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.

What is most needed in research on mathematics teacher education?

What would you say is most needed in research on mathematics teacher education? I pondered this in relation to an application recently. I immediately thought back to the 2005 review of research on mathematics teacher education by Adler, Ball, Krainer, Lin, and Novotna. They found the field characterized by “a predominance of small scale qualitative studies (how); teacher educators studying their own contexts (who); and a predominance of publications from countries where English is a national language” (p. 375).

Not much seems different 14 years later, if I must judge based on a current review of research on the role of practicum in mathematics teacher education, conducted within the TRACE project – except that Turkey seems to be doing a fair amount of research in the field.

Pondering our work on the desired teacher and work both in mathematics teacher education and teacher education more generally, I contemplated if a connecting focus could be the development of the new teachers' professional judgement. Perhaps like this?


Why these particular satellite topics?

As argued in the TRACE project, teacher education would benefit from a critical reflection on its applicability by improving our understanding of how teachers utilise learning from teacher education in their profession. Hence the dimension ‘tracing teacher education’.

International research has focused on student teachers, practising teachers, and even teacher educators learning to better notice learner thinking (e.g., Amador & Carter, 2016; Walkoe, Sherin, & Elby, 2019). This is utilised in formative assessment.

Key in professional judgement is the consideration of the context of learners and students. With the population diversification in Sweden, socio-cultural issues need to be addressed in teacher education. This applies equally to what student teachers learn about this, and to the approaches of teacher education itself. However, there is very limited international research on this in relation to mathematics teacher education. 

Our review of research on the practicum indicated that the quality of mentoring is vital to student teachers’ learning. The same review pointed to the importance of the meeting between theory and practice. However, research on ways to facilitate this meeting is diffuse, and the notion of theory within the field is extremely vague.

There are good indication that engagement with the visions of good teaching and a critical engagement with the nature of mathematics are pivotal to student teacher learning (Corey, Peterson, Lewis, & Bukarau, 2010; Jankvist, Clark, & Mosvold, 2019).

There is much to continue contemplating ...

Iben

References

Amador, J. M., & Carter, I. S. (2016). Audible conversational affordances and constraints of verbalizing professional noticing during prospective teacher lesson study. Journal of Mathematics Teacher Education, https://doi.org/10.1007/s10857-016-9347-x 

Corey, D. L., Peterson, B. E., Lewis, B. M., & Bukarau, J. (2010). Are There Any Places that Students Use Their Heads? Principles of High-Quality Japanese Mathematics Instruction. Journal for Research in Mathematics Education, 41(5), 438-478.

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.

Walkoe, J., Sherin, M., & Elby, A. (2019). Video tagging as a window into teacher noticing. Journal of Mathematics Teacher Education, 1-21.