Saturday, February 4, 2017

CERME 10 Day 4

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...

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.

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... :-)

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.