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141 people found this helpful

ByPhilon September 16, 2001

I found Pollya's "heuristic" approach to problem-solving applicable to both mathematical and non-mathematical problems. The goal of the heuristic approach is to study (and use!) the methods and rules of discovery and invention.

Here are just some of the questions that Pollya teaches as tools:

1. What is the unknown? What is the data? What conditions does the solution need to satisfy?

2. Do you know a related problem? Look at the unknown and try to think of a familiar problem having the same or a similar unknown.

3. Can you restate the problem? Can you solve a part of the problem.

4. Can you think of other data appropriate to determine the unknown?

5. Can you check the result?

6. Can you look back and use the result or the method for some other problem?

Overall, the author provides a systematic way to creatively solve problems. This volume has withstood the test of time for nearly 50 years. I recommend it highly.

Here are just some of the questions that Pollya teaches as tools:

1. What is the unknown? What is the data? What conditions does the solution need to satisfy?

2. Do you know a related problem? Look at the unknown and try to think of a familiar problem having the same or a similar unknown.

3. Can you restate the problem? Can you solve a part of the problem.

4. Can you think of other data appropriate to determine the unknown?

5. Can you check the result?

6. Can you look back and use the result or the method for some other problem?

Overall, the author provides a systematic way to creatively solve problems. This volume has withstood the test of time for nearly 50 years. I recommend it highly.

43 people found this helpful

ByAbhion November 20, 2012

Good aspects of this book have been said by most of the other reviewers. The main problem with such books is that for slightly experienced problem solvers, this book probably does not provide a whole lot of information as to what needs to be done to get better. For instance, for a kid who is in 10th grade struggling with math, this is a very good book. For a kid who is in his 11th grade trying for math Olympiad or for people looking at Putnam, this book won't provide much help.

Most people simply say that "practice makes perfect". When it comes to contest level problems, it is not as simple as that. There are experienced trainers like Professor Titu Andreescu who spend a lot of time training kids to get better. There is lot more to it than simply trying out tough problems.

The most common situation occurs when you encounter extremely tough questions like the Olympiad ones. Most people simply sit and stare at the problem and don't go beyond that. Even the kids who are extremely fast with 10th grade math miserably fail. Why?

The ONE book which explains this is titled "Mathematical Problem Solving" written by Professor Alan Schoenfeld. It is simply amazing. A must buy. In case you have ever wondered why, in spite of being lightning fast in solving textbook exercises in the 10th and 11th grade, you fail in being able to solve even a single problem from the IMO, you have to read this book. I am surprised to see Polya's book getting mentioned so very often bu nobody ever mentions Schoenfeld's book. It is a must read book for ANY math enthusiast and the math majors.

After reading this book, you will possibly get a picture as to what is involved in solving higher level math problems especially the psychology of it. You need to know that as psychology is one of the greatest hurdles to over when it comes to solving contest problems. Then you move on to "Thinking Mathematically" written by J. Mason et al. It has problems which are only few times too hard but most of the times, have just enough "toughness" for the author to make the point ONLY IF THE STUDENT TRIES THEM OUT.

The next level would be Paul Zeitz's The Art and Craft of Problem Solving. This book also explains the mindset needed for solving problems of the Olympiad kind. At this point, you will probably realize what ExACTLY it means when others say that "problem solving is all about practice". All the while you would be thinking "practice what? I simply cannot make the first move successfully and how can I practice when I can't even solve one problem even when I tried for like a month". It is problem solving and not research in math that you are trying to do. You will probably get a better picture after going through the above three books.

Finally, you can move on to Arthur Engel's Problem Solving Strategies and Titu Andreescu's Mathematical Olympiad Challenges if you managed to get to this point. There is also problem solving through problems by Loren Larson. These are helpful only if you could solve Paul Zeitz's book successfully.

To conclude, if you are looking for guidance at the level of math Olympiad, look for other books. This book won't be of much assistance. On the other hand, if you are simply trying to get better at grade school math, this book will be very useful.

Most people simply say that "practice makes perfect". When it comes to contest level problems, it is not as simple as that. There are experienced trainers like Professor Titu Andreescu who spend a lot of time training kids to get better. There is lot more to it than simply trying out tough problems.

The most common situation occurs when you encounter extremely tough questions like the Olympiad ones. Most people simply sit and stare at the problem and don't go beyond that. Even the kids who are extremely fast with 10th grade math miserably fail. Why?

The ONE book which explains this is titled "Mathematical Problem Solving" written by Professor Alan Schoenfeld. It is simply amazing. A must buy. In case you have ever wondered why, in spite of being lightning fast in solving textbook exercises in the 10th and 11th grade, you fail in being able to solve even a single problem from the IMO, you have to read this book. I am surprised to see Polya's book getting mentioned so very often bu nobody ever mentions Schoenfeld's book. It is a must read book for ANY math enthusiast and the math majors.

After reading this book, you will possibly get a picture as to what is involved in solving higher level math problems especially the psychology of it. You need to know that as psychology is one of the greatest hurdles to over when it comes to solving contest problems. Then you move on to "Thinking Mathematically" written by J. Mason et al. It has problems which are only few times too hard but most of the times, have just enough "toughness" for the author to make the point ONLY IF THE STUDENT TRIES THEM OUT.

The next level would be Paul Zeitz's The Art and Craft of Problem Solving. This book also explains the mindset needed for solving problems of the Olympiad kind. At this point, you will probably realize what ExACTLY it means when others say that "problem solving is all about practice". All the while you would be thinking "practice what? I simply cannot make the first move successfully and how can I practice when I can't even solve one problem even when I tried for like a month". It is problem solving and not research in math that you are trying to do. You will probably get a better picture after going through the above three books.

Finally, you can move on to Arthur Engel's Problem Solving Strategies and Titu Andreescu's Mathematical Olympiad Challenges if you managed to get to this point. There is also problem solving through problems by Loren Larson. These are helpful only if you could solve Paul Zeitz's book successfully.

To conclude, if you are looking for guidance at the level of math Olympiad, look for other books. This book won't be of much assistance. On the other hand, if you are simply trying to get better at grade school math, this book will be very useful.

ByPhilon September 16, 2001

I found Pollya's "heuristic" approach to problem-solving applicable to both mathematical and non-mathematical problems. The goal of the heuristic approach is to study (and use!) the methods and rules of discovery and invention.

Here are just some of the questions that Pollya teaches as tools:

1. What is the unknown? What is the data? What conditions does the solution need to satisfy?

2. Do you know a related problem? Look at the unknown and try to think of a familiar problem having the same or a similar unknown.

3. Can you restate the problem? Can you solve a part of the problem.

4. Can you think of other data appropriate to determine the unknown?

5. Can you check the result?

6. Can you look back and use the result or the method for some other problem?

Overall, the author provides a systematic way to creatively solve problems. This volume has withstood the test of time for nearly 50 years. I recommend it highly.

Here are just some of the questions that Pollya teaches as tools:

1. What is the unknown? What is the data? What conditions does the solution need to satisfy?

2. Do you know a related problem? Look at the unknown and try to think of a familiar problem having the same or a similar unknown.

3. Can you restate the problem? Can you solve a part of the problem.

4. Can you think of other data appropriate to determine the unknown?

5. Can you check the result?

6. Can you look back and use the result or the method for some other problem?

Overall, the author provides a systematic way to creatively solve problems. This volume has withstood the test of time for nearly 50 years. I recommend it highly.

BySandra Federon December 17, 2004

Are you like a dog with a bone when you're working on a brain teaser? After pages of scribbles, do you get a big grin on your face when you turn to the answers and say: "I'm right!" Then this book is for you.

And if you're not yet a die-hard problem-solver? You should step right up, too. You may get hooked.

G. Polya's book is based on the fact that, if we study how someone does something successfully, we can learn to do it successfully as well. How To Solve It is an application of 'heuristics' to solving problems.

There are certain mental operations useful in solving problems, any sorts of problems. Polya (who was an eminent mathematician and former Professor of Mathematics at Stanford University) describes and illustrates the most usual and useful of these operations, in a way that is irresistible and eye-opening.

These useful mental operations are organized according to when they come into play during the four steps to solving a problem. 1. You have to understand the problem. (Not as easy as it sounds.) 2. Find the connection between the data given and the unknown. Conceive the idea of a plan for the solution. 3. Carry out the plan. 4. Examine the solution obtained.

If you take some time and try to solve the problems selected to illustrate each mental operation, you will be well-rewarded. You will likely discover something surprising about your own problem-solving methods, and improve them in the process. You will definitely discover many new ideas and techniques to add to your arsenal.

For example, a first impulse when confronted with a problem is often to try to 'swallow it whole' -- to try to meet all of the conditions of the problem at once. G. Polya suggests keeping only part of the condition, and dropping the other part. This can lead you straight to a solution you might otherwise have completely missed.

His techniques help you to stand back and get to the heart of the problem, rather than getting lost in it.

Something else I liked very much about his book is his encouragement to guess, or to reason 'plausibly.' While the final proof must be strictly logical, "Anything is right that leads to the right idea." Problem-solving has every right to be fun, as well as purposeful.

And if you're not yet a die-hard problem-solver? You should step right up, too. You may get hooked.

G. Polya's book is based on the fact that, if we study how someone does something successfully, we can learn to do it successfully as well. How To Solve It is an application of 'heuristics' to solving problems.

There are certain mental operations useful in solving problems, any sorts of problems. Polya (who was an eminent mathematician and former Professor of Mathematics at Stanford University) describes and illustrates the most usual and useful of these operations, in a way that is irresistible and eye-opening.

These useful mental operations are organized according to when they come into play during the four steps to solving a problem. 1. You have to understand the problem. (Not as easy as it sounds.) 2. Find the connection between the data given and the unknown. Conceive the idea of a plan for the solution. 3. Carry out the plan. 4. Examine the solution obtained.

If you take some time and try to solve the problems selected to illustrate each mental operation, you will be well-rewarded. You will likely discover something surprising about your own problem-solving methods, and improve them in the process. You will definitely discover many new ideas and techniques to add to your arsenal.

For example, a first impulse when confronted with a problem is often to try to 'swallow it whole' -- to try to meet all of the conditions of the problem at once. G. Polya suggests keeping only part of the condition, and dropping the other part. This can lead you straight to a solution you might otherwise have completely missed.

His techniques help you to stand back and get to the heart of the problem, rather than getting lost in it.

Something else I liked very much about his book is his encouragement to guess, or to reason 'plausibly.' While the final proof must be strictly logical, "Anything is right that leads to the right idea." Problem-solving has every right to be fun, as well as purposeful.

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ByA customeron July 7, 1996

How to Solve It is the most significant contribution to heuristic since Descartes' Discourse on Method. The title is accurate enough, but the subtitle is far too modest: the examples are drawn mostly from elementary math, but the method applies to nearly every problem one might encounter. (Microsoft, for instance, used to and may still give this book to all of its new programmers.) Polya divides the problem-solving process into four stages--Understanding the Problem, Devising a Plan, Carrying out the Plan, and Looking Back--and supplies for each stage a series of questions that the solver cycles through until the problem is solved. The questions--what is the unknown? what are the data? what is the condition? is the condition sufficient? redundant? contradictory? could you restate the problem? is there a related problem that has been solved before?--have become classics; as a computer programmer I ask them on the job every day.

The book is short, 250 large-print pages in the paperback. Its style is clear, brilliant and does not lack in humor. Here is Polya's description of the traditional mathematics professor: "He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and turn his back on the class. He writes A; he says B; he means C; but it should be D." Behind the humor, though, lurks a serious complaint about mathematical pedagogy. Fifty years ago, when Polya was writing, and today still, mathematics was presented to the student, under the tyranny of Euclid, as a magnificent but frozen edifice, a series of inexorable deductions. Even the student who could follow the deductions was left with no idea how they were arrived at. How to Solve It was the first and best attempt to demystify math, by concentrating on the process, not the result. Polya himself taught mathematics at Stanford for many years, and one can only envy his students. But the next best thing is to read his book.

The book is short, 250 large-print pages in the paperback. Its style is clear, brilliant and does not lack in humor. Here is Polya's description of the traditional mathematics professor: "He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and turn his back on the class. He writes A; he says B; he means C; but it should be D." Behind the humor, though, lurks a serious complaint about mathematical pedagogy. Fifty years ago, when Polya was writing, and today still, mathematics was presented to the student, under the tyranny of Euclid, as a magnificent but frozen edifice, a series of inexorable deductions. Even the student who could follow the deductions was left with no idea how they were arrived at. How to Solve It was the first and best attempt to demystify math, by concentrating on the process, not the result. Polya himself taught mathematics at Stanford for many years, and one can only envy his students. But the next best thing is to read his book.

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ByAmazon Customeron December 28, 2005

Polya struck gold with this book! "How To Solve It" contains a simple, 5 step method for solving problems that's applicable in multitudes of disciplines. While the emphasis of this book is on story problems; Polya's method for problem solving is useful in areas such as computer programming, automotive troubleshooting, electronics repair, heating and cooling services, research writing, and much more.

I am constanly recommending this book to anyone in college. You can read the method and all the examples, or just read the method and a few examples. This book is easy to read, extremely relevant to today's promising careers, and can be understood in only 4 hours.

Marty A. Nickison II, BSCS Net+

author: Beyond the Books (Lulu)

I am constanly recommending this book to anyone in college. You can read the method and all the examples, or just read the method and a few examples. This book is easy to read, extremely relevant to today's promising careers, and can be understood in only 4 hours.

Marty A. Nickison II, BSCS Net+

author: Beyond the Books (Lulu)

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It's delightful to see this book is still in bookstores after 60 years, and I can still remember how much fun it was to read it 30 years ago. I came across it recently in a local bookstore, and after poring over it again, I was inspired to write a little review about it.

The most important thing about the book is Polya's little heuristic method for breaking down math problems and guiding you thru the process of solving them. Try to visualize the problem as a whole. Diagram it at first, even if you don't have all the details. Just initially try to get the most important parts of the problem down. Then try to get some sense of the relationship of the parts to the whole. Then tackle each of the component parts. If you get stuck, ask yourself if you could approach it another way, what could be missing, and so on. To this end, the questions at the back of the book are worth their weight in gold.

Polya's little heuristic and methods book is a timeless classic. This and Lancelot Hogben's "Mathematics for the Millions" have done more good for suffering math students than all the the dry textbooks put together that really don't teach you "how to solve it."

The most important thing about the book is Polya's little heuristic method for breaking down math problems and guiding you thru the process of solving them. Try to visualize the problem as a whole. Diagram it at first, even if you don't have all the details. Just initially try to get the most important parts of the problem down. Then try to get some sense of the relationship of the parts to the whole. Then tackle each of the component parts. If you get stuck, ask yourself if you could approach it another way, what could be missing, and so on. To this end, the questions at the back of the book are worth their weight in gold.

Polya's little heuristic and methods book is a timeless classic. This and Lancelot Hogben's "Mathematics for the Millions" have done more good for suffering math students than all the the dry textbooks put together that really don't teach you "how to solve it."

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ByDennis Mittonon December 22, 2005

A really helpful book that goes way beyond math. In our hurry to get through algebra we almost completely sidestep problem solving in teaching. It's true that we have `story problems' but they're usually just applications that restate the chapter's contents. That's why so many people get to a class like physical chem in college and flounder: they've never really been taught how to solve a problem only to regurgitate what's been read. This book by Polya fills in that gap.

More than that, though, this book is useful for life in general. Much of life is a leap from one decision to another and this book will help in looking at all problems and decision from new angles. (See Polya's book "Plausible Thinking" for lots more detail on this subject.)

Like in his other books the author has a knack for making the difficult seem a whole lot easier but parts can still be heavy going. Well worth the trudge, though.

More than that, though, this book is useful for life in general. Much of life is a leap from one decision to another and this book will help in looking at all problems and decision from new angles. (See Polya's book "Plausible Thinking" for lots more detail on this subject.)

Like in his other books the author has a knack for making the difficult seem a whole lot easier but parts can still be heavy going. Well worth the trudge, though.

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ByAbhion November 20, 2012

Good aspects of this book have been said by most of the other reviewers. The main problem with such books is that for slightly experienced problem solvers, this book probably does not provide a whole lot of information as to what needs to be done to get better. For instance, for a kid who is in 10th grade struggling with math, this is a very good book. For a kid who is in his 11th grade trying for math Olympiad or for people looking at Putnam, this book won't provide much help.

Most people simply say that "practice makes perfect". When it comes to contest level problems, it is not as simple as that. There are experienced trainers like Professor Titu Andreescu who spend a lot of time training kids to get better. There is lot more to it than simply trying out tough problems.

The most common situation occurs when you encounter extremely tough questions like the Olympiad ones. Most people simply sit and stare at the problem and don't go beyond that. Even the kids who are extremely fast with 10th grade math miserably fail. Why?

The ONE book which explains this is titled "Mathematical Problem Solving" written by Professor Alan Schoenfeld. It is simply amazing. A must buy. In case you have ever wondered why, in spite of being lightning fast in solving textbook exercises in the 10th and 11th grade, you fail in being able to solve even a single problem from the IMO, you have to read this book. I am surprised to see Polya's book getting mentioned so very often bu nobody ever mentions Schoenfeld's book. It is a must read book for ANY math enthusiast and the math majors.

After reading this book, you will possibly get a picture as to what is involved in solving higher level math problems especially the psychology of it. You need to know that as psychology is one of the greatest hurdles to over when it comes to solving contest problems. Then you move on to "Thinking Mathematically" written by J. Mason et al. It has problems which are only few times too hard but most of the times, have just enough "toughness" for the author to make the point ONLY IF THE STUDENT TRIES THEM OUT.

The next level would be Paul Zeitz's The Art and Craft of Problem Solving. This book also explains the mindset needed for solving problems of the Olympiad kind. At this point, you will probably realize what ExACTLY it means when others say that "problem solving is all about practice". All the while you would be thinking "practice what? I simply cannot make the first move successfully and how can I practice when I can't even solve one problem even when I tried for like a month". It is problem solving and not research in math that you are trying to do. You will probably get a better picture after going through the above three books.

Finally, you can move on to Arthur Engel's Problem Solving Strategies and Titu Andreescu's Mathematical Olympiad Challenges if you managed to get to this point. There is also problem solving through problems by Loren Larson. These are helpful only if you could solve Paul Zeitz's book successfully.

To conclude, if you are looking for guidance at the level of math Olympiad, look for other books. This book won't be of much assistance. On the other hand, if you are simply trying to get better at grade school math, this book will be very useful.

Most people simply say that "practice makes perfect". When it comes to contest level problems, it is not as simple as that. There are experienced trainers like Professor Titu Andreescu who spend a lot of time training kids to get better. There is lot more to it than simply trying out tough problems.

The most common situation occurs when you encounter extremely tough questions like the Olympiad ones. Most people simply sit and stare at the problem and don't go beyond that. Even the kids who are extremely fast with 10th grade math miserably fail. Why?

The ONE book which explains this is titled "Mathematical Problem Solving" written by Professor Alan Schoenfeld. It is simply amazing. A must buy. In case you have ever wondered why, in spite of being lightning fast in solving textbook exercises in the 10th and 11th grade, you fail in being able to solve even a single problem from the IMO, you have to read this book. I am surprised to see Polya's book getting mentioned so very often bu nobody ever mentions Schoenfeld's book. It is a must read book for ANY math enthusiast and the math majors.

After reading this book, you will possibly get a picture as to what is involved in solving higher level math problems especially the psychology of it. You need to know that as psychology is one of the greatest hurdles to over when it comes to solving contest problems. Then you move on to "Thinking Mathematically" written by J. Mason et al. It has problems which are only few times too hard but most of the times, have just enough "toughness" for the author to make the point ONLY IF THE STUDENT TRIES THEM OUT.

The next level would be Paul Zeitz's The Art and Craft of Problem Solving. This book also explains the mindset needed for solving problems of the Olympiad kind. At this point, you will probably realize what ExACTLY it means when others say that "problem solving is all about practice". All the while you would be thinking "practice what? I simply cannot make the first move successfully and how can I practice when I can't even solve one problem even when I tried for like a month". It is problem solving and not research in math that you are trying to do. You will probably get a better picture after going through the above three books.

Finally, you can move on to Arthur Engel's Problem Solving Strategies and Titu Andreescu's Mathematical Olympiad Challenges if you managed to get to this point. There is also problem solving through problems by Loren Larson. These are helpful only if you could solve Paul Zeitz's book successfully.

To conclude, if you are looking for guidance at the level of math Olympiad, look for other books. This book won't be of much assistance. On the other hand, if you are simply trying to get better at grade school math, this book will be very useful.

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Bynightowl03don December 16, 2006

This is a wonderful book. On a first reading it may seem a little confusing because the heuristics are organized alphabetically, rather than pedantically. This is not hard to overcome because Polya helpfully boldfaces related heuristics and by following the suggested threads you can get a fairly smooth read on the first reading. The way I dealt with it was I put sticky notes on the topics as I read them, so I could skim the ones I had already looked when threads hit a topic repeatedly.

I read this book many years ago, but it is still by my work station. I consult it when I get stuck on a problem, the heuristics do work.

For a first reading, the flipping around is annoying, but for reference purposes, the alphabetical order of the heuristics is quite convenience. I suspect that the first reading will not be your last, so the author's tradeoff was the right one.

I read this book many years ago, but it is still by my work station. I consult it when I get stuck on a problem, the heuristics do work.

For a first reading, the flipping around is annoying, but for reference purposes, the alphabetical order of the heuristics is quite convenience. I suspect that the first reading will not be your last, so the author's tradeoff was the right one.

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ByJames Joneson August 10, 2000

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.

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ByBill Butleron April 7, 2000

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!

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