Why use historical sources? Janet mentioned several reasons: Decrease risk of trivializing history when used as a teaching tool, help students see how to develop and reason with ideas on their own and help students develop mathematical competencies (not just techniques). In this work, history is used as a tool, both a cognitive tool and a motivational tool.
In the example here, the goal is to develop an understanding of elementary set operations and their basic properties. They start off with DeMorgan, but quickly go on to Boole's "Laws of Thought" (1854). One great thing about this source is that Boole is so explicit about all the choices one has to do about how to use symbols. His definition of addition of sets gains him in terms of algebra (giving x+y=z -> x=z-y), but gives notational inefficencies - and this can be used to discuss choices of notation with the students. After lots of work on Boole, they go on to Venn and Pierce.
Of course, the particular example in this talk is not directly applicable to Norwegian teacher students, but when looking at the reasoning and the design, there are many interesting points to bring home.
Tinne Hoff Kjeldsen's talk on "Uses of History for the Learning of and about Mathematics: Towards a Theoretical Framework for Integrating History of Mathematics in Mathematics Education" was a contribution in the direction of helping analysing teaching with history.
The first part concerned how history is used, the second concerned the roles of history in mathematics education. On how history is used, Tinne echoed Fried in warning against a Whig interpretation of history. She pointed to how trying to understand what mathematicians wrote, from their points of view, can give rise to many interesting questions, such as " Why did he choose that definition?"
She referred to Jensen's model of ways of using history:
- pragmatic (what can we learn from history) vs. scholarly history (past on its own terms)
- lay vs. professional history
- actor (used to orient oneself or act) vs. observer (enlightening purpose) history
- neutral vs. identity history
Tinne talked on the first and third pair of these.
On the roles of history, she referred to Niss and Sfard. Niss' model has eight "main competencies" and three "meta-levels". Sfard's theory is used to argue that HM can play a role in revealing "meta-discursive rules" and make students discuss these (Tinne also discussed this in Seoul). That may happen when different discursant have different meta-discursive rules. (It occurs to me that this is pretty close to what Evelyne refers to as dépaysement - reorienting.)
Tinne used two examples. One was on Egyptian mathematics in 10th grade. Analyzing the teachers' comments, Tinne was able to characterize the goals of the teachers based on the framework above. The second example was project work at Roskilde University on "Physics' influence on the development of differential equations". Again, using the framework gives a useful starting point for discussing the examples.
The Danish at ICME and here (as well as at earlier conferences) have been quite eager in promoting the use of theoretical models from general mathematics education in our discussions. I, for one, is partly convinced by their arguments, but of course there are also sceptics who feel we are better off creating our own models, to avoid inheriting problems from the other models.
In the discussion, Evelyne proposed also invokong Bakhtin's ideas of seeing the mathematical text itself as a dialogue - to get the students involved in dialogue.
For the oral presentations, David Guillemette talked on "Bridging Theoretical and Empirical Account of the Use of History in Mathematics Education? A Case Study", based on his master's degree. He tried to teach calculus with history, and wondered whether it was problematic to learn both concepts and the history at the same time. He wondered what meta-issues reflections could come out of such work. He chose to work on a part of Fermat's work on maximum and minimum. In the project, there were 20 students, 17-18 years old, students who had failed the course in the fall. Data collection was short interviews with very open questions.
He thought that history is important to go "beyond the here and now", cultivate the capacity to be astonished, developing our sensitivity in mathematics, developing a way of being-in-mathematics etc. He borrowed some ways of looking at his material from Uffe Jankvist (tool vs. goal) and Evelyne Barbin (cultural comprehension, repositioning, reorientation).
He then illustrated these theoretical ideas with quotes from his own students, and then started expanding on them, by referring to for instance Fried. He pointed out that we have lots of work left before we can establish a "common" framework for discussing.
In the discussion, Anne referred to "the unreasonable effectiveness of dépaysement" (reorientation). Only when you have seen a city from every viewpoint, you know the city. Thus, there was some discussion on the relationship between the ideas of Barbin and Bakhtin...
Rene Guitart discussed "Misuses of Statistics in a Historical Perspective: Reflexions for a Course on Probability and Statistics". He claimed that to understand well the concepts of probability and statistics, you have to go back to the historical sources. There is a "mathematical pulsation" between statistics and probability, and for a deep understanding of the two subjects, this pulsation should be studied.
In this talk, he went through a whole reading list of historical sources. First, the concept of "average". It should be stressed that the average is not "natural" and obvious, it is based on a decision. A good discussion is given in Bertrand (1889). He went on to discuss the concept of probability and the relativity of probability in time, and of the law of large numbers. In similar way, he discussed other key issues in probability and statistica, but I am unable to repeat them here.
After lunch, I made the final touches to my workshop. Based on the interesting discussions in ICME and HPM so far, I decided to make my own "framework" for discussing the design of historical materials for the teacher education classroom. It is interesting to me to notice that several frameworks for using history of mathematics in education, are too simplistic when applied to teacher education. For instance, Jankvist looks at history as a tool and history as a goal, but teacher students should probably also be able to use history both as a tool and as a goal, so in teacher education we can use history as a tool for teaching students how to use history as a goal, for instance. We always have to remember that teacher students are both learning the subject matter of mathematics (including history of mathematics) and how to teach mathematics.
As I only spent one or two hours creating the framework, it is certainly not a final version. Part of that time was even used for creating the powerpoint, which was a bit of a hassle as the computer I used had all its menues in Korean...
I had planned to hear George Heine's talk, but due to some changes in the programme, he was finishing as I entered the room. So the only thing remaining of Day 2, then, was my workshop.
A workshop at HPM is supposed to be a place where the partipicants work on something. Thus, the only way to make a truly terrible workshop is to talk too much - given the impressive knowledge of the participants, they will certainly find something interesting to talk about even if the materials they are given are silly. My workshop was based on four different activities I have used with my students, and the participants were asked to form groups, choose one of the activities and then discuss the activity - particularly which of the many goals we keep talking about will be touched upon. I was impressed by the discussions I heard as I was walking around. Of course, I don't know what everybody thought about the workshop, but at least some of them enjoyed it (if I am to believe what they said to me afterwards - and as a teacher, you should always believe on praise, as it is so rare...) So I'm happy about the outcome.
After the workshop, a few of us went out to have something ot eat and drink. My duties here in Daejeon were now done (except chairing a session on Thursday, that is), so now it was "holiday"...
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