Rolfe Schmidt

John Baez started an interesting discussion at the n-category cafe about Why Mathematics is Boring.   I don’t have much to say about Math journal articles — I read very few and when I do read them they are in areas where I’ve picked up the culture.  But it did remind me of some thoughts I had about how Math is taught.  I think John is right when he says there are ’stories’ missing when we communicate Math.

One of the most important stories rarely told is how we ended up with our canon of Real Analysis, Complex Analysis, and Algebra.  As an undergraduate or early graduate student in Math, you spend a lot of time mastering these subjects.  There are superb books to guide you on the way.  You are mastering powerful tools.  But why?  Why did anyone develop these tools in the first place?

The problem is that nobody talks about the problems.  The Prime Number Theorem.  Fermat’s Last Theorem.  Navier-Stokes.  The list goes on.  Sure people talk about them some, but these problems are hard, not for you, not now.  Young people shouldn’t waste their time on such things when there is so much to learn.

What isn’t said is that the tools we all have to learn were developed to solve real problems that students can understand.   Sometimes they worked, sometimes they didn’t.  How much of modern Algebra grew out of efforts to prove Fermat’s Last Theorem?  How much of it ‘worked’?  How much of the machinery of Geometry came from efforts to understand Physics?  I know I’m oversimplifying things, but I’m ranting here, not writing a book.

I’m not saying that we should teach everything from a historical perspective.  That would be far too inefficient. But some of this inefficiency has value.  It lets a student feel like a part of Math research.  They realize that other people had ideas pretty similar to theirs when they first saw a problem.  They know that even Gauss struggled and worked hard.  They understand why they are learning the tools they must learn.

Why not ask students to make their best estimate of how many prime numbers there are less than N?  I think you would get some surprisingly good answers.  Once they’ve thought about that for a little while they will be much more interested in how other people approach the problem.  Why not try to make sense of Maxwell’s Equations and produce the machinery of differential forms on the way?  That will probably be less boring than “an n-form is an n-linear antisymmetric function on…”.

So throw in some history.  Ask students the hard problems.  Make them take Number Theory and Geometry alongside Algebra and Analysis.  Make them learn some Physics. Then they will realize early on that they are Mathematicians.  And they will not find that boring.