The second day of
ICME13 started with a plenary lecture by Bill Barton, with the title
"Mathematics, education and culture: A contemporary moral
imperative". He started by remembering his first ICME (Adelaide) where Ubi
d'Ambrosio was the very first speaker he heard. This inspired him. His other
starting point was that pleasure is an important part of mathematics - the
pleasure of viewing fractals or of seeing a beautiful proof. (Hm - I often
think more of the fun of mathematics than of the pleasure. Maybe there is more
to be done in categorizing the pleasures of mathematics?) He asked us to look
at the Klein project - and to contribute with translations.

He went back to
Ubi's 1984 talk on ethnomathematics. Later, Ubi has spoken about mathematics as
the dorsal spine of modern civilization. Our goals should we responsible
creativity and ethical citizenship. But how? According to Ubi, we need to
include the cultural roots.

Barton invoked
Ecological Systems Theory (Bronfenbrenner). He used five levels: microsystem
institutions directly involved), mesosystem (links between microsystems),
exosystem (wider social setting - for instance financial world...), macrosystem
(cultural context etc), chronosystem (pattern of environmental events and
transitions, as well as socio-historical circumstances). He also invoked
Ecological Humanity - seeking to bridge the divide between science and the
humanities. The links between organisms define how the whole system works. He
also added Deep Ecology (Arne Næss) into the mix.

Barton
then distilled this into three principles: perspective,
reflexive, pleasure. The perspective principle: be
aware of other ways of understanding. The reflexive principle: do unto others
as you would have them do unto you. The pleasure principle: act so as to
increase pleasure. (All of these on both individual and system level.)

Was
the mathematical community responsible for the global financial crisis? It can
be argued that it was triggered by the models. Do
mathematicians love maths education as much as maths
educators are expected to love mathematics? And do we as maths educators
nurture our love for the subject? Do school teachers have the time and
resources to nurture their love for mathematics?

We all need to make
efforts to ensure mathematics education benefits from multiple points of view.
ICMEs should be more language friendly.

He critizised
streaming, which research shows increases the effect of socio-economic factors
(and so on). He compared this to having a (public) cooking test at a certain
age, and the outcome of the test decides who you can eat with and what you can
eat for the rest of your life. Fo 14 years we ask students to practice basics
without being offered the opportunity to experience the pleasure of
mathematics. And we know the effects of high stakes testing. (Of course, Barton
speaks in headlines, nuances are not included. But I guess that is okay from
time to time, too often we speak with all nuances and no headlines.)

After a coffee
break, there were lots of parallell invited lectures. I decided on Anthony A.
Essien's on "Preparing pre-service mathematics teachers to teach in
multilingual classrooms: a community of practice perspective". This is
obviously very relevant to my context in Oslo, Norway, and too little of what I
know about this is situated in mathematics learning. (Moreover, I do like
Wenger's perspectives.) Essien works at Wits University in Johannesburg.

There is a triple
challenge in multilingual maths classrooms: paying attention to mathematics,
paying attention to English (or other language of instruction), paying
attention to the mathematical language. (I would perhaps add attention to the
other languages the pupils know to the extent possible.) How do we prepare PSTs
(pre-service teachers) for this?

Essien uses Dowling
and Brown (2010)'s idea of organisational language, relating to empirical field
and theoretical field. Wenger (1998)'s communities of practice can be looked at
through three dimensions mutual enterprise, joint enterprise, shared repertoire.
But Wenger did not develop his theories for teacher education in mathematics.
COP theory does not have a coherent theory of language in use. Essien uses
Mortimer and Scott (2003) for this: meaning making as a dialogic process (DP). He developed cathegories based on literature
and data (not possible to summarize here...)

There are
interacting identities at play: becoming a teacher of mathematics/in
multilingual classrooms, becoming learners of maths/of maths practices,
becoming proficient English users for the purpose of teaching/learning
mathematics. Code-switching is part of this (maths practices). He studied
privileged practices-in-use. He showed how different universities teach in much
the same way, but still encourage development of different identities. (For
instance: when defining, one does it purely mathematically, the other focusing
on how it could be explained to children. Thus, the second may be expected to
support the development of an identity as a mathematics teacher.) The
perspective of multilingualism was lost somewhere on the way, however.

A question from the
room concerned whether the teacher educators were explicit in what role the
PSTs should assume (as in "consider this as a teacher" or
"consider this as a mathematics learner") The person asking suggested
that PSTs may easily be confused if this is not clear and explicit. I think
this is one of the most important points for getting students to move from
their role of pupil into their role as prospective teacher. (By the way, it was
really refreshing that there was time for questions, after a day of
conferencing with little interaction. The discussion was interesting and fun.)

(An idea I got for
an early task with my new students this autumn: "What is the most
difficult thing you know well in mathematics? How would you explain this to a
learner who don't know this too well in advance?" (And as an added
perspective "who knows another language better than Norwegian?"))

(Based on
yesterday's talk on design research, I also get inspired to look at the design
on history of mathematics-based mathematics lessons, by way of analysis of
published lessons or by interviews with lesson designers.)

Much of the ICME
programme is filled with the TSGs (Topic Study Groups). I usually attend the
TSG on history of mathematics, but this time I attended TSG47 (Pre-service
mathematics education), because this is where the project I will present on
Friday fits in. For a conference this site, it is important that people have a
"home" where they meet the same people more than once, but still I'm
a bit uncertain about the weight the TSGs have got in this conference - for
instance, most of Tuesday was filled with TSG sessions - mostly filled with
ridicuously short presentations (10 minutes each). We'll see how it works. (Of
course, the weight of the TSGs is a function of the number of the contributions
they receive and accept ("my" TSG has 66 accepted contributions),
which again is a function of how broad their themes are. So it is not an easy
problem to solve.)

In this first
session, there were four talks. Fou-Lai Lin (with Hui-Lai Lin) presented a talk
on "Using mathematics-pedagogy to facilitate professional growth of
elementary pre-service teachers". Hui-Lai started by discussing a game
where the point is to arrange coins as a rectangular form. (I didn't get the
details of the game, but I see ways of doing this so that lots of concepts will
be involved, for instance where children get points based on how high the
lowest factor can be.) Fou-Lai started by telling that Taiwanese mathematics
teachers are not good in mathematics, sometimes even fearful of it. She gave a
list of principles for designing PST-courses, which I didn't manage to note
down. She showed an example with binary numbers. PSTs should see analogy to
elementary student learning. She also stressed the importance of not only that
students should read textbooks themselves, but analyse them and conjecture how
pupils will struggle with the textbooks.

Secondly, Roland
Pilous talked about "Investigating the relationship between prospective
elementary teachers' math-specific knowledge domains." The focus of the
study was relationships between the domains (and in talking about domains, it
seems he was inspired by Ball et all's domains). The project was based on six
students and three lecturers. Four domains were identified: curricular
knowledge, content knowledge, teaching-related knowledge and knowledge of
student cognition. (The problem of such a tiny number of informants is of
course that whole domains of knowledge may be missing. If, for instance,
history of mathematics is not mentioned by the twelve informants, that may be
lost to the model.) He very quickly mentioned some of the relationships, but of
course time was too short to get an understanding of this.

Thirdly, Jane-Jane
Lo had a talk with the title "A self-study of integrating computer
technology in a geometry course for prospective elementary teachers". This
is a study of her own teaching, analysing the lesson plans and reflecting on
her choices in hindsight. You can use tools for technical activity or as
conceptual activity. Dick and Bos discuss pedagogical fidelity, mathematical
fidelity and cognitive fidelity. Use of technology should not compromise your
pedagogical principles or mathematical correctness, and the external
representations should facilitate the mathematizing processes. Her teaching was
first using Geometer's Sketchpad and Scratch. For the next iteration she used
lots of apps etc (for instance from learner.org). The problem is that there is
a learning curve for each app or software. Now she has gone back to Geogebra
exclusively. (And there are lots of things available at Geogebra Gallery.) Her
main challenges is the connection between physical, virtual and mental representations
- how much is needed of each - and to facilitate mathematics discourse when
each student has a computer in front of them.

Fourthly, Ryan Fox
discussed "Pre-service elementary teachers generation of multiple
representations of word problems involving proportions". He used Ball and Rowland for background
literature. He stressed the difference between the way PSTs were taught in
school and the way they will be expected to teach. (This will certainly be the
case - no PST can foresee which school cultures they will work in during their
careers.) This study was included four participants, all in their second year
of teacher education. They were interviewed four times, totalling 2-3 hours per
participants. The interviews concerned how the PSTs would teach certain topics.
His students mostly went for one solution method - one struggled a lot, finally
found a solution, stuck with that but forgot it by the next interview, another
never struggled, but kept using the same solution method. But the fourth
student managed to use four representations for the same problem.

After lunch, the TSG continued with two more talks. Gabriel Huszar talked on "An exploratory study about the reseponses of the prospective primary teachers using the concepts of measurement in maths". He started by referring to PISA and other studies, showing the need for intervention in Spanish education. This project tried to identify the most common conceptual problems and mistakes among PSTs in measurement. 81 students took part in the study, and the measurement instrument was a questionnaire of 12 tasks taken from PISA and PIAAC. Sadly, it was hard to see the graphs with the results in this talk. Of course, tasks which does not only ask for a repetition of algorithms, were more difficult for them.

Then, Mine
Işıksal-Bostan talked on "Prospective middle school mathematics teachers'
knowledge on cylinder and prism: generating definitions and relationship".
This was a video presentation, probably due to the difficult situation in
Turkey, due to mr. Erdogan, although this was not mentioned. She discussed the
importance of definitions in teacher knowledge. She referred to
Tall&Vinner, and Zazkis&Leikin. In this talk, the focus is cylinder and
prism. Of course, the formal definitions are different from the definitions
used in textbooks in primary school. She gave examples of how cylinders and
prisms are introduced in Turkish textbooks. The project were based on
questionnaires and interviews. The answers were catagorized according to whether
they gave sufficient and minimal conditions (she did not explain why this is an
interesting analysis to do - for primary school it may be more interesting that
the definition is understood and usable than that it is minimal). They were
also analyzed for prototype/non-prototype and other characteristics. In
conclusion, the concept image seemed more "correct" than the concept
definitions.

(Could I create an
activity in which students get a long list of definitions, in which they
discuss what are the differences between them?)

After another short
break, there were five more presentations in the TSG. My colleague Annette
Hessen Bjerke talked on "Measuring self-efficacy in teaching
mathematics". She noted that mathematics teaching self-efficacy is
particularly interesting, as many express low self-efficacy in maths teaching.
She refered to Bandura's concepts of outcome expectancy vs. personal
self-efficacy - her research looks at personal self-efficacy in tutoring
children. She developed a new instrument for this, with 20 items, 10 concerning
"rules" and 10 concerning "reasoning". She mapped
self-efficacy through the compulsory mathematics course in Norway. Rasch
analysis was used with an instrument that had been validated. She described how
it can be shown that the PSTs' self-efficacy improved, but also that there are
some students that have decreased self-efficacy at the end of thr course.
Further research (ongoing) will investigate the sources of self-efficacy.

Lenni Haapasalo
discussed "Assessing teacher education through NCTM standards and
sustainable activities". He had developed instruments to look at
self-confidence of different groups of students, and showed how they developed.
However, in the short time, it was difficult to understand details of the
instruments, whether they had been validated and so on. He added some comments
on the sad state (according to him) of the Finnish educational system and how
his colleagues (still according to him) earned money by travelling around the
world talking about how good the system was.

Siyin Yang had a
talk titled "A comparison of curriculum structure for prospective
elementary math teacher programs between the United States and China." She
studied two different elementary educational programs. She described the
different structures of the programs she had chosen to look at. Of course, the
details of the comparisons must be looked up in her papers.

Gabriela Valverde
Soto's talk was "Enhancing the mathematics competencies of future
elementary teachers: review of a design research". Her chosen topic was
ratio and proportionality. She described the phases of design research. This
was a rich presentation from a big research project including more than a
hundred audio recording, and the analysis looked for productive exchanges,
among other things. Again, I can't try to summarize.

Finally, Oleg
Ostrovskiy talked on "Visual representations of word problems".
External representations can help us solving word problems partly because word
problem solving correlates with working memory. He pointed to the Singapore
method for solving word problems. The project involved 17 PSTs, using the book
"Elements of modelling". Pre-tests and post-tests showed that more
PSTs used visual representations to solve the problems, and more adequate ones,
after the intervention than before. However, it takes a long time to learn
these ways of illustrating - it is not enough to just show the students the
method a few times.

In all, this was 11
TSG-related talks in one day, most of them ten minutes long, therefore quite
hurried while still missing out on significant points of the projects. I'm
sceptical of the ten-minute format, even though they can of course be seen as a
huge number of samples to inspire us to read the papers (even though the papers
are only four pages long, most of them, so they are also ridiculously lacking
in detail).

After this, there
were another seven hours of partly mathematics-related discussions before I got
back to my hotel, but I won't try to summarize these hours later, important
though they are.

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