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

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

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

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

## Wednesday, August 4, 2010

## Tuesday, August 3, 2010

### Euclid in color

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

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

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

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

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

## Sunday, August 1, 2010

### ESU6 Day 4

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

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

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

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

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

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

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

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

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

coffee break area

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

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

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

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

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

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

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

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

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

coffee break area

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

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