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How to Solve It: A New Aspect of Mathematical Method
A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out―from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft―indeed, brilliant―instructions on stripping away irrelevancies and going straight to the heart of the problem.
288 pages, Paperback
First published November 30, 1944
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January 3, 2013
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, the book probably won't teach you anything that you didn't know already. However, I have to say it twice, the book is written in a style so instructive that I'm pretty sure just about anybody could benefit from it.
In my opinion, this is definitely one of those books that every mathematician and everyone using mathematics (or even dealing with difficult problems of non-mathematical nature) should read and even perhaps have one lying around... just in case you feel like solving the Riemann hypothesis :P (or something wee bit easier for that matter XD)
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, the book probably won't teach you anything that you didn't know already. However, I have to say it twice, the book is written in a style so instructive that I'm pretty sure just about anybody could benefit from it.
In my opinion, this is definitely one of those books that every mathematician and everyone using mathematics (or even dealing with difficult problems of non-mathematical nature) should read and even perhaps have one lying around... just in case you feel like solving the Riemann hypothesis :P (or something wee bit easier for that matter XD)
November 27, 2010
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, but for solving any problem in any field.
Polya's general steps for solving problems include the following four steps: 1. understand the problem, 2. devise a plan, 3. carry out the plan, and 4. look back and examine the solution. These are simple and easy to remember steps, but powerful in their applicability to the most basic to the most complex problems that we face and are at the heart of learning. Over the years, different writers have revised these steps (added, taken away, shifted the wording and emphasis) the essential points still hold.
In addition, to the overall framework of Polya's heuristic and its generalizable nature, what I really like about this work is the fact that I can revisit it for nuggets of wisdom. The third section and roughly half of the book is taken up with "A Short Dictionary of Heuristic" which is a great resource. Each entry is a short essay on a given topic that weighs on either the nature of problem solving or the history of problem solving. One useful framework, that I took away immediately is the difference between "Problems to Solve" and "Problems to Prove." Making a distinction between these two types of problems it is easy to see that we often focus in education on problems to solve, but I and many students love finding out why (problems to prove).
So that said, I think this is a book that I will come back to and reference: a true classic in the educational literature.
Polya's general steps for solving problems include the following four steps: 1. understand the problem, 2. devise a plan, 3. carry out the plan, and 4. look back and examine the solution. These are simple and easy to remember steps, but powerful in their applicability to the most basic to the most complex problems that we face and are at the heart of learning. Over the years, different writers have revised these steps (added, taken away, shifted the wording and emphasis) the essential points still hold.
In addition, to the overall framework of Polya's heuristic and its generalizable nature, what I really like about this work is the fact that I can revisit it for nuggets of wisdom. The third section and roughly half of the book is taken up with "A Short Dictionary of Heuristic" which is a great resource. Each entry is a short essay on a given topic that weighs on either the nature of problem solving or the history of problem solving. One useful framework, that I took away immediately is the difference between "Problems to Solve" and "Problems to Prove." Making a distinction between these two types of problems it is easy to see that we often focus in education on problems to solve, but I and many students love finding out why (problems to prove).
So that said, I think this is a book that I will come back to and reference: a true classic in the educational literature.
August 12, 2020
"When Caesar's given your opponent the thumbs-down, it's easy to forget you still have to administer the coupe-de-grâce."
-Ian Stewart
Majority of the people who are aware of the Mathematics Events around the world have heard of George Polya even if they are not that much into mathematics themselves, for his name is on a par with legends like Gauss and Leibnitz now. Okay so we are all more or less familiar with his genius, but how many with his meant-to-be-simple-yet-intuitive works?
As a matter of fact I was a bit let down when I looked up the contents of the book after buying it for the first time. Why to waste money on some information you are already familiar with? But the book is quite something else to tell the truth. Firstly, it tells you the difference between a teacher and a researcher. You don't need to be a know-it-all to be a good teacher. However being a good teacher is probably even tougher, it demands time, practice, devotion and sound principles. He should try to teach the student discreetly, unobtrusively. You should know how to approach a problem, could it into something interesting for the listening fellas, and most importantly be an enthusiastic listener and observer. Saying which the book is not meant only for math-lovers, but for everyone else out there who is either a student or a teacher.
Meanwhile you may have heard of the age old Reductio Ad Absurdum, but have you ever thought of an interpretation like this?
" Irony adopts, to all appearance, a certain opinion and stresses it and overstresses it till it leads to a manifest absurdity."
The best part of the book, though is the third one: SHORT DICTIONARY OF HEURISTIC. Polya's heuristic strategies are undoubtedly quite well known. And though many of them does sound familiar enough, I don't think there's another author who can put it forward that simply. If yes, most probably he's paying a homage to Polya.
So, conclusion? Well it's not just an 'A New Aspect of Mathematical Method' as said. It's also a great example of lucid literature. Also I didn't think my money wasted. Neither will anyone,or so I believe. It's considered by many an equivalent to The Bible in matters of teaching. Well,I hope that makes some sense.
-Ian Stewart
Majority of the people who are aware of the Mathematics Events around the world have heard of George Polya even if they are not that much into mathematics themselves, for his name is on a par with legends like Gauss and Leibnitz now. Okay so we are all more or less familiar with his genius, but how many with his meant-to-be-simple-yet-intuitive works?
As a matter of fact I was a bit let down when I looked up the contents of the book after buying it for the first time. Why to waste money on some information you are already familiar with? But the book is quite something else to tell the truth. Firstly, it tells you the difference between a teacher and a researcher. You don't need to be a know-it-all to be a good teacher. However being a good teacher is probably even tougher, it demands time, practice, devotion and sound principles. He should try to teach the student discreetly, unobtrusively. You should know how to approach a problem, could it into something interesting for the listening fellas, and most importantly be an enthusiastic listener and observer. Saying which the book is not meant only for math-lovers, but for everyone else out there who is either a student or a teacher.
Meanwhile you may have heard of the age old Reductio Ad Absurdum, but have you ever thought of an interpretation like this?
" Irony adopts, to all appearance, a certain opinion and stresses it and overstresses it till it leads to a manifest absurdity."
The best part of the book, though is the third one: SHORT DICTIONARY OF HEURISTIC. Polya's heuristic strategies are undoubtedly quite well known. And though many of them does sound familiar enough, I don't think there's another author who can put it forward that simply. If yes, most probably he's paying a homage to Polya.
So, conclusion? Well it's not just an 'A New Aspect of Mathematical Method' as said. It's also a great example of lucid literature. Also I didn't think my money wasted. Neither will anyone,or so I believe. It's considered by many an equivalent to The Bible in matters of teaching. Well,I hope that makes some sense.
August 6, 2007
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).
January 21, 2011
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 problems in the back, presented to test your polished problem solving skills, are pretty awesome - definitely try to solve them yourself! One of my favorites: "A bear, starting at point P, walked one mile due south. Then he changed direction and walked one mile due east. Then he turned again to the left and walked one mile due north, and arrived exactly at the point P he start from. What was the color of the bear?" And no, this isn't a trick question - the answer makes perfect sense!
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 problems in the back, presented to test your polished problem solving skills, are pretty awesome - definitely try to solve them yourself! One of my favorites: "A bear, starting at point P, walked one mile due south. Then he changed direction and walked one mile due east. Then he turned again to the left and walked one mile due north, and arrived exactly at the point P he start from. What was the color of the bear?" And no, this isn't a trick question - the answer makes perfect sense!
November 29, 2015
I recently finished reading How To Solve It - A New Aspect Of Mathematical Method - by George Polya.
Below are key excerpts from this book that I found particularly insightful:
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.
Studying the methods of solving problems, we perceive another face of mathematics. Yes, mathematics has two faces; it is the rigorous science of Euclid but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. But the second aspect is new in one respect; mathematics "in statu nascendi,' in the process of being invented, has never before been presented in quite this manner to the student, or to the teacher himself, or to the general public.
Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution.
Where should I start? Start from the statement of the problem. What can I dot Visualize the problem as a whole as clearly and as vividly as you can. Do not concern yourself with details for the moment. What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its purpose on your mind. The attention bestowed on the problem may also stimulate your memory and prepare for the recollection of relevant points.
It would be a mistake to think that solving problems is a purely "intellectual affair"; determination and emotions play an important role. Lukewarm determination and sleepy consent to do a little something may be enough for a routine problem in the classroom. But, to solve a serious scientific problem, will power is needed that can outlast years of toil and bitter disappointments.
If you cannot solve the proposed problem do not let this failure afflict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again. Do not forget that human superiority consists in going around an obstacle that cannot be overcome directly, in devising some suitable auxiliary problem when the original one appears insoluble.
The future mathematician should be a clever problem-solver:; but to be a clever problem-solver is not enough. due time, he should solve significant mathematical problems; and first he should find out for which kind of problems his native gift is particularly suited.
In closing:
Going around an obstacle is what we do in solving any kind of problem: the experiment has a sort of symbolic value. The hen acted like people who solve their problem muddling: through, trying again and again, and succeeding eventually by some lucky accident without much insight into the reasons for their success. The dog who scratched and jumped and barked before turning around solved his problem about as well as we did ours about the two containers. Imagining a scale that shows the waterline in our containers was a sort of almost useless scratching, showing only that what we seek lies deeper under the surface. We also tried to work forwards first, and came to the idea of turning round afterwards. The dog who, after brief inspection of the situation, turned round and dashed off gives, rightly or wrongly, the impression of superior insight. No, we should not even blame the hen for her clumsiness. There is a certain difficulty in turning round, in going away from the goal, in proceeding without looking continually at the aim, in not following the direct path to the desired end. There is an obvious analogy between her difficulties and our difficulties.
A highly recommended read in the area of problem solving.
Below are key excerpts from this book that I found particularly insightful:
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.
Studying the methods of solving problems, we perceive another face of mathematics. Yes, mathematics has two faces; it is the rigorous science of Euclid but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. But the second aspect is new in one respect; mathematics "in statu nascendi,' in the process of being invented, has never before been presented in quite this manner to the student, or to the teacher himself, or to the general public.
Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution.
Where should I start? Start from the statement of the problem. What can I dot Visualize the problem as a whole as clearly and as vividly as you can. Do not concern yourself with details for the moment. What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its purpose on your mind. The attention bestowed on the problem may also stimulate your memory and prepare for the recollection of relevant points.
It would be a mistake to think that solving problems is a purely "intellectual affair"; determination and emotions play an important role. Lukewarm determination and sleepy consent to do a little something may be enough for a routine problem in the classroom. But, to solve a serious scientific problem, will power is needed that can outlast years of toil and bitter disappointments.
If you cannot solve the proposed problem do not let this failure afflict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again. Do not forget that human superiority consists in going around an obstacle that cannot be overcome directly, in devising some suitable auxiliary problem when the original one appears insoluble.
The future mathematician should be a clever problem-solver:; but to be a clever problem-solver is not enough. due time, he should solve significant mathematical problems; and first he should find out for which kind of problems his native gift is particularly suited.
In closing:
Going around an obstacle is what we do in solving any kind of problem: the experiment has a sort of symbolic value. The hen acted like people who solve their problem muddling: through, trying again and again, and succeeding eventually by some lucky accident without much insight into the reasons for their success. The dog who scratched and jumped and barked before turning around solved his problem about as well as we did ours about the two containers. Imagining a scale that shows the waterline in our containers was a sort of almost useless scratching, showing only that what we seek lies deeper under the surface. We also tried to work forwards first, and came to the idea of turning round afterwards. The dog who, after brief inspection of the situation, turned round and dashed off gives, rightly or wrongly, the impression of superior insight. No, we should not even blame the hen for her clumsiness. There is a certain difficulty in turning round, in going away from the goal, in proceeding without looking continually at the aim, in not following the direct path to the desired end. There is an obvious analogy between her difficulties and our difficulties.
A highly recommended read in the area of problem solving.
January 2, 2015
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 insight. But that alone would fill less than five pages of the text. The level of pedantism regarding terminology here that I found boringly intolerable and eventually I dreaded picking the book back up because I got it already. Ultimately I failed to find in this book what has made it so successful. "The List" is a great problem-solving approach, but that's just the pre-introduction page, and doesn't justify the remaining 253.
October 26, 2008
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, then they can go on to things such as philosophy or politics.
The first part of How to Solve It are essays on how to teach and how to approach problems in general. His view on teaching is leading a student to think. Giving the student problems where the answer is not the goal, but the experience in seeing a new type of problem. This is contrasted with viewing teaching as a series of cookbook or algorithms to be taught. It also means the role of the teacher is to provide the problem, then give only what is necessary to nudge the student in the direction needed for the student to discover the method of solution. And presumably, to be able to develop methods for other problems the student has not seen before. Very much like what graduate school is supposed to be.
The bulk of How to Solve It describes a wide range of approaches to problem solving. Some are familiar to a variety of disciplines such as business, crisis management, or general analysis. Some are more familiar to those in sciences or mathematics. But the illustrations are understandable to anyone past a first or second year of high school mathematics, making them much more understandable then, say, a graduate course in real analysis.
If I was in the position of working with first year graduate students in anything, I would recommend this book as something to read before they arrive on campus. It provides a good first exposure to many problem-solving approaches and an exhortation on how to think logically and analytically, that will suit them well when they are faced with the complicated subject matter that is ahead of them.
The first part of How to Solve It are essays on how to teach and how to approach problems in general. His view on teaching is leading a student to think. Giving the student problems where the answer is not the goal, but the experience in seeing a new type of problem. This is contrasted with viewing teaching as a series of cookbook or algorithms to be taught. It also means the role of the teacher is to provide the problem, then give only what is necessary to nudge the student in the direction needed for the student to discover the method of solution. And presumably, to be able to develop methods for other problems the student has not seen before. Very much like what graduate school is supposed to be.
The bulk of How to Solve It describes a wide range of approaches to problem solving. Some are familiar to a variety of disciplines such as business, crisis management, or general analysis. Some are more familiar to those in sciences or mathematics. But the illustrations are understandable to anyone past a first or second year of high school mathematics, making them much more understandable then, say, a graduate course in real analysis.
If I was in the position of working with first year graduate students in anything, I would recommend this book as something to read before they arrive on campus. It provides a good first exposure to many problem-solving approaches and an exhortation on how to think logically and analytically, that will suit them well when they are faced with the complicated subject matter that is ahead of them.
May 21, 2020
I'm conflicted about this book. There is a lot of good advice around the art of problem solving, but my god is there a lot of shit too. The layout is mostly a big alphabetical glossary of _math things_ --- everything from leading questions to notions of symmetry to anecdotes about absentminded professors --- and the layout doesn't particularly help. It's not organized by topic or ordered by first things first, it's just plopped down alphabetically. As such, it's hard to get into the flow.
This book however is lacking primarily in that it deals with how to solve "well-posed questions," which is to say, toy problems. There is very little about conducting your own open-ended research, and about how to turn wisps of ideas into well-posed ones.
This book however is lacking primarily in that it deals with how to solve "well-posed questions," which is to say, toy problems. There is very little about conducting your own open-ended research, and about how to turn wisps of ideas into well-posed ones.
June 11, 2015
This is an important book. Possibly historical in its utility and impact. I'm proud to have this on my shelf and will likely reference it every so often for the rest of my life.
September 7, 2022
A bit repetitive and so could be shorter but still packed with insight that may be rather invaluable, I also like how the author communicates with the reader.
Some great problem solving tips were also given in 3blue1brown's channel
{1. use the defining features of the set up
2. give things meaningful names
3. leverage symmetry
4. try describing one object in two different ways
5. draw a picture (have coordinates if you have some numbers for example)
6. ask a simpler version of the problem
7. read a lot and think about problems a lot}
Which is almost a summary of the most important things Polya mentions.
...
Wonders if I'll ever be smart enough to do anything with even the smallest significance in mathematics
Some great problem solving tips were also given in 3blue1brown's channel
{1. use the defining features of the set up
2. give things meaningful names
3. leverage symmetry
4. try describing one object in two different ways
5. draw a picture (have coordinates if you have some numbers for example)
6. ask a simpler version of the problem
7. read a lot and think about problems a lot}
Which is almost a summary of the most important things Polya mentions.
...
Wonders if I'll ever be smart enough to do anything with even the smallest significance in mathematics
October 11, 2014
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 through most of it, only stopping at the particular instances I found new/ relevant/ interesting. I skipped a lot of the examples (I know, I know, I'm still feeling guilty about it!) , regardless I found many good gems in the long stretch of text. The creator's paradox, for example, is something I've learned the hard way, but seeing it packed into a concept helped organize my thinking a lot.
I would have given it a much higher rating as a middleschool to highschool student.
I like the chapter at the end though. Fun questions.
Still, the way I went at the book is that I skimmed through most of it, only stopping at the particular instances I found new/ relevant/ interesting. I skipped a lot of the examples (I know, I know, I'm still feeling guilty about it!) , regardless I found many good gems in the long stretch of text. The creator's paradox, for example, is something I've learned the hard way, but seeing it packed into a concept helped organize my thinking a lot.
I would have given it a much higher rating as a middleschool to highschool student.
I like the chapter at the end though. Fun questions.
December 21, 2009
این کتاب ترجمه کار کلاسیک جرج پولیا: How to solve it هست.
به نظرم شاید برای خوانندهای که هنوز چندان با مسئلههای ریاضی کلنجار نرفته خیلی جالب نباشه، اما برای معلمان ریاضی و کسانی به صورت جدیتر درگیر حل مسائل ریاضی هستند کتاب تامل برانگیز و آموزندهای هست که کمک میکنه با دید بازتر راهی که در حل مسائل میرند رو بازبینی کنند و نسبت به فرآیند حل مسئله خودآگاهتر بشند. این خودآگاهی و توصیههای راهیابانه کتاب میتونه به بهتر شدن مهارت حل مسئله افراد کمک کنه.
به نظرم شاید برای خوانندهای که هنوز چندان با مسئلههای ریاضی کلنجار نرفته خیلی جالب نباشه، اما برای معلمان ریاضی و کسانی به صورت جدیتر درگیر حل مسائل ریاضی هستند کتاب تامل برانگیز و آموزندهای هست که کمک میکنه با دید بازتر راهی که در حل مسائل میرند رو بازبینی کنند و نسبت به فرآیند حل مسئله خودآگاهتر بشند. این خودآگاهی و توصیههای راهیابانه کتاب میتونه به بهتر شدن مهارت حل مسئله افراد کمک کنه.
February 6, 2014
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 :)
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 :)
December 8, 2017
Pólya is the teacher I never had.
Now you might get scared with the word "Mathematical" in the title - don't be. It is a general guide to how to solve a problem. Starting from establishing the question, gathering the known and to find the unknown. The method you are using to find the minimum distance between two points can also be used to find the most convenient road to your nearest grocery store. This book shows you -"How to Solve it".
Now you might get scared with the word "Mathematical" in the title - don't be. It is a general guide to how to solve a problem. Starting from establishing the question, gathering the known and to find the unknown. The method you are using to find the minimum distance between two points can also be used to find the most convenient road to your nearest grocery store. This book shows you -"How to Solve it".
May 10, 2014
The whole content of this book is presented in first two pages at start of it. All other text is explanation of these 2 pages, illustrated with examples, etc. I think, that it could be very useful for teachers...
May 24, 2017
you know if you should read this and a review on this site will be meaningless
August 18, 2019
One of the most useful books I've read in my life.
It's an authentic treasure to everyone interested in problem-solving (and every one of us has enough problems to be interested in with problem-solving).
I think this should be one of the obligatory lectures for high-school or college education because it states clearly the steps one intuitively follows when trying to solve a problem mathematical or not.
The key takeaways for me are:
1. If you conceive a plan to solve the problem you're almost done.
2. Going around obstacles is a good way of solving problems.
3. If you cannot solve the problem you have right now you could try several distinct strategies:
- Change the problem
- Change the expected result
- Change the conditions
- Change everything -> Anything that could make you improve your understanding of the problem could help.
It's an authentic treasure to everyone interested in problem-solving (and every one of us has enough problems to be interested in with problem-solving).
I think this should be one of the obligatory lectures for high-school or college education because it states clearly the steps one intuitively follows when trying to solve a problem mathematical or not.
The key takeaways for me are:
1. If you conceive a plan to solve the problem you're almost done.
2. Going around obstacles is a good way of solving problems.
3. If you cannot solve the problem you have right now you could try several distinct strategies:
- Change the problem
- Change the expected result
- Change the conditions
- Change everything -> Anything that could make you improve your understanding of the problem could help.
August 18, 2023
I was looking forward to reading this book, because it is considered a classic and because I wanted to learn more about a systematic approach to mathematical problems.
The author had obviously a very high mathematical and intellectual capability but I found the book very difficult to read.
I found the structure of the main section as a “dictionary” very tricky to follow, and the cross referencing of entries doesn’t help. There are many repetitive parts and messages as well.
Many of the examples are geometry related which is unhelpful for someone more numerically inclined.
It’s still a good book and it provided some interesting perspectives on problem solving, but was a bit disappointing for me!
The author had obviously a very high mathematical and intellectual capability but I found the book very difficult to read.
I found the structure of the main section as a “dictionary” very tricky to follow, and the cross referencing of entries doesn’t help. There are many repetitive parts and messages as well.
Many of the examples are geometry related which is unhelpful for someone more numerically inclined.
It’s still a good book and it provided some interesting perspectives on problem solving, but was a bit disappointing for me!
November 2, 2021
It's quite a good book for the average person.
I picked it up because I've found it as a recommendation in a stackoverflow comment on a topic regarding learning programming algorithms.
It's a good book because it introduces you in the right mental framework to approach the problem of learning something complex (such as algorithms).
It's great for those who are in a teaching proffesion, especially mathematics, and especially with children.
George Polya must have been a great teacher.
Not being in that niche, 3 stars from me.
I picked it up because I've found it as a recommendation in a stackoverflow comment on a topic regarding learning programming algorithms.
It's a good book because it introduces you in the right mental framework to approach the problem of learning something complex (such as algorithms).
It's great for those who are in a teaching proffesion, especially mathematics, and especially with children.
George Polya must have been a great teacher.
Not being in that niche, 3 stars from me.
August 22, 2021
I've finally read Polya's "How to Solve It." Polya describes methods for solving math problems, yet I can relate that it works in software development, too.
One of the surprising things is that Polya mentioned the sense of progress. Some could miss this sense and give up, but a successful problem-solver feels progress. It encourages to keep up and solve it eventually.
I want everyone to develop this sense of progress during solving problems and to succeed at problem-solving and creating software.
One of the surprising things is that Polya mentioned the sense of progress. Some could miss this sense and give up, but a successful problem-solver feels progress. It encourages to keep up and solve it eventually.
I want everyone to develop this sense of progress during solving problems and to succeed at problem-solving and creating software.
Read
August 21, 2023This the best book I have read about this topic.
I think the main ideas should be taught to every student or even every person for that matter; being able to solve problems is such an important skill.
Of course it has a mathematical tone, which may or may not be suited to you, but the principles in their generality are applicable to any category of life in which problems arise.
I think the main ideas should be taught to every student or even every person for that matter; being able to solve problems is such an important skill.
Of course it has a mathematical tone, which may or may not be suited to you, but the principles in their generality are applicable to any category of life in which problems arise.
August 10, 2020
Hands down one of the best books I have ever read (and a couple of years late too), and I plan to reread this every so often.
I stumbled on this book after watching Rich Hickey’s talk titled “Hammock driven development”, and if you know Rich Hickey then you know there is some weight behind the recommendation. I saw the book getting recommended again in other popular software talks and conferences and I decided to treat myself to the book. 100% worth it.
Although the direct audience is math/logic/science teachers and students, the principles laid down in this book are generally applicable in other forms knowledge work e.g policy, entrepreneurship, academia and so on and so forth.
It might be a bit intimidating to start the book without a technical background because the author relies on mathematical examples to make his points, but you can gloss over these examples and try to create your own examples or try and apply it to a problem that you are facing at work.
For the principles in this book to stick you have to be consistent. Here is a quote from the book that highlights this well
“Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them.”
The principles actually work, and I have definitely seen the quality of my work improve.
GO GET THIS BOOK
I stumbled on this book after watching Rich Hickey’s talk titled “Hammock driven development”, and if you know Rich Hickey then you know there is some weight behind the recommendation. I saw the book getting recommended again in other popular software talks and conferences and I decided to treat myself to the book. 100% worth it.
Although the direct audience is math/logic/science teachers and students, the principles laid down in this book are generally applicable in other forms knowledge work e.g policy, entrepreneurship, academia and so on and so forth.
It might be a bit intimidating to start the book without a technical background because the author relies on mathematical examples to make his points, but you can gloss over these examples and try to create your own examples or try and apply it to a problem that you are facing at work.
For the principles in this book to stick you have to be consistent. Here is a quote from the book that highlights this well
“Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them.”
The principles actually work, and I have definitely seen the quality of my work improve.
GO GET THIS BOOK
September 9, 2022
I feel like I am a bit late reading it, after so many years in academia, with a focus that deals with Math one way or the other. That is why I can't fully appreciate how much the tips would not be obvious to a beginner. I recommend this to mainly Math teachers, or people who would like to self teach Math, but even then, the book is not a cookbook manual to solve questions, and it does not try to be, as its title implies. Some of the tips are really abstract, some points are more philosophical than practical, but all in all, I think a topic that is this general will not be covered with more success than this.
One point that really bothered me was how the book tries to be a dictionary, while I think a more structured argument would be easier to follow. The book also refers to its other chapters pretty frequently, but this is more inline with hyperlinking in web rather than, the usual build up that we are accustomed from Math books. Maybe there exists a wikilike version of this? If not, I think there should be.
I am teaching myself Computer Science in a small but steady approach, and this book appeared in more than one suggested readings list, so I read it. If you already have experience in Math, this book is not that useful for learning CS.
One point that really bothered me was how the book tries to be a dictionary, while I think a more structured argument would be easier to follow. The book also refers to its other chapters pretty frequently, but this is more inline with hyperlinking in web rather than, the usual build up that we are accustomed from Math books. Maybe there exists a wikilike version of this? If not, I think there should be.
I am teaching myself Computer Science in a small but steady approach, and this book appeared in more than one suggested readings list, so I read it. If you already have experience in Math, this book is not that useful for learning CS.
January 16, 2022
Polya’s book on the meta-theory of analyzing mathematical problems, and (hopefully) solving them is a rare (semi) technical text that can be read fairly well both on audio and by eye, and still get something out of both. Of course, to really get the full instructional value, you must read this text visually, on account of the mathematical notation. Still Polya’s commentary on the nature of heuristics and how they are applied to problem solving shines through even with just the audio.
In many ways, this book covers the standard material that many other modern textbooks aimed at introduction to proofs and logic also survey, which includes the standard construction techniques of indirect proofs, the reductio ad-absurdum, the workhorse of direct proofs (of well-ordered objects), the induction, as well as a smattering of examples taken from what we would now call “school mathematics”, the fields of geometry, counting, and (classical) algebra. However, unlike modern textbooks designed for the standard bridge-course into formalism, there is much less focus in the practice of procedure (solving problems via exercises, to which there are none in this book, at least numbered ones), and more focus on the method of procedure, that is a commentary in english on why we are doing what we are doing in solutioning/analysis, and how to think about it as a human being.
This is a critical piece of knowledge that is often missing in modern text for this subject matter, and is an essential prerequisite to that thing we call “mathematical maturity” i.e. the ability to think, reason, and extend mathematical objects to our purposes. Many modern textbooks have a tendency to routinize this process of discovery/solutioning and this is a mistake. Although there may be a prescription that yields solutions “most of the time” for many classes of problems, to learn the process of interacting with mathematics in this way is to learn to be a computer. That is, it is to learn to divorce oneself with their ‘humanness’ in the way of thinking about mathematical things, and it is practically diminishing one’s capacity to efficiently solve problems as a human vis-a-vis the machine. That is, it is unlikely a human will be able to compete with a machine in traversing the problem in that avenue of approach.
Instead, the way many mathematicians have approached these abstract objects is to leverage their intuition. Just as a Chess player does not literally ken all of the enumerations of steps that can take place, but instead has a meta-idea of how to think of the board, and deconstructs it using that “theory” of the board to inform their next move, a human mathematician will have to leverage their intuition to devise heuristics to assess the object at hand, then attack the problem with that assessment. Poyla teaches it’s readers how to go about this process methodically. I found re-visiting this subject matter after having left it for a few years to do professional work, it has awakened many of my dormant capacities, and more importantly, helped me to extend previously weaker parts of my skill-sets.
This is a great example of a supremely well-written (somewhat) technical book, which rivals (possibly exceeds) the few others that also do this intersection well, such as Richard Feynman’s more technical instructional writings. Highly recommended to anyone learning (or relearning) this subject as a more holistic commentary on craft in mathematics.
In many ways, this book covers the standard material that many other modern textbooks aimed at introduction to proofs and logic also survey, which includes the standard construction techniques of indirect proofs, the reductio ad-absurdum, the workhorse of direct proofs (of well-ordered objects), the induction, as well as a smattering of examples taken from what we would now call “school mathematics”, the fields of geometry, counting, and (classical) algebra. However, unlike modern textbooks designed for the standard bridge-course into formalism, there is much less focus in the practice of procedure (solving problems via exercises, to which there are none in this book, at least numbered ones), and more focus on the method of procedure, that is a commentary in english on why we are doing what we are doing in solutioning/analysis, and how to think about it as a human being.
This is a critical piece of knowledge that is often missing in modern text for this subject matter, and is an essential prerequisite to that thing we call “mathematical maturity” i.e. the ability to think, reason, and extend mathematical objects to our purposes. Many modern textbooks have a tendency to routinize this process of discovery/solutioning and this is a mistake. Although there may be a prescription that yields solutions “most of the time” for many classes of problems, to learn the process of interacting with mathematics in this way is to learn to be a computer. That is, it is to learn to divorce oneself with their ‘humanness’ in the way of thinking about mathematical things, and it is practically diminishing one’s capacity to efficiently solve problems as a human vis-a-vis the machine. That is, it is unlikely a human will be able to compete with a machine in traversing the problem in that avenue of approach.
Instead, the way many mathematicians have approached these abstract objects is to leverage their intuition. Just as a Chess player does not literally ken all of the enumerations of steps that can take place, but instead has a meta-idea of how to think of the board, and deconstructs it using that “theory” of the board to inform their next move, a human mathematician will have to leverage their intuition to devise heuristics to assess the object at hand, then attack the problem with that assessment. Poyla teaches it’s readers how to go about this process methodically. I found re-visiting this subject matter after having left it for a few years to do professional work, it has awakened many of my dormant capacities, and more importantly, helped me to extend previously weaker parts of my skill-sets.
This is a great example of a supremely well-written (somewhat) technical book, which rivals (possibly exceeds) the few others that also do this intersection well, such as Richard Feynman’s more technical instructional writings. Highly recommended to anyone learning (or relearning) this subject as a more holistic commentary on craft in mathematics.
Read
April 21, 2022I flunked an exam, asked for book recs to get good at math, and was recommended this book. I have now gotten (slightly) gooder at math and this book helped. Mainly because it slapped me in the face by exposing how/where I was falling short in mathematical reasoning.
"But why must I ponder the god damn question statement for five minutes?" I said.
"Because you are a dummy and will make silly little mistakes like assuming what you have to prove by induction :)" said the book, and I felt very seen.
Highlights:
- Ponder! (Strategise your 'plan of attack' for the problem.)
- Attack + sanity check every step
- If you get stuck, start running the questions (have I seen proofs like these? What do the definitions mean? What theorems does this rely on?)
- Finish and Ponder (understand how you messed up -- what are the logic mistakes you keep on making and what are the gaps in your knowledge.)
"But why must I ponder the god damn question statement for five minutes?" I said.
"Because you are a dummy and will make silly little mistakes like assuming what you have to prove by induction :)" said the book, and I felt very seen.
Highlights:
- Ponder! (Strategise your 'plan of attack' for the problem.)
- Attack + sanity check every step
- If you get stuck, start running the questions (have I seen proofs like these? What do the definitions mean? What theorems does this rely on?)
- Finish and Ponder (understand how you messed up -- what are the logic mistakes you keep on making and what are the gaps in your knowledge.)
February 11, 2018
This is such a great book. Polya lays out the different approaches you can take to solve a problem. He primarily uses Euclidean geometry to explain the possible ways to approach different problems.
This is especially great when you have to analyze a problem that you haven't seen before - where and how to begin? He goes through the problem-solving process in detail, beginning with the detailed analysis of the question and then the systematic synthesis of the answer.
This book really changed how I see math, especially abstract math. I have a much deeper understanding and appreciation for abstract math after reading this book. The problems at the end of the book are challenging but fun when you finally (if at all) figure out what the trick is.
This is especially great when you have to analyze a problem that you haven't seen before - where and how to begin? He goes through the problem-solving process in detail, beginning with the detailed analysis of the question and then the systematic synthesis of the answer.
This book really changed how I see math, especially abstract math. I have a much deeper understanding and appreciation for abstract math after reading this book. The problems at the end of the book are challenging but fun when you finally (if at all) figure out what the trick is.
April 7, 2019
The book, written in 1950, still applies today and attempts to find a method to discover the solution to any mathematical problem we may encounter in our life. Although sometimes repetitive, the author is very pedagogical and presents various examples to illustrate his method. Despite the fact the examples are mainly on the Math field, he manages to translate them into other domains (like engineering, physics, psychology, etc.). A book nice to have in our library :)
December 7, 2023
se pudesse, obrigava todos os professores do mundo a ler isto. e depois a toda a gente que gosta de aprender coisas. obrigado lina oliveira do departamento de matemática do técnico por mo recomendar :))
October 10, 2022
This should be an obligatory textbook in high school.
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