ICME Day 3

Etienne Ghys held the most mathematical plenary at this ICME, with the subject "The Butterfly Effect". I will not try to summarize his popularization of chaos theory, but his underlying point was that mathematicians have a duty to popularize mathematics - and that the popularization of chaos theory has so far failed, in that it has been oversimplified in the popular culture. In particular, Ghys claims that chaos theory has often been painted as a purely negative "theory", saying that the future is impossible to predict, instead of as a positive theory making a contribution to our understanding. He showed how chaos theory often gives quite accurate prediction of the long-term behaviour of a phenomena, even though the behaviour at a particular moment can not be predicted.

It is interesting to notice that Ghys' talk was actually on the history of mathematics - the history of chaos theory, including Maxwell, Hadamard, Poincare, Lorenz, Smale. It should also be mentioned that Ghy's talk was the most beautiful and visually impressive so far at the conference, with great animations included.

After the talk, I wondered if teachers' behaviours are chaotic or, as Schoenfeld seemed to suggest, deterministic. I would guess that it is chaotic - that a slight change in the mood of the teacher or of the answer of a student can lead to a very different outcome in a particular case, even if the set of possible outcomes can be predicted.

Topic study group 20 continued with three talks. First, Jeffrey J. Wanko talked on "Understanding historical culture through mathematical representation". He talked on a course he has held - with students in general, not with pre-service teachers - entitled "Outside Euclid's Window", using book "Mathematics in Historical Context". The course is taught in Luxembourg, with study tour to important mathematical cities in the area. Examples of stuff covered was Gauss' calculation of the sum 1 + 2 + ... + 100, using diagrams to compute polynomials (in the US they learn about the FOIL method (first, outer, inner, last), adding and subtracting integers, understanding infinity (using Hilbert's ideas), building Platonic and Archimedean solids. The outcomes of the course were engaged students, who discovered mathematics and developed an understanding that it has evolved.

Andre Cauty talked about "Lab work of epistemology and history of sciences: How to transform an Aztec Xihuitl (a 18 periods year) into a calendar?" The Maya and Aztek had 260 days named by a complex expression of a number from 1 to 13 (or 2 to 14) and a sign of day (20 different). He went through many details on how these were written and several other cycles in the area, and historical attempts to harmonize these into a calendar. Sadly, the "lab work" part of the title was not touched upon, thus leaving the connection to the teaching of mathematics to the audience.

Maria del Carmen Morilla's title was "Visualization of the Archimedes mechanical demonstration to find the volume of the sphere using 3D dynamic geometry". She referred to Hanna, and the distinction between mathematical proofs than prove and mathematical proofs that explain. Using dynamic geometry helped explore Archimedes' demonstration of the relation between the volume of the sphere and the volume of the cone and the cylinder. The talk illustrated how dynamic geometry software can be a useful additional tool for explaining complicated geometrical arguments, to supplement the careful step-by-step exposition in dialogue with students.

After lunch, "Adjacent schools with infinite distance - narratives from north Korean mathematics classrooms" was Jung Hang Lee's title. I knew fairly little about North Korean mathematics teaching in advance (having read just one article about it before, which happened to be written by Jung Hang Lee and his supervisor), so this was a reasonable choice (even though there were many other tempting offers as well - for instance, I missed Luis Radford's lecture for this.

The research is based on interviews with refugees from the North Korean oppressive regime, 5 teachers and 10 students, three interviews with each. The teachers had teaching experience ranging from 6 to 25 years, all had taught in secondary school.

The North Korean school system wants to homogenize the students. Schools didn't even have school identities or school songs - on their uniforms they were only allowed to have a picture of the leader. But until the 1980s, many saw the system as a good idea, with free education and free health service. Then, however, conditions worsened.

The vice principal of the school are in charge of ideology in the school, and he manages the teachers. He is obviously member of the party. Positions were partly handed out based on family background check.

They do have a system of teacher development. Every week, one of the teachers present a new topic, and then discuss pedagogy and improvements to the topic. North Korean teachers don't have school holidays, but has to attend professional development sessions.

There is only one set of textbooks in North Korea, with a note on how much time should be spent on each subtopic. Throughout secondary school, mathematics is the subject with most hours, 6-7 hours s week in theory. There was a rigid class schedule, where each mathematics lesson should include five different sections, including Reinforcing the Policy of the Party, which they were meant to spend 20% of their time on.

After the "March of Suffering", in which 10 per cent of the population died, they mostly gave up on providing education for all, and concentrated on the most gifted students. Afternoon classes were often cancelled because the teachers had to go elsewhere to try to find food. Some teachers also brought goods to school to try to sell it to their students, even though this was an immense loss of face.

Probability theory has only recently (in 2002) been introduced in North Korea's curriculum. After all, many of the uses of probability theory are not relevant to North Korea (insurance, gambling, stock markets...) It replaced "computer and programming".

Wednesday's final plenary activity was a panel discussion on "Teacher education and development study: Learning to teach mathematics (TEDS-M)", with participants Konrad Krainer (Austria, Chair), Feng-Jui Hsieh (Taiwan), Ray Peck (Australia), Maria Teresa Tatto (USA). "Panel discussions" in such conferences tend to degenerate into a series of individual lectures which are not relating to each other, so it's always interesting to see what happens. In this case, the panellists had divided the topic neatly between them, presenting a lecture divided in four parts.

Konrad Krainer started by painting a picture by mentioning Hattie, Adler and Shulman. Then he introduced TEDS-M, whoch includes 23.000 teacher students from 17 countries. Norway is among the underachieving countries if you do a regression in relation to the Human Development Index. 

Ray Peck talked on "teaching and teacher knowledge". three MPCK subdomains: curricular, planning, enacting. He also showed some items and explained the concept of "Anchor Points" (which I can certainly not repeat here).

Feng-Jui Hsieh followed with the theme "teacher education and education system". She stressed that teaching is attractive in Taiwan, because of income, working hours, job security and status. Thus, the process to become a teacher is highly competitive. Taiwan surprisingly ranked as number one - surprisingly, because Taiwanese teachers are generalists at the primary level. However, based on items where they do badly, Taiwan have already taken steps to improve the teacher education. Taiwan also has a sharp decline between secondary and tertiary education.

At this point, Liv Sissel Grønmo got the chance to complain that too little is happening in Norwegian teacher education in response to TEDS (a fairly unreasonable complaint as the TEDS results have just been published). She also complained about the increased place of general pedagogy in Norwegian teacher education. In the end, she said that Norwegian teacher education at times seems like it is trying to educate teachers for mathematics without them having to know much mathematics. I am certainly more optimistic than Liv Sissel, and thinks that the new 5-10 education (which I am partly responsible for implementing at my institution) is a major improvement, because students will learn mathematics and didactics more directly related to the grades they will teach - and of course many will also have double the amount of mathematics compared to before. In 1-7 education,  I think a more tailored curriculum will also help. It would surprise me greatly if this reform would not contribute to an increase in the TEDS score on the primary and secondary level if TEDS is repeated some years from now. How we will improve the results on tertiary level, I am more unsure.

Maria Teresa Tatto talked on new research in teacher education based on TEDS-M. She gave examples of anchor points, and then went on to present the results of the study.

Like TIMSS and PISA, the main contribution by TEDS is probably not the rankings, but rather results on individual items that can help us get more knowledge about our own outcomes, and inform discussions on what we would like the outcomes to be and how to get there. It helps that the databases are, apparently, available on the TEDS website.

Of course, I am interested to look for traces of for instance history of mathematics in the TEDS items. It seems, though, that the study has too limited a concept of mathematics to include the history of mathematics. Thus, there are important bits of MCK and MPCK missing - unless it is found somewhere in the material that I have not found so far.

Finally, there was a meeting of the HPM Group. Here, the activities of the HPM Group were presented, including the HPM Newsletter, of which I've been an editor for the previous eight years. This meeting will be followed up with another session on Friday.