How to Solve It

Although the jacket blurb for this blog admits the possibility of the occasional rant, I do my best to confine my thoughts here to those that highlight some specific idea, either mathematical, technological, or otherwise scientific.  My usual yardstick for determining whether a topic is worthwhile is that I should have something to contribute, be it a truly new result, or perhaps just an interesting different perspective on existing ideas.  Cane-shaking is for politics and religion.

But this week, after finishing a couple of great books, I plan to do nothing more than gush a bit about reading interesting words that are not my own.  I suppose I could hide behind the guise of a book review as a flimsy attempt at justifying not having much more to say than, “What are you doing still reading here?  You should be reading over there.”

The first book is A Stubbornly Persistent Illusion, edited by Stephen Hawking.  It is a collection of “The Essential Scientific Works” of Albert Einstein, containing not just his technical papers, but some fascinating public addresses and writings for a more general audience.  I found particularly interesting the rather intimate “self-obituary” in which he describes his progression from youth, through the past 67 years of his life, to his then-current perspective on science as a mode of thought:

“As the first way out there was religion, which is implanted into every child by way of the traditional education-machine.  Thus I came– despite the fact that I was the son of entirely irreligious (Jewish) parents– to a deep religiosity, which, however, found an abrupt ending at the age of 12.  Through the reading of popular scientific books I soon reached the conviction that much in the stories of the Bible could not be true…  Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking.”

I am always fascinated, and inspired, by these “human” views into the life of a man known to many people only by his equations.  Even the people who fundamentally change our understanding of the world– and Einstein was certainly one of them– are not cold, unfeeling machines or superhumans churning out equations.  They are simply men and women, whose very human and common desire to understand the world, and awe at the prospect of doing so, is just as inspiring to me as their very unique ability to actually realize that understanding.

The second book is the motivation for the title of this post, How to Solve It, by George Polya.  This is actually my second time reading this book, but I suppose I feel more influence from it now that I spend more time trying to teach.

This book is striking to me for two reasons.  First, it is inspiring in the same way that the previous book is: Polya knows the “Aha!” moment, knows how it is shared by both the student and the teacher.  But he suggests that those moments need not always be produced (i.e., proved) by the teacher to be consumed (i.e., verified) by the student.  It is possible for the student not just to learn solutions presented by others, but to learn how to solve it himself:

“The author remembers the time when he was a student himself, a somewhat ambitious student, eager to understand a little mathematics and physics.  He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: ‘Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution? … And how could I invent or discover such things by myself?'”

This phenomenon is what makes teaching such an exciting thing to do.  It is certainly rewarding when a student says, “I understand what you did there.”  But it is infinitely more rewarding when a student says, “I wondered what would happen if I did this, and look what I found…”

But the second striking feature of this book is the extent to which it succeeds at its goal.  Polya accomplishes something that at first glance seems oxymoronic: he systematizes heuristic.  More simply, he describes, with precision and clarifying examples, the common approaches to solving mathematical problems that may otherwise seem to have nothing in common.  In hindsight, much of the guidance seems like common sense.  But there seems to be something uncannily useful in making that common sense explicit.  As I read it, I find myself thinking that this is a book that everyone should read, not just teachers and students of mathematics.

Finally, it’s simply a fun book to read.  Polya knows how to write with some wit, such as when describing an example of how not to do it:

“The traditional mathematics professor of popular legend is absentminded.  He usually appears in public with a lost umbrella in each hand.  He prefers to face the blackboard and to turn his back on the class.  He writes a, he says b, he means c, but it should be d.”

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