Wednesday, July 16, 2014

ESU Day 2

The second day of the conference started with Evelyne Barbin's plenary talk titled "Epistemologies and Theories behind History and Education: thirty years after Hans Freudenthal". Of course, Evelyne is an important figure in the HPM group, as Jan van Maanen stressed in his introduction of her. She started by referring to Hans Freudenthal, who protested the teaching of ready-made concepts instead of the development of the concepts. The phrase he coined for this, and which was mentioned throughout the talk, was "antididactical inversions".

Evelyne referred to two different ways of doing history; disorienting history and rational history, where the former strives to understand the facts of history, while the latter strives to give "big" theories. The former is looking at differences between authors, the latter concentrating more on similarities. I understood Evelyne's talk as a strong criticism of many attempts at the latter, where broad generalizations on the development of mathematics are made without sufficient regard for the details of the actual development.

Most of her talk focussed on particular, important examples from the history of mathematics (finding of tangents of curves, Leibniz' definitions of functions, conceptions of tangents etc) and examples of researchers giving statements of the development of mathematics which do not fit with the particular details. It would be dangerous for me to try to repeat concrete examples, as I do not have Evelyne's knowledge of the details.

Evelyne ended by warning against the "danger of a terrible didactical inversion: the didactical inversion of history of mathematics".

For the day's two hour workshop, I chose to listen to Jan van Maanen. It is more than fifteen years since I heard him the first time, and I always grab the opportunity when I have the chance. His title today was "'Telling mathematics' revisited". This was a development of a talk he held fifteen years ago (in Louvain), and two of the people present had also heard the talk then.

As a teacher, Jan spent time on problems that the pupils brought, and after a while, he realized that they had a lot of history in them. These stories have travelled miles and years.

For instance, he told us that he had talked to a Dutch girl, who had been told a problem by a girl while waiting for a ski lift in AustriaAustria 2010: "If I give you 2 sheep, you will have twice as many sheep as I have. If you give me two of your sheep, we have the same number of sheep. How many do we have?" The same story can be found, in variants, for instance in Nine chapters, 179 AD, and was included in Euler's Elements of Algebra 1770.

He gave lots of examples of such problems, such as the broken bamboo, the hundred birds problem, the two brothers (one of which always lies and the other who always tells the truth), "as I was going to St. Ives", 3-4-5 triangles in Groningen doors (Frank Swetz p 119).

Why does "telling mathematics" make sense also within the school?

  • Because of the challenge - not neccessarily using exactly what they learned that lesson.
  • It combines the usual with the unexpected.
  • It is the student who decides!
Presently, Jan has had two workshops with mathematics teachers to let them exchange mathematics that they knew from being told, asking them to make notes, then analyzing these notes in order to sort and cathegorize the entries, and to trace the entries back to their source.

He did the same with us, and "my" group (Mustafa, David and me) came up with three examples: - weights (one lighter than the others) - area with given perimeter - three people with black or white star on their foreheads. The winner is the person who knows his own color. They have to clap when they see a white star. When person A open his eyes, he sees two white stars. All three persons clap. But noone claims to know their own color. Then A knows. Why?

There were discussion at the end about how to "use" these problems. If the teacher gives the problem, should they be part of a structured plan of teaching? Jan's main point is perhaps to have the students contribute to the content of the classes, which means they can not necessarily fit nicely into the overall plan of the teacher. The point of giving students a chance to participate is more important.

Personally, I'm a bit depressed that I - from my memory - almost didn't work on anything outside the textbooks for my twelve years in school.

One sidenote: Jan mentioned a mathematical walk of Groningen - maybe this would also be a good idea for Oslo?

The second workshop of the day, this time a three-hour one, was Desiree Krüger and Sara Confalonieri on German and French textbooks' handling of negative numbers. They gave a background of the "elementary" textbooks that they wanted us to work on. The textbooks in question were written for universities, but it could not be assumed that the students had any prior knowledge of mathematics. In France, though, they had military schools that included mathematical teaching.

Then we worked in groups on selected original sources in the form of textbooks. My group (Andreas and me) worked on Kässtner's "Anfangsgrunde der Arithmetik Geometrie ebenen und sphärischen Trigonometrie, und Perspectiv". For me, both the typeface (fraktur) and language (German) was difficult, so there were some layers of difficulties to get through in addition to the unfamiliar definitions and examples (and context, of course). Particularly interesting was that the textbook stressed that you can choose for yourself what should be considered positive and what should be considered negative whenever you have two opposing magnitudes. For instance, a debt is negative if you consider wealth as the positive, but is positive if you choose to see the debt as positive. We wondered if this way of viewing it may be better than the normal way of doing it in Norwegian textbooks today.

Then on to oral presentations. First, David Guillemette talked on "Sociocultural approaches in mathematics education for the investigation of the potential of history of mathematics with pre-service secondary school teachers." He takes disorientation (dépaysement) (Barbin) as a starting point. He sees a gap between theoretical and empirical research in HPM. David wants to get deeper into the concept of disorientation and see how it turns up in real situations. He will use the theory of objectivation. (Learning maths is not just learning to do maths, but to be in mathematics (Radford)). He wants to give the pre-service teachers a voice.

He will also discuss whether the disorientation is an individual experience or could/should be seen in a social context. The narratives will be collected into a polyphonic narration, trying to integrate different points of view. One example of a preliminary results: a feeling of otherness and empathy towards the authors and the pupils are present.

Afterwards, there was a short discussion on the best translation of dépaysement (or whether it should be left untranslated). Alienation has also been used.

Secondly, René Guitart talked on "History in Mathematics According to André Weil." Weil used original sources to explain today's comprehension, reading the original sources with the help of today's knowledge, but without pretending that those authors had the same knowledge. Weil suggested historical notes in the papers of the Bourbaki group. A historical problem is (only) considered important if it generates a method or general theory. In two letters to his sister, he explained the meaning of his mathematical work by means of its history.

Thirdly, Leo Rogers' theme was "Historical Epistemology: Contexts for contemplating classroom activites." His talk concerned the connection between the origin of mathematics in ancient times and the connection to teaching of kids today, with lots of theories involved and lots of practical examples of what mathematics ancient cultures were doing. Sadly, my notes are not sufficient to give a detailed summary, so it's best to wait for Rogers' article (as is of course the case of most of the talks and workshops I mention).

One good quote from Neugebauer which I will keep in mind, was about how we do not gain a "historical perspective" as time goes by, we are just less burdened by evidence, which means we make bolder generalizations without fear that historical sources will contradict us. With this quote, Rogers' talk ended the day on the same note as the day started.

After this, the Advisory Board of the HPM had its traditional dinner, which included great food and wine and many friendly conversations - as well as a little official business.

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