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:

  1. Use descriptive, accurate terminology.  Go back to definitions and terminology. 
  2. Think aloud, ask yourself questions.
  3. Look for, and/or remember, similar problems/solutions.
  4. Vary the problem.
  5. Come up with a helper problem.
  6. Give yourself time and rest when you need it.
  7. Work efficiently.  
    1. That is, when you’re having trouble solving a problem, be clever about how you deal with it.  
    2.  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.
    3. “Set [yourself] a new question about the problem.”  This was the most useful advice to me.
  8. 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.

    "Josef K. was dreaming."

    Last fall, at long last, I got a copy of Kafka’s Complete Short Stories.  (That would be most everything except The Trial, The Castle, and Amerika.)  It’s a book to be savored slowly, piece by piece, while imagining it to be twice its length (~ 450 p.).  I quickly found the best way to read it is jumping back and forth between the longer stories in the front and the micro fiction in the back.

    Franz Kafka – most people love his books or despise them.  That’s pretty understandable.  He’s not the most accessible of authors.  On my part, I fell for his writing after listening to The Metamorphosis; since then, I keep coming back to his books.  Back to their chilling simplicity, back to their gloomy, frequently vulgar depiction of society.  Back to the endless plots that lead nowhere good!

    But of course, there’s more to it than that.  There is a lot of truth in Kafka’s world.  Absurdity, isolation, irony, and confusion.  The real world is not so far off; sometimes it is identical – mazes of bureaucracy and words, the sheer audacity of words.  Kafka’s vocabulary is simple, but his sentences are intricate.  His paragraphs are monstrosities, and he is making a point the whole time.  You feel the claustrophobia in those long, long paragraphs, just as you feel the futility of the protagonists’ repeated attempts at arriving at the solution.  You sink into their struggle to follow protocol and responsibilities, while vague frustrations meet them at every turn.  The reader need not like the protagonists; I rarely do.  It’s the setting that is fascinating, and it’s the conflict that motivates the stories.

    I’ll be very sorry to get to the end of Kafka’s writings – that is why I’m glad to be reading this one slowly.