How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library Book 85) by G. Polya (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I’m a math professor and this has completely changed the way I approach my lectures. I’ve gotten positive feedback from these changes which is great. I highly recommend this book for anyone who has to do critical thinking (that should be everyone!).

⭐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.

⭐A truly Brilliant book. Very informal. Dripping with insight and advice. How to go about solving problems. Mostly elementary math problems ( arithmetic and geometric ) but not exclusively. And also some sage advice on how to best teach mathematics – as exploring and solving new problems, not as the acquisition of a cookbook of recipes.Also some keen thoughts Towards the Development of an art and science of problem solving in general. Some historical examples from Pappus, Descartes, Leibniz et al. Leibniz always had a great focus on method. As did Descartes. Very easy to read and yet brilliantly suggestive and evocative. What are you waiting for?

⭐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…

⭐This book should be required reading for all teachers. Polya’s work follows neatly on the heels of Piaget and Vygotsky, and leverages the best of constructivism with still-relevant teaching methods. Yes, his language is appropriate to his era and is very male centric. Ignore that and learn the lessons. He is amazing.

⭐If you want instructions on how to become a genius, read and practice this book. If you don’t want to become a genius, but want to become a killer engineer, accountant, physicist, doctor, scientist, teacher or any other professional using math, read and practice this book.Modern Math texts cite this book constantly. They elevate the 5 step process to the word of the (something). Unfortunately, the rest of the text is about performing step 3, solving the algebraic equation. Step 2, writing the equation is the harder part for most students. Practice step 2 every day, and you will become master of time and space. We got computers to do step 3, that’s not the hard part.I tell students this book is about how to solve word problems. It is not about math, but how to use it.I found a copy of it in a stack of books in a sandwich shop on Main street. It belongs in every stack of books everywhere. It will improve the world.

⭐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 is a very good book that takes a lot of time to digest. So, get it well before your exam or you start your job as a trainee mathematical modeller! What I mean by that is, I read it, cover to cover, then returned to my university studies with renewed excitement and found I still got stuck. Par for the course in mathematics, but disappointing nonetheless. However, once I had solved the problem (or given up and checked the solution), then reflected on why I got stuck, a heuristic from this book often jumped out at me. Ah… if only I’d thought to try this it would’ve made the way forward so obvious!And that is essentially what the book is. You will not get a list of algorithms to take the creativity and hard work out of mathematical problem solving (frankly, that’d no more be a good thing than taking the creativity and hard work out of sport). What you will get is a discussion of thought processes that professional mathematicians use, probably unconsciously at that stage of education, that may help you make headway on your problem. Essentially, ‘What are fruitful questions to ask when I don’t know how to proceed or even begin?’ (Hint: don’t just sit there and stare at it waiting for the muse to strike you.) In fact, you probably use some of these already but don’t even realise you’re using the same strategy over and over again. In this way, Polya has done what mathematicians do: he has abstracted, generalised and systematised a hitherto hodgepodge of problem-solving recipes used implicitly in particular situations.A simple example for when you get stuck: ‘Can you rewrite the equation?’ I cannot count how many times I have fallen into this trap, realising after I’ve given up on a problem that the way to proceed would’ve jumped out at me had I only thought to rewrite it in a different form. On one level, it might simply be that you, personally, are really quite uncomfortable with a particular form of notation. Rewriting things might well put you at psychological ease with more familiar forms, or forms you’re much better practised at manipulating. Hate Leibniz notation for your calculus? Why not rewrite it as Newtonian to solve your problem, then translate back into Leibniz? This strategy, in fact, is what we do all the time – when you learn trig identities or to move between forms of vectors, say, you are implicitly learning the strategy: ‘Rewrite the equation to make it easier to deal with.’The other reason might simply be that by rewriting it, the solution jumps out at you. This is what happens whenever you multiply out, factor, substitute into equations etc. Just because your question doesn’t specify you need to do something, doesn’t mean you aren’t allowed to try it! But it has to occur to you first to rewrite your problem into an equivalent form, in order for the light at the end of the tunnel to reach you. And really this is something you do already: whenever you look up a word in a dictionary, you’re essentially seeing the term which you don’t understand rewritten as something you do understand. Then you can proceed with your paragraph, just like you can then proceed with your mathematical problem.

⭐I bought this book only for the bullet points about problem-solving but read a few additional chapters out of curiosity. I did not finish it, though, as it is geared mostly towards maths problems. However, the little I have read has provided me with some good insight on how to analyse problems and evaluate solutions to pick the right one.

⭐This was my maths teacher.

⭐If you’re looking for a book that gives you a framework with which you can begin to systematically approach problem solving, this is it. But in the end, its still over to you to put in the effort to try the method, work with it, absorb it and finally make it second nature. Well worth the effort though.

⭐The preface is so hard to understand. What is the guy saying?The contents are also hard to understand.I want to throw this book.

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