The 7th day of the
ICME13 conference was short. First a plenary panel on "transitions in
mathematics education". Panellists were Ghislaine Gueudet, Marianna Bosch, Andrea diSessa, Oh
Nam Kwon, and Lieven Verschaffel. The panel's theme - transitions - has many
interpretations, including transitions between themes (arithmetic to algebra),
transition to formal proof, transitions between school levels, transitions
between contexts, for instance language contexts, transitions between
curricula. In this panel, they focused on transitions as conceptual change and
on transitions of people as they move between social groups. To look at this,
they had epistemological, cognitive and socio-cultural perspectives. The result
of the work is a survey: "Transitions in

Mathematics
Education".

diSessa started by
talking about "Continuity versus Discontinuity in Learning Difficult
Concepts". Bachelard talked about epistemological obstacles, while others
talks about "pieces and processes" instead, arguing for more
continuous change. Misconceptions belong to the left side of this divide. The
answer to the discussion may be found in microgenetic perspective (J. Wagner) -
some research find incremental learning across many contexts, for instance when
trying to learn the law of large numbers. diSessa thinks the continuist side
will win, which will mean we will look at resources more than obstacles or
misconceptions. (Personally, I'm not sure I believe any side is 100 percent
right. Why can't both be right in different contexts? That is, that some
development is gradual or stepwise, while other develop is mora abrupt and
discontinuous?)

Kwon talked on
"Double discontinuity between Secondary School Mathematics and University
Mathematics". The double disconuinity concerns first moving from secondary
school mathematics to university mathematics and then back (as a teacher). She
then discussed Shulman, Ball and then Heinz et al ("School Related Content
Knowledge") and Thompson (Mathematical Meaning for Teaching Secondary
Mathematics).

While listening to
her talk, I wondered if it could it argued that the Norwegian system, in which
teachers for (lower) secondary schools are educated in teacher education
programmes in which mathematics and mathematics education courses are merged,
and where the mathematics is not "university mathematics" as such,
avoids such double discontuinities? In fact, the suggestions she ended her talk
with seemed to align almost exactly with the Norwegian model. But it must be
stressed that for upper secondary schools, the problem is present also in
Norway.

Borsch talked on
"Transitions between teaching institutions". Individual trajectories
are shaped by the institutions they enter, their activities and settings. The
transitions between primary and secondary and between secondary and tertiary
education, are much studied. There are many different levels of analysis which
are present in the literature. The main differences between primary and
secondary are pedagogy (interaction, autonomy, transmissionist) and discipline
(specialist teacher, more division between subjects). Many proposals for
smoothing the transition are found in the literature (and in the report).

The differences
between secondary and tertary education are similar to differences between
primary and secondary. In addition, teachers are researchers. There is more
research on particular topics (i.e. Algebra) and there are more proposals.
"Bridging courses" are discussed, but university mathematics content
is rarely questioned. It is not clear whether all perspectives are equally
represented in the literature.

Verschaffel talked
on "Transitions between in- and out-of-school mathematics". Much
learning and use og mathematics take place outside school. While this has
previously been "romanticized", currently, more interest is in what
happens at the boundaries. Of course, part of this is research on the
didactical contract when playing the game of word problems, in which
out-of-school experiences are often unwelcome. There are efforts to facilitate
and exploit transitions, for instance RME, "funds of knowledge",
Greer. Then the panel ended with a discussion, based on question from an
internet forum, at which point I stopped making notes.

Then, there was the
closing ceremony. Secretary general of ICMI, Abraham Arcavi, had a warm and
pleasant speech closing the conference. Then there were many speeches of thanks
to different contributors. There was a presentation of the next venue for ICME
(Shanghai ). And finally, a wonderful musical number.

What did I get out
of ICME13? (And why will I go to ICME14 in Shanghai in 2020?)

As I noted earlier,
there are many possible outcomes from such a conference, and I can summarize
some of them, in no particular order of importance:

- Meeting new (or old) people with which future collaborations are possible. At this conference, I can think of at least four people I didn't know before, with whom there can possibly be some sort of collaboration some time in future.
- Getting ideas for future research projects: During the conference, I wrote a list of nine research and development projects that I would like to do. Not all are new ideas, but many have been expanded while I've been here. And some are directly adapted from talks here. I only hope I can follow up on some of them when I get back home... Hugh Burkhardt's talk on design research was very inspiring, and I want to do something in that direction (more systematically than I've done before).
- Getting ideas for my own teaching (which can of course also turn into research projects): Marjolein Kool's project on making students creating non-routine mathematics problems, Megan Shaugnessy's and others' ideas on simulations of classroom situations (some of these concerning the notion of "noticing").
- Getting an overview of a field which will make it easier for me to read more about it later: For instance, I hope it will be easier for me to face Brousseau now that I've had a tiny introduction and some context.
- Listening to lectures and realizing that I do actually know something: it would be impolite to point out which talks contributed to this realization. But perhaps it is a good sign that for every ICME I go to, I think "been there, done that" a bit more frequently. This is not to critizise ICME (too much), of course not everything can be new to everyone.
- Socialize with new or old colleagues: of course, the tendency to cluster based on nationalities may be seen as a problem (and meetings like the LGBT get-together may counteract that a little), but it is a reality that many countries have few meeting-places, so that socializing with colleagues from the same country does make some sense. There's been a lot of that here.

The main problem
with ICME is of course its size - it's complete lack of intimacy. The trick is
of course to find a home in a TSG, but still there will be situations when
you're all alone and see hundreds of strangers walk past - which is a challenge
for smalltalk-challenged people like me. But still, there is every chance that
I'll go to ICME again in four years' time.

But before that,
we'll arrange our own conference (ESU8) in Oslo in 2018. That will be fun - and
intimate. And I hope I'll go to CERME in February, 2017.