The first TWG session on Saturday consisted of four ten-minute presentations, followed by discussions. As I had one of the presentations, it's a bit hard to give details on them (one does get a bit too occupied with one's own presentation in such circumstances). They were:

• Rodolfo Fallas-Soto: "Variational strategies on the study of the existence and uniqueness theorem for ordinary differential equations"

• Me: "Design research with history in mathematics education"

• Antonio Oller-Marcén: "Analyzing some algebraic mistakes from a XVI century Spanish text and observing their persistence among present 10th grade students"

• Katalin Gosztonyi: "Understanding didactical conceptions through their history: a comparison of Brousseau's and Varga's experimentations"

In the discussion, some of the points were:

• Tradition and contextualisation are important - the traditions researchers come from are important (in the case of my design research project). It is important to be clear about the context of them (but on the other hand, it is also important for design research projects to consider and describe which context they may be relevant for).

• There was a chicken-and-egg-discussion on what comes first in historical research - the question and/or method or the data. (Arguably, all the world is data - or; you can say that they only become data when they can be helpful in answering a question someone poses.)

• In what way do theoretical frameworks "work"?

• What to do once epistemological obstacles are identified? Should we face or avoid them (until students are "hungry" - why feed them if they're not?).

• Design research - can it be called a "theoretical framework" (as the chairs did in their framing of question for the group discussion). (My answer would be no. A participant also said that it could rather be seen as a framework of aspects to be thought of in such projects.)

The next part of the programme was a plenary panel. The panellists were Marianna Bosch (Spain), Tommy Dreyfus (Israel), Caterina Primi (Italy) and Gerry Shiel (Ireland). The topic of the panel was "Solid findings in mathematics education: what are they and what are they good for?" Marianne Bosch was the chair. The background for the panel was EMS' series of articles on "Solid findings in mathematical education". "Solid findings" are defined as important contributions, which are trustworthy and that can be applied. The panel wanted to examine the notion of "solid finding" and consider possible utilities and weaknesses.

Tommy Dreyfus pointed out that there are not many review articles in the field of mathematics education. The European Mathematical Society (EMS) decided to help remedy this. (The articles are in the Newsletter of the EMS issues 81-94.)

One example: we know that many students "prove" a universal statement by providing examples, across many age levels and countries, including teachers. We call this "empirical proof schemes". But to be called "solid", an explanation is also needed, and here the explanations are varied. But the main criteria for being "solid" holds. Another example: concept image. Students tend to think with their personal image rather than the definition. This occurs at all levels, in many countries, for almost 40 years and across many topics of mathematics. These are often formed by prototypes. Instruction plays a (limited) role. These findings can be considered "solid".

Solidity cannot be "proved", expert opinion is crucial, and experts from several fields should be consulted.

Caterina Primi talked about how psychometrics could contribute to solid findings in mathematics education. We often measure something else than the trait we are interested in - for instance signs of anxiety, even though it is the unobservable trait anxiety we are interested in. Of course, we can create instruments to try to measure the trait based on them, and these can also be used to find differences between groups. (And so on. It is hard to see how this rather elementary discussion of psychometrics contributes much to the general discussion of solid results - unless her talk is an implicit argument that psychometrics are more important than other research approaches to get solid results - as many would of course say about their own pet approach.)

Gerry Shiel's perspective was whether outcomes of international assessments (PISA) can contribute to evidence-based decision-making. Are PISA findings solid? On the one hand, it is huge (more than 500 000 students have contributed to it). He gave an introduction to PISA and how it tries to be an evidence-based series of studies including testings. He gave an example of how Ireland's performance in TIMSS changed over time, with a significant dip in 2009. This dip has not been explained. Ireland rebounded, while other countries had a dip in 2015 when digital testing was done. Also, Ireland has an increase in the gender difference between boys and girls, which is hard to explain. PISA results are used to inform policy - and PISA surprisingly tries to impact teaching directly by publishing their speculations on what can be inferred by the data.

In the discussion (which did not work very well, because of a somewhat confusing combination of "questions" from the floor and "questions" sent electronically), it was asked "solid for whom" - implying that what is solid for researchers may not be solid for teachers (and vice versa). This is an interesting point. Gabrielle Keiser mentioned that we need some methodology for writing review papers - it is a very difficult task, and for instance quantitative analyses are not always helpful.

(But in hindsight, it is easy to see that this topic invites people to promote their own research or conception of research...)

The last part of Saturday (before the gala dinner) was the last session of the TWG. First, there was a part where participants talked about planned or ongoing projects with calls for cooperation. Then we talked about future conferences, where I presented the plans for ESU8 in July, 2018. Plans for the HPM satelite conference to the ICME conference in Shanghai 2020 were presented - it will be somewhere in Asia. Then the process of the proceedings were discussed, and finally there was discussion on the report of the conference, the result of which will of course be seen in the proceedings of the conference.

Due to travel arrangements, for me the conference ended with the gala dinner on Saturday evening (which had much Irish music and rather less talk). Thus, this is the place for summarizing the experience. This was my first CERME conference, and I realized that CERME is not really one conference, it is rather ~25 mini-conferences under one roof and with shared amenities and a few common talks. This means that it in one sense is an intimate conference in the same way as smaller conferences are. However, getting the intimate feel demands some consicous choices - not to switch groups no matter how interesting the talks going on elsewhere are, and to try to socialize with people in the group and not be tempted to only socialize with the people you already know. Then, the CERME experience is quite different than for instance ICME, which is a smorgasbord of interesting talks where you risk never running into the same people twice (even though even ICME has some working groups, of course, so I am exaggerating a bit).

Dublin was great, the LGBT guided tour was great and the atmosphere throughout was also great. I did learn some new things during the conference, of course, but most importantly, I think, it solidified my determination to try to focus more in the future. I want to spend my research time to get deeper knowledge in some areas rather than having many parallell projects with different foci. I'll see how this works out...

## Saturday, February 4, 2017

### CERME 10 Day 3

Day 3 consisted solely of TWGs and an excursion. The first TWG session was devoted to discussion on the draft chapter on this group for an ERME book. It was introduced by Uffe Jankvist, who has written the chapter with Jan van Maanen. I did not note down anything from that discussion - but I was perplexed to be put in an "old-timers" group despite this being my first CERME. :-) (My feeling of being "young" was destroyed due to my participation in similar conferences since 2000...)

The rest of the morning session was spent on participants sharing informaion on important publications that the others should know of. I have notes of this somewhere, but we were also promised an email later summarizing this.

The second session started off with Renaud Chorlay's paper, about using parts of Nine Chapters in teacher training. He has three goals for working with this problem (which may be a problem, as students often focus on at most one). Liu Hui gave two justifications for multiplication of fractions, the second of which could probably be used in teaching, in my opinion. The use of a semantic embedding (word problem) is a resource, but also a worry as it can decrease the generality. Renaud argued convincingly that this example can be useful for discussion with teacher students, even though (according to him) perhaps not useful for direct work with children. I am a big fan of Renaud's work and am happy that he is now working in teacher education, as it means that his work - which is as always historically solid - now includes sharp analyses of what might be the use of the historical examples in teacher education.

Next, Regina Moeller and Peter Collignon talked on their paper which concerns the work on infinity with children. The concept has a long history, while teacher education students tend to have only the epsilon-delta based concept. (Of course, this is context-dependent - most Norwegian teacher education students would look at you wide-eyed if you mention epsilon or delta.) In their opinion, teachers need to know other conceptions that may be closer to the steps children go through. They look especially at Hilbert and Cantor - including the hotel of Hilbert, of course. The work can make students more aware that there exists different conceptions that they have not learned and to be more open-minded.

Then, Rui Candeias presented "Mathematics in the initial pre-service education of primary school teachers in Portugal: analysis of Gabriel Gonçalves' proposal for the concept of unit and its application in solving problems with decimals". This is part of a larger research project comparing different textbooks for teacher training. He presented in detail the steps adviced by Gonçalves. (Which makes me think that it could be a good idea to study historical teacher guides in Norway to point out to students the evolution of the field of mathematics education when it comes to concrete advice given to students.)

Maria Sanz gave the last presentation of the day; "Classification and Resolution of the Descriptive Historical Fraction Problems". She proposes a classification of the problems based on which methods can be used to solve them. It is unclear to me what this classification brings to the table - other aspects (known/unknown context, size of numbers, distractors included and so on) could be as important for practical use in classrooms. In the discussion, she was asked about connection to the mathematics education research on the same issues. It was also mentioned that in some countries they are "banned" from textbooks, while in others they are obviously not banned.

Some comments that turned up:

• What can these examples bring to teacher training? The common denominator seems to be that they are in a preliminary phase - but they can work to show students that problems are not something to be solved but rather something to be analysed to decide whether and how to use in their teaching.

• Could students solve and classify problems in the way of Maria themselves? Would that be more useful than being presented with a classification?

• History can be a good tool to connect algebra without the symbolism with algebra with symbols.

• A book by Brian Clegg on infinity was recommended.

I do think that a closer collaboration between maths ed people and history of mathematics people is called for. In some cases, we see discussions on how historical sources can be used in teaching of subjects where there exist a huge amount of literature in the field of mathematics education, but where this work is disregarded. This is every bit as bad as the huge number of papers in mathematics education that completely disregards the history of the subjects that they want to discuss.

This was the end of the third day. Well, not quite. I was lucky enough to take part on the "lavender walking tour", which was a walking tour of Dublin LGBT History. We saw the Oscar Wilde monument, the Parliament, Dublin Castle, the national library and many other places of importance. We got detailed and enthusiastic information on the liberation fight, including the disgraceful attitudes of the government when activists tried to save lives by distributing condoms (which were illegal at the time). Today, Ireland has moved in a liberal direction and is one of the few countries where gay marriage has been decided in a referendum - although relgious fundamentalists still have a role. The tour ended at a gay pub where we got to continue the discussion over some Irish refreshments.

CERME is the second big international mathematics education conference in less than a year with something concerning LGBT issues on or near the programme. I do hope that this is an emerging trend.

The rest of the morning session was spent on participants sharing informaion on important publications that the others should know of. I have notes of this somewhere, but we were also promised an email later summarizing this.

The second session started off with Renaud Chorlay's paper, about using parts of Nine Chapters in teacher training. He has three goals for working with this problem (which may be a problem, as students often focus on at most one). Liu Hui gave two justifications for multiplication of fractions, the second of which could probably be used in teaching, in my opinion. The use of a semantic embedding (word problem) is a resource, but also a worry as it can decrease the generality. Renaud argued convincingly that this example can be useful for discussion with teacher students, even though (according to him) perhaps not useful for direct work with children. I am a big fan of Renaud's work and am happy that he is now working in teacher education, as it means that his work - which is as always historically solid - now includes sharp analyses of what might be the use of the historical examples in teacher education.

Next, Regina Moeller and Peter Collignon talked on their paper which concerns the work on infinity with children. The concept has a long history, while teacher education students tend to have only the epsilon-delta based concept. (Of course, this is context-dependent - most Norwegian teacher education students would look at you wide-eyed if you mention epsilon or delta.) In their opinion, teachers need to know other conceptions that may be closer to the steps children go through. They look especially at Hilbert and Cantor - including the hotel of Hilbert, of course. The work can make students more aware that there exists different conceptions that they have not learned and to be more open-minded.

Then, Rui Candeias presented "Mathematics in the initial pre-service education of primary school teachers in Portugal: analysis of Gabriel Gonçalves' proposal for the concept of unit and its application in solving problems with decimals". This is part of a larger research project comparing different textbooks for teacher training. He presented in detail the steps adviced by Gonçalves. (Which makes me think that it could be a good idea to study historical teacher guides in Norway to point out to students the evolution of the field of mathematics education when it comes to concrete advice given to students.)

Maria Sanz gave the last presentation of the day; "Classification and Resolution of the Descriptive Historical Fraction Problems". She proposes a classification of the problems based on which methods can be used to solve them. It is unclear to me what this classification brings to the table - other aspects (known/unknown context, size of numbers, distractors included and so on) could be as important for practical use in classrooms. In the discussion, she was asked about connection to the mathematics education research on the same issues. It was also mentioned that in some countries they are "banned" from textbooks, while in others they are obviously not banned.

Some comments that turned up:

• What can these examples bring to teacher training? The common denominator seems to be that they are in a preliminary phase - but they can work to show students that problems are not something to be solved but rather something to be analysed to decide whether and how to use in their teaching.

• Could students solve and classify problems in the way of Maria themselves? Would that be more useful than being presented with a classification?

• History can be a good tool to connect algebra without the symbolism with algebra with symbols.

• A book by Brian Clegg on infinity was recommended.

I do think that a closer collaboration between maths ed people and history of mathematics people is called for. In some cases, we see discussions on how historical sources can be used in teaching of subjects where there exist a huge amount of literature in the field of mathematics education, but where this work is disregarded. This is every bit as bad as the huge number of papers in mathematics education that completely disregards the history of the subjects that they want to discuss.

This was the end of the third day. Well, not quite. I was lucky enough to take part on the "lavender walking tour", which was a walking tour of Dublin LGBT History. We saw the Oscar Wilde monument, the Parliament, Dublin Castle, the national library and many other places of importance. We got detailed and enthusiastic information on the liberation fight, including the disgraceful attitudes of the government when activists tried to save lives by distributing condoms (which were illegal at the time). Today, Ireland has moved in a liberal direction and is one of the few countries where gay marriage has been decided in a referendum - although relgious fundamentalists still have a role. The tour ended at a gay pub where we got to continue the discussion over some Irish refreshments.

CERME is the second big international mathematics education conference in less than a year with something concerning LGBT issues on or near the programme. I do hope that this is an emerging trend.

### CERME 10 Day 2

The second day of CERME 10 started where the first one ended - with a TWG (topic working group session). Please excuse my extremely short descriptions of the papers - the authors were just given ten minutes to remind participants of their papers as a basis for discussion, and I do not have the time to go back to the papers to give more detailed accounts. First, Kathy Clark talked on the very interesting TRIUMPHS project, a big design research project based on original sources. At this time, the project reports on a pilot study in the first year. I notice an inteesting focus on meta-discursive rules and on views of mathematics. They use Törner's aspects and his instrument - but the number of students included in the analysis at this point was small. It will be interesting to follow the project in years to come!

Rainer Kaenders talked about "Historical Methods for Drawing Anaglyphs". In this project, students draw 3d drawings using historical methods. The point was not to learn the methods, but to understand the mathematical principles in order to be able to do the drawings. Again, this was an interesting project giving ideas for working on geometry in new ways. Kaenders had used this in extracurricular activities with students, for which it seemed well suited.

Thirdly, Rita (Areti) Panaoura talked about the paper "Inquiry-based teaching approach in mathematics by using history of mathematics - a case study". In Cyprus, which has a centralized school system, history of mathematics is seen as a tool to investigate the mathematical concepts. She reiterated Siu's reasons that teachers hesitate in using HM. She gave examples of teachers' attitudes and knowledge. Teachers could not connect the HM and the inquiry-based teaching approach which was also mandated. Understanding what teachers need in order to include history of mathematics in their teaching, is very important in order to implement HM in teaching. As such, I find this paper interesting. A participant questioned whether the use of Egyptian multiplication is helpful. I think that depends on the goal. According to Rita, there are no teacher guide saying what the point is, therefore it is difficult to see if the example is well-chosen or not - and difficult for teachers to use it in a meaningful way. Thus, this paper shows the problem of giving teachers resources without giving them the reasoning behond them.

The fourth presentation was of the paper "Teaching kinematics using mathematics history" (Alfredo Martinez). This is a paper concerning a reconstruction of a method of measuring time which Galileo may have used. Students were able to measure time using a rhythm, thereby being able to recreate Galileo's results. It is a bit unclear to me if this really fits in the history of mathematics group or would rather fit in a history of science group (at some unspecified conference), though.

Then there was a group discussion and sharing. Some points:

• It is a shame that the scaffolding was not there for the teachers or students in the Egyptian multiplication example to see the connection to our algorithms.

• What "scaffolding" is needed? Notes to teachers and workshops are parts of the project Kathy talked about. Also, use of history of mathematics should also be included in teacher training.

• A historical document is not necessary, historical problems (without giving the actual source) worked on with students are also useful. But what difference does the source make? (Of course, many authors have written extensively on this.)

• Can all topics be taught using history? Are there too big obstacles in some cases?

• Can we do good history and good mathematics at the same time? (My answer would be that we are never "perfect" in the classroom, teaching is always full of compromises. So there is a question of what is good enough.)

• The geographical and cultural distance is important. Is Greek mathematics more motivating for pupils in Greece?

• How much of the original context must a teacher understand?

• Choice of examples: should they be "exemplary" or could we have "fringe" examples? Papers that are most interesting from a historical point of view, may not be the best ones from an educational point of view.

• How do teachers come to have materials that they can use? And how do they (learn to) orchestrate the classroom experience?

Then, there was time for another plenary: Lieven Verschaffel on "Young children's early mathematical competencies: analysis and stimulation". Researchers today believe that children have a "starter kit", object tracking system and approximate number system (ANS). Gradually, there is a development towards a symbolic representation. There are significant correlations between numberical magnitude understanding and early mathematical achievement.

The ordinality aspect of number is neglected in the cognitive neuroscientific work. But research suggest stronger correlation/predictability between ordinal aspect and mathematical skill. For instance Hyman Bass argues for developing number based on measurement. Basing the number concept on cardinality means that later developments, such as fractions, will be more difficult.

There is also more interest in children's understanding of basic arithmetic concepts and relations. There is little research on the consequences of this for later mathematics learning. Nunes et al (2015) is an exception.

Other researchers have looked at pattern and structures. Mulligan et al (2015) is the most comprehensive, looking at children's awareness of mathematical patten and structure (AMPS). A related intervention study shows no improvement in general mathematics achievement.

The research studies mentioned so far look at children's abilities, not their dispositions. (I.e. Asking children to look for a pattern, not measuring whether they see the pattern without a prompt.)

SFOR (spontaneous focusing on quantitative relations) - individual differences, and has a direct effect on mathematical results at end of elementary school. Several other such FLAs (four letter acronyms) were also mentioned- we do not know much about their development and interrelationship.

Then he went on to talk on domain-general (not domain-specific) abilities, such as attention, flexibility, inhibition, working memory etc. There is evidence of these abilities' importance - to a greater degree than domain-specific abilities.

Other aspects mentioned in the talk was the role of parents and early caregivers, preschool to elementary school transition, and the professional development of caregivers and teachers. He concluded by listing a whole range of important aspects which need to be further developed in years to come.

For the third session of the TWG, the first person was Luciane de Fatima Bertin, presenting the paper "Arithmetical problems in primary school: ideas that circulated in São Paulo/Brazil in the end of the 19th century". She highlighted the notion of appropriation and the notion of purpose. The word "problem" is undefined, but seems to be synonymous with "exercise", so it has no connection to the modern understanding connected to "problem solving". There was no discussion in the journals analysed on the use of problems in teaching.

Asger Senbergs talked on his article "Mathematics at the Royal Danish Military Academy of 1830". His article is based on his Master thesis. The research was based on his curiosity about why mathematics became the main topic when Denmark created a military academy. The value of mathematics as a goal in itself was prominent - not just as a tool for action on the field.

Ildar Safuanov's paper "The role of genetic approach and history of mathematics in works of Russian mathematics educators (1850-1950)" was up next. The paper details early Russians ideas on the genetic approach. The genetic approach was connected to the idea that pupils should not just witness but also create mathematics, and was included in the guidelines for mathematics teaching after the 1917 revolution.

Tanja Hamman talked about ""Sickened by set theory?" - About New Math in German primary schools". The title is from Der Spiegel from March 1974 ("Macht Mengenlehre krank?"). She has looked at textbooks and teacher guides from West Germany to see whether the main ideas were present in the textbooks. Traditional education did influence the implementation, it is not possible to create a clean slate when dealing with teaching.

Then, it was time for group discussions. Here are some points from the discussion:

• Do we see history of mathematics education mainly as part of general history, part of mathematics education or as part of history of mathematics?

• It is interesting to look at historical cases to investigate conditions for ("successful") implementation of educational reforms. (Which is part of the value of history of mathematics education for teacher education?)

• How does it matter that a subject has a history? Does it provide a knowledge base to look at your subject?

• Who decides what are popular and unpopular subjects? What are the forces behind which topics are in vogue at a given time?

• When you know more about the past, you have more tools to deal with the present.

• New Math - was it never, anywhere, implemented as intended, with the intended outcomes?

Thus ended the second day of CERME. Although most participants probably continued their discussions into the early hours of the next day, I returned to my hotel room to prepare for the university board meeting next week. It is necessary to mention this, as some colleagues have developed an unhealthy interest in my nightlife while in Dublin... :-)

Rainer Kaenders talked about "Historical Methods for Drawing Anaglyphs". In this project, students draw 3d drawings using historical methods. The point was not to learn the methods, but to understand the mathematical principles in order to be able to do the drawings. Again, this was an interesting project giving ideas for working on geometry in new ways. Kaenders had used this in extracurricular activities with students, for which it seemed well suited.

Thirdly, Rita (Areti) Panaoura talked about the paper "Inquiry-based teaching approach in mathematics by using history of mathematics - a case study". In Cyprus, which has a centralized school system, history of mathematics is seen as a tool to investigate the mathematical concepts. She reiterated Siu's reasons that teachers hesitate in using HM. She gave examples of teachers' attitudes and knowledge. Teachers could not connect the HM and the inquiry-based teaching approach which was also mandated. Understanding what teachers need in order to include history of mathematics in their teaching, is very important in order to implement HM in teaching. As such, I find this paper interesting. A participant questioned whether the use of Egyptian multiplication is helpful. I think that depends on the goal. According to Rita, there are no teacher guide saying what the point is, therefore it is difficult to see if the example is well-chosen or not - and difficult for teachers to use it in a meaningful way. Thus, this paper shows the problem of giving teachers resources without giving them the reasoning behond them.

The fourth presentation was of the paper "Teaching kinematics using mathematics history" (Alfredo Martinez). This is a paper concerning a reconstruction of a method of measuring time which Galileo may have used. Students were able to measure time using a rhythm, thereby being able to recreate Galileo's results. It is a bit unclear to me if this really fits in the history of mathematics group or would rather fit in a history of science group (at some unspecified conference), though.

Then there was a group discussion and sharing. Some points:

• It is a shame that the scaffolding was not there for the teachers or students in the Egyptian multiplication example to see the connection to our algorithms.

• What "scaffolding" is needed? Notes to teachers and workshops are parts of the project Kathy talked about. Also, use of history of mathematics should also be included in teacher training.

• A historical document is not necessary, historical problems (without giving the actual source) worked on with students are also useful. But what difference does the source make? (Of course, many authors have written extensively on this.)

• Can all topics be taught using history? Are there too big obstacles in some cases?

• Can we do good history and good mathematics at the same time? (My answer would be that we are never "perfect" in the classroom, teaching is always full of compromises. So there is a question of what is good enough.)

• The geographical and cultural distance is important. Is Greek mathematics more motivating for pupils in Greece?

• How much of the original context must a teacher understand?

• Choice of examples: should they be "exemplary" or could we have "fringe" examples? Papers that are most interesting from a historical point of view, may not be the best ones from an educational point of view.

• How do teachers come to have materials that they can use? And how do they (learn to) orchestrate the classroom experience?

Then, there was time for another plenary: Lieven Verschaffel on "Young children's early mathematical competencies: analysis and stimulation". Researchers today believe that children have a "starter kit", object tracking system and approximate number system (ANS). Gradually, there is a development towards a symbolic representation. There are significant correlations between numberical magnitude understanding and early mathematical achievement.

The ordinality aspect of number is neglected in the cognitive neuroscientific work. But research suggest stronger correlation/predictability between ordinal aspect and mathematical skill. For instance Hyman Bass argues for developing number based on measurement. Basing the number concept on cardinality means that later developments, such as fractions, will be more difficult.

There is also more interest in children's understanding of basic arithmetic concepts and relations. There is little research on the consequences of this for later mathematics learning. Nunes et al (2015) is an exception.

Other researchers have looked at pattern and structures. Mulligan et al (2015) is the most comprehensive, looking at children's awareness of mathematical patten and structure (AMPS). A related intervention study shows no improvement in general mathematics achievement.

The research studies mentioned so far look at children's abilities, not their dispositions. (I.e. Asking children to look for a pattern, not measuring whether they see the pattern without a prompt.)

SFOR (spontaneous focusing on quantitative relations) - individual differences, and has a direct effect on mathematical results at end of elementary school. Several other such FLAs (four letter acronyms) were also mentioned- we do not know much about their development and interrelationship.

Then he went on to talk on domain-general (not domain-specific) abilities, such as attention, flexibility, inhibition, working memory etc. There is evidence of these abilities' importance - to a greater degree than domain-specific abilities.

Other aspects mentioned in the talk was the role of parents and early caregivers, preschool to elementary school transition, and the professional development of caregivers and teachers. He concluded by listing a whole range of important aspects which need to be further developed in years to come.

For the third session of the TWG, the first person was Luciane de Fatima Bertin, presenting the paper "Arithmetical problems in primary school: ideas that circulated in São Paulo/Brazil in the end of the 19th century". She highlighted the notion of appropriation and the notion of purpose. The word "problem" is undefined, but seems to be synonymous with "exercise", so it has no connection to the modern understanding connected to "problem solving". There was no discussion in the journals analysed on the use of problems in teaching.

Asger Senbergs talked on his article "Mathematics at the Royal Danish Military Academy of 1830". His article is based on his Master thesis. The research was based on his curiosity about why mathematics became the main topic when Denmark created a military academy. The value of mathematics as a goal in itself was prominent - not just as a tool for action on the field.

Ildar Safuanov's paper "The role of genetic approach and history of mathematics in works of Russian mathematics educators (1850-1950)" was up next. The paper details early Russians ideas on the genetic approach. The genetic approach was connected to the idea that pupils should not just witness but also create mathematics, and was included in the guidelines for mathematics teaching after the 1917 revolution.

Tanja Hamman talked about ""Sickened by set theory?" - About New Math in German primary schools". The title is from Der Spiegel from March 1974 ("Macht Mengenlehre krank?"). She has looked at textbooks and teacher guides from West Germany to see whether the main ideas were present in the textbooks. Traditional education did influence the implementation, it is not possible to create a clean slate when dealing with teaching.

Then, it was time for group discussions. Here are some points from the discussion:

• Do we see history of mathematics education mainly as part of general history, part of mathematics education or as part of history of mathematics?

• It is interesting to look at historical cases to investigate conditions for ("successful") implementation of educational reforms. (Which is part of the value of history of mathematics education for teacher education?)

• How does it matter that a subject has a history? Does it provide a knowledge base to look at your subject?

• Who decides what are popular and unpopular subjects? What are the forces behind which topics are in vogue at a given time?

• When you know more about the past, you have more tools to deal with the present.

• New Math - was it never, anywhere, implemented as intended, with the intended outcomes?

Thus ended the second day of CERME. Although most participants probably continued their discussions into the early hours of the next day, I returned to my hotel room to prepare for the university board meeting next week. It is necessary to mention this, as some colleagues have developed an unhealthy interest in my nightlife while in Dublin... :-)

### CERME 10 Day 1

CERME 10 was my first CERME, taking place at Croke Park in Dublin. With a capacity of more than 80000, the stadium had plenty of space for the 800 participants. The opening ceremony included short adresses from various dignitaries (of course, including the leaders of the groups actually doing the work of preparing the conference). For instance, we learned how Hamilton got a key insight (concerning quaternions) by the Royal Canal (which passes just outside the stadium). In addition, there was some beautiful Irish music, of course.

The first plenary lecturer was Elena Nordi. Her title was "From Advanced Mathematical Thinking to University Mathematics Education: A story of emancipation and enrichment". She opened with an image from the Coen film "A serious man" - pointing out the popular conception of what university mathematics teaching look like: a professor filling a blackboard. University mathematics teaching today is much more varied than that - the demands on the teachers are quite varied. In her talk, she wanted to give an overview of the CERME work on university mathematics since the first CERME, in a way she called "impressionistic" and personal.

She pointed out that the field is quite young, for instance important papers such as Yackel & Cobb ("Sociomathematical Norms, Argumentation, and Autonomy in Mathematics") arrived in 1996. She pointed out that research on university mathematics education has in this period been moving away from being a "hobby" done by mathematics professors without a connection to the general mathematics education research. However, she also mentioned how her field differs from other fields in that there is a less clear distinction between teacher and researcher - the university lecturers are also often researchers. However, she did not fully go into the implications of this.

Her (rapid) talk discussed a huge number of papers from different CERME conferences, pointing out developments. For me, who is not doing research on or teach advanced mathematics, the talk was so full of unfamiliar names and developments that I will not attempt to summarize here. Sadly, the speed of her talk also excluded some participants - not all of which speak English on a daily basis. (In fact, 50 countries were represented in the conference.)

The main feature of the CERMEs are the TWGs (Topic Working Groups), which one is supposed to stay loyal to throughout the conference and which takes up most of the conference time. The first session of the TWG took place at the end of the first day. Renaud Chorlay gave a quick introduction to the working of the group.

After we had all introduced ourselves, we were ready for the first paper. That was Elizabeth de Freitas' paper called "A course in the philosophy of mathematics for future high school mathematics teachers". She talked about a course she has given for three years ar Adelphi University in New York, which was actually an alternative to a history of mathematics course. One important aspect is the philosophical paper students have to write - where they have to take a stand and defend a position on one central question from the philosophy of mathematics. Maurice O'Reilly presented his paper on "Multiple perspectives on working with original mathematical sources from the Edward Worth Library, Dublin". He stressed the scaffolding of students' work - helping and encouraging the students reading unfamiliar sources (to them) in foreign languages. These were short presentations as we had all read the papers in advance. Then we started discussing the expected and actual impact of the teaching projects. The discussion centered on whether there are ways of collecting data and convince others of the potential value of such approaches. Here are some points:

• The researchers had some data that could have been analysed to shed light on the potential. However, as some of the assumed values concerns students' long-term approach to and image of mathematics, maybe longitudinal studies are neccessary?

• In some cases, The visceral reactions of the students are powerful but not measurable? Some participants in the group recognized their own reaction in students' reaction.

• The role of the teacher seemed to be different here than in "usual" teaching. The projects can give ideas on how to teach to avoid the students' imitation.

• There is a pull to prove effectiveness, but also a danger of being drawn into the metrics. We need more research that convinces others than ourselves, but we also need development and ideas that can later be explored more. So papers such as these are valuable although they may not convince others.

That was already the end of the first day at CERME. Three more blog posts will follow.

The first plenary lecturer was Elena Nordi. Her title was "From Advanced Mathematical Thinking to University Mathematics Education: A story of emancipation and enrichment". She opened with an image from the Coen film "A serious man" - pointing out the popular conception of what university mathematics teaching look like: a professor filling a blackboard. University mathematics teaching today is much more varied than that - the demands on the teachers are quite varied. In her talk, she wanted to give an overview of the CERME work on university mathematics since the first CERME, in a way she called "impressionistic" and personal.

She pointed out that the field is quite young, for instance important papers such as Yackel & Cobb ("Sociomathematical Norms, Argumentation, and Autonomy in Mathematics") arrived in 1996. She pointed out that research on university mathematics education has in this period been moving away from being a "hobby" done by mathematics professors without a connection to the general mathematics education research. However, she also mentioned how her field differs from other fields in that there is a less clear distinction between teacher and researcher - the university lecturers are also often researchers. However, she did not fully go into the implications of this.

Her (rapid) talk discussed a huge number of papers from different CERME conferences, pointing out developments. For me, who is not doing research on or teach advanced mathematics, the talk was so full of unfamiliar names and developments that I will not attempt to summarize here. Sadly, the speed of her talk also excluded some participants - not all of which speak English on a daily basis. (In fact, 50 countries were represented in the conference.)

The main feature of the CERMEs are the TWGs (Topic Working Groups), which one is supposed to stay loyal to throughout the conference and which takes up most of the conference time. The first session of the TWG took place at the end of the first day. Renaud Chorlay gave a quick introduction to the working of the group.

After we had all introduced ourselves, we were ready for the first paper. That was Elizabeth de Freitas' paper called "A course in the philosophy of mathematics for future high school mathematics teachers". She talked about a course she has given for three years ar Adelphi University in New York, which was actually an alternative to a history of mathematics course. One important aspect is the philosophical paper students have to write - where they have to take a stand and defend a position on one central question from the philosophy of mathematics. Maurice O'Reilly presented his paper on "Multiple perspectives on working with original mathematical sources from the Edward Worth Library, Dublin". He stressed the scaffolding of students' work - helping and encouraging the students reading unfamiliar sources (to them) in foreign languages. These were short presentations as we had all read the papers in advance. Then we started discussing the expected and actual impact of the teaching projects. The discussion centered on whether there are ways of collecting data and convince others of the potential value of such approaches. Here are some points:

• The researchers had some data that could have been analysed to shed light on the potential. However, as some of the assumed values concerns students' long-term approach to and image of mathematics, maybe longitudinal studies are neccessary?

• In some cases, The visceral reactions of the students are powerful but not measurable? Some participants in the group recognized their own reaction in students' reaction.

• The role of the teacher seemed to be different here than in "usual" teaching. The projects can give ideas on how to teach to avoid the students' imitation.

• There is a pull to prove effectiveness, but also a danger of being drawn into the metrics. We need more research that convinces others than ourselves, but we also need development and ideas that can later be explored more. So papers such as these are valuable although they may not convince others.

That was already the end of the first day at CERME. Three more blog posts will follow.

## Sunday, July 31, 2016

### ICME13 Day 7 #icme13

The 7th day of the
ICME13 conference was short. First a plenary panel on "transitions in
mathematics education". Panellists were Ghislaine Gueudet, Marianna Bosch, Andrea diSessa, Oh
Nam Kwon, and Lieven Verschaffel. The panel's theme - transitions - has many
interpretations, including transitions between themes (arithmetic to algebra),
transition to formal proof, transitions between school levels, transitions
between contexts, for instance language contexts, transitions between
curricula. In this panel, they focused on transitions as conceptual change and
on transitions of people as they move between social groups. To look at this,
they had epistemological, cognitive and socio-cultural perspectives. The result
of the work is a survey: "Transitions in

Mathematics
Education".

diSessa started by
talking about "Continuity versus Discontinuity in Learning Difficult
Concepts". Bachelard talked about epistemological obstacles, while others
talks about "pieces and processes" instead, arguing for more
continuous change. Misconceptions belong to the left side of this divide. The
answer to the discussion may be found in microgenetic perspective (J. Wagner) -
some research find incremental learning across many contexts, for instance when
trying to learn the law of large numbers. diSessa thinks the continuist side
will win, which will mean we will look at resources more than obstacles or
misconceptions. (Personally, I'm not sure I believe any side is 100 percent
right. Why can't both be right in different contexts? That is, that some
development is gradual or stepwise, while other develop is mora abrupt and
discontinuous?)

Kwon talked on
"Double discontinuity between Secondary School Mathematics and University
Mathematics". The double disconuinity concerns first moving from secondary
school mathematics to university mathematics and then back (as a teacher). She
then discussed Shulman, Ball and then Heinz et al ("School Related Content
Knowledge") and Thompson (Mathematical Meaning for Teaching Secondary
Mathematics).

While listening to
her talk, I wondered if it could it argued that the Norwegian system, in which
teachers for (lower) secondary schools are educated in teacher education
programmes in which mathematics and mathematics education courses are merged,
and where the mathematics is not "university mathematics" as such,
avoids such double discontuinities? In fact, the suggestions she ended her talk
with seemed to align almost exactly with the Norwegian model. But it must be
stressed that for upper secondary schools, the problem is present also in
Norway.

Borsch talked on
"Transitions between teaching institutions". Individual trajectories
are shaped by the institutions they enter, their activities and settings. The
transitions between primary and secondary and between secondary and tertiary
education, are much studied. There are many different levels of analysis which
are present in the literature. The main differences between primary and
secondary are pedagogy (interaction, autonomy, transmissionist) and discipline
(specialist teacher, more division between subjects). Many proposals for
smoothing the transition are found in the literature (and in the report).

The differences
between secondary and tertary education are similar to differences between
primary and secondary. In addition, teachers are researchers. There is more
research on particular topics (i.e. Algebra) and there are more proposals.
"Bridging courses" are discussed, but university mathematics content
is rarely questioned. It is not clear whether all perspectives are equally
represented in the literature.

Verschaffel talked
on "Transitions between in- and out-of-school mathematics". Much
learning and use og mathematics take place outside school. While this has
previously been "romanticized", currently, more interest is in what
happens at the boundaries. Of course, part of this is research on the
didactical contract when playing the game of word problems, in which
out-of-school experiences are often unwelcome. There are efforts to facilitate
and exploit transitions, for instance RME, "funds of knowledge",
Greer. Then the panel ended with a discussion, based on question from an
internet forum, at which point I stopped making notes.

Then, there was the
closing ceremony. Secretary general of ICMI, Abraham Arcavi, had a warm and
pleasant speech closing the conference. Then there were many speeches of thanks
to different contributors. There was a presentation of the next venue for ICME
(Shanghai ). And finally, a wonderful musical number.

What did I get out
of ICME13? (And why will I go to ICME14 in Shanghai in 2020?)

As I noted earlier,
there are many possible outcomes from such a conference, and I can summarize
some of them, in no particular order of importance:

- Meeting new (or old) people with which future collaborations are possible. At this conference, I can think of at least four people I didn't know before, with whom there can possibly be some sort of collaboration some time in future.
- Getting ideas for future research projects: During the conference, I wrote a list of nine research and development projects that I would like to do. Not all are new ideas, but many have been expanded while I've been here. And some are directly adapted from talks here. I only hope I can follow up on some of them when I get back home... Hugh Burkhardt's talk on design research was very inspiring, and I want to do something in that direction (more systematically than I've done before).
- Getting ideas for my own teaching (which can of course also turn into research projects): Marjolein Kool's project on making students creating non-routine mathematics problems, Megan Shaugnessy's and others' ideas on simulations of classroom situations (some of these concerning the notion of "noticing").
- Getting an overview of a field which will make it easier for me to read more about it later: For instance, I hope it will be easier for me to face Brousseau now that I've had a tiny introduction and some context.
- Listening to lectures and realizing that I do actually know something: it would be impolite to point out which talks contributed to this realization. But perhaps it is a good sign that for every ICME I go to, I think "been there, done that" a bit more frequently. This is not to critizise ICME (too much), of course not everything can be new to everyone.
- Socialize with new or old colleagues: of course, the tendency to cluster based on nationalities may be seen as a problem (and meetings like the LGBT get-together may counteract that a little), but it is a reality that many countries have few meeting-places, so that socializing with colleagues from the same country does make some sense. There's been a lot of that here.

The main problem
with ICME is of course its size - it's complete lack of intimacy. The trick is
of course to find a home in a TSG, but still there will be situations when
you're all alone and see hundreds of strangers walk past - which is a challenge
for smalltalk-challenged people like me. But still, there is every chance that
I'll go to ICME again in four years' time.

But before that,
we'll arrange our own conference (ESU8) in Oslo in 2018. That will be fun - and
intimate. And I hope I'll go to CERME in February, 2017.

## Saturday, July 30, 2016

### ICME13 Day 6 #icme13

On day 6 (Saturday)
of ICME13, the plenary lecture was by Deborah Loewenberg Ball, whose work I
have referred to a lot in my own articles lately. She talked on
"Uncovering the special mathematical practices of teaching". As
usual, I was eager to get signs of her definition of "mathematical"
(see yesterday's blog). She organised her story in three parts: a journey,
getting lost, finding the way. At some point, many years ago, there was a
movement from wondering what mathematics teachers need in teaching to what they
use. The "getting lost" part was building tools to
"measure" teacher knowledge. On the positive side, ways of studying
outcomes of teacher education and professional development were developed, but
on the negative side we fell back from understanding practice to looking at
knowledge. Also, it may have contributed to separating aspects of teaching, for
instance equity. So in a way, we got microscope images of many aspects, but not
understanding it from the inside. She argued that we have to work on the
"mathematical work of teaching" mathematics. (Still not defining
"mathematical" or "mathematics", but at least referring to
previous talks this week.)

The work of teaching
is taking responsibility for maximising the quality of the interactions in a
classroom in ways that maximize the probability that learners learn. (This fits
quite well with Biesta, doesn't it?) Using "work" is to focus on what
the teacher DOES, not on the curriculum, what students are doing etc.
"Mathematical" listening, speaking, interacting, acting... are part
of the work of teaching. A fundamental part of the work is attuning to other
people and being oriented to others' ideas and ways of thinking and being.

She then showed us
examples from a grade 5 classroom with 29 pupils in a low-income community.
Showing us different sequences, she asked us to pinpoint what the teacher was
doing. (What I saw: Getting people to come up. Classroom management.
Monitoring. Steering discussion (sociocultural norms), respond to question,
saying " it doesn't have to be right") She pointed out how the
teacher had walked around the classroom, reading nearly 30 student answers,
choosing which to look into) Then Ball mentioned concepts from Cohen and
Lohan(?) on assigning mathematical competence, including developing competence.
The work of the teacher is partly noticing everything that is not the wrong
answer. Translation from learning goals to meaningful language for children.
The work of teaching is discursively intensive work: talking while monitoring
if anyone understands what you're saying and understanding the answers.

I liked the talk
very much. As I saw it, it was a step away from "the ball", which has
worked to privilege some kinds of knowledge and hide away others, not to
mention skills. The criticism of testing for MKT is also welcome. What comes
instead, though, is not clear. I'm afraid "the ball" is too simple
and pretty to be replaced by anything messier or more filled with doubt and
judgements...

The first invited
lecture I chose today was Ansie Harding's talk entitled "The role of
storytelling in teaching mathematics". Her talk was geared towards tertary
education, which is interesting, but I also hoped to learn things for primary
education. Of course, storytelling is also an important way of including
history of mathematics in primary school (re discussion group at previous
ICME), so also from that point of view it should be interesting, as well as for
my cooperation with the L1 teacher of my students is next academic year.

Harding teaches
400-500 students at a time at the University of Pretoria. Here, she first gave
the story of storytelling, from about 50.000 BC. (Already at that point someone
challenged the veracity of the story, which raises the question of the role of truth
in storytelling - which is also part of the discussion concerning history of
mathematics in teaching; which role does myths have in teaching?) In her story,
she included movies, TV and the internet (not including games) - but the
examples she used did not include movies - she stressed the value of low-tech
storytelling. Instead, her first story example was the story of the length of a
year, leap years and so on. (It is a bit difficult to see the difference
between her story and normal lecturing, so maybe normal lecturing should be
defined as well.) Features of storytelling: are essy to remember, are
compelling, are for all ages, embed value, travel, fuel conversation. A story
often has a character, ambition, problem, outcome as well as an emotional
connection.

The value of
storytelling in education is to entertain, inspire and educate. With tertary
students, stories can be used as an intro (but is a bit like having the dessert
before the main course), as a "by the way", as a commercial break, as
a reward (which is how she uses it). You need to get everyone involved and
create a "class feeling".

Ingredients: maths
connection, human element, tale (start, flow, end), humour. Her second example
was "How mathematics burned down the houses of parliament".

With engineering
students, she uses the last ten minutes every week (which contains 200 minutes
a week) for story telling. She showed a list of examples, much of which the
usual history of mathematics "stories". Telling stories takes some
effort (you must make it your own), it is a culture that you foster, there are
unexpected rewards.

5-10 students leave
before the stories all the time. She asked students for comments, and 30
students responded. Categories: emotional impact, reward, motivation, subject
impact, appreciation (more approachable), bigger picture (more to maths than
this course).

She ended her talk
with two stories, one on De Moivre and one called "The story of one, five,
seven" on van Gogh's sunflowers.

For me, this talk
was an inspiration to dare telling stories more in my own teaching, and to put
stress on them, not to tell them apologetically and quickly. However,
"making it my own" will probably mean making sure they are relevant
to the mathematics and often the history of mathematics that we are working on.

After a break, I was
back to TSG47, where there were three talks:

Caroline Lajoie
talked on "Learning to act in-the-moment: prospective elementary teachers'
roleplaying on numbers". The observation was made in one lesson in teacher
education. Her key concept was knowing to act in-the-moment. (See Mason and Spence
1999.) in this context students take
role of the teacher and 1-3...students, in front of the whole class. Role play
involves introduction time, preparation time (everybody prepares all roles),
play time and discussion time. She gave examples on how students, even though
prepared, need to improvise. One took a risk in a situation where she was not
certain of the answer, while another chose to switch to an explaining mode when
he saw that he didn't know what would happen. This came up in the discussion,
but that was because students were willing to discuss and "criticize"
each other.

Pere Ivars on
"The role of writing narratives in developing pre-service primary teachers
noticing". "Narratives" in this talk refers to stories about
classroom interactions. The hypothesis is that writing narratives will help
noticing. Students wrote two narratives, with feedback after the first
narrative (and some detail of the feedback given were provided here). The
development from the first to the second narrative was more evidence of
students understanding in more detail (it is important also to notice that
students wrote the first narrative based on observation, the other based on
what they themselves taught).

Lara Dick on
"Noticing and deciding the "next steps" for teaching: "a
cross-university study with elementary pre-service teachers." Noticing is
a skill teachers need and that they need to develop. Much work have been done
with using video to work on noticing, but little on use of student work (such
as the previous one). The current project was a three-hour lesson on
multiplication. The students saw student examples and analyzed one each and
made posters. The students were asked to attend to the mathematics, interpret
what they see and make an instructional decision - based on looking at all the
posters. Four major themes: gravitation towards traditional teaching ideas,
vague decisions, desire for written number sentences, focus on strategy
progression. (This seems like a lesson it would be useful to try to copy with
my students as well.)

(About TSG
proceedings: there is a format suggestion from the ICME , but it is unclear how
much flexibility there is. The suggestion was few but extended papers, while
the TSG organizers would prefer shorter but more papers.)

These talks remind
me that key concepts to meet for new teacher students are noticing and building
on what students know instead of figuring out what they don't know.

After an extended
break, I attended, of course, the HPM meeting. Luis Radford gave an
introduction to the HPM group, its
research and publications, while Fulvia Furinghetti presented the history of
the group. Then, Kathy Clark, the new chair of the HPM, gave a presentations
with her own background and some views on the HPM. She mentioned two projects:
ÜberPro project (Germany; Übergangsproblematik) and TRIUMPHS (US; transforming
instruction in undergraduate mathematics primary historical sources).

(The proceedings of
the 2016 HPM conference are online here: http://www.mathunion.org/fileadmin/ICMI/files/Digital_Library/History_and_Pedagogy_of_Mathematics_Proceedings_of_2016_ICME_Satellite_Meeting.pdf
)

This concluded the
last whole day of the conference. However, discussions on mathematics education
and other interesting topics continued for many more hours...

## Friday, July 29, 2016

### ICME13 Day 5 #icme13

In case you are
desperately looking for the Day 4 post, please note that Thursday was excursion
day. I chose to make this a Excursion-alone day, as there was a Manet
exhibition in the Kunsthalle (right next to my hotel) that I did not want to
miss. (It was great - and as so often, a good guide, in this case a multimedia
guide, made all the difference. I also don't have much of a memory, so I was
surprised at some of the cool paintings in the permanent exhibition, even
though I saw them three years ago.) Moreover, of course I had to prepare for my
talk on Friday, prepare for my workshops next weekend (at LAMIS in Ålesund) and
also generally wake up after Wednesday night. So I spent some time with a cup
of coffee and my paperwork in Lange Reihe, but also had dinner with colleagues
in the evening.

Of course, the
excursion day is also a day when you reflect on the conference so far. The
simple question "Was it a good conference?" can be unpacked into
subquestions such as Did you meet new (or old) colleagues that you may
collaborate with in the future? Did you get ideas for new projects or for your
teaching? Did you get an overview of topics that will make it easier to study
them further when you get home? Did you get that warm feeling of hearing a talk
and realizing that you actually know that topic quite well already? On at least
those counts, this conference has so far been a success.

Friday started with
a plenary lecture by Berinderjeet Kaur titled "Mathematics classroom
studies - multiple windows and perspectives". The TIMSS Video Studies
(1995 and 1999) was an inspiration for further classroom studies. They found
that teaching varied a lot between different
cultures - and we are not aware of the differences. The 1995 study
showed different patterns for different cultures. The 1999 study introduced a
new terminology of wide-angle lense perspective and close-up lense perspective,
and argued that both are needed. (By the way, I was lucky enough to get access
to data from this study to do a study of history of mathematics in the seven
countries. But that is many years ago now.)

In the late 1990s,
there were a couple of studies involving Singapore, the Kassel project and a
study of Grade 5 mathematical lessons. In the Kassel study, there were lesson
observations. There was a need of a shared vocabulary when talking about
classroom activities (for instance, she mentioned that the term "Singapore
maths" does not make sense to her). The study gave a wide-angle view of
Singapore (pretty much in line with what
one could expect). The next study consisted of just five lessons, also without
many surprising results.

In 2004, Singapore
joined the Learner's Perspective Study (LPS). This study investigated from the
perspective of students. Three teachers participated in Singapore, a ten-lesson
sequence for each were recorded. To code lessons, they wanted to take into account
the instructional objectives of the teacher (which seems to make sense), but a
teacher always have more than one objective to a lesson (which many school
administrators don't seem to understand). This made for complicated analysis.
The close-up lens gave a different picture than the wide-angle lense. Lessons
were well structured, objectives were clear, examples carefully selected,
student work carefully selected for classroom discussion. In addition, students
were interviewed after the lessons. This resulted in a long list of
characteristics that the students valued (which seems quite similar to what
teachers try to achieve).

She ended by arguing
that the stereotypes about Singaporean mathematics are not correct, especially
when you use a close-up lense. (This is not fully convincing to me, at least
not if based mainly on the study of three highly qualified teachers.) She also
noted 2010 survey data that showed that teaching for understanding is strong in
Singapore, and the analysis shows clear links between the different
instructional practices.

Then I attended
Michael Fried's talk on "History of mathematics, mathematics education,
and the liberal arts". I have heard Michael several times before, and knew
I couldn't go wrong there. He started by pointing out that he will not talk
about history of mathematics as a tool (referring to Jankvist). Here, the point
is history of mathematics as a object of study, not as something to use.

First, he started
with D. E. Smith, who was interested in history of mathematics and also set in
motion the idea of ICMEs. In another writing, Smith claimed that the motives
for teaching arithmetic is either for its utility or for its culture. History
of mathematics should be at the very heart of culture. He used both the
"parallelism argument" and history mathematics as a filter showing us
the importance of each part of mathematics. Fried argued that Smith's view of
mathematics is ahistorical - he looks at mathematics as a set of eternal
truths. Hence we see that the view of mathematics is not the result of not
knowing enough history. This view of mathematics is what allows it to be a
tool. And oppositely: allowing history to be a tool, makes it non-historical.

He then looked at
"religio historici", where original texts are at the heart of doing
history. It matters how you consider the past (of course, we know that parts of
the past have been deemed as uninteresting by historians at times). The past seen
as just what lead to the present (a Whig interpretation of history) is
uninteresting, "practical history". Oakeshott and Butterfield wanted
us to see the otherness of the past.

This unhistorical
history is particularly tempting in history of mathematics, because mathematics
is seen as eternal. He stressed the different concerns of historians of
mathematicians and teachers of mathematics. The demands on teachers means that
teachers will not put modern mathematics aside to teach history. Thus, teachers
needs to economize and make history of mathematics useful - thereby almost
neccessarily embrace a Whig view.

In 2001 Fried
claimed there was a clear choice between teaching Whig history or to drop the
idea of being useful. This damns all hope of including history of mathematics
in mathematics education.

However, in
principle, one can ask how mathematics education can be conceived to include
history of mathematics as an integral part. Here he named people like Radford,
Barbin, Jahnke, Jankvist, Clark, Guillemette as examples of people who can be
seen to be working on this. (This means fighting simplistic uses of the word
"mathematics" as learning methods, which can be glimpsed in some
research articles. I have often questioned what people mean by
"mathematical" in the phrase "mathematical knowledge for
teaching". In my opinion, the history of mathematics is an integral part
of mathematics, in the same way that you can't separate literatur from its
history in a meaningful way.)

Then he went on to
speak of the history of the "liberal arts", where mathematics was a
self-evident part. (He spent a significant time detailing this, including
pictures, which I cannot describe here.) The liberal arts was supposed to be
what you needed to become fully human. History was never considered part of
them. Today, history is considered part of the liberal arts, while mathematics
is missing. This could be fixed, and history of mathematics can be seen as a
way of solving it. Mathematics education then also becomes a way of reflecting
on ourselves.

(In Norwegian
teacher education, I'm not sure the argument is exactly the same. We talk about
so-called theoretical subjects and so-called "practical and aesthetical
subjects". But similar arguments can be made - of course, we know that
mathematics is also practical and aesthetical, as history of mathematics can
also show.)

This was a
thought-provoking talk. I believe that I will - as usual - end up in a
pragmatic view where including history of mathematics in different ways and
with different goals will still be better than doing nothing. I believe in
teaching history of mathematics as a goal, but that this can still be
"useful", and that teaching history of mathematics as a tool can
still instill a sense of "real" history. Teaching is never perfect,
it is always in need of judgment and a balancing act between different
concerns, so why should this area be different?

Then it was back to
the TSG. Derya Çelik talked on "Preservice mathematics teachers' gains for
teaching diverse students". As she could not come, the talk was sent as a
video. She talked about a project with 11 researchers. They analysed PSTs opinions
on how often the program provided opportunities to learn about teaching
students with diverse needs. 1386 PSTs took part. "Teaching for
Diversity" scale was used. In general, the results showed few
opportunities, although there were some (significant) differences between
regions, with more developed regions scoring higher. This fits with TEDS-M
results (also for Norway).

Then I presented the
work of Eriksen, Solomon, Rodal, Bjerke and me, with the title ""The
day will come when I will think this is fun" - first-year pre-service
teachers' reflections on becoming mathematics teachers". I did not make any
notes during this talk, as everything felt quite familiar... :-) The people
present seemed interested, though.

Oğuzhan Doğan talked
on "Learning and teaching with teacher candidates: an action research for
modeling and building faculty school cooperation". He started by stressing
the importance in teacher education in improving teaching practices. High quality
field experiences can contribute, while poor quality field experiences will
support imitation. The aim in this project is to find ways of improving. They
planned a hybrid course where teacher educators and PSTs plan mathematics tasks
and activities and apply them in a real elementary classroom. (He gave some details on the design of the
course which I can't repeat here.) At first, they applied tasks that the
teacher educators had made, but then they applied tasks that they had made
together. They saw that teacher candidates left the idea that the answers were
the most important. They also went from exercises towards discovery, and saw
the important role of manipulatives. The teacher involved also used the tasks
given in other concepts. For the future, they plan to do something like this in
the compulsory course, even if this course does not go on.

Finally, Wenjuan Li
talked on "Understanding the work of mathematics teacher educators: a
knowledge of practice perspective". She pointed out that there has been a
lot of research on PSTs and teachers, but not on what MTEs (mathematics teacher
educators) need to know. She used a distinction between knowledge for/in/of
practice. Of course, we need to include both mathematics and mathematics
education knowledge. The study further hopes that studying what MTEs do, will
inform us on what they need to know. (This is a bit doubtful, and moreover,
just as with mathematical knowledge for teaching, what the particular MTEs
don't know, will be missed in the model even if useful.) They included only six
MTEs in this project. All had school experiencem but they had limited knowledge
of research. (Which means that the results will probably be different from
results in a similar study in another context, with MTEs with another
background. And, remember, parts of the rationale of having teacher education in
universities, is that the MTEs are actively doing research and development
work, so that they will have very different specialities which (hopefully)
contributes to their teaching in different ways. Of course, the same criticism
can be (and probably has) been directed to Ball's work etc. But even so, the
results could be interesting to build upon with data from other contexts.)

After lunch, I heard
Ronald Keijzer's talk on "Low performers in mathematics in primary teacher
education". In the Netherlands, there is a third year national mathematics
test (because of PISA and TIMSS results, comparable to in Norway). The project
investigated characteristics of the students who did not pass this test. (Could
it be seen already in the first year?) Both interviews (n=12) and a
questionnaire (n=265) were used. Previous mathematics scores predicted score in
third year test. Low achievers specialize in teaching 4-8 year olds (not 8-12).
Many similarities with other students, but they are less enthusiastic than
those who passed the tests. They
disagreed that they were low achievers or did not work enough or that they
didn't get enough support. Rather, they blamed the test (interface, content,
preparation, feedback) and personal factors (dyscalculia, stress, concentration). So the conclusion is that low
performers are often that for a long time, they often blame the test or
personal factors. There was a lack of self-reflection. Comment from the room:
mindset treatment as a way of increasing self-reflection.

After this, I needed
to meet "my old group", TSG25 (on History of mathematics), where I
heard several talks throughout the evening:

ChunYan Qi talked on
"Research on the problem posing of the HPM". She referred to MKT, in
particular SCK, and she focused on problem posing based on HM. The research
look at 68 problems based on (one?) problem from Chinese history. She listed three
strategies for making problems based on history (but I didn't - in the very
short time - quite understand why these three were chosen or how we would know
that the results would be problems). Then she discussed which strategies were
preferred by students and other findings that turned up when analysing their
problems.

Tanja Hamann gave a
presentation on "A curriculum for history of mathematics in pre-service
teacher education". She started by referring to Jankvist, noting that
using history as a goal is perhaps a bit more important for teacher education
than at other levels, while history as a tool is more a general concern of all
levels. But of course, you often work on both at the same time. She argues that
HM is important for teachers to influence beliefs, give background knowledge
for teaching, building up diagnostic competences, providing an overview of
mathematics, being able to identify fundamental ideas, building language
competencies.

They try to
implement history in a long-term curriculum, including history throughout the
topics as small parts. They also have a special lecture on mathematics in
history and daily life. Finally, they use history in exercises and tasks. For
all of these three components she conjectured which goals they could contribute
to.

And after a break:

Jiachen Zou spoke on
"The model of teachers' professional development on integrating the
history of mathematics into teaching in Shanghai". He presented a model of
how teachers, researchers and designers can work together to design teaching
integrating history of mathematics. The model was in three dimensions with many
colors and many hermeneutical circles, and can not be described in words. It
was unclear to me how the model was developed (based on what data), maybe that
would have given me better understanding of the model. However, as always, time
is limited. (He was also asked how the ("somewhat formalistic") model
were developed, but it was not clear, except that the model had been used to
describe three different teachers' paths.)

ZhongYu Shen on
"Teaching of application of congruent triangles from the perspective of
HPM". His project concerns 7th grade in Shanghai, with two teachers
involved. The stories of Thales (finding the distance of a ship at sea from the
coast) and Napoleon (a similar method) was used as a starting point, in
addition to a Chinese measuring unit for length. They designed a lesson based
on this, and had a simple survey asking whether they understood and liked this,
and not as many liked it as understood it.

Fabián Wilfrido
Romero Fonseca on "The socioepistemologic approach to the didactic
phenomenon: an example". The example was three moments from history of
Fourier series: the problem of the vibrating string, Fourier's work on heat
propagation and development of engineering as a science. He showed how
activities were based on the analysis of the historical background, where
Geogebra were used to investigate. However, the activities have not actually
been used, just developed as part of a master thesis.

Thomas Krohn's talk
was "Authentic & historic astronomical data meet new media in
mathematics education". The students in question were in 10th, 11th and
12th grade. An authentic problem was to describe the movement of comets, and
this can be used by giving students some historical/astronomical background, as
well as the neccessary tools. The astronomical data are often presented in
lists in ancient books which are on the internet. As in ancient times, they
first made a projections from 3d to 2d, and then they tried to find a
reasonable function (often preferring to investigate instead of leaving it to a
computer to find.

Slim Mrabet, whose
title was supposed to be "The development of Thales' Theorem throughout
history", did not turn up.

Thus ended the
Formal parts of day five.

Subscribe to:
Posts (Atom)