How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

Math is very important, and this book shows how you can successfully solve math equations.

Joseph R. Dell'Aquila, Ph.D. My first exposure to this book was probably as a young college student. When I started teaching physics and mathematics at the college and university level, I recommended this book to all of my students. Why? The table-like pages xvi - xvii are an excellent reminder of fruitful ways to understand, think about, attack and solve problems. Although I am a PhD in theoretical physics, I still dip into it occasionally when I need some insight or want to recall what I knew about approaching a problem. Is the book at that high a level? Of course not. It is a basic introduction to the fundamentals of problem solving. But remember that Michael Jordan, in "I Can't Accept Not Trying," always thanked Dean Smith, his famous college coach - who would bench Jordan if he got sloppy - for teaching him the fundamentals and Jordan said within a page of that: "fundamentals, that's what made Larry Bird such a great player." That is all this book is trying to give, fundamentals, and it does so brilliantly. To those whose reviews said it was not helpful and wanted to know where was the graduate level analysis, if you want to stick with Polya try "Inequalities" by G. H. Hardy, J. E. Littlewood, G. Pólya, all great to exceptional mathematicians with plenty of analysis to share and, for more specialized work, Isoperimetric Inequalities in Mathematical Physics by Polya and Szego. Note that one computer scientist/programmer disliked the book but another lauded it. I would never want to restrict dialogue on review but please check out the appropriateness and level of any book you buy. I have rarely written negative reviews on Amazon or elsewhere because I do my homework: using the Internet to find information on the work and even going to a library to see whether I like what their copy offers (I'm phrasing it this way because different editions, perhaps the library's edition vs. the one you're considering purchasing, can be quite different). Buy and use Polya if it is appropriate to your needs.

I don’t read very many math books, but when I first picked up How to Solve It by G. Polya, I realized that this wasn’t your typical theoretical math book. I had only assumed so by flipping through the pages and seeing various figures and expressions… It is actually very little theory and more of how to actually approach hard problems. The very first part of the book lays it down on what Polya will drill upon the reader. The essential idea is basically a framework laid upon the reader on how to solve difficult problems — particularly in the realm of mathematics and logic. Why is this valuable? We tend to flail our hands and throw down the pen when we encounter a hard problem. Wouldn’t it be nice to have a systematic way, or yet a solution in which we can see on the horizon and eventually reach? This is the book that will teach us for that very purpose! The main idea is basically of when attempting to solve a hard problem, we must consider the following and ask ourselves the following: 1. What is the unknown? 2. What is the data that is presented? 3. Did we make use of all the conditions presented by the problem? These three questions can virtually help us self reflect on how we solve problems and in time, with much practice — aid us in actually becoming smarter problem solvers. Are geniuses born, or bred? Is it nature, or nurture? Well, with this framework in mind, acquiring the persona of the genius interpreted by others becomes more nurture than anything. So what if we are still stuck on the problem? First thing is first, the point in which Polya makes is that we should not rush. We should not attempt to solve a problem when we have an incomplete understanding of the problem, or task. Before declaring ourselves stuck.. we must ask ourselves if we truly have a grasp of the problem in front of us. Ask ourself again… have we seen a similar problem before in the past? Better yet, have we solved a similar problem in the past? Can we somehow use that prior knowledge and integrate it into the process of attempting to solve the current problem? Finding sub-problems, or problems within the problem in which we can solve can possibly help us with the overall problem. Can we find the connection between the data presented and the unknown? Notice, and I agree with Polya in that we tend to not have a thorough understanding of the problem if we cannot answer these questions. The most important takeaway I received from reading this book was this: If I find myself making progress on a problem, I should keep working on each step in a precise and detailed manner. I must be sure I can give justification on why I have approached each step the way I chose to. If I achieve the result, make sure I can check the result. Can I go back and reproduce it? Can I devise a similar example with a set of parameters to produce a predictable result? Upon reading all this, I had the realization that not only is this is the basis for problem-solving — it is the key to solving algorithms problems. Confirming the result is one thing, but Polya makes the key suggestion in that we must STOP! We should not move on. A difficult problem requires reflection. We must take time to reflect on the thought process we have taken to work out the problem. This will help us remember how we were able to solve the problem using the specific tools in our mental toolkit. It will help us with future problems. At some point during attempts to solve a difficult problem, we may get discouraged. We can’t give up! If we make one small step towards our solution, we need to appreciate the advancement. We need to be patient and take each step as a piece of the overall composition of the essential idea. Take our guesses seriously, and don’t rush. Being aware of a “hunch” and keeping it in consideration may lead to a serious breakthrough. Well, just as long as we are cautious! We need to examine any guesses critically and see if they can be of use to us. How to Solve It was amazing in drilling to me the overall problem solving process and caused me to self reflect on how I should approach hard problems. I don’t think I was that terrible at working on problems before — but now I truly believe I can become better at problem-solving and analysis if I take a step back and actually self-reflect on various points of the problem solving process. Developing such a habit and practicing it as if it was nature is key. This was overall a great read. It took me about a week to read and was a bit more difficult to go through — partly because it was so thought provoking! The only downside was that I believe that the book went a little too long and the pace changed 75% of the way through. I believe the examples presented either went over my head purely due to lack of interest, or by then, I had already become convinced with the philosophy drilled by G. Polya on how to problem solve.

Great descriptions of problem solving techniques. Some were not new to me, but having them written out and analyzed has made my approach to problems more logical and successful. It is not a straight through read, as it includes a dictionary (kind of) that makes reading cover to cover a disjointed task. The gems it holds are worth digging out individually and making sure you take the time to fully enjoy each one. Great for teachers as it focuses not only on how students should solve problems, but also on the role of the teacher in guiding learners.

This book is a classic on heuristics and should be required reading for math majors and others wanting to improve their problem-solving skills.

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.

After more than 50 years, Polya's advice on tackling problems is still worth reading. But be warned, this is not the latest, brightest, trendiest, best-selling "problem solving book" out there that target MBAs or a kind of personal self-development book. Its author had contributed to important fields of mathematics and he had been through many problems, many difficulties, many students and many different questions by those students. If you're a young but eager student who faces problems in math (or in natural sciences), a person trying to solve some puzzles or practical problems, or a researcher about to start a long and unguaranteed journey in order to solve a big problem then you owe yourself to have this classic on your bookshelf, or better on your table. I'd like to quote some important passages from the book but last time I checked my notes they are about as long as the book. So maybe it is better to let Polya do the talking...

I'll be concise. THE GOOD: Everything, essentially. He goes into REALLY, REALLY, REALLY deep explanations of methods of problem solving, when to use what techniques, how to use them, and how they work. Sometimes, the explanations go on for pages so you forget what you were originally reading about - but that's good! He also provides examples of his problem solving techniques and how to use them. What I love was that he repeatedly goes back to a big list of problem solving techniques he has in the beginning of the book. He keeps asking the questions necessary to solve an example problem, referring back to the list, so you get a hang of when to ask yourself what. The only bad thing is that it's old. It's not as easy to follow as a modern book. I don't get why things were written in this weird English compared to now, but it's not exactly a simple read. Give yourself time, read it more than once, really study it.

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数学の問題を解く方法が、学生を指導する、という視点で体系的に述べられています。 解答を導き出すための学生への質問集も用意されています。 考え方にびっくりするほどの斬新さは感じられませんが、数学教師や数学を志す人には役立つと思います。 では数学以外の一般的問題解決やプロジェクトなどの問題分析に役立つかというと、プロセスマネジメントが発達した現代ではそこまでの応用力を求めるのは無理かな、と感じました。 文章は簡潔で英語は読みやすいです。 今まで馴染みのなかった数学英語に接することも出来ました。また、錆付いた数学知識を（ある程度）思い出すきっかけになったと思います。 ガリレオ時代のフランスの学者 Descartes,Reneが引用されていたりもします。

Este libro es un pequeño tratado de heurística. Describe generalmente el proceso seguido para resolver problemas, aplicable a cualquier tipo de problemas, particularmente matemáticos (en los cuales se centra el autor). El libro esta escrito por un importante matemático de origen húngaro, y aunque ya han pasado unos 40 años de su primera impresión, su contenido no está obsoleto ni ha perdido vigencia. Resultará de interés a cualquier estudiante de ciencias (naturales,pricipalmente) o ,sobre todo, de matemáticas (entre otras cosas trata la intuición en matemáticas y la importancia de las demostraciones formales). La edición es buena y relativamente robusta.

This is an interesting read

This book is life changer . Even If you are an adult but still love to understand the mathametical concept buy it . A great mental exercise into problem solving