Why? Why teach the formal definition? And how do you motivate it? We began this unit by doing an empirical investigation of iterates of the function f(x) = rx(1 - x) for 0 ≤ x ≤ 1 in the context of rabbit populations: if x represent this year's population density in a particular warren, f(x) represents next year's population density in that warren. Students quickly discovered that a variety of long-term behaviors are possible (convergence to a single limit, oscillations between 2 or 4 points, and apparently "random" behavior that we couldn't quite nail down), and depend mostly on r. So then as we want to refine our ideas and start writing proofs about limits, we realized that we needed a formal definition--otherwise, as I put it in class, "we're not doing mathematics, we're doing what those people across the hall [the science teachers] do."
So that's the class's official motivation. But why slog through this? A traditional answer is this: if you're going to go on in proof-based mathematics, you need to be able to write proofs using the formal definition of limit. A less-traditional answer is that we're going to need some formal properties of limits, continuous functions, and sequences in this very class, and this is our first opportunity to start working those muscles. But the least-traditional, and most important answer is this: I'm using limits to teach mathematical thinking.
What? A fundamental mathematical activity is defining, and it's very hard. A good mathematical definition describes a phenomenon precisely, excluding everything else. In lower-level classes, I tell students that, unlike in English, a mathematical definition is like a poem: every word is crucial in exactly the place that it is. When possible, we actually analyze those poems: why do we define a trapezoid as "a quadrilateral with at least one pair of parallel sides", or an isosceles trapezoid as "A quadrilateral with at least one pair of parallel sides, such that the two angles formed by one of the parallel sides with the other two sides are congruent"? We drop conditions, try to draw things that "break" the revised definition, argue about the merits of an inclusive versus exclusive definition (what theorems about parallelograms follow from our definition of trapezoid?).
But it's rare that students get to construct their own definition, and so that's exactly what we did yesterday. We iterated through four stabs ("It's getting pretty violent in here!" quipped one student). Each time we started by taking the fuzzy new idea and rephrasing it in mathematical language. For example, when a student proposed "The terms in the sequence get closer to the limit," we rephrased as "|xn – L| decreases." But each time, one student or another would come up with an objection: "Look, we're saying that these terms are getting closer to 1733 1/3 ... but they're also getting closer to 2000, 2100, or anything bigger than that!" Then we rephrased: "What do you mean by closer?" "As small as you want." "Okay, then, so how do we say that mathematically?" "Smaller than any number." "Okay, then we're going to have to name that any number...."
Who? It helped that a few students had done the formal definition in a class the previous year, but I found an interesting way to handicap them: I told them they could only give two kinds of contributions, "genuine questions" and "counterexamples" in response to other students' proposals. That restriction didn't totally quiet them (although it did, somewhat) -- it forced them to think through what they had learned last year, and to apply the underlying ideas to the definition we were working with. At least two key ideas (and one major counterexample) were found by students who had never studied limits formally.
How? I've already said that I hamstrung students who had already studied the topic in a way that made them think mathematically without taking the work away from other students. But I made a few other crucial decisions that really made this half-hour of discussion go well:
- We started with a rich set of concrete examples--a bunch of sequences with a variety of long-term behaviors--on which we could draw as we worked on our definition. This context put the meat of the activity--asking whether our current stab included the things we wanted to include, and excluded the things we wanted to exclude--within just about every student's grasp.
- We started with sequences. In the past, I've found that limits of sequences are much easier for students to get going on than the limit of a function at a point. Sequences are simpler: the values are discrete, they only go in one direction, and the definition only involves one set of absolute values, not two. Moreover, there are lots of relatively interesting (nonconstant) sequences whose N's can be explicitly calculated from their ε's. Conceptually, too, there's a natural quality to "we're wondering what happens as time goes on" that "we're wondering what happens when x gets close to, but not exactly equal to, a" seems to lack.
Well? Did it work? I'm not sure yet. The kids were engaged and spent half an hour discussing definitions before we settled on what we called the "working draft" we'd use until further notice. Most of the students contributed to the discussion at least once, so that's something. And when we went through a proof, together, the kids seemed to follow. But I'll know more tomorrow, after kids try one on their own. I can tell you this, though: it was way more engaging for everyone, way more interesting, than starting class by writing the definition on the board and then slogging through a bunch of examples.