# Reconceptualizing Mathematics

## for Elementary School Teachers Author(s): Judith Sowder; Larry Sowder; Susan Nickerson; Ian Whitacre Fourth Edition ©2023 Publication Date: March 28, 2023 E-book ISBN: 9781319511845

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...

ISBN: 9781319511821
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ISBN: 9781319511845
Reconceptualizing Mathematics

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Part I Reasoning About Numbers and Quantities
Chapter 1
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

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

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