ICME14 Day 6

The sixth day started with the final session of TSG12 (on statistics education). The first talk of the day was Saleha Naghmi Habibullah, with the title "Implementation of a course on Tidyverse in Pakistan under the ASA Educational Ambassarod Program". She discussed a course in the in Tidyverse - preparation, implementation and experiences. The workshop was based on the UN Sustainable Development Goals, using Tidyverse to study MICS data, and was done in a competitive manner, with teams presenting their data analyses in the end.

Michal Dvir and Dani Ben-Zvi: "Young learners' reasoning with informal statistical models and modeling". They define an Informal Statistical Model (ISM) as a "purposeful, not necessarily mathematical, representation of the process by which the observed variability was generated and includes both deterministic and stochastic components". Dvir went on to define "The Integrating Modeling Approach" (which I cannot summarize here, but had to do with switching between the real world and the probabilistic model). When student generated several samples, it spurred them to compare between multiple samples and make predictions. (Very interesting - I need to read their article(s) to get more detail than I did from a 20-minute talk. Here's one: Dvir, M., & Ben-Zvi, D. (2021). Informal statistical models and modeling. Mathematical Thinking and Learning. https://doi.org/10.1080/10986065.2021.1925842)

Von Bing Yap: "The binomial model: coin tosses or clay pots?" Two different models for binomial distribution: making n coin tosses (the number of heads has a binomial distribution) and removal of clay pots (which are high-grade and low-grade) from the kiln (not binomial, as there is lots of dependence). Many real populations are more like clay pots than coin tosses. We need to introduce simple random sampling. He gave examples of exam tasks where the binomial model is not correct, for instance when it is stated that "80 % of people..." and it is then assumed that the probability for each person is 0.8. Usually, people have different probabilities, he argued.

Orlando González: "Variability modeling and data-driven decision-making using socially open-ended problems: a comparative study of high school students in Thailand, Brunei and Zambia". The starting point of the talk was a "Darts game" - students got the dart boards of two teams competing, and were asked who were the winners. Students had many (six) different ways of scoring the dart games, implicitly creating their own rules. Most created a scoring function based on all shots, while there were also some who considered only some of the shots. (It seems like an interesting task, but it is not clear to me which important part of statistics knowledge in particular is worked on. Rather, I imagine using it to have a discussion for more general purposes.)

(Mara Magdalena Gea, Jocelyn D. Pallauta, Pedro Arteaga, Carmen Batanero: "Algebraization levels of statistical tables in secondary textbooks". Here, I was distracted by some other work for a moment, and can not reasonably give an idea of the contents.)

Stine Gerster Johansen: "Data modelling with young learners as experiences of allgemeinbildung". Johansen first discussed the concept allgemeinbildung and Lehrer&English's model for datamodelling with young learners. Modelling can contribute to bildung, but is demanding. Here, she discussed work with a Danish 3rd grade class (ages 9-10), 4 sessions, 90 minutes each. She gave examples of how children were discussing parents' quarreling, and the potential for bildung, including discussing what the use of this statistics work could be for other children.

The next item on the agenda was invited lectures, and I went for Reidar Mosvold: "Trends, Emphases, and Potential Shifts in Research on Discussion in Mathematics Teaching". His talk was based on a review of the literature. He discussed the concept of discussion, stressing that it needs a subject ("a question of common concern") and that it concerns reaching a decision or exchange ideas (based on the OED definition and on Dillon (1994)), in contrast to idle talk. He also discussed the concept "teaching", leaning, among others, on "work of teaching" (Ball & Forzani, 2009), where core components can be described as problems (Lambert, 2001), predicaments (Cohen, 2011) and tasks (Ball, Thames, & Phelps, 2008). He discussed the problem of having systematic searches for "discussion" (which after all is a quite usual word), and the rest of the review issues. He ended up with a total of 72 studies; empirical articles on discussion and mathematics.

Some of the results he mentioned in his talk was that the studies varied in focus, surprisingly few define what they mean by discussion, and a majority were small scale studies. The problems of studies for instance were about teachers' or students' actions, experiences, learning, demands. Key issues were orchestration (more than half), talk moves, norms, demands... For instance, only two or three articles had any significant discussion of norms. (This is surprising to me, as norms seem to be so central these days. I have more than three articles on norms on the reading list for a course I'm teaching, but obviously they do not all fill the criteria of Mosvold's search.) He also found that there was a surprising variety of literature refernced. Of course, Lampert (1990), Yackel/Cobb, Ball and Stein/Grover/Henningsen, were referenced by many (but only 5-10 references to each, which is surprising). Also, surprisingly few referenced studies on discussion outside of mathematics education. 

One of his takeaways from the review, was that there should be more investigations on what might be involved in establishing a classroom climate for discussion. (Personally, I use Makar/Bakker/Ben-Zvi (2015) with my students. It's title is "Scaffolding norms of argumentation‑based inquiry in a primary
mathematics classroom" and would not be included in Mosvold's survey, as the word argumentation is used, instead of discussion. However, it is an overlap between norms of argumentation and norms of discussion, I would think.) In the discussion, he was asked about other terms such as "discourse", and he argued that it was necessary to limit the number of articles and at the same time "discussion" is an often used word. I asked my question ("I believe that the literature on argumentation, norms of argumentation and so on, would partly also be about discussions without actually using that word. Any comments on that?"), but Mosvold actually already answered that before the question was posted (one of the problems of sending questions in the chat without the possibility of withdrawing it...)

The final part of the sixth day was the final part of the TSG55 (The history of the teaching and the learning of mathematics). 

Sian E. Zelbo: "Building an American mathematical community from the ground up: Artemas Martin and the Mathematical Visitor". This talk is on Artemas Martin (1835-1918) from the US and his journal "Mathematical Visitor", which also included contributions from leading mathematicians of the time. Zelbo stressed that the aims were different than the Journal of Mathematics, to which it has been described as simply a precursor. The aim was to reach young people, and also teachers and administrators contributed.

Elisabete Zardo Búrigo: "The discarding of the rule of three in the 1960s: changes in elementary education in France and Brazil". The rule of three was discarded in the elementary school in the late 1960s, but was later reinstated. The rule of three was taught both for use in life and for use on entrance exams to post-elementary schools. However, as compulsory schooling was expanded, entrance exams were abolished and "New Math" targeted the rule of three as "too mechanical", rule of three was removed about 1970, in both countries. In France, a functional approach replaced the rule of three, while in Brazil, the study of proportions were delayed until later grades. The rule of three was reintroduced in France in 1985 (and was still there in many places from 1995), while in Brazil, it was mentioned in the 1997-8 plans, but from 2017 it was explicitly discouraged.

Yana Shvartsberg: "Mathematics education for young women during progressive era: historical overview". Her period was 1890-1920 and her geographical focus was the US. High school curricula were differentiated, according to the kind of industry/position the students were planning to go to. Mathematics became an elective topic for many students. Shvartsberg's research is looking at how this influenced women. There was a belief that women did not need the same education as boys, but at the same time, and many high schools for girls did not offer the same courses as high schools for boys, and more boys than girls selected mathematics courses. However, some educators argued that the same opportunities should be available for women.

Alexei Volkov and Viktor Freiman: "David Eugene Smith (1860-1944) and his work on mathematics education". Here's yet another talk on the US, but Smith also had international consequences, of course, being one of the founders of ICMI. This talk focused on his early didactical works. Of course, I will not try to summarize his life and works. In the talk, the connections and reactions to different European theorists (for instance Busse, Pestalozzi, Gruber, Tanck, Knilling...), were pointed out.

Alexander Karp: "College entrance exams in mathematics in Russia before the second world war: development, role, objectives". He stressed the importance of exams, also in terms of dictating an unofficial curriculum and thereby influencing how the subject is taught in schools. He gave rich examples of how the entrance exams were regarded through contemporary sources. Critics argued that the entrance exams should just include mathematics that is actually included in the secondary curriculum and not "extremely artificial techniques". In the discussion, it was mentioned that this situation, of entrance exams taking priority over the actual curriculum of lower school levels.

That concludes the 6th day of the ICME. One day to go.

ICME14 Day 5

It is day 5 of ICME14. The weather here in Oslo is incredible, with temperatures close to 30C in the afternoons, so it feels almost like being in Shanghai. On the other hand, sitting alone in my office at home has a different feel to it than attending a conference in person...

The day started off with invited lectures. This is perhaps the most difficult part of attending an ICME - outstanding scholars from around the world are holding talks simultaneously, and all you have to go by are the titles (and perhaps abstracts), and your prior knowledge of their work. But even people who write wonderfully can at times have terrible presentations (I have experienced people who spend most of their talk discussing with themselves that they have prepared far too much and that there is so much of interest that they will have to skip. Good idea: focus on claiming that what you are discussing (and have time for) is interesting...) So this year I went for an outstanding scholar which I also know give good presentations: Tinne Hoff Kjeldsen. Her title was "What Can History Do for the Teaching of Mathematical Modelling in Scientific
Contexts: Why and How?"

Kjeldsen has previously worked with how history can be used to reveal metadiscursive rules andd make them explicit objects of reflection and to provoke commognitive conflicts, and to provide a window into mathematics in the making. In this talk, she focused on how history can help in teaching modelling. Mathematical modelling has much to offer to other disciplines, on the other hand, different disciplines have different ideas of what models should be. She gave three examples: John von Neumann's model in economics, Vito Volterra's predatory-prey model and Nicolas Rashevsky's model on cell division. (I am not able to summarize what she said about these...) In the example of von Neumann, we see that the mathematician's purpose can be different from an economist's purpose, which may be to solve concrete problems in practice. In the example of Volterra, D'Ancona claimed that the model could lead to new insights even when the model could not be confirmed by data. In the example of Rashevsky, biologists questioned Rashevsky's assumptions. He investigated possible explanations, while biologists wanted the explanations based in empirical data. (The three cases are analysed further in Jessen & Kjeldsen, forthcoming.) 

Kjeldsen referred to Axel Gelfert (2018) on explorative modelling - providing "potential explanations of general patterns". He mentions three functions of explorative models: aiming at a starting point, proofs of principles and potential explanations. Kjeldsen shows how these functions fit with the examples she has given. Kjeldsen pointed out that there are many elements in modelling that are not explicitly captured in "modelling cycle" models for modelling (ex. Blomhøj and Jensen, 2006), such as the purpose of the modelling, the function of the model and so on. She showed that Bouman (2005)'s account of modelling could be a good supplement, although there are still issues not explicitly captured. 

At the end, she discussed an example of use of the example of Raschevky with students, and how looking at the contemporary discussion of his model, improved students' own modelling competence. Moreover, history of mathematics can provide a window into mathematical modelling "in the making", and let them reflect on how scientists get ideas, which strategies they use, which choices they make and how they argue and learn, and discussions about what counts as valid arguments and what mathematical modelling can provide in scientific contexts. She also discussed the concept of "historical awareness" and how the Raschevsky project helped students develop this.

I really like Kjeldsen's approach, giving in-depth discussions on how history of mathematics can provide specific awareness and knowledge, which is valuable contributions to the literature on history of mathematics in mathematics education.

(One thought after listening to Kjeldsen, is that teaching of modelling is necessarily limited by the teachers' knowledge of pertinent subjects where mathematical modelling is used. This also reminds me of Trude Sundtjønn's work, where attempts to make mathematics relevant for students in vocational education, met hurdles connected to the teachers' (lack of) knowledge of the vocations in question. Too often, examples of modelling in textbooks are simplistic, perhaps because authors try not to presuppose knowledge that teachers and students do not have. A good thing with history is that both the mathematics and the science involved may be simpler than using real-world examples from today.)

The next item on the agenda was a plenary lecture: Mercy Kazima: "Mathematical Work of Teaching in Multilingual Context". Kazima based the lecture on what we know about teaching and learning in a language different from the home language, and on the work on "mathematical work of teaching" (Ball, Thames & Phelps, 2008). She pointed out that although Ball et al is framing their theory in a general way, it is important to investigate in different contexts - for instance they may not fully cover the issues in multilingual classrooms. (As a matter of fact, it is rather usual that Western researchers formulate general theories based on local empirical data.) She also referred to Sorto et al (2018), including the knowledge of obstacles encountered by ELLs, knowledge of resources that ELLs draw upon in learning mathematics, and knowledge of instructional strategies that help ELLs in mathematics. (ELL means "English Language Learners", thus is based in a context where students have one home language while school is conducted in English. Hopefully, these are also relevant to learners that have no interest in learning English in their context...)

She gave some context: teaching in grades 1-4 are in Chichewa or other local languages (with textbooks in Chichewa), while in grades 5-7, teaching is in English. Teachers generally know at least the two languages Chichewa and English. This differentiates Malawi from countries in which teachers only know the language of instruction and not the students' home languages.

She gave some lessons from Malawian studies: 

• She pointed out that students' meanings for mathematical terms are oftten different from the mathematical meanings, and that these are influenced by home languages.
• Code switching can be used effectively to make mathematics accessible to students.
• Bilingual approach where use of home language is planned and proactive can be effective in making mathematics accessible to learners.
• Teacher education does not prepare teachers for the teaching in multilingual contexts.

The first of these means that the teachers should know the corresponding words in home language, different meanings of these words and how home language can be used to improve understanding of the mathematical terms. Based on similar discussions of the other three lessons, she discussed four types of mathematical work of teaching related to teaching in a multilingual context: 1) identifying resources in home language; 2) identifying obstacles in home language; 3) identifying obstacles in English; and 4) identifying strategies. She then gave examples from Malawi, which I do not try to summarize here. Just one example, though: the equal sign is translated as zikhala, which literally means "will become", which is of course an unfortunate understanding of the equal sign. Teachers can choose to instead explain using the word chimodzimodzi, meaning "the same as". Such vocabulary work is part of the mathematical work of teaching.

After lunch, there was the second (of three) plenary panels: "Mathematics Education Reform Post 2020: Conversations towards Building Back Better". As it does not make sense to sit looking at a computer for days on end, I decided to skip this one. 

The final item on the day's agenda was Topic Study Groups. I returned to TSG12 (on statistics):

Gail Burrill: "Margin or error: connecting chance to plausible". Burrill talked about ways of teaching margins of errors and confidence intervals. (My notes below will probably mostly be useful for myself as a way to remember some of her points... ) Both teachers and researchers have problems interpreting margin of errors and confidence intervals. To see the mean as a balance point helps students look at deviation from the mean. Using simulations helps students discuss what is the probablilty of getting particular outcomes when drawing a sample of a certain sizes. They learn the difference between sample size and numbers of samples. But what happens when we are supposed to say something about the population using our sample? 

Task: draw a sample of 30 M&Ms, to estimate the true proportion of blue M&Ms in the bag. Handing out M&M bags with different, but known, proportions of blue M&Ms. After a while, students go on to simulating. Then we can ask them: who have bags where 8 blue M&M would be plausible? A range of numbers are given, and this gives a starting point for what a margin of sampling error might be. Back to M&Ms: all bags now have 40 % blue M&Ms, they set a margin of error, and it turns out that (often) at least one of the groups do not have 40 % within their margin of error. So students learn that the margin of error is not giving an absolute bound.

I didn't get manage to note all the rich ideas, but I noted the use of StatKey as useful software to sample distributions. In addition she used TI Inspire (from Texas Instruments) and applets from Building Concepts Statistics and Probability.  (I did get a little lost in thought, as I tried to think how I can use some of this in teaching quantitative methods to my 400-500 students this fall.)

Cindy Alejandra Martínez-Castro, Lucía Zapata-Cardona & Gloria Lynn Jones: "Critical citizenship in statistics teacher education". Zapata-Cardona discussed the concepts of "critical citizenship" and "statistical investigations". She argued that investigations of crises in society would contribute to critical citizenship. (It is interesting to see the difference of the first two presentations today - the first one very detailed both on how the teaching was done and on the resulting mathematical understandings of students, the other one being less detailed on the statistical content, and more occupied with the general critical citizenship potentially promoted by such work.) In the discussion, a resource for working on critical citizenship was shared: http://iase-web.org/islp/pcs/.

Adam Molnar and Shiteng Yang: "Mathematics ability and other factors associated with success in introductory statistics". This talk was about a diagnostic test to study factors associated with success in a course in university. A methodological point of interest is that the students who answered the test, tended to do better than those who did not (that is; people who don't like mathematics, tend not to like to do a test either). The main finding was that "College GPA" (which seems to be the grade point average from college in the US) is highly corellated to the results in the introductory statistics course, and that adding the diagnostic test results, didn't really improve the model by a lot.

(By the way, a feature of online conferences, is that the discussions about one talk continues in the chat after the time for live discussion is up and during the following talks. I'm not sure if this should really be seen as a feature or as a bug - it does of course tend to take the attention away from the following paper, while adding flexibility.)

Karoline Smucker and Azita Manouchehri: "Elementary students' responses to quantitative data". Her research was on five third grade students (8-9 years of age), and she wanted to look at students prior to explicit instruction. The activity was collecting "wingspans" from fellow students. During graph creation, they were focused on creating the "right" graph and to follow rules. They had trouble graphing the quantitative data, but eventually, they decided to create groups, which meant that they made something close to histograms. There were several interesting findings connected to these "histograms", one of which was that some students were careful to include all the original data in the diagram. In their "analysis", the shapes of the diagrams, the center and the variability (what other third grade classes would look like) were included. Thus, we see that third grade students can get quite far in working on quantitative data without explicit instruction. 

In the discussion it was pointed out that these third grade students did better than teacher students do in research: http://dx.doi.org/10.1080/00207390902759584. It is highly interesting why this is. Maybe, it was pointed out, there was one student who had a good idea that the class ran with. On the other hand, as Dani Ben-Zvi argued, teacher students have gone through years of schooling where they learn that bar charts is the way to present data. Also, it was discussed what would have happened if the students had access to TinkerPlots instead of paper and pencils.

ICME14 Day 4

On the fourth day of the conference, apart from Thematic Afternoon in the morning (Norwegian time), and China Art and Culture Performance in the afternoon (which I will not blog about), the only ingredient was a plenary lecture by Robyn Jorgensen titled "Equity in Mathematics: What Does It mean? What Might It Look like?" (As I am currently doing some work myself on diversity in mathematics education, including race, gender, sexuality, class, functionality, culture, geography and ethnicity, I'm especially interested in this topic.)

She started the talk with addressing the myth of ability. We know that people's lack of success is often not connected to innate qualities. There is a difference between equality and equity - we do not want to treat everybody equally; people need to be treated differently. During the thirty years she has been working on equity (and even before that), there has been many concepts that have been in play: ability, giftedness, talent, ability grouping, access, sucsess, achievement and participation, recognition of difference, ethnomathematics (D'Ambrosio, Gerdes...), socially critical theories, habitus, Bernstein, power, hegemony, ideology, groups (race, gender, class, language...), "post" theories, identity, race theory, linguistic diversity, culturally responsive pedagogy... She stressed the importance of the organization and conferences "Mathematics Education and Society" (which I will attend for the first time this year - it will be interesting).

She claimed that the field has done a lot of work, but has not made a lot of impact in the 50 years. Therefore she wanted to point to problematic practices. For instance, the programme Direct Instruction in Remote Indigenous contexts has had questionable impact, and seems not to be culturally responsive. We need to approach a strength-based pedagogy, instead of identifying gaps and trying to fill them with scripted instruction. Different people have different world-views, and these need to be taken into account. 

She then went on to talk about a project called "Remote Numeracy Project", a project researching successful learning (instead of "what do teachers/leaders... do wrong?"). Approximately 40 schools were included, all over Australia. (This seems like a very good approach, different to most research, as seen in Smestad/Gillespie 2019...) Some schools that were doing well in mathematics, "protested" that they did not focus on mathematics at all - they focussed on well-being, safety, getting the students to come to school and to have healthy food... The schools seemed to have ways of enabling their teachers, who were often novice teachers. The researchers therefore looked at the envisioned practices, the enabled practices and the enacted practices. (Which is an interesting variant on Goodlad.) She discussed each of these in some detail, which I'll not try to summarize here. She stressed the importance of "embedding mathematics" - both in the contexts and cultures and in the brain. Also, mathematics is as much about language as it is about mathematical concepts, and for instance using the home language in mathematics was important. Also, being explicit about what was expected and how lessons were structured and so on. 

Although Jorgensen gave a complex picture that I have in no way been able to convey here, much of it was what is today generally considered good mathematics teaching, but with some extra elements concerning the vision and leadership of the school and the extra focus on the home language and culture. But there is surely more detail to be found in the articles, reports and case studies produced in the project.

ICME14 Day 3

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: 

ICME14 Day 2 Part 2

After lunch on the 2nd day, I attended TSG12, the topic study group on teaching and learning statistics. This is not because I am doing research on it (I'm not), but rather because it would be nice to learn more about this, and some of the talks seemed interesting. 

The first talk was by Lonneke Boels, presenting "Designing embodied tasks in statistics education for grade 10-12". Her research question was "How can literature—on misinterpreting histograms and on embodiment—in combination with results from an eye-tracking study inform embodied task design in statistics education?" Many students confuse bar graphs and histograms, and many think one of the axes should be time. Boels discussed an experiment with eye tracking, asking students to find the means from different diagrams. Eye tracking made it very obvious that the students were interpreting the histograms as if they were bar charts, and also that the students carefully read the text and the labels on the axes). The results were depressing. Thereafter, they did some experiments where students had to make boxplots themselves by dragging each data point to its correct position. They also had tasks where students had to find the balancing point (which is the mean) of the boxplots. (In a way, it doesn't make sense to summarize in this way, because the devil is of course in the details in such tasks. For instance, the software gave feedback when the balancing point was set correctly. Such details will influence the learning in some way. However, the point of my blog is obviously not to replace the article but to give a small taste which may lead you to look at the article...) Working with students merging and splitting classes seems to be a useful strategy.

(Being a novice in this area, I notice words such as CODAP, iNZigt, Tinkerplots and VUStat, only some of which I understand. I should investigate...)

In the discussion, it was asked what an "embodied task" is - is that dependent on the task or on your perspective when looking at the task? That is a good question, and I would perhaps answer that no mathematics task can be "non-embodied", but of course the way and degree to which a task is embodied, is an important consideration in task design. (This reminds me of a recent fad: to call everything "semiotic". I have read about "semiotic representations", but never of "non-semiotic representations". But of course, I am not saying that looking at embodied learning is an unimportant "fad".)

The next talk was Hanan Innabi, "Teaching statistics and sustainable learning". Her starting point was variation theory. The idea is that (based on Marton) students work on variations lead to sustainable learning. She shortly presented a research project with Marton, and mentioned the special thing that variation is an inherent thing in statistics, because uncertainty is an important part of statistics. (So in a sense, it is hard to teach statistics without much variation - although textbooks of course have statistics tasks where the data are given and variation is lost.) At the end she mentioned a couple of examples of how to work on this using variation.

The third talk of the TSG session, was Daniel Frischemeier's "Reading and interpreting distributions of numerical data in primary school". After giving a background of previous research, he presented his project of supporting primary school students to read the data and read between the data (not including the third level, reading beyond the data), with 19 primary school students (age 10-11), with use of real data, TinkerPlots and collaboration. The teaching was based on the PPDAC cycle (another FLA that I didn't know before). The conclusion was that ... and here the time was up, sadly. I suppose an article will be available at some point in the future. (In the discussion afterwards - in the chat - the distinction between stacked dotplots (such as in TinkerPlots) and messy dotplots (such as in Minitools) was stressed. Children find stacked dotplots more difficult to understand than messy dotplots.)

Then followed Carlos Monteiro and Karen Francois' "Statistical literacy as central competence to critically understand big data". Monteiro pointed out that students need to understand that they are also producers of big data, understand the power issues and to be able to analyse them critically. This is not an issue of how to handle the data technically, but to understand the origin of the data. 

(At this point, a small misunderstanding regarding time was sorted out, and Frischemeier got five more minutes to finish his presentation. The posttests were, not surprisingly, better than the pretests.)

The final presentation of the first session of TSG12 was Florian Berens, Kelly Findley and Sebastian Hobert's "Students beliefs about statistics and their influence on the students' attitudes towards statistics in introductory courses". In their experience, students have negative attitudes towards statistics when entering university, even though they have not had statistics before. They measured attitudes by the six-dimensional SATS-36 instrument, and created their own instrument for measuring the beliefs about statistics (descriptive perspective, investigative perspective, confirmation perspective and a rules-based perspective). They did find some correlations between attitudes and beliefs of statistics. In particular rules-based beliefs are connected to negative attitudes, while investigative beliefs are connected to positive attitudes. This raises interesting questions for teaching of statistics - probably also on lower levels than university.

Then the day ended with the Plenary panel on "Actors for Math Teacher Education: Joint Actions versus Conflicts". To be honest, I have attended very few good panel discussions in my life, mostly because they often end up not being panel discussions but instead turn into a series of barely related short lectures. Thus, I decided to take some time off from my blog-writing duties to just follow the plenary panel away from my keyboard. Instead, I watched it with pencil in hand, making some notes as it went along. Here it is:

That was the second day - and the first full day - of the ICME14. Five more days to go.

ICME14 Day 2 Part 1

The first item on the agenda was the first session of the Topic Study Groups. While usually going to the "The role of history of mathematics in mathematics education" TSG, but this year I joined TSG55, which is in the history of mathematics education. (The names seem alike, but the difference should be obvious...)

Alexander Karp welcomed us and mentioned that the field of history of mathematics education is getting recognition, for instance there are two series on history of mathematics education on Springer now, and Gert Schubring of course received a prestigious prize yesterday. The format of this TSG is talks of at most 10 minutes, followed by five minutes of discussion.

Vasily Busev and Alexander Karp: "Pafnuty Chebyshev and the mathematics education of his time". Chebysev was of course a major Russian mathematician who was also interested in mathematics education. Vasily Busev will publish a complete publication of Chebyshev's educational notes, which will give an interesting view of mathematics education at Chebyshev's time. For instance, he reviewed textbooks, and he was arguing that if a mathematical result can't be proved rigorously within the capabilities of the students, then students should be told so directly, instead of giving an almost rigorous proof.

Dirk De Bock: "Frédérique Papy-Lenger, the mother of modern mathematics in Belgium". De Bock has written an impressive book on New Math in Belgium, and this is especially interesting to me as I am doing some work on New Math in the Nordic countries currently. The talk is based on Lenger's collected writings and the reactions to these. Lenger vas part of the CIEAEM community. Already in the mid-50s, she thought that the focus on relations and structures in modern research mathematics could be a model for mathematics in school. She married the mathematician Papy in 1960, and became an important researcher in New Math in Belgium. Among her most important work was "Les Enfants et la Mathematique". She also worked with the Comprehensive School Mathematics Project in the US. Later, she worked on mathematics education for disabled children. Her work was important, but she did not get the recognition that she deserved. One possible reason was that she focused on the development, not on theory (which is an usual way to get overlooked). Also, she was overshadowed by her husband. Moreover, she was so connected to New Math that when New Math lost its position, so did many of its proponents.

Ildar Safuanov: "The history of mathematics education of Tartar nation". Safuanov discussed the history from medieval times until the 20th century. (As I know little about this in advance, it is difficult to give even a short summary of its history.)

Maria José Madrid, Carmen León-Mantero, Alexander Maz-Machado: "Mathematics and mathematics education in the 18th century Spanish journal "Semanario de Salamanca". The study of journals is an interesting supplement to studies of textbooks and other books. The journal in question was published twice a week, including both scientific articles and news from the city. For instance, it included mathematical problems, book reviews and job offers. Thus, we see how the study of journals can give an insight into the history of mathematics (including the role of mathematics in society). 

Maja Cindric: "Arithmetic textbooks in Croatia in the premodern period". Cindric talked about the history of Croatia and how it relates to neighbouring countries. In the period in question (in the 18th century), schools were run by Jesuits, not compulsory. There was no compulsory secular education until 1774. She looked at two arithmetics textbooks, from 1758 (by Mihajl Šilobold) and 1766 (by Mate Zoričić). Cindric gave details on differences between these books (which I am unable to detail here). As far as I understood, Šilobold's book was more focused on what we would call "problem solving". Cindric stressed how misconceptions in school can often be collected to terminology, and that the etymology of terminology can be traced by studying textbooks.

Karolina Karpińska: "Gnomonics in mathematics secondary school education on the territories of Poland in the 17th-20th century". Gnomonics is the theory behind the constriction of sundials. (I cannot think of sundials without thinking of my wonderful colleague Peter Ransom, who has often talked on sundials in lectures and outside of lectures.) She discussed the basic theory behind sundials, combining astronomy and mathematics. She discussed how gnomonics was taught in Poland - including different kinds of sundials and of course also whether they were treated generally (regardless of geometrical position). From 1812, gnomonics was introduced as an obligatory part of mathematics in 5th grade (in physics and chemistry). Later, it was part of geography lessons.

Shinnosuke Narita, Naomichi Makinae, Kei Kataoka: "Approach of an early 1940s Japanese secondary mathematics textbook to teaching the fundamental theorem of calculus". Calculus was introduced into textbooks in the 1940s, in 10th grade (4th grade in junior high). Narita detailed how the textbook introduced calculus, showing that students were expected to develop their knowledge through solving problems, before giving definitions later in the textbook. (Which seems reasonable - but it must be stressed that the role of the teacher is difficult to establish, so how free students actually were to develop their calculus.)

This was a demanding start of a day - two hours of presentations and discussions without breaks, on a wide variety of topics. In addition, Zoom is for some reason more demanding than physical meetings to me, particularly for my neck - it has something to do with having to sit in front of the camera in a particular way. Nonetheless, these first two hours of TSG55 showed clearly the variety and the tensions in history on mathematics education - sometimes very focused on the mathematics, sometimes on the institutions or on the discussions on mathematics in society and so on. Some researchers are very detailed on one little piece of a large puzzle, without including the context very much, while others give a lot of attention to context and less to detail. Of course this is also a matter of the limits imposed: four-page papers and 7 or 10-minute presentations. This is not conductive to including both detail and context sufficiently. But even disregarding these limits, I believe still researchers have different leanings, and that is fine. For instance: detailed accounts of a series of textbooks can be very valuable to another researcher, who can build upon them further.

The second item of the day was the plenary lecture by Lingyuan Gu: "45 Years: An Experiment on Mathmeatics Teaching Reform." This detailed mathematics reforms near Shanghai from 1977 to 2022. Giving just the highlights will seem like a string of buzzwords: from 1977 to 1992, the stress was on "affection, progression, attempt and feedback", from 1992 to 2007, the stress was on "comprehension" and "experiencing variation", while from 2007 to 2022, the focus was on "inquiry and creativity". This may seem silly, as everyone in mathematics education - from 1977 till today - will surely agree that affection, progression, attempt, feedback, comprehension, variation, inquiry and creativity are important factors. So the point is obviously not to jump from one to another of these (or to pretend - as Norwegian directorate of education tends to do - that every new curriculum includes some new focus that has never been thought about before). Instead, the point is to do a careful analysis of the current situation and try to establish what part of this mix of "buzzwords" (which are, after all, important concepts) need to be prioritized for a period of time going forward.  Such an approach, with careful research, testing and so on, has been followed in Shanghai. They combined macro studies with quantitative means, with micro studies with observations of low-scoring students and their interactions with their teachers.

I did enjoy his example of introducing parallell lines. Some teachers will just give a definition and many students will be able to resite it when asked. Other teachers give students an example ("are like two tracks of train"), thereafter opening up to a discussion about what else is or is not parallell lines ("Are they still parallel lines if the train makes a turn?". In this process, both what is and what is not (but are "nearly") will be included in that process. That's a nice close-up of some of the principles at work in a classroom.

It was interesting to see that even though inquiry had been an underlying principle for most of the time, results were not good by 2007. Therefore, inquiry was one of the main foci after 2007. That is perhaps not very surprising, I think - it is hard to break the habits of minds in classrooms, where everyone expects that no matter what the teacher says, the teacher still has the a script of what is supposed to be thought in a classroom. (Yes, I mean "thought", not just "taught"...) Results now are promising. 

One of the interesting points of going to international conferences is seeing that the problems people struggle with on one continent is often the same as on other continents, even though contexts are very different. Gu's talk was impressive while the main concepts were familiar. Lastly, he stressed the importance of video in educating teachers, to make available the analysis of classroom occurences in detail - with the teachers or teacher students. (Of course, then we start getting into the Lesson Study field, which I remember hearing a lot about at the 2000 ICME and which has been worked on a lot also in Norway, but has still not - as far as I know - become a normal part of education for all teacher students.) I also liked that Gu stressed that the point of this line of research is not to write "the perfect paper" but to improve teaching, which means that there will always be mistakes and wrong turns, but still, in the long run, improvement.

Already, this blog post is getting rather long, so I think I will end it here, and have the rest of the day in a brand new blog post...

ICME14 Day 1

The first day of ICME14 was a short one - lasting from 1:30PM to 5PM (local time in Norway). It consisted of the opening ceremony and a plenary lecture by Cédric Villani.

Needless to say, attending an opening ceremony is a bit different when you have travelled to another continent, are adjusting to a new time zone, and can congratulate yourself that you have been able to find your way around the conference area. At least you have found the room that is easiest to find: the place for the plenaries. As these often have a capacity of thousands of people, you may get the feeling of attending something important. At my first ICME, there was even an address from the US President at the time, Bill Clinton - quite surprisingly, as the conference was in Japan and had little to do with the US. (The address was on video, of course.) The sense of fulfillment in finding my office-at-home and managing to log in to the conference platform is not comparable.

The opening ceremony followed the usual format, including the mentioning of individuals and organizations that have been important in getting the conference in place, information about the city (Shanghai) and the conference, well-chosen words on the importance of mathematics and mathematics education as well as the awarding of prizes. To me, who is neither a native English or Mandarin speaker, having the English sound overlaid with simultaneous translation into Mandarin, made it rather difficult to follow what was said Some of the addresses were subtitled, however, and I did, for instance, notice that more than 120 countries are represented in ICME this year. Towards the end of the opening ceremony, more than 15000 viewers were logged on, which is surely a new record (and also probably was a reason for some transmission interruptions).

Two of the ICME awards, the Felix Klein award and the Hans Freudenthal award, are biannual, thus this year there was both the 2017 and 2019 awards to celebrate. The Emma Castelnuovo award is just awarded every four year. For more on the awards, see ICMI website. The winners were: 

2017 Felix Klein award: Deborah Ball

2017 Hans Freudenthal award: Terezinha Nunes

2019 Felix Klein award: Tommy Dreyfus

2019 Hans Freudenthal award: Gert Schubring

2020 Emma Castelnuovo award: NCTM (USA's National Council for Teachers of Mathematics)

These are all well known names for the mathematics education community. Personally, I've perhaps had most to do with the work of Gert Schubring, who has been so important in strengthening history of mathematics education as a research field. Of course, Deborah Ball are famous for the Mathematical Knowledge of Teaching oval, but has since moved on to other (and more fertile) grounds - on the moment-to-moment dilemmas teachers face in classrooms. Therezina Nunes and Tommy Dreyfus are important names that I have not personally been as occupied with. Nunes is of course co-author of "Street Mathematics", and have later worked further on mathematical thinking. Tommy Dreyfus has for instance worked on AIC, abstration in context. NCTM, of course, has had huge influence in the US, as well as internationally, through their journals, guidelines for teaching and many other publications.

It is nice to hear how these important figures in mathematics education are pointing out the importance of teamwork - new findings in mathematics education are rarely the product of an individual mind.

The opening ceremony finished about 30 minutes late, and then we moved almost straight on to the first plenary lecture of ICME14: Cédric Villani on "Mathematics in the Society" (which is a promising title, given that I will be going to the Mathematics Education and Society conference for the first time this autumn). Villani is a mathematician (Fields medalist!) and a member of the French parliament. He started by saying that he went to a television show to talk about the essence of mathematics, and he brought three objects: Euclid's Elements, a gömböc and a smartphone. In this way he brought forward how mathematics is about reasoning, but at the other hand has wonderful applications. Mathematics is both beautiful and useful. Also, mathematicians bring progress by introducing new ideas and looking at things from another viewpoint. Also, he talked about the connection between new problems and new concepts: new problems lead us to find new concepts to solve them, but new concepts give rise to new problems.

In mathematics, you are not obliged to believe the teacher: you can find errors in what the teacher is saying, and if the teacher is a good one, the teacher will be convinced of the reasoning if you are right. This does not work in the same way in other sciences. He claimed that the three most important parts of mathematics work is tenacity, imagination and rigour (with rigour third).

He talked, inspiringly, about the work he has done after his Fields medal, including a book about what it is to be a mathematician, a comic book and a photo book. Towards the end of his talk, he discussed his work as a politician (in the Scientific Parliamentary Office - I wonder how many other countries have them...) and on AI. He ended by saying how important it is to remember the past and learn from the past, which is a welcome message.

Of course, this was (as usual) just some of the points that I liked and managed to write down, it is not meant to be a full discussion of the lecture, which was, by the way, entertaining and lively, although it was a bit unusual, as it took the form of a autobiography more than a classical plenary lecture.

ESU6 Day 5

The last day of the ESU was also the shortest. I skipped the plenary lecture to secure more time for packing, checking out of the hotel and so on. The lecture I missed was Fulvia Furinghetti and Livia Giacardini’s “From Rome to Rome: Events, People, and Numbers during ICMI’s First Century”.

I did, however, take part in Snezana Lawrence’s workshop “Digitising the past mathematics by the future mathematicians”. It concerned how working on earlier textbooks may be interesting for today’s schoolkids. For instance, they may be interested to see that earlier textbooks (based on Euclid), actually gave definitions in a way that are almost gone now. Nowadays, textbooks give explanations with pictures and words, perhaps making it less clear what is the core definition and what is examples or illustrations. The workshop did not, however, give a very clear idea of what Snezana thinks that the role of “digitising” should be in this. While I do see that students could learn from scanning and transcribing short portions of text and then making it publicly available, I’m not convinced that it is worth the time. (And the other members of my group in this workshop was rather convinced it was not worth the time.) However, the work we did in the workshop was a good illustration of some of the points Michael Glaubitz made in his plenary lecture.



That marks the end of the conference. It was hard work. I leave the conference inspired and with a wish to continue working on history of mathematics. However, I also think that I was a bit too busy at this conference. In the end, I got to choose five of the workshops, but none of the oral presentations (I was always either speaking or being a chair). Having presentations on three of the days (one oral presentation, one workshop and one panel) is a bit too much also, both in the weeks ahead of the conference and in the conference itself.

I look forward to the HPM Meeting in Korea in 2012 and the ESU7 in Spain (maybe) in 2014.

ESU6 Day 4

Michael Glaubitz’ plenary lecture, which marked the start of the fourth day of the conference, was titled “The Use of Original Sources in the Classroom – Empirical Research Findings”. He set up a teaching experiment using original sources in two different ways: a genetic approach and a hermeneutic approach. Then he also had a “conventional” approach.

While a genetic approach is usually considered a means of introducing a topic (the pupils will meet the topic in the same order as it was developed in history), a hermeneutic approach (as suggested by Jahnke) concerns reading original sources when already having pre-knowledge about the topic concerned.





Glaubitz set up three teaching sequences of equal length which were then taught in different groups of students by different teachers. The genetic approach failed miserably, while the hermeneutic approach did significantly better than the conventional approach. Of course, such an experiment can never be a proof of the non-feasibility of an idea, so the results for the genetic approach are not the most interesting ones. However, it is a bit impressive that teachers used to the conventional approach can be guided into teaching successfully with original sources (using the hermeneutic approach) in such a short time. (Although we may of course, as always, have the effect of trying something new.)

To me, this is very promising results. The genetic approach seems too ambitious to me – in a way, you need to take into account all the history of mathematics at the same time, which puts enormous demands on the teachers’ knowledge. The hermeneutic approach delves into one particular point of time and one particular context. While that is certainly also demanding, it seems more like something many teachers could be willing to try – given the right materials.

The second plenary lecture was Raffaele Pisano’s “Which is the cultural and interdisciplinary role played by physical and mathematical sciences? Epistemological Reflections”. This talk concerned the connections between physics and mathematics. To be perfectly honest, I have never had a physics course in my life, and therefore found most of the lecture too advanced for me. (Which doesn’t say much, of course.)



Then there was another panel. The title was “The role of the history and epistemology of mathematics in pre-service teachers training”. The panelists were Evelyne Barbin, Fulvia Furinghetti, Snezana Lawrence and myself. I did not take notes during this panel (except notes on what I was going to say), so I’m not the right person to try to summarize it. However, for me personally, the main outcome of the panel itself and the preparations for it was a realization that it is time that I do something a bit more substantial when it comes to making history of mathematics available to teachers. It’s been ten years since my first HPM conference (in which I complained about the quality of Norwegian textbooks), six years since my second HPM conference (and first ESU) (in which I complained about the quality and quantity of history of mathematics in the TIMSS Video Study materials), three years since my second ESU (in which I did not complain actually – but showed examples of history of mathematics I’ve used in teacher training) and two years since my third HPM (in which I discussed teachers’ conceptions of history of mathematics). Isn’t it time for me to come to a HPM or ESU with a textbook or webpage and discuss why I have created this wonderful book/webpage in this way? Maybe in 2014?

I could mention that my part of the panel discussion was taking Deborah Ball’s model of teacher competence in mathematics as a starting point. My own preparations on this – as well as a question in the discussion – made me realize that I have to revisit Ball’s articles to see what her (and her colleagues’) conception of “mathematics” is.

After lunch, I attended Kristín Bjarnadóttir’s workshop “Arithmetic textbooks in 18th century Icelandic manuscripts”. I think I would have found such a workshop painstakingly boring ten years ago, but I’ve grown to appreciate the study of textbooks as a valuable part of research on history of mathematics. Moreover, the kind of “detective work” that Kristín has been doing here and which she invited us to take part in in this workshop, is rather fascinating. Her object of study is an Icelandic manuscript, and she wants to find out as much as possible about which textbooks influenced it. The “detective work” consists partly in finding out which textbooks were known in Iceland at that time, and of comparing “her” text with these. From time to time, there are clear “hits”, when numerical examples are exactly the same in different textbooks – they must clearly have a common source.



In Norway, there has been published a book on the first Norwegian mathematics (arithmetic) textbook: Tyge Hanssøn’s “Arithmetica Danica” from 1645. While the author (Geir Botten) has done lots of work on this book, he has not done research on the influences of the textbook (as far as I know). It would surely be interesting to see the results of such a study at one point.

coffee break area

The evening program was again oral presentations. Again, I was chairing (or cheering). There were three talks: Oscar Abdounur talked about the University of Sao Paulo, and how the invitation of European scholars to the university influenced it. Andreas Christiansen (whose name was terribly mispronounced by the chair – it’s a surprise that Christiansen (not Christensen) managed to speak after such an insult) spoke about Bernt Michael Holmboe’s textbooks and in particular how the concept of irrational number was presented in different editions, compared to how mathematicians elsewhere defined irrational numbers. Thirdly, Nuno Dias talked about the Portugese mathematics education, in particular after the Jesuits (who controlled Portugese education) were expelled in 1759.

ESU6 Day 3

Before going on to describe day 3 of the ESU6, I should perhaps remind you that proceedings of this conference will be available in about one year’s time. So if talks or workshops seem interesting, either because of the title or my description (or even both!) that’s the place to look for more.

The plenary lecture of day 3 (Wednesday) was Marc Moyon’s “Practical Geometries in Islamic Countries: the Example of the Division of Plane Figures”. Again, this is so far from my tiny area(s?) of competence that I will write little about it – in an attempt to make no errors (except errors of omission, of course). However, one main point is that mathematicians in these countries knew how to divide plane figures in different ways (for instance to divide three equal squares so that they could be put together to form one larger square). Craftsmen also knew how to do this – in ways that were not mathematically correct, but that were visually correct. Part of this lecture was about the relationship between the two.




Michael N. Fried asking Marc Moyon a question. Andreas and Uffe are listening.

Sunwook Hwaan presented the plans for the HPM 2012 conference which will take place in Daewoon, an hour or two from Seoul (where the ICME conference will be). The information given in such a presentation is always less important than just assuring the audience that the planning process is going smoothly, and we certainly got a good impression from this presentation.

Then there was a panel discussion on “The history of mathematics in school textbooks”. The panelists were Anne Boyé, Adriano Dematté, Ewa Lakoma and Costantinos (Costas) Tzanakis. Part of the discussion confirmed that the situations in different countries are quite similar in many respects. I started my HPM “career” in 2000 by describing how history of mathematics is treated in Norwegian textbooks (after the 1997 reform of the curriculum). There were lots of errors, biographical details with no real significance and meaningless tasks for the children to do.



In this discussion, Dematté noted that there is too much “reading about” and too little “doing mathematics” when history of mathematics is in the textbooks. (According to publishers and teachers, history of mathematics is for reading.) Ewa Lakoma noted that history of mathematics tends to become a “set of curious details” instead of a source of stimulating contexts. Costas (quoting someone, I’m sorry I missed whom) asked if the history of mathematics in the textbooks makes the students spectators. Boyé gave the example of a task from a textbook: “Archimedes is a famous ancient greek mathematician. He was born in Syracuse in 287 B. C. and died in 212 B. C. How many years did he live?” This sort of meaningless exercise was also in the 1997 material of Norwegian textbooks.

This was a very well prepared panel, in that the main contributions of the panel were published in advance and people were asked to prepare comments. I had prepared a comment on a section on different ways of including history of mathematics in textbooks. I find the question a bit strange, as if you have the history of mathematics in your left hand and your textbook in your right hand and you want to include some of the history in the textbook that already exists. Contrary to this, one may say that as history of mathematics is an integral part of mathematics, history of mathematics should be a natural, explicit and integral part of the textbooks. I gave the example of textbooks on literature: I do not think that textbook writers on literature first write about the literature, for instance “Romeo and Juliet”, and then sit down and wonder “Maybe we should mention Shakespeare? How could we include the history of literature in this literature textbook?” But, on the other hand, a textbook has to have an audience; otherwise it will not be very useful. To write a textbook that is brilliant but that no one will ever read is a futile exercise.

That, surprisingly, was the end of the third day. After getting used to working till at least 8 at night, it was welcome to finish before lunch – if only to get time for some meetings that would be difficult to arrange otherwise. In the afternoon, there were sightseeings. I very much doubt that you will want the details of that. And then there was a reception in the “Rathauskeller” (town hall cellar), which was a splendid place.

ESU6 Day 1

I am currently attending the European Summer University on the History and Epistemology of Mathematics Education. The conference takes place in Vienna, Austria. As usual at such conferences, I will blog about what I learn from it.

The conference was opened by Mannfred Kronfeller, the main organisor of the conference, and by the Vice Rector of the Technical University of Vienna, Hans Kaiser.

There were also short introductions by Evelyne Barbin, the chair of the HPM group, who outlined the history of the ESUs, and by Costas Tzanakis, who mentioned that representatives of 28 countries are present at the conference. That's not bad.

The first plenary speaker was Uffe Jankvist from Denmark. He talked about a teaching experiment that he has conducted (and which formed part of his PhD), where the history of mathematics played a major part. The topic was error-correcting codes and cryptography. But while this was the background of his talk, his real concern was to get more specific about what role history of mathematics can have in teaching. In particular, he studied whether he could see signs that the students would have meta-issue reflections on mathematics that were anchored in the mathematical in-issues that they were working on.

I find Uffe's approach very interesting. Instead of just claiming that "history of mathematics can make students reflect on the epistemology of mathematics", for instance, he sets out to study particular interpretations of this claim. It makes for a more nuanced and informed view of the part history of mathematics can play.



Uffe argues powerfully for the use of empirical studies (in this sense) in research on history of mathematics. I agree that this is one important path to follow. However, I believe that we still need the other paths as well. While studies such as this are convincing, they do too little to provide examples and materials for teachers to trz. For many teachers, successful attempts at including history of mathematics in their own classrooms are more powerful than researchers' results.

Then there was time for workshops. Workshops form a major part of the ESUs, as the ESU is supposed to give more "hands-on-experiences" than ordinary conferences. I attended Masami Isoda's workshop on how he has used the dbook software to create digital editions of textbooks with appended interactive materials. The example given was of historical instruments (from van Schooten's book). We were also given the chance of recreating the historical instruments by means of paper (see the wonderful instrument created by the group consisting of Jan van Maanen and me - I must admit that Jan did most of the work...)




I think this is an interesting way of connecting the textbook with more interactive media.

Then I attended a three-hour workshop with Uffe Jankvist. There have been discussions during the conference of whether the three-hour format is too long. In my opinion, Uffe's workshop provides an existence proof that three hours may at times be about the right amount of time. Uffe wanted the participants to analyze some bits of transcripts with the same analytical tools that he had used in his PhD. In the beginning, the participants were quite hesistant, as both the mathematical and historical topic under consideration was unfamiliar and the analytical tools were new to us. However, by the end of the workshop, it seemed to me that everybody was participating in trying to find the "potential anchoring points" that Uffe was interested in.

As mentioned above, I think that there is great potential in getting more detailed concepts of how history of mathematics may be included in teaching, making it possible for teachers to be more conscious about exactly which effects they are trying to produce in a particular circumstance.

Oral presentations are not such a high priority at the ESUs, so on Monday they were taking place from 1830 to 2000 at night. Gerlinde Faustmann talked about "Classroom experiences with history of mathematics", following the path mentioned above of providing examples and ideas for teaching mathematics using history. Then I had my presentation on the Norwegian literature on history of mathematics, where I outlined several points of potential improvements. (I keep thinking I should write a book on the subject - but where is the time for it...) And finally, Po-Hung Liu discussed "the evolution of college students' epistemological views of mathematics in a history-based class". This was a combination of teaching ideas with a somewhat detailed look at what students did get out of it.



Of course, giving such a short summary of the first 11 hours of the conference does not allow for too many details. The proceedings of the conference will be available perhaps a year from now, so more details will be obtainable there...

I will continue with a discussion of day 2 in a later post.

Mathematical teaching in North Korea

This blog has been silent lately - but things will get better, as I am planning to blog from the ESU6 conference in Vienna.

However, I just read an interesting article that I would like to tell you about:

Contents or ideology? A case study of mathematical teaching in North Korea. Written by Alexander Karp and Jung Hang Lee, published in Asia Pacific Journal of Education, 30: 1, 1-13.

The article gives the results of an interview study where both teachers and students from North Korea that have now found refuge in South Korea. The article deserves to be read in full, but I still want to mention a few points:

Ideology is a part of mathematics teaching in North Korea. An example mentioned in the article is text problems where collective farms have exceeded their goals "due to the beneficial influence of the Great Leader". In fact, mathematics lessons follow the same sequence as almost everywhere in the world, except for one detail: a 45-minute lesson should include Review, Checking Homework, Reinforcing the Policy of the Party, Delivering New Knowledge and Practice. "Reinforcing the Policy of the Party" is not a standard part of lessons in most countries, and some countries (such as Norway) has also almost done away with the "Checking Homework" part.

We see that Tim Rowland's concept of "contingency" has a part to play also in this context. The authors tell of a new teacher who came to the classroom to find a frog inside a box, with a sign from the frog's neck saying "Let's study math!" The teacher managed to turn this into a teachable moment, telling them about frogs' importance for the crops and telling them that they would, due to the leadership of the Great Leader, produce more crops in the future. (Sadly, we know that the "Great Leader" has mismanaged the country, also when it comes to crops.)

The article does not just tell us about the North Korean system, but reminds us that Stieg Mellin-Olsen long ago told us that mathematics will always include some politics and ideology. The authors also cite Gert Schubring and say that a difference between "what is officially proclaimed and what actually goes on" is nothing special for North Korea.

An interesting article.

Article: The emerging practice of a novice teacher: The roles of his school mathematics images

Jeppe Skott: The emerging practice of a novice teacher: The roles of his school mathematics images, Journal of Mathematics Teacher Education (2001) 4: 3-28.

From a project with four novice teachers including classroom observation and interviews, the author has chosen two episodes from one of the teachers, which sheds light on the connection between the teacher's "school mathematics images" (SMI) ("teachers' idiosyncratic priorities in relation to mathematics, mathematics as a school subject and the teaching and learning of mathematics in school") and their teaching.

The episodes can be interpreted to show that a teacher may very well do things during his teaching that clashes with his stated SMI, because other concerns are more pressing. In this article, one such concern was to build the confidence of a particular student - thus getting the right answer got more important than a full understanding of the process. On the other hand, the author worries that the actions in such situations may be copied to situations in which the more pressing concerns are not present.

The author also introduces another TLA (three letter acronym): CIP. A CIP is a "critical incident of practice", "an instance of [his] decision making in which multiple and possibly conflicting motives of his activity evolved; that were critical to his SMIs; and that were critical to the future development of the classroom interaction and for the students' learning opportunities".

Article: Teachers' opinions about some teaching material involving history of mathematics

Barry J. Fraser and Anthony J. Koop: Teachers' opinions about some teaching material involving history of mathematics, International Journal of Mathematical education in science and technology (1978) 9: 147-151.

In this article, the authors describe research in which they have given 39 mathematics teachers access to two different kinds of teaching materials involving history of mathematics: a play about Thales and an article related to the history of conics. The teachers were asked to read the materials and then answer a questionnaire. I find this particularly interesting, because this could be seen as research on the teachers' attitudes - without the teachers being influenced by enthusiastic teacher-researchers.

Interestingly, teachers were quite positive to the materials, particularly the play. (A theory might be that a play is more instantly usable than an article?) However, a significant number of the teachers would nonetheless not use it in their own lessons. There are many who thinks that the play would take too much time, for instance.

Also interestingly, almost all teachers agreed that "materials like this are not readily available elsewhere". Of course, that situation has changed a bit since 1978, but it is still an obstacle to teachers' use of history of mathematics.

Article: Didactics and History of Mathematics: Knowledge and Self-Knowledge

Michael N. Fried: Didactics and History of Mathematics: Knowledge and Self-Knowledge, Educational studies in mathematics (2007) 66: 202-223.

How to include history of mathematics in mathematics education in a way that is true to the history? This is an old question in the HPM community, and it is not resolved.

In this article, the author looks to Saussure and the theory of semiotics to argue that history of mathematics has an essential role to play in mathematics education. "The historian's and the working mathematician's ways of knowing are complementary."

First, he discusses what is the problem: "If, in the working mathematician's view, the otherness of a historical text is something illusory or merely superficial, the historical point of view is precisely the opposite. From tihs latter pole, a mathematical text is a cultural product, the product of a particular human being or group of human beings living in a particular time."

He gives a short (and welcome) introduction to Saussure, and describes two ways of knowing: synchronic and diachronic. Seen at one single point of time (synchronic), a language (or mathematics) seems static, and you can not see the social forces at play. When you look at a period of time (diachronic), you see how language (or mathematics) is evolving and ever-changing.

The author then gives an example: How Apollonius of Perga's Conics has been interpreted in different ways. Heath wrote that "[Apollonius'] method does not essentially differ from that of modern analytical geometry except that in Apollonius geometrical operations take the place of algebraic calculations" - an interpretation based squarely in the modern mathematics. The author of the article shows how Apollonius could rather give us an alternative way of viewing conics, which makes us more aware of our own modern view.

So what about mathematics education. The author argues that the object of mathematics education should not be only to learn mathematical methods, but "our self-knowledge as mathematical beings". As such, the history is essential to give another perspective and let us learn more about our own views.

Article: On the argument of simplicity in Elements and schoolbooks of Geometry

Evelyne Barbin: On the argument of simplicity in Elements and schoolbooks of Geometry, Educational studies in mathematics (2007) 66: 225-242.

We all want to present the mathematics in a simple way to our students, don't we? Sure we do, but what do we mean by "simple" exactly? In this article, the author very interestingly traces the concept of "simplicity" through several editions of Euclid's Elements as well as in some recent French textbooks. And it turns out that "simplicity" is not as simple as we thought...

Proclus, in his commentary to Euclid, proposed two "orders of simplicity": simplicity of a figure and simplicity of theorems. "A figure is more complex than another if it is obtained from the former with additions of lines or circles." A theorem is simpler than another if it comes earlier in the order of logical deduction.

Peletier preferred clearness of proofs rather than simplicity of figures.

Arnauld sought to follow the "natural order". "It is contrary to this natural order to prove, as Euclid does, propositions on perpendicular straight lines or on parallel straight lines (simple things) using triangles (compound things)." Therefore, the whole book was ordered according to the natural order. A consequence of this was that "perpendicular lines [were] studied without using angles and angles [were] studied without using triangles."

Hoüel wanted simplicity of "principles and proofs". He defined "the straight line by the motion of a point, and the plane by the motion of a straight line. Angles are defined by the ideas of motion and direction." (I find this last point particularly interesting, as it has been a problem in Norwegian textbooks that angles have been defined as static things only, without any dynamic dimension.)

Then, of course, in the 20th century, the notion of sets was regarded as "the simplest of all the notions." This had absurd consequences; "In the Mathématiques, classe de sixième, directed by Mauguin (1977), for eleven-year old pupils, the angle is defined as an equivalence class."

Finally, in a textbook from 1996, the angle is introduced with drawings of fans which are opened to different degrees. This quite good idea is however partly destroyed by a confusing use of colors.

I have just given a tiny idea of the contents of the article, of course. The author shows how these different ideas of simplicity is determining the structure of the books. Choose your notion of simplicity, and the structure will have to change. She ends with a reference to Descartes:

"As Descartes writes, arithmetic and geometry are more certain than the other sciences because their object is "so pure and so simple". [...] History of mathematics invites us to come back and to work with the notion of simplicity to construct the teaching of geometry."