Monday, July 14, 2014

ESU7: Day 1

The 7th European Summer University on the History and Epistemology of Mathematics takes place in Copenhagen this week. This is my 4th ESU, so it's already a tradition for me, and it is always nice to come back to meet many well-known faces - as well as many new ones. As usual, I will try to blog from the summer university, but as usual I will have to warn you that my impressions of what people were trying to get across, may be mistaken.

Conference organizer Uffe Jankvist gave a short introduction to the buildings and Evelyne Barbin an introduction to the history of ESUs. This is a special occation, as it is the last ESU Evelyne will take part in before she retires. Tinne Hoff Kjeldsen then gave an introduction to the themes of this conference.

The first plenary speaker was Adriano Demattè, with the title "History in the classroom: educational opportunities and open questions". I know him mostly from these conferences and his teaching materials, which are very inspiring. He started by referring to Schön, and claimed that teachers have to look for proper circumstances in order to reflect. In his talk, he went on to show how history of mathematics will help them.

Interestingly, he spent quite some time discussing his context, how many classes he teaches and so on. This is necessary, because contexts are so different in different countries. Even though he teaches pupils aged 14-19, the topics he teaches corresponds to what Norwegian students learn at age 16-20, for instance. But also, the fact that he teaches maybe five different classes at four different levels means that his teaching will be different from a professor who has one or two courses in a semester. It is not easy to be innovative all the time in such a context. And Demattè is not content with being innovative himself, he also want other teachers to be able to teach using historical materials.

Demattè discussed the genetic approach vs the hermeneutic approach. He referred to Jahnke's 2014 article on how history of mathematics can be used after students have had a first meeting with a new topic. He looked at how some of the steps in Jahnke 2014 would be difficult in a classroom.

He then described some teaching he had done with a short part of Pacioli's Summa (1494). He asked students to read and interpret. The problem was "Find for me a number that, if added to its square, makes twelve." Pacioli solved this by an algorithm. Of course, he ends up with square root of 12 1/4 minus 1/2, thus three. But there were no modern symbols, and the modern division into question and solution were not as obvious.

Not all students thought about reading the text several times to understand more, and many did not compare with the modern solution to try to understand words (such as "root"). There was no hermeneutic circle - students did not make guesses, and did not move between parts and the whole of the document. It seemed that only those students who was confidence that their answer would be accepted byt he rest of the class, had the willingness to guess.

Demattè's proposed compromise: a textbook where every chapter contains a "historical laboratory", with a list of prepared questions to guide the students in their interpretation, making sure they move between parts and the whole, work on central words in the text and so on. The questions are aimed at facilitating students' work, take into account typical student prerequisites, bridge between modern solution and Pacioli's solution. He also mentioned that these questions could be used for tests, which is really interesting - and controversial, I would think.

A comment from me (which I didn't raise there): A question is of course whether adopting the question-answer format of a traditional textbook helps students or just turns this into just another hunt for the right answers. But again, this is a case where such a compromise may be needed to make it feasable to use it in ordinary Italian classrooms. In Norway at least, we could add the fact that teachers might not fare so much better than the students in interpreting original sources.

I'm sure many would argue that the struggle with the text is an important part of students' work, and that removing the struggle also removes much of the point of working on original sources. Renaud Chorlay asked whether students improve in their reading/interpretation capabilities over time, but Adriano Dematté could not answer as he has changed his approach over time, so he does not have long-term experiences in this. Probably, what Chorlay was hinting at was precisely that the initial "suffering" of the students in working on a text without being shown the way, is worth it in the long(er) run.

So already there were big, open questions going through my mind...

The first workshop I attended was my own. I was very happy to have my workshop this early, as it is always a relief to have finished. The goal of the workshop was to have people familiarize themself with the framework on Mathematical Knowledge for Teaching (MKT), see several examples of how history can be used in mathematics education, and discuss different reasons for doing that - in light of MKT. I was quite happy with how the discussions unfolded through the two hours, although some participants were at times at a loss to understand what my aim was - as I didn't have any particular opinion to "sell".

One important discussion is how the HPM community should interact with the mathematics education community as a whole, for instance regarding MKT. Is MKT "bullshit" (reducing mathematics to solving problems and disregarding emotions, for instance), as someone said during group discussions, or is it an a useful framework also for us in discussing HPM? And is it important enough in the maths education world that we should point out the place of history of mathematics in it even if we're not fans?

One outcome of the discussions was to see how a single source could be viewed as contributing to knowledge in many different domains, depending on your context and your point of view. Several participants found MKT useful as a tool for discussing HPM outcomes in teacher education, as long as we don't see the domains as discrete and separate. The idea of using MKT for testing teachers' MKT, however, (as is one branch of the MKT research) met more scepticism.

Then there was a three-hour workshop by Susanne Spies. Gregor Nickel and Henrike Allmendinger from Germany: "Using original sources in teachers education - An analysis on possible effects and experiences." In a way, this workshop was a bit similar to my own, in that it also focussed on concrete examples of using history and on discussing possible outcomes. They started with the point that school education of mathematics is - and have almost always been - insufficient for starting on higher education in mathematics. In particular reflection and the ability to judge about mathematics is lacking in students, and these are fields in which history could be helpful.

They gave several ways in which historical sources could be a tool for teacher education (telling anecdotes (entertain, comfort) - can become heroic or jovial; genetic use (Toeplitz): implicit or explicit - can be confusing;defamiliarizing; paradigmatic use) and how it can be a discipline for reflection (historico-critical perspective; history of ideas and culture perspective (secondary sources?))

They listed many examples from high school teachers' education where they included historical sources (real analysis; lecture on different solutions of the isoperimetric problem; exercises on irrational numbers and incommensurability; worksheet on Bernoulli's way of finding the tangent of a parabola. Reflecting on different ways of argumentation; worksheet on different interpretations of the derivative; presentations of students based on historical sources: training text comprehension of original sources.

They use historical sources in courses on elementary mathematics, advanced mathematics, history of mathematics and pedagogy of mathematical, and were particularly interested in whether historical sources had different roles to play in the different kinds of courses.

Then we worked in groups on a part of Euclid's Elements: theorem 47 (Pythagoras' theorem). We worked in groups discussing which contexts the source could be used in and what impacts it could have. As someone commented also in my workshop, looking at single sources and their impact risks forgetting the more overarching impact of using history of mathematics repeatedly. For instance, will students really change their view of the epistemology of mathematics based on one example from history of mathematic?. Probably not - it is the 13th example that makes the difference (I'm joking about the number - I think...). Thus, when looking at single sources, we look at the particular impact of the source, and not on the general impact of HM.

Thus, we focused on what this particular part of Euclid could contribute, and all groups agreed (it seemed to me) that a likely output could be on understanding the concept of proof (through time) and on being able to interpret different texts (which a teacher needs to be able to do when his students do surprising things).

Then: the use of philosophical texts in maths education. We did a similar exercise with a work of Plato (seventh letter) on knowledge of objects (name, definition, image, knowledge - and the fifth). We discussed how this text could be used in different courses. Of course, it could inspire discussion about the insufficiency of just giving a definition to students, but an important question is why this discussion would be better using the historical source than without? (In a way, I wondered, is it the case that in research mathematics, a concept IS its definition to a higher degree than in school mathematics?)

The evening's oral presentation was chaired by me, and I seemed not to be able to chair and make notes at the same time, so my comments on those two presentations are a bit short. Panagiotis Delikanlis talked about how he works on the mathematical problem contained in Plato's Meno, while Fanglin Tian talked about how they work in a lesson study-like way on teaching logarithms based on the origin of logarithms. Although both were based on history, the second was much more connected to the historical context. I'm looking forward to reading her article with (probably) some more detail.

Then there were two poster presentations. The first, by Paulo Davidson, connected the themes HM, Funds of Knowledge and Culturally Relevant Pedagogy. It would be useful if this presentation was longer to get more details, for instance on how history was used implicitly to work on algebra in new ways. Moreover, I should read more about CRP, for instance an article cited by Ladson-Billings. The second, by Julio Corrêa, was on mathematics, education, modernism and war. In the short time he had, he could just sketch some of the connections between these concepts. The "Math Wars" in the US is an example, which is a debate (not really a "war", but with war-like rhetoric ("a nation at risk").)

Thus ended the scientific programme of the first day. But then there was Happy Hour, and I was happy to talk to many friendly and interesting people. It was particularly nice to talk to Torkel Heiede on his home turf - he noted that he'd worked in these buildings for thirty years. In the opening of the summer university, it was mentioned that the 2016 HPM conference will be held in Montpellier in France, so that will be my next chance of meeting many of these people.

I'll continue blogging tomorrow...

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