ESU5 Day 3

Day 3 started with a talk by Fritz Schweiger, entitled "The implicit grammar of mathematical symbolism". In a way, the talk was technical, so it will be difficult to discuss it here, in another way, it can be summed up quite effectively thus: It reminded me and made me more aware of the vast amount of implicit information there is in a mathematical text, which we (as educators) may at times not be good enough at discussing with our students. A simple example: 2x, 23 and 2 1/2 are read quite differently. Likewise, we have very clear ideas on what should be considered a good choice of letters: we will define a function f as f(x) = ax+b, but certainly not a function a as a(b) = fb+x. This is food for thought - what should we do to make the students see the importance and simplicity of such "rules"?

Thereafter, there was a panel discussion between Evelyne Barbin, Luis Radford, Fritz Schweiger and Frank Swetz. It is impossible to sum this up, but I'll try: Barbin talked about "perennial notions", how some notions include both epistemological depth, possibility of conceptual changes, links with other fields and historical and cultural interest. Her idea is that these notions are particularly suited for educational purposes.
Schweiger talked about "fundamental ideas", surely a concept of the same kind as "perennial notions". His definition is that fundamental ideas recur in the historical development of mathematics (a time dimension), recur in different areas of mathematics (horizontal dimension) and are anchored in everyday activities (human dimension).
Luis Radford, on the other hand, did not as much discuss the "How?" as the "Why?" of historical dimensions in mathematics teaching. His answer: no history means no understanding of reality.
Finally, Swetz said that we teach too much mathematics, and not enough ABOUT mathematics. His answer to this is: include the historical and cultural dimension and focus on problem solving. Historical problems give information about the society at the time, and are therefore a good tool.
The plenary discussion after this panel discussion was on "is mathematics universal?", "what is mathematics?" and "knowing=being?". No conclusion was reached.

In the afternoon, I took part in the workshop of Michael Fried and Bernard Alain. The title was "Reading and doing mathematics: Ancient and modern issues." The first part of the workshop was on Euclid, and then on some of Proclus' comments. To me, this served as a further reminder of whatever Euclid skills I may have had once. While useless for my own teaching in Norway, it is nonetheless important for me as a teacher educator to know a little about the work which more than any other have influenced mathematics teaching for the past 2000 year. The second part of the workshop was on the topic of Paideia (translated by Cicero as "Humanitas"). To me, it was interesting to see the way in which Euclid was considered a training of the mind - a point of view that fell out of favour some time ago. (I was also made aware of an article called "In Defence of Lecturing" by Mary Burgan. For me, who used to be a fanatical opponent of lecturing (even though I myself seem to have benefited from that kind of teaching - as well as others), it would be interesting to read this article.

After this, I decided to give my mind some rest. After writing this, I will go have a look at the town and prepare for the conference dinner which is this evening. I always get a bad feeling about missing a lot when I skip parts of the programme at such conferences, but on the other hand, it is important to have energy to actually take part in the workshops and listen attentively at the lectures - not just be there...