Tsang-Yi Lin held the first plenary talk, on the subject "Using History of Mathematics in High School Classroom: Some Experiments in Taiwan". HPM Tongxun was started in 1998 by Wann-Sheng Horng, and has been become important in spreading HPM in Taiwan. Lin is a high school teacher in Taiwan, and his students are interested to find out how mathematicians managed to find the formulas they did.

He gave several examples. The first was how to connect conic sections to the modern definitions in the textbook, using Dandelin's theorem. The second was on Apollonius' work on conic sections, but I must admit I did not follow all the details of the examples. The third was on tables of logarithm (mentioning the book by Dava Sobel: Longitude). It Taiwan, students memorize 0.301 as an approximation of log2, but how could we find the approximation? He introduces Napier and Briggs, and used Briggs' method.

The fourth example was on Cramer's rule, comparing Maclaurin's and Cramer's explanations. In this example, Cramer's symbols and notations brought cognitive obstacles to the students, and as the students knew Cramer's rule in advance, they were not as interested in new ways of looking at it.

Lin pointed out that there is a need of more articles - for teachers - about the history of topics in high school math textbooks. We also need teaching projects with guidelines in detail.

I missed Uffe Jankvist's regular lecture at ICME (because I felt I had to hear Marja van den Heuvel-Panheuzen's), but luckily I got another chance to hear him here. His title here was "A Historical Teaching Module on "The Unreasonable Effectiveness of Mathematics" - the Case of Boolean Algebra and Shannon Circuits." The topic was a teaching module used in upper secondary school in Denmark. The main idea was to have one original source for each of the three dimensions history, applications and philosophy, have guided readings and then essay assignments. (Guided reading means to give short parts followed by work to be sure that students have understood.) The readings chosen were Boole 1854, Shannon 1938 and Hamming 1980.

The students gave different reasons to prefer different texts - more mathematical (Shannon), more open (Hamming), more easy to follow (Boole). Different texts speak to different students. There were students responding positively to each of the three dimensions. Referring to Barbin's Dépaysement, vicariante and culturel, Uffe argued that all three of these were included. (This reminds me, of course, that I should go back and reread Barbin.

It is great to see Uffe's high-quality work which combines the development of resources, the actual teaching and development of theory. So of course I will read his paper carefully when time permits.

Jerry Lodder's talk was titled "Historical Projects in Discrete Mathematics". This turned out to be the same talk that he had last week at ICME, so I won't summarize it here. However, this time I got the URL to the webpage he was referring to: URL

After a nice lunch in which the Nordic delegation stuck together (Kristin, Andreas, Uffe, Tinne, Johan and myself) it was time for another round of oral presentations. Shu Chun Guo presented "A Discussion on the Meaning of the Discovery of Mathematics in the Warriers and the Han Dynasty." The focus was on bamboo strips with mathematical contents, many found in 1983, but originating from 157 BC, apparently. The strips are still being researched, but they include multiplication tables, calculation with fractions and much else. The talk was a bit difficult to follow for me, as the slides were mostly in Chinese and my understanding of Chinese is currently woefully inadequate.

The second oral presentation of this session was by Mustafa Alpaslan, presenting a paper of himself, Mine Isiksal and Cigdem Haser on "Relationship Between Pre-service Mathematics Teachers’ Knowledge of History of Mathematics and Their Attitudes and Beliefs towards the Use of History of Mathematics in Mathematics Education." They did quantitative research to fill a gap in the literature, as big quantitative studies on attitudes to HM are few and far between. They asked 1593 pre-service mathematics teachers and used two instruments: a knowledge in history of mathematics test and an attitudes to history of mathematics in mathematics education questionnaire. Based on Pearson product-moment correlation analysis, there were found lots of statistically significant correlations, which supports the idea that beliefs/attitudes and knowledge has an interplay.

Of course, correlation does not say anything about what is the cause and what is the effect - we don't know if we must improve students' knowledge to improve their beliefs and attitudes or vice versa - or indeed if beliefs and attitudes are easy to change as late as in teacher education.

Kathleen M. Clark then presented her paper "The Influence of Solving Historical Problems on Mathematical Knowledge for Teaching". Clark took Ball et all (Mathematical Knowledge for Teaching" as a starting point. She has taught a course on "Using History in Teaching Mathematics" four times, with 20-25 students each time. As part of the course, they had a "historical problems portfolio", where they had to choose ten problems/tasks/activities that they had worked on, restated, solved, and described the course objectives addressed and provided a reflection. She used this both for investigating how the students understanding of mathematical concepts were informed and what their work reveal about their beliefs about mathematics.

19 portfolios were analyzed. Here, she looked at two of the problems chosen; Method of false position and Method of completing the square. On the method of false position, she found that students struggled with alternative solution methods, they found older methods harder, failed to connect the method with linear solution methods they knew, and were unable to evaluate their success.

On the method of completing the square, she found that the students was helped by connecting geometry and algebra, their "awareness" heightened (making them aware that HM can be a good resource) and gave them alternative conceptions.

But the students were persistent in their critique of the way the original problems were posed, and wanted to convert the historical algorithm into the modern ones. From the fall of 2012, she will be a new project to look at the ways a course in HM can influence the knowledge the students will need for teaching.

The last thing on the programme on Monday was workshops, and I chose Peter Ransom's, as it was the most relevant one for my students' target age group, and as Peter's workshops are always enjoyable. As workshops are quite interactive things, I cannot lean back and write notes, so I cannot summarize it "as it happens". However, I can safely say that it kindled an interest in both fortifications and proportional dividers...

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