Friday, May 05, 2006

Studying original sources in mathematics education Day 5

The week in Oberwolfach is drawing to a close. Today was the last day of the academic programme, and there is just one breakfast left of the great food here. It will soon be the time to go home and try to keep up the motivation which has been building this week.

Today's first talk was by Hans Niels Jahnke. He talked about working with Bernoulli's lectures on the differentual calculus (1692) with 16/17-year olds. Halfway through the talk we were ourselves faced with trying to understand the mathematical ideas. Remember that an important idea was that of infinitely small quantities - and then even infinitely smaller quantities. This was the "foundation" for calculus for lots of years, even though a shaky one. (I wrote a Master thesis on part of the British side of this story.) It is always funny to try to understand the way these people were thinking on these things, and the students seem to have though so as well. I would certainly have tried to copy Jahnke in this if I had students in calculus.

The second talk was by Jan van Maanen. He was, with Jahnke and Fulvia Furinghetti, the organizers of this conference. His talk was titled "Original sources, projects, lessons and lectures". He has (as I knew before) a soft spot for original sources as in the real thing - he thinks the experience of actually handling a 1700th century book is an important part of the whole thing. That is of course good as far as it goes, but if studying original sources is to become something that the average pupil in Norway does, we will have to loosen the definition of "original sources" a bit (as van Maanen is fully aware of, of course).

van Maanen first gave a couple of examples of his own teaching through the years. Then he went on to talk about Iris van Gulik-Gulikers' PhD thesis, consisting of (among other things) teaching experiments on the history of similarity (similar triangles) and on non-Euclidean geometry. Both sounded fascinating, and I promptly mentioned that I would really want a copy of the thesis... The study was a fairly large one, being tested on about 800 pupils. I find it good to see that such solid, large-scale studies are now being done.

The rest of the day consisted of some round-table discussions and of some participants presenting things that they had not had time for earlier in the week. Among other things, I was reminded of a 1654 letter from Pascal to Fermat about a dice problem that I should really try to incorporate in my own teaching, and I was also forced to try to understand the algorithm for extracting cube roots. I think I got the main point of it in the end.

I think that I now should end these writings from Oberwolfach. The first four days are treated here:
Day 1
Day 2
Day 3
Day 4

Thursday, May 04, 2006

Studying original sources in mathematics education Day 4

A new day, new possibilities for learning!

The first lecture of the day was Katja Peters' talk "Perceiving history of mathematics". She has done a project where 18 year old students from a German intensive course visited a library for three consequetive days to study original sources. This was a way for the library to teach students about old books. Sadly (but understandably) most libraries do not want schoolstudents anywhere near their old books, so such projects are not possible anywhere. On the other hand, more and more old books are available online, which means that variants of this project may be easier to do in future. (Anyone who knows sites where old mathematical texts suitable for 6-16 year olds are available, are urged to tell me...)

Peters also talked about her way of choosing the most suitable books, and this was maybe the most valuable part of the talk for me. That is because the "upscaling problem" is very much a problem in this case - it is not clear how this experience can be brought into anything like the average classroom.

Kathy Clark talked about "Use of original sources: One Teacher's Experience with Personal Study and Curricular Inclusion". The design of her research was very interesting. She gave five teachers professional development sessions on the historical development of logarithms, and then studied how the teachers taught logarithms thereafter. Obviously, not all five changed their way of teaching significantly, but more than one did. That is actually quite good news - at times, I get the feeling that what teachers learn in professional development courses are left behind before they go back into their classrooms.

Evelyn Barbin's second talk of the workshop was titled "Reading in a historical context: "depaysement". The example of The Geometry of Descartes" (sorry for leaving out at least one French point on an "e" in there...) The main point was to show one example of students interpreting a historical source in its historical context, and one example of them comparing different texts. The importance of The Geometry as a step towards our way of using algebra for solving geometrical problems was underlined - today's students may not see the genius of Descartes because what he does has become so "normal" today...

Barbin also underlined that there has been a change in how source books are produced (at least by IREM). Before, source books were perhaps more source books for studying history of mathematics. Today, they are made to be used in teaching - the point is to understand mathematics by historical texts. She showed some wonderful new books which sadly are in a language I don't read well enough to appreciate them fully (French, of course).

What should be noted over and over again in these notes from this workshop is of course that all interpretations are mine - the speakers' own account of what they really meant to say is best found in their papers when they are published. While waiting for that, the Oberwolfach Report should give some idea, that should be available within a few months' time.

Michael R. Glaubitz also had a very interesting talk. His title was "Reading Al-Khwarizmi's Treatise On Quadratic Equations with 9th-graders. An Empirical Study". He had produced a wonderful workbook on this and two related subjects, and had done a teaching experiment in seven classes with a control group of an additional three classes. He did not do the teaching himself, which would obviously have interfered with the value of the project. The results of the project are not clear yet, but it is surely an interesting way of doing it. One interesting outcome is obviously an answer to the question "Do the students learn quadratic equations better?" As interesting are the questions "What else do they learn in the process?" and "What kind of knowledge of quadratic equations do they get?"

The last talk of the day was Wann-Sheng Horng who talked about different proofs of Heron's formula (for the area of a triangle. This talk focused on the dual role as both a teacher and a historian. Certainly, that is a very interesting combination, but a question I always come back to is this: to what degree does the teacher have to know the history of mathematics to be able to enrich his teaching with history of mathematics. It is obviously a question in the same family as lots of other questions of the same sort: Do you have to know the name of all flowers to talk about flowers to your students? I believe (or at least hope) that it is possible to start including history of mathematics in the mathematics lessons without being a historian. No, that was too weak - I know that this is possible.

Wann-Sheng Horng also mentioned another interesting point: by looking at the history of Heron's formula in China in the (a bit) broader context of the history of formulas for the area of a triangle (still in China), there is a "vertical integration" pointing to things from primary, secondary and high school mathematics.

After dinner, we had an "extra session" with a more free-flowing discussion on several issues, although the main concern of everyone was how we should improve the situation concerning the topic of the workshop: "Studying original sources in mathematics education".

I have long pages with ideas of what I should do when I get back home - I don't think I will keep mentioning them here. But such a workshop is certainly inspiring, and at the moment I feel it is likely that I will both go to a conference in Prague next summer and two others in Mexico and thereabout the summer after that. I just hope I won't work too much, and that I will also have enough time for my boyfriend back home... (I miss him...)

Studying original sources in mathematics education Day 3

Day 3 was in one way a very short day, in another a very long one - depending on how you look at it. There were only two talks, but there were a long walk in addition - two hours the one way and one and a half hour back. It was pleasant and the weather was great, and now everybody is probably ready for a new full day of work.

The first talk was by Costas Tzanakis, who is also the chair of the HPM group. He had three examples. The first one showed an example of using ancient Greek mathematical texts in the teaching of Euclidean geometry in the Greek Lyceum. This showed many aspects of the work they had done, but for me the most interesting part was the discussion of "why do we have to prove this" which always comes up in the classroom. This was treated by looking at the Epikureans' criticism of Euclid, and seems a fertile starting point for discussions.

His second example was on the concept of (instantaneous) speed. Here, a quote from Newton showed with all necessary clarity how confused this concept was at the time. A discussion on Newton and his immediate followers (and the criticism of him) could be very interesting. Working on Zeno's paradox is another option.

His third example was on Hamilton's quaternions, and Karin Reich followed up with a talk on "The historical roots of vector calculus: J. W. Gibbs (1839-1902)". Sadly, vector calculus is not in the Norwegian curriculum for 6-16-year-olds, so there was little in these two talks that I can take directly into my classroom. However, the point that we should some of the time look at "original sources" not as the ones that does neccessarily contain the most "original" ideas, but as some "old", authentic sources where mathematical ideas are treated, seems worth making.

The days here in Oberwolfach is making me start thinking of what I should do when I get back. One simple idea is just to use history of mathematics in my teaching at about the same level as before (or a bit more), and at the same time "tracking" the students' idea of the "value" of history of mathematics through the year. That may be an interesting starting point for a discussion. (On the other hand, I already have an ongoing interview study, and I will also do two studies on Norwegian mathematics textbooks, one on the historical contents and the other on the "family concept" in them, so perhaps I have enough to do for the next year...)

Tuesday, May 02, 2006

Studying original sources in mathematics education Day 2

Another interesting day at the mini-workshop at the MFO. The day started with a talk by Peter Rasfeld, where he told us about a project he has done on the problem of points (the famous problem worked on by lots of mathematicians, for instance Luca Pacioli, Girolamo Cardano, Niccolo Tartaglia, Blaise Pascal and Pierre de Fermat). He described a four-week project with his pupils aged 16, where the pupils worked on original sources much of the time - after initially trying to solve the problem themselves.

I am very happy to be able to hear this kind of presentations, as I have on previous conferences and meetings of this kind had an impression that most people have more interest in things for older students. For me, who is teaching teachers-to-be for 6-16-year-olds, it is important that the mathematics is at a reasonably low level...

The second talk was my talk. I would have liked to be able to say that it was absolutely brilliant, but I am in no position to know that. However, I think I managed to get the point across that including history of mathematics in the average classroom takes a lot more than just putting it into the curriculum. I used a study on the TIMSS 1999 Video Study material, a study on the Norwegian textbooks in elementary and secondary school, and a study that I am doing at the moment where I interview teachers about their attitudes to history of mathematics.

The third talk was by Adriano Dematte (with a sign over the last "e" that I am unable to conjure up on this computer). He talked on a new collection of materials for secondary school students (12 to 18) that is coming out in Italy in a few months. I'm looking forward to getting hold of a copy and considering what I could do to get this kind of material available in Norwegian for Norwegian teachers. Together with the Historical Modules from the US and with a book I just heard of today, this new book certainly will give me some ideas (although things will certainly have to be adjusted to fit the (not too impressive) level of the Norwegian mathematical curriculum.

The fourth talk (after lunch) was by Caroline Bardini and Luis Radford, with the title "Unknown, Variables and Parameters". I think I have tended to think that the difference between these should not be stressed too much to the students, but I believe that I should think it over more thoroughly. It is obvious from the discussion in the lecture hall and outside that there are important questions to be answered here. The video segments from classrooms were also interesting.

The fifth talk was Evelyn Barbin's talk titled "The different readings of original sources: an experience in pre-service teaching". The talk included an amazing number of ideas for historical sources that can be used in a mathematics course - her own present experience was from a course for pre-service teachers who were aiming for primary school. (French primary schools, that is, which is at another level altogether than Norwegian ones.) Maybe the most important part of her lecture, however (which was exemplified throughout the rest of the lecture), was an attempt to cathegorize the different ways students can engage with a historical source - from "interpret in our mathematical language" at the one extreme to "interpret in its own historical context" at the other.

My main decisions coming out of this day were these:
- I will put more effort into making available for Norwegian teachers materials on history of mathematics (for use in the classroom) in Norwegian.
- I will look again at my own teaching to think of how history of mathematics could enhance it more than presently, and think of the different ways students could engage with the materials.

In addition to this, there has been wonderful weather, good food, a nice walk and lots of friendly people. I'm looking forward to the next three days. And I'm glad I've had my presentation, so I don't have to think of that any more...

Monday, May 01, 2006

Studying original sources in mathematics education

I have been terrible at posting in this blog, but I'll try to improve. I am currently at a mini-workshop at the Mathematisches Forschungsinstitut in Oberwolfach on the topic of "Studying original sources in mathematics education". The place is wonderful - the place is so green and beautiful, and there are all neccessary amenities to have a great, productive time.

On this first day of the mini-workshop, I have heard two great talks. The first one was by Abraham Arcavi. He told us about a teacher workshop he conducted with Masami Isoda, where teacher students were confronted with problems from the Rhind Papyrus, on multiplication and on solving an equation (as we will see it). The starting point of the work was that whether you are learning history or you are learning to teach, you have to learn to interpret. And maybe it is easier to learn to interpret by working on historical sources than by working on student answers, as it is harder to dismiss a historical source as being "wrong".

Arcavi gave some details on this particular workshop, not least on how "scaffolding" was necessary to ensure that the teacher students managed to work through the text in a meaningful way. The point noted in the end of the talk, however, seems to be one of the most important - how can you "upscale" such an experiment? This particular workshop was conducted by two prominent researchers in the field of "history and pedagogy of mathematics" (HPM) - how can we make "ordinary" teachers do the same (or similar things)?

One small exercise: Ron is given the following exercise: "3/5 of a number is 12, what is the number?" Ron writes: "12*2 = 24 24:6 = 4 24-4 = 20" Is Ron correct? How would you as a teacher react to his answer?
(This should show the need of a teacher to listen attentively, not just "evaluative".

The other talk was by David Pengelley, who showed us how he works with Pascal's "Treatise on the Arithmetical Triangle" with his students. I did not know this text before (although I have referred to it many times), and it was interesting to look at it a bit closer. As mentioned in an earlier posting, I tend to use exercises on Pascal's triangle in my teaching, but surely I learned something new today - even though some of the more advanced stuff, such as Pascal's use of inductive proofs, is probably not suitable for our first course at my institution...

Obviously, people are speaking to the converts at such a workshop. As mentioned already, the important thing is to (to use Pengelley's words) "convince teachers that they want to do it". Developing materials seem to be the way to do that, and I'm sad to say that materials in Norwegian are few and far between. I should do something to remedy that at some point.