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.