PME37 Day 4 #pme37

The first talk of Wednesday (one I also chaired) was Shin-Yi Lee's talk on metacognitive strategies for improving problem solving abilities in probability. She had done an experiment in which 9th grade students were encouraged to use a metacognitive-strategy worksheet while working on probabilities. The most striking result to me was that the control group, getting traditional instruction, had a significant decrease in their results from the pre-test to the post-test. But in the discussion we concluded that this may often be the case: if students use their common sense, they may go a long way in simple probability tasks, but if they are given a few formulas, they will try putting everything into them, leaving their common sense. The study had its limitation in that we knew little about the experimental group and the control group, and many ideas for further studies popped up in the discussion.

Then I went to Gert Schubring's talk "From 'armchair pedagogy' to experimental research and to case studies". He discussed the history of empirical research in mathematics education. At an early stage, teaching was considered as a treatment with learning as an effect, and research was done to show the effect. There were many small-scale studies with little connection to theory. Around 1960, large-scale studies were made possible by grants from science funds. In 1969, at ICME1, Begle called out for more empirical research, less research based on opinions. He repeated his criticism in 1979.

Later, case studies got more prominent and the interest was more in qualitative than quantitative studies, although TIMSS and PISA have been important exceptions. Schubring claimed, however, that these studies perhaps grew more from outside the mathematics education research community than from within. Of course, as anyone who has ever attended a talk by Schubring will understand, this short note is nowhere near doing justice to Schubring's talk, full of information as they always are. The best thing I can hope is to have given a sense of what the topic was...

By the way, the term "armchair pedagogy" reminded me of an earlier discussion within the HPM community (where Gert Schubring is of course a prominent member) about "armchair research". There, I have argued that the field of HPM needs both research and development (in Norwegian: FoU - forskning og utviklingsarbeid). If by "armchair research" is meant the development of "good ideas" and teaching sequences by experienced educators with an interest in the history of mathematics, this is an invaluable part of the development of the HPM field, making it possible for teachers to take up HPM practices as well as providing materials that researchers can research. To do research on education, you need someone to provide the actual education you want to study, and it is not necessarily the best researcher who has these ideas...

Then, C. Miguel Ribeiro talked on "Characterizing prospective teachers' knowledge in/for interpreting students' solutions". In the programme, Arne Jacobsen was given as the presenting author, but there was evidently a change. Of course, an important part of being a mathematics teacher is to interpret and try to make sense of students' solutions. This project used Ball's MKT "ball" as a starting point, here looking particularly at common content knowledge and specialized content knowledge. They see the ability to make sense of students' knowledge as part of special content knowledge. In their materials, which concern fractions, they do see a tendency that teachers mostly see as correct those of students' solutions that are similar to the teachers' own solutions, which suggests a need to focus on this in pre-service teacher education. One strength of e study, by the way, was that it involved three different countries.

As so often happens in these cases, there was a lot of discussion on the quality of the tasks and how they should be interpreted. One comment was on the concept of "babbling", where the language of pupils is interpreted as the unqualified use of language in a context where the pupil is busy working out the mathematics, so that some of their utterances are not well thought through. Another comment was that the correct answer to the question "What amount of chocolate would 6 children get if we divide the 5 bars equally among them?" is 5...

Then it was time for the conference's excursion. I had chosen to go to Lübeck. I did not expect this excursion to be as fascinating as the North Korea border excursion at ICME, but it would still be a welcome diversion, both from mathematics education and from private problems...

PME37 Day 3 #pme37

The first thing on my agenda on Tuesday was the Research Forum on "strong" discursive research. For this, we were supposed to read some articles and transcripts in advance, one of which was a chapter by Anna Sfard, in which she seems to claim that good old quantitative research has no place in mathematics education. For instance; "When numerous cases that have nothing in common with each other except a certain superficial feature coalesce under a single numerical label, our access to the diverse factors responsible for individual learners’ success or failure is lost forever. To produce a picture that can count as truly helpful, an incomparably higher resolution is needed than can ever be attained in randomized experiments."

I disagree completely. I believe that both quantitative and qualitative research can be "truly helpful" in our field, and that they give us quite different kinds of insights. Thus, I looked forward to this research forum with some worries - but of course the best way of approaching it would be to see it as offering one more potential way of doing research in mathematics education, disregarding its claim to be the best or only way.

However, even during conferences life may disturb. In my case, some urgent private business had to be taken care of, so I missed both this research forum and almost all the rest of the day's activities. I had no way of concentrating on mathematics education. However, after doing what I could to sort the personal stuff out, it was good to get back to a lecture hall to try to think about something else.

So in the evening I heard Cynthia E. Taylor talk on the title "Facilitating Prospective Teachers' Knowledge of Student Understanding: The Case of One Mathematics Teacher Educator". She went through some of the theory connected to pedagogical content knowledge (PCK) from Shulman onwards. Taylor has studied the actions of a teacher educator in whole-class work - the teacher educator she studied was her close colleague, who was also her supervisor, which of course must have lead to a lot of Issues. The colleague even participated in creating the analythical cathegories. There was one nice methodological touch: "Extended video talks", in which the teacher educator watched a 20-minute video of her teaching (albeit three years later), stopped it where she wanted and commented, with no questions from the researcher before at the end. In this way, the researcher got a rich material with lots of explanations for what the teacher educator did. This particular teacher educator gave lots of examples of what typical pupils would do, both to teach students about misconceptions and to teach them that there are lots of different answers to the same question, and they have to get used to seeing new answers. I did not note down much more of the findings, but they can probably be found in the article.

Lastly, Janne Fauskanger talked on "Teachers' Mathematical Knowledge for Teaching Equality", which is connected to the Stavanger work on MKT which have been presented elsewhere earlier. Here, she is looking at what can be learned about teachers' knowledge of the equal sign when analysing multiple-choice questions vs. free-text questions.

After repeating the contents of "Ball's egg" (which seems to pop out in a lot of talks here at PME), we learned that the data for this paper is 30 teachers' responses to five MKT items. She described the incorrect answers of the four teachers who did not have all multiple choice questions correct - they seemed to have typical misconceptions (for instance having only the operational understanding of the equal sign). The long responses give much richer understanding, including drawing upon different aspects of MKT than the items were developed to measure. And some teachers answered "I'm not sure" but gave very insightful answers in the free-text items, which casts a shadow of doubt over how "I'm not sure" should be interpreted when you have no free text to go with it.

That free text answers give richer insights than multiple choice answers, on the other hand, is hardky surprising. Isn't the point of multiple choice questions also that they can be administered to a larger group of teachers, giving information on how the group as such fares - or by making possible to give more items to each teacher without the analysis afterwards being prohibitively time-consuming? Moreover, they can be used to provoke discussion among teachers, just as one commenter to Janne mentioned. Janne's research gives insights to inform the discussions on how such multiple choice tasks can (and how they can not) be used.

PME37 Day 2 #pme37

The Monday started with Doug Clarke's talk with the title "Understanding, Assessing and Developing Children's Mathematical Knowledge". He started by noting that, for many reasons, there has been a shift away from pencil and paper assessment as the main way of assessing children in both New Zealand and Australia. This talk had to do with thousands of teacher-lead one-on-one face-to-face task-based interviews with children for assessment purposes. The research was based on an idea of "growth points", which can be seen as "key stepping stones in children's mathematical understanding". For instance, for addition and subtraction, these "growth points" are similar to "strategies" for addition and subtraction that we teach our own students.

The Early Numeracy interviews were based on 60 different tasks, the tasks actually given depending on the child's answers along the way. A Rational Number Interview was developed for older kids. Nice task: from six numbers given, create two fractions that added together will be close to 1. In the interviews, a lot of student thinking could be inferred from how they moved their cards with numbers.

He referred to The Interconnected Model of Teacher Growth (Clarke and Hillingsworth 2002), which I should look up... During the study, teachers developed more realistic understanding of their students' abilities. For instance, one kid in the study could read a 15-digit number, while many teachers focus on numbers from. 1 to 20 at that age. The project also included making teachers more aware of important strategies. The teachers also developed improved questioning techniques - modelling their behaviour in their classes on the interviews. The teachers also learned that they sometime had to give students a bit extra time to think for them to find the answer.

The obvious question to this is one of resources. Most teachers would love to have the time to have long one-on-one talks with each student in every single subject they teach. That is not possible, and if it was, there would still be a discussion on whether all the extra money could be better spent in another way.

To this plenary, Marja van den Heuvel-Panhuizen gave a prepared reaction (which was of course based on the article in the proceedings, not on the actual plenary talk). She asked: what was the proof that it worked? The evaluations were based on self-reporting of teachers and improvement of students' achievement, but the results of control geoups were not given in the paper. She also went on to describe problems in Clarke's choice of words and so on. She then invoked Hans Freudenthal to say that observing learning processes is (or should be) at the core of teacher education. She seemed to use this as a reason why interview-based assessment is a good idea, and that starting from a typical learning path is a good thing.

Then there was a discussion group on Mathematics Teacher Educators' Knowledge for Teaching. This sounded interesting to me because we of course discuss our students' (prospective teachers') mathematical knowledge for teaching when educating teachers, and it is only fair that we look at our own knowledge as well. Moreover, I'm interested in reminding people that the history of mathematics probably should have a part in what mathematics teachers' should know, as well as (of course) what mathematics teacher educators should know. The first 25 minutes of the discussion group was spent on every participants' saying a little about themselves, which proves that the number of participants were not too small and that the participants were not too silent...

Then we went on to small group discussions, where one was to discuss what are the differences between mathematical knowledge for teaching for teachers compared to for teacher educators, whether it is different for different types of mathematics teacher educators, how we can do research on thism how to educate mathematics teacher researchers and how to ensure resources for this work. Splitting into groups were a challenge, but once we had settled in a room, interesting discussions unfolded. We discussed the first question: what is the difference between being a mathematics teacher and being a mathematics teacher educator? Some of the contributors had the experience of being both at the same time, which is an interesting experience. Someone felt that there was a similar leap from being a teacher to being a teacher educator as the leap from being a student of mathematics to being a teacher.

One topic we touched upon was research. While teachers of mathematics do usually not keep up to date on the most recent research, teacher educators certainly should. Also teacher educators need to know something about the (hypothetical) learning trajectory of teachers. An example is how we teach fractions in a way to make the teacher students understand that what they thought they knew well, they didn't know nearly well enough.

A "technical" question is whether one should try to adopt the cathegories of Ball or wrap Ball's egg within a new layer.

It is also interesting whether mathematics teachers all need to know the same, and the same goes for mathematics teacher educators. Of course, we tend to divide our teaching tasks based on our areas of expertise, so maybe it is enough that half of mathematics teacher educators know their history of mathematics, for instance? Could the same be said of teachers in school, or is that a "lonlier" profession in the sense that they tend to have sole responsibility for mathematics teaching in their group of pupils?

Today's discussion group reminded me that discussion groups are among the most difficult things to plan (as I learned at ICME, although it did turn out okay in the end) - you don't know in advance if there will be 10 or 100 people, and you don't know what the participants' previous knowledge may be. On the other hand, it could also be said that as long as you manage to set people together in groups and they are fairly interested in the subject, they will probably have a good time discussing the subject with each other, who are after all experts from around the world.

Olive Chapman's talk later in the afternoon had the title "Facilitating prospective secondary mathematics teachers' learning of problem solving for teaching". She noted that students' unguided reflection on problem solving often reduced to mentioning general steps (maybe similar to what textbooks often do?) or to describe which steps worked in their problem solving. Chapman wanted to help teachers become aware of different ways of engaging in and reflecting on problem solving and to develop deeper understanding of problem solving. The idea was to help teachers to use personal experiences to build their reflection on, by having them develop rich descriptions, through two approaches; a narrative approach and a "stuck-aha-approach" (focusing on turning points). In the talk, she discussed differences between these two approaches. The narrative approach was more affective, while the stuck-aha-approach was more cognitive. The narratives gave a good view of the process and real world problem solving. When looking at the stuck-aha the cyclic process is clearer. Both approaches were adequate to give room for reflection. Sharing, discussing and unpacking was necessary and gave the teachers the opportunity to a richer experience. (The sharing was done in groups immediately after writing the texts, and they were then challenged to discuss what they learned about problem solving by reading these texts, and what they learned about teaching problem solving.)

One person in the audience had the interesting comment that she has sometimes experienced that students reflect on the problem solving process and realize that their mathematical knowledge is insufficient (for instance that they don't remember Pythagoras' theorem, and they do need that to solve the problem), but that they don't follow up the reflection by actually trying to learn the mathematics in question. So there is the problem of reflection for reflection's sake.

Then there was a talk on teachers' emotions, by Pietro di Martino and others. They talked about the "math-redemption phenomenon". This has to do with the fact that many students who have had problematic experiences with mathematics, still look forward to teaching mathematics, and want to improve their relationship to mathematics. In addition to survey data, they also interviewed students. Students often identified "crises" in their relationship to mathematics, and these were always connected to the teachers they had. This turned into a fear of becoming the same kind of teacher as their previous crisis-inducing teachers. This also seemed to be a major motivator to reconstruct their relationship with mathematics. However, for some students this was seen as unobtainable (by themselves).

The concept of "maths-redemption" seems to be a powerful one, and could possibly also be used to motivate our students in Oslo. By the way, we also have lots of data on student emotions in Oslo, which it could be interesting to reanalyze in light of this talk/article.

Finally, starting at 6:20 pm, was the presentation by my colleagues Annette Hessen Bjerke and Yvette Solomon on our project in Oslo. The project is looking at how teacher students experience their transitions between their teacher education on campus and their school placements, and (in this paper) particularly at what they and their mentors think that the students have learned in the different arenas. For the content of this, I refer to the article in the proceedings.

Simon Goodchild commented that first year students could only hope to "cope" and that we would expect them to show more of their competence later. This is a valid point, of course; by choosing to focus on the first year students, we get another project than if we had focused on third-year students, for instance.

Thus ended the first full day of the confence. (Except, of course, that we had some reading to do before Tuesday.)

PME37 Day 1 #pme37

After attending four ICMEs and four HPMs, now is finally the time for me to attend my first PME conference as well, as it happens to be held close to home. Compared to ICMEs, PME 37 has a rather small attendance - from the list of participants, it seemed to be about 600 people.

As other conferences, it started with a series of welcome messages by people who are expected to say something at such an occation.

Then there was the first plenary lecture - Kristina Reiss gave a talk with the title "You can't teach an old dog new tricks? Developing mathematical competence over the life span." Her topic could also be framed as: What is lifelong learning of mathematics? I it important for non-mathematicians? And is it possible? Of course, she gave many reasons why it can be important, both personally and professionally. One example; wireless plan.

She described an experiment from Wynn (1992) - where she showed that four months' old children "understood" that 1+1=2. The experiment is also replicated on monkeys. She also described how such understanding is a predictor of later knowledge in mathematics. Thus, she gave examples of how matematics education research has developed to make us better at teaching mathematics to children (although it is still an important area of research, of course). The natural question is then what we know about how to develop mathematical knowledge with adult learners. Cohen (2003) says that adult education is under-researched, under-theorized and under-developed.

She noted that many "standards" that have developed are wonderful if you have wonderful students, great teachers and perfect equipment, but that they are less helåful if you have diverse classrooms including children with learning problems and so on. She also points out the paradox that mathematics is a unique subject in being axiomatic and based on proof, while school mathematics use quite different approaches. According to Reiss, we lose far too many students by making them believe that mathematics is too difficult. Felix Klein described a "double discontinuity", and (if I got this right) she borrowed his concept to describe the situations for prospective mathematics teachers in that they go from school mathematics to a more rigorous and formal approach and then back to school mathematics to become teachers.

She noted that adult learners tend to learn things when they are particularly motivated for it, for instance if they need it, while younger learners tend to learn because they "have to" learn it. However, I think that although we have the "power" to make younger students learn because we decide they have to, it may be an idea to get even them to learn because they are interested and see that they need it... (Not that this point necessarily is at odds with Reiss' opinion, of course.)

After this plenary, there was a social programme that I will not blog about. Thus ended the short first day of the conference. Tomorrow will be the first full day, with a scientific programme from 9 to 19...

HPM Day 5

So we have come to the final day of the 2012 HPM conference.

Hong Sung Sa discussed "Theory of equations in the history of Shosun mathematics". He compared Eastern mathematics to Western mathematics, and noted that in the East, rational numbers, not real numbers, was the basic field of equations, and they did not work on factorization, like they did in the West. Solutions were rational approximations, not solutions with irrational numbers. This is interesting, as the solution formulas were of course important topics in Western mathematics, and played a part in the development of algebra. For details of the history on these methods, however, I have to refer you to the proceedings.

Yoichi Hirano gave a "Remark on the Notion of Golden Ratio - Concerning "Divine Proportion" in the Renaissance". He claimed that the topic of "golden ratio" is often understood only fragmentary by teachers. He talked on the history of the golden ratio, from before Euclid, then through Fibonacci and Leonardo, throuch Descartes, Durer and Simpson, Ohm, Binet, Cook and Thompson. Golden ratio is seen both in mathematics, in human culture and in nature, and Hirano went through many of the well-known examples of these. Of course, Leonardo's Vitruvian man was included. Leonardo's (remarkably good) friend Luca Pacioli gave the name "divina proportione" for this ratio, published in 1509. Leonardo da Vinci provided the drawings of the figures in this book. The name is believed to have connection with the platonic solids. 

Hirano suggested that Pacioli may not self have found this name, and that Leonardo was the real author. As a reason for this, he mentions that an earlier book by Pacioli seems to be a plagiarization of an earlier work. This argumentation was not altogether convincing, in my opinion, and we should certainly guard against attributing to Leonardo more than we can prove.

Leo Corry's talk on "Euclid's Proposition II.5: A View through the Centuries-Geometry, Algebra and Teaching" was next. He presented the proposition II.5,which has been interpreted algebraically as the conjugate sentence. (The square of (a-b) is the square of a plus the square of b, minus 2ab.) Tannery, Zeuthen, Heiberg and Heath were among the people interpreting it in that way.  In his talk, he went through how different editions/translations of Euclid throughout the centuries were formulated in this regard.

The final talk of this session was Qing-jian Wang's "The New "Curriculum Standard" and the New Mathematics - the Union of History of Mathematics and Mathematics Education". The new curriculums in China means that it will be necessary to teach teachers how to include history of mathematics into mathematics teaching. In Taiwan, there has been a HPM Newsletter for a long time (under the leadership of Wann-Sheng Horng), but in neighbouring China, a 2002 conference was the first one with papers combining the aspects of history and pedagogy. Wang described the history of the HPM activities in China since that time.

After these conferences, I would like to summarize. First, what are the main ideas I take with me from these conferences? Well, I think the discussions, both in the conference halls and over a beer, about theoretical frameworks for discussing the goals and/or outcomes on teaching with history of mathematics were most important to me. At the end of the conferences, I am motivated to work more on this.

At ICME, I heard lots of talks that broadened my overall knowledge of mathematics education. Alan Schoenfeld's talk has stuck, mostly as a reminder of the roles our beliefs, knowledge and goals play when we design our teaching, for instance with history of mathematics.

At both ICME and HPM, I've learned about other efforts of including HM in teacher education around the world, which I should keep up to date on.

Secondly, what are the best moments of these more than two weeks in Seoul?
1. Crossing the border to North Korea was a special and strange moment. To step from one of the most successful democracies into one of the worlds' worst tyrannies, that to this day operates concentration camps, was thought-provoking.
2. Having the first sip of beer with Andreas and Johan after the hot excursion day - and a long walk in the sun to find someone who sold beer - felt quite right.
3. Holding a talk at ICME with people sitting on the floor and standing in the doorway wss new and kind of cool - even though it was a small room...
4. Seeing some of my favorite HPM people engaging in my materials at my HPM workshop was also great.
5. Many of the dinners and/or beers with great people were wonderful. The contents of the conference may tempt me to come back, but it is the people that makes it unthinkable not to come back.
6. The view from the hotel room in Seoul was breathtaking - literaly.

So, there we are. I have sign on to a contract making me "Head of Studies" for the next four years, which does not leave much time for research and development on HPM. But I do hope to set aside some of my spare time from time to time to work on it. And anyway, I'll be back 100 % from 2016...

HPM Day 4

Anne Michel-Pajus' title was "A voyage into the literary mathematical universe". Her talk was on mathematics in literature. D'Alembert stressed the importance of imagination in mathematics, and for teachers, imagination is also very important when designing teaching.

Anne discussed lots of examples of mathematics in literature, for instance The Birds, where the mathematician is driven off the stage. Or Thomas Pynchon's Against the day, 2006. She gave a number of categories - "Literary modes for mathematical tracks" - and gave examples of each:
- Literal insertion (without real relationship to the unfolding of the narrative).
Use in teaching: discussion of interpretation, introduction of topics...
- Popularization
Allows interdisciplinary work and gives pleasant contexts
- Mathematics in the structure
- A mathematical object as character
Charles Perrault: The Loves of the Ruler and the Compass.
Use in teaching: Ask students to write poems or short stories whose characters are mathematical objects.
- analogy (or transfer) (transfer of mathematical reasoning to non-mathematical objects)
Analogies are a fertile tool in mathematics.
- an important character is a mathematician

Throughout the talk, literature examples were read/played by the "actors" Peter Ransom, Frédéric Métin and David Pengelley.

Then Johan Prytz gave a talk on "Social Structures in Mathematics Education. Researching the History of Mathematics Education with Theories and Methods from Sociology of Education."  He saw two main motives for studying the history of mathematics education:
- contribution to educational history in general. By comparing his own studies and the studies of Lövheim, he shows that the study of the professional debate among mathematicians/teachers gives s different picture than studying the general, political debates.
- contribution to research on mathematical education. Often too much focus on the big reforms in the 1960s. Håstad (1978) calls everything before 1960 "tradition". The critics of the reforms are ignored, and the changes in mathematics education before 1960 are not discussed.

Why a sociological perspective? History of mathematics education is often purely textual. A purely textual study cannot explain why some texts and authors are more influential than others. Johan considers different arenas: political, national level. Central school administration. Teachers. What's between the central arenas and the teachers? For instance, those who produce educational texts will form one such arena. One of Johan's results is that this group functioned as a field. The relationships between different arenas is also of interest.

In the next session, I attended Evelyne Barbin and Michael Fried's talks, but took a break from blogging.

"Empirical Research on History in Mathematics Education: Current and Future Challenges for Our Field." was the title of the second panel discussion at ICME. Panelists: Uffe Thomas Jankvist (Denmark), Yi-Wen Su (Taiwan), Isoda Masami (Japan), David Pengelley (USA). The focus was on the relationship between history in mathematics education and general mathematics education research. It is important to get the general community's attention, and one way of doing this is empirical research. In a survey of the literature, Uffe has found about 100 empirical studies on the HPM in its 40 years' history.

Masama Isoda talked on lesson study and technology, in particular his work with dbook (see also his talk at an earlier conference). He proposed that lesson study can be seen as a kind of empirical research.

David Pengelley focused on original sources. One way of evaluating is through open responses from students on the benefits and disadvantages of using original sources. It is more difficult to prove benefits with statistical analysis. He referred to Glaubitz' study (which Glaubitz presented in Vienna two years ago), in which the deep analysis of one text gave very good results. Some methods of using HM seem to have positive effects, while others have negative results. 

He also discussed recruitment, transition and retention, claiming that HM offers students more reality, less fantasy ("mathematics drops down from the sky"). There's often a disconnect between what students think mathematics is, and what it really is.

Yi-Wen talked about Taiwanese experiences. Currently, there have been 15 Master theses on HM in Taiwan. She gave examples of work on the old problem: "how to measure an elephant on a boat?" In a three-year project students create animations and worksheets. In the third year of the project, they will be used in practice. The students improve their ability to search relevant materials (although it is unclear by what methods this result was established, and the exact connection to HM).

Uffe then had the last of the panel presentations. He quoted Katz: "Too much H and M, not enough P" Uffe claimed that theoretical constructs from the rest of mathematics education would be useful, both internally and externally. He referred to Kjeldsen, Barnett and others at this conference, showing how different projects presented at this conference could be analysed using the Niss competencies, for instance. He also discussed Ball's "egg" on MKT, showing that HM could also fit into all of the parts of that.

Most of the discussion afterwards was on the topic of theoretical constructs from outside HPM, and whether these could be expected to be useful in our context as they often do not include HM in a good way. In particular, Evelyne Barbin was sceptical of the use of the word "competencies", as we want our students to get more than that, for instance certain attitudes and beliefs. As I have written before (and said in Vienna two years ago), I similarly believe Ball's "egg" has a too limited view of mathematics to be directly useful for our purposes.

A connected topic was whether we should "compete on their terms" as someone phrased it - should we try to show that teaching mathematics with history is as "effective" as teaching without, using the definitions of "effectiveness" from without HPM? Michael Fried was sceptical of this, as teaching with HM has its advantages that need to be taken into account when performing tests, for instance. David Pengelley and Peter Ransom, however, disagreed. Based on long experience in teaching with HM, they had no problem claiming that their teaching was as "effective" as that of their colleagues, but with the added bonuses that teaching with HM brings.

Only two more oral presentations were left. I was chairing the session where Francois Platade gave a talk on 70 letters between Mittag-Leffler and Houël. The letters concerned mathematics and how to teach it, educational policy and mathematical journals, among other things. Most interesting for me were their discussions on how to teach complex functions - I'm sure that someone teaching complex functions could make good use of these original sources to illustrate different ways of looking at these. 

Finally, I got the opportunity to hear most of Andreas Christiansen's talk (as one of the speakers where I was a chair did not turn up. Andreas gave a well-structured and interesting talks about very different ways of defining the basic concepts in geometry in three Norwegian textbooks.

Finally, there was a HPM Meeting, where Evelyne presented the future plans for the HPM group while I said a few words about the newsletter. The next ESU will be in Barcelona in 2014, while the next HPM will be in Europe in 2016, not too far from the ICME in Hamburg.

An idea that came up was that the HPM website should include lists of resources on HPM. Perhaps one comprehensive one and a short one for beginners? It was also pointed out that there should be separate lists for different levels of students. Who will take this idea further, is unclear.

HPM Day 3

The HPM conference has a peculiar design, in that each major theme are treated in order, each getting a little less than one day. This has the obvious drawback that if you are particularly interested in one or two themes, you will miss most of that, as there are many presentations on that theme at the same time. If you happen to have a talk on the theme as well, there is pretty little left for you to hear. On the other hand, when you come to themes you are not that interested in, you will have plenty of those to choose from...

Personally, I'm most interested in talks of how to include history of mathematics in teacher education for prospective primary or lower secondary school teachers. But if the talks touch upon use of history in mathematics in general, or on the history of some topic related to the primary or secondary school curriculum, I'm happy...

On the third day of the conference, we were approaching the more "purely" historical parts of the conference. The plenary talk was Dominique Tournè's "Mathematics of the 19th Century Engineers: Methods and Instruments". He started by talking about Lagrange's numerical methods for solving equations, and Fourier and Sturm's criticism that the methods were not easy to use in practice. Lalanne also pointed out that many of the methods were not practicable. In this situation, engineers created their own methods for finding solutions quickly. They did not need a high accuracy, but needed speed, and it was important that they could be done in the field and not only in the office.

He went on to consider the "cut and fill" problem (where the volume of mass you cut out for a road should equal the volume of mass you fill in at another place). "Hair planimeters" and other instruments were developed to find the areas (on drawings) in practice. A new mathematical discipline, nomography, were developed. Another was "graphic statics", in which metal structures were constructed with drawings on paper instead of by calculations. Example: the Garabit Viaduct and the Eiffel Tower. More than 1700 drawings were made for the "backbone" of the Eiffel tower, with an additional 3600 drawings for the execution. 

Ballistics was another area in which the mathematician's solutions were too cumbersome. Firing tables were needed. There were also designed curves which would give the needed information directly.

The French model of École Plytechnique gave a mathematical, theoretical grounding followed by "practice" in engineering. This contributed to a marhematization of the engineering art.

The rest of the talk was devoted to the example of nomography. Lalanne, Massau, Lallemand, d'Ocagne and Soreau were the main characters in the development of this field. The solution of equations were in this way reduced to reading graphs. The graphs were eventually reduced to three lines next to each other, where you could find the value of the third variable by identifying the values of the first two and drawing a line between them and on to the third line. In this way, the problem of "cut and fill", for instance, was very much simplified.

It is quite obvious that in studying the history of mathematics, the development of pure mathematics has been prioritized, while the mathematics of the engineers have not been given so much attention. Tourné points out that this mathematics could be even more fruitful for the HPM community. This is a point I understand - and fits well into the overall pattern that primary school teachers (and their teacher trainers) are mostly purely "academic", not knowing anything about most of the occupations the children will take up, be it bakers, carpenters or fishermen.

"Why Do We Require a “History of Mathematics” Course for Mathematics Teacher Candidates? (And What Might Such a Course Look Like?)" This was the theme of the first panel discussion, with Mustafa Alpaslan, Sang Sook Choi-Koh, Kathleen Clark,  Ewa Lakoma and Frédéric Métin. 

Frédéric described how in France you have primary school from age 3 to 11 (elementary 6 to 11), secondary from 11 to 18 (college 11-15, lycee 15-18). To become a primary school teacher, you have three years of general studies (or maths, to become maths teacher at higher levels) + one year theory + one year practice (exams in each year). Then you become a civil servant. 

In the University of Burgundy, there is no history until they have an optional course in the first year of their master, "A short course on several mathematical ideas", or, for primary teachers, an optional course on "(Re)discovering Maths".

The main justification for these courses are to show students that they can do something else than traditional teaching and show a more cultural approach. It is also important to make the familiar unfamiliar.

Mustafa is teaching students who should teach students aged 12-14. They have pure maths ++ for the first two years, then two years of pedagogy etc. From 2007, there are courses on history of science, history of mathematics and philosophy of mathematics. These have centrally decided guidelines. (see his talk at ICME). 

Sang Sook: in Korea, elementary school from age 7, then middle school from 13, high school from  16, university from 19. 4 years of teacher education or 4 year maths program + 20 credits in the education department. There is a highly competitive teacher examination, which leads into the public school system. 

Their curriculum does not have explicit HM contents. Sang Sook argued based on the genetic principle that HM should be included in the teaching, and used this when teaching the concept of function. In textbooks in Korea, HM is used as motivational tool, but far less usual is HM used to teach concepts. Teachers also report that they don't know how to use HM in this way.

Ewa described the situation in Poland: the educational system is 6+3+3. You can become a teacher for grade 1-3 by taking courses in "pedagogy". You can become a mathematics teacher by taking mathematics courses and then some pedagogy. 

HM is not included in the key competencies, but implicit in curriculum proposals, explicit in school textbooks and other didactical materials. Significantly, there is no HM in the exams. At university, there is a course on HM, but this is not for teachers especially. 

Kathy noted that there may be important differences between HM courses given by maths departments and courses given by education departments. Personally, I'll say that a key is whether it is a course /for teacher students/ or just a general course. 

Then there was a discussion which I can certainly not summarize here. However, a lack of resources were mentioned by some people - we always tend to end up there. Mannfred mention how it is important that history gives a new way of approaching mathematics, for students who have spent many years working on mathematics. Michael Fried mentioned how it is also another way of thinking - and the students have little training in historical thinking. This ended the third day of the conference - the rest of the day was a nice excursion. The main point of an excursion - apart from providing some fresh air and a pause from lectures - is the opportunity to have long conversations with your colleagues, and thus get to know them in another way than in the confence halls. 

After this trip, we were ready for the last two days of the conference...