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

Selection of content in mathematics teacher education

The issue of what we need to include in a mathematics teacher education and how to sequence it is something I have grabbled with for some time now. It is all good to say that all the aspects in Ball et al.’s famous egg are justified content components. But few programs last long enough to cover this in a reasonable sense. My standard response to the selection question is that we must select content that is in some way exemplary – by which I mean that the content deals with a specific aspect that reflects and exemplifies greater issues.

 But in what ways? Exemplary to the main ‘knowledge of students and content’ (KSC) components? Exemplary to various pedagogies? Exemplary to the mathematical content? The challenge becomes to select content and related activities that illustrate several aspects of the complexity of mathematics teaching. But also to decide that some aspects are so overarching or so important that they need to become a main idea, running through courses. Which, then, are these?

Five years ago, I moved from a university in South Africa, to a university in Sweden. Fascinating to me was that the issue of KSC received relatively little attention at my new institution. What in the contexts made it so different what teacher educators thought their students needed? Horizon content knowledge was not widely engaged either, but all students had to learn to engage with research both as consumers of published research and as producers of small research studies, and curriculum theory was addressed within mathematics education courses as well. On the other hand, the psychological perspectives that had been included in the mathematics education courses at my South African institution are here mostly addressed in generic pedagogical courses.

The ’flower’ in an earlier blog was an attempt to identify a main idea, namely that teachers must utilize professional judgement. But the various ‘petals’ in the flower are of such different nature that they cannot guide selection of content.

One possible starting point could be the main professionally informed tasks or activities in which mathematics teachers engage. Then noticing, identifying key mathematical ideas, and guiding mathematical explorations are perhaps focal points?

Another potential starting point is to use a particular view of mathematics. To choose the perhaps most current, commognition. What would that mean to what happens in a classroom and what a teacher needs to know and be able to do?

Yet another way would be to use a particular mathematics education theory as a way to select content. The best candidate for this is perhaps ATD with its ecology, both for mathematical content and for education. The main challenge here is that each mathematics education task does not neatly correspond to one or a few techniques.

Add to this the issue of different competencies being more or less difficult to acquire. For instance, a recent paper in a special issue of ZDM suggests five levels of difficulties, where differentiation of instruction first comes in at level 4, and more explorative learner activities only became more dominant at level 5 (Kyriakides, Creemers, & Panayiotou, 2018). How does our ideal teacher then fit with what it is reasonable to expect from all our students? And what does this mean to the sequencing of content?

Kyriakides, L., Creemers, B. P., & Panayiotou, A. (2018). Using educational effectiveness research to promote quality of teaching: The contribution of the dynamic model. ZDM, 50(3), 381-393.

From teacher to teacher educator - what is good to know?

It is not unusual for teacher educators to be recruited amongst teachers. Adverts for new teacher educators may even specify school teaching experience as a requirement. However, besides possibly being required to engage in research, there is a difference between teaching mathematics and teaching others to teach mathematics. I was curious as to teacher educators’ experiences of this difference and what – over and above teaching experience – they felt they had needed. So I posted this as a question on ResearchGate.
 

Several of the responses reiterated the importance of teaching experience. If the sense I made of the Spanish is correct, one reply also pointed to the need for greater knowledge for the teacher educator, the sharing of research as a different practice, and the (greater) need for entrepreneurship.
 

Mentoring/feedback from students or colleagues also came up. Michael A. Buhagiar commented: “Something which I truly wished I had when I became a teacher educator was some form of mentoring - someone who is there to guide you as you enter this new world. I felt at a loss with regard to what was expected from me as a member of a Faculty of Education. I had no clue of the 'do' and 'do nots' both with regard to colleagues and students. It is as if people expect you to know what to do when in reality you do not know. I'm thinking in particular to course structures, assessment procedures, admin work and also what i was entitled to as a university staff member. I felt deskilled even if I had been a part-timer for a number of years.”

Lisa Österling mentioned co-teaching on existing courses as a good way to transition, and I guess this is a form of mentoring? 

Some mentioned the mental shift from teacher to academic required, and how different the teaching role must be as treating student teachers as learners does not facilitate their growth as professionals.

Personally, I feel at loss at times with things that would be a lot easier to me if I was teaching mathematics. For instance, I can generate or adapt mathematics tasks that simultaneously guide learners to ‘see’ some connections and to practice mathematical reasoning. Tasks where the task situation provides a substantial amount of the feedback to the learners, very much in the tradition of didactical situations. But it is much harder to do the same in teaching mathematics teachers, because reasoning is not sufficient to solve a teaching task in a reasonable way; it requires feedback from a broader context which does not necessarily "behave itself".

Michael A. Buhagiar reflected on the ways to introduce theory to students: “With regards to teaching, I gradually learned that student teachers want their lecturers to be practical rather than theoretical. As I'm a strong believer in a strong theoretical basis to support teacher practices, I had to learn how to introduce theory through practice. And this seems to work as my students continually comment that they like this approach.” 

Very much in line with the realistic teacher education approach! (Korthagen et al., 2001). But it leaves unanswered which theories, why these, and what situations/practices generate the need for some theory. What tasks must we generate and set for students to look at practice differently?

Another example. Variation theory tells us lots about how to vary our examples in teaching. I can do that when I teach mathematics, and even when I teach research methods. But when it comes to teacher education, I stumble. I am no longer after conceptual understanding but after developing a basis for professional judgement. What examples work for aspects of that?

So moving into teacher education myself meant confronting the difficulties adjusting well-tested theories to the new "content" as well as to the adult students who have made teaching their career choice.

One response made a distinction between a teacher trainer and a teacher educator, where the former deals with specific and relatively clearly demarcated aspect of teaching mathematics. In contrast, the teacher educator must consider the interplay of all the different relevant aspects, and the journey that the student teacher must traverse.

My own view on this is that the teacher trainer may have in mind a particular approach that ‘should’ be implemented in the classroom (see also the view on theory discussed elsewhere). On the other hand, the teacher educator needs to engage more deeply with the so-called "mathematical knowledge for teaching teachers" (Jankvist et al., 2019; Zopf, 2010) or perhaps “mathematics education knowledge for teaching mathematics teachers”.

How then do we develop this vast knowledge of the philosophy of mathematics, curriculum theory, subject specific education, general pedagogy, etc.? How do we select relevant areas to transpose didactically? How do we sequence this into coherent and engaging programs? How do we obtain reasonably well-founded answers to these questions?
 

Reference

Jankvist, U. T., Clark, K. M., & Mosvold, R. (2019). Developing mathematical knowledge for teaching teachers: potentials of history of mathematics in teacher educator training. Journal of Mathematics Teacher Education, 1-22.

Korthagen, F. A., Kessels, J., Koster, B., Lagerwerf, B., & Wubbels, T. (2001). Linking practice and theory: The pedagogy of realistic teacher education. Routledge.

Zopf, D. (2010). Mathematical knowledge for teaching teachers: The mathematical work of and knowledge entailed by teacher education. Unpublished doctoral dissertation. http://deepblue.lib.umich.edu/bitstream/handle/2027.42/77702/dzopf_1.pdf.