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...

HPM Day 2

Janet Heine Barnett's talk "Bottled at the Source: The Design and Implementation of Classroom Projects for Learning Mathematics via Primary Historical Sources" was about a project with - among others - David Pengelley. The goal of the project is to develop and disseminate projects based on original sources to support learning in core material connected to discrete mathematics. The example used here was one on Boolean algebra.

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"...

HPM Day 1

About 18 hours after the closing of ICME, there was the opening ceremony of HPM in Daejeon. More than 100 participants, coming from more than 25 countries, had found their way here. Evelyne Barbin and Sunwook Hwang gave welcome remarks, followed by a fine musical performance.

Tsang-Yi Lin held the first plenary talk, on the subject "Using History of Mathematics in High School Classroom: Some Experiments in Taiwan". HPM Tongxun was started in 1998 by Wann-Sheng Horng, and has been become important in spreading HPM in Taiwan. Lin is a high school teacher in Taiwan, and his students are interested to find out how mathematicians managed to find the formulas they did.

He gave several examples. The first was how to connect conic sections to the modern definitions in the textbook, using Dandelin's theorem. The second was on Apollonius' work on conic sections, but I must admit I did not follow all the details of the examples. The third was on tables of logarithm (mentioning the book by Dava Sobel: Longitude). It Taiwan, students memorize 0.301 as an approximation of log2, but how could we find the approximation? He introduces Napier and Briggs, and used Briggs' method. 

The fourth example was on Cramer's rule, comparing Maclaurin's and Cramer's explanations. In this example, Cramer's symbols and notations brought cognitive obstacles to the students, and as the students knew Cramer's rule in advance, they were not as interested in new ways of looking at it.

Lin pointed out that there is a need of more articles - for teachers - about the history of topics in high school math textbooks. We also need teaching projects with guidelines in detail. 

I missed Uffe Jankvist's regular lecture at ICME (because I felt I had to hear Marja van den Heuvel-Panheuzen's), but luckily I got another chance to hear him here. His title here was "A Historical Teaching Module on "The Unreasonable Effectiveness of Mathematics" - the Case of Boolean Algebra and Shannon Circuits." The topic was a teaching module used in upper secondary school in Denmark. The main idea was to have one original source for each of the three dimensions history, applications and philosophy, have guided readings and then essay assignments. (Guided reading means to give short parts followed by work to be sure that students have understood.) The readings chosen were Boole 1854, Shannon 1938 and Hamming 1980.

The students gave different reasons to prefer different texts - more mathematical (Shannon), more open (Hamming), more easy to follow (Boole). Different texts speak to different students. There were students responding positively to each of the three dimensions. Referring to Barbin's Dépaysement, vicariante and culturel, Uffe argued that all three of these were included. (This reminds me, of course, that I should go back and reread Barbin.

It is great to see Uffe's high-quality work which combines the development of resources, the actual teaching and development of theory. So of course I will read his paper carefully when time permits.

Jerry Lodder's talk was titled "Historical Projects in Discrete Mathematics". This turned out to be the same talk that he had last week at ICME, so I won't summarize it here. However, this time I got the URL to the webpage he was referring to:  URL

After a nice lunch in which the Nordic delegation stuck together (Kristin, Andreas, Uffe, Tinne, Johan and myself) it was time for another round of oral presentations. Shu Chun Guo presented "A Discussion on the Meaning of the Discovery of Mathematics in the Warriers and the Han Dynasty." The focus was on bamboo strips with mathematical contents, many found in 1983, but originating from 157 BC, apparently. The strips are still being researched, but they include multiplication tables, calculation with fractions and much else. The talk was a bit difficult to follow for me, as the slides were mostly in Chinese and my understanding of Chinese is currently woefully inadequate.

The second oral presentation of this session was by Mustafa Alpaslan, presenting a paper of himself, Mine Isiksal and Cigdem Haser on "Relationship Between Pre-service Mathematics Teachers’ Knowledge of History of Mathematics and Their Attitudes and Beliefs towards the Use of History of Mathematics in Mathematics Education." They did quantitative research to fill a gap in the literature, as big quantitative studies on attitudes to HM are few and far between. They asked 1593 pre-service mathematics teachers and used two instruments: a knowledge in history of mathematics test and an attitudes to history of mathematics in mathematics education questionnaire. Based on Pearson product-moment correlation analysis, there were found lots of statistically significant correlations, which supports the idea that beliefs/attitudes and knowledge has an interplay. 

Of course, correlation does not say anything about what is the cause and what is the effect - we don't know if we must improve students' knowledge to improve their beliefs and attitudes or vice versa - or indeed if beliefs and attitudes are easy to change as late as in teacher education.

Kathleen M. Clark then presented her paper "The Influence of Solving Historical Problems on Mathematical Knowledge for Teaching". Clark took Ball et all (Mathematical Knowledge for Teaching" as a starting point. She has taught a course on "Using History in Teaching Mathematics" four times, with 20-25 students each time. As part of the course, they had a "historical problems portfolio", where they had to choose ten problems/tasks/activities that they had worked on, restated, solved, and described the course objectives addressed and provided a reflection. She used this both for investigating how the students understanding of mathematical concepts were informed and what their work reveal about their beliefs about mathematics.

19 portfolios were analyzed. Here, she looked at two of the problems chosen; Method of false position and Method of completing the square. On the method of false position, she found that students struggled with alternative solution methods, they found older methods harder, failed to connect the method with linear solution methods they knew, and were unable to evaluate their success.

On the method of completing the square, she found that the students was helped by connecting geometry and algebra, their "awareness" heightened (making them aware that HM can be a good resource) and gave them alternative conceptions. 

But the students were persistent in their critique of the way the original problems were posed, and wanted to convert the historical algorithm into the modern ones. From the fall of 2012, she will be a new project to look at the ways a course in HM can influence the knowledge the students will need for teaching.

The last thing on the programme on Monday was workshops, and I chose Peter Ransom's, as it was the most relevant one for my students' target age group, and as Peter's workshops are always enjoyable. As workshops are quite interactive things, I cannot lean back and write notes, so I cannot summarize it "as it happens". However, I can safely say that it kindled an interest in both fortifications and proportional dividers...

ICME Day 7

Last day! The last day of an ICME is always a bit difficult. The people who have not already left, tend to have their minds on their flight back home, their next conference and so on. You get the sense that the rich mathematical didactical environment that has been at the venue for a week is slowly ebbing away, and that it will suddenly be replaced by something entirely different - a Health conference or whatever. But of course the scientific work will go on in countless locations around the world, just a little bit better coordinated (and more informed) than before.

Werner Blum had the task of firing up a homebound audience with the title "Quality Teaching of Mathematical Modeling - What Do We Know, What Can We Do?" He gave a few starting examples of modelling: Hassel pick-axe - how tall would a giant have to be to fit the pick-axe? And: is it worthwhile to go down to town to buy a t-shirt where it is cheaper? He used such examples to go through familiar steps of modelling processes.

Modelling competency is at the heart of PISA. Modelling is a cognitively demanding activity, because it involves several competencies, both mathematical and non-mathematical (including ethical considerations). Each step in the modelling process is a potential cognitive barrier for the students: 1. understanding the situation and constructing a situation model. Students have learned that they can solve problems without considering the words. "suspension of sense-making". "an orchestra needs 40 minutes to play Beethovens 6th symphony. How much for 9th?" 2. Simplifying and structuring - students are loath to take decisions on their own, for instance to round off numbers. 6. Validating - is seen as the teachers' job Students tend not to transfer. Which makes it doubtful if there is such a thing as general modelling competency.

He listed four kinds of justifications for modelling: pragmatic, formative, cultural, psychological. Based on these justifications, several "perspectives" of modelling can be seen, when combined with characterizations such as "authentic" or "mathematically rich" - I will not go into that here. But it is an interesting way of structuring the discussion, and I wonder if a similar framework would be useful in the field of HPM as well (and of course it could be that such a framework has already been proposed in articles I haven't read).

Finally, Blum gave ten rules for teaching modelling (based on empirical evidence): 1. Effective and learner-oriented classroom necessary 2. Activate learners cognitively also necessary 3. Activate learners meta-cognitively also necessary 4. Variety of suitable examples (real world contexts and mathematical contexts and topics) - transfer cannot be expected. Real world contexts help reducing the "suspension of sense-making. 5. Teachers ought to encourage individual solutions (they tend to favour their own solution) 6. Competencies evolve in long-term learning processes. Repeating and practicing is necessary 7. Assessment must reflect the aims of modelling appropriately 8. Parallell development of competencies and beliefs and attitudes 9. Digital technologies can be powerful tools: experiments, investigations, simulations, visualisations or calculations 10. Mathematical modelling can be learned by students supposed there is quality teaching.

Dongchen Zhao presented, on behalf of himself and Yunpeng Ma, a paper with the title "An analysis of the characteristics and strategies of the excellent teachers in mathematics lessons in primary school". There was  a curriculum reform in China in 2001, fully implemented from 2005. The previous one was from 1992. Zhao gave some short comments on the 1992 curriculum and then gave an introduction to the new curriculum. 

The lessons analysed in the project were prize-winning lessons from the National Contest in Exemplary Lessons. Unsurprisingly, the prize-winning lessons were found to comply with the new curriculum (which might be why they won prizes in the first place). He gave examples of how this was done. More interesting, perhaps, is other findings when looking at these videos. All lessons were dominated by public interaction (again not surprising, I guess, because it must be difficult to make an impressive video of students working individually, for instance). For us, the most interesting finding is perhaps that lots of the student speaking was done in chorus - up to 66 percent in one of the lessons... This seems to suggest that the teachers didn't often ask students to bring forward their way of thinking, but rather were asking rhetorical questions with only one valid answer. Moreover, students rarely raised questions by themselves - the teacher did most of the asking. It is interesting that lessons that are judged as good lessons in some respects, have such worrying characteristics in other respects.

I am thinking that such a lesson contest could be a cool idea also in Norway - not least because it would have teaching experts discuss - in concrete cases - what constitutes good teaching. After all, our discussions are so often concerning only hypothetical situations, and not real-life lessons with all their quirks.

This ended the ICME12. I hope to be back at ICME13 in Hamburg in July of 2016. But first, it is the HPM conference in Daejeon beginning on Monday...

ICME Day 6

The plenary on Saturday morning was cancelled as Jo Boaler could not come, and instead, Survey Team 2 was asked to give a presentation on "Gender and Mathematics Education", chaired by Gilah Leder.

The situation in Africa was described by Nouzha El Yacoubi. There are obviously big gender differences in Africa, for instance when it comes to literacy. More boys than girls go to school. With this general background, it goes without saying that boys are also doing better in mathematics. I'm a bit unsure, though, if mathematics has to be addressed specifically, or if the main thing is to get everybody to school first.

Next, Maria Trigueros talked about the situation in Mexico. There are gender differences (boys doing better) in 6th grade. Also, boys display better self-confidence from about that time. Technology helps improve attitudes among girls. The gender gap remains at the university level. There are promising policies, for instance revision of textbooks to avoid a bias in gender.

Helen Forgasz claimed that there is a downward trajectory in Australia. In TIMSS and PISA, there have been an increase in the gender differences, with statistically significant differences, boys doing better than girls. More boys than girls enter the "Mathematical Methods" course. The gender gap widens. In public perceptions, it is still believed that boys are better at mathematics than girls (a view that is of course also supported by TIMSS and PISA - you can hardly fault the public for being up to date on the latest international comparative research).

Lovisa Sumpter took a European perspective. She has done a literature survey, which confirmed the standard views of gender differences. Much of the research are published in general journal (not concerned with mathematics in particular). 

Sarah Theule Lubienski (the sixth woman on this six-person panel) talked on the US. She looked at how gender gaps vary from item to item in tests. Moreover, the gap in confidence is bigger than the gap in achievement or interest.  More women than men go to college, but more men study mathematics.

To me it does not seem equitable that one person talks about almost 50 countries, while others only talk about one small one. The unintended message is that Australia (or Mexico, or USA) is as important as the entire continent of Africa... However, the topic is an important one, which deserves all the attention it can get.

The last session of the TSG20 started with a talk by Jerry Lodder on "Primary historical sources in the classroom: Discrete mathematics". His example was a module on logic, in particular the truth table. He used sources related to Chrysippus, Boole and Frege. In particular, he showed how Frege tried to write Chrysippus' rules with Frege's notation. Lodder lets his students work on this, and then on Russel and Whitehead's work. The study showed statistically significant increase in student's attitude to the subject.

Secondly, Anne Michel-Pajus talked on "Historical algorithms in the classroom and in teacher-training". She has been giving in-service teacher training since the 1980s with an emphasis on historical sources, to enrich their culture in HM, deepen their understanding and help them build historically based activities. In the new curriculum in French high schools, students are to understand, describe, modify and explain algorithms. She looks at Heron's formula for the area of a rectangle with sides given. In the original source, this was not a formula, it was an algorithm. 

Another example is Chinese, Indian and medieval algorithms for finding a square root. Aryabhata's is a very short description of an algorithm. Diophantus gave a general formula for polygonal numbers in rhetorical language.

 In is way, Anne has illustrated a table of different ways of expressing a formula/algorithm. Then she went on to different levels of justification (checking a few examples, justifying procedure mathematically etc.) She shows Heron's explanaition of how to find the square root, using particular numbers to check the algorithm. Euler gives the same algorithm in symbolic language.

Al-Khwarizmi gives a geometrical proof for his algorithm for solving quadratic equation. Diaphantus gives an example of a formal reversal of an algorithm. Jordanus de Nemore: uses letters instead of numbers, but introduces a new letter for every step of the algorothm.

The relationship between formula and algorithm was thus very nicely illustrated, and I'll try to use some of these examples in my work in the future.

Finally, Uffe presented his, Reidar Mosvold, Janne Fauskanger and Arne Jakobsen's paper titled "Mathematical knowledge for teaching in relation to history in mathematics education". I'm happy to see that there are attempts to include HM in the framework of mathematical knowledge for teaching, as we were talking about in the conference in Vienna two years ago (see my part of the plenary lecture there). Uffe stressed that it is important for HPM to relate to the broader mathematics education research community.

Uffe gave a short introduction to "the egg", and then gave examples on negative numbers (based on Arcavi 1982) and on number systems (Heiede). He then made an attempt to connect the examples to the egg, which I cannot repeat here. But Uffe uses Ball's definition in which CCK (common content knowledge) is seen as only being about the 'pure' mathematics, so that in this case it would include knowing the definition of negative numbers and knowing that different number systems exist. I strongly disagree with this way of looking at mathematics and CCK, as it marginalizes history of mathematics into being not a part of what mathematics is, but only a tool for lesrning the mathematics. In this particular paper, Uffe et al are using the present version of "the egg" to analyse the outcome of HPM research. The goal is also to make HPM results more accessible and relevant to the mathematics educational community by connecting to one of the present fads. This is laudable, but we should also challenge this model by Ball et al to get a wider concept of mathematics, for instance to include parts of history of mathematics in CCK - that is, as a part of the mathematics curriculum that all children should be supposed to learn as part of a sound mathematics education. Not just as a means to learn mathematics, but as a goal inits own right.

After lunch, Adnan Baki had a regular lecture with the title "Integrating technologies into mathematics teaching: past, present and future". He traced his own story since he started working with technology in the early 80s, mentioning how he had trouble changing his teaching practice in a more constructivist direction, and began to see what Seymor Papert meant about computer's potential for changing teachers' role and classroom practice.

In the second part of his talk he gave several examples of working on geometry in LOGO and in GeoGebra. These are singularly unsuitable for summarizing here, being quite visual.

Finally, there was the second part of the discussion group on use of history of mathematics with children age 6-13. I thought this went quite well (many thanks to Kathy Clark). This is the first time in ICME that I have been formally in charge of anything more than a talk on my own research, and it pleases me a lot that it was not a dismal failure. An account on what went on in the group will be found in the next issue of the HPM Newsletter, I would think.