The plenary lecture of the day was Bernard Hodgson, titled "Whither the mathmatics/didactics interconnection? Evolution and challenges of a kaleidoscopic relationship as seen from an ICMI perspective". He was concerned with the divide between mathematicians and didacticians of mathematics - a divide partly due to they being connected to different paradigms, with different sets of rules for what is valid knowledge and so on.

He described how the mathematics education field is populated with mathematicians who are also teaching mathematics, as well as didacticians who are primarily doing their research in mathematics education. Of course, there is also the complication that even many mathematicians have a limited view of mathematics - we should include historians of mathematics in the discussion, for instance, but Hodgaon did not mention this.

Hodgson went on to talk about three mathematicians with thoughts on teaching: Archimedes (with his "Method" - with a clear idea of the difference between a proof and just a reasonable indication), Euler (with all his textbooks) and Pólya (ten commandments for teachers).

Hyman Bass has talked about the Klein era and the Freudenthal era of the ICMI. In the Klein era, ICMI was populated by mathematicians, while in Freudenthal's era, didactics became a subject in its own right. The Freudenthal era also saw the launching of Educational Studies in Mathematics. Also, plenary lectures at ICME has turned from mathematics (for instance fractals) to didactics.

Hodgson ended by listing some challenges for mathematicians, here only summarized in short: acknowledge education as a responsibility, career issues, presence at ICME, level of rigor. Challenges to didacticians: keep research accessible to outsiders, collaborate in mathematically-oriented fora, acknowledge importance of creative mathematical work by teachers, must know mathematics.

The divide Hodgson talked about is of course present also in the Norwegian context, to a certain degree. It is evident whenever a new course description is to be agreed upon, a conference is to be planned, and so on.

For the TSG20, Peter Ransom was presenting Snezana Lawrence's talk on a project he has been involved in himself. They are trying to make the subject engaging by providing historical background, beginning with "historical appreciative session" on a topic. They do work on historical sources and explore which could give access to 'rich' tasks in the mathematics classroom. They introduce modern technologies in relation to the historical mathematics. For instance, he gave examples from work with quadratic curves, where history and different technologies came together.

Then I had my talk, discussing teacher students' attitudes towards history of mathematics based on a questionnaire I used in connection with a teaching sequence I developed. The full paper is available in my Academia.edu account. (I will insert a link here when I get the time, of course.)

In a way it was nice that the room was too small, as it gave me the rare experience of lecturing with eager listeners sitting on the floor and standing in the hallway, listening through the open door.

Then there was Tinne Kjeldsen's "Genuine history and the learning of mathematics: The use of historical sources as a means for detecting students’ meta-discursive rules in mathematics". HM is part of the curriculum in Denmark. Tinne took Anna Sfard as a startong point, and argued that HM can have a profound role in becoming a participant in a mathematical discourse; to learn the meta-discursive rules. "Commognitive conflict" can give change in meta-rules. Historical texts can play the role of discursants.

She showed an example from a teaching module on the concept of a function, and in this talk focused on the norm that a variable should take alle values, not be restricted to an interval, looking at Euler (1748). The students first worked in groups on four worksheets, then "expert groups" (with representatives from all four of the earlier groups), which wrote a paper. Analysis of the papers and the discussions showed that there was indeed some discussion on meta-level rules. (but it is impossible to quote the transcripts here...)

She also mentioned in passing an interesting confusion by the students between Dirichlet's use of a and b and the modern textbooks' use of y= ax+ b. The students apparently saw a and b as connected to linear functions only, not thinking of a as a constant but as the slope of a linear function.

Next regular lecture was Terezinha Nunes' "Number, quantities and relations: Understanding mathematical reasoning in primary school". She claims that cultural tools are essential for thinking, and that we use quantities, numbers, operations and relations when we use mathematics. She defines operations as transformations applied to quantities or relations. Numbers have two types of meaning: formal and extrinsic. These have to be coordinated. There are also two kinds of relations: necessary relations (i.e. 5 is 2 more than 3) and contextual relations (i.e. "Per has three apples more than Nils"). She showed examples of pupil errors that were not really mathematical, but just contextual - not having understood the context fully.

Nunes' main point seemed to be that while we have spent lots of effort working on representations of quantities and operations, we have paid little attention to the representation of contextual relations to support reasoning. Problems involving relations are always more difficult than problems with only numbers and operations.

Children must learn how to use iconic models - the models are not obvious. Children's drawings will often be nice but not actually illustrate the relations, and will therefore not help in solving these problems. Numes and colleagues are now doing a study about the use of representations for relations in primary school, with three small groups (intervention group, control group and "unseen" group. Preliminary results seem to show that the models help, but pupils obviously tried easier ways of doing it when they could, which was unhelpful for learning to use the models...

While pupils in Singapore may work so much on the models that they become very proficient, pupils n England had a hard time learning to use them, which reminds us that this is not a routine to learn, but rather a tool one can learn to master. (A teacher from Singapore in the audience confirmed that also in Singapore, learning to use the representations was very difficult.

The rest of the day was spent on Discussion Group 5, of which I was the originator. I cannot summarize that here, but I'm sure we will provide a summary of the discussions among the 20+ participants in due time.

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