Day 4 of the scientific activities of ICME (which was the 5th day of the conference, as the fourth day was excursion day), started (for me) with a talk by Caroly Kieran; "Conceptualizing the learning of algebraic technique: Role of tasks and technology". She had convincing examples of how CAS (computer algebra systems) can be used to improve both the students' technical aptitude and conceptual understanding. This is not self-evident, indeed, I have myself been suspicious about CAS, thinking that they will only help students avoid doing the computations themselves. But of course, just as with calculators, CAS can also be used in an investigative manner - temporarily removing the students' need to do the algebraic manipulations themselves make them better able to make conjectures and check their conjectures on new examples. One of her examples was to let the students do (x-1)(x+1)= and (x-1)(x*x+x+1)= and then conjecture what (x-1)(x*x*x+x*x+x+1) and so on is (sorry for the terrible ASCII notation here). The use of CAS made the students confident that they had the right answer, and therefore the confidence needed to make hypotheses. I will certainly consider using this at some point in the future when teaching algebra to my students. (Kieran stressed, however, the importance of task design - she did not hide the fact that CAS can also be used terribly...)

The next part of my program was the TSG23. Uffe Thomas Jankvist held a talk titled "On empirical research in the field of using history in maths education". He argues that the HPM group has had too little "empirical research". Moreover, he has had a very interesting example of such research, which seems very good. His paper had a combination of being practical and theoretical at the same time, which was very good. My only "protest" was that he used the phrase "armchair research" as the opposite of "empirical research". I think this gives the wrong impression. In the history of HPM, it is true that there has not been much empirical research, but the papers have been divided into (at least) TWO other cathegories - the papers that have indeed been empirical (but not research) and the once that may have been research, but not empirical. Thave been lots of papers discussing individual experiences from the classrooms, as well as lots of papers discussing theoretical issues without the important empirical components. A discussion of the virtues of empirical research should at least take both of these other types of work into account.

There was also a paper by Lenni Haapasalo from Finland.

I had actually planned to skip the next plenary session, but ended up going anyway. I'm glad I did. The two professors Fujii and Even (Fujii from Japan and Even from Israel) had a talk each on the topic "Knowledge for teaching mathematics". Besides being genuinely funny, Fujii's talk was also a very interesting glimpse into the world of Japanese Lesson Study. He gave many interesting examples of mathematical discussions this led to. The main realization for me was that even though we do ask our teacher students (in Norway) to write detailed plan of every lesson, we never ask them to write in these plans what they expect their pupil's (mathematical) reaction to be. They write a lot on the pupils' actions (what they are supposed to DO), but not on which strategies they will probably use or the errors they will probably do. This is an important weakness. In Japan, the anticipation of such difficulties is an important part of lesson study, including (of course) the planning of how to approach (or even make use of) them, should they occur.

Prof. Even had a similar point of view, although from another view point. Her work with teachers has (among other things) included teachers reading research papers and

*replicating* the work in the papers (for instance giving their pupils the same tasks as discussed in the research papers). The teachers are often astounded that the misconceptions they read about in the research papers, are often present in their own classrooms as well. This made the teachers more capable of understanding students' thinking, but Even also stressed the importance on working on how to translate that understanding into teacher practice.

The last post of the programme was the ASG for the HPM (the 2nd part). First, Ubiratan d'Ambrosio held a very interesting talk on Julio Gonzalez Cabillon. (I'm sorry I can't get the name right on this keyboard.) Cabillon was the moderator - for years - of the Historia Mathematica email list, which was an amazing resource for researchers worldwide. As a moderator, Cabillon both provided a lot of the answers, but also intervened in a diplomatic manner when discussions got heated. Ubi has researched who Cabillon is, and has at least found him and talked to him. Today, Cabillon is, luckily, still very much alive, but has just decided to persue other interests than this list, which took so much effort for such a long time. It is a pity, but understandable, that noone has managed to create a list to replace the HM list.

The rest of the ASG was spent on Costas Tzanakis giving a report on the work of the HPM for the previous four years and on discussions on the future. These discussions will continue in Mexico City next week.

Thus ended the evening at ICME - and I went out with some colleagues for dinner. Another nice day at ICME11.

Labels: conference, ICME, mathematics