Comments on the course of Ed Dubinsky in Joensuu on July 30 - Aug 3, 1998

1. First lecture and related

I found the first lecture concerning the motivation and the pedagogical theory very interesting. The main reason for the interest most apparently lied on the fact that I have lectured several years at the department of mathematics in Joensuu my using more or less traditional methods with varying amount of frustrating experiences.
On the other hand the underlying ideas of APOS-theory etc. were somewhat familiar to me because of some experiments related to our basic algebra course during the autumn season 1997. The lecture however gave me more practical overview about the whole business, which I had earlier regarded as somewhat theoretical and unpractical. After all e.g. the ACE-cycle corresponds closely to my personal ideal about teaching mathematics although I haven't regarded the use of computers that important. To be honest, I haven't regarded the use of computers important at all. However, after some
experience of my own on ISETL my attitude to computers is more positive. I also felt that the use of computers can be useful just for practical reasons; the group has something concrete (typing code) to start with and the atmosphere in the group (as well as in the whole class) gets easily relaxed and active. However, I'm not sure if this activity necessarily implies some deeper thinking; here the choice of activities plays an important role.
The idea of cooperative learning has fascinated me already for some time and I actually believe that most people learn mathematics better in groups if the learning/teaching practices can be arranged properly. However, it's not at all clear to me how cooperative learning should be arranged in practice
if the class is big and the use of computers is either not possible or there are no suitable programs to use. I would be interested to hear about some cooperative learning experiments of this sort in the future.

2. About ISETL and MAPLE

To get started with ISETL it was a good idea to begin with some elementary calculus. However, I didn't get the feeling that ISETL would be necessarily very good for learning analysis. The feeling I had was that the issue of convergence can be easily underestimated. Of course, it's impossible to give an objective judgement after only some exercises. The experience on MAPLE was a sort of disappointment although I can understand that the more interesting exercises would have required much more time. Hopefully we will obtain a sort of short course on the use of MAPLE in near future. The second day on laboratory with some abstract algebra was very interesting to me due to the fact that I have been teaching basic abstract algebra for several years. According to my experience ISETL seems to fit very well for learning elementary abstract mathematics. It was a special pleasure to see how easily one could deal with permutation groups and quotient groups since these two topics are very clumsy to consider by using pencil and paper only. The general feeling about the ISETL-syntax was that it takes some time to learn the syntax well enough and therefore ISETL can not reasonably be used just for a short period of time (if the students are not already familiar with ISETL). On the other, it's just fair to admit that the feelings about the mystic behaviour of ISETL or difficulties with syntax might be related to the fact that my experience with computers is very limited. Most likely the students of today are more familiar with computers and do not find these difficulties overwhelming.
After all, I think it would be a good idea to make use of ISETL for teaching elementary (linear) algebra at our faculty.

3. An idea

I would like to take this opportunity to express my idea (too wild, I'm afraid) concerning our courses in algebra and linear algebra. The idea is based on the feeling that one could learn elementary abstract algebra much quicker by using ISETL than by traditional methods. This feeling can be justified by a following simple observation: one can deal with the main examples just like that once using paper and pencil only one quite elementary example easily takes something like one hour. As a consequence, students could learn during one 4-credit course based on ISETL almost the same amount basic algebra and linear algebra they now master after two separate 4-credit courses usually taken during first two years. Suppose that this combined basic algebra and linear algebra would be replaced to the spring term of the first year. Then students would learn both ISETL and cooperative learning from the very beginning. Also there are 4 credits left to obtain and this could be done by taking a somewhat deeper course either in linear algebra or in algebra later (say during third or fourth year). This program would in total imply better understanding of (linear) algebra and also the familiarity to ISETL and cooperative learning which could be utilized in the process of studies in mathematics.