2022 update: I’ve added a couple of sentences about the context of this event to emphasize how positive this experience was.
In my second year of undergrad I had a formative experience with Delta-Epsilon proofs that stuck with me for a long time. Last week I was able to provide a similar experience for some first year calculus students.
At the beginning of class the students had handed in their assignments for the week and the instructor (Ross Stokke) began lecturing; it started with a proof of a limit using the Delta-Epsilon definition. Ross gives some explanation and motivation, and then the TA unexpectedly walks in, ready to collect the assignments for grading.
On the board, Ross begins the proof as always: Let . Of course the next step is to provide a . (Normally you find this by doing a couple minutes of calculation on the back of your page. Students know how to do this, but it takes some time.)
Ross asks the class if they know what should be. We all shrug.
Ross then turns to the distracted TA who is collecting the assignments and asks him what the should be. He looks up at the limit and says “” and goes back to collecting assignments. Ross confirmed this was a good choice and continued on with the proof.
We were all flabbergasted. How did he know?
Of course, the instructor knew (or had a strong suspicion) that the TA would know the answer; the instructor was setting up the TA for success.
Now I know he isn’t actually a wizard…
I was shocked, but I knew that the TA wasn’t some kind of super-genius. Rather than be dismayed by this impressive feat I was motivated to learn this skill for myself. Suddenly, mastering these types of problems wasn’t something that only super-smart brainiacs could do.
This was an important moment in my undergrad. It jarred me into the mindset that I too could be the master of a skill.
Of course, in the remainder of the class the instructor taught us exactly how to choose these deltas.
Also, it’s worth emphasizing that this wasn’t an intimidating or scary move by the instructor. It came across as magical; or rather, it dispelled a myth I had about knowing math. It appeared to be a positive moment for everyone involved.
History repeats itself
Flashforward to this school year and one of the calculus instructors (Alfonso) is holding a massive office hour in the Huron lounge for about 25 students (I’m working on a research problem with my colleague Ivan, who is a calculus instructor this year). After about 90 minutes of Alfonso working through problems on the board, the students want to see a “piecewise limit”. So he invents this one:
- if .
He starts writing the proof (“Let “) and I interrupt him.
I ask if any of his students know what the should be. He asks and they shrug their shoulders.
So I yell over to Ivan (who has been helping a student, and hasn’t been listening at all) and ask him what the delta should be. He snaps out of his discussion, looks up, pauses and says ““.
Alfonso confirms that this will work, and goes on to write the details.
Is this a good teaching strategy?
On the surface this “magic” trick seems like one that should be used judiciously and infrequently. I want math to seem like something we can all do (given enough time, effort and struggle), and this trick seems to go against that principle. This trick could possibly serve to make students feel more unable to do math by pointing out how much better the instructors are at calculus than them.
On the other hand, I like that as a student this trick feels like something you can’t do at the moment but could possibly do in the future. I feel like this is the correct use of this kind of trick; show them something cool, then give them the tools to get to that level (which was done in my first story).
What do you think?