NERA 2015 Day 2 #NERA2015GU

The second day of NERA started off with parallel sessions, I chose the one on "Teacher's work and teacher education". First, my colleagues Kirsten Thorsen and Hanne Christensen talked on "Identity forming in teacher education". They are part of a project TPQ (as am I), and their talk is based on data from two of the sub-projects. The data involve surveys  and in-depth interviews with students, mentors and campus teachers. Students describe practice as the terrain - authentic and unpredictable - in particular in the first year. They describe campus learning as the map. The project started out with looking for "the gap" to be able to bridge it - but the "gap" metaphor doesn't really work if it is the terrain and the map. Peers are very important in both areas. The practice mentors are seen as giving solutions, while teacher educators give them theory.

Practice teachers do not feel competent in theoretical themes, so they do not connect practice to theory very much. They tend to focus on practical advice and discussions on what works in practice.

Secondly, Roald Tobiassen had a talk on "Portfolio as practice in teaching practicum: promoting reflection and constructing teacher identity". The project was connected to the teacher education that is called PPU in Norway, and a part-time model. The portfolio was meant to scaffold students' practicum learning and reflection. During the first year, the students had six pedagogical texts to write, starting with "My pedagogical creed". The students meet in groups of 4-6 students which discuss the tasks. This particular project looked at six of the students.

Tobiassen went quickly through a lot of the theory on portfolio. Then to the findings: these six students were positive with regards to the portifolio - they saw it as a way of developing teacher identity - they valued the authenticity. "Portfolios helped me to see where I wanted to go and how to get there". The six students were happy with the structure as well. They saw the portfolio as a tool for connecting what they read about being a teacher and their own experience.

Finally, Jóhanna Karlsdóttir's theme was "Storyline som metode i inklusiv læring og undervisning i praktik hos lærerstuderende". (Even though the title is in Swedish, most of the words are understandable for the English reader, I guess...) The study is based on one course in teacher education in Iceland, with 21 participants (in 6th semester) - the data are interviews, notebooks, discussions etc. The course is focused on inclusive education, which is apparently not taught systematically in teacher education in Iceland. She presented an eclectic mix of theories as a foundation for her work, including Gardner, Johnson & Johnson etc. She then described the storyline process (in the classical way). Her project goes on for one more year, but she has found some room for improvement, for instance in getting to know the method better before using it themselves. But storyline apparently is useful for inclusive education as it is building on pupils' resources.

I do think that I see more of why I do not like such a "broad" conference as much as more focused conferences. I think the variety of participants makes it almost impossible for those giving talks to present their projects effectively - it is impossible to know which theories and concepts and contexts are well-known to the people present. Thus, most presentations either spend too much time on stuff I already know well or jump too quickly over theories I do not know. This is in marked contrast to for instance the HPM conferences I go to every four years.

Jane Kenway, Monash University, had a plenary on "The emotional life of markets in education". She presented a project looking at elite schools around the world (but not generalizing educational policy based on these schools like yesterday's speaker). Elite schools may be fee-based or merit-based (and grant-funded). She talked on concepts such as "emotional geography" (how do feelings connect to places) and "economies of emotion" (the emotions of buying a Nike shoe is not connected to the smell and sound of the shoe factory). What emotions are evoked by certain schools to make parents send their pupils there?

She mocked the way elite schools market themselves with cliche slogans. (It does remind me of a student saying that she was so amazed of how teachers in her school in Uganda always reminded their pupils that they are the future of Uganda. Cliche, yes, but still an important reminder for the children. Isn't there too little - rather than too much - talk of the importance of schooling in western societies?)

While the subject of elite schools is a bit interesting, to me it feels like it is on the edge of what I'm interested in. I'm far more interested in how similar mechanisms are working (or not) when "ordinary" schools are concerned. How do competition for pupils influence the internal life of "ordinary", "non-elite" schools. Perhaps ideas for studying this can be found based on the study of the elite schools? Other characteristica of elite schools are certainly less interesting in that context, for instance how parents use relocation services to set up meetings with potential schools - not a very common practice among parents relocating within Oslo, for instance. The high pressure for performance in elite schools - the fail anxiety - is also something that is far less usual in "ordinary" schools, I would think.

A somewhat relevant feature of elite schools is the development of new "departments" tasked with producing emotions, for instance marketing departments. This is also the case for many Norwegian institutions, for instance my own, with its "avdeling for samfunnskontakt" which is trying to give HiOA a good image and avoid bad press. (While trying to remember that an important part of being a good institution in higher education is to encourage discussions and enjoy the benefits of free speech, even when uncomfortable.)

After lunch, I went to the parallel session on Classroom Research. Ingvill Krogstad Svanes presented "Teachers' instructional practices during students' individual seatwork in primary school". Her research is on six teachers in 3rd grade in Norwegian - one week each. She presented an analytical framework developed through the project. The main codes were instructional support, organizational support, emotional support, monitoring and "no direct interaction with students". One main finding is that there is a huge variation between the teachers in how they spend their time. Two of the six teachers give more instructional support than anything else. Others spend most of their time on organizational support. This seem connected to the clarity of the initial instructions and the choice of activities (scissors and glue lead to more need of practical help, for instance). Emotional support was almost not present, but that may be because only what was spoken was coded. Further subdivisions in the codes show that even within the categories there are important differences - some teachers are mostly telling the pupils, while others are challenging them more. (For the sake of openness: I am Svanes' boss as well.)

Next, Malin Norberg talked about "How do children in primary school make use of illustrations in mathematics textbooks?" She chose subtraction as an example (an interesting choice, as it is fairly difficult to illustrate in a static picture). 1742 illustrations from 21 textbooks were the data, in addition to discussions with twelve students about five illustrations. She has looked at two subtraction situations: decrease and compare. 86.5 % of illustrations were illustrating decrease. Sometimes the students need the illustration to do the subtractions, in other instances the illustrations are just illustrating a process.

Students sometimes read more into the illustrations than intended, and sometimes they are able to solve the mathematical task with symbols but not with the illustrations. (Not all illustrations are very good...) The teacher's role is important. In her PhD, she will work on teachers' role and teacher's guides.

For the last scheduled talk in this session I decided to just listen instead of writing notes - sorry...

After a coffee break, I went for the parallel session on Gender and Education again. The first talk was supposed to be by Jenny Bengtsson and Eva Bolander: "What the school can (not) do: Education markets and negotiation on sex, risk and schooling". However, this talk was cancelled...

The second talk of this session was Chie Nakazawa's "Adolescents' norms, attitudes and values regarding sexual and reproductive behaviours from a gender perspective - a comparison between Japan and Sweden." The talk is based on surveys of university students, about 500 in Sweden and about 2500 in Japan. One indication of the difference: about a quarter of Japanese women in the study claim never to have been interested in sex, while only 3.6 per cent of Sweden women claim the same. I am quite sure that this is a cultural and not a biological difference...

Also, only 2.4 per cent of Swedish men say they are homosexual or bisexual, compared to 10.4 per cent of Swedish women. This question was not even asked in Japan. Moreover, it is clear that Swedes find sex to be more enjoyable and less dirty and embarrassing than Japanese, if the survey answers are to be believed. Also on gender roles, the differences are significant.

Thirdly, there was the talk "A hotbed of heterosexuality? On the reproduction of notions of sexuality in language instruction" by Angelica Simonsson. Her talk is based on her ongoing PhD project. Her research question is whether sexuality and gender normativity is constructed in language education instruction in secondary school. She is looking both at teaching materials and at how pupils and teachers relate to these. It is based on two classes (grade 8) in two different schools, and the teaching aids used. Subjects: Swedish and English.

Her findings are that sexuality figure in teaching aids, and these are all hetero. This goes for both non-fiction films, fictional films, short stories and textbooks. There was a total lack of non-heterosexual relationships. (This is quite surprising, as my research show quite a bit of homosexuality in Norwegian textbooks. Does this mean that Norwegian textbooks are more inclusive, or would a study of Norwegian classrooms show a similar pattern, i.e. that teachers choose the heterosexual examples?) (in a comment at the end of the session, it was argued that even when gay characters are portrayed in literature for youth, they are portrayed in a very stereotypical way. I have not looked at that in my material.)

Finally, the fourth talk of the day was Per Nordén's "First Generation Rainbow Children Speak Their Minds - How queer kinship structures matters in education". With "rainbow children", he refers to children with one or more LGBT parent. The talk is based on interviews with 28 rainbow people from age 15-37. He gave long quotes from different stages of history. It is fascinating to see how different experiences are, and that these are intimately connected to changes in law and society. Sadly, he did not have the time to come to the part of their school experiences.

That's the end of the second day of NERA. Time for the last preparations for Friday morning's talk... 

NERA 2015 Day 1 #NERA2015GU

I will "report" on my first NERA conference in the same way as I report on other conferences I attend - through quick notes written throughout the conference. The conference lasts for three days, and I am having a presentation on Friday morning (and an early version is available on YouTube...) Hesitatingly, I will do the notes in English, as most of the presentations will be in English, even though most participants are Swedish and Norwegian.

After registration and lunch, the first thing on the programme was the opening, of course. Dean Roger Säljö welcomed us and gave a brief introduction to the Faculty of Education at the University of Gothenburg. There was also an introduction into the research activities in the faculty.

The first plenary talk was by professor Hugh Lauder, titled "The Repositioning of Education in the 21st Century and what Can Be Done About It". His starting point was the tightening bond between the economy and education in the 20th century - which he claimed is now eroding. In some countries, education's only goal (in political documents) is the economy. It has been assumed that education leads to upwards mobility, but that is dependent on an increasing number of jobs at the top (or downwards mobility on the top). Education is also seen as important for global citizenship, but this is dependent on children seeing a future, which they don't always see.

Now, there is an increasing polarization in wealth and income, and there is an increasing competition for a decreasing number of top jobs - the corporate ladders are now flat. We see the emergence of a global education system which educates multi-lingual multi-cultural candidates for transnational corporations. Those not "talented" are hitting a glass ceiling. (The argument is unconvincing, as education could certainly be economically worthwhile even though it does not give top jobs in multi-national corporations...)

32 percent of the poorest 10 per cent of British people are graduates - meaning that graduation is not guarantee against poverty. An increasing percentage of graduates are poor. (But again, this does not mean that education is not worthwhile for the individual. And it certainly doesn't mean that education is not good for the economy as a whole.) There is an insufficient supply of high skilled work.

(By the way, Lauder is the kind of lecturer who has a Powerpoint with huge amounts of text which he shows as he is saying something else. This doesn't work very well for me, maybe because I'm not a native English speaker and reader...)

He ended by saying that we cannot claim that education is there for the economy, what is then the purpose of education and how can we then get funding for it? (But of course humankind have discussed for thousands of years what education is for, so we do have plenty of non-economic answers to that question.)

As I have said, I don't find the arguments in this talk convincing - the numbers don't seem to add up to the conclusions he is stating. The data are on the outcome for the individuals, not for society. Perhaps the reasoning is better explained in his book(s). (But as was commented later in the day - in that case it would be good if he put his best arguments into his talk.)

Then, after a coffee break with a conspicious lack of coffee, there was the first parallel sessions. I went to the ICT & Education session, where three of my colleagues from Oslo presented. But first, Ann-Katrin Perselli had a talk entitled "From computer room to one-to-one". She described a phenomenological study with four teachers from two upper secondary schools, in which all students had their own computers. Among her findings: each student having a PC meant less fighting for the computer lab, while PCs were also disturbing. Teachers based their teachings on tips from colleagues, trial and error and websites - this was time-consuming for inexperienced teachers. Good relationships with students and teachers were a help. The study apparently shows how teachers' "lived experience" influence their approach to using IT in teaching. School leaders need to be aware that teachers are different.

Then, Bård Ketil Engen and Louise Mifsud presented work on an online course on collaborative learning activities - on master level. The course has been held for four years. They discussed different ways of engaging students online, now using Adobe Connect and Second Life. Asynchronous student collaboration was mediated via Etherpad and Wiki. The semester is designed with student activities alternating with online synchronous lectures, and finally there is an exam where students write a paper on the work they have been doing.

Technology influences communication. Students often get more passive online, and maybe a bit uncertain. Asynchronously, we see that students start out cooperating (dividing the labour) instead of collaborating. Overall, students are learning about CSCL activities while learning about CSCL. (I happen to be the boss of these fine teacher educators, a fact there's no reason to hide.)

Finally, Marianne Vinje had a talk with the title "Teacher Strategies for Meaningful Learning in a Blended Environment". Many challenges are facing higher education - resources are moving away from teaching, and research is rated higher than teaching. The role of instruction gets less important, teaching complex (higher-order) thinking is more important. Technologies give endless opportunities which have to be developed. One such opportunity is blended learning, which is what Vinje has been working on, using a community of inquiry (CoI)framework.

Teaching online is something else than traditional teaching - other factors are important than in traditional teaching. Studies show a change of many teachers from an instructional mode of teaching to a more Socratic mode of learning. Also, many teachers are more precise in their messages/information. Vinje thought blended learning gave her more classroom time to get to know her students.

As the last stop of the day, I chose the parallel session on Gender and Education. The first talk here was by Ingólfur Ásgeir Jóhannesson on "Does the National Curriculum Guide 2011 pave way for gender and queer studies in Icelandic schools?" The Icelandic National Curriculum has six fundamental pillars, one of them is equality. This talked is based on text studies and interviews of teachers. There is a focus on equal opportunities in the rest of the curriculum, but rarely on gender studies. Sexual orientation is mentioned, queer and queer studies are not mentioned. One of the books analysed was "I, You and We All" (for 6th to 8th grade) in this, intersectionality is clearly used, and there is a social understanding of gender.

Gender studies is an elective course in some upper secondary schools in Iceland. This is a course without a textbook or final exam. Queer studies is not a specific course anywhere, but is a part of gender studies. The inclusion of gender studies as an elective course is the result of a spontaneous movement among upper secondary teachers, supported by student interest. In one school, Pink Holocaust is taught.

The second part of that parallel session was Anja Kraus' talk on "'Gender' as a Tacit Dimension of Pedagogy". Her starting point was that the the aim of pedagogy is to set people free, the idea of Bildung. Gender can be seen as an analytical tool, helping to understand the constitution of practices and knowledge domains. Traditional pedagogy tends to rely on logic and on concepts that are supposed to exactly fit reality, she argued. Postmodern approaches rely more on self-interpretations. "Bring the body into the discussion" was one expression used. Queer theory, Butler's performativity and body-phenomenological concepts were contrasted, and apparently the latter was the preferred one.

As far as I understand it, a text is seen as problematic because it imposes logic on the world, which makes it a bit problematic to give a talk (a text) on a postmodern approach (or on anything, really).

Thus ended the first day of NERA 2015. A very varied day - a bit more varied than I prefer, I guess. But I do bring some interesting points with me from this day.

I'm not happy with the way I'm referred to...

Some time ago, Google Scholar noted that there was a reference to me in a new paper, The Need for the Inclusion of History of Mathematics into Secondary School Curriculum: Perceptions of Mathematics Teachers by Habila Elisha Zuya. It is published in the journal International Journal for Innovation Education and Research (IJIER), a journal that is peer-reviewed, but states that "The review process takes maximum two weeks." Hm...

The article refers to me in this way:

A number of researchers have pointed out that teachers' interest in mathematics increased when introduced to the history of mathematics (e.g. Smestad, 2009; Siu, 2004; Phillippou & Chritou, 1998; Stander, 1989). However, these researchers maintained that teachers found no interest in using the history of mathematics within the curriculum.

I do not think my data can be used to claim that "teachers' interest in mathematics increased when introduced to the history of mathematics" or that I claim that in the article. Neither can my article be used to suggest that "teachers found no interest in using the history of mathematics within the curriculum". Of course, I may be wrong, maybe indeed my article does suggest something else than what I intended. But I am more tempted to believe that this is a case of trying to fit too many references into too short a paper, so that the actual point of view of each reference is not retained.

I just wanted to get it off my heart. I don't think the journal would be interested in publishing a note to this effect, at least not without me paying for the privilege...

Comments on Gert Biesta

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

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

My "manuscript" can be found at Academia.edu.

Post-conference blues

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

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

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

ESU7 Day 5

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

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

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

The aims of the implementation was

• Knowledge of the original sources
• Recognition of the most significant changes in the discipline of mathematics
• To emphasize the socio-cultural relations of mathematics with politics, religion, philosophy and culture.
• (The most important). To encourage students to reflect on the development of mathematical thought and the transformations of natural philosophy.

There was an introduction to Chinese sources, including Nine Chapters, and then we looked at chapter 9, being given the problem, answer, algorithm and comment (by Lui Hui). (In both the original Chinese and in a French translation...) Students could do activities, for instance doing geometrical proofs on paper. (It was unclear to me, however, what the students were asked to do with the sources and what instructions they were given.)

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

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

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

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

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

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

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

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

• The same original source can, depending on the context, be used for many different purposes, and depending on these purposes, the design of the teaching will vary. The assessment of whether the teaching was successful, will also vary. Understanding better how these things work constitute major "open questions".
• In particular, in some contexts it is meaningful to give students an original source and a set of tasks to "guide" their reading. When this is appropriate, how these can best be designed and what effect this has on the outcome, form a subset of these open questions.

Of course, these questions was actualized already in the first plenary, so I'll send special thanks to Adriano Dematte for his plenary. But I've come across things reminding me of this every day of the conference.

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

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

ESU7 Day 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Four examples of websites on mathematics (which are not PLEs): Mathisgoodforyou.com NRICH MacTutor - StAndrews Mathigon.org

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

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

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

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

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