The first day of
ICME13 (The thirteenth international congress on mathematical education) was
Monday 25th of July, 2016. This conference takes place in Hamburg, Germany. As
usual, I will try to blog from the conference (as I noted yesterday).

I missed the
conference opening ceremony, but got there in time for the plenary panel on
"international comparative studies in mathematics: lessons for improving
students' learning". (Not that difficult, as the conference was at that
time 40 minutes behind schedule.) The panel participants were Jinfa Cai, Ida
Mok, Vijay Reddy and Kaye Stacy. The panel was based on a paper to be published
(open-access) shortly.

The panelists each
talked on one lesson to be learned about international studies, defined as
studies involving at least two "countries", with an intention to
compare them. Such studies are good to understand ourselves and to understand
different possibilities.

Lesson 1: examining
the dispositions and experiences of mathematically literate students. Kaye
Stacy talked about large international comparisons, PISA in particular. She
showed how PISA 2012 were trying to unpack the scores by looking at the
processes of doing mathematics. Different countries had different patterns -
for instance, Japan had a high score on formulating, lower score on employing
and interpreting. Netherlands and the UK had very different profiles.
(English-culture countries were best in interpreting.) This fitted not very
well with students' answers about their exposure to different kinds of tasks.
She also showed graphs showing huge gaps in confidence between boys and girls
in some countries (i.e. Australia, but not Shanghai). We need to draw evidence
from different sources to inform policy.

Lesson 2:
Understanding students' thinking. Jinfa Cai talked about small-scale studies,
trying to understand students' thinking when trying to solve tasks. He showed
two tasks, one where average is demanded, the other where the average is given.
About a fourth of students in both countries got the first task right and not
the second task. He showed some of the
different errors in the second task - most concerned adding and dividing, as in
the (correct) solution of the first task.

He also showed the
"pizza ratio problem", where he also gave different solutions, some
of which were almost only used in one of the two countries studied. Thus,
comparative studies gives a broader sense of possible ways of thinking.

Lesson 3: Changing
classroom instruction. Ida Ah Chee talked about TIMSS Video Study and Learner's
perspective study. They have complimentary roles. TIMSS Video Study showed that
teaching was a cultural activity. LPS compares lesson structures and lesson
events and looks also on practitioners' thoughts about their lessons. Lesson
events are as different as lesson structures. Analysing structure and events in
light of teacher interviews give an even richer picture than the TIMSS Video
Study did.

Lesson 4: Making Global Research Locally Meaningful:
TIMSS in South Africa. Vijay Reddy talked about the sociopolitical
perspectives. She stressed how big differences is a fruitful ground for
research. In the Apartheid system, bad education in mathematics for large
groups was explicit policy. After 1994, improved access to education has been a
priority, but improvement is slow. Mathematics results is a proxy for equality.
She discussed how TIMSS can be used to understand the local development. South
Africa had big inequalities in the TIMSS scores, reflecting social
inequalities. Finally, in 2012, results are better. TIMSS also shows that the
level of violence is higher on SA than in any other country.

The
"panel" did give some ideas for how to use international studies for
better understanding, and there are surely lots of data out there that can be
looked at, as an alternative to always collect your own data...

After lunch, I
attended the talk by Emma Castelnuovo-Award awardees of 2015, Hugh Burkhardt
and Malcolm Swan (in absentia). It is particularly interesting to see
proponents of what we in Norway call "utviklingsarbeid" awarded and
given prominence at such a conference. Their work have resulted in many
beautiful tasks and sets of tasks, carefully designed to facilitate particular
learning outcomes.

First he gave
information on ISDDE, which has goals of building a design community, raising
standards, increase influence on policy. Educational Designer is a journal and
the website is isdde.org.

The talk was very
inspiring, but difficult to write notes about, as many of the points mentioned
were illustrated beautifully with examples of tasks and student responses. But
here are a few points:

- The Shell Centre wanted impact through useful materials, with focus on design. Materials need to fit the system they aim to rearch. It doesn't help if materials are brilliant if they don't work. Therefore, engineering research involves design research and systematic development.
- Tasks can provide 'microworlds' for learning.
- They talk about novice tasks, intermediate tasks and expert tasks. Expert tasks involves complexity and unfamiliarity, which means that the technical demand cannot also be high. He gave many examples of the three levels of tasks.
- The usual role of a teacher is to manage, explain, set tasks. Certain tasks facilitates "role shifting", and role shifting changes the classroom culture.
- In general, what this team wants to achieve is technical fluency, conceptual understanding, strategies, appreciation.
- Mostly, lessons that they develop are either concept focused or problem solving focused. Concept focused lessons are developed in the "diagnostic teaching" way. He gave examples of different kinds of the two kinds of lessons.

This talk was a
whole lot more inspiring than what these points suggest. He suggested making
small units of lessons so that a teacher could substitute their textbook for a
few weeks, without having to make a bigger commitment. I would be interested in
joining such an effort...

Then there was a
discussion on a survey on competences, led by Mogens Niss, joined by Regina
Bruder, Nuria Planes and Ross Turner. The main issue is "What does it mean
to master mathematics?" (Which also begs the question what we mean by
mathematics.) We may talk about the products of mathematics or about the
enactment of mathematics. Knowing and doing is not the same thing.

Classically, the
answers were given in terms of content and related facts and procedural skills.
Criticised in Spens report 1938, by Pólya in 1945 and so on. The first IEA
study in the 1960s included several components. Also, Papert in the 1970s
commented. Since the 1980s, there has been a trend focusing on enactment.

Ross Turner asked
what are the relationships between mathematical literacy, numeracy,
quantitative literacy, mathematical competence/competencies, mathematics. How
would a Venn diagram look like? Or, if you put them in a diagram with products
and applications on two axes, how would it look? (As usual, I'm trying to
figure out where the history of mathematics and other integral cultural parts
of mathematics fit into such a diagram - I think it shows that the concepts
here are too narrow. - at least, to "products" should be added
"background". Turner, however, put history into applications.)

Regina Bruder talked
about two types of research: research where the construction competencies are
an object and research where it is a means. One important discussion is whether
it makes sense to discuss mathematical competencies without specifying which
domains they are related to. Also, it is discussed if affective considerations
should be included. Also, can they be detected empirically? In general, yes,
but they are often overlapping and are not developed or enacted in isolation.

Mogens Niss and the
others then discussed how competencies have played roles in different national
curricula. Nuria Planes explained that in Spain, competencies have had a big
inpact on paper but not in actual implementation and practice. In Latin America,
processes have been included in curricula. Ross Turner told us that in many
countries in South East Asia, doing mathematics is clearly included in
curricula. But of course, the main question is how these words are implemented.
Regina Bruder told much the same story for Germany. In Austria, personal and
social competencies are included.

Then, challenges to
implementation were discussed, but this seemed to be the usual list of points.
The conclusion was that bridges between action and research is needed - which
ties this panel nicely to the previous talk - research and development (in Norwegian:
FoU) both are needed, and need to work together.

That concluded the
first day of ICME 13. A day with a varied scientific programme, with a big
delay in the morning and with serious temperature problems in the afternoon
(the buildings obviously not built for full auditoriums in late July). My choice of outfit (shorts + t-shirt) will be
repeated for the days to come.

And as usual, most
important is what I do not mention here, lots of talk with colleagues over
coffee, lunch, beer and dinner and in-between.

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