Solving 'How to Solve It'
I hit the ground running when I started George Pólya's How to Solve It: A New Aspect of Mathematical Method. Somewhere in the middle, the momentum disappeared, and months later, I feel so relieved to have finished it. For all that, I give it 5 out of 5 stars...yes, indeed, why??
This is a math/logic/philosophy classic from 1945, dealing with heuristic, "the study of the methods and rules of discovery and invention." More particularly, it is a comprehensive guide to problem-solving. The first 40 pages or so are strictly about "How to Solve It" and classroom strategies, while the rest of the book elaborates on these themes in the "Short Dictionary of Heurstic." The back of the book has some sample problems/solutions, which, if I had more time and energy, I wouldn't mind trying.
Hopefully the word "math" does not turn you away! That is the one weakness of the book - most of the examples are in algebra and geometry, which, even for me, were often hard to follow. However, the heart of it is absolutely universal to all types of problem-solving where there are definite choices and paths to follow (e.g. not moral dilemmas, naturally). A few points I took away from it:
- Use descriptive, accurate terminology. Go back to definitions and terminology.
- Think aloud, ask yourself questions.
- Look for, and/or remember, similar problems/solutions.
- Vary the problem.
- Come up with a helper problem.
- Give yourself time and rest when you need it.
- Work efficiently.
- That is, when you're having trouble solving a problem, be clever about how you deal with it.
- As much as you can, work from question to conclusion, even if it's only a tiny part of the problem. Finish each work session with some sense of achievement.
- "Set [yourself] a new question about the problem." This was the most useful advice to me.
- When you start a problem, picture success - the end result!
Now, as to why this book was such a struggle to read. As you've probably read on my about, I'm working on a computer science degree, a division of applied mathematics. For that reason, most of the material in How to Solve It is not new to me; I've had a lot of classroom experience with these techniques. The sad thing is, there are many professors who do not apply these methods correctly. By the middle of the book, I started remembering their approach, and I thought, "Oh, this is that book, the source of my problems..."
Later, I gave it some objective thought and recalled my worst and best professors. I came to the realization that nearly all of them follow this pattern of teaching! What, then, is the defining difference? The most unhelpful professors are the ones unable to relate to students. It leads me to think they have forgotten there is a problem to solve. In contrast, the most illuminating professors come down from their dirigible of knowledge and join their students at square one - with a guidebook in hand, so we don't get lost.
In the end, it's not this book's fault. How to Solve It is a good book with good advice. If you feel comfortable reading math problems (even if you don't fully understand them), I recommend reading it before college, or anytime for your own personal benefit.