Saturday, July 28, 2012

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...

Friday, July 27, 2012

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.

Wednesday, July 25, 2012

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...

Thursday, July 19, 2012

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"...

Wednesday, July 18, 2012

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...

Sunday, July 15, 2012

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...

Saturday, July 14, 2012

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.

Friday, July 13, 2012

ICME Day 5

I have kept boasting that this is the fourth time that I've taken part in both the ICME and the HPM conferences. Among the things I've learned is that you should not feel bad about taking breaks - having a full programme of talks from morning to evening for two full weeks, simply does not make sense. The brain needs some spare time to think about something else - or maybe process what it's alteady learned...

So I took the morning of the fifth day off, sleeping late, even though Thursday (day 4) was excursion day. In fact, I was perhaps even more tired after the excursion day than after any other day - it was a day of many thoughts as we crossed the line to North Korea, if only for a few minutes...

Thus, day 5 started, for me, with Marja van den Heuvel-Panhuizen's talk, named "Freudenthal's work continues". She has been a big influence to Norwegian mathematics education lately, and I keep seeing her name, so it was about time I heard her "live".

She has worked at the Freudenthal Institute for 25 years, and started by talking about the history of this important research institute. It has a history going back to 1971. In November 2010, it was decided that primary mathematics education was not a part of the core goals of the science faculty, so the FI was divided. 

She talked on three projects connected to primary school: the didactical use of picture books, mathematical potential of students in special education, and textbook analyses.

First, the project on the didactical use of picture books in kindergartens. Freudenthal himself was sceptical to limited and isolated worksheets, and wanted to give children context-rich experiences in which the children could discover mathematical connections. Picturebooks can give rise to meaningful activities, offer cognitive hooks (Lovitt and Clarke) and give opportunities for practice. In a 2008 paper, van den Heuvel-Panhuizen showed that almost half of children's utterances when read a particular book, was mathematics-related. Among the other subprojects, they also found significant improvement in the children's Score on a PICO test (whatever that is). 

Then she went on to the project on special education, with work on subtraction and combinatorics. The project was motivated by policies claiming that weak students should not discover strategies by themselves, but be told how to do things. Of course, to accept this would mean that weak students can never appreciate mathematics as a creative field of knowledge.

van den Heuvel-Panhuizen presented a model with 12 (3x4) different methods for subtracting, and studied whether special education students could, on their own, use indirect addition to solve problems suited to that method (without being taught it first).  It turned out that the students could do that.

The third project was the textbook analysis project. Textbook analyses give us a first inside view of how mathematics is taught, and is therefore relevant to teacher educators. It focussed on content, learning facilitators and knowledge presupposed. Among the clear findings is that realistic textbooks have more didactical support - like the use of context, models, textual instruction etc.

Then there was the third session of the TSG20. First, Mustafa Alpaslan talked on “"History of mathematics” course for pre-service mathematics teachers: A case study", a paper written with Cigdem Haser. HM got mentioned in the curriculum of 2005. Textbooks had small snippets of historical information. For pre-service teachers, a course on HM was introduced by the government. In his masters thesis, Alpaslan showed an improved knowledge of HM in teacher students. In the present project, he and his colleague wanted to study such a course for students who will teach ages 12-14. The data collection included classroom observation for 5 weeks, examination of materials and semi-structured interviews.

The course was partly presentations by the teacher and then by the students, who had by then done small "projects". The course had nothing about the use of HM in maths education. Students found that doing mathematics "within its history" was enjoyable. But students wanted to know how to transfer this to the students. 

The course was based on group work, but this was not effective - they did not receive sufficient help on where to look for reliable sources, for instance.  Students were also busy studying for the general examination at the end of their teacher education.  The study suggested changes in the courses, and his doctoral work now will be dedicated to try to create a better course. I will be very interested in hearing more about this as it progresses.

Xuhua Sun was supposed to be the third speaker, but as the second speaker did not turn up, she was up next. The topic was "The systematic model Lu of JiuZhangSuanShu and its educational implication in fractional computation". She pointed out the two different traditions, the proof-based (of for instance Euclid) and the "problem-based" (of many civilizations), where solutions are given without proof. She then went on to show how 3/4 : 3/8 can be done in two ways: "flip and multiply" and transforming to the same denominator. She then connected this to the idea of Lu in Chinese mathematics. In Chinese mathematics one used different methods of multiplying with the common denominator to solve the problems in fractions.

Then there was the HPM group meeting. Evelyne Barbin talked first, on "Reading of original texts and 'dépaysement': from the teachers to the classroom". She referred to her article in For the learning of mathematics, 11, 1991 for the aims of the work on original sources. A key word for Barbin is 'dépaysement', or reorientation. By meeting an unusual take on something familiar, we will reconsider it. Barbin used the method of tangents by Roberval as her example. His idea of a tangent was the line which a point would follow if it detatched from its curve (which it was moving along). This idea was used to establish the tangent to the parabola and to the cycloid. 

Barbin showed two ways of using this in the classroom. The first is based on Frederic Vivien, who chose to translate it into the language of vectors. Then the pupils are asked to translate Robertval's reasoning in some examples into vectors. Barbin seemed very sceptical of this approach - it's not clear what you will learn about Roberval from this, as his context is disregarded, and what you learn about vectors, you could probably learn as well in another way. (this is my interpretation, not Evelyne's words, of course). The second is based on André Stoll, who used it with students who already knew how to find vectors. The ideas of Roberval can then be used to introduce the differensiation of vectors. He also uses this method to work on the cycloid.

Thus, an original text can be used both to read something known or to understand something else. But it is not necessary to read the text of Roberval to learn about vectors - for that, the examples could be used without the original source. To get good use of the original sources, it is interesting to discuss the context of the sources.

Then Fulvia Furinghetti talked on the history of mathematics in teacher training. She stressed that in teacher education, it is important to challenge teacher's existing beliefs. The aim is to make teachers reflective practitioners.

She agreed with Evelyne about the importance of replacement and reorientation, making the familiar unfamiliar, building cultural understanding. Finally, working on history of mathematics contributes to a construction of meaning of mathematical objects.

Fulvia then went on to describe a course on history of mathematics, consisting of theory and laboratory. The course is parallel with a course on mathematics education. It aims to create a community of practice, and is based on work on original sources: Fermat, Roberval, Barrow.

For me, it is interesting to see different ways of designing history of mathematics courses presented at ICME. Many of the courses presented would be utterly inappropriate in Norway, because of Norwegian students' lack of knowledge in mathematics. But there are also other differences between the courses than what can be ascribed to knowledge of the different knowledge levels of the students, of course. It would be very interesting to try to analyse different courses in relation to the designers' knowledge/resources, beliefs and goals (referring to Schoenfeld's talk on Monday). For instance, I'm concerned with showing students a variety of ways of including history of mathematics, while there are other courses that does not include that at all. 

Then there was the new chair of the HPM, Luis Radford. It was a rather informal talk, describing his path into the HPM group. At the end, he talked about how to use epistemology (based on Artigue) as a prism to critically observe and understand the objects of knowledge in the curriculum, and as a means to understand the development of the objects of knowledge. Radford is concerned with epistemological obstacles (both within mathematics and as related to the social, cultural and historical circumstances). His talk will hopefully appear somewhere.

Thursday, July 12, 2012

ICME Day 3

Etienne Ghys held the most mathematical plenary at this ICME, with the subject "The Butterfly Effect". I will not try to summarize his popularization of chaos theory, but his underlying point was that mathematicians have a duty to popularize mathematics - and that the popularization of chaos theory has so far failed, in that it has been oversimplified in the popular culture. In particular, Ghys claims that chaos theory has often been painted as a purely negative "theory", saying that the future is impossible to predict, instead of as a positive theory making a contribution to our understanding. He showed how chaos theory often gives quite accurate prediction of the long-term behaviour of a phenomena, even though the behaviour at a particular moment can not be predicted.

It is interesting to notice that Ghys' talk was actually on the history of mathematics - the history of chaos theory, including Maxwell, Hadamard, Poincare, Lorenz, Smale. It should also be mentioned that Ghy's talk was the most beautiful and visually impressive so far at the conference, with great animations included.

After the talk, I wondered if teachers' behaviours are chaotic or, as Schoenfeld seemed to suggest, deterministic. I would guess that it is chaotic - that a slight change in the mood of the teacher or of the answer of a student can lead to a very different outcome in a particular case, even if the set of possible outcomes can be predicted.

Topic study group 20 continued with three talks. First, Jeffrey J. Wanko talked on "Understanding historical culture through mathematical representation". He talked on a course he has held - with students in general, not with pre-service teachers - entitled "Outside Euclid's Window", using book "Mathematics in Historical Context". The course is taught in Luxembourg, with study tour to important mathematical cities in the area. Examples of stuff covered was Gauss' calculation of the sum 1 + 2 + ... + 100, using diagrams to compute polynomials (in the US they learn about the FOIL method (first, outer, inner, last), adding and subtracting integers, understanding infinity (using Hilbert's ideas), building Platonic and Archimedean solids. The outcomes of the course were engaged students, who discovered mathematics and developed an understanding that it has evolved.

Andre Cauty talked about "Lab work of epistemology and history of sciences: How to transform an Aztec Xihuitl (a 18 periods year) into a calendar?" The Maya and Aztek had 260 days named by a complex expression of a number from 1 to 13 (or 2 to 14) and a sign of day (20 different). He went through many details on how these were written and several other cycles in the area, and historical attempts to harmonize these into a calendar. Sadly, the "lab work" part of the title was not touched upon, thus leaving the connection to the teaching of mathematics to the audience.

Maria del Carmen Morilla's title was "Visualization of the Archimedes mechanical demonstration to find the volume of the sphere using 3D dynamic geometry". She referred to Hanna, and the distinction between mathematical proofs than prove and mathematical proofs that explain. Using dynamic geometry helped explore Archimedes' demonstration of the relation between the volume of the sphere and the volume of the cone and the cylinder. The talk illustrated how dynamic geometry software can be a useful additional tool for explaining complicated geometrical arguments, to supplement the careful step-by-step exposition in dialogue with students.

After lunch, "Adjacent schools with infinite distance - narratives from north Korean mathematics classrooms" was Jung Hang Lee's title. I knew fairly little about North Korean mathematics teaching in advance (having read just one article about it before, which happened to be written by Jung Hang Lee and his supervisor), so this was a reasonable choice (even though there were many other tempting offers as well - for instance, I missed Luis Radford's lecture for this.

The research is based on interviews with refugees from the North Korean oppressive regime, 5 teachers and 10 students, three interviews with each. The teachers had teaching experience ranging from 6 to 25 years, all had taught in secondary school.

The North Korean school system wants to homogenize the students. Schools didn't even have school identities or school songs - on their uniforms they were only allowed to have a picture of the leader. But until the 1980s, many saw the system as a good idea, with free education and free health service. Then, however, conditions worsened.

The vice principal of the school are in charge of ideology in the school, and he manages the teachers. He is obviously member of the party. Positions were partly handed out based on family background check.

They do have a system of teacher development. Every week, one of the teachers present a new topic, and then discuss pedagogy and improvements to the topic. North Korean teachers don't have school holidays, but has to attend professional development sessions.

There is only one set of textbooks in North Korea, with a note on how much time should be spent on each subtopic. Throughout secondary school, mathematics is the subject with most hours, 6-7 hours s week in theory. There was a rigid class schedule, where each mathematics lesson should include five different sections, including Reinforcing the Policy of the Party, which they were meant to spend 20% of their time on.

After the "March of Suffering", in which 10 per cent of the population died, they mostly gave up on providing education for all, and concentrated on the most gifted students. Afternoon classes were often cancelled because the teachers had to go elsewhere to try to find food. Some teachers also brought goods to school to try to sell it to their students, even though this was an immense loss of face.

Probability theory has only recently (in 2002) been introduced in North Korea's curriculum. After all, many of the uses of probability theory are not relevant to North Korea (insurance, gambling, stock markets...) It replaced "computer and programming".

Wednesday's final plenary activity was a panel discussion on "Teacher education and development study: Learning to teach mathematics (TEDS-M)", with participants Konrad Krainer (Austria, Chair), Feng-Jui Hsieh (Taiwan), Ray Peck (Australia), Maria Teresa Tatto (USA). "Panel discussions" in such conferences tend to degenerate into a series of individual lectures which are not relating to each other, so it's always interesting to see what happens. In this case, the panellists had divided the topic neatly between them, presenting a lecture divided in four parts.

Konrad Krainer started by painting a picture by mentioning Hattie, Adler and Shulman. Then he introduced TEDS-M, whoch includes 23.000 teacher students from 17 countries. Norway is among the underachieving countries if you do a regression in relation to the Human Development Index. 

Ray Peck talked on "teaching and teacher knowledge". three MPCK subdomains: curricular, planning, enacting. He also showed some items and explained the concept of "Anchor Points" (which I can certainly not repeat here).

Feng-Jui Hsieh followed with the theme "teacher education and education system". She stressed that teaching is attractive in Taiwan, because of income, working hours, job security and status. Thus, the process to become a teacher is highly competitive. Taiwan surprisingly ranked as number one - surprisingly, because Taiwanese teachers are generalists at the primary level. However, based on items where they do badly, Taiwan have already taken steps to improve the teacher education. Taiwan also has a sharp decline between secondary and tertiary education.

At this point, Liv Sissel Grønmo got the chance to complain that too little is happening in Norwegian teacher education in response to TEDS (a fairly unreasonable complaint as the TEDS results have just been published). She also complained about the increased place of general pedagogy in Norwegian teacher education. In the end, she said that Norwegian teacher education at times seems like it is trying to educate teachers for mathematics without them having to know much mathematics. I am certainly more optimistic than Liv Sissel, and thinks that the new 5-10 education (which I am partly responsible for implementing at my institution) is a major improvement, because students will learn mathematics and didactics more directly related to the grades they will teach - and of course many will also have double the amount of mathematics compared to before. In 1-7 education,  I think a more tailored curriculum will also help. It would surprise me greatly if this reform would not contribute to an increase in the TEDS score on the primary and secondary level if TEDS is repeated some years from now. How we will improve the results on tertiary level, I am more unsure.

Maria Teresa Tatto talked on new research in teacher education based on TEDS-M. She gave examples of anchor points, and then went on to present the results of the study.

Like TIMSS and PISA, the main contribution by TEDS is probably not the rankings, but rather results on individual items that can help us get more knowledge about our own outcomes, and inform discussions on what we would like the outcomes to be and how to get there. It helps that the databases are, apparently, available on the TEDS website.

Of course, I am interested to look for traces of for instance history of mathematics in the TEDS items. It seems, though, that the study has too limited a concept of mathematics to include the history of mathematics. Thus, there are important bits of MCK and MPCK missing - unless it is found somewhere in the material that I have not found so far.

Finally, there was a meeting of the HPM Group. Here, the activities of the HPM Group were presented, including the HPM Newsletter, of which I've been an editor for the previous eight years. This meeting will be followed up with another session on Friday.

ICME Day 2

Before going on to describe day 2, I should mention that I'm still meeting "new" Norwegians in almost every break. I think the number of Norwegians must at least have doubled since 2008. So I wonder how many will be in Hamburg in 2016...

The plenary lecture of the day was Bernard Hodgson, titled "Whither the mathmatics/didactics interconnection? Evolution and challenges of a kaleidoscopic relationship as seen from an ICMI perspective". He was concerned with the divide between mathematicians and didacticians of mathematics - a divide partly due to they being connected to different paradigms, with different sets of rules for what is valid knowledge and so on.

He described how the mathematics education field is populated with mathematicians who are also teaching mathematics, as well as didacticians who are primarily doing their research in mathematics education. Of course, there is also the complication that even many mathematicians have a limited view of mathematics - we should include historians of mathematics in the discussion, for instance, but Hodgaon did not mention this. 

Hodgson went on to talk about three mathematicians with thoughts on teaching: Archimedes (with his "Method" - with a clear idea of the difference between a proof and just a reasonable indication), Euler (with all his textbooks) and Pólya (ten commandments for teachers). 

Hyman Bass has talked about the Klein era and the Freudenthal era of the ICMI. In  the Klein era, ICMI was populated by mathematicians, while in Freudenthal's era, didactics became a subject in its own right. The Freudenthal era also saw the launching of Educational Studies in Mathematics. Also, plenary lectures at ICME has turned from mathematics (for instance fractals) to didactics.

Hodgson ended by listing some challenges for mathematicians, here only summarized in short: acknowledge education as a responsibility, career issues, presence at ICME, level of rigor. Challenges to didacticians: keep research accessible to outsiders, collaborate in mathematically-oriented fora, acknowledge importance of creative mathematical work by teachers, must know mathematics.

The divide Hodgson talked about is of course present also in the Norwegian context, to a certain degree. It is evident whenever a new course description is to be agreed upon, a conference is to be planned, and so on.

For the TSG20, Peter Ransom was presenting Snezana Lawrence's talk on a project he has been involved in himself. They are trying to make the subject engaging by providing historical background, beginning with "historical appreciative session" on a topic. They do work on historical sources and explore which could give access to 'rich' tasks in the mathematics classroom. They introduce modern technologies in relation to the historical mathematics. For instance, he gave examples from work with quadratic curves, where history and different technologies came together.

Then I had my talk, discussing teacher students' attitudes towards history of mathematics based on a questionnaire I used in connection with a teaching sequence I developed. The full paper is available in my account. (I will insert a link here when I get the time, of course.)

In a way it was nice that the room was too small, as it gave me the rare experience of lecturing with eager listeners sitting on the floor and standing in the hallway, listening through the open door. 

Then there was Tinne Kjeldsen's "Genuine history and the learning of mathematics: The use of historical sources as a means for detecting students’ meta-discursive rules in mathematics". HM is part of the curriculum in Denmark. Tinne took Anna Sfard as a startong point, and argued that HM can have a profound role in becoming a participant in a mathematical discourse; to learn the meta-discursive rules. "Commognitive conflict" can give change in meta-rules. Historical texts can play the role of discursants.

She showed an example from a teaching  module on the concept of a function, and in this talk focused on the norm that a variable should take alle values, not be restricted to an interval, looking at Euler (1748). The students first worked in groups on four worksheets, then "expert groups" (with representatives from all four of the earlier groups), which wrote a paper. Analysis of the papers and the discussions showed that there was indeed some discussion on meta-level rules. (but it is impossible to quote the transcripts here...)

She also mentioned in passing an interesting confusion by the students between Dirichlet's use of a and b and the modern textbooks' use of y= ax+ b. The students apparently saw a and b as connected to linear functions only, not thinking of a as a constant but as the slope of a linear function.

Next regular lecture was Terezinha Nunes' "Number, quantities and relations: Understanding mathematical reasoning in primary school". She claims that cultural tools are essential for thinking, and that we use quantities, numbers, operations and relations when we use mathematics. She defines operations as transformations applied to quantities or relations. Numbers have two types of meaning: formal and extrinsic. These have to be coordinated. There are also two kinds of relations: necessary relations (i.e. 5 is 2 more than 3) and contextual relations (i.e. "Per has three apples more than Nils"). She showed examples of pupil errors that were not really mathematical, but just contextual - not having understood the context fully.

Nunes' main point seemed to be that while we have spent lots of effort working on representations of quantities and operations, we have paid little attention to the representation of contextual relations to support reasoning. Problems involving relations are always more difficult than problems with only numbers and operations.

Children must learn how to use iconic models - the models are not obvious. Children's drawings will often be nice but not actually illustrate the relations, and will therefore not help in solving these problems. Numes and colleagues are now doing a study about the use of representations for relations in primary school, with three small groups (intervention group, control group and "unseen" group.  Preliminary results seem to show that the models help, but pupils obviously tried easier ways of doing it when they could, which was unhelpful for learning to use the models...

While pupils in Singapore may work so much on the models that they become very proficient, pupils n England had a hard time learning to use them, which reminds us that this is not a routine to learn, but rather a tool one can learn to master. (A teacher from Singapore in the audience confirmed that also in Singapore, learning to use the representations was very difficult.

The rest of the day was spent on Discussion Group 5, of which I was the originator. I cannot summarize that here, but I'm sure we will provide a summary of the discussions among the 20+ participants in due time.

Monday, July 9, 2012

ICME Day 1

The first day of ICME was a light one. It started with a two-hour opening ceremony, with a combination of welcome addresses and cultural performances, which was an eclectic combination of Korean and Western style, just as modern-day Korea. It is always nice to attend these opening ceremonies, as they try to convince us of the importance of our work in general and the conference in particular. This time, there was a recorded address from the president of S. Korea. Thereafter, the Felix Klein and the Hans Freudenthal awards for the last four years were awarded. Of course, I was particularly pleased to see Luis Radford get his well-deserved prize.

The first plenary lecture was by Don Hee Lee. She spoke on the topic "Mathematics education in the national curriculum system". She discussed the place of mathematics in the educational system, taking Plato as her starting point. Is mathematics taught mainly for the intellectual development of "the liberal man" (learning for its own sake) or for the development of society?

She quoted a Korean professor of mathematics (without necessarily agreeing) claiming that there is no mathematics that is both easy and interesting at the same time. Mathematics is based on thousands of years of intellectual work, and therefore advanced. While this is an interesting point of view to share with pupils who have unrealistic ideas of the amount of work (or lack thereof) necessary to learn mathematics, it should probably not be a credo for a mathematics teacher. 

For regular lecture today, I chose Alan Schoenfeld's "How we think: A theory of human decision-making, with a focus on teaching". Alan Schoenfeld received his Felix Klein Award in the morning, and the lecture hall was so crowded that the organizers immediately decided to ask him for a rerun later in the week. 

The starting point of the talk was the question "If I know enough about you, can I explain every action you do and every decision you make?" He claimed that the answer was yes, with the important caveat that he is talking about situations where you are an expert. Thus, it holds for the cooking of a chef, the teaching of a teacher and the setting of diagnoses of a doctor. So what do you need to know about a person to be able to explain his actions? You need to know about his knowledge/resources, his goals and his orientations (beliefs/values...)

The word "explain" is not meant as a mere verbal explanation, but the ability to set up a model which can - to a degree - predict the actions.

Why look at teaching? As he put it: "If you can model teaching, you can model just about everything." Teaching is highly complex; it is highly social and it is ever-changing.

He gave a few examples of how he has worked with teachers to model their behaviour. He gave one example of teacher who believes that he cannot tell students anything unless it's based on something the students already have said - and therefore is lost when the whole class agrees on a wrong answer. His actions were to a large degree explained by this belief. Another example - a teacher who teaches by raises issues, asks for student suggestions, clarifies and then moves on. He transitions to the next issue when goals are met. So what does he do when things don't go as he wants ? According to Schoenfeld, he "doesn't have a choice" - he will still build on the students' questions/input and discuss them with the class. His beliefs determines his actions.

His third example was on Deborah Ball teaching a 3rd grade class. She did some surprising things, including at one point asking a question that many would consider a mistake, as it derailed the discussion with the class. Schoenfeld didn't understand what Deborah was doing. Could it be modelled? And what about Deborah's "mistake"? After years of studying it and discussing it, he came up with a model here as well. Yes, her behaviour was consistent with a model. Within the model, her "mistake" was based on a need to understand her students' thinking before going on to the next topic.

So what's the point? Why are models important? I would ess that they are not, but that they serve to point out the importance of a teacher's resources, goals and beliefs (etc). If two experienced teachers do things very differently, we should not assume that it is because of some insignificant chance. It may very well be because of some fundamental difference. And of course it doesn't help much to give teacher students the necessary knowledge and skills to teach well, if we don't also consider their beliefs. As Schoenfled put it: a teacher who believes that only the best students can do problem solving, will not even try to give his poorer students problems to solve.

Schoenfeld's way of thinking seems highly relevant to the research project I'm currently involved in, and I will keep it in mind...

The rest of the day,  I got prepared for Tuesday - the day of my little talk.

ICME Day 0

This will be my fourth ICME (International Conference on Mathematics Education), after Tokyo, Copenhagen and Monterrey. As always, I arrived a few days early. It's nice to get used to the temperature and the time before the conference starts. Seoul reminds me of Taipei (a city I've spent some time in). There are the same smells, a similar mix of local and US stores and culture, similar traffic and so on. The subway system is comprehensive and easy to use - like in Taipei. So I immediately took a liking to the city.

It's also nice to meet up with old colleagues. On the day before the conference, I met lots of Norwegians; Andreas, Marianne (with husband), Torgeir, Liv Sissel, Odd Helge, Kristin (with husband), Ole Einar and more. And still, I have met only perhaps half of the Norwegians. In Tokyo in 2000, I don't believe I met anyone else from Norway.

My expectations this time are diverse. I'm looking forward to hearing some of the "big names" - people that I've read but never seen. And the Topic Study Group will be interesting, I'm sure. But I must admit that at the moment I'm thinking most of my own contributions - a talk on Tuesday, leading a Discussion Group on Tuesday and Saturday and saying some words on the HPM Newsletter on Wednesday. I never stop being nervous, even after all these years, and even if the groups I'm talking to are rather small. What if I don't get my points across? What if I mess everything up? Well, I'll do my best not to...

After ICME, I am, again for the fourth time, going on to the HPM conference, which is in Daejeon this time. I'm looking forward to that, partly because I have fewer responsibilities there (just a workshop), but mostly because the scale is smaller and it is (therefore?) friendlier. At ICME, you can meet and get to know someone on the first day and then never see them again - it's that big. (Or they may be consciously avoiding you, of course.)

It will be two interesting weeks, for sure.