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How to Solve It: A New Aspect of Mathematical Method
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How to Solve It: A New Aspect of Mathematical Method

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4.15 of 5 stars 4.15  ·  rating details  ·  1,708 ratings  ·  78 reviews
George Polya was a Hungarian mathematician. He wrote this, perhaps the most famous book of mathematics ever written, second only to Euclid's "Elements." "Solving problems," wrote Polya, "is a practical art, like swimming, or skiing, or playing the piano: You can learn it only by imitation and practice. This book cannot offer you a magic key that opens all the doors and sol ...more
Paperback, 280 pages
Published June 1st 2009 by Ishi Press (first published November 30th 1944)
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Ivan Vukovic
This book contains no magic, no tricks. It's not one of those "esoteric knowledge revealed" books nor a book which promises you'll get an Abel prize or a Fields Medal someday.

What this books is, is a systematic and incredibly instructive overview of guidelines in mathematical problem solving, which are, as the author put it - "natural, simple, obvious, and proceed from plain common sense."

If you've ever put yourself against a serious problem which you really, really, really wanted to have solved
...more
Jeff
George Polya's classic How to Solve It is a seminal work in mathematics education. Written in 1945 and referenced in almost every math education text related to problem solving I've ever read, this book is a short exploration of the general heuristic for solving mathematical problems. While the writing is a bit clunky (Polya was a mathematician and English was not his first language), the ideas are so deeply useful that they continue to have relevance not just for solving mathematical problems, ...more
Stefan Kanev
This is a great book.

It teaches solving mathematical problems. It is mostly focused on high-school problems, but it is applicable to most types of mathematical problems out there. The author has developed a nice heuristic framework for tackling problems and has done a wonderful job of explaining it. It's not just the methods – exposition is also a great takeaway from this read.

On the downside, the book was written in 1945 and sometime it shows. It's more cute than a nuisance, though :)
Thai Son
Hailed as the classic guide to problem solving, this book did quite a good job at categorizing the ways of looking at a problem, and some general methods of solving and treating them. However, I think I read this at the wrong time - it could have fascinated me much more had I read it in the early 2000s (then again, there was not any translation to Vietnamese back then, and I suspect my mediocre English back then would not let me finish it).

Still, the way I went at the book is that I skimmed thro
...more
Jessica
Geometry and Discrete Math are the only high school math classes I aced. Likely, it had to do with some teaching and presenting, as well as the interest I mustered not being totally repelled by them in the hoisting of what curriculum mandates must-be-learned.

This book takes a simple, interesting approach and though it's written in the 40s, many benefits remain to-be-had from popularity outside its field. For me, beginning this book, I recalled how as an undergrad tutor for ESL students, our cla
...more
Greg Talbot
Elegance in solving problems is not strictly a mathematical skill set. Polya wisely formats word problems, critical thinking problems, and yes mathematical problems that occasionally are intimidating.

But one of the big takeaways is that problems are only as hard as they are unresolved. Not only does Polya give excellent ideas for solving problems: creating auxiliary problems, using heuristics, working backwards.

Each example that Polya gives takes concentration and critical analysis. But when yo
...more
Louis
Oct 26, 2008 Louis rated it 5 of 5 stars  ·  review of another edition
Recommends it for: Anyone who has to analyze situations not seen before
Shelves: math-stats
This is a book I wish I had read at the beginning of grad school. How to Solve It is not as much about methods of solving mathematical problems as it is about various approaches to solving problems in general. The method he uses to teach problem solving is to apply the approaches to problems of geometry. This is actually in line with the ancient greek (Aristotle) opinion that the young should learn geometry first, then when they have learned logic and how to prove things with physical reality, t ...more
Ari
Aug 09, 2007 Ari rated it 5 of 5 stars  ·  review of another edition
Recommends it for: everyone
Shelves: owned
I don't remember when I first encountered this book -- I think it was early in my time at Cornell. It's had a great deal of influence on how I approach math. It's one of the best math books I've ever read, and quite possibly the best book on mathematical problem solving ever written.

There are two copies of it floating around my lab at Berkeley, evidence, i think, that I'm not the only one who appreciates it.

Polya was a first rate mathematician, and his book is devoted to explaining simply and u
...more
Mirek Kukla
Polya tries to explain how to become a better 'problem solver', and how to guide others to better solve problems themselves. The core of the content is terrific, and gets you thinking about 'how to best think'.

Unfortunately, almost everything gets repeated numerous times, and as a whole the books ends up being thoroughly redundant. You don't really need to read beyond the first 36 pages (the rest of the book consists of a 'problem solving dictionary', and here's where the redundancy begins).

The
...more
Saharvetes
This book may not necessarily make you a better problem solver—that comes only from practice—but it is a useful first step in examining the types of creativity that go into problem-solving. Thus it's almost more a philosophical or psychological work than a how-to guide. The examples are all at middle-school or high-school level; the real point of the book is the enumeration of problem-solving strategies. It may even help you get unstuck when you're overlooking some trick you've used in the past.
Andrew
Here's how you solve it:

First.
You have to understand the problem.
UNDERSTANDING THE PROBLEM
What is the unknown? What are the data? What is the condition?
Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
Draw a figure. Introduce suitable notation.
Separate the various parts of the condition. Can you write them down?
Second.
Find the connection between the data and the unknown.
You may be obliged to consi
...more
Ben
This is a cool book. It's a comparatively rare thing for a first-class mathematician to write really seriously and thoughtfully about math education; at least it's not something I've come across too much. And it's been about a year and a half since I read anything about math education I didn't feel like I already knew. (Props to the _Young Mathematicians at Work_ series by Fosnot and Dolk, which was the last thing I read like that...)

The thing that makes this book unusual is that it's about the
...more
Michael
This book was used as a reference in several of the other books I have read, and I understood it to be more of a general methodology of problem solving when I decided to read it. It is written in a somewhat awkward style, to an audience that is difficult to discern, and with enough repetition that I had to skip pages at a time to get to the next topic. This was frustrating as I really wanted to like this book. When Polya does focus on the generalized concepts of problem solving, he has wonderful ...more
Paul
Gave me new tools for solving problems, and crystalized others. I appreciated the variety of sample problems used to illustrate "the list" that is the core of the book, from simple plane & solid geometry, to crossword puzzles & anagrams. It was also interesting to learn his philosophy on teaching, which echoes in the structure of the book & example problems.
Randy
This book culminates G. Polya many years of work and research into distilling the essential components of problem solving. Being a mathematician, mathematics is the context in which most discussion occurs and in which most examples are given. However, in some sense the focus on mathematics is just one long running example. What the book is really trying to do is teach problem solving. It is trying to give you the essential concepts of successful problem solving as well as giving you a bag of tri ...more
Krollo
I quite liked this book. It provides a no-nonsense guide to thinking through problems and working out what to do. Some bits of the dictionary are pretty damn fun too. However, most of the useful content is in the first dialogue section. The problems at the back should not be missed - it's a good way to check how much you have learned.
Dzmitry Kishylau
There are two problems with this book:
1. The language is really dry.
2. The author tried to be as thorough as possible, which means most of the time spent talking about fairly trivial things.
As a result, reading it is no fun. The best part is the problems and puzzles in the end.
Steven Ramirez
This is, possibly, one of the best books on logic I have ever read. I wish I had read this in college. Hell, I wish I had read this in high school. It has given me a new perspective and understanding in solving complex problems. A lot of what they cover I had developed intuitively from learning how to program but seeing the process more formally broken down was a real eye opener. Plus, the "Questions" section at the end is exactly the kind of thing I was hoping for. This is the book I will have ...more
Fredrik Hallgren
Many of the techniques he mentions are obvious or naturally acquired as part a good undergraduate education in the natural sciences, but many of the points he makes are worth keeping in mind.
Mitchell
Nov 01, 2014 Mitchell rated it 4 of 5 stars  ·  review of another edition
Recommends it for: math students and teachers
This was a pretty good book. I think I should read it a second time to get more out of it. The structured approach to problem-solving it presents seems like it would be helpful in math classes.
Shifting Phases
A lovely collection of mathematical lore. Not highly applicable to my students, unfortunately. A few ideas translate, for example
- Know the purpose in each step of your problem-solving
- If you can't solve a problem, solve a different problem that is similar enough to help you
- Account for all the data
- Can you check the result?
- Can you derive it differently?

I also really enjoyed the section on proverbs (p. 221):
Who understands ill, answer ill.
A fool looks to the beginning, a wise man reg
...more
Matt
The information in this book was extremely useful and extremely difficult to approach. I read the Kindle edition which may be part of the problem, but the majority of the book is structured as a "dictionary of essays." I read the book from cover to cover (an odd phrase for an e-book) and the dictionary format felt very disjointed. What was worse about the format is that the overview explaining how everything was held together was also an entry in this weird dictionary!

I'm giving this three stars
...more
Bashnev
A useful book. Definitely one to read if you are an educator -- particularly if you are teaching mathematics. Quite repetitive after some point. Polya details his 4 steps of problem solving to the reader in the form of questions to put to a class. Chunks of the text are dialogues. Most of the book, however, is a "dictionary" of problem solving terms etc. This dictionary is not so interesting to read just like any other dictionary. The book finishes with some puzzles together with hints and solut ...more
Jenny GB
A perfect book for problem solving that lays out a general method that can be used to solve problems (math or otherwise). The best part for me was the beginning of the book where he lays out his four steps (understand the problem, plan how to solve it, carry out your plan, and check your solution). It may be a little tough for a beginner to cut through all his writing to the heart of it, but it's well worth the read. I was less impressed with the Dictionary of Heuristics and admittedly skipped a ...more
Marjorie Gruen
Helped me get through my degree. Greatest math book I've ever owned.
Prathik M
Decent book, best for students or teachers, nothing mind blowing or something that will make you go 'Wow! I've never thought about it this way', but it would help the younger folks and help teachers on how to teach.
Jimp
There are few books I remember from college but this is one that remains on my shelf and I even recommend it periodically. A great approach to how to approach math problems and how to teach an approach that builds understanding and not just methods in solving problems. I've applied the methods of the book to other technical areas both in how I approach problem solving and occasionally how I help others solve problems. A must read for any math teacher and useful for any teacher.
Justin TerAvest
This is my favorite mathematical method book, and it's the only one that sounds at all like the techniques you actually use to figure problems out.

This book is unique in that it describes the actual method of problem solving, and walks through generic techniques that apply strongly, not just in mathematics, but anywhere.

This book is a foundation in my reasoning through problems at work. I've lent out multiple copies, never seen them again, and it makes me happy.
David
This is one of those mathematical "classics" that those of us with a training in math are supposed to love. Fact of the matter is that it is poorly written and pedestrian in nature. If you are seeking insight into how mathematicians think and approach problem-solving, give this one a miss. You'd be far better off to read Hardy's "A Mathematician's Apology" (dated, but still charming), or Ian Stewart's recent "Letters to a Mathematician" (charming and not dated at all).
Rejeev Divakaran
Good book if you like solving problems (puzzles and geometrical problems in particular). Author discusses about the importance of heuristic approaches in mathematics and problem solving. Typically finding a solution is heuristic even though explaining it is analytic and using rigorous proof. It also explains lot of typical tips for solving problems like testing at boundary conditions, test by dimension, using analogy, using symmetry etc.
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