Day 3: The first item on the agenda is the lectures of awardees. The somewhat strange idea is that highly competent people have awarded five people/organizations for their lifelong, important contributions to the field, and they are then giving one-hour lectures in parallell. Needless to say, it is difficult to choose: should you go to learn more about someone you are not that familiar with, or should you go for the awardee that is closest to your own field (and who you therefore know).
I decided to go for Gert Schubring. I have heard him a number of times before (always impressive in the depth of his knowledge) and read many of his writings, of course. (As usual, I will only note down here some short points that caught my attention, and as always with the caveat that I may misunderstand. Always take what I write only as a pointer to be able to go to the source.) Schubring has of course been important to the field of history of mathematics education, both in his own research, in heading conferences, editing the journal on history of mathematics education and so on.
As Schubring received the Hans Freudenthal, it was fitting that he talked about his relation with Freudenthal throughout the years. Schubring himself started studying mathematics, and did not hear anything about the history of mathematics or its teaching. He defended his PhD in 1977, on the genetic principle ("Das genetische Prinsip in der Mathematik-Didaktik"). He praised the context at Bielefeld university, with interdisciplinary research and with theorists such as Niklas Luhmann. (It is interesting how such elements are (seen as) important to the path into history of mathematics education.) Thus, he was introduced to history of science, as a complex system, a sort of social history of science.
He discussed several of his early research projects, including the problems of obtaining and reading sources in those times. (Of course, the problems are still much the same, even though you may be lucky to find some sources online these days.) In 1979, he was invited to the first congress on social history of mathematics, meeting Henk Bos, Herbert Mehrtens and Ivo Schneider (among others). Gradually he developed two research strands: analysis of the development of mathematicsal concepts (the development of negative numbers as a major focus) and The history of the teaching and learning of mathematics. The second one is quite complex, as it has to do with for instance the education system, the labour market, the sciences, and so on. He did work on teacher education of mathematics teachers, on history of teaching mathematics and also the sciences in general.
He stressed the importance of comparative research. Much research on history of mathematics education (and history of education in general) is considering one nation state at a time, where one may easily take the national characteristics for granted instead of researching them. (I am paraphrasing quite heavily here.) Schubring worked on both Germany and Italy, and in his talk, he detailed some of the important differences in mathematical histories which influenced mathematics education heavily - and the functions of mathematics teaching. He also argued against the traditional practice of history of mathematics teaching as a history of the curriculum, syllabus, textbooks etc. Schubring stresses the analysis of texts combined with contextual analyses. An example is the analysis of changes within various editions of one textbook, finding corresponding changes in other textbooks, and connecting these changes to changes in the context (syllabus, debates, ...)
In the last part of his part, he talked about the development of the field into a broad international area of research. Elements to this development was the Topic Study Group (from 2004) with proceedings in Paedagogica Historica, the International Journal for the History of Mathematics Education (2006-16), a bi-annual series of conferences (proposed by Kristin Bjarnadottir - the next one in Mainz in 2022) and The Handbook on the History of Mathematics Education (2014) and two Springer series.
In the end, he discussed colonial/decolonial perspectives, which I find interesting.
After this, I went to the part of the ICME where affiliated organizations are presented. As usual, I went for HPM (International Study Group on the Relations between the History and Pedagogy of Mathematics) - a group I've been a part of for about 20 years now. This was a two-hour session with short presentations by leading researchers in the field, followed by a discussion session.
After a short welcome by Snezana Lawrence, Ysette Weiss gave a presentation of history of HPM, its relation to ICMI and some of its most recent activities. It was interesting to note that at the meeting in 1976, history's relation to the New Math reform was a topic of discussion.
Secondly, David Guillemette provided some theoretical perspectives of HPM. His starting point was two papers: Barbin, Guillemette, Tzanakis (2019) and Clark, Kjeldsen, Schorcht, Tzanakis and Wang (2016). Both call for more empirical research to study effectiveness, but also deeper understanding of theoretical issues. The field is in search of theoretical and conceptual frameworks. Fried et al (2016) points out that the nature of mathematics itself must be problematized, as must our view of history and the nature of matematics education.
Guillemette then went on to discuss five perspectives: the genetic perspective, the humanist perspective, the hermeneutic perspective, the discursive and pragmatic perspective; and the dialogical and ethical perspective. The humanist perspective he connected to Fried (2001, 2007), with mathematics contributing to students growing into "whole human beings". The hermeneutic perspective is connected to Jahnke (1994, 2014), the discursive and pragmatic to Sfard (2008) and Kjeldsen. Finally, the dialogical and ethical perspective he collected to Radford (2012, 2013, 2018). He also included his own work on "otherness", "empathy", "nonviolence" (Guillemette, 2018). He stressed the importance to situate ourself, both epistemologically and methodolically.
After a (rare) coffee break, Alexander Karp talked about the history of mathematics education. Obvioiusly, this overlapped somewhat with his introduction to the TSG55 and with Gert Schubring's talk this morning (see above). He mentioned two important surveys: Karp/Schubring: "Handbook on the History of Mathematics Education" and Karp/Furinghetti: "History of Mathematics Teaching and Learning". (The second one being freely available.) He stressed that history of mathematics education is a part of general social history, and that everything can be a source (not just textbooks...) He mentioned three examples of important questions in history of mathematics education: Why was commercial arithmetic so popular in 15-16 century? Why was not discrete mathematics represented in Soviet curriculum? Why was mathematics curriculum in Western Europe reorganized so strongly since 1960? (As examples of questions showing that history of mathematics education is part of social history.)
Lastly, Desiree van den Bogaart-Agterberg talked about history of mathematics in the classroom. She started by referring to Jankvist's division between history of mathematics as a tool or as a goal. She also referred to the seminal 2000 ICME Study, discussing different ways of including history of mathematics in classrooms. More practically, she referred to a forthcoming article by herself, where she identifies four formats: specks, stamps, snippets and stories - focusing on the size (but also the function) of the HM inclusion. (It is interesting with these different ways of analysing how HM is included in mathematics textbooks. However, it would be interesting as well to study how these different ways are actually used in the literature - which are the most fertile ones?) Bogaart-Agterberg then went on to discuss the use of original sources, which is also an important way of including HM. (Here, it is also tempting to mention my own article on different ways of including HM, which for instance also includes plays, which seem to be missing in some other "lists".) In the end, she mentioned the TRIUMPHS project, which has important resources, and also the importance of HM in mathematics education.
Thereafter, there was a discussion. Granted, the webinar format (in Zoom) is not very interactive. (As a participant, I don't even have an idea of the number of participants.) But there were some questions from the auditorium in Shanghai, which were discussed by the participants. Also, I added a question: "Could I challenge Desiree and David on the connections between their two talks: Are there perspectives discussed by David that are particularly useful in practical classroom implementation? Are there elements of what Desiree discussed that have particularly interesting theoretical connections?" David argued that we have to be careful in how we introduce history, because we can introduce history in a way that shows how mathematics is evolving, and that mathematicians in the past were also struggling and arguing. Desiree argued, in the chat, that the genetic principle and the humanistic principles are good places to start when thinking of including history of mathematics into teaching. (Sorry for suddenly using first names - it is a bad habit that happens sometimes when I mention people I know.)
(Personally, I'm not too happy with using the genetic principle, as it can be viewed quite narrowly. In this talk, I have to point out, Guillemette was quite explicit in defining it broadly. Still, it is connected to the simplistic idea that pupils' learning recapitulates humankinds' evolution of mathematics, and this idea is, to me, fundamentally problematic. Pupils are so different and there is no one "humankinds' evolution of mathematics" - the evolution of mathematics have been different in different cultures and different localities. A very broad concept of genetic principle - something like "there are often some resemblance between a particular students' learning of mathematics and the development of mathematical concept in some culture - becomes so general that I'm not sure it is helpful. I therefore prefer concepts such as "epistemological obstacle" (which Guillemette also mentioned).)
After lunch, there was the the second session of TSG55.
Antonio M. Oller-Marcén: "The beginning of modern mathematics in Spanish primary education: a look through textbooks and curriculum." Although Spain did not attend in Royaumont, New Math was fully implemented in the 1970 LGE (General Law of Education), after being introduced in the offical syllabus for ages 10-14 in 1967. But already in 1965, there were elements of modern mathematics in primary school. There has not been done much research on New Math in Spain, and Oller-Marcén's objective is to analyze the 1963 and 1965 sullaby and analyze two editions of textbooks. In 1965 syllabus, the idea of set is included in grade 1, the communitative and associative properties of addition and multiplication are explicitly included in grades 2 and 3. In the textbooks, one-to-one-correspondence was included, and numbers are defined in terms of sets. Moreover, sets and symbols from set theory are far more frequent than the syllabus would suggest. The set theories ideas were absent from the other parts of the textbooks, which is different from textbooks in the 1970s.
Dirk De Bock asked whether there was an explanation for why New Math arrived so early in textbooks in Spain - it is a bit strange given that the Spanish were not prominent in later meetings. We do not have an explanation for this currently. (This talk is actually a very good example that such careful analysis of textbooks is important, also in giving rise to further questions that can be analysed further.)
Johan Prytz asked whether textbooks had to be approved by the government, and Oller-Marcén clarified that they did. So these textbooks were approved.
María Santágueda-Villanueva and Bernardo Gómez-Alfonso: "Missing arithmetic methods: On the rules for the mixing of analogous things". The kind of problem looked at is "In what ratio must a grocer mix two varieties of tea costing Rs. 15 and Rs. 20 per kg respectively, so as to get a mixture worth Rs. 16.50 per kg?" They looked at different methods given for solving such problems in different textbooks.
Pilar Olivares-Carrillo and Dolores Carrillo-Gallego: "The calculation in the first commercialized Decroly's games". The games in question were made for special education students, but later published for both "anormal" and "normal" children. They included sensory games, calculation games and reading games. There were three purposes: motivate, learn, and to assess the learning. Olivares-Carrillo described the numerical games (impossible to summarize here).
Yoshihisa Tanaka, Eiji Sato and Nobuaki Tanaka: "Mathematical activities focusing on Japanese elementary arithmetic and secondary mathematics textbooks in the early 1940s". The activities were analysed using a method by Simada (1997); "A model of mathematical activity" (which seems like a model for modelling). He showed a number of problems and how they can be solved. The highlighted processes were to see similar cases and to generalize solutions.
Zhang Hong: "Development history and course setting of mathematics department in early universities in Sichuan province in modern times (1896-1937)". This was actually the first of the presentations done from the auditorium in Shanghai. After some initial feedback problems, the sound was clear, and Hong presented the history of the Sichuan University (and others) - which again is very difficult to summarize here. However, one main point was how mathematics education went from being inspired by Japanese ME to being inspired by European and US ME.
Li Wei Jun: "A probe into compiling mathematics textbooks by Christian missionaries in Late Qing Dynasty". The missionaries introduced Western ideas by compiling and translating textbooks. One example of such a missionary was Calvin Wilson Mateer, born 1836, who established schools in China and wrote several textbooks. Another example was John Fryer, who translated a lot of mathematics works with Chinese colleagues. A third example was Alexander Wylie. The translated textbooks introduced new methods, symbols and mathematical concepts to China.
Alexander Karp at the end stressed the importance of clarifying what is the original contribution, by describing what was already known and what the research has added. (While I of course agree to that in a general sense, I believe that in the format of ICME, with four-page papers and seven-minute talks, there is a lot that have to be left out, both in terms of theoretical assumptions, previous research, methodological considerations and implications for later research. These short talks can at best be an inspiration to look at the researchers' work in more detail when it is eventually published (or by getting in touch with the researcher).
The final part of Day 3 was discussion groups and workshops. I chose the Discussion Group 1: "Computational and Algorithmic Thinking, Programming and Coding in the School Mathematics Curriculum: Sharing Ideas and Implications for Practice". Again, I got tired of sitting at the keyboard, and changed format:
Some resources:
- DG1 website
- https://onlinelibrary.wiley.com/toc/14679639/2021/43/S1
- https://iase-web.org/islp/pcs/
- https://hal.archives-ouvertes.fr/hal-02423497/document
Bonus photo: me at ICME14:
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