Simmons S. Rediscovering Mathematics

Washington: The Mathematical Association of America, 2011. — 240 p. — ISBN: 978-0-88385-912-4.Rediscovering Mathematics is aimed at a general audience and addresses the question of how best to teach and study mathematics. The book attempts to bring the exciting and dynamic world of mathematics to a non-technical audience. With so much focus today on how best to educate the new generation and make mathematics less rote and more interactive, this book is an eye-opening experience for many people who suffered with dull math teachers and curricula. Rediscovering Mathematics is an eclectic collection of mathematical topics and puzzles aimed at talented youngsters and inquisitive adults who want to expand their view of mathematics. By focusing on problem solving, and discouraging rote memorization, the book shows how to learn and teach mathematics through investigation, experimentation, and discovery. Rediscovering Mathematics is also an excellent text for training math teachers at all levels. Topics range in difficulty and cover a wide range of historical periods, with some examples demonstrating how to uncover mathematics in everyday life, including: number theory and its application to secure communication over the Internet, the algebraic and combinatorial work of a medieval mathematician Rabbi, and applications of probability to sports, casinos, and gambling. Rediscovering Mathematics provides a fresh view of mathematics for those who already like the subject, and offers a second chance for those who think they don't.Acknowledgements
A Guide for the Reader
Introduction: How to Read Mathematics
An Example of Mathematical Writing—The Birthday Paradox
Our Reader Tackles the Birthday Paradox
Challenges
Mathematical Discovery in the Classroom
A Simple Lesson
The Math Teacher as Conductor
The Effective Teacher
Challenges
Don’t Reach for Your Calculator (Yet)
A Magic Trick
Clever Calculations
Pythagorean Triples
Challenges
Have Another Piece of Pie, Zeno?
A Simpler Way
Euclid, Proofs, and Writing Mathematics
Challenges
Thinking Like a Mathematician—Lessons from a Medieval Rabbi
Rabbi Levi ben Gershon and his Sums
The Number of Squares and Rectangles in a Grid
The Triangle Puzzle—A Model Mathematical Problem
Challenges
b]What is Mathematics Good For?[/b]
Knock Hockey
Basketball
Rates of Growth
Challenges
Three Averages
Average I. The Arithmetic Average
Average II. The Harmonic Average
Average III. The Geometric Average
Exploring Averages
Challenges
Algorithms—The Unexpected Role of Pure Mathematics
A Solution to the Two Jug Puzzle
Euclid’s Algorithm
Other Methods for Calculating Greatest Common Divisors
The Efficiency (or Speed) of an Algorithm
The Egyptian Multiplication Algorithm
The Fast Modular Exponentiation Algorithm
Greatest Common Divisors, Algorithms, and E-Commerce
E-Commerce and Cryptography
The Rest of the Story—The RSA Algorithm
Challenges
Pythagoras’ Theorem and Math by Pictures
A Proof of Pythagoras’ Theorem
A Personal Experience
The Height of a Pyramid
Challenges
Memorizing Versus Understanding
Example 1: FOIL
Example 2: Square Roots
Example 3: The Quadratic Equation
A “Real Life” Quadratic Equation
Challenges
Games and Gambling
A Carnival Game
Craps
The Careless Casino
Challenges
Soccer Balls and Counting Tricks
Euler’s Formula
Platonic Solids
Counting
The Puzzle of the Soccer Ball
Challenges
Pizza Pi and Area
Pi in the Bible
The Area of a Circle
A Hard Puzzle Involving Areas
Challenges
Back to the Classroom
A Classroom Experience
Emphasizing Understanding
What Next?
Resources for Rediscovering Mathematics
Further ReadingAbout the Author