A New Aspect of Mathematical Method (Princeton Science Library)

As a professional LSAT tutor/blogger in NYC, I found this book to offer simple advice on problem solving and logical thinking. It's useful because it gives you a framework to identify and analyze the relationship between evidence and conclusion.

The book gives you some questions to ask yourself about any Logic Game or Logical Reasoning stimulus.

When to read it: Before you begin studying or when you need a break.

Yes, this is a classic. Yes, I refer back to it frequently and always find something new which informs my teaching. But it is not the first book that I would recommend to a student (even the highly-selective students at a certain Norwegian liberal arts college in the upper Midwest) - the language can be quite difficult to hack through.
My favorite thing about this book is the historical perspective. Many of the ideas Polya puts forth are similar to those in the NCTM Standards, for example, and as you know there is far from unanimous support for what some perceive as the latest incarnation of "new math". Reading this book, one sees it's not new math at all. Pappus, one learns, introduced "analysis and synthesis" (what I might call "working backwards") as a follow-up to Euclid's Elements. That was a while ago...
Fascinating, makes me want to know more history so that I can weave it into my classes, but this book is not for everybody.

This book catalogs a list of techniques for attacking problems. The author is a mathematician and most examples are from this domain. The techniques work well. The book is organized as a glossary, which makes it easy to read piecemeal and a tad repetitive for one sitting.
The author's experience as an educator shines throughout. He clearly presents not just "how to solve it", but also "how to help another solve it" - an important part of any manager's portfolio.
Recommended.

Critical thinking and problem solving are not stressed in any kind of modern education. Students tend to take a cookbook approach to problem solving and wonder how is it that some people can think circles around them. Polya introduces the reader, in a simple yet rigorous manner with heuristics and examples, to how genius does problem solving. Its fascinating what truly learned people are able to articulate for the rest of us. Learn how to think outside of the box.

"Schule des Denkens" means school of thought. The title of the original manuscript reveals the motivation for this book: to teach better. Polya wrote it as a guide on how to lecture mathematics. But only the publicized edition, after Polyas migration to the United States tells in plain English what its good for a broad audience: How to Solve It. I am not a teacher, the didactic musings of a lecturer would produce not more than a spark of curiosity in me. But "How to Solve It" describes a general procedure for problem solving. This freaks me out! Polya hides this endeavor in favor of the didactic justification, and only later in the book will reveal that its content is a Modern Heuristic to "understand the process of solving problems, especially the mental operations typically useful in this process" [p. 129]. The book is divided into four parts, I will discuss them briefly.

Part I: In the Classroom
This 32 pages are all you need to grasp his algorithm of problem solving. Good ideas are simple and the procedure proposed is nothing counterintuitive. You could easily come a similiar conclusion by your own: What is the first step of problem solving? (1) Understanding the problem. What is the next step? (2) Devise a plan. Then?(3) Execute the plan. And finally(4) Look back.
Mightily impressed? Then you are a lobotomized PowerPoint disciple! But follow Polya in a Socratic dialog with the classroom and look into the train of thoughts of an educated problem solver. There are many subtilities to discover.By reading this chapter, more than once I had moments of Heureka!, when Poly guides you to ask the so called right questions and instructs you how to take a different point of view of the problem.

Part II: How to Solve it - a Dialog
The second part compresses the problem solving procedure, the ars inveniendi, in a summary of two pages. I did not gain from this, but it might be helpful as a short rehearsal when time passes by.

Part III: Dictionary of Heuristic
This is a 200 pages collection of heuristics to use as a pattern language for problem solving. The autor advices to take your time read this piece by piece when you are struggling with problems. Which I do.

Part IV: Problems, Hints, Solutions
These 8 pages are filled with exercises and smart hints how to approach the individual problems.

Conclusions
Polya opens your mind for solutions. I will tackle future hard problems only with Polya's algorithm and benefit from the careful order he imposes to the confused mind.Albeit written for teaching mathematics, I suspect that Polyas work is useful not only for quantitativ problem solving, but for qualitative problems too. Here, I have no proof and only the application of it will tell. The book is easy to read, and you might master the first fundamental part "In the Classroom" in 3-5 hours. It is also a really cheap book, 13-something Euro, and if you are a problem solver you will need and enjoy it.

Pardon the cliche, but: no home should be without this book. Polya's examples are mathematical, but the principles of heuristics and problem solving are universally applicable, whether you are a programmer, a mechanic, a TV repairman, or anyone trying to solve a problem. This is to problem solving what Strunk and White are to writing. BUY IT. I've lost track of my copy, and once I finish this review, I'll be replacing it.

I know the original version seemed like it would be great for someone that study math or physics and was heavily involved with proofs. However after reading the whole book and with some time I was able to really apply it to how I solve engineering problems. I will be honest it took some trial and error but within a year I was able to completely changed the way I solved problems (i.e. it needed an overhaul). The area I use it the most is thermal fluid systems and have had many complements on the way I articulated or solved problems, and I'm not a genius. The book is helpful and for the price you could take away a couple of items with out a hit in the pocket and just be that much better at figuring out problems.

This is a classic math teacher's resource book; but could be read and enjoyed by countless math afficianados. Polya gives people methodologies for solving problems in many possible ways. I recommend this book for all who would like to delve a bit into problem solving; I know math teachers all probably have this book already. Five out of five stars.

The issue is that solving problems is not made interesting and fulfilling experience.
This book beautifully explains the process of problem-solving. It starts from simple problems, lays down the fundamentals and leads to more complex problems.
One of the gems is the simple formula:
1. Understand the problem
2. Devise a plan (seeing how various items connect
3. Carry out the plan
4. Look back at the completed solution, review and discuss it.
It is also a good reference to teach kids how to approach problems.
Buy it and it will be a very handy reference.

This is a book on problem solving, in its widest meaning, though mathematics dominates, given the author's education. It's a feast! Prepare yourself for many hours of fun and education. If you'll accept an advice, study every single book written by Polya.

The books is insightful in ways that you will grasp immediately and others that will take will require months or years to understand. The structure of the text, which is very systematic and reads more like an encyclopedia than anything, surprised me at first but rest assured, you have the right book.

In fact, do you want to be a robot? I talked to a woman who took a whole semester in computer science and came out learning nothing. She told me this. My love affair with Real Math started with this book in a library. I was reading a book which had a bunch of interviews with the most successful programmers in the world. One was Czech and I do not remember his name. But he was asked the following question. "What in your opinion is the biggest mistake that programmers are doing in their educations or their work today?" He answered, "It's simple. They don't know how to solve problems. At our company, we have some simple books that tell you how to do this. The best is Polya's 'How to Solve It'. It has a little diagram in the back that completely runs you through a series of questions on solving math problems. But even in schools, they don't take this approach. Everything is by rote and repetition! You solve a problem and YOU DON'T KNOW WHAT YOU SOLVED! We have a lot of these little books." The late Isaac Asimov wrote a beautiful little book called "The Realm of Algebra". It's out of print. But he explains the entire realm of algebra in something like 150 pages. The best book I've ever seen about math. Math can be fun. Programming can be fun. But only if you ask Polya's questions in the back of this book. "What do I have to do to make this problem complete?" "What is missing from this problem?" "What could I add to make this problem solved?" A two page diagram in the back. And everybody knows that programming is just "crummy mathematics". BUY THE BOOK! BUY THE BOOK! BUY THE BOOK!. 2 pages in the end of this book and at least 50% of your math/programming problems are down the drain. Buy the books for your son if you are a Betty Crocker. Or your daughter. Or they will end up in the "Valley of the Dead". Solving problems in school for years and years and simply not knowing what they did! Good luck. Oh yes. One last thing. BUY THE BOOK!

There seems to be a cultish following for Polya's book, so I decided to pick it up even though I'm not a mathematician. I'm a philosophy PhD with an interest in "business strategy" (as they call it). The book's a little bit tough to move through, since he chose to write it as a glossary for the bulk of the text. That makes it boring. The more fundamental issue of course is that he's thinking about math when he gives his ideas for solving problems, and more specifically about TEACHING kids to solve math problems. Now, this is useful. And the general tenor is applicable to all kinds of problem solving. But I think it's not the holy grail it's meant to be -- there are other books on problem solving that make more practical sense if you are working on non-formal mathematical puzzles.

You know? I was looking for the book for 6 months. It's a very old book(Second Edition 1956). I guess no one in Bangladesh have the book. Not in any University Library or in general. No way to be found in New Market. I asked some one will visit Calcutta, India. He was my teacher. He failed to bring it for me.
At last, one of my friends at Toronto order the book for me, she got after 15 days and send it to me after 30 days through one of her relative.
I felt I must read it. After having and reading, I found that the book is really a very basic book to open your mind for facing problems. Which helps to understand, think, analyze and solve problems. I'm a programmer in C. I love to think about problem.

In retrospect, I regret very much not having read Polya's "How to solve it" before I went to grad school (or even better undergrad). Almost every paragraph leaves me with a feeling of deep wonder how he describes so insighfully and precisely what took me years of experience to understand.

I expected this second-hand book to be used. No problem at all, but this time I even could not see the difference between second-hand and new! Including shipping still cheaper as it had been, ordering it in the Netherlands,...

By Patricia's son Chris. My late kind Father gave me the book to study. I have gone over it at least once over the years and reviewed portions again and again; not because I love the book, but because it is not particularly effective. The title seems to me to be a misrepresentation--perhaps if I was in my 80's I would get the gist of it! Many readers, unless they are interested in probability and statistics may not be aware that Polya has a famous "Polya Urn problem" that is at the very heart of combinatorial probability theory and though it may be simply stated it can take on many different forms depending on the level of sophistication applied to carve out and describe all of the model's characteristics for any given realization.

The reason why I'm here is to find someone that makes a referral to another much better text on mathematical proof and logic that I can give to my pre-highschool niece and nephew as Polya at this point is simply out of the question and in all likelihood would do much more harm than good.

Therefore I am very reluctant to recommend this book; though for those that are interested it's highly likely that out of desperation it will someday become an inevitable purchase.

(It also should be said that in regards to both Solow and Polya's text's it would be so much more effective if some real machinery were introduced, for example the "open ball"--use the traditional delta epsilons as a momentary starting point. Also it would be useful if abstract concepts from set theory where used to introduce the notion of a "formula". I just don't see any reason to hold back on useful "Fregeian" machinery early on, or as my probability professor used to say "handy gadgets"--The author should have spent more time drawing on his strengths to find catchy ways of introducing the basic point set theory etc. and maybe the books title would been appropriate)

Heavily recommended by many, I decided to give How to Solve It a go, and real analysis, or should I say the science and art of proofing, is my worst nightmare. Although I do understand and appreciate the concept of proofs and consider it to be the most fundamental and mathematical aspect of mathematics, trying to prove already proven theorems, postulates, propositions, lemmas, and whatever else is a very tedious and mind-numbing task that robs the fun out of mathematics for the sake of learning. At least that's how I felt when I took the upper level college courses and amazingly survived them. In my opinion, it's the professors that needlessly made the experience of many students' to be harrowingly miserable due to their stilted, ad verbatim pedagogy with no real connection between prior knowledge and real analysis being taught. With that in mind, I've seen my math classes turned into the American Gladiators race, watching the large number of students from the beginning of the semester to dwindle down to a paltry few, and have essentially become the survival of the fittest. Why? Real Analysis, or should I say, "The science of proving without the art involved?" So, I decided to read How to Solve It to see if it could help me out while I was in one of those classes, and it amazingly did not. In fact, the book is so obscenely basic that even I could have written the same thing myself, and the knowledge contained in the book could not even guide me through whatsoever the material in the advanced classes. Maybe at just the very, very beginning of the whole journey, the book might work, and by that time, an ordinary mathematics student, falsely lured by the feeling of having fun while doing algebra, would be persuaded greatly to skip away from majoring in mathematics and find something else that is more realistic and not so painful. For me, anytime I get a problem with a proof, How to Solve It is a place I would never go to. All in all, How to Solve It is a vastly overrated book that is too simplistic and unhelpful.

Mathematics education is a backward field. George Polya, a fossil, fits right in. The bulk of human processing is hidden from our awareness. It's mostly subconscious. Problem solving is human processing and it's mostly hidden. Polya's writings on education leaves one with the impression that he believes mathematical processing is conscious thought. The elephant's tail is not the elephant. Polya's thinking does not take into consideration relatively recent discoveries from the field of cognitive science. These researchers tell us that the human brain has highly limited capacities for taking in and processing information. This is fundamental. In fact being good at honoring highly limited capacities appears to be central to intellectual power or to mathematical expertise. Their is no reason to believe that the un-hidden portion of human processing is the critical part although virtually all mathheads believe this. Reading mathematics is mostly a subconscious process. Poor problem solvers always read poorly. Polya doesn't say much about reading or limited capacities because his work isn't closely tied to cognitive reality.

This is a great book about mathematics and mathematical problem solving (and, to some extent, problem solving in general). There are many great ideas for teachers of mathematics, particularly those who teach in high school.

This is a classic math teacher's resource book; but could be read and enjoyed by countless math afficianados. Polya gives people methodologies for solving problems in many possible ways. I recommend this book for all who would like to delve a bit into problem solving; I know math teachers all probably have this book already. Five out of five stars.