Friday, August 2, 2013

PME37 Day 5 #pme37

Tim Rowland's talk on "Developing one-to-one teacher-student interaction in post-16 mathematics research" started the fifth day of PME37. He talked about the work of Clarissa Grandi. Her research was motivated by a feeling of mismatch between her intentions and her practice, where she felt she helped the students "too much", instead of scaffoldig. Rowland noted that "teaching by telling" is very prevalent, has been widely critizised but that there are not necessarily easily available and easily implementable alternatives for a teacher who wants to do something else.

Grandi analysed the form and function of her utterances during one-to-one interviews with students (and she had a few other data sources as well). It is an interesting choice to do this in the form of interviews instead of "real" classroom situations. I can see how the analysis of her utterances in interviews can give insights that are transferable to her work in the classroom. In the end, she had 12 codes showing different functions.

In the first cycle, there was a preponderance of "telling". Grandi was not happy with this. She used "telling" to demonstrate, direct, explain and funnel, not only to confirm, discuss conventions or "parallell model" (involving the solution of a related, but often simpler, example), in which cases telling was more meaningful, according to her. In the second cycle, she demonstrated and explained considerably less.

Through the research, Grandi realized that telling is sometimes consistent with a social constructivist agenda, but that she had also been able to modify her behaviour in the direction she wanted to.

Of course, some of the discussion afterwards was on the problem of doing research on oneself, and the problems you then have of distancing yourself from the situations.

In short oral presentations afterwards, first Jajaluxmi Naidoo presented a study on pre-service teachers' understanding of effective teaching strategies. The study showed that students with no teaching experience thought they should use the same strategies as their teachers had used in school. They did not have much belief in the more learner-centered strategies - it wouldn't work in real life. Which may be correct, South African classes often have more than 90 pupils, often with more ages in the same class and with many of the children hungry and unconcentrated. Of course, this is also relevant for other countries, although not as extreme- many students regard the strategies that teacher education advocates as unrealistic in that they do not take into account all the difficulties present in a classroom. This is a significant challenge for teacher education. We cannot include all the complexities of the classroom all the time in our discussions, but neither should we disregard them completely.

Charlotte Krog Skott gave a talk on "Lesson study in teacher education". This is based on an attempt to adopt the Japanese practice of lesson study to teacher education, where teacher students experiment with and discuss teaching. In analysis, they used cathegories from the knowledge quartet and others, and also used elements from the theories of communities of practice and situated learning. The pre-service students guided the students in a certain direction, in spite of their intentions. It also became obvious that the pre-service teachers didn't discuss the mathematics in the lesson study, instead focusing on how they could have asked questions to have the students think instead. They also gave a good example where PSTs designed a lesson where students should rotate to see four different tasks, and the PSTs were very good at picking different representations and making varied activities, that sadly did not connect to each other mathematically. Another result was that there is a potential to develop a community of practice around lesson study.

In just a few weeks' time, there is the ECER conference in Istanbul. As we would have to leave Kiel Friday morning, we spent the rest of Thursday planning the presentation in Istanbul. We had a very productive meeting at the hotel balcony, and now feel (almost) ready for ECER...

It was nice to be at PME - it was a lot friendlier than I thought it would be, and Germany is a very nice country (I'm already looking forward to ICME in 2016). I will always remember this conference for my struggles with Norwegian authorities during the conference, I'm afraid, but the conference as such was quite good.

Wednesday, July 31, 2013

PME37 Day 4 #pme37

The first talk of Wednesday (one I also chaired) was Shin-Yi Lee's talk on metacognitive strategies for improving problem solving abilities in probability. She had done an experiment in which 9th grade students were encouraged to use a metacognitive-strategy worksheet while working on probabilities. The most striking result to me was that the control group, getting traditional instruction, had a significant decrease in their results from the pre-test to the post-test. But in the discussion we concluded that this may often be the case: if students use their common sense, they may go a long way in simple probability tasks, but if they are given a few formulas, they will try putting everything into them, leaving their common sense. The study had its limitation in that we knew little about the experimental group and the control group, and many ideas for further studies popped up in the discussion.

Then I went to Gert Schubring's talk "From 'armchair pedagogy' to experimental research and to case studies". He discussed the history of empirical research in mathematics education. At an early stage, teaching was considered as a treatment with learning as an effect, and research was done to show the effect. There were many small-scale studies with little connection to theory. Around 1960, large-scale studies were made possible by grants from science funds. In 1969, at ICME1, Begle called out for more empirical research, less research based on opinions. He repeated his criticism in 1979.

Later, case studies got more prominent and the interest was more in qualitative than quantitative studies, although TIMSS and PISA have been important exceptions. Schubring claimed, however, that these studies perhaps grew more from outside the mathematics education research community than from within. Of course, as anyone who has ever attended a talk by Schubring will understand, this short note is nowhere near doing justice to Schubring's talk, full of information as they always are. The best thing I can hope is to have given a sense of what the topic was...

By the way, the term "armchair pedagogy" reminded me of an earlier discussion within the HPM community (where Gert Schubring is of course a prominent member) about "armchair research". There, I have argued that the field of HPM needs both research and development (in Norwegian: FoU - forskning og utviklingsarbeid). If by "armchair research" is meant the development of "good ideas" and teaching sequences by experienced educators with an interest in the history of mathematics, this is an invaluable part of the development of the HPM field, making it possible for teachers to take up HPM practices as well as providing materials that researchers can research. To do research on education, you need someone to provide the actual education you want to study, and it is not necessarily the best researcher who has these ideas...

Then, C. Miguel Ribeiro talked on "Characterizing prospective teachers' knowledge in/for interpreting students' solutions". In the programme, Arne Jacobsen was given as the presenting author, but there was evidently a change. Of course, an important part of being a mathematics teacher is to interpret and try to make sense of students' solutions. This project used Ball's MKT "ball" as a starting point, here looking particularly at common content knowledge and specialized content knowledge. They see the ability to make sense of students' knowledge as part of special content knowledge. In their materials, which concern fractions, they do see a tendency that teachers mostly see as correct those of students' solutions that are similar to the teachers' own solutions, which suggests a need to focus on this in pre-service teacher education. One strength of e study, by the way, was that it involved three different countries.

As so often happens in these cases, there was a lot of discussion on the quality of the tasks and how they should be interpreted. One comment was on the concept of "babbling", where the language of pupils is interpreted as the unqualified use of language in a context where the pupil is busy working out the mathematics, so that some of their utterances are not well thought through. Another comment was that the correct answer to the question "What amount of chocolate would 6 children get if we divide the 5 bars equally among them?" is 5...

Then it was time for the conference's excursion. I had chosen to go to Lübeck. I did not expect this excursion to be as fascinating as the North Korea border excursion at ICME, but it would still be a welcome diversion, both from mathematics education and from private problems...

PME37 Day 3 #pme37

The first thing on my agenda on Tuesday was the Research Forum on "strong" discursive research. For this, we were supposed to read some articles and transcripts in advance, one of which was a chapter by Anna Sfard, in which she seems to claim that good old quantitative research has no place in mathematics education. For instance; "When numerous cases that have nothing in common with each other except a certain superficial feature coalesce under a single numerical label, our access to the diverse factors responsible for individual learners’ success or failure is lost forever. To produce a picture that can count as truly helpful, an incomparably higher resolution is needed than can ever be attained in randomized experiments."

I disagree completely. I believe that both quantitative and qualitative research can be "truly helpful" in our field, and that they give us quite different kinds of insights. Thus, I looked forward to this research forum with some worries - but of course the best way of approaching it would be to see it as offering one more potential way of doing research in mathematics education, disregarding its claim to be the best or only way.

However, even during conferences life may disturb. In my case, some urgent private business had to be taken care of, so I missed both this research forum and almost all the rest of the day's activities. I had no way of concentrating on mathematics education. However, after doing what I could to sort the personal stuff out, it was good to get back to a lecture hall to try to think about something else.

So in the evening I heard Cynthia E. Taylor talk on the title "Facilitating Prospective Teachers' Knowledge of Student Understanding: The Case of One Mathematics Teacher Educator". She went through some of the theory connected to pedagogical content knowledge (PCK) from Shulman onwards. Taylor has studied the actions of a teacher educator in whole-class work - the teacher educator she studied was her close colleague, who was also her supervisor, which of course must have lead to a lot of Issues. The colleague even participated in creating the analythical cathegories. There was one nice methodological touch: "Extended video talks", in which the teacher educator watched a 20-minute video of her teaching (albeit three years later), stopped it where she wanted and commented, with no questions from the researcher before at the end. In this way, the researcher got a rich material with lots of explanations for what the teacher educator did. This particular teacher educator gave lots of examples of what typical pupils would do, both to teach students about misconceptions and to teach them that there are lots of different answers to the same question, and they have to get used to seeing new answers. I did not note down much more of the findings, but they can probably be found in the article.

Lastly, Janne Fauskanger talked on "Teachers' Mathematical Knowledge for Teaching Equality", which is connected to the Stavanger work on MKT which have been presented elsewhere earlier. Here, she is looking at what can be learned about teachers' knowledge of the equal sign when analysing multiple-choice questions vs. free-text questions.

After repeating the contents of "Ball's egg" (which seems to pop out in a lot of talks here at PME), we learned that the data for this paper is 30 teachers' responses to five MKT items. She described the incorrect answers of the four teachers who did not have all multiple choice questions correct - they seemed to have typical misconceptions (for instance having only the operational understanding of the equal sign). The long responses give much richer understanding, including drawing upon different aspects of MKT than the items were developed to measure. And some teachers answered "I'm not sure" but gave very insightful answers in the free-text items, which casts a shadow of doubt over how "I'm not sure" should be interpreted when you have no free text to go with it.

That free text answers give richer insights than multiple choice answers, on the other hand, is hardky surprising. Isn't the point of multiple choice questions also that they can be administered to a larger group of teachers, giving information on how the group as such fares - or by making possible to give more items to each teacher without the analysis afterwards being prohibitively time-consuming? Moreover, they can be used to provoke discussion among teachers, just as one commenter to Janne mentioned. Janne's research gives insights to inform the discussions on how such multiple choice tasks can (and how they can not) be used.

Tuesday, July 30, 2013

PME37 Day 2 #pme37

The Monday started with Doug Clarke's talk with the title "Understanding, Assessing and Developing Children's Mathematical Knowledge". He started by noting that, for many reasons, there has been a shift away from pencil and paper assessment as the main way of assessing children in both New Zealand and Australia. This talk had to do with thousands of teacher-lead one-on-one face-to-face task-based interviews with children for assessment purposes. The research was based on an idea of "growth points", which can be seen as "key stepping stones in children's mathematical understanding". For instance, for addition and subtraction, these "growth points" are similar to "strategies" for addition and subtraction that we teach our own students.

The Early Numeracy interviews were based on 60 different tasks, the tasks actually given depending on the child's answers along the way. A Rational Number Interview was developed for older kids. Nice task: from six numbers given, create two fractions that added together will be close to 1. In the interviews, a lot of student thinking could be inferred from how they moved their cards with numbers.

He referred to The Interconnected Model of Teacher Growth (Clarke and Hillingsworth 2002), which I should look up... During the study, teachers developed more realistic understanding of their students' abilities. For instance, one kid in the study could read a 15-digit number, while many teachers focus on numbers from. 1 to 20 at that age. The project also included making teachers more aware of important strategies. The teachers also developed improved questioning techniques - modelling their behaviour in their classes on the interviews. The teachers also learned that they sometime had to give students a bit extra time to think for them to find the answer.

The obvious question to this is one of resources. Most teachers would love to have the time to have long one-on-one talks with each student in every single subject they teach. That is not possible, and if it was, there would still be a discussion on whether all the extra money could be better spent in another way.

To this plenary, Marja van den Heuvel-Panhuizen gave a prepared reaction (which was of course based on the article in the proceedings, not on the actual plenary talk). She asked: what was the proof that it worked? The evaluations were based on self-reporting of teachers and improvement of students' achievement, but the results of control geoups were not given in the paper. She also went on to describe problems in Clarke's choice of words and so on. She then invoked Hans Freudenthal to say that observing learning processes is (or should be) at the core of teacher education. She seemed to use this as a reason why interview-based assessment is a good idea, and that starting from a typical learning path is a good thing.

Then there was a discussion group on Mathematics Teacher Educators' Knowledge for Teaching. This sounded interesting to me because we of course discuss our students' (prospective teachers') mathematical knowledge for teaching when educating teachers, and it is only fair that we look at our own knowledge as well. Moreover, I'm interested in reminding people that the history of mathematics probably should have a part in what mathematics teachers' should know, as well as (of course) what mathematics teacher educators should know. The first 25 minutes of the discussion group was spent on every participants' saying a little about themselves, which proves that the number of participants were not too small and that the participants were not too silent...

Then we went on to small group discussions, where one was to discuss what are the differences between mathematical knowledge for teaching for teachers compared to for teacher educators, whether it is different for different types of mathematics teacher educators, how we can do research on thism how to educate mathematics teacher researchers and how to ensure resources for this work. Splitting into groups were a challenge, but once we had settled in a room, interesting discussions unfolded. We discussed the first question: what is the difference between being a mathematics teacher and being a mathematics teacher educator? Some of the contributors had the experience of being both at the same time, which is an interesting experience. Someone felt that there was a similar leap from being a teacher to being a teacher educator as the leap from being a student of mathematics to being a teacher.

One topic we touched upon was research. While teachers of mathematics do usually not keep up to date on the most recent research, teacher educators certainly should. Also teacher educators need to know something about the (hypothetical) learning trajectory of teachers. An example is how we teach fractions in a way to make the teacher students understand that what they thought they knew well, they didn't know nearly well enough.

A "technical" question is whether one should try to adopt the cathegories of Ball or wrap Ball's egg within a new layer.

It is also interesting whether mathematics teachers all need to know the same, and the same goes for mathematics teacher educators. Of course, we tend to divide our teaching tasks based on our areas of expertise, so maybe it is enough that half of mathematics teacher educators know their history of mathematics, for instance? Could the same be said of teachers in school, or is that a "lonlier" profession in the sense that they tend to have sole responsibility for mathematics teaching in their group of pupils?

Today's discussion group reminded me that discussion groups are among the most difficult things to plan (as I learned at ICME, although it did turn out okay in the end) - you don't know in advance if there will be 10 or 100 people, and you don't know what the participants' previous knowledge may be. On the other hand, it could also be said that as long as you manage to set people together in groups and they are fairly interested in the subject, they will probably have a good time discussing the subject with each other, who are after all experts from around the world.

Olive Chapman's talk later in the afternoon had the title "Facilitating prospective secondary mathematics teachers' learning of problem solving for teaching". She noted that students' unguided reflection on problem solving often reduced to mentioning general steps (maybe similar to what textbooks often do?) or to describe which steps worked in their problem solving. Chapman wanted to help teachers become aware of different ways of engaging in and reflecting on problem solving and to develop deeper understanding of problem solving. The idea was to help teachers to use personal experiences to build their reflection on, by having them develop rich descriptions, through two approaches; a narrative approach and a "stuck-aha-approach" (focusing on turning points). In the talk, she discussed differences between these two approaches. The narrative approach was more affective, while the stuck-aha-approach was more cognitive. The narratives gave a good view of the process and real world problem solving. When looking at the stuck-aha the cyclic process is clearer. Both approaches were adequate to give room for reflection. Sharing, discussing and unpacking was necessary and gave the teachers the opportunity to a richer experience. (The sharing was done in groups immediately after writing the texts, and they were then challenged to discuss what they learned about problem solving by reading these texts, and what they learned about teaching problem solving.)

One person in the audience had the interesting comment that she has sometimes experienced that students reflect on the problem solving process and realize that their mathematical knowledge is insufficient (for instance that they don't remember Pythagoras' theorem, and they do need that to solve the problem), but that they don't follow up the reflection by actually trying to learn the mathematics in question. So there is the problem of reflection for reflection's sake.

Then there was a talk on teachers' emotions, by Pietro di Martino and others. They talked about the "math-redemption phenomenon". This has to do with the fact that many students who have had problematic experiences with mathematics, still look forward to teaching mathematics, and want to improve their relationship to mathematics. In addition to survey data, they also interviewed students. Students often identified "crises" in their relationship to mathematics, and these were always connected to the teachers they had. This turned into a fear of becoming the same kind of teacher as their previous crisis-inducing teachers. This also seemed to be a major motivator to reconstruct their relationship with mathematics. However, for some students this was seen as unobtainable (by themselves).

The concept of "maths-redemption" seems to be a powerful one, and could possibly also be used to motivate our students in Oslo. By the way, we also have lots of data on student emotions in Oslo, which it could be interesting to reanalyze in light of this talk/article.

Finally, starting at 6:20 pm, was the presentation by my colleagues Annette Hessen Bjerke and Yvette Solomon on our project in Oslo. The project is looking at how teacher students experience their transitions between their teacher education on campus and their school placements, and (in this paper) particularly at what they and their mentors think that the students have learned in the different arenas. For the content of this, I refer to the article in the proceedings.

Simon Goodchild commented that first year students could only hope to "cope" and that we would expect them to show more of their competence later. This is a valid point, of course; by choosing to focus on the first year students, we get another project than if we had focused on third-year students, for instance.

Thus ended the first full day of the confence. (Except, of course, that we had some reading to do before Tuesday.)

Sunday, July 28, 2013

PME37 Day 1 #pme37

After attending four ICMEs and four HPMs, now is finally the time for me to attend my first PME conference as well, as it happens to be held close to home. Compared to ICMEs, PME 37 has a rather small attendance - from the list of participants, it seemed to be about 600 people.

As other conferences, it started with a series of welcome messages by people who are expected to say something at such an occation.

Then there was the first plenary lecture - Kristina Reiss gave a talk with the title "You can't teach an old dog new tricks? Developing mathematical competence over the life span." Her topic could also be framed as: What is lifelong learning of mathematics? I it important for non-mathematicians? And is it possible? Of course, she gave many reasons why it can be important, both personally and professionally. One example; wireless plan.

She described an experiment from Wynn (1992) - where she showed that four months' old children "understood" that 1+1=2. The experiment is also replicated on monkeys. She also described how such understanding is a predictor of later knowledge in mathematics. Thus, she gave examples of how matematics education research has developed to make us better at teaching mathematics to children (although it is still an important area of research, of course). The natural question is then what we know about how to develop mathematical knowledge with adult learners. Cohen (2003) says that adult education is under-researched, under-theorized and under-developed.

She noted that many "standards" that have developed are wonderful if you have wonderful students, great teachers and perfect equipment, but that they are less helåful if you have diverse classrooms including children with learning problems and so on. She also points out the paradox that mathematics is a unique subject in being axiomatic and based on proof, while school mathematics use quite different approaches. According to Reiss, we lose far too many students by making them believe that mathematics is too difficult. Felix Klein described a "double discontinuity", and (if I got this right) she borrowed his concept to describe the situations for prospective mathematics teachers in that they go from school mathematics to a more rigorous and formal approach and then back to school mathematics to become teachers.

She noted that adult learners tend to learn things when they are particularly motivated for it, for instance if they need it, while younger learners tend to learn because they "have to" learn it. However, I think that although we have the "power" to make younger students learn because we decide they have to, it may be an idea to get even them to learn because they are interested and see that they need it... (Not that this point necessarily is at odds with Reiss' opinion, of course.)

After this plenary, there was a social programme that I will not blog about. Thus ended the short first day of the conference. Tomorrow will be the first full day, with a scientific programme from 9 to 19...