What should I try to learn from Polya's "How to Solve It"?
I think Pólya wanted to teach thinking about mathematical thinking. I read the book as an algorithm for solving algorithms, an exploration in the meta processes involved in reducing problems from unsolved to solved, iteratively turning the apparent intractable to the real tractable.
In essence, Pólya took a step back and said focus on the machinery around the problem before trying to think about the problem in its specifics. This, I concur, is necessary for working with abstract concepts. As Thurston wrote in On Proof and Progress in Mathematics "people have very different ways of understanding particular pieces of mathematics", which the "popular model" for mathematical thinking only caricatures:
(D) Mathematicians start from a few basic mathematical structures and a collection of axioms "given" about these structures, that (T) there are various important questions to be answered about these structures that can be stated as formal mathematical propositions, and (P) the task of the mathematician is to seek a deductive pathway from the axioms to the propositions or to their denials.
You'll find modern authors producing essays similar to Pólya's. See for example Triki and Cut the Knot. Unsurprisingly, as is fitting for the contemporary sensibility of bite-size information, these modern forms are link dumps, blog posts, and short how-to guides.
To answer your question,
What should I try to learn?
I don't think there is one unique answer for everyone. In fact, what you-of-today learn will be different than what you-of-six-months-hence learn, precisely because thinking about math changes how one thinks about math. Read it today and absorb what sticks. Think about it, then come back and re-read it. I guarantee the new you will see insights the old you missed.
I also read the book "How to solve it?", but in German (title "Schule des Denkens", literaly translation: "School of Thinking"). I will explain my personal impression and how you should read any book like this.
Impression: You should definitly learn the 4 steps of the problem-solving-process:
- Understanding the problem (What is unknown? What is given?)
- Devise a plan (Can you express the task in other words? Do you know a similar question?)
- Exercise the plan (Can you see that every step is correct?)
- Review and Reflection (Can you controll the result? Is it possible to derive it in another way?)
This process is also given implicitly in some calculus textbooks for school in Germany (e.g. Lambacher Schweizer). Further more it is the basis of some mathematic didactic literature and studies. The process is idealized and linear.
- How to read books like this. I found some interesting hints in the famous book "How to win friends and influence people?" by Dale Carnegie (which is by the way also a recommendable book). I just adapted the priciples to the case at hand:
• Develop a deep, driving desire to master proofs and mathematical problems,
• Read each chapter twice before going on to the next one.
• As you read, stop frequently to ask yourself how you can apply
each suggestion.
• Underscore each important idea.
• Review this book each month.
• Apply these principles at every opportunity. Use this volume as a
working handbook to help you solve your daily problems.
• Check up each week on the progress you are making. Ask
yourself what mistakes you have made, what improvement, what
lessons you have learned for the future.
• Keep notes showing how and when you
have applied these principles.
I think this suggestions are also helpful in the case of Polya's book, because it is written very action-orientedly, s.t. you can apply the principles in your every day life or you can apply them to eveluate problem solving strategies of other people. It really depends on your interest and motivation why you read the book.