Mexico 2008 – and lovely randomness

This summer, I will be staying in Mexico for 20 days. I will be attending both the ICME (International conference on mathematics education) in Monterrey, and the HPM (the meeting of The International Study Group on the Relations between History and Pedagogy of Mathematics) in Mexico City.

The trip here went well. However, I met two different approaches to randomness which were interesting. In Chicago, I was pulled aside for a routine random check of my luggage. Of course, I knew that nothing was wrong, so my only thought was “Why did they pick me?” I was stressed, since I didn’t have much time, and didn’t know exactly how much time all the procedures in transit in Chicago would take.

In Monterrey, much the same happened to my colleague – she was picked randomly to be checked. How? It was simple: every passenger picking the green line (Nothing to declare) had to press a big button. Then someone (probably a computer) decided if you should get a green or a red light – if you got a red light, you were chosen for random check.

The difference is not so large, but for the passenger, it does feel better, and the people working there will be able to defend themselves against the charges of racial profiling and all such charges that are usually impossible to defend yourself against. (However, they should still be able to check people who they found particularly suspicious…)

Then we were to take a taxi to town. We went to a counter and was presented by a screen. The woman behind the counter told us that we could pick any taxi company we wanted – or press the big “random” button. We chose to press “random”, paid our taxi ride and went out to find the company. An altogether better solution than to have every single taxi company having their own counter in the arrivals area – and the alternative would have been that we had walked out of the airport and chosen a taxi at random outside.

So already on the first day: two nice ways to use randomness to solve problems.

(Written 4th of June 2008.)

(The blog posts from Mexico will be posted with a delay of 20 days. This is simply because it is not considered advisable to advertise to the world that you are staying away from your apartment for weeks…)

Leonardo and Pacioli

Who would you have chosen to illustrate your mathematical text if you could choose anyone - including people long dead? Maybe not Picasso, who might confuse the readers. What about Leonardo da Vinci?

It has long been known that Leonardo da Vinci collaborated with (and was a very good friend of) the mathematician Luca Pacioli. This week, newspapers and magazines (for instance National Geographic) have written about Pacioli's "De Ludo Schaccorum" ("Of the Game of Chess"), which apparently is an important treatise for chess historians and which was just rediscovered last year.



The interesting (if unsurprising) find is that the chess diagrams may have been drawn by Leonardo da Vinci. That's cool! I wonder if Leonardo was a good chess player - I would be very surprised if he wasn't...

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...