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The Heart of Mathematics: An Invitation to Effective Thinking [Hardcover]

Edward B. Burger , Michael Starbird
5.0 out of 5 stars  See all reviews (3 customer reviews)

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Book Description

28 Sep 2004 1931914419 978-1931914413 2nd Revised edition
The power and attractiveness of the subject of mathematics is often hidden from students who are in introductory courses. In this new, innovative overview textbook, the authors put special emphasis on the deep ideas of mathematics, and present the subject through lively and entertaining examples, anecdotes, challenges and illustrations, all of which are designed to excite the student's interest. The underlying ideas include topics from number theory, infinity, geometry, topology, probability and chaos theory. Throughout the text, the authors stress that mathematics is an analytical way of thinking, one that can be brought to bear on problem solving and effective thinking in any field of study.
--This text refers to an out of print or unavailable edition of this title.

Product details

  • Hardcover: 784 pages
  • Publisher: Key College Publishing; 2nd Revised edition edition (28 Sep 2004)
  • Language: English
  • ISBN-10: 1931914419
  • ISBN-13: 978-1931914413
  • Product Dimensions: 24 x 19.2 x 3.5 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Bestsellers Rank: 1,226,042 in Books (See Top 100 in Books)

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From the reviews of the second edition: "In this book, the reader will see that mathematics is a network of intriguing ideas a ] . It is a really nice book. a ] This attractive book contains a lot of well-known and interesting problems a ] . It is really true that the ideas presented in this book are some of the most fascinating and beautiful ones around. a ] The reader will have something to explore, to learn, to think, to enjoy, and to add new aspects to his view of everything." (Valentina DagienA, Zentralblatt MATH, Vol. 1065, 2005) --This text refers to an out of print or unavailable edition of this title.

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3 of 3 people found the following review helpful
5.0 out of 5 stars One of the most mathematical maths text books 11 April 2009
This is a fantastic introduction to all things mathematical. The emphasis is on mathematical *thinking* rather than methods, and numerous puzzles and props allow the maths to be really understood.

But... the book is an introduction to the methods of a professional mathematician, so the topics aren't algebra, trigonometry, calculus. Rather there are loads of interesting and challenging examples from knot theory, topology, chaos theory, equivalence classes, number theory, voting maths. Comes with 3D glasses, non transitive dice, CD-Rom etc.

Well worth the not-insubstantial price. Perhaps accompanied with the other expensive but excellent book Mathematics from the Birth Of Numbers, which is far more 'traditional' (lots of algebra, proof as well as history of maths).
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3 of 3 people found the following review helpful
5.0 out of 5 stars Mathmatical thinking 14 May 2011
By Millie
The Heart of Mathematics is one of the best texts designed to improve mathematical thinking I have come across. It is packed with interesting problems and resources.
I am not sure that I would have paid the new price, but the used copy that I purchased was much cheaper and in very good condition.
I have only had the book for few weeks, but have already used a number of the tasks in my lessons (KS3 and 4). It has worked particularly well with getting some of the lowest ability students motivated and interested. They recently came to a lesson armed with problems that they had researched themselves to challenge me. This is a group that would normally be reluctant to do any work at all!
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1 of 1 people found the following review helpful
5.0 out of 5 stars The Heart of Maths 22 Nov 2011
Format:Hardcover|Verified Purchase
It is a thoroughly interesting and enjoyable book. It introduces mathematical concepts in simple language. Very well presented with great illustrations
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Most Helpful Customer Reviews on (beta) 4.0 out of 5 stars  22 reviews
110 of 113 people found the following review helpful
5.0 out of 5 stars An Unusual Book of Wide Interest 21 July 2002
By -_Tim_- - Published on
Format:Hardcover|Verified Purchase
The Heart of Mathematics is an unconventional math survey aimed primarily at social science and humanities students. While students in "soft" majors are the primary intended audience, math majors and others who have already progressed beyond the introductory level are likely to find this book of great interest as well.
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.
70 of 71 people found the following review helpful
5.0 out of 5 stars Math Appreciation, not computation or rigor 23 Sep 2004
By Ted R. Shoemaker - Published on
Format:Hardcover|Verified Purchase
_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 different 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 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 any 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.
102 of 107 people found the following review helpful
5.0 out of 5 stars Without Any Doubt, One of the Best Math Books I've Ever Read 28 May 2001
By Rawitat Pulam - Published on
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.
46 of 48 people found the following review helpful
4.0 out of 5 stars Great adult self ed resource 29 July 2004
By Julie Brennan - Published on
Format:Hardcover|Verified Purchase
The disturbing reviews indeed completely miss the point. The goal of this book is not to turn you into a mathematician. It is to help you appreciate what mathematics is.

I am planning on using this text for an adult self ed study group this fall. The goal is not to try to prove Cantor's method. You explore it and gain some understanding, but it isn't a mastery course that you come out of passing a test for, unless you are sitting in a classroom designed with that in mind, and the larger audience for this book is not in that narrow context. If you come out of it learning how to think mathematically, learning different ways to approach solving problems, learning that there is fun, beauty, art, order and sense to math, if you begin to *see* math in the world you live in, in nature, in ways you never noticed before - that is the goal. It is also threaded with history and the human drama that created math.

Both negative reviews were so poorly written and clearly missed the point that I dismissed them, but others I've recommended the book to have been confused, so I felt the need to respond.

I have also watched the video/DVD series these two authors put out through the Teaching Company, the Joy of Thinking, and I love what they are doing. Is every lecture perfect and resonating with everybody? No, but most resonate with most people. It certainly opened my eyes to things I never understood. Much of this book covers the same type of material.

Some people will find it more interesting than others, that is the nature of personal preference certainly. But the negative feedback indicates the book is flawed based on specific use in college classroom context, and it appears the reviewers did not understand the purpose of the book.

The four vs. five stars reflects the fact this is a first ed and could be just little more user friendly for lay people vs. college course users. I look forward to seeing the 2nd edition.
35 of 37 people found the following review helpful
5.0 out of 5 stars Gets to the heart of mathematics! 28 May 2004
By A Customer - Published on
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 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!
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