Taking Action Implementing Effective Mathematics Teaching Practices in Grades 9-12 ( etc.).pdf

Taking Action
Implementing Effective Mathematics
Teaching Practices
Grades 9-12
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Melissa Boston
Frederick Dillon
Margaret S. Smith
Series Editor
Stephen Miller

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)
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Copyright © 2017 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved.
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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
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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

iv Taking Action Grades 9–12
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 Shalunda Shackelford Video Clip 1
ATL 2.2 Shalunda Shackelford Transcript 1
ATL 5.2 Jamie Bassham Video Clip
ATL 5.2 Jamie Bassham Transcript
ATL 5.4 Debra Campbell Video Clip
ATL 5.4 Debra Campbell Transcript
ATL 9.1 Jeff Ziegler Video Clip 1
ATL 9.1 Jeff Ziegler Transcript 1
ATL 9.1 Jeff Ziegler Video Clip 2
ATL 9.1 Jeff Ziegler Transcript 2
ATL 10.1 Wobberson Torchon Video Clip
ATL 10.1 Wobberson Torchon Transcript

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 grade-
band 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

vi Taking Action Grades 9–12
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

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.

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.

2 Taking Action Grades 9–12
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

Setting the Stage 3
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 high-
quality 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

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.

Setting the Stage 5
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/

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:

Setting the Stage 7
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

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

Setting the Stage 9
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).

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?
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Exploring Exponential Relationships:
The Case of Vanessa Culver
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.
The Pay It Forward Task
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.

Setting the Stage 11
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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.
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.]
Ms. C.: Can someone explain why y 5 3x is the best choice? Missy, can you explain
how you were thinking about this?
Missy: 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.
Ms. C.: 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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Darrell: 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.
Ms. C.: Other comments?
Kara: 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.
Chris: 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.
Ms. C.: So let’s take another look at group 3’s poster. Does the middle column help
explain what is going on? Devon?
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.
Angela: 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.
Ms. C.: 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.
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.

Setting the Stage 13
Group 1
(equation—incorrect)
Group 2
(a table like groups
6’s & 7’s and
an equation)
Group 3
(a diagram like
group 4’s and a table)
y 5 3x
At every stage there
are three times as many
good deeds as there
were in the previous
stage.
y 5 3x
x
(stages)
y
(deeds)
1 3 3
2 3 3 3 9
3 3 3 3 3 3 27
4 3 3 3 3 3
3 3
81
5 3 3 3 3 3 3
3 3 3
243
Group 4
(diagram)
Group 5
(a table like groups
6’s & 7’s and a graph)
Groups 6 and 7
(table)
3 3 3 3 3 3 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.
x
(stages)
y
(deeds)
1 3
2 9
3 27
4 81
5 243
Fig. 1.3. Vanessa Culver’s students’ work

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.

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

• 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.
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?

Establish Mathematics Goals to Focus Learning 17
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

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 whole-
group 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).

The Bicycle 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.
Time (in seconds)
Distance
from
start
of
road
(in
feet)
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.

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.

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.
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.
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 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

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

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 sense-
making 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

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

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.

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’ self-
assessment 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.

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.

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

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.
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 Task B: Pay It Forward
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]).
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

Implement Tasks That Promote Reasoning and Problem Solving 31
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’

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

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 high-
level-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)

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.
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

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

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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