Unexpected moments

Students can learn things we don’t think we’re teaching.

A year or two ago one of our tutor’s students had an unexpected moment in mathematics.  She was slogging through trigonometric identities: equations in which one side, made up of some unlikely combinations of trig functions, is equal to the other side, an equally unlikely combination that looks quite different.  The student’s task is to prove it.  There is no question that it’s true, and no value of x or theta that we’re trying to find.  The point of the exercise is to get the student familiar with the various things that one can do with trig functions.  The application comes later on, when they appear in other equations and especially in calculus.  Trig identities really have no purpose in themselves.

But this student saw in the sequence of transformations that almost magically produce an exact result something, well, almost beautiful.  She had trouble expressing it, but we’re pretty sure we know what she meant.  What brought her to mind was a more recent student, who was facing complicated operations with matrices, a different sort of mathematical object.  There is a great deal of tedium in working with these by hand, and many opportunities for small mistakes to derail the process altogether.  He wasn’t optimistic about his chances of passing a test on them.  But after working through several problems he found himself getting the right answer more often.  And he saw how a strange series of unlikely operations would almost magically produce the desired result: a matrix with ones along the main diagonal and zeroes everywhere else.  He didn’t use the word “beauty,” but he might have.

It has been said often that mathematics and art are similar in some deep way, that in a sense the beauty of a master’s painting and a powerful theorem are the same.  There is much to be said for and against that idea, and modifying it; we won’t get into that now.  We think these students were approaching it, but at a lower level.  They were getting the same sort of satisfaction that a craftsman does when producing some object, not really new or of wonderful design, but simply well-made.  It’s the satisfaction of being skillful, of having techniques work out well.

We think more students experience this satisfaction than express it.  Most are focused intensely on getting the right answer, as a means to a good grade in the class.  An observation on the beauty of mathematics would be met with derision from a classmate who is struggling.  And carried to extremes an advanced attitude towards math can be damaging: one student our tutor had would get so interested in one or two of the problems in a standardized test that she’d play with them, working out structures and possibilities, and never finish the rest.  (He finally got her to focus on doing the test, saving her curiosity until later.  She’s now at Caltech.)

Should we work more on getting students to see this craftmanship-beauty?  It would certainly help students’ motivation.  But we’re not sure how to do it.