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The Heart of Mathematics: An Invitation to Effective Thinking

Edward B. Burger and Michael Starbird
Key College
Publication Date: 
Number of Pages: 
Hardcover with CDROM
[Reviewed by
Allen Stenger
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This book (HoM for short) teaches how mathematicians work on problems, unlike most liberal arts math books that teach some of the accomplishments of mathematics. Each chapter and major section starts out with one or several interesting and challenging problems, usually not obviously math-related, then shows how to attack them. The book is aimed at college liberal arts students who will probably never take another math course.

The authors are well-known proponents of discovery learning (inquiry based learning) in mathematics. Is this a discovery book? Yes, mostly. Chapter 1 is a series of Conundrums, followed by Nudges and then by Punch Lines. In a more mundane discovery book these would be called Problems, Hints, and Solutions. The remaining chapters start with one or several problems, but the authors give you the solution without forcing you to struggle with the problem yourself. But at the end we have the Mindscapes: additional challenging problems based on the ideas used in the section.

This second edition is similar to the first, with the text carried over with little change. The interior layout has been redone and is much more attractive, and most of the illustrations have been redrawn or replaced. The second edition has an excellent new last chapter on Deciding Wisely (quantifying risks, fair division, election paradoxes). This is presented as the capstone of the book, although most of the material could be presented much earlier. I think this is the most interesting chapter of the book and you should arrange your course so that you cover at least part of it.

The writing is lively, exuberant, and even silly at times. The deep philosophical pronouncements and platitudes are there too, but isolated in the introduction and the marginal notes so you can easily skip over them. It's a fun book!

Who is "the competition" for HoM? I browsed through several popular liberal arts and math appreciation texts but did not find any that competed head-on with HoM. We may actually have a unique and original book here, instead of a re-hash of existing texts! Other texts fall into a few categories:

  • Relevant Math: Books that show how math is relevant to everyday life, especially by emphasizing applications to the social sciences rather than the more traditional emphasis on the physical sciences. COMAP's For all Practical Purposes (Freeman, 7th edition 2006) was the pioneer here and still seems the dominant text.
  • Core Curriculum Math: Math facts everyone should know, whether they like it or not. Popular texts here are Staszkow & Bradshaw, The Mathematical Palette (Brooks Cole, 3rd edition 2005); Berlinghoff & Grant & Skrien, A Mathematics Sampler (Ardsley House, 5th edition 2001)
  • Fun and Amazing Math: These tend to be mathematically rigorous, although leaving out many details, and are admired by mathematicians. They have fallen out of favor for this kind of course because of a perception (probably accurate) that they are too difficult for the audience. Some prominent books here include: Courant & Robbins & Stewart, What is Mathematics? (Oxford, 2nd edition 1996); Herstein & Kaplansky, Matters Mathematical (Harper & Row 1974, AMS Chelsea reprint 1978); Stein, Mathematics: The Man-made Universe (original publication McGraw Hill 1963, Dover reprint 1998 of 1994 edition); Davis & Hersh, The Mathematical Experience (Birkhäuser 1981, Houghton Mifflin paperback reprint 1982); and Rademacher & Toeplitz, The Enjoyment of Mathematics (Princeton 1957, Dover reprint 1990).

HoM does not resemble any of these books very much. It's closest to the last category. It might be thought of as a discovery version of those books, that starts with the problems rather than the results.

A number of supplements are available for HoM:

  • Each copy comes with a CD-ROM and a set of 3D glasses bound in. The CD-ROM has an extensive set of Java applets that act as manipulatives. They include the Monty Hall problem, the counterfeit coin problem, RSA encryption, and many more. The applets are very well done and add value to the text. One drawback of the computer approach is that there's no transparency in the random simulations — how do we know the computer program is not rigging the results to support the orthodox mathematical opinion? We would have more confidence if we were flipping the coins ourselves.
  • The instructor's manual (co-authored with Deborah Bergstrand) provides some additional mathematical background, guidance on topics that have been troublesome, ideas for classroom exercises, and estimates of time needed for each section. It also has solutions for all problems. The supplement is well-done, although experienced and mathematically-knowledgeable teachers would probably not get a lot of extra value out of it.
  • There's an optional set of manipulatives that has a few items. I thought this was probably not very useful, since each item would only be used in one or two spots in the text, and most can be constructed from common household materials (constructing them has some value for the students too). One novelty is that the regular polyhedra are built as wire-frame models using pipe cleaners and drinking straws rather than the more traditional method of tracing a squashed version on a piece of paper and folding and gluing it together.
  • The companion web site  is primarily a marketing platform for the book but does contain some supplemental material for teachers and students, as well as the complete set of Java applets (mentioned above) so they can be run in a web browser if they cannot be installed on a local computer.
  • Various instructor videos and a test bank. I did not examine these.

This is a wonderful book! It does have a few weaknesses, or at least things to be wary of. It uses very little mathematical jargon and notation, and this may give some onlookers the false impression that it's not a "real" math book. If your students were given a traditional test on math concepts after the course they probably would do poorly because they wouldn't know the standard methods and vocabulary (this is a problem with all discovery courses). HoM is is certainly not a broad survey of mathematics. It concentrates on geometric problems, or at least problems that have helpful pictures, and is limited to discrete math except for a few intermediate-value arguments. The book is vague about its prerequisites. At various spots it assumes a good command of logarithms, of exponent math, and of complex numbers. The writing and group activity exercises are fairly generic and seem to be tacked on, not integrated with the text. But except for these quibbles it is a wonderful book.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.


Chapter One: Fun and Games: An Introduction to Rigorous Thought

1.1 Silly Stories each with a Moral: Conundrums That Evoke Techniques of Effective Thinking

1.2 Nudges: Leading Questions and Hints for Resolving the Stories

1.3 The Punch Lines: Solutions and Further Commentary

1.4 From Play to Power: Discovering Strategies of Thought for Life


Chapter Two: Number Contemplation

2.1 Counting: How the Pigeonhole Principle Leads to Precision Through Estimation

2.2 Numerical Patterns in Nature: Discovering the Beauty of the Fibonacci Numbers

2.3 Prime Cuts of Numbers: How the Prime Numbers Are the Building Blocks of All Natural Numbers

2.4 Crazy Clocks and Checking Out Bars: Cyclical Clock Arithmetic and Bar Codes

2.5 Public Secret Codes and How to Become a Spy: Encrypting Information Using Modular Arithmetic and Primes

2.6 The Irrational Side of Numbers: Are There Numbers Beyond Fractions?

2.7 Get Real: The Point of Decimals and Pinpointing Numbers on the Real Line


Chapter Three: Infinity

3.1 Beyond Numbers: What Does Infinity Mean?

3.2 Comparing the Infinite: Pairing Up Collections via a One-to-One Correspondence

3.3 The Missing Member: Georg Cantor Answers: Are Some Infinities Larger Than Others?

3.4 Travels Toward the Stratosphere of Infinities: The Power Set and the Question of an Infinite Galaxy of Infinities

3.5 Straightening Up the Circle: Exploring the Infinite Within Geometrical Objects


Chapter Four: Geometric Gems 

4.1 Pythagoras and His Hypotenuse: How a Puzzle Leads to the Proof of One of the Gems of Mathematics

4.2 A View of an Art Gallery: Using Computational Geometry to Place Security Cameras in Museums

4.3 The Sexiest Rectangle: Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle

4.4 Smoothing Symmetry and Spinning Pinwheels: Can a Floor Be Tiled Without Any Repeating Pattern?

4.5 The Platonic Solids Turn Amorous: Discovering the Symmetry and Interconnections Among the Platonic Solids

4.6 The Shape of Reality? How Straight Lines Can Bend in Non-Euclidean Geometries

4.7 The Fourth Dimension: Can You See It?


Chapter Five: Contortions of Space

5.1 Rubber Sheet Geometry: Discovering the Topological Idea of Equivalence by Distortion

5.2 The Band That Wouldn’t Stop Playing: Experimenting with the Möbius Band and Klein Bottle

5.3 Feeling Edgy? Exploring Relationships Among Vertices, Edges, and Faces

5.4 Knots and Links: Untangling Ropes and Rings

5.5 Fixed Points, Hot Loops, and Rainy Days: How the Certainty of Fixed Points Implies Certain Weather Phenomena


Chapter Six: Chaos and Fractals

6.1 Images: Viewing a Gallery of Fractals

6.2 The Dynamics of Change: Can Change Be Modeled by Repeated Applications of Simple Processes?

6.3 The Infinitely Detailed Beauty of Fractals: How to Create Works of Infinite Intricacy Through Repeated Processes

6.4 The Mysterious Art of Imaginary Fractals: Creating Julia and Mandelbrot Sets by Stepping Out in the Complex Plane

6.5 Predetermined Chaos: How Repeated Simple Processes Result in Utter Chaos

6.6 Between Dimensions: Can the Dimensions of Fractals Fall Through the Cracks?


Chapter Seven: Taming Uncertainty

7.1 Chance Surprises: Some Scenarios Involving Chance That Confound Our Intuition

7.2 Predicting the Future in an Uncertain World: How to Measure Uncertainty Using the Idea of Probability

7.3 Random Thoughts: Are Coincidences as Truly Amazing as They First Appear?

7.4 Down for the Count: Systematically Counting All Possible Outcomes

7.5 Collecting Data Rather Than Dust: The Power and Pitfalls of Statistics

7.6 What the Average American Has: Different Means of Describing Data

7.7 Parenting Peas, Twins, and Hypotheses: Making Inferences from Data


Chapter Eight: Deciding Wisely: Applications of Rigorous Thought

8.1 Great Expectations: Deciding How to Weigh the Unknown Future

8.2 Risk: Deciding Personal and Public Policy

8.3 Money Matters: Deciding Between Faring Well and Welfare

8.4 Peril at the Polls: Deciding Who Actually Wins an Election

8.5 Cutting Cake for Greedy People: Deciding How to Slice Up Scarce Resources