A Mind for Mathematics

—CAROL ANN TOMLINSON, William Clay Parrish, Jr. Professor, University of Virginia “Every teacher of mathematics, from the elementary grades to the university level, should read this book and act on its recommendations. The result will likely be engaged students, enhanced learning, and more confident problem solvers for life.”

—JAY MCTIGHE,

Coauthor, Understanding by Design

N

anci N. Smith has been on a quest to establish what it takes to be a good mathematics teacher who actively engages students and addresses their learning differences. In A Mind for Mathematics: Meaningful Teaching and Learning in Elementary Classrooms, Smith breaks down the complex components of teaching mathematics and divides them into practical strategies. She fuses mathematics research, useful classroom strategies, and examples from K–6 classrooms to help teachers influence students to work hard, grapple with challenging problems, and ultimately value mathematics.

A MIND FOR MATHEMATICS

“A Mind for Mathematics is about teaching mathematics in rich, meaningful ways while contributing to the success of a variety of learners. Rooted in the author’s considerable classroom experience, this stellar resource guides teachers who aspire to make mathematics a positive force in the minds and lives of their students.”

MEANINGFUL TEACHING AND LEARNING IN ELEMENTARY CLASSROOMS

Readers will: Study the aspects of instruction, assessment, and learning that they must cultivate to develop mathematical minds

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Gain a mental picture of the essential elements of an effective mathematics classroom

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Observe classroom examples and vignettes that illustrate the concepts in each chapter

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Answer reflection questions so they can relate the strategies in this book to those in their own classrooms

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Perform call-to-action tasks that will help them take the next essential steps in teaching mathematics

NANCI N. SMITH

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ISBN 978-1-943874-00-2 90000

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NANCI N. SMITH

Copyright © 2017 by Solution Tree Press Materials appearing here are copyrighted. With one exception, all rights are reserved. Readers may reproduce only those pages marked “Reproducible.” Otherwise, no part of this book may be reproduced or transmitted in any form or by any means (electronic, photocopying, recording, or otherwise) without prior written permission of the publisher.

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TABLE OF CONTENTS

About the Author . . . . . . . . . . . . . . . . . . . . . . . xiii Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The Good Mathematics Teacher. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

CH APT E R 1

Creating the Mathematical Environment. . . . . . 5 Classrooms That Foster Mathematical Learning. . . . . . . . . . . . . . . 6 A Growth Mindset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Productive Struggle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Beliefs and Strategies of Successful Mathematics Classrooms. . . 11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Questions for Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Call to Action: Create a Graphic Organizer. . . . . . . . . . . . . . . . . . 16

CH APT E R 2

Engaging Mathematical Minds. . . . . . . . . . . . . . 17 Fostering Student Engagement . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Choosing Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Managing Engagement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Questions for Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Call to Action: Begin a Task Bank . . . . . . . . . . . . . . . . . . . . . . . . 38

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CH APTE R 3

Reaching Different Mathematical Minds. . . . . . 39 The Meaning of Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 40 The Framework of Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . 43 The Differentiated Classroom. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Questions for Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Call to Action: Differentiate a Lesson. . . . . . . . . . . . . . . . . . . . . . 67

CH APTE R 4

Challenging Student Mathematicians. . . . . . . . 69 Mathematical Understanding. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Standards for Mathematical Practice. . . . . . . . . . . . . . . . . . . . . . 80 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Questions for Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Call to Action: Challenge Your Students. . . . . . . . . . . . . . . . . . . . 98

CH APTE R 5

Monitoring Mathematical Assessment . . . . . . . 99 A Broad Look at Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Preassessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Formative Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Summative Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Student Self-Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Questions for Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Call to Action: Create an Assessment Matrix . . . . . . . . . . . . . . . 126

CH APTE R 6

Balancing It All . . . . . . . . . . . . . . . . . . . . . . . . . 127 Students and Families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Mathematical Minds Mobile. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Ta b l e o f C o n t e n t s

Work and Home. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Sound Advice From Teachers. . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Questions for Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Call to Action: Build a Mathematical Minds Mobile. . . . . . . . . . . 140

References & Resources. . . . . . . . . . . . . . . . . . 141 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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ABOUT THE AUTHOR

Nanci N. Smith, PhD, is a full-time national and international consultant and featured conference speaker in the areas of mathematics, curriculum, assessment, differentiated instruction, and professional learning communities.

Her work includes professional development in forty-five states and nine countries. Nanci has taught mathematics at the high school and university levels, and differentiated instruction as a master’s course at Arizona State University and Northern Arizona University, and for the teachers at Singapore American School through Buffalo State University.

She was the mathematics consultant, designer, and author of the Meaningful Mathematics: Leading Students Toward Understanding and Application DVD series, and she developed a National Science Foundation–funded CD and DVD professional development series for middle school mathematics teachers. Her mathematics classroom is featured in the video series At Work in the Differentiated Classroom.

Nanci has authored several chapters and books in the areas of differentiation, effective mathematics instruction, curriculum design, and standards implementation. She has given interviews for publications and NPR and has been a featured speaker for the National Council of Teachers of Mathematics national conference and numerous other conferences in the United States and internationally.

Nanci received her doctorate in curriculum and instruction for mathematics education from Arizona State University. She is board certified in the United States in adolescent and young adult mathematics. She lives in Phoenix, Arizona, with her husband, Russ, and three cats. Her passions are her family (especially her eight grandchildren), travel, and knitting.

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To learn more about Nanci’s work, visit Effective Classrooms Educational Consulting, LLC (www.e2c2.com) or follow @DocNanci on Twitter.

To book Nanci N. Smith for professional development, contact pd@SolutionTree .com.

INTRODUCTION

Underpinning the learning revolution is a growing recognition that facing the challenges of the 21st century will require more than minor adaptations to current practice. —Steven E. Jungst, Barbara L. Licklider, and Janice A. Wiersema

In my first year of teaching, I was accused of teaching Mickey Mouse mathematics.

I knew then that I was different. You see, I disliked the text series I was supposed to follow, and I kept venturing off into new territory. I made up games (and I was teaching high school—unheard of!) so my students could practice skills and worked with the geography teacher to have my students use fractal reasoning and fractions to estimate the border lengths of countries they studied. This was when almost every mathematics classroom followed the same pattern: the teacher modeled the problem to be learned; students tried the examples the teacher presented; and then the teacher showed students the correct method for completing the examples. Finally, students completed thirty more problems for homework.

And I had the audacity to ask students to make posters and play games? Without knowing any different or any better, in my first years of teaching I shaped my belief that mathematics could and should be fun.

The Good Mathematics Teacher

What I did not know those first years was how complicated it truly is to be a good mathematics teacher. I engaged my students in games because that was my personality. I did not fully grasp the depth of mathematics, the conceptual understandings of mathematics, or the connections within mathematical topics or the connections mathematics has with other content areas, even though I was

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a mathematics major in college. I was good at mathematics and never struggled much doing problems. How little that actually helps one be a mathematics teacher!

I had to learn classroom management while I actively engaged my students. I had to go beyond just teaching them to do problems correctly with proper vocabulary.

I had to recognize and address students’ differences, not pretend they didn’t exist because it might hurt students’ self-esteem or I found it inconvenient or harder. I have been on this quest for almost twenty-five years as a classroom teacher, a university professor, a researcher, and ultimately a mathematics consultant working with schools and districts across the United States and around the world. I have learned over the years that our struggles in effectively teaching mathematics are universal. I have worked to shape curriculum and instruction alongside excellent teachers and administrators, to whom I will always be thankful.

Teachers face more challenges in the 21st century than probably any time in history. Content is expansive and robust, and we teach diverse groups of students.

Classroom structures and management are ever changing and complicated. And overriding all these challenges are the high-stakes tests that frustrate and intimidate teachers and students alike. Yet teaching in the 21st century has so many incredibly positive aspects. I am more convinced than ever that all students can become the mathematicians we long for them to become. We have more research than ever before from the fields of sociology, psychology, and cognitive science as well as education to help identify what does and does not maximize student learning and increase interest and motivation in learning mathematics. We have rigorous standards to increase learning and make us internationally competitive. We have structures to develop comprehensive and conceptual curricula and design instruction to reach all students. But how do we put this all together?

Overview of the Book

This book is designed to break down the complex components of teaching mathematics into workable parts, with each chapter adding another layer. It provides an easy-to-read and easy-to-implement blend of research, big ideas, connections between mathematics concepts and instruction, and practical strategies and examples for K–6 classrooms.

The chapters in this book all follow the same pattern. Each chapter and many sections within the chapters begin with a vignette or scenario that provides a mental picture of what is to come. Each chapter ends with reflection questions and a call to action. The reflection questions are designed to invite readers to

I n t ro d u c t i o n

make the material their own and relate it to their professional setting. The calls to action provide specific tasks that help readers take the next steps to implement the learning in each chapter. The book is designed for various professional development settings, including: }}

Collegial book studies

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Mentorships between experienced and new teachers

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Individual study and reflection

To mentor the mathematical minds in your care, you must carefully develop many aspects of instruction, assessment, classrooms, and learning. You will find answers to the following questions in this book, including specific and practical examples for reflection and learning. }}

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}}

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Chapter 1: How do I get my students to work hard, struggle with problems, and persevere to keep learning? What are the components of an effective mathematical learning environment?

Chapter 2: How do I hook and engage students in learning mathematics? What does that look like? What tasks are most likely to engage students while maintaining the necessary rigor?

How can I make skill development and practice fun? How can I make learning social? How do I manage an active classroom?

Chapter 3: How do I address all my students’ differences? What does differentiation really mean, and how do I implement it in my mathematics classroom on top of everything else I have to do? What learning differences do I need to attend to? How do I really attend to them? What does a differentiated classroom look like?

Chapter 4: What does it mean to understand mathematics?

How do I use mathematical understanding to drive instruction, and how do I present that to students? What does it look like when students understand? What should students be able to model and demonstrate as they gain mathematical understanding?

Chapter 5: How can I use assessment to truly further learning?

What types of assessments should be present in every unit? How does assessment inform my instruction? How involved should

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students be in the assessment process? How can assessments act as learning and reflection tools for students? }}

Chapter 6: How do I begin to do all of this? What role should families play in this, and how should I communicate with them? How can I remember to keep all the pieces in balance?

How can I keep my life in balance?

As we consider our changing technological world, it is critically important that our students see the value and joy in learning and applying mathematics inside and outside the classroom. Mathematics enables us to figure simple problems and decisions in everyday life and opens doors in career opportunities. However, the ultimate goal of this book is to help students learn mathematics more effectively and for teachers to increase their appreciation and love for mathematics. I hope this book becomes a go-to tool for you in planning and teaching meaningful mathematics in your classroom.

CHAPTER 4

The deeper we search, the more we find there is to know, and as long as human life exists I believe it will always be so.

—Albert Einstein

I often find myself on an airplane chatting with the person next to me. I have begun to think of creative ways to say what I do without actually saying what I do.

My heart hurts and my head aches with all the stories of adults who hate mathematics, were never very good at mathematics, and mostly can’t understand why we teach mathematics now in such crazy ways! I want to ask why we should teach mathematics in the same old ways when these adults are living proof that they didn’t work well. I try to explain why we teach mathematics differently today, but too often, their eyes start glazing over, and the mathematics coma begins to set in.

Many adults tell me that mathematics has just never made sense to them, and if they could have stopped with arithmetic, they would have been okay. Fractions and algebra were just too challenging and confusing. Too often, I see and hear the same things from students. Mathematics just doesn’t make sense. I find this heartbreaking. However, we all have experienced joy when our students’ eyes light up because they have obviously made a connection and they truly get it.

John Van de Walle (2007) notes that “the most basic idea in learning mathematics is that mathematics makes sense” (pp. 13–14). I wonder how many students and teachers today would agree with that. In order for mathematics to make sense, students need to understand mathematics instead of memorizing it as a set of facts, definitions, and steps.

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CHALLENGING STUDENT MATHEMATICIANS

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According to Jo Boaler (2008, as cited in Hull, Miles, & Balka, 2015):

The curriculum classes I took in college taught me that I could not use the word understand in an objective. After all, you can’t measure understanding. However, as teachers, we can certainly recognize the difference between students who know how to do mathematics and students who understand mathematics.

I met a fifth-grade student, Ronald, who understood mathematics. Ronald saw connections beyond what his teacher saw. For example, he explained that if a numerator and denominator were only one digit apart, the fraction was automatically in lowest terms. He went on to explain the rules of divisibility as a justification to his claim. I think he lost most of his class in the explanation, and perhaps his teacher as well, but his recognition of patterns supported by mathematical reasoning and explanation provides a clear example of understanding number relationships, as opposed to simply knowing the rules of how to reduce or simplify fractions. We want Ronalds in our mathematics classes. To enable this, we need to make sure that we provide opportunities to understand mathematics, not just learn how to do mathematical problems.

Mathematical Understanding

Perhaps you have not really considered the difference between knowing steps and facts in mathematics, being able to do mathematics, and understanding mathematics. Often these phrases and words are used as synonyms in documents, books, and our own conversations. However, they are not the same. Students who know and do mathematics without developing an understanding of mathematics most often hit a wall in learning. So what does it mean to understand mathematics? What does that look like?

In the introduction to the Common Core State Standards for mathematics, the National Governors Association Center for Best Practices and the Council of Chief State School Officers (NGA & CCSSO, 2010) describe what makes understanding distinct from knowing and doing:

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Children begin school as natural problem solvers and many studies have shown that students are better at solving problems before they attend math classes. They think and reason their way through problems, using methods in creative ways, but after a few hundred hours of passive math learning students have their problem-solving abilities drained out of them. They think that they need to remember the hundreds of rules they have practiced and they abandon their common sense in order to follow rules. (p. 72)

Challenging Student Mathematicians

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One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. . . . Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. (p. 4)

We can all recognize the students in our class who know how to do a problem correctly versus the students who have a depth of understanding. On his blog Granted, and . . ., Wiggins (2014) describes mathematical understanding as follows: Conceptual understanding in mathematics means that students understand which ideas are key (by being helped to draw inferences about those ideas) and that they grasp the heuristic value of those ideas.

They are thus better able to use them strategically to solve problems— especially non-routine problems—and avoid common misunderstandings as well as inflexible knowledge and skill.

According to John Barell (1991), students who understand mathematics can: • Explain it clearly, giving mathematical explanations and examples • Compare and contrast it with other concepts or procedures • Relate it to other topics, other subjects, and personal life experiences • Transfer it to unfamiliar settings and see the concept embedded within a novel problem • Generate questions and hypotheses that lead to new knowledge and further inquiries and combine it appropriately with other understandings • Pose new problems that exemplify or embody the concept • Create analogies, models, metaphors, symbols, or pictures of the concept • Generalize and justify from specifics to form connections • Use the knowledge to appropriately assess [their] performance, or that of someone else

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Grant Wiggins and Jay McTighe (1998) make understanding an attainable, measurable, plannable, and assessable goal through their work Understanding by Design. They change the word understand from a “do not use” status to a “must use and plan for” status in curriculum and instruction if we want our students to go beyond knowing and doing in mathematics.

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Making understanding explicit from the unit standards can be, and usually is, a mind-stretching task. Some standards are written at a conceptual level and are easier to use in discerning appropriate understandings for the unit, while others are more procedural. It is often up to the teacher to construct the essential understandings that undergird the topics. The most powerful understandings apply horizontally (in more than one unit) and vertically (year after year). In fact, one characteristic of an understanding is that it can be revisited, and students will be able to answer questions, relate examples, and make more connections with greater experience and information when they absorb that understanding.

Table 4.1 shows examples of understandings derived from standards. While the standards listed in the table come from the Common Core State Standards, conceptual understandings hold true for any set of standards. This table lists the understandings at a specific grade level to link to a certain standard, but true to the nature of understandings, they also relate to all or most grade levels and multiple additional standards.

Table 4.1: Examples of Understandings From Standards Standard

Understanding

K.CC.A.1: Count to 100 by ones and by tens.

Our numbers follow a specific pattern, based on tens, and can be counted using a variety of patterns.

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Fostering students who truly understand mathematics does not come without specific planning and intentionality in designing curriculum and instruction, beginning with the standards you address. You will uncover the essential understandings by unpacking the standards, and you will end up with about two to five understandings in a unit. Understandings are broad conceptual statements that get to the heart of the unit and provide the true reasons for learning the content.

Understandings are purposeful. They focus on the key ideas that require students to make sense of information and make connections while evaluating the relationships that exist within the understandings. Understandings show what the mathematician values or reasons out within a context.

Challenging Student Mathematicians Understanding

1.OA.A.1: Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, such as by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

We can solve mathematical problems using a variety of strategies, models, and representations.

Efficient application of computation strategies is based on the numbers and operations in the problems.

2.OA.A.1: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, such as by using drawings and equations with a symbol for the unknown number to represent the problem.

Addition and subtraction give a final count of items that are alike.

Only items that are alike can be added or subtracted.

3.NF.A.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Fractions always show a relationship of parts to the whole.

4.NBT.B.4: Fluently add and subtract multidigit whole numbers using the standard algorithm.

We can explain and justify the steps used in the standard algorithms for addition and subtraction by using properties of these operations and the understanding of place value.

Among all the techniques and algorithms you can choose for accurately performing multidigit computations, choose some procedures with which all students should be fluent for efficiency and communication.

5.OA.A.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

There is a meaningful order for simplifying and evaluating numerical expressions based on the meaning of operations.

2.OA.B.2: Fluently add and subtract within 20 using mental strategies. By end of grade 2, know from memory all sums of two one-digit numbers.

Source: Adapted from NGA & CCSSO, 2010; Sparks, 2012.

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Standard

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Understandings to Drive Unit Development

For example, consider the understanding suggested for second grade in table 4.1: “Addition and subtraction give a final count of items that are alike. Only items that are alike can be added or subtracted.” This understanding provides a conceptual basis for the operations of addition and subtraction that remains consistent throughout all of mathematics. For example, in kindergarten, we begin to develop the concept of addition by combining counters or other objects. Putting together two balloons and three balloons results in five balloons. The likeness of the balloons allows us to add them together. We wouldn’t add four rocks and three stars to get seven rock stars!

By second grade, we add multidigit numbers, and we tell students to line up the digits from the right. This is because we align place value, and in a number, the place value determines what is alike. This also holds true for decimals. We teach students to line up the decimal points. The reason is exactly the same as for multidigit numbers—place value determines what is alike in a number, and we can only add like items. With fractions, we need a common denominator to add or subtract but not to multiply or divide. Why? The denominator determines what is alike in a fraction, and we can only add like items. In algebra, we can only add like terms. It continues in this way with every topic in mathematics.

There is truly only one way to add or subtract: count the number of like items.

The question then should not be, How do you add or subtract these numbers? but rather, Are these alike? If they are alike, we can count them. If not, can we make them alike without changing the value, such as using an equivalent fraction? If we can make a quantity or value be like another one without changing its value (equivalency), then we can combine or add the quantities. If we cannot make quantities alike, we cannot add or subtract them.

If we always taught addition and subtraction as counting the like items, from kindergarten through college, then each new number system would add another layer to what students already learned, rather than a new topic and skill unrelated

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Taking the time to develop understandings from standards facilitates a cohesive approach to learning and developing schema for our students. We tend to teach mathematics topic by topic, and in fact, I have heard teachers tell me that if they did not teach a topic a day, they would not finish the curriculum. That is a very poor way to learn mathematics, and deep understandings and connections are very hard to grasp in this manner. Understandings provide the cohesive glue to learning many topics, skills, and concepts.

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to previous learning. This is the power of learning with understandings guiding the development of knowledge and skill.

As you look back over the sample understandings in table 4.1, you can see that you write understandings as complete sentences that can be preceded by the phrase, “Students will understand that . . .” The word that at the end of the phrase carries importance because it forces the statement to be specific enough to become an understanding. For example, let’s suppose we want students to understand fractions. Now put that into the statement: “Students will understand that fractions.” It doesn’t work. Fractions are a topic, not a specific understanding.

“Students will understand that fractions always show a relationship between a number of parts and a whole” is an understanding.

Writing understandings to make sense of the learning in a unit is necessary if we want students to go beyond memorization in mathematics. However, these understandings are not enough to design complete units and lessons. Students do need to know facts and skills in mathematics. For example, in order to truly reach a depth of understanding, students must know the facts and skills to make sense of an understanding. It doesn’t help students to know how to explain why you can only add like items if they can’t finish a problem because they don’t know that 2 + 3 = 5. A student might understand that the top number in a fraction shows how many parts are used and the bottom number shows how many parts are in the whole, but that is not enough. Students must know the vocabulary of numerator and denominator in order to communicate mathematically. These facts undergird the understanding that fractions always show a relationship of parts to a whole.

Skills are involved as well. Students must know how to add in various contexts.

They need to know how to simplify fractions, find equivalent fractions, and perform operations with fractions. In planning a unit, I list both the facts (for example, number facts, vocabulary words and definitions, formulas) and the steps for skills (for example, how to simplify a fraction) together as what students need to know.

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To write understandings is not an easy task. You write them from standards to give meaning to or make sense of the mathematics to learn. Understandings do not consist of step lists or rules, definitions, or facts. Understanding in mathematics is not synonymous with knowing how to do mathematics. Factual content— that which can be found in a book or memorized, such as the multiplication table or the definition of a triangle—is necessary but not enough. Understandings are conceptual and not factual. They grow over time and can be revisited to form deeper understanding with each additional layer of experience.

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Please note that the do does not describe a specific activity, such as: Complete the problems on page 128. Rather, it describes the essential evidence of learning, such as: Explain why common denominators are needed to add fractions, which students can demonstrate in a variety of ways. The teacher designs the specific manner or task in which students will display learning in an activity or assessment. Thus, this explicit method for clarifying content and learning—what a student should know, understand, and be able to do—serves as the basis for developing tasks and assessments (Tomlinson & Imbeau, 2014; Tomlinson & Moon, 2013).

While a unit typically has two to five understandings, the number of knows (facts and skills) or do’s (evidence of learning) has no limit. Table 4.2 shows a sample KUD for a second-grade unit on addition and subtraction, and table 4.3 (page 78) shows a sample KUD for a fifth-grade unit on measurement (volume).

Table 4.2: KUD for Unit on Addition and Subtraction From SecondGrade Standards Standards • 2.OA.A.1: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, such as by using drawings and equations with a symbol for the unknown number to represent the problem.

• 2.NBT.B.5: Add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

• 2.OA.B.2: Fluently add and subtract within 20 using mental strategies. By end of grade 2, know from memory all sums of two one-digit numbers.

• 2.MD.B.6: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, . . . , and represent whole-number sums and differences within 100 on a number line diagram (use number line as a strategy).

• 2.NBT.B.9: Explain why addition and subtraction strategies work, using place value and the properties of operations.

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Together, what students will come to know (the facts and skills) and understand by the end of the unit combine to make a robust, rigorous, and explicit curriculum. Finally, we must consider what students should be able to do as a result of their learning. We use the term do as a way to describe what students should be able to show and perform if they truly know and understand the content. Tables 4.2 and 4.3 provide complete examples of a unit design that unpacks standards into what students will come to know, understand, and be able to do (KUD).

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• 2.NBT.B.7: Add and subtract within 1,000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Understand

Be Able to Do

K1: We must know definitions of the following terms: • Adding to • Taking from • Putting together • Taking apart • Sums and differences • Addends • Equation • Place value • Odd and even • Relationship • Mental mathematics • Number line • Manipulatives • Strategies • Compose and decompose • Fluently • Regroup K2: Add and subtract within 20 fluently.

K3: There are multiple strategies for addition and subtraction.

• How to regroup a group of ten • Mental mathematics • Counting on • Doubles • Fact family K4: A number line represents whole numbers with equally spaced points corresponding to the numbers 1, 2, 3, . . .

K5: Even numbers can be counted by twos, and odd numbers cannot.

U1: We can only add or subtract items that are alike. Addition and subtraction give a final count of items that are alike.

U2: Addition and subtraction are inverse operations (they undo each other), so we can use these operations to solve each other.

U3: We can use many different strategies to add and subtract numbers (such as drawings, models, equations with symbols, number lines, fact families, and counting objects), and each strategy reveals aspects of the meaning of the operation.

U4: We can represent or describe numbers many different ways. Sometimes we need a different representation in order to add or subtract.

D1: Explain why we can only add or subtract items that are alike and how to decide if what we are adding is alike or not.

D2: Model and explain the role of place value in adding and subtracting multidigit numbers.

D3: Explain why addition and subtraction strategies work, including inverse operations and properties.

D4: Use addition and subtraction within 100 to solve one- and two-step word problems.

D5: Correctly use vocabulary in discussions and explanations.

D6: Add and subtract up to and including 20 based on place value.

D7: Use mental mathematics strategies to solve problems, including addition and subtraction, within 20.

D8: Use a variety of strategies, including number lines, manipulatives, and models, to complete addition and subtraction problems.

D9: Compose or decompose tens or hundreds.

D10: Explain how to use addition to subtract and how to use subtraction to add.

Source: NGA & CCSSO, 2010.

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Table 4.3: KUD for Unit on Volume From Fifth-Grade Standards Standards

Know

Understand

Be Able to Do

K1: We must know definitions of the following terms: • Edge • Face • Vertex • Net (if time allows) • Volume • Unit • Cubic unit • Base • Height • Length • Right rectangular prism K2: Volumes fill K3: V = bh or V = lwh

U1: Sometimes we can separate irregular shapes or solids into familiar shapes or solids. This allows us to find irregular areas and volumes.

U2: Area can be helpful when finding the volume of solid figures.

U3: We can describe solids and categorize them by their general shapes and by their faces, edges, and vertices.

D1: Find volumes of right rectangular prisms and composite shapes made of right rectangular prisms with correct units.

D2: Create a threedimensional figure with a given volume.

D3: Label threedimensional shapes with edges, faces, and vertices.

D4: Explain how area relates to volume.

D5: Explain how to use a unit cube in describing volume. Describe why it is in cubic units, as opposed to area, which is square units.

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• 5.MD.C.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. 5.MD.C.3.A: A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. 5.MD.C.3.B: A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

• 5.MD.C.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

• 5.MD.C.5: Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume. a. 5.MD.C.5.A: Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold wholenumber products as volumes, such as to represent the associative property of multiplication. b. 5.MD.C.5.B: Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems. c. 5.MD.C.5.C: Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve realworld problems.

Challenging Student Mathematicians Understand

Be Able to Do

K4: A cube with a side length of 1 unit is called a unit cube. This is the basic unit of volume.

K5: Volume is always measured in cubic units.

K6: We find volume through multiplication.

K7: We can find the volume of shapes consisting of two right rectangular prisms by adding the volume of each of the prisms making up the composite shape.

U4: The types of units we use describe what we measure, and what we measure has a specific type of unit.

U5: Some attributes of objects are measurable, and we can quantify them using unit amounts.

D6: Apply volume formulas to solve real-world problems.

D7: Create and solve original volume problems found in students’ lives.

D8: Model when we can and cannot add or multiply measurements.

Source: NGA & CCSSO, 2010.

The work of unpacking standards and making explicit what students will come to know, understand, and be able to do by the end of a unit is worth the effort, regardless of the resources or program used. The process promotes clarity and consistent priorities. It is foundational to establishing a viable curriculum, pacing, and purposeful instruction and assessment. It develops deep understanding for the teachers involved and ownership of the curriculum they teach (DuFour, DuFour, Eaker, & Many, 2010). If you can accomplish this process collaboratively, ideally in a collaborative team, it turns out even better. However, the process holds so much importance that it is worth tackling as an individual teacher if necessary.

Understandings are multifaceted, not linear, and resemble organized and structured webs or networks of ideas. The more connections that exist among the facts, ideas, and procedures involved in the concept, the more robustly we would define the understanding. In fact, as students gain and develop deeper and deeper understandings, they transition from novices to experts in their work (Williams, 1998).

Students build their conceptions on intuitions—the mental representations of facts that to the learner seem obvious. These beliefs and intuitions result from personal experience but can be correct or incorrect, or ignored, based on previous classroom experiences. One of our goals of instruction becomes a return to establishing and enlarging student intuition, sense making, and development of a correct feeling for concepts (Dreyfus & Eisenberg, 1982).

The KUD, and especially the understandings for a unit, becomes the foundation for assessment and instructional choices and design. Teachers design specific lesson tasks and unit assessments that address what students should be able to

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do (from the do list in the KUD) based on student evidence that they have come to know and understand. The KUD chart for a unit also becomes the source of understandable and purposeful learning targets.

From KUDs to Student-Friendly Learning Targets

A district or school might determine targets for students or sometimes leave that up to the teacher. Often, targets follow the phrasing of I can statements, such as, “I can rewrite numbers in order to make a ten when adding any two numbers.” I have had the pleasure to work with educators and students at the Killeen Independent School District in Killeen, Texas, where they write their learning targets according to this phrasing: “Today, I will so that I can . I’ll know it when .” For our example, we would write the learning target as follows: “Today, I will rewrite numbers strategically in order to make a ten so that I can add any two numbers mentally, quickly, and accurately. I’ll know it when I can add any two numbers using this strategy and explain why it works mathematically. I will be able to add numbers in many daily situations.”

Students can then concentrate on what they need to learn, understand why it is beneficial to learn, and understand how they can self-assess whether they have learned. The learning target has focus and purpose and has the correct rigor to address the curriculum standard, all through unpacking the unit standards with a KUD.

Standards for Mathematical Practice

The deeper conceptual and more rigorous content expectations expressed in the standards, unpacked through a KUD and communicated to students through learning targets, will certainly challenge students to have a growth mindset and value productive struggle, as described in chapter 1. The call to

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Teachers are commonly expected to post a learning target for students for each lesson. Learning targets have many formats, language preferences, and names; however, all should be based on the unit’s essential learning as found in the KUD. As you plan the lesson for each day and you note the specific area of K, U, or D, the learning target becomes obvious. For example, if today’s lesson centers on the mental strategy of making a ten in order to add two numbers (see K3 and U4 in table 4.2 on page 76), then the learning target focuses on the specific mental skill of making a ten and rewriting numbers in equivalent ways to make a ten.

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change mathematics instruction and learning in the United States does not solely depend on content. In fact, just moving content standards around will have no more impact on learning than any of the many standards documents that have already come and gone.

The eight Standards for Mathematical Practice in the Common Core mathematics describe how we should engage students with learning as mathematicians (NGA & CCSSO, 2010). Whether or not your state has adopted or follows the Common Core mathematics standards, the Standards for Mathematical Practice are appropriate for all mathematics learners. “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These Mathematical Practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education” (NGA & CCSSO, 2010, p. 6). The Standards for Mathematical Practice consist of the eight following skills (NGA & CCSSO, 2010).

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

When considering the Standards for Mathematical Practice, keep in mind that these practices do not describe teacher instruction but the actions students should exhibit in learning and making sense of mathematics. These actions do

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As I talk with mathematicians and mathematics leaders and teachers across the United States, most agree that the most impactful change occurring in mathematics classrooms is the manner in which we ask students to learn the content.

However, the reality is that many teachers tend to focus on the content standards primarily and trust that the thinking processes and habits of mind will naturally occur. That is not the case. The way we ask students to participate and engage in learning, with collaboration, discourse, and deep understanding, is not how mathematics has generally been done in the past. This is another area in which we should encourage growth mindset and productive struggle.

—CAROL ANN TOMLINSON, William Clay Parrish, Jr. Professor, University of Virginia “Every teacher of mathematics, from the elementary grades to the university level, should read this book and act on its recommendations. The result will likely be engaged students, enhanced learning, and more confident problem solvers for life.”

—JAY MCTIGHE,

Coauthor, Understanding by Design

N

anci N. Smith has been on a quest to establish what it takes to be a good mathematics teacher who actively engages students and addresses their learning differences. In A Mind for Mathematics: Meaningful Teaching and Learning in Elementary Classrooms, Smith breaks down the complex components of teaching mathematics and divides them into practical strategies. She fuses mathematics research, useful classroom strategies, and examples from K–6 classrooms to help teachers influence students to work hard, grapple with challenging problems, and ultimately value mathematics.

A MIND FOR MATHEMATICS

“A Mind for Mathematics is about teaching mathematics in rich, meaningful ways while contributing to the success of a variety of learners. Rooted in the author’s considerable classroom experience, this stellar resource guides teachers who aspire to make mathematics a positive force in the minds and lives of their students.”

MEANINGFUL TEACHING AND LEARNING IN ELEMENTARY CLASSROOMS

Readers will: Study the aspects of instruction, assessment, and learning that they must cultivate to develop mathematical minds

•

Gain a mental picture of the essential elements of an effective mathematics classroom

•

Observe classroom examples and vignettes that illustrate the concepts in each chapter

•

Answer reflection questions so they can relate the strategies in this book to those in their own classrooms

•

Perform call-to-action tasks that will help them take the next essential steps in teaching mathematics

NANCI N. SMITH

•

ISBN 978-1-943874-00-2 90000

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NANCI N. SMITH