Monday, October 27, 2014

Comments on Gert Biesta

Today, professor Gert Biesta was visiting HiOA, He presented his article "How does a competent teacher become a good teacher? On judgement, wisdom, and virtuosity in teaching and teacher education." (in R. Heilbronn & L. Foreman-Peck (Eds.), Philosophical perspectives on the future of teacher education. Oxford: Wiley Blackwell.)

As part of the seminar, I was asked to prepare a comment from the point of view of Norwegian teacher education. The seminar was not filmed, but I have made a video with my comments:
My "manuscript" can be found at

Friday, July 18, 2014

Post-conference blues

Last week, I sat in solitary confinement in my office.
I struggled with the contents of my workshop - who would care?
Then here - the adrenaline made my senses aware
of every single sigh, shaking head or smile.

That done, every waking hour was spent with soul mates -
breakfast, lunch and dinner and all the hours in between.
People all around the world care about what I do!
People share their thoughts and listen to my thoughts, too.

Then it ends. The plenary room is suddenly silent.
The echo of many "see you in two years" fade.
Life gets back to normal: and suddenly I see:
again, the only one here who care what I do - is me.

ESU7 Day 5

The final day of ESU7 started with Bjarne Toft's "Peter and Sylvester - the emergence of graph theory". The first paper on graph theory was by Danish mathematician Julius Petersen. Of course, in hindsight, precursors of graph theory can be seen in The Bridges of Konigsberg problem, for instance. Road networks, social (or sexual) relationships and website connections are examples of applications today.

Petersen was a teacher, teaching 35-36 hours a week, and publishing a new textbook every other year. He wrote a much used and acclaimed collection of geometrical problems. He finished his PhD in 1871, and published a lot in the following years. Then he got in touch with Sylvester, and they met in Copenhagen (in Tivoli!). In a letter of October 18th, 1889, Sylvester explained to Petersen what "graphs" are. In 1890, Petersen went to England to collaborate with him. The collaboration didn't work out, though, and there were some amusing letters going in several directions. They showed both how mathematicians are human beings with mood swings like everybody else, and that mathematicians are fallible, unlike mathematics books...

Fàtima Romero Vallhonesta and others then had a "workshop" on "Teacher Training in History of Mathematics" (but it turned out to be more of a lecture). They are a group of teacher making materials for the classroom which will be a publication in two year's time. They had a brief introduction, mentioning the difference between explicit and implicit use of history of mathematics. They pointed out that with teacher students, they used sources explicitly, while with other students, they used the sources mostly implicitly.

The aims of the implementation was

  • Knowledge of the original sources
  • Recognition of the most significant changes in the discipline of mathematics
  • To emphasize the socio-cultural relations of mathematics with politics, religion, philosophy and culture.
  • (The most important). To encourage students to reflect on the development of mathematical thought and the transformations of natural philosophy.
There was an introduction to Chinese sources, including Nine Chapters, and then we looked at chapter 9, being given the problem, answer, algorithm and comment (by Lui Hui). (In both the original Chinese and in a French translation...) Students could do activities, for instance doing geometrical proofs on paper. (It was unclear to me, however, what the students were asked to do with the sources and what instructions they were given.)

Al-Khwarizmi was then the next example - the usual solving of quadratic equations, but with both of al-Khwarizmi's ways of solving (and this time in an English translation). The goal of this particular task was that students should be better at algebra, and the original source is (as I mentioned) only read by teacher students, not with the mathematics students. First, students are asked to research al-Khwarizmi's life (some basic questions), then they learn both algebraic and geometric methods. The geometric method they learn by getting the complete geometric proof, but without numbers added to the figure, so that they themselves just provide the details, not the steps.

Their way of using al-Khwarizmi with mathematics students is quite unhistorical (as was pointed out by people attending) and I'm not sure that it could be called implicit use of history; rather it is teaching loosely inspired by history. Teacher students, on the other hand, are given the original sources and given the task of making activities based on them, and then the different ways are compared. I would be careful about doing that with my students, as there is a danger that all of them would make unhistorical materials so that it could be difficult or at least time-consuming to avoid ending the project without having reached anywhere meaningful.

Finally, there was an example from Pedro Núnez and from Viète. There was a fascinating dispute in the end of the workshop where an equation in the style of Viete was written also in the style of Descartes, where the variable A was changed into the variable x, but the z from Viete was kept. It was pointed out that this would be confusing, as z was a constant with Viete, but a variable with Descartes, so in the "Descartes version" of the equation, it should be c...

Kristin Bjarnadottir's workshop was a follow-up on her plenary the day before, with the opportunity to go into details. First, we had a look at another problem on measurements, which - among other things - showed Icelanders familiarity with different measurement units, as well as payments based on gender. Then we all collaborated on doing a Geogebra simulation of the movement of the sun in the Icelandic sky at different times of the year. This ended up with the function f(x)= -(90-65)* cos (x*2*pi/360) + a, where a is a value between -23.44 and 23.44 (on a slider), and where 65 is the latitude of the farm of Torsteinn. (Reykjavik 64, Torsteinns place 65, Rome 42 and so on.)

Then there was a little on the dominical letters, summer's extra week, the first week of summer and so on. I cannot possibly summarize this.

It was unavoidable that this conference too would end. The closing session was a time for praise for the organisers, but also to look forward, to the deadline for the proceedings papers (10-15 pages by November 15th), to the next HPM conference (Montpellier July 18th-22nd, 2016) as well as to the next ESU (Rethymnon, Greece, July 2018).

What were the highlights in the week for me? To be honest, the highlights for me personally was a conversation with Renaud Chorlay over a beer after the excursion, a late-night discussion with Francesco Maria Atzeni, breakfast talks with Costas Tzanakis, a few short lunch chats with Torkel Heiede, the traditional dinner with Kristin Bjarnadottir and Andreas Christiansen (traditional since the last ESU...) and coffee break talks with Mustafa Alpaslan. I could go on and on. The most important part of any such conference is the face-to-face discussions with other people who spend their life striving for some of the same goals as yourself, having some of the same interests and strangenesses... Sadly, my blog posts don't catch all of this very well, although my opinions throughout are of course enhanced by the conversations I've had throughout the conference and at previous conferences.

It is difficult for me to pinpoint a highlight from the scientific programme. Single ideas, such as David's use of the concept of "violence" to describe working with original sources, were thought-provoking. However, if I am to point out two main ideas that have (re)formed in my head during this conference, it is these:

  • The same original source can, depending on the context, be used for many different purposes, and depending on these purposes, the design of the teaching will vary. The assessment of whether the teaching was successful, will also vary. Understanding better how these things work constitute major "open questions".
  • In particular, in some contexts it is meaningful to give students an original source and a set of tasks to "guide" their reading. When this is appropriate, how these can best be designed and what effect this has on the outcome, form a subset of these open questions.
Of course, these questions was actualized already in the first plenary, so I'll send special thanks to Adriano Dematte for his plenary. But I've come across things reminding me of this every day of the conference.

In addition, my plan of writing a book for teachers (in Norwegian) on different ways of including history of mathematics in teaching, remains on the to do-list. More resources for teachers are always needed.

So, as I leave for a few days of holiday in Paris, I know there will be work waiting for me back at work for years to come (and in particular when I'm back in my research and teaching position from 2016...)

Thursday, July 17, 2014

ESU7 Day 4

Thursday started with familiar faces. First, there was Kristin Bjarnadottir (from Iceland), whom I have met in many conferences before, and whom I have blogged about before, I believe. Even though her field of research is far from mine, I remember being very inspired by her workshop at a previous conference. Now she had a plenary on "Calendars and currency - Embedded in culture, nature, society and language".

She started out by talking about ethnomathematics and presenting the history of Iceland. Due to a lack of metal, exchange of goods were the norm instead of buying with money. A hundred was equivalent to a cow or 240 fishes. In 1880: Danish krone adopted. Kristin gave an example of quite involved textbook problems using these exchange "rates".

The Misseri calendar was adopted in 930, and this was adjusted to the Julian calendar in the 1200s. The calendar was needed to establish the meeting times of the parliament. The practical problems with errors in the calendar (which meant that soon the parliament had to meet in the busiest parts of the farming year) was a powerful motivation for figuring out how to improve on the calendar. Later, other errors were found in similar ways. But in adjusted versions, the Misseri calendar has survived almost to our day.

Secondly, there was Gert Schubring's plenary "New approaches and results in the history of teaching and learning mathematics". He stressed that he avoided the word "education" In the title to avoid confusion with "mathematics education" (matematikdidaktik). He started by outlining the history of research on history of mathematics in school. New trends now: more specialized studies of specific countries, attempts at an international history and new methodological approaches, going beyond the administrative documents and using insights for instance from sociology. These efforts were mostly individual, but have recently also become institutional, thanks to the topic study groups of ICME from 2004 and launching the journal IJHME in 2006. In 2014 was published Handbook on the History of Mathematics Education.

If history of school mathematics is reduced to the history of curricula, the social reality is missed. Also, mathematics never occures in an isolated way, but must be seen in connection with other disciplines. Mathematics education became needed by state administrations in the same way that numeral systems and calculation with these were neccessiated by state administrations. Shubring went into some details on the origin of mathematics education in for instance China, which I cannot possibly summarize here.

Another question is how and why mathematics turned from something you should learn for state administration and into something everybody should learn. It turns out that France was the first country where this happened, after the revolution of 1789. Latin and mathematics were the two essential disciplines. Again, the amount of detail from different countries cannot be done justice here.

The second panel of the conferences had the title "The question of evaluation and assessment of experiences with introducing history of mathematics in the classroom". The panelists were Leo Rogers, Janet Heine Barnett, Ysette Weiss-Pidstrygach, David Guillemette and Frederic Metin.

Leo Rogers spent some time outlining the vast canvas we are moving about on: different target groups, different goals, different ways of including history of mathematics... These different contexts we are in, give different answers to questions such as why to assess, what to assess, how to assess, what data to collect and how to analyse and how to respond to the analyses. He also walked us through a lot of the common problems with testing/assessment, which are often of different forms when it comes to history of mathematics than in mathematics. (Of course, research on assessment in mathematics and history could have much input to us here.)

Janet talked on her work on the projects she has been involved it (with resources available online). The primary source projects have student tasks, which may be relevant for assessment. The primary goal Here is for students to learn core material in contemporary mathematics courses. But there are also hopes that these original sources support a development of a deeper level of understanding. The examination is mathematical. They also have homework where they get feedback and can rework it.

Frédéric talked about the French system, where there is a competative examination after the first year of teacher training (fourth year of university studies). Therefore, history of mathematics must also be suited to help students pass the exam. Major aims are then: - Enightening students knowledge in mathematics - Putting distance between students and mathematical knowledge - Link mathematical knowledge to other school disciplines - Training through (useful and job connected) research

Then: why assess, how assess, and is there any point in assessing the HM components, if the goal is not to learn HM? He gave examples of tasks he gives students and where it is quite simple to give similar tasks for the purpose of assessment - but still without being certain that it makes sense.

Ysette talked about maths camps where they got very few applicants, because of using the words "for gifted students". Asking students, they connected "gifted students" to the grades. These were attitudes that are problematic in teachers. History of mathematics must combat this, not strengthen it.

Concretely, comparing different textbooks' presentation of historical facts was an example on how to work on history of mathematics, which can be done in many different ways depending on students' background. They assess using lesson planning based on an historical excerpt.

Finally, David talked on experiences with preservice secondary school students, where the goal was to obtain deorientation (see his oral presentation). Assessment is problematic. Reading of original sources is a hermeneutic extreme sport without helmet. Students suffer the original sources, it includes shock and violence. Otherness is linked to empathy, keeping the subjectivity of the other alive. How to have the students suffer and maintain the empathy, how to assess the students?

Often, the evaluation is done by giving another text of one of the same authors they have worked on, trying to see if they understand the new text. Then the adventure ends. How to avoid that the relationship to the history ends?

Ewa Lakoma gave some input from her experiences in Poland, where examinations were given asking about historical facts, and group examinations where social skills were taken into account.

Evelyne reacted to the question of whether history of mathematics is a Humanities subject. HM is not in the tradition of mathematics, and helps us to speak mathematics and talk about mathematics with others. Mathematics is otherwise often taught by writing mathematics on the blackboard and by correcting papers with small signs on papers that have been handed in.

Janet pointed out that when we read mathematics, we have some techniques (for instance, finding an example) which students do not have. Therefore, many have turned to "guided reading" through tasks. (Actually, an interesting research project would be to study how different authors design the "guided reading" tasks and how they describe their reasoning between these tasks in their papers in HPM. Interviews would of course also be a possibility. It could also be compared to how this is done in other fields, such as Norwegian (L1) education.)

Tinne pointed out that the learning goals could both be on the big scale and on the small scale (the single session), and the assessments will vary according to these.

After lunch (and a much needed Starbucks ice coffee from the local minimarket), I chose the workshop of Caroline Kuhn and Snezana Lawrence on "Personalised Learning Environment and the History of Mathematics in the Learning of Mathematics". Personal is in the sense that the learner does design his own learning environment, choose tools etc.

After an introduction round, we had a round about our expectations. By this time, half an hour had passed. Then Caroline discussed different definitions of PLE. Ownership and agency are interesting in this context. Building a PLE is a good learning activity.

Four examples of websites on mathematics (which are not PLEs): NRICH MacTutor - StAndrews

PLE outside mathematics: graasp Plenk 2010: (Not PLE but a MOOC, which is simething different altogether.) Some other PLE: - works, but it is not so easy for students. (Valtonen Teemu)

After the break, in which a few of us was honestly still at a loss to understand exactly what a PLE was, we were put in groups and asked to craft an example of a teaching strategy for the tangent line problem, and then given a few sources (Apollonius and Fermat) and asked how knowledge of the sources could enhance the teaching. We all agreed that it couldn't really enhance the particular teaching ideas that we had thought of at first. This is, of course, often a problem for teachers: even though there are millions of sources online, the sources you find may not be interesting for your immediate purpose.

After the workshop, my thoughts are that my iPad (as well as my PC) can be regarded as my personal learning environment, with selected websites (bookmarks) and programs/apps, as well as resources I have made on my own. However, to be more useful in education, such a PLE needs an easy way for the teacher to suggest resources for the students, and - more importantly - it needs an ecosystem into which students may put their products and where other students may find these. A PLE with unlimited and uncurated access to "everything" will be fine for the expert, but certainly not for the student.

In the workshop, BBC's A Brief History of Mathematics was mentioned. I will have to check that out.

Thus ended Day 4. I skipped the oral presentations that started at 5:30, and waited for the Happy Hour instead...

Wednesday, July 16, 2014

ESU7 Day 3

The first of Wednesday's talks was Cécile de Hosson's on "Promoting interdisciplinary teaching through the use of elements of Greek and Chinese early cosmologies." Her talk discussed the connections between physics and mathematics, and how the history of physics could be used as part of physics education. She gave examples, for instance Eratosthenes' calculation of the circumference of the Earth, which is of course a good example that should be part of mathematics education. She noted that students had trouble accepting that the sun's rays are "nearly parallell" - mathematically, "nearly parallell" is quite different from "parallell"! This is of course a central part of the tension between mathematics and physics. The talk was one of several at the conference highlighting not just that the use of history will have to be different for different target groups, but also how it will have to be different.

Then there was the first panel; "Computational Technology: Historical and philosophical approaches to technics and technology in mathematics and mathematics education". I'm always very sceptical of panels, as they tend to disintegrate into four or five small talks that does not neccessarily connect very much to the professed theme of the panel and very rarely touch upon what the others have said. This, however, was beautifully organized, in that each panelist gave just a short talk (5 minutes?), followed by an answer by one of the other panelists before going on to the next short talk. All panelists were in active dialogue with the others and the theme of the panel.

Mario Sánchez Aguilar discussed how computer technology changes the way pupils and teachers work on mathematics. One way is through the use of non-traditional sources for help in working on the mathematics - nowadays, students can get so much help in different websites, for instance. For teachers, there is the possibility to enrich the instructional techniques, bringing videos into the classroom (YouTube) and by bringing the teaching into pupils' homes.

In response to this, Per Jönsson asked what much of mathematics needs to be internalized and how much can be outside you for you to know mathematics. He also asked if the teacher is needed at all in this system.

Per then went on to discuss "what is mathematics?" Computing have changed problem solving dramatically (in the research on mathematics). Mathematics should now be learned with more focus on problem solving and computing.

Mikkel Willum Johansen reacted by pointing out that the role computers play is as a tool for mathematicians.

Mirko Maracci: computers are used for educational purposes. Two ways of looking at it is that they mediate learning processes or that they embody knowledge. When they're seen as mediating, it can be either that the artifact may permit the transformation of the object and/or permit the subject's concious-raising of the object.

Morten reaction to Mirko: important to be critical - tool also changes and transforms mathematics. When a tool is used, it can change things in different ways. Important to study of the whole ecosystem.

Mikkel pointed put that mathematics is impossible without the right tools - it is a tool-driven practice. Tools have consequences, however. So even though computers are "just another tool", they have an impact on the content of mathematics. Computer-assisted proofs, mass cooperation and experimental mathematics are examples of how mathematics is changing.

Morten said that in teaching, computers challenge the existing mathematics education practices, for instance in something so basic as "what is a good task?" Mathematics education is squeezed between research mathematics, school traditions, the applications in specialized domains and children's everyday domains. New tools will change all four of these domains.

Evelyne raised the obvious question of gender (in this all-male panel). What are the gender aspects of the influences computers have? There was quite a bit of discussion on this, although mostly anecdotal in nature.

Another comment from the audience: we have to decide what we are going to teach - we cannot go on teaching what we have done. And we have been in this situation before through history.

It was nice to have a panel about computers in mathematics education without focusing all the time on the problems they bring by stealing students' attention. Of course Facebook was mentioned, but I was on Facebook at the time, so I didn't hear the complaints so well... (I had just blogged about yesterday's experiences at the conference, and noticed that a Swedish colleague who took part in HPM in Daejon but couldn't be here, thanked me for these updates...)

After this panel, the academic program was over, but the day had just begun. We had a bus ride to Nyhavn, a boat trip on the canals (with lunch), a guided tour of Christiania, a couple of hours on our own and finally a dinner which transformed into a party and lasted into the small hours of the night. There were no presentations during all that time, but I had more academic conversations than for the rest of the conference combined. I had interesting chats with people from France, Norway, China, Italy, Sweden, Denmark, Germany and probably many others that I have forgotten by now. The programme is usually too tight to have long conversations in the breaks, so this day was welcome also for giving us time to discuss. And of course we managed to squeeze in some non-academic discussions as well...

ESU Day 2

The second day of the conference started with Evelyne Barbin's plenary talk titled "Epistemologies and Theories behind History and Education: thirty years after Hans Freudenthal". Of course, Evelyne is an important figure in the HPM group, as Jan van Maanen stressed in his introduction of her. She started by referring to Hans Freudenthal, who protested the teaching of ready-made concepts instead of the development of the concepts. The phrase he coined for this, and which was mentioned throughout the talk, was "antididactical inversions".

Evelyne referred to two different ways of doing history; disorienting history and rational history, where the former strives to understand the facts of history, while the latter strives to give "big" theories. The former is looking at differences between authors, the latter concentrating more on similarities. I understood Evelyne's talk as a strong criticism of many attempts at the latter, where broad generalizations on the development of mathematics are made without sufficient regard for the details of the actual development.

Most of her talk focussed on particular, important examples from the history of mathematics (finding of tangents of curves, Leibniz' definitions of functions, conceptions of tangents etc) and examples of researchers giving statements of the development of mathematics which do not fit with the particular details. It would be dangerous for me to try to repeat concrete examples, as I do not have Evelyne's knowledge of the details.

Evelyne ended by warning against the "danger of a terrible didactical inversion: the didactical inversion of history of mathematics".

For the day's two hour workshop, I chose to listen to Jan van Maanen. It is more than fifteen years since I heard him the first time, and I always grab the opportunity when I have the chance. His title today was "'Telling mathematics' revisited". This was a development of a talk he held fifteen years ago (in Louvain), and two of the people present had also heard the talk then.

As a teacher, Jan spent time on problems that the pupils brought, and after a while, he realized that they had a lot of history in them. These stories have travelled miles and years.

For instance, he told us that he had talked to a Dutch girl, who had been told a problem by a girl while waiting for a ski lift in AustriaAustria 2010: "If I give you 2 sheep, you will have twice as many sheep as I have. If you give me two of your sheep, we have the same number of sheep. How many do we have?" The same story can be found, in variants, for instance in Nine chapters, 179 AD, and was included in Euler's Elements of Algebra 1770.

He gave lots of examples of such problems, such as the broken bamboo, the hundred birds problem, the two brothers (one of which always lies and the other who always tells the truth), "as I was going to St. Ives", 3-4-5 triangles in Groningen doors (Frank Swetz p 119).

Why does "telling mathematics" make sense also within the school?

  • Because of the challenge - not neccessarily using exactly what they learned that lesson.
  • It combines the usual with the unexpected.
  • It is the student who decides!
Presently, Jan has had two workshops with mathematics teachers to let them exchange mathematics that they knew from being told, asking them to make notes, then analyzing these notes in order to sort and cathegorize the entries, and to trace the entries back to their source.

He did the same with us, and "my" group (Mustafa, David and me) came up with three examples: - weights (one lighter than the others) - area with given perimeter - three people with black or white star on their foreheads. The winner is the person who knows his own color. They have to clap when they see a white star. When person A open his eyes, he sees two white stars. All three persons clap. But noone claims to know their own color. Then A knows. Why?

There were discussion at the end about how to "use" these problems. If the teacher gives the problem, should they be part of a structured plan of teaching? Jan's main point is perhaps to have the students contribute to the content of the classes, which means they can not necessarily fit nicely into the overall plan of the teacher. The point of giving students a chance to participate is more important.

Personally, I'm a bit depressed that I - from my memory - almost didn't work on anything outside the textbooks for my twelve years in school.

One sidenote: Jan mentioned a mathematical walk of Groningen - maybe this would also be a good idea for Oslo?

The second workshop of the day, this time a three-hour one, was Desiree Krüger and Sara Confalonieri on German and French textbooks' handling of negative numbers. They gave a background of the "elementary" textbooks that they wanted us to work on. The textbooks in question were written for universities, but it could not be assumed that the students had any prior knowledge of mathematics. In France, though, they had military schools that included mathematical teaching.

Then we worked in groups on selected original sources in the form of textbooks. My group (Andreas and me) worked on Kässtner's "Anfangsgrunde der Arithmetik Geometrie ebenen und sphärischen Trigonometrie, und Perspectiv". For me, both the typeface (fraktur) and language (German) was difficult, so there were some layers of difficulties to get through in addition to the unfamiliar definitions and examples (and context, of course). Particularly interesting was that the textbook stressed that you can choose for yourself what should be considered positive and what should be considered negative whenever you have two opposing magnitudes. For instance, a debt is negative if you consider wealth as the positive, but is positive if you choose to see the debt as positive. We wondered if this way of viewing it may be better than the normal way of doing it in Norwegian textbooks today.

Then on to oral presentations. First, David Guillemette talked on "Sociocultural approaches in mathematics education for the investigation of the potential of history of mathematics with pre-service secondary school teachers." He takes disorientation (dépaysement) (Barbin) as a starting point. He sees a gap between theoretical and empirical research in HPM. David wants to get deeper into the concept of disorientation and see how it turns up in real situations. He will use the theory of objectivation. (Learning maths is not just learning to do maths, but to be in mathematics (Radford)). He wants to give the pre-service teachers a voice.

He will also discuss whether the disorientation is an individual experience or could/should be seen in a social context. The narratives will be collected into a polyphonic narration, trying to integrate different points of view. One example of a preliminary results: a feeling of otherness and empathy towards the authors and the pupils are present.

Afterwards, there was a short discussion on the best translation of dépaysement (or whether it should be left untranslated). Alienation has also been used.

Secondly, René Guitart talked on "History in Mathematics According to André Weil." Weil used original sources to explain today's comprehension, reading the original sources with the help of today's knowledge, but without pretending that those authors had the same knowledge. Weil suggested historical notes in the papers of the Bourbaki group. A historical problem is (only) considered important if it generates a method or general theory. In two letters to his sister, he explained the meaning of his mathematical work by means of its history.

Thirdly, Leo Rogers' theme was "Historical Epistemology: Contexts for contemplating classroom activites." His talk concerned the connection between the origin of mathematics in ancient times and the connection to teaching of kids today, with lots of theories involved and lots of practical examples of what mathematics ancient cultures were doing. Sadly, my notes are not sufficient to give a detailed summary, so it's best to wait for Rogers' article (as is of course the case of most of the talks and workshops I mention).

One good quote from Neugebauer which I will keep in mind, was about how we do not gain a "historical perspective" as time goes by, we are just less burdened by evidence, which means we make bolder generalizations without fear that historical sources will contradict us. With this quote, Rogers' talk ended the day on the same note as the day started.

After this, the Advisory Board of the HPM had its traditional dinner, which included great food and wine and many friendly conversations - as well as a little official business.

Monday, July 14, 2014

ESU7: Day 1

The 7th European Summer University on the History and Epistemology of Mathematics takes place in Copenhagen this week. This is my 4th ESU, so it's already a tradition for me, and it is always nice to come back to meet many well-known faces - as well as many new ones. As usual, I will try to blog from the summer university, but as usual I will have to warn you that my impressions of what people were trying to get across, may be mistaken.

Conference organizer Uffe Jankvist gave a short introduction to the buildings and Evelyne Barbin an introduction to the history of ESUs. This is a special occation, as it is the last ESU Evelyne will take part in before she retires. Tinne Hoff Kjeldsen then gave an introduction to the themes of this conference.

The first plenary speaker was Adriano Demattè, with the title "History in the classroom: educational opportunities and open questions". I know him mostly from these conferences and his teaching materials, which are very inspiring. He started by referring to Schön, and claimed that teachers have to look for proper circumstances in order to reflect. In his talk, he went on to show how history of mathematics will help them.

Interestingly, he spent quite some time discussing his context, how many classes he teaches and so on. This is necessary, because contexts are so different in different countries. Even though he teaches pupils aged 14-19, the topics he teaches corresponds to what Norwegian students learn at age 16-20, for instance. But also, the fact that he teaches maybe five different classes at four different levels means that his teaching will be different from a professor who has one or two courses in a semester. It is not easy to be innovative all the time in such a context. And Demattè is not content with being innovative himself, he also want other teachers to be able to teach using historical materials.

Demattè discussed the genetic approach vs the hermeneutic approach. He referred to Jahnke's 2014 article on how history of mathematics can be used after students have had a first meeting with a new topic. He looked at how some of the steps in Jahnke 2014 would be difficult in a classroom.

He then described some teaching he had done with a short part of Pacioli's Summa (1494). He asked students to read and interpret. The problem was "Find for me a number that, if added to its square, makes twelve." Pacioli solved this by an algorithm. Of course, he ends up with square root of 12 1/4 minus 1/2, thus three. But there were no modern symbols, and the modern division into question and solution were not as obvious.

Not all students thought about reading the text several times to understand more, and many did not compare with the modern solution to try to understand words (such as "root"). There was no hermeneutic circle - students did not make guesses, and did not move between parts and the whole of the document. It seemed that only those students who was confidence that their answer would be accepted byt he rest of the class, had the willingness to guess.

Demattè's proposed compromise: a textbook where every chapter contains a "historical laboratory", with a list of prepared questions to guide the students in their interpretation, making sure they move between parts and the whole, work on central words in the text and so on. The questions are aimed at facilitating students' work, take into account typical student prerequisites, bridge between modern solution and Pacioli's solution. He also mentioned that these questions could be used for tests, which is really interesting - and controversial, I would think.

A comment from me (which I didn't raise there): A question is of course whether adopting the question-answer format of a traditional textbook helps students or just turns this into just another hunt for the right answers. But again, this is a case where such a compromise may be needed to make it feasable to use it in ordinary Italian classrooms. In Norway at least, we could add the fact that teachers might not fare so much better than the students in interpreting original sources.

I'm sure many would argue that the struggle with the text is an important part of students' work, and that removing the struggle also removes much of the point of working on original sources. Renaud Chorlay asked whether students improve in their reading/interpretation capabilities over time, but Adriano Dematté could not answer as he has changed his approach over time, so he does not have long-term experiences in this. Probably, what Chorlay was hinting at was precisely that the initial "suffering" of the students in working on a text without being shown the way, is worth it in the long(er) run.

So already there were big, open questions going through my mind...

The first workshop I attended was my own. I was very happy to have my workshop this early, as it is always a relief to have finished. The goal of the workshop was to have people familiarize themself with the framework on Mathematical Knowledge for Teaching (MKT), see several examples of how history can be used in mathematics education, and discuss different reasons for doing that - in light of MKT. I was quite happy with how the discussions unfolded through the two hours, although some participants were at times at a loss to understand what my aim was - as I didn't have any particular opinion to "sell".

One important discussion is how the HPM community should interact with the mathematics education community as a whole, for instance regarding MKT. Is MKT "bullshit" (reducing mathematics to solving problems and disregarding emotions, for instance), as someone said during group discussions, or is it an a useful framework also for us in discussing HPM? And is it important enough in the maths education world that we should point out the place of history of mathematics in it even if we're not fans?

One outcome of the discussions was to see how a single source could be viewed as contributing to knowledge in many different domains, depending on your context and your point of view. Several participants found MKT useful as a tool for discussing HPM outcomes in teacher education, as long as we don't see the domains as discrete and separate. The idea of using MKT for testing teachers' MKT, however, (as is one branch of the MKT research) met more scepticism.

Then there was a three-hour workshop by Susanne Spies. Gregor Nickel and Henrike Allmendinger from Germany: "Using original sources in teachers education - An analysis on possible effects and experiences." In a way, this workshop was a bit similar to my own, in that it also focussed on concrete examples of using history and on discussing possible outcomes. They started with the point that school education of mathematics is - and have almost always been - insufficient for starting on higher education in mathematics. In particular reflection and the ability to judge about mathematics is lacking in students, and these are fields in which history could be helpful.

They gave several ways in which historical sources could be a tool for teacher education (telling anecdotes (entertain, comfort) - can become heroic or jovial; genetic use (Toeplitz): implicit or explicit - can be confusing;defamiliarizing; paradigmatic use) and how it can be a discipline for reflection (historico-critical perspective; history of ideas and culture perspective (secondary sources?))

They listed many examples from high school teachers' education where they included historical sources (real analysis; lecture on different solutions of the isoperimetric problem; exercises on irrational numbers and incommensurability; worksheet on Bernoulli's way of finding the tangent of a parabola. Reflecting on different ways of argumentation; worksheet on different interpretations of the derivative; presentations of students based on historical sources: training text comprehension of original sources.

They use historical sources in courses on elementary mathematics, advanced mathematics, history of mathematics and pedagogy of mathematical, and were particularly interested in whether historical sources had different roles to play in the different kinds of courses.

Then we worked in groups on a part of Euclid's Elements: theorem 47 (Pythagoras' theorem). We worked in groups discussing which contexts the source could be used in and what impacts it could have. As someone commented also in my workshop, looking at single sources and their impact risks forgetting the more overarching impact of using history of mathematics repeatedly. For instance, will students really change their view of the epistemology of mathematics based on one example from history of mathematic?. Probably not - it is the 13th example that makes the difference (I'm joking about the number - I think...). Thus, when looking at single sources, we look at the particular impact of the source, and not on the general impact of HM.

Thus, we focused on what this particular part of Euclid could contribute, and all groups agreed (it seemed to me) that a likely output could be on understanding the concept of proof (through time) and on being able to interpret different texts (which a teacher needs to be able to do when his students do surprising things).

Then: the use of philosophical texts in maths education. We did a similar exercise with a work of Plato (seventh letter) on knowledge of objects (name, definition, image, knowledge - and the fifth). We discussed how this text could be used in different courses. Of course, it could inspire discussion about the insufficiency of just giving a definition to students, but an important question is why this discussion would be better using the historical source than without? (In a way, I wondered, is it the case that in research mathematics, a concept IS its definition to a higher degree than in school mathematics?)

The evening's oral presentation was chaired by me, and I seemed not to be able to chair and make notes at the same time, so my comments on those two presentations are a bit short. Panagiotis Delikanlis talked about how he works on the mathematical problem contained in Plato's Meno, while Fanglin Tian talked about how they work in a lesson study-like way on teaching logarithms based on the origin of logarithms. Although both were based on history, the second was much more connected to the historical context. I'm looking forward to reading her article with (probably) some more detail.

Then there were two poster presentations. The first, by Paulo Davidson, connected the themes HM, Funds of Knowledge and Culturally Relevant Pedagogy. It would be useful if this presentation was longer to get more details, for instance on how history was used implicitly to work on algebra in new ways. Moreover, I should read more about CRP, for instance an article cited by Ladson-Billings. The second, by Julio Corrêa, was on mathematics, education, modernism and war. In the short time he had, he could just sketch some of the connections between these concepts. The "Math Wars" in the US is an example, which is a debate (not really a "war", but with war-like rhetoric ("a nation at risk").)

Thus ended the scientific programme of the first day. But then there was Happy Hour, and I was happy to talk to many friendly and interesting people. It was particularly nice to talk to Torkel Heiede on his home turf - he noted that he'd worked in these buildings for thirty years. In the opening of the summer university, it was mentioned that the 2016 HPM conference will be held in Montpellier in France, so that will be my next chance of meeting many of these people.

I'll continue blogging tomorrow...