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.