Proof-Generated Definitions!

Okay, for a long time, I didn’t know what to blog about. Now, I have decided to write about my teaching ideas that take ages to turn into a piece of research. That is why I have started with this strange title for my first blog. The title comes from the exact same phrase coined by Lakatos in his book Proofs and Refutations.  Here is how I used the idea recently.

Last week, I was going to teach the definition of a convergent sequence. Previously, we had played a lot with sequences geometrically: how they are represented on a number line and how they are graphed as a function on the coordinate plane. We hadn’t done any calculations; just we drew a lot of figures to see how different sequences (i.e., convergent, divergent, bounded, increasing, decreasing) behave graphically.  The question I (as the lecturer) had to answer was which “theorem” could justify and motivate the definition of a convergent sequence. Indeed, at that stage of students’ knowledge, there was perhaps only one choice; but, surprisingly that choice seemed to be a perfect fit:  if a sequence converges, then its limit is unique. There was only one small problem, students have no reason to think otherwise: how on earth a convergent sequence could have two limits? Thus, I needed to raise the question and also play devil’s advocate. Here is how the plan goes in the class.

I: How do we know that this sequence has only one limit (I drew the following figure)?

Students: What? Obviously, it has only one; see, it is getting closer and closer to the red line.

I: But, what if there is another line very close to the red line that the sequence is getting close to it too (I didn’t draw that other line on the graph).

Students: It (the sequence) is going to pass that line; see (one of the students drew the following (blue) line):

I: What if the line I thought of is closer to the red line.

Students: After a few other terms, it is going to pass that line as well.

Here, suddenly one of the students said: “What if the sequence also passes the red line?” As a result, I changed my role from someone who was “against” uniqueness to someone who was defending it.

I: But, it can’t go that far from the red line, can it?

Then, I cheated and I told the rest of the story! After all, it was the first time that I tried to generate the definition of convergnce of a sequence via a theorem in the class. I more motivated the definition and less generated it, though, we were very close to hit the target and generated a number of key ideas that were used in the definition. Next time that I teach sequences, I’ll try to go all the way and report it in another post.

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