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# Reconceptualizing Mathematics

## for Elementary School TeachersFourth Edition| ©2023 Judith Sowder; Larry Sowder; Susan Nickerson; Ian Whitacre

*Reconceptualizing Mathematics*, founded on research and studies of mathematics teaching and learning, promotes an active experience in learning by doing in math. The esteemed author team has written the only textbook of its kind to both incorporate aspects of student-centered learning int...

*Reconceptualizing Mathematics*, founded on research and studies of mathematics teaching and learning, promotes an active experience in learning by doing in math. The esteemed author team has written the only textbook of its kind to both incorporate aspects of student-centered learning into lessons and model the teaching that will be expected of their students. This active approach is brought to life online in Achieve with flexible tools like hands-on activity guides, Desmos-powered interactives, and a variety of assessment types designed to coach students toward a deeper conceptual understanding.

ISBN:9781319483050

Access all your course tools in one place!

ISBN:9781319483135

Take notes, add highlights, and download our mobile-friendly e-books.

ISBN:9781319303730

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ISBN:9781319499280

This package includes Achieve and Loose-Leaf.

*Reconceptualizing Mathematics*, founded on research and studies of mathematics teaching and learning, promotes an active experience in learning by doing in math. The esteemed author team has written the only textbook of its kind to both incorporate aspects of student-centered learning into lessons and model the teaching that will be expected of their students. This active approach is brought to life online in Achieve with flexible tools like hands-on activity guides, Desmos-powered interactives, and a variety of assessment types designed to coach students toward a deeper conceptual understanding.

Features

A "Goals for Reconceptualizing Mathematics" section begins each chapter, providing the student with an overview of the chapter content, to stimulate student interest.

- Discussions are intended primarily for whole-class discussion. These provide more opportunities to converse about the mathematics being learned, to listen to the reasoning of others, and to voice disagreement when appropriate.
- Activities are intended to be worked on in small groups or pairs, providing some hands-on experience with the content. In most instances, they can be completed and discussed in class. Discussion of activities is worthwhile because many times, other groups will take a different approach.
- Links to media interactives, including Dynamic Figures and Interactive Figures (software apps) in Activities and Learning Exercises, provide a platform for students to solve problems with software tools such as Desmos, Geogebra, and virtual manipulatives.
- Key Concepts and definitions are highlighted in every chapter. This information needs to be considered thoughtfully to fully understand it, with examples provided to help the student make sense of the concepts.
- Examples provide needed clarification and opportunities to explore meaning and demonstrate procedures.
- Learning Exercises for homework or classroom discussion provide additional opportunities for problem solving and practice. They are intended primarily to help students think through the section content, to note the relationships, and to extend what they have learned.
- Issues for Learning sections offer engaging, informative highlights of some of the research about children’s learning of topics associated with the chapter content.
- Think About questions appear throughout the text, asking the student to pause and think about the content being presented.
- Take-Away Messages fall at the end of each section, summarizing what the student needs to know.
- Elementary Textbook Pages show how the mathematics being presented is connected to the classroom.
- Skills refreshers for basic arithmetic, protractor use, and working in Excel are available online, to help the student in working with these tools.

New to This Edition

- NEW! All online resources, assessment and e-book housed in Achieve.
- Increased use of technology throughout. This includes links to external sites or to newly developed digital manipulatives with Desmos or Geogebra.
- Increased focus on in-class activities, adding more depth to current activities. Select Discussions and Activities from the text have been adapted to Instructor Activity Guides, housed online in Achieve.
- Within Chapter 4 Some Conventional Ways of Computing, sections have been created to separate addition and subtraction (Section 4.1) and multiplication and division (Section 4.2). Chapter 4’s Issues for Learning feature deepens the discussion of the Role of Algorithms, emphasizing the importance of understanding the standard algorithms conceptually.
- New activities and interactives added for the following chapters:
- Chapter 7 Making Sense of Fraction Operations
- Chapter 15 Further Topics in Algebra and Change
- Chapter 22 Transformation Geometry
- Chapter 23 Measurement Basics
- Chapter 28 Determining More Complicated Probabilities

**
Reconceptualizing Mathematics**

Fourth Edition| ©2023

Judith Sowder; Larry Sowder; Susan Nickerson; Ian Whitacre

# Digital Options

## Achieve

Achieve is a comprehensive set of interconnected teaching and assessment tools that incorporate the most effective elements from Macmillan Learning's market leading solutions in a single, easy-to-use platform.

## E-book

Read online (or offline) with all the highlighting and notetaking tools you need to be successful in this course.

**Reconceptualizing Mathematics**

Fourth Edition| 2023

Judith Sowder; Larry Sowder; Susan Nickerson; Ian Whitacre

## Table of Contents

Part I Reasoning About Numbers and QuantitiesChapter 1

Reasoning About Quantities

Goals for Reconceptualizing Mathematics

1.1 Ways of Thinking About Solving Story Problems

1.2 Quantitative Analysis

1.3 Problem Solving

1.4 Issues for Learning: Ways of Illustrating Story Problems

Review

Chapter 2

Numeration Systems

Goals for Reconceptualizing Mathematics

2.1 Ways of Expressing Values of Quantities

2.2 Place Value

2.3 Bases Other Than Ten

2.4 Operations in Different Bases

2.5 Issues for Learning: Understanding Place Value

Review

Chapter 3

Understanding Whole Number Operations

Goals for Reconceptualizing Mathematics

3.1 Ways of Thinking About Addition and Subtraction

3.2 Children’s Ways of Adding and Subtracting

3.3 Ways of Thinking About Multiplication

3.4 Ways of Thinking About Division

3.5 Children Find Products and Quotients

3.6 Issues for Learning: Developing Number Sense

Review

Chapter 4

Some Conventional Ways of Computing

Goals for Reconceptualizing Mathematics

4.1 Addition and Subtraction on Whole Numbers and Decimal Numbers

4.2 Issues for Learning: The Role of Algorithms

Review

Chapter 5

Using Numbers in Sensible Ways

Goals for Reconceptualizing Mathematics

5.1 Mental Computation

5.2 Computational Estimation

5.3 Estimating Values of Quantities

5.4 Using Scientific Notation for Estimating Values of Very Large and Very Small Quantities

5.5 Issues for Learning: Mental Computation

Review

Chapter 6

Meanings for Fractions

Goals for Reconceptualizing Mathematics

6.1 Understanding the Meanings of

6.2 Comparing Fractions

6.3 Equivalent (Equal) Fractions

6.4 Relating Fractions, Decimals, and Percentages

6.5 Issues for Learning: Understanding Fractions and Decimals

Review

Chapter 7

Making Sense of Fraction Operations

Goals for Reconceptualizing Mathematics

7.1 Problem Solving with Fractions

7.2 Making Sense of Addition and Subtraction of Fractions

7.3 Making Sense of Multiplication of Fraction

7.4 Making Sense of Division of Fractions

7.5 Issues for Learning: Teaching Fraction Operations Review

Review

Chapter 8

Multiplicative Comparisons and Multiplicative Reasoning

Goals for Reconceptualizing Mathematics

8.1 Quantitative Analysis of Multiplicative Situations

8.2 Fractions in Multiplicative Comparisons

8.3 Issues for Learning: Standards for Learning

Review

Chapter 9

Ratios, Rates, Proportions, and Percentages

Goals for Reconceptualizing Mathematics

9.1 Ratio as a Measure

9.2 Comparing Ratios

9.3 Percentages in Comparisons and Changes

9.4 Issues for Learning: Developing Proportional Reasoning

Review

Chapter 10

Integers and Other Number Systems

Goals for Reconceptualizing Mathematics

10.1 Big Ideas About Signed Numbers

10.2 Children’s Ways of Reasoning About Signed Numbers

10.3 Other Models for Signed Numbers

10.4 Operations with Signed Numbers

10.5 Multiplying and Dividing Signed Numbers

10.6 Number Systems

10.7 Issues for Learning: Open Number Sentences

Review

Chapter 11

Number Theory

Goals for Reconceptualizing Mathematics

11.1 Factors and Multiples, Primes and Composites

11.2 Prime Factorization

11.3 Divisibility Tests to Determine Whether a Number Is Prime

11.4 Greatest Common Factor, Least Common Multiple

11.5 Issues for Learning: Understanding the Unique Factorization Theorem

Review

Part II Reasoning About Algebra and Change

Chapter 12

What Is Algebra?

Goals for Reconceptualizing Mathematics

12.1 Algebraic Reasoning in Elementary School

12.2 Numerical Patterns and Algebra

12.3 Functions and Algebra

12.4 Algebra as Generalized Arithmetic

12.5 Algebraic Reasoning About Quantities

12.6 Issues for Learning: The National Assessment of Educational Progress and Achievement in Algebra

Review

Chapter 13

A Quantitative Approach to Algebra and Graphing

Goals for Reconceptualizing Mathematics

13.1 Using Graphs and Algebra to Show Quantitative Relationships

13.2 Understanding Slope: Making Connections Across Quantitative Situations, Graphs, and Algebraic Equations

13.3 Linear Functions and Proportional Relationships

13.4 Nonlinear Functions

13.5 Issues for Learning: Algebra in the Elementary Grades

Review

Chapter 14

Understanding Change: Relationships Among Time, Distance, and Rate

Goals for Reconceptualizing Mathematics

14.1 Distance–Time and Position–Time Graphs

14.2 Using Motion Detectors

14.3 Graphs of Speed Against Time

14.4 Interpreting Graphs

14.5 Issues for Learning: Common Graphing Errors

Review

Chapter 15

Further Topics in Algebra and Change

Goals for Reconceptualizing Mathematics

15.1 Finding Linear Equations

15.2 Solving Two Linear Equations in Two Variables

15.3 Different Approaches to Problems

15.4 Average Speed and Weighted Averages

15.5 More About Functions

15.6 Issues for Learning: Topics in Algebra

Review

PART III Reasoning About Shapes and Measurement

Chapter 16

Polygons

Goals for Reconceptualizing Mathematics

16.1 Review of Polygon Vocabulary

16.2 Organizing Shapes

16.3 Triangles and Quadrilaterals

16.4 A Focus on Problem-Solving Strategies

16.5 Issues for Learning: Some Research on Two-Dimensional Shapes

Review

Chapter 17

Polyhedra

Goals for Reconceptualizing Mathematics

17.1 Shoeboxes Have Faces and Nets!

17.2 Introduction to Polyhedra

17.3 Representing and Visualizing Polyhedra

17.4 Congruent Polyhedra

17.5 Some Special Polyhedra

17.6 Issues for Learning: Dealing with 3D Shapes

Review

Chapter 18

Symmetry

Goals for Reconceptualizing Mathematics

18.1 Symmetry of Shapes in a Plane

18.2 Symmetry of Polyhedra

18.3 Issues for Learning: What Is Geometry in the Pre-K–8 Curriculum?

Review

Chapter 19

Tessellations

Goals for Reconceptualizing Mathematics

19.1 Tessellating the Plane

19.2 Tessellating Space

Review

Chapter 20

Similarity

Goals for Reconceptualizing Mathematics

20.1 Similarity and Dilations in Planar Figures

20.2 More About Similar Figures

20.3 Similarity in 3D Figures

20.4 Issues for Learning: Similarity and Proportional Reasoning

Review

Chapter 21

Curves, Constructions, and Curved Surfaces

Goals for Reconceptualizing Mathematics

21.1 Planar Curves and Constructions

21.2 Curved Surfaces

21.3 Issues for Learning: Standards for Mathematical Practice

Review

Chapter 22

Transformation Geometry

Goals for Reconceptualizing Mathematics

22.1 Some Types of Rigid Motions

22.2 Finding Images for Rigid Motions

22.3 A Closer Look at Some Rigid Motions

22.4 Composition of Rigid Motions

22.5 Transformations and Earlier Topics

22.6 Issues for Learning: Promoting Visualization in the Curriculum

Review

Chapter 23

Measurement Basics

Goals for Reconceptualizing Mathematics

23.1 Conceptualizing and Measuring Length

23.2 Key Ideas of Measurement

23.3 Conceptualizing and Measuring Angle

23.4 Further Exploration of Angles

23.5 Issues for Learning: Measurement of Length and Angle Size

Review

Chapter 24

Area, Surface Area, and Volume

Goals for Reconceptualizing Mathematics

24.1 Area and Surface Area

24.2 Volume

24.3 Issues for Learning: Measurement of Area and Volume

Review

Chapter 25

Counting Units Fast: Measurement Formulas

Goals for Reconceptualizing Mathematics

25.1 Circumference, Area, and Surface Area Formulas

25.2 Volume Formulas

25.3 Issues for Learning: What Measurement Topics Are in the Curriculum?

Review

Chapter 26

Special Topics in Measurement

Goals for Reconceptualizing Mathematics

26.1 The Pythagorean Theorem

26.2 Some Other Kinds of Measurements

Review

PART IV Reasoning About Chance and Data

Chapter 27

Quantifying Uncertainty

Goals for Reconceptualizing Mathematics

27.1 Understanding Chance Events

27.2 Methods of Assigning Probabilities

27.3 Simulating Probabilistic Situations

27.4 Issues for Learning: Research on the Understanding of Probability

Review

Chapter 28

Determining More Complicated Probabilities

Goals for Reconceptualizing Mathematics

28.1 Tree Diagrams and Lists for Multistep Experiments

28.2 Probability of One Event or Another Event

28.3 Probability of One Event and Another Event

28.4 Conditional Probability

28.5 Issues for Learning: What Does it Mean to Understand Probability?

Review

Chapter 29

Introduction to Statistics and Sampling

Goals for Reconceptualizing Mathematics

29.1 What Are Statistics?

29.2 Sampling: The Why and the How

29.3 Simulating Random Sampling

29.4 Types of Data

29.5 Conducting a Survey

29.6 Issues for Learning: Sampling

Review

Chapter 30

Representing and Interpreting Data with One Variable

Goals for Reconceptualizing Mathematics

30.1 Representing Categorical Data

30.2 Representing and Interpreting Measurement Data

30.3 Examining the Spread of Data

30.4 Measures of Center

30.5 Deviations from the Mean as Measures of Spread

30.6 Examining Distributions

30.7 Issues for Learning: Understanding the Mean

Review

Chapter 31

Dealing with Multiple Data Sets or with Multiple Variables

Goals for Reconceptualizing Mathematics

31.1 Comparing Data Sets

31.2 Lines of Best Fit and Correlation

31.3 Issues for Learning: More Than One Variable

Review

Chapter 32

Variability in Samples

Goals for Reconceptualizing Mathematics

32.1 Having Confidence in a Sample Statistic

32.2 Confidence Intervals

32.3 Issues for Learning: What Probability and Statistics Should Be in the Curriculum?

Review

Chapter 33

Special Topics in Probability

Goals for Reconceptualizing Mathematics

33.1 Expected Value

33.2 Permutations and Combinations

33.3 Issues for Learning: Children Finding Permutations

Review

**Reconceptualizing Mathematics**

Fourth Edition| 2023

Judith Sowder; Larry Sowder; Susan Nickerson; Ian Whitacre

## Authors

### Judith Sowder

**Judith Sowder**is a Professor Emerita of Mathematics and Statistics at San Diego State University. Her research has focused on the development of number sense and on the instructional effects of teachers mathematical knowledge at the elementary and middle school level. She served from 1996 to 2000 as editor of the Journal for Research in Mathematics Education and served a three-year term on the National Council of Teachers of Mathematics Board of Directors. She has directed numerous projects funded by the National Science Foundation and the Department of Education. In 2000 she received the Lifetime Achievement Award from the National Council of Teachers of Mathematics.

### Larry Sowder

**Larry Sowder**is Professor Emeritus of Mathematics and Statistics at San Diego State University. He taught mathematics to preservice elementary school teachers for more than 30 years. His work in a special program in San Diego elementary schools also shaped his convictions about how courses in mathematics for preservice teachers should be pitched, as did his joint research investigating how children in the usual Grades 4-8 curriculum solve "story" problems. He served on teh National Research Council Committee that published

*Educating Teachers of Science, Mathematics, and Technology*(NRC, 2001).

### Susan Nickerson

**Susan Nickerson**is an Associate Professor in San Diego State Universitys Department of Mathematics and Statistics. Her research interest is long-term professional development of elementary and middle school teachers. In particular, her focus is describing, analyzing, and understanding effective contexts that promote teachers knowledge of mathematics and mathematics teaching.

### Ian Whitacre

**Ian Whitacre** is an Associate Professor in Elementary Mathematics Education in the School of Teacher Education at Florida State University. He studies mathematical thinking and learning at the elementary and middle-school levels. He has also done extensive research specifically focused on preservice elementary teachers’ development of number sense. He regularly teaches courses for preservice elementary teachers and is always looking for ways to better support their learning.

**Reconceptualizing Mathematics**

Fourth Edition| 2023

Judith Sowder; Larry Sowder; Susan Nickerson; Ian Whitacre

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Judith Sowder; Larry Sowder; Susan Nickerson; Ian Whitacre | Fourth Edition | ©2023 | ISBN:9781319508913

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**Reconceptualizing Mathematics**

Fourth Edition| 2023

Judith Sowder; Larry Sowder; Susan Nickerson; Ian Whitacre

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