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# Mathematize It! [Grades K-2]

** Mathematize It!** shares a reasoning approach that takes the initial focus off specific numbers and computations and put it on the actions and relationships expressed in the word problem.

**Grade Level:**PreK-12**ISBN:**9781544389851**Published By:**Corwin**Series:**Corwin Mathematics Series**Year:**2020**Page Count:**232**Publication date:**May 06, 2020

**Price: $35.95**

## Description

*“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 SanGiovanni

Coordinator, Elementary Mathematics

Howard County Public School System, Ellicott City, MD

**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, which 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 K-2** 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 K-2

· 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 addition and subtraction of whole numbers as well as early thinking for multiplication and division

· Demonstrates the use of concrete manipulatives 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! *

## Author(s)

### Kimberly Morrow-Leong

**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 5â9 classroom teacher, Kâ8 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

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*.

### Linda M. Gojak

**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

**Publisher's Acknowledgements**

**Table of Contents**

**Video List**

**About the Authors**

**Introduction: Why You Need to Teach Students to Mathematize**

**Problem Solving Strategies Gone Wrong**

**What is Mathematizing? Why Is It Important**

Focus on Operation Sense

Using Mathematical Representations

**Teaching Students to Mathematize**

Building your Understanding of Operations and Related Problem Situations

Playing in the Mathematizing Sandbox: A Problem-Solving Model

**Final Words Before You Dive**

**Chapter 2: Moving From Counting to Addition & Subtraction**

**Thinking About Counting, Addition, and Subtraction**

Sandbox Notes: Explore Your Thinking

Students and Teachers Think About the Problem

The Development of Courting

Counting or Add-To?

Representing Problem Situations or Representing Answers

**Using Children's Literature to Explore add-To and Take From Situations**

Transcribe the Action or Relationship

**Moving beyond Twenty**

Students and Teachers Think About the Problems

Representing Problem Situations Multiple Ways

Translating the Five Relationships: Try It Out

Teaching Students to Use Concrete and Pictorial Models

**Key Ideas**

**Try It Out!**

Identify the Principle

Write the Problem

Change It Up

**Reflect**

**Chapter 3: Add-To Problems: Locating the Change**

**Thinking About Active Addition Situations**

**Thinking About Active Addition Situations**

Sandbox Notes: Explore Your Thinking

Students and Teachers Think About the Problems

Finding the Unknown, Three Story Structures

**Story Structures: Implications for Teaching**

**Modeling the Active Problem Situation**

Model Actions

Distinguish Pictures from Pictorial Representation

Identify Important Features

Creatively Meet Students' Needs

**Using Children's Literature to Explore Add-To Problems**

Make Predictions

Transcribe the Action or Relationship

**Digging Deeper into Start Unknown Situations**

Complicating Things: The Start Unknown Variation

Strategies for Making Sense of Start Unknown Problem Situation

Separating Computation from Operation

Moving to Equations

**Key Ideas**

**Try It Out!**

Identify the Problem Situation

Write the Problem

Change It Up

**Reflect**

**Chapter 4: Take-From Problems: Locating the Change**

**Thinking About Active Subtraction Situations**

Sandbox Notes: Explore Your Thinkin

Students and Teachers Think About the Problems

Finding the Unknown, Three Story Structures

Story Structures: Implications for Teaching

Modeling the Active Problem Situation

**Digging Deeper into the Start and Change Unknown Situations**

Moving from Concrete to Symbolic Representation

**Using Children's Literature to Explore Take-From Situations**

Create Another Outcome

Finding the Unknown Quantity

**Key Ideas**

**Try It Out!**

Identify the Problem Situatoon

Write the Problem

Change It Up

**Reflect**

**Chapter 5: Part-Part-While: Understanding the Relationship**

**Thinking About Part-Part-Whole Situations**

Sandbox Notes: Explore Your Thinking

Students and Teachers Think About the Problems

Defining the Part-Part-Whole Situation

Modeling Relationships vs. Action

Moving from Counters to Bar Models

The Special Case of Both Parts Unknown

Modeling Measurement Problems

A Note About the Commutative Property

**Using Children's Literature to Explore part-Part-Whole Situations**

Make Predictions

**Moving to Larger Whole Numbers**

Students and Teachers Think about the Problems

Modeling Measurement Problems

Writing Equations: Addition or Subtraction

Writing Equations: Addition or Subtraction

Finding the Equation in the Model

**Key Ideas**

**Try It Out!**

Identify the Problem Sitaution

Write the Problem

Change It Up

**Reflect**

**Chapter 6: Additive Comparison: Another Kind of Relationship**

**Thinking About Additive Comparison Situation**

Students and Teachers Think about the Problems

Early Years Comparisons

Additive Comparison Situations

**Building Models for Comparisons**

Students and Teachers Think About the Problems

Making Use of Models For Thinking

**The Language of Comparisons**

Language Can Get Tricky

Building Bar Models for Comparisons

Problem Posing as an Instructional Strategy

**Using Children's Literature to Explore Additive Comparison Situations**

Create Another Outcome

**Key Ideas**

**Try It Out**

Identify the Problem Situation

Write a Problem

Change It Up

**Reflect**

**Chapter 7: Early Multiplication and Division: Patterns and Predictions**

**Thinking about Early Multiplicative Thinking**

Sandbox Notes: Explore Your Thinking

Students and Teachers Think About the Problem

Modeling Even and Odd Numbers

Equal Groups Problem Situation

Using Patterns to Identify Even and Odd Numbers

**Using Children's Literature to Explore Early Multiplication**

Transcribe the Action or Relationship

**Bringing Another Dimension with Arrays**

Recognizing Area/Array as a Problem Situation

Building an Understanding of Arrays as Structures

**Using Children's Literature to Explore Arrays**

Transcribe the Action or Relationship

**Setting the Stage Division**

Equipartitioning

Two Models for Division

**Using Children's Literature to Explore Early Division**

Transcribe the Action or Relationship

**Key ideas**

**Try It Out**

Identify the Problem Situation

Write the Problem

Change It Up

**Reflect**

**Chapter 8: Changing How You Teach Word Problems**

**Getting into the Mathematizing Sandbox**

**8 Shifts in Instruction for Building Students' Problem-Solving Skills**

Do Word Problems for Sense-Making

Treat Context and Computation Separately

Create More and Varied Representations

Explore All the Work Operations can Do

Add Operation Sense Routines with a Variety of Problem Situatons

Listen to Students and Be Curious

Make Time for Mathematizing in the Sandbox

**Guidance for Moving Forward: FAQ's**

Finding Unexpected Mathematics in Stories

## Reviews

“One of the most important goals of K–2 math education should be developing a robust understanding of addition and subtraction that can serve as a foundation for later learning. In Mathematize It! Morrow-Leong, Moore, and Gojak help teachers transform solving word problems from an activity focused on getting answers to an activity that supports students’ understanding of the mathematical principles of the operations. Teaching

suggestions are integrated throughout the book in a manner that allows teachers to make adaptations to meet the needs of their students. This book belongs on every primary grade teachers’ bookshelf.”

Linda Levi“One of the most important goals of K–2 math education should be developing a robust understanding of addition and subtraction that can serve as a foundation for later learning. In Mathematize It! Morrow-Leong, Moore, and Gojak help teachers transform solving word problems from an activity focused on getting answers to an activity that supports students’ understanding of the mathematical principles of the operations. Teaching

suggestions are integrated throughout the book in a manner that allows teachers to make adaptations to meet the needs of their students. This book belongs on every primary grade teachers’ bookshelf.”

Director, CGI Math Teacher Learning Center, LLC

“Kim Morrow-Leong, Sara Delano Moore, and Linda Gojak have written an essential guide for teachers of grade K–2 mathematics—both those new to the field and those with many years of experience. Read this book and learn how to help students develop a deep understanding of the mathematical principles behind each operation—in other words, effectively help students learn to mathematize problem situations.”Beth Skipper

Elementary Education Consultant, Reston, VA

“Help your students get into the messy fun of true problem solving, where the goal is to make sense of the world—not just arrive at an answer! Mathematize It! will engage you in the nuance and power of teaching through context. You’ll be surprised and delighted by the student examples, inspired by the teacher commentary, and ready to craft story problems that uncover the big ideas of your grade.”Berkeley Everett

UCLA Mathematics Project (University of California, Los Angeles Mathematics Project)

“Mathematize It! is a must-have resource for teachers and teacher leaders. In this comprehensive resource the authors not only introduce a new verb—mathematize—to the primary teacher’s vocabulary, they also provide teachers with a ‘go-to guide’ for teaching problem solving. Features such as the Mathematizing Sandbox as a problem-solving model, the emphasis on the use of multiple representations, and the integration of children’s literature provide teachers with instructional tools to help students develop a strong operation sense.”Latrenda Knighten

Elementary Mathematics Instructional Specialist, Baton Rouge, LA

“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

Author of Pi of Life: The Hidden Happiness of Mathematics and Math Recess: Playful Learning in an Age of Disruption

“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

National Board Certified Teacher Co-Founder & CEO, KP Mathematics

“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

Math Interventionist and MTSS Coordinator Metcalf Elementary School, Exeter, RI Presidential Award for Excellence in Mathematics and Science Teaching Awardee, Former Member of the NCTM Board of Directors

“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

Past President, Virginia Council for Mathematics Supervision (VCMS), Mathematics Specialist for Virginia Council of Teacher of Mathematics (VCTM)

“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

Mathematics Coach, 2004 Presidential Award for Excellence in Mathematics and Science Teaching Awardee, Riffa Views International School, Bahrain

this clear and practical guide needs to be required reading and discussion fodder for every elementary teacher of mathematics. It’s just that clear, informative, and insightful!”

“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—Steve Leinwand

this clear and practical guide needs to be required reading and discussion fodder for every elementary teacher of mathematics. It’s just that clear, informative, and insightful!”

Principal Researcher American Institutes for Research, Washington, DC

“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

Math Specialist, Atlanta, GA

“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

Director, Center for Mathematics Achievement Lesley University, Cambridge, MA Author of Adding Parents to the Equation

“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

Chief Curriculum Officer, Cosenza & Associates, LLC Past President, Texas Council of Teachers of Mathematics NCTM Representative for NCSM: Leadership in Mathematics Education

“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 McNamara

Associate Professor, Author of Beyond Pizzas & Pies (With Meghan Shaughnessy) and Beyond Invert & Multiply, California State University, East Bay, Hayward, CA

“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 SanGiovanni

Coordinator, Elementary Mathematics, Howard County Public School System, Ellicott City, MD

“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

Associate Professor, Mathematics and Education, Arizona Christian University, Glendale, AZ, Author of Every Math Learner

“Leaving behind the procedures and the gimmicks, Mathematize It! will effectively help develop teachers’ understanding of problem types and the pedagogical techniques necessary to teach students how to solve problems through understanding. Using comprehension strategies and varied representations are key to student success with contextualized computation. The sandbox notes, structured techniques, and videos make this book very teacher friendly. The ideas are easily transferable to the mathematics classroom.”Natalie Crist

Coordinator of Elementary Mathematics, Baltimore County Public Schools

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### Related Resources

- Moving From Counting to Addition and Subtraction [Lessons and strategies]
- Why You Need to Teach Students to Mathematize [Lessons and strategies]