Relationship between Birth Month and Mathematics Performance in Norway

I'm happy that the article that Annette Hessen Bjerke, Elisabeta Eriksen, André Rognes and myself have published, is online. In it, we investigate how students' mathematics score correlate with whether they are born early or late in the year. The effect of being the youngest in a class seems to last for a long time.

Relationship between Birth Month and Mathematics Performance in Norway: (2021). Relationship between Birth Month and Mathematics Performance in Norway. Scandinavian Journal of Educational Research. Ahead of Print.

ICME14 Day 7

Seven days - that's too much. Of course, it makes sense to make the most out of it when thousands of mathematics educators come from all over the world and assemble in one city once every four years, and then, also there's the added value of experiencing a new city. When instead sitting in an office at home, a whole week is a lot. Another conference I went to this year solved this by expanding the conference from one to two weeks - with shorter days (which automatically meant that when things did not interest you, you suddenly had a whole day for other work. 

And still, I should not complain, as ICME is arranged in a way that means most items on the agenda is between 8.30 AM and 5 PM. People in other parts of the world have more difficult time problems. 

There are some issues that need to be thought about after such a hybrid event:

- Video recorded talks work fine - but why are we all watching them at the same time? There is no real reason to include the plenary talks in the time table: they could be posted and seen at the viewers' own discretion. With a time slot at the end of the conference for interaction with plenary speakers, communication would still be taken care of. In CERME, most working groups have the rule that everybody should read everybody else's papers before the sessions. We could then have a more compact conference and more dialogue.

- Zoom meetings (with 10-30 participants) work fine. The dialogue is possible to arrange. Webinars are more challenging - the dialogue via questions in the chat is problematic, it never turns into a real discussion. 

- The hybrid mode works better than expected, but it does not work well. There are far too many problems with the sound and so on.

For the future, I guess there will still be huge international (and local) conferences where you will meet people in person and start collaborations that would not be possible online only. But I think we will see a lot more short conferences that will be online only. If all talks are prerecorded and you can see all talks (and/or read papers) in advance, two or three days of 3-4 hours of actual dialogue, could be really valuable and time well spent. (I'm excited that the MES conference this autumn has chosen a format where the time table is 22PM-01AM, 6AM-9AM and 3PM-6PM (Norwegian time). The idea is that everybody should be able to attend to at least two of the three time slots. For Norway, it is possible to choose between the 22-01 or the 6-9 time slots (in addition to the perfectly acceptable 3-6PM time slot), but I do not advice to attend both the 22-01 and the 6-9 for days on end. Sleep is important.)

Oh, and I miss the social events and the excursions, of course. I have fond memories of the border between North and South Korea, the pyramids of Mexico, and then, partly as a result of the excursions and programmed social events; dinners with new and old friends. Of course, I cannot say I have any new friends after this ICME - although I certainly have a few more names and faces which will make it easier to get in touch either by email or in person in future.

Enough of that. Today's first item was invited lectures, and I chose Susanne Prediger: "Enhancing Language as a Catalyst for Developing Robust Understanding – A Topic-specific Research Approach". When I did an overview of the research on language and mathematics exam tasks a few years ago, Prediger's name was everywhere, so I wanted a chance to connect a face and a voice to the words - and of course to hear of recent developments. 

She talked about 12 years of projects in the research group MuM - Mathematics learning under conditions of language diversity" (as always my notes here can not possibly convey the content of the talk, but may inspire someone to look up the relevant articles). We see, all over the world, that schools fail to provide equitable access to mathematics for students with low academic language proficiency. This can be multilingual students, but also monomingual students, for instance with low socioeconomic background. This is mainly concerned with conceptual understanding, which has to do with rich relationships between concepts. It is also important that in school, academic language proficiency is needed, not just everyday language, and it is important that children get opportunities to learn this. Of course, stress on language in mathematics teaching and working on different representation is important.

She discussed design principles for topic-specific design research. She gave examples from two Grade 6 boys in work on fraction, where the boys' language hinders the explication of (and development of) meaning. Academic language demand mediating between the other registers. Thus, one important point is to have students connect language registers and representations (not just to talk and write much). She gave a series of examples on how traditional learning trajectorys for concept learning can be tweaked into developing discoursive practices. (Or rather: to combine a conceptual learning trajectory with a language learning trajectory.)

She went on to describe a large-scale (38 classes, 655 seventh graders), with language-responsive intervention, with pre-test and post-test (with control group). The language-responsive intervention group learned significantly more than the control group. The results held also (and were even higher) for students with high language proficiency. And they were largest for non-routine items. Thus, it works for all students. But there was a huge variation between classes. She pointed out that it is needed to look at teachers' enactment. (I would argue that one should also look at the norms of the classrooms.) In looking at teachers' enactment, they used the TRU Framework, slightly adapted. (See Prediger et al, in ZDM 2021.) 18 classrooms were video-recorded, and quality dimensions were rated every 5 minutes. They found high variance in Use of Contributions, Equitable Access and Discursive Demand, and these were related to the results. 

It is quite amazing to see top researchers reflect on their research journeys and how their findings have spurred new questions and how these new questions build on other parts of mathematics education research and use multiple methods to get even further. She actually shared a link to her video to the participants at the end, and I think this will be really interesting to discuss with colleagues when the holiday is over.

Final reflections

As often happens at week-long conferences, I run out of steam and can't keep up the blogging throughout the last day. I have already given some reflections on the form of the conference at the top of today's post. Now I will try to summarize for myself some of the key points to keep thinking of. 

From TSG55, I am intrigued by the great variety in the papers. I did note Alexander Karp's repeated calls for contextualization and to get beyond doing studies of one textbook at a time in isolation, but in my opinion, sometimes the smaller pieces that will later fit in a larger puzzle, can be valuable research projects in themselves. I see this particularly in working with New Math in the Nordic countries, that there are so many detailed investigations that have to be done to better be able to compare and contrast different periods and/or different countries. Some of these detailed investigations can possibly fit into the rather insane four-page limit of ICME, but not if they are also to include enough of the context and research background and implications...

Being a novice to TSG12 (statistics), it was interesting to get a quick idea of where the research front is; what are the issues being discussed and what are seminal works that everybody refer to, and what are the key competencies that students are able to understand at different age levels. I was particularly impressed by some of the work on the youngest students in school. I now have several tools to look into and several articles to read to get a clearer understanding of this. In the same way, attending the discussion group on algorithmic thinking gave me an insight into where the problems are. They were not altogether surprising.

The plenary panel on the collaboration and conflict between mathematicians and mathematics (teacher) educators was interesting, although in my context, I believe the division between mathematicians and mathematics teacher educators is rather fuzzy, and the collaboration and conflicts with other actors (pedagogues, politicians, administrators) are rather more frustrating and time consuming.

I did enjoy the HPM session, although I've taken part in quite a number of such sessions through the years. I also enjoyed getting an overview of Gert Schubring's research through his awardee lecture. In fact, there were a number of lectures like this (including Prediger's lecture today), with a prominent researcher reflecting on many years of research, which is a nice occation to see people's work as a whole instead of reading one article at a time. (However, one does wonder how many twists and turns are left out - from my own experience, as a quite non-prominent researchers - I know that I happen to take part in many research projects that does not fit neatly into an overall story. But that may be one reason why I am so non-prominent.)

Perhaps I should now reflect on my own contribution to the conference. Blogging is one way for me to keep alert and to actively process what is happening. Even in physical conferences, my contribution in terms of questions and comments are not that frequent (although before I've always had one presentation, and I've also contributed to the HPM session some times, and also led a discussion group once). This time, I think I contributed with just two or three questions during the whole week. But in normal circumstances, we also discuss what we have heard during lunches and dinners, and during walks around the town and so on. I miss that. 

This is the end of my ICME blogging this time. I hope to be back for ICME in 2024 (and of course for many other conferences before that). (Oh, I notice that next ICME will be in Sydney, Australia, from July 7th to July 14th, 2024. As I was originally planning to skip this ICME because I want to cut down on flying, it seems unlikely that I will be going to the ICME in Sydney. So perhaps I'll rather go to the 2028 one.)

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.

A small, free course in quantitative methods for teacher students

Last autumn, I was in charge of the quantitative part of a course in theory of science and research methods for teacher education students (MGVM4100) at OsloMet - Oslo Metropolitan University. Because of the COVID-19 pandemic, all lectures were turned into digital format, and I made a series of smaller videos instead of a few long ones. The videos were published on YouTube. But only today (half a year later) have I had the time to provide all the videos with subtitles. That means that the videos should make sense for people no matter what language they prefer (via YouTube's automatic translation) - although the text in the slides used are mostly in Norwegian.

The students are supposed to read Peter M. Nardi's book "Doing Survey Research", 4th edition. That is a very good book, in my opinion, but the examples there are not from education (for instance science education). My videos are based on teaching the course a couple of times before, and my aim has been to give enough background in quantitative methods for the students to relate to quantitative methods in their work as teachers as well as in their work on previous research when doing their master thesis. Moreover, I hope they will have a foundation that will make it reasonable easy to study further if they want to use quantitative methods in their own research in their master thesis. As far as I have been able, I have used examples from different school subjects, and also created my own examples based on real data. By showing how things can be done in SPSS, I also hope to demystify that, so that students dare to use SPSS if they need to. Still, the course is not a SPSS course per se.

• Introduction: 0 0 Kvantitativ metode: Innledning

Uke 1

• Introduction first week: 1 0 Kvantitativ metode: Innledning uke 42 
• Central concepts: 1 1 Kvantitativ metode: Sentrale begreper
• Making a survey: 1 2 Kvantitativ metode: Å lage spørreskjema
• Sample and research design: 1 3 Kvantitativ metode: Utvalg og forskningsdesign
• Descriptive statistics: 1 4 Kvantitativ metode: Deskriptiv statistikk
• Hypothesis testing: 1 5 Kvantitativ metode: Hypotesetesting
• Normal distribution and Z-test: 1 6 Kvantitativ metode: Normalfordeling og Z-test
• Example: School Emphasis on Academic Success: 1 E1 Kvantitativ metode: Eksempel: Læringstrykk
• Example: Reliability: 1 E2 Kvantitativ metode: Eksempel reliabilitet
• Example: Survey: 1 E3 Kvantitativ metode: Eksempel spørreskjema
• Example: Sample: 1 E4 Kvantitativ metode: Eksempel på utvalg 

Uke 2

• Introduction second week: 2 0 Kvantitativ metode: Innledning uke 43
• Bivariate analysis: 2 1 Kvantitativ metode: Bivariate sammenhenger
• Example 1: Chi-square: 2 E1 Kvantitativ metode: Eksempel 1: Kjikvadrat
• Example 2: Chi-square: 2 E2 Kvantitativ metode: Eksempel 2: Kjikvadrat
• Compare means: 2 2 Kvantitativ metode: Sammenlikne gjennomsnitt
• ANOVA: 2 3 Kvantitativ metode: ANOVA
• Example: ANOVA: 2 E3 Kvantitativ metode: Eksempel på ANOVA

Uke 3

• Introduction third week: 3 0 Kvantitativ metode: Innledning uke 44
• Two-way ANOVA: 3 1 Kvantitativ metode: Toveis ANOVA
• Multiple R: 3 2 Kvantitativ metode: Multippel R
• Multiple regression: 3 3 Kvantitativ metode: Multippel regresjon
• The last phase of a project: 3 4 Kvantitativ metode: Sluttfasen av et prosjekt
• Repetition: 3 5 Kvantitativ metode: Repetisjon
• Example: Multiple regression: 3 E1 Kvantitativ metode: Eksempel: Multippel regresjon

Of course, this is just a first, short course in quantitative methods. Many would have prioritized this in another way. But I still hope the videos may be of use to some. 

 

Of course, there will also be errors in the videos. So far, my 400 students have just found one error: that is in the video about Hypothesis testing, and there is a correction in the description of the video on YouTube as well as in the subtitles.

Feel free to let me know if you have had use of the videos or if you have other input. (I am thinking of making videos about effect sizes - as my videos are as of know only occupied with significance. I am also considering more info on scales, on SEM and perhaps also on the racism involved in the beginning of the history of quantitative methods.)

 

 

 

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: