Let me say what I’m going for here. The – limit is a notoriously difficult definition.1 How to scaffold my class to handle this difficulty? I am banking on the following strategy: make them need the definition. Make them unsatisfied with anything less. Continue poking holes in their current understanding, continue showing them inconsistencies between what they believe and the language they have to describe it, till they have no choice but to try to build something new. Then, let them try to build it. If they build the very thing I’m going for, rejoice. If they build something equally precise and powerful, rejoice. If they cannot build either (the most likely outcome, since the “right answer” took the world mathematical community 150 years to come up with), then it will still make powerful sense to them because it satisfactorily answers a question they were already engaged in trying to answer. That’s the plan anyway.
I will leave you with the two problem sets from the last class, and the readings and presentation from this one. I am very proud of the presentation. After that, I’ll write down one new thought for where to take this.
We engaged people’s attempts to define infinite decimals from the previous class, then abruptly shifted topics:
I let them work long enough so everyone got to do the first section of problems. My goals were:
1) Make participants recognize that they believe the speed of a moving object is something that exists in a particular moment of time.
2) Make them recognize that their naive definition of speed (distance / time) doesn’t actually handle this case.
3) Realize that we thus have a similar definitional problem as with repeating decimals.
We got this far. Then, with just 7 or so minutes left, I gave them another problem set:2
This problem set was designed to get somebody who has never studied calculus basically to take a simple derivative, to bring them into the conversation, and to refresh everyone else’s memory about the basic idea of derivatives. The last problem was on there just so that the calculus folks had a challenge available if they wanted it. Anyway, I had people finish the “Algebra Calisthenics” and “Speed” sections for homework.
This class, we began by engaging this homework, getting a feel for the standard calculus computation in which you identify the speed of an object in a moment as the value toward which average speeds seem to be headed as you look at smaller and smaller intervals. Then we began to press on what this really means.
I handed out a xerox of the scholium from the end of the first section of Book 1 of Newton’s Principia. (The last page of this pdf.) This is where Newton tries to explain what the hell he’s even talking about. I directed their attention to this telling sentence:
An in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities, not before they vanish, nor afterwards, but with which they vanish.
Then, I showed them the following presentation. Wanting to share this with you is the real reason for this blog post. I had a lot of fun making it.
What’sCalculusReallyDoing (as pdf)
What’sCalculusReallyDoing (as powerpoint)
Then I passed out a choice excerpt from the awesome criticism of early calculus by Bishop George Berkeley. (Specif, section XIV.)
I asked for the connection between the definitional problem we have here and the definitional problem we had 2 classes ago regarding infinite decimals. (“They both involve getting closer and closer to something but never getting there.”) Then I asked them to try to come up with definitions to address these problems.
This is such a non-sequitur but here’s my one additional thought. I’ve been thinking about how to push participants to recognize a definition as unsatisfying. Tonight, reading Judith Grabiner’s 1983 essay in the AMM about Cauchy and the origins of the – limit (here it is as a pdf), I had an idea that is totally new to me. Retrospectively I think it’s sort of obvious, but I totally never thought of it before:
To get people to recognize that a definition is mathematically inadequate, have them try to use the definition, for example to prove something! In my case, all of them think that 1/3 = 0.333… Great. So, if we have a candidate definition of the meaning of limits or convergence, can we use it to prove 1/3 = 0.333…? If not, maybe we need a better definition.
(I had this idea when I read Grabiner’s statement that thought Cauchy gave the definition of the limit purely verbally and a bit vaguely, he translated it into the more rigorous language of inequalities when he actually started using it to prove theorems.)
 This is for at least 2 distinct (though related reasons): first of all, it’s got three nested quantifiers. “For all , there exists a , such that for all satisfying …” That just makes it inherently confusing. Secondly, it does not in any way psychologically resemble the intuitive image it is intended to capture. This is the definition of the limit. When I think of limits I have these beautiful visual images of little points getting closer to something. When I try to identify a limit, I just imagine the thing that they’re getting closer to. That’s the whole story. When I try to get rigorous, I replace this beautiful and simple image with three nested quantifiers. Yuck.
 You will notice some interconnections in the sequence of problems. After a few good experiences with this last year and then hearing how much fun everyone had at PCMI, I am beginning to feel like these sequences of densely but subtly interconnected problems are really, really awesome. Constructing them is a deep art and I am a tiny apprentice. But you can get started humbly and still see payoff: it was certainly a cool moment today in class when we went over these problems and a number of folks who had done out Speed problems #1-3 “the long way” realized that they could have applied their answer to Algebra Calisthenics #2 to do these three problems in moments in their heads.