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.