In case you are
desperately looking for the Day 4 post, please note that Thursday was excursion
day. I chose to make this a Excursion-alone day, as there was a Manet
exhibition in the Kunsthalle (right next to my hotel) that I did not want to
miss. (It was great - and as so often, a good guide, in this case a multimedia
guide, made all the difference. I also don't have much of a memory, so I was
surprised at some of the cool paintings in the permanent exhibition, even
though I saw them three years ago.) Moreover, of course I had to prepare for my
talk on Friday, prepare for my workshops next weekend (at LAMIS in Ålesund) and
also generally wake up after Wednesday night. So I spent some time with a cup
of coffee and my paperwork in Lange Reihe, but also had dinner with colleagues
in the evening.

Of course, the
excursion day is also a day when you reflect on the conference so far. The
simple question "Was it a good conference?" can be unpacked into
subquestions such as Did you meet new (or old) colleagues that you may
collaborate with in the future? Did you get ideas for new projects or for your
teaching? Did you get an overview of topics that will make it easier to study
them further when you get home? Did you get that warm feeling of hearing a talk
and realizing that you actually know that topic quite well already? On at least
those counts, this conference has so far been a success.

Friday started with
a plenary lecture by Berinderjeet Kaur titled "Mathematics classroom
studies - multiple windows and perspectives". The TIMSS Video Studies
(1995 and 1999) was an inspiration for further classroom studies. They found
that teaching varied a lot between different
cultures - and we are not aware of the differences. The 1995 study
showed different patterns for different cultures. The 1999 study introduced a
new terminology of wide-angle lense perspective and close-up lense perspective,
and argued that both are needed. (By the way, I was lucky enough to get access
to data from this study to do a study of history of mathematics in the seven
countries. But that is many years ago now.)

In the late 1990s,
there were a couple of studies involving Singapore, the Kassel project and a
study of Grade 5 mathematical lessons. In the Kassel study, there were lesson
observations. There was a need of a shared vocabulary when talking about
classroom activities (for instance, she mentioned that the term "Singapore
maths" does not make sense to her). The study gave a wide-angle view of
Singapore (pretty much in line with what
one could expect). The next study consisted of just five lessons, also without
many surprising results.

In 2004, Singapore
joined the Learner's Perspective Study (LPS). This study investigated from the
perspective of students. Three teachers participated in Singapore, a ten-lesson
sequence for each were recorded. To code lessons, they wanted to take into account
the instructional objectives of the teacher (which seems to make sense), but a
teacher always have more than one objective to a lesson (which many school
administrators don't seem to understand). This made for complicated analysis.
The close-up lens gave a different picture than the wide-angle lense. Lessons
were well structured, objectives were clear, examples carefully selected,
student work carefully selected for classroom discussion. In addition, students
were interviewed after the lessons. This resulted in a long list of
characteristics that the students valued (which seems quite similar to what
teachers try to achieve).

She ended by arguing
that the stereotypes about Singaporean mathematics are not correct, especially
when you use a close-up lense. (This is not fully convincing to me, at least
not if based mainly on the study of three highly qualified teachers.) She also
noted 2010 survey data that showed that teaching for understanding is strong in
Singapore, and the analysis shows clear links between the different
instructional practices.

Then I attended
Michael Fried's talk on "History of mathematics, mathematics education,
and the liberal arts". I have heard Michael several times before, and knew
I couldn't go wrong there. He started by pointing out that he will not talk
about history of mathematics as a tool (referring to Jankvist). Here, the point
is history of mathematics as a object of study, not as something to use.

First, he started
with D. E. Smith, who was interested in history of mathematics and also set in
motion the idea of ICMEs. In another writing, Smith claimed that the motives
for teaching arithmetic is either for its utility or for its culture. History
of mathematics should be at the very heart of culture. He used both the
"parallelism argument" and history mathematics as a filter showing us
the importance of each part of mathematics. Fried argued that Smith's view of
mathematics is ahistorical - he looks at mathematics as a set of eternal
truths. Hence we see that the view of mathematics is not the result of not
knowing enough history. This view of mathematics is what allows it to be a
tool. And oppositely: allowing history to be a tool, makes it non-historical.

He then looked at
"religio historici", where original texts are at the heart of doing
history. It matters how you consider the past (of course, we know that parts of
the past have been deemed as uninteresting by historians at times). The past seen
as just what lead to the present (a Whig interpretation of history) is
uninteresting, "practical history". Oakeshott and Butterfield wanted
us to see the otherness of the past.

This unhistorical
history is particularly tempting in history of mathematics, because mathematics
is seen as eternal. He stressed the different concerns of historians of
mathematicians and teachers of mathematics. The demands on teachers means that
teachers will not put modern mathematics aside to teach history. Thus, teachers
needs to economize and make history of mathematics useful - thereby almost
neccessarily embrace a Whig view.

In 2001 Fried
claimed there was a clear choice between teaching Whig history or to drop the
idea of being useful. This damns all hope of including history of mathematics
in mathematics education.

However, in
principle, one can ask how mathematics education can be conceived to include
history of mathematics as an integral part. Here he named people like Radford,
Barbin, Jahnke, Jankvist, Clark, Guillemette as examples of people who can be
seen to be working on this. (This means fighting simplistic uses of the word
"mathematics" as learning methods, which can be glimpsed in some
research articles. I have often questioned what people mean by
"mathematical" in the phrase "mathematical knowledge for
teaching". In my opinion, the history of mathematics is an integral part
of mathematics, in the same way that you can't separate literatur from its
history in a meaningful way.)

Then he went on to
speak of the history of the "liberal arts", where mathematics was a
self-evident part. (He spent a significant time detailing this, including
pictures, which I cannot describe here.) The liberal arts was supposed to be
what you needed to become fully human. History was never considered part of
them. Today, history is considered part of the liberal arts, while mathematics
is missing. This could be fixed, and history of mathematics can be seen as a
way of solving it. Mathematics education then also becomes a way of reflecting
on ourselves.

(In Norwegian
teacher education, I'm not sure the argument is exactly the same. We talk about
so-called theoretical subjects and so-called "practical and aesthetical
subjects". But similar arguments can be made - of course, we know that
mathematics is also practical and aesthetical, as history of mathematics can
also show.)

This was a
thought-provoking talk. I believe that I will - as usual - end up in a
pragmatic view where including history of mathematics in different ways and
with different goals will still be better than doing nothing. I believe in
teaching history of mathematics as a goal, but that this can still be
"useful", and that teaching history of mathematics as a tool can
still instill a sense of "real" history. Teaching is never perfect,
it is always in need of judgment and a balancing act between different
concerns, so why should this area be different?

Then it was back to
the TSG. Derya Çelik talked on "Preservice mathematics teachers' gains for
teaching diverse students". As she could not come, the talk was sent as a
video. She talked about a project with 11 researchers. They analysed PSTs opinions
on how often the program provided opportunities to learn about teaching
students with diverse needs. 1386 PSTs took part. "Teaching for
Diversity" scale was used. In general, the results showed few
opportunities, although there were some (significant) differences between
regions, with more developed regions scoring higher. This fits with TEDS-M
results (also for Norway).

Then I presented the
work of Eriksen, Solomon, Rodal, Bjerke and me, with the title ""The
day will come when I will think this is fun" - first-year pre-service
teachers' reflections on becoming mathematics teachers". I did not make any
notes during this talk, as everything felt quite familiar... :-) The people
present seemed interested, though.

Oğuzhan Doğan talked
on "Learning and teaching with teacher candidates: an action research for
modeling and building faculty school cooperation". He started by stressing
the importance in teacher education in improving teaching practices. High quality
field experiences can contribute, while poor quality field experiences will
support imitation. The aim in this project is to find ways of improving. They
planned a hybrid course where teacher educators and PSTs plan mathematics tasks
and activities and apply them in a real elementary classroom. (He gave some details on the design of the
course which I can't repeat here.) At first, they applied tasks that the
teacher educators had made, but then they applied tasks that they had made
together. They saw that teacher candidates left the idea that the answers were
the most important. They also went from exercises towards discovery, and saw
the important role of manipulatives. The teacher involved also used the tasks
given in other concepts. For the future, they plan to do something like this in
the compulsory course, even if this course does not go on.

Finally, Wenjuan Li
talked on "Understanding the work of mathematics teacher educators: a
knowledge of practice perspective". She pointed out that there has been a
lot of research on PSTs and teachers, but not on what MTEs (mathematics teacher
educators) need to know. She used a distinction between knowledge for/in/of
practice. Of course, we need to include both mathematics and mathematics
education knowledge. The study further hopes that studying what MTEs do, will
inform us on what they need to know. (This is a bit doubtful, and moreover,
just as with mathematical knowledge for teaching, what the particular MTEs
don't know, will be missed in the model even if useful.) They included only six
MTEs in this project. All had school experiencem but they had limited knowledge
of research. (Which means that the results will probably be different from
results in a similar study in another context, with MTEs with another
background. And, remember, parts of the rationale of having teacher education in
universities, is that the MTEs are actively doing research and development
work, so that they will have very different specialities which (hopefully)
contributes to their teaching in different ways. Of course, the same criticism
can be (and probably has) been directed to Ball's work etc. But even so, the
results could be interesting to build upon with data from other contexts.)

After lunch, I heard
Ronald Keijzer's talk on "Low performers in mathematics in primary teacher
education". In the Netherlands, there is a third year national mathematics
test (because of PISA and TIMSS results, comparable to in Norway). The project
investigated characteristics of the students who did not pass this test. (Could
it be seen already in the first year?) Both interviews (n=12) and a
questionnaire (n=265) were used. Previous mathematics scores predicted score in
third year test. Low achievers specialize in teaching 4-8 year olds (not 8-12).
Many similarities with other students, but they are less enthusiastic than
those who passed the tests. They
disagreed that they were low achievers or did not work enough or that they
didn't get enough support. Rather, they blamed the test (interface, content,
preparation, feedback) and personal factors (dyscalculia, stress, concentration). So the conclusion is that low
performers are often that for a long time, they often blame the test or
personal factors. There was a lack of self-reflection. Comment from the room:
mindset treatment as a way of increasing self-reflection.

After this, I needed
to meet "my old group", TSG25 (on History of mathematics), where I
heard several talks throughout the evening:

ChunYan Qi talked on
"Research on the problem posing of the HPM". She referred to MKT, in
particular SCK, and she focused on problem posing based on HM. The research
look at 68 problems based on (one?) problem from Chinese history. She listed three
strategies for making problems based on history (but I didn't - in the very
short time - quite understand why these three were chosen or how we would know
that the results would be problems). Then she discussed which strategies were
preferred by students and other findings that turned up when analysing their
problems.

Tanja Hamann gave a
presentation on "A curriculum for history of mathematics in pre-service
teacher education". She started by referring to Jankvist, noting that
using history as a goal is perhaps a bit more important for teacher education
than at other levels, while history as a tool is more a general concern of all
levels. But of course, you often work on both at the same time. She argues that
HM is important for teachers to influence beliefs, give background knowledge
for teaching, building up diagnostic competences, providing an overview of
mathematics, being able to identify fundamental ideas, building language
competencies.

They try to
implement history in a long-term curriculum, including history throughout the
topics as small parts. They also have a special lecture on mathematics in
history and daily life. Finally, they use history in exercises and tasks. For
all of these three components she conjectured which goals they could contribute
to.

And after a break:

Jiachen Zou spoke on
"The model of teachers' professional development on integrating the
history of mathematics into teaching in Shanghai". He presented a model of
how teachers, researchers and designers can work together to design teaching
integrating history of mathematics. The model was in three dimensions with many
colors and many hermeneutical circles, and can not be described in words. It
was unclear to me how the model was developed (based on what data), maybe that
would have given me better understanding of the model. However, as always, time
is limited. (He was also asked how the ("somewhat formalistic") model
were developed, but it was not clear, except that the model had been used to
describe three different teachers' paths.)

ZhongYu Shen on
"Teaching of application of congruent triangles from the perspective of
HPM". His project concerns 7th grade in Shanghai, with two teachers
involved. The stories of Thales (finding the distance of a ship at sea from the
coast) and Napoleon (a similar method) was used as a starting point, in
addition to a Chinese measuring unit for length. They designed a lesson based
on this, and had a simple survey asking whether they understood and liked this,
and not as many liked it as understood it.

Fabián Wilfrido
Romero Fonseca on "The socioepistemologic approach to the didactic
phenomenon: an example". The example was three moments from history of
Fourier series: the problem of the vibrating string, Fourier's work on heat
propagation and development of engineering as a science. He showed how
activities were based on the analysis of the historical background, where
Geogebra were used to investigate. However, the activities have not actually
been used, just developed as part of a master thesis.

Thomas Krohn's talk
was "Authentic & historic astronomical data meet new media in
mathematics education". The students in question were in 10th, 11th and
12th grade. An authentic problem was to describe the movement of comets, and
this can be used by giving students some historical/astronomical background, as
well as the neccessary tools. The astronomical data are often presented in
lists in ancient books which are on the internet. As in ancient times, they
first made a projections from 3d to 2d, and then they tried to find a
reasonable function (often preferring to investigate instead of leaving it to a
computer to find.

Slim Mrabet, whose
title was supposed to be "The development of Thales' Theorem throughout
history", did not turn up.

Thus ended the
Formal parts of day five.

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