posted: November 6, 2008 - 3:02am by peter.krautzberger
tags: Mathematics, university level teaching

My last blog entry ended in too much of a rant; sorry about that. The reason was primarily that it had gotten too long and I had to stop. So this post is about the first few ideas I tried to implement and the problems that arose.

The primary problem that I have encountered among the students is lethargy.

So I asked myself what one wants to achieve in teaching a (set theory) course. The following are some basic observations.

1) Identify what kind of mathematical problems this field deals with

2) Identify the tools developed to approach these problems

3) Teach the students to analyze the new kinds of problems, identify their core and show how the tools of the lecture can solve them

4) Get the students to communicate their own approach, ideas and intuitions

The first two issues can be dealt with by good preparation on the part of the lecturer and TA.

The third and fourth issues however cannot be addressed without an active participation of the students. Even though I am responsible for creating an atmosphere where the students feel able to do so.

Mathematics is all about communication. And that communication is much like the communication between artists of any kind -- exchanging thoughts, ideas, worries and convictions so as to get our (mathematical) creativity to the point where it can finally reach out to inexplicably touch the core of the problem and solve it -- we simply do not have the luxury of the natural sciences to be inspired by external input in the form of, say, experimental data.

All we have is our thoughts and maybe some good notation to put them on a blackboard -- and a blackboard is only good for a very quiet conversation... Hence lethargy endangers everything.

So what is the situation?

Students are sitting in my tutorial waiting for a standad solution and the assurance that this is enough to earn their credit points -- a perfect reminder of Polya's introduction to >>How to solve it<<. So I have tried the following ideas.

As I described in my last entry, over the course of the semester students are required to present their own solutions in front of the class (depending on the size of the class between 2-5 times each). However, this usually ends up in a dialogue between the TA/GSI and the student. Therefore I am working on ways to get them to talk and listen to each other.

Towards this end, I have started by picking two students instead of one -- but not to let them present as a team. Instead, I pick one student with a solution (the solver) and one student without a solution (the counterpart).

The solver presents the solution while explaining it specifically to the counterpart. Accordingly, the counterpart is in charge of checking that solution. He/she >>simply<< tries to understand it -- tries to not just scribble down some notes, but to actively gain an understanding of the problem, the solvers approach to it and the eventual chain of thought that lead to a solution.

The counterpart is important because he/she failed to solve the problem. Not solving a problem is the major everyday experience in mathematics. We fail all the time! Hardly any idea of a solutions works. But this failure is the most important part. Every approach that does not solve the problem identifies additional structure and difficulties, sheds light on aspects that were not yet considered or expected; it questions, alters and renews our intuitions.

This is both the only way to get to a solution as well as the most beautfiul experience -- for at the end not even the most intricate proof can tell you how wonderful complex the journey towards fully understanding that problem was.

After the first two tutorials of the semester, I have had two very different experiences with this strategy. In the first tutorial, the students embraced the concept quickly and willingly. It was astounding, really. The discussion between solver and counterpart quickly started a debate among more students, asking me and each other rather a lot of questions and above all loosing the unnecessary shyness to ask >>embarrassing<< questions -- which I believe was the best sign for an open learning environment.

However the second tutorial was nearly back to >>normal<<. The students were unwilling to participate, the exchange among them never really got off the ground, the communication between solver and counterpart was very basic and slow.

So I have decided to do two things. On the one hand, I will be patient and I will keep at it -- there's no need to panic after two classes, right? On the other hand, I will try to tweak a couple of things. Above all, I have to reduce my own interaction with the solvers. As I must admit I have been too involved, telling them to reevaluate their steps or their setup or whatnot. This certainly seems to prevent the counterparts from making their own kind of sense of the solution.

Additionally, I will have to increase the teamwork among them, maybe have two counterparts so that they have more time to react.

In any case, I will hopefully write again how things develop. Hopefully, I come up with some more ideas, too -- just in case...

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