The Heart of Mathematics: An Invitation to Effective Thinking

The Heart of Mathematics: An Invitation to Effective Thinking

by Edward B. Burger, Michael Starbird
     
 

Make mathematics fun and satisfying for everyone

Math can be a living source of powerful ideas that transcend mathematics; a window into mind-opening philosophical concepts such as infinity, fourth dimensions, chaos, and fractals; and a practical training ground for developing skills in analysis, reasoning, and thought—if you have the right approach

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Overview

Make mathematics fun and satisfying for everyone

Math can be a living source of powerful ideas that transcend mathematics; a window into mind-opening philosophical concepts such as infinity, fourth dimensions, chaos, and fractals; and a practical training ground for developing skills in analysis, reasoning, and thought—if you have the right approach and the right guide.

The Heart of Mathematics: An Invitation to Effective Thinking­—now in its third edition—transforms mathematics into an engaging, relevant experience even for the most math-phobic student. Infusing this book with humor and enthusiasm, Edward B. Burger and Michael Starbird—both recipients of the Mathematical Association of America’s foremost national teaching award and countless state, regional, and campus-wide teaching honors—introduce students to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking.

Richer and more rewarding than ever, this new edition features:

  • An emphasis on mathematical methods of investigation
  • Visualization techniques that make key concepts easier to understand
  • Accessible, friendly writing style that encourages critical thinking
  • "Life Lessons"-effective methods of thinking that students will retain and apply beyond the classroom

End of section Mindscape activities for the development of application, problem-solving, and argumentation skills

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Product Details

ISBN-13:
9781559534079
Publisher:
Springer-Verlag New York, LLC
Publication date:
10/28/1999
Edition description:
Older Edition
Pages:
646
Product dimensions:
7.53(w) x 9.30(h) x 1.09(d)

Table of Contents

CHAPTER ONE: Fun and Games: An introduction to rigorous thought

CHAPTER TWO: Number Contemplation
Section 2.1. Counting [Pigeonhole principle].
Section 2.2. Numerical Patterns in Nature: [Fibonacci numbers].
Section 2.3. Prime cuts of numbers [Prime numbers].
Section 2.4. Crazy clocks and checking out bars [Modular arithmetic].
Section 2.5. Secret Codes and How to Become a Spy [RSA public key cryptography].
Section 2.6. The irrational side of numbers [Irrational numbers].
Section 2.7. Get real [The real number line].

CHAPTER THREE: Infinity
Section 3.1. Beyond Numbers [An introduction to one-to-one correspondence].
Section 3.2. Comparing the Infinite [Examples of one-to-one correspondences].
Section 3.3. The Missing Member [Cantor's diagonalization proof that |N|<|R|].
Section 3.4. Travels Toward the Stratosphere of Infinities [Power set theorem].
Section 3.5. Straightening up the circle [Geometrical correspondences].

CHAPTER FOUR: Geometric Gems
Section 4.1. Pythagoras and his hypotenuse [Blaskara's elegant proof].
Section 4.2. A view of an art gallery [A view-obstruction question from computational geometry].
Section 4.3. The sexiest rectangle [The Golden Rectangle].
Section 4.4. Soothing symmetry and spinning pinwheels [Aperiodic tilings].
Section 4.5. The Platonic Solids Turn Amorous [Symmetry and duality in the Platonic Solids].
Section 4.6. The shape of reality? [Non-Euclidean geometries].
Section 4.7. The Fourth Dimension [Geometry through analogy].

CHAPTER FIVE: Contortions of Space
Section 5.1. Rubber sheet geometry [Topological equivalence by distortion].
Section 5.2. The Band That Wouldn't Stop Playing [Möbius Band and Klein Bottle].
Section 5.3. Circuit training. [The Euler circuit theorem].
Section 5.4. Feeling edgy? [The Euler characteristic].
Section 5.5. Knots and links [A little knot theory].
Section 5.6. Fixed Points, Hot Loops, and Rainy Days [The Brouwer Fixed Point Theorem].

CHAPTER SIX: Fractals and Chaos
Section 6.1. Images [A gallery of fractals].
Section 6.2. The infinitely detailed beauty of fractals [Creating fractals through repeated processes].
Section 6.3. Between dimensions [Fractal dimension].
Section 6.4. The mysterious art of imaginary fractals [Julia and Mandelbrot Sets].
Section 6.5. The Dynamics of Change [Repeated applications of simple processes].
Section 6.6. Predetermined chaos [Deterministic chaos].

CHAPTER SEVEN: Taming Uncertainty
Section 7.1. Chance surprises [Unexpected scenarios involving chance].
Section 7.2. Predicting the future in an uncertain world [Probability].
Section 7.3. Random thoughts [Coincidences].
Section 7.4. Down for the count [Systematic counting].
Section 7.5. Dizzling, Defending, and Doctoring [Probability of Precipitation, game theory, Bayesian probability]

CHAPTER EIGHT: Meaning from Data
Section 8.1. Stumbling Through a Minefield of Data [Pitfalls of statistics].
Section 8.2. Getting Your Data to Shape Up [Organizing, describing, and summarizing data]
Section 8.3. Looking at Super Models [Mathematically described distributions]
Section 8.4. Go Figure [Making inferences from data, hypothesis testing]
Section 8.5. War, Sports, and Tigers [Cause and effect and correlation, Simpson's Paradox, famous applications of inference]

CHAPTER NINE: Deciding Wisely
Section 9.1. Great Expectations [Expected value]
Section 9.2. Risk [Deciding personal and public safety]
Section 9.3. Money Matters [Compound interest]
Section 9.4. Peril at the polls [voting]
Section 9.5. Cutting cake for greedy people [fair division]

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