id id

A Series for Teaching Mathematics

Developing Essential Understanding of Addition and Subtraction • Pre-K–Grade 2

Developing Essential Understanding

What is the relationship between addition and subtraction? How do you know
whether an algorithm will always work? Can you explain why order matters in
subtraction but not in addition, or why it is false to assert that the sum of any two
whole numbers is greater than either number?
Developing
Essential
How much do you know … and how much do you need to know?

Helping your students develop a robust understanding of addition and subtraction

requires that you understand this mathematics deeply. But what does that mean?
This book focuses on essential knowledge for teachers about addition and subtrac-
Understanding
tion. It is organized around two big ideas, supported by multiple smaller, intercon- of
nected ideas—essential understandings. Taking you beyond a simple introduction to

Addition &
these operations, the book will broaden and deepen your mathematical understand-
ing of one of the most challenging topics for students—and teachers. It will help
you engage your students, anticipate their perplexities, avoid pitfalls, and dispel
misconceptions. You will also learn to develop appropriate tasks, techniques, and
tools for assessing students’ understanding of the topic.

Subtraction
Focus on the ideas that you need to understand thoroughly to teach confidently.

Move beyond the mathematics you expect your students to

learn. Students who fail to get a solid grounding in pivotal
concepts struggle in subsequent work in mathematics and related
Essential
Understanding
Series
iea
d
disciplines. By bringing a deeper understanding to your teaching, you can help
students who don’t get it the first time by presenting the mathematics in multiple ways.
The Essential Understanding Series addresses topics in school mathematics that are
Pre-K–Grade 2
critical to the mathematical development of students but are often difficult to teach.
Each book in the series gives an overview of the topic, highlights the differences between
what teachers and students need to know, examines the big ideas and related essential
understandings, reconsiders the ideas presented in light of connections with other
mathematical ideas, and includes questions for readers’ reflection.

12-10/2.5/VP
ISBN 978-0-87353-664-6
13792
NCTM

9 780873 536646 13792

 Essential
Understanding
Series
iea
d
Developing
Essential Understanding
of

Addition and Subtraction

for Teaching Mathematics in
Prekindergarten–Grade 2

Janet H. Caldwell
Rowan University
Glassboro, New Jersey

Karen Karp
University of Louisville
Louisville, Kentucky

Jennifer M. Bay-Williams
University of Louisville
Louisville, Kentucky

Ed Rathmell
Volume Editor
University of Northern Iowa
Cedar Falls, Iowa

Rose Mary Zbiek

Series Editor
The Pennsylvania State University
University Park, Pennsylvania

Copyright © 2011 by the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in other formats without written permission from NCTM.
 Copyright © 2011 by
The National Council of Teachers of Mathematics, Inc.
1906 Association Drive, Reston, VA 20191-1502
(703) 620-9840; (800) 235-7566; www.nctm.org
All rights reserved

Library of Congress Cataloging-in-Publication Data

Developing essential understanding of addition and subtraction for teaching math-

ematics in prekindergarten-grade 2 / Janet H. Caldwell ... [et al.].
p. cm.
ISBN 978-0-87353-664-6
1. Addition--Study and teaching (Preschool) 2. Addition--Study and teaching
(Primary) 3. Subtraction--Study and teaching (Preschool) 4. Subtraction--Study and
teaching (Primary) I. Caldwell, Janet H.
QA135.6.D486 2010
372.7’2--dc22
2010036661

The National Council of Teachers of Mathematics is a public voice of mathematics

education, supporting teachers to ensure equitable mathematics learning of the
highest quality for all students through vision, leadership, professional development,
and research.

For permission to photocopy or use material electronically from Developing

Essential Understanding of Addition and Subtraction for Teaching Mathematics
in Prekindergarten–Grade 2, please access www.copyright.com or contact the
Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
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It will be the responsibility of the user to identify such materials and obtain the
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The publications of the National Council of Teachers of Mathematics present

a variety of viewpoints. The views expressed or implied in this publication, unless
otherwise noted, should not be interpreted as official positions of the Council.

Printed in the United States of America

e-Book ISBN: 978-0-87353-798-8
 Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Why Addition and Subtraction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Understanding Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
The Big Ideas and Essential Understandings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Benefits for Teaching, Learning, and Assessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Ready to Begin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Addition and Subtraction: The Big Ideas and Essential Understandings
Representing and Solving Problems: Big Idea 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Building on sequential counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
The inverse relationship of addition and subtraction . . . . . . . . . . . . . . . . . . . . . 14
Representing situations with addition or subtraction . . . . . . . . . . . . . . . . . . . . . 16
Part-part-whole relationships and number sentences . . . . . . . . . . . . . . . . . . . . . 25
Representing situations in different ways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Procedures for Adding and Subtracting: Big Idea 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 28
The commutative and associative properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Subtraction: Neither commutative nor associative . . . . . . . . . . . . . . . . . . . . . . . . 32
Place value and composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Properties and algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Connections: Looking Back and Ahead in Learning
Links to Big Idea 1: Extending Models beyond Whole Numbers . . . . . . . . . . . . . 50
Part–part–whole and part–whole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Discrete and continuous models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A foundation for future mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Links to Big Idea 2: Place Value and Properties across Content . . . . . . . . . . . . . . 54
Place value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Measurement concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Data representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A link to later topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Challenges: Learning, Teaching, and Assessing
Setting the Stage for Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Representations and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
The Structure of Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Key words: A failed shortcut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Reasoning about actions and relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
The Structure of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Using Appropriate Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Student-Invented Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Assessing Evidence of Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Is Subtraction Harder than Addition? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
 v

Foreword
Teaching mathematics in prekindergarten–grade 12 requires a
special understanding of mathematics. Effective teachers of mathe-
matics think about and beyond the content that they teach, seeking
explanations and making connections to other topics, both inside
and outside mathematics. Students meet curriculum and achieve-
ment expectations when they work with teachers who know what
mathematics is important for each topic that they teach.
The National Council of Teachers of Mathematics (NCTM) pres-
ents the Essential Understanding Series in tandem with a call to
focus the school mathematics curriculum in the spirit of Curriculum
Focal Points for Prekindergarten through Grade 8 Mathematics: A
Quest for Coherence, published in 2006, and Focus in High School
Mathematics: Reasoning and Sense Making, released in 2009. The
Essential Understanding books are a resource for individual teach-
ers and groups of colleagues interested in engaging in mathematical
thinking to enrich and extend their own knowledge of particular
mathematics topics in ways that benefit their work with students.
The topic of each book is an area of mathematics that is difficult
for students to learn, challenging to teach, and critical for students’
success as learners and in their future lives and careers.
Drawing on their experiences as teachers, researchers, and
mathematicians, the authors have identified the big ideas that are
at the heart of each book’s topic. A set of essential understandings—
mathematical points that capture the essence of the topic—fleshes
out each big idea. Taken collectively, the big ideas and essential
understandings give a view of a mathematics that is focused, con-
nected, and useful to teachers. Links to topics that students encoun-
ter earlier and later in school mathematics and to instruction and
assessment practices illustrate the relevance and importance of a
teacher’s essential understanding of mathematics.
On behalf of the Board of Directors, I offer sincere thanks
and appreciation to everyone who has helped to make this series
possible. I extend special thanks to Rose Mary Zbiek for her leader-
ship as series editor. I join the Essential Understanding project team
in welcoming you to these books and in wishing you many years of
continued enjoyment of learning and teaching mathematics.

Henry Kepner
President, 2008–2010
National Council of Teachers of Mathematics
 vii

Preface
From prekindergarten through grade 12, the school mathematics
curriculum includes important topics that are pivotal in students’
development. Students who understand these ideas cross smoothly
into new mathematical terrain and continue moving forward with
assurance.
However, many of these topics have traditionally been chal-
lenging to teach as well as learn, and they often prove to be barriers
rather than gateways to students’ progress. Students who fail to get
a solid grounding in them frequently lose momentum and struggle
in subsequent work in mathematics and related disciplines.
The Essential Understanding Series identifies such topics at all
levels. Teachers who engage students in these topics play critical
roles in students’ mathematical achievement. Each volume in the
series invites teachers who aim to be not just proficient but out-
standing in the classroom—teachers like you—to enrich their under-
standing of one or more of these topics to ensure students’ contin-
ued development in mathematics.
How much do you need to know?
To teach these challenging topics effectively, you must draw on a
mathematical understanding that is both broad and deep. The chal-
lenge is to know considerably more about the topic than you expect
your students to know and learn.
Why does your knowledge need to be so extensive? Why must
it go above and beyond what you need to teach and your students
need to learn? The answer to this question has many parts.
To plan successful learning experiences, you need to under-
stand different models and representations and, in some cases,
emerging technologies as you evaluate curriculum materials and
create lessons. As you choose and implement learning tasks, you
need to know what to emphasize and why those ideas are math-
ematically important.
While engaging your students in lessons, you must anticipate
their perplexities, help them avoid known pitfalls, and recognize
and dispel misconceptions. You need to capitalize on unexpected
classroom opportunities to make connections among mathematical
ideas. If assessment shows that students have not understood the
material adequately, you need to know how to address weaknesses
that you have identified in their understanding. Your understanding
must be sufficiently versatile to allow you to represent the math-
ematics in different ways to students who don’t understand it the
first time.
In addition, you need to know where the topic fits in the full
span of the mathematics curriculum. You must understand where
viii Preface

your students are coming from in their thinking and where they are
heading mathematically in the months and years to come.
Accomplishing these tasks in mathematically sound ways is
a tall order. A rich understanding of the mathematics supports the
varied work of teaching as you guide your students and keep their
learning on track.
How can the Essential Understanding Series help?
The Essential Understanding books offer you an opportunity to
delve into the mathematics that you teach and reinforce your con-
tent knowledge. They do not include materials for you to use
directly with your students, nor do they discuss classroom manage-
ment, teaching styles, or assessment techniques. Instead, these books
focus squarely on issues of mathematical content—the ideas and
understanding that you must bring to your preparation, in-class in-
struction, one-on-one interactions with students, and assessment.
How do the authors approach the topics?
Big ideas and For each topic, the authors identify “big ideas” and “essential
essential understandings.” The big ideas are mathematical statements of over-
understandings are arching concepts that are central to a mathematical topic and link
identified by numerous smaller mathematical ideas into coherent wholes. The
icons in the books. books call the smaller, more concrete ideas that are associated with

iea
d
each big idea essential understandings. They capture aspects of the
corresponding big idea and provide evidence of its richness.
The big ideas have tremendous value in mathematics. You
can gain an appreciation of the power and worth of these densely
marks a big idea, packed statements through persistent work with the interrelated
and essential understandings. Grasping these multiple smaller concepts
and through them gaining access to the big ideas can greatly in-
crease your intellectual assets and classroom possibilities.
marks an essential
In your work with mathematical ideas in your role as a teacher,
understanding.
you have probably observed that the essential understandings are
often at the heart of the understanding that you need for present-
ing one of these challenging topics to students. Knowing these ideas
very well is critical because they are the mathematical pieces that
connect to form each big idea.
How are the books organized?
Every book in the Essential Understanding Series has the same
structure:
• The introduction gives an overview, explaining the reasons
for the selection of the particular topic and highlighting some
of the differences between what teachers and students need
to know about it.
• Chapter 1 is the heart of the book, identifying and examining
the big ideas and related essential understandings.
Preface ix

• Chapter 2 reconsiders the ideas discussed in chapter 1 in

light of their connections with mathematical ideas within
the grade band and with other mathematics that the students
have encountered earlier or will encounter later in their study
of mathematics.
• Chapter 3 wraps up the discussion by considering the chal-
lenges that students often face in grasping the necessary
concepts related to the topic under discussion. It analyzes the
development of their thinking and offers guidance for pre-
senting ideas to them and assessing their understanding.
The discussion of big ideas and essential understandings in
chapter 1 is interspersed with questions labeled “Reflect.” It is im-
portant to pause in your reading to think about these on your own
or discuss them with your colleagues. By engaging with the material
in this way, you can make the experience of reading the book par-
ticipatory, interactive, and dynamic.
Reflect questions can also serve as topics of conversation
among local groups of teachers or teachers connected electronically
in school districts or even between states. Thus, the Reflect items
can extend the possibilities for using the books as tools for formal
or informal experiences for in-service and preservice teachers, indi-
vidually or in groups, in or beyond college or university classes.
A new perspective
The Essential Understanding Series thus is intended to support you
in gaining a deep and broad understanding of mathematics that
can benefit your students in many ways. Considering connections
between the mathematics under discussion and other mathematics
that students encounter earlier and later in the curriculum gives the
books unusual depth as well as insight into vertical articulation in
school mathematics.
The series appears against the backdrop of Principles and
Standards for School Mathematics (NCTM 2000), Curriculum Focal
Points for Prekindergarten through Grade 8 Mathematics: A Quest
for Coherence (NCTM 2006), Focus in High School Mathematics:
Reasoning and Sense Making (NCTM 2009), and the Navigations
Series (NCTM 2001–2009). The new books play an important role,
supporting the work of these publications by offering content-based
professional development.
The other publications, in turn, can flesh out and enrich the
new books. After reading this book, for example, you might select
hands-on, Standards-based activities from the Navigations books
for your students to use to gain insights into the topics that the
Essential Understanding books discuss. If you are teaching students
in prekindergarten through grade 8, you might apply your deeper
understanding as you present material related to the three focal
x Preface

points that Curriculum Focal Points identifies for instruction at your

students’ level. Or if you are teaching students in grades 9–12, you
might use your understanding to enrich the ways in which you can
engage students in mathematical reasoning and sense making as
presented in Focus in High School Mathematics.
An enriched understanding can give you a fresh perspective
and infuse new energy into your teaching. We hope that the under-
standing that you acquire from reading the book will support your
efforts as you help your students grasp the ideas that will ensure
their mathematical success.

We appreciate the thoughtful comments and suggestions offered by

the following individuals who reviewed an earlier version of this
book: Linda Coutts, Terry Crites, Corey Drake, Walter Seaman, and
Erica Guenther Steele.
 1

Introduction
This book focuses on ideas about addition and subtraction, includ-
ing connections to many mathematical ideas that are common in
primary school curricula. These are ideas that you must understand
thoroughly and be able to use flexibly to be highly effective in your
teaching of mathematics in prekindergarten through grade 2. The
authors have assumed that you have had a variety of mathematics
experiences that have motivated you to delve into—and move be-
yond—the mathematics that you expect your students to learn.
The book is designed to engage you as a learner of mathemat-
ics, focusing on content and curricular considerations that will bet-
ter equip you to plan and implement lessons and assess your stu-
dents’ learning in ways that reflect the full complexity of addition
and subtraction. A deep, rich understanding of these operations and
their properties will enable you to communicate their influence and
scope to your students, showing them how these ideas relate to the
mathematics that they have already encountered—and will continue
to encounter—throughout their school mathematics experiences.
The understanding of addition and subtraction that you gain
from this focused study thus supports the vision of Principles and
Standards for School Mathematics (NCTM 2000): “Imagine a class-
room, a school, or a school district where all students have access
to high-quality, engaging mathematics instruction” (p. 3). This
vision depends on classroom teachers who “are continually grow-
ing as professionals” (p. 3) and routinely engage their students in
meaningful experiences that help them learn mathematics with
understanding.

Why Addition and Subtraction?

Like the topics of all the volumes in the Essential Understanding
Series, addition and subtraction compose a major area of school
mathematics that is crucial for students to learn but challenging for
teachers to teach. Students in prekindergarten through grade 2 need
to understand these operations well if they are to succeed in these
grades and in their subsequent mathematics experiences. Learners
often struggle with ideas about addition and subtraction. What is
the relationship between addition and subtraction? How do prob-
lem solvers know whether an algorithm will always work? Many
students mistakenly believe that these operations have identical
properties, and they often compute as though there were a universal
associative property. The importance of addition and subtraction
and the challenge of understanding them well make them essential
for teachers of prekindergarten through grade 2 to understand in
great depth.
2 Introduction

Beyond having a solid understanding of addition and subtrac-

tion—operations that are complex in and of themselves—you must
also know how concepts and properties of addition and subtraction
relate to other mathematical ideas that your students will encounter
in the lesson at hand, the current school year, and beyond. Addition
and subtraction are essential to understand deeply because they
are foundational to many other mathematical concepts within the
prekindergarten–grade 2 curriculum and beyond. Having a rich un-
derstanding of addition and subtraction will influence your instruc-
tional decisions in such areas as the following:
✓ Selecting tasks for a lesson
✓ Developing key questions to pose to students
✓ Choosing materials for exploring relevant mathematics
✓ Ordering topics and ideas over time
✓ Assessing the quality of students’ work
✓ Devising ways to challenge and support students’ thinking
This checklist, although not exhaustive, can serve as a self-
assessment or guide as you read. Asking yourself how the particular
mathematical idea under discussion can influence your work with
students will maximize the impact of this book and provide a clear
answer to the question, “Why develop essential understanding of
addition and subtraction?”

Understanding Addition and

Subtraction
Teachers teach mathematics because they want others to understand
it in ways that will contribute to success and satisfaction in school,
work, and life. Helping your young students develop a robust and
lasting understanding of addition and subtraction requires that you
understand this mathematics deeply. But what does this mean?
It is easy to think that understanding an area of mathemat-
ics, such as addition and subtraction, means only knowing certain
facts, being able to use basic techniques to solve particular types
of problems, and mastering relevant vocabulary. For example, at
the primary level, you are expected to know such facts as “addition
of whole numbers is a commutative operation.” You are expected
to be skillful in solving problems that involve multiple steps. Your
mathematical vocabulary is assumed to include such terms as sum,
difference, inverse, fact family, and whole number.
Obviously, facts, vocabulary, and techniques for solving certain
types of problems are not all that you are expected to know about
addition and subtraction. For example, in your ongoing work with
Introduction 3

students, you have undoubtedly discovered that you need to distin-

guish among different addition and subtraction strategies, such as
knowing the difference between “counting all” and “counting on.”
It is also tempting to focus only on a long list of mathematical
ideas that all teachers of mathematics in prekindergarten through
grade 2 are expected to know and teach about addition and sub-
traction. Often state mathematics standards are examples of such
lists. However important the individual items might be, these lists
cannot capture the essence of a dynamic understanding of the topic.
Understanding these operations deeply requires that you not only
know important mathematical ideas but also recognize how these
ideas relate to one another. For example, knowing the properties
of addition is important, but so is realizing how they are applied in
algorithms and generalizations that students generate (for example,
that + 9 is the same result as + 10 – 1). Your dynamic understand-
ing continues to grow with experience and as a result of opportuni-
ties to embrace new ideas and find new connections among familiar
ones.
Furthermore, your understanding of addition and subtraction
should transcend the content intended for your students. In some
cases, it is clear why you need to know more mathematics than
your students. For instance, your understanding of how addition
and subtraction properties connect with multiplication and division
properties—mathematics that your students will encounter later—can
influence your choice of instructional tasks and the terminology
that you use. You might have your students explore when commu-
tativity and associativity hold (with addition, but not subtraction)
and why this is the case—foundations that they can later build on
and extend by exploring with multiplication and division.
Other differences between the understanding that you need to
have and the understanding that you expect your students to ac-
quire are less obvious, but your experiences in the classroom have
undoubtedly made you aware of them at some level. For example,
students commonly ask, “Can I do it this way?” or “Does this al-
ways work?” or “Is this the same as that?” They naturally attempt
to make generalizations or connections. In such instances, how
many times have you been grateful to have an understanding of
addition and subtraction that enables you to recognize the merit
in a student’s unanticipated mathematical question or claim? How
many other times have you wondered whether you could be missing
an opportunity because of a gap in your knowledge?
As you have almost certainly discovered, knowing and being
able to do familiar mathematics are not enough when you’re in
the classroom. You also need to be able to identify and justify or
refute novel claims—or, better yet, assist students in thinking about
these claims in guided conversations. Such claims and justifications
4 Introduction

might draw on ideas or techniques that are beyond the mathemati-

cal experiences of your students and current curricular expectations
for them. For example, you may need to be able to refute the often-
asserted, erroneous claim that the sum of any two whole numbers
is greater than either of the two addends (take, for example, 6 + 0),
or you may need to explain to a student why order matters in sub-
tracting whole numbers but not in adding them.

The Big Ideas and Essential

Understandings
Thinking about the many particular ideas that are part of a rich
understanding of addition and subtraction can be an overwhelming
task. Articulating all of those mathematical ideas and their connec-
tions would require many resources. To choose which ideas to in-
clude in this book, the authors considered a critical question: What
is essential for teachers of mathematics in prekindergarten through
grade 2 to know about addition and subtraction to be effective in
the classroom? To answer this question, the authors drew on a va-
riety of resources, including personal experiences, the expertise of
colleagues in mathematics and mathematics education, the wisdom
of classroom teachers, and the reactions of reviewers and profes-
sional development providers, as well as ideas from curricular mate-
rials and research on mathematics learning and teaching.
As a result, the mathematical content of this book focuses
on essential ideas for teachers about addition and subtraction. In
particular, chapter 1 is organized around two big ideas related
to this important area of mathematics. Each of these big ideas is
supported by more specific mathematical ideas called essential
understandings.

Benefits for Teaching, Learning, and

Assessing
Understanding addition and subtraction can help you implement
the Teaching Principle enunciated in Principles and Standards for
School Mathematics. This Principle sets a high standard for instruc-
tion: “Effective mathematics teaching requires understanding what
students know and need to learn and then challenging and support-
ing them to learn it well” (NCTM 2000, p. 16). As in teaching about
other critical topics in mathematics, teaching about addition and
subtraction requires knowledge that goes “beyond what most teach-
ers experience in standard preservice mathematics courses” (p. 17).
Chapter 1 comes into play at this point, offering an overview
of addition and subtraction that is intended to be more focused
Introduction 5

and comprehensive than many discussions of the topic that you are
likely to have encountered. This chapter enumerates, expands on,
and gives examples of the big ideas and essential understandings
related to the two operations and their properties, with the goal of
supplementing or reinforcing your understanding. Thus, chapter 1
aims to prepare you to implement the Teaching Principle fully as
you provide the support and challenge that your students need for
robust learning about addition and subtraction.
Consolidating your understanding in this way also prepares
you to implement the Learning Principle outlined in Principles and
Standards: “Students must learn mathematics with understanding,
actively building new knowledge from experience and prior knowl-
edge” (NCTM 2000, p. 20). To support your efforts to help your
students learn about addition and subtraction in this way, chapter
2 extends your understanding of these operations by identifying
specific ways in which the big ideas and essential understandings
connect with mathematics that students typically encounter earlier
or later in school. This chapter supports the Learning Principle by
emphasizing longitudinal connections. For example, as their math-
ematical experiences expand, students gradually develop an under-
standing of the connections between addition of whole numbers
and addition of decimals, and they can use various representations
as needed.
The understandings that chapters 1 and 2 convey can strength-
en another critical area of teaching. Chapter 3 addresses this area,
building on the first two chapters to show how an understanding of
addition and subtraction can help you select and develop appropri-
ate tasks, techniques, and tools for assessing your students’ under-
standing of addition and subtraction. An ownership of the big ideas
and essential understandings related to addition and subtraction,
reinforced by an understanding of students’ past and future experi-
ences with these ideas, can help you ensure that assessment in your
classroom supports the learning of significant mathematics.
Such assessment satisfies the first requirement of the
Assessment Principle set out in Principles and Standards (NCTM
2000): “Assessment should support the learning of important math-
ematics and furnish useful information to both teachers and stu-
dents” (p. 22). An understanding of addition and subtraction can
also help you satisfy the second requirement of the Assessment
Principle, by enabling you to develop assessment tasks that give
you specific information about what your students are thinking and
what they understand. For example, the following task challenges
students to think about the meaning of the equals sign and the
meaning of an equation in a task like the following:
What number do you think belongs in the box?
8+4=☐+5
6 Introduction

This kind of task taps children’s thinking and provides information

that can point to better ways to support their learning.

Ready to Begin
The introduction has painted the background, preparing you for
the big ideas and associated essential understandings that you will
encounter in chapter 1. Reading the chapters in the order in which
they appear can be a very useful way to approach the book. Read
chapter 1 in more than one sitting, allowing time for thoughtful
examination of the Reflect questions embedded in the text. Absorb
the ideas—both big ideas and essential understandings—related to
addition and subtraction. Appreciate the connections among these
ideas. Discuss them with colleagues. Carry your newfound or rein-
forced understanding to chapter 2, which guides you in seeing how
the ideas related to addition and subtraction are connected to the
mathematics that your students have encountered earlier or will en-
counter later in school. Then read about teaching, learning, and as-
sessment issues in chapter 3 and test these ideas in your classroom.
Alternatively, you may want to take a look at chapter 3 before
engaging with the mathematical ideas in chapters 1 and 2. Reading
the book in this way, with the challenges of teaching, learning, and
assessment clearly in mind, along with possible approaches to them,
will enable you to gain a different perspective on the material in
the earlier chapters. Use the checklist provided earlier as a guide for
thinking about new knowledge you have acquired across these
dimensions of teaching.
No matter how you read the book, let it serve as a tool to
expand your understanding, application, and enjoyment of addition
and subtraction.
 1
Chapter

Addition and Subtraction:

The Big Ideas and Essential
Understandings
Young children begin learning mathematics before they enter
school. They learn to count, and they can solve simple problems by
counting. In the primary grades, mathematics instruction focuses
on the development of number sense, understanding of numerical
operations, and fluency in performing computations. Principles and
Standards for School Mathematics (National Council of Teachers of
Mathematics [NCTM] 2000) describes the development of these skills
and concepts, asserting that by the end of grade 2, students should—
• understand numbers, ways of representing numbers, relation-
ships among numbers, and number systems;
• understand meanings of operations and how they relate to
one another;
• compute fluently and make reasonable estimates. (NCTM
2000, p. 78)
Curriculum Focal Points for Prekindergarten through Grade 8
Mathematics: A Quest for Coherence (NCTM 2006) continues to
emphasize the importance of developing both conceptual under-
standing and procedural understanding of addition and subtraction.
Building on Principles and Standards, Curriculum Focal Points rec-
ommends that instruction focus on developing this understanding
throughout the early grades. Activities in kindergarten should center
on joining and separating sets:
Children use numbers, including written numerals, to repre-
sent quantities and to solve quantitative problems, such as . . .
modeling simple joining and separating situations with objects.
They choose, combine, and apply effective strategies for an-
swering quantitative questions, including quickly recognizing
the number in a small set, counting and producing sets of
given sizes, counting the number in combined sets, and counting
backward. (NCTM 2006, p. 12)

7
8 Addition and Subtraction

In grade 1, instruction should focus on developing students’ un-

derstanding of addition and subtraction as well as related facts and
strategies associated with these operations:
Children develop strategies for adding and subtracting whole
numbers on the basis of their earlier work with small numbers.
They use a variety of models, including discrete objects, length-
based models (e.g., lengths of connecting cubes), and number
lines, to model “part-whole,” “adding to,” “taking away from,”
and “comparing” situations to develop an understanding of the
meanings of addition and subtraction and strategies to solve
such arithmetic problems. Children understand the connections
between counting and the operations of addition and subtrac-
tion (e.g., adding two is the same as “counting on” two). They
use properties of addition (commutativity and associativity) to
add whole numbers, and they create and use increasingly so-
phisticated strategies based on these properties (e.g., “making
tens”) to solve addition and subtraction problems involving ba-
sic facts. By comparing a variety of solution strategies, children
relate addition and subtraction as inverse operations. (NCTM
2006, p. 13)
In grade 2, the instructional focus should shift to helping students
develop quick recall of addition and related subtraction facts, as
well as fluency with multi-digit addition and subtraction:
Children use their understanding of addition to develop quick
recall of basic addition facts and related subtraction facts. They
solve arithmetic problems by applying their understanding of
models of addition and subtraction (such as combining or sep-
arating sets or using number lines), relationships and proper-
ties of number (such as place value), and properties of addition
(commutativity and associativity). Children develop, discuss,
and use efficient, accurate, and generalizable methods to add
and subtract multidigit whole numbers. They select and apply
appropriate methods to estimate sums and differences or cal-
culate them mentally, depending on the context and numbers
involved. They develop fluency with efficient procedures, in-
cluding standard algorithms, for adding and subtracting whole
numbers, understand why the procedures work (on the basis
of place value and properties of operations), and use them to
solve problems. (NCTM 2006, p. 14)
This chapter discusses in detail the mathematical concepts that
these publications outline. In addition, it relates various mathemati-
cal processes to these concepts by exploring—
• situations for which addition and subtraction can be used to
solve problems;
The Big Ideas and Essential Understandings 9

• ways to represent addition and subtraction;

• ways to reason with addition and subtraction; and
• connections and relationships among these and other
mathematical topics.
It also examines numerical relationships that arise from studying
multiple representations and the reasoning required for the mean-
ingful use and understanding of computational algorithms, written
and mental, standard and nonstandard. The representations have
been chosen primarily for their usefulness in illustrating the math-
ematical concepts. Most of the early examples use counters, since
these constitute the most elementary representation, but later dis-
cussions involve the use of other representations for addition and
subtraction, such as the number line, a hundreds chart, and base-
ten place-value blocks.
“Unpacking” ideas related to addition and subtraction is a
critical step in establishing deeper understanding. To someone with-
out training as a teacher, these ideas might appear to be simple to
teach. But those who teach young students are aware of the subtle-
ties and complexities of the ideas themselves and the challenges of
presenting them clearly and coherently in the classroom. Teachers
of young students also have an idea of the overarching importance
of addition and its inverse operation, subtraction:
Overarching idea: Addition and its inversely related operation,
subtraction, are powerful foundational concepts in mathemat-
ics, with applications to many problem situations and connec-
tions to many other topics. Addition determines the whole in
terms of the parts, and subtraction determines a missing part.
This overarching idea anchors teachers’ understanding and
their instruction. It incorporates two big ideas about addition
and subtraction that are crucial to understand. The first relates to
when to use each operation, and the second deals with how to get
answers efficiently and accurately. Each of these two big ideas in-
volves several smaller, more specific essential understandings.
These big ideas and essential understandings are identified here
as a group to give you a quick overview and for your convenience
in referring back to them later. Read through them now, but do not
think that you must absorb them fully at this point. The chapter
will discuss each one in turn in detail.
10 Addition and Subtraction

iea
d Big Idea 1. Addition and subtraction are used to represent
and solve many different kinds of problems.

Essential Understanding 1a. Addition and subtraction of whole

numbers are based on sequential counting with whole numbers.

Essential Understanding 1b. Subtraction has an inverse relation-

ship with addition.

Essential Understanding 1c. Many different problem situations

can be represented by part-part-whole relationships and
addition or subtraction.

Essential Understanding 1d. Part-part-whole relationships can

be expressed by using number sentences like a + b = c or
c – b = a, where a and b are the parts and c is the whole.

Essential Understanding 1e. The context of a problem situation

and its interpretation can lead to different representations.

Big Idea 2. The mathematical foundations for understand-

iea
d ing computational procedures for addition and subtraction of
whole numbers are the properties of addition and place value.

Essential Understanding 2a. The commutative and associative

properties for addition of whole numbers allow computations
to be performed flexibly.

Essential Understanding 2b. Subtraction is not commutative or

associative for whole numbers.

Essential Understanding 2c. Place-value concepts provide a

convenient way to compose and decompose numbers to
facilitate addition and subtraction computations.

Essential Understanding 2d. Properties of addition are central in

justifying the correctness of computational algorithms.
The Big Ideas and Essential Understandings 11

Representing and Solving Problems:

Big Idea 1
Big Idea 1. Addition and subtraction are used to represent and solve
many different kinds of problems. iea
d
Many different types of problems can be represented by addition
or subtraction. It is important to learn how to recognize these situ-
ations and represent them symbolically, building on counting with
whole numbers. By understanding these situations and their repre-
sentations well, teachers can provide students with many different
examples of addition and subtraction problems. The discussion be-
low of Big Idea 1 presents and examines fifteen examples that illus-
trate situations that can be represented by addition or subtraction.

Building on sequential counting

Essential Understanding 1a. Addition and subtraction of whole
numbers are based on sequential counting with whole numbers.

Situations that can be represented by addition or subtraction can

be considered as basic applications of counting forward or back. For an extended
Even very young children can solve simple addition and subtrac- discussion of
tion story problems by counting concrete objects (e.g., Starkey and counting strategies
Gelman 1982; Carpenter and Moser 1983). They establish a one- and number ideas,
to-one correspondence by moving, touching, or pointing to each see Developing
object that they are counting as they say the corresponding number Essential Under-
words. The following two examples demonstrate how counting re- standing of Number
lates to addition and subtraction situations. and Numeration
for Teaching
Example 1 lends itself to a number of simple counting
Mathematics in Pre-
strategies:
kindergarten–Grade
Example 1: Max has 2 apples. He picks 5 more. How many 2 (Dougherty et al.
apples does Max have now? 2010).

This problem can be represented with concrete objects by first plac-

ing 2 counters (the quantity that Max starts with) and then placing
a second group of 5 more counters. Figure 1.1 illustrates the two
groups of counters.

Fig. 1.1. Counters representing 2 and 5

12 Addition and Subtraction

A variety of counting strategies might be used to find the total

number of counters:
• Count all: Count each of the counters: 1, 2 [pause] 3, 4, 5, 6, 7.
• Count on from the first number: A more efficient way to find
the total is to count on, beginning with the first quantity
given in the problem (in this case, 2): 2 [pause], 3, 4, 5, 6, 7.
• Count on from the larger number: A still more efficient way
to find the total is to count on, beginning with the larger
number (5, in this case) and counting on the smaller number
(2): 5, [pause] 6, 7.
Reflect 1.1 explores possible ways to use counters with these
strategies.

Reflect 1.1
What counters might a child point to as she uses each of the counting strategies
shown above?
Does a child need all of the counters for counting on?

Each of the “counting on” strategies is more efficient when

problem solvers recognize the first number that they use without
counting. This process reduces the difficulty of many tasks and is
frequently useful in playing games, counting coins, or other simple
everyday tasks. Recognizing patterns on number cubes and dominoes,
such as in figure 1.2 is particularly helpful.

Fig. 1.2. Number cubes and dominoes

Relating numbers to the benchmark quantities 5 and 10 helps

students see the relative sizes of numbers and can therefore support
their transition from counting to later work in addition and subtrac-
tion. In the five-frame on the left in figure 1.3, we not only recog-
nize the three counters without counting, but we also note without
counting that there are two empty spaces, so 3 is 2 less than 5, or
3 + 2 = 5, or 5 – 2 = 3. In the ten-frame on the right, we see that
6 is 1 more than 5, or 6 = 5 + 1, and that 6 is 4 less than 10, or
6 + 4 = 10, or 6 = 10 – 4.
The Big Ideas and Essential Understandings 13

Fig. 1.3. Five-frame and ten-frame representing 3 and 6, respectively

Working with dot patterns, whether on a ten-frame or a num-
ber cube, can help students recognize the number of objects without
counting. Some representations are more useful for building recog-
nition of multiples—especially doubles. The arrangement of six dots
on a number cube is similar to the arrangement of counters on the
ten-frame shown in figure 1.4. This arrangement leads to thinking
of 6 as two rows of 3, or 6 = 3 + 3. Reflect 1.2 explores extending
this thinking to other arrangements of dots.

Fig. 1.4. Another way to show 6

Reflect 1.2
What number relationships might students perceive from the standard
arrangements of dots on a number cube?

Example 2 lends itself to a different counting strategy:

Example 2: Sari has 5 apples. Three are red. The rest are
yellow. How many of Sari’s apples are yellow?
One way of using counting to solve this problem is to lay out 5
counters, separate (perhaps by circling) the 3 that represent red
apples, and then count the remaining counters. Figure 1.5 depicts
this situation.

Fig. 1.5. Counters representing 5 with 3 as one part

We might also solve this problem by “counting on.” We could
lay out 3 counters for the 3 red apples and then count on until we
had counters for 5 apples, as shown in figure 1.6.
14 Addition and Subtraction

3 4 5

Fig. 1.6. Counting on to represent 5 with 3 as one part

Alternatively, we might represent the problem by “counting
back.” We could start with 5 counters and then count back 3 for the
3 red apples, as illustrated in figure 1.7. Consider the question in
Reflect 1.3 to compare “counting on” and “counting back.”

5 4 3 2

Fig. 1.7. “Counting back” to represent 5 with 3 as one part

Reflect 1.3
Why is “counting back” so much more difficult than “counting on”?

The inverse relationship of addition and

subtraction
Essential Understanding 1b. Subtraction has an inverse relationship
with addition.

The chart in figure 1.8 shows the input and the output for the
algebraic rule “add 2.” The output number is always two more than
the input number.

Rule: Add 2
Input Output
1 3
5 7
8 10
11 13

Fig. 1.8. Input/output table

The relationship in figure 1.8 is a function and can be repre-
sented by a function machine, as shown in figure 1.9. We can re-
verse the action of adding 2 by subtracting 2.
The Big Ideas and Essential Understandings 15

For a discussion
of the inverse
x Add 2 x+2
relationship
between
Fig. 1.9. A function machine for “add 2” multiplication
and division, see
Example 2, about Sari’s apples, showed a problem situation Developing Essential
that some people would represent by addition, while others would Understanding of
use subtraction. The “counting on” subtraction strategy described Multiplication and
above is grounded in the fact that 5 – 3 = ☐ is equivalent to 5 = 3 Division for Teach-
+ ☐. The result of subtracting b from a, a – b, is formally defined ing Mathematics in
as the number y where a = b + y. This definition builds logically Grades 3–5 (Otto
et al. 2011).
on what students already know about addition, demonstrating why
some problems can be solved by either operation.
Understanding the relationship between addition and subtrac-
tion reduces the number of facts that students must “know” by
giving them a consistent, reliable strategy for subtraction: use the
related addition fact. These related facts then form fact families.
The third column in figure 1.10 shows a more formal algebraic
description of a fact family.

2+3=5 4+4=8 b+y=a

3+2=5 y+b=a
5–2=3 8–4=4 a–b=y
5–3=2 a–y=b

Fig. 1.10. Examples of fact families

The language describing subtraction is often very difficult for For a discussion of
using appropriate
students to comprehend and use correctly. We may read the expres-
terminology and
sion 5 – 3 in many ways. Thinking about this expression in terms
representing word
of parts and wholes may be helpful, since 5 (the minuend) is the problems with
whole, and 3 (the subtrahend) is a part. “Five minus 3” is the way multiplication and
that many adults would read the expression. Students might read division, see
it as “5 take away 3,” but they might also say it as “3 taken away Developing Essential
from 5.” The same expression might be read either as “5 subtract Understanding of
3” or as “3 subtracted from 5.” It also might be read as “5 less 3” Multiplication
or as “3 less than 5.” Note that in these phrases, the order of the and Division for
numbers shifts, and some expressions include a preposition (from Teaching
or than). It is very difficult for children to distinguish differences Mathematics in
among the meanings of these phrases, and this confusion leads Grades 3–5 (Otto
et al. 2011).
them to make frequent reversal errors. Chapter 3 describes strategies
for helping students make sense of actions in word problems and
the language of addition and subtraction.
16 Addition and Subtraction

Representing situations with addition or

subtraction
Essential Understanding 1c. Many different problem situations can
be represented by part-part-whole relationships and addition or
subtraction.

Part-part-whole relationships show how two numbers—the parts—

are related to a third number—the whole. For example, 2 and 3
are parts of the whole, 5. Addition and subtraction situations have
been analyzed extensively and categorized in several different ways
(see, for example, Carpenter [1985] and Schifter, Bastable, and
Russell [2000]). The Common Core State Standards for Mathematics
(Common Core State Standards Initiative 2010) include a table
(adapted from the National Research Council [2009, pp. 32, 33];
see fig. 1.11) showing one way of categorizing different types of
problems.
The first two types of problem situations identified in the
table—“add to” and “take from”—involve actions. “Adding to” prob-
lems involve increasing by joining, while “taking from” problems
involve decreasing or separating. Each of these situations can be
further categorized by considering what information must be found
(the result of the action, the change, or the start). Other problem
situations do not change the amounts in any set; these “no action”
situations may involve putting together collections of objects, tak-
ing apart a collection of objects, or comparing two collections of
objects.
This section considers each of these types of situations by
using part-part-whole representations. Reflect 1.4 invites further
thinking about such situations.

Reflect 1.4
How is the language used to describe a situation related to its concrete
representation?
Why is it important to understand what seem to be subtle distinctions in word
problems?

Examples 1 and 2 have already illustrated two different types

of problem situations, one of which was an action situation and the
other of which was not.
• In example 1, Max began with 2 apples and then picked 5
more. The concrete representation shows a group of 2 counters
and then a group of 5 counters. This “add to” action problem
involves an initial quantity (start) and then an action that
The Big Ideas and Essential Understandings 17

Fig. 1.11. Common addition and subtraction situations. Adapted from the Common Core
State Standards for Mathematics (Common Core State Standards Initiative 2010, p. 88,
from the National Research Council [2009, pp. 32, 33]).

joins to or increases that quantity, with the result unknown. It

also indicates a part-part-whole relationship in which one part
is the initial quantity and the other part is the quantity that is
added (the change). The whole is the unknown result in this
case. The common symbolic representation for this problem
involves addition: 2 + 5 = ☐.
• In example 2, Sari has 5 apples, 3 of which are red. The con-
crete representation of this problem begins with 5 counters,
which are then considered as two parts. This part-part-whole
problem involves no action; it is static, with no apples added
or taken away. The whole and one of the parts are given. The
other part is what must be found. This take-apart problem
might be considered to be a missing addend problem, because
18 Addition and Subtraction

it can be represented as 5 = 3 + ☐. Note that writing the

number sentence in this order models the situation directly
and helps students understand the meaning of the equals sign
as indicating that both sides represent the same amount. This
missing addend problem situation can also be represented as
subtraction: 5 – 3 = ☐.
The next five examples illustrate different types of “action”
problems. Examples 3 and 4 involve “adding to” by joining or in-
creasing, while examples 5–7 involve “taking from” by separating
or decreasing. Sample situations of all types (that is, with the result,
the change, or the start unknown) are provided.
Example 3 presents one type of “add to” situation:
Example 3: Juanita has 2 cookies. How many more cookies
does she need to have 5 cookies?
Making a concrete representation of this problem involves begin-
ning with 2 counters and then adding counters by a “counting on”
strategy until there are 5 counters, as shown in figure 1.12. This
“add to” problem involves an initial quantity (start) and then an
action that increases that quantity; it entails joining an unknown
number of cookies (the change) on to the starting set of 2 cookies.
The initial quantity, or start (2), and the unknown change are the
parts, and the result (5) is the whole. This problem is also a miss-
ing addend problem, which can be represented by either addition or
subtraction: 2 + ☐ = 5 or 5 – 2 = ☐.

2 3 4 5

Fig. 1.12. Counters representing counting on from 2 to

find a total of 5
Example 4 illustrates a different category of “add to” problem:
Example 4: Jose had some marbles. Angel gave him 3 marbles.
Now he has 5 marbles. How many marbles did Jose have to
start with?
Representing this “add to” problem concretely is difficult, since the
initial quantity (start ) is unknown. One way to model the problem
concretely is to begin with several counters, perhaps in a container,
and then make a separate group of 3 counters (to represent the
marbles that Angel gives Jose), as shown in figure 1.13.
The Big Ideas and Essential Understandings 19

Fig. 1.13. Counters representing an unknown amount and 3 more

After Jose receives 3 marbles from Angel, he has 5 marbles in
all. So, to continue to model the problem, we might add counters
from the first group to the ones in the second group until we reach
5, as shown in figure 1.14.

Fig. 1.14. Counters representing an unknown start and 3, with a

whole of 5
Consider another concrete model, which requires working back-
ward from the whole. This model involves setting out 5 counters (to
represent Jose’s total number of marbles) and then considering the
parts, as suggested in figure 1.15 by the circling of 3 (to represent
Jose’s marbles from Angel) as one part of the whole. This “join”
problem involves an unknown start (a part), an action increasing
that quantity (a change of 3, or a part), and a result (the whole, 5).
It is a missing addend problem, which can be represented either as
addition or subtraction: ☐ + 3 = 5, or 5 – 3 = ☐.

Fig. 1.15. Counters representing 5, with 3 as one part

Example 5 turns the discussion to subtraction situations:
Example 5: Ben has 5 cookies. He eats 2 cookies. How many
cookies does Ben have left?
This “take-away,” or “separation,” situation is familiar and easy
to represent concretely. We might simply lay out 5 counters and
then remove 2 of them. This action is frequently shown pictorially
by crossing out counters, as illustrated in figure 1.16.
20 Addition and Subtraction

Fig. 1.16. Counters representing 5 take away 2

Note that in our model, we can remove any two counters,
not just the ones on the right (or the left). Figure 1.17 shows an
alternative way in which we could remove two counters.

Fig. 1.17. Another way to show 5 take away 2

This “take from” problem involves an initial quantity, or start,
which is the whole. Then the problem describes a decrease, which
takes away, or separates, a part, and the problem solver must find
the missing part, or result. The situation depicted is a part-part-
whole situation, with one part as the unknown. Notice that this
subtraction story still depicts a part-part-whole relationship and
still is in the form of start–action or change–result, but now the
start is the whole, and the result is a part. This action situation is
most often represented by subtraction: 5 – 2 = ☐.
Example 6 illustrates a different type of “take from” situation:
Example 6: Mary has 5 flowers. She gives some to Anya. She
has 2 flowers left. How many flowers did Mary give to Anya?
To model this “take from” problem concretely, we might start with
5 counters (see fig. 1.18a) and remove counters until only 2 are left
(see fig. 1.18b). Like example 5, this problem involves a start, or
initial quantity, which is the whole, but then it describes decreasing
that quantity by taking away or separating an unknown part (the
change) to leave 2 counters (the result, or remaining part). This ac-
tion situation can be represented most directly from the
context and modeled as 5 – ☐ = 2. The addition number sentences
5 = 2 + ☐ and 5 = ☐ + 2 are equivalent mathematically but are
rarely used in this context.
Example 7 illustrates the third type of “take from” situation:
Example 7: EJ has some books. He gives 2 books away. He has
3 left. How many books did EJ have to start with?
To represent this “take from” situation concretely, we might set out
one group of 2 counters and another group of 3 counters and then
combine these parts to find the whole. This interpretation of the
The Big Ideas and Essential Understandings 21

(a)

(b)

Fig. 1.18. Counters showing (a) a start of 5 and (b) a result of 2 left
after an action (“take away”) separates some from 5
situation does not correspond to the way in which the situation is
described, however, and thus it can be particularly difficult for stu-
dents to understand. It requires working backward, putting together
rather than taking away. To make a more basic concrete representa-
tion, we might start with a pile of counters and separate 2 from it
(see fig. 1.19a). Then we might take counters away from the pile
until only 3 are left (see fig. 1.19b).

(a)

(b)

Fig. 1.19. Counters representing taking 2 from an unknown quantity,

with 3 remaining
This is a “take away,” or “separate,” action situation in which
the start, or initial quantity, is unknown. The action involves de-
creasing the initial quantity by 2, leaving a result of 3. The situation
may be represented symbolically as ☐ – 2 = 3, or as 2 + 3 = ☐.
Note that the first number sentence is tied more closely to the con-
text and model. Reflect 1.5 draws attention to how numbers can be
used in different types of problems.

Reflect 1.5
Using 2, 7, and 9, write “add to” and “take from” problems with the result
unknown, the change unknown, and the start unknown.

Six of the first seven examples have included an action (“add

to” or “take from”) and have illustrated how the unknown can be
22 Addition and Subtraction

either a part or a whole in addition and subtraction stories. Not

all addition and subtraction situations involve adding to or taking
away, however, as example 2 demonstrated. The next five examples
illustrate other no-action contexts for addition and subtraction.
In these situations, the classification of the unknown as the start,
change, or result is not appropriate. Deeply understanding addition
and subtraction involves recognizing that we can use one or both of
these operations to represent situations such as the following, while
also realizing that an equation such as 7 – 3 = ☐ can represent very
different situations. Students need to be able to model each type of
situation directly, but they do not need to know the labels for each
type of problem.
The situation in example 8 involves putting collections of ob-
jects together:
Example 8: Ming has 3 red balls and 2 blue balls. How many
balls does he have in all?
A concrete representation of this problem would involve setting out
two groups of counters (the parts), with the unknown whole. This
“putting together” situation can be represented by the number sen-
tence 3 + 2 = ☐. In this situation, the total is unknown.
Example 9 presents a situation that involves a comparison of
collections of objects:
Example 9: Karen has 5 books. Jenny has 2 books. How many
more books does Karen have than Jenny?
This “comparing” situation might be represented concretely, as
shown in figure 1.20, by setting out 5 counters to show Karen’s
books and 2 counters to show Jenny’s books, and then matching
them up. In this case, the difference is unknown. The larger quantity
(the number of Karen’s books) can be considered as the whole in
this problem, with the number of books that Jenny has as one part
and the number of other books (the difference) as the other part.
The situation can be represented symbolically by using either sub-
traction or addition: 5 – 2 = ☐ or 2 + ☐ = 5.

Karen’s books

Jenny’s books

Fig. 1.20. Counters representing 5 compared with 2

The situation in example 10 also involves comparing:
Example 10: What is the length, in inches, of the line segment
in figure 1.21?
The Big Ideas and Essential Understandings 23

In this situation, we can find the length of the line by determining

the distance between 2 and 5 on the number line, or the number of
units between those two numbers. We can consider the situation of
finding this difference as a comparison situation, in which we know
the whole (where the segment ends on the right), and we know one
part (where the segment starts on the left), and we must find the
part between the two (the difference). We can represent this situa-
tion symbolically by using either addition or subtraction: 2 + ☐ = 5
or 5 – 2 = ☐.

2 3 4 5

Fig. 1.21. Length as a comparison situation

The idea of using a number line to represent comparison situa-
tions is valuable. Reflect 1.6 explores possibilities.

Reflect 1.6
Think of some other situations involving comparisons.
How are these situations like finding the length of an object by using a ruler?
How are these situations like finding the difference between two numbers by
using a number line?

Example 11 shows a different type of comparison situation:

Example 11: Jim has 2 marbles. Sarah has 3 more than Jim.
How many marbles does Sarah have?
This example presents a comparison situation in which the dif-
ference is known and the larger quantity is unknown. Figure 1.22
shows that this situation can be represented concretely by laying
out 2 counters in one row (the starting part) and then setting out
counters to match the first 2 and 3 more (the other part).

Jim’s marbles

Sarah’s marbles

Fig. 1.22. Counters representing 3 more than 2

To answer the question, we must find the number of marbles
that Sarah has (the whole). The number of marbles that Jim has
composes one part, and the number of extra marbles that Sarah has
24 Addition and Subtraction

composes the other part. The usual symbolic representation is the

number sentence 2 + 3 = ☐, but the situation can also be repre-
sented by subtraction as ☐ – 2 = 3.
Example 12 shows yet another type of comparison situation:
Example 12: Ellen has 5 dolls. She has 2 more than Cindy.
How many dolls does Cindy have?
Like the situation in example 11, this is also a comparison situation
in which the difference is known, but this time the smaller quantity
is unknown. One way to represent this situation concretely is shown
in figure 1.23, which presents two rows of 5 counters, with the dif-
ference (2) removed from Cindy’s set of dolls. If the larger number
of dolls is considered as the whole, then the difference is one part,
and the other part is the number of dolls Cindy has.

Ellen’s dolls

Cindy’s dolls

Fig. 1.23. Counters representing 2 less than 5

Note that an alternative way to model this situation concretely
Essential might simply be to use one set of 5 counters (Ellen’s dolls) and
Understanding 1d then remove the difference. (Alternative representations will be
Part-part-whole discussed further in the next section, in connection with Essential
relationships can be Understanding 1d.) The symbolic representation might be either
expressed by using subtraction (5 – 2 = ☐) or addition (5 = ☐ + 2).
number sentences Each of the twelve examples presented so far illustrates a
like a + b = c or slightly different problem situation, but each can be interpreted as a
c – b = a, where a part-part-whole relationship. Each can be modeled both concretely
and b are the parts and pictorially. Each can also be represented symbolically by using
and c is the whole. either addition or subtraction.

Part-part-whole relationships and number

sentences
Essential Understanding 1d. Part-part-whole relationships can be
expressed by using number sentences like a + b = c or c – b = a,
where a and b are the parts and c is the whole.

In each of the examples discussed up to this point, we have seen

that we can describe the problem situation as involving two parts
that make up one whole. In each case, we have found that addition
or subtraction (or both) is an appropriate operation to use to solve
the problem. Each of these examples also meets two other impor-
tant criteria:
The Big Ideas and Essential Understandings 25

1. The parts are disjoint; that is, they do not overlap with each For more on these
other. ideas about
2. The parts are exhaustive; that is, the whole contains no ob- composing and
decomposing
jects other than the parts.
quantities and
The next two examples show situations that may not satisfy these part-whole
criteria. relationships, see
Consider the seemingly simple situation in example 13: Developing Essential
Understanding
Example 13: Two families are going to the shore. There are of Number
5 people in one family and 6 people in the other family. How and Numeration
many people are going to the shore? for Teaching
Mathematics in
Most people encountering this problem would think that 11 people
Prekindergarten–
are going to the shore, as shown in figure 1.24a, but this may not Grade 2 (Dougherty
be the correct answer. Suppose that one of the families includes a et al. 2010).
daughter of the other. Then, since the two groups overlap, as in fig-
ure 1.24b, we cannot just add to find the total. When adding or sub-
tracting, the parts must be non-overlapping, or disjoint, sets. Each
part is a set, and the union of these sets is the whole.

5 6 4 1 5

(a) (b)

Fig. 1.24. Showing two families as (a) disjoint groups and

(b) overlapping groups
Example 14 presents a situation involving possible arrange-
ments or combinations of parts in the whole:
Example 14: Hal has 5 balls. Some are red, and some are blue.
How many balls of each color might Hal have?
In this situation, Hal may have some yellow or green balls, or balls
of other colors, in addition to the red and blue ones. However, if we
assume that Hal has only red and blue balls, then we must find all
of the possible ways to “make 5.” Looking at the number 5 as the
whole and determining all of the ways in which 5 might be split, or
decomposed, into two parts, as shown below, lays the foundation
for understanding addition and subtraction:

5=4+1 5=3+2 5=2+3 5=1+4

Beyond knowing the different ways to decompose a particular num-
ber, such as 5, lies knowing what to expect when we decompose
any number, as suggested in Reflect 1.7.
26 Addition and Subtraction

Reflect 1.7
What patterns do you see in the number sentences:
5 = 4 + 1, 5 = 3 + 2, 5 = 2 + 3, and 5 = 1 + 4?
Why is it important to include both 5 = 4 + 1 and 5 = 1 + 4?
List all of the ways to make 10 as the sum of two counting numbers.
How many ways are there?

Representing situations in different ways

Essential Understanding 1e. The context of a problem situation and
its interpretation can lead to different representations.

In most of the examples discussed previously, more than one

number sentence is a reasonable choice for representing a problem
situation symbolically. Often, the more appropriate choice depends
on the way in which a problem situation is interpreted.
Consider the situation in example 15:
Example 15: Ana has 8 postcards to mail and only 6 postcard
stamps. How many more stamps does she need?
One interpretation of this problem might be that Ana has 6 stamps
and needs to have 8, so what number does she need to add to 6 to
get 8? Under this interpretation, the situation represents a “putting
together” part-part-whole relationship, where one part consists of
the 6 stamps that Ana has, and the other part consists of the num-
ber of stamps that she needs. Ana’s situation can thus be interpret-
ed as a missing addend situation that leads to the number sentence
6 + ☐ = 8. Alternatively, this situation might be interpreted as a
one-to-one correspondence, or a comparison situation: how many
more than 6 is 8? In this case, the appropriate symbolic representa-
tion might be 8 – 6 = ☐. A child who does subtraction by thinking
about a related addition fact might find the missing addend rep-
resentation to be more useful than the equivalent subtraction one.
Either addition or subtraction is an appropriate way to express a
part-part-whole relationship.
In some of the number sentences discussed above, the whole
appeared on the left side of the equals sign rather than the right
side. It does not matter on which side of the equals sign we place
the whole, since the expressions on each side are equivalent to
each other. Try to suspend this understanding temporarily to take a
child’s perspective as you respond to the question in Reflect 1.8.
The Big Ideas and Essential Understandings 27

Reflect 1.8
Many children think that the equals sign always comes before the answer.
How might such a child complete the following number sentence:
2 + 6 = ☐ + 4?
28 Addition and Subtraction

Procedures for Adding and

Subtracting: Big Idea 2

iea
d Big Idea 2. The mathematical foundations for understanding
computational procedures for addition and subtraction of whole
numbers are the properties of addition and place value.

Whether we compute by using traditional paper-and-pencil algo-

rithms, mental math, estimation, or invented algorithms, the proper-
ties of addition, along with place value, provide the basis for our
understanding of each procedure. Examination of the essential un-
derstandings related to Big Idea 2 highlights this fact.

The commutative and associative properties

Essential Understanding 2a. The commutative and associative
properties for addition of whole numbers allow computations to be
performed flexibly.

In example 14, discussed earlier in connection with Essential

Essential Understanding 1d, Hal has 5 balls, some of which are red and some
Understanding 1d of which are blue. Our list of possible combinations of red and blue
Part-part-whole balls shows that the expressions 1 + 4 and 4 + 1 can have different
relationships can be interpretations in this case, since one number represents the number
expressed by using of red balls, and the other, the number of blue balls. More generally,
number sentences however, it does not matter whether we put the red balls or the blue
like a + b = c or
ones first, since 1 + 4 gives the same result as 4 + 1. The commuta-
c – b = a, where a
tive property for addition of whole numbers is stated generally as
and b are the parts
and c is the whole. a + b = b + a. Although it may not be necessary for students to state
the commutative property by using variables or even to recognize it
by that name, it is essential that they be able to explain and use the
property in solving problems, as illustrated in the situation with the
balls.
The commutative property simplifies addition greatly, espe-
cially as children are learning their number facts. Because 3 + 8 is
the same as 8 + 3, they can find the sum by using the “counting
on” strategy, starting with the larger number. Use of the commuta-
tive property cuts the number of addition facts that children must
memorize from 100 to 55. The addition table in figure 1.25 illus-
trates this fact; it is symmetric about the diagonal (shaded), and the
answers (or wholes) for a + b and b + a appear directly across the
diagonal from each other.
Understanding and being able to use the commutative property
for addition are also very important in doing mental math with
larger numbers. For example, to find 17 + 135 mentally, starting
The Big Ideas and Essential Understandings 29

0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

4 4 5 6 7 8 9 10 11 12 13

5 5 6 7 8 9 10 11 12 13 14

6 6 7 8 9 10 11 12 13 14 15

7 7 8 9 10 11 12 13 14 15 16

8 8 9 10 11 12 13 14 15 16 17

9 9 10 11 12 13 14 15 16 17 18

Fig. 1.25. Addition facts table

with 135 and then counting on 17 is much easier than starting with
17 and then counting on 135.
Example 16 illustrates the usefulness of another property of
addition:
Example 16: Mrs. Brown rolls three number cubes each day
for her students to add. Today she rolled the numbers 5, 6, and
4. What is the sum?
One way to find the sum is to add the numbers in order (see fig.
1.26). Another way is to add the last two numbers first to make 10
(see fig. 1.27). The associative property for addition allows us to re-
group addends; it is generally stated formally as follows:
a + (b + c) = (a + b) + c
Being able to use the associative property fluently is important in
developing good number sense.
In developing an understanding of the counting numbers, stu-
dents learn that a number can be decomposed as a sum of two parts
in many ways; for example, 8 = 5 + 3, 8 = 4 + 4, and 8 = 2 + 6.
Combining this notion of decomposing a number with the com-
mutative and associative properties is foundational to most addition
and subtraction fact strategies. One strategy that children use for
30 Addition and Subtraction

5 6 4 5 + 6 + 4 = (5 + 6) + 4

11 = 11 + 4

15 = 15

Fig. 1.26. Adding (5 + 6) + 4

5 6 4 5 + 6 + 4 = 5 + (6 + 4)

10 = 5 + 10

15 = 15

Fig. 1.27. Adding 5 + (6 + 4)

the addition of small numbers is the “make 10” strategy. To model
this strategy concretely, consider the numbers 8 and 5, as shown on
ten-frames in figure 1.28.

Fig. 1.28. Modeling counting on a ten-frame

To find the sum of 8 and 5, we can split 5 into 2 + 3. We can
move 2 of the counters on the right to the ten-frame on the left.
Then, as shown in figure 1.29, we can see that 8 + 2 is 10, and 10 +
3 = 13. Representing our model formally in writing, we can see the
For a discussion of
the commutative decomposing of 5, followed by the use of the associative property
and associative for addition:
properties of 8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 10 + 3 = 13.
multiplication and
division and their
roles in computa-
tion strategies, see
Developing Essential
Understanding of
Multiplication Fig. 1.29. Using the “make 10” strategy
and Division
for Teaching We can use a similar strategy to simplify computations with
Mathematics in larger numbers. For example, one way of using the commutative
Grades 3–5 (Otto and associative properties of addition to find 78 + 26 involves
et al. 2011). working with 100 as a benchmark number by grouping 75 and 25:
The Big Ideas and Essential Understandings 31

78 + 26 = (3 + 75) + (25 + 1) Decomposition of each addend

= 3 + (75 + (25 + 1)) Associative property for addition
= 3 + ((75 + 25) + 1) Associative property for addition
= 3 + (100 + 1) Addition
= 3 + (1 + 100) Commutative property for
addition
= (3 + 1) + 100 Associative property for addition
= 4 + 100 = 104 Addition

Alternatively, we might use 80 as a benchmark number:

78 + 26 = 78 + (2 + 24) Decomposition of the second addend
= (78 + 2) + 24 Associative property for addition
= 80 + 24 Addition
= 80 + (20 + 4) Decomposition of the second addend
= (80 + 20) + 4 Associative property for addition
= 100 + 4 = 104 Addition

In fact, we can find many ways to obtain the sum 78 + 26 by work-

ing with multiples of 10 or 100 as benchmark numbers and us-
ing the properties of addition. Reflect 1.9 explores the use of this
strategy.

Reflect 1.9
Find at least two ways in which the commutative and associative properties of
addition can help you find each of the following sums:
97 + 105 347 + 454

The associative property is also the basis for the strategy of

using known facts to find sums of one-digit numbers. This strategy
is, in fact, a more general version of the “make 10” strategy but can
also be used with other facts. For example, suppose that we need to
find 7 + 8, and we know that 7 + 4 = 11. Then we can break 8 into
4 + 4 and add each part separately:
7 + 8 = 7 + (4 + 4) = (7 + 4) + 4 = 11 + 4 = 15
Another type of problem that students frequently solve by using
this strategy involves “near doubles”—two addends that nearly
duplicate each other. Because students usually remember addition
facts for doubles, such as 7 + 7 = 14, they can use these facts to
help find other sums. For example,
7 + 8 = 7 + (7 + 1) = (7 + 7) + 1 = 14 + 1 = 15.
32 Addition and Subtraction

The associative property for addition can also be extremely

helpful to students in grade 3 or 4 when they are adding a column
of several numbers. Consider the following sum:

44
65
36
77
28
+82

Students can use the “make 10” strategy to look for number combi-
nations in the ones column that add to 10:
4 + 6 = 10 and 8 + 2 = 10
Note that this approach to the digits in the ones column requires the
use of both the associative and commutative properties:

(4 + 5 + 6) + 7 + (8 + 2) = (4 + (6 + 5)) + 7 + 10
= ((4 + 6) + 5) + 7 + 10
= 10 + (5 + 7) + 10
= 10 + (12 + 10)
= 10 + (10 + 12) = (10 + 10) + 12 =
20 + 12 = 32

Continuing to the tens column in the same problem, students can

use an analogous “make 100” process, first looking for combina-
tions that make 100: 40 + 60 = 100, 30 + 70 = 100, and 20 + 80 =
100. They will now have 300 + 32 = 332. Note that the “make 100”
process illustrated here has many variations, all of which are
equally correct and valid.

Subtraction: Neither commutative

nor associative
Essential Understanding 2b. Subtraction is not commutative or
associative for whole numbers.

Subtraction does not have the same properties as addition. Consider

the two expressions 5 – 2 and 2 – 5. Although both involve finding
the distance between 2 and 5, they do not have the same meaning.
We might read the first as “5 minus 2” or “the difference between
5 and 2,” and the second as “2 minus 5” or “the difference between
2 and 5.” In the first case, we might interpret the number sentence
to mean, “Start with 5 counters, and then take away 2,” leaving 3
counters. And in the second case, we might interpret the sentence
The Big Ideas and Essential Understandings 33

2 – 5 in the same fashion to mean, “Start with 2 counters and then

take away 5.” Even young children can understand that in this situ-
ation, they can take away 2 but then still need to take away 3 more;
they learn much later that doing so is possible, but only by expand-
ing to negative numbers, when they discover that the correct an-
swer is –3. Subtraction does not have a commutative property; the
order of the terms in a subtraction expression is important. Reflect
1.10 investigates a closely related false claim that students frequent-
ly make about subtraction.

Reflect 1.10
What misunderstandings might students have if they make the assertion,
“You can’t take a bigger number from a smaller one?”

Subtraction also does not have an associative property. Because

a property must hold for every possible combination of numbers,
we can show that subtraction is not associative by producing a sin-
gle example that does not work. Consider the following two ways in
which we might use subtraction with the numbers 15, 8, and 5:
(15 – 8) – 5 = 7 – 5 = 2
15 – (8 – 5) = 15 – 3 = 12
The values of the two expressions (15 – 8) – 5 and 15 – (8 – 5) are
not the same, so subtraction is not associative.
The “make 10” strategy, shown earlier as an application of the
associative property of addition of whole numbers, can be applied
to subtraction. To find 15 – 7, we can start with 15, shown with
counters on two ten-frames as in figure 1.30a. By thinking of 7 as
5 + 2, we first take 5 away and have 10 left (fig. 1.30b). Then we
take away 2 more counters (fig. 1.30c): 10 – 2 = 8.

(a)

(b)

(c)

Fig. 1.30. Using the “make 10” strategy for 15 – 7

34 Addition and Subtraction

To show formally and symbolically why the “make 10” strat-

egy works with subtraction requires that we use integers and their
properties, which are familiar to us, if not to our students. Note that
such a justification is not appropriate for children in the primary
grades, since it requires the use of negative numbers. In this justifi-
cation, we subtract a quantity by adding the inverse, or opposite, of
the number; we will call this procedure the “definition of subtrac-
tion.” The opposite of a number is also equal to the product of the
number and –1: –x = –1(x). This property is called “multiplication
by –1” in the analysis below.

c – (a + b) = c + –(a + b) Definition of subtraction

= c + (–1)(a + b) Multiplication by –1
= c + ((–1) • a + (– 1) • b) Distributive property
for multiplication over
addition
= (c + (–1 • a)) + (–1 • b) Associative property for
addition
= (c – a) – b Multiplication by –1

Thus, 15 – 7 = 15 – (5 + 2) = (15 – 5) – 2 = 10 – 2 = 8.
As in the case of addition, we can extend the “make 10”
strategy to a “make 100” or “make 1000” or “make a multiple of 10”
strategy to subtract multi-digit numbers. Consider an example:
1002 – 17 = 1002 – (2 + 15) = (1002 – 2) – 15 = 1000 – 15 = 985
Solving mentally involves decomposing 17 as 2 + 15 and then dis-
tributing the multiplication by –1.

Place value and composition

Essential Understanding 2c. Place-value concepts provide a
convenient way to compose and decompose numbers to facilitate
addition and subtraction computations.
For more on ideas
about place
value, see Composing and decomposing numbers involves combining and
Developing Essential separating them to make parts and wholes. In our base-ten number
Understanding of system, when we accumulate 10 units, we can regroup them into a
Number and larger unit. Ten ones make 1 ten, and 10 tens make 1 hundred, and
Numeration so on. The size of each new group determines the new place value.
for Teaching A digit thus represents a different quantity, depending on its loca-
Mathematics in tion in a number. The digit 2 represents 2 ones in 562, 2 tens in
Prekindergarten– 526, and 2 hundreds in 256.
Grade 2 (Dougherty Place value can be very helpful when we are using mental
et al. 2010). math to add and subtract numbers. For example, if we need to find
The Big Ideas and Essential Understandings 35

46 + 30, it is helpful to recognize that we are simply adding 3 tens

to 46. We might then count on by 10s from 46: 56, 66, 76.
Similarly, to find 46 + 32, we might decompose each number
into tens and ones. When doing mental math, many people work
from left to right. We might think as follows:
46 + 32 = (40 + 6) + (30 + 2) = (40 + 30) + (6 + 2) = 70 + 8 = 78
Estimation relies heavily on place-value concepts. To estimate
a sum or difference, we want to replace the given numbers with
numbers that are close to them but allow us to do the computation
mentally. Thus, to find 479 – 186, we might round each number to
the nearest hundred: 500 – 200 = 300. We might round the numbers
to the nearest ten, but 480 – 190 is difficult to compute mentally, so
we might think of this as
480 – 200 + 10 = 280 + 10 = 290.
We might also recognize that 480 – 190 is very close to either
490 – 190 or 480 – 180. In each case, our understanding of place
value is critical in thinking about the numbers flexibly and decom-
posing and composing them to simplify computation. Reflect 1.11
invites you to consider possibilities.

Reflect 1.11
Find these sums and differences by composing and decomposing the numbers
in different ways:
48 + 23 93 – 38
376 + 127 738 – 129

Properties and algorithms

Essential Understanding 2d. Properties of addition are central in
justifying the correctness of computational algorithms.

Procedures for finding sums and differences of multi-digit numbers

are all based on the properties of addition, whether they involve
mental math, estimation, standard paper-and-pencil algorithms, or
other algorithms. The discussion of Essential Understanding 2d that
follows first considers addition and subtraction on the number line
and hundreds chart, and then it examines various addition and sub-
traction algorithms.
Working with a number line
Two useful models for developing understanding of addition and
subtraction are a number line and a hundreds chart. In each case,
36 Addition and Subtraction

For a discussion of students extend their use of the “counting on” and “counting back”
using the properties strategies to add and subtract two-digit numbers. Although a num-
of multiplication and ber line is not necessary for such reasoning, it can be helpful in
division to justify visualizing the actions of addition or subtraction. For example, as
the correctness of shown in figure 1.31, to add 38 + 16 on the number line, a student
algorithms, see might first add 38 + 10, then add 2 to get to a “nice” number (a
Developing Essential
multiple of 10), and finally add 4.
Understanding of
Multiplication
and Division for +10 +2 +4
Teaching
Mathematics in
Grades 3–5 (Otto et 38 48 50 54
al. 2011).
Fig. 1.31. A number line representing 38 + 16 as 38 + 10 + 2 + 4
Another way for a student to find this sum might be to begin
at 38, add 2 to get to a “nice” number, and finish by adding 14, as
illustrated in figure 1.32. In each case, the student is using decom-
position and the associative property for addition as he or she takes
the numbers apart and then puts them back together in different
ways.

+2 +14

38 40 54

Fig. 1.32. Another number line representation for 38 + 16

as 38 + 2 + 14
We can subtract either by counting back or counting on. To
find 62 – 25, one person might start at 62 and count back to sub-
tract 20, and then count back to subtract 2 to get to a multiple of
10, and finally count back to subtract 3. An illustration of this
strategy appears in figure 1.33.

–3 –2 – 20

37 40 42 62

Fig. 1.33. A number line representing counting back to find 62 – 25

Another person might first mark both numbers on the number
line and then start at 25 and count on, as illustrated in figure 1.34.
In using this strategy, it is critical to keep track of the differences
between the intermediate numbers. In the case shown in the figure,
the difference is 5 + 30 + 2 = 37.
The Big Ideas and Essential Understandings 37

+5 +30 +2

25 30 60 62

Fig. 1.34. A number line representation of counting on to find 62 – 25

Working with a hundreds chart

A hundreds chart (or 0–99 chart) like that shown in figure 1.35
shows place-value relationships by lining up the digits in the ones
column. It is different from the number line in that it shows only
the whole numbers, with no spaces between for fractions or deci-
mals. By using the hundreds chart to add and subtract, students can
build a visual model that can be very helpful in doing mental math.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Fig. 1.35. A hundreds chart

For example, to find 38 + 16, we can use the “counting on”
strategy, beginning with the larger number, 38. As shown in figure
1.36, we might first add 10 by moving down one row, and then
count on 2 to get to 50, and then count on 4 to finish up.
To find 62 – 25, we can either count back or count on. To
count back, we start at 62 and count back 25, as shown in figure
1.37. To do so, we can count back 2 rows to 42 and then count
back 5 ones to 37.
38 Addition and Subtraction

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

Fig. 1.36. A hundreds chart representing 38 + 16

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

Fig. 1.37. A hundreds chart representing 62 – 25 by counting back

To count on, we might start at 25, move down 3 rows to 55,

and then count on 7 more to get to 62, as illustrated in figure 1.38.
The difference is 30 + 7, or 37.

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

Fig. 1.38. A hundreds chart showing 62 – 25 by counting on

Alternatively, we might start at 25, move down 4 rows to 65

and then count back 3 to get to 62. The difference in this case is
40 – 3, or 37, and we are actually doing the following computation:
25 + 40 – 3 = 65 – 3 = 62. Reflect 1.12 invites an analysis of this
work with hundreds charts and number lines.
The Big Ideas and Essential Understandings 39

Reflect 1.12
What role do the properties of addition play in the computations performed
on a number line and a hundreds chart?

The standard algorithm for paper-and-pencil addition

Place-value blocks provide a concrete way to represent the addition
process known as “regrouping.” Again consider the case of
38 + 16, as illustrated in figure 1.39. First, we lay out blocks to
show 38 as 3 tens and 8 ones, and 16 as 1 ten and 6 ones. Next,
since we have more than 9 ones, we add these together to make 14,
or 1 ten (shown by the circled ones in the figure) and 4 ones. We re-
cord a 4 in the ones column of our addition, as shown on the right
in the figure. Then we record the 1 ten as a 1 above the tens place
in the number 38 in our computation, and we add the tens:
1 + 3 + 1 = 5. We record a 5 in the tens column, for a sum of 54.

1
38
+16 8 + 6 = 14 ones
54 or 4 ones + 1 ten
1 + 3 + 1 = 5 tens

Fig. 1.39. Place-value blocks modeling the standard algoritm for

addition
This process can be formally justified by using the commuta-
tive and associative properties:
38 + 16 = (30 + 8) + (10 + 6) Place value
= 30 + (8 + 10) + 6 Associative property for
addition (twice)
= 30 + (10 + 8) + 6 Commutative property for
addition
= (30 + 10) + (8 + 6) Associative property for
addition (twice)
= 40 + 14 Addition
= 40 + (10 + 4) Place value
= (40 + 10) + 4 Associative property for
addition
= 50 + 4 Addition
= 54 Place value
40 Addition and Subtraction

Note that this formal justification is intended for teachers; students

are unlikely to use this language or degree of formality in explaining
their work.

Other algorithms for addition

Another method for adding multi-digit numbers uses very much the
same properties as those shown formally above but does not
involve regrouping at all. This method, the “partial sums” algo-
rithm, adds each place value separately and then adds the sums.
This procedure may begin either on the left, as in examples (a) and
(b) in figure 1.40, or on the right, as in example (c).

2385
128
1239
38 258
+4528
+16 +691
22
40 900
130
+14 160
1000
54 + 17
+7000
1017
8152

(a) (b) (c)

Fig. 1.40. Adding by using the partial sums algorithm

Yet another addition algorithm that students sometimes learn is
called the “opposite change” (or “give and take” or “add-subtract”)
algorithm. In this procedure, the goal is to change one of the num-
bers to a multiple of 10 (or 100 or 1000) so that it is easy to add.
We can accomplish this by adding a number to one of the addends
and subtracting it from the other. This is similar to the “make 10”
strategy discussed earlier, as demonstrated in the following simple
example:
9 + 7 = (9 + 1) + (7 – 1) = 10 + 6 = 16
Figure 1.41 shows a slightly more advanced example; 4 is subtract-
ed from the first number and added to the second to make a mul-
tiple of 10. A still more complex example in figure 1.42 uses two
modifications instead of just one to determine the sum.

38 –4 34
+16 +4 +20
54

Fig. 1.41. Subtracting and adding 4 to simplify addition

The Big Ideas and Essential Understandings 41

+2 +70
428 430 500
–2 –70
+576 +574 +504
1004

Fig. 1.42. Using two steps in the “opposite change” algorithm

Why does this procedure give a correct answer? Consider the
general case of a + b, where we add and subtract a number c :

(a + c ) + (b – c ) = a + (c + (b – c )) Associative property for

addition
= a + ((c + b) + (–c )) Associative property for
addition and definition of
subtraction
= a + ((b + c) + (–c )) Commutative property for
addition
= a + (b + (c + (–c )) Associative property for
addition
= a + (b + 0) Additive inverse for addition
=a+b Addition identity

Two properties of addition are used here. The additive inverse For a discussion of
property for addition states that any number x has an opposite, –x, using properties of
such that the sum of the two numbers x + (–x) = x – x = 0. Later, multiplication and
students will paraphrase this property by saying, “The sum of any division to justify
number and its opposite is zero.” In the primary grades, however, alternative strategies,
since students do not use negative numbers, the idea is usually see Developing
Essential
stated simply as, “Any number minus itself is zero.” The addition
Understanding of
identity is zero, the number that can be added to any other number
Multiplication
without changing its value. We usually write this property of addi- and Division for
tive identity symbolically as 0 + x = x = x + 0. Teaching
Students first use these properties as they learn their addition Mathematics in
and subtraction facts, and then they continue to use them in the Grades 3–5 (Otto
later grades when they solve equations and simplify algebraic et al. 2011).
expressions. Reflect 1.13 explores the use of the partial sums and
opposite change algorithms.

Reflect 1.13
Use the “partial sums” and “opposite change” algorithms to find the following
sums:
36 + 58 296 + 375 1357 + 8642
42 Addition and Subtraction

The standard algorithm for subtraction

Place-value blocks may be useful in illustrating the algorithm for
subtraction that is most widely used at the present time. In using
this procedure, we begin computing with the ones column and
move to the left, regrouping as needed. An example appears in
figure 1.43.

Representation with Symbolic

Place-Value Blocks Representation Description in Words
Represent 42 with
42 place-value blocks.
–1 7

Because we need to
3 take away 7 ones but
4 12 only have 2 ones, we
–1 7 regroup 1 ten as 10
ones, so now we have
3 tens and 12 ones.
3 Subtract the ones.
4 12
–1 7
5

31
Subtract the tens.
4 2
–1 7
2 5

Fig. 1.43. Place-value blocks modeling the standard algorithm

for subtraction
This procedure, like the standard algorithm for addition, can be
justified by using the properties of addition and place value:
42 – 17 = (30 + 12) – (7 + 10) Facts and place value
= (30 + 12) – 7 – 10 c – (a + b) = c – a – b
(see discussion of Essential
Understanding 2c)
= (30 + 12) + (–7) + (–10) Definition of subtraction
= 30 + (12 + (–7)) + (–10) Associative property for
addition
= 30 + 5 + (–10) Addition
= 30 + (–10) + 5 Commutative property for
addition
= 30 – 10 + 5 = 25 Addition, subtraction, and
definition of subtraction
The Big Ideas and Essential Understandings 43

With larger numbers, the standard algorithm usually alternates be-

tween regrouping each place value and subtracting it. A variation
of this algorithm performs all regrouping first, beginning with the
ones column, as illustrated in figure 1.44a, and then subtracts each
place value, as illustrated in figure 1.44b.
Essential
Understanding 2c
6 11 10 11 6 11 10 11
Place-value concepts
7 2 1 1 7 2 1 1 provide a convenient
–3 6 5 4 –3 6 5 4 way to compose and
3 5 5 7 decompose numbers
to facilitate addi-
(a) (b) tion and subtraction
computations.
Fig. 1.44. Varying the standard algorithm by first (a) regrouping all
place values, and then (b) subtracting

Other algorithms for subtraction

Many other algorithms exist for subtraction. Some are used in
other cultures today, some were widely taught in the United States,
Canada, or Mexico in the past, and others are the invention of stu-
dents, past or present. Several of these alternative algorithms for
subtraction are useful to highlight.
The first of these algorithms, often known as the “partial
differences” algorithm, has two variations. In its basic form, this
algorithm proceeds from left to right, with users subtracting each
place value one by one. This procedure requires very good mental
math skills, especially in making multiples of 10, 100, and 1000.
Figure 1.45 shows the procedure for the problem 438 – 172. Each
step appears in the center, with two different ways of recording the
steps on each side. Some very able students (and adults) do these
computations mentally, recording only the result.

438 438
–172 –100
438 – 100 = 338 338
338
–72 338 – 70 = 268 –70
268 268 – 2 = 266 268
–2 –2
266 266

Fig. 1.45. Using the partial differences algorithm to

subtract 438 – 172
The second variation on the partial differences algorithm
sometimes uses negative numbers. Users of this procedure compute
each place value one at a time, beginning on the left. Figure 1.46
illustrates their process. If the top digit is larger than the bottom
44 Addition and Subtraction

one (as in the ones column in the figure), then they add the differ-
ence onto the partial difference. If the top digit is smaller than the
bottom one (as in the tens column), then they subtract the differ-
ence from the partial difference.

438
–172
400 – 100 = 300
300 Have 30, need to subtract 70,
–40 so subtract 40 from previous difference: 300 – 40 = 260;
260 8 – 2 = 6, so need to add 6.
+6
266
Fig. 1.46. A variation on the partial differences algorithm

It may be easier to see how this algorithm works by looking at

it horizontally:
438 – 172 = (400 + 30 + 8) – (100 + 70 + 2)
= (400 – 100) + (30 – 70) + (8 – 2)
= 300 – 40 + 6
= 266
The second step involves using the definition of subtraction and the
associative and commutative properties for addition as well as the
distributive property. Because this algorithm begins to lay a founda-
tion for negative numbers and provides ways to subtract a larger
digit from a smaller one, it is attractive to middle grades teach-
ers. However, although students sometimes invent the algorithm
on their own, it is rarely taught as the preferred algorithm for all
students.
The next algorithm builds on the procedure for using a number
line or hundreds chart for subtracting. This “counting on” procedure
uses “nice” numbers to count up from the subtrahend (the part, or
the number, being subtracted) to the minuend (the whole). This al-
gorithm is also often called the “cashier’s algorithm,” since it offers
an efficient way of making change. The following subtraction prob-
lem offers an example:
4000 – 2356 Start at 2356 and add 4 to get 2360 4
Add 40 to get 2400 40
Add 600 to get 3000 600
Add 1000 to get 4000 1000
Add the partial differences 1644

Note that this procedure does not require any regrouping. Reflect
1.14 explores the two alternative subtraction algorithms discussed
so far.
The Big Ideas and Essential Understandings 45

Reflect 1.14
Use the “partial differences” and “counting on” algorithms to find each of the
following differences:
73 – 38 642 – 135
702 – 187 2305 - 1268

Another algorithm, often called the “Austrian method,” is

dominant in some cultures and was widely used in the United States
prior to 1940 (Ross and Pratt-Cotter 1999). This algorithm, also
called the “additions,” “equal addends,” or “equal addition” method,
involves adding the same number (10, 100, 1000, etc.) to both the
minuend and the subtrahend, as illustrated in figure 1.47.

13 11 5 is larger than 1, so add 10 ones to 1 and 1 ten to 7 11 – 5 = 6

4 3 1 8 is larger than 3, so add 10 tens to 3 and 1 hundred to 2 13 – 8 = 5
23 7 8 5
4 hundreds – 3 hundreds = 1 hundred
1 5 6
Fig. 1.47. Using the Austrian method to subtract 438 – 275

An alternative notation for this algorithm sometimes shows

these additions by placing 1s on diagonals between the numbers.
Figure 1.48 illustrates this process for the subtraction 431 – 174.
First a 1 is placed on the diagonal between the 1 in the ones col-
umn and the 7 in the tens column. This 1 represents the 1 ten to
be added to both numbers. A 1 is also placed on the diagonal be-
tween the 3 in the tens column and the 1 in the hundreds column.
This 1 represents the 100 to be added to both of these numbers.
Subtraction proceeds as shown in the figure.

4 11 ones – 4 ones = 7 ones

1 3 11
–1 7 4 13 tens – 1 ten – 7 tens = 5 tens
2 5 7 4 hundreds – 1 hundred – 1 hundred = 2 hundreds

Fig. 1.48. The Austrian method for subtracting 431 – 174

Reflect 1.15 invites you to try out the Austrian method for sub-
traction. This algorithm is very similar to the next one—the last one
discussed here—which involves making the same change to both the
minuend and the subtrahend.
46 Addition and Subtraction

Reflect 1.15
Use the Austrian algorithm (also known as the “additions,” “equal addends,” or
“equal addition” algorithm) to find each of the following differences:
73 – 38 642 – 135
702 – 187 2305 - 1268

The “same change” algorithm for subtraction is more general

than the Austrian method, since any number can be added to the
numbers being subtracted. Usually, numbers are chosen to simplify
the problem and eliminate the need for regrouping. In the example
in figure 1.49, a 6 is first added to get a 0 in the ones column in
the number being subtracted. Then a 20 is added to get a multiple
of 100. Now the subtraction is very easy. This method requires only
addition and subtraction facts with sums to 10. Reflect 1.16 explores
the use of this algorithm.

431 +6 437 +20 457

–174 +6 –180 +20 –200
257

Fig. 1.49. The “same change” algorithm for subtracting 431–174

Reflect 1.16
Use the “same change” algorithm to find each of the following differences:
73 – 38 642 – 135
702 – 187 2305 – 1268

One way to justify this procedure involves considering the dif-

ference of two numbers (b – a) as the distance between those points
on a number line. If the same number — say, c — is added to both
numbers, then both slide by the same distance on the number line,
so they stay the same distance apart. Figure 1.50 illustrates this fact.

+c +c

a b

Fig. 1.50. Using the number line to represent the “same change”
algorithm for subtraction
The properties of addition and place value can be used to show
more formally why this procedure works. Suppose that in the case
The Big Ideas and Essential Understandings 47

of two whole numbers a and b, with b being subtracted from a, a

whole number c is added to both numbers.

(a + c) – (b + c) = (a + c ) + –(b + c ) Definition of subtraction

= (a + c ) + (–1)(b + c ) Multiplication by –1
= (a + c ) +(–1)(b ) + (–1)(c ) Distributive property for
multiplication over addition
= (a + c ) +(–b ) + (–c ) Multiplication by –1
= a + (c + (–b )) + (–c ) Associative property for addition
= a + (–b + c ) + (–c ) Commutative property for addition
= (a + (–b )) + (c + (–c )) Associative property for addition
= (a – b ) + (c – c ) Definition of subtraction
= (a – b ) + 0 Additive inverse
=a–b Zero as additive identity

Thus, the “same change” algorithm is the result of generalizing

an arithmetic pattern; adding the same number to both the minuend
and subtrahend does not change the difference between the two
numbers. Reflect 1.17 invites further investigation of the algorithm.

Reflect 1.17
Can you subtract the same number from both the minuend and subtrahend to
simplify subtraction?
If so, why does this work?
Solve the following problems by adding the same number and then try to solve
them by subtracting the same number.
73 – 59 523 – 378
4321 – 1987
Consider your results, and if possible, discuss them with your colleagues.

Conclusion
To understand and use addition and subtraction fluently,
students need to understand when to use each operation, how to
write number sentences to match specific problem situations, and
how to compute sums and differences. Students need to be able not
only to do computations but also to explain why these procedures
work, using words, diagrams, or models. The essential understand-
ings described in this chapter constitute a solid foundation for
teachers, enabling them to recognize addition and subtraction in
its many contexts, to apply and validate a variety of strategies to
48 Addition and Subtraction

solve addition and subtraction problems, and to use place value and
strategies such as “make 10” and using known facts to help students
make sense of basic facts and mental computations. For students in
prekindergarten through grade 2, these ideas are essential in and of
themselves as well as for understanding future mathematics.
 2
Chapter

Connections: Looking Back

and Ahead in Learning
This chapter focuses on the connections between foundational ideas
of addition and subtraction and more basic ideas about number
and numeration, as well as more advanced or complex topics, such
as rational number concepts and procedures, algebraic thinking,
measurement, and data representations. The big ideas discussed in
chapter 1 encompass and extend earlier, simpler concepts while
laying a reliable foundation for later, more sophisticated ideas. Big
Idea 1 is important for work not only with whole numbers but also
with other number forms, such as fractions and integers. The kinds
of problems discussed in chapter 1 apply to these other numbers,
supporting students’ growing understanding of operations on them
iea
d Big Idea 1
Addition
and subtrac-
as well as underlying concepts of algebra (see Barnett-Clarke et al.
tion are used to rep-
[2010]). Big Idea 2 makes a direct connection to earlier ideas, since resent and solve many
students put place-value concepts and the properties of addition to different kinds
use in both student-generated and standard algorithms for addition of problems.
and subtraction.
Big Idea 2 also connects to students’ later work in mathematics.
Place value and the properties of addition—first applied in whole
number addition and subtraction—connect to a myriad of other iea
d Big Idea 2
The mathe-
mathematics concepts across the five content strands—number and matical
operations, algebra, geometry, measurement, and data analysis and foundations for
probability (NCTM 2000). The properties of addition, for example, understanding com-
are connected to multiplication and division and the properties of putational procedures
those operations. Algebraic thinking has a strong connection to Big for addition and
Idea 2 because the properties of addition are generalized rules, and subtraction of whole
when students are analyzing general properties that work in addi- numbers are the
properties of addition
tion, they are using algebraic thinking. Measurement and data rep-
and place value.
resentations make use of place value and properties of addition.
The discussion that follows first considers connections related
to Big Idea 1—specifically, ideas associated with part-part-whole
and part-whole relationships, continuous and discrete models, and
measurement. The discussion then turns to connections to Big Idea

49
50 Addition and Subtraction

2, including ideas related to place value, multiplication and divi-

sion, algebra, measurement (revisited), and data representation.

Links to Big Idea 1: Extending Models

beyond Whole Numbers
The extension of When children first learn to add and subtract, they typically confine
whole number themselves to whole numbers or counting numbers. However, they
ideas to rational gradually discover the need to extend the set of numbers that they
numbers is detailed are using to include positive fractions, and they soon discover the
in Developing need to add and subtract with these numbers. Applying familiar
Essential
representations of addition and subtraction on whole numbers can
Understanding of
help them understand these operations on new numbers.
Rational Numbers
for Teaching
Mathematics in Part-part-whole and part-whole
Grades 3–5
(Barnett-Clarke et al. A major concept, captured in Essential Understanding 1c, is that
2010). addition and subtraction problems can be described by using
the language of part-part-whole number relations or sentences.
Interpreting addition and subtraction in terms of part-part-whole
relationships allows these operations to be extended to other forms
of numbers, such as fractions, decimals, and percents. For example,
the “action” and “no-action” types of problems discussed in chapter
1 also apply to rational numbers. The problem 7/8 – 3/4, for ex-
ample, can be thought of as the action of “taking away” 3/4 from
7/8, or of comparing the size (measure) of the two fractions. Giving
students problems with different parts missing (the start,
start action,
or result ), while less common in teaching rational numbers than
in teaching whole numbers, is important in developing students’
fluency in computing with rational numbers. Encountering these
varieties of problems with whole number addition and subtraction
can support students’ skill in working with fractions. Although it is
generally accepted that understanding whole numbers is important
to learning fractions, the ways in which students’ understanding
of whole numbers can support or inhibit their understanding of
fractions are not well understood (Mack 1995; Smith 1995; Van de
Walle, Karp, and Bay-Williams 2010). There are similarities and dif-
Essential ferences between whole numbers and fractions that are important to
Understanding 1c know and to share with students. One of those is the conceptual re-
lationship between “part-part-whole,” used to describe addition and
Many different
subtraction, and “part-whole,” used to describe ratios and fractions.
problem situations
can be represented As discussed in chapter 1, the notion of “part-part-whole” is
by part-part-whole that two parts are combined to compose a whole, meaning that the
relationships and two parts add to make the whole. Thus, the part-part-whole inter-
addition or pretation indicates an equation. “Part-whole,” by contrast, does not
subtraction. indicate an equation but the relationship of one part to the whole.
Connections: Looking Back and Ahead in Learning 51

Yet, in both part-part-whole and part-whole representations, the

part is being related to the whole. The fraction 5/12, for example,
can represent 5 inches, or part of 12 inches, a whole foot.
Figure 2.1 illustrates this use of 5/12. Notice that the figure
shows a “missing” part of the foot (7 inches), so we can relate the
situation to the idea of part-part-whole if we think about the whole
foot as 5 in. + ☐ in. = 12 in. Observe how closely the idea of part-
part-whole is to that of part-whole, with an important distinction:
the second “part” is not apparent in the notation 5/12 . With frac-
tions, part-whole can represent part of a region, part of a group (like
a classroom of people), or part of a length (like the example above).
Just as addition has other representations besides part-part-whole,
so fractions have other interpretations besides part-whole, but part-
whole is the most common one in textbooks and in classrooms.
Likewise, just as students benefit from many types of addition story
problems, so they would understand fractions better if instruction
placed more emphasis on other representations of them (Clarke,
Roche, and Mitchell 2008; Siebert and Gaskin 2006; Lamon 1999).

Whole (foot)

Part (5”)

Fig. 2.1. Representing 5/12 as part of a foot

In a decimal quantity such as 0.3, the part-whole relationship
is less apparent in the way in which the number is expressed be-
cause the whole is not explicit in the notation, but the expression
0.3 still means 3 parts of 10. In the case of 0.003, the meaning is 3
parts of the whole of 1,000. Similarly, 3% specifically means 3 parts
out of 100, but it implies that no matter what the size of the whole,
the part is just 3/100 of that whole.
Two points are important here. First, the part-part-whole repre- The meanings of
sentation for addition can serve as a foundation on which to build fractions, decimals,
part-whole concepts related to fractions, as long we make the dis- and percents and
tinction between the two. The example above of 5 inches can help ways to express
to illustrate how to make this connection. Any time that we give rational numbers
students a problem with a fractional amount (for example, 1/8 of are elaborated in
Developing Essential
mile walked), representing it as a part-part-whole relationship
Understanding of
(1 + 7 = 8) and connecting it to the parts (1/8 part of mile walked
Rational Numbers
and 7/8 part of mile not walked), can help students to see the for Teaching
connection. Mathematics in
The second point is equally important: using a part-whole Grades 3–5
interpretation is a way for students to see the connections among (Barnett-Clarke et al.
fractions, decimals, and percents. Students who focus on the big 2010).
52 Addition and Subtraction

question, “What part of the whole is ☐?” are able to see that “4
parts out of 5” is equivalent to “8 parts out of 10” and “80 parts
out of 100.” They see that these quantities are proportionally the
same and understand that they can write this part-whole relation-
ship in a variety of ways, including 4/5, 8/10 , 0.8, 0.80 and 80%.
Understanding proportionality and multiple representations are
essential to becoming mathematically proficient.

Discrete and continuous models

The terms continuous and discrete may bring back memories of high
school or college algebra discussions of functions, but the discus-
sion here focuses on models or types of manipulative or visual aids
used to illustrate addition and subtraction concepts. When applied
to a model or manipulative aid, discrete means that it cannot rep-
resent quantities between two numbers. Connecting cubes, for ex-
ample, are typically used with each cube representing one whole. It
is therefore difficult for students to see or represent quantities such
as 121/2 or 123.6 with connecting cubes. Counters, such as those
used on the ten-frames in chapter 1, are also discrete models. The
hundreds chart, even though it shows numbers in a line, is another
discrete model, though the hundreds chart lends itself to addition
and subtraction methods that mirror the use of the number line.
Discrete models allow grouping and regrouping. These models,
therefore, lend themselves to situations in which students are work-
ing with place value and are decomposing and recomposing num-
bers to add more efficiently. For example, in the equation 18 + 32,
students may use base-ten blocks to represent each number, and
then group the 8 and 2 to make another ten, arriving at the answer
of 5 tens, or 50. Mentally, with a foundation in discrete models, stu-
dents may decompose the problem into 18 + 30 + 2 and recompose
this sum to 20 + 30 to get 50. Through the model, students may see
the tens, add those, and then see the ones, add those, and then com-
bine to reach a solution. A contrasting approach with a continuous
model is discussed below.
The regrouping that discrete models allow can extend to mod-
eling decimals and decimal operations. In a set of base-ten blocks,
if the hundreds-block, or “flat” (10-by-10-by-1 piece), is designated
to represent one whole, then each tens block, or “rod” (10-by-1-
by-1 piece) becomes one-tenth (0.1) of the whole, and each ones-
block, or “unit” cube (or “single”; 1-by-1-by-1 piece), becomes
one-hundredth (0.01) of the whole. Alternatively, a thousands cube
(10-by-10-by-10 piece) could also be designated as one whole, in
which case each little unit cube would become one-thousandth
(0.001) of the whole. Even though the unit can be assigned to rep-
resent hundredths or thousandths, there are still quantities between
Connections: Looking Back and Ahead in Learning 53

hundredths or thousandths that cannot be represented by the model, For additional

and hence, the model is still discrete. discussion of
Continuous models, by contrast, offer students the chance to continuous and
“see” the magnitudes of numbers in a different way—along a con- discrete quantities
tinuum. By using a number line, students can see where 42 lies in and models, see
relation to 50, or 100, or 1000, for example. Continuous models—in Developing
Essential
particular, the number line—are useful for all types of numbers. The
Understanding
number line is the model that is commonly used with integers, with
of Number and
the line extending to the left of zero as well as to the right. Students Numeration for
can use it with fractions and for adding fractions with like or unlike Teaching
denominators (see the example in the next section, on measure- Mathematics in
ment). By working with the number line, they can develop ideas Prekindergarten–
about decimals much as they develop ideas about whole numbers. Grade 2 (Dougherty
For example, the “jumps” on the number line in figure 2.2 show et al. 2010).
18 + 32, but we can adapt the number line so that the tick marks
represent ones and are relabeled 0 through 10, and then the jumps
illustrate the problem 1.8 + 3.2.

0 10 20 30 40 50

Fig. 2.2. “Jumps” marked by a student to illustrate adding 18 + 32

by first counting by 10s and then adding 2 ones
As in adding and subtracting whole numbers, the use of both
discrete and continuous models in adding and subtracting frac-
tions and decimals allows students to use different strategies (e.g.,
decomposing and jumping), and develop a deeper understanding of
the operations.

Measurement
One of the most important advantages of the number line model is
that it allows students to integrate concepts related to measurement.
Measurement is closely linked to addition and subtraction, a fact
that becomes very obvious when students are using number line
models for learning to perform these operations. This chapter dis-
cusses measurement twice. It addresses the topic here, in relation to Essential
Big Idea 1, because measurement is often connected to comparisons, Understanding 1c
and comparison situations are one of the problem types described
Many different
in connection with Essential Understanding 1c (see example 10, for problem situations
example, on p. 23). Later, the chapter revisits measurement, in rela- can be represented
tion to Big Idea 2, as one of the content connections for addition by part-part-whole
and subtraction. relationships and
A ruler with subdivisions for halves (and eventually fourths addition or
and eighths) can help young children extend their knowledge of subtraction.
54 Addition and Subtraction

adding and subtracting whole numbers to rational numbers. For ex-

ample, if students use a ruler that has half inches marked, as shown
in figure 2.3, they can use a “jump” strategy to add sums not only
like 2 + 4, but also like 21/2 + 1, and so on, seeing and conceptual-
izing the lengths of a 1/2-unit jump and a whole-unit jump.

0 1 2 3 4

Fig. 2.3. A ruler with halves marked can be useful for linking
whole number addition and subtraction to addition and subtraction
of fractions.

A foundation for future mathematics

iea
d Big Idea 1
Addition and
In summary, Big Idea 1, which recognizes the usefulness of addition
and subtraction in representing and solving many kinds of prob-
subtraction
are used to lems, provides a foundation for an enormous amount of mathemat-
represent and solve ics. Students’ concepts of addition and subtraction as combining or
many different kinds taking away (“action”) or comparing (“no action”) can be developed
of problems. and reinforced through the use of different discrete and continuous
models. Teaching whole number addition and subtraction with both
kinds of models is very important, since both are used to represent
a range of topics, including operations with decimals, fractions, and
integers, as well as measurement concepts.

Links to Big Idea 2: Place Value and

iea
d Big Idea 2
The mathe- Properties across Content
matical
Big Idea 2 focuses on place value and the properties of addition as
foundations for
understanding fundamental to procedures for adding and subtracting whole num-
computational bers. The connection of these computational procedures to proper-
procedures for ties and place value must be made explicit in teaching. By doing so,
addition and subtrac- instruction builds connections and foundations for other
tion of whole mathematical ideas.
numbers are the
properties of addition
and place value. Place value
The discussion of place value that follows is brief, since the rela-
tionship between place value and addition and subtraction is an in-
tegral part of Big Idea 2 and has been discussed at length in chapter
1. However, place value itself is at the heart of many of the connec-
tions to other content areas and thus merits examination here as well.
Connections: Looking Back and Ahead in Learning 55

The notion of grouping by tens is the basis of the metric system. For
example, knowing that 78 is 7 tens and 8 ones helps students un-
derstand that 78 millimeters means 7 centimeters (or 70 millimeters)
and 8 millimeters. It is important to distinguish thinking of 78 as
7 tens and 8 ones from writing out 78 by rote as 70 + 8. The latter
may not lead to the realization that the 7 in 70 signifies 7 tens.
Multiple experiences with the “split” and “jump” strategies and
discrete and continuous models reinforce the important conceptual
ideas of place value. Students who have such experiences will ex-
tend their place-value concepts to include tenths, hundredths, and
smaller places. As students work with decimal values, their contin-
ued use of split and jump strategies can support their understanding
of decimal quantities (e.g., that 0.35 is 35 hundredths, or 3 tenths
and 5 hundredths, noted symbolically as 0.3 + 0.05). Essential
As expressed in Essential Understanding 2c, place-value con- Understanding 2c
cepts provide a convenient way to compose and decompose num-
bers to facilitate addition and subtraction computations. This is true Place-value concepts
provide a convenient
across many curriculum topics. The notion of grouping, for exam-
way to compose and
ple, carries over to, and connects directly with, work with measure-
decompose numbers
ment units, though the groupings may not be by tens. With inches to facilitate addi-
and feet, for instance, the basic idea of trading is the same, but now tion and subtraction
the groups consist of units of 12 instead of 10. For example, con- computations.
sider the sum
3 feet 6 inches + 2 feet 10 inches.
In this case, we can add feet with feet (like tens with tens) and inch-
es with inches (like ones with ones). When we reach 12 inches, we
can make a trade (composing a unit of higher quantity) for the next
higher unit—in this case, 1 foot (rather than 1 ten). The metric sys-
tem, already in base 10, is very clearly linked to place value (adding
like units).
This notion of adding the same-sized parts gains even more
importance as the foundation for understanding the addition of
fractions. Consider the fraction 1/3. It represents a unit, three of
which make a whole (1/3 + 1/3 + 1/3). Similarly, 1/4 represents a
different-sized unit, four of which make a whole (1/4 + 1/4 + 1/4 +
1/4 ). How can we combine 2/3 and 1/4? They aren’t pieces or units
of the same size. For us to combine them, the units (like the place
values of numbers or the units in measurements) must be the same.
In this case, we can use the foot as a model to help us solve the
problem: 2/3 of a foot is 8 inches, or 8/12 of a foot, and 1/4 of a foot
is 3 inches, or 3/12 of a foot. In our model, the sum is therefore 11/12
of a foot; the solution to our problem is hence 11/12. These frac-
tion equivalences (2/3 and 8/12, for example) represent trading—for
example, we trade every one third for four twelfths—and we make
these trades to obtain same-sized units for combining.
56 Addition and Subtraction

When students are older, they confront mathematical situa-

tions in which they need to add like terms in expressions such as
3x + 5y + 8x + 3x. It is their understanding of whole number addi-
tion—in particular, the importance of adding numbers of like size—
tens with tens, and ones with ones—that builds the foundation for
understanding that the x-terms are based on the same quantity and
can be combined, but the y-terms are not necessarily based on that
quantity and so cannot be combined with the x-terms. Adding func-
tions and vectors is similarly grounded in this notion of adding like
quantities.

Multiplication and division

In working with whole numbers, students encounter multiplication
as repeated addition and division as repeated subtraction, so the
connections of addition and subtraction with multiplication and
division are embedded in these descriptions of the operations. But
time and experience are necessary for those connections to become
explicit to students.
Adding doubles may help students make a first connection
from addition to multiplication, since it is the beginning of com-
bining equal-sized sets. Students might explore other situations in
which they are adding equal-sized sets (or groups). For example,
how many pieces of gum are in 3 packs if each pack has 5 pieces of
gum? Students might solve the problem by adding 5 + 5 + 5. They
might skip-count or add in their heads: 5, 10, 15. Symbolically, this
process is written as 3 3 5, and for students to make the connec-
tion between addition and multiplication, they might attach words
to this expression, such as “3 packs of 5,” “3 sets of 5,” or “3 groups
with 5 in each group.” Teachers often guide students in exploring
multiplication by using an area model. In a grid that is 7 by 6, for
example, a student can see 7 rows of 6 (or vice versa). The grid lets
the student see the repeating rows that are all the same length, with
the visual image reinforcing the connection between addition and
multiplication. Understanding this connection allows students to
decompose the rectangle into two smaller rectangles, one that is 7
by 5 and one that is 7 by 1, if they find it easier to calculate 7 3 5,
and then add on 7 more.
Division is usually defined as the inverse operation of multipli-
cation, but just as multiplication of whole numbers can be described
as repeated addition, division of whole numbers can be described as
repeated subtraction. Consider the following example:
Katie was preparing treat bags for her birthday party, and
she wanted to put four stickers in every bag. If Katie has 24
stickers, how many bags will she be able to fill?
Connections: Looking Back and Ahead in Learning 57

Notice that this wording does not describe a sharing (which might
be expressed by the question, “How can 24 stickers be equally
shared among 8 bags?”). Students might solve this problem by
counting down from 24 by fours (20, 16, 12, …), recording a tally to
keep track of how many bags are getting filled (equal subtraction),
or they might solve it by counting up to 24 by 4s (equal addition).
Story problems of this type (with the number of groups unknown)
are not often used in the teaching of division, so students frequently
solve such problems by using repeated subtraction, without making
the connection to division.
Algorithms for multiplication and division are grounded in the
properties of addition and multiplication—namely, the commutative,
associative, and distributive properties. For example, by applying
place-value concepts, students can think of the problem 4 3 18 as
4 3 (10 + 8). By applying the distributive property, they can see
that this means that there are four 10s and four 8s. Students can
compute these mentally and then add 40 and 32 to get 72. Students
can also solve many division problems mentally by applying prop-
erties. For example, they might solve 132 ÷ 12 by thinking about
the problem in the following way: 132 ÷ 12 = (120 + 12) ÷ 12 =
(10 + 1) = 11. Or they might use a “counting up” strategy, think-
ing that ten 12s are 120 and one more 12 makes 132, so count-
ing up to eleven 12s will get them to 132. As stated in Essential
Understanding 2a, knowing the relationships among addition, Essential
subtraction, multiplication, and division enable students to solve Understanding 2a
problems such as this one flexibly, by using any one or more of the
other operations. The commutative and
associative proper-
Addition and subtraction are also implicit in the standard al-
ties for addition of
gorithm for division. Consider the example of 134 ÷ 8, shown in
whole numbers allow
the table in figure 2.4. Column 1 shows the steps in the standard computations to be
algorithm, and column 2 shows the thought process involved at performed flexibly.
each step. Representing these steps as equations, as in column 3, il-
lustrates the use of the relevant properties. In division, the process
is carried out in partial products.
Learning basic multiplication facts is connected to addition
place value and properties. Consider the basic fact 6 3 7. This is For a discussion
a challenge for many students. If students understand that 6 3 7 of the properties
means 6 groups of 7, or 7 + 7 + 7 + 7 + 7 + 7, they can group the of division, see
sevens however they like. One common way is 5 3 7 + 7. But that Developing Essential
Understanding of
is not the only way. If a student knows that 3 3 7 is 21, for exam-
Multiplication
ple, then he can interpret the problem as (7 + 7 + 7) + (7 + 7 + 7) =
and Division
(3 3 7) + (3 3 7) = 21 + 21 = 42. Moving between addition expres- for Teaching
sions and multiplication expressions helps students understand Mathematics
the connections between the operations and use them in solving in Grades 3–5 (Otto
problems, eventually memorizing the facts. It is the properties of et al. 2011).
58 Addition and Subtraction

Accompanying Implicit equation(s), applying properties

Written calculation thought process of addition and multiplication

)
8 134 How many 8s are in
130, or how many 8s
134 ÷ 8 = ((13 × 10) + 4) ÷ 8

are in 13?
1 The 1 in the quotient is 134 ÷ 8 = (80 + 54) ÷ 8
)
8 134 in the tens place, and
ten 8s are 80; subtract
= (80 ÷ 8) + (54 ÷ 8)
= 10 + (54 ÷ 8)
−80
80 from the dividend
54 of 134; the difference
is 54.
16 The 6 in the quotient 134 ÷ 8 = 10 + (54 ÷ 8)
)
8 134 is in the ones place,
and six 8s are 48;
= 10 + ((48 + 6) ÷ 8)
= 10 + ((48 ÷ 8) + (6 ÷ 8))
−80
subtract 48 from the = 10 + (6 + (6 ÷ 8))
54 difference of 54; 6 is = (10 + 6) + (6 ÷ 8)
− 48 the remainder.
6
Answer: 16 R 6, 16 3/4, or 16.75
Fig. 2.4. Reasoning and equations representing properties of addition and
division in the standard algorithm for division in the case of 134 ÷ 8

Essential addition that justify the correctness of invented and standard

Understanding 2d algorithms (Essential Understanding 2d).

Properties of addition
are central in justify- Algebra
ing the correctness The role of algebra in the primary grades should go far beyond oc-
of computational
casional appearances in repeating and growing patterns. In fact, to
algorithms.
switch metaphors, we might describe algebra as a lens for magnify-
ing relationships and properties of number and operations. That is
certainly true in the case of addition and subtraction. Recall that
Essential Understanding 1b expresses the idea that subtraction is the
inverse of addition, an idea introduced in chapter 1 by an example,
part of which is shown in figure 2.5.
Essential The table and the function machine illustrate not only the re-
Understanding 1b versibility of addition and subtraction (Essential Understanding 1b),
but also the generalizability of number. In the table, students can
Subtraction has an see that the relationship from the input to the output is “plus 2,”
inverse relationship and that the relationship from the output to the input is “minus 2.”
with addition. Once students have recognized the rule, they can use examples of
larger numbers—for example, 100 or 500. As students see that
the rule always applies, they recognize that it applies in general,
not just in certain cases. In other words, they realize that the rule
Connections: Looking Back and Ahead in Learning 59

Input Output
1 3
5 7
8 10
11 13

x Add 2 x+2

Fig. 2.5. Add 2 as an addition operation

is a generalization. When students generalize a series of numerical

examples, they are doing algebra. Suppose that students are work-
ing with the following problem set:
14 – 9 = 12 – 9 = 15 – 9 = 18 – 9 =
Consider how using an algebraic lens on their thinking can support
students’ strategies for finding the differences and later memorizing
these facts.
Students may use a range of tools and strategies to solve these
problems. A hundreds chart is a nice tool for visualizing how the
initial value and the result relate to each other. Instead of stop-
ping after students obtain correct answers, the conversation needs
to continue to include patterns that students notice, testing those
ideas. These generalizations are algebraic thinking, and such rea-
soning is central to learning addition and subtraction well. Students
might notice that in all cases they could have used –10 + 1 instead
of – 9. The discussion of this set of problems should include (at
least) two important questions:
1. Does this –10 + 1 “rule” apply to all of our examples?
2. Will this –10 + 1 “rule” apply to any “minus 9” problem, or
when will this rule work?
In this case, this rule always works, and symbolically it is written as
n – 9 = n – 10 + 1.
The notion of decomposing and composing numbers runs
throughout this book, and it has its roots in algebraic thinking. If
7 is the whole, for example, in a part-part-whole situation, what
could the parts be? They could be 3 and 4, 1 and 6, and so on.
Students can be asked to find all the (whole number) ways of de-
composing 7. A popular context (Yackel 1997) involves monkeys in
trees: How can students distribute 7 monkeys in two trees. The more
challenging question that students must ultimately answer is, “How
60 Addition and Subtraction

do we know that we have found all of the solutions?” Students may

make a list or table like that in figure 2.6 to find all the ways.

Big tree Little tree

5 2
6 1
4 3

Fig. 2.6. Table representing ways to distribute 7 monkeys in 2 trees

As students are looking for patterns in the table and trying to
notice a rule for knowing when they have found all the ways, they
are generalizing, and therefore they are using algebraic thinking.
Students might notice, for example, that for every digit from 0 to 7
there is one way (0 + 7, 1 + 6, …, 7 + 0). Carpenter, Franke, and Levi
(2003) report that second graders noticed that there is one more
possible way than the size of the number—in this case, 8 ways.
Finally, as students explore addition and subtraction story
problems, using an algebraic lens can expand the way in which
they write equations. As suggested throughout chapter 1, numerical
relationships in story problems can be represented in several ways—
for example, as a + ☐ = b or b – a = ☐. The first equation shows a
missing addend, an unknown quantity that students will later repre-
sent with a variable. Asking students to solve equations in algebraic
Chapter 3 offers a form, like 5 + ☐ = 4 + 8, can support their understanding of the
full discussion of the equals sign, give them experience with sums, and help them under-
important stand number relationships. In the case of 5 + ☐ = 4 + 8, they can
issue of developing apply the idea that if the first addend on the left (5) is one less than
understanding of the first addend on the right (4), then the second addend on the left
the equals sign by must be one more than the second addend on the right (8). As the
working with, and
discussion in this section has illustrated, addition and subtraction
developing concepts
do not simply provide foundations for algebra (or vice versa); these
related to, addition
and subtraction. arithmetic operations and algebra are intimately interrelated and
mutually support learning in both domains.

Measurement concepts
Measurement, like algebra, is very closely linked to concepts of
addition and subtraction. The related concepts that students explore
in the measurement strand include measuring time, weight, and
length; comparing the lengths of two objects (subtraction); combin-
ing lengths in various contexts, including perimeter problems;
and combining areas of shapes (or angle measures). The following
Connections: Looking Back and Ahead in Learning 61

examples show how close the connection is between measurement

and addition and subtraction:
Example 1: Your brother says that you can play the video
game in 45 minutes. You look at the clock and see that it is
5:07. What time should you come back for your turn to play?

Example 2: Suppose that a snake grows from 5 inches at birth

to 17 inches as an adult. How many inches does the snake
grow from birth to adulthood?

Example 3: Suppose that an aerial view of one section of

fence is shown on the left below and one patio tile is shown
on the right:

How many fence sections do you need to enclose the patio

shown below? If the whole patio is also going to be covered in
tiles, how many will fit in the space?

Students’ probable approaches to these problems are discussed

below.
Students might solve the problem in example 1 in a variety
of ways. One would be to recognize that they are adding 45 and 7
and do so by first splitting 7 and then adding 45 + 5 + 2 = 52 min-
utes, to get 5:52 as the time to come back for the video game. Or
students might solve this problem by counting on, or jumping, four
times by 10 and then one time by 5: 5:17, 5:27, 5:37, 5:47, 5:52.
The first strategy shows that the students recognize that they are
adding minutes and are thinking about minutes as counting objects.
The second strategy is aligned conceptually with the disk of an ana-
log clock, which is really a segment (from 0 to 12) of the number
line curved into a circle.
Example 2 presents a comparison situation. Students are com-
paring the snake’s length as an adult to its length at birth. They
62 Addition and Subtraction

could make physical comparisons with drawings or with a manipu-

lative like connecting cubes or Cuisenaire rods (in fig. 2.7, the vari-
able b represents birth length, and the variable a represents adult
length). Students are likely to use a “counting up” strategy, because
of the context and the way in which the story problem is written.
However, they may also count down from 17. Encouraging multiple
ways to think about the problem is important in deepening student
understanding of addition and subtraction.

Fig. 2.7. Comparing measurements as a context for subtraction

Although area and perimeter may be content goals for later
grades, example 3 cleverly builds ideas that will support later learn-
ing of these concepts while simply providing a context for adding.
Students can find the perimeter by counting fence sections one by
one (possibly acting out the process with materials), or they can
find how many fence lengths are needed for each side and then add
those four quantities together. They can also determine how many
tiles will fit in the space by counting individual tiles (possibly us-
ing color tiles to make a model of the space) or by seeing that there
are three rows of 4 and using repeated addition. Young children can
For additional solve more challenging problems of the same type. For example,
discussion of students can cut out nets for open boxes and apply their addition
quantity and skills to figure out how many squares are on their surface and how
measure, see many cubes might fill them. For young students, these measurement
Developing Essential
problems provide an interesting context for measuring and adding—
Understanding of
not for finding surface area and volume formally. Students can em-
Number and Numer-
ation for Teaching ploy a hundreds chart, a number line, or base-ten blocks in figuring
Mathematics in out the total quantity as the numbers get larger.
Prekindergarten– Measurement and addition and subtraction are synergistic.
Grade 2 (Dougherty Measurement provides an interesting context for adding and sub-
et al. 2010). tracting, and using such contexts supports students’ understanding
of both measurement and the operations.

Data representation
Strategies for analyzing and displaying data are grounded in con-
cepts of addition and subtraction. When young children create bar
graphs, they should analyze what they see. They can compare data
(by considering such questions as, “How many more people liked
horses than liked sheep?”) or combine data (by considering ques-
tions like, “How many people picked a farm animal?”).
Connections: Looking Back and Ahead in Learning 63

Venn diagrams give young children a different way to look at

and compare data. Consider figure 2.8, which shows data gathered
by a group of 18 first graders about themselves. Initially, the teacher
might pose counting questions for the students to answer by exam-
ining the data, such as the following:
• “How many of you have both a sister and a brother?” (3)
• “How many students have sisters?” (7; requires combin-
ing the number of tokens in the overlap of the “sister” and
“brother” circles with the number of tokens in the “sister
only” section of the “sister” circle)
• “How many have brothers?” (10)
• “How many of you have a brother or a sister?” (14)
The “or” question is the most difficult and relates to an important
property for addition—that quantities to be added cannot overlap
(see example 13 in chapter 1; p. 25). Students will debate whether
the answer is 17, obtained by adding 7 + 10, or 14, obtained by
counting the total inside the two circles of the Venn diagram: 7 +
3 + 4. Having students stand up if they have a brother or sister (or
both) will illustrate that the correct response is 14. It is important
for teachers to show that the other answer counts 3 students (those
in the overlap) twice. This notion of distinct (or discrete) groups be-
comes more complicated in middle and high school, when students
use formulas for counting. Early experiences such as this one can
provide a conceptual foundation.

I have a brother. I have a sister.

iea
d Big Idea 2

The mathematical
Fig. 2.8. Data gathered from a group of grade 1 students foundations
for understanding
computational
A link to later topics procedures for
addition and
In summary, Big Idea 2, which recognizes place value and the ad- subtraction of
dition properties as fundamental to computational procedures for whole numbers are
addition and subtraction, relates to many critical topics in the the properties
mathematics curriculum. By working with contexts in algebra, mea- of addition and
surement, and data, and by making explicit connections to place place value.
64 Addition and Subtraction

value, students gain a deep understanding of addition and subtrac-

tion and are ready to explore more advanced topics that depend on
this understanding.

Conclusion
A quick glance back at the headings in this chapter gives an indi-
cation of the vast importance of addition and subtraction to other
mathematics. The connections described above collectively raise two
important points.
First, having a strong and deep conceptual understanding is
what allows addition and subtraction to serve as foundational top-
ics. For example, if students do not understand the part-part-whole
nature of addition and subtraction, they cannot use it as a founda-
tion to constructing the notion of part-whole. Similarly, students
who frequently use discrete and continuous models in addition and
subtraction can apply them to multiplication and division and to
operations with fractions and decimals. As described above, stu-
dents can apply their whole number knowledge of jumps to solving
decimal operations, but only if they have had such experiences with
whole number addition and subtraction.
Second, conceptual understanding and procedural proficiency
are intertwined, as described in Adding It Up (Kilpatrick, Swafford,
and Findell 2001). Connecting place value to addition and subtrac-
tion, connecting addition to multiplication (and division), and see-
ing the generalizability of procedures (algebra)—to name just a few
of the connections described above—require both conceptual under-
standing of and procedural proficiency with addition and subtrac-
tion. In other words, developing a sound understanding of addition
and subtraction supports and is supported by place-value concepts,
algebraic thinking, and ideas from other topics. When it comes to
computation, proficiency in addition and subtraction of whole num-
bers is essential to developing mathematical proficiency with other
number forms, such as fractions, decimals, and integers.
 3
Chapter

Challenges: Learning,
Teaching, and Assessing
Consider the following dialogue between two collaborating teach-
ers as they focus intently on which problem to use to begin their
students’ inquiry into subtracting with numbers in the teens. While
these teachers debate the merits of each combination, they are in-
tegrating their knowledge of content with their pedagogical knowl-
edge—their understanding and experience of how students learn—to
make the best professional decisions to enhance their students’
learning of subtraction.
Let’s listen in as these two teachers plan. Ms. Wilson and Mr.
Lee are negotiating about the best way to get students to think
about the inverse relationship of subtraction and addition. They
have made two decisions: (1) they want to use a context involving
a girl who has seven trading cards and needs to have a larger quan-
tity to complete the full set, and (2) they intend to ask, “How many
more does she need to collect?” They are debating whether to use
13 – 7 = ☐ or 12 – 7 = ☐ as the basis for the lesson. They want to
create opportunities for students to build on their prior knowledge
of place value and possibly identify and explore the use of 10 as
a benchmark in finding differences. What Ms. Wilson and Mr. Lee
have to say may surprise you:
Ms. Wilson: I think we need to set up the problem so that students
will investigate the idea that the difference between the
two numbers can be found by adding up in jumps and
using 10 as a landmark. This is the same reasoning that
will help them when working with larger numbers.
Mr. Lee: I agree, but I’m thinking that 12 might be a better
number to use, since they’re more likely to know the
combination of 3 plus 2 than 3 plus 3. At this point, I
want to keep the combinations as easy as possible, so
the students can focus on understanding the process of
subtraction rather than try to figure out more

65
66 Addition and Subtraction

challenging combinations. We can add the more com-

plex numbers later when the subtraction process seems
to be firmly in place. I also think it makes sense con-
textually that a full collection of trading cards would
be a dozen. That might be easier to relate to.
Ms. Wilson: I see your point, but I actually think that the students
will be more familiar with the “doubles” combination
of 3 plus 3, and that the symmetry, or evenness, of
the difference in relation to 10 will help them identify
this way of thinking about the problem. I can see them
grabbing their ten-frames here to visualize the com-
parison between 13 and 7. I think using 13 in the prob-
lem is more likely to be a good starting point for their
investigation. We will also need to connect this to the
corresponding missing addend problem of 7 + ☐ = 13.

Setting the Stage for Addition and

Subtraction
Like Ms. Wilson and Mr. Lee, as you set the stage for your students’
learning of addition and subtraction, you must consider their prior
knowledge carefully. You must work purposefully and strategically
to bridge the gap between the mathematics experiences that stu-
dents bring from home and the approaches to learning mathemat-
ics that you present in school. The foundation that you establish
for additive thinking will influence your students’ reasoning about
operations and their formation of the underlying mathematics con-
cepts that will develop into multiplicative thinking and provide the
scaffolding for more advanced learning.
When you are considering instructional tasks related to addi-
tion and subtraction, your students’ understanding should be a key
focus. Specifically, it is valuable to consider students’ mastery of
these topics in light of the five strands of mathematical proficiency
identified by Kilpatrick, Swafford, and Findell (2001): conceptual
understanding, computing (procedural fluency), applying (strategic
competency), adaptive reasoning, and engaging (productive dis-
position). The acronym formed by this list of the elements of pro-
ficiency—UCARE—is an easy way to remember these components
(National Research Council 2002). These five elements of proficien-
cy are interwoven through the exploration that follows of the peda-
gogical considerations that are related to addition and subtraction
in the primary grades.
Challenges: Learning, Teaching, and Assessing 67

Representations and Models

The concrete–semi-concrete–abstract (CSA) teaching sequence (also
known as the concrete-representational-abstract [CRA] instructional
approach) has been used in mathematics education in a variety of
forms for years (Heddens 1964; Witzel, Mercer, and Miller 2003).
This model reflects a continuum from concrete representations and
models to semi-concrete representations and images to symbols
and abstraction. The thinking that students develop while explor-
ing concrete representations often allows them to connect their
ideas to abstract representations. Built into this approach is a cy-
clic return through the components of the model as students learn
new concepts in ways that link to previous ideas. The CSA model
emphasizes conceptual understanding as a vehicle for develop-
ing mathematical proficiency. As students’ thinking is developing
through each representation, this process supports connections and
moves students to full understanding and mastery of addition and
subtraction.
It is important to recognize that the manipulative materials that
you choose to support your learning goals for teaching addition and
subtraction will influence your students’ thinking and shape the
conceptual models that the students develop. Whether you select
one-inch cubes, plastic counters, bundles of sticks, base-ten blocks,
hundreds charts, or number lines, these materials will provide your
students with rich visual representations for concepts and lay a
foundation for more abstract thinking. Although manipulative ma-
terials can be powerful tools to support students’ thinking, using
them in effective ways is a complex task (Leinhardt et al. 1991). As
Ma (1999) states, “A good vehicle, however, does not guarantee the
right destination. The direction that students go with manipulative
materials depends largely on the steering of their teacher” (p. 5).
For example, a teacher may use a large cube from a set of base-ten
materials to represent 1000 in an addition problem, but some young
students will see the squares etched on the six faces of the cube and
incorrectly assume the cube represents 600. The materials are often
more abstract than we think; as adults, we have made the concep-
tual leap.
As an example of the influence of manipulative materials,
consider what frequently happens when we wish to move students
beyond the use of a counting strategy to prepare them to deal ef-
fectively with larger numbers and we make the strategic choice of a
number line or empty number line as a model. Faced with this par-
ticular representation, students are likely to use a “jump” strategy,
discussed in chapter 2 and again below, as they “chunk” numbers
and jump to positions on the line.
68 Addition and Subtraction

Essential
The Structure of Word Problems
Understanding 1c As described in chapter 1 in relation to Essential Understanding 1c,
the structures of word problems in addition and subtraction have
Many different been analyzed and frequently categorized into two major groups:
problem situations
(1) part-part-whole, representing “join,” “combine,” “separate,” or
can be represented
“take away” situations, as well as some situations with no action;
by part-part-whole
relationships and and (2) comparison, representing situations focusing on the differ-
addition or ence of two sets, finding the small set, or finding the large set. What
subtraction. does this mean for teaching and assessment? The story problems
that you select should cover the spectrum of these situations; if they
do not, your students will struggle to find ways to solve the types
of problems that you omit. For example, if students encounter only
“take away” stories—the type most commonly presented by teachers
for subtraction—they won’t recognize that a problem comparing two
students’ heights presents a subtraction situation. In fact, compari-
son problems are less common in textbooks and in classrooms and
therefore are more likely to be missed by students on assessments.
In particular, comparison problems with the reference amount
unknown are the most difficult problems for students to decipher
and the least often highlighted in practice. Two examples of these
problems follow:
1. Courtenay has 10 music CDs. All of her CDs are either coun-
try music or hip-hop. She has 4 more country CDs than
hip-hop CDs. How many country CDs and how many hip-hop
CDs does Courtenay have?

2. Morgan has 7 apples and oranges in the refrigerator for

healthy snacks. He has 3 more apples than oranges. How
many apples and how many oranges does Morgan have?
It is important to use a variety of problem structures so that stu-
dents have opportunities to engage in discussions of models and
situations that arise in real-world settings.
Special education researchers suggest that students with dis-
abilities be explicitly taught these underlying structures so that
they can identify the features of the situations and judge when to
use addition or subtraction (Fuchs et al. 2008; Xin, Jitendra, and
Deatline-Buchman 2005). Fuchs and colleagues (2008) recommend
that teachers help students understand how to separate important
information in the problem from superficial details as well as how
to apply these structures to problems presented in tables or charts.
When these students are then exposed to novel problems, this em-
phasis on problem structure and relevant features will assist them
in generalizing from similar problems on which they have practiced
(Fuchs et al. 2004).
Challenges: Learning, Teaching, and Assessing 69

Furthermore, students encounter challenges as they try to

translate word problems into representations or number sentences.
For example, consider the following problem:
Liza had 8 apples. Then she picked 5 more apples. How many
apples did she have then? Can you show how you thought
about this problem?
This problem can be translated directly from the context as
8 + 5 = ☐. Contrast that to the situation in the following story
problem:
Ermando has 12 cents. He needs 25 cents to buy the trading
card that he wants. How much more money does he need?
This problem may be translated as 12 + ☐ = 25, aligned to the
meaning of the story, or it may be translated as 25 – 12 = ☐. It Essential
could be translated as 25 – ☐ = 12. As Essential Understanding 1d Understanding 1d
suggests, a story problem structured in this way is likely to lead to
a range of equations, an outcome that provides an excellent op- Part-part-whole
portunity for deepening students’ understanding of addition and relationships can
be expressed using
subtraction. Encouraging these different ways to write an equation,
number sentences
discussing their equivalence, and justifying how each equation
like a + b = c or
correctly represents the situation help to build the robust under- c – b = a, where a
standing captured in the UCARE components of proficiency. and b are the parts
Students can navigate more capably through the task of and c is the whole.
translating from language statements to symbolic representa-
tions in number sentences if they work explicitly and strategically
(Lochhead and Mestre 1988). Using information expressed in word
problems, students often mistakenly move from left to right, word
by word, as they try to craft an equation. This decoding of language
and conversion of it to numerical symbols are particularly challeng-
ing for English language learners (ELLs). Again, the underpinnings
of algebraic thinking come into play. Lochhead and Mestre (p. 133)
suggest a three-step process, involving students’ qualitative, quanti-
tative, and conceptual understanding, from which we can construct
the following model:
1. Explore students’ qualitative understanding. As you probe
your students’ grasp of the problem, be on the lookout for
foundations of understanding on which you can build by
asking students how well they comprehend the problem
(you can ask older students to restate it in their own words).
Then try to separate the students’ difficulties in reading or
interpreting (especially in the case of ELLs) from any pos-
sible mathematical confusion that they are experiencing,
remembering that sometimes both will come into play. A
question that you might ask is, “Are there more ________ or
70 Addition and Subtraction

________?” Or “Are the 8 apples part of the whole amount

of apples or all of them?”
2. Explore students’ quantitative understanding. If students
struggle in expressing a basic understanding of the problem
situation, then you might move to a simpler case of the
same problem to see how well they understand the numeri-
cal relationships. A question that you might ask is, “Suppose
there are 10 ________; how many more ________ would you
need?”
3. Explore students’ conceptual understanding. Finally, as
students write their equations, pose questions to probe their
understanding. A question that you might ask is, “What
would happen if we changed the number from ________ to
________?” Or you might ask, “What would happen if they
were getting more ________ instead of losing ________?”
Another possible question might be, “If you know you need
________, and you have ________, how can you find the
other part?” Or, to spark student-to-student discussion, you
might direct attention to a particular student’s work and say,
“Look, ________ wrote an equation that is different from
yours; can both equations be right?”
Note that in all three steps, you are not giving the “right” answer
but are helping your students move to a position where they are
confident of their approach. This shift refocuses the conversation on
the relationships among the numbers and keeps the students’ think-
ing from stopping once class members share the right answer.

Key words: A failed shortcut

Research indicates that it is wise to avoid teaching students to use a
key-word strategy in solving word problems. When young students
are taught to use such a strategy, they often misinterpret words and
shift from a focus on analyzing the problem and sense making to
a mechanical and potentially inappropriate reliance on key words
(Kenney et al. 2005). If students are taught to search for key words
to unravel word problems, they are likely to try to decipher all such
problems in that fashion, without attempting to interpret them
meaningfully in context. Not surprisingly, this can lead to misin-
terpretation and confusion. For example, consider the word more in
the following two situations:
1. Jessica had 9 pennies. She spent 5 pennies, and then she
lost 1 more. Now how many does she have?
2. MacKenna has 4 cartons of yogurt. How many more does
she need so that she can have 1 carton of yogurt for break-
fast for 7 days?
Challenges: Learning, Teaching, and Assessing 71

Students who think that more is the key word and means add will
mistakenly do just that. Other words and phrases, such as alto-
gether and in all, may also be misleading if students believe that
they always suggest a single path to addition. The same can be
said for left and fewer if students believe that they invariably sug-
gest subtraction. More important, many problems do not have key
words, leaving students who have learned only a key-word ap-
proach without a way of determining a solution strategy. Two-step
problems are almost impossible to solve through a key-word ap-
proach. Unfortunately, this is an approach that teachers often use
with students who are struggling, and such students are least likely
to be able to sort out the nuanced meanings of these words in word
problems.

Reasoning about actions and relationships

Clearly, key-word shortcuts create misunderstandings for students
that impede their progress toward mathematical proficiency, as re-
flected in the UCARE components. Understanding word problems
is much more complex than understanding the words and their ar-
rangement. Instead, it involves understanding the quantities and
their explicit and implicit relationships, framed within the context
of the story. Removing the story and the nuance from a word prob-
lem (to make the problem more accessible to students) actually re-
moves the student from the problem.
What is best to emphasize in solving word problems is investi-
gating the actions and incorporating reasoning rather than pulling
out specific words. The very same words may be used in different
types of problems (for example, in all may appear in addition, sub-
traction, multiplication, or division problems). Instead, we want to
give students experiences with many different contexts and types
of word problems, using different ways of indicating arithmetical
actions. One of the best approaches is to have young students dic-
tate or write their own word problems. You can even indicate what
operation you would like for them to highlight. As students begin to
craft the problems, they see the components and language required
to develop addition or subtraction situations, and you will be able
to assess their understanding.

The Structure of Equations

Research shows that the equals sign—in particular, where it occurs
in an equation—creates consistent problems for students (RAND
Mathematics Study Panel 2003). Even some older students—and
adults—harbor confusion about the meaning of the equals sign
and how number sentences can be written. Therefore, a diagnostic
72 Addition and Subtraction

interview might be an appropriate assessment tool. A diagnostic

interview is a formative assessment (usually one-on-one) that can
provide in-depth information about a student’s knowledge and
conceptual strategies. Consider the following interchange between a
teacher and a young student during a diagnostic interview regard-
ing a series of addition equations:
Ms. Klumb: [Pointing to the problem shown in fig. 3.1, 8 + 4 =
☐ + 5] Look at the first problem. What number do you
think belongs in the box?
Aniessa: Twelve [writing 12, as shown in fig. 3.1].

Fig. 3.1. A student’s confusion, as exhibited in a missing

addend problem
Ms. Klumb: How did you get that answer?
Aniessa: I started with the 8; then I counted 4 more to get 12.
It’s easier to start with the bigger number.
Ms. Klumb: [Pointing to the right side of the equals sign] Why does
the equation show to add 5 more?
Aniessa: I need to add 5 more?
Ms. Klumb: What does the equals sign stand for?
Aniessa: It means, “What is the answer?” So, when you see an
equals sign, you put the answer. That’s why I wrote 12.
Ms. Klumb: Let’s try the next problem [shows the problem in fig. 3.2,
☐ + 5 = 5 + 8]. What number belongs in the box?
Aniessa: [Writing 0 as shown in fig. 3.2] This is easy, 0 plus 5
equals 5.

Fig. 3.2. A student’s confusion about the equals sign

Ms. Klumb: [Pointing to the right side of the equals sign] Why does
it show + 8?
Aniessa: When the equals sign goes behind the last number, then
you write the answer after it.
As Aniessa’s responses suggest, if you ask a student what
the equals symbol means, you are likely to get an answer that it
means something like “makes.” In her conversation with Ms. Klumb,
Aniessa said that the equals sign means “What is the answer?”
Aniessa’s confusion was compounded by the difference in the format
Challenges: Learning, Teaching, and Assessing 73

of the equations from the classic form of a + b = ☐. When older

students explore higher-level algebraic thinking, they learn that
the equals sign means equivalence (and has other meanings, as well
[Verschaffel, Greer, and De Corte 2007]) in the sense that both sides Developing
of the equation are “balanced.” But to set the stage carefully for Essential
that knowledge, you should emphasize from the start that the equals Understanding
of Number and
sign is a relational symbol—not an operational symbol (like the op-
Numeration for
erators +, –, , ÷). What appears to the left of the equals symbol
Teaching
should be the same as, or equivalent to, what is on the right of the Mathematics in
symbol. By using a balance as a concrete representation (either an Prekindergarten–
actual balance or a virtual manipulative), students can place values Grade 2 (Dougherty
on either side and explore equations. Dougherty and colleagues et al. 2010) discusses
(2010) address the importance of the equals sign and its use in com- the importance of
posing and decomposing whole numbers. the equals sign.
In the same way that students should recognize multiple
contexts as indicating addition, they should recognize multiple
ways of writing addition and subtraction equations. The notation
that most students see or use initially for addition (and subtrac-
tion) is horizontal—for example, 8 + 3 = ☐. Young students rarely
see equations written in a vertical manner until they reach larger
(multi-digit) addition and subtraction problems, when the novelty of
the vertical arrangement may add to their difficulties in performing
these computations.
Even within the framework of horizontal equations, students
need to see multiple ways to record situations. Seeing equations
written only in the manner of 8 + 3 = ☐ (part + part = whole) or
8 – 3 = ☐ (whole – part = part) contributes to the kind of misun-
derstanding of the equals sign illustrated above (McNeil and Alibali
2005; Seo and Ginsburg 2003). To avoid the pitfalls of a single ap-
proach, you can use a variety of equations with the unknown quan-
tity in multiple positions, including equations that closely match a
story situation. For example, consider the following story:
Mark wants to know the total number of coins in his pocket.
He takes them all out and counts 5 nickels and 7 dimes and
puts them all back. How many coins does Mark have in his
pocket?
This situation could be recorded mathematically as ☐ = 5 + 7,
to align with the situation exactly as stated in the problem. This
mathematical statement is referred to as the semantic equation of
the problems, in that it precisely matches the order of the quantities
as presented in the problem. When a situation is translated into an
equation that has the unknown amount and no other quantities on
the right of the equals sign, the equation is known as the compu-
tational form (which is also the form in which it would be entered
into a calculator).
74 Addition and Subtraction

Equations can be written in several ways, and students should

also be familiar with equations that include an addition statement
on both sides of the equals sign. For example, students should
record relational statements, such as 3 + 4 = 4 + 3 or 5 = 5.
Students should also justify that a statement like the following is
true: 4 + 3 = 5 + 2. These number sentences indicate the sameness
in quantity of the two collections on either side of the equals sign,
and working with such statements will help students deepen their
understanding of the equals sign, as well as of properties such as
the commutative property.

Using Appropriate Language

As students in prekindergarten to grade 2 are in the process of de-
veloping their reading literacy, they are making parallel advances
in their mathematical literacy. Both reading and mathematics have
fundamental rules and conventions, depend on comprehension
(understanding a situation in a story or a word problem), and have
significant practical uses outside of the classroom. In each case, use
of appropriate terminology established in the primary years sets the
foundation for learning in the future.
Terminology should always support students’ understanding.
Vocabulary such as borrowing and carrying focuses on procedures
rather than understanding. These terms have the potential to create
problems for students, including the following:

1. Students think that they can arbitrarily change a number in

a subtraction problem if it is too small by “borrowing” from
another number.

2. Students focus on digits rather than number values.

3. Students are not reinforcing their understanding of the

base-ten number system.

Language that focuses on the concept of equivalent exchanges has

much more power to strengthen and support students’ developing
understanding. Ma (1999) suggests that the use of language related
to decomposing (or composing) a unit of higher value capitalizes on
the exchange of tens as a generalizable process that can be applied
to a variety of problem situations. By unbundling and bundling
materials, students can experience the structure of the base-ten
system. For example, when considering a problem such as
42 – 27 = ☐, students should begin to think, “Can I subtract a
number in the 20s from a number in the 40s,” instead of, “Can I
subtract 7 from 2?” Borrowing and carrying are strictly procedural
steps in which a number (e.g., 1) is moved either to the left or right.
Challenges: Learning, Teaching, and Assessing 75

Therefore, this commonly used, well-practiced, but ultimately inef-

fective language, however ingrained in us, should be replaced with
terminology that honors the concepts that relate to the operations.
Part of teaching addition and subtraction involves preparing
students to use these operations not only on whole numbers in a
part-part-whole structure but also on other numbers that they will
encounter later, such as rational numbers and integers, and expres-
sions in algebraic systems. Language such as “addition makes big-
ger,” and “subtraction makes smaller” (considering the first number
as the point of departure) gives students rules that they discover
no longer hold when they encounter integers or consider equa-
tions with zero. Many students who hear the incorrect but all-too-
common assertion that a calculation such as 5 – 7 “cannot be done”
eventually experience cognitive dissonance when they learn about
negative numbers. In the system of integers, addition can indeed
“make smaller,” and subtraction can indeed “make bigger.” This
seeming conflict can be anticipated, and language can be crafted
so that students’ previous knowledge can serve as a meaningful
foundation for later mathematics concepts. Then the extension of
addition and subtraction to other number systems can fit in the
logical progression to more sophisticated understandings. This ad-
justment in language allows new knowledge to be assimilated rather
than accommodated, to use Piaget’s (1928) descriptions of learning
processes. The challenge then is to teach elementary ideas about ad-
dition and subtraction without setting up conceptual conflicts with
these more advanced concepts.

Student-Invented Strategies
Just as it is important to value different ways to record equations
that represent story problems, so it is important to value student-
invented strategies for adding and subtracting. Invented strategies
are grounded in students’ understanding of number (as implied in
Essential Understanding 1a), and they support and reflect the stu-
dents’ mathematical proficiency (as elaborated in the five UCARE
components—understanding, computing, applying, reasoning, and
engaging). Introducing traditional algorithms too early may actu-
ally cause harm by initiating a misplaced focus on memorization
and procedural skill at the expense of conceptual understanding Essential
(Kamii and Dominick 1998). It is best to delay the introduction of Understanding 1a
traditional algorithms in the primary years, maintaining a number Addition and
orientation instead of a digit orientation. In other words, keeping subtraction of whole
students focused on the number value is much more productive in numbers are based
the long run than encouraging them to take one digit at a time. on sequential
Research suggests that students who use invented strategies prior to counting of whole
any standard algorithms have a better grasp of place value and are numbers.
76 Addition and Subtraction

more flexible in using prior knowledge to extend it to novel situa-

tions and problems (Verschaffel, Greer, and De Corte 2007).
Students need to formulate their own mental or invented strat-
egies with one-digit numbers to help them understand two-digit (or
higher) addition and subtraction. Just as many adults in real-world
situations use operation strategies that they were never formally
taught in school, students can begin to recognize when a process or
strategy is generalizable.
Three strategic approaches are likely to emerge as students be-
gin to invent strategies for addition and subtraction. These capital-
ize on the students’ number sense and their resulting ability to com-
pose and decompose numbers. We can group these strategies into
the categories discussed by Verschaffel, Greer, and De Corte (2007)
as “split,” “jump,” and “compensate.”
The “split” strategy emphasizes the decomposition and recom-
position of numbers largely by place value. Using this approach,
students might think of 54 – 32 = ☐ as 50 – 30 = 20 and 4 – 2 = 2,
for an answer of 22. This strategy aligns well with early use of con-
crete base-ten materials.
The “jump” strategy is illustrated for 54 – 32 = ☐ in figure 3.3:
54 – 32 = 54 – 10 – 10 – 10 – 1 – 1 = 22
Like a splitting approach, jumping also allows students to de-
compose numbers through chunking in a “jump-up” (addition) or
“jump-down” (subtraction) approach.

Fig. 3.3. Using the jump strategy for 54 - 32 = ☐

The third approach is the “compensation” strategy, which al-

lows students to capitalize on the derived strategies for basic facts
discussed in chapter 1 by creating other, more “friendly” numbers to
think about and work with in solving problems. For example,
to solve the addition problem 148 + 33 = ☐, a student might add
150 + 33 = 183 and then subtract 183 – 2 = 181.
Eventually, students will be ready to develop the standard
paper-and-pencil algorithms, linked to conceptual understandings
of how they work, as one way to perform addition and subtraction
problems. As discussed by Verschaffel, Greer, and De Corte (2007)
“standard algorithms tend to mask the underlying principles that
Challenges: Learning, Teaching, and Assessing 77

make them work” (p. 590). Students who memorize a standard

algorithm, but do not understand it, are more likely to make errors
and forget how to use it (Thompson 1999). This is particularly evi-
dent when students focus on digits rather than the number (Fuson
1992). A focus on number rather than digits also honors the in-
tersecting relationship of place value and multi-digit addition and
subtraction that is emphasized in Essential Understanding 2c. For Essential
example, when thinking about a problem like 60 – 58 = ☐, a stu- Understanding 2c
dent who is using a digits approach will carry out the algorithm by Place-value concepts
focusing first on the ones digits and then on the tens, while a stu- provide a convenient
dent who is using a number approach will think about the numbers way to compose and
and mentally “see” that the difference between 58 and 60 is 2. decompose numbers
There are cultural differences in the use of algorithms, and stu- to facilitate addi-
dents might bring to your classroom new algorithmic approaches tion and subtraction
that reflect their prior knowledge or their families’ experiences. computations.
These algorithms may also inform their current learning. For exam-
ple, consider the Austrian method, discussed in chapter 1 (pp. 45–46).
A popular subtraction algorithm used in many other countries and
frequently known as “equal addition,” or “add tens to both,” this
method is based on the fact that adding the same amount to the
minuend and the subtrahend will not change the difference between
the two. So, in finding the difference indicated by 43 – 28 = ☐, a
student using “equal addition” would mentally add 10 to the 3 in
the ones column of the 43, thereby obtaining 13 in the ones col-
umn. To counteract the effect of the addition of that 10, the student
would mentally add 10 to the 28 (subtrahend) and then subtract 38.
Because the “equal addition” process might not be familiar to
you, you might try it to experience how it works (for an example,
see fig. 1.47 and the discussion leading up to it on p. 45). You will
find that this approach (among others) can serve as a wonderful
context for exploring subtraction (or addition) and place value.
Your heightened awareness of cultural differences in addition and
subtraction algorithms can increase the comfort and sense of con-
nectedness experienced by ELLs, who bring different knowledge to
the classroom. These alternative algorithms also provide strategies
that may make more sense to struggling learners who do not under-
stand a traditional algorithm.

Assessing Evidence of Learning

Doug Clarke, a researcher in mathematics education, has teamed
with colleagues to describe a powerful trajectory of growth points
in the learning of addition and subtraction. This trajectory provides
a framework for creating diagnostic assessment that taps students’
fundamental understanding. The team’s Early Numeracy Research
78 Addition and Subtraction

Project (ENRP) includes developmental assessment tasks or diagnos-

tic interviews for each of the six “growth points” (Clarke 2001) that
are summarized in the chart in figure 3.4. The growth points mark
key transitions in students’ growth from one type of strategy to
another. These growth points are related to but not identical with
the strategies named and described in this book. For example,
students reach the “count-on” growth point when they move from a
“counting all” strategy to a “counting on” strategy.

Growth Point Description

Count all (two collections) Student counts all to find the total
of two collections

Count on Student counts on from one

number to find the total of two
collections (eventually counting
from the higher number)

Count back, count down to, count Given a subtraction situation,

up from student chooses appropriately
from strategies including counting
back, counting down to, and
counting up from

Basic strategies (doubles, Given an addition or subtraction

commutativity, add 10, tens facts, problem, student uses strategies
other known facts) such as doubles, commutativity,
adding 10, tens facts, or other
known facts

Derived strategies (near doubles, Given an addition or subtraction

add 9, build to next 10, fact problem, student uses strategies
families, intuitive strategies) such as near doubles, adding 9,
build to next 10, fact families,
or intuitive strategies

Extending and applying addition Given a range of tasks (including

and subtraction by using basic, problems with multi-digit
derived, and intuitive strategies numbers), student can complete
them mentally, using the
appropriate strategies and a clear
understanding of key concepts

Fig. 3.4. A chart summarizing Clarke’s (2001, p. 213) six key growth
points in the domain of addition and subtraction strategies

These growth points align with our discussion of mathematics

content in chapter 1. Diagnostic interviews are a means to assess
students as they move forward on this trajectory. These tools
Challenges: Learning, Teaching, and Assessing 79

can also explore the knowledge and thinking of students as they

consider place-value concepts and addition and subtraction beyond
the basic facts.
Diagnostic interviews also extend to more sophisticated under-
standing. For example, in a classic diagnostic interview, Ross (2002)
explored students’ understanding of the “ten as one” concept, which
is central to place value. A version of the original task follows:
Given a set of 36 counters, students are asked first to count
them and then to write the corresponding number. When they
have written 36, the interviewer circles the digit 6 and poses the
question, “Does this part of your 36 have anything to do with
how many counters there are?” Then the interviewer circles the
digit 3 and repeats the question.
A diagnostic interview is meant to be a snapshot of students’ un-
derstanding. The researchers evaluated the students’ knowledge ac-
cording to where their performances fell on a spectrum from merely
selecting 6 counters and then 3 counters to a full understanding of
the 3 as three tens and the 6 as six ones. For students to add or sub-
tract skillfully, especially in cases involving regrouping, they need
to demonstrate a full understanding of the value of each digit in the
number. In Ross’s study, many students did not fully understand the
single-digit number in the tens column as 30. A student who lacks
this understanding has little hope of carrying out addition and sub-
traction successfully and meaningfully.
Another diagnostic interview takes place when the teacher asks
a student to complete an addition or subtraction and talk about
what he or she is thinking in the process. For example, the teacher
might say, “Add 99 and 145.” A variety of performances might
result. Some young students might try to align these numbers in
columns (with some students misaligning the 99 over the 14 portion
of 145), while older or more capable students might treat the 99 as
100 and then add it with a subtraction of 1 at the end to counteract
their original jump to 100. This last example of evidence of student
thinking clearly shows links to the study of derived facts as dis- Essential
cussed previously in relation to Essential Understanding 2a. Understanding 2a
One interesting aspect of the use of diagnostic interviews is the
The commutative
positive disposition toward mathematics that students seem to de-
and associative
velop as they are asked to share and discuss their thinking individu-
properties for
ally with the teacher (Clarke 2001). As the students reflect on their addition of whole
thinking, their own awareness of their use of strategies is height- numbers enable
ened. This gives them increased confidence to express themselves computations to be
and believe that their effort and thinking are valued in the learning performed flexibly.
process. The interviews also provide teachers with precise evidence
of the students’ strengths and weaknesses, and this information can
help in tailoring instruction and supporting all learners.
80 Addition and Subtraction

Is Subtraction Harder than Addition?

Researchers who have explored whether subtraction is more dif-
ficult than addition have said yes, it is (Baroody 1984; Fuson 1984).
Consider the problem 9 + 3 = ☐. Counting on 3 more after 9 gives
10, 11, 12. This process corresponds to a visual display of 9 items
and the action of introducing 3 more. Both of these methods align
nicely with the concrete, semi-concrete (the extension of fingers or
tally marks) and oral methods that students use. The case of sub-
traction is a bit different. Verbal solution methods in this case can
be seen as “counting down to” a number, “counting down from”
a number and “counting up to” a number (Fuson 1984). First, it
should be noted that the usual concrete model for a problem such
as 12 – 4 = ☐ would be a display of 12 items from which 4 would
be removed, and then the remaining items would be recounted. This
model does not match the three methods of “counting down to,”
“counting down from,” or “counting up to.” When students use a
verbal solution approach, they must also decide whether to count
down starting at the first number in the problem or at the number
that is one less. So, for example, with 12 – 4 = ☐, students might
verbalize “counting down to” as “12, 11, 10, 9, and then the answer
is 8,” though they might articulate “counting down from” as “11,
10, 9, 8, for an answer of 8.” These distinctions are important since
students can accidentally move between these approaches and be-
come confused about the answer. This is the same confusion that
emerges on initial use of the number line, when students confuse
the unit of length with the points labeled by a number (and count
the points instead of the spaces between them, a misunderstanding
For an extended that Dougherty and colleagues [2010] address in their discussion
discussion of unit, of unit, count, measure, and number). In fact, Fuson suggests that
count, measure, students would benefit from the “counting up to” approach since
and number, see
they can apply it consistently and then “think addition” for subtrac-
Developing Essential
tion situations. Also, a “counting up to” approach supports students
Understanding
of Number and who have issues with working memory or some ELLs who find the
Numeration backward counting linguistically challenging. Furthermore, subtrac-
for Teaching tion may be harder because students calculate differences from their
Mathematics in knowledge of sums, making subtraction a two-part process for some
Prekindergarten– students. For example, in trying to find 15 – 7 = ☐, the first part is
Grade 2 (Dougherty identifying the corresponding addition problem of 7 + 8 = 15, and the
et al. 2010). second part is using that information to generate the answer of 8.

Conclusion
This chapter has examined the pedagogical considerations that
emerge from thinking about the ways in which addition and
subtraction are used to represent and solve many different kinds
Challenges: Learning, Teaching, and Assessing 81

of problems. Becoming mathematically proficient in addition and

subtraction (and developing proficiency in all five UCARE compo-
nents) requires a learning environment that focuses on meaningful
approaches to interpreting story problems, flexible ways to record
equations, multiple ways to solve addition or subtraction problems,
and numerous opportunities to connect concrete representations to
abstract concepts. Furthermore, only through thoughtful conversa-
tions during mathematics instruction and by listening carefully for
evidence of students’ understanding can you diagnose your students’
misunderstandings, identify their strengths, and help them move
ahead.
Together, the three chapters in this book can serve as a re-
source for developing your mathematical understanding of addition
and subtraction and helping you translate those understandings into
engaging learning experiences for your students. First and second
graders can often add quite well, using at least one algorithm or
particular model. If you take away (no pun intended!) just one idea
from this book, it should be that it is essential to engage students
in using different models and a range of strategies (algorithms) for
solving addition and subtraction problems. By doing so, you will
position your students to develop a deep understanding of addition
and subtraction, equipping them as completely as possible for
success in later mathematical topics and a lifetime of competence in
mathematics.
82

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id id
A Series for Teaching Mathematics

Developing Essential Understanding of Addition and Subtraction • Pre-K–Grade 2

Developing Essential Understanding

What is the relationship between addition and subtraction? How do you know
whether an algorithm will always work? Can you explain why order matters in
subtraction but not in addition, or why it is false to assert that the sum of any two
whole numbers is greater than either number?
Developing
Essential
How much do you know … and how much do you need to know?

Helping your students develop a robust understanding of addition and subtraction

requires that you understand this mathematics deeply. But what does that mean?
This book focuses on essential knowledge for teachers about addition and subtrac-
Understanding
tion. It is organized around two big ideas, supported by multiple smaller, intercon- of
nected ideas—essential understandings. Taking you beyond a simple introduction to

Addition &
these operations, the book will broaden and deepen your mathematical understand-
ing of one of the most challenging topics for students—and teachers. It will help
you engage your students, anticipate their perplexities, avoid pitfalls, and dispel
misconceptions. You will also learn to develop appropriate tasks, techniques, and
tools for assessing students’ understanding of the topic.

Subtraction
Focus on the ideas that you need to understand thoroughly to teach confidently.

Move beyond the mathematics you expect your students to

learn. Students who fail to get a solid grounding in pivotal
concepts struggle in subsequent work in mathematics and related
Essential
Understanding
Series
iea
d
disciplines. By bringing a deeper understanding to your teaching, you can help
students who don’t get it the first time by presenting the mathematics in multiple ways.
The Essential Understanding Series addresses topics in school mathematics that are
Pre-K–Grade 2
critical to the mathematical development of students but are often difficult to teach.
Each book in the series gives an overview of the topic, highlights the differences between
what teachers and students need to know, examines the big ideas and related essential
understandings, reconsiders the ideas presented in light of connections with other
mathematical ideas, and includes questions for readers’ reflection. NCTM