Joys of Pedagogy II
November 8th, 2007 by WaltThis comment by klein4g helped me clarify for myself exactly what my objection to the examples then definition style of teaching. It’s that the author or speaker is pretending that we’re collectively coming up with the common definition as an act of creativity, when in reality there’s a right answer and the author knows it. It’s the pretense that annoys me.
November 8th, 2007 at 11:49 pm
Can’t both be true? We’re collectively and creatively coming up with the right answer that the author already knows?
November 9th, 2007 at 7:43 am
You do realize that all the definitions were ultimately arrived at by acts of creativity and collective agreement. The answers are only “right” in the sense that they are the generally accepted definitions that the mathematical community has agreed upon. The good author will know this fact, and proceed accordingly.
November 10th, 2007 at 3:24 am
Yes, but the author didn’t always already know the right answer.
It’s true that a batch of examples condescendingly bolted on to the front of a nice dry set of definitions is - well - condescending (and unconvincing too, since the examples will have been carefully chosen to lead up to the definitions). But if the author (a) is a human being and (b) originally thought his way into the subject rather than having it injected into him ready-made, an indication of why the path we’re following is an interesting path to follow can be a great encouragement.
Since few authors actually satisfy both (a) and (b), a reasonable alternative is to start with the definitions and then immediately give a few examples of why these particular definitions are a Good Thing. That way you add a human touch without the whole thing seeming a fake.
November 10th, 2007 at 11:47 am
I think what I dislike about giving examples before a “broad intuitive definition” (rather than the precise and maybe dry definition) is that when you aren’t sure what point you’re meant to be taking away from an example it’s difficult to do more than follow what the expositor is saying, whereas if you’re told what property the examples show, you can think about whether this is true and an important feature for an example as it’s being explained. That’s why I’m unconvinced by the “it lets you come up with the definition principle yourself” argument: in books and lectures you’re necessarily having stuff put so obviously in your path you aren’t really figuring it out for yourself, whereas you could be sharpening your critical faculties more meaningfully. (Maybe computer instructional programs might have the flexibility to help here.)