Day 3 of ICME13 started off with Günter M. Ziegler's talk ""What is mathematics" - And why we would ask, where one should experience or learn that, and who can teach it". According to the twitterati, it was good and entertaining. I missed it.
In conferences such as this, it is important to break away from the programme to have time to reflect and to interact with other participants. (Unike some other conferences I go to, this conference does not have many workshops or discussion arenas.) Therefore, I skipped the plenary and an additional lecture, but was back on track after that, for four more TSG-talks:
Skip Fennell talked on "Preparing elementary school teachers of mathematics: a continuing challenge". He started by giving a brief historical background. He than pointed out that in many countries, elementary teacher education leads to teaching one or two subjects, while in the US (and Norway until now), teachers become generalists. Also, in many countries there are a limited numbers of programs, sometimes hard to get into, while in the US there are more than 1700 programmes, some online. There are voices arguing for having math specialists in elementary school. (In Norway, we are switching to a programme where you have to have a master's degree in one subject, but still will teach two or three other subjects.) However, in 20 states, there are math specialists, often called "maths coaches". These positions are, however, vulnerable in touch (economic) times. In the Q&A, he conjectured that there must be significant challenges in having teacher education programmes which provides everything except practice periods online. We all know the difficult negotiations between campus and practice, and these cannot be easier in such a context.
Marjolein Kools's "Designing non-routine mathematical problems as a challenge for high-performing prospective teachers". She started by giving a non-routine mathematical problem. In Netherlands, there is a national test in the third year of teacher education, and this project concerned creating non-routine problems for PSTs to work on. The got PSTs to help them, but it was very challenging for them to produce task of the right level of difficulty. Therefore, the project turned into a project trying to support the PSTs in creating, and this became a design research project. At the end, 4 of 8 students could design non-routine problems independently. Key to this result was master-classes where Ronald created problems while speaking aloud, trying them out and so on. (Although the starting point of this project was a national test, it seems worthwhile to work on this even in countries which are lucky enough not to have such a national test. It would be cool to do something like this in Norway as well...)
Eda Vula's "Preservice teachers' procedural and conceptual understanding of fractions". She referred to Ma, Shulman, Ball etc. as starting points for her talk. The research questions concerned representations and the relationship with and between procedural and conceptual understanding. 58 PSTs participated by doing a 20 item fractions knowledge test. The test obviously unearthed problems, for instance in the meaning of unit. PSTs had good procedural knowledge, but rarely good conceptual knowledge in addition.
Megan Shaughnessy's "Appraising the skills for eliciting student thinking that preservice teachers bring to teacher education". Of course teachers need to be able to find out what their students are thinking. This project takes this point into teacher education - how do we find out what our students think. The idea here is to use standardized simulations. PSTs get a student response and prepare an interaction with this "student", then they interact with his student (who is a teacher educator trained and using a response guideline). She showed a video to illustrate how this worked. All PSTs did this, and the videos were analysed, and the findings were really interesting (but I can't summarize them here). The study shows some of the moves that PSTs need to learn, what can be built upon and what requires unlearning. (And of course the videos themselves may be useful to work on thIs.)
After lunch, I attended the "thematic afternoon" on European didactic traditions. First, there was a "panel" with Alessandra Mariotti, Michéle Artigue, Marja Van den Heuvel-Panhuizen and Rudolf Strãsser. They had identified five characteristics of the the European traditions: strong connection to mathamatics and mathematicians, strong role of theory, role of design activities, key role of empirical research.
Mariotti talked on the role of mathematics and mathematicians. Historically, mathematicians were important figures in discussions on how to prepare mathematics teachers. In the beginning of the 20th century, Klein had a central role (Erlangen Program). Also, after WW2, mathematicians had key roles in the New Math reform(s). Later, mathematicians also played important roles in the French IREMs, for instance, where mathematicians, teachers and didacticians mixed. In Germany and Netherlands, on the other side, mathematicians played a smaller role in turning mathematics education into a research field. The case of proof as a theme of research have been a feature, and this may be influenced by the role of mathematicians.
Artigue talked on the role of theory. She started with the French tradition, which she claimed had a particular importsnce in building a theoretical foundation. These had a systemic vision of the field, a strong epistemological sensitivity, theories mainly conceived as tools for understanding. New development are connected to previous theories. In the Dutch tradition, Realistic Mathematics Education is the main contribution. This is a theory oriented towards educational design. The theories are still refined. In Italy there is s long tradition of action research and collaboration between mathematicians and teachers, now continued in Research for innovation. In the Germsn tradition, the theoretical landscspe is not so easy to synthesize. (The rest of Europe is left out in this survey.)
Van den Heuvel-Panheuzen talked about the role of design activities. The French tradition: essential component of research work and based on theoretical frameworks. In the German tradition, design activities took place in context of Stoffdidaktik, evaluation was often left out. Empirical turn from 1970s. Italy: empirical analysis as base for didactical innovations, pragmatic approach. From 1980s innovations have turned to research. In the Dutch tradition, the design aspect is the most significant characteristic. Theory development resulting from design activities, later theory was used as a guide for futher theory development.
Strässer talked on the role of empirical research. Different kinds of empirical research have developed because the complexity of the field. Examples: the large-scale study COACTIV, the case study MITHALAL. Of course, there are also a distinction between qualitative and quantitative methods. The purposes can be prescriptive or descriptive, and they may be developing vs illustrating theory. Action resesearch have been stronger in I and F, while fundamental research, strongest in F, G, N.
In addition to the content of the talk, what will surely be remembered about this evening is the heat and humidity of the auditorium, not least when it was decided to close all windows to close the curtain to improve the visibility of the screen slightly. It is a testament to the participants' dedication to the disipline that so many chose to come back to another session after the coffee break.
I know far too little about the French tradition, so I chose to go there. I'm also struggling to learn French, and hope that I will be able to read some of the French work later. Brousseau, Chevallard and Vergnaud were the stars of this bit of the program. IREMs were established from 1968, leading to a fruitful collaboration between mathematicians and teachers. From 1975: first doctoral programs, from 1980: journals.
The first theoretical pillar was the theory of didactical situations (Brousseau) - with the "revolutionary" idea that the pbject of research was the situation. Centre of observation for research created in 1972. The core concepts had been firmly established through the 1980s. The second pillar: the theory of didactic transposition and the ATD. (Chevellard) presented in 1980, followed by book in 1985. We "embed an activity into a whole whose ecology is studied". The third pillar was probably the theory of conceptual fields (Vergnaud), which seemed to be taken for granted in this meeting.
There was a talk on research on axial symmetry in France, to illustrate a development. There has been a development from the study of students' conceptions, combining the theory of conceptual fields and the theory of didactical situations to the study of teachers' practices and of their cognitive effects. Now, French research on symmetry is more occupied by the question of language. There was another talk on French research on school algebra. The first studies were in the 1980s, by Chevallard, in terms of didactic transpositions. Three questions:
- what is school algebra, and what is (or have been) considered as algebra? (Didactic transpositions, institutional constraints)
- What algebra could be taught? Under what conditions? How implement them? (Reference epistemological models (REM), didactic engineering (new ecologies))
- How do ICT modify the nature and the way of using, teaching and learning algebra? (Concept of ICT as part of the adidactic milieu, taking into account the instrumental genesis, didactic and computer transpositions)
At this point, the thing turned into a Q&A session in which the questions were unintelligible while the answers were moderately more intelligible. I chose this point of time to leave the auditorium. However, there was one interesting question that I was able to understand. It was something like this: "It seems that the research questions the French pose about algebra are quite similar to research questions elsewhere. In what ways do the answers differ?" I did not catch the answers, but I made up my own: the problem/challenge with having different theoretical perspectives and traditions are exactly that we may pose the same sorts of questions but look at them from different points of view, and we therefore have an inkling that the answers will be (slightly or significantly) different. However, since the answers are themselves given within different theoretical traditions, pinpointing exactly how they differ demands exactly the same comparison and analysis of perspectives (or perhaps even the creation of a "meta-theoretical perspective") that is some of the point of this thematic afternoon and similar work. If we could easily explain what the different traditions accomplishes, we would be at another point of the development than we are. Perhaps.
Thus ended the third day of the ICME13 conference. I had a very good experience in the TSG, getting inspiration for further work. The "thematic afternoon" was an interesting experiment, but the different backgrounds of the participants must have made it almost impossible to plan.
After this, there was social gathering again, followed by an (unofficial) LGBT get-together of ICME participants. I chose to post this before these events, though, to avoid the temptation to blog about them