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Mathematical Reasoningby Calvin T. Long
Synopses & Reviews
The fourth edition of Mathematical Reasoning has an increased focus on professional development and connecting the material from this class to the elementary and middle school classroom. The authors have provided more meaningful content and pedagogy to arm readers with all the tools that they will need to become excellent elementary or middle school teachers.
Thinking Critically. Sets and Whole Numbers. Numeration and Computation. Number Theory. Integers. Fractions and Rational Numbers. Decimals and Real Numbers. Algebraic Reasoning and Representation. Statistics: The Interpretation of Data. Probability. Geometric Figures. Measurement. Transformations, Symmetries, and Tilings. Congruence, Constructions, and Similarities.
For all readers interested in mathematical reasoning for elementary teachers.
The Fifth Edition of Mathematical Reasoning for Elementary Teachers, with new co-author Richard Millman, focuses on mathematical knowledge needed for teaching–demonstrating why future teachers are learning math content as well as when they will use it in the classroom. One of the most important aspects of teaching is being able to explain why students’ methods and ideas are either right or wrong. Imparting this skill to future teachers the emphasis of this fantastic text.
About the Author
Calvin Long received his B.S. in Mathematics from the University of Idaho. Following M.S. and Ph.D. degrees in Mathematics from the University of Oregon he worked briefly as an analyst for the National Security Agency and then joined the faculty at Washington State University. His teaching ran the gamut from elementary algebra through graduate courses and included frequently teaching the content courses for prospective elementary school teachers.
His other professional activities include serving on numerous committees of the National Council of Teachers of Mathematics and the Mathematical Association of America, and holding various leadership positions in those organizations. Professor Long has also been heavily engaged in directing and instructing in-service workshops and institutes for teachers at all levels, has given over one hundred presentations at national and regional meetings of NCTM and its affiliated groups, and has presented invited lectures on mathematics education abroad.
Professor Long has co-authored two books and is the sole author of a text in number theory. In addition, he has authored over ninety articles on mathematics and mathematics education, and also served as a frequent reviewer for both The Arithmetic Teacher and The Mathematics Teacher. In 1986, he received the Faculty Excellence Award in Teaching from Washington State University and, in 1991, he received a Certificate for Meritorious Service to the Mathematical Association of America.
Aside from his professional activities, Dr. Long enjoys listening to, singing, and directing classical music, reading, fly fishing, camping, and back packing.
Duane DeTemple received his B.S. with majors in Applied Science and Mathematics from Portland State College. Following his Ph.D. in Mathematics from Stanford University, he joined the faculty at Washington State University, where he is now Professor of Mathematics. He has been extensively involved with teacher preparation and professional development at both the elementary and secondary levels. Professor DeTemple has been a frequent consultant to projects sponsored by the Washington State Office of the Superintendent of Public Instruction, the Higher Education Coordinating Board, and other boards and agencies.
Dr. DeTemple has co-authored three other books and over 80 articles on mathematics or mathematics materials for the classroom. He is a member of the Washington State University President’s Teaching Academy, and in 2007 was the recipient of the WSU Sahlin Faculty Excellence Award for Instruction and the Distinguished Teaching Award of the Pacific Northwest Section of the Mathematical Association of America.
In addition to mathematics teaching and research, Duane enjoys reading, listening to and playing music, hiking, biking, canoeing, traveling, and playing tennis.
Richard Millman received a B.S. from the Massachusetts Institute of Technology and a Ph.D. from Cornell University in Mathematics. He is the Outreach Professor of Mathematics at the University of Kentucky, which supports pre-service and in-service teacher training for PreK-12 mathematics teachers. He is Principal Investigator and Project Director for ALGEBRA CUBED, a grant from the National Science Foundation to improve algebra education in rural Kentucky.
Dr. Millman has co-authored three books in mathematics, co-edited two others, and received nine peer reviewed grants. He has published over 40 articles about mathematics or mathematics education and has taught a wide variety of mathematics and mathematics education courses throughout the undergraduate and graduate curriculum, including those for preservice teachers. He received, with a former student, an Excel Prize for Expository Writing for an article in The Mathematics Teacher.
Rich enjoys traveling, writing about mathematics, losing golf balls, listening to music, and going to plays and movies. He also loves and is enormously proud of his grandchildren with whom he enjoys discussing the conceptual basis of mathematics, among other topics.
Table of Contents
1. Thinking Critically
1.1 An Introduction to Problem Solving.
1.2 Pólya's Problem-Solving Principles.
1.3 More Problem-Solving Strategies.
1.4 Algebra as Problem-Solving Strategy
1.5 Additional Problem-Solving Strategies
1.6 Reasoning Mathematically
2. Sets and Whole Numbers
2.1 Sets and Operations on Sets.
2.2 Sets, Counting, and the Whole Numbers.
2.3 Addition and Subtraction of Whole Numbers.
2.4 Multiplication and Division of Whole Numbers.
3. Numeration and Computation
3.1 Numeration Systems Past and Present.
3.2 Nondecimal Positional Systems.
3.3 Algorithms for Adding and Subtracting Whole Numbers.
3.4 Algorithms for Multiplication and Division of Whole Numbers.
3.5 Mental Arithmetic and Estimation.
3.6 Getting the Most Out of Your Calculator.
4. Number Theory
4.1 Divisibility of Natural Numbers.
4.2 Tests for Divisibility.
4.3 Greatest Common Divisors and Least Common Multiples.
4.4 Codes and Credit Card Numbers: Connections to Number Theory.
5.1 Representations of Integers.
5.2 Addition and Subtraction of Integers.
5.3 Multiplication and Division of Integers.
6. Fractions and Rational Numbers
6.1 The Basic Concepts of Fractions and Rational Numbers.
6.2 Addition and Subtraction of Fractions
6.3 Multiplication and Division of Fractions
6.4 Properties and Applications of Rational Numbers
7. Decimals, Real Numbers, and Proportional Reasoning
7.1 Decimals and Real Numbers
7.2 Computations with Decimals.
7.3 Proportional Reasoning
8. Algebraic Reasoning and Representation
8.1 Expressions, Functions, and Equations
8.2 Graphing Functions in the Cartesian Plane
9. Statistics: The Interpretation of Data
9.1 Graphical Representation of Data.
9.2 Measures of Central Tendency and Variability.
9.3 Statistical Inference and Sampling
10.1 Empirical Probability.
10.2 Principles of Counting.
10.3 Permutations and Combinations
10.4 Theoretical Probability
11. Geometric Figures
11.1 Figures in the Plane.
11.2 Curves and Polygons in the Plane.
11.3 Figures in Space.
12. Measurement: Length, Area, and Volume
12.1 The Measurement Process.
12.2 Area and Perimeter.
12.3 The Pythagorean Theorem.
12.4 Surface Area and Volume.
13. Transformations, Symmetries, and Tilings
13.1 Rigid Motions and Similarity Transformations.
13.2 Patterns and Symmetries.
13.3 Tilings and Escher-like Design.
14. Congruence, Constructions, and Similarity
14.1 Congruent Triangles.
14.2 Constructing Geometric Figures.
14.3 Similar Triangles.
Appendix A Manipulatives in the Mathematics Classroom
Appendix B Spreadsheets
Appendix C Graphing Calculators
Appendix D A Brief Guide to the Geometer’s Sketchpad
Appendix E Resources
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