ESU7 Day 5

The final day of ESU7 started with Bjarne Toft's "Peter and Sylvester - the emergence of graph theory". The first paper on graph theory was by Danish mathematician Julius Petersen. Of course, in hindsight, precursors of graph theory can be seen in The Bridges of Konigsberg problem, for instance. Road networks, social (or sexual) relationships and website connections are examples of applications today.

Petersen was a teacher, teaching 35-36 hours a week, and publishing a new textbook every other year. He wrote a much used and acclaimed collection of geometrical problems. He finished his PhD in 1871, and published a lot in the following years. Then he got in touch with Sylvester, and they met in Copenhagen (in Tivoli!). In a letter of October 18th, 1889, Sylvester explained to Petersen what "graphs" are. In 1890, Petersen went to England to collaborate with him. The collaboration didn't work out, though, and there were some amusing letters going in several directions. They showed both how mathematicians are human beings with mood swings like everybody else, and that mathematicians are fallible, unlike mathematics books...

Fàtima Romero Vallhonesta and others then had a "workshop" on "Teacher Training in History of Mathematics" (but it turned out to be more of a lecture). They are a group of teacher making materials for the classroom which will be a publication in two year's time. They had a brief introduction, mentioning the difference between explicit and implicit use of history of mathematics. They pointed out that with teacher students, they used sources explicitly, while with other students, they used the sources mostly implicitly.

The aims of the implementation was

• Knowledge of the original sources
• Recognition of the most significant changes in the discipline of mathematics
• To emphasize the socio-cultural relations of mathematics with politics, religion, philosophy and culture.
• (The most important). To encourage students to reflect on the development of mathematical thought and the transformations of natural philosophy.

There was an introduction to Chinese sources, including Nine Chapters, and then we looked at chapter 9, being given the problem, answer, algorithm and comment (by Lui Hui). (In both the original Chinese and in a French translation...) Students could do activities, for instance doing geometrical proofs on paper. (It was unclear to me, however, what the students were asked to do with the sources and what instructions they were given.)

Al-Khwarizmi was then the next example - the usual solving of quadratic equations, but with both of al-Khwarizmi's ways of solving (and this time in an English translation). The goal of this particular task was that students should be better at algebra, and the original source is (as I mentioned) only read by teacher students, not with the mathematics students. First, students are asked to research al-Khwarizmi's life (some basic questions), then they learn both algebraic and geometric methods. The geometric method they learn by getting the complete geometric proof, but without numbers added to the figure, so that they themselves just provide the details, not the steps.

Their way of using al-Khwarizmi with mathematics students is quite unhistorical (as was pointed out by people attending) and I'm not sure that it could be called implicit use of history; rather it is teaching loosely inspired by history. Teacher students, on the other hand, are given the original sources and given the task of making activities based on them, and then the different ways are compared. I would be careful about doing that with my students, as there is a danger that all of them would make unhistorical materials so that it could be difficult or at least time-consuming to avoid ending the project without having reached anywhere meaningful.

Finally, there was an example from Pedro Núnez and from Viète. There was a fascinating dispute in the end of the workshop where an equation in the style of Viete was written also in the style of Descartes, where the variable A was changed into the variable x, but the z from Viete was kept. It was pointed out that this would be confusing, as z was a constant with Viete, but a variable with Descartes, so in the "Descartes version" of the equation, it should be c...

Kristin Bjarnadottir's workshop was a follow-up on her plenary the day before, with the opportunity to go into details. First, we had a look at another problem on measurements, which - among other things - showed Icelanders familiarity with different measurement units, as well as payments based on gender. Then we all collaborated on doing a Geogebra simulation of the movement of the sun in the Icelandic sky at different times of the year. This ended up with the function f(x)= -(90-65)* cos (x*2*pi/360) + a, where a is a value between -23.44 and 23.44 (on a slider), and where 65 is the latitude of the farm of Torsteinn. (Reykjavik 64, Torsteinns place 65, Rome 42 and so on.)

Then there was a little on the dominical letters, summer's extra week, the first week of summer and so on. I cannot possibly summarize this.

It was unavoidable that this conference too would end. The closing session was a time for praise for the organisers, but also to look forward, to the deadline for the proceedings papers (10-15 pages by November 15th), to the next HPM conference (Montpellier July 18th-22nd, 2016) as well as to the next ESU (Rethymnon, Greece, July 2018).

What were the highlights in the week for me? To be honest, the highlights for me personally was a conversation with Renaud Chorlay over a beer after the excursion, a late-night discussion with Francesco Maria Atzeni, breakfast talks with Costas Tzanakis, a few short lunch chats with Torkel Heiede, the traditional dinner with Kristin Bjarnadottir and Andreas Christiansen (traditional since the last ESU...) and coffee break talks with Mustafa Alpaslan. I could go on and on. The most important part of any such conference is the face-to-face discussions with other people who spend their life striving for some of the same goals as yourself, having some of the same interests and strangenesses... Sadly, my blog posts don't catch all of this very well, although my opinions throughout are of course enhanced by the conversations I've had throughout the conference and at previous conferences.

It is difficult for me to pinpoint a highlight from the scientific programme. Single ideas, such as David's use of the concept of "violence" to describe working with original sources, were thought-provoking. However, if I am to point out two main ideas that have (re)formed in my head during this conference, it is these:

• The same original source can, depending on the context, be used for many different purposes, and depending on these purposes, the design of the teaching will vary. The assessment of whether the teaching was successful, will also vary. Understanding better how these things work constitute major "open questions".
• In particular, in some contexts it is meaningful to give students an original source and a set of tasks to "guide" their reading. When this is appropriate, how these can best be designed and what effect this has on the outcome, form a subset of these open questions.

Of course, these questions was actualized already in the first plenary, so I'll send special thanks to Adriano Dematte for his plenary. But I've come across things reminding me of this every day of the conference.

In addition, my plan of writing a book for teachers (in Norwegian) on different ways of including history of mathematics in teaching, remains on the to do-list. More resources for teachers are always needed.

So, as I leave for a few days of holiday in Paris, I know there will be work waiting for me back at work for years to come (and in particular when I'm back in my research and teaching position from 2016...)