Textbook

The Heart of Mathematics: An Invitation to Effective Thinking, 3rd Edition

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Transform your mathematics course into an engaging and mind-opening experience for even your most math-phobic students. The Heart of Mathematics: An invitation to effective thinking --now in its third edition--succeeds at reaching non-math, non-science-oriented majors and encouraging them to discover the mathematics inherent in the world around them. Infused throughout with the authors' humor and enthusiasm, The Heart of Mathematics introduces students to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking.

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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]

Author Information

Edward B. Burger is professor of mathematics in the Department of Mathematics and Statistics at Williams College. He graduated from Connecticut College in 1985, where he earned B.A. Summa Cum Laude with Distinction in Mathematics, and received his Ph.D. in mathematics from the University of Texas at Austin. He did his postdoctoral work at the University of Waterloo in Canada. Dr. Burger has received numerous awards including: the Award of Excellence, for "educational mathematics videos that break new ground from the Technology & Learning magazine, and the Distinguished Achievement Award, for Educational Video Technology from The Association of Educational Publishers. He was honored as one of the "100 Best of America", Listed in Reader's Digest's Annual Special Issue as Best Math Teacher. He also received the Residence Life Academic Teaching Award, University of Colorado at Boulder and the Robert W. Hamilton Book Award, for "The Heart of Mathematics".

Michael Starbird Michael Starbird is a University Distinguished Teaching Professor of Mathematics at The University of Texas at Austin. He has received more than a dozen teaching awards including the Mathematical Association of America’s 2007 national teaching award and several university-wide teaching awards based largely on his course in mathematics for liberal arts students. Starbird brings intriguing mathematics to general audiences through his classes, lectures, books, and video courses. In 1989, Starbird was UT’s Recreational Sports Super Racquets Champion.

New To This Edition

A new section (5.3) called Circuit Training: From the Königsberg Bridge Conundrum to graphs demonstrates how a specific question can lead to a whole new branch of mathematics. Important concepts arise from isolating essential features from specific conundrum.
The new Section 5.3 Circuit Training combines well with the next Section 5.4 Feeling Edgy?: Exploring relationships among vertices, edges, and faces to give a nice introduction to graph theory.
The second edition’s Chapter 6, Chaos and Fractals, has been re-organized to create the new Chapter 6, Fractals and Chaos. The re-organization allows the reader to more clearly appreciate the various features of fractals separately from the conundrums of chaos.
The second edition’s Chapter 7, Taming Uncertainty, which dealt with probability and statistics has been greatly expanded and split into two new chapters: Chapter 7, Taming Uncertainty, focuses on probability while Chapter 8, Meaning from Data, treats statistical reasoning.
In Chapter 7, Taming Uncertainty, readers will find intriguing illustrations of what to expect from randomness. Visual examples and results from coin flips illustrate the wisdom that from randomness, we should learn to expect the unexpected.
A new section in Chapter 7 discusses several real-life issues involving probability:
- the meaning of the surprisingly poorly understood phrase, “There is a 30% chance of rain tomorrow.”
- some game theory including an application to decision-making on the football field, and
- how a doctor can use probabilistic reasoning when new test results alter the diagnosis of a disease.
- Probabilistic reasoning can help us all understand our world in a more nuanced manner. The new features of Chapter 7 explore that perspective.
The new Chapter 8, Meaning from Data, begins by presenting statistical ideas through statistical pitfalls. These examples of dubious reasoning expose the themes that sound statistical reasoning develops.
Chapter 8 emphasizes the challenge of taking a set of data and finding strategies for extracting meaning from those data.
The themes of organizing, describing, and summarizing a collection of data lead naturally to the concepts of descriptive statistics.
The goal of understanding what we can infer about an entire data set when all we actually see is a sample from that data set leads us to concepts of inferential statistics.
Commonsense is a fundamental tenet of sound statistical reasoning, and commonsense is front and center in our presentation.
An entirely new section presents statistical challenges in real life from estimating the numbers of Nazi tanks in World War II to understanding how great baseball players of the past might have been.
Throughout the book, new expanded hints illustrate for the student how to apply sound methods of reasoning to selected Mindscapes.
The expanded hints and many additional Mindscapes allow students to translate ideas into action and effective learning.
New Mindscapes that bolster mathematical techniques allow instructors who wish to do so to include more quantitative skill development in the course along with conceptual richness.

Hallmark Features

accessible, student-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
activities that encourage collaborative learning and group work -hands-on manipulative kit that directs students to model their thinking and actively explore the world around them