Reading in Calculus Class

One classroom activity I struggle to make time for is reading.  As an activity that can be done on one’s own, I generally feel that reading is not an especially productive use of class time.  That being said, I do try to make space for it on occasion:  it’s a nice change-up from routine, good material can make for a good discussion, and students of all ages seem to enjoy being read to, by teachers or by peers.

Finding appropriate reading material for a math class can be difficult, but Steven Strogatz’s excellent book, “The Calculus of Friendship,” was a great fit for my calculus class.  The book is part memoir, part homage, and part introduction to advanced calculus.  As such, it offers a readable balance of engaging narrative and challenging mathematics.

As a class, we enjoyed reading together about Strogatz’s mathematical journeys as told through personal narrative, letters to and from his former teacher, and the presentation of some particularly interesting math problems.  We also enjoyed working through some of the more advanced mathematical material presented in the book, like Fourier Series and differential equations.

It was time well-spent with my senior class.  While giving them a little taste of what might lie ahead, it also prompted some reflection on where they had been, and it all happened in the framework of some great mathematics.

And who knows:  now that the idea is planted, maybe I’ll become involved in a fruitful mathematical correspondence with one of my students someday!

Math and Art: An Impossible Construction

A favorite pastime of mine is offering impossible problems to students as extra credit, like asking them to find the smallest perfect square that has a remainder of 3 when divided by 4.  I don’t tell them the problems are impossible, of course, as that would ruin the fun.  Usually it engages and confuses them, and it makes them suspicious of me.  That’s a win-win-win in my book.

So while discussing some three-dimensional geometry, I offered extra credit to anyone who could build a model of a Klein bottle.  The Klein bottle is a hard-to-imagine surface that has neither an inside nor an outside; it’s like a bag that is sealed up, but somehow the bag is inverted in on itself.  If you are familiar with the Mobius strip, the Klein bottle is basically a Mobius strip, one dimension up.

One reason that the Klein bottle is hard to visualize is that it can’t exist in three dimensions.  It needs a fourth dimension in order to twist around on itself, kind of like the way the Mobius strip (which itself is two-dimensional) needs that third dimension to twist through before you tape it back together.  So, I was pretty impressed with the student who made this:


Not bad at all, for someone who is dimensionally challenged.  Here’s a nice representation for comparison, although it’s still a cheat:  the Klein bottle really doesn’t intersect itself.

A nice example of student work!

Follow

Get every new post delivered to your Inbox

Join other followers: