Just an old sweet song keeps Georgia on my mind

Did you know that the official song of the state of Georgia, US is "Georgia on my mind"?

Actually, that song is the only thing I associate with Georgia. (Except, when realizing that the state capital is Atlanta, I suddenly also associate it with Coke and CNN, as well as olympics and a bomb.)

Why do I suddenly care about Georgia? Because I'm going there in June - for the InSITE 2009 conference, presenting a paper with a colleague.

Good ideas for things to see and do in Georgia are welcome!

HPM 2008 Day 4

Day 4 of the conference was actually just half a day. Glen Van Brummelen talked on “Crossing Cultures, Seas and the Cosmos – In Search of the Origins of Trigonometry”. Fascinating though the history of trigonometry may be, the most interesting part of this talk for me, was the discussion he had on how to look at history. He warned against reading history as “the royal road to us”, for instance by reading original sources, but only noticing the parts that are part of the history leading to our current knowledge. This seems to put quite high demands on the teachers – for me, it will not be a question of avoiding anachronisms, but rather how much anachronisms to avoid. Too high demands on teachers in this regard will effectively mean removing all history from schools.

Kristin Bjarnadóttir held a fascinating talk on “A Puzzle Rhyme from 1782”. It is interesting to see how the “same” problem crosses borders and get immersed in new cultures.

Finally, Cecilia Costa talked about “The Alto Duoro ‘wine coopers’ mathematics”. The point was to look at what kind of mathematics is involved in making the barrels used for wine production, and this was done by interviewing the craftsmen themselves. The talk ended up in a discussion on why the barrels are not sylindrical, in which David Pengelley argued that it could make it easier to avoid leaks, while Peter Ransom argued that non-sylindrical barrels could more easily be rolled (not only in straight lines, but also in curves). I’m sure both of them have a point.

Thereby ended the HPM 2008 conference. The next HPM conference will be held in 2012 somewhere near Korea (where ICME12 will be held). But before that, there is the CERME in Lyon in January-February 2009 and the ESU in the Netherlands or in Greece in July 2010. Moreover, there will probably be other opportunities to meet the HPM family as well. But at the moment of writing, it feels good that the conference is over – my brain can’t take more input at the moment, and it will be nice to have a few weeks of vacation before trying to work on some of the issues here.

HPM 2008 Day 3

Karen Parshall held the opening lecture of the third day of the HPM 2008. Her talk was on “The Evolution of a Community of Mathematical Researchers in North America 1636-1950”. For me, particularly interesting was the fact that North America for quite some time remained “loyal” to the English way of doing thing, which made them, for a time, slow to pick up on the progress in continental Europe. (My Master thesis concerned the 1600s and early 1700s, so it’s particularly interesting for me.)

Then recent Felix Klein award winner Ubiratan d’Ambrosio held a talk with the long title “The transmission and acquisition of Mathematics in Colonial and Early Independent Countries in the Americas, and a brief Reference to the 20th century.” Who else than Ubi has the knowledge to give a one-hour talk on the mathematics of a whole continent over a period of 500 years? It is always a pleasure to hear Ubi talk, and I hope I will get many more chances in conferences in years to come.

Jodelle Magner had a talk on “Napier’s Rods In Today’s Classrooms” which showed one way of working with different multiplication algorithms. Then Peter Ransom had a talk on sundials – decided on at short notice to fill the gap left by people who didn’t come to the conference. It is always nice to hear Peter talk – even though this was (of course) closely resembling what he did at ICME the week before.

David Pengelley and Janet Heine Barnett’s workshop had the title “Learning Discrete Mathematics and Computer Science via Primary Historical Sources: Student projects for the classroom.” It consisted of two sets of worksheets, which the participants worked on: one on the bridges of Koenigsberg, another on Pascal’s triangle (according to Pascal). This was quite interesting.

Ewa Lakoma talked on “On the role of the history of mathematics in mathematics education for the knowledge-based society”. She only mentioned in passing a survey that has been done in Poland, showing how history of mathematics is perceived in different groups, and I would like to know more about this aspect of her talk.

HPM 2008 Day 2

The second day of HPM 2008 started with Evelyne Barbin (the new chair of HPM) and a talk titled “Dialogism in mathematical writing: historical, philosophical and pedagogical issues”. She gave a quick introduction to some aspects of Bakhtine’s work, and in particular the three “bullet points” active responsive attitude of the listener, addressivity and speech genres. I see that Bakhtine can indeed be interesting in the context of HPM, and need to make sure to learn more about him.

I skipped the next plenary lecture (as I had to check my email) but then went to Edel M. Reilly’s talk “Mathematics Apart: Examining the History of Subject Isolation and Its Implications for Mathematics Education”. According to her abstract (and title) this should be a historical talk, but her actual talk focussed on the present and future. I will have to read the paper to see if the historical element is there… Anyway, it was interesting to hear to which degree the US school system has chosen to compartmentalize mathematics (in most of the rest of the world, students take “mathematics courses”, not “algebra” and “geometry” without connections between them…)

Mala Saraswathy Nataraj talked about “Using history of mathematics to develop student understanding of number system structure. She described ways in working with pupils to get them aquainted with our numeral system through work with “toothpicks”. It also had a very interesting idea: that the work on writing large numbers (using powers of 10) is a useful prerequisite for later understanding the notation in algebra. It seems very reasonable, I just hadn’t thought about it like that before.

Chorlay Renaud and Anne Michel-Pajus presented the paper “The multiplicity of viewpoints in elementary function theory: historical and didactical perspectives.” This was quite fascinating. One major point was that functions could be presented in “two different worlds”: the world of quantity and the world of sets. Each of these worlds have semiotic and conceptual coherence – however, mixing them may confuse the students. In France, it is obviously the “world of sets” which is the “correct” one – functions are seen as a relation between sets. In Norway, I guess we cling to the “world of quantity” almost until university, although some definitions or ways of formulating things from “the world of sets” may creep in here and there. It would have been interesting to look at this…

Staffan Rodhe talked about “Emanuel Swedenborg’s work on differential calculus” with a few more details than last time I heard his talk (four years ago).

In the lunch break, there was a meeting of the “advisory board” of HPM – there’s never any rest… 

After lunch, George W. Heine talked about “Euler’s Contributions to Mathematical Cartography”. The first part of the talk was particularly fascinating, in which Heine discussed Nicolas Delisle’s attempts (?) to make maps of Sibiria. Although given huge resources, Delisle never succeeded, and he stubbornly denied seeking help from Euler – which may be a sign that he was actually a spy in Moscow, just trying to keep the resources flowing in while at the same time shipping maps out of the country (which he apparently did).

Then today’s highlight: Peter Ransom’s act as a seafarer in the workshop “Yo Ho Ho-ratio: some mathematics of Trafalgar (How Lord Nelson inspired curriculum development in mathematics)”. Peter’s workshops are always good, so obviously we had a good time – while making things, doing calculations, learning about history… And not only had Peter put all the resources onto a CD which we got for free – he had also made pencils with the name of the workshop on it… That’s what I call dedication…

In the evening there was a very nice private social gathering, and then – on Wednesday – we had a “free day”, on which I joined many of the others to Teotihuacán – but I should write about that in another blog…

HPM 2008 Day 1

The first day of the HPM 2008 was very good. It takes place in a very nice building in the very centre of Mexico City. The first talk was David Pengelley's "The use of original sources in the teaching of mathematics pupils at all levels". He discussed his dream: that pupils at all levels should learn all of their mathematics from primary sources. I surely do not share that dream, but I certainly think that primary sources has their place in the mathematics classrooms, just as pupils in Norway will never be allowed to go through 10 years of education without reading Ibsen (instead of just short summaries of his works). David also came up with the idea of having the HPM maintain a bibliography of the HPM field. That is certainly both an incredibly big task, but it would also be very useful. Just imagine an online bibliography where you could search for particular mathematicians, levels (in school), type of paper (empirical research, ideas from the classroom...), language. It could certainly not be the work of one person, but still some person must start and dedicate a significant amount of time to it...

The next plenary lecture was Rosa María Fanfár, who talked on "Matemática educativa. La convergencia de series infinitas". I must admit that this area of mathematics is one I do not usually work on, so that it was a bit of an overload on my brain to try to understand all of the mathematical examples based on the simultanous translation into English - it didn't help that I hadn't made sure to sit close enough to the screens to actually see them. Luckily, the paper is in the proceedings.

Then it was time for parallell sessions, and I was chairing one of them. The talks there was good: Maria Christina Araújo de Oliveira and Ruy Pietropaoplo talked about a magazine for uneducated school teachers in Brazil in the 1950s and 1960s. Ildikó Pelczer gave "A historical overview of analysis exams in Rumania", which showed patterns that I suspect would also be found in Norwegian teacher education exams of today. Hans-Stefan Siller discussed the emergence of Informatics as a subject in his talk "Informatics - A subject developing out of mathematics", while Flávia Soares discussed examinations for teachers in the 1800s in Brazil.

In the second parallell session, Funda Gonulates discussed a project which she has done, where she tried to measure whether working on history of mathematics did change the students' attitudes to history of mathematics and their knowledge of how to include it in teaching. However, she did not find significant increases. This does point to one of the problems of doing empirical research in the HPM field - the number of students involved needs to be high, and even if there were significant results, critical voices would certainly demand a control group. And given a control group, some would also question if the teaching given to the control group was "good enough". While this kind of research is certainly useful, for the individual teacher who wants to try to include history of mathematics, I think it is even more useful to get examples which can motivate him. "The proof is in the pudding": if the teacher thinks that the students are benefitting from the history, he will keep including it. On the other hand, no teacher will include anything based only on research papers telling him that something works for some faraway students.

Bjørn Smestad (who - again - happens to be me) talked about an interview study he has done trying to find out things about teachers' conceptions on history of mathematics. This talk went much better than the talk in Monterrey - maybe because it was much better prepared. There were interesting questions afterwards as well.

Robert Peard gave interesting ideas of courses he has had for teacher students in which he did not try to teach them the "basics" which they had failed to learn through 12 years of schooling, but instead focused on a few examples in which mathematics is important for understanding the world around us - for instance the calendar.

Beverly M. Reed talked about "The effects of studying the history of the concept of function on student understanding of the concept". Her theoretical basis was APOS theory, based on Piaget's theories. Her approach was qualitative, and she had promising conclusions, although I guess critics would still argue that her results could also have been obtained without history of mathematics.

That concluded the formal part of day 1. A very good day indeed. And now I'm free to do "whatever I want" for the rest of my stay in Mexico, as my talks are finished...

ICME Day 5 Part 2

Day 5 became the last day of the conference for me, as I preferred to stay in my hotel to prepare Monday's talk at HPM instead of going to the conference on Sunday.

Thus, the last I got from ICME was the presentation of three new ICMI Studies, at least two of which I will certainly try to get time to read as they become available. ICMI Study 15 concerns "The Professional Education and Development of Teachers of Mathematics", while ICMI Study 17 is titled "Technology Revisited". The presentations actually concerned the making of the studies more than the actual ideas inside them.

So, that's the end of ICME11 - I hope I'll get to ICME12 in Seoul in four years' time. But now: HPM 2008...

ICME Day 5 (Part 1)

On Day 5 of ICME11, I've so far only heard one talk. That was Jan van Maanen's excellent talk "Professional development of teacher educators, the ELWIeR initiative". Why excellent? Because it was funny, based on an important work and gave me useful information. It was not surprising that it was funny - Jan's talks are almost always entertaining. The project was a large one, but Jan focussed on two aspects - the development of new "handbooks" and on some of the research in the initiative.

Apparently, the textbook situation in teacher education has been quite different from that in Norway. In Norway, there are new textbooks "all the time", while in the Netherlands, Van Dormolen's 1973 textbook "Didactiek van de wiskunde" is still in use. The creation of the new handbooks has involved most of the teacher educators in the Netherlands, and has led to considerable professional development within the community. This of course makes me think of whether there may be a similar project that Norwegian teacher educators could collaborate on, to get similar positive results. At the same time, of course, I'm eager to see the results of the Dutch efforts - that is, I want to read the finished handbooks...

He also discussed some research findings, but I will not try to give an impression of them here. However, it is interesting to see that this research also focussed on teacher students' ability to understand the pupils' ways of thinking (just as at least two of yesterday's speakers). I'm happy to say that we have had at least some work on that with the students at my institution, although we could certainly do more.

It the final session of the TSG23, Bjørn Smestad (who happens to be me) held a talk on three years of student projects on history of mathematics. I was not too happy with the outcome of the talk - it went just as badly as I feared in advance. My idea was to look at some student projects that I've done with my students. In these student projects, the students were given little input from me (even though they could certainly have asked for more), and I therefore thought that the products of these projects could give an idea of the sorts of problems that also ordinary teachers in school might have faced if they had taken the curriculum requirements to include history of mathematics in their teaching, seriously. The discussion afterwards focussed more on how such students projects could have been done, however, rather than how teachers in schools might better be helped. However, as I'm having a talk on Monday with a similar question in the end, I may hope that that will work better...

Afterwards, there was more discussion on the way to go on. For me, it is clearer than before that we have to treat two issues seperately: on the one hand how we - as educators who are very interested in the history of mathematics - may include history of mathematics in our teaching, and on the other hand how we can help other teachers include history of mathematics in their teaching, preferably on a large scale (that is, not only five or six teachers supported by an expert, but thousands of teachers...) Both issues are very interesting, but it is not at all obvious that what a special person such as Jan van Maanen can do in his classroom, can be replicated by less knowledgeable teachers. On the other hand, neither is it clear that history of mathematics has a place in the teaching of uninterested teachers.

After this discussion, I had lunch and went to the computer room to write this and check my email...

ICME Day 4

Day 4 of the scientific activities of ICME (which was the 5th day of the conference, as the fourth day was excursion day), started (for me) with a talk by Caroly Kieran; "Conceptualizing the learning of algebraic technique: Role of tasks and technology". She had convincing examples of how CAS (computer algebra systems) can be used to improve both the students' technical aptitude and conceptual understanding. This is not self-evident, indeed, I have myself been suspicious about CAS, thinking that they will only help students avoid doing the computations themselves. But of course, just as with calculators, CAS can also be used in an investigative manner - temporarily removing the students' need to do the algebraic manipulations themselves make them better able to make conjectures and check their conjectures on new examples. One of her examples was to let the students do (x-1)(x+1)= and (x-1)(x*x+x+1)= and then conjecture what (x-1)(x*x*x+x*x+x+1) and so on is (sorry for the terrible ASCII notation here). The use of CAS made the students confident that they had the right answer, and therefore the confidence needed to make hypotheses. I will certainly consider using this at some point in the future when teaching algebra to my students. (Kieran stressed, however, the importance of task design - she did not hide the fact that CAS can also be used terribly...)

The next part of my program was the TSG23. Uffe Thomas Jankvist held a talk titled "On empirical research in the field of using history in maths education". He argues that the HPM group has had too little "empirical research". Moreover, he has had a very interesting example of such research, which seems very good. His paper had a combination of being practical and theoretical at the same time, which was very good. My only "protest" was that he used the phrase "armchair research" as the opposite of "empirical research". I think this gives the wrong impression. In the history of HPM, it is true that there has not been much empirical research, but the papers have been divided into (at least) TWO other cathegories - the papers that have indeed been empirical (but not research) and the once that may have been research, but not empirical. Thave been lots of papers discussing individual experiences from the classrooms, as well as lots of papers discussing theoretical issues without the important empirical components. A discussion of the virtues of empirical research should at least take both of these other types of work into account.

There was also a paper by Lenni Haapasalo from Finland.

I had actually planned to skip the next plenary session, but ended up going anyway. I'm glad I did. The two professors Fujii and Even (Fujii from Japan and Even from Israel) had a talk each on the topic "Knowledge for teaching mathematics". Besides being genuinely funny, Fujii's talk was also a very interesting glimpse into the world of Japanese Lesson Study. He gave many interesting examples of mathematical discussions this led to. The main realization for me was that even though we do ask our teacher students (in Norway) to write detailed plan of every lesson, we never ask them to write in these plans what they expect their pupil's (mathematical) reaction to be. They write a lot on the pupils' actions (what they are supposed to DO), but not on which strategies they will probably use or the errors they will probably do. This is an important weakness. In Japan, the anticipation of such difficulties is an important part of lesson study, including (of course) the planning of how to approach (or even make use of) them, should they occur.

Prof. Even had a similar point of view, although from another view point. Her work with teachers has (among other things) included teachers reading research papers and replicating the work in the papers (for instance giving their pupils the same tasks as discussed in the research papers). The teachers are often astounded that the misconceptions they read about in the research papers, are often present in their own classrooms as well. This made the teachers more capable of understanding students' thinking, but Even also stressed the importance on working on how to translate that understanding into teacher practice.

The last post of the programme was the ASG for the HPM (the 2nd part). First, Ubiratan d'Ambrosio held a very interesting talk on Julio Gonzalez Cabillon. (I'm sorry I can't get the name right on this keyboard.) Cabillon was the moderator - for years - of the Historia Mathematica email list, which was an amazing resource for researchers worldwide. As a moderator, Cabillon both provided a lot of the answers, but also intervened in a diplomatic manner when discussions got heated. Ubi has researched who Cabillon is, and has at least found him and talked to him. Today, Cabillon is, luckily, still very much alive, but has just decided to persue other interests than this list, which took so much effort for such a long time. It is a pity, but understandable, that noone has managed to create a list to replace the HM list.

The rest of the ASG was spent on Costas Tzanakis giving a report on the work of the HPM for the previous four years and on discussions on the future. These discussions will continue in Mexico City next week.

Thus ended the evening at ICME - and I went out with some colleagues for dinner. Another nice day at ICME11.

ICME Day 3

The third day of the ICME started with José Antonio de la Peña’s talk on “Current trends in mathematics”. This seems as an almost impossible topic to cover in one hour, but la Peña had chosen to focus on just a few highlights, and to take a “light” approach. I think this was wise. He also pointed to the Mathematics in the movies site, which features video clips from movies. I’ll check it out…

Next, I heard an excellent talk by Jeremy Kilpatrick titled “A Higher Standpoint”. The point of departure was Felix Klein’s “Elementary Mathematics from a Higher Standpoint”, published in the first decade of the 20th century. Klein’s object was to “bring to the attention of secondary school teachers the significance for their professional work of their academic studies”. He pointed to a “double discontinuity”, whereas the student who goes on from school to higher education, feels that the mathematics is completely different, and at the same time the teacher who has finished his higher education and goes back to school to teach, does not see the connection either. Klein wanted to remedy this by updating the school curriculum and by revising the university instruction to take into account the needs of the school teacher. The talk went on to discuss the books in more detail, and also discussed Pólya and Freudenthal in the connection with Klein’s ideas.

Kilpatrick’s talk made me want to read Klein’s textbooks, which was certainly one of his objects. It also gave lots of food for thought.

At about this time in the conference, there was distributed a leaflet giving information on the 12th ICME, taking place in Seoul July 8th-15th, 2012. I’m already looking forward to it! The website is: http://icme12.org.

The next thing I attended was the TSG23. Louis Charbonneau talked about “Astronomical and mathematical instruments as pedagogical tools”, putting emphasis on the emotional aspects of being able to touch instruments that have been used to measure heaven and Earth… It was a very interesting and enthusiastic talk. Snezana Lawrence gave the answer to my concluding question in Saturday’s talk (which means I have to update my talk a little…) My question is what we can do to make teachers able to include history of mathematics in their teaching. Lawrence has used history of mathematics as the focus of a teacher development program that seemed very good. I really need to get to know more about this project (and I guess I can find out more on her website, mathisgoodforyou.com).

Liliana Milericich read the paper “The teaching and learning of integral calculus from a historical perspective”, which pointed to some pitfalls in the teaching of integral calculus. As this is not part of what I teach, I didn’t note down particular things that I need to remember from that talk.

Later in the day, I attended the SEG (Sharing Experiences Group) on interactive whiteboards (IWBs). This was incredibly interesting to me, as I’m just starting out in this area, and having written a paper for a conference just before leaving for Mexico. There were lots of interesting thoughts there, and seeing the webpage of these people was also very interesting. Moreover, I was pointed to an interesting conference in Cambridge in June 2009, which I will consider going to.

That ended day 3 of the conference. As day 4 was an excursion day, I will go on with this blog with day 5 later…

ICME Day 2 (part 2)

The first ASG meeting of the HPM group consisted of two interesting talks. Gert Schubring’s talk was titled “Researching into the History of Mathematics Education – an HPM perspective”. He discussed how teachers were educated in different states from the middle of the 18th century. (Before that, states took little interest in this.) The talk was packed with information, and I can obviously not repeat it here. However, it was interesting to hear how there was a huge effort after the French revolution to teach teacher educators, while in France in 1763, it was argued that it was needless to educate teachers, as good textbooks would make sure that the teachers could educate themselves.

Fulvia Furinghetti talked on “The emergence of women in the international arena of mathematics education – Just so stories”. Although it was a talk about the women who has played a role, the main picture is of course that there have been very few women who has been prominent in the field until recently. Lately, that has luckily changed.

ICME Day 2 (part 1)

Today's plenary speaker was Celia Hoyles, and the topic was "Technology and mathematics education: Transforming the mathematical practices of learners and teachers through digital technology". To me, it was a great talk, as it functioned as an introduction into a field in which I will work more in the coming years. She described six areas in which ICT is transforming mathematics education:

• dynamic and visual tools to explore in shared space
  
• tools to outsource processing power
  
• new representational infrastructures
  
• connections between school and learner's culture
  
• connectivity
  
• intelligent support for the teacher.
  

She had examples in all of these areas, and her talk certainly made me eager to get hold of the forthcoming ICMI Study on these topics.

She also referred to the National Centre for Excellence in the Teaching of Mathematics website, and in particular the Mathemapedia - a Wikipedia for mathematics, apparently. I haven't had the time to look at it yet, but should certainly check how this site compares to my plans for a teacher education wiki locally in Norway. Maybe there will be common areas of interest.

The next lecturer was Anna Sfard. Her talk was titled "Learning mathematics as developing a discourse". Her talk was - unsurprisingly - well attended, so the room was far too small, as was her time. Anyway, I found it most interesting. She discussed what she terms as a transition from the metaphor of "aquisition" to a metaphor of "participation" as far as learning is concerned - moving from a Piagetian world to a Vygotskyan. This has far-reaching consequences. She looks at mathematics as a discourse, and a number as a "discursive construct". This has far-reaching implications when it comes to interpreting children's approach to mathematical problems.

The third item on my agenda today was the TSG23 - the topic study group on history of mathematics in education (not to be confused with the topic study group on the history of mathematics education, of course). Due to some technical difficulties, one of the talks were postponed to tomorrow's session, and only Costas Tzanakis had his talk. This was a very interesting talk on how history of mathematics may throw light on the problems pupils face when trying to learn the variance concept. One particularly striking point (to me) was how educators have tried to make the topic "soft" by including examples from social sciences instead of from physics and geometry, but have ended up making it harder. This is because the terms mean - and even variance - have clear, concrete interpretations in certain examples from physics and geometry, and because they offer the possibility of experiments.

Now, I'm having lunch break, but will go to the ASG meeting for the HPM group later this evening. But it has already been a very interesting day. (And I've also had the time to look at my own talk, which will be on Saturday. I think it will be ok.)

(As mentioned before, these posts are delayed by 20 days.)

ICME Day 1

The ICME 11 has started today - the main conference for mathematics educators worldwide. As always, the first day is mainly spent on formalities and on setting the stage for things to come.

The opening consisted of a whole series of speeches - inevitable, I guess - and the distribution of the Felix Klein Awards and the Hans Freudenthal Awards for 2005 and 2007. This was a nice thing to see - both that everybody took the time to listen to praise for the lifetime achievements of some of our most wonderful colleagues, and that the prize winners showed that the prizes were actually quite important to them as a recognition of a lot of heavy work.

The winners were:
Felix Klein Award 2005: Ubitaran d'Ambrosio
Hans Freudenthal Award 2005: Paul Cobb
Felix Klein Award 2007: Jeremy Kilpatrick
Hans Freudenthal 2007: Anna Sfard

Although all four are well-known names, personally, I know the work of Ubi best, and found it particularly touching to see him thanking his wife for all her patience for 50 years...

The first two plenary lectures were on what has happened for the previous ten years and what should happen in the future. The venue for the plenary parts of the program is not very good - there is too much background noise, and it's difficult to see the screens from many positions. The solutions in the previous two ICMEs (which involved using actual buildings instead of a "tent") worked better. However, I thought that the first lecture (by Artigue and Kilpatrick) was quite inspiring and interesting. They tried to summarized the main trends in mathematics education over the previous ten years. Some keywords: more systemic approaches, a look at constraints, semiotic perspectives.

The panel on what we need to know, which was supposed to summarize the answers of educators around the globe, however, did not manage to inspire me. Well, Paul Kobb's interpretations of the (mainly US) answers from North America, were meaningful. The other three also discussed important questions, but in a way that made it look less like a panel than separate lectures. The format of plenary activities for thousands of listeners is a difficult one, indeed.

The highlight of the day was undoubtedly the workshop by one of my favorite colleagues, Peter Ransom. He and a colleague had a workshop on using sundials in the classroom. Peter is so enthusiastic that it's hard not to start loving sundials, and the activities were fun as well. I should try to find a way to include sundials in my teaching in Oslo...

That was the first day already. In between were coffee, lunches etcetera, of course. And lots of meetings with colleagues who I haven't met for 1, 4 or even 8 years. A nice start to the conference.

(As mentioned before, these posts are written during the conference, but posted afterwards.)

ESU5 Day 5

The first part of day 5 was a very interesting and engaging talk by Leo Corry on a bit more recent history of mathematics. He explored the connection between David Hilbert and the "new maths" in the US and "modern maths" in France. Hilbert has been portrayed as a proponent of "mathematics as a game" and axiomatics as the most important thing in mathematics. Leo Corry gave an important quote of Hilbert that suggests otherwise. I don't have the quote here, but the meaning was something like this: When mathematicians build their theories, they do not start by working for years on the foundations, before going on to build the theory on the foundations. Quite the contrary: mathematicians build beautiful spaces and corridors, and only when they start seeing signs that the foundations are not strong enough for further developments, they start to worry about them. Hilbert's opinion was never that axiomatics was the main thing in mathematics, but that axiomatics was important to put the beautiful theories already developed on a stronger foundation. Corry asked if this is maybe also what we should do in schools - develop (with our pupils) wonderful mathematics to let them see the beauty, and then only later worry about the details of the foundations. This is, of course, quite the contrary to the ideas of New Maths, which was so inspired by axiomatics.

The second thing I took part in today was my own workshop. It did turn out quite well, in my opinion - to the extent that the participants' activity is a measure of success. Most of the people there took part in the discussions, and some people told me that they appreciated a focus on primary and lower secondary pupils. What my workshop did was to look at some of the activities I do with my students, and to discuss whether they are meaningful. I added a subtitle to my talk: "If you can't do very much, can you still do something?", and I could actually also add: "If you don't KNOW very much, can you still do something?" After all, my knowledge of the history of mathematics is inferior to that of most people at this conference, but if I am to wait until I'm an expert before I start working on this with my students, they will never learn anything on history of mathematics. So my answer is clear: it is usually better to do something (and maybe make someone interested) than to wait until what you do is flawless...)

And of course, by presenting my stuff here, I've already removed some mistakes because of the comments of the participants...

Now, I've had a wonderful lunch, and will enjoy an afternoon of mathematics in the knowledge that I'm done with my presentation...

ESU5 Day 4

Day 4 was a short one, mathematically speaking. After lunch, I joined a guided tour of Prague - on foot. It was an interesting walk, which lastet for four hours. The guide was quite a character, and she spoke several languages - English, French, German, Italian, Spanish, Russian as well as Czech, of course. She had strong opinions on many things, and while I don't take her word as the truth on every issue, it was quite entertaining. And of course, it was another chance to talk a little to a few of the conference participants.

The morning sessions were devoted to history of mathematics education, which I must admit is not my favorite subject. I do know that it is good for me to know a little about it, but it does not contribute very much to my main goals: to see how history of mathematics may be taught in primary and secondary school both to give students a better cultural understanding and a better mathematical understanding.

The first hour was by Gert Schubring and Helene Gispert (I'm sorry for not being able to write the French names and words correctly on these keyboards). They talked about the developments in Germany and France in the 20th century. For the next two hours they were joined by Livia Giacardi for a discussion on "The emergence of mathematics as a major teaching subject in secondary schools", with Nikos Kastanis' contribution on the Greek situation being read, as he could not be present. It is indeed interesting to see how wars and the change of borders do influence mathematics teaching in different regions. Maybe the most interesting part of the morning was to see illustration from Nazi era textbooks for childern. Anyone who thinks mathematics cannot have a political dimension, would certainly change their mind after seeing these illustrations...

ESU5 Day 3

Day 3 started with a talk by Fritz Schweiger, entitled "The implicit grammar of mathematical symbolism". In a way, the talk was technical, so it will be difficult to discuss it here, in another way, it can be summed up quite effectively thus: It reminded me and made me more aware of the vast amount of implicit information there is in a mathematical text, which we (as educators) may at times not be good enough at discussing with our students. A simple example: 2x, 23 and 2 1/2 are read quite differently. Likewise, we have very clear ideas on what should be considered a good choice of letters: we will define a function f as f(x) = ax+b, but certainly not a function a as a(b) = fb+x. This is food for thought - what should we do to make the students see the importance and simplicity of such "rules"?

Thereafter, there was a panel discussion between Evelyne Barbin, Luis Radford, Fritz Schweiger and Frank Swetz. It is impossible to sum this up, but I'll try: Barbin talked about "perennial notions", how some notions include both epistemological depth, possibility of conceptual changes, links with other fields and historical and cultural interest. Her idea is that these notions are particularly suited for educational purposes.
Schweiger talked about "fundamental ideas", surely a concept of the same kind as "perennial notions". His definition is that fundamental ideas recur in the historical development of mathematics (a time dimension), recur in different areas of mathematics (horizontal dimension) and are anchored in everyday activities (human dimension).
Luis Radford, on the other hand, did not as much discuss the "How?" as the "Why?" of historical dimensions in mathematics teaching. His answer: no history means no understanding of reality.
Finally, Swetz said that we teach too much mathematics, and not enough ABOUT mathematics. His answer to this is: include the historical and cultural dimension and focus on problem solving. Historical problems give information about the society at the time, and are therefore a good tool.
The plenary discussion after this panel discussion was on "is mathematics universal?", "what is mathematics?" and "knowing=being?". No conclusion was reached.

In the afternoon, I took part in the workshop of Michael Fried and Bernard Alain. The title was "Reading and doing mathematics: Ancient and modern issues." The first part of the workshop was on Euclid, and then on some of Proclus' comments. To me, this served as a further reminder of whatever Euclid skills I may have had once. While useless for my own teaching in Norway, it is nonetheless important for me as a teacher educator to know a little about the work which more than any other have influenced mathematics teaching for the past 2000 year. The second part of the workshop was on the topic of Paideia (translated by Cicero as "Humanitas"). To me, it was interesting to see the way in which Euclid was considered a training of the mind - a point of view that fell out of favour some time ago. (I was also made aware of an article called "In Defence of Lecturing" by Mary Burgan. For me, who used to be a fanatical opponent of lecturing (even though I myself seem to have benefited from that kind of teaching - as well as others), it would be interesting to read this article.

After this, I decided to give my mind some rest. After writing this, I will go have a look at the town and prepare for the conference dinner which is this evening. I always get a bad feeling about missing a lot when I skip parts of the programme at such conferences, but on the other hand, it is important to have energy to actually take part in the workshops and listen attentively at the lectures - not just be there...

ESU5 Day 2 continues...

After lunch on Friday, I was going to attend a workshop on "Histories on zeros", but as this talk suddenly switched its language into French (which I, sadly, do not master), I went instead to Maria Menghini's "The 'Elements de Geometrie' of A. M. Legendre: an analysis of some proofs from yesterday's and today's point of view". I always find Euclid a bit unsatisfying, as I always am presented with just parts of the work - which means I'm not aware of what is already proved. Not knowings which tools are available is frustrating. I guess the same phenomenon also explains part of Norwegian students' frustration with proofs in geometry - as Norwegian students never get the opportunity to follow the logical building from the foundations. Anyway, given these problems, I found this particular workshop enlightening, especially concerning the problems Legendre and his time had with what we call rational and irrational numbers.

Later in the evening, I heard Ferreira Eduardo Sebastiani's talk on "Ethnomathematic's use in Indian teacher's formation" - another talk suddenly switched to French. Here, however, the transparents were in English, and I got the general idea of the talk - how ethnomathematics is used in working with the Waimiri-Atroari Tribe.

Finally, I heard Robin Wilson's talk on "Lewis Carroll in Numberland". The talk has the same title as a book which is to be published by Penguin later, and which I definitely will want to read. The talk had several hilarious quotes from Lewis Carroll, but it also pointed me to the book "Suggestions as to the best method of taking votes", which I certainly will want to read as well.

Thus ended the second day of the conference - except that food and drink was consumed in the evening.

ESU5 Day 2

The second day of the conference started with a talk by Ulrich Rebstock titled "Mathematics in the service of the Islamic community". I must admit that my knowledge of Islamic mathematics is even worse than my knowledge of for instance European mathematics, and every talk on the subject is therefore sure to give me new information. Rebstock showed how much of Arab mathematics focused on practical aspects of mathematics, and gave examples from a wealth of books on subjects like measuring, taxes and trade. But the more "theoretical" mathematics was never far from the surface, for instance, examples with made-up "practical interest" were many. I particularly liked this example: In a Turkish bath, one day 30 visitors had payed their entrance. The owner knew there had been three Jews, and that he had collected 30 dirhams. The prices were Muslims 1/2 dirham, Christians 2 dirhams, Jews 3 dirhams... Another interesting example of mathematics of the time were inheritance problems when the gender of one of the beneficiaries were uncertain (!)

Thereafter, I joined Chris Weeks´ talk on Condorcet´s paradox. Condorcet wrote a treatise (Essai sur l´application de l´analyse a la probabilité des décisions rendues a la pluralité des voix) on the problem of voting, showing that any voting systems has its problems (although it should be possible to create a voting system that doesn´t repeatedly elect Bush president...) The treatise is very interesting, and it is tempting to use it with my students this autumn - especially as there is a local election coming up.

Then there was lunch. 99 crowns for a three-course meal is great - maybe a fifth of what I would expect to pay in Norway...

ESU5 Day 1 continues

After lunch, I enjoyed a three-hour workshop by Renaud Chorlay and Philippe Brin, both French high school teachers. Apparently, French kids start learning probability quite late, but when they do, at age 16-18, thez do it seriously. Chorlay and Brin showed three examples of mathematical original sources that thez use in their classrooms: a discussion on life expectancy between the Huygens brothers, a note on a game of dice by Leibniz and some familiar texts on the "problem of points" by Pacioli, Pascal, Fermat and others. In all, there were several texts here that I haven´t seen before, and that I would like to use in my own teaching.
After this, I heard five 10-minute presentations, on a mathematical exhibition (Oscar Joao Abdounur), on mathematics theater (Funda Gonulates), on error correcting codes (Uffe Thomas Jankvist), on Ramanujan (Jim Tattersall) and on area and volume (Luciana Zuccheri and Paola Gallopin). All of these were interesting, but frankly, 10-minute talks only serve as an introduction which should be followed bz personal communication whenever particularly interesting things occur. (Even though some of the presenters tried to put as much content into the presentations as physicallz possible.)

This finished off the official part of the first daz, but then some of us went out to enjoy Prague´s food and drinks, of course. The prices are ridiculously low by Norwegian standards (as in manz parts of the world...)

ESU5 Day 1

I am attending the ESU5 - the 5th European Summer University On The History And Epistemology In Mathematics Education. This goes on in Prague this week, and I will report on it in this blog. More than 220 participants will choose from the 128 activities by presenters from 28 countries.

The first plenary speaker was Luis Puig. His title was "Researching the history of algebraic ideas from an educational point of view." He compared three approaches to algebra: the Babylonian, as interpreted bz Jens Hoyrup, al-Khwarizmi´s and Jordanus de Nemore´s. Especially interesting to me was Nemore "De Numeris Datis", 1225, who assigned letters to all quantities, with no distinction between known and unknown. If a times a was needed, it would be assigned a letter, say b. Puig noted parallells with how some pupils treat letters in the beginning, before they see the point of the letters.

Then I joined Jan van Maanen´s workshop on "The work of Euler and the current discussion on skills". We were given copies of part of Euler´s 1770 algebra textbook, and studied parts of it. Euler goes to great length to define + and -, for instance, so it is clear that Euler saw a great need to improve the "basic skills" of his fellow men. However, he did so while relating algebra to the real world. In the copies were also Euler´s explanation of the algorithm for extracting square roots. As I have never learned that, I should do that as homework...
It is terribly hot in Prague now, but huge doses of refreshing mathematics and history helps!