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# Reconceptualizing Mathematics

## Third EditionJudith Sowder; Larry Sowder; Susan Nickerson

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A new edition to further prepare you to impact your students’ mathematical understanding.

*Reconceptualizing Mathematics*, founded on research and studies of learning and mathematics teaching for many years, offers you an active part in learning and experience doing math. It’s the only textbook of its kind to both incorporate aspects of student-centered learning into lessons and model the teaching that will eventually be expected of you. To this end, it provides you with active learning through tasks, activities, and support for facilitating productive classroom discussions. Quantitative reasoning and problem solving are recurring themes in *Reconceptualizing Mathematics*, and it teaches you to understand the quantities embedded in a situation and how they relate to each other. The text also features learning exercises for you to practice what you’ve learned, and take-away messages summarizing each section that are ideal for helping you prepare for exams.

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**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