Sunday, July 31, 2016

ICME13 Day 7 #icme13

The 7th day of the ICME13 conference was short. First a plenary panel on "transitions in mathematics education". Panellists were Ghislaine Gueudet, Marianna Bosch, Andrea diSessa, Oh Nam Kwon, and Lieven Verschaffel. The panel's theme - transitions - has many interpretations, including transitions between themes (arithmetic to algebra), transition to formal proof, transitions between school levels, transitions between contexts, for instance language contexts, transitions between curricula. In this panel, they focused on transitions as conceptual change and on transitions of people as they move between social groups. To look at this, they had epistemological, cognitive and socio-cultural perspectives. The result of the work is a survey: "Transitions in
Mathematics Education".

diSessa started by talking about "Continuity versus Discontinuity in Learning Difficult Concepts". Bachelard talked about epistemological obstacles, while others talks about "pieces and processes" instead, arguing for more continuous change. Misconceptions belong to the left side of this divide. The answer to the discussion may be found in microgenetic perspective (J. Wagner) - some research find incremental learning across many contexts, for instance when trying to learn the law of large numbers. diSessa thinks the continuist side will win, which will mean we will look at resources more than obstacles or misconceptions. (Personally, I'm not sure I believe any side is 100 percent right. Why can't both be right in different contexts? That is, that some development is gradual or stepwise, while other develop is mora abrupt and discontinuous?)

Kwon talked on "Double discontinuity between Secondary School Mathematics and University Mathematics". The double disconuinity concerns first moving from secondary school mathematics to university mathematics and then back (as a teacher). She then discussed Shulman, Ball and then Heinz et al ("School Related Content Knowledge") and Thompson (Mathematical Meaning for Teaching Secondary Mathematics).
While listening to her talk, I wondered if it could it argued that the Norwegian system, in which teachers for (lower) secondary schools are educated in teacher education programmes in which mathematics and mathematics education courses are merged, and where the mathematics is not "university mathematics" as such, avoids such double discontuinities? In fact, the suggestions she ended her talk with seemed to align almost exactly with the Norwegian model. But it must be stressed that for upper secondary schools, the problem is present also in Norway.

Borsch talked on "Transitions between teaching institutions". Individual trajectories are shaped by the institutions they enter, their activities and settings. The transitions between primary and secondary and between secondary and tertiary education, are much studied. There are many different levels of analysis which are present in the literature. The main differences between primary and secondary are pedagogy (interaction, autonomy, transmissionist) and discipline (specialist teacher, more division between subjects). Many proposals for smoothing the transition are found in the literature (and in the report).
The differences between secondary and tertary education are similar to differences between primary and secondary. In addition, teachers are researchers. There is more research on particular topics (i.e. Algebra) and there are more proposals. "Bridging courses" are discussed, but university mathematics content is rarely questioned. It is not clear whether all perspectives are equally represented in the literature.

Verschaffel talked on "Transitions between in- and out-of-school mathematics". Much learning and use og mathematics take place outside school. While this has previously been "romanticized", currently, more interest is in what happens at the boundaries. Of course, part of this is research on the didactical contract when playing the game of word problems, in which out-of-school experiences are often unwelcome. There are efforts to facilitate and exploit transitions, for instance RME, "funds of knowledge", Greer. Then the panel ended with a discussion, based on question from an internet forum, at which point I stopped making notes.

Then, there was the closing ceremony. Secretary general of ICMI, Abraham Arcavi, had a warm and pleasant speech closing the conference. Then there were many speeches of thanks to different contributors. There was a presentation of the next venue for ICME (Shanghai ). And finally, a wonderful musical number.

What did I get out of ICME13? (And why will I go to ICME14 in Shanghai in 2020?)

As I noted earlier, there are many possible outcomes from such a conference, and I can summarize some of them, in no particular order of importance:
  • Meeting new (or old) people with which future collaborations are possible. At this conference, I can think of at least four people I didn't know before, with whom there can possibly be some sort of collaboration some time in future.
  • Getting ideas for future research projects: During the conference, I wrote a list of nine research and development projects that I would like to do. Not all are new ideas, but many have been expanded while I've been here. And some are directly adapted from talks here. I only hope I can follow up on some of them when I get back home...  Hugh Burkhardt's talk on design research was very inspiring, and I want to do something in that direction (more systematically than I've done before).
  • Getting ideas for my own teaching (which can of course also turn into research projects): Marjolein Kool's project on making students creating non-routine mathematics problems, Megan Shaugnessy's and others' ideas on simulations of classroom situations (some of these concerning the notion of "noticing").
  • Getting an overview of a field which will make it easier for me to read more about it later: For instance, I hope it will be easier for me to face Brousseau now that I've had a tiny introduction and some context.
  • Listening to lectures and realizing that I do actually know something: it would be impolite to point out which talks contributed to this realization. But perhaps it is a good sign that for every ICME I go to, I think "been there, done that" a bit more frequently. This is not to critizise ICME (too much), of course not everything can be new to everyone.
  • Socialize with new or old colleagues: of course, the tendency to cluster based on nationalities may be seen as a problem (and meetings like the LGBT get-together may counteract that a little), but it is a reality that many countries have few meeting-places, so that socializing with colleagues from the same country does make some sense. There's been a lot of that here.

The main problem with ICME is of course its size - it's complete lack of intimacy. The trick is of course to find a home in a TSG, but still there will be situations when you're all alone and see hundreds of strangers walk past - which is a challenge for smalltalk-challenged people like me. But still, there is every chance that I'll go to ICME again in four years' time.


But before that, we'll arrange our own conference (ESU8) in Oslo in 2018. That will be fun - and intimate. And I hope I'll go to CERME in February, 2017.

Saturday, July 30, 2016

ICME13 Day 6 #icme13

On day 6 (Saturday) of ICME13, the plenary lecture was by Deborah Loewenberg Ball, whose work I have referred to a lot in my own articles lately. She talked on "Uncovering the special mathematical practices of teaching". As usual, I was eager to get signs of her definition of "mathematical" (see yesterday's blog). She organised her story in three parts: a journey, getting lost, finding the way. At some point, many years ago, there was a movement from wondering what mathematics teachers need in teaching to what they use. The "getting lost" part was building tools to "measure" teacher knowledge. On the positive side, ways of studying outcomes of teacher education and professional development were developed, but on the negative side we fell back from understanding practice to looking at knowledge. Also, it may have contributed to separating aspects of teaching, for instance equity. So in a way, we got microscope images of many aspects, but not understanding it from the inside. She argued that we have to work on the "mathematical work of teaching" mathematics. (Still not defining "mathematical" or "mathematics", but at least referring to previous talks this week.)

The work of teaching is taking responsibility for maximising the quality of the interactions in a classroom in ways that maximize the probability that learners learn. (This fits quite well with Biesta, doesn't it?) Using "work" is to focus on what the teacher DOES, not on the curriculum, what students are doing etc. "Mathematical" listening, speaking, interacting, acting... are part of the work of teaching. A fundamental part of the work is attuning to other people and being oriented to others' ideas and ways of thinking and being.

She then showed us examples from a grade 5 classroom with 29 pupils in a low-income community. Showing us different sequences, she asked us to pinpoint what the teacher was doing. (What I saw: Getting people to come up. Classroom management. Monitoring. Steering discussion (sociocultural norms), respond to question, saying " it doesn't have to be right") She pointed out how the teacher had walked around the classroom, reading nearly 30 student answers, choosing which to look into) Then Ball mentioned concepts from Cohen and Lohan(?) on assigning mathematical competence, including developing competence. The work of the teacher is partly noticing everything that is not the wrong answer. Translation from learning goals to meaningful language for children. The work of teaching is discursively intensive work: talking while monitoring if anyone understands what you're saying and understanding the answers.

I liked the talk very much. As I saw it, it was a step away from "the ball", which has worked to privilege some kinds of knowledge and hide away others, not to mention skills. The criticism of testing for MKT is also welcome. What comes instead, though, is not clear. I'm afraid "the ball" is too simple and pretty to be replaced by anything messier or more filled with doubt and judgements...

The first invited lecture I chose today was Ansie Harding's talk entitled "The role of storytelling in teaching mathematics". Her talk was geared towards tertary education, which is interesting, but I also hoped to learn things for primary education. Of course, storytelling is also an important way of including history of mathematics in primary school (re discussion group at previous ICME), so also from that point of view it should be interesting, as well as for my cooperation with the L1 teacher of my students is next academic year.

Harding teaches 400-500 students at a time at the University of Pretoria. Here, she first gave the story of storytelling, from about 50.000 BC. (Already at that point someone challenged the veracity of the story, which raises the question of the role of truth in storytelling - which is also part of the discussion concerning history of mathematics in teaching; which role does myths have in teaching?) In her story, she included movies, TV and the internet (not including games) - but the examples she used did not include movies - she stressed the value of low-tech storytelling. Instead, her first story example was the story of the length of a year, leap years and so on. (It is a bit difficult to see the difference between her story and normal lecturing, so maybe normal lecturing should be defined as well.) Features of storytelling: are essy to remember, are compelling, are for all ages, embed value, travel, fuel conversation. A story often has a character, ambition, problem, outcome as well as an emotional connection.

The value of storytelling in education is to entertain, inspire and educate. With tertary students, stories can be used as an intro (but is a bit like having the dessert before the main course), as a "by the way", as a commercial break, as a reward (which is how she uses it). You need to get everyone involved and create a "class feeling".

Ingredients: maths connection, human element, tale (start, flow, end), humour. Her second example was "How mathematics burned down the houses of parliament".

With engineering students, she uses the last ten minutes every week (which contains 200 minutes a week) for story telling. She showed a list of examples, much of which the usual history of mathematics "stories". Telling stories takes some effort (you must make it your own), it is a culture that you foster, there are unexpected rewards.

5-10 students leave before the stories all the time. She asked students for comments, and 30 students responded. Categories: emotional impact, reward, motivation, subject impact, appreciation (more approachable), bigger picture (more to maths than this course).

She ended her talk with two stories, one on De Moivre and one called "The story of one, five, seven" on van Gogh's sunflowers.

For me, this talk was an inspiration to dare telling stories more in my own teaching, and to put stress on them, not to tell them apologetically and quickly. However, "making it my own" will probably mean making sure they are relevant to the mathematics and often the history of mathematics that we are working on.

After a break, I was back to TSG47, where there were three talks:
Caroline Lajoie talked on "Learning to act in-the-moment: prospective elementary teachers' roleplaying on numbers". The observation was made in one lesson in teacher education. Her key concept was knowing to act in-the-moment. (See Mason and Spence 1999.)  in this context students take role of the teacher and 1-3...students, in front of the whole class. Role play involves introduction time, preparation time (everybody prepares all roles), play time and discussion time. She gave examples on how students, even though prepared, need to improvise. One took a risk in a situation where she was not certain of the answer, while another chose to switch to an explaining mode when he saw that he didn't know what would happen. This came up in the discussion, but that was because students were willing to discuss and "criticize" each other.

Pere Ivars on "The role of writing narratives in developing pre-service primary teachers noticing". "Narratives" in this talk refers to stories about classroom interactions. The hypothesis is that writing narratives will help noticing. Students wrote two narratives, with feedback after the first narrative (and some detail of the feedback given were provided here). The development from the first to the second narrative was more evidence of students understanding in more detail (it is important also to notice that students wrote the first narrative based on observation, the other based on what they themselves taught).

Lara Dick on "Noticing and deciding the "next steps" for teaching: "a cross-university study with elementary pre-service teachers." Noticing is a skill teachers need and that they need to develop. Much work have been done with using video to work on noticing, but little on use of student work (such as the previous one). The current project was a three-hour lesson on multiplication. The students saw student examples and analyzed one each and made posters. The students were asked to attend to the mathematics, interpret what they see and make an instructional decision - based on looking at all the posters. Four major themes: gravitation towards traditional teaching ideas, vague decisions, desire for written number sentences, focus on strategy progression. (This seems like a lesson it would be useful to try to copy with my students as well.)

(About TSG proceedings: there is a format suggestion from the ICME , but it is unclear how much flexibility there is. The suggestion was few but extended papers, while the TSG organizers would prefer shorter but more papers.)

These talks remind me that key concepts to meet for new teacher students are noticing and building on what students know instead of figuring out what they don't know.

After an extended break, I attended, of course, the HPM meeting. Luis Radford gave an introduction to the HPM  group, its research and publications, while Fulvia Furinghetti presented the history of the group. Then, Kathy Clark, the new chair of the HPM, gave a presentations with her own background and some views on the HPM. She mentioned two projects: ÜberPro project (Germany; Übergangsproblematik) and TRIUMPHS (US; transforming instruction in undergraduate mathematics primary historical sources).



This concluded the last whole day of the conference. However, discussions on mathematics education and other interesting topics continued for many more hours...

Friday, July 29, 2016

ICME13 Day 5 #icme13

In case you are desperately looking for the Day 4 post, please note that Thursday was excursion day. I chose to make this a Excursion-alone day, as there was a Manet exhibition in the Kunsthalle (right next to my hotel) that I did not want to miss. (It was great - and as so often, a good guide, in this case a multimedia guide, made all the difference. I also don't have much of a memory, so I was surprised at some of the cool paintings in the permanent exhibition, even though I saw them three years ago.) Moreover, of course I had to prepare for my talk on Friday, prepare for my workshops next weekend (at LAMIS in Ålesund) and also generally wake up after Wednesday night. So I spent some time with a cup of coffee and my paperwork in Lange Reihe, but also had dinner with colleagues in the evening.

Of course, the excursion day is also a day when you reflect on the conference so far. The simple question "Was it a good conference?" can be unpacked into subquestions such as Did you meet new (or old) colleagues that you may collaborate with in the future? Did you get ideas for new projects or for your teaching? Did you get an overview of topics that will make it easier to study them further when you get home? Did you get that warm feeling of hearing a talk and realizing that you actually know that topic quite well already? On at least those counts, this conference has so far been a success.

Friday started with a plenary lecture by Berinderjeet Kaur titled "Mathematics classroom studies - multiple windows and perspectives". The TIMSS Video Studies (1995 and 1999) was an inspiration for further classroom studies. They found that teaching varied a lot between different  cultures - and we are not aware of the differences. The 1995 study showed different patterns for different cultures. The 1999 study introduced a new terminology of wide-angle lense perspective and close-up lense perspective, and argued that both are needed. (By the way, I was lucky enough to get access to data from this study to do a study of history of mathematics in the seven countries. But that is many years ago now.)

In the late 1990s, there were a couple of studies involving Singapore, the Kassel project and a study of Grade 5 mathematical lessons. In the Kassel study, there were lesson observations. There was a need of a shared vocabulary when talking about classroom activities (for instance, she mentioned that the term "Singapore maths" does not make sense to her). The study gave a wide-angle view of Singapore  (pretty much in line with what one could expect). The next study consisted of just five lessons, also without many surprising results.

In 2004, Singapore joined the Learner's Perspective Study (LPS). This study investigated from the perspective of students. Three teachers participated in Singapore, a ten-lesson sequence for each were recorded. To code lessons, they wanted to take into account the instructional objectives of the teacher (which seems to make sense), but a teacher always have more than one objective to a lesson (which many school administrators don't seem to understand). This made for complicated analysis. The close-up lens gave a different picture than the wide-angle lense. Lessons were well structured, objectives were clear, examples carefully selected, student work carefully selected for classroom discussion. In addition, students were interviewed after the lessons. This resulted in a long list of characteristics that the students valued (which seems quite similar to what teachers try to achieve).

She ended by arguing that the stereotypes about Singaporean mathematics are not correct, especially when you use a close-up lense. (This is not fully convincing to me, at least not if based mainly on the study of three highly qualified teachers.) She also noted 2010 survey data that showed that teaching for understanding is strong in Singapore, and the analysis shows clear links between the different instructional practices.

Then I attended Michael Fried's talk on "History of mathematics, mathematics education, and the liberal arts". I have heard Michael several times before, and knew I couldn't go wrong there. He started by pointing out that he will not talk about history of mathematics as a tool (referring to Jankvist). Here, the point is history of mathematics as a object of study, not as something to use.

First, he started with D. E. Smith, who was interested in history of mathematics and also set in motion the idea of ICMEs. In another writing, Smith claimed that the motives for teaching arithmetic is either for its utility or for its culture. History of mathematics should be at the very heart of culture. He used both the "parallelism argument" and history mathematics as a filter showing us the importance of each part of mathematics. Fried argued that Smith's view of mathematics is ahistorical - he looks at mathematics as a set of eternal truths. Hence we see that the view of mathematics is not the result of not knowing enough history. This view of mathematics is what allows it to be a tool. And oppositely: allowing history to be a tool, makes it non-historical.

He then looked at "religio historici", where original texts are at the heart of doing history. It matters how you consider the past (of course, we know that parts of the past have been deemed as uninteresting by historians at times). The past seen as just what lead to the present (a Whig interpretation of history) is uninteresting, "practical history". Oakeshott and Butterfield wanted us to see the otherness of the past.

This unhistorical history is particularly tempting in history of mathematics, because mathematics is seen as eternal. He stressed the different concerns of historians of mathematicians and teachers of mathematics. The demands on teachers means that teachers will not put modern mathematics aside to teach history. Thus, teachers needs to economize and make history of mathematics useful - thereby almost neccessarily embrace a Whig view.

In 2001 Fried claimed there was a clear choice between teaching Whig history or to drop the idea of being useful. This damns all hope of including history of mathematics in mathematics education.

However, in principle, one can ask how mathematics education can be conceived to include history of mathematics as an integral part. Here he named people like Radford, Barbin, Jahnke, Jankvist, Clark, Guillemette as examples of people who can be seen to be working on this. (This means fighting simplistic uses of the word "mathematics" as learning methods, which can be glimpsed in some research articles. I have often questioned what people mean by "mathematical" in the phrase "mathematical knowledge for teaching". In my opinion, the history of mathematics is an integral part of mathematics, in the same way that you can't separate literatur from its history in a meaningful way.)

Then he went on to speak of the history of the "liberal arts", where mathematics was a self-evident part. (He spent a significant time detailing this, including pictures, which I cannot describe here.) The liberal arts was supposed to be what you needed to become fully human. History was never considered part of them. Today, history is considered part of the liberal arts, while mathematics is missing. This could be fixed, and history of mathematics can be seen as a way of solving it. Mathematics education then also becomes a way of reflecting on ourselves.

(In Norwegian teacher education, I'm not sure the argument is exactly the same. We talk about so-called theoretical subjects and so-called "practical and aesthetical subjects". But similar arguments can be made - of course, we know that mathematics is also practical and aesthetical, as history of mathematics can also show.)

This was a thought-provoking talk. I believe that I will - as usual - end up in a pragmatic view where including history of mathematics in different ways and with different goals will still be better than doing nothing. I believe in teaching history of mathematics as a goal, but that this can still be "useful", and that teaching history of mathematics as a tool can still instill a sense of "real" history. Teaching is never perfect, it is always in need of judgment and a balancing act between different concerns, so why should this area be different?

Then it was back to the TSG. Derya Çelik talked on "Preservice mathematics teachers' gains for teaching diverse students". As she could not come, the talk was sent as a video. She talked about a project with 11 researchers. They analysed PSTs opinions on how often the program provided opportunities to learn about teaching students with diverse needs. 1386 PSTs took part. "Teaching for Diversity" scale was used. In general, the results showed few opportunities, although there were some (significant) differences between regions, with more developed regions scoring higher. This fits with TEDS-M results (also for Norway).

Then I presented the work of Eriksen, Solomon, Rodal, Bjerke and me, with the title ""The day will come when I will think this is fun" - first-year pre-service teachers' reflections on becoming mathematics teachers". I did not make any notes during this talk, as everything felt quite familiar... :-) The people present seemed interested, though.

Oğuzhan Doğan talked on "Learning and teaching with teacher candidates: an action research for modeling and building faculty school cooperation". He started by stressing the importance in teacher education in improving teaching practices. High quality field experiences can contribute, while poor quality field experiences will support imitation. The aim in this project is to find ways of improving. They planned a hybrid course where teacher educators and PSTs plan mathematics tasks and activities and apply them in a real elementary classroom.  (He gave some details on the design of the course which I can't repeat here.) At first, they applied tasks that the teacher educators had made, but then they applied tasks that they had made together. They saw that teacher candidates left the idea that the answers were the most important. They also went from exercises towards discovery, and saw the important role of manipulatives. The teacher involved also used the tasks given in other concepts. For the future, they plan to do something like this in the compulsory course, even if this course does not go on.

Finally, Wenjuan Li talked on "Understanding the work of mathematics teacher educators: a knowledge of practice perspective". She pointed out that there has been a lot of research on PSTs and teachers, but not on what MTEs (mathematics teacher educators) need to know. She used a distinction between knowledge for/in/of practice. Of course, we need to include both mathematics and mathematics education knowledge. The study further hopes that studying what MTEs do, will inform us on what they need to know. (This is a bit doubtful, and moreover, just as with mathematical knowledge for teaching, what the particular MTEs don't know, will be missed in the model even if useful.) They included only six MTEs in this project. All had school experiencem but they had limited knowledge of research. (Which means that the results will probably be different from results in a similar study in another context, with MTEs with another background. And, remember, parts of the rationale of having teacher education in universities, is that the MTEs are actively doing research and development work, so that they will have very different specialities which (hopefully) contributes to their teaching in different ways. Of course, the same criticism can be (and probably has) been directed to Ball's work etc. But even so, the results could be interesting to build upon with data from other contexts.)

After lunch, I heard Ronald Keijzer's talk on "Low performers in mathematics in primary teacher education". In the Netherlands, there is a third year national mathematics test (because of PISA and TIMSS results, comparable to in Norway). The project investigated characteristics of the students who did not pass this test. (Could it be seen already in the first year?) Both interviews (n=12) and a questionnaire (n=265) were used. Previous mathematics scores predicted score in third year test. Low achievers specialize in teaching 4-8 year olds (not 8-12). Many similarities with other students, but they are less enthusiastic than those who passed the tests.  They disagreed that they were low achievers or did not work enough or that they didn't get enough support. Rather, they blamed the test (interface, content, preparation, feedback) and personal factors (dyscalculia, stress,  concentration). So the conclusion is that low performers are often that for a long time, they often blame the test or personal factors. There was a lack of self-reflection. Comment from the room: mindset treatment as a way of increasing self-reflection.

After this, I needed to meet "my old group", TSG25 (on History of mathematics), where I heard several talks throughout the evening:

ChunYan Qi talked on "Research on the problem posing of the HPM". She referred to MKT, in particular SCK, and she focused on problem posing based on HM. The research look at 68 problems based on (one?) problem from Chinese history. She listed three strategies for making problems based on history (but I didn't - in the very short time - quite understand why these three were chosen or how we would know that the results would be problems). Then she discussed which strategies were preferred by students and other findings that turned up when analysing their problems.

Tanja Hamann gave a presentation on "A curriculum for history of mathematics in pre-service teacher education". She started by referring to Jankvist, noting that using history as a goal is perhaps a bit more important for teacher education than at other levels, while history as a tool is more a general concern of all levels. But of course, you often work on both at the same time. She argues that HM is important for teachers to influence beliefs, give background knowledge for teaching, building up diagnostic competences, providing an overview of mathematics, being able to identify fundamental ideas, building language competencies.

They try to implement history in a long-term curriculum, including history throughout the topics as small parts. They also have a special lecture on mathematics in history and daily life. Finally, they use history in exercises and tasks. For all of these three components she conjectured which goals they could contribute to.

And after a break:
Jiachen Zou spoke on "The model of teachers' professional development on integrating the history of mathematics into teaching in Shanghai". He presented a model of how teachers, researchers and designers can work together to design teaching integrating history of mathematics. The model was in three dimensions with many colors and many hermeneutical circles, and can not be described in words. It was unclear to me how the model was developed (based on what data), maybe that would have given me better understanding of the model. However, as always, time is limited. (He was also asked how the ("somewhat formalistic") model were developed, but it was not clear, except that the model had been used to describe three different teachers' paths.)

ZhongYu Shen on "Teaching of application of congruent triangles from the perspective of HPM". His project concerns 7th grade in Shanghai, with two teachers involved. The stories of Thales (finding the distance of a ship at sea from the coast) and Napoleon (a similar method) was used as a starting point, in addition to a Chinese measuring unit for length. They designed a lesson based on this, and had a simple survey asking whether they understood and liked this, and not as many liked it as understood it.

Fabián Wilfrido Romero Fonseca on "The socioepistemologic approach to the didactic phenomenon: an example". The example was three moments from history of Fourier series: the problem of the vibrating string, Fourier's work on heat propagation and development of engineering as a science. He showed how activities were based on the analysis of the historical background, where Geogebra were used to investigate. However, the activities have not actually been used, just developed as part of a master thesis.

Thomas Krohn's talk was "Authentic & historic astronomical data meet new media in mathematics education". The students in question were in 10th, 11th and 12th grade. An authentic problem was to describe the movement of comets, and this can be used by giving students some historical/astronomical background, as well as the neccessary tools. The astronomical data are often presented in lists in ancient books which are on the internet. As in ancient times, they first made a projections from 3d to 2d, and then they tried to find a reasonable function (often preferring to investigate instead of leaving it to a computer to find.

Slim Mrabet, whose title was supposed to be "The development of Thales' Theorem throughout history", did not turn up.


Thus ended the Formal parts of day five.

Wednesday, July 27, 2016

LGBT get-together at ICME13 #icme13

For most LGBT (lesbian, gay, bisexual, transgender) people born in the 20th century, "coming out" never ends. Of course, the first few people you tell may be the most difficult ones, but years later, meeting new people still involves small questions in the back of the mind: Should I be open? Will they mind? Should I use gender-neutral words ("my partner") to avoid having to face potential negative attitudes?

I still feel that way when meeting new people back home in "safe, liberal, politically-correct" Norway, but I feel so even more in international contexts, where people from the whole world sits around the same table. At international conferences, there is the added consideration: I'll only meet the people for a few days every few years, so why ruffle any feathers? Why risk causing someone to be upset? Couldn't we just stay away from "controversial" subjects such as the gender of my husband?

The concrete result of this is that I (and many others, I assume), feel less free to be ourselves, to mention my husband where it's otherwise natural in conversations and so on. I get (even) more reserved than I usually are. As a result, others get to know me a little bit less, and I get to know others a little bit less. (Anecdotal evidence: there are people who I've met and talked to at conferences over a period of ten years before we realized both were gay.)

This long preample is my answer to the question I asked myself when invited to a LGBT get-together at ICME. I thought "Cool. But why?" I thought it would be strange meeting people here based on sexual orientation or gender identity, and wondered what we could possibly have in common. Now I've realized that having such a meeting is just as silly an idea as having "gay pride" events: it is silly, but still neccessary, because not having such meetings means going on with the status quo, where everyone makes their own micro-decisions of not telling anyone so as not to cause (possible/imagined) offence.

The LGBT get-together was yesterday, and it was great. Surprisingly (?) I even met people interested in the same part of maths ed as myself. Many thanks to Pauline and Nils for getting the idea and making the idea reality, and to the HIV prevention center for the venue and help.


I hope I will see lots of the people who were there again, and that this event will contribute to a more inclusive ICME. I also hope these things will be repeated at future ICMEs, as long as neccessary. Silly or not.

ICME13 Day 3 #icme13

Day 3 of ICME13 started off with Günter M. Ziegler's talk ""What is mathematics" - And why we would ask, where one should experience or learn that, and who can teach it". According to the twitterati, it was good and entertaining. I missed it.

In conferences such as this, it is important to break away from the programme to have time to reflect and to interact with other participants. (Unike some other conferences I go to, this conference does not have many workshops or discussion arenas.) Therefore, I skipped the plenary and an additional lecture, but was back on track after that, for four more TSG-talks:

Skip Fennell talked on "Preparing elementary school teachers of mathematics: a continuing challenge". He started by giving a brief historical background. He than pointed out that in many countries, elementary teacher education leads to teaching one or two subjects, while in the US (and Norway until now), teachers become generalists. Also, in many countries there are a limited numbers of programs, sometimes hard to get into, while in the US there are more than 1700 programmes, some online. There are voices arguing for having math specialists in elementary school. (In Norway, we are switching to a programme where you have to have a master's degree in one subject, but still will teach two or three other subjects.) However, in 20 states, there are math specialists, often called "maths coaches". These positions are, however, vulnerable in touch (economic) times. In the Q&A, he conjectured that there must be significant challenges in having teacher education programmes which provides everything except practice periods online. We all know the difficult negotiations between campus and practice, and these cannot be easier in such a context.

Marjolein Kools's "Designing non-routine mathematical problems as a challenge for high-performing prospective teachers". She started by giving a non-routine mathematical problem. In Netherlands, there is a national test in the third year of teacher education, and this project concerned creating non-routine problems for PSTs to work on. The got PSTs to help them, but it was very challenging for them to produce task of the right level of difficulty. Therefore, the project turned into a project trying to support the PSTs in creating, and this became a design research project. At the end, 4 of 8 students could design non-routine problems independently. Key to this result was master-classes where Ronald created problems while speaking aloud, trying them out and so on. (Although the starting point of this project was a national test, it seems worthwhile to work on this even in countries which are lucky enough not to have such a national test. It would be cool to do something like this in Norway as well...)

Eda Vula's "Preservice teachers' procedural and conceptual understanding of fractions".  She referred to Ma, Shulman, Ball etc. as starting points for her talk. The research questions concerned representations and the relationship with and between procedural and conceptual understanding. 58 PSTs participated by doing a 20 item fractions knowledge test. The test obviously unearthed problems, for instance in the meaning of unit. PSTs had good procedural knowledge, but rarely good conceptual knowledge in addition.

Megan Shaughnessy's "Appraising the skills for eliciting student thinking that preservice teachers bring to teacher education". Of course teachers need to be able to find out what their students are thinking. This project takes this point into teacher education - how do we find out what our students think. The idea here is to use standardized simulations. PSTs get a student response and prepare an interaction with this "student",  then they interact with his student (who is a teacher educator trained and using a response guideline). She showed a video to illustrate how this worked. All PSTs did this, and the videos were analysed, and the findings were really interesting (but I can't summarize them here). The study shows some of the moves that PSTs need to learn, what can be built upon and what requires unlearning. (And of course the videos themselves may be useful to work on thIs.)

After lunch, I attended the "thematic afternoon" on European didactic traditions. First, there was a "panel"  with Alessandra Mariotti, Michéle Artigue, Marja Van den Heuvel-Panhuizen and Rudolf Strãsser. They had identified five characteristics of the the European traditions: strong connection to mathamatics and mathematicians, strong role of theory, role of design activities, key role of empirical research.

Mariotti talked on the role of mathematics and mathematicians. Historically, mathematicians were important figures in discussions on how to prepare mathematics teachers. In the beginning of the 20th century, Klein had a central role (Erlangen Program). Also, after WW2, mathematicians had key roles in the New Math reform(s). Later, mathematicians also played important roles in the French IREMs, for instance, where mathematicians, teachers and didacticians mixed. In Germany and Netherlands, on the other side, mathematicians played a smaller role in turning mathematics education into a research field. The case of proof as a theme of research have been a feature, and this may be influenced by the role of mathematicians.

Artigue talked on the role of theory. She started with the French tradition, which she claimed had a particular importsnce in building a theoretical foundation. These had a systemic vision of the field, a strong epistemological sensitivity, theories mainly conceived as tools for understanding. New development are connected to previous theories. In the Dutch tradition, Realistic Mathematics Education is the main contribution. This is a theory oriented towards educational design. The theories are still refined. In Italy there is s long tradition of action research and collaboration between mathematicians and teachers, now continued in Research for innovation. In the Germsn tradition, the theoretical landscspe is not so easy to synthesize. (The rest of Europe is left out in this survey.)

Van den Heuvel-Panheuzen talked about the role of design activities. The French tradition: essential component of research work and based on theoretical frameworks. In the German tradition, design activities took place in context of Stoffdidaktik, evaluation was often left out. Empirical turn from 1970s. Italy: empirical analysis as base for didactical innovations, pragmatic approach. From 1980s innovations have turned to research. In the Dutch tradition, the design aspect is the most significant characteristic. Theory development resulting from design activities, later theory was used as a guide for futher theory development.

Strässer talked on the role of empirical research. Different kinds of empirical research have developed because the complexity of the field. Examples: the large-scale study COACTIV, the case study MITHALAL. Of course, there are also a distinction between qualitative and quantitative methods. The purposes can be prescriptive or descriptive, and they may be developing vs illustrating theory. Action resesearch have been stronger in I and F, while fundamental research, strongest in F, G, N.

In addition to the content of the talk, what will surely be remembered about this evening is the heat and humidity of the auditorium, not least when it was decided to close all windows to close the curtain to improve the visibility of the screen slightly. It is a testament to the participants' dedication to the disipline that so many chose to come back to another session after the coffee break.

I know far too little about the French tradition, so I chose to go there. I'm also struggling to learn French, and hope that I will be able to read some of the French work later. Brousseau, Chevallard and Vergnaud were the stars of this bit of the program. IREMs were established from 1968, leading to a fruitful collaboration between mathematicians and teachers. From 1975: first doctoral programs, from 1980: journals.

The first theoretical pillar was the theory of didactical situations (Brousseau) - with the "revolutionary" idea that the pbject of research was the situation. Centre of observation for research created in 1972. The core concepts had been firmly established through the 1980s. The second pillar: the theory of didactic transposition and the ATD. (Chevellard) presented in 1980, followed by book in 1985. We "embed an activity into a whole whose ecology is studied". The third pillar was probably the theory of conceptual fields (Vergnaud), which seemed to be taken for granted in this meeting.

There was a talk on research on axial symmetry in France, to illustrate a development. There has been a development from the study of students' conceptions, combining the theory of conceptual fields and the theory of didactical situations to the study of teachers' practices and of their cognitive effects. Now, French research on symmetry is more occupied by the question of language. There was another talk on French research on school algebra. The first studies were in the 1980s, by Chevallard, in terms of didactic transpositions. Three questions:
  1. what is school algebra, and what is (or have been) considered as algebra? (Didactic transpositions, institutional constraints)
  2. What algebra could be taught? Under what conditions? How implement them? (Reference epistemological models (REM), didactic engineering (new ecologies))
  3. How do ICT modify the nature and the way of using, teaching and learning algebra? (Concept of ICT as part of the adidactic milieu, taking into account the instrumental genesis, didactic and computer transpositions)

At this point, the thing turned into a Q&A session in which the questions were unintelligible while the answers were moderately more intelligible. I chose this point of time to leave the auditorium. However, there was one interesting question that I was able to understand. It was something like this: "It seems that the research questions the French pose about algebra are quite similar to research questions elsewhere. In what ways do the answers differ?" I did not catch the answers, but I made up my own: the problem/challenge with having different theoretical perspectives and traditions are exactly that we may pose the same sorts of questions but look at them from different points of view, and we therefore have an inkling that the answers will be (slightly or significantly) different. However, since the answers are themselves given within different theoretical traditions, pinpointing exactly how they differ demands exactly the same comparison and analysis of perspectives (or perhaps even the creation of a "meta-theoretical perspective") that is some of the point of this thematic afternoon and similar work. If we could easily explain what the different traditions accomplishes, we would be at another point of the development than we are. Perhaps.

Thus ended the third day of the ICME13 conference. I had a very good experience in the TSG, getting inspiration for further work. The "thematic afternoon" was an interesting experiment, but the different backgrounds of the participants must have made it almost impossible to plan.


After this, there was social gathering again, followed by an (unofficial) LGBT get-together of ICME participants. I chose to post this before these events, though, to avoid the temptation to blog about them

Tuesday, July 26, 2016

ICME13 Day 2 #icme13

The second day of ICME13 started with a plenary lecture by Bill Barton, with the title "Mathematics, education and culture: A contemporary moral imperative". He started by remembering his first ICME (Adelaide) where Ubi d'Ambrosio was the very first speaker he heard. This inspired him. His other starting point was that pleasure is an important part of mathematics - the pleasure of viewing fractals or of seeing a beautiful proof. (Hm - I often think more of the fun of mathematics than of the pleasure. Maybe there is more to be done in categorizing the pleasures of mathematics?) He asked us to look at the Klein project - and to contribute with translations.

He went back to Ubi's 1984 talk on ethnomathematics. Later, Ubi has spoken about mathematics as the dorsal spine of modern civilization. Our goals should we responsible creativity and ethical citizenship. But how? According to Ubi, we need to include the cultural roots.

Barton invoked Ecological Systems Theory (Bronfenbrenner). He used five levels: microsystem institutions directly involved), mesosystem (links between microsystems), exosystem (wider social setting - for instance financial world...), macrosystem (cultural context etc), chronosystem (pattern of environmental events and transitions, as well as socio-historical circumstances). He also invoked Ecological Humanity - seeking to bridge the divide between science and the humanities. The links between organisms define how the whole system works. He also added Deep Ecology (Arne Næss) into the mix.

Barton then distilled this into three principles: perspective, reflexive, pleasure. The perspective principle: be aware of other ways of understanding. The reflexive principle: do unto others as you would have them do unto you. The pleasure principle: act so as to increase pleasure. (All of these on both individual and system level.)

Was the mathematical community responsible for the global financial crisis? It can be argued that it was triggered by the models. Do mathematicians love maths education as much as maths educators are expected to love mathematics? And do we as maths educators nurture our love for the subject? Do school teachers have the time and resources to nurture their love for mathematics?

We all need to make efforts to ensure mathematics education benefits from multiple points of view. ICMEs should be more language friendly.

He critizised streaming, which research shows increases the effect of socio-economic factors (and so on). He compared this to having a (public) cooking test at a certain age, and the outcome of the test decides who you can eat with and what you can eat for the rest of your life. Fo 14 years we ask students to practice basics without being offered the opportunity to experience the pleasure of mathematics. And we know the effects of high stakes testing. (Of course, Barton speaks in headlines, nuances are not included. But I guess that is okay from time to time, too often we speak with all nuances and no headlines.)

After a coffee break, there were lots of parallell invited lectures. I decided on Anthony A. Essien's on "Preparing pre-service mathematics teachers to teach in multilingual classrooms: a community of practice perspective". This is obviously very relevant to my context in Oslo, Norway, and too little of what I know about this is situated in mathematics learning. (Moreover, I do like Wenger's perspectives.) Essien works at Wits University in Johannesburg.

There is a triple challenge in multilingual maths classrooms: paying attention to mathematics, paying attention to English (or other language of instruction), paying attention to the mathematical language. (I would perhaps add attention to the other languages the pupils know to the extent possible.) How do we prepare PSTs (pre-service teachers) for this?

Essien uses Dowling and Brown (2010)'s idea of organisational language, relating to empirical field and theoretical field. Wenger (1998)'s communities of practice can be looked at through three dimensions mutual enterprise, joint enterprise, shared repertoire. But Wenger did not develop his theories for teacher education in mathematics. COP theory does not have a coherent theory of language in use. Essien uses Mortimer and Scott (2003) for this: meaning making as a dialogic process (DP).  He developed cathegories based on literature and data (not possible to summarize here...)

There are interacting identities at play: becoming a teacher of mathematics/in multilingual classrooms, becoming learners of maths/of maths practices, becoming proficient English users for the purpose of teaching/learning mathematics. Code-switching is part of this (maths practices). He studied privileged practices-in-use. He showed how different universities teach in much the same way, but still encourage development of different identities. (For instance: when defining, one does it purely mathematically, the other focusing on how it could be explained to children. Thus, the second may be expected to support the development of an identity as a mathematics teacher.) The perspective of multilingualism was lost somewhere on the way, however.

A question from the room concerned whether the teacher educators were explicit in what role the PSTs should assume (as in "consider this as a teacher" or "consider this as a mathematics learner") The person asking suggested that PSTs may easily be confused if this is not clear and explicit. I think this is one of the most important points for getting students to move from their role of pupil into their role as prospective teacher. (By the way, it was really refreshing that there was time for questions, after a day of conferencing with little interaction. The discussion was interesting and fun.)

(An idea I got for an early task with my new students this autumn: "What is the most difficult thing you know well in mathematics? How would you explain this to a learner who don't know this too well in advance?" (And as an added perspective "who knows another language better than Norwegian?"))

(Based on yesterday's talk on design research, I also get inspired to look at the design on history of mathematics-based mathematics lessons, by way of analysis of published lessons or by interviews with lesson designers.)

Much of the ICME programme is filled with the TSGs (Topic Study Groups). I usually attend the TSG on history of mathematics, but this time I attended TSG47 (Pre-service mathematics education), because this is where the project I will present on Friday fits in. For a conference this site, it is important that people have a "home" where they meet the same people more than once, but still I'm a bit uncertain about the weight the TSGs have got in this conference - for instance, most of Tuesday was filled with TSG sessions - mostly filled with ridicuously short presentations (10 minutes each). We'll see how it works. (Of course, the weight of the TSGs is a function of the number of the contributions they receive and accept ("my" TSG has 66 accepted contributions), which again is a function of how broad their themes are. So it is not an easy problem to solve.)

In this first session, there were four talks. Fou-Lai Lin (with Hui-Lai Lin) presented a talk on "Using mathematics-pedagogy to facilitate professional growth of elementary pre-service teachers". Hui-Lai started by discussing a game where the point is to arrange coins as a rectangular form. (I didn't get the details of the game, but I see ways of doing this so that lots of concepts will be involved, for instance where children get points based on how high the lowest factor can be.) Fou-Lai started by telling that Taiwanese mathematics teachers are not good in mathematics, sometimes even fearful of it. She gave a list of principles for designing PST-courses, which I didn't manage to note down. She showed an example with binary numbers. PSTs should see analogy to elementary student learning. She also stressed the importance of not only that students should read textbooks themselves, but analyse them and conjecture how pupils will struggle with the textbooks.

Secondly, Roland Pilous talked about "Investigating the relationship between prospective elementary teachers' math-specific knowledge domains." The focus of the study was relationships between the domains (and in talking about domains, it seems he was inspired by Ball et all's domains). The project was based on six students and three lecturers. Four domains were identified: curricular knowledge, content knowledge, teaching-related knowledge and knowledge of student cognition. (The problem of such a tiny number of informants is of course that whole domains of knowledge may be missing. If, for instance, history of mathematics is not mentioned by the twelve informants, that may be lost to the model.) He very quickly mentioned some of the relationships, but of course time was too short to get an understanding of this.

Thirdly, Jane-Jane Lo had a talk with the title "A self-study of integrating computer technology in a geometry course for prospective elementary teachers". This is a study of her own teaching, analysing the lesson plans and reflecting on her choices in hindsight. You can use tools for technical activity or as conceptual activity. Dick and Bos discuss pedagogical fidelity, mathematical fidelity and cognitive fidelity. Use of technology should not compromise your pedagogical principles or mathematical correctness, and the external representations should facilitate the mathematizing processes. Her teaching was first using Geometer's Sketchpad and Scratch. For the next iteration she used lots of apps etc (for instance from learner.org). The problem is that there is a learning curve for each app or software. Now she has gone back to Geogebra exclusively. (And there are lots of things available at Geogebra Gallery.) Her main challenges is the connection between physical, virtual and mental representations - how much is needed of each - and to facilitate mathematics discourse when each student has a computer in front of them.

Fourthly, Ryan Fox discussed "Pre-service elementary teachers generation of multiple representations of word problems involving proportions".  He used Ball and Rowland for background literature. He stressed the difference between the way PSTs were taught in school and the way they will be expected to teach. (This will certainly be the case - no PST can foresee which school cultures they will work in during their careers.) This study was included four participants, all in their second year of teacher education. They were interviewed four times, totalling 2-3 hours per participants. The interviews concerned how the PSTs would teach certain topics. His students mostly went for one solution method - one struggled a lot, finally found a solution, stuck with that but forgot it by the next interview, another never struggled, but kept using the same solution method. But the fourth student managed to use four representations for the same problem.

After lunch, the TSG continued with two more talks. Gabriel Huszar talked on "An exploratory study about the reseponses of the prospective primary teachers using the concepts of measurement in maths". He started by referring to PISA and other studies, showing the need for intervention in Spanish education. This project tried to identify the most common conceptual problems and mistakes among PSTs in measurement. 81 students took part in the study, and the measurement instrument was a questionnaire of 12 tasks taken from PISA and PIAAC. Sadly, it was hard to see the graphs with the results in this talk. Of course, tasks which does not only ask for a repetition of algorithms, were more difficult for them.

Then, Mine Işıksal-Bostan talked on "Prospective middle school mathematics teachers' knowledge on cylinder and prism: generating definitions and relationship". This was a video presentation, probably due to the difficult situation in Turkey, due to mr. Erdogan, although this was not mentioned. She discussed the importance of definitions in teacher knowledge. She referred to Tall&Vinner, and Zazkis&Leikin. In this talk, the focus is cylinder and prism. Of course, the formal definitions are different from the definitions used in textbooks in primary school. She gave examples of how cylinders and prisms are introduced in Turkish textbooks. The project were based on questionnaires and interviews. The answers were catagorized according to whether they gave sufficient and minimal conditions (she did not explain why this is an interesting analysis to do - for primary school it may be more interesting that the definition is understood and usable than that it is minimal). They were also analyzed for prototype/non-prototype and other characteristics. In conclusion, the concept image seemed more "correct" than the concept definitions.

(Could I create an activity in which students get a long list of definitions, in which they discuss what are the differences between them?)

After another short break, there were five more presentations in the TSG. My colleague Annette Hessen Bjerke talked on "Measuring self-efficacy in teaching mathematics". She noted that mathematics teaching self-efficacy is particularly interesting, as many express low self-efficacy in maths teaching. She refered to Bandura's concepts of outcome expectancy vs. personal self-efficacy - her research looks at personal self-efficacy in tutoring children. She developed a new instrument for this, with 20 items, 10 concerning "rules" and 10 concerning "reasoning". She mapped self-efficacy through the compulsory mathematics course in Norway. Rasch analysis was used with an instrument that had been validated. She described how it can be shown that the PSTs' self-efficacy improved, but also that there are some students that have decreased self-efficacy at the end of thr course. Further research (ongoing) will investigate the sources of self-efficacy.

Lenni Haapasalo discussed "Assessing teacher education through NCTM standards and sustainable activities". He had developed instruments to look at self-confidence of different groups of students, and showed how they developed. However, in the short time, it was difficult to understand details of the instruments, whether they had been validated and so on. He added some comments on the sad state (according to him) of the Finnish educational system and how his colleagues (still according to him) earned money by travelling around the world talking about how good the system was.

Siyin Yang had a talk titled "A comparison of curriculum structure for prospective elementary math teacher programs between the United States and China." She studied two different elementary educational programs. She described the different structures of the programs she had chosen to look at. Of course, the details of the comparisons must be looked up in her papers.

Gabriela Valverde Soto's talk was "Enhancing the mathematics competencies of future elementary teachers: review of a design research". Her chosen topic was ratio and proportionality. She described the phases of design research. This was a rich presentation from a big research project including more than a hundred audio recording, and the analysis looked for productive exchanges, among other things. Again, I can't try to summarize.

Finally, Oleg Ostrovskiy talked on "Visual representations of word problems". External representations can help us solving word problems partly because word problem solving correlates with working memory. He pointed to the Singapore method for solving word problems. The project involved 17 PSTs, using the book "Elements of modelling". Pre-tests and post-tests showed that more PSTs used visual representations to solve the problems, and more adequate ones, after the intervention than before. However, it takes a long time to learn these ways of illustrating - it is not enough to just show the students the method a few times.

In all, this was 11 TSG-related talks in one day, most of them ten minutes long, therefore quite hurried while still missing out on significant points of the projects. I'm sceptical of the ten-minute format, even though they can of course be seen as a huge number of samples to inspire us to read the papers (even though the papers are only four pages long, most of them, so they are also ridiculously lacking in detail).


After this, there were another seven hours of partly mathematics-related discussions before I got back to my hotel, but I won't try to summarize these hours later, important though they are.

Monday, July 25, 2016

ICME13 Day 1 #icme13

The first day of ICME13 (The thirteenth international congress on mathematical education) was Monday 25th of July, 2016. This conference takes place in Hamburg, Germany. As usual, I will try to blog from the conference (as I noted yesterday).

I missed the conference opening ceremony, but got there in time for the plenary panel on "international comparative studies in mathematics: lessons for improving students' learning". (Not that difficult, as the conference was at that time 40 minutes behind schedule.) The panel participants were Jinfa Cai, Ida Mok, Vijay Reddy and Kaye Stacy. The panel was based on a paper to be published (open-access) shortly.

The panelists each talked on one lesson to be learned about international studies, defined as studies involving at least two "countries", with an intention to compare them. Such studies are good to understand ourselves and to understand different possibilities.

Lesson 1: examining the dispositions and experiences of mathematically literate students. Kaye Stacy talked about large international comparisons, PISA in particular. She showed how PISA 2012 were trying to unpack the scores by looking at the processes of doing mathematics. Different countries had different patterns - for instance, Japan had a high score on formulating, lower score on employing and interpreting. Netherlands and the UK had very different profiles. (English-culture countries were best in interpreting.) This fitted not very well with students' answers about their exposure to different kinds of tasks. She also showed graphs showing huge gaps in confidence between boys and girls in some countries (i.e. Australia, but not Shanghai). We need to draw evidence from different sources to inform policy.

Lesson 2: Understanding students' thinking. Jinfa Cai talked about small-scale studies, trying to understand students' thinking when trying to solve tasks. He showed two tasks, one where average is demanded, the other where the average is given. About a fourth of students in both countries got the first task right and not the second task.  He showed some of the different errors in the second task - most concerned adding and dividing, as in the (correct) solution of the first task.

He also showed the "pizza ratio problem", where he also gave different solutions, some of which were almost only used in one of the two countries studied. Thus, comparative studies gives a broader sense of possible ways of thinking.

Lesson 3: Changing classroom instruction. Ida Ah Chee talked about TIMSS Video Study and Learner's perspective study. They have complimentary roles. TIMSS Video Study showed that teaching was a cultural activity. LPS compares lesson structures and lesson events and looks also on practitioners' thoughts about their lessons. Lesson events are as different as lesson structures. Analysing structure and events in light of teacher interviews give an even richer picture than the TIMSS Video Study did.

Lesson 4:  Making Global Research Locally Meaningful: TIMSS in South Africa. Vijay Reddy talked about the sociopolitical perspectives. She stressed how big differences is a fruitful ground for research. In the Apartheid system, bad education in mathematics for large groups was explicit policy. After 1994, improved access to education has been a priority, but improvement is slow. Mathematics results is a proxy for equality. She discussed how TIMSS can be used to understand the local development. South Africa had big inequalities in the TIMSS scores, reflecting social inequalities. Finally, in 2012, results are better. TIMSS also shows that the level of violence is higher on SA than in any other country. 

The "panel" did give some ideas for how to use international studies for better understanding, and there are surely lots of data out there that can be looked at, as an alternative to always collect your own data...

After lunch, I attended the talk by Emma Castelnuovo-Award awardees of 2015, Hugh Burkhardt and Malcolm Swan (in absentia). It is particularly interesting to see proponents of what we in Norway call "utviklingsarbeid" awarded and given prominence at such a conference. Their work have resulted in many beautiful tasks and sets of tasks, carefully designed to facilitate particular learning outcomes.

First he gave information on ISDDE, which has goals of building a design community, raising standards, increase influence on policy. Educational Designer is a journal and the website is isdde.org.

The talk was very inspiring, but difficult to write notes about, as many of the points mentioned were illustrated beautifully with examples of tasks and student responses. But here are a few points:
  • The Shell Centre wanted impact through useful materials, with focus on design. Materials need to fit the system they aim to rearch. It doesn't help if materials are brilliant if they don't work. Therefore, engineering research involves design research and systematic development.
  • Tasks can provide 'microworlds' for learning.
  • They talk about novice tasks, intermediate tasks and expert tasks. Expert tasks involves complexity and unfamiliarity, which means that the technical demand cannot also be high. He gave many examples of the three levels of tasks.
  • The usual role of a teacher is to manage, explain, set tasks. Certain tasks facilitates "role shifting", and role shifting changes the classroom culture.
  • In general, what this team wants to achieve is technical fluency, conceptual understanding, strategies, appreciation.
  • Mostly, lessons that they develop are either concept focused or problem solving focused. Concept focused lessons are developed in the "diagnostic teaching" way. He gave examples of different kinds of the two kinds of lessons.

This talk was a whole lot more inspiring than what these points suggest. He suggested making small units of lessons so that a teacher could substitute their textbook for a few weeks, without having to make a bigger commitment. I would be interested in joining such an effort...

Then there was a discussion on a survey on competences, led by Mogens Niss, joined by Regina Bruder, Nuria Planes and Ross Turner. The main issue is "What does it mean to master mathematics?" (Which also begs the question what we mean by mathematics.) We may talk about the products of mathematics or about the enactment of mathematics. Knowing and doing is not the same thing.

Classically, the answers were given in terms of content and related facts and procedural skills. Criticised in Spens report 1938, by Pólya in 1945 and so on. The first IEA study in the 1960s included several components. Also, Papert in the 1970s commented. Since the 1980s, there has been a trend focusing on enactment.

Ross Turner asked what are the relationships between mathematical literacy, numeracy, quantitative literacy, mathematical competence/competencies, mathematics. How would a Venn diagram look like? Or, if you put them in a diagram with products and applications on two axes, how would it look? (As usual, I'm trying to figure out where the history of mathematics and other integral cultural parts of mathematics fit into such a diagram - I think it shows that the concepts here are too narrow. - at least, to "products" should be added "background". Turner, however, put history into applications.)

Regina Bruder talked about two types of research: research where the construction competencies are an object and research where it is a means. One important discussion is whether it makes sense to discuss mathematical competencies without specifying which domains they are related to. Also, it is discussed if affective considerations should be included. Also, can they be detected empirically? In general, yes, but they are often overlapping and are not developed or enacted in isolation.

Mogens Niss and the others then discussed how competencies have played roles in different national curricula. Nuria Planes explained that in Spain, competencies have had a big inpact on paper but not in actual implementation and practice. In Latin America, processes have been included in curricula. Ross Turner told us that in many countries in South East Asia, doing mathematics is clearly included in curricula. But of course, the main question is how these words are implemented. Regina Bruder told much the same story for Germany. In Austria, personal and social competencies are included.

Then, challenges to implementation were discussed, but this seemed to be the usual list of points. The conclusion was that bridges between action and research is needed - which ties this panel nicely to the previous talk - research and development (in Norwegian: FoU) both are needed, and need to work together.

That concluded the first day of ICME 13. A day with a varied scientific programme, with a big delay in the morning and with serious temperature problems in the afternoon (the buildings obviously not built for full auditoriums in late July). My  choice of outfit (shorts + t-shirt) will be repeated for the days to come.


And as usual, most important is what I do not mention here, lots of talk with colleagues over coffee, lunch, beer and dinner and in-between.