I have heard the story (may be an urban legend?) of a top professor who occasionally wanted to teach freshman analysis. He believed in the method of letting students see how a mathematician's mind works, so he came to the lectures largely unprepared, and was proving theorems as they were met in the textbook. Consequently he would not always follow the straightest path to the destination. Occasionally he ran afoul full speed, and had to backtrack a bit or even start over.
Does this method help a large enough group of students to make it worth our while?
My views in what follows.
- Students do get to see how a trained mathematician approaches the task of proving a theorem.
- They learn that doing math is not a single-track exercise.
- Students see the real way of doing math as opposed to fully digested polished presentation in the book.
- This slows down the progress a bit, which may help some.
- Taking notes (if a student chooses to do so meticulously) becomes a nightmare, when the teacher is hopping from one step to another
- This slows down the progress a bit making it harder to cover all the material.
- Doesn't invite student participation (unless you are willing to invest even more time on it).
What I have tried/done
Accidentally doing this (krhm).
After a motivational example I go straight to the proof of the main result of the day's lecture. I keep adding the necessary assumptions to the statement of the theorem as we encounter them. Also I get these lightbulb moments "Ahh, this is why the colleague who wrote these lecture notes had that lemma a few pages back. Comes in handy here." Naturally I had skipped that lemma, but go through it now - distracting some, but also giving a more realistic impression of how math gets done.
Summary of the discussion
The views expressed by participants in this discussion converge to AFAICT:
- Letting students see dead-ends of trains of thought has, indeed, a lot of pedagogical merit, and could be turned into a tool of inviting student participation.
- But these wrong turns should be crafted as carefully as the polished examples. They should not be random by-products of un(der)preparation.
Good teachers get their results in the old-fashioned way of earning them.
All: Thanks for sharing your ideas and suggestions in how to best implement this in practice.