The situation in Africa was described by Nouzha El Yacoubi. There are obviously big gender differences in Africa, for instance when it comes to literacy. More boys than girls go to school. With this general background, it goes without saying that boys are also doing better in mathematics. I'm a bit unsure, though, if mathematics has to be addressed specifically, or if the main thing is to get everybody to school first.

Next, Maria Trigueros talked about the situation in Mexico. There are gender differences (boys doing better) in 6th grade. Also, boys display better self-confidence from about that time. Technology helps improve attitudes among girls. The gender gap remains at the university level. There are promising policies, for instance revision of textbooks to avoid a bias in gender.

Helen Forgasz claimed that there is a downward trajectory in Australia. In TIMSS and PISA, there have been an increase in the gender differences, with statistically significant differences, boys doing better than girls. More boys than girls enter the "Mathematical Methods" course. The gender gap widens. In public perceptions, it is still believed that boys are better at mathematics than girls (a view that is of course also supported by TIMSS and PISA - you can hardly fault the public for being up to date on the latest international comparative research).

Lovisa Sumpter took a European perspective. She has done a literature survey, which confirmed the standard views of gender differences. Much of the research are published in general journal (not concerned with mathematics in particular).

Sarah Theule Lubienski (the sixth woman on this six-person panel) talked on the US. She looked at how gender gaps vary from item to item in tests. Moreover, the gap in confidence is bigger than the gap in achievement or interest. More women than men go to college, but more men study mathematics.

To me it does not seem equitable that one person talks about almost 50 countries, while others only talk about one small one. The unintended message is that Australia (or Mexico, or USA) is as important as the entire continent of Africa... However, the topic is an important one, which deserves all the attention it can get.

The last session of the TSG20 started with a talk by Jerry Lodder on "Primary historical sources in the classroom: Discrete mathematics". His example was a module on logic, in particular the truth table. He used sources related to Chrysippus, Boole and Frege. In particular, he showed how Frege tried to write Chrysippus' rules with Frege's notation. Lodder lets his students work on this, and then on Russel and Whitehead's work. The study showed statistically significant increase in student's attitude to the subject.

Secondly, Anne Michel-Pajus talked on "Historical algorithms in the classroom and in teacher-training". She has been giving in-service teacher training since the 1980s with an emphasis on historical sources, to enrich their culture in HM, deepen their understanding and help them build historically based activities. In the new curriculum in French high schools, students are to understand, describe, modify and explain algorithms. She looks at Heron's formula for the area of a rectangle with sides given. In the original source, this was not a formula, it was an algorithm.

Another example is Chinese, Indian and medieval algorithms for finding a square root. Aryabhata's is a very short description of an algorithm. Diophantus gave a general formula for polygonal numbers in rhetorical language.

In is way, Anne has illustrated a table of different ways of expressing a formula/algorithm. Then she went on to different levels of justification (checking a few examples, justifying procedure mathematically etc.) She shows Heron's explanaition of how to find the square root, using particular numbers to check the algorithm. Euler gives the same algorithm in symbolic language.

Al-Khwarizmi gives a geometrical proof for his algorithm for solving quadratic equation. Diaphantus gives an example of a formal reversal of an algorithm. Jordanus de Nemore: uses letters instead of numbers, but introduces a new letter for every step of the algorothm.

The relationship between formula and algorithm was thus very nicely illustrated, and I'll try to use some of these examples in my work in the future.

Finally, Uffe presented his, Reidar Mosvold, Janne Fauskanger and Arne Jakobsen's paper titled "Mathematical knowledge for teaching in relation to history in mathematics education". I'm happy to see that there are attempts to include HM in the framework of mathematical knowledge for teaching, as we were talking about in the conference in Vienna two years ago (see my part of the plenary lecture there). Uffe stressed that it is important for HPM to relate to the broader mathematics education research community.

Uffe gave a short introduction to "the egg", and then gave examples on negative numbers (based on Arcavi 1982) and on number systems (Heiede). He then made an attempt to connect the examples to the egg, which I cannot repeat here. But Uffe uses Ball's definition in which CCK (common content knowledge) is seen as only being about the 'pure' mathematics, so that in this case it would include knowing the definition of negative numbers and knowing that different number systems exist. I strongly disagree with this way of looking at mathematics and CCK, as it marginalizes history of mathematics into being not a part of what mathematics is, but only a tool for lesrning the mathematics. In this particular paper, Uffe et al are using the present version of "the egg" to analyse the outcome of HPM research. The goal is also to make HPM results more accessible and relevant to the mathematics educational community by connecting to one of the present fads. This is laudable, but we should also challenge this model by Ball et al to get a wider concept of mathematics, for instance to include parts of history of mathematics in CCK - that is, as a part of the mathematics curriculum that all children should be supposed to learn as part of a sound mathematics education. Not just as a means to learn mathematics, but as a goal inits own right.

After lunch, Adnan Baki had a regular lecture with the title "Integrating technologies into mathematics teaching: past, present and future". He traced his own story since he started working with technology in the early 80s, mentioning how he had trouble changing his teaching practice in a more constructivist direction, and began to see what Seymor Papert meant about computer's potential for changing teachers' role and classroom practice.

In the second part of his talk he gave several examples of working on geometry in LOGO and in GeoGebra. These are singularly unsuitable for summarizing here, being quite visual.

Finally, there was the second part of the discussion group on use of history of mathematics with children age 6-13. I thought this went quite well (many thanks to Kathy Clark). This is the first time in ICME that I have been formally in charge of anything more than a talk on my own research, and it pleases me a lot that it was not a dismal failure. An account on what went on in the group will be found in the next issue of the HPM Newsletter, I would think.

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