HPM Day 3

The HPM conference has a peculiar design, in that each major theme are treated in order, each getting a little less than one day. This has the obvious drawback that if you are particularly interested in one or two themes, you will miss most of that, as there are many presentations on that theme at the same time. If you happen to have a talk on the theme as well, there is pretty little left for you to hear. On the other hand, when you come to themes you are not that interested in, you will have plenty of those to choose from...

Personally, I'm most interested in talks of how to include history of mathematics in teacher education for prospective primary or lower secondary school teachers. But if the talks touch upon use of history in mathematics in general, or on the history of some topic related to the primary or secondary school curriculum, I'm happy...

On the third day of the conference, we were approaching the more "purely" historical parts of the conference. The plenary talk was Dominique Tournè's "Mathematics of the 19th Century Engineers: Methods and Instruments". He started by talking about Lagrange's numerical methods for solving equations, and Fourier and Sturm's criticism that the methods were not easy to use in practice. Lalanne also pointed out that many of the methods were not practicable. In this situation, engineers created their own methods for finding solutions quickly. They did not need a high accuracy, but needed speed, and it was important that they could be done in the field and not only in the office.

He went on to consider the "cut and fill" problem (where the volume of mass you cut out for a road should equal the volume of mass you fill in at another place). "Hair planimeters" and other instruments were developed to find the areas (on drawings) in practice. A new mathematical discipline, nomography, were developed. Another was "graphic statics", in which metal structures were constructed with drawings on paper instead of by calculations. Example: the Garabit Viaduct and the Eiffel Tower. More than 1700 drawings were made for the "backbone" of the Eiffel tower, with an additional 3600 drawings for the execution. 

Ballistics was another area in which the mathematician's solutions were too cumbersome. Firing tables were needed. There were also designed curves which would give the needed information directly.

The French model of École Plytechnique gave a mathematical, theoretical grounding followed by "practice" in engineering. This contributed to a marhematization of the engineering art.

The rest of the talk was devoted to the example of nomography. Lalanne, Massau, Lallemand, d'Ocagne and Soreau were the main characters in the development of this field. The solution of equations were in this way reduced to reading graphs. The graphs were eventually reduced to three lines next to each other, where you could find the value of the third variable by identifying the values of the first two and drawing a line between them and on to the third line. In this way, the problem of "cut and fill", for instance, was very much simplified.

It is quite obvious that in studying the history of mathematics, the development of pure mathematics has been prioritized, while the mathematics of the engineers have not been given so much attention. Tourné points out that this mathematics could be even more fruitful for the HPM community. This is a point I understand - and fits well into the overall pattern that primary school teachers (and their teacher trainers) are mostly purely "academic", not knowing anything about most of the occupations the children will take up, be it bakers, carpenters or fishermen.

"Why Do We Require a “History of Mathematics” Course for Mathematics Teacher Candidates? (And What Might Such a Course Look Like?)" This was the theme of the first panel discussion, with Mustafa Alpaslan, Sang Sook Choi-Koh, Kathleen Clark,  Ewa Lakoma and Frédéric Métin. 

Frédéric described how in France you have primary school from age 3 to 11 (elementary 6 to 11), secondary from 11 to 18 (college 11-15, lycee 15-18). To become a primary school teacher, you have three years of general studies (or maths, to become maths teacher at higher levels) + one year theory + one year practice (exams in each year). Then you become a civil servant. 

In the University of Burgundy, there is no history until they have an optional course in the first year of their master, "A short course on several mathematical ideas", or, for primary teachers, an optional course on "(Re)discovering Maths".

The main justification for these courses are to show students that they can do something else than traditional teaching and show a more cultural approach. It is also important to make the familiar unfamiliar.

Mustafa is teaching students who should teach students aged 12-14. They have pure maths ++ for the first two years, then two years of pedagogy etc. From 2007, there are courses on history of science, history of mathematics and philosophy of mathematics. These have centrally decided guidelines. (see his talk at ICME). 

Sang Sook: in Korea, elementary school from age 7, then middle school from 13, high school from  16, university from 19. 4 years of teacher education or 4 year maths program + 20 credits in the education department. There is a highly competitive teacher examination, which leads into the public school system. 

Their curriculum does not have explicit HM contents. Sang Sook argued based on the genetic principle that HM should be included in the teaching, and used this when teaching the concept of function. In textbooks in Korea, HM is used as motivational tool, but far less usual is HM used to teach concepts. Teachers also report that they don't know how to use HM in this way.

Ewa described the situation in Poland: the educational system is 6+3+3. You can become a teacher for grade 1-3 by taking courses in "pedagogy". You can become a mathematics teacher by taking mathematics courses and then some pedagogy. 

HM is not included in the key competencies, but implicit in curriculum proposals, explicit in school textbooks and other didactical materials. Significantly, there is no HM in the exams. At university, there is a course on HM, but this is not for teachers especially. 

Kathy noted that there may be important differences between HM courses given by maths departments and courses given by education departments. Personally, I'll say that a key is whether it is a course /for teacher students/ or just a general course. 

Then there was a discussion which I can certainly not summarize here. However, a lack of resources were mentioned by some people - we always tend to end up there. Mannfred mention how it is important that history gives a new way of approaching mathematics, for students who have spent many years working on mathematics. Michael Fried mentioned how it is also another way of thinking - and the students have little training in historical thinking. This ended the third day of the conference - the rest of the day was a nice excursion. The main point of an excursion - apart from providing some fresh air and a pause from lectures - is the opportunity to have long conversations with your colleagues, and thus get to know them in another way than in the confence halls. 

After this trip, we were ready for the last two days of the conference...