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.

Sequencing in mathematics teacher education

All teacher education programs I am familiar with require students to attend courses. In our case, courses in mathematics, courses in general pedagogy, courses in practicing teaching, courses in ethics, courses in research methods, and courses in mathematics education. A lot of the sequencing of the content in the mathematics courses is given by the nature of the content. For sure, one can learn differentiation before integration, or integration before differentiation, but one needs to first have an understanding of what a function is in both cases.

That is not so for mathematics education, as I see it. What then could be viable principle(s) for the sequencing of content?

In a recent workgroup revising one of the programs at my institution, we considered this. And came up with these suggestions:

(a) We work from students working with part of a lesson, to a lesson, to a sequence of lessons, to term or year.

(b) We work from students working with teaching strongly classified mathematics to students working with teaching more interdisciplinary.

(c) We work from the elements that research identifies as easier for students to apply (such as exemplifying using variation theory) to the elements they find more difficult (such as engaging learners in reasoning).

(d) We consider carefully which "eye-openers" can be engaged when, so that we meet the students where they are in this respect. It does not make sense, for instance, to wait to challenge the idea that the teacher is one that must explain procedures to learners.

This resulted in us sequencing the content into four parts:

(1) Noticing learner thinking
(2) Making mathematical objects/concepts available to learners
(3) Giving everyone access to the world of mathematics (inclusion)
(4) Furthering mathematical thinking.

We would love to hear the views of others on these ideas.

/Iben