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# A Problem Solving Approach to Mathematics for Elementary School Teachers

**by**Billstein, Rick; Libeskind, Shlomo; Lott, Johnny

### 9780321756664

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## Summary

## Author Biography

**Rick Billstein **is a Professor of Mathematics at the University of Montana. He has worked in mathematics teacher education at this university for over 40 years and his current research is in the areas of curriculum development and mathematics teacher education. He teaches courses for future teachers in the Mathematics Department. He served as the site director for the *Show-Me Project*, an NSF-funded project supporting the dissemination and implementation of standards-based middle grades mathematics curricula. He worked on the NSF grant *Tinker Plots* to develop new data analysis software and he serves on the Advisory Boards for several other national projects. From 1992-1997, he directed the NSF-funded *Six Through Eight Mathematics* (STEM) middle school mathematics curriculum project and is now directing the *Middle Grades* MATH*Thematics Phase II Project. *Dr. Billstein has published articles in over 20 different journals, and has co-authored over 40 books, including ten editions of* A Problem Solving Approach to Mathematics for Elementary Teachers*. He typically does about 25 regional and national presentations per year and has worked in mathematics education at the international level. He presently serves on the Editorial Board of NCTM’s *Mathematics Teaching in the Middle School*. Dr. Billstein was recently awarded the George M. Dennison Presidential Faculty Award for Distinguished Accomplishment at the University of Montana.

**Shlomo Libeskind** is a professor in the mathematics department at the University of Oregon in Eugene, Oregon, and has been responsible there for the mathematics teaching major since 1986. In addition to teaching and advising pre-service and in-service teachers, Dr. Libeskind has extensive writing experience (books, articles, and workshop materials) as well as in directing mathematics education projects. In teaching and in writing, Dr. Libeskind uses a heuristic approach to problem solving and proof; in this approach the reasonableness of each step in a solution or proof is emphasized along with a discussion on why one direction might be more promising than another. As part of his focus on the improvement of the teaching of mathematics, Dr. Libeskind is also involved at many levels locally, nationally, and worldwide in the evaluation of mathematics teacher preparation programs. In his home state, he is actively involved in schools and councils, as well as in reviewing materials for the state standards for college admission. Most recently (spring 2008) he visited teacher colleges in Israel as a Fulbright Fellow. During this visit he conducted observations and critiques of the preparation of mathematics teachers at several colleges in Northern Israel. Dr. Libeskind received his Bachelor’s and Master’s Degrees in Mathematics at the Technion (Israel Institute of Technology) and his PhD in Mathematics at the University of Wisconsin, Madison.

**Johnny W. Lott** began his teaching career in the public schools of DeKalb County, Georgia, outside Atlanta. There he taught mathematics in grades 8-12. He also taught one year at the Westminster Schools, grades 9-12, and one year in the Pelican, Alaska, school, grades 6-12. Johnny is the co-author of several books and has written numerous articles and other essays in the "Arithmetic Teacher", "Teaching Children Mathematics", "The Mathematics Teacher", "School Science and Mathematics", "Student Math Notes", and "Mathematics Education Dialogues". He was the Project Manager for the "Figure This!" publications and website developed by the National Council of Teachers of Mathematics (NCTM) and was project co-director of the State Systemic Initiative for Montana Mathematics and Science (SIMMS) Project. He has served on many NCTM committees, has been a member of its Board of Directors, and was its president from April 2002-April 2004. Dr. Lott is Professor Emeritus from the Department of Mathematical Sciences at The University of Montana, having been a full professor. He is currently the Director of the Center for Excellence in Teaching and Learning, Professor of Mathematics, and Professor of Education at the University of Mississippi. Additionally, he is on the Steering Committee of the Park City Mathematics Institute, works with the International Seminar, the Designing and Delivering Professional Development Seminar, and is editor for its high school publications. His doctorate is in mathematics education from Georgia State University.

## Table of Contents

**1. An Introduction to Problem Solving**

1-1 Mathematics and Problem Solving

1-2 Explorations with Patterns

1-3 Reasoning and Logic: An Introduction

**2. Numeration Systems and Sets**

2-1 Numeration Systems

2-2 Describing Sets

2-3 Other Set Operations and Their Properties

**3. Whole Numbers and Their Operations**

3-1 Addition and Subtraction of Whole Numbers

3-2 Algorithms for Whole-Number Addition and Subtraction

3-3 Multiplication and Division of Whole Numbers

3-4 Algorithms for Whole-Number Multiplication and Division

3-5 Mental Mathematics and Estimation for Whole-Number Operations

**4. Number Theory**

4-1 Divisibility

4-2 Prime and Composite Numbers

4-3 Greatest Common Divisor and Least Common Multiple

Online Module: Clock & Modular Arithmetic

**5. Integers**

5-1 Integers and the Operations of Addition and Subtraction

5-2 Multiplication and Division of Integers

**6. Rational Numbers and Proportional Reasoning**

6-1 The Set of Rational Numbers

6-2 Addition, Subtraction, and Estimation with Rational Numbers

6-3 Multiplication and Division of Rational Numbers

6-4 Ratios, Proportions, and Proportional Reasoning

**7. Decimals: Rational Numbers and Percent**

7-1 Introduction to Decimals

7-2 Operations on Decimals

7-3 Nonterminating Decimals

7-4 Percents and Interest

**8. Real Numbers and Algebraic Thinking**

8-1 Real Numbers

8-2 Variables

8-3 Equations

8-4 Functions

8-5 Equations in a Cartesian Coordinate System

Online Module: Using Real Numbers in Equations

**9. Probability**

9-1 How Probabilities Are Determined

9-2 Multistage Experiments with Tree Diagrams and Geometric Probabilities

9-3 Using Simulations in Probability

9-4 Odds, Conditional Probability, and Expected Value

9-5 Using Permutations and Combinations in Probability

**10. Data Analysis/Statistics: An Introduction**

10-1 Designing Experiments/Collecting Data

10-2 Displaying Data: Part I

10-3 Displaying Data: Part II

10-4 Measures of Central Tendency and Variation

10-5 Abuses of Statistics

**11. Introductory Geometry**

11-1 Basic Notions

11-2 Linear Measure

11-3 Curves, Polygons, and Symmetry

11-4 More About Angles

Online Module: Networks

**12. Congruence and Similarity with Constructions**

12-1 Congruence through Constructions

12-2 Other Congruence Properties

12-3 Other Constructions

12-4 Similar Triangles and Similar Figures

Online Module: Trigonometry Ratios via Similarity

**13. Congruence and Similarity with Transformations**

13-1 Translations and Rotations

13-2 Reflections and Glide Reflections

13-3 Dilations

13-4 Tessellations of the Plane

**14. Area, Pythagorean Theorem, and Volume**

14-1 Areas of Polygons and Circles

14-2 The Pythagorean Theorem, Distance Formula, and Equation of a Circle

14-3 Geometry in Three Dimensions

14-4 Surface Areas

14-5 Volume, Mass, and Temperature

Online Appendices

Spreadsheets

Graphing calculators

Geometer’s Sketchpad

GeoGebra