So it is clear that Mathematics today can lay claim on six of the seven original Liberal Arts, at least if we allow ourselves some historical anachronisms. Because in antiquity it was only the Quadrivium that were mathematical (if I remember correctly the term was introduced by Boëthius in the 5:th century going back the Pythagorean Arkytas via 4:th century Martianus Capella). The Trivium was all Language – it is only through modern reinterpretation that Logic and Grammar can be construed as Mathematics.
But – and I think this a very important point – mathematics is a language in itself. Or it is more appropriate to say that mathematics has a language or that it has several languages as pointed out in an article by a Swedish mathematician, Christer Kiselman. To this I have to return to in another post. Because when I’m thinking of mathematics as a humanism, then it is the language aspects of it that I think are central.
But I wrote that just saying that Mathematics is Liberal Arts is begging the question. One reason is the fact that Mathematics always has been very successfully applied to the real world. This is true for as long as we have historical records – just think of time-keeping, astronomy, agriculture, administration et cetera in ancient cultures. The application of mathematics to the real world of physics became common during the scientific revolution (and to the real world of engineering and economics after the industrial revolution). It is impossible to imagine the breakthroughs of Galilei, Newton (and all their lesser known precursors and contemporaries) without mathematics. Not to speak of all the discoveries of electro-magnetism on which almost all of modern technology rests!
I think this spectacular success of applied mathematics has shifted the focus of almost everyone to such an extent that the fundamental humanistic nature of the subject has been almost forgotten. Or if it is not forgotten, the applications overshadows it. Anyway, over to the responses I’ve got.
Reactions to the Key Question
Two recurrent responses that I’ve got is: (1) There is more freedom (as compared to larger research universities) at Liberal Arts Colleges to plan your classes as you like. The smaller class size (10-25 students) also allows for more student-teacher interaction. There are no lecture-hall size lectures to hundreds of students. (2) There is more emphasis on depth and understanding rather than on computational techniques and skills. This is roughly what people at Carleton and Macalester said to me. But I got the impression that the road to understanding went through proofs. Hmm… there must be something else …
When I met with Paul Zorn in Northfield and posed the question to him, he said that he didn’t think it was such a big difference. In a way this was a surprising answer – but at the same time it confirmed my impressions so far – and of course it depends on how you interpret “big differences”. But he also said that you cannot nowadays teach the subject as if everyone was on the road to graduate school (PhD). Many students take a math course or two to satisfy the breadth requirement, or out of interest even though they are majoring in another subject.
Then the conversation drifted into the fact that today’s youth have a different view of learning than we had (when we were young) – we often suspended doubt and just went along with whatever was taught. That does not really work today. It is seen in how they learn to use new gadgets. They don’t read manuals (if there are any) they just play with the thing itself. It stills remains for teaching to adapt to that mentality. Or perhaps not adapt, but using it efficiently along with other teaching methods. And this leads into an aside:
I got the impression that at Carleton and Macalester, computer software was used as a matter of fact in many courses. There were no particular problems with students catching on. At my place in Sweden I don’t think it works that easily – I think we have something to learn here. I think it has to do with the determination in the use of software – it has to be used relentlessly in, if not all, but many courses. And, BTW, do away with the silly calculators!
Next day (over dinner) I asked Ted Vessey the Key Question referring to my impressions so far. His answer was crisp and clear quoting Lynn Steen: “Always let the best teacher teach the first course”. One thing that differs is actually the perception of the role of teaching. Teaching is central at Liberal Arts Colleges, research comes next. At bigger research universities it may be the other way around. This is something that I’ve heard confirmed later on.
Then Ted said that students don’t like “word problems” because that forces them to think. They have to translate the problem into mathematical formalism, do the computations, and then translate back. He then said that mathematics is the language you need in order to solve problems.
When I have more specifically asked about bringing up philosophical, historical and cultural aspects in the teaching, I have got the response that: Yes, it is indeed done now and then.
So perhaps the conclusion is that even though the typical Liberal Arts mathematics class is not explicitly built around humanistic ideas, there is nevertheless a subtle influence from the environment.