Reconceptualizing Mathematics

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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; 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; 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 Rela.

**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** 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** 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.

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