Hong Sung Sa discussed "Theory of equations in the history of Shosun mathematics". He compared Eastern mathematics to Western mathematics, and noted that in the East, rational numbers, not real numbers, was the basic field of equations, and they did not work on factorization, like they did in the West. Solutions were rational approximations, not solutions with irrational numbers. This is interesting, as the solution formulas were of course important topics in Western mathematics, and played a part in the development of algebra. For details of the history on these methods, however, I have to refer you to the proceedings.
Yoichi Hirano gave a "Remark on the Notion of Golden Ratio - Concerning "Divine Proportion" in the Renaissance". He claimed that the topic of "golden ratio" is often understood only fragmentary by teachers. He talked on the history of the golden ratio, from before Euclid, then through Fibonacci and Leonardo, throuch Descartes, Durer and Simpson, Ohm, Binet, Cook and Thompson. Golden ratio is seen both in mathematics, in human culture and in nature, and Hirano went through many of the well-known examples of these. Of course, Leonardo's Vitruvian man was included. Leonardo's (remarkably good) friend Luca Pacioli gave the name "divina proportione" for this ratio, published in 1509. Leonardo da Vinci provided the drawings of the figures in this book. The name is believed to have connection with the platonic solids.
Hirano suggested that Pacioli may not self have found this name, and that Leonardo was the real author. As a reason for this, he mentions that an earlier book by Pacioli seems to be a plagiarization of an earlier work. This argumentation was not altogether convincing, in my opinion, and we should certainly guard against attributing to Leonardo more than we can prove.
Leo Corry's talk on "Euclid's Proposition II.5: A View through the Centuries-Geometry, Algebra and Teaching" was next. He presented the proposition II.5,which has been interpreted algebraically as the conjugate sentence. (The square of (a-b) is the square of a plus the square of b, minus 2ab.) Tannery, Zeuthen, Heiberg and Heath were among the people interpreting it in that way. In his talk, he went through how different editions/translations of Euclid throughout the centuries were formulated in this regard.
The final talk of this session was Qing-jian Wang's "The New "Curriculum Standard" and the New Mathematics - the Union of History of Mathematics and Mathematics Education". The new curriculums in China means that it will be necessary to teach teachers how to include history of mathematics into mathematics teaching. In Taiwan, there has been a HPM Newsletter for a long time (under the leadership of Wann-Sheng Horng), but in neighbouring China, a 2002 conference was the first one with papers combining the aspects of history and pedagogy. Wang described the history of the HPM activities in China since that time.
After these conferences, I would like to summarize. First, what are the main ideas I take with me from these conferences? Well, I think the discussions, both in the conference halls and over a beer, about theoretical frameworks for discussing the goals and/or outcomes on teaching with history of mathematics were most important to me. At the end of the conferences, I am motivated to work more on this.
At ICME, I heard lots of talks that broadened my overall knowledge of mathematics education. Alan Schoenfeld's talk has stuck, mostly as a reminder of the roles our beliefs, knowledge and goals play when we design our teaching, for instance with history of mathematics.
At both ICME and HPM, I've learned about other efforts of including HM in teacher education around the world, which I should keep up to date on.
Secondly, what are the best moments of these more than two weeks in Seoul?
1. Crossing the border to North Korea was a special and strange moment. To step from one of the most successful democracies into one of the worlds' worst tyrannies, that to this day operates concentration camps, was thought-provoking.
2. Having the first sip of beer with Andreas and Johan after the hot excursion day - and a long walk in the sun to find someone who sold beer - felt quite right.
3. Holding a talk at ICME with people sitting on the floor and standing in the doorway wss new and kind of cool - even though it was a small room...
4. Seeing some of my favorite HPM people engaging in my materials at my HPM workshop was also great.
5. Many of the dinners and/or beers with great people were wonderful. The contents of the conference may tempt me to come back, but it is the people that makes it unthinkable not to come back.
6. The view from the hotel room in Seoul was breathtaking - literaly.
So, there we are. I have sign on to a contract making me "Head of Studies" for the next four years, which does not leave much time for research and development on HPM. But I do hope to set aside some of my spare time from time to time to work on it. And anyway, I'll be back 100 % from 2016...