by using group 4’s strategy, which would give them (25 3 24)/2. Ms. Daniel told students that their next task was to find the number of handshakes for any group of people and to write a generalization that would always work. She planned to connect the strategies of group 5 and group 4 later in the lesson, when they discussed the generalization. By the end of this lesson, students saw a range of solution methods involving a concrete representation (group 2’s solution), a table (group 1’s solution), a verbal description (that Belinda gave in lines 28–30 and Jamal gave in lines 46–50), a geometric representation (group 3’s solution), and an algebraic generalization (beyond the scope of the example but what students were pursuing at the end of the episode — lines 101–103). Emphasizing with students these representations and their connections is important, and encouraging students to use different representations is beneficial in solving future problems when they make sense in a particular situation. For example, the geometric method was fine for nine handshakes but not efficient or even doable for twenty-five or one thousand handshakes. The discussion of other methods gives students access to more generalizable methods so that they can begin to see how different representations may be more or less helpful in specific situations. Each representation provides different information and may offer insight when solving problems. Representations are essential for building students’ conceptual understandings in ways that support the retention of knowledge. In “The Case of Patrick Donnelly,” first explored in chapter 1, the use of different representations was also central to students’ learning and engagement. Drawings, tables, and numeric strategies gave individual students access to the Candy Jar task. The connections made by Mr. Donnelly during the Candy Jar lesson and Ms. Daniels during the Handshake lesson likely benefited students by helping them see the relationships among different approaches and representations, by expanding their repertoire of ways to approach problems, and by developing their flexibility. According to Lesh, Post, and Behr (1987), strengthening the ability to move between and among these representations improves the growth of children’s concepts. The teacher has an important role in helping students develop confidence and competence both in creating their own representations and in selecting from an extensive repertoire of conventional representations. Teachers should help students use representations meaningfully. By encouraging students to discuss the representations that they are using in their work, teachers can monitor students’ developing fluency with representations. When students see how others interpret what they have written and how others have represented the same ideas, they can evaluate representations and recognize characteristics that make a representation appropriate as well as useful. Through such a process, students should come to appreciate the effectiveness of conventional forms of representation and the role of representations in enabling communication with others.

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Using Questions to Connect Representations A key to helping students make connections among different representations is asking questions that highlight, connect, and contrast different representations. In “The Case of Terry Daniel,” the teacher continuously asked questions of her students to prompt them to make sense of their own solutions, as well as those of their peers (e.g., “Do you agree?” “Can someone explain?” “Can you share?” “Can someone repeat?” “Does that make sense?” “Could you come up and show us?” “What do you all think of __________?”). In ATL 6.3, you first generate different representations that students can use to complete a task and then consider the questions that you could ask to help students make connections among the different representations.

Analyzing Teaching and Learning 6.3 Generating and Connecting Representations • Solve the Pick Your Cable Provider task shown below and generate different representations that could be used to solve the task. • Imagine that the students in your class produced the solutions to the Pick Your Cable Provider task in figure 6.3 and created posters that showed their work.  —  In what ways might students benefit from seeing the different ways in which they might represent and solve the problem?  —  What questions could you ask to help students make connections among different representations that students used?

Pick Your Cable Provider In the past, cable television companies charged a flat rate for a cable television package. Recently, some cable companies are starting to offer packages where you pay a flat rate for access plus a fee per channel. • TV Party charges a $40.00 flat rate plus $1.50 per channel. • Cable Club charges a flat rate of $20.00 plus $3.00 per channel. Your friend Felicia argues that because TV Party’s flat rate is two times as much Cable Club and their price per channel is half of Cable Club’s price per channel, the cost will be the same for any number of channels. Explain why you agree or disagree with Felicia. Adapted from Institute for Learning (2013). Lesson guides and student workbooks are available at ifl.pitt.edu.

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Student Work for the Pick Your Cable Provider Task Student 1

Student 4

First I agree and then I disagreed. First I thought Felicia ujst made sense, but then I thought about what 10 channels cost and I saw it was two different costs so Felicia had to b e wrong. Then i wondered about whether they would cost the same so I graphed them on the calculator and saw they cross at (13.3, 60) si that means 13.3 channels, so I knew that they are venver really the same price, but from 0 to 13 channels cable club is cheaper and then after 14 channels and up TV Prty is cheaper. So I would get TV party becuase I like a lot of channels.

Student 2

Student 5

Student 3

Student 6

Fig. 6.3. Six student solutions to the Pick Your Cable Provider task

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Analysis of ATL 6.3: Generating and Connecting Representations This task provides an opportunity for students to connect physical, visual, symbolic, contextual, and verbal representations. As previously stated, the connections should go in both directions from each representational pair, as shown in figure 6.1. The teacher’s responsibility is to help the students make these connections through classroom discourse. Students must become comfortable in relating symbolic variable expressions to verbal, tabular, and graphical representations of numerical and quantitative relationships. Chapter 4 noted that a common approach to this task involves creating a system of linear equations and solving them algebraically to find the point of intersection. Although this process is mathematically valid, the intersection point (13.3, 60) does not immediately have meaning in the context of the problem. An approach using similar reasoning that could connect more closely to the context would be for students to make a case that that two different linear equations can at best have only one shared point, so Felicia must be wrong (no solving required). Alternatively, teachers might use this task to see what students know and how they reason about linear situations prior to having specific techniques for or experience with solving systems. Toward that end, students need to first decide whether they agree with Felicia and then create an argument supporting their position. All six students disagreed with Felicia and used various representations (i.e., tables, graphs, equations, and words) to support their claims, as figure 6.3 shows. A discussion should highlight several aspects of this task. First, Felicia is wrong, because only one point (13.3, 60) fits both situations. One can use tables, graphs, or equations to determine this point. Second, the teacher should make connections between and among tables, graphs, equations, and context. The teacher should therefore focus on several areas —  • how students can use different representations to find the point of intersection;

• what the slope means in context, how it distinguishes the graphs, and how it appears in the table;

• what the y-intercept means in context, how it distinguishes the graphs, and how it appears in the table.

Students do not make connections among representations without questions from the teacher that highlight these connections. A starting point would be to focus on the point of intersection through questions such as the following: • Is there a number of channels that costs the same for both companies?

• Where is this number found on the graph, in the table, or by using the equations?

For example, student 6 could explain the group’s process and why the students used two tables to focus on the intersection point. Students drew a line between 10 and 15 channels on

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the first table and then made a second table from 10 channels to 14 channels with a line drawn between 13 channels and 14 channels, where the rates first became cheaper for TV Party. Through this process, they determined that the point of intersection was between 13 and 14 channels, even though they did not find the precise point. Student 4 could explain how the group generated its graph and the location of the point of intersection. Student 1 could then add how the group determined the value of the point of intersection (13.3, 60) and how the graph that student 4 created shows when one cable company was cheaper than the other. Such experiences can lay a foundation for solving systems of simultaneous equations. A second set of questions focuses on the y-intercept. For example —  • What does the y-intercept mean in this context?

• Which representations make the intercepts obvious? Students 6 and 5 could discuss where the y-intercept occurs in their tables and what they mean in this context. They should compare their y-intercepts with Student 4’s graph. Student 2 also has a table, but the y-intercept is not apparent; a discussion should focus on where the intercept would be located in that table. Students 2 and 4 should explain how they generated their equations and where the y-intercept occurs in the equation. The teacher can ask all students to consider how the table, the graph, and the equation relate to one another. Another area for student discussion is the slope; the teacher can generate this discussion by posing such questions as the following to ask students to consider the steepness of the lines: • What happens to the graph for TV Party when the number of channels increases from five channels to ten channels or from two channels to three channels? • What happens to the graph for Cable Club when the number of channels increases from five channels to ten channels or from two channels to three channels? • How can you find the slope from each representation, equation, table, context, and graph? • How can slope help you solve the task?

Student 3 based the answer on reasoning solely on the basis of slope; the teacher should ask him or her to explain the meaning of “goes up $3.00 per channel” and “goes up $1.50 per channel.” Students 5, 6, and 2 should locate these “goes up” numbers in their tables. Students 1 and 4 should explain where their graphs reflect these numbers. A final area of questioning for student discussion is to focus again on the graphs and to discuss the speed at which the line that represents Cable Club catches up with the line that represents TV Party. This discussion connects back with the intuitive reasoning in Felicia’s original statement that notes the differences in flat rates and per-channel fees. Instead of focusing on an incorrect multiplicative relationship between them, a teacher might ask the following:

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• If Cable Club is $20 less expensive at the beginning, how quickly does that gap close as you add channels?

• How much less is the gap at one channel, at two channels, and at five channels? • Can you use this information to predict when the gap will close completely? • How does this information connect with Felicia’s original argument?

Using student 4’s graph makes this gap identifiable and describable, and one can use the table that student 2 produced to determine how much closer the costs for each of the two plans become for each channel added. Through this discussion, students develop an understanding of — and experience with — slope and y-intercept and their appearance in tables, graphs, and equations. Another important topic for class discussion is comparing and contrasting the merits of graphical, tabular, and symbolic representations. As the representational repertoire of students expands, they need to reflect on their use of representations to develop an understanding of the relative strengths and weaknesses of various representations for different purposes. A teacher might ask the following questions: • Which representation indicates the point at which the two cable companies intersect and TV Party becomes more economical? • Is it easier to see the slope from the graph or from the equation? • How can you determine the slope from the table?

Through discussion, students can identify the strengths and the limitations of different forms of representation. Graphs give a picture of a relationship. Equations typically offer easily interpreted descriptions of a relationship between variables. Students do not make connections among representations without help, and the questions that the teacher asks highlight these connections. Teachers should emphasize the importance of representing mathematical ideas in a variety of ways. Modeling this process as they debrief a problem with the class is one way that a teacher can encourage students to use and analyze different representations. Discussing why some representations are more effective than others in a particular situation enables students to critique aspects of their representations. Teachers should strategically choose for the wholeclass discussion which student representations to highlight and in what order. According to Knuth (2000, p. 53): An important aspect of developing a robust understanding of the notion of function means not only knowing which representation is most appropriate for use in different contexts but also being able to move flexibly in different translation directions. As teachers, we need to recognize this goal as being important for instruction. 118   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Using and Connecting Mathematical Representations: What Research Has to Say Mathematics is an abstract system that humans use to quantify, describe, and make sense of their world. From a cognitive standpoint, we as humans represent those abstract structures and concepts by using physical representations (National Research Council 2001). Representations are part of everything that we do in the name of mathematics. Lesh, Post, and Behr (1987) note that using the multiple representational forms shown in figure 6.1 and making connections among those representations supports deeper mathematical understanding. Asking students to describe their mathematical thinking with multiple representations and making connections among the representations gives a good sense of the internalized understandings that students hold about abstract mathematical ideas (Goldin and Shteingold 2001). For example, the symbol −1, a fish at 1 foot below sea level, a dot marking −1 on a standard number line, and a single red two-color counter all are representations of the integer negative one. The extent to which students see some, all, or none of them as a representation of negative one gives us important information regarding their abstract conception of negative integers and of negative one, in particular. Visual and physical representations are both particularly important in the middle grades mathematics classroom while students are continuing to develop their algebraic reasoning and spatial sense. Tasks that ask students to visualize two- and three-dimensional shapes and solids in multiple ways can support conceptual understanding of such geometric measurement concepts as area, surface area, and volume (Ben-Haim, Lappan, and Houang 1985; Ferrer et al. 2001). Using multiple representations enables students to develop meaning for those constructs and the formulas that help calculate those geometric measurement quantities. Physical and virtual manipulatives support the development of algebraic reasoning and the connections between the concrete and abstract as early as third grade (Suh and Moyer 2007). This work can lead to better strategic use of general algebraic strategies such as factoring and manipulating symbolic equations. These studies and many others suggest that early and consistent work moving across multiple representations, including context through the use of meaningful contextual tasks, helps students better understand why mathematics works and how to deploy mathematical tools effectively and efficiently. Mathematics instructional resources in the elementary grades often emphasize multiple representations, including pictures and hands-on manipulatives. The use of manipulatives and pictures should not fade in the middle grades but should continue alongside the development of more mathematically specific representations like symbolic expressions and equations, tables, and graphs (Tripathi 2008). In addition, virtual representations such as digital graphing tools, virtual manipulatives, and geometric and statistical visualization software should be used frequently as sense-making tools alongside traditional pencil-and-paper representations. Using multiple representations and having students move flexibly across them should include Use and Connect Mathematical Representations    119 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

different starting and ending points (e.g., the graph should not always be an end product). Teachers should employ them with meaningful mathematical discourse to better understand students’ emerging thinking and reasoning.

Promoting Equity by Using and Connecting Mathematical Representations High-cognitive-demand tasks, as discussed in chapter 3, allow the use and connection of multiple mathematical representations. These tasks, as well as the ability of students to use multiple representations to make sense of the underlying mathematics, are equitable because they afford a wide range of access to the mathematical ideas. Allowing entry at different levels — such as building with manipulatives, drawing pictures, or creating tables — can help students engage in the mathematical ideas in ways that make sense to them. For example, the pictures that students D and E produced in the Cars and Motorcycles task (shown in fig. 6.2) could serve as a starting point for almost any student, regardless of his or her previous mathematical background. The use of multiple mathematical representations also allows students to draw on multiple resources of knowledge, one of the five equity-based practices (Aguirre, Mayfield-Ingram, and Martin 2013). Teachers must explicitly value and encourage multiple mathematical representations that allow students to draw on their mathematical, social, and cultural competence. Promoting the creation and discussion of unique mathematical representations positions students as being mathematically competent. By connecting more informal representations with more formal mathematical representations over time, students can use their initial funds of knowledge to develop mathematical fluency and power in ways that the broader mathematical community shares.

Key Messages • Select tasks that allow students to decide which representations to use in making sense of the problems and in explaining and justifying their reasoning. • Allocate substantial instructional time for students to use, discuss, and make connections among representations and to consider the advantages or disadvantages of various representations in the context of the problem. • Focus students’ attention on the structure or essential features of mathematical ideas.

• Design ways to elicit and assess students’ abilities to use representations meaningfully to solve problems.

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• Use multiple forms of representations to help students make sense of and understand mathematics. • Contextualize mathematical ideas by connecting them with real-world situations.

Taking Action in Your Classroom: Using and Connecting Multiple Representations Next, consider the implications that the ideas discussed in this chapter have for your own practice. We encourage you to begin this process by engaging in in each of the following Taking Action in Your Classroom activities described below.

Taking Action in Your Classroom Using and Connecting Multiple Representations Find a task related to the content that you are currently teaching that gives students an opportunity to use different representations. • Anticipate the representations that a student might use. • For each representation, write a question that assesses what the student understands about the representation and a question that advances her or his understanding by making a connection with another representation. • Teach the lesson, and reflect on how the use of representations supported student learning and related to the lesson’s goals.

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CHAPTER 7

Facilitate Meaningful Mathematics Discourse The Analyzing Teaching and Learning (ATL) activities in this chapter engage you in exploring the effective teaching practice facilitate meaningful mathematics discourse. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 29): Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments. Mathematical discourse should build on and honor student thinking, provide students with opportunities to share ideas, clarify understandings, develop convincing arguments, and advance the mathematical learning of the entire class. When teachers and students engage in mathematical discourse, teachers can better understand students’ thinking and support them in clarifying, refining, and connecting their explanations. The opportunities that teachers offer for students to engage in mathematical discourse also have powerful implications for students’ identities as mathematical knowers and doers. In this chapter, you will —  • analyze a video and narrative case related to mathematics discourse;

• anticipate student thinking related to a task that promotes reasoning and problem solving to prepare for meaningful mathematics discourse;

• select, sequence, and connect student responses to foster classroom discussion that supports mathematical goals; • review key research findings related to meaningful mathematics discourse; and • plan for and enact a discourse-focused lesson in your own classroom.

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For each Analyzing Teaching and Learning activity, note your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, facilitate meaningful mathematics discourse. If possible, share and discuss your responses and ideas with colleagues. After you have written or shared your ideas, read the analysis section, in which we offer ideas relating the ATL to the focal teaching practice.

Exploring Classroom Discourse Analyzing Teaching and Learning activity 7.1, takes you into the classroom of Peter Dubno, where his eighth-grade students have solved the Counting Cubes task (see below). Mr. Dubno wanted his students to understand that —  • an equation can describe the relationship between two quantities (i.e., the number of cubes and the building number); • different but equivalent equations can represent the same situation;

• connections can be made between pictorial and symbolic representations; and

• variables must be clearly defined.

The Counting Cubes Task



1

2

3

  1. Describe a pattern that you see in the cube buildings.   2. Use your pattern to write an expression for the number of cubes in the nth building.   3. Use your expression to find the number of cubes in the fifth building. Check your results by constructing the fifth building and counting the cubes.   4. Look for a different pattern in the buildings. Describe the pattern, and use it to write a different expression for the number of cubes in the nth building. Adapted from Lappan et al. (2004a).

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Students worked in pairs to solve the task, and they posted the tables and equations that represented their work in the classroom. At the beginning of the video clip, pairs of students explain the thought processes that they used to connect the volume or number of cubes in each building with their equation. The students then point out differences and similarities in the equations generated.

Analyzing Teaching and Learning 7.1 more Analyzing Whole-Class Classroom Discourse Watch the video clip and download the transcript of the discussion of the Counting Cubes task in Peter Dubno’s classroom. While you watch the video, pay attention to the student discourse and the connections that students make between representations. Specifically —    1. What does the discourse reveal about students’ understandings of the connections between the pictorial and algebraic representations?   2. To what extent does the discourse facilitate students’ explanations or clarifications of their thinking?   3. To what extent does the discourse make mathematics more visible and accessible for student examination and discussion? You can access and download the videos and their transcripts by visiting NCTM’s More4U website (nctm.org/more4u). The access code can be found on the title page of this book.

Analysis of ATL 7.1: Exploring Classroom Discourse In the video clip of Mr. Dubno’s class, the discourse reveals a number of connections that students articulate between the visual and algebraic representations. Several groups connect the structure of the buildings to their algebraic expressions, instead of simply finding the number of cubes and working with decontextualized numbers and expressions (lines 10–14, 26–27, 46–48, 57–61 of the transcript). In sharing two solutions in which the students defined the variable differently, students also had to grapple with the requirement that the expression must also work with the first building in the sequence (lines 49–63). Throughout the clip, the nature of the discourse made it possible for students to clarify the definitions of their variables and use those definitions to build and interpret their expressions. Specifically, students did not initially see the differences in the ways that the variables were used across the two expressions 5n 1 1 and 5n 2 4 (line 44); but with some press from the teacher (line 45), students begin to explore what was different about the two expressions (lines Facilitate Meaningful Mathematics Discourse    125 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

46–48). Mr. Dubno’s questioning (lines 69–72) helped the students think specifically about what the variable in each expression represented, leading to a public clarification of the fact that in one expression the variable represented arm length (5n 1 1), and in the other expression the variable represented the building number (5n 2 4) (lines 73–78). The discourse made the mathematics more visible and accessible by providing insights into how students were thinking about the figures and defining the variable. Once the differences in the way that students were defining the variable were made public, students had the opportunity to make sense of them and to both clarify and productively challenge the mathematical arguments that their peers made. Specifically, the first group defined its variable differently from other groups (as the length of the “arm” on the visual pattern rather than as the building number), and the public sharing of multiple solutions gave other students a forum for publicly making comparisons among the strategies and the ways of defining the variables (lines 57–61, 67–68). Mr. Dubno asked several times for similarities and differences between and among the equations and expressions that the groups generated (lines 41–42, 45, 64–66). As a result, students were able to explain to one another the similarities and differences in thinking in each of the solution paths, as well as the equivalence of the algebraic expressions (lines 23–24, 57–61, 67–68, 73–78). Throughout this clip, Mr. Dubno allowed students to share diverse ways of thinking about the pattern, pressed students to compare those ways of thinking for similarities and differences and to further articulate and refine their explanations of the key mathematical ideas at play. The choices that he made were not arbitrary; those choices can have a powerful impact on students’ development of mathematical identities. Mr. Dubno took particular care to position his students as authors of mathematical ideas in the way that he orchestrated the sharing of solution paths. Placing his students at the center of the discussion and allowing them not only to present their mathematical ideas but also to clarify and defend their ideas in the face of questions from peers presented a powerful message to students that they could be the authors of meaningful mathematical ideas. By positioning his students as authors of the mathematical ideas, setting them up to debate seemingly contradictory solutions, and stepping out of the way as much as possible to allow that debate to unfold, Mr. Dubno empowered his students in important ways. In considering this brief clip, one might wonder whether the featured discussion was spontaneous or carefully orchestrated. Did Mr. Dubno just respond to what occurred “in the moment,” or was there more to it? We contend that Mr. Dubno’s class was the result of careful planning using practices that are accessible to any teacher. The sections that follow describe some of these practices.

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Anticipating Student Thinking The Five Practices for Orchestrating Productive Discussions (Five Practices; Smith and Stein 2011) support teachers in their planning and teaching while they seek to facilitate meaningful mathematics discourse as a part of ambitious teaching practice. These practices are intended to make student-centered instruction more manageable by moderating the degree of improvisation required during a discussion. Instead of relying on in-the-moment responses to students’ contributions, the practices instead emphasize the importance of planning. Through planning, teachers can anticipate likely student contributions, prepare responses that they might make to those lines of thinking, and make decisions about structuring students’ presentations to further the mathematical agenda for the lesson. The next two Analyzing Teaching and Learning activities provide opportunities to engage in aspects of the Five Practices related to lesson planning. The Five Practices are —   1. anticipating likely student responses to challenging mathematical tasks;

 2. monitoring students’ actual responses to the tasks (while students work on the tasks in pairs or small groups);

 3. selecting particular students to present their mathematical work during the whole-class discussion;  4. sequencing the student responses that will be displayed; and

 5. connecting different students’ responses and connecting the responses to key mathematical ideas. This chapter focuses in particular on anticipating, selecting, and sequencing. Monitoring connects to aspects of teacher questioning discussed in chapter 5 and to eliciting and using student thinking discussed in chapter 8. This chapter briefly discusses connecting; but it also draws in aspects of many practices, including representations (chapter 3), building procedural fluency from conceptual understanding (chapter 4), and questioning (chapter 5). (5 Practices for Orchestrating Productive Discussions by Smith and Stein [2011] includes more detailed information about the Five Practices.) ATL 7.2 asks you to consider the Hexagon task and anticipate the ways in which students might approach solving the task. This work of anticipating student thinking is an important first step in preparing to orchestrate productive discussions.

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Analyzing Teaching and Learning 7.2 Anticipating Student Solutions to a Pattern Task Solve the Hexagon task shown below. In focusing on question 4, consider a variety of approaches to solving the task (both correct and incorrect) that students might use. You might also ask colleagues to solve the task and draw from their approaches.

The Hexagon Task Trains 1, 2, 3, and 4 (shown below) are the first four trains in the hexagon pattern. The first train in this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon is added.

Train

Train 2

Train 3

Train 4

  1. Compute the perimeter for each of the first four trains.   2. Draw the fifth train, and compute the perimeter of that train.   3. Determine the perimeter of the tenth train without constructing it.   4. Write a description that that can be used to compute the perimeter of any train in the pattern.   5. Determine which train has a perimeter of 110. Based on your solution and the other solutions that you have considered, which solution paths do you think your students are most likely to use? Adapted from Foreman and Bennett (1995).

Analysis of ATL 7.2: Anticipating Student Solutions to a Pattern Task Students can solve the Hexagon task in a variety of different ways, depending on how they make sense of the visual pattern and the mathematical tools that they have at their disposal. Student work for this task shows that students dissect and explain the pattern in many different ways. Some begin by looking at the tops and bottoms of the hexagon as contributing

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to the perimeter and adding in the end sides. Others begin with the total number of sides in each hexagon (six) and take steps to eliminate the joined edges that do not contribute to the perimeter. Some look at the pattern in an iterative or recursive way, thinking about what each additional hexagon contributes to the perimeter, whereas others look at the pattern as a whole. Some students also begin by finding the first four perimeters and work strictly with the numerical pattern without connecting the numbers to the geometric characteristics of the hexagon trains. Misconceptions and incomplete solution paths include looking solely at the addition of four sides for each new train number and not including the two end sides or taking a scaling approach to find the tenth train by doubling the perimeter of the fifth train. Figure 7.1 is an excerpt from a lesson plan for the Hexagon task that appears in appendix A. This excerpt illustrates likely solution strategies and possible misconceptions. After teachers have identified these likely solution strategies, the next steps in planning include identifying questions for each strategy that assess and advance student thinking, as well as selecting and sequencing student responses to plan the class discussion.

Anticipated Likely Solutions Visual-geometric solution A Students see the train as tops and bottoms with two ends. The top and bottom of each hexagon contribute four sides to the perimeter. The ends each contribute two sides to the perimeter. y 5 4 units (number of hexagons) 1 2 (ends) y 5 4x 1 2 Visual-geometric solution B Students see the train as two hexagons on each end, each of which contribute five sides to the perimeter. The remaining hexagons — two less than the total number of hexagons in the train — each contribute four sides to the perimeter. y 5 4 units (number of hexagons 2 2) 1 5 units (2 end hexagons) y 5 4 (x 2 2) 1 10 continued on next page

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Visual-geometric solution C Students multiply the number of hexagons by 6 since each hexagon has six sides (6x). Then the vertical sides on the inside of the hexagon trains — where two hexagons are connected — are taken away since these sides do not count in the perimeter. The number of vertical sides taken away is the number of hexagons minus one multiplied by two, that is 2(x 2 1). y 5 6 sides (number of hexagons) 2 (x 21) the train number minus one times two vertical sides for each of these trains. 6x 2 2(x 2 1) Variations of this solution may be of the form (6x 2 2x) 1 2. Arithmetic-algebraic solution D Students may build a table of values from which an algebraic formula can emerge. Some students may use a recursive strategy of 14. If students use this recursive or additive reasoning, acknowledge its usefulness for small train numbers and then ask them how that reasoning will help them find the perimeter of large train numbers.

Train Number

Perimeter

1

 6

2

10

3

14

4

18

5

22

6

26

7

30

8

34

14 14 14 14

Arithmetic-algebraic solution E y 5 mx 1 b Students might create a linear equation using the data in the table. The constant increase of 4 for every additional hexagon indicates that the formula will be linear and that 4 is the slope (m) in the equation y 5 mx 1 b. If y (perimeter of the train) is the function of x, (the train number), then b 5 2.

Possible Errors and Misconceptions Incorrect definition of perimeter Students may need to revisit the definition of perimeter: a closed curve bounding a plane area, the length of such a boundary, the distance around a figure. This is best addressed in the launch but may need to be reinforced while students work in groups. continued on next page

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Assuming that the relationship is proportional Students may find the perimeter for the fifth train and double it to find the perimeter of the tenth train. This method works only if the relationship is proportional (this relationship is not proportional because of the two static end sides in the train pattern). Creating an incorrect formula based on recursion Students may identify the change of 14 for each successive train and write a formula similar to y 5 x 1 4. Creating a formula solely based on the numeric pattern Students may create the formula using the values for several perimeters and may not connect the features of the formula back to the visual. While not an error or misconception per se, making connections among different representations and to the visual is an important outcome in this task.

Fig. 7.1. Anticipated likely student solutions and possible errors and misconceptions for the Hexagon task

Supporting students’ engagement in a high-cognitive-demand task rests on carefully considering the range of mathematical thinking in which students might engage. Anticipating student thinking lays the groundwork for the other practices. As noted in chapter 5, identifying specific likely student solution strategies to the task allows you to prepare targeted assessing and advancing questions to ask while monitoring students’ work. The cases of Patrick Donnelly (in chapter 1) and Sandra Pascal (in chapter 3), reinforce this notion. While students worked, Mr. Donnelly asked questions that helped him understand students’ solution strategies; and during the whole-class discussion, he was intentional about selecting and sequencing students’ solutions. This work would be challenging, if not impossible, on the fly; and it is likely that Mr. Donnelly anticipated the strategies that students would share prior to teaching the lesson. Ms. Pascal’s lesson did not show clear evidence of anticipating student thinking, and it featured a funneling pattern of questioning and a much less mathematically rich discussion. As you will see in the next section, anticipating and monitoring student thinking also allows you to bring structure and coherence to the whole-class discussion of solutions.

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Supporting Classroom Discourse During Task Implementation A discussion of a mathematical task with multiple solution paths or approaches must move beyond being a “show and tell” of different strategies. A meaningful mathematical discussion must focus on building an understanding of the key mathematical ideas in the lesson by selecting, sequencing, and meaningfully connecting strategies in support of the learning goal (Smith and Stein 2011). In ATL 7.3, you will consider how to orchestrate the discussion of a high-cognitive-demand task by revisiting “The Case of Patrick Donnelly” from chapter 1.

Analyzing Teaching and Learning 7.3 Discourse in “The Case of Patrick Donnelly” Revisit “The Case of Patrick Donnelly” (chapter 1, page 10). What aspects of the Five Practices do you see Patrick Donnelly using that support students in engaging in meaningful mathematical discourse?

Analysis of ATL 7.3: Discourse in “The Case of Patrick Donnelly” Mr. Donnelly engaged many students during the discussion and pressed the students to make sense of and debate the ideas that their peers put forth. Much of the discourse was designed to help students make mathematical arguments and critique the arguments of others. Students engaged in argumentation when Mr. Donnelly made salient the differences between students’ solution paths, asked students to talk about the differences (lines 51–52), and asked students to agree or disagree with one another’s reasoning (line 57). He encouraged students to ask questions of one another to better understand the mathematical arguments in each solution path (lines 28–29). Mr. Donnelly drew out the important mathematical ideas during this discussion by making some specific moves of his own. The case shows Mr. Donnelly helping students make connections among different strategies and to the key mathematical ideas in the lesson (lines 31–32), which contribute to the goal of understanding the proportional relationship inherent in the task. Mr. Donnelly also pushed students’ thinking related to the key ideas in the lesson by posing a challenging question and then giving students time to talk to their partners before engaging the whole group in a discussion (lines 64–65, 74–76). In addition, Mr. Donnelly used the Five Practices (Smith and Stein 2011). This work likely began with anticipating student thinking. Although there is no direct evidence in the

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case regarding Mr. Donnelly’s planning related to student thinking, he must have anticipated how students would solve the tasks and the challenges that they were likely to face. Without this information, he would not have been able to interact with students as he did or engage in the subsequent practices. During the lesson, Mr. Donnelly monitored students while they worked, keeping track of who was doing what and asking questions as needed to assess what students knew and advance them toward the goal of the lesson (lines 13–16, 21–23). When the whole-class discussion began, Mr. Donnelly selected specific students to present responses because of the mathematics available in their responses (lines 23–26) and sequenced those solutions to engage the entire class and present a coherent mathematical storyline (lines 23–26, 29–32). Specifically, he selected groups 4, 5 and 2 to present (in that order) because each of the groups used one of the strategies that he was targeting (scaling up, scale factor, and unit rate, respectively) and moved from less sophisticated (group 4’s scaling up) to more sophisticated and generalizable (group 2’s unit rate) approaches to ensure that all students had access to the discussion. Finally, during the discussion, Mr. Donnelly shaped that mathematical storyline by connecting students’ responses to one another and to the fundamental ideas at the heart of the lesson (lines 58–59, 64–65, 78–80). The work of selecting and sequencing responses that the students share is integral to a meaningful mathematical discussion. By selecting students to share their solutions and sequencing those solutions in particular ways, teachers can exert influence on the content and trajectory of the mathematical ideas that students share, thereby allowing the teacher to craft a mathematical storyline that moves students toward the mathematical goal, providing opportunities to consider multiple mathematical approaches in specific ways, as well as allowing for deliberate and meaningful connections among mathematical ideas.

Orchestrating a Productive Whole-Class Discussion by Using Student Solutions In “The Case of Patrick Donnelly,” the teacher selected specific solutions for students to present, sequenced those solutions in a particular way, and asked questions aimed at connecting those solutions to build toward an understanding of important mathematical ideas. In ATL 7.4, we ask you to use the anticipated likely solutions from the Hexagon task explored in ATL 7.2 to identify solution paths that have the potential to be good candidates to share in a whole-class discussion.

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Analyzing Teaching and Learning 7.4 Selecting, Sequencing, and Connecting Student Solutions Imagine that you had determined the following learning goals for the lesson featuring the Hexagon task: Students will understand that —  a. an equation can describe the relationship between two quantities, the independent (x) and dependent (y) variables; b. linear relationships have a constant rate of change between the quantities, are depicted graphically by a line, and can be written symbolically as y 5 mx 1 b, where m is the constant rate of change and the slope of the line and b is the value of the y-quantity when x 5 0 (i.e., the y-intercept); c. different but equivalent equations can represent the same situation; and d. connections can be made between tables, graphs, equations and contexts. Your students produced Solutions A — E shown in figure 7.2. Consider the following: • Which of these five solution paths would you select for students to share in a whole-class discussion? (Select at least three.) • In what order would you sequence these solutions, and why? • What teacher moves would you make to highlight connections among these solutions and your lesson goals?

Analysis of ATL 7.4: Selecting, Sequencing, and Connecting with the Hexagon Task There are many possible ways to select and sequence the set of solutions to the Hexagon task (shown in figure 7.1). On the basis of the goals articulated for the lesson, however, a few criteria might immediately present themselves: • To meet goals (a) and (b), at least one solution should focus on the equation.

• To meet goal (a), discussion of the solutions should connect the quantities in the hexagon pattern to the concept of independent and dependent variables.

• To meet goal (b), at least one equation-focused solution should be in slope-intercept form. • To meet goal (c), at least two different but equivalent equations should be included. • To meet goal (d), discussion of the solutions should focus on connections among representations, and the table in solution D might be important to include.

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• To meet goals (b) and (d), a connection to graphing might be important even if students do not graph the relationship. In addition, the following considerations for selecting and sequencing solutions help in planning the discussion of task solutions (adapted from Smith and Stein 2011): • Present strategies in sequence from concrete to abstract.

• Present strategies that afford broad student access first, and then move to more unique or mathematically complex solutions. • Sequence solutions in ways that highlight the connections among solutions with common mathematical features.

Figure 7.2 shows one possible set of choices for selecting and sequencing solutions by using the lesson goals and these guidelines. On the left are the selected solutions in a suggested presentation order. On the right are questions that the teacher might ask to encourage connections among the solutions. Visual-geometric solution A is selected first, because it is a very common approach to solving this task. It also includes an algebraic representation of the form y 5 mx 1 b, and the two numeric quantities (4 and 2) in the equation can be very easily connected to the geometric pattern (top and bottom sides of the hexagon plus two end sides). Visual-geometric solution B is next. Its features are similar to those of solution A, but it takes the first and last hexagon as discrete quantities instead of including them as contributing to the 14 rate of change. This solution also allows for a discussion of how the variable is defined, why (x 2 2) makes sense as a part of this formula, and whether this formula is equivalent to the formula in solution A. Visual-geometric solution C is still grounded in the concrete aspects of the visual pattern, but it approaches that pattern in a very different way. This solution begins with considering the full set of sides on a hexagon and subtracting the sides that do not contribute to the perimeter. It includes subtracting sides that are double counted, whereas the previous visual solutions consider the addition of sides. It also introduces a third variable expression, (x 2 1), and essentially decomposes the 14 rate of change into components of 16 and 22. This visual approach is more complex than the previous two but affords opportunities to continue to develop the concepts of rate of change and equivalence. Finally, arithmetic-algebraic solution D has the least direct connection to the visual solution but instead approaches the generalization numerically. It may also result in the same equation as visual-geometric solution A. It allows an opportunity to connect back to solution A ( y 5 4x 1 2). It also allows connections to be made between the symbolic and tabular representations with respect to rate of change and the y-intercept or constant value. In addition, a teacher may wish to show a graph of the relationship to make connections among the equation, the table, and the graph, even if the teacher did not ask students to graph the relationship. The sequence described is but one of many possible ways to select Facilitate Meaningful Mathematics Discourse    135 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

and sequence a set of solutions. The selection and sequencing should connect back to the mathematical goal for the lesson, to students’ prior mathematical experiences, and to the mathematical storyline that a teacher wishes to build through the discussion. The selecting and sequencing should also be designed to allow all students entry and access to the discussion.

Sharing and Discussing the Task Selecting and Sequencing

Connecting Responses

Which solutions should students share during the lesson? In what order? Why?

• make sense of the mathematical concepts that you want them to learn? • make connections among the different strategies and solutions that are presented?

Visual-geometric solution A y 5 4x 1 2 Visual-geometric solution B y 5 4 (x − 2) 1 10

What specific questions will you ask so that students — 

(Possible student responses are shown in green.) What do the numbers represent in 4x 1 2? Students will say that the 2 represents the ends of the train. The four units are the two on the top and the two on the bottom of each hexagon, and the x represents the train number. Why are you subtracting 2, then multiplying by 4? The number of hexagons on the inside of the train is always 2 less than the total number of hexagons in a train, so I need to subtract 2 from the total number of hexagons (x 2 2). Then I multiply by 4 because each inside hexagon has a perimeter of 4 (2 on the top and 2 on the bottom). What does the 110 represent? The five sides on the two end hexagons. How are the first two formulas related? 4x 1 2 and 4(x 2 2) 1 10 are related because they both count the ends of the train (2 and 5) and then show the number of sides that are left to be counted (4x and 4[x 2 2]). Are these expressions equivalent? The perimeter for the hundredth train using both formulas is the same. The expressions are therefore equivalent. Which formula would you rather use to find the perimeter of the 3,132nd train number? Finding the perimeter of the 3,132nd train is easier by using the 4x 1 2 formula because it only requires multiplying 3,132 by 4 then adding 2.

continued on next page

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Visual-geometric solution C 6x 2 2(x − 1)

What does the 6x represent? The 6 represents the six sides to each hexagon and the x represents the number of hexagons. Why are you subtracting 2(x 2 1)? What does the x 2 1 represent? Can you connect this expression to the picture of the hexagon train? Two vertical sides of each hexagon need to be taken away when two hexagons are connected. There is one fewer point of connection between two hexagons than the number of hexagons in each train.

Making connections with the table: arithmeticalgebraic solution D

In the equation y 5 4x 1 2, where is the 4 in the table? Where is the 2?

Making connections with graphs

If you were to graph the perimeter of the trains as a function of the train number, what would that graph look like and why? Why is it a linear function?

The 4 is the difference between the perimeters of two successive trains. Two would be the perimeter for the zero train. This would mean a train with just two vertical sides. But this doesn’t make any sense.

Students should plot points on the basis of the table produced and then make connections between the slope of the line and the hexagon train and describe the rate of change as “four sides are added for each hexagon” which gives m 5 4/1 5 4.

Fig. 7.2. One possible way to select and sequence responses to the hexagon task

The lesson plan included in appendix A brings together all the work on the Hexagon task from ATL 7.2 and ATL 7.4 to demonstrate what a complete lesson plan that represents ambitious teaching of this task might look like. In considering the ways in which using the Five Practices in planning can support meaningful mathematics discourse, remembering the role of goals and tasks is important. Meaningful mathematics discourse is unlikely to occur unless teachers choose and use mathematical tasks that promote reasoning and problem solving. Limiting the tasks that students use in the classroom to rote application of procedures or memorized facts (see the categories of low-level tasks in fig. 3.1) will likely restrict mathematics discourse to students reciting the steps that they performed to solve a problem. Having a clear mathematical goal for that task guides decision making regarding the solution strategies to share, their order, and the important mathematical connections to make. Facilitate Meaningful Mathematics Discourse    137 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

High-cognitive-demand tasks can often lend themselves to multiple mathematical goals in different contexts. For example, a teacher could use the Hexagon task featured in this chapter to discuss the fundamentals of linear growth patterns, to develop symbolic models for those patterns, to discuss the connections between recursive and closed-form generalizations, or to make meaningful connections among equations, tables, and graphs. Identifying a goal during the planning phase of a lesson focuses and sharpens the work of anticipating student thinking, selecting and sequencing solutions, and making connections among those solutions.

Facilitate Meaningful Mathematics Discourse: What Research Has to Say A wide array of research over the previous two decades has underscored the important connections between mathematics classroom discourse that focuses on reasoning and problem solving and positive student learning outcomes (e.g., Carpenter, Franke, and Levi 2003; Michaels, O’Connor, and Resnick 2008). The typical instructional pattern that has dominated secondary mathematics classrooms in the past has been a routine of an opening warm-up activity, review of homework, lecture and worked examples on a new topic, and independent work on similar examples while the teacher circulates and provides feedback (Welch 1978). These classrooms tend to use procedural, low-cognitive-demand tasks, and the ensuing discourse around them follows an Initiate-Response-Evaluate (I-R-E) pattern first discussed in chapter 5, in which the teacher asks a question, the student responds, and the teacher evaluates whether the response is correct (Mehan 1979b). The I-R-E pattern does not provide students with opportunities to engage in mathematical reasoning and sense making and offers teachers little feedback regarding what students know and understand. Changing the I-R-E pattern in classrooms begins not only with selecting tasks that promote reasoning and problem solving but also with changing the interaction patterns between teachers and students and among students in the classrooms. Teachers must consider how they can position student thinking and discussion as the core activity in the classroom while still ensuring that mathematical ideas emerge that support progress toward their mathematical goals (Engle and Conant 2002). Making regular use of research-based tools like the Five Practices (Stein et al. 2008; Smith and Stein 2011) can help teachers move beyond discussions in which students share their strategies in a “show-and-tell” fashion without the teacher making any connections between strategies or to the mathematical ideas that are the focus on the lesson (Wood and Turner-Vorbeck 2001) toward discussions that instead build mathematical ideas in systematic ways. Tools like the Five Practices offer teachers a broad framework that supports planning and teaching. Changes to classroom norms must take place if teachers are to transform classrooms from traditional transmission modes of instruction to more discourse-based interactive spaces.

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Teachers and students must consistently negotiate what it means to know and do mathematics in the classroom, which often means challenging more didactic notions about learning mathematics that students may have when they enter the classroom (Hufferd-Ackles, Fuson, and Sherin 2004). Teachers also can benefit from routines and pedagogical moves that specifically support student discourse. As a teacher shifts from a traditional mode of teaching to a more discoursebased classroom, more routines that moderate contributions and discussion are necessary. Specific talk moves that teachers can make also support students in learning productive ways of talking and reasoning in mathematics (Chapin, O’Connor, and Anderson 2003). In a study of grade 4–7 classrooms, Chapin and colleagues found that the increased use of talk moves such as wait time and revoicing increased the quantity and quality of student talk, which translated to significant gains on assessments of students’ mathematical thinking and reasoning (Chapin, O’Connor, and Anderson 2003). Herbel-Eisenmann and colleagues have recently adapted these talk moves into a set of six teacher discourse moves that are specifically designed to support the demands of secondary mathematics classroom discourse (Herbel-Eisenmann, Steele, and Cirillo 2013). Together, teachers can use these frameworks for planning and implementing lessons, for developing productive classroom norms around discourse, and specific routines and pedagogical moves in concert to transform their classrooms from classrooms in which the teacher does most of the talking and thinking to ones in which the students do most of the talking and thinking. Through thoughtful use of these tools, student contributions can be moved to the center of classroom practice while still achieving meaningful and rigorous mathematical outcomes.

Promoting Equity through Facilitating Meaningful Mathematics Discourse The work of promoting meaningful mathematics discourse in the classroom has far-reaching implications for equity. In more traditional I-R-E-pattern classrooms, students who contribute correct answers receive immediate positive feedback; and students who provide incorrect, incomplete, or partial responses are likely to receive negative or — at best — neutral feedback. That feedback is limited to the accuracy of the students’ answers rather than the content of their mathematical thinking and reasoning. In a classroom that features meaningful mathematics discourse, students have opportunities to share their mathematical thinking in addition to their answers and have opportunities to hear criticism and feedback, critique the reasoning of others, and evolve their thinking in a visible, public space. A discoursebased mathematics classroom affords stronger access for every student — those who have an immediate answer or approach to share, those who have begun to formulate a mathematical approach to a task but have not fully developed their thoughts, and those who may not have

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an approach but can offer feedback to others. A discourse-based classroom also gives students access to the thinking of their peers, providing them with more possible approaches to a task than just the one that the teacher chooses to demonstrate during a lecture. As you plan for lessons using rich tasks that promote problem solving, consider how you will intentionally make teacher moves that support students’ access to mathematical ideas. Judicious use of waiting (Herbel-Eisenmann, Steele, and Cirillo 2013; Mehan 1979b) particularly after a student contribution gives all students time to process mathematical ideas. Probing student thinking through a teacher press (Herbel-Eisenmann, Steele, and Cirillo 2013; Kazemi and Stipek 2001) publicly clarifies mathematical ideas in ways that are accessible to all students. Inviting student participation and creating opportunities to engage with another’s reasoning (Herbel-Eisenmann, Steele, and Cirillo 2013) in strategic ways can help ensure that all students raise, unpack, and use important ideas, not just the students who first originated the idea publicly. In this way, meaningful mathematics discourse has the potential to challenge spaces of marginality (Aguirre, Mayfield-Ingram, and Martin 2013) by systematically including more student voices and giving all students access to important mathematical ideas. A discourse-based mathematics classroom also has profound implications for student positioning (Wagner and Herbel-Eisenmann 2009). Interactions between teachers and students in mathematics class constantly assign roles to one another, and these roles have implications for how students’ learning dispositions and identities develop (Anderson 2009; Gresalfi 2009). These interactions in a discourse-focused mathematics classroom must help students see themselves as people who can know, do, and make sense of mathematics, challenging aspects of marginality that lie within the students’ own identities. The language choices that teachers and students make, the norms that are set in the classroom, and the types of mathematical interactions that are or are not encouraged can all support or inhibit the development of productive attitudes and practices toward mathematics. For example, careful monitoring of how students are selected to share their thinking might reveal inherent biases (such as calling on male students first, calling on students perceived as high-achieving last) that provide access for some and not for others and send students messages about their status. Students may send messages to one another that reinforce fixed-mindset conceptions of mathematical ability in subtle ways, such as “We used Bailey’s answer in our group because Bailey usually gets it right.” Teachers should attend to the ways in which students position one another as capable or not capable of doing mathematics and should disrupt talk that may lead to unproductive conceptions of what it means to know and do mathematics. While teachers monitor small-group work and discussion, they can make many moves to position students in positive ways with respect to the mathematics and to challenge spaces of marginality. Inviting student participation in a small-group discussion can strategically target students who may not immediately offer their thinking. Inviting student participation during the small-group-work phase of a lesson reduces risk for students who may struggle with their

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confidence and identity in mathematics. Similarly, asking students to revoice another student’s idea in a small-group or whole-group setting positions students as mathematically capable of making sense of and critiquing another student’s reasoning. Teachers can use these moves in strategic moments during a discussion to provide students who have historically been marginalized in mathematics with opportunities for meaningful mathematical engagement. These moves must enable students to grapple with meaningful mathematics, not just give a simple answer or parrot another student’s strategy.

Key Messages • Meaningful mathematics discourse is not improvisational; rather, it is something for which teachers can plan.

• Preparing for meaningful mathematics discourse means anticipating how students will think about a mathematical task, what approaches the teacher wishes students to share publicly, and how to order and connect those solutions.

• Facilitating meaningful mathematics discourse rests on choosing a task with ample opportunities for discussion and asking questions that encourage students to engage in mathematical discussion and struggle productively. • Engaging students in meaningful mathematics discourse supports the development of their mathematical identities and dispositions.

• An equitable classroom is one in which all students have access and entry to meaningful mathematical conversations.

Taking Action in Your Classroom: Planning for Meaningful Discourse It is time to consider the implications that the ideas discussed in this chapter have for your own practice. We encourage you to begin this process by engaging in each of the Taking Action in Your Classroom activities described on page 142.

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Taking Action in Your Classroom Planning and Orchestrating a Meaningful Discussion Choose a task to use in your classroom that has the potential to elicit multiple solution paths (e.g., a doing mathematics task from chapter 3 or a task that uses multiple representations from chapter 5). • Anticipate the solution strategies, both correct and incorrect, that students might use. • Identify the solution strategies that you want to share to move your mathematical goal forward and the order in which you want to share them. • Note the specific pedagogical moves that you might use to support discussion related to specific solutions or connecting solutions. • Keep track of the solutions that you anticipated and any others that arise, and record talk moves you intend to use. (The monitoring chart discussed in chapter 8 may be helpful here.) • Teach the lesson, and reflect on how the planning for discourse supported student learning. Pay particular attention in your reflection to issues of equity. How did you balance the mathematical and social aspects of your classroom when you interacted with groups and conducted the whole-class discussion?

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CHAPTER 8

Elicit and Use Evidence of Student Thinking The Analyzing Teaching and Learning (ATL) activities in this chapter engage you in exploring the effective teaching practice, elicit and use evidence of student thinking. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 53) —  Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning. Evidence of student thinking and understanding is essential in helping teachers determine what students currently know and understand about mathematical ideas and in supporting their ongoing learning. Specifically, evidence provides a window into students’ thinking, helps the teacher determine the extent to which students are reaching the mathematics learning goals, facilitates making instructional decisions during the lesson, and helps the teacher prepare for subsequent lessons. According to Leahy and her colleagues (Leahy et al. 2005, p. 19), “Everything students do . . . is a potential source of information about how much they understand.” As you will see as you work through this chapter, this teaching practice is a form of formative assessment. According to Wiliam (2007, p. 1054), “formative assessment is an essentially interactive process, in which the teacher can find out whether what has been taught has been learned, and if not, to do something about it.” In this chapter, you will —  • analyze a task to determine its potential for revealing student thinking;

• analyze student work to determine what it can reveal about student thinking;

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• analyze a video clip to investigate how a teacher can elicit and use the thinking of the group to advance students’ understanding; • analyze a narrative case and consider what the teachers does to elicit and use student thinking during the lesson; • review key research findings related to eliciting and using thinking;

• reflect on ways in which you elicit and use evidence of student thinking in your classroom. For each Analyzing Teaching and Learning activity, note your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, elicit and use evidence of student thinking. If possible, share and discuss your responses and ideas with colleagues. After you have written or shared your ideas, read the analysis, in which we offer ideas relating the ATL to the focal teaching practice.

Investigating the Power of a Task to Elicit Student Thinking A first step in eliciting student thinking is engaging students in tasks that can provide insights into their thinking about key mathematical ideas (Wiliam 2011). As discussed in chapter 3, such tasks can be categorized as high level (see fig. 3.1). In Analyzing Teaching and Learning 8.1, you analyze the Bag of Marbles task (shown below) and consider what the task could elicit about students’ understanding of mathematics.

Analyzing Teaching and Learning 8.1 Considering the Potential of a Task Review the Bag of Marbles task shown below. • What mathematical understandings might the task elicit from students? • What misconceptions might be revealed about students’ understanding?

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The Bag of Marbles Task Ms. Rhee’s math class was studying statistics. She brought in three bags containing red and blue marbles. The three bags were labeled as shown below:

Bag X

Bag Y

Bag Z

Ms. Rhee shook each bag. She asked the class, “If you close your eyes, reach into a bag, and remove one marble, which bag would give you the best change of picking a blue marble?” Which bag would you choose? Explain why that bag gives you the best choice of picking a blue marble. You may use the diagrams above in your explanation. This task was adapted from the QUASAR Cognitive Assessment Instrument (Lane 1993).

Analysis of ATL 8.1: Considering the Potential of a Task The Bag of Marbles task is a doing mathematics task (see fig. 3.1). Although the context of the task is statistics, it is a proportional reasoning task. Proportional reasoning is a critical area of mathematics for middle school students because it moves students from thinking about additive relationships to considering relationships that are multiplicative. Lesh, Post, and Behr (1988, p. 93) call proportional reasoning “the capstone of children’s elementary school arithmetic” and “the cornerstone of all that is to follow.” The key to solving this task correctly is recognizing that comparing the three bags of marbles involves finding a common basis for making a comparison. The ways of accomplishing this task include the following — 

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• Constructing part-to-part ratios of blue to red or red to blue marbles and scaling the ratios up or down to establish a common number of red or blue marbles for comparison. For example, 25 blue to 75 red or 1 blue to 3 red Bag X Bag Y 20 blue to 40 red or 1 blue to 2 red Bag Z 25 blue to 100 red or 1 blue to 4 red

These ratios lead to the conclusion that the chance of choosing a blue marble from bag Y is greater because it has fewer red marbles for every one blue marble.

• Constructing part-to-whole ratios of blue marbles to the total number of marbles for each of the three bags and determining the greatest ratio using unit fractions, decimals, or percents. For example, Bag X 25 blue marbles/100 marbles, or 1/4, or .25, or 25% 20 blue marbles/60 marbles, or 1/3, or .333, or 331/3 % Bag Y Bag Z 25 blue marbles/125 marbles, or 1/5, or .20, or 20%

These ratios lead to the conclusion that the chance of choosing a blue marble from bag Y is greater because the unit fraction 1/3 is the largest of the three or because the percentage of blue marbles (331/3 percent) is the greatest.

Students’ solutions to the Bag of Marbles task can make salient whether students realize that making a comparison across bags requires finding a common basis for comparison; as well as how students draw on their knowledge of ratios, fractions, decimals, and percents in making comparisons. Not recognizing the need for a common basis for comparison can lead students to select the bag that has the most blue marbles (bags X and Z) or the most marbles overall (bag Z) without considering the relationship between the blue marbles and the red marbles in each bag. A common misconception that can surface in solving problems of this type is identifying the relationship between quantities as additive rather than multiplicative. This error can lead students to simply subtract the number of red and blue marbles and conclude that bag Y is the best bag to choose because the difference between the number of red marbles and the number of blue marbles is smallest: (bag X 5 75 2 25 5 50; bag Y 5 40 2 20 5 20; bag Z 5100 2 25 5 75). Although this approach leads to selecting the correct bag in this task, the rationale is flawed and the approach does not generalize to all situations. (This issue arose in chapter 5 in the discussion of the solution that Jordan, a student in Ms. Hanson’s class, presented.) This task can also reveal what students understand about part-to-whole ratios as opposed to part-to-part ratios. Students may be able to construct appropriate ratios but may not interpret them correctly. For example, students may construct part-to-part ratios but talk about them as if they were part-to-whole ratios and conclude that bag Y is 1/2 or 50 percent blue rather than 1/3 blue. 146   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

So, what about the Bag of Marbles task makes it a good candidate for making student thinking visible? First, because it is a task that promotes reasoning and problem solving, it requires students to think. Additionally, the requirement to “explain why” that is embedded in the task forces students to articulate their thinking. Without that prompt, students could choose the correct bag for the wrong reasons, and that thinking would not be visible to the teacher. Second, students can complete the task in many different ways (including building models or drawing pictures), so it is accessible to students. All students are likely to be able to do something that will make clear what they currently understand. Finally, the task can make a key misconception in the domain of proportional reasoning surface — the use of additive rather than multiplicative reasoning. Research has documented that middle school students frequently apply additive strategies to reason incorrectly about proportional situations (e.g., Karplus, Pulos, and Stage 1983a, 1983b). Bringing such reasoning to the surface is therefore essential so that teachers can challenge and guide it in a productive direction.

Using Student Thinking to Advance Understanding Although the Bag of Marbles task has the potential to elicit student thinking and reasoning, as discussed in chapter 3, the way in which students’ work on the task is supported during the lesson can limit this potential. If the teacher directs students to a particular method (as in “The Case of Sandra Pascal” in ATL 3.3), teachers will learn little about what students think or understand. ATL 8.2 involves analyzing four pieces of work that students who solved this task produced. It asks you to consider how you can use what you learn about each student’s understanding to advance his or her learning. Advancing student learning first requires being clear on the learning target of instruction. As mentioned in chapter 2, advancing learning without clarity regarding the learning goal (i.e., what you want students to learn as a result of engaging in a particular lesson) is impossible. While you engage in ATL 8.2, consider that the goal for the lesson is for students to understand the following:   1. Comparing different amounts of the same quantities requires establishing a common basis for comparison.   2. Using ratios, rates, and percents to make comparisons is possible.   3. Ratios are multiplicative rather than additive comparisons.

  4. Ratios can compare part-to-part, part-to-whole, or two different measures.   5. Creating different ratios from the same situation is possible.

These learning goals will guide your decision-making regarding how to support student learning.

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Analyzing Teaching and Learning 8.2 Investigating Students’ Thinking on a Proportional Reasoning Task Review the student work shown. • What misconceptions did the task elicit from students? • What questions (assessing and advancing) might you ask students to help them recognize and move beyond their misconceptions and toward the goals of the lesson? A

B

C

x

75 25

=

3 1

=3

y

40 20

=

2 1

=2

z

100 25

=

4 1

=4

Since the marbles in bag 2 total 125 I think your chances would be higher than the others.

D

Analysis of ATL 8.2: Investigating Students’ Thinking on a Proportional Reasoning Task Because each of the four responses to the Bag of Marbles task reveals some degree of faulty reasoning, the teacher is now able help students recognize and move beyond such reasoning and develop a sounder understanding of the relationships among the quantities in the task. As discussed in chapter 5, assessing and advancing questions are a basic tool that teachers can use to make student thinking public and move students to new understandings. Telling a student that he or she is wrong or giving him or her a particular strategy to use will not, in the long run, lead to a deeper understanding of mathematics. According to Principles to Actions (NCTM

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2014, pp. 35–36), “purposeful questions allow teachers to discern what students know and adapt lessons to meet varied levels of understanding.” Student A constructed a part-to-part ratio but incorrectly described the bags as if they were part-to-whole ratios (i.e., bag X as 1/3 blue, bag Y as 1/2 blue, and bag Z as 1/4 blue). The student did correctly select bag Y; however, the rationale “1/2 is a lot” is not clear and does not show a comparison across bags. The goal in interactions with that student should be to help the student clarify the distinction between ratios and fractions. Although both ratios and fractions can be used to make comparisons (goal 2), the student needs to clearly understand what a ratio is and what is being compared (goal 4). In questioning the student, the teacher first needs to assess what the student understands about what she or he did by asking such questions as “How did you get 1/3, 1/2, and 1/4 ?” and “What does each number in the comparison mean in terms of the problem?” After determining what the student understands, the teacher can begin to advance the student’s understanding by asking him or her to consider other situations (e.g., If you had a fourth bag with two red marbles and two blue marbles, how would you express the relationship between the marbles? One of the other students said that 1/4 of the marbles in bag X were blue; can both answers be right?). Student B used the previously described additive approach, finding the difference between the number of red marbles and the number of blue marbles in each bag. The goal in interactions with that student should focus on goal 1 (establishing the need for a common basis for making a comparison) and goal 3 (ratios are multiplicative rather than additive comparisons). A discussion with that student should begin by asking the student what he or she did, why, and how the student’s work helped him or her answer the question. The teacher could then ask the student whether the approach always works and give the student additional bags of marbles to consider (e.g., bag W has two hundred blue marbles and four hundred red marbles, that is, the same ratio as bag Y but a larger difference; bag V has three blue marbles and nine red marbles, that is, a smaller unit rate than bag Y and smaller difference) to help the student see that the method that he or she used is not sound. Student C constructed part-to-part ratios and compared the number of red marbles to the number of blue marbles in each bag. The student correctly determined the unit rates —  bag X 5 3/1; bag Y 5 2/1; bag Z 5 4/1 — but then chose bag Z because it had the largest number of marbles. How (or if ) this decision relates to the ratios is not clear. The focus of interactions with this student should be on goal 4 — gaining clarity regarding what the ratios are comparing. The first step with this student would be to determine what he or she understood about the ratios that he or she constructed and why the student chose bag Z. The teacher ultimately wants the student to compare the smaller bags denoted by the unit rates. Toward this end, the teacher might ask the student to draw a model of each of the smaller bags and consider which one of those smaller bags would offer the best chance of getting a blue marble. This question is intended to help the student see that bag Y has fewer red marbles for each blue marble and why that bag would be the optimal choice. Elicit and Use Evidence of Student Thinking    149 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Student D compared bags X and Z because they have the same number of blue marbles and concluded that bag X would give her or him a better chance of getting a blue marble because there are fewer red marbles. Although this conclusion is a sound one to reach on the basis of a comparison of the two bags, it does not consider bag Y. The focus of interactions with this student should be on goal 1, the need for a common basis for comparison. Here the teacher might begin by asking why the student compared bags X and Z and what she or he thought about bag Y. The teacher might ask the student to consider whether it would be possible to create a common number of blue or red marbles so that she or he could compare bag X and bag Y or compare all three bags. The written responses that each of these four students gave reveal much about what the students knew and understood related to the mathematical ideas that guide this lesson. But by engaging the students in a discussion of what they did and why, the teacher obtains a more detailed understanding of what the students are thinking that can then serve as the basis for further discussion and instruction.

Considering How to Elicit and Use Student Thinking during a Lesson In ATL 8.3, you revisit the eighth-grade classroom of Mrs. Elizabeth Brovey, whose prealgebra students are working on the Two Storage Tanks task shown below. Students in Mrs. Brovey’s class have been constructing linear equations to show the relationship between quantities represented in real-world contexts, geometric patterns, and graphs. The task featured in this lesson is the first time that students have worked on a system of equations that includes both increasing and decreasing relationships.

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The Two Storage Tanks Task Two large storage tanks, T and W, contain water. T starts losing water at the same time additional water starts flowing into W. The graph below shows the amount of water in each tank over a period of time. Assume that the rates of water loss and water gain continue as shown.

  1. When will the two tanks contain the same amount of water? Explain how you found your answer, and interpret your solution in terms of the problem.   2. If you have not already done so, write an equation for each storage tank that can be used to determine the amount of water in the tank at any given number of hours. • Explain what the different parts of each equation mean in terms of the problem. • Explain what the different parts of each equation mean in terms of the graph. Adapted from NAEP Released Items, 2003-8M10 #13. http://nces.ed.gov/NationsReportCard/nqt (National Center for Educational Statistics 2013)

Through their work on this task, Mrs. Brovey wanted her students to continue to build the following understandings: • Linear relationships have a constant rate of change among the quantities; are depicted graphically by a line; and can be written symbolically as y 5 mx 1 b, where m is the

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constant rate of change and the slope of the line, and b is the value of the y-quantity when x 5 0 (i.e., the y-intercept).

• Connections can be made among different representations (e.g., contexts, tables, graphs, and equations) with any representation as a starting point. • A line is increasing if it goes up from left to right. The slope is positive, i.e., m . 0. A line is decreasing if it goes down from left to right. The slope is negative, i.e., m , 0. • The solution to a system of two unique nonparallel linear equations in two variables is represented by the ordered pair (x, y) that makes both equations true statements; graphically, the solution is the point of intersection of the two equations.

With these clear goals for student learning, Mrs. Brovey can look for evidence that students are reaching these goals. For example, she wants to see if students can identify the amount of water in each tank at the start and relate that amount to the y-intercept on the graph and the b in the equation y 5 mx 1 b. She also wants to learn whether students can determine the rate of change from the graph, recognize that Tank T has a negative rate of change whereas Tank W has a positive rate of change, and form an equation that relates the starting value for each tank to the rate of change. Finally, she wants to learn whether students can explain how to find the point of intersection and what it means in the context of the problem. Mrs. Brovey began the lesson by reading the task aloud and asking students to work in small groups. The video clip shows the interactions that Mrs. Brovey had with one of the small groups. This group had determined equations for both tanks. The clip began with the students explaining one of their equations. This video clip shows how a teacher elicits and uses student thinking during a small-group discussion.

Analyzing Teaching and Learning 8.3 more Exploring How to Elicit and Use Student Thinking during Group Work Watch the video clip and download the transcript of the discussion of the Two Storage Tanks task in Elizabeth Brovey’s classroom. While you watch the video, consider each of the following questions: • What did Mrs. Brovey do to elicit students’ thinking? • How did Mrs. Brovey use students’ thinking to advance their learning toward the goals of the lesson? You can access and download the videos and their transcripts by visiting NCTM’s More4U website (nctm.org/more4u). The access code can be found on the title page of this book.

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Analysis of ATL 8.3: Exploring How to Use and Elicit Student Thinking during Group Work The first thing that Mrs. Brovey did to elicit student thinking was to engage the students in a task that would promote reasoning and problem solving. In addition to finding the point of intersection (which students could do by extending the lines, drawing successive slope triangles, making a table of known values and extending it by using the rate of change, or by using substitution or elimination) and writing equations, students had to explain their thinking and interpret their findings in terms of the problem context. The task had the potential to reveal what the students understood about the relationships portrayed on the graph and the extent to which they could create and explain equations that depicted those relationships. The teacher asked assessing and advancing questions that elicited how students had interpreted the graphs. For example, questions such as “What does the 900 mean for the problem?” (line 4 of the transcript) and “Where? I don’t see minus 50” (line 10) allowed students to explain that 900 was the number of gallons of water in Tank T (lines 5–9) and that “minus 50” was the number of gallons of water lost every hour (lines 11–30). Mrs. Brovey’s subsequent request to Leon to explain what was happening in Tank W (lines 31–32) showed his conclusion that Tank W was gaining 50 gallons of water per hour rather than 25 gallons per hour (lines 39–41). After Leon’s thinking was clear and was made public within his group, the teacher challenged Leon and his peers to “prove” that this interpretation was correct (lines 42–43, 47). When Leon explained why it was 50 gallons per hour (lines 48–53) and noted that “we add 50 more and it’s 400” (line 53), the teacher questioned “Is it 400?” (line 54). This question caused Leon, as well as the other students in the group, to ultimately realize that the rate of change was actually 25 gallons per hour (lines 60–82). Throughout this segment, the teacher elicited students’ thinking and used their thinking to determine the next course of action. She did not tell the group that its equation for tank W was incorrect but instead challenged the students to prove it. In so doing, she was supporting students’ productive struggle, building their capacity to persevere in the face of a challenge, and sending the implicit message that they were capable of figuring out the task on their own. She held students accountable by using advancing questions that left the group with something to do or think about and telling them that she would come back (lines 43, 101), reminding them that everyone in the group was responsible (lines 75–77), and telling the group that anyone in the group should be able to answer her questions (lines 101–102).

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Exploring Ways to Elicit and Use Students’ Thinking during Instruction In ATL 8.4, you visit the classroom of Deborah Dyson, who used the Building a Pizza task (shown below) to explore linear functions with her eighth-grade students. The case presents an excerpt from her classroom in which Ms. Dyson and her students discussed and analyzed the strategies used in solving the task.

The Building a Pizza Task You and your friends are going to buy pizza from Domino’s. From previous orders, you know that a medium pizza with two toppings costs $14.00 and that a medium pizza with five toppings costs $20.00. a) If Domino’s charges the same amount for each topping added to a plain cheese pizza, what is the cost per topping? b) If you wanted to order a medium cheese pizza with no additional toppings, how much would you expect to pay? c) Write a general rule that you can use to determine the price of any medium Domino’s pizza. For each part of the task, be sure to explain how you got your answer and why it makes sense. Adapted from Mathalicious. http://www.mathalicious.com/lessons/domino-effect

Analyzing Teaching and Learning 8.4 Exploring How to Elicit and Use Student Thinking during Instruction Read “The Case of Deborah Dyson” and consider the following:  1. What does the teacher do to elicit student thinking?  2. How does the teacher use student thinking during the lesson?  3. What might the teacher do in class the following day to build on what she learned during this lesson?  4. What did the teacher need to do to prepare for this lesson?

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1

The Case of Deborah Dyson

2 3 4 5 6

Mrs. Dyson wanted her students to use the concepts of slope and y-intercept in a problem context, write an equation to represent the relationship between a dependent variable and an independent variable, and gain facility in recognizing and expressing a linear function in a table, a graph, and an equation. She selected the Building a Pizza task for several reasons:

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

• It aligned with her goals for student learning. • It was challenging yet accessible. • It would motivate students.

• It would reveal students’ understanding and thinking. Mrs. Dyson began the process of planning for the lesson by first considering the approaches that students were likely to take while they worked on the task. For example, she decided that some students would reason with arithmetic, subtracting the costs of the two pizzas and the numbers of toppings and dividing, whereas others would make a table or a graph by using the two pizzas that she gave as starting points. She then generated questions that she could ask to assess what the students understood and to advance their understanding. She used this information to create monitoring charts in which she could record what students were doing and thinking (figs. 8.1 and 8.2). While students began working with their groups on parts (a) and (b) of the task, Mrs. Dyson, armed with her monitoring chart, walked around the room stopping at different groups to listen to their conversations and to make notes. When students struggled to begin, she asked, “What is different about the two pizzas? What happens to the price when you increase from a two-topping pizza to a five-topping pizza? How much does each topping cost?” When students simply found the difference between the two costs ($20 2 $14) without finding the difference between the numbers of toppings, she asked, “How much more does a five-topping pizza cost than a two-topping pizza? How many more toppings were on the five-topping pizza? How much do you think each topping costs?” When she encountered a group that was moving in a productive direction, she asked the students to explain the representation that they had used (e.g., table, graph, reasoning) and to consider how they could use their strategy to find the cost of any pizza no matter how many toppings it had. In addition, she left each group with something to think about. Consider, for example, her exchange with group 4 (Rashard, Hala, Michael, and Candace) about the graph that they had produced (see a portion of their graph in fig. 8.3).

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35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Mrs. D.: What did you do to get your graph? Hala:

We plotted the two points they gave us  — a two-topping pizza for $14 and a five-topping pizza for $20  —  and then connected them with a line.

Mrs. D.: Where did this triangle come from? Michael?

Michael: We decided to make a triangle that took us from the first point to the second point.

Mrs. D.: Why did you do that? Candace?

Candace: Well we want to see how much change there was from the first pizza to the second pizza. We could see that the number of toppings changed by three (pointing to the horizontal leg of the triangle) and the cost changed by $6 (pointing to the vertical leg of the triangle). So, this means that each topping cost $2.

Mrs. D.: How can you see the price per topping on the graph? Rashard?

Rashard: Well we went back to our first pizza (two toppings, $14) and could see that if we increase the number of toppings by one, then the price goes up by two (pointing to the graph). So, if we go from two toppings to three toppings, the price goes from $14 to $16 and it keeps going like this.

Mrs. D.: So how could you find the cost of a pizza no matter how many toppings it had? Rashard: Just extend the line. Hala:

Yeah. The line can go on forever . . . so you just extend it until you get to the number of toppings you want and then go up to find the cost.

Mrs. D.: I notice that you extended your line so that it intersects with the y-axis. What does this point (pointing to the point at which the line intersects the y-axis) mean? [Students in the group looked at each other and shrugged. The teacher waited ten seconds before continuing.]

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Mrs. D.: I am going to leave you to discuss what this point means in terms of the number of toppings and the cost of a pizza. I will stop back in a few minutes to see what the four of you come up with.

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After checking in on all five groups, Mrs. Dyson quickly reviewed the information that she had collected on her monitoring chart. Group 4 had produced a graph, whereas other groups reasoned with arithmetic (either initially or after a false start and a bit of redirection). She decided to ask Shawna from group 3 to present her group’s reasoning with the arithmetic approach first, followed by a table that Mrs. Dyson had prepared (since none of the groups had produced one), and ending with Hala from group 4 presenting her group’s graph. (Fig. 8.3 shows the work presented.) Mrs. Dyson wanted to be certain that students could see how the three representations connected with one another and ultimately, how students could use each representation to generate a rule that they could use to find the cost of a pizza, no matter how many toppings it had.

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During the discussion, Mrs. Dyson asked the presenters to explain what their group did and why, and she invited other students to consider whether the approach made sense and to ask questions. She asked questions that helped students make connections among the different strategies and with the primary ideas that she was targeting. In particular, she asked students where in the table and the graph the cost per topping appeared and where in the table and the graph the cost of a cheese pizza with no additional toppings appeared. When she showed the table that she had constructed, she asked the students to turn and talk to the members of their group for two minutes to see whether they could determine how she had constructed the table, whether that approach worked, and how it connected with the graph that group 4 constructed. When fifteen minutes of class remained, Mrs. Dyson asked the groups to review the three approaches that they had discussed and to see whether they could use one of the approaches to create a general rule that they could use to find the cost of any pizza (part [c] of the task). She again visited the groups, asking questions and recording information on her monitoring chart (shown in fig. 8.2). That information would help her in planning her lesson for the following day.

(In writing this case study, Margaret Smith (University of Pittsburgh), drew on lessons taught by secondary methods students at the University of Pittsburgh and on Silver and Smith [2015].)

Strategy Graph — Plot ordered pairs on a coordinate plane and draw a slope triangle or determine the ratio of rise to run between the two points. Connect the points with a line that includes (0,10).

Questions • Where does your graph represent the price per topping? • What does the cost per topping mean in terms of the graph? • What does the y-intercept mean in terms of the problem? • How could you use your graph to find the cost of any pizza?

Who/What

Order

Rashard, Hala, Michael, Candace (G4)

Hala (G4) — third

Used slope triangle; Talked about the “rate” of $6 for 3 toppings; saw equivalence to $2 for one topping; found y-intercept

Ask students how G4’s solution connects with the solution that G3 had presented.

Had trouble explaining the meaning of y-intercept but could explain that extending the graph to find any cost was possible.

continued on next page

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Table — Make a table that has toppings 0 through 5 and fill in the cost for two and five toppings.

• Where does your table represent the price per topping?

No one used this approach.

Ask students if the reasoning is sound and how it connects with the first solution.

• Where does your table represent the price of a plain pizza? • What would the graph of your points look like? • How could you use your table to find the cost of any pizza?

Reasoning with Arithmetic — Determine that if the five-topping pizza costs $20 and the twotopping pizza costs $14, then the difference in cost is $6 and the difference in the number of toppings is 3:

• How did you find the cost of a plain pizza?

Chris, Ashley, Tyronne, Mirah (G1);

• How could you find out the cost of any pizza?

Jennifer, Marko, Delmar, Shawna (G3);

• Could you use your method to find the cost per topping given the price of any two medium pizzas?

6  3 5 2, so each topping is $2.00. Subtract the Two Amounts — Note that one pizza cost $20 and the other cost $14, so subtract and calculate the cost per topping as 6/3 5 2.

• How much more does a five-topping pizza cost than a twotopping pizza? • How many toppings were added? • How much would a three-topping pizza cost? Why?

Table —  second

Shawna (G3) — first

G2 and G5 subsequently used this approach. Found the cost per topping but initially had trouble finding the cost of a pizza with no toppings. After getting on track, the groups were working on a generalization to find cost for any pizza. Aaron, Amber, Sheere, Tamika (G5) Initially stated the cost per topping to be $6, but after I asked a few questions realized that it would be the cost for three toppings. continued on next page

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Cannot Get Started

• What are you trying to find?

Yolanda, Jared, Mick, Leslie (G2)

• What is different about these two pizzas?

Were able to answer questions about what the differences meant and had some ideas on how to proceed. Finally used a “reasoning with arithmetic” approach.

• What happens to the price when you increase from two toppings to five toppings?

Fig. 8.1. Monitoring chart for parts (a) and (b) of the Building a Pizza task (From Silver and Smith 2015)

Strategy Symbolic Cost 5 $2t 1 $10 t 5 number of toppings

Questions • What does each part of equation mean in the context of the problem? • If the cost per topping increased, what would change in your equation? • If the cost of a cheese pizza increased, what would change in your equation? • What do you think your equation will look like when graphed? Why?

Who/What

Order

Jennifer, Marko, Delmar, Shawna (G3) Explained each part of equations in terms of context; knew what would change if different pricing. The group could describe what the equation looked like when graphed. G4 got this equation after revising its initial work.

continued on next page

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Narrative You take the number of toppings, multiply by $2, and add $10.

Algebraic— Incorrect C 5 $10x 1 $2

• How would you figure out the cost of a pizza with three toppings? Ten toppings? • Can you write this cost as an equation? How can you represent the number of toppings? How do the number of toppings and the cost per topping relate to each other? What happens to the $10? • How much does a one-topping pizza cost using your rule? How much does a two-topping pizza cost? • What changes and what remains the same each time?

Cannot Get Started

• How much would a pizza cost with one topping? • How much would a pizza cost with two toppings? • What changes when you add more toppings? What remains the same no matter how many toppings you have? Can you write down what you did?

Chris, Ashley, Tyronne, Mirah (G1); Yolanda, Jared, Mick, Leslie (G2) Had trouble using variables to expressing the relationship symbolically. G5 was able to write a rule in narrative form after the initial struggle to get started. Rashard, Hala, Michael, Candace (G4) When the students tried their rule for specific values, they found that it did not give them the same information that they found with their graph. They fixed it. Aaron, Amber, Sheere, Tamika (G5) Found cost per topping, but students were not sure how to generalize a rule. Ended up writing a rule in narrative form.

Fig. 8.2. Monitoring chart for part (c) of the Building a Pizza task (From Silver and Smith 2015)

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Reason with Arithmetic (Groups 1, 2, 3, 5)

Table (Teacher Constructed)

Graph (Group 4)

Fig. 8.3. Work produced by groups on the Building a Pizza task

Analysis of ATL 8.4: Exploring How to Elicit and Use Student Thinking during Instruction Mrs. Dyson selected a high-level task that aligned with her goals for student learning (see lines 2–5 in “The Case of Deborah Dyson”). It also could engage students in reasoning and problem solving. In addition, the task was motivating  —  presenting a familiar context and a question that students could not immediately answer  —  and that students could solve in different ways, making it accessible to them. Perhaps the most important aspect of the task Elicit and Use Evidence of Student Thinking    161 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

pertaining to this discussion is that it included the requirements that students explain how they solved the problem and why their solution made sense, thereby increasing the potential of the task to reveal students’ understanding and thinking about important mathematical ideas. Mrs. Dyson posed questions to students throughout the lesson to elicit their thinking about the problem so that she could determine the extent to which they were reaching the lesson goals and to provide support to help them to move forward. For example, in her interaction with group 4 (lines 35–63), she asked the students to explain how they constructed their graph, where the triangle came from, and how they could see the price per topping on the graph. When she asked students how they could find the cost of a pizza with any number of toppings (lines 52–53), Rashard indicated that you could “just extend the line” (line 54). When Mrs. Dyson asked students to explain what the y-intercept meant in terms of the context of the problem (lines 58–59), students were not able to do so. In that brief exchange with the students in group 4, the teacher collected considerable evidence regarding what they did and did not seem to understand at that point. The students could convert the information given in the problem (i.e., a two-topping pizza cost $14; a three-topping pizza cost $16) to points on a graph and use the graph to determine the cost per topping. They understood that the graph “can go on forever” although they had not yet determined a method for finding the cost of any pizza without the graph. Finally, she learned that the students could not yet explain the meaning of the y-intercept. She left students with the charge to continue to discuss the meaning of the y-intercept in terms of the number of toppings and the cost of the pizza. She knew that that aspect of understanding linear relationships in context was essential, and she wanted to hold students accountable for continuing to explore it. Mrs. Dyson learned a lot about the thinking of her students during their small-group work, as evidenced by her recordings in column 3 of the monitoring sheets (see figs. 8.1 and 8.2). In particular, she could see the strategies to which students were drawn (e.g., graph as opposed to reasoning with arithmetic) and with what students appeared to struggle (e.g., identifying the cost per topping for group 5; finding the cost of a pizza with no topping for groups 1 and 4. In turn, she used what she learned from her interactions with the groups to make decisions regarding the groups (and students) who would present their work during the whole-class discussion and the order in which students would present the solutions. Hence, the thinking of her students provided the foundation of the discussion that unfolded. Although the teacher did introduce a table during the discussion that students had not produced (lines 80–83), she used it to foster and bring to the surface additional student thinking about its construction and relationship to the graph. During the next class, Mrs. Dyson might want to discuss part (c) of the task by asking groups 5 and 3 to discuss the rules that they created. Because she had students present the narrative rule before the symbolic rule, she could challenge students to consider how the two rules relate to each other and how they relate to the graph that group 4 constructed. She could bring into the discussion students from different groups who struggled with aspects of 162   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

generalizing (e.g., group 4, which initially wrote the equation incorrectly; group 1, which had trouble writing the rule symbolically). Because this task presented a challenge to students and brought some confusion to the surface, Mrs. Dyson might also want to give students additional experience with the concepts in context and more practice with writing and graphing equations to express generalizations. For example, she could extend the lesson by giving students information about small and large pizzas and asking them again to create rules. She might also consider an extension question that would challenge groups that finished quickly to consider the graphs of the three rules and to explain how they could account for the difference in the steepness of the lines. Mrs. Dyson might also want to consider whether rearranging the groups would be useful while students worked on these extensions. For example, to offer more support for student learning, she might want to break up the group of students who had difficulty in starting on the task and intersperse these students with students from groups that had been more successful. Ms. Dyson carefully planned this lesson. She anticipated ways in which the students might solve the task, determined questions that she would ask about specific solutions that would allow her to assess what students knew and advance them toward the goal of the lesson, and created monitoring sheets (see figs. 8.1 and 8.2) that would consolidate this information and allow her to attend to students’ thinking during the lesson. (The monitoring sheets shown are extensions of the tools introduced to support questioning in chapter 5 and to structure meaningful mathematical discussions in chapter 7.) Through this planning process, Ms. Dyson prepared herself to support students without telling them how to solve the problem and thereby limiting their opportunities to think and reason. Ms. Dyson gives an example of a teacher who engaged in the five practices for orchestrating a productive discussion (Smith and Stein 2011) that chapter 7 discussed. It is similar to a formative assessment approach called the Math Forum, in which a teacher gains “a strong sense of individual students’ as well as the whole class’s understanding of mathematical concepts” (Suurtamm 2012, p. 31). Many similarities exist in the ways in which Deborah Dyson and Patrick Donnelly, whose classroom we first explored in chapter 1, facilitated their lessons. Mr. Donnelly also gave students a task that required them to think, reason, and explain; asked questions throughout the lesson that focused on explaining what they knew and what they thought about solutions that others produced; and planned a discussion in which he carefully selected and sequenced the work that students produced. In addition, Mr. Donnelly gave students an exit slip at the end of class (lines 78–82) that was intended to elicit their current understanding about the relationship between the candies. We would expect that his lesson the following day would be guided by what he learned from the exit slip. According to Marzano (2012, p. 80), “Exit slips are one of the easiest ways to obtain information about students’ current levels of understanding.”

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Elicit and Use Evidence of Student Thinking: What Research Has to Say Understanding how students are thinking about mathematical ideas and using that information to guide teaching is at the heart of what we do as teachers. Eliciting and using evidence of student thinking extends beyond the work of assessment to determine what students have learned at the close of a lesson. Teachers who consistently elicit student thinking during a lesson can use that evidence to adapt their instruction to better meet their students’ needs (Leahy et al. 2005). Research studies related to eliciting and using evidence of student thinking focus on two key areas: • How teachers interpret and make sense of student thinking

• How teachers use what they know and understand about student thinking before, during, and after a lesson As noted in chapter 7, interpreting student thinking begins with anticipating the range of correct and incorrect responses that students might produce given a particular mathematical task. This work of anticipating does not occur in a vacuum and goes beyond simply identifying an answer as correct or incorrect. Research-based learning trajectories for particular mathematical ideas (e.g., Clements and Sarama 2004; Sztajn et al. 2012) and progressions for broader mathematical topics (e.g., Common Core Standards Writing Team 2011; Wu 2013) provide teachers the tools for categorizing types of student thinking and situating them on a continuum that suggests ways to move students forward. Teachers can select a task and plan a lesson by using learning progressions or trajectories in ways that offer entry points to students with multiple different understandings, and using these learning progressions or trajectories can move student thinking forward according to the trajectories. For example, a teacher might approach a lesson on surface area and volume of rectangular prisms by using a contextual problem and unit cubes, so that students who are still building a more abstract conception of volume and surface area can physically build the prism, whereas other students may move directly to generalized formulas. The teacher can design questions that push the students who are still building physical models to see and use structure to press toward a generalization. The teacher might ask students who are already generalizing questions about how scaling dimensions affects surface area and volume (such as, What happens to the surface area and volume if the length of one side doubles?). On a smaller scale, research regarding misconceptions and reasoning strategies for a particular topic can give teachers similar tools that help them plan effectively for instruction that uses diverse student thinking (see Lamon 2007 for an example related to proportional reasoning). After selecting and enacting tasks that elicit meaningful student thinking, teachers have to consider how best to use that thinking both during and after the lesson. Thoughtfully planned questions that elicit important aspects of student thinking can lead to important mathematical 164   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

ideas being made public, and a teacher must then plan to bring those aspects of student thinking together in a discussion that brings to the surface key ideas and builds understandings for all students (Bray 2013; Smith and Stein 2011; Swan 2001). The ways in which students talk and write about a mathematical concept gives teachers immediate — and immediately useful — feedback about the extent to which a given lesson met the teacher’s instructional goals for students. If a teacher has considered the range of student responses before the lesson and has connected them with his or her instructional goals, the evidence of student thinking can offer a powerful assessment of student learning. In turn, teachers can use this information to plan subsequent lessons and build on student thinking. This cycle underscores the importance of soliciting high-quality student thinking in every lesson for the purpose of formative assessment. If teachers wait until the end of the week or the end of a unit to elicit and use evidence of student thinking, they have little clear information on which to base immediate instructional decisions in the lessons leading up to that assessment (Wiliam 2007). .

Promoting Equity by Eliciting and Using Evidence of Student Thinking Eliciting student thinking and using that thinking during a lesson can send important and powerful messages about students’ mathematical identities. By carefully listening to and interpreting student thinking, teachers can position students’ contributions as mathematically valuable and as contributing to a broader collective understanding of the mathematical ideas at hand (Davis 1997; Duckworth 1987; Harkness 2009). Teachers can use this everyday work of listening to student ideas and probing their thinking to highlight important mathematical ideas. This move can strengthen students’ identities as knowers and doers of mathematics, in addition to giving teachers a more nuanced view of their own students as learners (Crespo 2000). Until teachers elicit the ways in which students are thinking, they are blind to the ways in which students may be drawing on multiple sources of knowledge to think and reason mathematically (Aguirre, Mayfield-Ingram, and Martin 2013). After students make their thinking public, teachers can engage in a wide variety of moves that have implications for students’ mathematical identities. Teachers might choose to elevate a student to a more prominent position in the discussion by identifying her or his idea as one that is worth exploring. The teacher might invite broader participation by explicitly asking other students to comment on the work, a move that promotes a diversity of views and strategies in the discussion. The teacher might also ask students to connect their approach or response with the one being offered, which can support the class in moving toward a more convergent view of the important mathematical topics at the core of the lesson. Eliciting and using student thinking in ways that focus on the mathematics can also promote historically marginalized student populations or students who do not have a strong

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track record of success in mathematics. In particular, promoting a classroom culture that sees mistakes or errors as important reasoning opportunities can encourage a wider range of students to engage in mathematical discussions with their peers and the teacher. Students are more likely to make meaningful contributions to the classroom over time if they feel that their contributions, even if incorrect, are treated respectfully.

Key Messages • Mathematical tasks that promote reasoning and problem solving (and have characteristics of high-level tasks, as described in fig. 3.1) are necessary (but not sufficient) for eliciting student thinking.

• When teachers ask students to explain or justify their response to a task in writing or orally, they elicit student thinking.

• When teachers ask questions that build on what they learned about what students know and can do without telling them what to do or how to do it, they extend student thinking.

Taking Action in Your Classroom: Eliciting and Using Evidence of Student Thinking Next, consider the implications that the ideas that this chapter has discussed have for your own practice. We encourage you to begin this process by engaging in each of the following Taking Action in Your Classroom activities.

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Taking Action in Your Classroom Eliciting and Using Evidence of Student Thinking Choose a task that has the potential to elicit multiple solution paths to use in your classroom (i.e., a doing mathematics task from chapter 3 or one that uses multiple representations from chapter 5). • Anticipate the solution strategies, both correct and incorrect, that students might use. • Identify the solution strategies that you want to share to move your mathematical goal forward and the order in which you want to share them. • Create a monitoring sheet to help you keep track of the solutions that you anticipated and any others that arise, and note potential questions that you can ask. • Teach the lesson, and reflect on the extent to which you elicited and used student thinking. • Consider how you could use the information gathered in the monitoring tool to guide your practice beyond the current lesson.

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Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

CHAPTER 9

Support Productive Struggle in Learning Mathematics The Analyzing Teaching and Learning (ATL) activities in this chapter engage you in exploring the effective teaching practice, support productive struggle in learning mathematics. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 48): Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and support to engage in productive struggle as they grapple with mathematical ideas and relationships. According to Hiebert and Grouws (2007, p. 387) productive struggle is a key feature of teaching that “consistently facilitates students’ conceptual understanding.” In productive struggle, students must work to make sense of a situation and determine a course of action to take when a solution strategy is not stated, implied, or immediately obvious. Hence, the tasks that lead to productive struggle for students are those that are “within reach but that present enough challenge, so there is something new to figure out” (p. 388). In this chapter, you will —  • analyze a video clip to investigate how a teacher helps struggling students make progress on a task;

• analyze five teacher-student dialogues to determine the types of interactions that help or hinder students’ ability to persevere in the face of struggle; • analyze a teacher-student dialogue and identify the strategies that the teacher uses to support productive struggle; • determine a way to support a student’s productive struggle;

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• review key research findings related to productive struggle; and

• reflect on productive struggle in your own classroom.

For each Analyzing Teaching and Learning activity, note your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, support productive struggle in learning mathematics. If possible, share and discuss your responses and ideas with colleagues. After you have written or shared your ideas, read the analysis, in which we offer ideas relating the ATL to the focal teaching practice.

Supporting Students’ Efforts to Make Progress Analyzing Teaching and Learning 9.1 takes you into the classroom of Patricia Rossman, where students are working on the Hexagon task, a task that you first encountered in chapter 7. Ms. Rossman wanted her students to understand that —  • creating a generalization that describes the relationship between the train number and the perimeter of the train is possible;

• a constant rate of change exists between the quantities in the relationship — as the train number increases by one, the train’s perimeter increases by four; • a table shows the rate of change between the quantities as the first difference, a graph shows it as the slope of the line, and an equation shows it as the coefficient of the input quantity; and

• the different components of their generalization can be seen in the visual pattern.

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The Hexagon Task Trains 1, 2, 3, and 4 (shown below) are the first four trains in the hexagon pattern. The first train in this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon is added.

    Train 1

Train 2

Train 3

Train 4

  1. Compute the perimeter for each of the first four trains.   2. Draw the fifth train, and compute the perimeter of that train.   3. Determine the perimeter of the tenth train without constructing it.   4. Write a description that can be used to compute the perimeter of any train in the pattern.   5. Determine which train has a perimeter of 110. Adapted from Foreman and Bennett (1995).

Many of the students in Patricia Rossman’s class have recently arrived in the United States and are in a dual-language program. The lesson featured in the video clip is an introduction to a unit titled “Number Sense, Patterns, and Algebraic Thinking.” This lesson is the first time that the teacher has asked students to engage in a task that requires them to analyze and look for patterns in a visual diagram. Before the interaction shown in this video clip, Ms. Rossman launched the task, ensuring that students understood the words in the task and reminding them that all students in a group should understand the methods that they discuss. The video clip shows one small group working on its explanation and interacting with Ms. Rossman and ultimately presenting their solution to the entire class.

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Analyzing Teaching and Learning 9.1 more Determining How to Support Student Learning • Watch the video clip and download the transcript of a small group interacting with Ms. Rossman. • What does the teacher do to support students’ learning? You can access and download the videos and their transcripts by visiting NCTM’s More4U website (nctm.org/more4u). The access code can be found on the title page of this book.

Analysis of ATL 9.1: Determining How to Support Student Learning The small group that Ms. Rossman worked with initially (lines 1–59 in the transcript) identified the recursive pattern of 14 and found the perimeter of the tenth train by repeatedly adding four to the perimeter of the previous train. Because Ms. Rossman wanted students to be able to generalize the relationship between the train number and the perimeter (a stated goal of her lesson), she pressed students to find a way to describe the tenth train without knowing the perimeter of the train that preceded it. Students came to see that the first and last hexagons in a train each contributed five sides to the perimeter and the hexagons in the middle of the train each contributed four sides. They then went on to explain what the tenth train would look like. During the whole-group discussion that followed, Daniel (a student from the small group who had been very quiet but was the first to describe the tenth train — line 48) described to the class how the group had determined the perimeter of the train. The teacher subsequently asked another student from the small group to point out which hexagons in a train contributed four and five sides to the perimeter. Finally, another student in the class, not part of the featured small group, explained the strategy in his own words. So, what exactly did Ms. Rossman do to support the learning of the students in this small group? The answer to that question is a review of many of the practices discussed in previous chapters. Ms. Rossman initially had clear goals for students’ learning and used these goals as a target toward which she moved students. She selected a task that aligned with her goals; that promoted reasoning and problem solving; and that was, as discussed in chapter 3, a lowthreshold, high-ceiling task. That is, all students could enter and explore the task at some level, but the task could engage students in challenging mathematics. In this clip, the students in the small group began the task with a basic strategy of counting the sides of the hexagons that formed the perimeter of the trains and created an organized list that showed the train number and the perimeter.

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Students were able to use the diagram of the hexagon trains to physically point to and count the sides of the hexagons and later to connect the “plus 4” with the hexagons located in the middle of the train and “plus 5” with the hexagons located at each end of the train. The use of the representation gave students access to the task, in part bypassing barriers that may have existed if the task were more reliant on written and verbal language. The questions that Ms. Rossman asked focused on connecting the visual representation with the verbal descriptions and on probing students to assess what they knew (lines 3, 13, 16–17, 23, 25, 32, 34), and to advance their thinking (lines 5–9, 43, 45, 54–56). Throughout her initial questioning of the small group, Ms. Rossman elicited student thinking about the patterns that they noted and the extent to which they were making connections between their method and the visual diagram. She could therefore use that thinking and leverage those connections in the whole-class discussion by asking a student to share the group’s method and asking another student to explain it. This episode from Ms. Rossman’s class is an example of supporting productive struggle. By clearly identifying learning goals and selecting an aligned high-level task, Ms. Rossman was able to elicit students’ thinking through her use of questioning and representations and, in so doing, help students determine the perimeter of a train without counting. The key to supporting productive struggle is helping students make progress without telling them what to do and how to do it, one of the most significant challenges of ambitious mathematics teaching. According to NCTM (2000, p. 19): Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge. Although the students had not yet produced a generalization that would work in finding the perimeter of any train, a goal in this lesson, their identification of the underlying structure of a train (two end hexagons that each contributed five sides and the middle hexagons that each contributed four sides) left them well positioned to take on question 4 — writing a general description.

Considering the Role of Teacher Questioning in Supporting Productive Struggle Analyzing Teaching and Learning 9.2 asks you to consider five minidialogues in which students who were attempting to solve the Lifeguard task (shown below) had each reached an impasse — they got only so far in completing the task and were unable to make further progress. In each case, the teacher intervened in an effort to support them in making progress on the task. Support Productive Struggle in Learning Mathematics    173 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

The Lifeguard Task Benjamin’s friend, Maleeka, is taking a job as a lifeguard at the city pool. She will be paid a constant rate per hour. The graph below represents Maleeka’s potential earnings for three different numbers of hours of summer work.

  1. Maleeka’s short-term goal is to earn more than $80 for new Xbox One games. Describe the number of hours that she must work in order to make enough money for the games. Show all work; and represent your answer by using words, inequality notation, and a number line.   2. Maleeka’s long-term summer goal is to make more than $750. Determine the number of hours that she must work in order to meet this goal. Show all work; and represent your answer by using words, inequality notation, and a number line. Adapted from Institute for Learning (2015a). Lesson guides and student workbooks available at ifl.pitt.edu.

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Analyzing Teaching and Learning 9.2 Investigating Teacher Interventions Read the minidialogues that follow, then —  • Discuss the nature of each student’s struggle. • Identify what the teacher does to help students move beyond the impasse that they had reached. • Determine whether the teacher supported the students’ productive struggle. (Margaret Smith and Victoria Bill [Smith and Bill 2015a] developed dialogues 1–4 in 2015 for an Institute for Learning Professional Development Session in Syracuse, New York. The title of the session was “Supporting Students’ Productive Struggle in Learning Mathematics.”)

Minidialogues for The Lifeguard Task Dialogue 1 A student lists the ordered pairs: (2, 24) (4, 48) (9, 108). Teacher: Where did these numbers come from? Student: These are the labels for the points on the graph. Teacher: So how do they help you answer the question? Student: Maleeka wants to make $80, and I know that’s between four hours and nine hours, but I am not sure how to get closer than that. Teacher: How much money did Maleeka make in two hours? Student: $24. Teacher: So, how much money would she make in one hour? [Student divides 24 by 2 on her paper.] Student: $12? Teacher: That’s right. So now you know that she makes $12 in one hour, and we want to find out how many hours she needs to work to make more than $80. Do you remember how we used equations to solve problems like this? Student: 80  12? Teacher: Yes! Solve 12x 5 $80 by dividing 80 by 12. See what you get.

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Dialogue 2 A student’s graph is shown below.

Teacher: Tell me what you have here. Student: I drew a line and connected the three points that were given. Then I drew a horizontal line through $80 because that is the least amount of money she wants to make. Teacher: So, what did this tell you? Student: The point where the two lines intersect is the number of hours where she makes $80. So it is (x, 80). The value of x is somewhere between 6 and 7 hours, but closer to 7. Teacher: So, how can you figure it out more precisely? Student: I am not sure. Teacher: If you look at the three points that are given, how are hours worked related to the amount earned? Student: The hours worked times 12 gives you the amount she earned. Teacher: So, can this help you figure out how to find the x-coordinate of your point of intersection? Student: I can set up an equation x(12) 5 80 and solve for x. Teacher: That sounds like a good plan. I will check back in with you later.

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Dialogue 3 A student makes the table shown below. Hours

Earnings

2

24

4

48

9

108

Teacher: Tell me how you got the numbers in your table? Student: I read the points in the graph. Teacher: Does that help you answer the question? Student: Not really. She had to work more than four hours but less than nine. I am not sure where to go from here. Teacher: I am wondering if you could expand your table to include other values that were not on the graph. How could you get started? Student: I could put in hours 1 through 9 and then see if I could figure out how much she would make for that number. Teacher: Okay. So how will you figure out how much she earns for any number of hours? Student: Well, if I look at the three values in the table, it seems that each of the hours was multiplied by 12 to get the earnings. So that means she must make $12 an hour. So, I will just multiply all the hours in my new table by 12 and see if I can find what number of hours get her more than $80. Teacher: Sounds good!

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Dialogue 4 A student cannot get started. Teacher: What have you figured out so far? Student: Nothing. I am not sure what to do. Teacher: The first thing you need to do is to figure out how much money she makes every hour. Do you remember what operation you need to use to do that? Student: Divide? Teacher: What are you going to divide? Student: 2  24? Teacher: No, 24  2. Student: So, it’s 12. Teacher: So this is the amount she makes per hour. Now you need to divide 80 by 12.

Dialogue 5 A student cannot get started. Teacher: What have you figured out so far? Student: Nothing. I am not sure what to do. Teacher: Why don’t you start by reading the problem again and then looking really closely at the graph. I will be back in a few minutes.

Analysis of ATL 9.2: Investigating Teacher Interventions Each of the students portrayed in the dialogues has reached an impasse; that is, he or she has reached a point at which he or she is no longer sure what to do or how to move forward. In each situation, the teacher intervenes in an effort to help the student move beyond the impasse but does so in very different ways. In dialogue 1, the student has identified three points on the graph that represent the hours worked and earnings and listed the ordered pairs that correspond to these points. Although the student seemed to recognize that Maleeka had to work between four hours and nine hours, the student was not sure how to be more specific. The teacher began by asking the student how much money Maleeka made in two hours (a value on the graph) and then how much she made in one hour (a value not on the graph). The teacher then framed the problem that the student needed to solve (“she makes $12 in one hour and we want to find out how many hours 178   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

she needs to work to make more than $80”), asked the student to recall how the class had used equations to solve such problems, and reframed the students’ tentatively offered division statement as an equation. Although the teacher left the student with a clear pathway to follow to solve the task, and hence a way to move beyond the impasse, the pathway represented the teachers’ thinking rather than that of the student. In dialogue 2, the student connected the three points on the graph that were given, drew a horizontal line at y 5 80 to represent earnings of $80, and indicated that the point at which the two lines intersected would be the number of hours that Maleeka had to work to earn $80. The student determined that the number of hours that Maleeka needed to work to make $80 would be between six and seven hours. The student was not sure initially how to find the exact number of hours but came to realize that the number of hours worked could be determined by solving the equation x(12) 5 80. The teacher’s role here was minimal — she began by asking the student to tell her what she had done, what the graph told her, and how she could find the number of hours more precisely. The teacher then asked the student to look for a relationship between the number of hours worked and the amount of money earned. The first set of questions assessed what the student seemed to understand, and the final question asked the student to think about the relationship between hours and wages and how it could be useful. The student then determined a plan to follow. The teacher did not guide the student in any way or do any thinking for the student. In dialogue 3, the student identified three points on the graph that represented the hours worked and earnings and made a table with those values, but the student was not sure how to move forward. The teacher suggested that the student expand the table to include other numbers of hours and asked how the student could determine the wages earned for the hours not found on the graph. The student then identified the hourly wage as $12 and made a plan for moving forward. Although the teacher suggested that the student expand the table, she based this suggestion on the fact that the students had already created a table. The student determined the actual plan for creating the expanded table. The guidance from the teacher built on the student’s way of thinking and allowed the student to move beyond the impasse. In dialogue 4, the student did nothing and claimed not to know what to do. The teacher did not make any attempt to determine what the student understood about the task and began telling the student exactly what to do. Although the student may have been able to follow the directions that the teacher gave and finish with an answer to the question, it is not clear what sense the student made of the problem because we have no access to the thinking of the student. The student was no longer struggling, but the pathway that he or she was following was the teacher’s pathway. In dialogue 5, as in dialogue 4, the student did nothing and was not sure what to do. In this example, the teacher suggested general strategies of rereading the problem and looking more closely at the graph without attempting to determine what the student understood about the problem. Although the teacher was not directing the student to a particular pathway, this type Support Productive Struggle in Learning Mathematics    179 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

of support is unlikely to help a student move forward in a productive way. The student may have already taken those steps — he or she may have read the problem carefully and examined the graph. The feedback that the teacher gave, which was intended to be supportive, may have actually negatively affected the student’s mathematical identity and send an implicit message that he or she was not capable. Warshauer (2015b, p. 387) describes four types of teacher responses to struggle: • Telling

• Directed guidance

• Probing guidance • Affordance

Telling involves supplying students with information (e.g., a strategy to use) that removes the struggle, thereby allowing students to make progress on the task by following the prescribed strategy. Dialogue 4 is an example of telling. Directed guidance involves redirecting students to a different strategy, one that is consistent with the teacher’s way of thinking rather than the student’s way of thinking, thereby enabling the student to move beyond the impasse. Dialogue 1 is an example of directed guidance. Both telling and directed guidance tend to lower the cognitive demands of a task by turning a task that initially required reasoning and problem solving into one that requires only carrying out a procedure with no connection to meaning. Probing guidance involves determining what a student is currently thinking, encouraging a student’s self-reflection, and offering ideas that are based on the student’s thinking. Dialogue 3 is an example of probing guidance. Affordance involves asking students to articulate what they have done, encouraging their continued efforts with limited intervention, and allowing them time to continue their work. Dialogue 2 is an example of affordance. Both probing guidance and affordance tend to maintain the cognitive demands of the task by supporting productive struggle. What about dialogue 5? Dialogue 5 is an example of a fifth type of response, one that we have seen in classrooms in which we have worked, that we would categorize as unfocused or vague. In this type of response, the teacher does not direct students to a particular strategy or build on students’ thinking but instead provides a suggestion that is often too general to be helpful. How can a teacher know if the struggle is productive or unproductive? According to Warshauer (2015b, p. 390) Struggle is productive if —  • the intended goals and the cognitive demand of the task are maintained;

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• students’ thinking is supported by acknowledging effort and mathematical understanding; and

• students are able to move forward in the task execution through student actions. In dialogues 1 and 4, the struggle is no longer productive. The demands were not maintained, and the student moved forward primarily because of the teacher’s actions. In dialogues 2 and 3, the struggle is productive. The demands were maintained; the support provided built directly on the thinking of students and made it possible for the student to move forward on the basis of his or her own actions. As discussed in chapter 3 in the examples of Sandra Pascal and Patrick Donnelly, the telling and directed guidance that Ms. Pascal provided lowered the demands of the task and took the thinking opportunities away from students. By contrast, Mr. Donnelly supported productive struggle by providing probing guidance. Dialogue 5 is slightly different. Dialogues 1 and 4 focused the students on a particular pathway that would likely lead to successful completion of the task. Dialogue 5 indicates no push on the teacher’s part to focus students’ attention in a specific direction. Although the cognitive demand of the task was not lowered, the teacher did not sufficiently support student thinking and the student was unlikely to make much progress.

Other Teaching Moves That Support Productive Struggle ATL 9.3 involves analyzing a dialogue between Ms. Kaufmann, a seventh-grade teacher, and a small group of students in her class (Kaila, Jake, Jerome, and Chris). The class is working on an inequalities unit, and Ms. Kaufmann has selected the Deep Dark Secret task, shown on the next page, because it presents students with a situation that they have not yet encountered in their work in the unit — a negative rate of change.

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The Deep Dark Secret Task The deep sea is the lowest level of the ocean floor. Sunlight does not reach the deep sea, and scientists are discovering that it is home to amazing creatures that live in total darkness. Deep-sea creatures live more than 1000 feet below the water’s surface. Marine biologists in an underwater vessel are descending to study a new species of fish that was discovered in the deep sea. Their vessel is currently located 100 feet below the surface (2100 feet). From this location, the biologists start their timer at t 5 0 and begin their descent. They descend at a rate of 25 feet per minute. The vessel continues descending at this constant rate until it has reached a depth at which they can study the deep-sea creatures.   1. After how many minutes will the vessel be positioned to observe deep-sea creatures? Explain how you know.   2. Write an inequality to represent the depth of the vessel at any point in time, t, after the vessel has reached a depth at which the marine biologists can observe deep-sea creatures, where t represents the number of minutes since the vessel has left its original position of 2100 feet. Adapted from Institute for Learning 2015b. Lesson guides and student workbooks available at ifl.pitt.edu.

Analyzing Teaching and Learning 9.3 Identifying Strategies That Support Productive Struggle • Read the following dialogue, in which Ms. Kaufmann provides probing guidance to a group of students, who have arrived at an incorrect solution to the Deep Dark Secret task. • What did the teacher do to support the group’s productive struggle?

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1

Teacher-Student Dialogue for The Deep Dark Secret Task

2 3

Ms. K:

4

Kaila:

5

Ms. K:

6

Jake:

7

Ms. K:

8

Chris:

9

Ms. K:

10 11

So how long do you think it will take before they will see the creatures? Kaila? We think it would take 40 minutes.

It will take 40 minutes to do what? Jake? To get to the deep sea.

How did you get 40? Chris? We divided 1000 by 25.

Why did you do that? Jerome?

Jerome: It says that deep-sea creatures live more than 1000 feet below the surface. So, if the vessel goes down 25 feet per minute, you just divide 1000 by 25. Can you draw a picture of the situation to help explain what you did? I will be back in a few minutes to see what you came up with.

12 13

Ms. K:

14 15

[Teacher leaves to respond to other students and returns five minutes later. Members of the group discuss what the picture should look like. Kaila offers to draw.] So, tell me about your picture.

16

Ms. K:

17 18 19

Kaila:

20

Ms. K:

21 22 23

Jerome: The vessel kept going down 25 feet every minute. So, you need to find out how many 25s it takes to get to 1000. That will tell you how many minutes it would take to get to 1000. So, it takes 40 minutes, just like Kaila said.

Okay, see [referring to the drawing below], we started with the surface of the ocean and then showed where the deep sea started, which is 1000 feet below the surface.

So, how does this picture explain what you did? Jerome?

Okay. So, what else do we know about the situation? Jake, can you reread the problem?

24 25

Ms. K:

26 27

[ Jake reads the problem aloud and stops when he reaches the part that says “Their vessel is currently located 100 feet below the surface.”]

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Oh wait . . . our drawing isn’t right . . . the vessel starts 100 feet below the surface. We need to fix our picture.

28 29

Chris:

30

Ms. K:

31 32

Jake:

33

Jerome: Okay, so that means that they only went down 900 feet, not 1000.

Does everyone see what Chris is saying?

Yeah. The vessel was already 100 feet below the surface when they decided to go see the sea creatures.

34 35

Kaila:

[Quickly making a change, as in the sketch shown below.] Now this shows what Jerome just said.

36

Ms. K:

37 38

Chris:

So does the starting point make a difference? Chris?

39 40

Ms. K:

41

Jake:

42

Jerome: Which is 36.

Yeah, because like Jerome said, they only went down 900 feet from where they started, not 1000 feet.

So, what does the drawing suggest to you about your initial approach to the problem? So, it should be 900 divided by 25. [ Jerome quickly divides 900 by 25]. Okay, so what does 36 mean in this problem? Kaila?

43

Ms. K:

44

Kaila:

45

Ms. K:

46 47 48

Chis:

49

Ms. K:

50

Kaila:

51 52

Ms. K: Does everyone agree? Students: Yeah!

53 54 55

Ms. K:

It is the number of minutes it took to get to deep sea. Does everyone agree?

I think it takes 36 to get to 1000 feet but you need to get deeper than that to actually see sea creatures because the creatures are more than 1000 feet below the surface. So how long will that take? More than 36 minutes.

I liked the way you all worked together on this. I knew you could figure it out! Now you are ready to take on the second part of the task. I will check in with you a little later.

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Analysis of ATL 9.3: Identifying Strategies That Support Productive Struggle Most notably, Ms. Kaufman supported her students’ productive struggle by providing probing guidance. As discussed in the analysis of ATL 9.2, Ms. Kaufman determined what the students were currently thinking, engaged them in reflecting on their work, and offered ideas based on their thinking. Warshauer (2015a, p. 392) has identified four strategies for supporting productive struggle — question, encourage, give time, acknowledge — that when applied to this interaction between Ms. Kaufman and her students, give more insight into what the teacher actually did. First, the teacher asked questions that made the students’ thinking public (lines 1–9). The answers to this series of questions made clear to the teacher the source of the students’ error. She then asked students to draw and explain a picture that represented the situations (lines 12–13, 16) and to reread the problem aloud (lines 24–25). This series of interactions encouraged students to reflect on and reconsider what they had done and ultimately to revise their drawing (lines 34–35) and their solution (lines 42, 46–48). She also gave the group time to work to create the picture (lines 14–15) without hovering over them, sending the clear message that she had confidence that they could do the work without her assistance. In the end, she acknowledged the group’s effort and the students’ ability to work through the problem (lines 53–55). Through the use of these strategies, Ms. Kaufman was able to maintain the demands of the task and help students move their thinking forward. What is worth noting in the interaction between Ms. Kaufman and her students is that unlike in the five dialogues analyzed in ATL 9.2, the students were not struggling when Ms. Kaufman began her interaction with them. That is, they had found the answer of forty minutes and seemed quite confident with it. The teacher, however, recognized that the answer was incorrect and pressed the students to explain and reflect on what they had done so that they would recognize and move beyond the error. If the students had not corrected their initial error, their potential for success in the second question in the task would have been compromised. This type of persistent probing may be necessary to persuade students who are confident in their approach to reexamine or question their findings.

Supporting Students Who Cannot Get Started One challenge that teachers often encounter when they engage students in solving a highlevel task is the student (or students) who cannot get started on the task. This problem can sometimes be avoided by selecting a task that has multiple entry points and launching the task so that students are clear on what is happening in the problem and what the teacher is asking them to do.

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But what do you do when you approach a student who seems to have reached an impasse almost immediately? ATL 9.2 shows two approaches to dealing with this situation. In dialogue 4, the teacher told the students what to do, thereby taking over the thinking and eliminating the struggle; in dialogue 5, the teacher made a suggestion that was too vague and unfocused to be particularly helpful in getting beyond the impasse. In ATL 9.4, you consider the type of guidance that will support a student’s productive struggle on a task without lowering the cognitive demand of the task when the student is not clear on how to get started. This ATL revisits the Ducklings task first encountered in ATL 3.2.

Analyzing Teaching and Learning 9.4 Supporting a Student Who Cannot Get Started Imagine that the students in your class are starting work on the Ducklings task described below. You launch the task by asking students a series of questions: • What do you know about surveys? • What does typical mean? • Can you provide an example of something that is typical? • What does justifying an answer mean? Students had some experience in analyzing data the previous year, and the purpose of this task is to determine what they understand about basic statistics and data organization. What type of probing guidance could you give to students who could not get started on this task that will support their productive struggle without taking over the thinking for them?

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The Ducklings Task The local nature club is carrying out a survey of the number of ducklings in each family of ducks in the lake. Here are the results of the survey: 4, 7, 6, 5, 8, 7, 5, 4, 10, 4, 9, 6, 5, 4, 4, 5, 9, 8, 4 How many ducks are in a typical family? Use tables, graphs, or arithmetic to justify your answer. Adapted from http://www.insidemathematics.org/assets/common-core-math-tasks/ducklings.pdf (Noyce Foundation, 2012).

Analysis of ATL 9.4: Supporting a Student Who Cannot Get Started When a student cannot get started on a problem, it generally is not because he or she has no relevant knowledge to bring to bear on the situation. More often, the student, for some reason, is unable to connect what he or she does know with the current task. A first course of action with the student is to figure out what he or she understands about the problem by asking questions such as the following: What do the numbers in the problem represent? (they are the number of ducklings in a family); How many different families are there? (nineteen); What do you notice about the number of ducklings in the families? (one family has ten ducklings, five families have four ducklings each, most have four or five ducklings). After these basic questions, the teacher could ask students to consider the number of ducklings that they might expect to see in another duck family at the same lake and why. This query will give the teacher a sense of whether students identify a number between 4 and 10 (the range), determine that the next family will probably have four ducklings (the mode) because more families had four ducklings than any other number, or something else entirely. The teacher can follow up by asking what number would be unexpected in the next family and why, to encourage the students to articulate some conception of a measure of center in the data set. At this point, the teacher might suggest that students try to organize the data in some way so that the patterns they notice are clear to others. As mentioned in chapters 7 and 8, anticipating the ways in which students are likely to solve a task and the difficulties that they may encounter in so doing, as well as the ways that the teacher will respond when students run into roadblocks, are essential parts of the planning process. Such planning in advance of the lesson will better prepare the teacher to support struggling students during the lesson. In addition to specific strategies, teachers with whom we

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have worked have often added a “cannot get started” category to their monitoring sheets and planned questions specific to the task that will support students in making progress. One thing to remember is that the teacher’s goal in intervening when a student or group is struggling is not to make sure that every student has a correct and complete response prior to the whole-group discussion. Rather, the goal is to “support students’ fledgling efforts to make sense of the task before them and to make sure their thinking is headed in a productive direction” (Cartier et al. 2013, p. 88). Hatano and Inagaki (1991, p. 346) argue that during a whole-group discussion, all students have the opportunity to “collect more pieces of information about the issue of the discussion and to understand the issue more deeply.”

Supporting Productive Struggle in Learning Mathematics: What Research Has to Say Hiebert and Grouws (2007) have argued that developing conceptual understanding of mathematics requires productive struggle — giving students time so that they can wrestle with important mathematical ideas. Tasks that promote productive struggle — those that encourage reasoning and problem solving — should challenge students but be within their reach (Hiebert and Wearne 2003). Several studies give insight into productive struggle in middle school classrooms. Kapur (2010) found that seventh-grade students who persisted in solving complex problems (even when they were not successful) outperformed students who received only a lecture and practice intervention. In a study of classrooms in which sixth-grade and seventh-grade students engaged in solving high-level proportional reasoning tasks, Warshauer identified teacher actions that support productive struggle, as described previously in this chapter. In Warshauer (2015b), she identified two types of teacher responses that helped students move beyond an impasse: probing guidance and affordance. In both types of responses, the teacher provides support that honors and builds on the thinking of students without removing the demands of the task or doing the thinking for them. In Warshauer (2015a), she identified four strategies (question, encourage, give time, acknowledge) for supporting students’ productive struggle that helped students make progress. Smith (2000a) described the way that Elaine Henderson, a sixth-grade teacher, changed her teaching by redefining what it meant to be successful in her classroom. During the first year of implementing a curriculum that focused on reasoning and problem solving, the teacher initially simplified complex problems so that students would feel successful. She thereby lowered the demands of the tasks and eliminated the need for struggle. Over time, Ms. Henderson came to see that students needed to struggle to develop their ability to persevere and solve more challenging tasks, and she ultimately found ways to support their struggle without taking over their thinking. Supporting students’ struggle and ability to

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persevere led to increases in students’ learning, as evidenced by improved performance over the course of the year on items that measured reasoning and problem solving. Embracing productive struggle as a central component in learning mathematics with understanding requires that the teacher and students take on new roles in the classroom. The teacher is no longer the source of all knowledge, and the students are not passive recipients of knowledge who are just waiting for someone to tell them what to do. Figure 9.1 summarizes the way in which Elaine Henderson redefined success in her classroom with new expectations for students, new actions for the teacher, and new indicators of what counts as success (Smith 2000b). Teacher Actions Consistent with Expectations

Classroom-Based Indicators of Success

Most “real” tasks take time to solve; frustration may occur; perseverance in the face of initial difficulty is important.

Use “good” tasks; explicitly encourage students to persevere; find ways to support students without removing all the challenges in a task.

Students engaged in the tasks and did not give up too easily. The teacher supported students when they “got stuck” but did so in a way that kept the task at a high level.

Correct solutions are important, but so is being able to explain how you thought about and solved the task.

Ask students to explain how they solved a task. Make sure that the quality of the explanations is valued equally as part of the final solution.

Students were able to explain how they solved a task.

Students have a responsibility and an obligation to make sense of mathematics by asking questions when they do not understand and by being able to explain and justify their solution paths when they do understand.

Give students the responsibility for asking questions when they do not understand, and have students determine the validity and appropriateness of strategies and solutions.

With encouragement, students questioned their peers and provided mathematical justifications for their reasoning.

New Expectations for Students

Give students access to tools that will Students were able to use tools to Diagrams, sketches, and hands-on materials are important tools for stu- support their thinking processes. solve tasks that they could not solve dents to use in making sense of tasks. without them.

Communicating with others about your thinking during a task makes it possible for others to help you make progress on the task.

Ask students to explain their thinking, and ask questions that are based on students’ reasoning, as opposed to how the teacher is thinking about the task.

Students explained their thinking about a task to their peers and the teacher. The teacher asked probing questions based on the student’s thinking.

Fig. 9.1. Key elements in Elaine Henderson’s efforts to redefine success for herself and her students (Taken from Smith 2000b, p. 382)

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The elements that Smith identified in figure 9.1, along with the specific types of support (probing guidance and affordance) and strategies (question, encourage, give time, acknowledge) suggested by Warshauer (2015b), suggest concrete actions that teachers can take in creating classroom environments that encourage, value, and support struggle.

Promoting Equity by Supporting Productive Struggle in Learning Mathematics Achieving equity in mathematics instruction begins with a close examination of the nature of the mathematical tasks in which students have the opportunity to engage. Too frequently, historically marginalized students are relegated to classes in which they learn and practice procedures and never engage in the types of reasoning and problem-solving tasks described in this book. Central to equity-based effective teaching practices (Aguirre, Mayfield-Ingram, and Martin 2014) is the belief that strengthening mathematics learning and cultivating positive mathematical identities require engaging students in cognitively demanding tasks and promoting persistence, encouraging students to see themselves as competent problem solvers, and assuming that mistakes are sources of learning. The study by Boaler and Staples (2008), described previously, shows what can happen when every student has access to ambitious instruction as well as the opportunity to learn mathematics with understanding. Teachers at Railside (one of the three high schools that was the focus of the study) were able to reduce the achievement gap between different ethnic groups by giving all students access to challenging curricula and ambitious instruction in heterogeneous classes. By the end of the second year of the study, the students at Railside outperformed students at the other two high schools although they began year 1 achieving at significantly lower levels than students at the other two schools. According to the authors (p. 635): The Railside teachers held high expectations for students and presented all students with a common, rigorous curriculum to support their learning. The cognitive demand that was expected of all students was higher than other schools partly because the classes were heterogeneous and no students were precluded from meeting high-level content. Even when students arrived at school with weak content knowledge well below their grade level, they were placed into algebra classes and supported in learning the material and moving on to higher content. Teachers also enacted a high level of challenge in their interactions with groups and through their questioning. (Note: italics added by author.) The point here is that every student needs to have access to challenging mathematical work and the opportunity to engage in productive struggle with appropriate levels of

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support. According to NCTM (2000, p. 12), equity means that “reasonable and appropriate accommodations be made as needed to promote access and attainment for all students.” The use of other effective mathematics teaching practices — such as purposeful questioning, meaningful discourse, and tasks that promote reasoning and problem solving — must come together to influence classroom-level and school-level decisions to support and promote productive struggle for all students. Students may be more willing to engage in productive struggle if they feel that the teacher believes that they can learn and has their best interest at heart. Kleinfeld (1975) proposed the concept of a warm demander to describe the type of teacher who can bring out the best in students. Berry and Ellis (2013, p. 176) describe a warm demander as “a teacher who knows the culture of students, has strong relationships with students, and commands that everyone within the classroom will be respected and follow classroom norms.” According to Bondy and Ross (2008), a warm demander takes deliberate actions to build relationships with students, learn about students’ cultures, provide clear and consistent expectations, and communicate an expectation of success. High expectations are accompanied by enough support and scaffolding to facilitate students’ progress on challenging work (e.g., productive struggle), thereby convincing students that the teacher cares for and believes in them (Bondy and Ross 2008). As Principles to Actions (NCTM 2014) notes, for teachers to support productive struggle, they must examine their beliefs about students and adopt and practice a stance that all students can learn meaningful mathematics.

Key Messages • Engaging students in productive struggle is essential to developing conceptual understanding in mathematics

• Tasks that promote reasoning and problem solving are most likely to promote struggle and the need for perseverance • Teachers need to support productive struggle by providing probing guidance and affordance without taking over the thinking for students by telling or giving too much direction. • Teachers can support students’ productive struggle by asking questions, encouraging reflection, providing time, and acknowledging effort.

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Taking Action in Your Classroom: Supporting Students’ Productive Struggle Next, consider the implications that the ideas discussed in this chapter have for your own practice. We encourage you to begin this process by engaging in the following Taking Action in Your Classroom activities.

Taking Action in Your Classroom Supporting Student Struggle on a Challenging Mathematical Task Choose a task that promotes reasoning and problem solving (chapter 3) that you plan to implement in your classroom. For that task, do the following: • Describe what you will see students doing or hear students saying that would represent productive struggle with the task; • Describe what you will see students doing or hear students saying that would represent unproductive struggle with the task; and • Identify the ways in which you will use the four strategies discussed in the chapter to support the productive struggle of students so that they can make progress toward the mathematical goals of the lesson.

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CHAPTER 10

Pulling It All Together Working through chapters 2 through 9 offers an opportunity to explore each of the eight effective teaching practices individually and to develop a deeper understanding of what each practice entails and how it supports ambitious instruction. Each of the previous chapters gave insights indicating how the focal practice connected with other practices. This chapter discusses the eight effective teaching practices as a set of connected practices that form a framework for mathematics teaching and considers what is necessary to make these practices a reality in a classroom. Specifically, this concluding chapter refers back to activities explored in previous chapters, along with one new Analyzing Teaching and Learning (ATL) activity, and uses them to emphasize four essential points threaded throughout the book: • The eight effective teaching practices are a coherent and connected set of practices that when taken together create a classroom learning environment that supports the vision of mathematics teaching and learning that NCTM advocates and offers opportunities for students to achieve the world-class standards that states and provinces have put into place.

• Ambitious teaching requires thoughtful and thorough lesson planning that is driven by clear mathematics goals for students’ learning and considerable thought regarding what students are likely to do in response to a task and the range of support that the teacher can provide to ensure that students wrestle productively with challenging aspects of a task.

• Improving teaching over time requires deliberate reflection on whether students learned what the teacher was trying to teach and what the teacher did that may have supported or inhibited students’ learning and then making judicious adjustments to instruction on the basis of what the teacher learned through the reflection process.

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• Instruction must be equitable — each and every student must have the opportunity to learn mathematics with understanding. Toward this end, teachers must consider the five equity-based teaching practices hand-in-hand with the eight effective mathematics teaching practices.

A Coherent and Connected Set of Mathematics Teaching Practices “The Case of Patrick Donnelly,” revisited throughout this book, serves as a touchstone to which we related the new learning in each chapter. Hence, the case was intended to make salient the synergy of the effective teaching practices — that the success of Mr. Donnelly’s lesson was the result of integrating the practices in a coherent way rather than by attending to individual teaching practices. Figure 10.1 presents a framework for mathematics teaching that shows the relationships between and among the teaching practices and how they work together to support ambitious instruction as they did in Mr. Donnelly’s classroom.

Fig. 10.1. A framework for mathematics teaching that highlights the relationships between and among the eight effective teaching practices

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As discussed in chapter 2, learning goals focus and frame the teaching and learning that occur throughout a lesson. Hence “establish mathematics goals to focus learning” sits at the top of the framework to signify that setting goals is the starting point for all instructional decision making. The clarity and specificity of goals, as well as the ways that such clarity and specificity support subsequent instruction, is clear in almost every narrative and video case examined in this book (with the exception of “The Case of Sandra Pascal,” in which the absence of clear and specific learning goals may have contributed to the lack of success of the lesson). An example is Kelly Polosky’s lesson (introduced in chapter 2) featuring the triangle task. Ms. Polosky wanted her students to understand that —  • the area of a triangle is one-half of its length times its width;

• the relationship among the area, the length, and the width of a triangle can be generalized to a formula; and

• there are several equivalent ways of writing the formula for the area of a triangle and each can be related to a model. Ms. Polosky’s goals guided the decisions that she made before and during the lesson — from the task that she selected for students to work on, to the materials that she made available to support students’ work, to the questions that she asked to probe students’ thinking and press them to justify their conclusions, to the way that she synthesized student contributions and worked with students to connect the formula with the model. The goals were not just statements to record and forget — they served as a beacon that helped guide the lesson from beginning to end. If goals represent the destination for students’ mathematical learning in a given lesson, then tasks are the vehicles that move students from their current understanding toward those goals. Depending on the goals of a lesson (or sequence of lessons), teachers might select a task that promotes reasoning and problem solving or that engages students in building procedural fluency from conceptual understanding, the second level of the framework shown in figure 10.1. Tasks that promote reasoning and problem solving develop students’ conceptual understanding of mathematics and serve as a base on which to build procedural fluency. Hence, these two practices connect directly with the goals of the lesson and with each other. In each of the narrative and video cases examined throughout the book, the teacher selected a task that promoted reasoning and problem solving as the basis for instruction; and in all but one case (that of Sandra Pascal), the task was consistent with the goals for the lesson. For example, the triangle task that Ms. Polosky used is a doing mathematics task, on the basis of the criteria presented in the Task Analysis Guide (shown in fig. 3.1). Therefore, no prescribed pathway exists for solving the task, and students must explore and uncover the relationships (e.g., between rectangles and triangles; between length and width) to determine the generalization. Through their work on the task, Ms. Polosky’s students were able to construct formulas for the area of a triangle and relate them to the model, thereby achieving the goals for the lesson. Pulling It All Together    195 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Ms. Polosky’s lesson did not focus explicitly on developing procedural fluency; but as discussed in chapter 4, the conceptual understanding of the formula for finding the area of a right triangle that students developed during the triangle task can become the basis for developing procedural fluency in subsequent lessons. The heart of any lesson is the discussion (represented by the “facilitate meaningful mathematical discourse” rectangle in fig. 10.1). Discussions provide a forum in which students can share ideas and clarify understandings, develop convincing arguments regarding why and how things work, and develop a language for expressing mathematical ideas (NCTM 2000). In addition, discussions offer an opportunity for the teacher to move both small groups and the entire class toward the mathematical understandings that are the target of the lesson. The use of the four teaching practices situated within the large rectangle in figure 10.1 (pose purposeful questions, use and connect mathematical representations, elicit and use evidence of student thinking, and support productive struggle) facilitate discussions that take place in both small-group settings and whole-group settings. Together, these four practices focus students’ mathematical work and thinking on the goals of the lesson. While students work collaboratively in small groups, questions and representations can support productive struggle and elicit evidence of students’ thinking, which can guide the teacher’s planning of the whole-group discussion. During the whole-group discussion, a teacher might ask questions to elicit students’ thinking or to encourage students to make connections among representations. Together, the four practices in the “facilitate meaningful mathematical discourse” rectangle interact (in service of the goals and reliant on the task) to facilitate meaningful discourse. A clear understanding of the goals for the lesson provides a frame for the mathematical ideas that the teacher will elicit during the whole-group discussion and can help teachers determine the strategies, ideas, representations, etc., to select for presentation and discussion. Having goals in mind also supports teachers’ assessment of students’ learning by reminding them what to look for and listen for as evidence of students’ progress toward the goals. Hence, facilitating meaningful discourse makes students’ thinking public and accessible to the teacher, serving as a formative assessment that feeds back into teachers’ instructional decisions (e.g., goals and tasks) for subsequent lessons. The dashed line connecting the bottom of the model back to goals represents this connection in figure 10.1. The Kelly Polosky example shows how these four practices played out during a whole-class discussion. During the discussion of the triangle task, Ms. Polosky posed purposeful questions to move students’ thinking toward the goal of recognizing the equivalence of the two formulas that they had created (e.g., lines 52–53 of the transcript, “Do you think that means the same thing?”). Her questions elicited students’ thinking about the relationship between the area of a square and the area of a rectangle (e.g., line 17, “Why do you . . . I’m wondering why you need to do that [cut the square in half ]?”). She used students’ current ideas as the basis for subsequent questions, such as encouraging students to use the diagrams that she had provided (representation) to connect the formula with the model (lines 80–81, “How can we do half 196   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

times length times width?”). Although students were at times ambiguous or incorrect (e.g., lines 57–59, in which a student appeared to be confusing a lowercase L with the number 1), Ms. Polosky allowed the mathematical justification to unfold organically rather than correcting or clarifying the ideas herself. Soliciting input from different students allowed Ms. Polosky to support productive struggle by making a wide range of ideas available for students to consider while they clarified their own understandings. Hence, through these teaching practices (pose purposeful questions, elicit and use evidence of student thinking, use and connect mathematical representations, support productive struggle in learning mathematics), Ms. Polosky supported students’ reasoning and problem solving while they worked to make sense of the formulas that they developed, connect them with the models, and generalize the formula for all right triangles. These four practices also supported students’ work on tasks that promote reasoning and problem solving and ultimately accomplish the goals for the lesson in the narrative cases of Patrick Donnelly (chapter 1) and Deborah Dyson (chapter 8). Enacting the effective teaching practices requires purposeful actions and decisions on the part of the teacher, such as selecting the goals and task for the lesson, identifying questions that elicit specific aspects of students’ thinking (e.g., strategies, ideas, struggles, and misconceptions), and determining what connections and mathematical ideas should surface during the wholegroup discussion. Making these decisions while a lesson unfolds and reacting in ways that align with the effective teaching practices can place a high demand on teachers’ time, thinking, and processing during a lesson. Fortunately, teachers can engage in thinking through many of these actions and decisions before the lesson, during the lesson planning process. In so doing, teachers can purposefully plan to embed the effective teaching practices into individual lessons and sequences of lessons to support students’ learning of mathematics with understanding.

Thoughtful and Thorough Lesson Planning Good advance planning is the key to effective teaching. Good planning “shoulders much of the burden” of teaching by replacing “on-the-fly decision making” during a lesson with careful investigation into the what and how of instruction before the lesson is taught (Stigler and Hiebert 1999, p. 156). According to Fennema and Franke (1992, p. 156) During the planning phase, teachers make decisions that affect instruction dramatically. They decide what to teach, how they are going to teach, how to organize the classroom, what routines to use, and how to adapt instruction for individuals. The teachers featured in our examples throughout the book gave careful consideration to what they were going to teach and how they were going to support students before they set foot in the classroom. The lesson plan for the Hexagon task (appendix A) is an example of Pulling It All Together    197 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

the type of thoughtful and thorough lesson planning that is necessary to support ambitious instruction and the learning and engagement of each and every student. This lesson plan embodies many of the ideas discussed in previous chapters that relate to ambitious teaching and equity. The plan begins with identifying clear and specific learning goals, as well as evidence of what students will be doing or saying to indicate that they have met these goals. The plan then focuses on the task — identifying what task the teacher will use, what instructional supports students will receive while working on the task, what prior knowledge is necessary to enter the task, and how the task will be launched. In addition, the plan includes anticipated solutions, including misconceptions and questions that the teacher can ask to assess and advance student learning. The questions elicit students’ thinking so that the teacher can use their thinking to advance their understanding and support their productive struggle. The section labeled “sharing and discussing the task” is essentially a road map for facilitating the discussion. It is a plan for selecting, sequencing, and connecting student responses so that the important mathematical ideas that the teacher targeted in the lesson are public and explicit. Such planning also offers a useful framework within which to operate when unanticipated responses arise in class. A clear sense of the goals of the lesson and the mathematical storyline of the lesson through careful planning liberate a teacher’s resources to better make sense of unexpected responses and make principled decisions about how to handle them. A series of questions drives the lesson planning process. The goal is to prompt teachers to think deeply about a specific lesson and how to advance students’ mathematical understanding during the lesson. The emphasis in the plan is on what students will do and how to support them rather than on teacher actions. According to Smith, Bill, and Hughes (2008, p. 137), “By shifting the emphasis from what the teacher is doing to what students are thinking, the teacher will be better positioned to help students make sense of mathematics.” The teachers with whom we have worked have consistently commented that planning lessons in this way helps them enact lessons that maintain students’ opportunities for reasoning and problem solving, pose purposeful questions, and facilitate meaningful discourse. As one teacher commented, Sometimes it’s very time-consuming, trying to write these lesson plans, but it’s very helpful. It really helps the lesson go a lot smoother and even not having it front of me, I think it really helps me focus my thinking, which then [it] kind of helps me focus my students’ thinking, which helps us get to an objective and leads to a better lesson. (Smith, Bill, Hughes 2008, p. 137) Teachers with whom we have worked, including the one quoted above, have noted that this type of lesson planning is time-consuming. Working with colleagues might allow a teacher to divide and conquer the planning of multiple lessons. Over time, you and your colleagues will begin to accumulate a library of lesson plans. In the meantime, you might find that planning

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lessons by using a tool, such as the monitoring tool (discussed in chapter 8) or the lessonplanning template found in appendix B, can support your enactment of an effective teaching practice (or practices) and your students’ learning of mathematics. Finally, we also note the importance of moving beyond planning individual lessons to planning sets of related lessons that build student understanding over time. As stated by Hiebert and colleagues (1997, p. 31), Students’ understanding is built up gradually, over time, and through a variety of experiences. Understanding usually does not appear full-blown, after one experience or after completing one task. This means that the selection of appropriate tasks includes thinking about how tasks are related, how they can be chained together to increase the opportunity for students to gradually construct their understandings. Some examples of related lessons throughout this book include the four tasks on rigid motion (fig. 3.5), the lessons on finding the area of a triangle (chapters 2 and 4), and the discussion of sequences of tasks (including the Candy Jar task and the missing-value problem) that build up students’ understanding of proportional relationships and fluency in chapter 4.

Deliberate Reflection Similar to most worthwhile and complex endeavors, enacting the effective teaching practices will improve with time, experience, and deliberate reflection. Improvement requires identifying what is working or not working and then being willing to make the necessary changes. Reflecting on classroom experiences makes teachers aware of what they and their students are doing and how their actions and interactions are affecting students’ opportunities to learn. As first stated in chapter 1, cultivating a habit of systematic and deliberate reflection improves one’s teaching while sustaining lifelong professional development. Such reflection, however, is only the starting point for transforming teaching. According to Artzt and Armour-Thomas (2002, p. 7): Teachers must also be willing and able to acknowledge problems that may be revealed as a result of the reflective process. Moreover, they must explore the reasons for the acknowledged problems, consider more plausible alternatives, and eventually change their thinking and subsequent action in the classroom. Hiebert and colleagues (2007) suggest a framework for analyzing teaching that supports the reflection process: • Specify the learning goals for the instructional episode (What are students supposed to learn?)

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• Conduct empirical observations of teaching and learning (What did students learn?)

• Construct hypotheses about the effects of teaching on students’ learning (How did the teaching help (or not help) students learn?) • Use analysis to propose improvements in teaching (How could teaching more effectively help students learn?)

This framework focuses reflection on determining whether students learned what the teacher intended for them to learn, identifying how teaching might have supported or inhibited student learning, and then deciding how to improve the teaching. ATL 10.1 asks you to consider how a teacher, Beverly Allen, might use this framework to improve her teaching. In the lesson, Ms. Allen engages her seventh-grade students in solving the Mixing Juice task first discussed in chapter 5.

Analyzing Teaching and Learning 10.1 Reflecting on and Improving Teaching Read “Comparing Ratios: The Case of Beverly Allen.” • Use the analyzing teaching framework proposed by Hiebert and colleagues (2007) to determine what Ms. Allen might have concluded through her analysis of the Comparing Ratios lesson. • Identify the effective teaching practices that Ms. Allen might use to improve her teaching.

1

Comparing Ratios: The Case of Beverly Allen

2 3 4 5 6

Seventh-grade teacher Beverly Allen wants her students to develop the ability to recognize proportional situations and to make comparisons among quantities by using ratios, fractions, decimals, rates, unit rates, and percents. She selected the Mixing Juice task (shown below) for the lesson because it aligned with her goals, was cognitively challenging, and had multiple entry points and solution paths.

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7

The Mixing Juice Task

8 9 10 11 12

Arvind and Mariah attend summer camp. Everyone at the camp helps with the cooking and cleanup at meal times. One morning, Arvind and Mariah are in charge of making orange juice for all the campers. They plan to make the juice by mixing water and frozen orange juice concentrate. To find the mix that tastes best, they decide to test some recipes.

13

14

  1. Which recipe will make juice that is the most “orangey”? Explain.

15

  2. Which recipe will make juice that is the least “orangey”? Explain.

16

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Taken from Lappan et al. (2004).

Ms. Allen decided to launch the task by making juice from a can of concentrate. She thought that this demonstration would help students understand how concentrate and water combine to make juice. After asking students a few questions (e.g., How do you think that the juice would taste if we added more water or concentrate than the amount that the recipe suggests?), she told students that they would be looking at different recipes for orange juice and trying to determine which one was the most “orangey” and which one was the least “orangey.” While the students began working on the task in their groups, Ms. Allen walked around the room making sure that students were engaged in productive conversations. Her first impression was that students were not using strategies that would help them in finding and explaining the solution. For example, group 2 was in the process of finding the difference between the number of cups of concentrate and the number of cups of water in each mix, group 5 simply decided that the mix with the smallest amount of water had to be the most orangey, and group 3 was drawing pictures of each mix but not scaling the mixtures up or down to find a common basis for comparison. In addition, group 6, which had been trying to get her attention for several minutes, claimed to have no idea how to begin.

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34 35 36 37

38 39 40 41 42 43 44 45 46 47 48 49

With nearly half the class off to a rocky start, Ms. Allen decided to bring the class back together so that they could, as a class, figure out how to organize the given information. Ms. Allen worked with the class to create part-to-part ratios and part-towhole fractions for each of the four mixes (as shown in the chart below). Mix A

Mix B

Mix C

Mix D

Part-to-part ratios

2 c. concentrate 3 c. water

1 c. concentrate 4 c. water

4 c. concentrate 8 c. water

3 c. concentrate 5 c. water

Part-to-whole fractions

2 c. concentrate 5 cups in mix

1 c. concentrate 5 cups in mix

4  c. concentrate 12 cups in mix

3 c. concentrate 8 cups in mix

She then asked students how they could compare the fractions. Students suggested finding a common denominator or converting the fraction to a percent. Ms. Allen indicated that these methods were both good and told students to return to their groups to determine which mix was the most orangey and which one was the least orangey. Most of the groups that had previously struggled were now using one of the two identified strategies. Group 6 was multiplying 5 3 12 3 8 to find a common denominator; groups 3 and 5 were dividing the numerators of the fractions by the denominators to obtain a decimal, which they then converted to a percent by moving the decimal point two places to the right. Group 2 continued to focus on the difference until Ms. Allen suggested that one of the other methods would be better. After a few minutes, Ms. Allen asked students in group 6 to add their new fractions to the chart on the board and asked students in group 3 to add their percents to the chart. Mix A

Mix B

Mix C

Mix D

Part-to-part ratios

2 c. concentrate 3 c. water

1 c. concentrate 4 c. water

4 c. concentrate 8 c. water

3 c. concentrate 5 c. water

Part-to-whole fractions

2 c. concentrate 5 cups in mix

1 c. concentrate 5 cups in mix

4  c. concentrate 12 cups in mix

3 c. concentrate 8 cups in mix

Group 6’s fractions

192 480

 96 480

160 480

180

Groups 3’s percents

40%

20%

33.33%

37.5%

480

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50 51 52 53 54 55 56 57 58 59 60 61

Ms. Allen brought the whole class together and asked the following question: “Now that we have all this information, how can we tell which mix is the most or least orangey?” She asked students from group 6 which mix they believed was the most orangey. Sheila, a member of group 6, said, “we decided that mix A is most orangey because the 192/480 is the biggest fraction.” Justin, a member of group 3, jumped in: “Yeah, that is what we thought too — 40 percent is bigger than the other percents so that means it is orangiest.” Ms. Allen asked if everyone agreed. Heads nodded and thumbs went up, indicating consensus. Ms. Allen asked which was least orangey. The class replied in unison. “B!” With minutes left in the class, Ms. Allen told students that for homework they would complete a similar problem, in which they would compare mixtures of cranberry juice. She explained that they would find the details on their homework site.

(Margaret Smith and Victoria Bill [Smith and Bill 2015b] wrote this case for “Illuminating Student Thinking: Assessing and Advancing Questions,” an Institute for Learning Professional Development Session in Syracuse, New York, in 2015.)

Analyzing ATL 10.1: Reflecting on and Improving Teaching To deliberately reflect on one’s teaching, one must collect lesson artifacts to support reflection. These artifacts can include an audio or video recording of the lesson, samples of student work (collected or photographed), charts that the teacher or students produced, photographs of any board work, and lesson plans produced in preparation for the lesson. The detailed notes of a classroom observer — such as a principal, instructional coach, math supervisor, or colleague — can also offer evidence of what occurred during instruction and can help in reflecting on the lesson. In this example, Beverly Allen had three artifacts on which to draw in her reflection: a lesson plan prepared before the lesson, an audio recording of the lesson, and a photograph of the chart that the class created during the lesson. Ms. Allen might begin her analysis of the lesson by considering her goals for the lesson and what students appeared to learn related to those goals. She wanted her students to develop the ability to recognize proportional situations and to make comparisons among quantities by using ratios, fractions, decimals, rates, unit rates and percents (lines 2–4). In listening to the audio recording of the lesson and looking at the photograph of the chart that appeared on the board, she could conclude that students were able to determine that mix A was most orangey and mix B was least orangey by using fractions and percents to make comparisons among quantities (lines 51–56). Although only two groups (groups 3 and 6) presented and discussed the solutions, the agreement of the entire class with the conclusions reached (lines 56–59) could convince Ms. Allen that all students accepted and understood the solutions. Although students had arrived at a correct solution to the problem by using two of the methods that Ms. Allen had targeted in her lesson goals (using fractions and percents), the

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correct solution offers evidence of what students could do (a performance-goal outcome), not what they understood or even what they had learned (learning-goal outcomes). A further analysis of the lesson might lead Ms. Allen to recognize that these two strategies emerged only after she had helped the class create part-to-part ratios and part-to-whole fractions (lines 34–37) and explicitly asked students how to compare the fractions (line 38). After the suggestion that students either find a common denominator for the fractions or convert the fractions to percents, students had to do little more than perform a previously learned rule for manipulating the fractions. Ms. Allen might also have noticed that little to no discussion occurred regarding the meaning of the resulting fractions and percents in context, what a proportional situation is or is not, and what it meant for quantities to be in a proportional relationship. Ms. Allen may then have realized that she directed too much of what went on during the lesson and that she did too much of the thinking for students. In particular, the factors of maintenance and decline of high-level tasks (fig. 3.2) may have helped her recognize that she had removed the problematic aspects of the task when she led students through the production of the chart on the board (line 36–37) and validated the two methods that groups had suggested (lines 38–39, 46–47). She may have also noted that she did not give students sufficient time to wrestle with the demanding aspects of the task. As soon as she noticed that students were not using the strategies that she wanted them to use (lines 34–36), she brought the class together to create the chart (lines 36–37). Although Ms. Allen anticipated the correct approaches that students might use to solve the task, she had not considered the difficulties that they would encounter. If she had focused on both the correct approaches and the incorrect approaches during the planning process, she may have been better prepared to deal with her struggling students. She might then have reflected on how she could use the effective teaching practices to improve her instruction and student learning. She might have recognized that when students struggled with the task (lines 26–33), she needed to support their productive struggle without taking over the thinking for them. This support could have involved asking questions to elicit students’ thinking regarding what they understood about the mixing juice context and the relationship between juice and concentrate (e.g., How much juice does one recipe of each mix make? How would you make a larger batch of the mix if you wanted it to taste the same? How can you compare the mixes when the amounts of concentrate and water are different?). She could give probing guidance by encouraging students’ self-reflection and offering ideas on the basis of the students’ thinking. The representation that group 3 used (lines 30–31) did not seem helpful in moving them toward the instructional goals. Ms. Allen could have strategically provided a range of representational tools that would have encouraged the kind of reasoning that connected with her goals. For example, she could have also encouraged students to represent the situation in a concrete way by using two colors of tiles to represent concentrate and water that would have

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supported considering both the part-to-part and part-to-whole comparisons simultaneously. Many of these types of questions and suggestions could be prepared in advance of the lesson, as discussed in “The Case of Deborah Dyson” (chapter 8) and as shown in the Hexagon lesson plan (appendix A). Perhaps the most important takeaway from Ms. Allen’s reflection would be the realization that the goals of the lesson needed to guide her actions during the lesson. She needed to consider, every step of the way, how she was going to help students reach the goals without taking over the thinking for them. The bulk of the work that students did focused on finding equivalent fractions and converting fractions to percents instead of more directly focusing on proportional reasoning. The final discussion in the class did not bring out the thinking of students, and it did not explicitly address the goals that Ms. Allen had set. The student work produced during the class (both what they did and what they said) gave Ms. Allen almost no evidence that students had met her learning goals. With the realization that she had little evidence that students had met the learning goals, Ms. Allen was positioned to prepare her next lesson and to offer more support for her students, so that they would leave the lesson not only learning what she had intended but also feeling a sense of ownership over the ideas. Reflecting on a lesson should guide the planning of the next lesson, which in turn guides the enactment of the lesson and provides new teaching practices and evidence of students’ learning on which to reflect. The teaching cycle in figure 10.2 represents this process.

Teaching

Planning

Reflecting

Fig. 10.2. The teaching cycle

As with planning, reflecting on sequences of lessons, as well as individual lessons, is important to assess the teaching and learning that occurs over time. We encourage you to take advantage of opportunities for reflection individually, collaboratively with colleagues, or with the support of an instructional coach. As suggested by Hiebert and colleagues (2007), basing reflections on evidence of teaching practice and student learning (e.g., samples of student work,

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lists of questions asked during a lesson, audio or video recording, observer’s notes, exit tickets) helps teachers form hypotheses about the effects of teaching on students’ learning and propose improvements in teaching.

Instruction Must Be Equitable Chapters 2–9 have considered the specific ways in which the effective mathematics teaching practices can individually support equitable learning opportunities for students. Those chapters discussed the ways in which the practices can support the development of students’ identities as mathematical knowers and doers, how the practices can engage students who may have been historically marginalized, and the role of the teaching practices in creating publicly accessible thinking and reasoning spaces for students. Using the practices together and in combination over time creates powerful opportunities to support meaningful mathematical learning for all students, regardless of their history and prior experiences. Just as carefully analyzing students’ mathematical learning and development over the course of a year is important, documenting and reflecting on the development of students’ identities in the middle grades through the use of the effective mathematics teaching practices is equally important. For example, while teachers engage students with tasks that vary across the cognitive demand categories discussed in chapter 3, noting the ways in which students might engage with different types of tasks and how that engagement might shift over time is necessary. It is essential to monitor the ways in which students contribute to meaningful mathematics discourse and pose purposeful questions of their own during a lesson, as well as the ways in which they position themselves relative to their peers, the teacher, and the discipline of mathematics. A teacher might note, for example, that a student has strong mathematical reasoning skills when she or he works on her or his own but that she or he hesitates to share that thinking in small-group or whole-group discussions. Giving that student feedback that reinforces the validity of his or her thinking, pressing him or her to engage with other students, and creating opportunities for the student to pose and answer questions about mathematics can potentially change that student’s perspective of what it means to do mathematics and what is valued in the mathematics classroom. The development of students’ mathematical identities has powerful implications for their performance in mathematics class, and teachers should evaluate this development periodically and analyze it for evidence of change. Using the effective mathematics teaching practices consistently and in combination can provide meaningful opportunities for such an analysis. In addition to classroom-based equity concerns, systemic equity considerations in the middle grades also are important to challenge. The middle school is often the beginning of such practices as tracking and ability grouping, which separate students with previous track records of mathematical success from students who have not yet exhibited strong mathematical success. This separation is detrimental to student learning opportunities, because classes that aggregate 206   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

previously unsuccessful students almost always cover less mathematical content in more time. Teachers therefore must consider how they can use goals, tasks, discourse, and questioning to design instruction that can reach all students by providing students access to the task and support while they struggle productively. Teachers must become advocates for dismantling tracking systems that separate students and their opportunities to learn and must press their schools and districts for meaningful just-in-time supports that help all students succeed the first time in mathematics, instead of placing students into lower tracks or forcing them to repeat coursework. Too frequently, schools relegate historically marginalized students to classes in which they learn and practice procedures but never engage in the types of reasoning and problem-solving tasks described in this book. According to Principles to Actions (NCTM 2014, p. 60): Access and equity in mathematics at the school and classroom levels rest on beliefs and practices that empower all students to participate meaningfully in learning mathematics and to achieve outcomes in mathematics that are not predicted by or correlated with student characteristics. . . . Support for access and equity requires, but is not limited to, high expectations, access to high-quality mathematics curriculum and instruction, adequate time for students to learn, appropriate emphasis on differentiated processes that broaden students’ productive engagement with mathematics, and human and material resources. While you engage in the cycle of planning, teaching, and reflecting, it is essential to consider how your instruction will or did support the learning of each and every student. If each of your students is not engaging in reasoning and problem solving and making reasonable progress toward your lesson goals, then you need to reflect on the factors that may be affecting students’ lack of success and take corrective action. Unless all students are experiencing success in your classroom, you need to do more work.

Next Steps: Ongoing Work to Improve Practice Although you are at the end of this book, we hope that your journey in exploring the effective mathematics teaching practices in your own classroom is just beginning. If you have worked through the book alone, consider revisiting the activities with a colleague. If you have worked through the book with colleagues, consider how to continue to support one another to plan, teach, and reflect on your teaching in ways that highlight the effective teaching practices and support students’ learning. You might analyze additional episodes of teaching (live or in narrative or video form) or other artifacts of mathematics teaching (instructional tasks, sequences of tasks, and sets of student work) and discuss the extent to which the eight effective teaching practices are apparent in the lesson, tasks, student work, or teaching and

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the impact that they appeared to have on teaching and learning. Another opportunity for continued growth might include a book club in which a group of teachers can read about an effective teaching practice and then meet (face-to-face or virtually) to engage in a professional development module highlighting that practice (e.g., a chapter or activity in this book or resources available online through the Principles to Actions Toolkit). You might also consider planning lessons with colleagues that use the eight effective teaching practices as a framework. Invite your math coach (if you have one), department chair, administrator, or another instructional leader to participate. As suggested previously in this chapter, teams of teachers could collaborate to plan sequences of lessons and begin to develop a library of lessons that promote reasoning and problem solving or develop procedural fluency from conceptual understanding. When you have an opportunity to teach the coplanned lessons (or any lessons featuring tasks that promote reasoning and problem solving), observe one another and pay particular attention to the practices used in the lesson and how the practices do or do not support students’ learning. Several tools for analyzing tasks, task implementation, questions, and wholegroup discussion identified throughout this book could serve as a basis for collecting data that guide instructional improvements around the effective teaching practices. Instead of attending to all eight practices at once, you and your colleagues might select a focal practice (or practices) to be at the forefront of your teaching and reflecting, with connections to other practices acknowledged as playing supporting roles. When sharing your work with administrators or instructional leaders, you might engage them in considering how the effective teaching practices could become part of the school’s or district’s formal observation, feedback, and evaluation structures, as well as why this focus would be beneficial (e.g., because of the positive impact on students’ learning and engagement).

Final Thoughts As indicated previously in this chapter, enacting the effective mathematics teaching practices will improve with time, experience, thorough and thoughtful lesson planning, and deliberate reflection. Changing one’s teaching is hard work that takes sustained and meaningful effort over time. A simple introduction to the teaching practices does not mean that a teacher will immediately adopt and use them in ways that reflect ambitious mathematics instruction. In fact, these practices often bring to the surface important unproductive beliefs for teachers. The teacher must reflect on the following questions: • Do I truly believe that students can develop conceptual understanding before I introduce them to procedures?

• Do I believe that I can support meaningful student learning for a diverse group of students who have had very different mathematical experiences in their past?

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• What prompts me to intervene with a group rather than leave them to work, and are my interventions supporting productive struggle or taking it away? Teachers need time to reflect on their beliefs and current teaching practices and to make changes in practice that will better support students’ learning (Goldsmith and Schifter 1997). This process is a gradual one that is likely to occur over time. As a reader of this book, you have already taken an important step toward thinking more deeply about teaching and learning in your classroom. Persistence and commitment will help you continue the journey to instructional improvement. We close with a quote from The Teaching Gap (Stigler and Hiebert 1999, p. 179): The star teachers of the twenty-first century will be those who work together to infuse the best ideas into standard practice. They will be teachers who collaborate to build a system that has the goal of improving students’ learning in the “average” classroom, who work to gradually improve standard classroom practices. . . .The star teachers of the twenty-first century will be teachers who work every day to improve teaching — not only their own but that of the whole profession.

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Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

APPENDIX A

Lesson Plan for the Hexagon Task Learning Goals

Evidence

What understandings will students take away from this lesson?

What will students say, do, or produce that will give evidence of their understandings?

Students will understand that — 

• Look for equations that accurately depict the relationship between the train number and the perimeter of a train.

• an equation can describe the relationship between two quantities, the independent (x) and dependent (y) variables;

• different but equivalent equations can represent the same situation; and

• Look and listen for connections among representations. For example, in y 5 4x 1 2, 4 is the rate of change — the slope of the graph, the constant difference between y values in the table when x values are incremented by 1, and the number of sides that are added to the perimeter each time that another hexagon is added; 2 is the constant — the y-intercept, the value of the y when x 5 0, and the exterior vertical sides on the first and last hexagon in a train.

• connections can be made among tables, graphs, equations and contexts.

• Look and listen for students explaining how two or more formulas are equivalent.

• linear relationships have a constant rate of change between the quantities, are depicted graphically by a line, can be written symbolically as y 5 mx 1 b, where m is the constant rate of change and the slope of the line, and b is the value of the y-quantity when x 5 0 (i.e., the y-intercept);

    Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Task What is the main activity that students will be working on in this lesson? Students will be working on the Hexagon task, described in chapters 7 and 9.

Instructional Support —  Tools, Resources, Materials What tools or resources will be available to give students entry to — and help them reason through — the activity? Hexagon manipulatives, copies of the geometric sequence, graph paper

Prior Knowledge

Task Launch

On what prior knowledge and experience will students draw in their work on this task?

How will you introduce and set up the task to ensure that students understand the task and can begin productive work, without diminishing the cognitive demand of the task?

• The ability to define and use a variable in an equation or expression • Algebraic skills needed to transform equivalent expressions (e.g., distributive property, combining like terms) • An understanding of the definition of perimeter and how to identify sides that do and do not contribute to perimeter in a complex shape • The ability to use a table, graph, or equation to describe a linear relationship

• Show the first three hexagon trains on a document camera or interactive white board. • Ask students to build the first three trains at their desks and then privately write down their mathematical observations about the trains. (Provide sentence starters for struggling students as needed.) I noticed that ______. In the first train,______, but in the second and third trains, _______.

Essential Questions What are the essential questions that I want students to be able to answer over the course of the lesson? • What patterns do I see in the perimeters of hexagon trains as they get longer, and how can I express these patterns mathematically? • How can I show that two equations are equivalent? • What connections exist among the various representations of a linear relationship?

Train 1 is different from train 2 because_______. • Ask students to share their observations and record on chart paper what they observed. Attach students’ names to the observations for reference and to acknowledge individual contributions. Encourage students to reference the observations while they continue to work on the task. • Finally, ask for the perimeters of the first three trains, and address any lingering misconceptions about perimeter before groups begin work.

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Anticipated Likely Solutions and Instructional Supports What are the various ways that students might complete the activity? Be sure to include incorrect, correct, and incomplete solutions. What questions might you ask students that will support their exploration of the activity and bridge between what they did and what you want them to learn? These questions should assess what a student currently knows and advance her or him toward the goals of the lesson. Be sure to consider questions that you will ask students who cannot begin as well as students who finish quickly. Instructional Supports (Assessing and Advancing Questions)

Solutions

• What does the 2 mean? Where is it in the picture of the train?

Visual-geometric solution A

Students see the train as tops and bottoms with two ends. The top and bottom of each hexagon contribute four sides to the perimeter. The ends each contribute two sides to the perimeter.

• What does the 4 mean? Where is it in the fifth train? • How can you use the equation to solve for the perimeter of the tenth train? The hundredth train?

y 5 4 units (number of hexagons) 1  2 (ends) y 5 4x 1 2 • What does the 5 mean? Where is it in the train?

Visual-geometric solution B

• What does the “x” mean?

Students see the train as two hexagons on each end, each of which contribute five sides to the perimeter. The remaining hexagons — two less than the total number of hexagons in the train — each contribute four sides to the perimeter.

• Why are you subtracting 2, then multiplying by 4? • How can you use your formula to find the perimeter of the fifth train? The tenth train? The fifty-fifth train?

y 5 4 units (number of hexagons 2 2) 1 5 units (2 end hexagons) y 5 4 (x 2 2) 1 10

Appendix A: Lesson Plan for the Hexagon Task    213 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

• What does 6 represent?

Visual-geometric solution C

• What does (x 2 1) represent? Students multiply the number of hexagons by 6 since each hexagon has six sides (6x). Then the vertical sides on the inside of the hexagon trains — where two hexagons are connected — are taken away since these sides do not count in the perimeter. The number of vertical sides taken away is the number of hexagons minus one multiplied by two, that is 2(x 2 1).

• Why did you multiply (x 2 1) by 2? How can you use your formula to find the perimeter of the fifth train? The tenth train? The fifty-fifth train?

y 5 6 sides (number of hexagons) 2 6x 2 2(x 2 1) (x 2 1) the train number minus one times two vertical sides for each of these trains. Variations of this solution may be of the form (6x 2 2x) 1 2. Arithmetic-algebraic solution D

• What does you table tell you?

Students may build a table of values from which an algebraic formula can emerge.

• How much bigger is the perimeter of one train (e.g., train 4) than the perimeter of the train that came before it (e.g., train 3)? Is this true for any two consecutive trains?)

Train Number

Perimeter

1

 6

2

10

3

14

4

18

5

22

6

26

7

30

8

34

14

• How can you find the perimeter for a larger train? What if you do not know the perimeter of the train that preceded it?

14

• Can you show me where the 14 is in the picture of the trains?

14

• Is there a way to figure out how many 4s there are for each train?

14

Some students may use a recursive strategy of 14. If students use this recursive strategy or additive reasoning, acknowledge its usefulness for small train numbers and then ask them how that reasoning will help them find the perimeter of large train numbers.

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Arithmetic-algebraic solution E

• How did you get y 5 4x 1 2?

y 5 mx 1 b Students might create a linear equation by using the data in the table. The constant increase of 4 for every additional hexagon indicates that the formula will be linear and that 4 is the slope (m) in the equation y 5 mx 1 b. If y (perimeter of the train) is a function of x (the train number), then b 5 2. Possible Errors and Misconceptions Incorrect definition of perimeter Students may need to revisit the definition of perimeter: It is a closed curve bounding a plane area, the length of such a boundary, the distance around a figure. Defining perimeter is best in the launch of the task, but the teacher may need to reinforce the definition while groups work. English language learners: Distinguishing between a misconception and an Englishlanguage misunderstanding is important for all students, including English language learners. Have students use visuals when explaining their thinking. You should point to the hexagon trains when you ask clarifying questions.

• Where is the 4 in your table? Where is the 4 in the picture of the trains? • Where is the 12 in the table? Where is the 12 in the trains? • What would a graph of this equation look like? How do you know?

Instruction Supports (Assessing and Advancing Questions) • Can you show me how to find the perimeter of the fifth train by counting? • Imagine that each hexagon was a table. How many people could sit at five tables pushed together, as in train 5? Remind the student that the perimeter is the distance around the outside of the figure. • Can you recalculate the perimeters of the first five trains?

Assuming that the relationship is proportional

• Can you sketch the tenth train? Does the perimeter match what you calculated?

Students may find the perimeter for the fifth train and double it to find the perimeter of the tenth train. This method works only if the relationship is proportional (it is not proportional because of the two static end sides in the train pattern).

• When you doubled the fifth train, how many more sides did you add? • Take a look at train 2 and train 4. Can you double train 2 to get train 4? Can you use the visual pattern to explain why this method does not work?

Appendix A: Lesson Plan for the Hexagon Task    215 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Creating an incorrect formula based on recursion

• What do the x and y mean in your formula?

Students may identify the change of 14 for each successive train and write a formula similar to y 5 x 1 4.

• Can you show how your formula works for train 3? • Does your formula work if you do not know the perimeter of the previous train? • Show me the 14 on the visual pattern. How many 4s are counted in train 3? How can you account for all of them? • Are there sides in the perimeter that are not represented in your formula?

Creating a formula that is based solely on the numeric pattern

• Where do you see the 4 and the 2 in your equation in the picture of the traini?

Students may create the formula by using the values for several perimeters and may not connect the features of the formula back to the visual. While not an error or misconception per se, making connections among different representations and to the visual is an important outcome in this task.

• What would your equation look like when graphed? How are the 4 and the 2 evident in the graph?

Sharing and Discussing the Task Selecting and Sequencing

Connecting Responses

Which solutions should students share during the lesson?

What specific questions will you ask so that students — 

In what order? Why?

• make sense of the mathematical concepts that you want them to learn? • make connections among the different strategies and solutions that are presented? (Possible student responses are shown in green.)

Visual-geometric solution A y 5 4x 1 2

What do the numbers represent in 4x 1 2? Students will say that the 2 represents the ends of the train. The four units are the two on the top and the two on the bottom of each hexagon, and the x represents the train number.

216   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Visual-geometric solution B y 5 4 (x 2 2) 1 10

Why are you subtracting 2, then multiplying by 4? The number of hexagons on the inside of the train is always 2 less than the total number of hexagons in a train, so I need to subtract 2 from the total number of hexagons (x 2 2). Then I multiply by 4 because each inside hexagon has a perimeter of 4 (2 on the top and 2 on the bottom). What does the 110 represent? The five sides on the two end hexagons. How are the first two formulas related? 4x 1 2 and 4(x 2 2) + 10 are related because they both count the ends of the train (2 and 5) and then show the number of sides that are left to be counted (4x and 4[x 2 2]). Are these expressions equivalent? The perimeter for the hundredth train using both formulas is the same. The expressions are therefore equivalent. Which formula would you rather use to find the perimeter of the 3,132nd train number? Finding the perimeter of the 3,132nd train is easier by using the 4x 1 2 formula because it only requires multiplying 3,132 by 4 then adding 2.

Appendix A: Lesson Plan for the Hexagon Task    217 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Visual-geometric solution E 6x 2 2(x 2 1)

What does the 6x represent? The 6 represents the six sides to each hexagon and the x represents the number of hexagons. Why are you subtracting 2(x 2 1)? What does the x 2 1 represent? Can you connect this expression with the picture of the hexagon train? Two vertical sides of each hexagon need to be taken away when two hexagons are connected. There is one fewer point of connection between two hexagons than the number of hexagons in each train.

Making connections with the table: arithmetic-algebraic solution D

In the equation y 5 4x 1 2, where is the 4 in the table? Where is the 2? The 4 is the difference between the perimeters of two successive trains. Two would be the perimeter for the zero train. This would mean a train with just two vertical sides. But this does not make any sense.

Making connections with graphs

If you were to graph the perimeter of the trains as a function of the train number, what would that graph look like and why? Why is it a linear function? Students should plot points on the basis of the table produced and then make connections between the slope of the line and the hexagon train and describe the rate of change as “four sides are added for each hexagon” which gives m 5 4/1 5 4.

Homework/Assessment Ask students to do the following: (1) determine which train has a perimeter of 110 sides; and (2) decide which of the methods shared in class would be the best one to use in determining the train number, given a specific number of sides.

218   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

APPENDIX B

A Lesson Planning Template Learning Goals

Evidence

What understandings will students take away from this lesson?

What will students say, do, or produce that will give evidence of their understandings?

    Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Task What is the main activity that students will be working on in this lesson?

Instructional Support —  Tools, Resources, Materials What tools or resources will be available to give students entry to — and help them reason through — the activity?

Prior Knowledge

Task Launch

What prior knowledge and experience will students draw on in their work on this task?

How will you introduce and set up the task to ensure that students understand the task and can begin productive work, without diminishing the cognitive demand of the task?

Essential Questions What are the essential questions that I want students to be able to answer over the course of the lesson?

Anticipated Likely Solutions and Instructional Supports What are the various ways that students might complete the activity? Be sure to include correct, incorrect and incomplete solutions. What questions might you ask students that will support their exploration of the activity and bridge between what they did and what you want them to learn? These questions should assess what a student currently knows and advance her or him toward the goals of the lesson. Be sure to consider questions that you will ask students who cannot begin as well as students who finish quickly.

220   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Correct and Incomplete Solutions

Instruction Supports (Assessing and Advancing Questions)

Possible Errors and Misconceptions

Instruction Supports (Assessing and Advancing Questions)

Sharing and Discussing the Task Selecting and Sequencing

Connecting Responses

Which solutions do you want students to share during the lesson?

What specific questions will you ask so that students — 

In what order? Why?

 —  make sense of the mathematical ideas that you want them to learn  —  make connections among the different strategies and solutions that are presented?

Homework/Assessment

Adapted from Smith, Bill, and Hughes 2008.

Appendix B: A Lesson Planning Template    221 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

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232   Taking Action Grades 6–8 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Credits and Permissions The “Candy Jar Task” (pages 10, 30, 38, 53, 78–80) and “The Case of Marie Hanson” (pages 78–80) are reprinted/adapted by permission of the Publisher. From Margaret Schwan Smith, Edward A. Silver, and Mary Kay Stein. Improving Instruction in Rational Numbers and Proportionality: Using Cases to Transform Mathematics Teaching and Learning (Volume 1) New York: Teachers College Press. Copyright 2005 by Teachers College, Columbia University. All rights reserved. The “Unknown Triangles Activity Sheet” (Task D, page 62) is reprinted with permission from Illuminations, copyright 2008, by the National Council of Teachers Mathematics. All rights reserved. https://illuminations.nctm.org/uploadedFiles/Content/Lessons/Resources/6-8/ Bermuda-AS-UnknownTri.pdf “Mixing Juice” (pages 90, 201) is from the Connected Math Project Grade 7: Comparing and Scaling. © 1997 by Michigan State University, G. Lappan, J. Fey, W. Fitzgerald, S. Friel, and E. Phillips. Used by permission of Pearson Education, Inc. All rights reserved. The “Supreme Court Handshake” task (page 107) is adapted with permission from Illuminations, copyright 2008, by the National Council of Teachers of Mathematics. All rights reserved. http://illuminations.nctm.org/Lesson.aspx?id=2112

References   233 Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.