The Heart of Mathematics: An invitation to effective thinking

The Heart of Mathematics addresses the big ideas of mathematics (many of which are cutting edge research topics) in a non-computational style intended to be both read and enjoyed by students and instructors, as well as by motivated general readers. It features an engaging, lively, humorous style full of surprises, games, mind-benders, and all without either sacrificing good mathematical thought or relying on mathematical computation or symbols.

The authors are award-winning authors, holding awards such as: Distinguished Teaching Award (Burger, from the Mathematical Association of America); Chauvenet Prize (the best expository mathematics writer in the world, Burger, from the MAA) and many others.

The book gives readers a good feel for the variety of problems that mathematicians tackle. In fact, one of the book's great strengths is the range of topics it covers, from number theory and games, to topology, to chaos and fractals. It does this with little use of conventional mathematical notation or jargon, and the level of presentation is so elementary that the book can be "read" just as any non-technical book can be read. At the same time, the authors go to great lengths to encourage reader participation. Many hands-on demonstrations and experiments are provided, and the end-of-chapter exercises ask readers to discuss the material with others and write about their experiences.

The topics presented are fascinating. I read this book on my vacation and found several passages to read to my wife and daughter almost every day. (This provided a lot of amusement for everyone when my 12-year-old daughter would solve problems in a few seconds that I had been pondering without much success.)

The book's subtitle is "An Invitation to Effective Thinking," and the authors present problem-solving strategies that can be applied to problems within and outside the field of mathematics. While readers will no doubt be familiar with many of them already, it is difficult for me to imagine anyone who would not benefit from at least some of the strategies presented.

The authors' writing is very informal with a lot of corny humor - possibly too much for a lot of people - but at the same time you do get a sense of the authors as good guys who know some important things and want to share the wealth.

In summary, this is a most unusual and stimulating book. Highly recommended.


Without Any Doubt, One of the Best Math Books I've Ever Read   May 28, 2001
Rawitat Pulam (University of Tsukuba, Japan)
84 out of 87 found this review helpful

After browsing this book for a while in a book store somewhere near Tokyo station, I decided to buy it immediately, even it's price will about almost doubled the price I would get if I buy from internet. (I bought it for about $115).

But, it turned out to be one of the best decision I'd ever made in my life as a book-worm. (Well, I do love reading :-)

This book will change the way you look at the world, the way you look at yourself, the way you think, and so on. It is cleary one of the best books on Math I've ever read so far (I do love Math as well :)

Instead of throwing you the formulas, the authors lead you to the Mathematical Thinking. How to think this, how to solve that. What Mathematic really is and what are its applications, and why it had been used that way. All explained through simple but fun-to-think example/problems. Using non-technical language, and almost no formula(!). In fact, almost everyone with little math background can enjoy this book. It doesn't seem to require anything more than middle school math. So, you can really see the pictures, the very big pictures, clearly in your mind, which is really importand.

With the wide range of content, from Prime Numbers to Chaos, Fractals, Infinity, 4th Dimension, and more. The concept of each is well-explained, in a manner I mentioned earlier. This book will definitely help you to understand the making of "modern" science better.

One more thing is: Mathematic is not something that far from our everyday life as some might think. It's the alternative way to view our everyday life. The way with reason, pattern, logic, relation betweening things, and more. (Well, I realized this before start reading this book. But for those who haven't realize, this book will open the door for you).

Math can also be fun. (In fact, it IS .. That's why I love it :-)

And this book tell you why, and how to find and feel those "fun" and enjoy Math. Without any complicated computation, and limited only by your imagination.


Math Appreciation, not computation or rigor   September 23, 2004
Ted R. Shoemaker (Madison, Wisconsin, USA)
45 out of 46 found this review helpful

_The_Heart_Of_Mathematics_ was never intended as a traditional textbook to teach you how to calculate. If that is what you are looking for, you need a diferent book. Its value -- and this is the best book of its kind that I have found -- is in helping the reader gain an appreciation for mathematics. Its title could well have been _Math_Appreciation_ . It was most likely intended as a way of satisfying the "math requirement" for non-math majors who feel allergic to math.

I have read comments from several people debating the merits of this book. Perhaps it would help to use inject an analogy into the conversation. Suppose you wanted to learn (or teach) music. One teacher chooses to teach her students how to play the piano;
another has her students listen to CDs of great performances; another teaches his students how to read music; and another teaches the biographies of Beethoven and Mozart. Which of these teachers is right? Which kind of music do you want to learn?

The question itself is mistaken, if you think that it has exactly one correct answer. The best answer is: ALL OF THEM. The problem here is not in what one of these approaches will teach, but in what it omits.

Now, translating back from the metaphor: I want my children to learn how to compute AND how to love math. Which is right? Both of them.

This book shows you how to have fun with math. If you or your students end up learning something, and wanting to learn more -- that's the idea.


Learning To Think Mathematically   September 2, 2004
Mead C. Whorton Jr. (Bastrop, LA United States)
18 out of 18 found this review helpful

This is an excellent textbook in that its primary emphasis is on some of the great ideas in mathematics and effective thinking. Topics covered are numbers, infinity, geometry, topology, chaos and fractals, uncertainty, and decision making. The text is replete with myriad illustrations which assist the student in grasping key ideas. Text would be excellent for a liberal arts math course, an enrichment course for middle school and high school teachers, or as a course for advanced high school students. Comes with 3D glasses and exploration CD.
In summary, conveys the beauty of mathematics while teaching students to think.


Gets to the heart of mathematics!   May 28, 2004
26 out of 28 found this review helpful

I disagree strongly with the previous reviewer who found the text "disturbing." That person and his/her student friend missed the point entirely! This is not a textbook aimed at the traditional recipes for solving sets of mathematical problems. Rather, it is a survey of mathematical thought from ancient to modern times and the astonishing aspect is that it is within the grasp of all students to comprehend it!

For example, we don't just learn the Pythagorean formula for right triangles and apply it to specific problems. We discover with hands on clarity WHY Pythagoras' theorem is true! What could be more elegant that Euclid's easily understood proof that there are infinitely many prime numbers? Moreover, we get to see those abstract notions put to great use in encryption without which even amazon.com would not be the great success that it is! All of this is comprehensible to any student willing to read the text and to participate in classroom discussion.

The authors nurture creative thinking throughout keeping students alert to and on the lookout for patterns while encouraging them to try new methods of attacking problems. This is how REAL mathematics works! Also, they make it clear that mathematics is not a closed subject having solved all number problems. They provide many examples of problems that took centuries to solve (Fermat's Last Theorem) along with some that have yet to be cracked (Goldbach's Conjecture).

Things really start to get interesting when the text delves into the nature of infinity. The authors set this up very cleverly, first, with an early introduction of a simple and innocent looking game which is eventually used as a stepping stone into Cantor's proof and, second, with a highly visual analogy of numbers on a conveyor belt used to compare the cardinality of sets. Finally, they treat the student to an infinity of infinities! The student cannot help but grasp the essence of the great ideas and appreciate the thinking that yielded such marvelous concepts.

The text introduces many more areas of fascinating mathematics some which were touched on in earlier reviews here. I particularly enjoyed the discussions of the fixed point theorem as well probability and statistics in the final chapter where the student sees the need to question statistical data (polls). The student will acquire an appreciation of both the power and limitations of statistical inference.

Will the student leave the course laden with mathematical techniques and skills that will allow them to solve systems of partial differential equations or to model nonequilibrium chemical processes or to design the first interstellar space probe? Of course not. They will leave the course as better thinkers and with a much greater appreciation of mathematics!