Day 3 of ICME13
started off with Günter M. Ziegler's talk ""What is mathematics"
- And why we would ask, where one should experience or learn that, and who can
teach it". According to the twitterati, it was good and entertaining. I missed
it.

In conferences such
as this, it is important to break away from the programme to have time to
reflect and to interact with other participants. (Unike some other conferences
I go to, this conference does not have many workshops or discussion arenas.)
Therefore, I skipped the plenary and an additional lecture, but was back on
track after that, for four more TSG-talks:

Skip Fennell talked
on "Preparing elementary school teachers of mathematics: a continuing
challenge". He started by giving a brief historical background. He than
pointed out that in many countries, elementary teacher education leads to
teaching one or two subjects, while in the US (and Norway until now), teachers
become generalists. Also, in many countries there are a limited numbers of
programs, sometimes hard to get into, while in the US there are more than 1700
programmes, some online. There are voices arguing for having math specialists
in elementary school. (In Norway, we are switching to a programme where you
have to have a master's degree in one subject, but still will teach two or
three other subjects.) However, in 20 states, there are math specialists, often
called "maths coaches". These positions are, however, vulnerable in
touch (economic) times. In the Q&A, he conjectured that there must be
significant challenges in having teacher education programmes which provides
everything except practice periods online. We all know the difficult
negotiations between campus and practice, and these cannot be easier in such a
context.

Marjolein Kools's
"Designing non-routine mathematical problems as a challenge for
high-performing prospective teachers". She started by giving a non-routine
mathematical problem. In Netherlands, there is a national test in the third
year of teacher education, and this project concerned creating non-routine
problems for PSTs to work on. The got PSTs to help them, but it was very
challenging for them to produce task of the right level of difficulty.
Therefore, the project turned into a project trying to support the PSTs in
creating, and this became a design research project. At the end, 4 of 8
students could design non-routine problems independently. Key to this result
was master-classes where Ronald created problems while speaking aloud, trying
them out and so on. (Although the starting point of this project was a national
test, it seems worthwhile to work on this even in countries which are lucky
enough not to have such a national test. It would be cool to do something like
this in Norway as well...)

Eda Vula's
"Preservice teachers' procedural and conceptual understanding of
fractions". She referred to Ma,
Shulman, Ball etc. as starting points for her talk. The research questions
concerned representations and the relationship with and between procedural and
conceptual understanding. 58 PSTs participated by doing a 20 item fractions
knowledge test. The test obviously unearthed problems, for instance in the
meaning of unit. PSTs had good procedural knowledge, but rarely good conceptual
knowledge in addition.

Megan Shaughnessy's
"Appraising the skills for eliciting student thinking that preservice
teachers bring to teacher education". Of course teachers need to be able
to find out what their students are thinking. This project takes this point
into teacher education - how do we find out what our students think. The idea
here is to use standardized simulations. PSTs get a student response and
prepare an interaction with this "student", then they interact with his student (who is a
teacher educator trained and using a response guideline). She showed a video to
illustrate how this worked. All PSTs did this, and the videos were analysed,
and the findings were really interesting (but I can't summarize them here). The
study shows some of the moves that PSTs need to learn, what can be built upon
and what requires unlearning. (And of course the videos themselves may be
useful to work on thIs.)

After lunch, I
attended the "thematic afternoon" on European didactic traditions.
First, there was a "panel"
with Alessandra Mariotti, Michéle Artigue, Marja Van den
Heuvel-Panhuizen and Rudolf Strãsser. They had identified five characteristics
of the the European traditions: strong connection to mathamatics and
mathematicians, strong role of theory, role of design activities, key role of
empirical research.

Mariotti talked on
the role of mathematics and mathematicians. Historically, mathematicians were
important figures in discussions on how to prepare mathematics teachers. In the
beginning of the 20th century, Klein had a central role (Erlangen Program). Also,
after WW2, mathematicians had key roles in the New Math reform(s). Later,
mathematicians also played important roles in the French IREMs, for instance,
where mathematicians, teachers and didacticians mixed. In Germany and
Netherlands, on the other side, mathematicians played a smaller role in turning
mathematics education into a research field. The case of proof as a theme of
research have been a feature, and this may be influenced by the role of
mathematicians.

Artigue talked on
the role of theory. She started with the French tradition, which she claimed
had a particular importsnce in building a theoretical foundation. These had a
systemic vision of the field, a strong epistemological sensitivity, theories
mainly conceived as tools for understanding. New development are connected to
previous theories. In the Dutch tradition, Realistic Mathematics Education is
the main contribution. This is a theory oriented towards educational design.
The theories are still refined. In Italy there is s long tradition of action
research and collaboration between mathematicians and teachers, now continued
in Research for innovation. In the Germsn tradition, the theoretical landscspe
is not so easy to synthesize. (The rest of Europe is left out in this survey.)

Van den
Heuvel-Panheuzen talked about the role of design activities. The French
tradition: essential component of research work and based on theoretical
frameworks. In the German tradition, design activities took place in context of
Stoffdidaktik, evaluation was often left out. Empirical turn from 1970s. Italy:
empirical analysis as base for didactical innovations, pragmatic approach. From
1980s innovations have turned to research. In the Dutch tradition, the design
aspect is the most significant characteristic. Theory development resulting
from design activities, later theory was used as a guide for futher theory
development.

Strässer talked on
the role of empirical research. Different kinds of empirical research have
developed because the complexity of the field. Examples: the large-scale study
COACTIV, the case study MITHALAL. Of course, there are also a distinction
between qualitative and quantitative methods. The purposes can be prescriptive
or descriptive, and they may be developing vs illustrating theory. Action
resesearch have been stronger in I and F, while fundamental research, strongest
in F, G, N.

In addition to the
content of the talk, what will surely be remembered about this evening is the
heat and humidity of the auditorium, not least when it was decided to close all
windows to close the curtain to improve the visibility of the screen slightly.
It is a testament to the participants' dedication to the disipline that so many
chose to come back to another session after the coffee break.

I know far too
little about the French tradition, so I chose to go there. I'm also struggling
to learn French, and hope that I will be able to read some of the French work
later. Brousseau, Chevallard and Vergnaud were the stars of this bit of the
program. IREMs were established from 1968, leading to a fruitful collaboration
between mathematicians and teachers. From 1975: first doctoral programs, from
1980: journals.

The first
theoretical pillar was the theory of didactical situations (Brousseau) - with
the "revolutionary" idea that the pbject of research was the
situation. Centre of observation for research created in 1972. The core
concepts had been firmly established through the 1980s. The second pillar: the
theory of didactic transposition and the ATD. (Chevellard) presented in 1980,
followed by book in 1985. We "embed an activity into a whole whose ecology
is studied". The third pillar was probably the theory of conceptual fields
(Vergnaud), which seemed to be taken for granted in this meeting.

There was a talk on
research on axial symmetry in France, to illustrate a development. There has
been a development from the study of students' conceptions, combining the
theory of conceptual fields and the theory of didactical situations to the
study of teachers' practices and of their cognitive effects. Now, French
research on symmetry is more occupied by the question of language. There was
another talk on French research on school algebra. The first studies were in
the 1980s, by Chevallard, in terms of didactic transpositions. Three questions:

- what is school algebra, and what is (or have been) considered as algebra? (Didactic transpositions, institutional constraints)
- What algebra could be taught? Under what conditions? How implement them? (Reference epistemological models (REM), didactic engineering (new ecologies))
- How do ICT modify the nature and the way of using, teaching and learning algebra? (Concept of ICT as part of the adidactic milieu, taking into account the instrumental genesis, didactic and computer transpositions)

At this point, the
thing turned into a Q&A session in which the questions were unintelligible
while the answers were moderately more intelligible. I chose this point of time
to leave the auditorium. However, there was one interesting question that I was
able to understand. It was something like this: "It seems that the
research questions the French pose about algebra are quite similar to research
questions elsewhere. In what ways do the answers differ?" I did not catch
the answers, but I made up my own: the problem/challenge with having different
theoretical perspectives and traditions are exactly that we may pose the same
sorts of questions but look at them from different points of view, and we
therefore have an inkling that the answers will be (slightly or significantly)
different. However, since the answers are themselves given within different
theoretical traditions, pinpointing exactly how they differ demands exactly the
same comparison and analysis of perspectives (or perhaps even the creation of a
"meta-theoretical perspective") that is some of the point of this
thematic afternoon and similar work. If we could easily explain what the
different traditions accomplishes, we would be at another point of the
development than we are. Perhaps.

Thus ended the third
day of the ICME13 conference. I had a very good experience in the TSG, getting
inspiration for further work. The "thematic afternoon" was an
interesting experiment, but the different backgrounds of the participants must
have made it almost impossible to plan.

After this, there
was social gathering again, followed by an (unofficial) LGBT get-together of
ICME participants. I chose to post this before these events, though, to avoid
the temptation to blog about them

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