Q&A: How to Scaffold Epsilon-Delta Limit Proofs

by Justin Skycak (@justinskycak) on

Cross-posted from here.

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Question

I think it’s more common than not, for example, to preempt a discussion of epsilon-delta limit proofs with disclaimers like “it’s a concept most students struggle with” and “there’s no way to get used to it but by grinding through a lot of practice problems.”

If I don’t offer a warning about a difficult subject, the students may become stressed out, ascribing their slower progress to something about themselves rather than about the material or the circumstance.

On the other hand, I worry that this might prime them to give up or become frustrated with the material sooner than if I had said nothing.

Answer

I know this is generally accepted as normal behavior by teachers, but personally, if I ever find myself tempted to attach disclaimers like “don’t worry if you feel overwhelmed by this”, then I view it as an indication that I’m about to try to take too big a conceptual leap, and instead, I need to figure out a way to better scaffold the curriculum.

Here’s a rough outline of how I might scaffold epsilon-delta limit proofs:

  1. Before I even mention the general definition of $\lim\limits_{x \to a} f(x) = L,$ I instead tell students that "the general definition is easiest to understand if you first understand some concrete examples."
  2. I state the definition in the case of $\lim\limits_{x \to 0} 2x = 0,$
    $$\begin{align*} \forall \epsilon > 0 \,\, \exists \delta > 0: |x| < \delta \Rightarrow |2x| < \epsilon, \end{align*}$$
    and we play a game where I choose an $\epsilon$ and they win by stating a $\delta$ that makes $|x| < \delta \Rightarrow |2x| < \epsilon$ true.
  3. Once they find this game easy to win, I ask them to describe their "winning strategy" for general $\epsilon,$ i.e. create a formula for $\delta$ in terms of $\epsilon$ that will allow them to win regardless of what $\epsilon$ I choose.
  4. Then I say "you guys are too good at this, I'm tired of losing, let's make the game harder," and state the definition in a slightly more complicated case like $\lim\limits_{x \to 3} 2x = 6.$
  5. Once they find *that* game easy to win, we discuss the general definition of $\lim\limits_{x \to a} f(x) = L,$ and then we continue playing the game with progressively more complicated $f(x),$ discussing the different kinds of strategies that are necessary to win at each "higher level."
  6. And finally I say something like "like any game, you're going to need plenty of practice to become good at it," which conveys to students that they're going to need to grind practice problems, but doesn't prime them to give up or become frustrated.


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