I’d seen three interesting pieces about calculus in the last couple of weeks. The first two had me thinking about how you could share some of the basic ideas from calculus with kids, and the last one helped me organize my thoughts.
The three inspirations for today’s project are:
(1) Keith Devlin’s Letter to a Calculus Student as well as Letter to a Calculus Student – the Sequel which is where I saw the first article.
Devlin makes this point about how students see calculus:
Beauty is one of the last things you are likely to associate with the calculus. Power, yes. Utility, that too. Hopefully also ingenuity on the part of Netwon and Leibniz who invented the stuff. But not beauty. Most likely, you see the subject as a collection of techniques for solving problems to do with continuous change or the computation of areas and volumes. Those techniques are so different from anything you have previously encountered in mathematics, that it will take you every bit of effort and concentration simply to learn and follow the rules.
(2) Eugenia Cheng’s How to Bake Pi.
Cheng makes a similar argument to Devlin about calculus (I’d give the page reference, but I only have the audio book, the passage is near the beginning of the book and (probably) can be found by looking for “calculus” in the index.)
“Another popular moment where people reach their abstract limit is calculus, which involves a completely new and strange – and, frankly, a bit sneaky – way of manipulating and reasoning with “infinitesimally small” things.”
Both Devlin and Cheng point out that part of the struggle with learning calculus is learning to work with at least two new ideas – limits and “infinitesimally small” things. Their writing got me thinking about how you share some of those ideas with kids earlier so that they were not completely foreign ideas when students encountered them in calculus.
Then Dan Anderson’s blog post from a few days ago gave me an idea – 3D printing:
In this post Dan shows how to make some standard shapes we see in a calculus course using a new programming language specifically designed to make 3D printing easier. I do not have this new programming language, but I do have Mathematica and I spent some time yesterday learning how to replicated some of Dan’s shapes using the RegionPlot3D function in Mathematica. Here’s one example:
Having learned enough from Dan to be dangerous, I printed some Riemann sum-like approximations to a triangle and decided to have an informal talk with the kids about a few basic ideas from calculus this morning. We started by looking at a few of the shapes and hearing what the boys thought calculus was:
Next I used the four 3D printed “triangles” to introduce (informally) two basic ideas from calculus:
(i) Using something that you already know – in this case how to find the area of a rectangle – to find something that you do not already know.
The manipulatives seemed to help boys seemed to pick up on the ideas. They were even bothered right from the start about the idea of chopping up into infinitely many rectangles 🙂
Having computed the approximation of the area of our right triangle using 4 rectangles in the prior video, we extend the idea to n rectangles in this video. To understand the approximation in general, we need to understand the sum . We walk through this part slowly because the algebraic calculations are confusing to my younger son.
What I really liked here was their intuition that as n became large, the area of our approximation got closer and closer to 1/2. Right at the end, though, they wonder about what happens when n is infinity 🙂
Now, having showed that the idea of chopping up into rectangles to find the area seems to work well, I wanted to show them a situation where the idea does not seem to work well at all. The example here uses the same approximations to the triangle that we used previously, but this time we use our 3d printed shapes to try to determine the length of the hypotenuse.
It turns out that although it appears visually that the approximation is getting better and better, it is, in fact, not really getting better at all. This point was (I’m happy to report) very confusing to them.
The point of this example here is that approximations and limits can be tricky.
Finally, we move from tricky to super duper tricky to show that the ideas underlying calculus might be super subtle.
One my favorite math moments with the kids was watching Numberphile’s video showing that 1 + 2 + 3 + . . . . = -1/12. My younger son was actually screaming at the computer the first time he saw that video. BUT, we just used that series to show the area of our triangle was 1/2 🙂
As my older son concludes – “Things get weird when you go to infinity.”
Yes they do!!
So, a fun project inspired by Keith Devlin, Eugenia Cheng, and Dan Anderson. Even though it would be silly to take a deep dive into the details, some of the ideas really did grab them and made them wonder about what happens when you use infinitely small (or large) concepts in an argument. Definitely a fun little project 🙂