Elementary Mathematics Pedagogical Content Knowledge: Powerful Ideas for Teachers

Most chapters include Preview, Review, Self-Check, Selected References, Online and and Exercises Introduction Chapter 1: The changing landscape of elementary mathematics teaching and learning A distinction between conceptual knowledge and procedural knowledge Problem-focused teaching Communication in a mathematics classroom Reasoning Patterns and Change Mathematical connections Chapter 2: The Powerful ideas: Composition, Decomposition, Relationships, Representation, and Context 39 Composition Decomposition Relationships Representation Context Chapter 3: Powerful ideas related to the number system What is counting? (How Many?) What are numbers? How do we represent numbers? What is a number system? How have number systems developed differently in the past (and what's so special about today's number system?) The idea of base Basic Digits and Place Value Composition Chapter 4: Powerful ideas related to arithmetic Addition and Subtraction Multiplication Division Chapter 5: Powerful ideas related to geometry What is geometry? Why learn geometry? Why do people have difficulty with geometry? Spatial visualization and properties of shapes Translations, Rotations, Reflections (or Slides, Turns, and Flips) Similarity and Congruence Reasoning about types of shapes Deduction Chapter 6: Powerful ideas related to parts and wholes Reprise: One Whole or Many Parts? Decomposition and Partitioning What is the whole? (and Why does it matter?) Various meanings of fractions Equivalent Fractions and Simplest Form Adding and subtracting with fractions Multiplying with fractions Dividing with fractions Caution about a possible point of confusion Chapter 7: Powerful ideas related to the decimal numbers Relationship between decimal numbers and fractions Relationship between decimal numbers and place value concepts Special uses of decimal numbers: money and percent Arithmetic: Special concerns with decimal numbers Chapter 8: Powerful ideas related to measurement The concept of measurement Why does measurement matter? Linear measurement Linear measurement as compared with area measurement Measurement of volume Measurement of weight Measurement of time Units as standards The power of the metric system Uncertainty and degrees of precision Estimation Choosing the best measurement unit The use of formulas (or not?) Chapter 9: Powerful ideas related to data analysis Some general thoughts about statistics Posing statistical questions Collecting data Organizing and displaying data Measures of center The concept of distribution Descriptive Graphs Chapter 10: Powerful ideas related to probability What is probability? Finding the Number of Outcomes Some Problem Situations Formulas What about Odds? Some Common Misconceptions Chapter 11: Powerful ideas related to ratio and proportion Linear relationships Percent What about Non-linear Relationships Chapter 12: Powerful ideas related to reasoning Number sense Inductive reasoning Deductive reasoning The Yin and Yang of Using Inductive Reasoning and Deductive Reasoning Together Mathematical authority? Evaluation of mathematical thought. Chapter 13: Powerful ideas related to algebra Basics of Algebra Functions Graphing Functions on the Coordinate System Non-linear Functions Appendix A: Explanations of Complex Formulas The Combinations Formula Cross-multiplication for Solving Proportions The Lattice Method of Multiplication Appendix B: Answers to Self-Check and Activities