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.