Mathematize It! [Grades 6-8]: Going Beyond Key Words to Make Sense of Word Problems, Grades 6-8

by Kimberly Morrow-Leong, Sara Delano Moore, Linda M. Gojak

Overview

Help students reveal the math behind the words

"I don’t get what I’m supposed to do!" This is a common refrain from students when asked to solve word problems.

Solving problems is about more than computation. Students must understand the mathematics of a situation to know what computation will lead to an appropriate solution. Many students often pluck numbers from the problem and plug them into an equation using the first operation they can think of (or the last one they practiced). Students also tend to choose an operation by solely relying on key words that they believe will help them arrive at an answer, without careful consideration of what the problem is actually asking of them.

Mathematize It! Going Beyond Key Words to Make Sense of Word Problems, Grades 6–8 shares a reasoning approach that helps students dig into the problem to uncover the underlying mathematics, deeply consider the problem’s context, and employ strong operation sense to solve it. Through the process of mathematizing, the authors provide an explanation of a consistent method—and specific instructional strategies—to take the initial focus off specific numbers and computations and put it on the actions and relationships expressed in the problem.
 
Sure to enhance teachers’ own operation sense, this user-friendly resource for Grades 6–8:

·         Offers a systematic mathematizing process for students to use when solving word problems

·         Gives practice opportunities and dozens of problems to leverage in the classroom

·         Provides specific examples of questions and explorations for multiplication and division, fractions and decimals, as well as operations with rational numbers

·         Demonstrates the use of visual representations to model problems with dozens of short videos

·         Includes end-of-chapter activities and reflection questions

How can you help your students understand what is happening mathematically when solving word problems? Mathematize it! 

Product Details

ISBN-13:	9781071819876
Publisher:	SAGE Publications
Publication date:	08/21/2020
Series:	Corwin Mathematics Series
Sold by:	Barnes & Noble
Format:	eBook
Pages:	264
File size:	49 MB
Note:	This product may take a few minutes to download.

About the Author

Kimberly Morrow-Leong is an adjunct instructor at George Mason University in Fairfax, Virginia, a part-time researcher at American Institutes for Research, and a consultant for Math Solutions. She is a former grade 5–9 classroom teacher, K–8 mathematics coach, and coordinator of elementary professional development for the National Council of Teachers of Mathematics (NCTM). She recently completed an elected term as vice president and 2018 program chair for NCSM, Leadership in Mathematics Education. She holds a BA in French language and a masters in linguistics (TESOL). She also holds an MEd and PhD in mathematics education leadership from George Mason University. Kim is the 2009 recipient of the Presidential Award for Excellence in Mathematics and Science Teaching (PAEMST) from Virginia. She is happiest when working with teachers and students, putting pencils down and getting messy with manipulatives!

Sara Delano Moore is an independent mathematics education consultant at SDM Learning.  A fourth-generation educator, her work focuses on helping teachers and students understand mathematics as a coherent and connected discipline through the power of deep understanding and multiple representations for learning. Sara has worked as a classroom teacher of mathematics and science in the elementary and middle grades, a mathematics teacher educator, Director of the Center for Middle School Academic Achievement for the Commonwealth of Kentucky, and Director of Mathematics&Science at ETA hand2mind. Her journal articles appear in Mathematics Teaching in the Middle School, Teaching Children Mathematics, Science&Children, and Science Scope.

Winner of the Presidential Award for Excellence in Science and Mathematics Teaching, Linda M. Gojak directed the Center for Mathematics and Science Education, Teaching, and Technology (CMSETT) at John Carroll University for 16 years. She has spent 28 years teaching elementary and middle school mathematics, and has served as the president of the National Council of Teachers of Mathematics (NCTM), the National Council of Supervisors of Mathematics (NCSM), and the Ohio Council of Teachers of Mathematics.

Table of Contents

List of Videos xix

Preface xxi

Acknowledgments xxiv

About the Authors xxvii

Chapter 1 Introduction: Why you Need to Teach Students to Mathematize 1

Problem-Solving Strategies Gone Wrong 2

What Is Mathematizing? Why Is It important? 4

Focusing on Operation Sense 4

Using Mathematical Representations 5

Teaching Students to Mathematize 7

Building Your Understanding of the Operations and Related Problem Situations 7

Exploring in the Mathematizing Sandbox: A Problem-Solving Model 10

A Note About Negative Values 12

Final Words Before You Dive In 13

Chapter 2 Add-to-And Take-From 14

Thinking About Active Addition and Subtraction 14

Students and Teachers Think About the Problems 18

Finding the Unknown, Three Story Structures 18

Story Structures: Implications for Teaching 20

Focus on Take-From Situations 23

Complicating Things: The Start Unknown Variation 24

Teaching Students to Use Concrete and Pictorial Models 27

Extending to Negative Values 28

Students and Teachers Think About the Problems 29

Reasoning and Notating With Negative Values 33

Conceptual Models of Integer Contexts 34

The Importance of Clear Notation 37

Key Ideas 40

Try It Out! 41

Reflect 43

Chapter 3 Part-Part-Whole 44

Thinking About Part-Part-Whole Situations 44

Students and Teachers Think About the Problems 48

Defining the Part-Part-Whole Situation 49

Modeling Relationships Versus Action 49

Writing Equations: Addition or Subtraction 50

Extending to Negative Values 53

Students and Teachers Think About the Problems 54

Representing Part-Part-Whole With Two-Color Counters 55

Blurred Lines Between Part-Part-Whole and Take-From 58

The Special Case of Both Parts Unknown 60

Key Ideas 62

Try It Out! 63

Reflect 64

Chapter 4 Additive Comparison: Another Kind of Relationship 66

Thinking About Additive Comparison Situations 66

Students and Teachers Think About the Problems 70

Language Can Get Tricky 71

Extending to Negative Values 74

Students and Teachers Think About the Problems 75

A Note About Equations 76

Knowing When to Add and When to Subtract 77

Modeling the Difference When Working With Negative Values 80

Key Ideas 82

Try It Out! 83

Reflect 84

Chapter 5 Equal Groups Multiplication and Division: Two Factors, Different Jobs 86

Thinking About Equal Groups Situations 86

Students and Teachers Think About the Problems 90

Multiplier and Measure Factors 91

Making Sense of Factors Less Than One 93

A Progression of Equal Groups Understanding 98

Thinking About Equal Groups Division 101

Students and Teachers Think About the Problems 101

Partitive Division 102

Measurement (Quotitive) Division 103

The Unknowns in Partitive and Measurement Division 104

Matching Models to Contexts 105

Tracking the Unit Whole and the Referent Whole 107

Key Ideas 112

Try It Out! 113

Reflect 115

Chapter 6 Ratios and Rates: Describing Relationships 116

Thinking About Problem Situations With Ratios and Rates 116

Students and Teachers Think About the Problems 120

Defining Terms 121

Four Contexts of Ratios and Proportional Reasoning Problems 123

Associated Sets 124

Well-Chunked Measures 125

Stretchers and Shrinkers 129

Part-To-Whole 132

Translating the Five Representations: Try It Out 138

Key Ideas 141

Try It Out! 142

Reflect 143

Chapter 7 Multiplicative Comparison: More than Scale Factors 144

Thinking About Multiplicative Comparison Situations 144

Students and Teachers Think About the Problems 148

Additive Versus Multiplicative Comparison 149

Constant Ratio 150

Absolute and Relative Change 151

When Multiplication Makes a Quantity Smaller 152

The Dilation Ratio 155

Extending to Negative Values 156

Students and Teachers Think About the Problems 157

Negative Measure Factors and Vectors 159

Negative Scale Factors and Reflections 159

Key Ideas 163

Try It Out! 163

Reflect 165

Chapter 8 Area/Array: Two Factors, Same Job 166

Thinking About Area/Array Situations 166

Students and Teachers Think About the Problems 170

Symmetric Versus Asymmetric Multiplication 171

Representing Fraction and Decimal Division 174

Extending to Negative Values 176

Models of Thinking and Models for Thinking 177

Array as a Mathematical Structure 177

Key Ideas 181

Try it Out! 182

Reflect 184

Chapter 9 Probability and Cartesian Products: Combinatorics 186

Thinking About Combinatorics Situations 186

Students and Teachers Think About the Problems 190

Understanding of Probability is Based in Intuition 191

Experiencing Experimental Outcomes in Game Format 192

Analyzing and Making Sense of Experimental Outcomes 194

Building Intuition With Games 196

Using Models to Predict Number of Outcomes 199

The Path Toward Systematic Thinking 199

Key Ideas 202

Try It Out! 203

Reflect 204

Chapter 10 Changing How you Teach Word Problems 205

Getting Into the Mathematizing Sandbox 205

Eight Shifts in Instruction for Building Students' Problem-Solving Skills 207

Do Word Problems for Sense-Making 207

Treat Context and Computation Separately 208

Create More and Varied Representations 208

Explore All the Work Operations Can Do 209

Add Operation Sense Routines to the School Day 210

Offer Students Experiences With a Variety of Problem Situations 210

Listen to Students and Be Curious 211

Make Time for Mathematizing in the Sandbox 211

Guidance for Moving Forward: FAQs 212

Appendix 216

References 220

Index 224

Editorial Reviews

As our students begin to mathematize the world around them, it becomes extremely important that we listen to their thinking so that we can continue to move their understanding forward. What makes Mathematize It! such a useful tool for teachers is that it thoughtfully unpacks student strategies, which helps inform and guide our next move as a classroom teacher.

Graham Fletcher

This dynamic author trio brings years of classroom experiences to one of the central problems of teaching and learning mathematics: making sense of word problems. Focusing on the construct of ‘mathematizing’—drawing, constructing, describing, representing, and making sense of situations—this clear and practical guide needs to be required reading and discussion fodder for every middle school teacher of mathematics. It’s just that clear, classroom grounded, and insightful!

Steve Leinwand

Mathematize It! is a book that should be on the shelf of every classroom teacher and division leader who supports mathematics teaching and leading. This valuable resource helps educators to think about the what, why, and the how to make sense of word problems. It gives a framework and visuals on how to support teachers’ understanding around problem types and solving problems and excels in assisting teachers in how to make a commitment to teaching for greater understanding.

Spencer Jamieson

This book is a must-have for anyone who has faced the challenge of teaching problem solving. The ideas to be learned are supported with a noticeably rich collection of classroom-ready problems, examples of student thinking, and videos. Problem solving is at the center of learning and doing mathematics. And so, Mathematize It! should be at the center of every teacher’s collection of instructional resources.

John San Giovanni

Mathematize It! is a must-read for anyone who has struggled to teach word problems and is ready to figure out what really works. The authors present a plethora of strategies that help students focus on the thinking part of the problem-solving process while gently helping the reader understand that so many of our ‘tried-and-true’ methods, such as key words, really don’t work. They help us realize that the real work of solving word problems is in the sense-making phase—once students have made sense of a problem, calculating the solution is the simpler part of the process.

Kimberly Rimbey

The authors provide a detailed and practical guide on how to take a word problem, uncover the mathematics embedded in it, carefully consider representations, and use it all to solve the problem. The reader begins to realize that all models are not created equal. The authors’ careful attention to the nuances within mathematical relationships illustrates how mathematizing differs from answer getting, yet shows us that ideas like operation sense and computation are related. The authors’ plain language explanations empower us to leverage those relationships in order to help students become better mathematicians.

Paul Gray

When I read Mathematize It!, I experienced a journey of extending the way I think about mathematical ideas. Readers can develop their mathematical knowledge for teaching by reading this book. Also, with their sandbox approach, the authors describe a problem-solving process that is playful and exploratory. The sandbox process also orients problem solvers toward seeing mathematical structures and relationships. I recommend reading this book to experience deep mathematics and, in turn, learn more about how to support students with engaging in mathematizing.

Amanda Jensen

Mathematize It! is a must have for every middle school teacher! The authors provide clear guidance and suggestions to help our students become effective problem solvers. The opportunities provided to ponder and practice, as well as the student sample work and videos, make this a resource you will grow professionally from and positively impact student learning. I wish I had this resource when I started teaching 25 years ago!

Kevin Dykema

The list of generational math books to come along and truly synthesize what we know so far and what we need to know is a very short and exclusive list. Well, you can confidently add Mathematize It! to this collection. Written by three of the most respected math educators today, the book zeros in on that often poorly traveled journey between the question and answer in problem solving. Mathematize It! will be your go-to resource to install the mathematical play revolution in elementary classes everywhere!

Sunil Singh

Mathematize It! engages readers deeply in the mathematics content through an easy-to-use visual analogy: playing in a sandbox. The authors have found a way to make problem-solving seem like a fun task—one that is akin to something we’ve all been doing forever: playing. Their clever and applicable problem-solving model of thinking provides a structure teachers can use to support students in tackling word problems and actually enjoying the process. It’s time for you to play in the sandbox and more importantly, Mathematize It!

Hilary Kreisberg

Finally! An answer for equipping students in making sense of word problems. Mathematize It! clarifies the challenges in problem solving and gives concrete steps and advice on understanding problem contexts and the mathematics involved. The examples, student work, and videos throughout the book bring ideas to life, and make their implementation doable. This is a must-read for every math teacher who desires their students to truly understand the role of mathematics in the world.

Nanci N. Smith

I can’t wait to use Mathematize It! in my work with teachers and students! The excellent examples, including actual student work and teacher commentaries, highlight the complexity of the problem situations in a way that is clear and usable for classroom teachers and for those of us who support them. The focus on operation sense, understanding the role that each quantity plays, and connecting representations to problems makes this a must read for anyone helping students become successful problem solvers. I especially appreciate the inclusion of non-whole-number examples!

Julie Mc Namara

Mathematize It! addresses the complexity of problem solving more completely than any other individual resource. It is easy to say that we must teach students to ‘mathematize situations’ but this book helps us to actually help students learn to do it. The challenge and reflection pieces at the end of each chapter are a game changer for unveiling teaching opportunities, prompting discussion in your PLC, and moving this from a book on the professional shelf to a powerful tool to impact instruction.

Gina Kilday

This is a game changer … even after 20 years of supporting students and their sensemaking of word problems, I am thrilled to learn even more from this trio of authors. They offer practical suggestions, opportunities for practice, and relevant research in order to increase awareness of best practices surrounding word problems. The only key word in this case is MATHEMATIZE! To have this resource in your hands is to have an invitation to the ‘mathematizing sandbox’.

Beth Terry