How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) Paperback – October 27, 2014

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.

This work gives an excellent overview of his method and ends with some non-traditional problems to work with solutions. His heuristic dictionary has interesting portions, but does not form coherent approach being written in alphabetical order. It does form a useful collection of tips and hints for working through problems generally.

This book has a wonderful set of mental models to solve math problems. I would highly recommend it mainly to help the reader build some problem-solving heuristics.

The major disadvantage of this book is that it does not help the reader build muscle memory of the various problem-solving heuristics. It would have been better if the author taught one heuristic followed by a few examples and problem sets before moving on to the next heuristic. The other challenge is that the author often does not print diagrams for geometry problems, making it harder to focus on problems when the reader has to draw mental pictures after having to decipher the original problem statement. One could argue that this is part of the learning, but I have noticed that I learn faster when I can see information presented in different formats.

So my advice is by the book, become familiar with the heuristics and then find another source to practice these techniques more deliberately.

He was a great mathematician but this book is not very helpful.
I purchased this book because of reviews here and recommendations from Math Stack Exchange. I found the book ... puzzling. The entire message is in the chart at the beginning of the book. The rest of the book you expect would expand on, explain, or gives example of the technique outlined in the front pages. But it does not. Instead it repeats and repeats again, often using the exact English phrases over and over again. The majority of the book is a dictionary of heuristic thinking. But the definitions are often circular and thus perhaps not so useful.

The format of the book is strange and not logical. Fuzzy. This book reduces to a 3-4 page hand out - 1 page for the message and a few pages of examples; combined with a poorly done dictionary on a different (although related) subject. Perhaps a more accurate title might be "A Mathematician's Dictionary of Heuristic Science with a Few Notes on Problem Solving".

On to it's usefulness as a problem solving method. There may be some merit in his suggestions for problem solving as outlined in the front pages. I gave it a try on (for me) a moderately difficult problem. I found his way to be (as another reviewer noted) methodical without being helpful. Then I tried solving a similar problem without using his method but really thought about what I was doing to solve the problem. Used my normal more intuitive method I found I do incorporate many of his suggestions but not in such a linear way. Perhaps it is me. Perhaps it is how my brain makes connections but I found this book confusing and not helpful.

With that said, I have bought the book and I will keep it on my shelf. Perhaps I will pick it up in a year or so and at that time figure out what exactly everyone is raving about.