Anne discussed lots of examples of mathematics in literature, for instance The Birds, where the mathematician is driven off the stage. Or Thomas Pynchon's Against the day, 2006.
She gave a number of categories - "Literary modes for mathematical tracks" - and gave examples of each:
- Literal insertion (without real relationship to the unfolding of the narrative).
Use in teaching: discussion of interpretation, introduction of topics...
- Popularization
Allows interdisciplinary work and gives pleasant contexts
- Mathematics in the structure
- A mathematical object as character
Charles Perrault: The Loves of the Ruler and the Compass.
Use in teaching: Ask students to write poems or short stories whose characters are mathematical objects.
- analogy (or transfer) (transfer of mathematical reasoning to non-mathematical objects)
Analogies are a fertile tool in mathematics.
- an important character is a mathematician
Throughout the talk, literature examples were read/played by the "actors" Peter Ransom, Frédéric Métin and David Pengelley.
Then Johan Prytz gave a talk on "Social Structures in Mathematics Education. Researching the History of Mathematics Education with Theories and Methods from Sociology of Education."
He saw two main motives for studying the history of mathematics education:
- contribution to educational history in general. By comparing his own studies and the studies of Lövheim, he shows that the study of the professional debate among mathematicians/teachers gives s different picture than studying the general, political debates.
- contribution to research on mathematical education. Often too much focus on the big reforms in the 1960s. Håstad (1978) calls everything before 1960 "tradition". The critics of the reforms are ignored, and the changes in mathematics education before 1960 are not discussed.
Why a sociological perspective? History of mathematics education is often purely textual. A purely textual study cannot explain why some texts and authors are more influential than others. Johan considers different arenas: political, national level. Central school administration. Teachers. What's between the central arenas and the teachers? For instance, those who produce educational texts will form one such arena. One of Johan's results is that this group functioned as a field. The relationships between different arenas is also of interest.
In the next session, I attended Evelyne Barbin and Michael Fried's talks, but took a break from blogging.
"Empirical Research on History in Mathematics Education: Current and Future Challenges for Our Field." was the title of the second panel discussion at ICME. Panelists: Uffe Thomas Jankvist (Denmark), Yi-Wen Su (Taiwan), Isoda Masami (Japan), David Pengelley (USA). The focus was on the relationship between history in mathematics education and general mathematics education research. It is important to get the general community's attention, and one way of doing this is empirical research. In a survey of the literature, Uffe has found about 100 empirical studies on the HPM in its 40 years' history.
Masama Isoda talked on lesson study and technology, in particular his work with dbook (see also his talk at an earlier conference). He proposed that lesson study can be seen as a kind of empirical research.
David Pengelley focused on original sources. One way of evaluating is through open responses from students on the benefits and disadvantages of using original sources. It is more difficult to prove benefits with statistical analysis. He referred to Glaubitz' study (which Glaubitz presented in Vienna two years ago), in which the deep analysis of one text gave very good results. Some methods of using HM seem to have positive effects, while others have negative results.
He also discussed recruitment, transition and retention, claiming that HM offers students more reality, less fantasy ("mathematics drops down from the sky"). There's often a disconnect between what students think mathematics is, and what it really is.
Yi-Wen talked about Taiwanese experiences. Currently, there have been 15 Master theses on HM in Taiwan. She gave examples of work on the old problem: "how to measure an elephant on a boat?" In a three-year project students create animations and worksheets. In the third year of the project, they will be used in practice. The students improve their ability to search relevant materials (although it is unclear by what methods this result was established, and the exact connection to HM).
Uffe then had the last of the panel presentations. He quoted Katz: "Too much H and M, not enough P" Uffe claimed that theoretical constructs from the rest of mathematics education would be useful, both internally and externally. He referred to Kjeldsen, Barnett and others at this conference, showing how different projects presented at this conference could be analysed using the Niss competencies, for instance. He also discussed Ball's "egg" on MKT, showing that HM could also fit into all of the parts of that.
Most of the discussion afterwards was on the topic of theoretical constructs from outside HPM, and whether these could be expected to be useful in our context as they often do not include HM in a good way. In particular, Evelyne Barbin was sceptical of the use of the word "competencies", as we want our students to get more than that, for instance certain attitudes and beliefs. As I have written before (and said in Vienna two years ago), I similarly believe Ball's "egg" has a too limited view of mathematics to be directly useful for our purposes.
A connected topic was whether we should "compete on their terms" as someone phrased it - should we try to show that teaching mathematics with history is as "effective" as teaching without, using the definitions of "effectiveness" from without HPM? Michael Fried was sceptical of this, as teaching with HM has its advantages that need to be taken into account when performing tests, for instance. David Pengelley and Peter Ransom, however, disagreed. Based on long experience in teaching with HM, they had no problem claiming that their teaching was as "effective" as that of their colleagues, but with the added bonuses that teaching with HM brings.
Only two more oral presentations were left. I was chairing the session where Francois Platade gave a talk on 70 letters between Mittag-Leffler and Houël. The letters concerned mathematics and how to teach it, educational policy and mathematical journals, among other things. Most interesting for me were their discussions on how to teach complex functions - I'm sure that someone teaching complex functions could make good use of these original sources to illustrate different ways of looking at these.
Finally, I got the opportunity to hear most of Andreas Christiansen's talk (as one of the speakers where I was a chair did not turn up. Andreas gave a well-structured and interesting talks about very different ways of defining the basic concepts in geometry in three Norwegian textbooks.
Finally, there was a HPM Meeting, where Evelyne presented the future plans for the HPM group while I said a few words about the newsletter. The next ESU will be in Barcelona in 2014, while the next HPM will be in Europe in 2016, not too far from the ICME in Hamburg.
An idea that came up was that the HPM website should include lists of resources on HPM. Perhaps one comprehensive one and a short one for beginners? It was also pointed out that there should be separate lists for different levels of students. Who will take this idea further, is unclear.
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