Saturday, January 16, 2016

My Favorite Course (to teach): Calculus

Why is calculus my favorite? Let me count the ways ...
  1. It tells a story.
  2. It has cool historical connections,
  3. ... and great connections to science.
  4. It's a good time to help students start to see what proof means.
  5. I keep learning more.

Calculus Tells a Story...
...if we let it. And the conventional textbooks don't. So I used two different creative commons texts (Boelkins and Hoffman), some of my own materials, and a few things from some of my favorite bloggers, and I made a coursepack to use for the first three weeks. I gave a talk about it at the Joint Mathematics Meeting a week ago. As part of my preparation for that, I made a new blog page. Click 'calculus' above, and you'll see all of my materials, including the slides from my talk, links to the creative commons texts I used, and lots more.

What stories does calculus tell? It takes one of the central concepts from algebra, that of slope, and twists it so it will work for curves. To do that, we need to consider two points that are "infinitely close together," whatever that means. So we have to delve into the weirdness of "infinitely close." Once we get good at all that, we can find out where things reach their maximum and minimum values, and use that to graph all sorts of curves. We also use that to optimize, to get the most volume with the least surface area (when building boxes), for instance. And then we play with finding areas of strange shapes, and how that's connected to slopes.

Calculus has cool historical connections, and great connections to science.
Archimedes figured out all sorts of things that are really a part of calculus (call it proto-calculus), and used the 'method of exhaustion' which is a foundation for what we now do with limits. Newton and Leibniz are credited with inventing calculus, even though lots of what we do in Calculus I had already been figured out. The main thing they discovered was what we call the Fundamental Theorem of Calculus, which says that areas and rates of change are inverse functions. It makes sense that two different people invented calculus because it was needed at the time for the science questions that were being considered: lenses and light, paths of planets, gravity, angle to shoot a cannon, volume of the Earth. And then it took 150 years to get that limit thing just right, and another 150 years (in 1960 Abraham Robinson invented non-standard analysis) to prove that Newton's original conception (of fluxions) wasn't so far off.

It's a good time to help students start to see what proof means.
Did you realize that the two 'formulas' we all know for circles are very different sorts of creatures?  The first, C=2*pi*r, is really just a restatement of a definition. pi is defined to be C(ircumference) over D(iameter), so it takes 2 or 3 algebraic steps to get to C=2*pi*r. But the other, A = pi*r2, should be proved. The simplest almost-proof comes from cutting the circle up and rearranging it.

I keep learning more.
I learned two cool things while preparing for that talk: Newton had a clearer conception of limits than we usually think,  and Archimedes' calculation of an approximation for pi was easier to follow than I would have imagined, and really simple and beautiful (in our modern notation).

And to make this post a fun one for all you MTBOS folks, here's the worksheet I designed to share with my calculus class (.doc and .pdf), leading them through Archimedes' first few steps as he worked toward the 96-gon to approximate pi. Go ahead, try it and put your answer for the 96-gon in the comments. (I couldn't find it anywhere else online!)

*(There's a better way to show word docs, right? Someone tell me. I should know that after all these years of blogging!)

Saturday, January 2, 2016

Newton and the Notion of Limit (he knew more than I thought he did)

Preparing to give a math talk has been very educational for me. I posted about ten days ago about finally figuring out how Archimedes calculated pi with his 96-gon.

Now I just found out that Newton wrote more about limits than we're usually led to believe. In 1687, Newton wrote:

"Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...."

This quote comes from Bruce Porciau's paper, Newton and the Notion of Limit, in Historia Mathematica. He gives much more evidence that Newton understood the limit concept pretty well.

I guess I can still say that it took the best minds in all the world 150 years to come up with a precise definition of limit. But Bishop Berkeley's complaint ...
"And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?"
... now seems to me more the product of a small mind and less the careful quest for precision of a mathematician. Now I lean more toward thinking Newton (and Leibniz?) got it, but it took 150 years for a mathematician to create a precise definition that would convince all the other mathematicians.

Joint Mathematics Meetings in Seattle this coming week

I leave on Wednesday for the Joint Mathematics Meetings in Seattle. I'm giving a talk there on using creative commons textbooks in calculus. Friday, 1:20pm, room 620. I'd like to meet online friends there!
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