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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ISBN: 9781319511845
Reconceptualizing Mathematics


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                                                  
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                                                                
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                                                                 
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                                                                      
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                                                                         
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                                             
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                                                
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                                                                     
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                                                  
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                                                                   
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                            
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           
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                                                     
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                                                                  
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                                                                               
PART III Reasoning About Shapes and Measurement                                                                      
Chapter 16
       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                                     
Chapter 17
       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                                                                     
Chapter 18
       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?                                   
Chapter 19
       Goals for Reconceptualizing Mathematics                                                                                   
      19.1    Tessellating the Plane                                                                                                     
      19.2    Tessellating Space                                                                                                          
Chapter 20
       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                                               
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                                                
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                                        
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                                                 
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                                                    
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?                            
Chapter 26
Special Topics in Measurement                                                                                                         
      Goals for Reconceptualizing Mathematics                                                                                   
      26.1    The Pythagorean Theorem                                                                                               
      26.2    Some Other Kinds of Measurements                                                                                
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                                   
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?                                 
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                                                                                           
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                                                                   
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                                                                   
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?             
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                                                          

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