ICME Day 1

The first day of ICME was a light one. It started with a two-hour opening ceremony, with a combination of welcome addresses and cultural performances, which was an eclectic combination of Korean and Western style, just as modern-day Korea. It is always nice to attend these opening ceremonies, as they try to convince us of the importance of our work in general and the conference in particular. This time, there was a recorded address from the president of S. Korea. Thereafter, the Felix Klein and the Hans Freudenthal awards for the last four years were awarded. Of course, I was particularly pleased to see Luis Radford get his well-deserved prize.

The first plenary lecture was by Don Hee Lee. She spoke on the topic "Mathematics education in the national curriculum system". She discussed the place of mathematics in the educational system, taking Plato as her starting point. Is mathematics taught mainly for the intellectual development of "the liberal man" (learning for its own sake) or for the development of society?

She quoted a Korean professor of mathematics (without necessarily agreeing) claiming that there is no mathematics that is both easy and interesting at the same time. Mathematics is based on thousands of years of intellectual work, and therefore advanced. While this is an interesting point of view to share with pupils who have unrealistic ideas of the amount of work (or lack thereof) necessary to learn mathematics, it should probably not be a credo for a mathematics teacher. 

For regular lecture today, I chose Alan Schoenfeld's "How we think: A theory of human decision-making, with a focus on teaching". Alan Schoenfeld received his Felix Klein Award in the morning, and the lecture hall was so crowded that the organizers immediately decided to ask him for a rerun later in the week. 

The starting point of the talk was the question "If I know enough about you, can I explain every action you do and every decision you make?" He claimed that the answer was yes, with the important caveat that he is talking about situations where you are an expert. Thus, it holds for the cooking of a chef, the teaching of a teacher and the setting of diagnoses of a doctor. So what do you need to know about a person to be able to explain his actions? You need to know about his knowledge/resources, his goals and his orientations (beliefs/values...)

The word "explain" is not meant as a mere verbal explanation, but the ability to set up a model which can - to a degree - predict the actions.

Why look at teaching? As he put it: "If you can model teaching, you can model just about everything." Teaching is highly complex; it is highly social and it is ever-changing.

He gave a few examples of how he has worked with teachers to model their behaviour. He gave one example of teacher who believes that he cannot tell students anything unless it's based on something the students already have said - and therefore is lost when the whole class agrees on a wrong answer. His actions were to a large degree explained by this belief. Another example - a teacher who teaches by raises issues, asks for student suggestions, clarifies and then moves on. He transitions to the next issue when goals are met. So what does he do when things don't go as he wants ? According to Schoenfeld, he "doesn't have a choice" - he will still build on the students' questions/input and discuss them with the class. His beliefs determines his actions.

His third example was on Deborah Ball teaching a 3rd grade class. She did some surprising things, including at one point asking a question that many would consider a mistake, as it derailed the discussion with the class. Schoenfeld didn't understand what Deborah was doing. Could it be modelled? And what about Deborah's "mistake"? After years of studying it and discussing it, he came up with a model here as well. Yes, her behaviour was consistent with a model. Within the model, her "mistake" was based on a need to understand her students' thinking before going on to the next topic.

So what's the point? Why are models important? I would ess that they are not, but that they serve to point out the importance of a teacher's resources, goals and beliefs (etc). If two experienced teachers do things very differently, we should not assume that it is because of some insignificant chance. It may very well be because of some fundamental difference. And of course it doesn't help much to give teacher students the necessary knowledge and skills to teach well, if we don't also consider their beliefs. As Schoenfled put it: a teacher who believes that only the best students can do problem solving, will not even try to give his poorer students problems to solve.

Schoenfeld's way of thinking seems highly relevant to the research project I'm currently involved in, and I will keep it in mind...

The rest of the day,  I got prepared for Tuesday - the day of my little talk.

ICME Day 0

This will be my fourth ICME (International Conference on Mathematics Education), after Tokyo, Copenhagen and Monterrey. As always, I arrived a few days early. It's nice to get used to the temperature and the time before the conference starts. Seoul reminds me of Taipei (a city I've spent some time in). There are the same smells, a similar mix of local and US stores and culture, similar traffic and so on. The subway system is comprehensive and easy to use - like in Taipei. So I immediately took a liking to the city.

It's also nice to meet up with old colleagues. On the day before the conference, I met lots of Norwegians; Andreas, Marianne (with husband), Torgeir, Liv Sissel, Odd Helge, Kristin (with husband), Ole Einar and more. And still, I have met only perhaps half of the Norwegians. In Tokyo in 2000, I don't believe I met anyone else from Norway.

My expectations this time are diverse. I'm looking forward to hearing some of the "big names" - people that I've read but never seen. And the Topic Study Group will be interesting, I'm sure. But I must admit that at the moment I'm thinking most of my own contributions - a talk on Tuesday, leading a Discussion Group on Tuesday and Saturday and saying some words on the HPM Newsletter on Wednesday. I never stop being nervous, even after all these years, and even if the groups I'm talking to are rather small. What if I don't get my points across? What if I mess everything up? Well, I'll do my best not to...

After ICME, I am, again for the fourth time, going on to the HPM conference, which is in Daejeon this time. I'm looking forward to that, partly because I have fewer responsibilities there (just a workshop), but mostly because the scale is smaller and it is (therefore?) friendlier. At ICME, you can meet and get to know someone on the first day and then never see them again - it's that big. (Or they may be consciously avoiding you, of course.)

It will be two interesting weeks, for sure.

HPM Newsletter - February edition

A surprising treat for people interesting in the connection between the history and the pedagogy of mathematics (HPM): The HPM Newsletter website has published a few articles which will be included in the March paper/pdf issue of the newsletter. This is the start of a new habit, in which we hope to publish new materials every month, to be collected in the paper/pdf version three times a year.

New on the website today is for instance an article on BSHM Bulletin number 3/2010, including a link to the full papers of that issue of the Bulletin.

Enjoy!

HPM Newsletter 75

The newest issue of HPM Newsletter (no 75!) is now available at the HPM website.

It can also be read online at HPM Newsletter's website, which is currently in a test phase.

ESU6 Day 5

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

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



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

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

Euclid in color

I see that Taschen has reprinted Oliver Byrne's 1847 edition of Euclid, with a lots of use of color.

I hope I'll get my library to lend me a copy - it will be interesting to see how Euclid changes when a pedagogical use of colors are included. I fear that there may be some unintended consequences... (I haven't yet checked if any historian of mathematics has written anything on this particular book - I would surprised if noone has.)

Have a look at the book at Taschen's homepage (linked to above) - it's really unusual.

ESU6 Day 4

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

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





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

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

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



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

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

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



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

coffee break area

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