Reconceptualizing Mathematics

for Elementary School Teachers

Fourth Edition

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...

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Part I Reasoning About Numbers and Quantities
Chapter 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

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! 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

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.

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