Elementary Mathematics Pedagogical Content Knowledge: Powerful Ideas for Teachers » (1st Edition)

James Schwartz is currently an Associate Professor of Education at Ralph C. Wilson School of Education, St. John Fisher College. He has a rich experience as both an elementary teacher and teacher-educator. James has contributed to various texts including Essentials of Educational Technology 1/e, (Allyn and Bacon, 1999); Essentials of Elementary Mathematics 2/e, (Allyn and Bacon, 1999); and Teaching Elementary School Mathematics 6/e, (Allyn and Bacon, 1996).

“This book would be a great tool for helping [today’s future elementary teachers] acquire a ‘gut level’ understanding of mathematics concepts.”

- Hester Lewellen, Baldwin-Wallace College, OH

“The writing in this text is very clear and would easily be understood by the intended audience. The real-world examples put the various math concepts into a context that is easily understood. The vignettes at the beginning of each chapter are interesting and they get the reader to begin thinking about the math concepts that will follow. Each of the chapters seem to build on one another and the author often refers back to activities and concepts from previous chapters which is meaningful to the reader because it lets the reader know that the information they are learning builds their conceptual understanding of other mathematical concepts. “

- Melany L. Rish, University of South Carolina, Aiken

Organized around five key concepts or “powerful ideas” in mathematics, this book presents elementary mathematics content in a concise and nonthreatening manner for teachers.

Designed to sharpen teachers’ mathematics pedagogical content knowledge, the friendly writing style and vignettes relate math concepts to “real life” situations so that they may better present the content to their students. The five “powerful ideas” (composition, decomposition, relationships, representation, and context) provide an organizing framework and highlight the interconnections between mathematics topics. In addition, the book thoroughly integrates discussion of the five NCTM process strands.

Features:

• Icons highlighting the NCTM process standards appear throughout the book to indicate where the text relates to each of these.

• Practice exercises and activities and their explanations reinforce math concepts presented in the book and provide an opportunity for reflection and practice.

• Concise, conversational chapters and opening vignettes present math contents simply enough for even the most math-anxious pre-service teachers.

Most chapters include "Preview," "Chapter Review," "Self-Check," "Selected References, "Online Activities" and "Activities 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