So I took the morning of the fifth day off, sleeping late, even though Thursday (day 4) was excursion day. In fact, I was perhaps even more tired after the excursion day than after any other day - it was a day of many thoughts as we crossed the line to North Korea, if only for a few minutes...

Thus, day 5 started, for me, with Marja van den Heuvel-Panhuizen's talk, named "Freudenthal's work continues". She has been a big influence to Norwegian mathematics education lately, and I keep seeing her name, so it was about time I heard her "live".

She has worked at the Freudenthal Institute for 25 years, and started by talking about the history of this important research institute. It has a history going back to 1971. In November 2010, it was decided that primary mathematics education was not a part of the core goals of the science faculty, so the FI was divided.

She talked on three projects connected to primary school: the didactical use of picture books, mathematical potential of students in special education, and textbook analyses.

First, the project on the didactical use of picture books in kindergartens. Freudenthal himself was sceptical to limited and isolated worksheets, and wanted to give children context-rich experiences in which the children could discover mathematical connections. Picturebooks can give rise to meaningful activities, offer cognitive hooks (Lovitt and Clarke) and give opportunities for practice. In a 2008 paper, van den Heuvel-Panhuizen showed that almost half of children's utterances when read a particular book, was mathematics-related. Among the other subprojects, they also found significant improvement in the children's Score on a PICO test (whatever that is).

Then she went on to the project on special education, with work on subtraction and combinatorics. The project was motivated by policies claiming that weak students should not discover strategies by themselves, but be told how to do things. Of course, to accept this would mean that weak students can never appreciate mathematics as a creative field of knowledge.

van den Heuvel-Panhuizen presented a model with 12 (3x4) different methods for subtracting, and studied whether special education students could, on their own, use indirect addition to solve problems suited to that method (without being taught it first). It turned out that the students could do that.

The third project was the textbook analysis project. Textbook analyses give us a first inside view of how mathematics is taught, and is therefore relevant to teacher educators. It focussed on content, learning facilitators and knowledge presupposed. Among the clear findings is that realistic textbooks have more didactical support - like the use of context, models, textual instruction etc.

Then there was the third session of the TSG20. First, Mustafa Alpaslan talked on “"History of mathematics” course for pre-service mathematics teachers: A case study", a paper written with Cigdem Haser. HM got mentioned in the curriculum of 2005. Textbooks had small snippets of historical information. For pre-service teachers, a course on HM was introduced by the government. In his masters thesis, Alpaslan showed an improved knowledge of HM in teacher students. In the present project, he and his colleague wanted to study such a course for students who will teach ages 12-14. The data collection included classroom observation for 5 weeks, examination of materials and semi-structured interviews.

The course was partly presentations by the teacher and then by the students, who had by then done small "projects". The course had nothing about the use of HM in maths education. Students found that doing mathematics "within its history" was enjoyable. But students wanted to know how to transfer this to the students.

The course was based on group work, but this was not effective - they did not receive sufficient help on where to look for reliable sources, for instance. Students were also busy studying for the general examination at the end of their teacher education. The study suggested changes in the courses, and his doctoral work now will be dedicated to try to create a better course. I will be very interested in hearing more about this as it progresses.

Xuhua Sun was supposed to be the third speaker, but as the second speaker did not turn up, she was up next. The topic was "The systematic model Lu of JiuZhangSuanShu and its educational implication in fractional computation". She pointed out the two different traditions, the proof-based (of for instance Euclid) and the "problem-based" (of many civilizations), where solutions are given without proof. She then went on to show how 3/4 : 3/8 can be done in two ways: "flip and multiply" and transforming to the same denominator. She then connected this to the idea of Lu in Chinese mathematics. In Chinese mathematics one used different methods of multiplying with the common denominator to solve the problems in fractions.

Then there was the HPM group meeting. Evelyne Barbin talked first, on "Reading of original texts and 'dépaysement': from the teachers to the classroom". She referred to her article in For the learning of mathematics, 11, 1991 for the aims of the work on original sources. A key word for Barbin is 'dépaysement', or reorientation. By meeting an unusual take on something familiar, we will reconsider it. Barbin used the method of tangents by Roberval as her example. His idea of a tangent was the line which a point would follow if it detatched from its curve (which it was moving along). This idea was used to establish the tangent to the parabola and to the cycloid.

Barbin showed two ways of using this in the classroom. The first is based on Frederic Vivien, who chose to translate it into the language of vectors. Then the pupils are asked to translate Robertval's reasoning in some examples into vectors. Barbin seemed very sceptical of this approach - it's not clear what you will learn about Roberval from this, as his context is disregarded, and what you learn about vectors, you could probably learn as well in another way. (this is my interpretation, not Evelyne's words, of course). The second is based on André Stoll, who used it with students who already knew how to find vectors. The ideas of Roberval can then be used to introduce the differensiation of vectors. He also uses this method to work on the cycloid.

Thus, an original text can be used both to read something known or to understand something else. But it is not necessary to read the text of Roberval to learn about vectors - for that, the examples could be used without the original source. To get good use of the original sources, it is interesting to discuss the context of the sources.

Then Fulvia Furinghetti talked on the history of mathematics in teacher training. She stressed that in teacher education, it is important to challenge teacher's existing beliefs. The aim is to make teachers reflective practitioners.

She agreed with Evelyne about the importance of replacement and reorientation, making the familiar unfamiliar, building cultural understanding. Finally, working on history of mathematics contributes to a construction of meaning of mathematical objects.

Fulvia then went on to describe a course on history of mathematics, consisting of theory and laboratory. The course is parallel with a course on mathematics education. It aims to create a community of practice, and is based on work on original sources: Fermat, Roberval, Barrow.

For me, it is interesting to see different ways of designing history of mathematics courses presented at ICME. Many of the courses presented would be utterly inappropriate in Norway, because of Norwegian students' lack of knowledge in mathematics. But there are also other differences between the courses than what can be ascribed to knowledge of the different knowledge levels of the students, of course. It would be very interesting to try to analyse different courses in relation to the designers' knowledge/resources, beliefs and goals (referring to Schoenfeld's talk on Monday). For instance, I'm concerned with showing students a variety of ways of including history of mathematics, while there are other courses that does not include that at all.

Then there was the new chair of the HPM, Luis Radford. It was a rather informal talk, describing his path into the HPM group. At the end, he talked about how to use epistemology (based on Artigue) as a prism to critically observe and understand the objects of knowledge in the curriculum, and as a means to understand the development of the objects of knowledge. Radford is concerned with epistemological obstacles (both within mathematics and as related to the social, cultural and historical circumstances). His talk will hopefully appear somewhere.

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