The Early Numeracy interviews were based on 60 different tasks, the tasks actually given depending on the child's answers along the way. A Rational Number Interview was developed for older kids. Nice task: from six numbers given, create two fractions that added together will be close to 1. In the interviews, a lot of student thinking could be inferred from how they moved their cards with numbers.

He referred to The Interconnected Model of Teacher Growth (Clarke and Hillingsworth 2002), which I should look up... During the study, teachers developed more realistic understanding of their students' abilities. For instance, one kid in the study could read a 15-digit number, while many teachers focus on numbers from. 1 to 20 at that age. The project also included making teachers more aware of important strategies. The teachers also developed improved questioning techniques - modelling their behaviour in their classes on the interviews. The teachers also learned that they sometime had to give students a bit extra time to think for them to find the answer.

The obvious question to this is one of resources. Most teachers would love to have the time to have long one-on-one talks with each student in every single subject they teach. That is not possible, and if it was, there would still be a discussion on whether all the extra money could be better spent in another way.

To this plenary, Marja van den Heuvel-Panhuizen gave a prepared reaction (which was of course based on the article in the proceedings, not on the actual plenary talk). She asked: what was the proof that it worked? The evaluations were based on self-reporting of teachers and improvement of students' achievement, but the results of control geoups were not given in the paper. She also went on to describe problems in Clarke's choice of words and so on. She then invoked Hans Freudenthal to say that observing learning processes is (or should be) at the core of teacher education. She seemed to use this as a reason why interview-based assessment is a good idea, and that starting from a typical learning path is a good thing.

Then there was a discussion group on Mathematics Teacher Educators' Knowledge for Teaching. This sounded interesting to me because we of course discuss our students' (prospective teachers') mathematical knowledge for teaching when educating teachers, and it is only fair that we look at our own knowledge as well. Moreover, I'm interested in reminding people that the history of mathematics probably should have a part in what mathematics teachers' should know, as well as (of course) what mathematics teacher educators should know. The first 25 minutes of the discussion group was spent on every participants' saying a little about themselves, which proves that the number of participants were not too small and that the participants were not too silent...

Then we went on to small group discussions, where one was to discuss what are the differences between mathematical knowledge for teaching for teachers compared to for teacher educators, whether it is different for different types of mathematics teacher educators, how we can do research on thism how to educate mathematics teacher researchers and how to ensure resources for this work. Splitting into groups were a challenge, but once we had settled in a room, interesting discussions unfolded. We discussed the first question: what is the difference between being a mathematics teacher and being a mathematics teacher educator? Some of the contributors had the experience of being both at the same time, which is an interesting experience. Someone felt that there was a similar leap from being a teacher to being a teacher educator as the leap from being a student of mathematics to being a teacher.

One topic we touched upon was research. While teachers of mathematics do usually not keep up to date on the most recent research, teacher educators certainly should. Also teacher educators need to know something about the (hypothetical) learning trajectory of teachers. An example is how we teach fractions in a way to make the teacher students understand that what they thought they knew well, they didn't know nearly well enough.

A "technical" question is whether one should try to adopt the cathegories of Ball or wrap Ball's egg within a new layer.

It is also interesting whether mathematics teachers all need to know the same, and the same goes for mathematics teacher educators. Of course, we tend to divide our teaching tasks based on our areas of expertise, so maybe it is enough that half of mathematics teacher educators know their history of mathematics, for instance? Could the same be said of teachers in school, or is that a "lonlier" profession in the sense that they tend to have sole responsibility for mathematics teaching in their group of pupils?

Today's discussion group reminded me that discussion groups are among the most difficult things to plan (as I learned at ICME, although it did turn out okay in the end) - you don't know in advance if there will be 10 or 100 people, and you don't know what the participants' previous knowledge may be. On the other hand, it could also be said that as long as you manage to set people together in groups and they are fairly interested in the subject, they will probably have a good time discussing the subject with each other, who are after all experts from around the world.

Olive Chapman's talk later in the afternoon had the title "Facilitating prospective secondary mathematics teachers' learning of problem solving for teaching". She noted that students' unguided reflection on problem solving often reduced to mentioning general steps (maybe similar to what textbooks often do?) or to describe which steps worked in their problem solving. Chapman wanted to help teachers become aware of different ways of engaging in and reflecting on problem solving and to develop deeper understanding of problem solving. The idea was to help teachers to use personal experiences to build their reflection on, by having them develop rich descriptions, through two approaches; a narrative approach and a "stuck-aha-approach" (focusing on turning points). In the talk, she discussed differences between these two approaches. The narrative approach was more affective, while the stuck-aha-approach was more cognitive. The narratives gave a good view of the process and real world problem solving. When looking at the stuck-aha the cyclic process is clearer. Both approaches were adequate to give room for reflection. Sharing, discussing and unpacking was necessary and gave the teachers the opportunity to a richer experience. (The sharing was done in groups immediately after writing the texts, and they were then challenged to discuss what they learned about problem solving by reading these texts, and what they learned about teaching problem solving.)

One person in the audience had the interesting comment that she has sometimes experienced that students reflect on the problem solving process and realize that their mathematical knowledge is insufficient (for instance that they don't remember Pythagoras' theorem, and they do need that to solve the problem), but that they don't follow up the reflection by actually trying to learn the mathematics in question. So there is the problem of reflection for reflection's sake.

Then there was a talk on teachers' emotions, by Pietro di Martino and others. They talked about the "math-redemption phenomenon". This has to do with the fact that many students who have had problematic experiences with mathematics, still look forward to teaching mathematics, and want to improve their relationship to mathematics. In addition to survey data, they also interviewed students. Students often identified "crises" in their relationship to mathematics, and these were always connected to the teachers they had. This turned into a fear of becoming the same kind of teacher as their previous crisis-inducing teachers. This also seemed to be a major motivator to reconstruct their relationship with mathematics. However, for some students this was seen as unobtainable (by themselves).

The concept of "maths-redemption" seems to be a powerful one, and could possibly also be used to motivate our students in Oslo. By the way, we also have lots of data on student emotions in Oslo, which it could be interesting to reanalyze in light of this talk/article.

Finally, starting at 6:20 pm, was the presentation by my colleagues Annette Hessen Bjerke and Yvette Solomon on our project in Oslo. The project is looking at how teacher students experience their transitions between their teacher education on campus and their school placements, and (in this paper) particularly at what they and their mentors think that the students have learned in the different arenas. For the content of this, I refer to the article in the proceedings.

Simon Goodchild commented that first year students could only hope to "cope" and that we would expect them to show more of their competence later. This is a valid point, of course; by choosing to focus on the first year students, we get another project than if we had focused on third-year students, for instance.

Thus ended the first full day of the confence. (Except, of course, that we had some reading to do before Tuesday.)

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