Time of Covid and time of exams

 This time of the year, the pressure is on for both students and teachers, with both exams and Covid. For the first time, a group of student teachers I never met IRL are finalizing their first year of teacher education. Fortunately, students have been amazingly creative, even though teacher education online due to a pandemic is new to all of us. 

Student teachers doing their practicum in schools were not only subjected to online teaching as students, the also had the experience of teaching and observing online teaching in schools. In Sweden, secondary education this spring was a mix of in-school and online education, where different solutions of one week home- one week online were common, to facilitate social distancing. Several student teachers took to opportunity to use their didactic tools to compare in-school and online teaching. As an example, the milieu  or didactic variables of Brosseau proved helpful to describe how mainly the opportunities for students and teachers to engage in discussions online were reduced, and as a consequence, students were left to work alone with little support. For this reason, it was difficult to organize for "a-didactical situations" online, where the teachers would delegate responsibility for working with a mathematical activity to learners. As a consequence, much time for learning mathematics was lost.

The positive thing is that we all are more confident in digital tools for education. The main concern for me was never my own digital competence, but the effort to make sure how the counterpart could participate. Now, systems have improved, and everyone is more or less familiar with their use. This will probably impact the way we work and meet in the future. 

Another positive thing is the movies and materials produced. Below is a screen-shot from a digitalization of a task designed by Malcom Swan (https://www.educationaldesigner.org/ed/volume1/issue1/article3/) which I moved into a digital board, where students could copy-paste digitally. It actually was easier to move the pieces around, not having everything blowing around when someone opens a window! I really think that much of the things we produced are possible to reuse later on. 

I wish I had time to engage in deeper research on the different adaptations of mathematics teaching to the present situation, how teachers, student teachers and students experience the situation. I see publications of this kind will begin to appear, and I wonder if the pandemic will also alter the direction of research in mathematics education, 

Practical rationality and constructive ambiguity

30 years ago, I came across the work of Magdalene Lampert and was impressed. By the writing, the clarity, the insistence on immense respect for the complexity of teachers' work. Then time came and went, kids that were born during academic reading and writing grew up amidst more academic reading and writing, and I sort of but never quite forgot the work of Lampert. How much fun it was to re-read her 1985 text How do teachers manage to teach? ! I love the way she hooks me on a narrative which she returns to several times before providing the closure - the need for which kept me reading. I love the way in which the word "manage" takes on a different meaning after reading through the paper. I love the way she equates managing teaching with having internal debates with oneself about ways to manage problems that cannot be solved. The ambiguity a teacher feels confronted with difficult choices can be used constructively to manage the situation rather than pick one of several problematic choices. A teacher may draw on research for this, but research can never suggest a best way out of the difficult situations.

Some of these ideas have been picked up on by Patricio Herbst and Daniel Chazan in their notion of practical rationality. The fundamental idea, as I understand it, is that teaching actions are justified with respect to (a) the norms of teaching this particular content, (b) the professional obligations on the teacher - of course impacted by the teacher's knowledge, competencies, beliefs (or patterns of participation), and identity. They use this to aim for descriptions of mathematics teaching activity generally, not specific to individual teachers. As they so clearly write, it is a way out of only doing applied experimental work on mathematics education research (Herbst & Chazan, 2011).

Clearly, these are useful perspectives in engaging in research on the practice of mathematics teachers. I'm busy writing up a paper on "Tanja" who breaks with the norm I had in mind when she changes strategy in the middle of a lesson shaped around an exploration to almost give learners a procedure to follow. Unpacking her reference to the obligations she mentions in interviews I conducted with her (as part of TRACE) helps me understand the dilemma she may have felt. It also helps me engage her more in the future, because she chose a way out rather than manage the situation (in my current interpretation at least).

That way, it also helps me in thinking about teacher education. When Grossman and colleagues write that

“… professional education must help novices attend to the complexities of interaction, whether in a classroom, congregation, or therapist’s office, and to respond in the moment under conditions of uncertainty.” (Grossman et al. 2009, p. 2060).

there is no distinction made between responding to uncertainty through making difficult choices or through managing in Lampert's sense. If indeed teacher education pedagogies must include representing teachers' practice to student teachers, do we represent this aspect of managing, do we deal with dilemmas, internal debates and conflicting sides of one's identity? If such pedagogies must include decomposing practices so that they can be analysed and understood, do we analyse situations to which we cannot as teacher educators offer solutions? And if we do so, do we do it with more than the standard answer that the way forward "depends"? And if such pedagogies include engaging students in approximations of practice, do we put students in situations of dilemmas under more controlled circumstances? Which dilemmas could we work with constructively?

Some people say that a classic in literature is one that can be read again and again as one ages, offering something new at each read. To me, Lampert's text may well be a classic, not because it offers me something new every time, but because it reminds me of some key issues that I tend to forget in my attempt at making the profession accessible to students. Even though I deal with dilemmas in my own teaching regularly.

Perhaps we should make our students read her text?

Have a lovely summer if you are on the Northern hemisphere. Otherwise, enjoy your winter break.

/Iben

References 

Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P. (2009). Teaching practice: A cross-professional perspective. Teach. Coll. Rec., 111(9), 2055-2100.

Herbst, P., & Chazan, D. (2011). Research on practical rationality: Studying the justification of actions in mathematics teaching.The Mathematics Enthusiast, 8(3), 405–462. 

Lampert, M. (1985). How do teachers manage to teach? Harvard Educational Review, 55(2), 178-194

Selecting content for a teacher education programme

It's been a while! Last time, I wrote about the sequencing of content, but not about the selection of the content. This, however, was actually our starting point. We had a two-dimensional view of the content as our starting point, but as the work continued, we developed our view. I am hoping that by sharing some of that journey, we can engage in dialogue about the principled thinking other mathematics teacher educators engage in programme development.

The starting point

Previously, the four mathematics education courses had been arranged according to mathematical topics. It was important to still cover the key ideas within topics, but we decided to use mathematics education ideas as the organising principle. In order to cover both, we worked from a matrix:

	Learner thinking	Representations	Discursive perspectives	...
Number/arithmetic
Geometry
Algebra
...

The idea was to decide on a sequence of the mathematics education perspectives and then have a primary and a secondary mathematics topic connected to each of these perspectives. And this is how we worked.

It gave rise to these four courses:

(1) Noticising learner thinking, with a focus on number and probability

(2) To make mathematical objects/concepts accessible to learners (hereunder representations), with a focus on geometry and algebra

(3) To open mathematics to all (inclusion), with a focus on statistics and measurement

(4) To further mathematical thinking and communication (hereunder reasoning), with a focus on algebra and geometry

Adding the pedagogies

Stumbling across an article for other purposes added another dimension.

Taking clinical practice seriously will require us to add pedagogies of enactment to our existing repertoire of pedagogies of reflection and investigation. (Grossman, Hammerness & McDonald, 2009, s. 274)

Of course we were aware of using different pedagogies, and as most teacher education programmes, ours have to include practica which are pedagogies of enactment. But the quote draw our attention to how these pedagogies could be used more as planning tools, to ensure that the different pedagogies are linked. In addition, we felt that a pedagogy was missing, namely the reproducing or acquisition pedagogy, where students engage texts in order to learn about existing concepts, research results and theories.

This pointed us to a three-dimensional model, where each activity in the courses would address (mainly) one mathematics education point, one mathematics topic, and use on pedagogy. And where the activities would link through the different pedagogies. For the mathematics dimension, we added a distinction between focusing on mathematical objects/concepts or mathematical practices/discourses. Of course other approaches are possible, but we felt this was a powerful planning tool. Here's an example of more detailed planning:

The colour codes made it easy to get an overview of which dimensions had been covered. It is the same yellow throughout here, which indicates that the focus is on learner thinking throughout. It is the same green, which indicates that the focus is on number sense and conceptual understanding. It is the same red/pink, which indicates that the focus is on number. Only the blue column changes, and this reflects the changes in pedagogy.

Transformations between theory, practice and the empirical

Incidentally, by including an analytic element - nothing new in doing that - we also ensure that teacher education engages theory as more than a way to inform practice, a normative theoretical perspective. Working with actual learner tasks, classroom videos, teaching materials etc. and analysing these, we bring in the empirical dimension (something discussed very insightfully in Carlsen and von Oettingen, 2020). As Carlsen and von Oettingen point out, these three dimensions offer different perspectives on the same incident, and thus allows one to see the old as unfamiliar or the new as familiar. It is in the interactions and not the least the transformations of one perspective to another that the real learning may happen, the one that also transforms the self.

References
Carlsen, D., & von Oettingen, A. (2020). Universitetsskolen–et bud på en didaktisk orienteret forskningsbasering af læreruddannelsen. Acta Didactica Norden, 14(2).

Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re‐imagining teacher education. Teachers and Teaching: theory and practice, 15(2), 273-289.