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How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Paperback – February 23, 2009

ISBN-13: 978-0521719780 ISBN-10: 052171978X

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How to Think Like a Mathematician: A Companion to Undergraduate Mathematics + How to Prove It: A Structured Approach, 2nd Edition + How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)
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Product Details

  • Paperback: 274 pages
  • Publisher: Cambridge University Press (February 23, 2009)
  • Language: English
  • ISBN-10: 052171978X
  • ISBN-13: 978-0521719780
  • Product Dimensions: 7.4 x 0.8 x 9.7 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (19 customer reviews)
  • Amazon Best Sellers Rank: #87,526 in Books (See Top 100 in Books)

Editorial Reviews

Review

"In this book, Houston has created a primer on the fundamental abstract ideas of mathematics; the primary emphasis is on demonstrating the many principles and tactics used in proofs. The material is explained in ways that are comprehensible, which will be a great help for people who seem to hit the wall regarding what to do when confronted with the creation of a proof... In this book, Houston takes a systematic and gentle approach to explaining the ideas of mathematics and how tactics of reasoning can be combined with those ideas to generate what would be considered a convincing proof."
Charles Ashbacher, Journal of Recreational Mathematics

"The author provides concise, crisp explanations, including definitions, examples, tips, remarks, warnings, and idea-reinforcing questions. Houston expresses thoughts clearly and concisely, and includes succinct remarks to make points, clarify arguments, and reveal subleties."
W.R. Lee, Choice Magazine

Book Description

Looking for a head start in your undergraduate degree in mathematics? This friendly companion eases beginning students into real mathematical thinking, unlocking important techniques for effective mathematics so you can communicate with clarity, solve problems, and explore the world of definitions, theorems and proofs with real confidence.

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I have just started reading this book, in which the author strongly encourages the reader to do all of the exercises.
Marcia H Cupery
"How to Think Like Mathematician" by Dr. Kevin Houston is a very engaging, readable and pragmatic text on mathematical "technique".
Edmon Begoli
This book and the paper "Lockhart's Lament" really helped me understand what real Math is about, at least at a very basic level.
Joshua S. Infiesto

Most Helpful Customer Reviews

47 of 47 people found the following review helpful By Edmon Begoli on December 14, 2009
Format: Paperback
I am reading a great book that is in author's words intended for undergraduate students of mathematics, but that in my opinion offers much more to a motivated reader.

"How to Think Like Mathematician" by Dr. Kevin Houston is a very engaging, readable and pragmatic text on mathematical "technique". It is a non-pompous, well written, valuable, easy to follow and understand valuable set of lessons and tips on understanding and adopting mathematical method, language, theorems, proofs and techniques.

It is an introductory text, so for more in depth treatment of the subjects such as proofs and number theory you may need to look further into books such as Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition) orThe Princeton Companion to Mathematics but if you are looking to understand the mathematical method and how to be able to read and write "serious" math this is an ideal book.

I specially have to point out Dr. Houston's writing style - it is engaging, humorous, but substantial, pedagogical and never trivial.
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36 of 37 people found the following review helpful By Marcia H Cupery on January 4, 2013
Format: Paperback Verified Purchase
I have just started reading this book, in which the author strongly encourages the reader to do all of the exercises. Upon finishing the first exercise, I went to check my answers, and found that there were no solutions provided in the book to any of the exercises that appear in any of the chapters. After doing some research online, I found that the author has his own website and has since provided a PDF with the solutions to many of the exercises. I'm not sure if all of them are covered, but the list of solutions does appear to be quite extensive. So, in case you were like me and were wondering if your solutions were correct, you can check your answers by downloading the PDF at [...]
So far, I am really enjoying the read!
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17 of 17 people found the following review helpful By Joshua S. Infiesto on April 29, 2012
Format: Paperback Verified Purchase
This book and the paper "Lockhart's Lament" really helped me understand what real Math is about, at least at a very basic level. I was ineptly fumbling through my math classes before I read both. Lockhart's Lament is freely available from the AMA and I highly recommend it to anyone starting in math.
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Format: Paperback
To be successful in mathematics, your mind must perform operations that are unlike the operations needed to do most other things. You must be able to hold abstract ideas, sometimes several at a time, as well as see the relationships between multiple concepts. Furthermore, those abstract ideas are built on other abstract ideas; for example, most of mathematics is built on the fundamental abstract idea of the use of a variable.
However, being different and at times being hard does not mean that the ideas of mathematics are incomprehensible. Humans excel at understanding abstract ideas, a strong argument can be made that such a skill is the very definition of human intelligence. In this book, Houston has created a primer on the fundamental abstract ideas of mathematics; the primary emphasis is on demonstrating the many principles and tactics used in proofs. The material is explained in ways that are comprehensible, which will be a great help for people who seem to hit the wall regarding what to do when confronted with the creation of a proof.
Many students are capable of leaping the comprehension hurdle; yet hit a wall when it comes time to apply the concepts in order to generate a proof. In this book, Houston takes a systematic and gentle approach to explaining the ideas of mathematics and how tactics of reasoning can be combined with those ideas to generate what would be considered a convincing proof.

Published in Journal of Recreational Mathematics, reprinted with permission
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10 of 10 people found the following review helpful By KHuereca on August 2, 2010
Format: Paperback Verified Purchase
Loved the book, It's definately going to be a go-to book for my other advanced classes. It is wrote in an extraordinarily user-friendly way and is actually somewhat fun to read.
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6 of 6 people found the following review helpful By F. Pieterse on October 26, 2011
Format: Kindle Edition
An excellent book that will introduce a complete novice to mathematical thinking.
There is a chapter also on Set theory which is a nice touch (and it is used illustratively so it doesn't feel like you are studying sets specifically). The book gives an excellent outline on how to approach a proof and how to study mathematics which often perplexes even the most talented of mathematicians (problems or theory etc.?)

That said it is extremely basic. Anyone who has competed in math olympiads for a year or two will know the proof techniques and be familiar to some extent on how to set them out. Also the problem solving component is so-so. I suppose the study tips could be useful but I feel that they are self-explanatory to some extent if one has a certain degree of mathematical experience.

A good book for neophytes. But if you are in your second year at University/College then this book will NOT help you. For first-year it will be of limited value, as one learns best by DOING.
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4 of 4 people found the following review helpful By B. Walter on October 4, 2013
Format: Paperback
I'm very pleased to be using this text in the college course I teach for math majors. It's exceptionally clear and contains exactly the most important topics for new mathematicians to learn. I have a few small disagreements with Houston (for example, he says 0 isn't a natural number), but these are far outweighed by the quality of his explanations and the excellent homework exercises he provides. Any student who understands the material in this text is very, very well prepared for study of upper-division pure mathematics.

My one real complaint is that this is the worst-bound book I've ever owned; it started falling apart the very first time I used it. My students this quarter have been working through this text for a week now, and almost every one of their copies already has pages falling out of it. Maybe the hardbound version doesn't have this problem? I hope Cambridge Univ. Press learns about modern binding practices soon...

Besides that, though, this is an excellent text - head and shoulders above the similar texts I've considered for adoption in my courses.
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