Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 9-12 0873539761, 9780873539760

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Taking Action Implementing Effective Mathematics Teaching Practices

Melissa Boston Frederick Dillon Margaret S. Smith Series Editor

Stephen Miller

Grades 9-12

more More resources available online www.nctm.org/more4u Look inside for your access code

Taking Action: Implementing Effective Mathematics Teaching Practices

in Grades 9–12 Melissa Boston Duquesne University Frederick Dillon Institute for Learning, University of Pittsburgh Margaret S. Smith Series Editor University of Pittsburgh Stephen Miller Akron Public Schools (Retired)

more www.nctm.org/more4u Access code: TAI15201

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 Sixth Printing August 2019

Library of Congress Cataloging-in-Publication Data Library of Congress Cataloging-in-Publication Data Names: Boston, Melissa. Title: Implementing effective mathematics teaching practices in grades 9-12 / Melissa Boston, Duquesne University [and three others]. Description: Reston, VA : National Council of Teachers of Mathematics, [2017] | Series: Taking action | Includes bibliographical references. Identifiers: LCCN 2016057989 (print) | LCCN 2017008090 (ebook) | ISBN 9780873539760 (pbk.) | ISBN 9780873539999 (ebook) Subjects: LCSH: Mathematics--Study and teaching (Secondary)--United States. Classification: LCC QA135.6 .I4645 2017 (print) | LCC QA135.6 (ebook) | DDC 510.71/273--dc23 LC record available at https://lccn.loc.gov/2016057989 The National Council of Teachers of Mathematics advocates for high-quality mathematics teaching and learning for each and every student.

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 9–12 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-forprofit 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.

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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. . . . . . . . . . . . . . . . . . . . . . . . . . 49 Chapter 5 Pose Purposeful Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter 6 Use and Connect Mathematical Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter 7 Facilitate Meaningful Mathematical 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.

Chapter 8 Elicit and Use Evidence of Student Thinking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Chapter 9 Support Productive Struggle in Learning Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Chapter 10 Pulling It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Appendix A Proof Task Lesson Plan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Appendix B A Lesson Planning Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Accompanying Materials at More4U ATL 2.2 ATL 2.2

Shalunda Shackelford Video Clip 1 Shalunda Shackelford Transcript 1

ATL 5.2 ATL 5.2

Jamie Bassham Video Clip Jamie Bassham Transcript

ATL 5.4 ATL 5.4

Debra Campbell Video Clip Debra Campbell Transcript

ATL 7.1 ATL 7.1

Shalunda Shackelford Video Clip 1 Shalunda Shackelford Transcript 1

ATL 7.2 ATL 7.2

Shalunda Shackelford Video Clip 2 Shalunda Shackelford Transcript 2

ATL 9.1 ATL 9.1 ATL 9.1 ATL 9.1

Jeff Ziegler Video Clip 1 Jeff Ziegler Transcript 1 Jeff Ziegler Video Clip 2 Jeff Ziegler Transcript 2

ATL 10.1 Wobberson Torchon Video Clip ATL 10.1 Wobberson Torchon Transcript

iv   Taking Action Grades 9–12 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 9–12 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 (co-chair), Melissa Boston, Fredrick Dillon, Amy Hillen, DeAnn Huinker, Stephen Miller, 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 high school classroom where students are working on a statistics unit in which they are fitting a function to data and then using the function they created to solve a problem. As the class begins, the teacher asks the class what they know about bungee jumping. Students indicate that it involves jumping off something high, like a bridge, while connected to an elastic cord. As one student explains, “You jump off and you fall, but then the cord springs you back up again and again.” The teacher then asks, “What happens if the cord is too short or too long?” Students respond that if the cord is too short, it might not be much fun because you wouldn’t fall very far and then you wouldn’t spring back much. But if the cord is too long, you could crash into the ground. The teacher then shows a YouTube video of a bungee jump at Victoria Falls (https://www.youtube.com /watch?v5UQFMy9Tz8dY), which captivates students’ attention and leaves many exclaiming, “Cool. I want to try that!” (This lesson is adapted from NCTM Illuminations, https://illuminations.nctm.org/Lesson.aspx?id52157.) The teacher then explains that they are going to model a bungee jump using Barbie dolls and rubber bands: “You will conduct an experiment, collect data, and then use the data to predict the maximum number of rubber bands that should be used to give Barbie a safe jump from 400 cm.” She provides each group of students with a Barbie and 20 rubber bands and indicates that other supplies they need (e.g., a large piece of paper, measuring tool, tape) can be found on the resource table at the back of the room. She then asks the class: “What is it you need to figure out?” Students respond that they need to figure out how far Barbie will fall as the number of rubber bands increases. The teacher then demonstrates how to attach the rubber band to Barbie’s feet and how to attach one rubber band to the next so that they all do it the same way.

    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.

As students begin their work, the teacher monitors the activity, intervening as needed to ensure that they are constructing the bungee cord correctly, using measuring tools appropriately, and keeping track of the data as they continue to add rubber bands to the bungee cord. As students conclude their data collection, the teacher reminds them that they need to create a scatterplot of the data and determine a line of best fit, which they could check using a web-based applet. (See http://illuminations.nctm.org/Activity.aspx?id54186 for an applet that can support this investigation.) She explains that once they have their line of best fit, they need to predict the maximum number of rubber bands they will need for Barbie’s 400 cm jump. When predictions have been finalized, the teacher explains that they are going to reconvene on the second floor stairwell, where she has already marked a height of 400 cm. She explains that they will test their conjectures with the number of rubber bands they predicted and determine how close they come to 400 cm. The class ends with students returning to the classroom and discussing as a group how accurate their predictions were, why some lines of best fit might have been more accurate than others, and what the slope and y-intercept of the equations actually mean in the bungee Barbie context.

A Vision for Students as Mathematics Learners and Doers The lesson portrayed in this opening scenario exemplifies the vision of school mathematics that the National Council of Teachers of Mathematics (NCTM) has been advocating for in a series of policy documents over the last 25 years (1989, 2000, 2006, 2009a). 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 cannot be answered quickly 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 other students’ perspectives. In the “bungee Barbie” scenario, students were faced with a problem, and they needed to collect and analyze data in order to solve it. All students could enter the problem by creating bungees of different lengths and dropping Barbie to see how far she fell, measuring the length of each jump, recording data, and constructing a scatterplot. Students were able to make a

2   Taking Action Grades 9–12 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.

guess at the line of best fit and then check their guess through the use of the applet. During the discussion, students reported on the accuracy of their predictions, reflected on why some predictions were better than others, and were pressed to consider what the line of best fit equation meant in the context of the bungee Barbie task. 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 was generated by the applet). The vision for student learning advocated for by NCTM, and represented in our opening scenario, has gained growing support over the past decade as 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 will require teachers to engage in what has been referred to as “ambitious teaching.” Ambitious teaching stands in sharp contrast to the well-documented routine found in many classrooms that consists of homework review and 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 (tell students what to do); We Do (practice doing it with students); and You Do (practice doing it on your own) (Santos 2011). In instruction that uses this approach, the focus is on learning and practicing procedures with limited connection to meaning. Students have limited opportunities to reason and problem-solve. While they may learn the procedure as intended, they often do not understand why it works and apply the procedure in situations where it is not appropriate. According to W. Gary Martin (2009, p. 165), “Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results” — that is, at times students get answers that make no sense, yet they have no idea how to judge correctness because they are mindlessly applying a procedure they do not really 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 provide an appropriate level of support to diverse learners. The goal is to ensure that each and every student succeeds in doing highquality academic work, not simply executing procedures with speed and accuracy. In our opening scenario, we see a teacher who 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, use of an applet for investigating line of best fit, partners with whom to exchange ideas), monitored students while

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they worked and provided support as needed, and orchestrated a discussion in which students’ contributions were key. However, what we don’t see in this brief scenario is exactly how the teacher is eliciting thinking and responding to students so that every student is supported 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 is intended to support teachers in meeting the challenge of ambitious teaching by describing and illustrating a set of teaching practices that will facilitate the type of “responsive and discipline-connected instruction” that is at the heart of ambitious teaching.

Support for Ambitious Teaching Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014) provides guidance on what it will take to make ambitious teaching, and the rigorous content standards it targets, a reality in classrooms, schools, and districts in order to support mathematical success for each and every student. At the heart of this book, Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 9–12, is a set of eight teaching practices that provide a framework for strengthening the teaching and learning of mathematics (see fig. 1.1). These teaching practices describe intentional and purposeful actions taken by teachers 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). Each of these teaching practices is examined in more depth through illustrations and discussions in the subsequent chapters of this book.

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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. 1.1. The 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 are provided to students, particularly to historically underserved and/or marginalized youth (i.e., students who are Black, Latina/

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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 will relate the eight effective teaching practices to specific equity-based practices that have been shown to strengthen mathematical learning and cultivate positive student mathematical identities (Aguirre, Mayfield-Ingram, and Martin 2013). Figure 1.2 provides a list of five equity-based instructional practices, along with brief descriptions.

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. 1.2. The Five Equity-Based Mathematics Teaching Practices (Adapted from Aguirre, Mayfield-Ingram, and Martin 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, Mayfield-Ingram, and Martin 2013, p. 14) define mathematical identities as 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 students see themselves as “not good at math” and approach math with fear and lack of confidence. Their identity, developed through earlier years of schooling, has the potential to affect their school and career choices. Anthony and Walshaw (2009, p. 8) argue:

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Teachers are the single most important resource for developing students’ mathematical identities. By attending to the differing needs that derive from home environments, languages, capabilities, and perspectives, teachers allow students to develop a positive attitude to mathematics. A positive attitude raises comfort levels and gives students greater confidence in their capacity to learn and to make sense of mathematics. The effective teaching practices discussed and illustrated in this book are intended to help teachers meet the needs of each and every student so that all students develop confidence and competence as learners of mathematics.

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 education professionals working with teachers would also benefit from the illustrations and discussions of the effective teaching practices. This book can be used in several different ways. Teachers can read through the book on their own, stopping to engage in the activities as suggested or trying things out in their own classroom. Alternatively, and perhaps more powerfully, teachers can work their way through the book with colleagues in professional learning communities, in department meetings, or when time permits. We feel that there is considerable value added by being able to exchange ideas with one’s peers. Teacher educators or professional developers could 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 interested in improving their instructional practices. In this book we provide a rationale for and discussion of each of the eight effective teaching practices and connect them to the equity-based teaching practices when appropriate. We provide examples and activities intended to help high school teachers develop their understanding of each practice, how it can be enacted 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 are presented throughout the book: Analyzing Teaching and Learning (ATL) and Taking Action in Your Classroom. Analyzing Teaching and Learning activities invite the reader to actively engage with specific artifacts of classroom practice (e.g., mathematics tasks, narrative cases of classroom instruction, video clips, student work samples). Taking Action in Your Classroom provides specific suggestions regarding how a teacher can begin to explore specific teaching practices in her or his classroom. The ATLs are drawn, in part, from activities found

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in the Principles to Actions Professional Learning Toolkit (http://www.nctm.org/PtAToolkit/). Additional activities beyond what can be found in the toolkit have been included to provide 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 are 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, provide images of aspects of effective teaching. As such they are examples to be analyzed rather than models to be copied. (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.) As 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 are posed, as well as make note of issues and new ideas that emerge. These written records can serve as the basis for your own personal reflections, informal conversations with other teachers, or 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 ways in which the effective teaching practices are interrelated. (Note that this order differs from the one shown in fig. 1.1 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 a chapter by asking the reader to engage in an Analyzing Teaching and Learning (ATL) 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 related 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 in which the reader is encouraged

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to purposefully relate the teaching practice being examined to her or his own classroom instruction. While we are presenting 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 are related and how they work in concert to support student 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 critical components of the teaching cycle and 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, the Case of Vanessa Culver, which takes you into Ms. Culver’s classroom where algebra 1 students are exploring exponential relationships. The case presents an excerpt from a lesson in which Ms. Culver and her students are discussing and analyzing the various strategies students used to solve the Pay It Forward task. (Note: This case, written by Margaret Smith [University of Pittsburgh], is based on a lesson planned and taught by Michael Betler, a student completing his secondary mathematics certification and MAT degree at the University of Pittsburgh during the 2013–2014 school year.) When new teaching practices are introduced in chapters 2–9, we relate the new practice to some aspect of the Case of Vanessa Culver. In so doing, we are using the case as a touchstone to which we can relate the new learning in each chapter. 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).

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Analyzing Teaching and Learning 1.1 Investigating Teaching and Learning in an Algebra Classroom As you read the Case of Vanessa Culver, consider the following questions and record your observations in your journal or notebook so that you can revisit them when we refer to the Pay It Forward task or lesson in subsequent chapters: • What does Vanessa Culver do during the lesson to support her students’ engagement in and learning of mathematics? • What aspects of Vanessa Culver’s teaching are similar to or different from what you do? • Which practices would you want to incorporate into your own teaching practices?

2

Exploring Exponential Relationships: The Case of Vanessa Culver

3 4 5 6 7 8 9

Ms. Culver wanted her students to understand that exponential functions grow by equal factors over equal intervals and that, in the general equation y 5 bx, the exponent (x) tells you how many times to use the base (b) as a factor. She also wanted students to see the different ways the function could be represented and connected. She selected the Pay It Forward task because it provided a context that would help students in making sense of the situation, it could be modeled in several ways (i.e., diagram, table, graph, and equation), and it would challenge students to think and reason.

10

The Pay It Forward Task

1

11 12 13 14 15 16 17 18

In the movie Pay It Forward, a student, Trevor, comes up with an idea that he thinks could change the world. He decides to do a good deed for three people, and then each of the three people would do a good deed for three more people and so on. He believes that before long there would be good things happening to billions of people. At stage 1 of the process, Trevor completes three good deeds. How does the number of good deeds grow from stage to stage? How many good deeds would be completed at stage 5? Describe a function that would model the Pay It Forward process at any stage.

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19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Ms. Culver began the lesson by telling students to find a function that models the Pay It Forward process by any means necessary and that they could use any of the tools that were available in the classroom (e.g., graph paper, chart paper, colored pencils, markers, rulers, graphing calculators). As students began working in their groups, Ms. Culver 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 the number of good deeds increase at each stage? How do you know?). When students struggled to figure out what to do, she encouraged them to try to visually represent what was happening at the first few stages and then to look for a pattern to see if there was a way to predict the way in which the number of deeds would increase in subsequent stages. As she made her way around the room, Ms. Culver also made note of the strategies students were using (see fig. 1.3) so she could decide which groups she wanted to have present their work. She decided to have the strategies presented in the following sequence. Each presenting group would be expected to explain what they did and why and to answer questions posed by their peers. Group 4 would present their work first since their diagram accurately modeled the situation and would be accessible to all students. Group 3 would go next because their table summarized numerically what the diagram showed visually and made explicit the stage number, the number of deeds, and the fact that each stage involved multiplying by another 3. Groups 1 and 2 would then present their equations one after the other. At this point Ms. Culver decided that she would give students 5 minutes to consider the two equations and decide which one they thought best modeled the situation and why. Below is an excerpt from the discussion that took place after students in the class discussed the two equations that had been presented in their small groups.

44 45 46

Ms. C.:

47 48

Ms. C.:

49 50 51

Missy:

52 53 54

Ms. C.:

So who thinks that the equation y 5 3x best models the situation? Who thinks that the equation y 5 3x best models the situation? [Students raise their hands in response to each question.] Can someone explain why y 5 3x is the best choice? Missy, can you explain how you were thinking about this?

Well, group 1 said that at every stage there are three times as many deeds as the one that came before it. That is what my group (4) found too when we drew the diagram. So the “3x” says that it is three times more. Does everyone agree with what Missy is saying? [Lots of heads are shaking back and forth indicating disagreement.] Darrell, why do you disagree with Missy?

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I agree that each stage has three times more good deeds than the previous stage, I just don’t think that y 5 3x says that. If x is the stage number like we said, then the equation says that the number of deeds is three times the stage number — not three times the number of deeds in the previous stage. So the number of deeds is only 3 more, not 3 times more.

55 56 57 58 59

Darrell:

60

Ms. C.:

61 62 63

Kara:

64 65 66

Chris:

67 68

Ms. C.:

69 70 71

Devon:

72 73 74 75 76

Angela:

77 78 79

Ms. C.:

80 81 82 83 84 85 86 87 88 89

At this point Ms. Culver asked group 5 to share their graph and proceeded to engage the class in a discussion of what the domain of the function should be, given the context of the problem. The lesson concluded with Ms. Culver telling the students that the function they had created was called exponential and explaining that exponential functions are written in the form of y 5 bx. She told students that in the 5 minutes that remained in class, they needed to individually explain in writing how the equation related to the diagram, the table, the graph, and the problem context. She thought that this would give her some insight regarding what students understood about exponential functions and the relationship between the different ways the function could be represented.

Other comments?

I agree with Darrell. y 5 3x works for stage 1, but it doesn’t work for the other stages. If we look at the diagram it shows that stage 2 has 9 good deeds. But if you use the equation, you get 6 not 9. So it can’t be right.

y 5 3x is linear. If this function were linear, then the first stage would be 3, the next stage would be 6, then the next stage would be 9. This function can’t be linear — it gets really big fast. There isn’t a constant rate of change. So let’s take another look at group 3’s poster. Does the middle column help explain what is going on? Devon?

Yeah. They show that each stage has 3 times more deeds than the previous one. For each stage, there is one more 3 that gets multiplied. That makes the new one three times more than the previous one.

So that is why I think y 5 3x best models the situation. Stage 1 had 3 good deeds, stage 2 had three people each doing three deeds so that is 32, stage 3 had 9 people (32) each doing 3 good deeds, so that is 33. The x tells how many 3’s are being multiplied. So as the stage number increases by 1, the number of deeds gets three times larger.

If we keep multiplying by another three like Angela described, it is going to get big really fast like Chris said. Chris also said it couldn’t be linear, so take a minute and think about what the graph would look like.

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Group 1 (equation—incorrect)

Group 2 (a table like groups 6’s & 7’s and an equation)

y 5 3x

y 5 3x

At every stage there are three times as many good deeds as there were in the previous stage.

3

3

3

3

3

x (stages)

3

3

3

So the next stage will be 3 times the number there in the current stage so 27 3 3. It is too many to draw. You keep multiplying by 3.

y (deeds)

1

3

3

2

333

9

3

33333

27

4

33333 33

81

5

333333 333

243

Group 5 (a table like groups 6’s & 7’s and a graph)

Group 4 (diagram)

3

Group 3 (a diagram like group 4’s and a table)

Groups 6 and 7 (table) x (stages)

y (deeds)

1

3

2

9

3

27

4

81

5

243

Fig. 1.3. Vanessa Culver’s students’ work

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Moving Forward There are many noteworthy aspects of Ms. Culver’s instruction and examples of her use of the effective teaching practices. However, we are not going to provide an analysis of this case here. Rather, as you work your way through chapters 2 through 9, you will revisit the case of Ms. Culver and consider the extent to which she engaged in the focal practice and the impact it appeared to have on student learning and engagement. As you progress through the chapters, you may want to return to the observations you made during your initial reading of the case and consider the extent to which you are now seeing things in 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 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 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. Goals should set the course for a lesson and provide support and direction for teachers’ instructional decisions. For example, the selection of instructional tasks should follow from the stated goals, hence providing a road map for the lesson. Goals can help guide teachers’ decision making during a lesson, such as determining which questions to ask or identifying which student-generated strategies and ideas to pursue. Additionally, goals are part of a progression of learning. Goals are a key part in determining what tasks are relevant to the planned learning progression, what representations might be highlighted during a lesson or sequence of lessons, and what will be the focus of mathematical discourse in a lesson. In this chapter, you will — • explore and compare different goal statements created for a lesson on exponential functions;

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

• review key research findings related 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 (ATL) activity, make note of 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. Once you have written down or shared your ideas, then read the analysis where we offer ideas relating the Analyzing Teaching and Learning activity to the focal teaching practice.

Exploring Lesson Goals We begin the chapter by asking you to engage in Analyzing Teaching and Learning 2.1. Here you will compare two different goal statements that might be written for a lesson on exponential functions.

Analyzing Teaching and Learning 2.1 Comparing Goal Statements   1. Review goal statements A and B and consider these questions: • How are they the same and how are they different? • How might the differences matter? Goal A: Students will identify a function of the form y 5 bx as an exponential function where x is the exponent and b is the base. Students will be able to substitute values for x and b to evaluate exponential functions. Goal B: Students will understand that exponential functions grow by equal factors over equal intervals and that, in the general equation y 5 bx, the exponent (x) tells you how many times to use the base (b) as a factor.   2. If needed, read (or reread) the Case of Vanessa Culver in chapter 1. In what ways does goal B (lines 3–5 in the case) align with Ms. Culver’s teaching practice?

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Analysis of ATL 2.1: Comparing Goal Statements ATL 2.1 asks you to consider how goals A and B are similar and different. While both goals address the same mathematical content, goals A and B expect much different types of mathematical work and thinking from students. To meet goal A, students need to identify y 5 bx as an exponential function, substitute values into the exponential function, and evaluate the function. Notice that the underlined verbs imply memorization (e.g., identifying a function of a given form) and executing procedures (e.g., substituting and evaluating). Tasks aligned with goal A might provide students with values for x and b (perhaps embedded in a word problem) and ask students to create and evaluate exponential functions. Prior to completing such tasks, students are often provided with the definition and form of an exponential function. While germane to students’ mathematical learning, these skills do not invoke conceptual understanding, thinking, and reasoning around exponential growth and the behavior of exponential functions. In goal B, students are expected to understand exponential growth and what it means for x to be the exponent and b to be the base in an exponential function y 5 bx. Such an understanding is essential for recognizing when real-world or mathematical relationships can be modeled with exponential functions. Tasks aligned with goal B, such as the Pay It Forward task in Ms. Culver’s lesson, provide scenarios where students can model and understand exponential growth in a variety of ways on the basis of their prior knowledge of the growth of linear functions and ways of representing linear relationships. The differences between goals A and B matter because they require very different mathematical activity from students, which in turn generates differences in the nature of students’ mathematical learning.

Exploring How Lesson Goals Support Teaching and Learning in the Case of Vanessa Culver Ms. Culver identified goal B as her intention for students’ learning in the lesson featured in the case. She used the mathematical goal to focus learning in several ways. Ms. Culver selected a task that would help students meet her goals for students’ learning (lines 10–18). The Pay It Forward task provided a context that supported students in making sense of exponential functions, could be modeled in several ways (that would make exponential growth apparent), and promoted thinking and reasoning. Ms. Culver made tools available to help students explore exponential growth, such as graph paper and graphing calculators (lines 21–22). She asked questions to help students attend to how the pattern was growing, such as, “How do the number of good deeds increase at each stage? How do you know?” (lines 24–25). Ms. Culver sequenced the presentations to build up students’ understanding of exponential growth (lines 32–39), particularly in selecting different representations of exponential growth and progressing

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from diagrams (group 4) to tables (group 3) to equations (groups 1 and 2) to graphs (group 5). In the whole-group discussion, Ms. Culver provided opportunities for a number of students to explain exponential growth using the context of the problem and representations shared by different groups (lines 44–89). Students used their developing understanding of exponential growth to determine which function ( y 5 3x or y 5 3x) correctly modeled the Pay It Forward situation. Hence, Ms. Culver’s mathematical goals for the lesson provided direction for determining what task to use, what questions to ask throughout the lesson, and how to structure the wholegroup discussion in order to focus students’ learning on understanding exponential growth. Having goals (and a task) that focused her instructional decisions on promoting students’ understanding of mathematics, rather than rote procedures or facts without understanding, was an essential first step.

Considering How Lesson Goals Support Teaching and Learning With ATL 2.2, we will go into the classroom of Shalunda Shackelford, where students are examining graphs that model the speed of a bike and truck over a given time period (see the next page).

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The Bicycle and Truck Task

Distance from start of road (in feet)

A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.

Time (in seconds)   1. Label the graphs appropriately with B(t) and K(t). Explain how you made your decision.   2. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description.   3. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words.   4. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack. Taken from Institute for Learning (2015a). Lesson guides and student workbooks are available at ifl.pitt.edu.

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Note that some inconsistencies exist in how the graphs model the real-life movement of a bike and truck (e.g., a vehicle would not come to an immediate stop at 9 seconds). Consistencies and inconsistencies in how the graphs model the real-life movement of a bike and truck can foster productive mathematical discussion among students. A teacher might ask students to identify ways in which the graphs are not realistic and discuss why, or he or she might ask them to consider how they would change the graph to better model the real-life movement of a bike and a truck. A graph that more realistically depicts the movement of a bike and truck is available at http://www.nctm.org/PtAToolkit. Ms. Shackelford has three content goals for her students. She wants them to understand the following:   1. The language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values.   2. Context is important for interpreting key features of a graph portraying the relationship between time and distance.

  3. The average rate of change is the ratio of the change in the dependent variable to the change in the independent variable for a specified interval in the domain. According to Principles to Actions (NCTM 2014), “Teachers need to be clear about how the learning goals relate to and build toward rigorous standards” (p. 12). It is important to note that the Bike and Truck task fits within a sequence of lessons on creating and interpreting functions. Ms. Shackelford incorporated the lessons into her curriculum to engage students with mathematical ideas aligned with the mathematics standards adopted by her state. Specifically, the Bike and Truck task provides students with opportunities to explore rigorous standards related to functions and modeling, as identified in figure 2.1.

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Examples of Rigorous State and National Standards

Connection to the Bike and Truck Task

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Questions 1–3 ask students to interpret key features of the graph portraying the relationship between time and distance traveled for a bike and a truck.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

Question 3 asks students to use the domain of the function to determine which vehicle was the first to reach 300 feet.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*

Question 4 asks students to consider the average rate of change for the bike and truck on the basis of the graph.

Question 4 also provides opportunities to relate the domain of the function to its graph as students consider changes in time when determining average rate of change.

Question 1 may prompt intuitive discussions of rate of change as students identify which graph represents the bike (steady rate) and which graph represents the truck. Question 3 may also encourage intuitive discussions of average rate of change as students determine whether (and why) the bike or the truck arrived at 300 feet first.

*Indicates that the standard provides opportunities for mathematical modeling

Fig. 2.1. Aligning the Bike and Truck task to rigorous standards

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Prior to the Bike and Truck lesson, Ms. Shackelford has been working on facilitating mathematical discussions and targeting mathematical practices in very deliberate ways. Hence, in addition to content goals, Ms. Shackelford also has process goals for her students to engage in mathematical discourse, problem solving, mathematical argumentation, and mathematical modeling. The video clip from Ms. Shackelford’s classroom has three segments of discussion (see More4U at nctm.org/more4u for the video clip). First, Ms. Shackelford introduces a common misconception, framing it as a question from her “imaginary friend Chris.” She draws students’ attention to the “flat” horizontal portion on the graph of the truck K(t) between 9 and 12 seconds. Ms. Shackelford explains that “Chris” thinks that the truck was traveling on a “straight path” during that interval, and she asks for students to come to the front of the room and explain why they agree or disagree with Chris. Second, a student presents an explanation for question 3 (page 19). Third, while the class verbally agreed with the student’s explanation for question 3, Ms. Shackelford checks for additional questions and misunderstandings.

Analyzing Teaching and Learning 2.2 more The Case of Shalunda Shackelford Watch the video clip of the discussion of the Bike and Truck task in Ms. Shackelford’s classroom. Consider the extent to which the content and process goals she has established for the lesson are evident in the discussion and what she did to keep students focused on the main points of the lesson. Identify specific instances in which Ms. Shackelford makes an instructional decision that is directly related to her goals and what students say or do as a result of that move. You can access and download the video and its transcript 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 2.2 Ms. Shackelford used the mathematical content and process goal to focus learning. She selected a task and asked additional questions that would address content goals 1, 2, and 3 and maintain students’ perseverance in solving and making sense of problems (process goal). (Note that evidence of content goal 3 is not present in this video clip. When you view clip 2 in chapter 7, watch for Ms. Shackelford’s students to discuss average rate of change.) Ms. Shackelford introduced a misconception (e.g., interpreting the graph as the path of the truck; lines 1–6) so that students must consider how time and distance vary together (content goal 1). Ms. Shackelford also pressed students for their misconceptions regarding how the graphs model

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which vehicle “got there first (lines 68–90).” These instructional moves provided opportunities for students to use the context to interpret key features of the graph (content goal 2) and to consider how the path of the bike and truck are modeled with mathematics. Modeling with mathematics (process goal) also occurred as students used mathematical representations (graph) and concepts/ideas (rate of change, domain, range) to make sense of the path of the bike and truck, the associated changes in time and distance, and how this related to speed. (Note that some features of the graph do not model the real-life movement of a bike and truck, and asking students to identify and explain these inconsistencies could serve these goals as well.) Ms. Shackelford supported students as they persisted in their problem-solving and sensemaking efforts (process goal) by pressing them to clarify their own thinking and understanding and to ask questions if they disagreed or did not understand. For example, even after students expressed verbal agreement with one student’s use of the graph to explain question 3, Ms. Shackelford asked students to share what still wasn’t making sense to them about the situation (lines 66–68). Next, Ms. Shackelford created an opportunity for students to present and defend opposing opinions (process goal) when Jacobi and Charles came to the front of the room (lines 7–52) and when students were asked to explain what ideas they agreed or disagreed with at the end of the clip (lines 73–90). She positioned students to construct viable arguments, explain and defend their ideas, and critique the reasoning of their classmates. Finally, Ms. Shackelford provided opportunities for students to engage in mathematical discourse (process goal), including defending their position; she asked questions and pressured students for explanations and meaning (e.g., “You agree, why?”; lines 23, 88). In the video clip, we see Ms. Shackelford make several purposeful moves aligned with her goals for the lesson. According to Principles to Actions (NCTM 2014), “The establishment of clear goals not only guides teachers’ decision making during a lesson but also focuses students’ attention on monitoring their own progress toward the intended learning outcomes” (emphasis added; p. 12). Ms. Shackelford expects students to monitor their own learning, and she communicates this by pressing students to express whether they agree or disagree with “Chris” (e.g., content goal 1, understanding how two quantities vary together) and what they do not understand following the explanation of question 3 (e.g., content goal 2, using the context to interpret key features of the time/distance graph). In these ways, the goals also support students’ monitoring of their own learning and understanding.

Establish Mathematics Goals to Focus Learning: What Research Has to Say The cases in this chapter provide examples of teachers using goals to inform instructional decisions and focus students’ learning. Teachers’ goals addressed important aspects of students’ understanding of mathematics and aligned with state and national standards. It is important

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to note that teachers’ goals (and the tasks selected to accomplish those goals) did not exist in isolation (e.g., a fun or interesting task; an activity for a Friday or the day before winter break). Rather, the goals supported teachers’ decision making because they were embedded within sequences of learning progressions (Daro, Mosher, and Corcoran 2011) and intended to develop students’ understanding of important mathematical ideas (Charles 2005). According to Principles to Actions (NCTM 2014), goals connected to learning progressions and big mathematical ideas help teachers consider how to support students as they make transitions from prior knowledge to more sophisticated mathematical understandings (Clements and Sarama 2004; Sztajn et al. 2012). Stein (2017) indicates that goals impact teaching and learning (1) by guiding teachers’ instructional decisions and (2) by impacting the nature and focus of students’ work. First, as illustrated by the cases in this chapter, goals for students’ mathematical learning should support teachers’ decisions in selecting tasks, asking questions, and framing the direction of whole-group discussions. In student-centered lessons, students often suggest or develop a wide array of mathematical ideas and strategies. Mathematical goals can help teachers determine which ideas and strategies to pursue and serve as “reference points” for guiding mathematical discussions (Ball 1993; Stein 2017). In fact, goals are identified as an important first step for teachers in considering how to select and sequence students’ mathematical work and ideas when orchestrating mathematics discussions (Stein et al. 2008). Hence, teachers with a sound understanding of instructional goals and the multiple pathways that students can (and cannot) take to reach them are better equipped to support students’ learning of mathematics (Leinhardt and Steele 2005). Second, research indicates that teachers’ use of goals to guide instruction supports students’ ability to monitor their own mathematical learning (Clarke, Timperley, and Hattie 2004; NCTM 2014; Zimmerman 2001). When teachers explicitly refer to goals during a lesson, students are better able to self-assess and focus (or refocus) their learning, which 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).

Promoting Equity by Establishing Mathematics Goals to Focus Learning Teachers’ use of goals to focus instructional decisions supports students’ learning of mathematics in general, and specific types of goals can enhance opportunities to learn mathematics for traditionally marginalized students. Principles to Actions identifies “high expectations” as one of several required supports for promoting access and equity in learning meaningful mathematics. Research indicates significant gains in students’ learning and reductions in achievement gaps when teachers communicate clear expectations, express

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challenging but attainable goals, and create an environment in which students feel supported to attain high goals (Boaler and Staples, 2008; Marzano 2003; McTighe and Wiggins 2013). “High expectations” do not imply difficult or complex mathematical procedures and concepts beyond students’ reach. Rather, goals and expectations should establish learning progressions that build up students’ mathematical understanding, increase students’ confidence in their own ability to do mathematics, and, in doing so, support students’ identities as mathematical learners. Too often, instructional tracking and deficit beliefs regarding the mathematical abilities of students of color, students who are poor, or students for whom English is not their first language lead to different opportunities to engage with interesting and rigorous mathematical content ( Jackson et al. 2013; Phelps et al. 2012; Walker 2003). When students’ mathematical abilities are underestimated, students [receive] fewer opportunities to learn challenging mathematics. Low-track students encounter a vicious cycle of low expectations: Because little is expected of them, they exert little effort, their halfhearted efforts reinforce low expectations, and the result is low achievement (Gamoran 2011). (NCTM 2014, p. 61) Furthermore, if mathematical goals and expectations focus primarily on rote skills and procedures, without attention to meaningful mathematics learning, low-track and marginalized students will not develop a deep understanding of mathematics (Ellis 2008; Ellis and Berry 2005). Instead, instructional goals (and the tasks aligned with those goals) should promote students’ reasoning and problem solving (e.g., goal B in ATL 2.1; Ms. Shackelford’s content and process goals). Such goals communicate the belief and expectation that all students are capable of participating and achieving in mathematics; in other words, such goals communicate a growth mindset (Boaler 2015; Dweck 2006). Hence, goals can support equitable instruction by setting clear and high expectations, promoting students’ mathematical reasoning and problem solving, and communicating the growth mindset that all students are capable of engaging in meaningful mathematical activity.

Relating “Establish Mathematics Goals to Focus Learning” to Other Effective Teaching Practices Establish mathematics goals to focus learning is closely connected to several other effective teaching practices. Since this is the first chapter in the book that is focused on an effective teaching practice, in this section we connect establish mathematics goals to focus learning to other effective teaching practices. Specifically, we discuss the synergy between goals and tasks, questions, and facilitating discourse. In subsequent chapters, the connections between the focal effective teaching practice and other practices are woven throughout the chapter. Establish Mathematics Goals to Focus Learning    25 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.

Implement Tasks that Promote Reasoning and Problem Solving

If goals represent the destination for students’ mathematical learning from a given lesson, then tasks are the vehicles that move students from their current understanding toward those goals. Tasks provide opportunities for students to learn and understand the mathematical content and processes necessary to achieve learning goals. Specifically, goals for students’ reasoning and problem solving require tasks that promote reasoning and problem solving. (Such tasks are discussed in chapter 3.) If the tasks students encounter in mathematics class only provide procedural practice, students are not going to attain goals for thinking, reasoning, and understanding mathematics (Stein et al. 2009). Finally, if tasks and goals align and focus on promoting students’ reasoning and problem solving, using goals to inform instructional decisions could also support implementing tasks in ways that provide and maintain students’ opportunities for reasoning and problem solving throughout a mathematics lesson.

Pose Purposeful Questions

With clear goals in mind, teachers can ask questions that prompt students to engage with the mathematical ideas aligned with those goals. (Posing purposeful questions is discussed in chapter 5.) Knowing the goals for a lesson can help teachers craft questions before and during a lesson by considering (or responding to) students’ specific ideas, strategies, or misconceptions. In this way, goals can support lesson planning just as they support instructional decisions during a lesson. Teachers’ questions can help focus students’ work and thinking on important aspects of the task or mathematics, thus supporting students’ attainment of the lesson goals. Students’ responses allow teachers to assess students’ progress toward the intended goals and to determine next instructional steps. Teachers’ questions can also support students’ selfassessment of their own progress.

Facilitate Meaningful Mathematical Discourse

Goals can inform instructional planning and decisions around facilitating mathematical discourse. A clear focus on goals provides a clear frame for the mathematical ideas to be elicited during the whole-group discussion and can help teachers determine what strategies, ideas, representations, and so forth to select for presentation and discussion. (Facilitating meaningful mathematical discourse is discussed in chapter 7.) Having goals in mind also supports teachers’ assessment of students’ learning by enabling them to know what to look and listen for as evidence of students’ progress toward the goals.

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Key Messages • Establish clear goals to focus students’ learning, and explicitly communicate these goals to students. • Establish goals that promote mathematical understanding, reasoning, and problem solving. • Create goals within learning progressions that build students’ understanding of important mathematical ideas. • Use goals to guide instructional decisions and focus students’ learning.

• Support students’ use of goals to monitor and assess their own progress.

Taking Action in Your Classroom: Establishing Mathematics Goals To Focus Learning The Taking Action in Your Classroom activity provides an opportunity to apply some of these key findings in your classroom.

Taking Action in Your Classroom 2.1 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 you wanted students to learn is explicit. • How might your more explicit goal statement guide your decision making before and during the lesson? Consider a lesson you will teach in the near future and the current learning goals for this lesson. • What type of mathematical work and thinking does the goal expect of students: memorization and procedures or thinking, reasoning, and sense making? If needed, rewrite the goal to require thinking, reasoning, and sense making in students’ mathematical activity. • Consider how to use your goal statement to guide your instructional decisions before and during the lesson. Consider whether the new goal aligns with the task you planned to use or whether a new task is needed.

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CHAPTER 3

Implement Tasks That Promote Reasoning and Problem Solving The Analyzing Teaching and Learning 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 strategies. Tasks set the stage for a lesson and for the enactment of the other effective teaching practices. The tasks a teacher selects must encourage thinking, reasoning, and problem solving; have multiple pathways; and allow students to decide which representations to use. Successful implementation of tasks includes helping students make connections among different representations, posing questions that foster students’ understanding, supporting productive struggle, and facilitating discourse. In this chapter, you will — • solve and compare mathematical tasks;

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

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

For each Analyzing Teaching and Learning (ATL), make note of 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

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and discuss your responses and ideas with colleagues. Once you have written down or shared your ideas, then read the Analysis where we offer ideas relating the Analyzing Teaching and Learning activity to the focal teaching practice.

Comparing the Cognitive Demands of Tasks In ATL 3.1, you will solve and compare two mathematical tasks that focus on exponential functions. As you solve each task, consider the strategies you are drawn to and the extent to which the strategies you use are ones that make sense to you and that you can explain mathematically.

Analyzing Teaching and Learning 3.1 Comparing Two Tasks   1. Solve tasks A and B in figure 3.1. Consider solution paths that are suggested by the tasks. If no path is suggested, try to come up with at least one other way to solve the task.   2. Compare the two tasks. How are they the same? How they are different?   3. Which task is more likely to promote reasoning and sense making? Why?

Task A: The Petoskey Population The population of Petoskey, Michigan, was 6,076 in 1990 and was growing at the rate of 3.7% per year. The city planners want to know what the population will be in the year 2025. Write and evaluate an expression to estimate this population. (Source: Holt Algebra 2 [Schultz et al. 2004, p. 415]).

Task B: Pay It Forward In the movie Pay It Forward, a student, Trevor, came up with an idea that he thought could change the world. He decided to do a good deed for three people, and then each of the three people would do a good deed for three more people and so on. He believed that before long there would be good things happening to billions of people. At stage 1 of the process, Trevor completed three good deeds. How does the number of good deeds grow from stage to stage? How many good deeds would be completed at stage 5? Describe a function that would model the Pay It Forward process at any stage.

Fig. 3.1. Two tasks that involve exponential relationships

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Analysis of ATL 3.1: Comparing Two Tasks The two tasks in figure 3.1 address the same mathematical topic, exponential relationships. Both tasks are set in a context, and students could perceive each task as a plausible real-world situation. In both tasks, students are provided with an initial value and a growth rate and asked to develop a formula to represent the situation (e.g., “Write an expression” or “Describe a function”). Even with these similarities, solving each task requires very different types of thinking from students. In Task B, the Pay It Forward task, students are given two pieces of information. At the first stage there are three good deeds; the three people for whom good deeds were done will perform three additional good deeds each. The action in the problem, and the prompt for students to consider the growth from stage to stage, supports students in making sense of and modeling the exponential growth situation. Pay It Forward is an open-ended task and does not suggest a solution path. Students are free to develop drawings, graphs, or tables in their initial approaches to solving the problem, and the context of the problem makes it likely that students might use one or more of these representations. In solving the task, students would be expected to describe the growth, determine how many good deeds occur at stage 5, and generalize this growth into a function. While the task requests a function to model the process, it doesn’t specify how the function may be written, freeing students to use symbols or words. In Task A, the Petoskey Population task, students are also given an initial value and a growth rate — each piece of information needed to create the exponential expression. The prompts for the task request a very specific format for students’ solutions: write and evaluate an expression. Students can solve the task by fitting the given values into the correct places in an exponential growth expression and evaluating the expression for t 5 35. While the Petoskey Population task does not specifically suggest a solution path, the context and prompts in the problem do not support students in making sense of and modeling exponential growth. Students are not asked to consider the growth of the situation or to generalize the growth pattern into an expression. Task A is not an introductory problem. It presupposes that the students already know about exponential functions and probably that they have examined growth functions. Frequently, word problems such as the Petoskey Population task occur within traditional textbook lessons after students have been exposed to the formula, requiring students to produce answers rather than engage in reasoning and problem solving. The Pay It Forward task makes different entry levels possible for students. For students who struggle, drawing pictures or making a tree diagram may be a good starting strategy. More adept students may think of multiplication and exponents right away. For students who finish the initial task rapidly, there is the implied question “When will there be one billion good deeds?” to consider. When students begin to discuss their solutions, the teacher will be able to sequence the solutions to help students make sense of common errors. One common error occurs when students think that the process is additive, particularly following students’

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exploration of linear relationships. Even if no students use this approach in a given lesson, the teacher can still ask about an additive (linear) function (e.g., if each of the three people did one good deed each at each stage) and compare it with the multiplicative (exponential) process in Pay It Forward to gain an understanding of how linear and exponential functions differ. Students would have an opportunity to recognize that each stage is multiplied by a consistent factor in an exponential relationship. As an introduction to exponential functions, Pay It Forward can support students in developing the general form of an exponential growth function through their own reasoning and problem solving. The Pay It Forward and Petoskey Population tasks have many similar surface-level features but differ in the types of thinking they require from students. These differences matter because they provide different opportunities for students’ learning of mathematics. Smith and Stein (1998) developed the Task Analysis Guide in figure 3.2 to classify mathematical tasks according to the level and types of thinking a task requires from students. Based on the criteria listed in the guide, we would classify Pay It Forward as a “doing mathematics” task and Petoskey Population as a “procedures with connections” task. We will use the Task Analysis Guide to classify the tasks presented in Analyzing Teaching and Learning 3.2.

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.

continued on next page

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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. • 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. 3.2. The Task Analysis Guide (TAG)—Characteristics of mathematical tasks at four levels of cognitive demand (from Smith and Stein, 1998, p. 348)

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Classifying Mathematical Tasks For ATL 3.2, you will use the Task Analysis Guide to classify four problems from a geometry unit on transformations by level and type of thinking required to complete the task.

Analyzing Teaching and Learning 3.2 Classifying Mathematical Tasks   Read the four tasks in figure 3.3. Use the Task Analysis Guide in figure 3.2 to classify each task according to the level of cognitive demand. What features of the Task Analysis Guide apply to each task? Write down your ideas, and discuss with colleagues if possible, before reading the analysis that follows.

Task 1: What are the three types of rigid geometric transformations? What does it mean to be a rigid transformation?

Task 2: Describe the rotation that moves ∆DEF onto ∆D’E’F’.

continued on next page

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Task 3: What transformation or series of transformations move ∆DEF onto ∆D’E’F’? Is ∆DEF  ∆D’E’F’? How do you know?

Task 4: Predict the effect of rotating quadrilateral QUAD 90° around point D. Sketch the rotated image, Q’U’A’D’, and justify your work.

Taken from Institute for Learning (2015b). Lesson guides and student workbooks are available at ifl.pitt.edu.

Fig. 3.3. Tasks at different levels of cognitive demand

Analysis of ATL 3.2: Classifying Mathematical Tasks According to the Task Analysis Guide, task 1 is an example of a “memorization” task. The task asks students to recall, state, or identify what a rigid transformation is and then choose which of the rigid transformations was used. Task 2 is a “procedures without connections” task. It is algorithmic (or computational, procedural), requires little cognitive effort, and does not require students to make connections to underlying mathematical concepts. The student merely determines how the figure was transformed, and then identifies which transformation has been used. Note that tasks 1 and 2 do not provide opportunities for reasoning and problem solving or for students to develop their own understanding of mathematical ideas. Instead, the tasks require producing correct answers (“answer-getting”). If a student did not know the answer or process when first encountering the task, he or she could not solve the task. Task 3 is an example of a “doing mathematics” task. Students are pulling together their knowledge to create solutions, using a variety of reasoning methods. Students cannot

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mindlessly use an algorithm or rule. Instead, they must connect to their prior knowledge about transformations — whether they decide to use a single transformation (a rotation of 90° counterclockwise or a rotation of 270° clockwise) or a series of transformations (such as a reflection of ∆DEF over the line y 5 x and then an additional reflection of ∆D'E'F' over the y-axis) or some combination. Then, the student must determine whether (and why or why not) the corresponding parts are the same. The student can also use the distance formula and SSS to prove that the two triangles are congruent or use some other method, such as measuring one angle and finding the lengths of the two sides that create the angle to use SAS. Task 4 is an example of a “procedures with connections” task, as students are explaining the conceptual underpinnings of the solution process. “Procedures with connections” tasks often ask students to use and connect multiple representations to explain, develop, or uncover mathematical relationships. Here the students extend their previous knowledge of transformations, including rotation around the point (0, 0), to determine what will happen in this task. Students may see a way to translate QUAD and then use previous knowledge, or students may create a visual method using models to solve the problem. Selecting a task with the potential to engage students in reasoning and problem solving is the first part of the focal teaching practice in this chapter. Additionally, teachers must consider how to implement tasks in ways that maintain students’ opportunities for reasoning and problem solving throughout the mathematics lesson. We will consider important factors in implementing tasks that promote reasoning and problem solving in Analyzing Teaching and Learning 3.3.

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Implementing Teaching and Learning Research has identified a set of factors (shown in fig. 3.4) that are associated with the maintenance and decline of high-level tasks (Henningsen and Stein 1997). Specifically, in classrooms where tasks decline during implementation, a subset of the factors listed on the left of figure 3.4 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 3.4 are often at play. In ATL 3.3, you will analyze the implementation of the Pay It Forward Task in two different algebra classrooms and determine whether the demands of the task were maintained and what factors account for the outcome.

Analyzing Teaching and Learning Activity 3.3 Comparing Instruction in Two Classrooms • Read or reread the Case of Vanessa Culver in chapter 1. What factors from the framework in figure 3.4 describe the teaching and learning in the case? Cite line numbers from the case to provide examples of the factors you identify. • Read the Case of Steven Taylor (following). What factors from the framework in figure 3.4 describe the teaching and learning in the case? Cite line numbers from the case to provide examples of the factors you identify. • How are Vanessa Culver’s and Steven Taylor’s lessons the same? How do they differ? How do the factors in the framework appear to impact student learning?

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Factors Associated with Decline

Factors Associated with Maintenance

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).

1. Scaffolding (i.e., teacher simplifies task so that students can solve it; teacher maintains complexity but makes greater resources available). Can occur during whole-class discussion, presentations, group work, or pair work.

2. The teacher shifts emphasis from meaning, concepts, or understanding to correctness or completeness of the answer. 3. The teacher does not allow enough time for students to wrestle with the demanding aspects of the task or allows too much time and students flounder or drift off task. 4. Classroom-management problems prevent sustained engagement. 5. Task is inappropriate for the group of students (because of, for example, lack of interest, lack of motivation, lack of prior knowledge needed to perform, or because task expectations are not clear enough to put students in the right cognitive space).

2. Students have a way to monitor their own progress (e.g., the teacher discusses rubrics and uses them to judge performance; teacher makes explicit and uses means for testing conjectures). 3. The teacher or capable students model high-level performance. 4. The teacher maintains sustained press for justifications, explanations, meaning through teacher questioning, comments, and feedback. 5. The teacher selects tasks that build on students’ prior knowledge. 6. The teacher draws frequent conceptual connections. 7. The teacher provides sufficient time to explore (not too little, not too much).

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

Fig. 3.4. Factors of maintenance and decline of high-level tasks (Adapted from Stein, Smith, Henningsen, and Silver, 2009, p. 16)

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Exploring Exponential Relationships: The Case of Steven Taylor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Mr. Taylor wanted to engage his students in a lesson that would give them a chance to reason abstractly and quantitatively. He began the lesson by showing students a small portion of the movie Pay It Forward in which the protagonist (Trevor) explains how he could change the world by doing good deeds. Some students in the class had seen the movie, but that didn’t seem to matter — all students were intrigued to be watching a movie in math class. They wondered how this was going to relate to algebra. Mr. Taylor stopped the video before giving away any methods or hinting at a solution. He gave students a copy of the task and had one student read it aloud. He then instructed students to begin work on the problem with their partners. As students worked on the task, Mr. Taylor walked around the room, stopping at different groups to listen in on their conversations. Students were confused about what was happening at each of the stages and unsure of what it meant to describe a function. He decided to bring the class back together. Mr. Taylor began by pointing out that the task asked them to describe a function that would model the Pay It Forward process at any stage. He then engaged them in the following exchange:

17

Mr. T:

18

Chris:

19 20

Mr. T:

21 22

Colin:

23

Mr. T:

24

Samantha: The stage number?

25

Mr. T:

26

Derek:

27 28 29

Mr. T:

So who remembers what a function is? Chris? [Chris did not raise his hand.] I don’t know.

What did we write in our notebooks a few weeks ago? Can someone find it? [A minute passes while students look through their notebooks.] A function relates input and output. For every input, there is only one output. So what do we need to do here? What is the input? Yes. What is the output? The number of deeds?

Good. So we need to describe the relationship between the stage number and the number of deeds. We could do that using an equation. So let’s start by looking at what happens at each stage.

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30 31 32 33 34 35 36 37

Mr. Taylor then drew a stick figure to represent Trevor and each of the three people for whom Trevor did good deeds (see fig. 3.5). He then asked students how many good deeds were completed at stage 1, and everyone shouted “3!” He then asked students how many deeds each of the three people would do at stage 2. Again they shouted “3!” He told students to continue the diagram for stages 2 and 3 and to record their information in a table so they could keep track of the number of deeds in each stage. With a clear idea of what to do, students continued to work in their groups to complete the diagram and table.

Fig. 3.5. The diagram Mr. Taylor drew on the board 38 39 40 41 42

As Mr. Taylor resumed his visits to the groups, he now noticed that they all had completed the diagram (see fig. 3.6 for example) and had made a table showing stage number and the corresponding number of deeds. He began to ask students if they could now write an equation for the function. Most groups said that it would be y 5 3x, since each stage was three times the one before it.

Stage #

# of Deeds

1

3

2

9

3

27

Fig. 3.6. An example of the diagram and table generated by students

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43 44 45

Once again Mr. Taylor decided to bring the class together for a discussion. He put a copy of the table shown below (minus the last row and the third column) on the document camera. He began:

46

Mr. T:

47

Ss:

48

Mr. T:

49

Ss:

50 51

Mr. T:

52

Ss:

53

Mr. T:

54

Ss:

55 56 57

Mr. Taylor went on to explain that at each stage they needed to look at the number of threes they are multiplying together. He added a third column to the table and elicited from students the number of times you multiply three at each stage.

How much bigger is the number of deeds in stage 2 than in stage 1? 3 times bigger!

How much bigger is the number of deeds in stage 3 than in stage 2? 3 times bigger!

How much bigger do you think the deeds in stage 4 will be when compared to stage 4? 3 times bigger!

How much is 3 times 27?

81! [Mr. Taylor adds stage 4 and 81 deeds to the table.]

Stage #

58 59 60 61

# of Deeds

3 by 3

1

3

3

2

9

333

3

27

33333

4

81

3333333

He then asked students how they can represent the number of threes that get multiplied together without writing them all out. Camilla yelled out “use an exponent.” Mr. Taylor told her, “That is exactly right,” and wrote y 5 3x on the board. At that point the bell rang. Mr. Taylor told his class that they would pick up on this tomorrow. This case, written by Margaret Smith at the University of Pittsburgh, was based on patterns of instruction documented in the research literature and observed in dozens of classrooms.

Analysis of ATL 3.3: Comparing Instruction in Two Classrooms Analysis of the Case of Vanessa Culver

Ms. Culver makes many of the teaching moves associated with maintaining high-level demands. Ms. Culver selected Pay It Forward because it provided a context for her students that she felt would help them make sense of the concepts that build understanding of exponential functions. She selected a task that builds on students’ prior knowledge of multiplication and exponents, as well as their knowledge of different ways to solve problems,

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that is, using diagrams, tables, graphs, or symbols (line 8). She also knew that this task would push her students beyond their current knowledge and would be a challenge for them. The class started with Ms. Culver asking her students to explore different ways to solve the problem (lines 19–22). As students worked, she circulated about the class providing scaffolding and asking questions that required explanations and justifications (lines 24–25). She also used student work time to take note of student responses so she could sequence them into a lesson that would make connections about the mathematics clear (30–32). She was aware of class progress, so she was able to ensure that she had given the students sufficient time to think about the problem as well. As the students shared their solutions, Ms. Culver had students ask questions and provide explanations (lines 33–34). The first group to share had a diagram and a verbal rule, while the second group had a table but no description of a function. Then, Ms. Culver had the students consider two different forms of the function (linear and exponential) and explain what was happening. Throughout the discussion, Ms. Culver asked questions to enable the students to make sense of the mathematics (44–79). At the end of the excerpt, Ms. Culver restated what students had said and prepared students to differentiate the graph of a linear function and the graph created from the problem (lines 77–79). Once again, she drew on students’ prior knowledge by centering the discussion on the domain for the function (group 5 had a continuous graph, but it should be discrete; lines 80–). The class ended with the students making connections among the different representations that had been used and explaining their reasoning (lines 84–86). The students were then responsible for reflecting on what they had learned.

Analysis of the Case of Steven Taylor

Mr. Taylor had process goals for his class (reason abstractly and quantitatively) but no clear content goals. As the class progressed, Mr. Taylor noted that students were struggling (lines 11–12). Though the amount of time the students were able to grapple with the problem was not enough for them to consider the different aspects it contained, he pulled the class together to discuss the problem (line 13). Mr. Taylor’s questions turned the task into a routine: students were told to consider the stage number as an input and the number of good deeds as the output (lines 23–28). The emphasis went from problem solving to obtaining a correct answer when he started a solution path for the students —creating a diagram (30–37). When many students thought the function was y 5 3x, Mr. Taylor did not ask for explanations of their solutions. Instead, he took over student thinking by creating a chart that the students completed by answering short low-level demand questions (lines 46–54). As the class ended, students were not asked to reflect on their work, nor were any connections made among the representations that had been used.

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Comparing Ms. Culver’s and Mr. Taylor’s classes

Both Ms. Culver’s and Mr. Taylor’s classes started with students attempting to solve the problem as the teacher circulated to take note of student understanding. A major difference between the classes occurred in lines 11–12 of Mr. Taylor’s class. When the students were confused, he called the class together and asked them questions focused on a particular solution path, looking for patterns at each stage. After the class went back to working in groups, he noticed a common error: students were considering the function y 5 3x (lines 41–42). Again, Mr. Taylor called them back together and asked leading questions to move students toward an answer (lines 43–54). In contrast, Ms. Culver allowed the students to wrestle with the problem. She made notes about the student strategies (lines 30–31) and then sequenced the work that would be shared to build student understanding of the concepts, including the same y 5 3x solution (line 44) that was seen in Mr. Taylor’s class. Mr. Taylor turned what was a “doing mathematics” task into a “procedures without connections” task that “[required] limited cognitive demand for successful completion.” By starting students on a particular solution path, Mr. Taylor left no ambiguity about what needed to be done or how to do it. The short low-level demand questions in lines 46–54 did not require any explanations from the students and did not make connections between multiplication and exponential functions. In contrast, Ms. Culver led a discussion in which the students shared solutions and reasoned about the answers they had created in order to build an understanding of exponential growth (lines 44–79). Just as different tasks provide different opportunities for students’ learning, the differences between these two lessons are important because of the impact on students’ learning. The next section presents research findings related to implementing tasks that promote reasoning and problem solving.

Implement Tasks that Promote Reasoning and Problem Solving: What Research Has to Say Throughout this chapter, we have provided activities and tools to support teachers in considering the importance of tasks that promote reasoning and problem solving and how such tasks are implemented during instruction. A rich history of research spanning several decades has established the importance of such tasks in supporting students’ learning of mathematics. Principles to Actions highlights three major research findings generated by this research. First, “Not all tasks provide the same opportunities for student thinking and learning” (NCTM 2014, p. 17). Building on the work of Doyle (1988), researchers in the Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) project (Stein et al. 2009) developed the Task Analysis Guide (fig. 3.2) as a way of classifying mathematical tasks according to the level or type of thinking that tasks could potentially elicit from students.

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For example, a task that can be solved through the rote application of a set procedure, such as task A in figure 3.1, requires a very different level and type of thinking than a task that requires reasoning and problem solving, such as task B in figure 3.1. Second, “Student learning is greatest in classrooms where tasks consistently encourage high-level student thinking and reasoning and least in classrooms where tasks are routinely procedural in nature” (NCTM 2014, p. 17). A number of studies have examined teaching and learning in classrooms using different types of mathematics curricula. These studies consistently associate higher student achievement with mathematics curricula that contain a predominance of cognitively challenging tasks; for research specific to high school mathematics curricula (including algebra), see Boaler and Staples (2008), Cai and colleagues (2011), Grouws and colleagues (2013), and Schoen and colleagues (1999). Large national studies (e.g., Cobb and Smith 2008; Kane and Staiger 2012) and international studies (e.g., Hiebert et al. 2003) consider the implementation of cognitively challenging tasks as a key indicator of mathematics instruction that promotes students’ learning. In fact, in the Trends in International Mathematics and Science (TIMSS) 1999 Video Study, the implementation of tasks in ways that maintained students’ opportunities for reasoning and problem solving was the most significant factor associated with higher student achievement between countries (Stigler and Hiebert 2004). Third, “Tasks with high cognitive demands are the most difficult to implement well and are often transformed into less demanding tasks during instruction” (NCTM 2014, p. 17). Research consistently indicates both the importance and the complexity of maintaining highlevel demands throughout a lesson. Stein and colleagues (Henningsen and Stein 1997; Stein et al. 2009) describe how students’ engagement with a task that could potentially promote reasoning and problem solving might result in actual reasoning and problem solving (e.g., cognitively demanding mathematical work and thinking) or might decline into less rigorous mathematical activity (e.g., applying prescribed procedures) as the task is implemented during instruction. The “productive struggle” often accompanying cognitively challenging tasks might lead students to press the teacher for step-by-step directions. Conversely, by asking questions, encouraging conceptual connections, and holding students accountable for explanations and meaning, teachers can maintain students’ engagement in reasoning and problem solving throughout a lesson (Henningsen and Stein 1997). This body of research has provided valuable insights into the implementation of cognitively challenging tasks, such as the “Factors associated with maintenance and decline of high-level demands” (Henningsen and Stein 1997) provided in figure 3.4, and ways of orchestrating mathematical discussions (Stein et al. 2008) that will be presented in chapter 6.

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Promoting Equity by Implementing Tasks that Promote Reasoning and Problem Solving Principles to Actions identifies “high-quality mathematics curriculum” as one of several supports necessary to promote access and equity in learning meaningful mathematics. Research presented in the previous section consistently indicates that cognitively challenging tasks have a significant positive impact on students’ mathematical learning, especially when such tasks are implemented in ways that maintain students’ opportunities for reasoning and problem solving. Hence, the nature of mathematical tasks and how tasks are implemented during instruction are critical components of students’ opportunities to learn mathematics (Boston and Wilhelm 2015; Perry 2013), particularly for students who have been traditionally marginalized in mathematics (e.g., children of minority groups, children for whom English is not their first language, children of poverty, and children who have previously struggled in mathematics). Opportunity gaps (Flores 2007) are created when some groups of students are provided different opportunities to learn mathematics than other students, such as differences in the nature of mathematical tasks and how tasks are implemented during instruction. All students need exposure and access to rigorous tasks (e.g., “procedures with connections” and “doing mathematics” tasks as described in the Task Analysis Guide, fig. 3.2). As stated in chapter 1, challenging yet attainable goals (and the tasks aligned with those goals) support students’ mathematical work and thinking and communicate the growth mindset that all students are capable mathematicians. Engaging students in high-level tasks affords them the opportunity to go deep with the mathematics, one of the five equity-based mathematics teaching practices (Aguirre, Mayfield-Ingram, and Martin 2013). Principles to Actions asserts, “Inequitable learning opportunities can exist in any setting, diverse or homogenous, whenever only some, but not all, teachers implement rigorous curricula” (NCTM 2014, p. 60). For example, a common misconception is that tasks that promote reasoning and problem solving will be too difficult for students who have previously struggled in mathematics or students in remedial or “low-tracked” classes. As summarized by Rubel and Chu (in press), however, research finds that traditional direct instruction “is not broadly effective with students” (e.g., Boaler 2015; Boaler and Greeno 2000) and “differentially less effective for students from marginalized and under-served groups” (e.g., Franke, Kazemi, and Battey 2007; Ladson-Billings 1997; Spencer 2009). In contrast, “When there are many ways to be successful, many more students are successful” (Boaler and Staples 2008, p. 16). When students are provided opportunities to develop and understand a variety of strategies, representations, and ways of thinking and reasoning, more students are likely to be mathematically successful than when the only option for success is memorizing and applying a prescribed method created and demonstrated by someone else. Specific types of rigorous tasks have been identified as especially effective in providing equitable and accessible mathematics learning opportunities. First, tasks considered as “low Implement Tasks That Promote Reasoning and Problem Solving    45 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.

threshold, high ceiling” (McClure 2011) allow multiple points of access (low threshold) and can be explored or extended in multiple ways (high ceiling). For example, the Pay It Forward task can be approached through a variety of representations, including diagrams that directly model the first few stages of the situation. This “low threshold” provides initial access to the task and underlying mathematical ideas from which students can develop and construct more sophisticated mathematical ideas. The task can be extended in numerous ways, such as finding when one million good deeds would occur or considering the impact of changing the number of people or the time interval at each stage. In providing equitable learning opportunities (e.g., giving each student the support needed to grow his or her mathematical understanding), teachers can support students who excel at mathematics through “high-ceiling” tasks. Second, tasks that are culturally relevant or that promote learning mathematics for social justice have been shown to increase students’ engagement and motivation in mathematics. Culturally relevant tasks use students’ funds of knowledge (composed of prior experiences, culture, language, interests, etc.) as the basis of mathematical tasks (Aguirre, Turner et al. 2013). Tasks that promote social justice engage students in using mathematics to understand and eradicate social inequities (Gutstein 2006). Both types of tasks portray mathematics as useful and important in students’ lives and promote students’ lived experiences as important in mathematics class. Third, tasks considered “group-worthy” support students in collaborating with peers around important mathematics ideas (Lotan 2003). Group-worthy tasks contain many of the characteristics already identified as important (e.g., open-ended, opportunities for problem solving, multiple points of entry), while also requiring productive interactions between students, group and individual accountability, and clear criteria for evaluation of the group’s product. Lotan suggests that group-worthy tasks provide genuine dilemmas and authentic problems for which students must struggle together to explore pathways and develop and defend solutions. Research indicates that tasks that promote reasoning and problem solving play an important role in developing students’ identity and agency as mathematical thinkers. D. B. Martin (2012) defines mathematical identity as “dispositions and deeply held beliefs that individuals develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics to change the conditions of their lives” (pp. 57–58). In other words, mathematical identity comprises how students see themselves in relation to mathematics and their ability to engage in mathematics. Boaler and Staples (2008) consider mathematical agency as students’ opportunities to develop their own ideas and to ask their own genuine questions. By providing multiple pathways for success and encouraging students’ own ideas and strategies, tasks that promote reasoning and problem solving can enhance students’ positive identities and sense of agency in mathematics. Tasks and instruction that promote reasoning and problem solving provide greater access, interest, and opportunity to learn mathematics for all students, and this is essential to support the mathematical learning of traditionally marginalized populations of students. By providing 46   Taking Action Grades 9–12 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.

mathematical work that is engaging and relevant, teachers can support students’ understanding of mathematics and develop students’ identities as capable mathematicians.

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 needed to fully develop students’ understanding of mathematical ideas.

• Tasks that promote reasoning and problem solving can provide access to all students and promote equity by providing students with multiple entry points and ways to demonstrate competence.

Taking Action in Your Classroom: Implementing Tasks that Promote Reasoning and Problem Solving It is now time to consider what implications 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 below.

Taking Action in Your Classroom 3.1 Use the Task Analysis Guide (fig. 3.2) to analyze the tasks you have used in one of your classes over the last few weeks. • Were the tasks high-level demand tasks? If so, did your implementation of the tasks provide your students with the opportunity to reason and problemsolve, that is, did you maintain a high level of cognitive demand for the tasks? How? • Identify tasks in your textbook (or in other available resources) that align with your mathematical and process goals and that can provide additional opportunities for students to reason and problem-solve. In what ways can you be prepared to implement the tasks to support your students and place the tasks at a high cognitive level?

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Taking Action in Your Classroom 3.2 Reflect on a lesson that you have previously taught that was based on a highdemand task. Was your lesson more like Vanessa Culver’s or more like Steven Taylor’s? What actions or interactions might account for the outcome of the lesson? Which of the factors from figure 3.4 best characterize the lesson?

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CHAPTER 4

Build Procedural Fluency from Conceptual Understanding The Analyzing Teaching and Learning 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. Developing procedural fluency is an important mathematical goal for all students. 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 at play. Conceptual understanding and procedural fluency are essential and integrated components of mathematical proficiency. In order for students to be able 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 rests a set of mathematical procedures. In this chapter, you will — • establish the connection between procedural fluency and conceptual understanding;

• explore the ways in which tasks can be sequenced to promote conceptual understanding first and then build procedural fluency;

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

• reflect on building procedural fluency from conceptual understanding in your own classroom. For each Analyzing Teaching and Learning (ATL), make note of 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. Once you have written down or shared your ideas, read the analysis, where we offer ideas relating the Analyzing Teaching and Learning activity to the focal teaching practice.

The Relationship between Conceptual Understanding and Procedural Fluency Conceptual understanding and procedural fluency are critical and connected components of students’ mathematical proficiency. But a central question in this chapter is exactly how are procedural fluency and conceptual understanding related? We take the stance that conceptual understanding must come first and serve as the foundation on which to build procedural fluency. In the activities in this chapter, we explore this complex relationship and provide you with tools to think about how to support your students in developing conceptual understanding and building procedural fluency. In Analyzing Teaching and Learning 4.1, we begin by considering how the sequence of tasks, beginning with Pay It Forward and progressing to Petoskey Population, could build procedural fluency from conceptual understanding.

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Analyzing Teaching and Learning 4.1 The Relationship between Conceptual Understanding and Procedural Fluency How would you expect the students in Vanessa Culver’s class to approach the Petoskey Population task (from fig. 3.1) after having had the opportunity to explore exponential growth relationships in a variety of contexts? “The population of Petoskey, Michigan, was 6,076 in 1990 and was growing at the rate of 3.7% per year. The city planners want to know what the population will be in the year 2025. Write and evaluate an expression to estimate this population.” (Source: Holt Algebra 2 [Schultz et al. 2004, p. 415])

Analysis of ATL 4.1: The Relationship between Conceptual Understanding and Procedural Fluency In solving the Pay It Forward task, Ms. Culver’s students were able to represent the exponential relationship by using a diagram, table of values, and graph. By considering the way the number of good deeds “grew” from stage to stage, students were able to reason that the equation y 5 3x represented the pattern of growth (instead of y 5 3x). Following the Pay It Forward Task, Ms. Culver likely provided additional opportunities for students to explore more tasks and situations with exponential growth relationships of the form y 5 bx and then progressing to situations modeled by y 5 (a)bx (where a  1). On the basis of students’ representations and strategies from the Pay It Forward task, it appears that Ms. Culver’s students might decide to make a table (see fig. 4.1) to represent the growth from year to year in the Petoskey Population task. In constructing the table (perhaps with the use of technology), they could begin to see and express regularity in repeated reasoning and to make use of structure — each new table entry is the previous entry times 1.037. The tediousness of extending the table to 35 entries (for the year 2025) might prompt students to look for a pattern of growth or change, as they did in previous tasks. In Pay It Forward, the number of deeds at any step was three times greater than the previous step, and “3 times greater” was represented by 3x. In the Petoskey Population Task, the number of people each year is 3.7% greater, or 1.037 times greater, than the previous year. Students making this connection might reason that the equation involves 1.037x. Also, while the number of good deeds started at 1 in Pay It Forward, the population starts at 6,076 in this task. So, the equation would need to begin at 6,076 and multiply by 1.037 at each step, or y 5 (6,076) 3 1.037x, where x is the number of years after 1990.

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Year

x 5 number of years after 1990

y 5 Population

1990

0

6,076



3 1.037

1991

1

(6,076) 3 1.0375 6,301



3 1.037

1992

2

(6,301) 3 1.0375 6,534



3 1.037

1993

3

(6,534) 3 1.0375 6,776

1994

4

(6,776) 3 1.0375 7,027

... 2025

3 1.037

... 35

(6,076) 3 1.03735

Fig. 4.1. Possible table for the Petoskey Population task

Students could also generate a graph (perhaps using technology) and use the graph to determine the population in any given year. Figure 4.2 shows a sample graph, created using the free online graphing tool Desmos (https://www.desmos.com).

Fig. 4.2. Possible graph for the Petoskey Population task

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Students might also compute the population for the first few years following 1990 and look for patterns: 1991: (6,076 1 6,076 3 .037) 5 (6,076) 3 1.037 5 6,301 1992: (6,076) 3 1.037 3 1.037 5 6,534 1993: (6,076) 3 1.037 3 1.037 3 1.037 5 6,776 Even if students did not organize their work in this way initially, purposeful questions from Ms. Culver could focus students’ attention on the pattern of growth being “1.037 times greater” in each step. From this numerical work, students could then generalize that in each year following 1990, the initial population of 6,076 is multiplied by a factor of 1.037, generating the equation y 5 (6,076)(1.037x). Ms. Culver’s students may also model the situation with a diagram (see fig. 4.3). While the task does not lend itself to a literal drawing of the situation (even for the initial steps, as students could do for the Pay It Forward task), students might choose to represent the original population with a diagram. For example:

Fig. 4.3. Possible diagram for the Petoskey Population Task

Once again, using repeated reasoning, the structure of the model or computations, and their prior experiences with exponential growth, students could see the pattern of growth as “1.037 times greater” each year, and they could use this as the basis for generalizing the equation. A sequence of tasks beginning with Pay It Forward (and other situations that can be modeled by y 5 bx) to several experiences with exponential growth relationships of the form y 5 (a)bx, to more complex exponential growth relationships (such as the Petoskey Population task) would provide a foundation for students to reason about common formulas Build Procedural Fluency from Conceptual Understanding    53 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 exponential growth and decay, such as A(t) 5 a(11r)t, where A(t) is the amount after t time periods, a is the initial amount (a0 in some books), r is the rate of growth or decay (written as a negative) per time period, and t is the number of time periods. Such a progression of tasks and experiences could build a conceptual foundation for procedures by helping students understand what the numbers in the formula mean and what each part of the formula does (e.g., the reasons behind the procedures). Using the right formula to solve the problem at hand requires a deep understanding of what the formula means and the characteristics of situations in which it can be used.

The Role of Task Sequences Consider a sequence of tasks to develop students’ understanding of volume formulas for prisms, cylinders, pyramids, and cones, beginning with the lesson Popcorn Prisms, Anyone? from NCTM Illuminations (http://illuminations.nctm.org/Lesson.aspx?id=2927). In the Popcorn Prisms, Anyone? exploration, students are challenged to create rectangular prisms using 8.5- by 11-inch pieces of paper, first using 8.5 inches as the height and then using 11 inches as the height (see fig. 4.4). Students compare the volume by using popcorn to fill the two prisms. The task prompts students to explain why the volumes are not the same and to generalize what they experienced by representing the volume algebraically: “Explain why the prisms do not hold the same amount. Use the formula for the volume of a prism to guide your explanation.”

Fig. 4.4. Prisms created in the Popcorn Prisms, Anyone? Exploration (Reprinted with permission from Illuminations, copyright 2009, by the National Council of Teachers of Mathematics. All rights reserved. Available at http://illuminations.nctm.org/Lesson.aspx?id=2927)

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The same activity is then repeated with students creating cylinders in Popcorn Cylinders, Anyone? (fig. 4.5), and again students are asked, “Explain why the cylinders do or do not hold the same amount. Use the formula for the volume of a cylinder to guide your explanation.”

Fig. 4.5. Cylinders created in the Popcorn Cylinders, Anyone? Exploration (Reprinted with permission from Illuminations, copyright 2009, by the National Council of Teachers of Mathematics. All rights reserved. http://illuminations.nctm.org/Lesson.aspx?id52927)

Students are then asked to generalize across the explorations of prisms and cylinders (NCTM 2009b): Which measurement impacts the volume more: the radius or the height? Work through the example below to help you answer the question. • Assume that you have a cylinder with a radius of 3 inches and a height of 10 inches. Increase the radius by 1 inch and determine the new volume. Then using the original radius, increase the height by 1 inch and determine the new volume. • Which increased dimension had a larger impact on the volume of the cylinder? Why do you think this is true? By how much would you have to decrease the height of Cylinder B to make the volumes of the two prisms equal? Compare and contrast your results from the prism activity and the cylinder activity. What conclusions can you make about the relationship between dimensions, area, and volume? Build Procedural Fluency from Conceptual Understanding    55 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 Popcorn, Anyone? explorations serve as the focus of Analyzing Teaching and Learning 4.2. Before engaging in ATL 4.2, solve the Popcorn Prisms, Anyone? and Popcorn Cylinders, Anyone? activities, and discuss your mathematical work and thinking with colleagues.

Analyzing Teaching and Learning 4.2 Considering Sequences of Tasks Consider how the Popcorn, Anyone? activities might be used to develop procedural fluency from conceptual understanding: • How does the sequence of activities help students develop a conceptual understanding of the volume formulas for rectangular prisms and cylinders? • What next steps or follow-up tasks might support students in developing procedural fluency with the volume formulas for rectangular prisms and cylinders? Consider how the exploration of rectangular prisms and right cylinders in the Popcorn, Anyone? activities might be extended to other shapes identified in rigorous standards for high school geometry (e.g., pyramids, cones, and spheres).

Analysis of ATL 4.2: Considering Sequences of Tasks Students might initially assume that the two prisms (or two cylinders) created from two sheets of paper of the same size will have the same volume. As they explore the task, students realize that the dimensions of each solid are indeed different, thus producing different volumes. Students may also realize that, as they experiment with changing the dimensions, (1) increasing the area of the base appears to impact the volume more than increasing the height; and (2) volume continues to increase as the dimensions get closer together (e.g., as the prism or cylinder becomes more cube-like). The questions in the tasks serve to draw students’ attention to the area of the base of each solid. In both activities, the general volume formula, V 5 (area of the base) 3 height, helps highlight the relationship between the dimensions of the figure and its volume in rectangular prisms and right cylinders. Sequences of tasks and experiences that develop procedural fluency have three essential components (Briars 2016): • First, the sequence should begin with tasks that develop conceptual understanding by building on students’ informal knowledge, providing experiences where students engage 56   Taking Action Grades 9–12 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.

only conceptually on the basis of their prior knowledge. In the Popcorn, Anyone? activities, students explore the volume of different rectangular prisms and cylinders visually and conceptually, using popcorn to “measure” volume.

• Second, the next tasks should support students in developing informal strategies to solve problems, where they use their own invented strategies and shortcuts and engage with a variety of strategies created by their peers. The Popcorn, Anyone? activities support students in developing an understanding of volume formulas as “area of the base times the height” and considering how changes in the dimensions of the prisms and cylinders affect the volume. • Third, students need to refine informal strategies to develop fluency with standard methods and procedures (algorithms or formulas). Engage students in considering how to make invented strategies or shortcuts more efficient; explicitly comparing and contrasting different methods to solve the same problem helps students build fluency. Prompt students to create generalized procedures or formulas by looking across several examples, contexts, and experiences, and to justify the generalizations by “looking for and making use of structure” or identifying “regularity in repeated reasoning.” At the end of the Popcorn, Anyone? activities, students are asked to compare prisms and cylinders and make generalizations. Extensions to the task (e.g., to right prisms, pyramids, or cones) could prompt students to generalize their findings to the volume formulas of other shapes.

As next steps, students might be asked to explore other right prisms (e.g., triangular, hexagonal, or octagonal prisms) and determine whether the general formula still holds. These experiences help students visualize why the volume formula works, perhaps seeing the area of the base as “one layer” and then multiplying by h layers, where h is the height of the prism or cylinder. Ideally, middle school experiences that build on this conceptual basis would engage students in exploring other shapes (e.g., right cones and pyramids). If the area of the “layer” varies from the base to the height of the figure, as in right pyramids or cones, then the volume formula of “area of the base times the height” would need to be adjusted as well. Students could create a right rectangular pyramid and a rectangular prism with the same base and height and compare the volumes, perhaps by measuring the amount of rice needed to fill both figures. After exploring several examples, students could be prompted to notice that the volume of the pyramid appears to be one-third the volume of the prism. Similarly, students could create right cones and compare with the volume of right cylinders with the same height and diameter. As students progress to high school, they can consider the volume of a pyramid by looking at numerical approaches similar to limits, using the volumes of associated “cube pyramids” (for an example, see Navigating through Measurement in Grades 9–12 [Albrecht et al. 2005]).

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Consider how this progression of tasks that promote reasoning and problem solving changes students’ learning experiences from the typical sequence of tasks that promote rote procedures and memorization of volume formulas. Often, students are provided with a completed example of how to use the formula and procedure, shown a two-dimensional picture of a three-dimensional shape already marked with the appropriate dimensions, and required only to substitute values into the appropriate places in the formula and compute. Note that this progression of tasks aligns with rigorous state and national standards across seventh grade (volume of right prisms), eighth grade (volume of cones, cylinders, and spheres), and high school (use volume formulas for cylinders, pyramids, cones, and spheres to solve problems). Students who have multiple opportunities to investigate volume conceptually develop an understanding of how the volume formula works procedurally. A conceptual understanding is the first step in building procedural fluency and fluidity with using volume formulas to solve problems. A foundation of conceptual understanding provides students with tools (other than memorization) to make sense of and reconstruct the volume formula in problem-solving situations or on common assessment items on standardized tests (fig. 4.6).

Fig. 4.6. Example of a common assessment item for volume of prisms (Source: The Improving Mathematics Education in Schools [TIMES] Project, Australian Mathematical Sciences Institute [AMSI], copyright 2010 by the University of Melbourne. Retrieved from http://amsi.org.au/teacher_modules/area_volume_surface_area.html#Volume_of_a_Prism)

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Developing an Understanding of Procedures In Analyzing Teaching and Learning 4.3, we ask you to consider progressions of tasks that would support students’ conceptual understanding and build procedural fluency for different methods of working with quadratic equations and formulas (e.g., factoring, completing the square, and quadratic formula). Here we consider the way in which Nicole Bartlone has constructed opportunities for her Algebra 2 students to develop an understanding of the procedural methods for working with quadratics.

Analyzing Teaching and Learning 4.3 Developing an Understanding of Procedures Read the Case of Nicole Bartlone. • How did the sequence of tasks and experiences help students build procedural fluency from conceptual understanding for methods used to work with quadratic functions? • How does Ms. Bartlone’s assignment contribute to the development of procedural fluency?

Working with Quadratic Functions: The Case of Nicole Bartlone 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ms. Bartlone stared at a page in her algebra 2 textbook with 60 review problems from a chapter titled “Quadratic Functions.” The set of exercises was intended to provide students with practice in solving quadratic equations using the quadratic formula, completing the square, and factoring. Directions for different subsets of problems included “Find the solutions to these equations using the quadratic formula”; “Find the solutions to these equations using completing the square”; and “Find the solutions for these equations using factoring.” Additionally, some problems were marked with red stars to indicate that they were especially challenging. Ms. Bartlone reflected on the lessons over the last unit. Students had worked with factoring and been introduced to both the quadratic formula and completing the square in algebra 1 two years ago. Though she anticipated that her students had some memory of the procedures, Ms. Bartlone examined their answers to her questions in class and on a few review questions to determine how much they truly understood about quadratic equations and functions. She decided to do work to reinforce the conceptual basis for the procedures of factoring, completing the square, and applying the quadratic formula.

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16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Her first set of lessons looked at functions such as y 5 x2 1 6x 2 16 to find the zeros (for example, solving x2 1 6x 2 16 5 0). Ms. Bartlone had the class investigate graphs of the functions to find the x-intercepts. She then used the Missing Function activity (chapter 5) to make the connection between the graphic solutions and the factored form of an equation/function. Ms. Bartlett used prompts similar to those used by Ms. Bassham (featured in the Missing Function video), such as, “If a function is x2 1 6x 2 16 and one of its factors is x 2 2, what is the other factor?” and “What do the factors tell us about the zeros of the function?” In this way, she helped students see a connection between factoring and graphing. Then, students looked at how to graph quadratic functions based on knowing the zeros and using a table of values. They made connections between the graph of a function, the factored form of the function, the zero factor property, and the zeros (x-intercepts) of the graph. By tying the review to new material about graphing quadratic functions, Ms. Bartlone was able to have the class review previous materials in the context of new learning. Next, Ms. Bartlone had her students think about finding the vertex of a quadratic function that was written in the form ƒ(x) 5 x2 1 6x 2 16. Students were quick to use technology to graph and find the vertex, but they were not sure about a method for finding the vertex symbolically. Ms. Bartlone had them work with pictures and algebra tiles to consider rewriting the function in what the class called “graphing form,” that is, y 5 a(x 2 h)2 1 k, where (h, k) is the vertex. By using pictures, students made “squares” to find the vertex of ƒ(x) 5 x2 1 6x 2 16 (completing the square).

37

38

Fig. 4.7. Completing the square

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39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

Students were then able to graph quadratic functions by finding zeros (based on their previous work) and then finding the vertex. Initially, the students all used pictures, but eventually some students started finding shortcuts. Students said things such as, “I always split the x term in half, then square its coefficient.” Through class discussion, Ms. Bartlone’s class made connections between their pictures and the reasoning behind completing the square. Once again, this was connected to graphing (and to contextual problems that required a graph) so students could see their work as more than a symbolic manipulation. Though the students remembered the quadratic formula fairly well, Ms. Bartlone had a problem set that started with functions such as g(x) 5 4x2 1 6x 2 3, for which she had students find the vertex using completing the square and then the zeros. That lesson culminated with students using completing the square to find the vertex and zeros of k(x) 5 ax2 1 bx 1 c, thus proving the quadratic formula (when finding the zeros) (fig. 4.8). After these lessons, Ms. Bartlone had students work on tasks in which they reviewed and applied the three methods for working with quadratic functions as discussed in the chapter. For example, students found solutions of equations such as 5x2 1 4x 2 2 5 0, where using the quadratic formula would be the most efficient method, or for functions such as 0 5 x2 2 8x 1 12, which would be easy to solve using factoring. At the end of those lessons, Ms. Bartlone engaged students in a conversation about which method would be more efficient on the basis of the situation. Similarly, during the graphing parabolas lesson, Ms. Bartlone engaged students in finding the vertex or the x-intercepts. For example, given a function such as ƒ(x) 5 x2 1 7x 1 9, students may find it easiest to use completing the square to find the vertex, but they could also use any of their three learned methods to find the x-intercepts. Again, students compared the methods they had learned and discussed which method would be the most efficient for a given situation. For this review, Ms. Bartlone wanted to do more than assign students the typical practice set of “2–60 even.” She decided to make copies of the practice problems, cut apart the 30 different problems, and then place each set in an envelope. As students entered the classroom, she planned to hand each pair of students an envelope and the following directions: • Sort the problems into three piles according to which method you would use to solve them: the quadratic formula, completing the square, or factoring. You must use each method at least three times.

• Once you have sorted the problems, take turns selecting a problem from each pile and explaining to your partner why the method of solution you selected fits the particular question. Identify (or determine) the specific parts of the problem that make it more efficient to use the method you have chosen.

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77 78 79 80 81

As homework, Ms. Bartlone planned to ask students to solve at least 3 problems from each pile until they were confident in using each method. She would provide students with the answer key from the teacher’s edition so they could check their answers, but she would indicate that “being able to explain the method used and why it was used” was more important than the answers. This case, written by Frederick Dillon, is based on his 35 years of experience teaching high school mathematics.

Fig. 4.8. Completing the square to obtain the quadratic formula

Analysis of ATL 4.3: Developing an Understanding of Procedures How does the sequence of tasks and experiences help students build procedural fluency from conceptual understanding for methods used to work with quadratic functions? In the initial lessons, Ms. Bartlone supported students’ understanding by allowing them to test new procedures based on their own prior knowledge. She provided opportunities for students to use and connect a variety of representations (as discussed in chapter 6); for example, the 62   Taking Action Grades 9–12 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.

x-intercepts, zeros, and factors of a quadratic equation are seen graphically and symbolically. Common procedures associated with quadratic functions, such as completing the square and the quadratic formula, arose organically as students needed them in the context of their work in graphing quadratic functions (lines 39–51). Ms. Bartlone supported students as they developed these procedures in ways that made sense, using manipulatives, diagrams, or their own previous knowledge. Finally, she engaged students in comparing and contrasting different strategies for finding solutions in different situations (lines 70–81). From these experiences, students would be able to reconstruct the methods and when to use them from a conceptual basis. How does Ms. Bartlone’s assignment serve to build procedural fluency? First, students engage with the procedures conceptually by sorting the functions into groups. Students do not spend the majority of their time engaging in rote practice, blindly substituting values into the rules, or performing complex symbol manipulations. Instead, students make sense of when and how to use the different methods. The assignment contains just enough problems to provide practice at applying the methods while still maintaining a focus on understanding. In other words, students are not assigned so many practice problems that they do not have the time (or interest) to think about the mathematics. As described in Principles to Actions (NCTM 2014), Ms. Bartlone provided opportunities for students to demonstrate that they understood how to use the formula or procedure and to explain how the formula or procedure works. The ideas from the sequence of tasks in the Popcorn, Anyone? activities and in the Case of Nicole Bartlone can be used to consider other sequences of tasks and experiences that would lead students to build procedural fluency from conceptual understanding. Here are some examples: • Pythagorean theorem, distance formula, formula of a circle: Principles to Actions (NCTM 2014, p. 44) provides an example of how students can understand, re-create, and develop fluency with the distance formula as an application of the Pythagorean theorem. This sequence can be taken a step further to help students develop the formula of a circle in the coordinate plane (x2 1 y2 5 r 2) by considering situations, such as which trees can be watered by an 8-foot sprinkler (anchored at the origin; see the lesson In, On, or Out in “Orchard Hideout,” Interactive Mathematics Program Year 3 [Fendel et al. 2012]), which flowers are safe from a goat tied to a stake in the yard by a 15-foot rope, etc. By considering the distance of a specific point from the origin, students could use the distance formula or the Pythagorean theorem. By considering the boundary of points that are within reach of the sprinkler or the goat (e.g., within the length of the radius from the origin), students can relate the hypotenuse of the triangle (“c” in the Pythagorean theorem) to the radius of the circle and the x and y distances to the legs of the circle (“a” and “b” in the Pythagorean theorem); while individual points can be tested using a2 1 b2 5 c2, the entire “boundary” can be mapped out by x2 1 y2 5 r2. • Distributive property, factoring, completing the square: Visual representations (e.g., area models) and manipulatives (algebra tiles) can support students’ initial understanding

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of the distributive property, multiplying binomials (fig. 4.9a), and factoring (fig. 4.9b). These experiences can help students “see” and understand common procedures and shortcuts such as First-Outside-Inside-Last (FOIL), multiplying perfect square binomials (fig. 4.9c), and “completing the square” (fig. 4.8 from Ms. Bartlone’s lesson). For example, teachers and textbooks often provide a shortcut for multiplying perfect square binomials such as (x 1 3)2: square the first term, multiply the terms and double, and square the last term. Figure 4.9c shows why this shortcut works. By engaging with a visual representation such as algebra tiles or an area model, students can explore and generalize this shortcut on their own. The virtual algebra tiles used in figure 4.9 are available at NCTM Illuminations (figs. 4.9a and 4.9b) (https://illuminations.nctm .org/activity.aspx?id=3482) and the National Virtual Manipulatives Library (NVLM; fig. 4.7c; http://nlvm.usu.edu/en/nav/frames_asid_189_g_1_t_2.html?open=activities).

(a) Multiplying binomials

(b) Factoring (x 1 3)2 5 ? Square the first term (x2), multiply the terms together and double (6x), square the last term (9) (x 1 3)2 5 x2 1 3x 1 3x 1 9 5 x2 1 6x 1 9

(c) Perfect squares Fig. 4.9. Visual models for algebraic procedures

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• Qualitative graphs, slope, and average rate of change: Consider how exploring qualitative graphs can lead students to develop an understanding of slope and average rate of change. Analyzing change in graphs that model real-life situations can help students develop procedural fluency for finding slope and average rate of change and can also serve as the conceptual basis of understanding derivatives. In the Bike and Truck task, Ms. Shackelford asked students two questions that required them to reason about slope or average rate of change or both: (1) On which interval did the truck go the fastest? (2) Did the bike or the truck go faster after 17 seconds (the point where the lines intersect)? Students compared the ratio of the change in distance with the change in time to answer these questions. From repeated experiences of analyzing change in graphs of real-life situations, students will begin to generalize and make sense of slope and average rate of change. Historically, students’ experiences in high school mathematics have been characterized by procedures without connection to meaning or understanding. Often, “understanding” is associated with memorization or rote execution of procedures. Consequently, mathematics teachers lament students’ lack of “understanding” from unit to unit and year to year — when students have simply forgotten procedures and formulas they never truly understood and that remained disconnected from prior knowledge. As stated by W. G. Martin (2009, p. 165): Students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results. In this chapter, we suggested ways to develop students’ conceptual understanding through visual models, representations, and drawing on students’ prior knowledge before moving to more formal methods and procedures. While these methods may take more time, the time invested up front to develop students’ understanding will pay off in the time saved reteaching forgotten procedures. In addition to building conceptual foundations in this chapter, we also noted the importance of moving to procedural fluency, where students can apply (or re-create) important mathematical procedures when needed, can access or invent a variety of procedures for solving problems, and can think and work flexibly with numbers in problem-solving situations.

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Build 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 they have not been given opportunities to develop an understanding of when to use a procedure and why, students will often apply procedures in ways that are inappropriate to the mathematical task at hand (W. G. Martin 2009). Conversely, when procedures are connected to conceptual understanding, students are better able to retain the procedures and make more appropriate decisions in how they apply them to new situations (Fuson, Kalchman, and Bransford 2005). While procedural fluency and conceptual understanding are often contrasted in terms of teaching practice, they are not mutually exclusive. Hiebert and Grouws (2007) synthesize the findings of research 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, only procedural fluency was developed. In other words, procedural fluency can be developed through conceptually focused experiences like the ones described in this chapter, not just via 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 then attempting to connect to conceptual understandings later. Most commonly, students who learn the procedure first will continue to apply the procedure without engaging in the challenging work of making meaning for the procedure. Monique Butler is a well-intentioned teacher — she wants her students to learn the FOIL method of multiplying two binomials in order to be prepared for items on a standardized test. But when she returns the day after teaching them the procedure and asks them to demonstrate connections to visual area representations using algebra tiles, students do not engage in the ways she wants them to. This is because students had no motivation to make use of the visual, as they could use the procedure they had memorized to attempt to solve the tasks. Students also had no understanding behind the procedure, which led 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. Research also sheds light on the role of procedural practice (and homework) in high school mathematics. Distributed practice, where students practice on smaller sets of carefully selected problems and receive feedback, has been shown to be effective for supporting fluency (NCTM 2014). Practice of a specific procedure should (1) occur after students have developed a conceptual basis and (2) should be spaced or distributed over time (Pashler et al. 2007; Rathmell 2005; Rohrer 2009; Rohrer and Taylor 2007).

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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. Hence, a rush to procedural fluency can damage students’ identities as mathematical knowers and doers and promote increased mathematical anxiety (Ashcraft 2002; Ramirez et al. 2013). 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 focuses instead on building conceptual understanding through the use of multiple mathematical representations and multiple solution paths, students have a wider range of options for entering a task and building mathematical meaning. If procedures are subsequently built on this strong foundation, a student can always return to the conceptual understanding to confirm or regenerate the procedure if it is not remembered. For example, a student in Ms. Bartlone’s class could use factoring, completing the square, or the quadratic formula when working with quadratic functions. While we hope for all students to 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 identified by Aguirre, Mayfield-Ingram, and Martin (2013) have strong connections to building procedural fluency from conceptual understanding. When students begin the exploration of a new mathematical idea with conceptually focused instruction, this supports them in going deep with mathematics. Given the fact that procedures without a conceptual grounding are more likely to be misapplied or forgotten, going deep with the mathematics through a conceptual focus early on in an instructional sequence supports all students in gaining 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 get started productively on a task and ultimately work their way toward procedural fluency. Building that fluency from conceptual understanding also connects to the equity-focused practice of leveraging multiple mathematical competencies. When a procedure is memorized without a conceptual backing, 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. Bartlone’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. This again affords broader access to students in order to demonstrate their mathematical competence in diverse situations. Build Procedural Fluency from Conceptual Understanding    67 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.

Key Messages • When students learn mathematical procedures without a sound conceptual underpinning, they often misapply and forget those procedures.

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

Taking Action in Your Classroom: Building Procedural Fluency from Conceptual Understanding In Taking Action in Your Classroom 4.1, we invite you to explore how you can develop sequences of tasks that provide a conceptual basis from which students can build procedural fluency.

Taking Action in Your Classroom 4.1 Identify two or three procedures that are essential for a unit of study in your course or grade level. Discuss with peers ways to build conceptual understanding of each procedure. • What is students’ prior knowledge related to this procedure? • Develop a series of tasks that support students in building procedural fluency from conceptual understanding. Use resources (e.g., NCTM’s Developing Essential Understandings series, Standards progressions documents) for ideas on ways to use representations, diagrams, or manipulatives to develop a conceptual basis for the procedure. • As you develop a sequence of tasks and experiences, consider the progression described in this chapter: begin with experiences where students engage conceptually, progress to tasks that elicit informal and student-invented strategies, and then move to tasks where students can generalize and apply formal algorithms and procedures.

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In Taking Action in Your Classroom 4.2, we invite you to consider the role of practice problems and homework in promoting procedural fluency.

Taking Action in Your Classroom 4.2 Identify a procedure that students will master with fluency within an upcoming unit of study. Consider the homework provided by the textbook: • How many problems are students asked to solve? • To what extent does the homework include tasks that promote thinking and reasoning or tasks that promote rote procedures and memorization? • If the homework focuses on rote practice, revise the homework directions to include a smaller set of problems that will elicit students’ conceptual understanding and build toward procedural fluency.

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CHAPTER 5

Pose Purposeful Questions The Analyzing Teaching and Learning 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 only tool teachers have to 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 two narrative cases and compare the questions posed by each teacher;

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

• write questions that could be used to assess and advance students’ understanding based on samples of students’ work; • analyze a video and identify the type of questions used during the whole-group discussion;

• review key research findings related to posing purposeful questions; and • reflect on questions used in your own classroom.

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For each Analyzing Teaching and Learning (ATL), make note of 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. Once you have written down or shared your ideas, read the Analysis, where we offer ideas relating the Analyzing Teaching and Learning activity to the focal teaching practice.

Comparing Teachers’ Questions In Analyzing Teaching and Learning 5.1, we compare the questions asked by Ms. Culver and Mr. Taylor during the Pay It Forward lessons.

Analyzing Teaching and Learning 5.1 Comparing Teacher Questions Consider the Case of Vanessa Culver (chapter 1) and the Case of Steven Taylor (chapter 3), both of which feature the Pay It Forward task. Figure 5.1 lists the questions Ms. Culver asked during the whole-group discussion (lines 44–79) and the questions asked by Mr. Taylor during the whole-group discussion (lines 17–29 and 46–61). Compare the questions asked by each teacher: • What did each teacher learn about students’ thinking? • How are the types of questions asked by Ms. Culver and Mr. Taylor the same and how are they different? • Do the differences in the types of questions matter?

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Ms. Culver’s whole-group discussion

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Mr. Taylor’s whole-group discussion

Fig. 5.1. Questions asked by Ms. Culver and Mr. Taylor

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Analysis of ATL 5.1: Comparing Teacher Questions What did each teacher learn about students’ thinking? During the whole-group discussion in Ms. Culver’s lesson, students provided their thinking and reasoning about (1) whether the function should be written as y 5 3x or y 5 3x (Missy, Darrell, and Kara, lines 49–63); (2) how linear functions grow at a constant rate compared with exponential functions growing “really big fast” (Chris, lines 64–66); and (3) why the change of “3 times larger” is modeled by an exponential function (Angela, lines 72–76). By asking, “Can someone explain why y 5 3x is the best choice?” (line 47) and “Darrell, why do you disagree with Missy?” (lines 53–54), Ms. Culver provided an opportunity for students to share and clarify their thinking and make sense of the thinking and reasoning of others. She was not focused on eliciting correct answers, as she provided an opportunity for students to reason through the incorrect answer, y 5 3x, and make sense of why it did not model the mathematics in the problem situation (e.g., the number of deeds becoming three times larger). Through this discussion, Ms. Culver learned that four or five students had at least an initial understanding of exponential relationships and could explain why y 5 3x modeled the situation. To ascertain what other students understood, Ms. Culver asked the students to “explain in writing how the equation related to the diagram, the table, the graph, and the problem context” during the last 5 minutes of class. During the whole group discussion (lines 17–29 and 46–61) in Mr. Taylor’s lesson, students’ contributions included (1) reading the definition of function from the textbook; (2) identifying the “stage number” as the input and the “number of deeds” as the output; (3) correctly responding “3” or “3 times bigger” to Mr. Taylor’s questions regarding the diagram and table; and (4) one student responding “use an exponent” when Mr. Taylor asked students how they could represent “the number of threes that get multiplied together without writing them all out.” Mr. Taylor did not learn a great deal about students’ thinking during the discussion, as he did not hear students explain their own thinking and reasoning about the mathematics in the task. At best, he had evidence that students realized the numbers in the table were “3 times bigger” and that exponents are a condensed way of expressing repeated multiplication. How are the types of questions asked by Ms. Culver and Mr. Taylor the same and how are they different? Do the differences in the types of questions matter? To begin with, Ms. Culver asked open questions. In trying to determine what students understood about the exponential relationship in the problem, she asked questions that required students to explain their own thinking as they worked in small groups and during the whole-group discussion. In other words, she asked questions to make students’ thinking visible (e.g., lines 47–48, 52–54). She also asked questions to engage students in exploring mathematics (e.g., pursuing an incorrect solution first [lines 44–45]) and asking students to consider the graph [lines 78–79]). Finally, Ms. Culver asked questions that invited students to participate (e.g., asking who agreed or disagreed, line 52).

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In contrast, Mr. Taylor asked questions that required brief answers (lines 46, 48, 50–51), had a single right or wrong answer (lines 23, 25 53), and often elicited choral responses from students (lines 46–54). He also asked questions that students were likely to answer correctly by following along with his representation (e.g., diagram, table; lines 46–52) and his thinking (lines 58–59), rather than questions that elicited students’ own representations, reasoning, or ways of thinking. The differences in the teachers’ questions matter because they provide different opportunities and present different expectations for students’ mathematical thinking and reasoning. In Ms. Culver’s lesson, students were afforded the opportunity to formulate and express their own thinking, to make sense of the reasoning of others, and to make sense of mathematical ideas — and the expectation that they would do so. In Mr. Taylor’s lesson, the questions he asked provided students with the opportunity to show that they were following along with his thinking. Questioning in Ms. Culver’s lesson provided her with a window into students’ thinking and understanding in order to plan her next instructional moves. In this way, Ms. Culver’s questions provided formative assessment throughout the lesson. Principles to Actions (NCTM 2014) 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 are not given an opportunity to make connections or build their own understanding. Focusing involves the teacher honoring what the students are thinking by pressing them to communicate their thinking clearly and asking them to reflect on their thinking and the thinking of their classmates. Mr. Taylor’s pattern of questioning fits the funneling pattern; his questions elicited brief responses pertaining to his ideas and representations, and students were not given opportunities to share their own ideas and representations. In contrast, Ms. Culver’s pattern of questioning fits the focusing pattern; she used questions to elicit students’ own ideas and strategies and to provide opportunities for students to reason through the mathematics at the heart of the lesson. Principles to Actions (NCTM 2014) also identifies the importance of attending to different types of questions asked throughout a lesson. According to Principles to Actions, there are four types of questions that are important in teaching mathematics: • Gathering information questions ask students to recall facts, definitions, or procedures. • Probing thinking questions ask students to explain, elaborate, or clarify their thinking.

• Making the mathematics visible questions ask students to discuss mathematical structures and make connections among ideas and relationships. • 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.

Note that question types 2, 3, and 4 would be considered “open” questions, while “gathering information” questions typically have one correct answer. In Mr. Taylor’s lesson, the majority of 76   Taking Action Grades 9–12 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.

questions serve the purpose of gathering information: “So who remembers what a function is?” (line 17). “What is the input? . . . What is the output?” (lines 23, 25). In Ms. Culver’s classroom, we also see examples of the other types of questions: • Probing thinking: “Can someone explain why y 5 3x is the best choice? Missy, can you explain how you were thinking about this?” (lines 47–48)

• Making the mathematics visible: “So let’s take another look at Group 3’s poster. Does the middle column help explain what is going on?” (lines 67–68) • Encouraging reflection and justification: “Does everyone agree with what Missy is saying? Darrell, why do you disagree with Missy?” (lines 52–54) Throughout this chapter, you will have the opportunity to analyze additional high school mathematics lessons and students’ work in order to identify and craft purposeful questions, such as the types of questions used by Ms. Culver. As a starting point, the non-task-specific questions shown in figure 5.2 can support teachers’ attempts to foster an interactive classroom environment where discourse is encouraged and supported and where students explain and reflect on their understanding and ask questions of both their peers and their teacher. • 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 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 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?

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• 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 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. 5.2. General questions that foster discussion (Source: NCTM 1991, pp. 3–4)

Posing Questions to Move the Mathematics Forward In Analyzing Teaching and Learning 5.2 (page 81), we visit the classroom of Jamie Bassham (see More4U at nctm.org/more4u for the video clip). Ms. Bassham’s class is working on the Missing Function task shown on the next page.

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The Missing Function Task

Taken from Institute for Learning (2013a). Lesson guides and student workbooks are available at ifl.pitt.edu.

Ms. Bassham’s lesson takes place in an algebra 2 classroom with 16 students. By engaging in the Missing Function lesson, Ms. Bassham intends for students to —   1. explore the meaning of multiplying functions by use of tables and graphs; and

  2. develop an understanding that the x-intercepts of a quadratic function (when they exist) consist of the x-intercepts of the two linear functions whose product defines it. The Missing Function task occurs within a sequence of related lessons on building polynomial functions, created by the Institute for Learning, University of Pittsburgh (2013a). By engaging in this set of lessons, students will — • deepen understanding of how multiple representations are used to represent the same relationship between two variable quantities;

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• learn how to combine polynomial functions arithmetically using multiple representations; and

• understand the relationship between factors and zeros of a polynomial function (Institute for Learning 2013a, p. 7). Prior to the Missing Function lesson, students developed methods of determining the sum and difference of polynomial functions using a graph, table, and symbolic representations. Students used arithmetic to combine polynomial functions (represented in tables, graphs, and equations) and then explored the relationship between the original and new functions. For example, students were asked to consider the sum and difference of a linear function and quadratic function represented in a table of values and to justify the general statement about the sum of different polynomial functions (e.g., “The graph of the sum of linear and quadratic function is always a parabola”). At the beginning of the Missing Function lesson, the class generated the chart in figure 5.3 to identify what information they could gather from the table of values for f (x).

Fig. 5.3. Chart generated by Ms. Bassham’s students

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Analyzing Teaching and Learning 5.2 more Posing Questions to Move the Mathematics Forward For this activity, you will watch the video clip of Jamie Bassham’s class, where students are working on the Missing Function task in small groups and then discussing it as a whole class. But before watching the video, solve the Missing Function task in more than one way. If possible, compare and discuss your strategies with a colleague. • What do you notice about the questions the teacher asked? • How do the teacher’s questions move the mathematics in the lesson forward? You can access and download the video and its transcript 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: Posing Questions to Move the Mathematics Forward Students in Ms. Bassham’s class were grappling with determining how to represent a function given two other functions. Groups of students used the graph to find values for h(x), and then determined values of g (x) by dividing specific h(x) values by the corresponding f (x) values. (This may be the result of students’ previous experiences in using tables of values to determine the resulting function when adding or subtracting two polynomials.) Next, students plotted the x-values and the results of the division as the new function g (x). They analyzed the graph to determine the equation of the resulting function g (x). Some students determined the equation of h(x); for example, one group indicated that they used the graphing calculator to determine h(x). In the final presentation, a student multiplied f (x) and g (x) using FOIL (First-OutsideInside-Last) and compared the resulting expression to h(x) as determined by the graphing calculator. She also commented that her group plotted the points of h(x) and they “worked.” In the video, Ms. Bassham visited three groups of students with the intention of asking purposeful questions. She began her interaction with each group by asking a question to help her probe, understand, and assess students’ thinking about the task (e.g., “What are we thinking about here?” [line 1 in the transcript]; “What are we doing here?” [line 15]; “How are we doing over here, girls?” [line 29]). Through these questions, Ms. Bassham learned how students were approaching the task.

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According to NCTM (2000), asking questions that reveal students’ knowledge about mathematics allows teachers to design instruction that responds to and builds on this knowledge. Note the role that probing thinking questions play in providing formative assessment information for the teacher by revealing students’ knowledge and thinking. Hence, we also refer to these questions as assessing questions because they enable the teacher to determine what the student knows and understands about key mathematical ideas, problemsolving strategies, or representations. Assessing questions are closely tied to the mathematical work and thinking produced by the student. When Ms. Bassham left a group, she often asked a question (or made a comment) to advance or encourage students’ thinking (lines 13–14, 27–28, 52–54). For example: • Group 1 decided to look for the points that make up g (x). Ms. Bassham asked, “How are you going to get that [ g (x)]?” (line 13), and then walked away.

• Ms. Bassham asked the second group, “We want to see what we’re going to multiply by, what you can come up with this” (lines 21–22) and encouraged them to continue their line of thinking as she walked away.

• Ms. Bassham challenged group 3 to clarify their thinking on their use of multiplication and division. When students began to articulate the way they determined the values of g (x), the teacher stepped away from the group. The purpose of advancing questions is to support students in moving forward in solving the task beyond their current work and thinking or to explore the underlying mathematics more deeply. These questions move students toward the targeted goal of the lesson. Assessing questions enable teachers to elicit student thinking, and advancing questions position teachers to use student thinking to move students toward the goals of the lesson. Assessing and advancing questions asked during small-group work support students’ readiness to engage in mathematical discourse during the whole-group discussion. The characteristics of assessing and advancing questions are shown in figure 5.4.

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Assessing Questions

• Based closely on the work the students have produced • Clarify what the students have done and what they understand about what they have done • Provide information to the teacher about what the students understand

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 them to extend what they know to a new situation • Press students to think about something they are not currently thinking about Teacher WALKS AWAY, leaving students to figure out how to proceed.

Fig. 5.4. Characteristics of assessing and advancing questions (Developed by Victoria Bill and Margaret Smith 2008)

During the whole-class discussion, Ms. Bassham’s questions moved students’ thinking forward toward the two main goals of the lesson. Many of these questions exemplify the making the mathematics visible type of questions. First, Ms. Bassham asked questions to focus students’ learning on the goal of “exploring the meaning of multiplying functions by use of tables and graphs.” She asked students to explain how they determined the values for g (x) and highlighted the “inverse” relationship between multiplication and division (lines 91–98, 166). Second, Ms. Bassham asked questions to focus students’ learning on the goal of “understanding that the x-intercepts of a quadratic function (when they exist) consist of the x-intercepts of the two linear functions whose product defines it.” Students were asked to examine “key points” on the graph, especially those with “special names” (lines 111, 118–21). Ms. Bassham wanted students to conclude that each x-intercept of the quadratic function provides the x-intercept of each of the linear functions. Students articulated that connection through the teacher’s questions. Ms. Bassham continued to ask students to consider the relationship between the equations and the graphs, with particular attention to where the graphs cross the x-axis (lines 130–31, 144–46). Toward the close of the lesson, she also asked a question to encourage reflection and justification: “Why do you think this is g (x)?” (line 152).

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Ms. Bassham’s questions were directly tied to her students’ thinking. She supported productive struggle through the questions she asked, particularly sequences of assessing and advancing questions. As Ms. Bassham circulated among small groups, she never told students that they were right or wrong but asked questions and indicated that they were “on the right track.” Ms. Bassham elicited and made use of evidence of student thinking and contributions. She actively listened to what students were saying and formed the next question or teaching move on the basis of their responses. Consider what it takes on the part of the teacher to ask questions that respond to students’ current thinking and move that thinking toward the mathematical goals of the lesson. As Ms. Bassham engaged in planning the Missing Function lesson, she considered ways her students might approach the task and crafted questions to support students’ thinking toward the goals of the lesson. During the lesson, even though she needed to listen to and interpret students’ ideas and strategies “on the spot,” Ms. Bassham had already anticipated many of these strategies and how she would respond in order to move students’ thinking forward.

Planning Assessing and Advancing Questions In ATL 5.3, you will develop questions that assess and advance students’ thinking on the basis of a set of students’ work for the Comparing Products task, which follows the Missing Function task in the sequence of lessons (Institute for Learning 2013a, p. 43). In this task, students generate and compare several examples of the product of two linear functions and make generalizations about characteristics of the product function that are determined by the relationship between the linear factors. The goals of the Comparing Products lesson are for students to develop the understanding that (1) polynomials can be written as the product of linear factors, and (2) the product of two linear functions is always a quadratic function. Before engaging in ATL 5.3, solve the Comparing Products task on the next page. If possible, discuss with colleagues your strategies and other strategies you predict students might use.

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Analyzing Teaching and Learning 5.3 Planning Assessing and Advancing Questions Review the three samples of students’ work shown in figure 5.5. For each response consider the following: • What does the student appear to know and understand about mathematics? (Be sure to justify your conclusions.) • What questions could you ask in order to— — assess the student’s thinking? — advance the student’s thinking? — support the student in making mathematical connections? Once you (and your colleagues) have developed assessing and advancing questions, revisit your list of questions and identify examples of the four types of questions: (1) gathering information, (2) probing thinking, (3) making the mathematics visible, and (4) encouraging reflection and justification.

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The Comparing Products Task

Taken from Institute for Learning (2013a). Lesson guides and student workbooks are available at ifl.pitt.edu.

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Student A

Student B

Student C

Fig. 5.5. Samples of students’ work for the Comparing Products task

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Analysis of ATL 5.3: Planning Assessing and Advancing Questions The Comparing Products task provides an opportunity for students to investigate the function that results from multiplying two linear functions. Students are asked to choose any two linear functions. The task then directs them to compare the resulting function when the original functions form parallel, perpendicular, and intersecting lines. By examining multiple examples within their group, students should conclude that the product of two linear functions forms a quadratic function, as long as the functions are not constant functions (such as y 5 2). By examining the products of specific types of lines, students should discover that the slopes of the two lines, when multiplied together, form the coefficient a of the resulting quadratic function, y 5 ax2 1 bx 1 c. Hence, the product of the slopes determines whether the resulting parabola will open up (a  0) or down (a  0). The product of linear functions that represent parallel lines will always result in a positive leading coefficient and a parabola that has a minimum (and hence opens “up”), because the slopes of parallel lines are equal. If the linear functions represent perpendicular lines, the leading coefficient will be negative (the slopes of perpendicular lines have opposite signs), and the resulting parabola will have a maximum (and hence, opens “down”). (Consider also what happens when the same line is used for both factors.) Students may have a more difficult time generalizing when the two linear functions are intersecting and oblique (i.e., not perpendicular). If the slopes are both positive or both negative, the resulting parabola will have a minimum. If the slope of one line is positive and the slope of the other line is negative, the resulting parabola will have a maximum. The results mentioned in the previous paragraph remain consistent. (Consider why these generalizations hold.) In figure 5.5, student A investigates two perpendicular lines, f (x) 5 x 1 4 and g (x) 5 2x 1 4. Notice that the student has incorrectly identified g (x) 5 x 2 4. The student has correctly graphed the product of (x 1 4)(x 2 4) though the symbolic form of the function is not apparent in the student’s work. A teacher may begin to assess this student by asking, “Tell me about your lines. How did you determine the function for each line?” If the student self-corrects, the teacher may advance the student’s thinking by asking, “How will the change in g (x) affect the product? How will this affect your graph?” The teacher might also press the student to identify additional characteristics of the parabola or to compare the graph and parabola to those of other students. Student B investigates parallel lines. Using the student’s labels, “line x” is y 5 x 1 2 and “line y” is y 5 x. The student lists the y-intercept and x-intercept for each line and identifies the product as y 5 x2. The student has also made the (albeit vague) generalization regarding maximums and minimums. Since the intercepts are not labeled, the teacher might assess the student by asking, “Tell me about what you have written: y 5 2, x 5 22, y 5 0, and x 5 0.

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What do they represent?” A follow-up question could center on the incorrect product (y 5 x2): “How did you determine the product of your two linear functions?” The teacher may want to advance this student’s thinking by asking, “You say that some products can have a maximum and others have a minimum; how can we predict when each will occur?” Student C chooses to investigate intersecting lines using the expressions x 1 3 and 2x. The student has correctly shown the steps in multiplying the two linear functions to find the product, 2x2 1 6x. The student also makes generalizations regarding the parabolas formed by different types of lines. However, the student’s graphs do not display y 5 x 1 3 or y 5 2x. The teacher may choose to assess the student’s understanding of graphing by asking, “How did you determine how to graph the linear functions you chose?” The teacher could then advance the student’s thinking by asking, “How will the results change based on your new graphs?” or “How did you determine your generalizations? Why are they true?” In ATL 5.3, you developed questions that serve to assess and advance students’ thinking. As in Ms. Bassham’s lesson, assessing and advancing questions are particularly useful as students work in small groups. After the teacher asks an assessing question, it is important that he or she listen to the student’s thinking in order to understand where the student is mathematically. Only then can the teacher ask an appropriate question to move the student’s thinking forward. After asking an advancing question, however, the teacher might walk away (to the next group) and leave the student or group to ponder and investigate the question using their own resources. It is helpful for teachers to keep student work that illustrates typical strategies, unique strategies, and common misconceptions to assist in planning future lessons focused on the same task or similar tasks. These student-work artifacts not only help teachers design potential questions but also help them in their whole-class debriefing of strategies or misconceptions that have not been revealed in the current class summary.

Patterns of Questions In ATL 5.4 you will engage in solving the Playground task and analyzing a video clip (see More4U at nctm.org/more4u) of a lesson featuring this task in order to identify different types of questions asked by the teacher. The video is from the classroom of Debra Campbell and features students in her geometry class. In using the Playground task, Ms. Campbell intends for students to understand the following:   1. The points equidistant from two given points in a plane form a line.

  2. The line that is formed bisects the segment connecting the two given points.

  3. The line that is formed is perpendicular to the segment connecting the two given points. Pose Purposeful Questions   89 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.

  4. Previously learned definitions, postulates, and theorems may be used to prove that the “new” line is the perpendicular bisector of the segment connecting the two given points. Prior to the lesson, Ms. Campbell’s students have worked with midpoint, distance, perpendicularity, and related angle and linear concepts. Students have also proved triangles congruent with SAS (Side-Angle-Side), SSS (Side-Side-Side), and ASA (Angle-Side-Angle). In the video clip, Ms. Campbell is working with small groups of students, asking questions to ensure that students continue working on the task as well as to help students focus their thinking and progress toward a solution.

Analyzing Teaching and Learning 5.4 more Patterns of Questions Begin by solving the Playground task on the next page. If possible, discuss with colleagues your strategies and other strategies you predict students might use. Watch the Playground task video. Highlight on the transcript the questions Ms. Campbell asks as she interacts with pairs and groups of students: • Note when she uses an assessing question and how the question is tied directly to the students’ current work and thinking. • Note when she uses an advancing question and how the question serves to move students’ current work and thinking forward. • Identify examples of the four types of questions: (1) gathering information, (2) probing thinking, (3) making the mathematics visible, and (4) encouraging reflection and justification. You can access and download the video and its transcript 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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The Playground Task

Taken from Institute for Learning (2013b). Lesson guides and student workbooks are available at ifl.pitt.edu.

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Analysis of ATL 5.4: Patterns of Questions As with other cases in this chapter, Ms. Campbell used a task that promoted reasoning and problem solving, and this task provided opportunities for her to ask purposeful questions that moved students toward the mathematical goals of the lesson. In turn, Ms. Campbell’s questioning throughout the lesson supported students’ engagement in thinking and reasoning and hence maintained high-level cognitive demands as the task was implemented. Ms. Campbell posed purposeful questions throughout the video. She asked many assessing questions to prompt students to explain what they did, how they did it, and why they did it. These questions often fall into the categories of gathering information and probing thinking, as they served to probe and clarify students’ thinking and make it visible and accessible to the teacher and to other students. Notice how closely these questions were tied to students’ current work and thinking (line numbers are in the transcript) : • Line 1: “So, okay, how did you come up with 3 points?” (probing thinking)

• Line 11: “How did you draw a square? Where’s your square at?” (gathering information) • Lines 20–21: “Okay, so how are you using that midpoint formula to make sense of this problem?” (probing thinking)

• Line 47: “But how did you construct that line? How do you construct that line?” (probing thinking) • Line 60: “How did you know how to draw your line?” (probing thinking)

• Line 61: “What do you mean ‘perpendicular’?” (gathering information) • Lines 79–85: “Could I do that?” (probing thinking)

Ms. Campbell also asked advancing questions that served to move students beyond their current thinking, make connections, justify why something does or does not work, and explain a pattern. These questions often exemplify the categories of making the mathematics visible and encouraging reflection and justification. After Ms. Campbell asked an advancing question, she often moved on to the next group. Also notice how Ms. Campbell “made good” on her promises to return to groups to check their progress. • Lines 17–18: “Could you prove to me — I’m going to come back and see if you guys can prove to me that there could be more points?” (encouraging reflection and justification)

• Lines 28–29: “So what does that line mean? I’m going to come back to that. I want you to be able to tell me what does that little line mean.” (making the mathematics visible; encouraging reflection and justification) • Lines 43–44: “So you’re going to have to tell me how you can use that slope. I’m going to come back and ask you how can you use that slope.” (making the mathematics visible)

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• Lines 56–58: “Is there a pattern that you’re noticing if there’s more than three points? Be ready to tell me about what pattern do you see. If there’s more than three points, what can you tell me about the pattern of seeing those three points?” (making the mathematics visible) Ms. Campbell used a consistent pattern of question types. She began with questions to probe thinking or gather information. Once she understood students’ thinking and strategies, she asked questions to encourage reflection and justification or to make the mathematics visible. For example, we see this pattern in the first interaction with students in the video. Ms. Campbell began by asking questions to probe students’ thinking and gather information about the group’s strategy of drawing a square (lines 1–11), such as “How did you draw a square? Where’s your square at?” (line 11). She followed up with questions to move students’ thinking forward by encouraging reflection and justification (lines 14, 17–18). Once she asked the group an advancing question, “Could you prove to me — I’m going to come back and see if you guys can prove to me that there could be more points?” (lines 17–18), she then left to interact with the next group. (Although Ms. Campbell used the phrase “Could you prove . . . ?” she was asking for a justification rather than an argument that meets the higher standard of proof.) Ms. Campbell’s pattern of questioning (e.g., beginning with probing thinking or gathering information questions and then moving to questions that encourage reflection and justification or make the mathematics visible) enables a teacher to elicit and use students’ thinking. Questions serve to make a student’s thinking visible for the teacher, the student, and other students in the group. In the process of responding to probing thinking or gathering information questions, students often “see” a next step, make a connection, or realize an error on their own as they respond to the question. Questions that encourage reflection and justification or make the mathematics visible support students in taking the next step toward the goals of the lesson on the basis of their current mathematical work and thinking, using ideas and strategies that make sense to them. In this way, as described by the research presented in the next section, teacher’s questions can serve to support students’ mathematical engagement, understanding, and achievement.

Pose Purposeful Questions: What Research Has to Say The questions a teacher asks as 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. Teacher’s questions can support high-level thinking, prompt for explanations and connections, and encourage students to delve more deeply into mathematics. Teachers’ questions can also serve to elicit facts, procedures, or calculations.

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As noted in Principles to Actions, research has identified the importance of types of questions and patterns of questioning. Several frameworks have been developed to categorize types of teachers’ questions (e.g., Boaler and Brodie 2004; Chapin and O’Connor 2007; Sorto et al. 2009; Wimer et al. 2001). These frameworks call attention to ways that teachers’ questions elicit different levels and types of students’ mathematical knowledge and thinking and prompt students to engage with mathematics in different ways. For example, consistent with ideas in Principles to Actions that teachers’ questions should prompt students to “explain and reflect on their own thinking,” Wimer’s (2001) categories of higher-order and lower-order questions identify differences in the cognitive processes elicited by different types of questions. Higherorder questions ask students to think deeply about mathematics (e.g., these questions often contain the word “why?”), make mathematical connections, or provide mathematical reasoning. Lower-order questions ask students to demonstrate recall of facts, properties, or procedures. Similarly, the framework featured in Principles to Actions (NCTM 2014, pp. 36–37) and discussed throughout this chapter identifies different purposes for questions (e.g., gathering information, probing thinking, making the mathematics visible, and encouraging reflection and justification). Frameworks can help teachers analyze their own questioning types, raise awareness of the types of questions they are asking, and identify which types of questions are asked to which students. Frameworks can also help teachers analyze their patterns of questioning. Different types of questions might occur in patterns that help teachers understand and respond to students’ thinking (Walsh and Sattes 2005). For example, a question intended to gather facts or information, while considered a lower-order question, might help a teacher assess a student’s prior knowledge or current understanding before the teacher is able to ask a deeper conceptual question. Similarly, as presented in this chapter, teachers may need to ask a number of assessing questions before asking advancing questions that move students’ thinking forward. Research indicates that assessing and advancing questions are particularly beneficial in supporting students during small-group work and in preparing students to engage in mathematical discourse during whole-group discussions (Smith and Stein 2011; Stein et al. 2008; Stein et al. 2009). Traditionally, mathematics lessons featured a questioning pattern of “Initiate-ResponseEvaluate” or I-R-E (Mehan 1979), where 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. Incorporating the open-ended types of questions discussed in this chapter (e.g., probing thinking, making the mathematics visible, encouraging reflection and justification) can help teachers move beyond the I-R-E pattern and create space for studentgenerated ideas, strategies, and representations in mathematics lessons.

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Promoting Equity by Posing Purposeful Questions Through the use of questioning, teachers can create equitable learning opportunities in mathematics classrooms. The questions a student is asked, and how a teacher follows up on the student’s response, can support the student’s development of a positive mathematical identify and sense of agency as a “thinker and doer of mathematics” (Aguirre, Mayfield-Ingram, and Martin 2013). By making an effort to listen to and understand students’ thinking (by asking assessing questions, probing students’ thinking, and encouraging reflection and justification), teachers indicate that students’ thinking makes sense. Communicating a respect for students’ thinking, even thinking that is still in development or only partially correct (such as when Ms. Culver asked the student who agreed with y 5 3x to explain her thinking), invites more students to participate and to feel capable of participating. In contrast, classrooms characterized by I-R-E patterns of questioning reward correctness and often position students who are quickest to achieve the correct answer as “smart” mathematically. Funneling patterns of questioning can also position “smart” students as those who use strategies similar to the one being advocated by the teacher (often the most mathematically efficient strategy) at the expense of strategies that develop conceptual understanding or serve as “stepping-stones” to procedural fluency. Asking questions that provide space for students’ ideas, strategies, representations, and explanations expands opportunities for students to be positioned as capable in a mathematics lesson. This does not imply that “anything goes” or that mathematical correctness or precision does not matter. Instead, teachers ask questions that support productive struggle and honor the thinking process as well as the final result. Teachers can sequence the questions they ask and the order in which ideas and strategies are shared to build toward the mathematical goals of the lesson (e.g., the focusing pattern; using questions that make the mathematics visible). Students’ efforts to think through a problem and explain and justify their own thinking come to be seen as characteristics of “smartness” in the mathematics classroom. Furthermore, teachers can follow up with students’ contributions in ways that assign competence (Cohen et al. 1999) to particular students, such as verbally marking a student’s ideas as interesting, noting a student’s effort, or identifying an important aspect of a student’s strategy (e.g., Ms. Culver refers to ideas offered by Angela and Chris). While the suggestions regarding purposeful questioning throughout this chapter would serve to benefit all students, we encourage teachers to pay particular attention to who is being asked which types of questions, who is being positioned as competent, and whose ideas are featured and privileged. Supporting a strong mathematical identity and sense of agency and belonging in mathematics is especially important for our students of historically underserved populations (e.g., black, Latino/a, American Indian, or members of other minorities; students of poverty; students who are English language learners; students who have not been successful

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in school and in mathematics; and students whose parents have had limited access to educational opportunities [NCTM 2014]). Equitable opportunities for participation and 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 students’ thinking, but do not take over or funnel students’ thinking, serve to 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 them. (Chapter 7 will provide additional opportunities to consider the importance of lesson planning.)

Taking Action In Your Classroom: Analyzing Questions And Responses In Taking Action in Your Classroom 5.1 and 5.2, we invite you to explore the questions you asked during a lesson and to consider what insights these questions give you about students’ understanding of mathematics.

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Taking Action in Your Classroom 5.1 Teach a lesson around a high-level task. Videotape a lesson, or have a colleague observe you, and list all of the questions you ask during small-group work and during the whole-group discussion. Reflect on the questions you asked and consider the extent to which the questions— • 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. Reflect on your questions through a lens of equity: • Who is being asked each type of question? • In what ways are your questions marking students’ thinking as valid or assigning competence? • In what ways are your questions serving to hold students accountable for high-level products and processes, while also supporting students to achieve at high levels?

Taking Action in Your Classroom 5.2 Plan a lesson around a high-level task that has the potential to elicit a variety of strategies. Solve the task using the different strategies you anticipate that students might use. (Compare and discuss the strategies with a colleague, if possible.) For each strategy, prepare questions you can ask to assess and advance students’ learning toward the goals of the lesson. As you teach the lesson, circulate between small groups, and ask the questions you have planned (and other assessing and advancing questions based on students’ specific work and responses). Take note of which questions you asked and how the questions served to— • make students’ thinking visible (assessing questions); or • move students’ thinking forward (advancing questions).

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CHAPTER 6

Use and Connect Mathematical Representations The Analyzing Teaching and Learning 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. Principles and Standards for School Mathematics refers to representations as the process “of capturing a mathematical concept or relationship in some form and to the form itself ” (NCTM 2000, p. 67). As shown in figure 6.1, representations refer to visual diagrams, symbolic notation, verbal descriptions, contextual situations, and physical models. For example, the numeric symbol “2” is a representation, as well as the traditional algebraic notation of variables, such as x and y. Algebra tiles, graphs, and tables are representations frequently used in high school math classes. However, there is a danger of these representations being taught as discrete skills and as an end to themselves. Hence, making connections between different representations is crucial. The arrows that connect each pair of representations in figure 6.1 show the bidirectional nature of the relationship—you can start with any of the representational forms and relate it to the others. Representations play multiple roles in learning. First and foremost, they are the basis of student understanding of mathematical ideas. Additionally, representations help students make sense of mathematical contexts, recognize connections among related mathematics concepts, and communicate mathematical understanding and reasoning.

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Visual (diagrams, graphs and pictures)

Symbolic

Physical

(algebraic and numeric)

(manipulatives and models)

Contextual

Verbal

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

In this chapter, you will— • explore different representations used to solve problems;

• consider student representations for problems and how each contributes to student understanding; • analyze a task and an instructional episode in which students share different representations;

• develop questions that can be asked to help students make connections between and among different representational forms; • review key research findings related to mathematical representations; and • reflect on representation use in your own classroom.

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For each Analyzing Teaching and Learning (ATL), make note of 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 and ideas with colleagues. Once you have written down or shared your ideas, read the analysis, where we offer ideas relating the Analyzing Teaching and Learning activity to the focal teaching practice.

Exploring Different Representations In Analyzing Teaching and Learning 6.1, you will be solving and analyzing solutions to a task so that you can consider how different representations offer access and reasoning opportunities for students. The task is from Anthony Bokar’s geometry class. Mr. Bokar wanted a task that showed connections between algebra and geometry, specifically the use of related rates, concepts of angles and circles, and the creation of equations. He chose the Clock Hands task because it was accessible to all students since it could be modeled physically or solved symbolically, and it made connections to concepts about circles, such as central angles and degrees in sectors of a circle.

Analyzing Teaching and Learning 6.1 Exploring Different Representations Solve the Clock Hands task on the next page. Use as many different representations as you can to solve the problem. Consider the student responses in figure 6.2. • What does each representation tell you about what the student understands about the situation? Are students making a connection between geometric concepts of degrees and circles and rates of change and algebraic representations? • How are the different representations connected to each other?

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The Clock Hands Task The hands of an analog clock are overlapping at midnight. When is the next time they overlap? How many times will the minute hand and the hour hand overlap in the twelve-hour span between midnight and noon? Justify your solutions.

Student A

Student B

Student C

Student C Table

continued on next page

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Student D

Student E

Fig. 6.2. Student solutions to the Clock Hands task

Analysis of ATL 6.1: Exploring Different Representations The problem of the Clock Hands task can be solved in a variety of ways. Possibilities include making drawings, considering the related rates of the two hands arithmetically, creating and solving an equation algebraically, and acting it out physically. Figure 6.2 shows a variety of solutions that reflect different degrees of sophistication in approaching the problem and in students’ abilities to generalize their thoughts and use a range of different representations. Student A used drawings of a clock (three of the drawings are shown) to show the hands meeting at about 1:05, 2:10, . . . , 11:55, and then noon. The student understood that both hands were moving but did not take into account that the hour hand continued moving after 1:00 or consider the different rates at which the hands travel. This is an accessible approach that relied only on the drawing. Student B used the physical model of a clock to consider the problem and correctly concluded that the hands meet 10 times. The physical model shows more understanding of the movement of the clock hands than that of student A. The connection between the drawings (student A) and the physical model (student B) reveals important characteristics of the problem: the hands move at different rates, and the hour hand continues to move as the minute hand completes one rotation. That is, the student using the real clock sees that the hour hand is continually moving so that the overlap does not occur at exactly 1:05 but a little bit afterward. This understanding leads to a connection with student C’s work. Use and Connect Mathematical Representations    103 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 C calculated the rate of change of each hand in terms of a rotation. This student recognized that the minute hand made one rotation in an hour but that the hour hand moved 1 /12 of a rotation. The student states that in a twelve-hour period, the minute hand completed 12 rotations around the clock face but that the hour hand only went around once. The student acknowledged the covariation of the hands—the minute hand moved 12 times as fast as the hour hand. For the case of finding the overlap after midnight, the student understood that the minute hand would move a little more than one rotation. Using previous knowledge that a circle contains 360°, student C connected with the statement about the related rates (the hour hand moves 1/12 of a rotation when the minute hand moves one full rotation) to determine that the minute hand moved an additional 30° before overlapping the hour hand. While the minute hand is moving 30°, the hour hand is moving about 1/12 of 30°, so the student estimated that the hands met a little after 1:05.41. The student then reasoned that the hands would meet again every 1 hour and 5.41 minutes (“1:05.41” in the student’s notation), and she created a table to show all the times the hands met. There is evidence that the student saw the pattern of the same time interval occurring for each intersection and recognized that the hands do not meet between 11 and 12. Student D created an equation based on the concept of the minute hand traveling 360° in an hour, while the hour hand only travels 30°. The student made a connection between the degrees in a circle and rate of change. Then the student solved the equation by combining like terms and concluded that the hands met at 1:05 5/11. Although the student found this answer, he was not able to use the same reasoning to find the next solution and so could not generalize. The student did not seem to consider using 1:05 5/11 as the starting point and then using the same solution pattern as before. If he had done this, he would have found that the hands met after 30° and a little bit more each time. He could have used the same equation over and over by just adjusting the variable to stand for time elapsed after midnight (start time), then time elapsed since 1:05 5/11 (first overlap), time elapsed since 2:10 10/11 (second overlap), and so forth. Because the student couldn’t find another solution, he couldn’t generalize the solution. This representation connects to that of student C because they were both looking at the rotation as being 360°. The connection between 30° and 360° to 1/12 of a rotation and one complete rotation, respectively, allows students to consider that the relative rates of change are the same no matter which measurement is used. The students can see how their estimates are not as precise as using related rates and that they can obtain a greater degree of accuracy by using an equation. By combining the reasoning in the table (student C’s solution) with the solution by student D, all students can understand how to generalize the solution. Student E made an equation using the concept of rotation (1 rotation per hour for the minute hand) and solved that for the first overlap. Then, she used a similar equation for the next meeting but used 2 rotations instead of one and found the time to be 2:10 10/11. When the student compared the two answers, she saw that the difference from the first overlap to

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the next was 5 5/11 minutes. Though there were only two examples, the student generalized to count 10 overlaps. This equation connects to that of students C and D—first by considering the rotations and then by recognizing the pattern by adding to 1:05 5/11. The representations used to solve the Clock Hands task show different levels of sophistication, ranging from an easily accessible method that showed a basic understanding of the problem but did not attend to the related rates of the hands (student A) to the algebraic approaches that culminated in a complete solution (student E). While the use of related rates and the use of an equation were the intended targets of instruction, the approaches used by students A, B, and C reveal their thinking about the situation and can ultimately be linked to the algebraic representation. The different representations allowed all students to enter the problem and to make sense of the relationship between the movements of the hands. 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. Clearly, classroom discourse centered on helping students make connections among representations is needed. When students are discussing different representations, it is important that they consider what is shown by each representation. That is, the representation is not an end in itself but a tool for making sense of the problem and then working toward a solution. The representations and the connections among them help build an understanding of mathematical concepts and ideas. The classroom discussion should focus on enabling students to understand that written representations of mathematical ideas are an essential part of learning and doing mathematics. It is important to encourage students to represent their ideas in ways that make sense to them. 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.

How might a teacher use the different representations produced by students during an actual lesson? What purpose would these representations serve? To answer these questions, we look at a discussion in Monica Tienda’s Algebra 2 class in which students are exploring a problem involving conditional probability.

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Exploring the Use of Representations during Instruction In ATL 6.2, we drop in on a discussion in Monica Tienda’s algebra 2 class where students are exploring a conditional probability problem. Ms. Tienda had three goals for her students’ learning in this lesson. She wanted them to— • see situations in which the sample space for a probability problem is restricted to create a conditional probability situation;

• consider probability in a real context to determine whether it is a case for simple probability, compound probability, or something else; and

• understand that different mathematical representations can be used to make sense of and solve a task and understand the relationship between different representations. First, Ms. Tienda wanted students to see situations in which the sample space for a probability problem in context is restricted to create a conditional probability situation. That is, considering the probability of event A occurring when event B has already occurred means to look only at the events in the sample space for event B, and then to calculate the probability requested in terms of the sample space of B as opposed to the entire sample space. Ms. Tienda was aiming for students to recognize this important fact about conditional probability. Next, she wanted her students to consider probability in a real context and to determine whether it is a case for simple probability (a ratio of desired outcomes to total outcomes in the sample space), compound probability (the probability of event A or event B, which may involve a case where the outcomes for two events are counted, taking into account that some items may be double-counted), or something else. In this problem, the result is “something else,” which will be defined as conditional probability. Finally, she wanted her students to understand that different mathematical representations can be used to make sense of and solve a task, and she also wanted them to understand the relationship between different representations. Ms. Tienda selected a task that aligned well with her goals for this lesson, a critical factor in the potential for success in a lesson.

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Analyzing Teaching and Learning 6.2 Exploring the Use of Representations during Instruction Solve the Student Homework problem shown below. Read the excerpt from the discussion that took place in Ms. Tienda’s classroom. • What different representations did Ms. Tienda’s students produce? • What did Ms. Tienda do to help her students make connections among the different representations? • How do you think the use of these different representations supported students learning?

The Student Homework Problem Math teachers frequently claim that doing homework helps students get better grades in their math classes. To test this claim, a survey of high school math students was conducted from which the following results were obtained: • 48% completed math homework regularly. • 55% have a B average or better in math class. • 40% do not complete math homework regularly AND have less than a B average in math class. Do these data support the claim that students who complete math homework regularly are more likely to get a B or better in math class? Justify your answer using mathematics. Source: Oregon Department of Education (2016).

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The Use of Representations in Algebra 2 The Case of Monica Tienda Monica Tienda distributed copies of the Student Homework problem to her students. After some initial discussion of key parts of the problem, such as the difference between the first two statements and the combined statement, she told students to solve the task any way they wished. While students began work on the task, she monitored the activities of each group, intervening as needed and keeping track of who did what. When it seemed that the groups had all arrived at an answer, she called them together for a whole-group discussion. Below is an excerpt of what transpired. Diamond [in group 4], will you use the document camera and explain what your group did?

1 2

Ms. T:

3 4 5 6 7

Diamond: We made a table that showed different percents of students who got each part. We had two different things to consider. One of them was whether a student got a B or higher or not. We made those the row labels on our table. But we also had to consider whether students regularly did homework or not. We used that for the column headings.

Fig. 6.3. Group 4’s solution

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8 9 10 11 12 13

Diamond: We knew that 40% of the students who didn’t do homework regularly got less than a B, so we filled in the cell for “less than B/Don’t” with 40/100. Since 55% of the students had a B, we knew 45% altogether got less than a B. So, that left 5/100 for the “less than B/Do” cell. Then, since 48% of the students were in the “Do” column altogether, 43/100 for the “B or more/Do” cell. Then there was only 12/100 for the last cell. Does anyone have any questions for Diamond?

14

Ms. T:

15

Ernesto: How does that show us the answer?

16 17 18

Diamond: Only 12% of the students who didn’t do homework got a B or higher, but if you look at the 48% of students who did homework, 43% of them got a B or higher. Did 12% of students who didn’t do homework get a B or higher, or did 12% of all students not do homework and get a B or higher?

19 20

Ms. T:

21 22

Diamond: I’m not sure which. I think 12% of all the students both did not do homework and got a B or higher. Yeah, I can see that on the table.

23 24 25

Ms. T:

Several groups used a Venn diagram that showed us some of the same information as in group 4’s table. Chris [group 2], please share your group’s work with us.

Fig. 6.4. Group 2’s solution 26 27 28 29 30 31 32 33 34

Chris:

We started our Venn diagram by thinking about 100 people and where the 40 who didn’t do homework and who didn’t get at least a B would go. We decided it made sense to put that outside our circles. We knew doing homework and getting a B or more had to intersect because we had more than 100 people. So, we made two intersecting circles. Since 48 doing homework plus 55 getting a B plus 40 not doing homework and getting less than a B makes 143, we had 43 too many people. That meant we had counted them twice, so we put 43 in the intersection and subtracted to get the other values.

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[To the entire class] How do the values in the Venn diagram relate to the table? Who notices something about both representations? [Wait time]

35 36

Ms. T:

37 38 39

Pat:

40 41 42 43 44 45

Chris:

46

Ms. T:

47 48 49 50

Ernesto: They kind of say the same thing, except Diamond’s group didn’t figure out the percents. But they still talked about 43% out of 48% and 12% of people in the survey. They just used percent instead of making it out of 100 people. It’s the same thing.

51 52 53 54

I see the 43 right from the table. The 43 in the table is in the “B and more/ Do” square. It’s the same as the intersection of the circles. That’s because they stand for the same thing!

I see the 5 and 12 from the diagram, now. But we didn’t solve it like group 4. We said 43 of the 48 people who did homework made 89.6%. So there is like a 90% probability of getting a B or more if you do homework. But only 12 people out of 100 got a B or higher who didn’t do homework. So the people who don’t do homework only have a 12% chance of getting a B or higher. Can someone compare these two answers? [Wait time]

Diamond: In the diagram from Chris’s group, she really only used the numbers that are in the circle for Homework and then made the probability. In our table, we really only used the column for Homework and then used that number to make the same probability. We didn’t use all of the population. Remember we called the entire population the sample space.

55

Ms. T:

56

Diamond: So, we didn’t use the entire sample space, right?

57 58 59

Ms. T:

Yes! Sometimes mathematicians say we used a restricted sample space. Group 5 did something a little different looking. Kalei [group 5], please share what your group did.

Fig. 6.5. Group 5’s solution

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We made a tree diagram. It was kind of easy at first, but then it got a little harder. We started off with two choices, either did homework or not. So, our first branch shows that with .48 on the “does” branch and .52 on the “doesn’t” branch. Then we made two branches on each point to show less than B and B or more. This got tougher. We knew that “doesn’t and less than B” had to be .40, and “doesn’t and B or more” had to be .12. We worked backwards to get the numbers on the branches. We had multiplied to get the final values. That meant that .52 times the missing branch made .4, so the missing branch value was about .77. Then the other branch for “doesn’t” had to be .23. Then we did the same thing on the “does” branches. We followed the branches, saw the .896 for “does and B,” which we knew was higher than the .12 who didn’t do homework but got a B.

60 61 62 63 64 65 66 67 68 69 70 71

Kalei:

72

Ms. T:

73 74

Chris:

75 76

Ms. T:

77 78

Blake:

79 80

Diamond: Even though we didn’t divide anything out, we still explained our answer. It’s the same.

81 82 83

Ms. T:

Chris, I see you have something to say.

We had 89.6% in our Venn diagram work. The 43 out of 48 is the same as the branch of people who did homework and got a B or more. [To whole class] What other connections do you see from the three methods? [Wait time]

The 12 is in all three telling us how many people did not do homework and did get a B or more. No one multiplied or anything to get that.

I know group 1 used a different method than group 5, even though they also started with a tree diagram. Matthew, will you show us how your group solved this?

Fig. 6.6. Group 1’s solution

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84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Matthew: We started with a tree diagram, too. But realized something was different. We could tell that we couldn’t just use our format of wanted outcomes over total outcomes because we were looking at probability of getting a B or higher given that we did homework. That didn’t sound like our way of doing compound probability, which is getting a B or higher AND doing homework, so we looked in our book and found this was something called conditional probability. We found a formula for conditional probability and used it to determine the probability of getting a B or more given that someone had done homework. Those meant we had to find the probability of getting a B and did homework and divide it by all the people who did homework. That was .43 and .48, giving us a probability of .896, or 89.6% chance of getting a B if you did your homework. That came right off the table from group 4, but we got to use the stats formula. Ms. T:

Let’s discuss why you knew part of this problem was not a compound probability question.

The group began discussing the specifics of their decisions, but the class was drawing to an end. Ms. Tienda had the students write an exit slip comparing the different representations and explaining what they found helpful about them. She also asked students to continue the discussion from the end of class about how simple and compound probabilities (studied earlier) differ from conditional probability. Because the final group had looked up the formula, she knew the class was ready to consider how the formula was derived and how to apply it. She planned to look at more conditional probability questions that would lead to a better understanding of the concept and ways to solve problems using it. Additionally, students were now ready to consider independence and how that is related to conditional probability, such as considering the conditional probability of A given B is the same as the probability of A when the events A and B are independent. This case, written by Frederick Dillon, is based on his 35 years of experience teaching high school mathematics.

Analysis of ATL 6.2: Exploring the Use of Representations during Instruction Ms. Tienda noticed different levels of sophistication among her students’ responses, as well as opportunities to sequence student representations in a way to help students see connections among the different representations and solutions. She decided to start with the table (lines 1–2). The table was based on previous work students had done using tables to analyze information, but this table took that understanding to a new level. However, Ms. Tienda felt that the table was accessible to all students and, for that reason, was a good starting point. Ms. Tienda was mindful of Diamond’s explanations, especially about the values in each cell of the

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table. Ms. Tienda knew that these same values would occur in the other representations that would be shared, so ensuring that students understood how each value was derived and what it actually stood for was important. A key bit of dialogue occurred when Ms. Tienda asked Diamond about the meaning of 12 percent in the table. The question in lines 19–20, “Did 12% of students who didn’t do homework get a B or higher, or did 12% of all students not do homework and get a B or higher?” makes a distinction between a compound probability (happening in this part of the problem) and a conditional probability (happening in another part of the problem). Ms. Tienda then moved to the Venn diagram (lines 23–25) as several groups had used it, and the specific numeric value from the table (group 4’s solution) could be easily seen. The student discussion and Ms. Tienda’s questions helped students see how the value of 43 for getting a B or higher and doing homework regularly was found and where these values could be located in each representation so that students could see how both the table and the Venn diagram were displaying the same data but in slightly different ways. Ms. Tienda then decided to highlight two approaches she felt used representations that were related to each other. These two solutions are the tree diagram (fig. 6.5) and the formula (fig. 6.6). Though it is not shown in the work here, the students using the formula started with a tree diagram (the tree diagram is the same as the one shown in fig. 6.5). Because the formula uses the tree diagram, Ms. Tienda chose to use the tree diagram next. There are some sophisticated calculations used to find the values on the branches that required careful explanations. Kalei delineated the steps necessary to find those values, particularly the .896. Because the value of .896 occurred in earlier representations (line 41), the stage was set to make connections back to the table (by completing the 43/48 calculation) and the Venn diagram. One of Ms. Tienda’s goals for the class was for them to recognize that the problem was not only a simple or compound probability question, but that it also contained something new, a conditional probability. The last group she asked to share had started with the tree diagram but extended it to include use of the formula. Matthew, the student explaining group’s 1 solution, stated that the students knew from their previous work that not all parts of the problem fit with their understanding of probability, so they knew to investigate further. Ms. Tienda saw this as a bridge to goals for the next few days—specifically, recognizing conditional probability and its differences from compound and simple probability and understanding that, for some cases, a formula for conditional probability was an appropriate method. Students could then debate the relative merits of each representation (in this case, the formula was ”discovered” by a group but was not needed to solve the problem). The exit slips were clear on what students recognized about simple and compound probabilities but indicated that some students were unclear about what some of the key components of conditional probability were or meant, such as “restricted sample space” and “the probability of A given that B has occurred.” Ms. Tienda could tell from the exit slips and the discussion (such as Diamond’s comments in lines 51–54, 56) that she needed to employ situations that required Use and Connect Mathematical Representations    113 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 use of the concepts of conditional probability. Additionally, she felt that the work and discussion from the last group meant she could use problem contexts to develop the conceptual underpinnings for the conditional probability formula and then use the formula to find probabilities and to discuss independence. The sequencing of solutions that utilized different representations was also an important component of the Case of Vanessa Culver (introduced in chapter 1). In that case, Ms. Culver sequences the different representations she observed in class to build an understanding not only of the concept of exponential functions but also of different ways to represent them. She began with the most accessible, the pictures of the different stages of the Pay It Forward task. From there she was able to discuss the functions that the students created, both y 5 3x and y 5 3x. Consideration of those functions was tied to the tables and the graphs that students had made so that students were able to see how the different representations were connected. In comparison, Mr. Taylor (chapter 3) was not able to help students see connections among a myriad of representations because he took over the thinking for students by introducing a table as the method to use for solving the task. It was not clear whether students saw the connection between the two tables and the symbolic exponential function. An analysis of Ms. Tienda’s classroom provides an example of how a lesson that makes connections between different representations can be used to build students’ understanding. Now you are ready to think about ways in which you might use a task to do the same.

Connecting Representations A key to helping students make connections among different representations is asking questions that highlight, connect, and contrast different representations. Tasks need to be chosen that not only meet the mathematical goals of the lesson but also have the possibility of being solved with a variety of representations. The goals for the Squares from Tiles task featured in ALT 6.3 on the next page include students recognizing patterns and creating rules to describe them for constant functions, linear functions, and quadratic functions. Inherent in those goals is that students will consider the rates of change in the data they discover and will use the rates of change to differentiate among linear and quadratic functions. In ATL 6.3, you will solve a task using as many representations as you can. Then, you will consider the sequence you would use to present the representations during a whole-class discussion and the questions you would ask to help students make connections among the solutions and to the mathematical goals that are driving the lesson. Keep in mind that you are anticipating possible student approaches to solving this task. The goal is not for students to use multiple representations but for them to reason about a mathematical task. The teacher needs to consider possible representations and how to connect them in order to ensure that all students understand the mathematics in the task.

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Analyzing Teaching and Learning Activity 6.3 Connecting Representations • Begin by solving the Squares from Tiles task below. Solve the task using as many different representations as you can. If possible, discuss your solutions, and other representations you predict students might use, with colleagues. • Imagine that the students in your class produced the solutions to the Squares from Tile task and created posters showing their work, which were hung in the classroom. — What questions could you ask to help students make connections between the different representations? — In what ways might students benefit from seeing the different ways in which the problem could be represented and solved?

The Squares from Tiles Task A computer animation program shows a square (called the main square) that grows into larger squares by increasing the length of each side by one unit each step of the animation. The main square is made of smaller congruent squares that stay the same size. The main square is created by using green squares (each of which has two edges on the perimeter), blue squares (each of which has one edge on the perimeter), and red squares (with no edges on the perimeter). Every large square must have at least one red square in it. Write rules that determine how many green, blue, and red squares are needed to make a main square with any given side length.

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Student A

Student B

continued on next page

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Student C

Student D Desmos.com

Student E

Student F

Fig. 6.7. Student work on the Squares from Tiles task

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Analysis of ATL 6.3: Connecting Representations The Squares from Tiles task is accessible to all students. It has an entry point (drawing a picture or building a physical model) that comes naturally to students. From the picture or model, students can transition to a variety of different representations to make sense of the situation and to arrive at a solution. The first solution in figure 6.6 shows drawings student A created to start the problem. The three drawings from student A all show four green tiles in the corners of the square, regardless of the size of the square, while the number of red and blue tiles increase as the square gets larger. The teacher might start by asking the student to consider patterns in the drawings. “What do you notice in the pictures? What do you think happens in the next picture? Why?” The questions should push the student to consider generalizing the patterns that are becoming evident from the pictures. Student B made a table to look for a pattern for each different colored square. The teacher could ask the student to explain how the rate of change was calculated for each entry. Then, the teacher can refer back to the drawings from student A: “For the green tiles, how can we see the rate of change of zero from the first picture to the second picture?” The teacher can ask similar questions about the rate of change for the blue tiles, pressing the students to identify the growth of four blue tiles for each one unit added per large square side. Because the student states that the change is always four, the teacher can ask for justification of that, which can again refer back to the diagrams for evidence. The red tiles offer the opportunity to delve a bit deeper into understanding. Not only can the teacher ask about how the rate of change is found, but she or he can also ask about the statement “Not the same” and what the student infers from that. The columns for rate of change connect to other student work that comes later (students C, D, and F), so the teacher should have questions ready to ensure that students see each pattern and where it comes from in the pictures of large squares. The teacher can lay the groundwork for later work by asking students to determine what the rate of change means for green tiles and blue tiles—for example, “What does it mean when a set of data has no change in its dependent values? How can you describe what you are seeing?” This leads directly into the work of student C, who used formal function notation rules for each color of tiles. Students may not expect to see a horizontal line as part of a solution generated with patterning. A key part of the Squares from Tiles task is that a constant function (G(n) 5 4) is part of the solution. Additionally, the teacher should be asking questions that tie the rules back to both the pictures and the table: “How does the table help you create the rule for the blue tiles? Where is the rate of change of four in your rule? Where is it in the pictures?” The teacher may gain understanding of students’ concept of variables by asking, “Why did you use n 5 the number of tiles on a side? Are there other ways to write this function? What would happen if you let n 5 the number of blue tiles per side?” Student C did not simplify either function for blue or red. Other student work may have other forms of the same expressions,

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enabling the teacher to ask about equivalent forms of the expressions and how the students know that those representations are the same. For example, “I notice some of you have R(n) 5 (n  2)2, while others have R(n) 5 n2  4n 1 4. Are these the same? How can I tell? How are these represented in the picture?” This type of question can come up again with the work from student F. Student D chose to use graphs. In the graph titled Graph for Green, student D had a set of points that appear to be on a horizontal line. The teacher should be asking how the graph relates to student A’s picture and student B’s chart with rates of change. “I notice you have four for each number of tiles. Where can you see that in the drawings?” “How does your set of points reflect student B saying, ‘no change in green’?” The graph entitled Graph for Blue shows a relationship between side length and the number of blue tiles with points in a linear pattern. The teacher can ask whether the student thinks this set of points will always be linear and why, or he or she can relate back to the table from student B by asking where the rate of change of four can be found in the graph. There are two different graphs for side length versus the number of red tiles, one with the same scale as the blue tile graph and one with a zoomed-out view. Questions should include asking why the two graphs were needed (perhaps the student was unsure whether the pattern was linear or not with the first graph) and what pattern the student discerns from the second graph that is zoomed out. Since the student has not declared a rule, the teacher can explore student understanding by asking whether the relationship of side length to red tiles appears to be linear and why (or why not). The teacher should also explore the possibility of the relationship being exponential or not (hoping to hear that the table did not show a multiplicative pattern in the rate of change). The teacher can then draw the students’ attention to the rate of change, which is not indicating linear or exponential patterns, and then ask the students what pattern the graph and table may be. Student E used an extension of the table for red to create a system of equations, which was then solved to find a function for red tiles. The teacher’s first questions should focus on why the student assumed that the rule would be quadratic (a typical answer may be that the red tiles always formed a square in the middle of the large shape or that the pattern of the graph looked like a quadratic). Then the teacher needs to understand how the student created the system of equations. “You have R(4) 5 16A 1 4B 1 C 5 4. Where are the coefficients of A, B, and C coming from? How does this connect to the picture or table?” The subsequent answer should indicate that the input of 5 for the sides on the large square has a solution of 9, and the coefficient of A is 52 or 25. Questions can then make links between the drawings or the table and the coefficients of the equations. The final sample, student F, shows symbolic and functional notation. Questions for student F need to determine where the rules green 5 4 and blue 5 4(n 2 2) came from, and how the student created the rule for the total number of squares and then used that to create a function for red squares. This representation can be linked to all of the other representations as it used pictures, the results from the table or graph, and symbolic notation. The student appears 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.

to indicate connections among the picture and the rules from students A and C, but questions need to ensure that is the case. Questions such as “How does your work relate to other pictures? Show me where the parts of your subtraction red 5 n2 2 4(n 2 2) 2 4 come from,” ask the student to make connections to the drawings or tables, while a question such as, “How did you know blue 5 4(n 2 2)?” can focus on the tables or rules portions. The Squares from Tiles problem was used to have students consider rules for constant functions, linear functions, and quadratic functions. There are multiple access points for students—from making drawings (or using manipulatives) through systems of equations. The teacher needs to plan for all of these solutions and to have questions ready to press students to explain not only how their representations relate to the problem context but also how the representations connect to each other. The sequence modeled here began with what may be the most accessible solution path (a drawing). It built on that to reach the solutions with higher levels of abstraction so that the connections were made apparent through the questions and the flow of the work. Questioning is an important way to help students connect representations. Chapter 4 describes how representations and the questions asked around their use can help students build conceptual understanding and the basis for procedural fluency (e.g., multiplying binomials with algebra tiles). An additional example of using questions to help students make connections can be seen in the Missing Function lesson discussed in chapter 5. The lesson started with a table and a graph and then asked students to find a missing factor for a polynomial product. Students reasoned about the representations and also made connections to symbolic representations. The teacher, Ms. Bassham, helped the students discuss mathematics and make connections among the representations through the questions she asked, as seen in the analysis of the lesson.

Use and Connect 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 do this by representing those abstract structures and concepts using physical representations (National Research Council 2001). As such, representations are part and parcel of everything we do in the name of mathematics. Lesh, Post, and Behr (1987) note that making use of the multiple representational forms shown in figure 6.1 and making connections between those representations support deeper mathematical understanding. By asking students to describe their mathematical thinking using multiple representations and making connections between them, we gain a better sense of the internalized understandings that students hold about abstract mathematical ideas (Goldin and Shteingold 2001). For example, an exponential growth pattern for a bacteria population that

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doubles every day can be represented as f (x) 5 2x, the corresponding graph of the exponential function, and a table of values showing the growth pattern iterated by whole number values of x. These are all representations of the same underlying mathematical relationship and structure. The extent to which students see some, all, or none of these as a representation of the relationship represented by f (x) 5 2x gives us important information about their abstract conceptions of multiplicative relationships and of exponential functions in particular. In the study of algebra and functions, symbolic, tabular, and graphical representations are staples of the high school curriculum. How and when these representations are used as reasoning tools is an important consideration. Knuth (2000) notes that high school students often exhibit an overreliance on symbolic representations, even when a corresponding graphical representation of the function or relationship would be more appropriate for thinking and reasoning. Knuth’s study also notes that students’ connections between algebraic and graphical representations can be very superficial; students understand that the x and y values generated from an equation can be plotted on a Cartesian plane to produce the graph, but they may not understand that any point on the graph produces a set of values that satisfies the equation. One potential reason for these disconnects is that tables and graphs are often positioned in mathematics curricula as end products rather than as starting points and representational tools to use in reasoning and problem solving (Leinhardt, Zaslavsky, and Stein 1989). Teachers should be deliberate about using multiple starting representations and providing students with opportunities to make sense of mathematical relationships using multiple representations, rather than simply engaging students in exercises in which they are given an equation, complete a table of values, and sketch a graph—in that order. The use of representations is critical to fostering meaningful understandings in high school geometry as well. Hollebrands (2003) notes that dynamic geometry software (such as The Geometer’s Sketchpad and GeoGebra) supports deeper understanding of transformational geometry concepts and the connections between transformations and functions. Similar positive student outcomes are evident with the use of statistics software that supports visualization and analysis (Chance et al. 2007). These tools, in addition to more ubiquitous graphing calculators or software, promote students’ opportunities to use and connect mathematical representations and strengthen student learning outcomes.

Promoting Equity by Using and Connecting Mathematical Representations High-cognitive demand tasks, as discussed in chapter 3, afford the opportunity to use and connect multiple mathematical representations. These tasks are more equitable as they afford a wider 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

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the mathematical ideas in ways that make sense to them. For example, the pictures produced by student A in the Squares from Tiles task (see fig. 6.10) could serve as a starting point for almost any student, regardless of his or her previous mathematical background. In addition, providing color tiles to students to support their work on this task would also provide an entry point. The students solving the Student Homework problem (figs. 6.3–6.6) entered with many different ways of finding a solution. From a simple table to a more complex use of a statistical formula, students had access to the problem on the basis of their own backgrounds. The use of multiple mathematical representations also allows students to draw on multiple resources of knowledge, one of the five equity-based mathematics teaching practices (Aguirre, Mayfield-Ingram, and Martin 2013). It is important for a teacher to explicitly value and encourage multiple mathematical representations that allow students to draw on their mathematical, social, and cultural competence. By promoting the creation and discussion of unique mathematical representations, teachers can position students as being mathematically competent. By connecting more informal representations to more formal mathematical representations over time, students can make use of their initial funds of knowledge to develop mathematical fluency and power in ways that are shared by the broader mathematical community.

Key Messages • Use and connect multiple representations to deepen student understanding of concepts and procedures. • Select tasks that allow students to decide which representations to use in making sense of the problems and to explain and justify 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. • Success in problem solving is related to students’ ability to move flexibly among representations (NCTM 2014, p. 26).

• Students need to use representations meaningfully and make connections among different representations.

• Contextualize mathematical ideas by connecting them to real-world situations.

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Taking Action in Your Classroom: Using and Connecting Representations In Taking Action in Your Classroom, we invite you to explore the questions you asked during a lesson and to consider what insights these questions give you about students’ understanding of mathematics.

Taking Action in Your Classroom 6.1 Plan a lesson around a high-level task that has the potential to elicit a variety of strategies. Solve the task using different representations and strategies you anticipate that students might use. (Compare and discuss the strategies with a colleague, if possible.) • Anticipate different representations students will use, and design different ways to elicit students’ abilities to use different representations for this task. • Design different ways to assess students’ abilities to use representations to solve this task. • Teach the lesson, and reflect on how the use of representations supported student learning related to the lesson’s goals

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CHAPTER 7

Facilitate Meaningful Mathematical Discourse The Analyzing Teaching and Learning activities in this chapter engage you in exploring the effective teaching practice, facilitate meaningful mathematical 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 provide students with opportunities to share ideas, clarify understandings, develop convincing arguments, and advance the mathematical learning of the entire class. During a mathematical discussion, “a teacher can use a student’s connection with some mathematics to teach the student while also teaching the class as a whole” (Lampert 2001, p. 174). The opportunities that teachers provide 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 (ATL), make note of your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, facilitate meaningful mathematical discourse. If possible, share and discuss your responses and ideas with colleagues. Once you have written down or shared your ideas, read the analysis, where we offer ideas relating the Analyzing Teaching and Learning activity to the focal teaching practice.

Encouraging Classroom Discourse In Analyzing Teaching and Learning 7.1, we revisit the classroom of Shalunda Shackelford where students are engaged in discussing the Bike and Truck task (see the next page). Ms. Shackelford wanted her students to understand the following: • The language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values.

• Context is important for interpreting key features of a graph portraying the relationship between time and distance. • The average rate of change is the ratio of the change in the dependent variable to the change in the independent variable for a specified interval in the domain.

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The Bike and Truck Task A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.

  1. Label the graphs appropriately with B(t) and K(t). Explain how you made your decision.   2. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description.   3. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words.   4. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack. Adapted from Institute for Learning (2015a). Lesson guides and student workbooks are available at ifl.pitt.edu.

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Recall from chapter 2 that Ms. Shackelford posed a misconception to the class about what is happening when the graph of the truck is a horizontal line, using her “imaginary friend Chris.” In ATL 7.1, we consider how Ms. Shackelford encouraged and supported mathematics discourse around this misconception and other mathematical ideas in the Bike and Truck task.

Analyzing Teaching and Learning 7.1 Encouraging Classroom Discourse Rewatch video clip 1 of the discussion of the Bike and Truck task in Shalunda Shackelford’s classroom (from ATL 2.2). As you watch the video this time, pay attention to the ways in which the teacher facilitates discourse. Specifically— • How does the teacher support students to share and defend their own ideas? • How does the teacher provide students with the opportunity to clarify understandings? You can access and download the video and its transcript 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: Encouraging Discourse In video clip 1, Ms. Shackelford encouraged discourse in a number of ways. First, she supported students in sharing and defending their own ideas. Ms. Shackelford introduced the idea that the truck was “moving in a straight path” when the truck’s movement was represented by a horizontal line on the graph. By asking students to defend whether they agreed or disagreed with her “imaginary friend Chris” (lines 1–8 in the transcript), Ms. Shackelford was setting up the poles of a debate and using a potential disagreement to engage students in mathematical discourse (Chazan and Ball 1999). Jacobi and Charles volunteered to defend each pole of the debate (why they agreed or disagreed about whether the truck was moving in a straight path; lines 5–40 and 47–52), thus providing an opportunity for all students to construct viable arguments and critique the reasoning of others. In addition, Ms. Shackelford provided several other opportunities for students to share and explain their thinking (lines 18–19, 21, 23, 29–32, 46, 73–75). For example, a student provided a thorough explanation to question 3 in the task, using the graph projected to the class (lines 56–63). Toward the end of the clip, Ms. Shackelford presses a student to explain, “You agree, why?” (line 88). Second, Ms. Shackelford provided students with the opportunity to clarify their understandings. Students are expected to listen for understanding and ask questions if they disagree, as evidenced in her telling the class, “If you disagree, say something” (lines 9–10).

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Ms. Shackelford used a scenario to help Jacobi understand that time is moving even though she was standing still, and she then asked Jacobi to explain his new understanding (lines 47–52). Even after the class “agrees” (chorally) with the explanation provided in lines 56–63, Ms. Shackelford asked students to “have what they’re thinking in their head” (lines 67–68). When two students express confusion about how “the truck got there first” (lines 69–72), Ms. Shackelford asked for someone to explain “to see if we can get them to understand what’s really going on” (line 75). Following the explanation, she returned to the original students and asked, “What are you thinking now?” (lines 82–83). Ms. Shackelford did not accept “we all agree” as evidence of students’ understanding. Instead, she encouraged students to express their disagreement, continued to press for explanations, and provided students the opportunity to develop convincing arguments (e.g., “Make sure you justify your reasoning” [lines 8–9]). She often asked students to reexplain ideas to the class (lines 29–32, 46, 73–75). Ms. Shackelford made several talk moves to support students’ engagement in mathematical discourse. In particular, she used “accountable talk” (Michaels et al. 2002/2010), specifically the talk moves of “linking” and “press.” Based on the work of Michaels and colleagues, Boston (2012) defines “linking” as talk moves that create accountability to the learning community, such as revoicing (O’Connor and Michaels 1996); prompting students to extend, analyze, or critique the mathematical work and thinking of others; and fostering connections and comparisons between the work or ideas of different students. Often, linking moves create opportunities for student-to-student discourse. When making a linking move, the teacher positions students as the authors of ideas and invites other students to respond, reexplain, counter, or build on these ideas. Examples of linking moves made by Ms. Shackelford can be found in lines 21, 29–32, and 64 (e.g., “Charles, did you hear what he had to say? . . . All right, do you still disagree with him?” [lines 19, 21]). “Press moves” promote talk that is accountable to the mathematics (or, more generally, to the rigor and justification practices of the discipline), such as when teachers prompt students to explain their thinking, validate the accuracy of their computations, and justify their claims (Boston 2012). Press moves often sustain teacher-student discourse but also provide other students with the opportunity to hear the reasoning or justification behind another student’s idea or strategy; this is similar to the probing thinking or making the mathematics visible question types identified in chapter 5. Press moves typically involve a teacher asking a student to unpack, justify, support, or “say more” about an idea he or she has offered. Ms. Shackelford made press moves in lines 23, 46, and 88 (e.g., “You agree, why?” [line 88]). In summary, Ms. Shackelford made many important moves to facilitate meaningful discourse (e.g., setting up the poles of a debate, linking, and press). Throughout the video clip, she also posed purposeful questions to move students’ thinking toward the goals of the lesson. Key questions asked by Ms. Shackelford are explored in ATL 7.2.

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Facilitating Discourse by Posing Purposeful Questions In Analyzing Teaching and Learning 7.2, you will have the opportunity to watch and analyze more of the discussion of the Bike and Truck lesson in Ms. Shackelford’s classroom. In video clip 2, Ms. Shackelford asks two purposeful questions: • “Between what two seconds did the truck drive the fastest?” (line 1 in the transcript)

• “Is the bike or the truck moving faster after 17 seconds?” While we do not hear Ms. Shackelford ask this question during the clip, at the transition in the video on line 39, students discuss whether the bike or truck was going faster after the 17-second mark on the graph.

Analyzing Teaching and Learning 7.2 more Facilitating Discourse by Posing Purposeful Questions Watch video clip 2 of the discussion of the Bike and Truck task in Shalunda Shackelford’s classroom. • How do the two questions posed by Ms. Shackelford support the class in engaging in a discussion that furthers their understanding of the mathematical ideas? • How does Ms. Shackelford provide students with the opportunity to develop convincing arguments? • Identify instances of linking and press moves. How did each type of talk move support students’ engagement in mathematical discourse? You can access and download the video and its transcript 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 7.2: Facilitating Discourse by Posing Purposeful Questions The two questions posed by Ms. Shackelford provided an opportunity for students to pull together their ideas about the relationship (and covariation) between time and distance on the graph in relation to the movement and speed of the bike and the truck. Several students indicated that the truck went the fastest over the interval in which the line was the “steepest” (lines 3–4, 10, 13 in the transcript). Three students proposed that you can determine the interval on which the truck went the fastest by comparing the distance traveled over the range of time (lines 21, 26, 34–38). Note that there is a transition between lines 39 and 40 on the transcript when students went to the board (off-camera) to find the average rate of change of 130   Taking Action Grades 9–12 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 truck between 7 and 9 seconds and 16 and 18 seconds (the calculations are visible at 2:12 minutes into the video clip): 160 − 120 40 20 = = 9−7 2 1 300 − 220 80 40 Truck on interval from 16 to 18 seconds: = = 18 − 16 2 1

Truck on the interval from 7 to 9 seconds:

Using this same idea, students then determined whether the bike or the truck was moving faster after 17 seconds (the intersection point on the graph). A student demonstrated her calculations for finding the average rate of change of each vehicle on the overhead (2:12 to 2:30 in the video clip): Truck on interval from 17 to 18 seconds: Bike on interval from 17 to 19 seconds:

300 − 260 40 = 18 − 17 1 300 − 260 40 = 19 − 17 2

Ms. Shackelford asked her, “So explain to me what all that means then” (line 39). By asking these questions and pressing students to justify “fastest” or “steepest” with calculations, Ms. Shackelford provided students with the opportunity to develop convincing arguments and to understand the reasoning of others (e.g., line 48, “Tell me what you understand about what she just said”). As evidence, one student comments that she understands now that she “got the whole equation of it” (line 44). The questions also allowed Ms. Shackelford to assess students’ understanding of the mathematics again at this point, and she returns to three students to determine whether these students had worked through their previous misconceptions (lines 49–51). All three students responded in ways that demonstrated at least a developing understanding of the relationship between time and distance on the graph (lines 52–54, 55–56, 58–59, and 63–64). In this way, Ms. Shackelford used mathematical discourse and purposeful questions to advance students’ understanding of the mathematical ideas toward the goals of the lesson. Ms. Shackelford made linking moves when she asked students what they understood about what other students said (lines 24, 42, 48, 62). For example, in line 24, she asked, “Chelsea, what do you understand about what Ne’Kail said?” She also had a few instances of revoicing that served to prompt students to refine or clarify their ideas (lines 14, 18, 23). Press moves included asking students “why” or “what do you mean” or to explain (lines 1–2, 11, 15, 27, 39). Press moves are similar to probing or making the mathematics visible questions (described in chapter 5); however, press moves follow a students’ response (rather than serving as an initial question). In fact, both press moves and linking moves serve to disrupt the I-R-E pattern of discussion (Initiate-Response-Evaluate [Mehan 1979]) by following up on a student’s

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response (R) to a question (I) in ways that do not evaluate (E) but instead request greater detail or justification (press) or allow students to engage with the ideas of their peers (linking). Notice that almost all of the talk provided by the teacher served to generate mathematical thinking and reasoning from students; hence, all of the mathematical thinking and reasoning that surfaces during the discussion is offered by students. The mathematical discourse supported students’ engagement in thinking and reasoning throughout the implementation of the lesson (which was initially made possible by the teacher selecting a task that promoted thinking and reasoning). Ms. Shackelford and most of the other teachers we have visited so far in this book prepared themselves to engage students in high-level tasks in ways that did not take over students’ opportunities for thinking and reasoning. In the next section, we explore how classroom discourse supports task implementation.

Supporting Classroom Discourse During Task Implementation Orchestrating a discussion that maintains students’ engagement in thinking and reasoning and advances students’ learning toward the goals of the lesson is a challenging and complex task of teaching (Boerst et al. 2011; Franke, Kazemi, and Battey 2007). The Five Practices for Orchestrating Productive Mathematics Discussions (Smith and Stein 2011) are designed to support teachers in planning and facilitating meaningful mathematical discourse as a part of ambitious teaching practice. These are the five practices:  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 in a specific order

 5. Connecting different students’ responses and connecting the responses to key mathematical ideas Through these practices, teachers can reduce the improvisation and in-the-moment decisions on how to respond to students’ contributions required during a discussion. Through planning, teachers can anticipate likely student contributions, prepare responses and questions they can use while monitoring students’ work, make decisions about how to structure students’ presentations (e.g., what strategies to select and how to sequence them), and plan questions to ask during the discussion to support students in connecting mathematical strategies and ideas in ways that advance the mathematical goals of the lesson. (For more detailed information about the five practices, please refer to 5 Practices for Orchestrating Productive Mathematics Discussions by Smith and Stein [2011].) 132   Taking Action Grades 9–12 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.

Anticipating Student Solutions The next three Analyzing Teaching and Learning activities provide opportunities to engage in aspects of the five practices related to lesson planning. In Analyzing Teaching and Learning 7.3, we ask you to consider the Proof 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.

Analyzing Teaching and Learning 7.3 Anticipating Student Solutions Solve the Proof task shown below. Create a variety of approaches for solving the task that students might use. In addition to possible strategies, anticipate what misconceptions students might have or incorrect solutions they might pursue. You might also ask colleagues to solve the task and draw from their approaches. Based on your solution and the other solutions you have considered, which solution paths do you think are most likely to be used by your students?

The Proof Task Prove that for every integer n ( . . . −3, −2, −1, 0, 1, 2, 3, 4 . . . ), the expression n2 1 n will always be even. Source: Adapted from Morris (2002).

Analysis of ATL 7.3: Anticipating Student Solutions Reasoning and proving is a topic known to be difficult for students in high school mathematics (Coe and Ruthven 1994; Healy and Hoyles 2000; Knuth et al. 2002). Students might approach the Proof task in a number of ways (see fig. 7.2). One approach to “proving” is to test a limited number of numeric examples, illustrating the misconception that empirical examples are sufficient to prove that a statement is true. This approach is illustrated in argument 5 in figure 7.2, where the student claims to have proven that n2 1 n is always even on the basis of the results of testing a variety of different numbers. To make a more general argument, students might also draw on the properties of certain types of numbers, such as consecutive numbers (argument 1) or the products and sums of odd and even numbers (arguments 1, 2, and 3). Note how arguments 1, 2, and 3 are not tied to specific numbers but reference odd and even numbers in general. Arguments 1 and 2 reason logically about properties of numbers, and argument 3

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represents even and odd numbers algebraically. In argument 4, the student uses a diagram to support the argument. While the diagrams address specific values of n, the student also presents a general argument (based on the diagram) for any odd or even number.

Anticipated likely solutions to the Proof task Argument 1: Consecutive integers Since n2 1 n 5 n(n 1 1), and this is the product of two consecutive numbers. Consecutive numbers means that one of the numbers is even and the other is odd. The product of an odd and even number is even, since one of the numbers is divisible by 2. In other words, n or n 1 1 is divisible by 2 with no remainder, so the product n(n 1 1) is also divisible by 2. Since all even numbers are divisible by 2, n2 1 n is always even. Argument 2: Logical argument using two cases (even and odd) If n is an odd counting number, then n2 will be odd, because odd 3 odd 5 odd. Since an odd 1 odd 5 even, and n2 and n are odd, then n2 1 n is even. If n is an even counting number, then n2 will be even, because even 3 even 5 even. Since even 1 even 5 even, and n2 and n are even, then n2 1 n is even. Since all counting numbers are either even or odd, I have taken care of all numbers. Therefore, I’ve proved that for every counting number n, the expression n2 1 n is always even. Argument 3: Algebraic argument using two cases (even and odd) A number is even if it can be written as 2m, where m is any integer. That is, a number is even if it is a multiple of 2. A number is odd if it can be written as 2m 1 1, where m is any integer. That is, a number is odd if it is a one more than a multiple of 2. If n is even, then you get (2m)2 1 2m 5 4m2 1 2m 5 2(2m2 1 m). Since both are even then the sum is also even. The sum is even because it is a multiple of 2. If n is odd, then you get (2m 1 1)2 1 2m 1 1. This gives you: 4m2 1 4m 1 1 1 2m 1 1 5 4m2 1 6m 1 2 5 2(2m2 1 3m 1 1). This has to be even because 2 times any number is even, because it is a multiple of 2. So it always works.

continued on next page

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Argument 4: Visual argument So if I start with a square, say 5 by 5, and add it to the number 5:

Ok now I will match up the columns so that all but one column has a pair (the darkest purple one). The darkest purple column will be matched with the light purple 5-column that is added to the square. So that will make the whole thing even because you can divide the entire thing into two equal pieces.

Let me try another one (6 3 6):

The columns in the 6 by 6 match up perfectly with none left over, and the added part 6 folds in half. So every number is paired which makes 62 1 6 an even number. If the square is an odd by an odd like 5 3 5, then there will always be a column left over since an odd number does not divide by 2 evenly. The leftover column of an oddsided square will always match with the added column part. If the square is even by even, then every column has a match. The added part for an even by even will also be even based on the problem. And an even number divides two with nothing left over or folds perfectly. So it does not matter the counting number that you start with when you square it and add it to itself, it will always result in an even number. continued on next page

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Argument 5: Empirical examples Let n 5 1.  Then n2 1 n 5 12 1 1 5 2.  2 is even, so this works. Let n 5 2.  Then n2 1 n 5 22 1 2 5 6.  6 is even, so this works. Let n 5 3.  Then n2 1 n 5 32 1 3 5 12.  12 is even, so this works. Let n 5 101.  Then n2 1 n 5 1012 1 101 5 10,201 1 101 5 10,302.  10,302 is even, so this works. Let n 5 3,056.  Then n2 1 n 5 3,0562 1 3,056 5 9,339,136 1 3,056.  9,342,192 is even, so this works. I randomly selected several different types of numbers. Some were high, and some were low. Some were even and some were odd. Some were prime and some were composite. Since I randomly selected and tested a variety of types of counting numbers, and it worked in every case, I know that it will work for all counting numbers. Therefore, n2 1 n will always be even.

Fig. 7.2. Anticipated solutions to the Proof task

Anticipating student thinking lays the groundwork for facilitating rich mathematical discourse. Once you have identified specific solution strategies students are likely to use, you can prepare targeted assessing and advancing questions to ask while monitoring their work. The Case of Vanessa Culver (chapter 1) and that of Steven Taylor (chapter 3) reinforce this notion. Ms. Culver anticipated that her students would solve the Pay It Forward task in multiple ways (diagrams, tables, graphs, equations) (lines 6–9). Ms. Culver asked questions as she monitored students’ work (lines 22–25). These questions helped her understand students’ solution strategies and ways of thinking. She purposefully selected and sequenced the solutions students would present during the whole-class discussion (lines 30–39). She was intentional about ordering the solutions and asking questions during the whole-group discussion that supported students in thinking, reasoning, making connections, and building up their understanding of exponential relationships. We contend that this work would be challenging, if not impossible, to enact “on the fly” if Ms. Culver had not anticipated strategies students would use and planned the questions she would ask in advance of the lesson. In contrast, Mr. Taylor’s lesson did not show clear evidence of anticipating student thinking. He did not ask assessing and advancing questions as he monitored student work, and the whole-group discussion featured a funneling pattern of questioning and a discussion that was not rich mathematically. Anticipating and monitoring student thinking also allows you to bring structure and coherence to the whole-class discussion of solutions. Anticipating likely

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strategies and planning questions prior to a lesson alleviates some of your “cognitive load” and “in-the-moment” decision making as a lesson unfolds, so that your focus can be on eliciting, understanding, and connecting students’ thinking.

Monitoring The assessing and advancing questions you plan to ask while monitoring students’ work can be organized in a “monitoring chart” (Smith and Stein 2011), along with a column for your notes about who created which type of solution, what solutions you will select to be shared, and how you will sequence the presentations during the whole-group discussion. The beginning of a monitoring chart is provided in figure 7.3, and ATL 7.4 provides the opportunity for you to complete this monitoring chart for the Proof task.

Anticipated Solutions to the Proof Task

Assess and Advance Questions

Select and Sequence

1. Consecutive integers 2. Logical argument (two cases) 3. Algebraic argument (two cases) 4. Visual argument 5. Empirical examples 6. Can’t get started

Fig. 7.3. Monitoring chart for the Proof task

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Analyzing Teaching and Learning 7.4 Monitoring Complete the monitoring chart in figure 7.3 by writing at least two questions that might be used to assess students’ understanding of the mathematical ideas in the task and two questions that might be used to advance students’ understanding of the mathematical ideas in the task (revisit chapter 5 for ideas on assessing and advancing questions). These questions will support you in monitoring students’ work on the task. As you write your questions, consider these goals for the Proof task: Students will understand— a. for an argument to be a proof, it must show that the conjecture or claim is true for all cases; b. the statements and definitions that are used in the argument must be ones that are true and accepted by the community because they have been previously justified; c. proofs can take different forms and utilize different representations (pictures, words, symbols), but all of the different representations can be connected; and d. examples can help you see patterns, but they alone do not constitute proof. In addition, students will— e. see structure in the expression n2 1 n and use the structure to write the expression in an equivalent form n(n 1 1); f. interpret the expression as the product of two consecutive numbers and/or of an odd and even number; and g. understand how the expression can be physically modeled.

Analysis of ATL 7.4: Monitoring As general assessing questions that apply more broadly, students should be asked to explain something directly represented in their written work. For example, “How did you get . . . ?” or “Why did you . . . ?” or “What do those numbers mean/represent?” Specific to the Proof task, students who generated argument 1 might be asked, “How did you get n(n 1 1)?” Students who generated argument 2 or 3 might be asked why they considered two different cases. Students who produced each argument might be asked how they know that their argument is a valid proof; in other words, how have they shown that their argument works in all cases. In argument 4, we would want to see whether students could articulate a general argument based

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on their diagram. In argument 5, we especially would want to press students to consider what constitutes a proof and whether they have truly proven their argument to be true for all cases. Figure 7.4 provides a set of possible assessing and advancing questions that could be asked of students who produced the different arguments for the Proof task.

Anticipated solutions 1. Consecutive integers

Assessing and Advancing Questions • How did you get n(n 1 1)? • How do you know that one of your numbers must be odd and one must be even? • Can you be sure that there isn’t a number for n that would make the claim false? So, is this a proof? Can you make a list of things your argument does that would allow us to count it as a proof?

2. Logical argument (two cases)

• Why did you start by looking at two different cases? Is that necessary? • You seem to be convinced that you have proven the claim. What makes you so sure that your argument is a proof? Can you make a list of things your argument does that would allow us to count it as a proof?

3. Algebraic argument (two cases)

• Why did you start by looking at two different cases? Is that necessary? • Why did you specify that 2m was even and that 2m 1 1 was odd? • How do you know that 2(2m2 1 m) and 2(2m2 1 3m 1 1) are even numbers? An advanced response would look like this: If 2m2 1 m 5 k (and k is an integer by closure of addition and multiplication of integers), then 4m2 1 2m 5 2k, so the sum is even. • When you say “It always works,” what do you mean? So, is this a proof? Can you make a list of things your argument does that would allow us to count it as a proof? continued on next page

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Anticipated solutions 4. Visual argument

Assessing and Advancing Questions • How does drawing the vertical or horizontal line show dividing by two? • What do you mean you can “match up” the columns or the rows? Why does this matter? • How does the “matching up” relate to drawing the horizontal or vertical line? You have shown that this works when n 5 5 and n 5 6, but why are you so sure that this will work for every number? Can you add to your argument so that we will be convinced that it will always work? Does this argument work for all integers?

5. Empirical examples

• You have tried lots of different numbers. How do you know that there isn’t a number you haven’t tried yet that will not work? Is there something about how odd and even numbers work when squared and added that would help convince us? Would a picture help explain what is happening? What other way can you convince us?

6. Can’t get started

• What is the problem asking you to do? • What does it mean to be even? Can you try to put some numbers in to see if it seems to work? Then think about what you know about even and odd numbers that might make it work or not work.

Fig. 7.4. Assessing and advancing questions for the Proof task

Selecting, Sequencing, Connecting At this point, you have engaged in two of the five practices: anticipating and (planning for) monitoring. ATL 7.5 provides an opportunity to consider the remaining practices: selecting, sequencing, and connecting.

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Analyzing Teaching and Learning 7.5 Selecting, Sequencing, Connecting Review the solutions to the Proof task and select the approaches you would have shared during the whole-group discussion. Sequence the solutions in the order you would have students present them so that they build up students’ understanding of the mathematical ideas in the task. What specific questions will you ask so that students make connections— • among the different strategies/solutions that are presented? • to the mathematical ideas that you want them to learn (e.g., the goals of the lesson)?

Analysis of ATL 7.5: Selecting, Sequencing, and Connecting There are many possible ways to select and sequence the possible solutions to the Proof task. On the basis of the goals articulated for the lesson, however, a few criteria might immediately present themselves: • To meet goal (e), at least one solution should focus on the equivalent form n(n 1 1). • To meet goal (f ), at least one solution should base the argument on the properties of n and n 1 1 as consecutive integers or odd/even integers.

• To meet goals (c) and (g), at least one solution should include a diagram or model, and questions should be asked to make connections between this model and the other solutions. • To meet goals (a) and (d), the discussion of each solution should include questions about whether the argument has proven the statement to be true for all cases.

• To meet goal (b), students should be pressed to explain any statements or definitions they are using within their arguments. In addition, the following considerations for selecting and sequencing solutions help in planning the discussion of task solutions (adapted from Smith & 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 so that the connections among solutions with common mathematical features can be highlighted.

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Using the lesson goals and these guidelines, the teacher might use the set of choices for selecting and sequencing solutions shown on the next page in figure 7.5. On the left are the selected solutions in a suggested presentation order. On the right are questions the teacher might ask to encourage connections between the solutions. First, begin with solution 2, the logical argument using two cases, because it is straightforward and uses students’ prior knowledge of the product and sum of even and odd numbers (goal f ). The argument provides an opportunity for students to use and explain statements and definitions that the class has previously accepted as true (goal b) and to justify why their argument holds for all cases (goals a and d). Second, have students share solution 3, the algebraic argument, because of the connections to the logical argument presented first (goals c and f ). By juxtaposing the two arguments, students will be able to see the similarities — the logical argument says in words what the algebraic argument is showing numerically. Again, students should be pressed to justify why this argument holds for all cases (goals a and d). Third, use solution 4, the visual argument. This argument provides a way to “show” visually what happens when n2 and n are added together (goal g). It connects (goal c) with the logical argument by providing a visual of the work, and it connects to the algebraic argument because dividing by 2 (or factoring out the 2) is the symbolic/algebraic method of creating two columns or two rows (literally dividing the diagram into 2 parts). Fourth, have students share solution 5, the argument based on empirical examples. Ideally, the group that created and presented this argument would have also generated a more complete argument (perhaps as a result of being asked advancing questions as they worked in small groups). If not, you could indicate that many students approached the Proof task in this way, or you could present “proof by empirical argument” as a strategy used by students in another class. This solution will help you assess whether students are starting to see what is required for proof (goal a). Last, have students present solution 1, consecutive integers (goal e). This strategy is last because it requires seeing the symbolical structure differently as the first step. At this point in the discussion, students will have a good idea of what constitutes a proof. Also, this argument connects well (goal c) to the visual argument since n(n 1 1) are the dimensions of the rectangle. The sequence described is but one of many possible ways to select 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 story line that a teacher wishes to build through the discussion. The selecting and sequencing should also be designed so that all students have entry and access to the discussion.

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Sharing and Discussing the Proof Task Selecting and Sequencing

Connecting Responses

Which solutions do you want to have shared 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; and • make connections among the different strategies/ solutions that are presented?

FIRST: Solution 2

• Is this a proof? Why or why not?

Logical argument using two Cases

• What about it makes you think it is a proof?

SECOND: Solution 3

• How is the algebraic argument similar to or different from the logical argument?

Algebraic argument

THIRD: Solution 4 Visual argument

• Is it a proof? What about it makes you think it is a proof? • Is this argument similar to either of the other arguments? In what ways? • Is this a proof? Why or why not? (If the students have not clearly generalized from their example so that it is clear that this works for all cases, it is not a proof. If this is the case, press the class to consider what could be added to the argument to make it a proof.)

FOURTH: Solution 5

• Is this a proof? Why or why not?

Empirical examples

• Why did the visual argument count as a proof when they used examples, but examples don’t count as a proof in this situation?

FIFTH: Solution 1

• How is this the same or different from the other arguments that have been made?

Consecutive integers

• How does it relate to the visual argument? • Does it count as a proof? Why or why not? • Is it a proof? What about it makes it a proof?

Fig. 7.5. Possible sequence of solutions and connect questions for the Proof task

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Appendix A provides a lesson plan for the Proof task. The lesson plan provides examples of assessing and advancing questions for each solution, a sequence for sharing solutions during the whole-group discussion, and questions to ask to connect students’ responses together and to key mathematical ideas. The lesson plan in appendix A brings together all the work on the Proof task from ATL 7.4 and 7.5 to demonstrate what a complete lesson plan representing 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, remember the role of goals and tasks. Meaningful mathematics discourse is unlikely to occur unless teachers choose and use mathematical tasks that promote reasoning and problem solving. If the tasks teachers use in the classroom are limited to the rote application of procedures or memorized facts (see the characteristics of low-level tasks in fig. 3.2), mathematics discourse will likely be restricted to students reciting the steps they performed to solve the problem. Having a clear mathematical goal for the task guides your decision making regarding which solution strategies to share and in what order and what important mathematical connections to make. 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 between those solutions.

Facilitate Meaningful Mathematical Discourse: What Research Has to Say A wide array of research over the past 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., Boaler and Staples 2008; Schoen et al. 1999; Silver 1996). In their research involving three high schools, Boaler and Staples (2008) found that instruction characterized by opportunities for rich mathematics discourse was associated with increased learning gains for students. Coupled with cognitively challenging tasks (chapter 3) and consistent accountability and support to achieve at high levels, engaging students in mathematical discourse was identified as essential in raising the mathematics achievement of students at Railside High School (1) beyond the other two schools in the study and (2) in ways that minimized previous achievement gaps. Similarly, Schoen and colleagues (1999) and Silver (1996) identified the connection between opportunities for students to communicate their thinking and reasoning during mathematics lessons and higher student achievement on measures of mathematical reasoning and problem solving. As discussed in chapter 5, the prevalent Initiate-Response-Evaluate (I-R-E) pattern of discourse (Mehan 1979) does not provide students with opportunities to engage in mathematical reasoning and sense making and provides teachers with little feedback regarding what students know and understand. Changing the I-R-E pattern in classrooms begins with

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selecting tasks that promote reasoning and problem solving — tasks that elicit more from students than correct answers and procedures and hence provide something for students to think and reason about during the discussion. In addition to tasks, interaction patterns between teachers and students and among students in classrooms must change as well. As noted earlier in this chapter, 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; Michaels et al. 2010). When students offer ideas, rather than evaluating the ideas as right or wrong, teachers must consider how they can support the students to deepen their understanding or explanation (press), support other students to grapple with and understand the ideas (linking), and ask purposeful questions that support students’ progress toward the mathematical goals. HerbelEisenmann and colleagues have recently identified a set of six teacher discourse moves intended to support the demands of secondary mathematics classroom discourse (Herbel-Eisenmann, Steele, and Cirillo 2013; pp. 183–84): (1) waiting; (2) inviting student participation; (3) revoicing; (4) asking students to revoice; (5) probing a student’s thinking; and (6) creating opportunities to engage with another’s reasoning. Making regular use of research-based tools such as those suggested by Herbel-Eisenmann and colleagues (2013) or the five practices (Smith and Stein 2011; Stein et al. 2008) can help teachers move beyond a discussion that is simply a show-and-tell reporting out of discrete strategies (Wood and Turner-Vorbeck 2001) toward discussions that instead build mathematical ideas in systematic ways. Together, these frameworks for planning and implementing lessons, for developing productive classroom norms around discourse and specific routines and pedagogical moves can be used in concert by teachers to transform their classrooms from ones in which the teacher does most of the talking and thinking to ones where 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 by Facilitating Meaningful Mathematical Discourse The work of promoting meaningful mathematics discourse in the classroom has far-reaching implications for equity. By eliciting students’ own ideas and strategies and creating space for students to present their ideas and strategies during the whole-group discussion, the teacher communicates that students’ ideas matter. Students come to realize that their work and their thinking serve an important role in the lesson and that they are being recognized as knowers and doers of mathematics. Such moves foster a positive identity (D. B. Martin 2012) and a sense of agency as mathematicians (Berry and Ellis 2013; Boaler and Staples 2008) as students

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come to feel capable of engaging in and generating mathematical activity on their own (e.g., forming ideas, posing questions, creating representations, evaluating correctness or validity, justifying, arguing, solving). In more traditional I-R-E pattern classrooms, students who contribute correct answers are provided with immediate positive feedback, and students who provide incorrect, incomplete, or partial responses are likely to be provided with negative or, at best, neutral feedback. In a classroom that features meaningful mathematics discourse, the I-R-E pattern is broken by purposeful questions and talk moves that keep the conversation going and extend the opportunities for student participation. Students have opportunities to share their mathematical thinking in addition to their answers, to hear criticism and feedback, and to critique the reasoning of others. As such, a discourse-based mathematics classroom provides 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 an approach but can provide feedback to others. 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). 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 students and not others and send students messages about their status. Students may send messages to one another that reinforce fixed-mindset conceptions of mathematical ability, 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 they should disrupt talk that may lead to unproductive conceptions of what it means to know and do mathematics. A discourse-based classroom also gives students access to their peers’ thinking, providing them with more possible approaches to a task than just the one the teacher or textbook chooses to privilege. Hearing an explanation unpacked or revoiced by another student provides additional opportunities for a student to understand and learn. Also, knowing that asking for clarification is valued, and that taking time to make sense of ideas is an expected and natural part of mathematical thinking and reasoning, can provide students with the space and confidence to ask questions that enhance their own mathematical learning. 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.

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Key Messages • Meaningful mathematical 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, the approaches that you wish to have publicly shared, and the ordering and connecting of 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 productively struggle. • 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 now time to consider what implications 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 the next page.

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Taking Action in Your Classroom 7.1 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, one that uses multiple representations from chapter 6). • Anticipate the solution strategies, both correct and incorrect, that students might use. • Identify the solution strategies you want to share to move your mathematical goal forward and the order in which you want to share them. • Make note of specific talk moves you might use related to specific solutions or connecting solutions. • Create a monitoring sheet to help you keep track of the solutions you anticipated and any others that arise and that notes talk moves you intend to use. • Teach the lesson and reflect on how the planning for discourse supported student learning.

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CHAPTER 8

Elicit and Use Evidence of Student Thinking The Analyzing Teaching and Learning 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 critical in helping teachers determine what students currently know and understand about key mathematical ideas and in supporting students’ ongoing learning. Specifically, evidence provides a window into students’ thinking and helps the teacher determine the extent to which students are reaching the math learning goals; evidence can be used to make instructional decisions during the lesson and to 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 shares many features 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 can be revealed about student thinking;

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• analyze a narrative case and consider what the teacher does to elicit and make use of student thinking during the lesson; • review key research findings related to eliciting and using thinking; and

• reflect on ways you elicit and use evidence of student thinking in your own classroom. For each Analyzing Teaching and Learning (ATL), make note of 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. Once you have written down or shared your ideas, read the analysis, where we offer ideas relating the Analyzing Teaching and Learning activity 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 have the potential to provide insights into their thinking (Wiliam 2011). As we discussed in chapter 3, such tasks can be categorized as high-level demand (see fig. 3.2). In ATL 8.1, you will analyze the Four Situations task (fig. 8.1) and consider what the task could elicit about students’ understanding of mathematics.

Analyzing Teaching and Learning 8.1 Investigating the Potential of a Task Solve the Four Situations task.   1. Why is this task appropriate for eliciting evidence of student thinking?   2. What mathematical understandings might the task reveal?   3. What misconceptions might the task reveal?

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Four Situations 1. Sketch a graph to model each of the following situations. Think about the shape of the graph and whether it should be a continuous line or not. A: Candle Each hour a candle burns down the same amount. x 5 the number of hours that have elapsed. y 5 the height of the candle in inches.

B: Letter When sending a letter, you pay quite a lot for letters weighing up to an ounce. You then pay a smaller, fixed amount for each additional ounce (or part of an ounce.) x 5 the weight of the letter in ounces. y 5 the cost of sending the letter in cents. C: Bus A group of people rent a bus for a day. The total cost of the bus is shared equally among the passengers. x 5 the number of passengers. y 5 the cost for each passenger in dollars. D: Car value My car loses about half of its value each year. x 5 the time that has elapsed in years. y 5 the value of my car in dollars.

Student materials

Representing Functions of Everyday Situations © 2015 MARS, Shell Center, University of Nottingham

Fig. 8.1. Four Situations task (Source: Mathematics Assessment Resource Service 2015. Available at http://map.mathshell.org/lessons.php?unit59260&collection58.) Elicit and Use Evidence of Student Thinking    151 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.

Analysis of ATL 8.1: Investigating the Potential of a Task For several reasons, the Four Situations task is a good candidate for making student thinking visible. This is a high-level task requiring students to use non-algorithmic thinking, to understand mathematical concepts and relationships, and to expend considerable effort in creating a solution. It is an example of “doing math” (see fig. 3.2) — that is, the task promotes reasoning and problem solving and thereby requires students to think. Students must access prior knowledge about linear functions, including the meaning of possible x- and y-intercepts, rate of change (is it constant or not?), whether the input/domain values are discrete or continuous, and how the function acts as x gets very large (end behavior). Ryan Eller used the Four Situations task in his pre-calculus class in order to gather information on his students’ knowledge of functions, which they had studied in previous courses. In figure 8.2, we see the work of one of his students, Willow. Willow has correctly answered all four parts of the task. Her work, as well as the questions Mr. Eller asks and her responses to them, allows us to inspect more closely the mathematics in each part of the task.

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Willow

Four Situations

1. Sketch a graph to model each of the following situations. Think about the shape of the graph and whether it should be a continuous line or not. A: Candle Each hour a candle burns down the same amount. x 5 the number of hours that have elapsed. y 5 the height of the candle in inches.

B: Letter When sending a letter, you pay quite a lot for letters weighing up to an ounce. You then pay a smaller, fixed amount for each additional ounce (or part of an ounce.) x 5 the weight of the letter in ounces. y 5 the cost of sending the letter in cents. C: Bus A group of people rent a bus for a day. The total cost of the bus is shared equally among the passengers. x 5 the number of passengers. y 5 the cost for each passenger in dollars. D: Car value My car loses about half of its value each year. x 5 the time that has elapsed in years. y 5 the value of my car in dollars.

Student materials

Representing Functions of Everyday Situations © 2015 MARS, Shell Center, University of Nottingham

Fig. 8.2. A student’s solutions to the Four Situations task

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In situation A (candle), Willow correctly created a continuous, linear function. The function starts at a point on the y-axis (the height of the candle when lit) and ends at a point on the x-axis (the candle is at height zero at the time it stopped burning). A student may stop the graph above the x-axis, stating that the candle was snuffed out before melting all the way down. Willow drew a linear function because she inferred a constant rate of decrease from the statement in the task, “Each hour a candle burns down the same amount.” Some of Mr. Eller’s students labeled the axes, but Willow did not, which is fine, as the prompt says to sketch a model based on the situations. In situation B (letter), Willow created a step function. She correctly identified this situation as one that is not continuous. Willow selected an initial height for the cost of a letter or parcel that weighs between 0 and 1 ounce. She used open points at each end of the segment for this constant price as there is no 0-ounce letter and the 1 ounce letter will cost more (the prompt says “up to an ounce”). The initial height is relatively high compared with the jump between costs for successive sizes of letters. Willow demonstrated this by having the jump between costs for 1 to 2 ounces, 2 to 3 ounces, etc., as a fixed height difference that is smaller than the initial height. Willow has shown a good understanding of (1) the different rates for each ounce class; (2) the rate of change being zero over an ounce weight class; and (3) the relative differences between the different costs per ounce class. In situation C (bus), Willow created a discrete graph. When questioned by Mr. Eller, Willow explained that this is because you cannot have a fraction of a person. Her graph also shows a decreasing rate of change between prices as more people are added to the group. When asked why she did this, Willow told Mr. Eller that she experimented with several numerical values to see what happened, and she noticed that the price decrease seemed to slow down as the number of people increased. Willow has shown an understanding that the rate of change is not constant and that the nonlinear graph started with a rapid rate of change that then became less rapid. Willow also did not start on the y-axis. She explained that the first person pays the entire cost of the bus and that there is no “zero person price.” As part of this solution, Willow has shown that the price never reaches a value of zero. Mr. Eller wanted to see a rational function of the form a/x, which Willow has drawn. His questions will help him decide whether Willow knows that this is a rational function or whether she is considering it to be some other type of function. Willow used a continuous function for situation D (car value). Her work shows an initial value for the car (its value at purchase, or time zero) that has a steep initial decrease followed by slower decreases. The value of the car keeps decreasing but is never zero. When asked to explain her solution, Willow stated that this was an exponential decay that showed an initial rapidly decreasing rate of change followed by a slower and slower rate of change, approaching an almost horizontal curve, which has very little change occurring as time increases. The task does not ask students to create or consider function rules but instead asks them to consider the shape of a graph. This requires students to consider the salient characteristics 154   Taking Action Grades 9–12 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 each situation and determine how these characteristics should be represented graphically. Because this approach is asking for a sketch, it is likely that all students will be able to produce something that will make clear what they currently understand. Finally, the task has the potential to expose what students understand about the nature of functions. For example, can students distinguish between linear and nonlinear situations, between contexts that represent discrete versus continuous data, or between increasing and decreasing functions? Can students specifically identify exponential and rational functions and describe the characteristics of such functions? Once the thinking of students has been made public through the use of a task such as Four Situations, the teacher is then positioned to use that thinking to advance their understanding. Mr. Eller now has several options he can consider on the basis of what he learned from his students’ work on this task. He can evaluate the work to determine student strengths and weaknesses. Since he has his students work in groups, he can circulate about the room, employ questioning (chapter 5), and engage students in discussion (chapter 7) to learn about their understandings about the shapes of graphs and continuity. Then, he can adapt what is happening in the class to focus on misconceptions, or he can plan additional activities that will foster better acquisition of concepts and extend students’ learning.

Using Students’ Thinking to Advance their Understanding Although the Four Situations task has the potential to elicit student thinking and reasoning, as noted in chapter 3, the potential can be limited by the way in which students’ work on the task is supported during the lesson. If students are directed to a particular method (as we saw in the Case of Steven Taylor in ATL 3.3), little will be learned about what students think or understand. In ATL 8.2 you will analyze four pieces of work (fig. 8.3) produced by Mr. Eller’s students. Here we ask you to consider how you can advance each student’s learning on the basis of what you discover about the student’s understanding. Setting clear mathematical learning goals (as discussed in chapter 2) is a key part of being able to advance students’ learning, as instruction should be tailored to meet your learning goals. As you engage in ATL 8.2, use the following goals to guide your decision making as you plan how to support students’ learning: by engaging in the Four Situations task, students will understand (1) that functions may be linear or nonlinear; (2) how rate of change affects the graph for a contextual situation (linear and nonlinear); (3) what the initial value means in a context situation (generally the y-intercept); and (4) domain and range in terms of continuous or discrete.

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Analyzing Teaching and Learning 8.2 Using Students’ Thinking to Advance Understanding Review the student work shown below. • What misconceptions did the task elicit from students? What did the students appear to understand? • 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?

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Anup

Four Situations

1. Sketch a graph to model each of the following situations. Think about the shape of the graph and whether it should be a continuous line or not. A: Candle Each hour a candle burns down the same amount. x = the number of hours that have elapsed. y = the height of the candle in inches.

B: Letter When sending a letter, you pay quite a lot for letters weighing up to an ounce. You then pay a smaller, fixed amount for each additional ounce (or part of an ounce.) x = the weight of the letter in ounces. y = the cost of sending the letter in cents. C: Bus A group of people rent a bus for a day. The total cost of the bus is shared equally among the passengers. x = the number of passengers. y = the cost for each passenger in dollars. D: Car value My car loses about half of its value each year. x = the time that has elapsed in years. y = the value of my car in dollars.

Student materials

Representing Functions of Everyday Situations © 2015 MARS, Shell Center, University of Nottingham continued on next page

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Chris

Four Situations

1. Sketch a graph to model each of the following situations. Think about the shape of the graph and whether it should be a continuous line or not. A: Candle Each hour a candle burns down the same amount. x = the number of hours that have elapsed. y = the height of the candle in inches.

B: Letter When sending a letter, you pay quite a lot for letters weighing up to an ounce. You then pay a smaller, fixed amount for each additional ounce (or part of an ounce.) x = the weight of the letter in ounces. y = the cost of sending the letter in cents. C: Bus A group of people rent a bus for a day. The total cost of the bus is shared equally among the passengers. x = the number of passengers. y = the cost for each passenger in dollars. D: Car value My car loses about half of its value each year. x = the time that has elapsed in years. y = the value of my car in dollars.

Student materials

Representing Functions of Everyday Situations © 2015 MARS, Shell Center, University of Nottingham continued on next page

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Brie

Four Situations

1. Sketch a graph to model each of the following situations. Think about the shape of the graph and whether it should be a continuous line or not. A: Candle Each hour a candle burns down the same amount. x = the number of hours that have elapsed. y = the height of the candle in inches.

B: Letter When sending a letter, you pay quite a lot for letters weighing up to an ounce. You then pay a smaller, fixed amount for each additional ounce (or part of an ounce.) x = the weight of the letter in ounces. y = the cost of sending the letter in cents. C: Bus A group of people rent a bus for a day. The total cost of the bus is shared equally among the passengers. x = the number of passengers. y = the cost for each passenger in dollars. D: Car value My car loses about half of its value each year. x = the time that has elapsed in years. y = the value of my car in dollars.

Student materials

Representing Functions of Everyday Situations © 2015 MARS, Shell Center, University of Nottingham continued on next page

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Dott

Four Situations

1. Sketch a graph to model each of the following situations. Think about the shape of the graph and whether it should be a continuous line or not. A: Candle Each hour a candle burns down the same amount. x = the number of hours that have elapsed. y = the height of the candle in inches.

B: Letter When sending a letter, you pay quite a lot for letters weighing up to an ounce. You then pay a smaller, fixed amount for each additional ounce (or part of an ounce.) x = the weight of the letter in ounces. y = the cost of sending the letter in cents. C: Bus A group of people rent a bus for a day. The total cost of the bus is shared equally among the passengers. x = the number of passengers. y = the cost for each passenger in dollars. D: Car value My car loses about half of its value each year. x = the time that has elapsed in years. y = the value of my car in dollars.

Student materials

Representing Functions of Everyday Situations © 2015 MARS, Shell Center, University of Nottingham

Fig. 8.3. Student work for the Four Situations task

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Analysis of ATL 8.2: Using Students’ Thinking to Advance Understanding Although each of the four responses to the Four Situations task reveals some degree of faulty reasoning, the teacher is now in the position to help students recognize and move beyond incorrect reasoning and develop a sounder understanding of the functional relationships in the task. As we discussed in chapter 5, assessing and advancing questions provide a key tool teachers can use to make student thinking transparent and to move students to new understandings. Telling a student she is wrong or giving him a particular strategy to use will not, in the long run, lead to a deeper understanding of mathematics. According to Principles to Actions (NCTM 2014, p. 36), “Purposeful questions allow teachers to discern what students know and adapt lessons to meet varied levels of understanding.” Anup’s responses show some understanding that should be acknowledged. He correctly identified increasing and decreasing functions for all cases. He also recognized that the domain for B is discrete, even though the graph is not entirely correct. Mr. Eller may ask, “I noticed that you have not connected the values. What happens between those values?” anticipating that Anup may not have considered weights between the whole number values or that Anup may not sufficiently understand what should be done when the price is the same over an interval. Mr. Eller should also ask about the graph starting at the origin: “Is it possible for a letter or parcel to weigh 0 ounces? Would a package that weighed 1/2 ounce cost nothing to mail?” This may prompt Anup not only to consider the value at x 5 0 but also to give another way to look at fractional values for weights. For both situations C and D, Anup thought that the functions decreased the same amount each time, thus resulting in a linear function. Mr. Eller may ask Anup to consider using arithmetic to investigate what may be happening. Anup may make a table, graph the points he calculated, or just look at the differences in successive y-values to come to the conclusion that these situations are not linear. Then, Mr. Eller may ask Anup to reconsider his sketch to see whether more questioning is needed to support Anup’s understanding of the general shape of the function. Mr. Eller can also ask about the domain and range in each case, especially if an arithmetic approach is used. Brie also has responses that the teacher should ask about in order to give positive feedback. For example, Mr. Eller might ask, “I notice you have a step function in B. Why?” He might say, “Explain why you have open points on your graph for situation B.” Acknowledging the correct reasoning to start the problem (if the explanation is valid) is a good way to begin a conversation with a student about his or her work. Then, Mr. Eller may follow up by asking why the postal rate for weights between zero and one is $0 and redirect Brie to consider how that would change the graph. Mr. Eller should also question why the difference between successive steps is the same and whether that fits with the statement in the task: “You pay quite a lot for letters weighing up to an ounce.” The point is to help the student recognize that the initial segment should start relatively high above the x-axis (compared with the constant

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differences of successive steps). Brie used a nonlinear function for situation A. Mr. Eller should ask Brie to consider what it means to “[burn] down by the same amount.” He might also ask about situations involving linear decreases that the class has seen before. In her solution for situation C, Brie shows understanding of the domain of the function and the shape of the graph. However, Brie should be asked to explain her reasoning for situation C and for the even differences between prices as the number of people increases. The solution for situation D is similar to the response from Anup. Both Anup and Brie indicated that the graph is linear with a slope of 21/2. As with Anup, Mr. Eller should ask Brie to consider an arithmetic solution or, in this specific case, what it actually means to decrease by 1/2 of an amount each time. For example, “What does it mean to take half of a quantity and then take half again? How does that relate to this problem?” are questions that can get Brie to consider this part of the task. Chris viewed each of the functions as linear. A starting question could be, “What does it mean for a function to be linear? How would you describe the rate of change?” On the basis of Chris’s response, Mr. Eller may need to consider whether he understands that a linear function requires a constant additive change. Chris may have the concept of linear function correct, at which point Mr. Eller needs to probe further to ascertain what Chris is thinking in the nonlinear cases. Since the sketch provided shows an increasing function, Mr. Eller needs to determine how Chris thought about situation A. Possible questions might include, “What do the values on your x- and y-axes stand for? How is this reflected in your graph?” Perhaps Chris thinks that the y-axis is the amount of the candle that has burned off, or perhaps he is not reading the context properly. The student’s response will determine the teacher’s next actions. For situation B, Chris has an initial value that is not at zero (a correct item to point out: “I like that you saw that the initial price is above 0”). A misreading of the problem may also be the issue with C, as Chris has an increasing rather than a decreasing function. A starting point might be a general question such as, “What does your graph represent?” If Chris responds with “the number of people,” Mr. Eller may ask about the domain (which should be discrete), and he may ask Chris to reconsider what the prompt was asking before he suggests that Chris try again. If Chris says that the graph represents the price per person, Mr. Eller should have questions prepared to have the student consider what happens to each equal-sized piece of a quantity as more pieces are partitioned off, such as, “If there are 10 people, how does the price change when you add one person? Two people?” As with Anup and Brie, Chris views the last part of the task as a decreasing linear function. Similar questions as previously stated should focus on the context and what it means to halve something. Mr. Eller may start making positive comments about Dott’s solutions for situations A, C, and D, as she appears to have these parts correct. He should then probe for understanding with questions such as, “Please explain what the graph in situation A is telling me?” and “Why does the graph for situation C start away from the x- and y-axes? Will the graph ever reach zero? Why?” and “You have a continuous graph for situation C. Is it possible to have a fraction of a person?” and “What does your scale for situation D indicate? You show the graph leveling 162   Taking Action Grades 9–12 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.

off as x increases. What does that mean in the context of the problem?” Dott’s response for situation B has a high starting value and then increases in increments smaller than the initial value, but the graph is linear and continuous. Mr. Eller’s questions can be used to find out why she started with a horizontal segment but then made a continuous, linear graph. Mr. Eller could say, “I like that you realized that the price between 0 and 1 ounces was constant. What does your graph indicate is happening as the weight of letter increases after a weight of 1 ounce? The task stated that you pay a ‘fixed amount for each additional ounce.’ What does that say about the price of a letter that weighs 1.2 ounces and a letter that weighs 1.4 ounces? How can you represent that?” The teacher can get Dott to consider what is occurring between ounces 1 and 2 and how the situation can be represented. The teacher also has some decisions to make for the class as a whole. The four solutions shown here are a sample of the class’s work. Using students’ thinking to inform instructional decisions is an important step in formative assessment. Questions about how students interpreted the task can help the teacher determine next steps. After reviewing the work of the entire class, the teacher may find that most students are comfortable with the linear decreasing function in situation A. The idea of a decrease is evident in the answers to situations C and D, but students may need to consider what the rate of change actually means in each question. The teacher may want to have students look at exponential decay and simple rational functions both symbolically and numerically to increase familiarity with this type of problem and also to help them reason about what it means to have a linear decrease. The solutions for situation B indicate that students have some recognition of the concept of a step function, but students need to have more tasks that produce step functions to help them recognize contexts that are represented with step functions and what a step function means. Another takeaway for the teacher is that students need more experience with determining whether a domain is continuous or discrete. Teacher questioning may reveal that students were not recalling previous situations in which a discrete domain occurred or that students did not consider a discrete domain. Tasks involving rates for parking ($8 dollars for the first hour and $7 for each additional hour or fraction of an hour) and other contextual situations could be used. We can also investigate the use of eliciting and using student thinking in the lessons taught by Vanessa Culver (chapter 1) and Steven Taylor (chapter 3). Ms. Culver asks questions of the student groups as she moves around the room so that she can determine what they know and understand and provide additional support as needed. She also uses this time to select and sequence student work in order to create a whole-class discussion that brings out student thinking about exponential functions and about what the task is asking. She learns about student understanding (eliciting student thinking) and is able to adapt her lesson as student responses are made (using student thinking). By contrast, Mr. Taylor does little to elicit students’ thinking. The lesson is built on his thinking, and therefore he does not get a picture of what his students understand. Elicit and Use Evidence of Student Thinking    163 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 work by itself gives an idea of what the students understand and what needs more attention. However, by engaging the students in a discussion of what they did and why, the teacher gets a more detailed understanding of what the students are thinking that then serves as the basis for further discussion and possible instruction.

Considering How Student Thinking Is Elicited and Used During a Lesson In ATL 8.3 we go into the algebra 2 and trigonometry classroom of Barbara Lynch. Ms. Lynch is a 25-year teacher who has spent her entire career teaching mathematics at Woodlake High School. Her 10th- and 11th-grade students have completed units in which they used graphing calculators, tables, and symbolic representations to explore how changes in parameters (e.g., m and b in y 5 mx 1 b) affect the graphs of functions (including linear, absolute value, quadratic, and square root functions, all of which are referred to as “parent functions”). Recently, the students in Ms. Lynch’s class have been studying trigonometry, working with the sine, cosine, and tangent functions. The class made connections to right triangles, worked on application problems, drew angles on the coordinate plane in standard position, defined sine and cosine in terms of x and y coordinates, considered how to describe clockwise and counterclockwise rotations (Ms. Lynch told them about the convention of positive and negative angle measures), and solved simple trigonometric equations (e.g., cosx 5 .66 in right triangles). In the lessons immediately preceding the one depicted in the case, Ms. Lynch’s students studied the graphs of y 5 sinx and y 5 cosx. Through observations and discussion, the students reached the following conclusions: • The graphs of y 5 sinx and y 5 cosx are repeating the same shape over and over (the student discussion led to them creating and formalizing the definition of “periodic” and that the period length is 2π). • There is a maximum value of y 5 1 and a minimum value of y 5 21 for both graphs.

• There is a line halfway between the maximum and minimum values that is situated at y 5 0 (the students named this “the midline”); there is a set distance from the midline to the maximum or minimum value, which, the students were told, is called the “amplitude.” The task featured in this lesson is the first time students have worked on transformations of trigonometric graphs. Through their work on this task, Ms. Lynch wants her students to continue to build their understanding regarding function transformation. Specifically, she wants students to understand the following: • Adding/subtracting a constant to the function shifts the graph up or down the y-axis. • Multiplying the function by a positive constant compresses or dilates the function.

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• Multiplying by a negative constant reflects the function over the x-axis.

• Adding/subtracting a positive constant to the argument of a function causes a horizontal shift.

• Symbolically, linear combinations of the transformations affect the graphs in the same way as doing each transformation separately. • Periodic functions have unique characteristics (period, midline, amplitude). Transformations affect these characteristics in the graph as well.

With these clear goals for student learning, Ms. Lynch can look for evidence that students are reaching these goals.

Analyzing Teaching and Learning 8.3 Exploring How Student Thinking Is Elicited and Used during Instruction Read the Case of Barbara Lynch and consider these questions: • What does the teacher do to elicit student thinking? • How does the teacher use student thinking during the lesson? • What might the teacher do in class the following day to build on what she learned in this lesson?

The Case of Barbara Lynch 1 2 3 4 5 6 7 8

As the students entered the room, Ms. Lynch had them sit in their preassigned small groups. She then asked the students to review the homework assignment on transformations (shown in fig. 8.4). She circulated around the room as students shared their work with their peers. She noticed that students were able to create functions that reflected over the x-axis and that they were able to shift a function vertically without any difficulty. She also learned that there was some confusion about the effect of a parameter such as a in y 5 aƒ(x) and with horizontal translations. She made a note to include more investigations about those two transformations in future lessons.

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Homework Assignment 1. Draw any function you want on your graph paper. Call the function g(x). a. Graph the function h(x) 5 g(x) 1 3. b. Graph the function j(x) 5 2g(x). c. Graph the function k(x) 5 g(x 1 1). d. Graph the function m(x) 5 2g(x). 2. ƒ(x) is a function on the coordinate plane. a. Write a function a(x) that translates ƒ(x) right three units. b. Write a function b(x) that translates ƒ(x) down six units. c. Write a function c(x) that reflects ƒ(x) over the y-axis. d. Write a function d(x) that contracts ƒ(x) by a scale factor of 1/2.

Fig. 8.4. Homework assignment for the Case of Barbara Lynch 9 10 11 12 13 14

After a few minutes, Ms. Lynch called the class together, thanked them for their hard work, and asked them to consider what they had just discussed in terms of the graphs of y 5 sinx and y 5 cosx. She distributed the Investigate the Graphs of Sine Waves task (shown in fig. 8.5). The class read the investigation and took a few moments to consider it. By Ms. Lynch’s established classroom procedure, the students asked any questions they had about the instructions and task before starting to work.

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Investigate the Graphs of Sine Waves Investigation: For each of the forms of sine functions below, you will explore the graph, its location on the coordinate plane, and how the new graphs are related to the graph of the parent function y 5 sinx. y 5 asinx

y 5 sinx 1 c

y 5 asinx 1 c

Before you start, individually predict how you think the parameters will affect the graphs. Test your predictions using your graphing calculator. Substitute different values for a and/or c. Use a variety of values including ones that are greater than 1, between 0 and 1, and positive and negative. Record your observations. As a group, use your graphs to answer these questions: a. Were your predictions correct? State clearly what you expected and what your experiments showed you. b. How do changes in a and c affect the parent function y 5 sinx?

Fig. 8.5. Investigate the Graphs of Sine Waves Task 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

With the class still arranged in small groups, Ms. Lynch told them they would have 15 minutes to work, after which there would be a whole-class discussion about their findings. As students worked on the task, Ms. Lynch walked around the classroom, first making sure that everyone was getting started without any difficulty and then monitoring the progress of each group.

Group 1: Ms. Lynch stopped at group 1 as students were discussing their predictions and results for y 5 asinx (fig. 8.6). Chris was commenting that the graphs of the sine waves kept getting taller. Ms. Lynch asked, “What do you mean by ‘taller’?” Chris explained that when a became larger, the curve became narrower, or more stretched, similar to what the class had noticed with y 5 ax 2. Ms. Lynch asked others in the group, “What do you think? Did you get the same results as Chris?” Alex replied, “Yeah, I noticed that the max and min got further from the axis.” Pat added, “Yep, that’s what I found, too. Let’s go on to the next one.” “Whoa, wait a minute,” Ms. Lynch commented as she was looking at the group’s recording sheet. “I have a couple of questions. Let’s look at the values you used for a.” She continued, “Hmm, you all agreed the amplitude became larger as a got larger. So I’m wondering, what’s the smallest a value you could use?” She did not specifically remind the students of their previous work with transformations and using different parameter values. Chris commented

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33 34 35 36 37 38 39 40 41 42

that 2 was a good initial value for a, since for y 5 sinx the coefficient of sinx, that is a, is 1. Ms. Lynch then decided to ask Jordan, a student in the group who had been quiet. Jordan responded, “Well, I guess a could be a negative number. We found a pattern when multiplying by a negative with other function families.” Pat stated that they should “start with 22 and see what happened.” Ms. Lynch replied, “What do you think will happen to the graph when you use negative values for a?” All but one of the students said the graph would get taller and flip (the other student thought the graph would move down). Ms. Lynch told them to test their ideas and said, “Are there other values of a you could try also? How do you think you could make the amplitude shorter?” as she moved on to the next group.

Form

Value of a

Value of c

Equation graphed

2

0

y 5 2sinx

5

0

y 5 5sinx

3

0

y 5 3sinx

10

0

y 5 10sinx

y 5 sinx

Fig. 8.6. Group 1’s table 43 44 45 46 47 48 49 50

Group 4: Ms. Lynch noticed that group 4 had completed the chart in figure 8.7. She asked Shayla, “What does your group mean by ‘flipped over’?” Shayla explained that “flipped over” meant that the graph reversed over the x-axis. Ms. Lynch said, “Let’s talk about the ‘flipped over’ thing. Shayla said the graph is ‘reversed.’ What does that mean to the rest of you?” D’Juan recalled “reflection over the x-axis” from their discussion of absolute value and parabolic functions on the previous night’s assignment. Ms. Lynch responded, “So, when and how does the sine wave get reflected? Look at your chart. What do you think causes the sine wave to be reflected?”

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Fig. 8.7. Group 4’s table 51 52 53 54 55 56 57 58 59 60 61 62 63 64

Cristel answered, “It seems like it flips over, I mean gets reflected, when a is negative.” Ms. Lynch asked, “What do the rest of you think about Cristel’s conjecture?” Taylor said, “This is just like our other functions.” The group agreed. Ms. Lynch said, “I have another question. You mentioned the amplitude, but you were not specific. Where is the amplitude? You noted the amplitude getting larger. Can you make a smaller amplitude? If so, how?” Ms. Lynch continued circulating around the room, stopping to ask each group questions about their graphs. She often had to prompt students to try positive and negative values for a and to test values between 21 and 1. She made notes to herself as she visited each group to remind herself to address specific ideas during the wholegroup discussion. Even though students did not have the chance to explore equations of the form y 5 asinx 1 c, Ms. Lynch noticed that students could easily see and describe the effect c had on the graph of the sine wave. She decided it was time to pull the class together for a discussion.

65

The Discussion

66 67 68 69 70 71

It had become a norm in Ms. Lynch’s class, after many struggles and much persistence, for students to share and discuss their work publicly. When errors were made, Ms. Lynch referred to them as “learning opportunities” and stressed that students often learned more about mathematics by discussing the errors than from reviewing correct solutions. In fact, Ms. Lynch modeled this by encouraging her students to point out “learning opportunities” she might make during a lesson.

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72 73 74 75 76 77 78

Ms. Lynch began the whole-group discussion by asking group 1 to share and explain their group’s graphs (fig. 8.8). Howie explained that they graphed y 5 2sinx, y 5 23sinx, y 5 4sinx and y 5 210sinx and found that the larger they made a, the larger the amplitude became, saying, “The amplitude is the same as a.” Tamara added that when a values were negative, the sine wave was flipped, or reflected over the x-axis. Ms. Lynch asked them where the amplitude was. Tamara pointed to the distance from the midline to 3 on y 5 23sinx saying, “That is the amplitude of 3.”

y 5 sinx, y 5 2sinx, y 5 23sinx window [22π. 2π] by [23,3}

y 5 sinx, y 5 4sinx, y 5 210sinx window [22π. 2π] by [212,12}

Fig. 8.8. Graphs produced by group 1 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 96

Next, Ms. Lynch asked the class about Howie’s statement and wrote on the board under a heading WHAT WE DISCOVERED, “The amplitude is a,” and “A negative a value causes the sine function to reflect over the x-axis.” She then asked the class, “Do you agree with Howie and Tamara’s group?” to which the majority of the students nodded. After a few moments of silence, Pat mentioned that the negative sign only caused the graph to flip but that a distance (the amplitude) could not be negative. Howie immediately said, “You know what I meant,” so Ms. Lynch asked him to correct the statement. He pondered for a few seconds and said, “The amplitude is the positive of a.” Pat then asked, “Isn’t it really the absolute value?” A brief discussion on definitions and terminology led to the class agreeing to use “The amplitude is equal to the absolute value of a.” Ms. Lynch asked, “How does this fit your predictions? Does the amplitude always get larger?” She then asked group 6 to share their graphs (fig. 8.9). Renée showed their graphs of y 5 sinx (dark) y 5 2sinx, y 5 1/2 sinx in the first picture and y 5 sinx (dark), y 5 21/4 sinx and y 5 23sinx (on the next page). “We used the same viewing window for all the graphs. We think that when a is a fraction, the graph gets a smaller amplitude,” she explained.

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y 5 sinx, y 5 2sinx, y 5 1/2 sinx window [22π. 2π] by [23,3}

y 5 sinx, y 5 21/4 sinx, y 5 23sinx window [22π. 2π] by [23,3}

Fig. 8.9. Graphs produced by group 6 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

Renée went on to say, “See when a is 1/2 and 21/4 the amplitude is less than 1. It is the absolute value of a.” Ms. Lynch asked the class, “Do you agree with Renée?” Students were nodding, so she continued, “They said when a is a fraction, the amplitude shrinks. Is that always true?” Taylor, from another group, remarked that he also used a value of 1/3 for a and that the sine wave got “shorter,” so he thought it was probably always true. “Anyone else want to comment?” Ms. Lynch asked. Since no one volunteered to respond, she continued. “Something is confusing me. You agreed with Howie and Tamara that as the a value got larger, the amplitude got larger, and you agreed with Renée and Taylor that when a is a fraction, the graph gets ‘shorter.’ What would happen if a had a value of, say, 21/2? Would the amplitude be larger or smaller than the graph of y 5 sinx?” Many students answered that the amplitude would be larger, and Ms. Lynch asked them to explain their thinking. “Because 21/2 is more than 2 and we know the sine wave gets a larger amplitude the larger a is,” Shayla answered. “But a is a fraction,” Ms. Lynch responded. “I thought you said the amplitude got smaller when a was a fraction.” Many students responded that they meant fractions that were less than 1. “What about 231/4?” Ms. Lynch inquired. “It’s less than 1.” “You know what we mean,” the students shouted. “You’re always so picky!” Ms. Lynch continued with the discussion, prompting the students to use precise language and notation when answering questions. She made certain that students discussed the fact that values of a between 21 and 1 resulted in a sine wave that was contracted — that is, had a smaller amplitude, and that values of a greater than 1 or less than 21 resulted in dilated (or stretched) sine waves. She also asked students to represent the relationship symbolically and added to the list of properties, “If 21 , a , 1, the sine wave has a smaller amplitude than y 5 sinx and if a . 1 or a , 21, the sine wave has a larger amplitude

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121 122 123 124 125 126 127 128 129 130 131 132 133 134 135

than y” and drew rough sketches of sine waves to represent each case. Several students then noted that the amplitude of y 5 sinx was one, and that the amplitude of y 5 asinx would then be a because each y-value of the function was multiplied by a. Since there were only a few minutes left in the class period, Ms. Lynch asked the students to begin the discussion about the effect of “1 c ” in their small groups and told them they would continue the discussion tomorrow. She also asked the students to think about the equations of the form y 5 asinx 1 c and to be prepared to discuss their conclusions about the effects of the changes in the parameters on both a sine wave and a cosine wave tomorrow. Ms. Lynch noted that students were connecting the effects of 1 c on the sine waves to what happened with graphs of previously studied functions. After class, she knew students understood what multiplying by a constant did to a graph, and she decided that there would not be a need for more than a quick discussion about the effect of adding a constant. She determined that the next class should pick up with y 5 asinx 1 c. She added several questions reviewing horizontal shifts of functions to help work on student confusion about that material.

This case, written by Frederick Dillon, is based on his 35 years of experience teaching high school mathematics.

Analysis of ATL 8.3: Exploring How Student Thinking Is Elicited and Used during Instruction The first thing Ms. Lynch did to elicit student thinking was to implement a task that promoted reasoning and problem solving and to provide students with the opportunity to discuss their thinking publicly. Ms. Lynch planned this lesson carefully. She anticipated ways in which students might solve the task, determined questions 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 considered ways in which she would monitor student progress during the lesson. Through this planning process, Ms. Lynch prepared herself to support students without telling them how to solve the problem and thereby limiting their opportunities to think and reason. Ms. Lynch provides an example of a teacher who engages in the five practices for orchestrating a productive discussion (Smith and Stein 2011) that we discussed in chapter 7. She followed this up by checking students’ answers to their homework task as the students discussed it. She noted that students were confident in their understanding of what signified a vertical translation and what caused a reflection over the x-axis. She also noticed that students were confused about the effect of multiplying by a constant and by horizontal shifts. She was able to emphasize the former in the day’s lesson and prepared to continue to address horizontal shifts in future lessons (lines 6–8 and 134–135). The homework had the potential to reveal what the students understood about the different transformations they had studied and their areas of misunderstanding or confusion. 172   Taking Action Grades 9–12 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.

When working with small groups, Ms. Lynch pressed students to explain their graphs and what they were discovering. She asked group 1 to explain what they meant by “taller” and then advanced their thinking by having them consider what happened with other values of the multiplier (lines 22, 40–42). In both groups, Ms. Lynch discovered that students were not using precise mathematical language when describing what they were noticing. She asked questions that created a need for precise terminology in the context of the lesson (lines 45–47, 85–89, 113–115). With group 4, Ms. Lynch heard students discussing reflections, and she asked questions that helped students connect back to their prior learning. She was able to determine that the students in group 1 had not considered a full range of values for the multiplier of the sine function, so she asked them to consider how to make the amplitude smaller to prod them into thinking about other multipliers (lines 29–31). In whole-group discussion, Ms. Lynch was able to understand student thinking by their explanations and by asking questions to focus their attention. She asked questions to determine whether students understood what happened with positive and negative multipliers (lines 72–78), what it meant to multiply by a fraction (lines 98–100), and what students really meant by “fraction” (lines 103–107). Throughout the lesson, Ms. Lynch elicited students’ thinking and used their thinking to determine the next course of action. She didn’t tell the groups whether they were correct. Rather, she had them discuss their ideas, and she asked clarifying questions when needed. 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 this out on their own (lines 41–42, 57–59). She ended the class with students discussing more of the task and telling them they would continue the discussion the next day, implying that she expected students to keep thinking about the lesson and that they were capable of making conjectures and conclusions about the prompts (lines 124–129). After class, Ms. Lynch made decisions for the next day’s instruction on the basis of what she had learned about her students’ thinking in the lesson. How might Ms. Lynch use student thinking to inform the next lesson? For class the following day, students might be asked to write functions that demonstrate how y 5 sinx was transformed in each picture (see fig. 8.10). On the basis of students’ work from the current day’s lesson, this would set the stage for further exploration of transformations of the sine graph.

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Fig. 8.10. Problems to start the lesson

Ms. Lynch might plan to have students investigate linear combinations of the form y 5 asinx 1 c, which is a continuation of work started in the current class, and to see how the students succeed with those before moving on. Her discussions with the class suggest that this part of the lesson will not take the entire class period. She may plan to have the class look at the graph of y 5 3sinx 1 5 and to ask the students, as individuals, to determine how that differs from the graph of y 5 sinx. She might have the class discuss their solutions as pairs and then as a whole class. She could use the results of this discussion to ask more questions about the form y 5 asinx 1 c. She might choose to demonstrate with Desmos (the online graphing

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capability), using sliders for the different parameters so that students could see the effects of different transformations (fig. 8.11).

Fig. 8.11. Desmos screen shot showing sliders for y 5 2sinx 21

Ms. Lynch considers questions and tasks to move the class forward in case the class struggles with y 5 asinx 1 c. One possible backup plan is to have the class experiment with different parameter changes on their calculators or have the class split into groups of four students in which each pair creates a transformed graph for the other pair to investigate and determine the transformations. Ms. Lynch also anticipates that the class will be able to look at y 5 sinbx and how the parameter b affects the graph of y 5 sinx. Since the primary effect of b is to change the length of the period of the sine wave, Ms. Lynch could decide to restrict initial studies to nonnegative real numbers in order to focus on what is most important about the transformation. In their previous families of graphs, her students have seen a compression or dilation of the graph, but the addition of a period for the function is new to their studies. When looking at the graphs of y 5 0.1x3 and y 5 2x3, Ms. Lynch may anticipate that her students will recall that the smaller coefficient (0.1) made the graph stretch out, while the larger coefficient (2) made the graph narrower, or compressed (figs. 8.12 and 8.13).

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Fig. 8.12. Graph of y 5 0.1x3

Fig. 8.13. Graph of y 5 2x3

As mentioned earlier in this section, the students already know that the period of the sine function is 2π. As a next step, Ms. Lynch wants the students to consider the period of 2π and what it means for a parameter to cause it to stretch or compress. She could plan a series of questions about graphs, such as y 5 sin2x, y 5 sin 1/2 x and y 5 sin3x that are graphed alongside y 5 sinx, particularly as one of her goals is for students to see how many times a graph “repeats” when compared with y 5 sinx. That is, when y 5 sin2x (solid) is graphed alongside y 5 sinx (dashed), the graph of y 5 sin2x repeats twice for one period of y 5 sinx (fig. 8.14).

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Fig. 8.14. Graphs of y 5 sinx (dashed) and y 5 sin2x (solid)

Another goal may be to have students predict how the parameter b will change the period length and what that will look like on a graph (that is, that the period length 2π is divided by b to find the new period length). She could have students experiment with their calculators and by making tables of key values (such as π and 2π). Ms. Lynch might create exit slips and class opening questions that ask students to explain the transformation created by b in y 5 sinbx, as well as group exercises and Desmos explorations. Her end goal for this section is for students to predict changes to y 5 sinx, to recognize the transformations to y 5 sinx when shown a transformed graph, and to graph functions of the form y 5 asinbx 1 c without technology. Ms. Lynch may plan several formative assessment questions that will be employed over the course of the lessons so that she can be sure that her students understand the concepts and skills she has as her goals. On the basis of the evidence she collects from her formative assessments, she may adapt her instruction to the needs that become apparent.

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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 making use of that information to inform 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 learned at the close of a lesson. Teachers who consistently elicit student thinking during a lesson can make use of 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, and how teachers make use of what they know and understand about student thinking before, during, and after a lesson. Interpreting student thinking begins with anticipating the range of correct and incorrect responses that students might produce given a particular mathematical task, as noted in chapter 7. 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 State Standards Writing Team 2011; Wu 2013) provide teachers with the tools to categorize types of student thinking and situate them on a continuum that suggests ways to move students forward. Teachers can select a task and plan a lesson using learning progressions or trajectories in ways that offer entry points to students with multiple different understandings and that have the potential to 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 still building a more abstract conception of volume and surface area could physically build the prism while other students may move directly to generalized formulas. The questions a teacher might ask using that task could be designed to push the students still building physical models to see and make use of structure to press toward a generalization. The teacher might ask those students who are already generalizing questions about how scaling dimensions would impact surface area and volume, such as what would happen to the surface area and volume if we doubled the length of one side. On a smaller scale, research regarding misconceptions and reasoning strategies for a particular topic can provide teachers with similar tools to effectively plan for instruction that makes use of 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 ideas being made public, and a teacher must then plan to bring those aspects of student thinking together in a discussion that surfaces key ideas and builds understandings for 178   Taking Action Grades 9–12 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.

all students (Bray 2013; Smith and Stein 2011; Swan 2001). The ways in which students talk and write about a mathematical concept give the teacher 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 in advance of the lesson and connected them to his or her instructional goals, the evidence of student thinking can provide a powerful assessment of student learning. In turn, the teacher 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 purposes 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 making use of that thinking during a lesson can send important and powerful messages about students’ mathematical identities. By carefully listening to and interpreting student thinking, teachers position students’ contributions as mathematically valuable and 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 has the potential to strengthen students’ identities as knowers and doers of mathematics, as well as providing 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). Once student thinking is made public, teachers can engage in a wide variety of moves that have implications for students’ mathematical identities. The teacher might choose to elevate a student to a more prominent position in the discussion by identifying his or her idea as one 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 to 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 serve to promote historically marginalized student populations or students who do not have a strong track record of success in mathematics. In particular, promoting a classroom culture in which mistakes or errors are viewed as important reasoning opportunities can encourage a wider range

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of students to engage in mathematical discussions with their peers and the teacher. When student thinking is placed at the center of classroom activity, it is more likely that students who have felt evaluated or judged in their past mathematical experiences will make meaningful contributions to the classroom over time.

Key Messages • Mathematical tasks that promote reasoning and problem solving (and have characteristics of high-level demand tasks as described in fig. 3.2) are necessary (but not sufficient) for eliciting student thinking.

• Student thinking is elicited when teachers ask students to explain or justify their response to a task in writing or orally.

• Student thinking is extended when teachers ask questions that build on what they learned about what students know and can do without telling students what to do or how to do it.

Taking Action in Your Classroom: Eliciting and Using Evidence of Student Thinking It is now time to consider what implications 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 the next page.

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Taking Action in Your Classroom 8.1 Eliciting and Using Evidence of Student Thinking Choose a task that has the potential to elicit multiple solution paths to use in your classroom (e.g., a “doing mathematics” task from chapter 3; one that uses multiple representations from chapter 5). • Anticipate the solution strategies, both correct and incorrect, that students might use. • Identify the solution strategies 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 you anticipated and any others that arise and that notes potential questions you can ask. • Teach the lesson and reflect on the extent to which you elicited and made use of student thinking. • Consider how the information gathered in the monitoring tool could be used to inform your practice beyond the current lesson.

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CHAPTER 9

Support Productive Struggle in Learning Mathematics The Analyzing Teaching and Learning 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). Tasks that promote reasoning and problem solving, as discussed in chapter 3, provide such opportunities. 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 the teacher uses to support productive struggle;

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• determine a way to support productive struggle when a student immediately reaches an impasse;

• review key research findings related to productive struggle; and • reflect on productive struggle in your own classroom.

For each Analyzing Teaching and Learning (ATL), make note of 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. Once you have written down or shared your ideas, read the analysis, where we offer ideas relating the Analyzing Teaching and Learning activity to the focal teaching practice.

Supporting Students’ Efforts to Make Progress In Analyzing Teaching and Learning 9.1, we go into the classroom of Jeffrey Ziegler where his 11th- and 12th-grade students are working on the S-pattern task (see the next page). The students in this class have struggled with mathematics throughout high school, and Mr. Ziegler is trying to help them develop the capacity to persevere when they encounter a challenge. Through their work in this lesson, Mr. Ziegler wants his students to understand the following:   1. An equation can be written that describes the relationship between two quantities.

  2. Different but equivalent equations can be written that represent the same situation.   3. The symbolic and pictorial representations can be connected.

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The S-pattern Task

  1. What patterns do you notice in the set of figures?   2. Sketch the next two figures in the sequence.   3. Describe a figure in the sequence that is larger than the 20th figure without drawing it.   4. Determine an equation for the total number of tiles in any figure in the sequence. Explain your equation, and show how it relates to the visual diagram of the figures.   5. If you knew that a figure had 9,802 tiles in it, how could you determine the figure number? Explain.   6. Is there a linear relationship between the figure number and the total number of tiles? Why or why not? Adapted from Foreman and Bennett (1995).

The lesson begins with Mr. Ziegler engaging students in a brief discussion of the task. They establish the fact that the S-pattern is growing in two dimensions, getting both “taller” and “bigger.” Before they begin their work, Mr. Ziegler tells students (Smith, Bill, and Ziegler 2015, pp. 9–10 ): Now there are 6 prompts. . . . The first one, the second one, third one is to kind of get you started but it is on you guys to work with your groups to come up with a way to find the patterns. You don’t necessarily have to wordfor-word answer these questions, but they’re there to help you maybe get started. Before watching the video clips, solve the task in more than one way. If possible, compare and discuss your strategies with a colleague.

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Analyzing Teaching and Learning 9.1 more Determining How Student Learning Can Be Supported Watch the two video clips in which we see Mr. Ziegler visiting two different small groups of students who are struggling to form a generalization of the pattern they observe. Consider the teacher’s actions and interactions with groups 1 and 2 in both clips. • What does the teacher do to support his students’ learning? • In what ways does Mr. Ziegler use the other effective teaching practices in his effort to support his students’ productive struggle? You can access and download the videos and theirs 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 Student Learning Can Be Supported So what exactly did Mr. Ziegler do to support the learning of his students while they worked in small groups? Perhaps most important, he did not tell students what to do or how to do it. While being more directive about a pathway to pursue would have allowed students to make progress more quickly, it would have lowered the cognitive demands of the task and sent students the message that they were not capable of solving the task without the teacher’s direct assistance. While students may have arrived at a correct answer, they would feel no ownership over the work and would have learned nothing about the importance of perseverance and the satisfaction they would feel when they accomplished something they thought was beyond their reach. Instead, Mr. Ziegler determined what students currently understood about the task and provided suggestions for them to pursue on the basis of the approach they had selected. In other words, he supported students’ productive struggle. The answer to the question “So what exactly did Mr. Ziegler do to support the learning of his students while they worked in small groups?” is a review of many of the practices we have discussed in previous chapters. (This analysis is based in part on an analysis of group 1’s efforts that appeared in Smith et al. [2015]). Mr. Ziegler established clear goals for student learning that he used to guide his decision making during the lesson. These goals focused on what students would understand about mathematics as a result of engaging with the task — not on what students would do. Specifically, in his interactions with group 1, Mr. Ziegler pressed students to come up with a generalization (goal 1) by encouraging them to find a way to determine the total number of tiles without counting (lines 39–40), and then later (lines 152–54), to connect the equation they found to the visual representation of tiles (goal 2). He consistently pressed

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group 2 to connect their equation with the picture (lines 82–83, 103, 133–34). (Goal 3 was not observable in the small-group work.) Mr. Ziegler selected and implemented a task that promoted reasoning and problem solving. This high-level “doing mathematics” task could not be solved by application of a known procedure, and it required considerable effort to determine the underlying structure of the pattern. The figures in the sequence provided a way for students to use and make connections between mathematical representations, in this case between the symbolic equations and the physical arrangement of tiles. Mr. Ziegler consistently pressed students to connect their equations with the physical arrangement of tiles. Finally, the six questions in the task provided additional support for students who needed it. In particular, questions 1–3 may have helped some struggling students gain entry into the task, and questions 5–6 may have provided extensions for students who finished the problem prior to the whole-class discussion. Hence, the task itself had what we referred to in chapter 3 as a “low-threshold, high ceiling,” thus providing all students with entry into the problem as well as the opportunity to engage in rigorous thinking and reasoning. The teacher’s charge to students that “you don’t necessarily have to word-for-word answer these questions” allowed students to decide where they needed to enter the problem. Mr. Ziegler posed purposeful questions. While some questions served to gather information and required little thinking (e.g., lines 28–38), in other questions Mr. Ziegler asked one student to explain what another student had said (lines 68, 145), challenged students to explain what they were doing and why it worked (e.g., lines 162–67), and pressed students to make connections between the equation and the diagram (lines 82–83, 103, 133–34, 152–54). Mr. Ziegler elicited and made use of student thinking. He began his interactions with both groups by assessing what they currently knew and understood (e.g., lines 8–9, 54–58, 140–44) and then advancing them beyond where they currently were (e.g., lines 39–40, 82–85, 152–54, 165–67). By leaving a group with something additional to think about and work on without his guidance, Mr. Ziegler was sending the message that the students were capable of making progress without him. There is limited evidence of meaningful mathematical discourse in these two clips because of the nature of the clips — we can only see what the groups were doing when the teacher was with them, and his interactions with the group were more of a question-and-answer exchange than a discussion. However, during the whole-group discussion at the end of the lesson, Mr. Ziegler had four groups present their solutions to the task in the following order:  1. n 5 (x 1 1)(x − 1) 1 2 (Group 1)  2. n 5 (x − 1)2 1 2x

 3. n 5 x2 1 1

 4. n 5 (x 1 1)2 − 2x

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During this time, Mr. Ziegler facilitated the discussion by (1) asking students to make connections between the physical arrangement of tiles, the equations, and the figure numbers; (2) pressing students to discuss the equivalence of the equations; and (3) having students explain why the function was not linear. What was particularly noteworthy about the discussion was the presentation from group 1. Recall that when Mr. Ziegler first visited group 1, students were struggling to find a way to determine the number of tiles in a figure without counting. In his second visit to the group, they shared a generalization (n 5 x2 1 1) but could not connect it to the picture. However, when Mr. Ziegler visited the group a third time (not shown in the video clips), he found that they had created a generalization [n 5 (x 1 1)(x 2 1) 1 2] that stemmed from the physical arrangement of tiles. He invited one of the members of the group to present this solution during the whole-group discussion. As the presenter noted (Smith et al. 2015, p. 12): We found out that you can turn all these S’s into rectangles as you can see right here [referring to each of the figures in the sequence]. For example, we used 5. We used 5 and then you go down, make a rectangle out of it. If you notice, you have 2 left over and if you count, you have 6 going down, 4 going across [see fig. 9.2]. That leaves you with the 4 by 6. When you multiply those together, you get 24. Is that correct? [Multiple students talking at once].

6 going down

4 going across

Fig. 9.1. Drawing that supported group 1’s presentation

Now we do that, you notice we have 2 out, so what you want to do is, you want to plus 2 because you have 2 extra. Because you have 2 extra and that leaves you with 26. So what we did was, if you notice, when you make your S into a square, you subtract 1 from 5 and you add 1 to 5 so what you get is, you get x 1 1 and you get an x 2 1 and since you’re multiplying this, you have to put it in parentheses. And on the outside, you want to add 2 and that’s your equation. 188   Taking Action Grades 9–12 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.

Although the presenter refers to the rectangle as a square, it is clear that the student understands how the equation can be used to find the total number of tiles in a figure and how the equation related to the physical arrangement of tiles — two goals of the lesson. The integrated use of all of these practices supported students’ productive struggle in Mr. Ziegler’s class. Group 1 stands as a particularly strong example of the progress students can make over the course of a lesson with appropriate levels of support. 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 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 (emphasis added).

Considering the Role of Teacher Questioning in Supporting Productive Struggle In Analyzing Teaching and Learning 9.2, you are asked to consider the types of teacher interventions that do and do not support students’ productive struggle. In each of the five mini-dialogues in ATL 9.2, students who were attempting to solve the Amazing Amanda task (see the next page) 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. The students had previously proved that the measure of the interior angles of a triangle sum to 180 degrees, and they had established the fact that the sum of the angles around a point in the plane is 360 degrees. These ideas were reviewed before they began work on the task. Before you analyze the mini-dialogues, solve the task in more than one way. If possible, compare and discuss your strategies with a colleague.

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The Amazing Amanda Task Amanda claims to have an amazing talent. “Draw any polygon. Don’t show it to me. Just tell me the number of sides it has, and I can tell you the sum of its interior angles.” Is Amanda’s claim legitimate? Does she really have an amazing gift, or is it possible for anyone to do the same thing?

  1. Working individually, investigate the sum of the interior angles of at least two polygons with 4, 5, 6, 7, or 8 sides. Use a straight-edge to draw several polygons. Make sure that some are irregular polygons. Subdivide each polygon into triangles so you can use what you already know about angle measures to determine the sum of the interior angles of your polygon. Organize and record your results.   2. As a group, combine your results on a single recording sheet and answer these questions: • How did group members subdivide their polygons into triangles? Did everyone do it in the same way? If different, how did that affect your calculations? • Does whether the polygon is regular or irregular affect the sum of the angle measures? Why or why not? • What patterns did you notice as you explored this problem? • What is the relationship between the number of sides of the polygon and the sum of the measures of the interior angles of the polygon? Express this relationship algebraically and explain how you know that your expression will work for any convex polygon. Adapted from “Amazing Amanda,” copyright 2007 by the University of Pittsburgh.

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Analyzing Teaching and Learning 9.2 Investigating Teacher Interventions After reading the mini-dialogues shown below: • Discuss the nature of each student’s struggle. • Identify what the teacher does to help students move beyond the impasse they have reached. • Determine whether the teacher supported students’ productive struggle.

Mini-Dialogues for the Amazing Amanda Task Dialogue 1 A student made the drawing shown below.

T: What did you do here? S: I drew a polygon with 5 sides. T: Then what? S: I divided it into triangles. And I got 4 triangles. But I don’t think it is right because when I asked around, no one else had 4. T: Your triangles can’t go outside the polygon. If you take your picture and just get rid of one of your diagonals, you will have the right number of triangles. [Student erases one of the diagonals.]

T: That’s right. So how many triangles do you have now? S: 3. T: Okay. So now you just need to multiply 3 by 180 and you will be set. So now try another one using this method.

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Dialogue 2 A student made the table shown below. # of Sides

Degrees of Interior Angles

3

180

4

360

5

540

6

720

7

900

8

1,080

T: Tell me how you constructed your table. S: I decided to try all of the polygons from 3 to 8. I knew that the 3-sided polygon—a triangle—had angles that summed to 180 degrees because we did that last week. Then I drew polygons with more sides on scrap paper. I subdivided each polygon into non-overlapping triangles. Then I counted the number of triangles in each polygon and multiplied by 180. T: Why did you multiply by 180? S: Because the angles of each triangle sum up to 180 so to find the sum of all the angles in a polygon you need to multiply the number of triangles in the polygon by 180. T: So how does this help you determine the relationship between the number of sides of the polygon and the sum of measures of the interior angles? S: I am not sure. I know that you multiply the number of triangles in the polygon by 180 like I said, so I guess I need to figure out how many triangles there are in each polygon. Maybe I will add a column to the table to keep track of this. T: That sounds like a good plan. I will check back in with you later.

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Dialogue 3 A student made the drawing shown below.

T: So tell me about your drawing? S: I made a 5-sided polygon and subdivided into 5 non-overlapping triangles. T: And then what? S: Well, since each triangle has angles that sum to 180 degrees, I multiplied 180 by 5 and got 900. [Student sounds unsure of herself.] T: So what is the problem? S: I think it is too big. I took out my protractor and did a rough measure of the angles and I got closer to 500. T: Nice way to check if your answer is reasonable. So let’s take a closer look at your diagram. Can you show me where the angles of the triangles are? [Student points to the angles in each triangle.] T: So are all the angles you just pointed to included in the interior angles of the polygon? S: No. All these [points to the angles formed around the center point] are not included in the interior angles. Oh, so somehow I need to figure out how not to count these. T: I will leave you to figure out what you know about the angles around a point and how this can help you solve your problem. I will check in with you later.

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Dialogue 4 A student can’t get started. T: What have you figured out so far? S: Nothing. I am not sure what to do. T: The first thing I want you to do is to draw a polygon with 4 sides. [Student draws a square.] T: Now you need to divide it into triangles, starting at one of the vertices. [Student divides the square into two triangles by drawing the diagonal.] T: Okay. So you have two triangles. What is the sum of the angles of a triangle equal to? S: 180? T: So if you have two triangles, what would the sum of the angles be? S: 360? T: Yes! So the angles of a 4-sided polygon sum up to 360 degrees. Now try a fivesided polygon and use the same method of breaking it up into triangles that we just did.

Dialogue 5 A student can’t get started. T: What have you figured out so far? S: Nothing. I am not sure what to do. T: Go back through your notes and review the work you did when we proved that the sum of the angles of a triangle sum to 180.

Analysis of ATL 9.2: Investigating Teacher Interventions All of the students portrayed in the dialogues have reached an impasse: they have reached a point at which they are no longer sure what to do or how to move forward. In each case, 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 drawn a 5-sided polygon and made it into triangles. (Note that the student drew a concave polygon rather than a convex polygon. This is not addressed iin the dialogue.) The student thinks this is incorrect because the other students he conferred with did not get 5 triangles in a 5-sided polygon. The teacher immediately tells the student that his triangles can’t go outside the polygon and tells him to remove one of the diagonals from his drawing. The teacher then reframes the problem as “You just need to multiply 3 by 180.” While 194   Taking Action Grades 9–12 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 teacher left the student with a clear pathway to follow to solve the task, which would serve to move him beyond the impasse, the pathway represented the teacher’s thinking rather than the student’s. In dialogue 2, the student has created a table that shows the number of sides and the corresponding sums of the degrees in the interior angles for polygons with 3, 4, 5, 6, 7, and 8 sides. The student knows that the sum of the interior angles can be found by multiplying by 180 the number of triangles into which the polygon has been divided, but she has not figured out how to find the number of subdivided triangles without drawing. The teacher’s role here is minimal. She asks questions in order to understand what the student has done and then asks the student to consider how what she has done will help her determine the relationship between the number of sides of the polygon and the sum of measures of the interior angles. The student then reframes the problem as “I need to figure out how many triangles there are in each polygon” and defines a pathway to pursue: “I will add a column to the table to keep track of this.” The teacher did not guide the student in any way or do any thinking for the student. In dialogue 3, the student drew a 5-sided polygon, subdivided it into 5 non-overlapping triangles, and then multiplied 5 by 180 to determine that the sum of the interior angles was 900 degrees. The student’s suspicion that 900 was too big was verified when he used a protractor to measure the angles. Although the student knew that he was not correct, he was not sure why. The teacher asked two key questions that focused the student’s attention on the angles formed around the point on the interior of the polygon —“Can you show me where the angles of the triangles are?” and “So are all the angles you just pointed to included in the interior angles of the polygon?” This led the student to conclude that he needed to find a way not to count the angles of the triangles in the interior of the polygon. The guidance provided by the teacher built on the student’s way of thinking and allowed the student to move beyond the impasse. In dialogue 4, the student has done nothing and claims not to know what to do. The teacher does not make any attempt to determine what the student understands about the task and begins telling the student exactly what to do. While the student may be able to follow the directions given by the teacher and end up with an answer to the question, it is not clear what sense the student has made of this problem because we have no access to her thinking. The student is no longer struggling, but the pathway she is following is the teacher’s. In dialogue 5, as in dialogue 4, the student has done nothing and is not sure what to do. In this case, the teacher suggests the general strategy of reviewing his notes on previous work related to the sum of the interior angles of a triangle. While the teacher is not directing the student to a particular pathway, it is unlikely that this type of support will help the student move forward in a productive way. It may also be the case that the student has already taken this step. In that case, the feedback the teacher gives, which is intended to be supportive, may actually negatively impact the student’s mathematical identity and send an implicit message that he is not capable. Support Productive Struggle in Learning Mathematics    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.

Warshauer (2015b, p. 387) describes four types of teacher responses to struggle: (1) telling, (2) directed guidance, (3) probing guidance, and (4) affordance. Telling involves supplying students with information (e.g., a strategy to use) that removes the struggle, thus allowing them to make progress on the task by following the prescribed strategy. Directed guidance involves redirecting students to a different strategy, one that is consistent with the teacher’s rather than the student’s way of thinking, enabling the student to move beyond the impasse. Probing guidance involves determining what a student is currently thinking, encouraging the student’s self-reflection, and offering ideas based on his or her thinking. Affordance involves asking the student to articulate what he or she has done, encouraging continued efforts with limited intervention, and allowing the student time to continue to work. Before reading further, revisit each of the dialogues above and decide how you would classify each one. We would categorize dialogue 4 as an example of telling and dialogue 1 as 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 to one that requires only carrying out a procedure with no connection to meaning. We would categorize dialogue 3 as probing guidance and dialogue 2 as affordance. Both probing guidance and affordance tend to maintain the cognitive demands of the task by supporting productive struggle. What about dialogue 5? We see dialogue 5 as being an example of a fifth type of response, one that we have seen in classrooms in which we have worked. We would categorize this response as unfocused or vague: the teacher doesn’t direct the student to a particular strategy or build on the student’s thinking but instead provides a suggestion that is often too general to be helpful. How will you know if students’ 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;

• the student’s thinking is supported by acknowledging effort and mathematical understanding; and

• the student is able to move forward in the task execution through his or her actions.

We would argue that 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 are maintained; the support provided builds directly on the thinking of the student and makes it possible for the student to move forward based on his or her own actions. As we discussed in chapter 3 when we investigated the cases of Vanessa Culver and Steven Taylor, the telling and directed guidance provided by Mr. Taylor lowered the demands of the task and took away the students’ thinking opportunities. By contrast, Ms. Culver supported productive struggle by providing what we could now describe as probing guidance.

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Dialogue 5 is slightly different. Dialogues 1 and 4 focus the students on a particular pathway that will be likely to lead to successful completion of the task. In dialogue 5, there is no push on the teacher’s part to focus the student’s attention in a specific direction. So while the cognitive demand of the task was not lowered, student thinking was not sufficiently supported, and it is unlikely that the student would make much progress.

Other Teaching Moves That Support Productive Struggle In ATL 9.3, you analyze a dialogue between Ms. Blanchard, a geometry teacher, and a small group of students in her 10th-grade class (Michelle, Tim, Anthony, and Stephanie). The class is working on the coordinate geometry unit. Ms. Blanchard has selected the As the Crow Flies task, shown on the next page. The purpose of this task is to help students develop an understanding of the meaning of the distance formula, based on their knowledge of the Pythagorean theorem and graphing on a coordinate system. Rather than presenting the formula as a rule to be memorized, Ms. Blanchard wanted her students to develop conceptual understanding of the formula on which she can later derive and build fluency with the procedure. Before reading further, solve the As the Crow Flies task in more than one way. If possible, compare and discuss your strategies with a colleague. After making sure that students understood the task, Ms. Blanchard gave the groups 5 minutes to complete questions 1 and 2. She then brought the students back together to discuss their answers. Students determined that the distance to their friend’s house was 16 blocks and that the path could either be 6 blocks south and 10 blocks east (shown in fig. 9.2) or 10 blocks east and 6 blocks south. They also determined that the helicopter path would be a straight line connecting the two houses (shown in fig. 9.2 as the hypotenuse of the right triangle). They then set up the coordinate-axis system and got to work on the remaining components task.

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As the Crow Flies Task

  1. Draw a path that you could drive from your house to your friend’s house using the fewest number of turns. Calculate the distance in blocks.   2. Draw a path that you could take if you could use a helicopter to fly straight to your friend’s house “as the crow flies.”   3. Establish a coordinate-axis system, using the school as the origin. Determine the coordinates for the two houses. Use this to help you find the distance between your house and your friend’s house “as the crow flies.”   4. How can you use your findings in part 3 to determine the distance between any two points? Reprinted with permission from the NCTM Reasoning and Sense Making Task Library, copyright 2011, by the National Council of Teachers of Mathematics. All rights reserved. Available at http://www.nctm.org /rsmtasks/.

Fig. 9.2. Example of the pathways drawn by Ms. Blanchard’s students and the location of the x and y axis

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Analyzing Teaching and Learning 9.3 Identifying Strategies That Support Productive Struggle • Read the dialogue below in which Ms. Blanchard provides probing guidance to a group of students who got the correct answer but did not explore the problem as intended. Their solution would not lead them to a generalized approach for finding distance given two points. • What did the teacher do to support the group’s productive struggle?

So what did you do?

1

T:

2

Michelle: We used the Pythagorean theorem and got 2√34 or about 11.6. How did you get that? Tim?

3

T:

4 5 6

Tim:

7

T:

8 9

Stephanie: It is the number of blocks between the two houses if you go in a straight line between them.

Well we knew that the legs of the triangle were 6 and 10 from what we did in the first part of the task, so we did 62 1 102 and got 136. So we took the square root of that using our calculator and got our answer.

So what does 11.6 mean? Stephanie?

Nice work. So now I am wondering how you would find the distance if you only had the coordinates of the two houses? Anthony?

10 11

T:

12 13

Anthony: Well we can’t. In order to use the Pythagorean theorem, you need to have the length of the legs. Good point. I would like you to see if you can find the coordinates of the houses, using the school as the origin, and see if that could help you find the lengths of the legs of your triangle. I will be back in a few minutes to see what you come up with.

14 15 16 17

T:

18 19 20

[Teacher leaves to respond to other students and returns 5 minutes later. During her absence the group determined the coordinates of the points. Michelle recorded the ordered pairs on their diagram as shown in figure 9.3.] So tell me what you found.

21

T:

22 23 24

Michelle: Okay, see [referring to fig. 9.3] we found that our house would be (22, 5) and our friend’s house would be (8, 21). We also labeled the corner of the triangle where the right angle is. That would be (22, 21).

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Fig. 9.3. Coordinates of houses 25

T:

26 27 28

Anthony: Well if we look at the vertical leg, we know that it has a length of 6 (because we can count it) but that is also the difference in the y values in the ordered pairs. 5 2 (21) 5 6

29 30 31 32 33 34

T:

So how do the coordinates help you find the length of the legs?

It is the difference between the y values in what two ordered pairs? Stephanie?

Stephanie: Oh wait . . . I was going to say between the ordered pairs (22, 5) and (22, 21) but I now see that is the same as the difference in the y values in (2, 5) and (8, 21). T:

Does everyone see what Stephanie is saying? Tim?

35 36

Tim:

37

T:

38 39 40

Michelle: Well we know that the horizontal leg has a length of 10 (because we can count it) but that is also the difference in the x values in the ordered pairs. 8 2 (22) 5 10.

41 42 43 44 45 46 47 48

T:

Yeah. The y value for the corner point is the same as the y value for the friend’s house. So how did you find the length of the other leg? Michelle?

The differences in which pairs? Anthony?

Anthony: Well it is the difference between the x values in (8, 21) and (22, 5) but again since the corner has the same x value as our house, then you can think of the distance that way too. T:

Okay, so how does this help us? Stephanie?

Stephanie: Well I think we could find the distance between any two points by finding the vertical difference and the horizontal difference using the coordinates of the two points and then using the Pythagorean theorem.

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49 50 51 52 53

T:

54

Tim:

55 56

T:

I liked the way you all worked together on this so far. Now I would like you to see if you can find the distance between two points using the method Stephanie’s just described. Suppose your aunt’s house is at coordinates (10, 7). Can you find the “as the crow flies” difference without drawing a triangle and counting to find the length of the legs? Probably.

Great. You have done a really nice job working through this together as a team. I will check in with you a little later.

Analysis of ATL 9.3: Identifying Strategies That Support Productive Struggle Most notably, Ms. Blanchard supported her students’ productive struggle by providing probing guidance. As discussed in the analysis of ATL 9.2, probing guidance involves determining what the students are currently thinking, engaging them in reflecting on their work, and offering ideas based on their thinking. Warshauer (2015a, p. 392) has identified four strategies for supporting productive struggle — question, encourage, give time, acknowledge. Identifying the strategies Ms. Blanchard uses with her students gives us more insight into what she actually did. First, she asked questions that made students’ thinking public (lines 1–9). The answers to this series of questions made it clear that, while the students had a correct solution, their method would not help them in determining the distance between any two points. Ms. Blanchard did, however, learn that the students recognized the utility of the Pythagorean theorem in finding distance. She then asked students whether they could find the distance between the houses if they only had the coordinates of the houses (lines 10–11). She challenged them to find the coordinates and see whether the coordinates could help determine the legs of the triangle (lines 14–16). This interaction encouraged students to reflect on and reconsider what they had done and, ultimately, to find the coordinates of the houses (lines 22–24) and determine the relationship between horizontal distance and the differences in the x values of the ordered pairs and the vertical distance and the difference in the y values in the ordered pairs (lines 38–40, 26–28). She also gave the group time to work to find the coordinates and consider how the coordinates could be used to find length (lines 18–20) without hovering over them. This sent the clear message that she had confidence that they could do this without her assistance. In the end she acknowledged the group effort and their ability to work through the problem (lines 55–56). Through the use of these strategies, Ms. Blanchard 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. Blanchard and her students is that unlike the five dialogues we analyzed in ATL 9.2, the students were not struggling when she

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began her interaction with them. That is, they had come up with the answer of 11.6 blocks and seemed quite confident in it. However, the teacher recognized that the process they used would not generalize to situations where any two points were given, and, therefore, they did not meet the mathematical goals of the lesson. She helped students recognize the limitations of their method and therefore created a need for finding another method. Students’ potential for success in the fourth question in the task would be compromised if they had not determined the relationship between the coordinates and the lengths of legs of the triangle. Since the teacher’s goal was for her students to develop conceptual understanding of the distance formula, determining this relationship was an essential step in the process.

Supporting Students Who Can’t Get Started One challenge teachers often encounter when they engage students in solving a high-level task is the student (or students) who can’t get started on the task. This can sometimes be alleviated by selecting a task that has multiple entry points and launching the task so that students are clear on what is going on in the problem and what they are being asked to do. But, at times, despite your best efforts to make the task accessible, some students require additional support and assistance to be able to begin the task. But what do you do when you approach a student who seems to have reached an impasse almost immediately? We saw two approaches to dealing with this situation in ATL 9.2. In dialogue 4, the teacher told the student what to do, thus taking over the thinking and eliminating the struggle; in dialogue 5, the teacher provided a suggestion that was too vague and unfocused to be particularly helpful in getting beyond the impasse. In ATL 9.4, you will 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.

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Analyzing Teaching and Learning 9.4 Supporting a Student Who Can’t Get Started Imagine that the students in your class are starting work on the Eruptions: Old Faithful Geyser task shown on the next page. You launch the task by asking students what they know about Old Faithful and showing them a video of the geyser erupting (https://www.youtube.com/watch?v=wE8NDuzt8eg). You then distribute the task and discuss the information that is in the table. You randomly assign each student two rows of data, which provide the times between eruptions on two different days, to work with. You explain that they will have about 15 minutes to work on questions 1 and 2 individually. They will then convene as a group and complete questions 3 and 4 together. Students have had some prior experience in analyzing data and graphing data. The purpose of this task is to make inferences about population parameters based on a random sample from that population. What type of probing guidance could you provide to students who cannot get started on this task that will support their productive struggle without taking over the thinking for them?

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Eruptions: Old Faithful Geyser Task

  1. Look over the data. Is there anything you notice or anything you wonder about in your two samples of data? Jot down some of your “notices” and “wonders.”   2. Create at least one type of graphical representation for each of the two days of data to help you visualize any patterns in the wait time. Jot down any additional notices and wonders that occur to you.   3. Share the graphical representation of your data in your group. What do you notice, wonder about, as you look through your group’s graphical representations?   4. Accept the challenge to act as “data detectives.” Agree as a group on a graphical way to display your data. On the basis of your data, make a group decision about how long you would expect to wait between blasts of Old Faithful if you showed up at Yellowstone Park when Old Faithful had just finished erupting. Be prepared to present your graph to the other groups in the class and to defend your group’s data-based prediction for the expected wait time. Reprinted with permission from the NCTM Reasoning and Sense Making Task Library, copyright 2012, by the National Council of Teachers of Mathematics. All rights reserved. Available at http://www.nctm.org /rsmtasks/.

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Analysis of ATL 9.4: Supporting a Student Who Can’t Get Started When a student can’t get started on a problem, it generally isn’t the case that he or she has no relevant knowledge to bring to bear on the situation. It is more often that, for some reason, the student 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 “What do the numbers in the rows represent?” (“It is the amount of time between eruptions of Old Faithful.”) “How many different data points are there for each day?” (“18.”) “What do you notice about time between eruptions?” (“They are not the same. On day 5 the least amount of time between eruptions was 50 minutes, and the most amount of time was 87 minutes. On day 12 the least was 49 minutes, but the most was 93 minutes.”) From these basic questions, you could ask the student how he or she could organize the data in order to get a better sense of how much time you could expect to wait between eruptions so that the patterns the student notices would be clear to someone else. He or she may suggest putting the numbers in order or re-creating intervals (e.g., 50–60) that would include multiple values. While the ultimate goal is for the student to create a graph to represent the data, organizing the data is a first step needed in order to accomplish that goal. Leaving the student to undertake this task on his or her own and then checking on the student later sends the message that you are confident that this student can do it on his or her own. As we have mentioned in chapters 7 and 8, anticipating the ways in which students are likely to solve a task and the difficulties they may encounter in doing so, as well as the ways you will respond when students run into roadblocks, is a critical part of the planning process. Such planning in advance of the lesson will better prepare you to support struggling students during the lesson. In addition to specific strategies, teachers we have worked with have often added a “can’t get started” category to their monitoring sheets and preplanned questions specific to the task that will support students in making progress. One thing to keep in mind is that the teacher’s goal in intervening when a student or group is struggling is not to make sure that each and 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 should have the opportunity to “collect more pieces of information about the issue of the discussion and to understand the issue more deeply.”

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Support 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 to 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 provide insight into productive struggle. 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. (While the majority of research related to productive struggle has been conducted with teachers and students at the middle school level, what has been learned through these studies is also applicable to high school students.) In a study of classrooms in which sixth- and seventh-grade students engaged in solving high-level proportional reasoning tasks, Warshauer identified teacher actions that supported productive struggle (described earlier in this chapter). In Warshauer (2015b), she identified probing guidance and affordance as two types of teacher responses that could help students move beyond an impasse. In both cases, 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 help students make progress. Smith (2000a) describes the way in which 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, Ms. Henderson initially simplified complex problems so that students would feel successful. In so doing, she lowered the demands of the task and eliminated the needed for struggle. Over time, she came to see that students needed to struggle in order 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. This 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 just waiting to be told what to do. Figure 9.4 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).

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New Expectations for Students

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 solutions and 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.

Diagrams, sketches, and hands-on materials are important tools for students to use in making sense of tasks.

Give students access to tools that will support their thinking and processes.

Students were able to use tools to solve tasks that they could not solve 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. 9.4. Key elements in Ms. Henderson’s efforts to redefine success for herself and her students (Smith 2000b, p. 382) Support Productive Struggle in Learning Mathematics    207 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 elements identified by Smith in figure 9.4, along with the specific types of support (probing guidance and affordance) and strategies (question, encourage, give time, acknowledge) suggested by Warshauer (2015b), provide teachers with concrete actions they can take in creating classroom environments where struggle is encouraged, valued, and supported.

Promoting Equity by Supporting Productive Struggle in Learning Mathematics As stated in the opening of this chapter, productive struggle comprises the work that students do to make sense of a situation and determine a course of action when a solution strategy is not stated, implied, or immediately obvious. From an equity perspective, this implies that every student must have the opportunity to struggle with challenging mathematics and to receive support that encourages persistence without removing the challenge. 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” (Hiebert and Grouws 2007, p. 388). Too frequently, historically marginalized students are overrepresented in classes that focus on memorizing and practicing procedures and rarely provide opportunities for students to think and figure things out for themselves. When students in these classes struggle, the teacher often tells them what to do without building their capacity for persistence. Central to the equity-based mathematics teaching practices (Aguirre, Mayfield-Ingram, and Martin 2013) is the belief that strengthening mathematics learning and cultivating positive mathematical identities requires 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. Jeffrey Ziegler, featured in the opening video in this chapter, takes the development of identity seriously. He begins the year by having students write their math autobiography so that he can understand how students see themselves as learners of mathematics and what experiences have shaped the development of this identity. He then uses what he has learned to help students come to see themselves differently. As he describes: I had the students write their math autobiography. So they had to talk about what their math identity is. So, you know, where are you now with mathematics? Why do you think you are there? What do you think has happened from the time that you can remember with mathematics until now? When did you find yourself shaping into the person that says, “You know, I really love math. I think math is exciting. I think math is fun.” Or that person that says, “You know, I really kind of get that pit in my stomach when I have to come to math class every day.” But there’s a reason for that. Something happened along the way. Something happened multiple times that got you to this point. What was that? Because, you know, it’s

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my responsibility to continue to shape their math identity. So if they really love math and they really enjoy math, so it’s my responsibility to challenge them, but to hopefully keep them in that space. And students that come in struggling or really disliking it, how do I create an environment, a space, a year which helps reshape that or form it in other ways that hopefully when they leave, you know, they’re not just saying, “I love Mr. Ziegler’s class,” or “I thought this class was great,” but that “Math, I really got a new insight on what mathematics is and how I really do mathematics and how much fun it could be.” A study by Boaler and Staples (2008) provides evidence of what can happen when every student is given access to ambitious instruction and the opportunity to learn mathematics with understanding. Teachers at Railside (one of the three high schools that were the focus of the study) were able to reduce the achievement gap between different ethnic groups by providing all students with 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 despite the fact that 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 (emphasis added). 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. 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 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- and school-level decisions in order 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

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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 just 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 a teacher to support productive struggle, the teacher must examine his or her 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 providing too much direction.

• Productive struggle can be accomplished by asking questions, encouraging reflection, providing time, and acknowledging effort.

Taking Action in Your Classroom: Supporting Students’ Productive Struggle It is now time to consider what implications the ideas discussed in this chapter have for your own practice. We encourage you to begin this process by engaging in the Taking Action in Your Classroom 9.1 and 9.2 activities.

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Taking Action in Your Classroom 9.1 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— • 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 identified in the chapter to support students’ productive struggle so that they can make positive progress toward the mathematical goals of the lesson.

Taking Action in Your Classroom 9.2 Supporting Productive Struggle Over the course of a week, note how you interact with students who are struggling. Perhaps take turns with a colleague observing each other’s classrooms, videotape, or audiotape if possible. Then consider these questions: • How did you respond when a student could not get started? • What advice did you give when students were stuck? • How did you encourage students who had found an answer but did not engage with the mathematical goals of the task?

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CHAPTER 10

Pulling It All Together As you worked through chapters 2 through 9, you had the 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. While we provided some insight in each chapter regarding how the focal practice connected to other practices, in this chapter, we discuss the eight effective teaching practices as a set of connected practices that form a framework for mathematics teaching, and we consider what it will take to make these practices a reality in your classroom. Throughout this concluding chapter, we refer back to activities that you explored in earlier chapters, along with one new ATL, and we use these activities to emphasize four key points that have been threaded throughout the book:   1. 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 advocated by NCTM and provide opportunities for each and every student to achieve the world-class standards that have been put into place by states and provinces.

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

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

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

A Coherent and Connected Set of Mathematics Teaching Practices The Case of Vanessa Culver, which we revisited throughout the book, served as a touchstone to which we related the new learning in each chapter. The case was intended to make salient the synergy of the effective teaching practices — the success of Ms. Culver’s lesson was the result of integrating the practices in a coherent way rather than by attending to individual teaching practices. In figure 10.1, we present a framework for mathematics teaching that shows the relationship between and among the teaching practices and how they work together to support ambitious instruction, as they did in Ms. Culver’s classroom.

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

As we discussed in chapter 2, learning goals serve to focus and frame the teaching and learning that occur throughout a lesson. Hence, establish math goals to focus learning sits at the top of the framework, signifying that setting goals is the starting point for all instructional decision making. The clarity and specificity of goals, and how such clarity and specificity supported subsequent instruction, are clear in nearly every narrative and video case we have examined (with the exception of the Case of Steven Taylor, where the absence of clear and specific learning goals may have contributed to the lack of success of his lesson). Take, for example, Shalunda Shackelford’s lesson (introduced in chapter 2) featuring the Bike and Truck task. Ms. Shackelford wanted her students to understand that (1) the language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values; (2) context is important for interpreting key features of a graph portraying the relationship between time

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and distance; and (3) the average rate of change is the ratio of the change in the dependent variable to the change in the independent variable for a specified interval in the domain. Ms. Shackelford’s goals guided the decisions she made before and during the lesson — from the task she selected for students to work on, to posing a common misconception from her “imaginary friend Chris,” to the questions she asked to foster student–student discourse and to press students to reason about mathematical relationships, to the way she revisited students with misconceptions in order to give them the opportunity to explain their new reasoning. The goals were not just statements to be recorded and forgotten; 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 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 serve to develop students’ conceptual understanding of mathematics and provide a base on which to build procedural fluency. Hence, these two practices are directly connected to the goals of the lesson and to each other. In each of the narrative and video cases that we 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 (Steven Taylor, where his goal was very vague), the task was consistent with the goals for the lesson. For example, the Bike and Truck task used by Ms. Shackelford is a “doing mathematics” task based on the criteria presented in the Task Analysis Guide (fig. 3.2). As such there is no prescribed pathway for solving the task, and students must explore and uncover the relationships (e.g., between the graph and the movement of the bike and truck, between distance and time) in order to understand covariation and average rate of change. Through their work on the task, Ms. Shackelford’s students were able to reason about changes in time and distance and how these changes were connected to the context and represented on the graph, thus achieving the goals for the lesson. Ms. Shackelford’s lesson did not focus explicitly on developing procedural fluency, but, as we discussed in chapter 4, the conceptual understanding of average rate of change can serve as the basis for developing procedural fluency in subsequent lessons. The heart of any lesson is the discussion (represented by the large rectangle in fig. 10.1). Discussions provide the opportunity for students to 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 provide 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. Discussions that take place in both small- and whole-group setting are facilitated by use of the four teaching practices situated within the discussion rectangle in figure 10.1: pose purposeful questions, use and connect mathematical 216   Taking Action Grades 9–12 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.

representations, elicit and use evidence of student thinking, and support productive struggle in learning mathematics. Together, these practices serve to focus students’ mathematical work and thinking on the goals of the lesson. As students work collaboratively in small groups, questions and representations can be used to support productive struggle and elicit evidence of students’ thinking, which can inform the teacher’s planning of the whole-group discussion. During the whole-group discussion, a teacher might ask questions for the purpose of eliciting students’ thinking or encouraging connections between representations. Together, these practices 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 to be elicited during the wholegroup discussion and can help teachers determine what strategies, ideas, and representations to select for presentation and discussion. Having goals in mind also supports teachers’ assessment of students’ learning; it means that they know what to look 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 the next lesson or lessons. This is represented in figure 10.1 by the dashed line connecting the bottom of the model back to goals. To consider how these four practices play out during a whole-class discussion, we return to Shalunda Shakelford’s classroom. During the discussion of the Bike and Truck task, Ms. Shackelford posed purposeful questions to move students’ thinking toward the goals of understanding how time and distance change together (covariation), connecting the context and the graph, and determining average rate of change. In particular, she introduced a misconception from her “imaginary friend Chris” (clip 1 transcript, lines 1–4), she asked students to explain when the truck was moving the fastest (clip 2 transcript, lines 1–2), and she asked students to explain what was happening with the bike and truck after the 17-second mark (clip 2, discussion beginning on line 40). Her questions also served to elicit students’ thinking (e.g., clip 1, line 82, “What are you thinking now?”) and provide opportunities for student to justify and clarify their own ideas (clip 1, lines 21–23, “. . . do you still disagree with him? . . . go ahead and tell me why.”). At the end of the second video, we see Ms. Shackelford returning to three students who had previously expressed misconceptions and asking them to explain their new understanding. By soliciting input from different students, she is supporting productive struggle by making a wide range of ideas available for students to consider as they clarify their own understandings. 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. Shackelford supported students’ reasoning and problem solving as they worked to make sense of changes in time and distance, how these changes were represented on the graph and in context, and how to determine average rate of change. We can also see how these four practices were used (in both smalland whole-group discussion) to support students’ work on tasks that promote reasoning and Pulling It All Together    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.

problem solving and to ultimately accomplish the goals for the lesson in the narrative case of Vanessa Culver (chapter 1) and the video case of Debra Campbell (chapter 5). Enacting the effective teaching practices requires purposeful actions and decisions on the part of the teacher, such as selecting the goals and tasks for the lesson, identifying what questions are essential to ask and what aspects of students’ thinking should be elicited (e.g., strategies, ideas, struggles, and misconceptions), and determining what connections and mathematical ideas should surface during the whole-group discussion. Making these decisions in the moment as a lesson unfolds, and reacting in ways aligned with the effective teaching practices, would 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 in 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 advanced 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 prior to setting foot in the classroom. The lesson plan for the Proof task (appendix A) provides an example of the type of thoughtful and thorough lesson planning that is needed in order to support ambitious instruction and the learning and engagement of every student. This lesson plan embodies many of the ideas regarding ambitious teaching and equity that we have discussed in earlier chapters. 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 will be used, what instructional supports will be provided to students for working on the task, the prior knowledge needed to enter the task, and how the task will be launched. In addition, the plan provides anticipated solutions, including misconceptions, and questions that can be asked to assess and advance student learning. The questions serve to elicit students’ thinking so it can be used to advance their understanding 218   Taking Action Grades 9–12 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 support their productive struggle. The section labeled “Sharing and Discussing the Task” is essentially a road map for facilitating the discussion. It provides a plan for selecting, sequencing, and connecting student responses so that the key mathematical ideas targeted in the lesson are public and explicit. Such planning also provides a useful framework within which to operate when unanticipated responses arise in class. With a clear sense of the goals of the lesson and the mathematical story line of the lesson through careful planning, a teacher’s resources are liberated to make better sense of unexpected responses and make principled decisions about how to handle them. As you can see, the lesson plan is driven by a series of questions. 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 consistently comment 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 kind of helps me focus my students’ thinking, which helps us get to an objective and leads to a better lesson. (Smith et al. 2008, p. 137) Teachers we have worked with, including the one quoted above, have noted that this type of lesson planning is time-consuming. Working with colleagues might allow you 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 lessons using a tool such as the “monitoring chart” (discussed in chapter 7) or the lesson planning 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 the planning of 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. Pulling It All Together    219 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.

Some examples of related lessons throughout this book include the tasks on building polynomial functions (figs. 5.3 and 5.6); the lessons on finding the volume of prisms, cylinders, cones, and pyramids (chapter 4); and the discussion of sequences of tasks that build procedural fluency from conceptual understanding in chapter 4.

Deliberate Reflection Like 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 we stated in chapter 1, cultivating a habit of systematic and deliberate reflection may hold the key to improving one’s teaching as well as 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 his colleagues (2007) suggest a framework for analyzing teaching that supports the reflection process: • Specify the learning goal(s) for the instructional episode (What are students supposed to learn?)

• 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 was intended, identifying how teaching might have supported or inhibited student learning, and then deciding how the teaching could be improved. In ATL 10.1, we ask you to consider how a teacher, Mr. Torchon, might use this framework to reflect on a lesson. In the lesson, Mr. Torchon engages his algebra 1 students in solving the Calling Plans task (Achieve, Inc. 2002).

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The Calling Plan Task 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 they charge you 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?

Analyzing Teaching and Learning 10.1 more Reflecting on and Improving Teaching Watch the Calling Plans video: • Using the analyzing teaching framework proposed by Hiebert and colleagues (2007), determine what Mr. Torchon might have concluded through his analysis of the Calling Plans video clip. • Identify the effective teaching practices Mr. Torchon used during the portion of the lesson portrayed in the video clip. • Identify the effective teaching practices that Mr. Torchon might use in order to enhance student learning during the portion of the lesson portrayed in the video clip. You can access and download the video and its transcript by visiting NCTM’s More4U website (nctm.org/more4u). The access code can be found on the title page of this book.

Analyzing ATL 10.1: Reflecting on and Improving Teaching In order to deliberately reflect on one’s teaching, it is necessary to 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 produced by the teacher or students, 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 provide evidence of what occurred during instruction and aid in reflecting on the lesson. For our purposes, we will assume that Mr. Torchon had three artifacts on which to draw in his reflection: a lesson plan prepared prior to the lesson, an audio recording of the lesson, and the charts created by students during the lesson. Note that the video clip portrays only a brief “snapshot” of Mr. Torchon and his students engaging in the Calling Plans lesson. We are using this snapshot of instruction to model the Pulling It All Together    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.

process of reflecting on a lesson through the lenses of student learning (Hiebert and colleagues’ framework) and the effective teaching practices. Some of the teaching practices noted as possible “enhancements” may occur in other parts of the lesson not featured on the video clip or may occur consistently during other lessons in Mr. Torchon’s classroom.

Specify the learning goals

Using Hiebert and colleagues’ framework, Mr. Torchon might begin his analysis of the lesson by considering his goals for the lesson and what students appeared to learn related to those goals. He wanted students to understand that —    1. the point of intersection is a solution to each equation (plans A and B);   2. a system of equations may be solved with a table or a graph; and

  3. when the graph of one function is below the graph of another function, the function with the lower graph has y-values that are less than the other function. In his lesson plan, Mr. Torchon also identified the following standards as central to this lesson: • Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. • For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Conduct empirical observations of teaching and learning

In the video, we hear two students present the work of their group, and we see on the posters (at 1:53 in the video) that many groups used tables to solve the problem. The poster of the second presenter also included a graph (at 2:12). During the whole-group discussion, students were able to identify the point of intersection as important (line 48 in the transcript), and several students contributed ideas to explain that this point represents where both plans are the same price for the same number of minutes (lines 48, 50, 60, 63, and particularly 64–66). A student was also able to explain how the graph shows when one calling plan is cheaper than the other (lines 67–68), and Mr. Torchon invited another student to use “the same line of reasoning” to show when one plan is more expensive than the other (lines 71–75). Students used tables and graphs to solve the problem (goal 2), identified the importance and meaning of the point of intersection in the context of the problem (goal 1), and explained how the relative position of the lines indicated which calling plan was more or less expensive than the other (goal 3). In doing so, students demonstrated the ability to “solve systems of linear equations . . . approximately (e.g., with graphs)” and to “interpret key features of graphs and tables in terms of the quantities.”

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However, Mr. Torchon might also note that only one group included equations on their poster (presentation 2; lines 32–45; around 2:30 in the video), and these equations were incorrect. Also, while students engaged with representations (graph, tables) and a context that would support their conceptual understating of systems of equations, the use of equations and systems of equations was not present in students’ work or the whole-group discussion featured in the video clip. For example, while students discussed the point of intersection in the context of the problem, in the segment we see in the video, students did not identify the point of intersection as the “solution to each equation” (goal 1). The aspect of the standard regarding solving systems of equations by “focusing on pairs of linear equations in two variables” was not addressed in the part of the lesson we observed.

Construct hypotheses

Students appeared to understand and be able to use tables and graphs well, and their work in small groups indicated that they could correctly determine the price for a given number of minutes for each calling plan. Mr. Torchon was careful to ask purposeful questions in small groups that did not direct students’ thinking or suggest a pathway, such as “How would I find that out?” (line 4) and “How would you show that?” (line 23). However, Mr. Torchon might have asked more advancing questions in this part of the lesson to move students toward creating equations to represent each calling plan. Similarly, during the whole-group discussion in the video clip, his questions and students’ responses remained focused on tables and graphs and did not make connections to equations.

Use analysis to promote improvements

Students’ difficulties in representing the calling plans by using equations might raise two issues for Mr. Torchon to address in future lessons. First, he might consider providing additional experiences for students to represent contextual situations using linear equations. Second, he might consider how to support students in developing equations in small groups so that he could draw on students’ own work during the discussion. Having the calling plans scenarios represented as equations (in addition to tables and graphs) would have enabled Mr. Torchon and his students to make stronger connections to solutions of systems of equations. For example, he might decide to plan questions that would prompt students to make generalizations based on their calculations. He could also consider what he might do in subsequent lessons to build on the conceptual basis provided by the context and representations, so that students would have the opportunity to consider what the point of intersection means for a system of equations and to develop ways to “solve systems of linear equations exactly” as stated in the standard. Mr. Torchon demonstrated several effective teaching practices. He had clear goals for the lesson, and he used many aspects of his goals to focus students’ learning. He selected a task that promoted reasoning and problem solving, and he implemented the task in ways that supported

Pulling It All Together    223 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.

students’ engagement in high-level thinking. From the chart listing “Factors associated with the maintenance or decline of high-level cognitive demands” (fig. 3.4), we see Mr. Torchon “[pressing] for justifications, explanations, meaning through teacher questioning, comments, feedback” (lines 4, 7, 14, 17, 25) and “[drawing] frequent conceptual connections” (lines 19–21, 23, questions about the point of intersection in context and on the graph). Throughout the lesson, the focus remains on meaning and understanding rather than correctness of procedures and answers. Mr. Torchon also asked purposeful questions (lines 46–47, 51–52, 58–59), elicited and used evidence of student thinking (lines 61–62, 71–75), and supported students in using and connecting mathematical representations (lines 55–56, 67–68). The effective teaching practices could be used as a lens to consider ways to enhance students’ learning in the Calling Plans lesson as well. As stated previously, posing purposeful questions specifically intended to support students’ development of equations to represent each calling plan and to discuss and compare the calling plans as systems of equations may serve to move students’ learning toward fulfilling more aspects of the goals of the lesson. Using the goals to frame instructional decisions might have prompted Mr. Torchon to connect the representations of tables, graphs, and context to equations and systems of equations. Finally, this lesson could be the first in a sequence of lessons in which students build procedural fluency based on the conceptual foundation fostered by connections between representations and purposeful questions. Reflecting on a lesson should inform the planning of the next lesson, which in turn informs the enactment of the lesson and provides new teaching practices and evidence of students’ learning on which to reflect. This process is represented by the teaching cycle in figure 10.2.

Teaching

Planning

Reflecting Fig. 10.2. The teaching cycle

224   Taking Action Grades 9–12 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.

As with planning, we note the importance of reflecting on sequences of lessons, as well as individual lessons, to assess the teaching and learning that occur 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, 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 In chapters 2 through 9, we considered the specific ways in which the effective mathematics teaching practices can individually support equitable learning opportunities for students. Throughout these chapters, we discussed the ways in which the practices can support the development of students’ identities as mathematical knowers and doers, how the practices can promote students who may have been historically marginalized, and the role of the teaching practices in creating publicly accessible thinking and reasoning spaces for students. When the practices are used together and in combination, powerful opportunities are created over time to support meaningful mathematical learning for all students, regardless of their history and prior experiences. Just as it is important to carefully analyze students’ mathematical learning and development over the course of a year, it is equally important to document and reflect on students’ identity development through the use of the effective mathematics teaching practices. For example, as we engage students with tasks that vary across the cognitive demand categories discussed in chapter 3, it is important to note the ways in which they might engage with different types of tasks and how that engagement might shift over time. It is critical 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 over time. A teacher might note, for example, that a student has strong mathematical reasoning skills when he or she works independently but is hesitant to share that thinking in small- or whole-group discussions. Providing that student with feedback that reinforces the validity of his or her thinking, pressing the student to engage with other students, and creating opportunities for him or her 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 the performances they display in mathematics class and, as such, should be evaluated periodically and analyzed for evidence of change. Using the effective mathematics teaching practices consistently and in combination can provide meaningful opportunities for such an analysis. Pulling It All Together    225 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 addition to classroom-based equity concerns, there are also systemic equity considerations that are important to challenge. High school mathematics coursework is often characterized by practices such as tracking and ability grouping that 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, as classes that aggregate previously unsuccessful students almost always cover less mathematical content in more time and tend to use the same didactic teaching and procedurally focused tasks that may have inhibited students’ success and contributed to their disengagement in mathematics in the first place. As such, it is critical for teachers to consider how they can use goals, tasks, discourse, and questioning to design instruction that has the potential to reach all students by providing students access to the task and support as they struggle productively. Teachers must become advocates for the dismantling of tracking systems that separate students and their opportunities to learn, and they must press their schools and districts for meaningful just-in-time supports that will help all students succeed the first time in mathematics, rather than placing some students into lower tracks or forcing them to repeat coursework. Too frequently, historically marginalized students are relegated to classes where they learn and practice procedures and never engage in the types of reasoning and problemsolving tasks that we have described. 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. As you engage in the cycle of planning, teaching, and reflecting, it is critical to consider how your instruction supported 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, you need to reflect on the factors that may be impacting students’ lack of success and take corrective action. Unless all students are experiencing success in your classroom, there is more work to be done.

226   Taking Action Grades 9–12 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.

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 another colleague. If you have worked through the book with colleagues, consider how to continue to support each other to plan, teach, and reflect on your teaching in ways that highlight the effective teaching practices and support students’ learning. You might analyze additional “cases” (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 practices are apparent in the lesson, tasks, student work, or teaching and what impact they appear to have on teaching and learning. Another opportunity for continued growth might include a book club where 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 Professional Learning Toolkit [http://www.nctm.org/PtAToolkit/]). You might also consider co-planning lessons with colleagues using the eight effective teaching practices as a framework. Invite your math coach (if you have one), department chair, administrator, or other instructional leader to participate. As suggested earlier 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 and develop procedural fluency from conceptual understanding. When you have an opportunity to teach the co-planned lessons (or any lessons featuring tasks that promote reasoning and problem solving), observe each other with particular attention to what practices are being used in the lesson and how the practices did or did not support students’ learning. Several tools for analyzing tasks, task implementation, questions, and whole-group discussion identified throughout this book could serve as a basis for collecting data that inform instructional improvements around the effective teaching practices. Rather than attending to all eight practices at once, you and 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 and why it would be beneficial to do so (e.g., the positive impact on students’ learning and engagement).

Pulling It All Together    227 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.

Final Thoughts As we said earlier 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. Simply being introduced to the teaching practices does not mean that a teacher will immediately adopt them and use them in ways that reflect ambitious mathematics instruction. In fact, these practices often bring to the surface important unproductive beliefs for teachers: “Do I truly believe that students can develop conceptual understanding before being introduced 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?” “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 is a gradual process 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.

228   Taking Action Grades 9–12 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

Proof Task Lesson Plan Learning Goals (Residue) What understandings will students take away from this lesson? This lesson primarily targets standard for mathematical practice 3—construct viable arguments and critique the reasoning of others. In particular, it focuses on what it takes for an argument to be classified as a proof. Through this lesson, students will understand the following: • For an argument to be a proof, it must show that the conjecture or claim is true for all cases. • The statements and definitions that are used in the argument must be ones that are true and accepted by the community because they have been previously justified. • Proofs can take different forms and utilize different representations (pictures, words, symbols), but all of the different representations can be connected. • Examples can help you see patterns, but they alone do not constitute proof.

Evidence What will students say, do, produce, etc., that will provide evidence of their understandings? Students should ultimately be able to distinguish between arguments that are and are not proofs. They will be asked after each student presentation to determine whether the presented argument is a proof and to explain why or why not. As an exit ticket at the end of class, students will be asked to explain, in their own words, how they would describe the characteristics of a proof to a student who was not in class during the discussion. I will expect them to say things like “you can’t just use examples,” “you can use words, pictures, and algebra,” “you have to show that it works for all cases.” I will review the exit slips and pick one that is underdeveloped or incorrect and use that to launch the following lesson.

In addition, students will see structure in the expression n2 1 n, use the structure to write the expression in an equivalent form n(n 1 1), interpret the expression as the product of two consecutive numbers or of an odd and even number, and understand how the expression can be physically modeled.

    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? Prove that for every integer n (. . ., –3, –2, –1, 0, 1, 2, 3, 4 . . .), the expression n2 + n will always be even. Task adapted from Anne K. Morris, “Mathematical Reasoning: Adults’ Ability to Make the Inductive-Deductive Distinction,” Cognition and Instruction 20, no. 1 (2002): 79–118.

Instructional Support— Tools, Resources, Materials What tools or resources will be made available to give students entry to, and help them reason through, the activity? Grid paper, colored pencils, square tiles (These resources along with other basic tools—rulers, scissors, calculators, protractors—will be placed in boxes on tables where groups of 4 can easily access them.) Print copies of the task and exit slip

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?

Definitions of integer, odd and even numbers. • Integer: the whole numbers {0, 1, 2, 3, 4,. . .} and their opposites • Even number: an integer that can be divided by two without a remainder. Often represented as 2m (where m is an integer). • Odd number: an integer that when divided by 2 yields a remainder of 1. Often represented as 2m 1 1 (where m is an integer).

Ask students, “What does it mean to prove something?” Elicit the idea that proof requires evidence that something is always true. If students relate proof to what is needed to convict a defendant in a court of law, make the argument that in the court system you must prove “beyond a reasonable doubt.” In math it is different— there is no room for doubt! Tell students that they are going to prove that n2 1 n will always be even. Ask students what it means to be even or odd. Give students a minute to figure out what happens when you add or multiply even and odd numbers together. Ask students to share out, make a list on chart paper. Odd 1 Odd 5 Even; Even 1 Even 5 Even; Odd 1 Even 5 Odd Odd 3 Odd 5 Odd; Even 3 Even 5 Even; Odd 3 Even 5 Odd

230   Taking Action Grades 9–12 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 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 him or her toward the goals of the lesson. Be sure to consider questions that you will ask students who can’t get started as well as students who finish quickly. [This task can also be solved using mathematical induction. However, since this lesson is an introductory proof lesson, students will not have the experiences on which to draw in order to use this method.] GENERAL QUESTIONS FOR ALL RESPONSES • When I first visit students as they are working on the task, begin my interactions with them by saying, “Tell me what you have figured out so far.” Once I have a clear sense of what they did, I can ask assessing questions that probe their work more directly and specifically. • The bold questions are intended to advance students’ understanding. When I ask these questions, I will walk away and let students collaborate on answering them. 1. Consecutive Integers

• How did you get n(n 1 1)?

Recognizes that n2 1 n can also be written as n(n 1 1) and that n and n 1 1 are consecutive numbers. Builds an argument that one of the numbers must be even, and the other one must be odd, and that the product of an odd and even number is even. Since the product is even, it is divisible by 2 since all even numbers are divisible by 2. So n2 1 n is always even.

• How do you know that one of your numbers must be odd and one must be even? • Can you be sure that there isn’t a number for n that would make the claim false? • So, is this a proof? Can you make a list of things your argument does that would allow us to count it as a proof?

Appendix A: Lesson Plan    231 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.

2. Logical Argument: Two cases Looks at two cases—when n is odd and when n is even. Argues that if n is an odd counting number, then n2 will be odd, because odd 3 odd 5 odd. An odd plus an odd is even, so since n2 and n are odd, n2 1 n is even. If n is an even counting number, then n2 will be even, because even 3 even 5 even. An even plus an even number is even, so since n2 and n are even, n2 1 n is even.

• Why did you start by looking at two different cases? Is that necessary? • You seem to be convinced that you have proven the claim. What makes you so sure that your argument is a proof? • Can you make a list of things your argument does that would allow us to count it as a proof?

Since all counting numbers are either even or odd, I have taken care of all numbers. Therefore, I’ve proved that for every counting number n, the expression n2 1 n is always even. 3. Algebraic Argument: Two cases A number is even if it can be written as 2m, where m is any integer. That is, a number is even if it is a multiple of 2. A number is odd if it can be written as 2m 1 1, where m is any integer. That is, a number is odd if it is one more than a multiple of 2. Looks at two cases: when n is odd and when n is even. Argues that if n is even, then you get (2m)2 1 2m. Since both are even, then the sum is also even. 4m2 1 2m 5 2(2m2 1 m). Then, the sum is even because it is a multiple of 2. If n is odd, then you get (2m 1 1)2 1 2m 1 1. This gives you 4m2 1 4m 1 1 1 2m 1 1. When you combine like terms you get 4m2 1 6m 1 2; then you can factor to get 2(2m2 1 3m 1 1). This has to be even because 2 times any number is even, because it is a multiple of 2.

• Why did you start by looking at two different cases? Is that necessary? • Why did you specify that 2m was even and that 2m 1 1 was odd? • How do you know that 2(2m2 1 m) and 2(2m2 1 3m 1 1) are even numbers? An advanced response would look like this: If 2m2 1 m 5 k (and k is an integer by closure of addition and multiplication of integers), then 4m2 1 2m 5 2k, so the sum is even. • When you say “It always works” what do you mean? • So, is this a proof? Can you make a list of things your argument does that would allow us to count it as a proof?

So it always works.

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4. Visual Argument Starts with an odd number and represents n2 1 n as a picture. For example, here n 5 3.

• How does drawing the vertical or horizontal line show dividing by two? • What do you mean you can “pair up” the columns or the rows? Why does this matter? • How does the “pairing up” relate to drawing the horizontal or vertical line?

Says, “Now add the n to the square which makes it an n 3 (n 1 1) rectangle—in this case, a 3 3 4 rectangle. You can divide the new rectangle into two equal pieces, which means that it is divisible by 2.”

• You have shown that this works when n = 3 and n = 4, but why are you so sure that this will work for every number? Can you add to your argument so that we will be convinced that it will always work? Does this argument work for all integers?

Now try it for an even number. Here n 5 4.

Now add the n to the square, which makes it an n 3 (n 1 1) rectangle. In this case a 4 3 5 rectangle. You can divide the new rectangle into two equal pieces, which means that it is divisible by 2.

So when you add n 3 1 to the n 3 n rectangle and get a rectangle that is n 3 (n 1 1), you will always be able to divide it in half.

Appendix A: Lesson Plan    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.

4. Visual Argument (continued) That means when n is odd, your n 3 (n 1 1) rectangle will have an even number of columns, which means you can always divide it in half. You can always pair up the columns. When n is even, your n 3 (n 1 1) rectangle will have an even number of rows, which means that you can always divide it in half. You can always pair up the rows. So it does not matter the counting number that you start with, when you square it and add it to itself, it will always result in an even number. 5. Empirical Examples Student randomly selects several different numbers. Some have large magnitude, and some are closer to zero. Some are even, and some are odd. Some are prime, and some are composite. Student then argues that since a variety of different numbers were tried and worked in every case, this will work for all counting numbers. Therefore, n2 1 n will always be even.

6. Can’t get started

• You have tried lots of different numbers. How do you know that there isn’t a number you haven’t tried yet that will not work? • Is there something about how odd and even numbers work when squared and added that would help convince me? Would a picture help explain what is happening? What other way can you convince me?

• What is the problem asking you to do? • What does it mean to be even? • Can you try to put some numbers in to see if it seems to work? Then think about what you know about even and odd numbers that might make it work or not work.

234   Taking Action Grades 9–12 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.

Sharing and Discussing the Task Selecting and Sequencing

Connecting Responses

Which solutions do you want to have shared during the lesson?

What specific questions will you ask so that students—

In what order? Why?

• make sense of the mathematical ideas you want them to learn; and • make connections among the different strategies/solutions that are presented. For each presentation, the author is asked to explain what he or she did. Clarifying questions are asked by the teacher and students as needed; the questions below are asked of the entire class.

FIRST Logical Argument: 2 Cases (solution 2) Start with this one because it is basic and uses what students know about the product and sum of even and odd numbers. SECOND Algebraic Argument (solution 3) Go to this one next because it connects to the logical argument. By juxtaposing the two arguments, students will be able to see how they are similar—the first argument says in words what the second is showing numerically. THIRD Visual Argument (solution 4) This argument provides a ways to “show” visually what happens when n2 and n are added together. It connects with the first argument by providing a visual of the work and connects to the algebraic argument because factoring the 2 is the algebraic method of creating two columns or two rows.

• Is this a proof? Why or why not? • What about it makes you think it is a proof?

• How is the algebraic argument similar to or different from the logical argument? • Is it a proof? What about it makes you think it is a proof?

• Is this argument similar to either of the other arguments? In what ways? • Is this a proof? Why or why not? (If the students have not clearly generalized from their example so that it is clear that this works for all cases, it is not a proof. If this is the case, press the class to consider what could be added to the argument to make it a proof.)

Appendix A: Lesson Plan    235 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.

FOURTH Empirical Examples (solution 5)

• Is this a proof? Why or why not?

(It would be best to have a group present this if they were able to move on to a more complete argument. Otherwise, present this as something that you saw in another class. HAVE ONE READY TO PRESENT IN CASE I NEED IT.)

• Why did the visual argument count as a proof when they used examples, but examples don’t count as a proof in this situation?

This solution will help put to the test whether students are starting to see what is required for proof. FIFTH Consecutive Integers (solution 1) This will be last because it is less likely that students will do this, and by the time we get here, they will have a good idea of what constitutes a proof. Also, it connects well to the visual since n(n 1 1) are the dimensions of the rectangle.

• How is this the same or different from the other arguments that have been made? • How does it relate to the visual argument? • Does it count as a proof? Why or why not? • Is it a proof? What about makes it a proof?

Based on Margaret S. Smith, Victoria Bill, and Elizabeth Hughes, “Thinking Through a Lesson Protocol: A Key for Successfully Implementing High-Level Tasks,” Mathematics Teaching in the Middle School 14, no. 3 (2008): 132–38.

236   Taking Action Grades 9–12 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 (Residue)

Evidence

What understandings will students take away from this lesson?

What will students say, do, produce, etc., that will provide evidence of their understandings?

Task

Instructional Support— Tools, Resources, Materials

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

What tools or resources will be made available to give students entry to, and help them reason through, the activity?

    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.

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?

Anticipated Solutions

Instructional Support—Teacher

What are the various ways that students might complete the activity?

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?

Be sure to include incorrect, correct, and incomplete solutions.

These questions should assess what a student currently knows and advance him or her toward the goals of the lesson. Be sure to consider questions that you will ask students who can’t get started as well as students who finish quickly.

Use the monitoring tool to provide the details related to Anticipated Solutions and Instructional Support. Sharing and Discussing the Task Selecting and Sequencing

Connecting Responses

Which solutions do you want to have shared during the lesson?

What specific questions will you ask so that students—

In what order? Why?

• make sense of the mathematical ideas you want them to learn; and • make connections among the different strategies/solutions that are presented.

238   Taking Action Grades 9–12 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.

Monitoring Tool (purple: complete prior to lesson; gray: complete during the lesson) Anticipated Solutions

Instructional Support Assessing Questions

Who/What

Order

Advancing Questions

Unanticipated Solutions

Based on Margaret S. Smith, Victoria Bill, and Elizabeth Hughes, “Thinking Through a Lesson Protocol: A Key for Successfully Implementing High-Level Tasks,” Mathematics Teaching in the Middle School 14, no. 3 (2008): 132–38.

Appendix B: Lesson Planning Template    239 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.

References Achieve, Inc. Foundations for Success: Mathematics Expectations for the Middle Grades. Washington, D.C.: Achieve, Inc. (2002). Aguirre, Julia, Karen Mayfield-Ingram, and Danny Bernard Martin. The Impact of Identity in K–8 Mathematics. Reston, Va.: National Council of Teachers of Mathematics, 2013.

Aguirre, Julia M., Erin E. Turner, Tonya G. Bartell, Crystal Kalinec-Craig, Mary Q. Foote, Amy Roth McDuffie, and Corey Drake. “Making Connections in Practice: Developing Prospective Teachers’ Capacities to Connect Children’s Mathematical Thinking and Community Funds of Knowledge in Mathematics Instruction.” Journal of Teacher Education 64, no. 2 (2013): 178–92. Albrecht, Masha R., Maurice J. Burke, Wade Ellis Jr., Dan Kennedy, and Evan M. Maletsky. Navigating through Measurement in Grades 9–12. Reston, Va.: National Council of Teachers of Mathematics, 2005.

Ames, Carole, and Jennifer Archer. “Achievement Goals in the Classroom: Students’ Learning Strategies and Motivation Processes.” Journal of Education Psychology 80, no. 3 (1988): 260–67.

Anderson, Kate T. “Applying Positioning Theory to the Analysis of Classroom Interactions: Mediating Micro-Identities, Macro-Kinds, and Ideologies of Knowing.” Linguistics and Education 20, no. 4 (2009): 291–310. Anthony, Glenda, and Margaret Walshaw. Effective Pedagogy in Mathematics. Educational Practices Series–19. Brussels, Belgium: Internationa1 Academy of Education, 2009.

Artzt, Alice F., and Eleanor Armour-Thomas. Becoming a Reflective Mathematics Teacher. Mahwah, N.J.: Lawrence Erlbaum Associates, 2002. Ashcraft, Mark H. “Math Anxiety: Personal, Educational, and Cognitive Consequences.” Current Directions in Psychological Science 11, no. 5 (2002): 181–85.

Ball, Deborah Loewenberg. “With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics.” Elementary School Journal 93, no. 4 (1993): 373–97.

Berry, Robert Q., and Mark W. Ellis. “Multidimensional Teaching.” Mathematics Teaching in the Middle School 19, no. 3 (2013): 172–78.

    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.

Boaler, Jo. Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching. San Francisco, Calif.: John Wiley & Sons, 2015. Boaler, Jo, and Karin Brodie. “The Importance, Nature, and Impact of Teacher Questions.” In Proceedings of the Twenty-Sixth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, vol. 2, edited by Douglas E. McDougall and John A. Ross, pp. 774–90. Toronto, Ontario: Ontario Institute for Studies in Education at the University of Toronto, 2004. Boaler, Jo, and James G. Greeno. “Identity, Agency, and Knowing in Mathematics Worlds.” In Multiple Perspectives on Mathematics Teaching and Learning, edited by Jo Boaler, pp. 171–200. Westport, Conn.: Ablex Publishing, 2000. Boaler, Jo, and Megan Staples. “Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School.” Teachers College Record 110, no. 3 (2008): 608–45.

Boerst, Timothy A., Laurie Sleep, Deborah Loewenberg Ball, and Hyman Bass. “Preparing Teachers to Lead Mathematics Discussions.” Teachers College Record 113, no. 12 (2011): 2844–877. Bondy, Elizabeth, and Dorene D. Ross. “The Teacher as Warm Demander.” Educational Leadership 66, no. 1 (2008): 54–58. Boston, Melissa. “Assessing Instructional Quality in Mathematics.” The Elementary School Journal 113, no. 1 (2012): 76–104. Boston, Melissa, and Anne Garrison Wilhelm. “Middle School Mathematics Instruction in Instructionally Focused Urban Districts.” Urban Education (March 2015): 1–33. doi:10.1177/0042085915574528.

Bray, Wendy S. “How to Leverage the Potential of Mathematical Errors.” Teaching Children Mathematics 19, no. 7 (2013): 424–31. Briars, Diane. “Supporting Teachers in Building Procedural Fluency from Conceptual Understanding.” Presentation at the annual meeting of the Pennsylvania Council of Teachers of Mathematics, Seven Springs, Pennsylvania, August 2016.

Cai, Jinfa, Ning Wang, John C. Moyer, Chuang Wang, and Bikai Nie. “Longitudinal Investigation of the Curricular Effect: An Analysis of Student Learning Outcomes from the LieCal Project in the United States.” International Journal of Educational Research 50, no. 2 (2011): 117–36. Cartier, Jennifer L., Margaret S. Smith, Mary Kay Stein, and Danielle K. Ross. 5 Practices for Orchestrating Productive Task-Based Discussions in Science. Reston, Va.: National Council of Teachers of Mathematics, 2013.

Chance, Beth, Dani Ben-Zvi, Joan Garfield, and Elsa Medina. “The Role of Technology in Improving Student Learning of Statistics.” Technology Innovations in Statistics Education 1, no. 1 (2007). https://escholarship.org/uc/item/8sd2t4rr#page-1.

Chapin, Suzanne H., and Catherine O’Connor. “Academically Productive Talk: Supporting Students’ Learning in Mathematics.” In The Learning of Mathematics, 2007 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by W. Gary Martin, pp. 113–28. Reston, Va.: NCTM, 2007.

242   Taking Action Grades 9–12 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.

Chapin, Suzanne, Catherine O’Connor, and Nancy Anderson. Classroom Discussions: Using Math Talk to Help Students Learn, Grades 1–6. Sausalito, Calif.: Math Solutions, 2003. Charles, Randall I. “Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics.” Journal of Mathematics Education 7, no. 3 (2005): 9–24.

Chazan, Daniel, and Deborah Ball. “Beyond Being Told Not to Tell.” For the Learning of Mathematics 19, no. 2 (1999): 2–10. Clarke, Shirley, Helen Timperley, and John Hattie. Unlocking Formative Assessment: Practical Strategies for Enhancing Students’ Learning in the Primary and Intermediate Classroom. Auckland, New Zealand: Hodder Moa Beckett, 2004. Clements, Douglas H., and Julie Sarama. “Learning Trajectories in Mathematics Education.” Mathematical Thinking and Learning 6, no. 2 (2004): 81–89.

Cobb, P., and T. Smith. “The Challenge of Scale: Designing Schools and Districts as Learning Organizations for Instructional Improvement in Mathematics.” In International Handbook of Mathematics Teacher Education 3, edited by Dina Trosh and Terry Wood, pp. 231–254. Rotterdam, The Netherlands: Sense Publishers, 2008. Coe, Robert, and Kenneth Ruthven. “Proof Practices and Constructs of Advanced Mathematics Students.” British Educational Research Journal 20, no. 1 (1994): 41–53.

Cohen, Elizabeth G., Rachel A. Lotan, Beth A. Scarloss, and Adele R. Arellano. “Complex Instruction: Equity in Cooperative Learning Classrooms.” Theory into Practice 38, no. 2 (1999): 80–86. Common Core State Standards Writing Team. Progressions for the Common Core State Standards in Mathematics (Draft). Grades 6–8, Statistics and Probability. Tucson, Ariz.: Institute for Mathematics and Education, University of Arizona, 2011.

Crespo, Sandra. “Seeing More Than Right and Wrong Answers: Prospective Teachers’ Interpretations of Students’ Mathematical Work.” Journal of Mathematics Teacher Education 3, no. 2 (2000): 155–81.

Daro, Phil, Frederic A. Mosher, and Tom Corcoran. Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. CPRE Research Report# RR-68. Madison, Wis.: Consortium for Policy Research in Education, 2011.

Davis, Brent. “Listening for Differences: An Evolving Conception of Mathematics Teaching.” Journal for Research in Mathematics Education 28, no. 3 (1997): 355–76. Doyle, Walter. “Work in Mathematics Classes: The Context of Students’ Thinking during Instruction.” Educational Psychologist 23, no. 2 (1988): 167–80.

Driscoll, Mark, Johannah Nikula, and Jill Neumayer DePiper. Mathematical Thinking and Communication: Access for English Learners. Portsmouth, N.H.: Heinemann, 2016. Duckworth, Eleanor. The Having of Wonderful Ideas: and Other Essays on Teaching and Learning. New York: Teachers College Press, 1987. Dweck, Carol. Mindset: The New Psychology of Success. New York: Random House, 2006.

Ellis, Mark W. “Leaving No Child Behind Yet Allowing None Too Far Ahead: Ensuring (In) Equity in Mathematics Education through the Science of Measurement and Instruction.” Teachers College Record 110, no. 6 (2008): 1330–356.

References  243 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.

Ellis, Mark W., and Robert Q. Berry III. “The Paradigm Shift in Mathematics Education: Explanations and Implications of Reforming Conceptions of Teaching and Learning.” Mathematics Educator 15, no. 1 (2005): 7–17. Engle, Randi A., and Faith C. Conant. “Guiding Principles for Fostering Productive Disciplinary Engagement: Explaining an Emergent Argument in a Community of Learners Classroom.” Cognition and Instruction 20, no. 4 (2002): 399–483. Fendel, Dan, Diane Resek, Lynne Alper, and Sherry Fraser. Interactive Mathematics Program Year 3. 2nd ed. Mount Kisco, N.Y.: It’s About Time, 2012. Fennema, Elizabeth, and Meghan Franke. “Teachers’ Knowledge and Its Impact.” In Handbook of Research on Mathematics Teaching and Learning, edited by Douglas Grouws, pp. 147–64. Indianapolis, Ind.: Macmillan, 1992.

Flores, Alfinio. “Examining Disparities in Mathematics Education: Achievement Gap or Opportunity Gap?” High School Journal 91, no. 1 (2007): 29–42. Foreman, Linda C., and Albert B. Bennett. Visual Mathematics Course 2. Salem, Ore.: Math Learning Center, 1995.

Franke, Megan L., Elham Kazemi, and Daniel Battey. “Mathematics teaching and Classroom Practice.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester Jr., pp. 225–56. Charlotte, N.C.: Information Age; Reston, Va.: National Council of Teachers of Mathematics, 2007. Fuchs, Lynn S., Douglas Fuchs, Karin Prentice, Mindy Burch, Carole L. Hamlett, Rhoda Owen, and Katie Schroeter. “Enhancing Third-Grade Students’ Mathematical Problem Solving with SelfRegulated Learning Strategies.” Journal of Educational Psychology 95, no. 2 (2003): 306. Fuson, Karen C., Mindy Kalchman, and John D. Bransford. “Mathematical Understanding: An Introduction.” In How Students Learn: History, Mathematics, and Science in the Classroom, edited by M. Suzanne Donovan and John D. Bransford, Committee on How People Learn: A Targeted Report for Teachers, National Research Council, pp. 217–56. Washington, D.C.: National Academies Press, 2005.

Goldin, Gerald, and Nina Shteingold. “Systems of representation and the development of mathematical concepts.” In The Roles of Representations in School Mathematics, 2001 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Albert A. Cuoco, pp. 1–23. Reston, Va.: NCTM, 2001. Goldsmith, Lynn T., and Deborah Schifter. “Understanding Teachers in Transition: Characteristics of a Model for the Development of Mathematics Teaching.” In Mathematics Teaching in Transition, edited by Elizabeth Fennema and Barbara Scott Nelson, pp. 19–54. Mahwah, N.J.: Lawrence Erlbaum Associates, 1997. Gresalfi, Melissa S. “Taking Up Opportunities to Learn: Construction Dispositions in Mathematics Classrooms.” Journal of the Learning Sciences 18, no. 3 (2009): 327–69. Grouws, Douglas A., James E. Tarr, Óscar Chávez, Ruthmae Sears, Victor M. Soria, and Rukiye D. Taylan. “Curriculum and Implementation Effects on High School Students’ Mathematics Learning from Curricula Representing Subject-Specific and Integrated Content Organizations.” Journal for Research in Mathematics Education 44, no. 2 (2013): 416–63.

244   Taking Action Grades 9–12 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.

Gutierrez, Rochelle. “Why (Urban) Mathematics Teachers Need Political Knowledge.” Journal of Urban Mathematics Education 6, no. 2 (2013): 7–19. Gutstein, Eric. Reading and Writing the World with Mathematics: Toward a Pedagogy for Social Justice. New York: Taylor & Francis, 2006.

Harkness, Shelly S. “Social Constructivism and the Believing Game: A Mathematics Teacher’s Practice and Its Implications.” Educational Studies in Mathematics 70, no. 3 (2009): 243–58. Hatano, Giyoo, and Kayoko Inagaki. “Sharing Cognition through Collaborative Comprehension Activity.” In Perspectives on Socially Shared Cognition, edited by Lauren B. Resnick, John M. Levine, and Stephanie D. Teasley, pp. 331–480. Washington, D.C.: American Psychological Association, 1991. Healy, Lulu, and Celia Hoyles. “A Study of Proof Conceptions in Algebra.” Journal for Research in Mathematics Education 31, no. 4 (2000): 396–428.

Henningsen, Marjorie A., and Mary Kay Stein. “Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning.” Journal for Research in Mathematics Education 28, no. 5 (1997): 524–49. Herbel-Eisenmann, Beth A., and M. Lynn Breyfogle. “Questioning Our Patterns of Questioning.” Mathematics Teaching in the Middle School 10, no. 9 (2005): 484–89.

Herbel-Eisenmann, Beth A., Michael D. Steele, and Michelle Cirillo. “(Developing) Teacher Discourse Moves: A Framework for Professional Development.” Mathematics Teacher Educator 1, no. 2 (2013): 181–96. Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, N.H.: Heinemann, 1997.

Hiebert, James, Ronald Gallimore, Helen Garnier, Karen Bogard Givvin, Hilary Hollingsworth, Jennifer Jacobs, Angel Miu-Ying Chui, et al. Highlights from the TIMSS 1999 Video Study of EighthGrade Mathematics Teaching. Washington, D.C.: U.S. Department of Education, National Center for Education Statistics, 2003. Hiebert, James, and Douglas A. Grouws. “The Effects of Classroom Mathematics Teaching on Students’ Learning.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester Jr., pp. 371–404. Charlotte, N.C.: Information Age; Reston, Va.: National Council of Teachers of Mathematics, 2007. Hiebert, James, Anne K. Morris, Dawn Berk, and Amanda Jansen. “Preparing Teachers to Learn from Teaching.” Journal of Teacher Education 58, no. 1 (2007): 47–61.

Hiebert, James, and Diana Wearne. “Developing Understanding through Problem Solving.” In Teaching Mathematics through Problem Solving: Grades 6–12, edited by Harold L. Schoen and Randall I. Charles, pp. 3–13. Reston, Va.: National Council of Teachers of Mathematics, 2003. Hollebrands, Karen. “High School Students’ Understandings of Geometric Transformations in the Context of a Technological Environment.” Journal of Mathematical Behavior 22, no. 1 (2003): 55–72. http://www.sciencedirect.com/science/article/pii/S073231230300004X.

References  245 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.

Institute for Learning, University of Pittsburgh. Algebra 2 Set of Related Lessons: Building Polynomial Functions. Pittsburgh, Pa.: Institute for Learning, 2013a.

———. Geometry Set of Related Lessons: Investigating Coordinate Geometry and Its Use to Solve Problems. Pittsburgh, Pa.: Institute for Learning, 2013b. ———. Algebra 1 Creating and Interpreting Functions: A Set of Related Lessons. Pittsburgh, Pa.: Institute for Learning, 2015a. ———. Geometry Set of Related Lessons: Investigating Congruence in Terms of Rigid Motion. Pittsburgh, Pa.: Institute for Learning, 2015b. Jackson, Kara, and Paul Cobb. “Refining a Vision of Ambitious Mathematics Instruction to Address Issues of Equity.” Presentation at the annual meeting of the American Educational Research Association, Denver, Colo., April, 30–May 4, 2010.

Jackson, Kara, Anne Garrison, Jonee Wilson, Lynsey Gibbons, and Emily Shahan. “Exploring Relationships between Setting Up Complex Tasks and Opportunities to Learn in Concluding Whole-Class Discussions in Middle-Grades Mathematics Instruction.” Journal for Research in Mathematics Education 44, no. 4 (2013): 646–82.

Kane, Thomas J., and Douglas O. Staiger. Gathering Feedback for Teaching: Combining High-Quality Observations with Student Surveys and Achievement Gains. Research Paper, MET Project. Seattle, Wash.: Bill & Melinda Gates Foundation, 2012. http://files.eric.ed.gov/fulltext/ED540960.pdf. Kapur, Manu. “Productive Failure in Mathematical Problem Solving.” Instructional Science 38, no. 6 (2010): 523–50.

Kleinfeld, Judith. “Effective Teachers of Eskimo and Indian Students.” The School Review 83, no.2 (1975): 301–44. Knuth, Eric. “Understanding Connections between Equations and Graphs.” Mathematics Teacher 93, no. 1 (2000): 48–53.

Knuth, Eric J., Jeffrey Choppin, Marian Slaughter, and Jamie Sutherland. “Mapping the Conceptual Terrain of Middle School Students’ Competencies in Justifying and Proving.” In Proceedings of the 24th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, edited by Denise S. Mewborn, Paola Sztajn, Dorothy Y. White, H. G. Weigel, R. L. Bryant, and K. Nooney, vol. 4, pp. 1693–1670. Athens, Ga.: Clearinghouse for Science, Mathematics, and Environmental Education, 2002.

Ladson-Billings, Gloria. “It Doesn’t Add Up: African American Students’ Mathematics Achievement.” Journal for Research in Mathematics Education 28, no. 6 (1997): 697–708. Lamon, Susan J. “Rational Numbers and Proportional Reasoning: Toward a Theoretical Framework for Research.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester Jr., pp. 629–667. Charlotte, N.C.: Information Age; Reston, Va.: National Council of Teachers of Mathematics, 2007.

Lampert, Magdalene. Teaching Problems and the Problems of Teaching. New Haven, Conn.: Yale University Press, 2001. Lampert, Magdalene, Heather Beasley, Hala Ghousseini, Elham Kazemi, and Megan Loef Franke. “Using Designed Instructional Activities to Enable Novices to Manage Ambitious Mathematics Teaching.” In Instructional Explanations in the Disciplines, edited by Mary Kay Stein and Linda Kucan, pp.129–41. New York: Springer, 2010. 246   Taking Action Grades 9–12 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.

Leahy, Siobhan, Christine Lyon, Marnie Thompson, and Dylan Wiliam. “Classroom Assessment: Minute by Minute, Day by Day.” Educational Leadership 63, no. 3 (2005): 18–24.

Leinhardt, Gaea, and Michael D. Steele. “Seeing the Complexity of Standing to the Side: Instructional Dialogues.” Cognition and Instruction 23, no. 1 (2005): 87–163. Leinhardt, Gaea, Orit Zaslavsky, and Mary Kay Stein. “Functions, Graphs, and Graphing: Tasks, Learning, and Teaching.” Review of Educational Research 60, no.1 (1989): 1–64.

Lesh, Richard, Tom Post, and Merlyn Behr. “Representations and Translations among Representations in Mathematics Learning and Problem Solving.” In Problems of Representation in the Teaching and Learning of Mathematics, edited by Claude Janvier, pp. 33–40. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1987. Lotan, Rachel A. “Group-Worthy Tasks.” Educational Leadership 60, no. 6 (2003): 72–75.

Marshall, Anne Marie, Alison Castro Superfine, and Reality S. Canty. “Star Students Make Connections.” Teaching Children Mathematics 17, no. 1 (2010): 39–47.

Martin, Danny Bernard. “Learning Mathematics While Black.” The Journal of Educational Foundations 26, nos. 1/2 (2012): 47–66.

Martin, W. Gary. “The NCTM High School Curriculum Project: Why It Matters to You.” Mathematics Teacher 103, no. 3 (2009): 164–66.

Marzano, Robert J. What Works in Schools: Translating Research into Action. Alexandria, Va.: Association of Supervision and Curriculum Development, 2003. Mathematics Assessment Resource Service. “Four Situations Task.” University of Nottingham and UC Berkeley, 2015. http://map.mathshell.org/lessons.php?unit=9260&collection=8. McClure, Lynne. “Using Low Threshold High Ceiling Tasks in Ordinary Classrooms.” Cambridge, England: Nrich Maths Project, University of Cambridge, 2011. http://nrich.maths.org/7701. McTighe, Jay, and Grant P. Wiggins. Essential Questions: Opening Doors to Student Understanding. Alexandria, Va.: Association for Supervision and Curriculum Development, 2013.

Mehan, Hugh. Learning Lessons: Social Organization in the Classroom. Cambridge, Mass.: Harvard University Press, 1979.

Michaels, S., M. C. O’Connor, M. W. Hall, and L. B. Resnick. Accountable Talk® Sourcebook: For Classroom Conversation That Works. Pittsburgh, Pa.: Institute for Learning, 2010. Morris, Anne K. Mathematical reasoning: Adults’ ability to make the inductive- deductive distinction. Cognition and Instruction 20, no. 1 (2002): 79–118. 

National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989. ———. Professional Standards for Teaching Mathematics. Reston, Va.: NCTM, 1991. ———. Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

———. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. Reston, Va.: NCTM, 2006. ———. Focus in High School Mathematics: Reasoning and Sense Making. Reston, Va.: NCTM, 2009a. ———. “Popcorn, Anyone?” NCTM Illuminations, 2009b. http://illuminations.nctm.org/Lesson .aspx?id=2927.

References  247 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.

———. Principles to Actions: Ensuring Mathematical Success for All. Reston, Va.: NCTM, 2014.

National Governors Association Center for Best Practices and Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Common Core State Standards (College-and Career-Readiness Standards and K–12 Standards in English Language Arts and Math). Washington, D.C.: NGA Center and CCSSO, 2010. http://www.corestandards.org. National Research Council. Adding It Up: Helping Children Learn Mathematics. Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, eds., Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, D.C.: National Academies Press, 2001. O’Connor, Mary Catherine, and Sarah Michaels. “Shifting Participant Frameworks: Orchestrating Thinking Practices in Group Discussion.” In Discourse, Learning, and Schooling, edited by D. Ghicks, pp. 63–103. New York: Cambridge University Press, 1996.

Oregon Department of Education (ODE), “Mathematics Work Sample Assessment: Student Homework Problem,” 2016. http://www.ode.state.or.us/wma/teachlearn/subjects/mathematics/ assessment/worksamples/statistics_homeworkandgrades_task.pdf.  Pashler, Harold, Patrice M. Bain, Brian A. Bottge, Arthur Graesser, Kenneth Koedinger, Mark McDaniel, and Janet Metcalfe. Organizing Instruction and Study to Improve Student Learning. IES Practice Guide (NCER 2007-2004). Washington, D.C.: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education, 2007.

Perry, Ayanna Dara-Franklin. Equitable Spaces in Early Career Teachers’ Mathematics Classrooms. Doctoral diss. (unpublished), North Carolina State University, 2013. Phelps, Geoffrey, Douglas Corey, Jenny DeMonte, Delena Harrison, and Deborah Loewenberg Ball. “How Much English Language Arts and Mathematics Instruction Do Students Receive? Investigating Variation in Instructional Time.” Educational Policy 26, no. 5 (2012): 631–62.

Ramirez, Gerardo, Elizabeth A. Gunderson, Susan C. Levine, and Sian L. Beilock. “Math Anxiety, Working Memory, and Math Achievement in Early Elementary School.” Journal of Cognition and Development 14, no. 2 (2013): 187–202. Rathmell, Edward C. “Basic Facts: Questions, Answers, and Comments.” Cedar Falls, Iowa: Thinking with Numbers, 2005.

Rohrer, Doug. “The Effects of Spacing and Mixed Practice Problems.” Journal for Research in Mathematics Education 40, no. 1 (2009): 4–17. Rohrer, Doug, and Kelli Taylor. “The Shuffling of Mathematics Problems Improves Learning.” Instructional Science 35, no. 6 (2007): 481–98.

Rubel, Laurie, and Haiwen Chu. “Equity-Directed Teaching Practices for Secondary Mathematics in Urban Schools.” Journal for Research in Mathematics Education (in press). Santos, Sherri Dobbs. I Do We Do You Do: An RTI Intervention for Math Problem Solving Grades 1-5. Author, 2011.

Schoen, Harold L., James T. Fey, Christian R. Hirsch, and Arthur F. Coxford. “Issues and Options in the Math Wars.” Phi Delta Kappan 80, no. 6 (1999): 444–53.

248   Taking Action Grades 9–12 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.

Schultz, James E., Wade Ellis Jr., Kathleen A. Hollowell, Paul A. Kennedy, and Martin Engelbrecht. Holt Algebra 2. New York: Holt, Rinehart and Winston, 2004. Silver, Edward A. “Moving Beyond Learning Alone and in Silence: Observations from the QUASAR Project Concerning Communication in Mathematics Classrooms.” In Innovations in Learning, New Environments for Education, edited by Leona Schauble and Robert Glaser, pp. 127–60 . Mahwah, N.J.: Lawrence Erlbaum Associates, 1996.

Smith, Margaret S. “Balancing Old and New: An Experienced Middle School Teacher’s Learning in the Context of Mathematics Instructional Reform.” Elementary School Journal 100, no. 4 (2000a): 351–75. ———. “Reflections on Practice: Redefining Success in Mathematics Teaching and Learning.” Mathematics Teaching in the Middle School 5, no. 6 (2000b): 378–82, 386.

Smith, Margaret S., Victoria Bill, and Elizabeth K. Hughes. “Thinking through a Lesson Protocol: A Key for Successfully Implementing High-Level Tasks.” Mathematics Teaching in the Middle School 14, no. 3 (2008): 132–38.

Smith, Margaret S., Victoria Bill, and Jeffrey Ziegler. “Supporting Productive Struggle in Mathematics Classrooms.” New England Mathematics Journal 47 (2015): 7–17.

Smith, Margaret S., and Mary Kay Stein. “Selecting and Creating Mathematical Tasks: From Research to Practice.” Mathematics Teaching in the Middle School 3, no. 5 (1998): 344–49.

Smith, Margaret S., and Mary Kay Stein. 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, Va.: National Council of Teachers of Mathematics, 2011. Sorto, M. Alejandra, Terence Mccabe, Max Warshauer, and Hiroko Warshauer. “Understanding the Value of a Question: An Analysis of a Lesson.” Journal of Mathematical Sciences & Mathematics Education 4, no. 1 (2009): 50–60. Spencer, Joi A. “Identity at the Crossroads: Understanding the Practices and Forces That Shape African American Success and Struggle in Mathematics.” In Mathematics Teaching, Learning, and Liberation in the Lives of Black Children, edited by D. B. Martin, pp. 200–30. New York: Routledge, 2009. Stein, Mary Kay. “The Nature and Role of Goals in and for Mathematics Instruction.” In Enhancing Classroom Practice with Research behind Principles to Actions, edited by Denise A. Spangler and Jeffrey J. Wanko, pp. 1–11. Reston, Va.: National Council of Teachers of Mathematics, 2017.

Stein, Mary Kay, Randi A. Engle, Margaret S. Smith, and Elizabeth K. Hughes. “Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move beyond Show and Tell.” Mathematical Thinking and Learning 10, no. 4 (2008): 313–40. Stein, Mary Kay, Margaret S. Smith, Marjorie A. Henningsen, and Edward A. Silver. Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development. Reston, Va.: National Council of Teachers of Mathematics, 2000/2009.

Stigler, James W., and James Hiebert. The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom. New York: Simon and Schuster, 1999. ______. “Improving Mathematics Teaching.” Educational Leadership 61, no. 5 (2004):12–16.

Suurtamm, Christine. “Assessment Can Support Reasoning.” Mathematics Teacher 106, no. 1 (2012): 28–33. References  249 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.

Swan, Malcolm. “Dealing with Misconceptions in Mathematics.” In Issues in Mathematics Teaching, edited by Peter Gates, pp. 147–65. New York: Routledge, 2001.

Sztajn, Paola, Jere Confrey, P. Holt Wilson, and Cynthia Edgington. “Learning Trajectory Based Instruction: Toward a Theory of Teaching.” Educational Researcher 41, no. 5 (2012): 147–56.

Wagner, David, and Beth Herbel-Eisenmann. “Re-Mythologizing Mathematics through Attention to Classroom Positioning.” Educational Studies in Mathematics 72, no. 1 (2009): 1–15. Walker, Erica N. “Who Can Do Mathematics?” In Activating Mathematical Talent, edited by Bruce R. Vogeli and Alexander Karp, pp. 15–27. Boston: Houghton Mifflin and National Council of Supervisors of Mathematics, 2003. Walsh, Jackie Acree, and Beth Dankert Sattes. Quality Questioning: Research-Based Practice to Engage Every Learner. Thousand Hills, Calif.: Corwin Press, 2005.

Warshauer, Hiroko Kawaguchi. “Strategies to Support Productive Struggle.” Mathematics Teaching in the Middle School 20, no. 7 (2015a): 390–93. ———. “Productive Struggle in Middle School Mathematics Classrooms.” Journal of Mathematics Teacher Education 17, no. 4 (2015b): 375–99.

Weiss, Iris R., and Joan D. Pasley. “What Is High-Quality Instruction?” Educational Leadership 61, no. 5 (2004): 24–28. Wiliam, Dylan. “Keeping Learning on Track: Classroom Assessment and the Regulation of Learning.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester Jr., pp. 1053–1098. Charlotte, N.C.: Information Age; Reston, Va.: National Council of Teachers of Mathematics, 2007. ______. Embedded Formative Assessment. Bloomington, Ind.: Solution Tree Press, 2011.

Wimer, Jeffrey W., Carolyn S. Ridenour, Kelli Thomas, and A. William Place. “Higher Order Teacher Questioning of Boys and Girls in Elementary Mathematics Classrooms.” Journal of Educational Research 95, no. 2 (2001): 84–92. Wood, Terry, and Tammy Turner-Vorbeck. “Extending the Conception of Mathematics Teaching.” In Beyond Classical Pedagogy: Teaching Elementary School Mathematics, edited by Terry Wood, Barbara Scott Nelson, and Janet E. Warfield, pp. 185–208. Mahwah, N.J.: Lawrence Erlbaum Associates, 2001.

Wu, Hung-Hsi. Teaching Geometry According to the Common Core Standards. 2013. https://math.berkeley .edu/~wu/Progressions_Geometry.pdf. Zimmerman, Barry J. “Theories of Self-Regulated Learning and Academic Achievement: An Overview and Analysis.” In Self-Regulated Learning and Academic Achievement: Theoretical Perspectives, edited by Barry J. Zimmerman and Dale H. Schunk, pp. 1–65. Mahwah, N.J.: Lawrence Erlbaum Associates, 2001.

250   Taking Action Grades 9–12 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.