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- Reconceptualizing Mathematics

# Reconceptualizing Mathematics

## for Elementary School TeachersThird Edition| ©2017 Judith Sowder; Larry Sowder; Susan Nickerson

*Reconceptualizing Mathematics*, founded on research and studies of learning and mathematics teaching for many years, is designed for use in classrooms in which students take an active part in learning and experience doing math. The esteemed author team has written the only textbook of...

*Reconceptualizing Mathematics*, founded on research and studies of learning and mathematics teaching for many years, is designed for use in classrooms in which students take an active part in learning and experience doing 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. To this end, the authors provide worthwhile tasks, activities, and support for facilitating discussions.

Quantitative reasoning and problem solving are recurring themes in *Reconceptualizing Mathematics*. The authors approach problem solving that teaches students to understand the quantities embedded in the situation and how they relate to each other.**Maximize Teaching and Learning with WebAssign Premium**

Macmillan Learning and WebAssign have partnered to deliver WebAssign *Premium* – a comprehensive and flexible suite of resources for your math for teachers course. Combining the most widely used online homework platform with authoritative textbook content and Macmillan’s esteemed Mathtools, WebAssign *Premium* extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at www.webassign.net/whfreeman.

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##
**This new edition further prepares future teachers to impact students’ mathematical understanding.**

**This new edition further prepares future teachers to impact students’ mathematical understanding.**

*Reconceptualizing Mathematics*, founded on research and studies of learning and mathematics teaching for many years, is designed for use in classrooms in which students take an active part in learning and experience doing 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. To this end, the authors provide worthwhile tasks, activities, and support for facilitating discussions.

Quantitative reasoning and problem solving are recurring themes in *Reconceptualizing Mathematics*. The authors approach problem solving that teaches students to understand the quantities embedded in the situation and how they relate to each other.**Maximize Teaching and Learning with WebAssign Premium**

Macmillan Learning and WebAssign have partnered to deliver WebAssign *Premium* – a comprehensive and flexible suite of resources for your math for teachers course. Combining the most widely used online homework platform with authoritative textbook content and Macmillan’s esteemed Mathtools, WebAssign *Premium* extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at www.webassign.net/whfreeman.

Features

• **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** intended to be worked in small groups or pairs, providing some hands-on experiences with the content. In most instances, they can be completed and discussed in class. Discussion on activities is worthwhile because many times other groups will take a different approach.

•** Definitions/Key Concepts** are boxed and contain definitions and other important ideas. This information needs to be considered very thoughtfully to fully understand it. Examples are provided to help the student make sense of them.

• **Examples** provide needed clarification and opportunities to explore meaning and demonstrate procedures.

• ** Learning Exercises** for homework or classroom discussion provide opportunity for practice. They are intended primarily to help students think through section content, note the relationships, and 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 and ask students to pause and think about the content being presented.

•

**Take-Away Messages**fall at the end of each section and summarize what the student should know.

•

**Elementary Textbook Pages**show how the mathematics being presented is connected to the classroom.

•

**Notes columns**appear alongside text in each chapter.

•

**Skills refreshers**for basic arithmetic; protractor use; and instructions TI-73, and Excel are available online to help brushing up.

New to This Edition

**OVERALL CHANGES**•

**NEW “Focus on SMP,” based on Standards for Mathematical Practice sections**now explicitly discusses and points to places where students are engaging in the mathematical practices, and addresses one of the most difficult-to-understand aspects of the CCSS.

•

**Coverage of problem solving**has been expanded throughout the book. The focus on quantitative reasoning continues, but has been expanded to discuss other problem-solving strategies. New sections devoted to discussing and giving examples of problem-solving strategies have been added to each of the parts.

•

**New examples of student work**provide contemporary examples to connect students with classroom learning.

•

**Updated design**is brighter and more engaging for students, with more visuals added full-colored art (instead of the current 2-color art).

• The classification and purpose of the features has been made clearer so students know their intentions.

**CONTENT CHANGES**•

**Greater incorporation of mathematical models**keeps with the emphasis in CCSS. The models are rectangular array/area, number line, base-ten blocks, ratio tables) into the examples, activities, and exercises, as well as line plots (or dot plots).

•

**A rewrite of Chapter 3**now explicitly discusses problem types of CGI and Common Core.

•

**Chapter 6**now discusses the progression of fraction development in the Standards.

•

**Part II**begins with a focus on early algebra—children's algebraic thinking.

- Chapter 12 has significant changes in line with the new focus on early algebra.
- More examples in the treatment of graphing in Chapters 13 and 14.
- Chapter 20’s discussion of Transformation ties more tightly to Common Core.

• Part III has been elaborated in many ways, including more (non-) examples and richer treatment of Pythagorean theorem, and more references to dynamic geometry software.

• Part IV data sets and examples were updated.

LaunchPad, our online course space, combines an

**interactive e-Book**with high-quality multimedia content and ready-made assessment options, including LearningCurve adaptive quizzing and coded machine-gradable exercises from the textbook. Content is easy to assign or adapt with your own material, such as readings, videos, quizzes, discussion groups, and more. LaunchPad also provides access to a Gradebook that offers a window into your students' performance—either individually or as a whole. Use LaunchPad on its own or integrate it with your school’s learning management system so your class is always on the same page. To learn more about LaunchPad for**, or to request access, go to launchpadworks.com.***Reconceptualizing Mathematics,*Third Edition
Look Inside

**
Reconceptualizing Mathematics**

Third Edition| ©2017

Judith Sowder; Larry Sowder; Susan Nickerson

# Digital Options

## Launchpad

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**Reconceptualizing Mathematics**

Third Edition| 2017

Judith Sowder; Larry Sowder; Susan Nickerson

## Table of Contents

**PART I: REASONING ABOUT NUMBERS AND QUANTITIES**

Chapter 1: Reasoning About Quantities

1.1 Ways of Thinking About Solving Story Problems

Chapter 1: Reasoning About Quantities

1.2 Quantitative Analysis

1.3 Problem Solving

1.4 Issues for Learning: Ways of Illustrating Story Problems

1.5 Check Yourself

**Chapter 2: Numeration Systems**

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

2.6 Check Yourself

**Chapter 3: Understanding Whole Number Operations**

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

3.7 Check Yourself

**Chapter 4: Some Conventional Ways of Computing**

4.1 Operating on Whole Numbers and Decimal Numbers

4.2 Issues for Learning: The Role of Algorithms

4.3 Check Yourself

**Chapter 5: Using Numbers in Sensible Ways**

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

5.6 Check Yourself

**Chapter 6: Meanings for Fractions**

6.1 Understanding the Meanings of a/b

6.2 Comparing Fractions

6.3 Equivalent (Equal) Fractions

6.4 Relating Fractions, Decimals, and Percents

6.5 Issues for Learning: Understanding Fractions and Decimals

6.6 Check Yourself

**Chapter 7: Computing with Fractions**

7.1 Adding and Subtracting Fractions

7.2 Multiplying by a Fraction

7.3 Dividing by a Fraction

7.4 Issues for Learning: Teaching Calculation with Fractions

7.5 Check Yourself

**Chapter 8: Multiplicative Comparisons and Multiplicative Reasoning**

8.1 Quantitative Analysis of Multiplicative Situations

8.2 Fractions in Multiplicative Comparisons

8.3 Issues for Learning: Standards for Learning

8.4 Check Yourself

**Chapter 9: Ratios, Rates, Proportions, and Percents**

9.1 Ratio as a Measure

9.2 Comparing Ratios

9.3 Percents in Comparisons and Changes

9.4 Issues for Learning: Developing Proportional Reasoning

9.5 Check Yourself

**Chapter 10: Integers and Other Number Systems**

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

10.8 Check Yourself

**Chapter 11: Number Theory**

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

11.6 Check Yourself

**PART II: REASONING ABOUT ALGEBRA AND CHANGE**

Chapter 12: What is Algebra?

12.1 Algebraic Reasoning in Elementary School

Chapter 12: What is Algebra?

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

12.7 Check Yourself

**Chapter 13: A Quantitative Approach to Algebra and Graphing**

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

13.6 Check Yourself

**Chapter 14: Understanding Change: Relationships Among Time, Distance, and Rate**

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

14.6 Check Yourself

**Chapter 15: Further Topics in Algebra and Change**

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

15.7 Check Yourself

**PART III: REASONING ABOUT SHAPES AND MEASUREMENT**

Chapter 16: Polygons

16.1 Review of Polygon Vocabulary

Chapter 16: Polygons

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

16.6 Check Yourself

**Chapter 17: Polyhedra**

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

17.7 Check Yourself

**Chapter 18: Symmetry**

18.1 Symmetry of Shapes in a Plane

18.2 Symmetry of Polyhedra

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

18.4 Check Yourself

**Chapter 19: Tessellations**

19.1 Tessellating the Plane

19.2 Tessellating Space

19.3 Check Yourself

**Chapter 20: Similarity**

20.1 Similarity and Dilations in Planar Figures

20.2 More About Similar Figures

20.3 Similarity in Space Figures

20.4 Issues for Learning: Similarity and Proportional Reasoning

20.5 Check Yourself

**Chapter 21: Curves, Constructions, and Curved Surfaces**

21.1 Planar Curves and Constructions

21.2 Curved Surfaces

21.3 Issues for Learning: Standards for Mathematical Practice

21.4 Check Yourself

**Chapter 22: Transformation Geometry**

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

22.7 Check Yourself

**Chapter 23: Measurement Basics**

23.1 Key Ideas of Measurement

23.2 Length and Angle Size

23.3 Issues for Learning: Measurement of Length and Angle Size

23.4 Check Yourself

**Chapter 24: Area, Surface Area, and Volume**

24.1 Area and Surface Area

24.2 Volume

24.3 Issues for Learning: Measurement of Area and Volume

24.4 Check Yourself

**Chapter 25: Counting Units Fast: Measurement Formulas**

25.1 Circumference, Area, and Surface Area Formulas

25.2 Volume Formulas

25.3 Issues for Learning: What Measurement is in the Curriculum?

25.4 Check Yourself

**Chapter 26: Special Topics in Measurement**

26.1 The Pythagorean Theorem

26.2 Some Other Kinds of Measurements

26.3 Check Yourself

**PART IV: REASONING ABOUT CHANCE AND DATA**

**Chapter 27: Quantifying Uncertainty**

27.1 Understanding Chance Events

27.2 Methods of Assigning Probabilities

27.3 Simulating Probabilistic Situations

27.4 Issues for Learning: Research on the Learning of Probability

27.5 Check Yourself

**Chapter 28: Determining More Complicated Probabilities**

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 Probability and Problem Solving

28.6 Check Yourself

**Chapter 29: Introduction to Statistics and Sampling**

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

29.7 Check Yourself

**Chapter 30: Representing and Interpreting Data with One Variable**

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

30.8 Check Yourself

**Chapter 31: Dealing with Multiple Data Sets or with Multiple Variables**

31.1 Comparing Data Sets

31.2 Lines of Best Fit and Correlation

31.3 Issues for Learning: More Than One Variable

31.4 Check Yourself

**Chapter 32: Variability in Samples**

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?

32.4 Check Yourself

**Chapter 33: Special Topics in Probability**

33.1 Expected Value

33.2 Permutations and Combinations

33.3 Issues for Learning: Children Finding Permutations

33.4 Check Yourself

**Appendix A: Video Clips Illustrating Children’s Mathematical Thinking**

Appendix B: Summary of Formulas

Appendix C: Using the Table of Randomly Selected Digits (TRSD)

Appendix D: Data Sets in Printed Form

Master for Nets

Appendix B: Summary of Formulas

Appendix C: Using the Table of Randomly Selected Digits (TRSD)

Appendix D: Data Sets in Printed Form

**Available Online or In Print via Custom**

Appendix E: About the Geogebra Lessons

Appendix F: A Review of Some Rules

Appendix G: Using a Protractor to Measure Angle Size

Appendix H: Using the TI-73

Appendix I: Using Excel

Appendix J: Using the Illuminations Website

Masters for Base Materials, Pattern Blocks, and Dot Paper

Appendix E: About the Geogebra Lessons

Appendix F: A Review of Some Rules

Appendix G: Using a Protractor to Measure Angle Size

Appendix H: Using the TI-73

Appendix I: Using Excel

Appendix J: Using the Illuminations Website

Look Inside

**Reconceptualizing Mathematics**

Third Edition| 2017

Judith Sowder; Larry Sowder; Susan Nickerson

## 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 University's 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.

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**Reconceptualizing Mathematics**

Third Edition| 2017

Judith Sowder; Larry Sowder; Susan Nickerson

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**Reconceptualizing Mathematics**

Third Edition| 2017

Judith Sowder; Larry Sowder; Susan Nickerson

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