### 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...)

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...)

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