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Pedagogical Acumen: What should you know before you’re allowed in the classroom?
Helen Chick
University of Tasmania [email protected]
Presentation at the “Conversations on Knowledge for Teaching” Conference
11-‐13 February 2015, Launceston, Tasmania Note 1: The presentation was given with the title shown in the first slide below (reflecting the theme I had been asked to address initially), but the main point of the talk is better reflected in the title given above. Note 2: The slides are as presented; the accompanying notes are my post hoc recollections of what I thought I said, what I meant to say, and/or what I actually meant by what I did say!
The prevalence of teachers teaching “out of area”, notably in mathematics, is a long-‐standing and wide-‐ranging issue, occurring not only within Australia but other countries throughout the world, and the situation does not seem to be getting better. These people are qualified teachers, but working in an area for which they have no subject/discipline qualification. Are they, then, really maths teachers?
In an attempt to address the issue in my home state, my institution has been asked by the state Department of Education, to develop a one-‐term intensive program for these
teachers (they will study four units — two in maths, two in science — with these being equivalent to a full-‐time study load for a semester/half a year). The big questions here are: What are the most important things to include in such a program that will have the most impact on these teachers’ capacity to teach maths? What kind of knowledge do these teachers need? These are complex questions, involving recognition of the complex nature of the work of teaching a particular discipline area. To highlight this, let’s consider the following scenario, arising in a Grade 5 classroom during some work on addition of fractions.
Here, the student has used materials to represent fractions and the addition process, and the materials seem to justify/confirm/explain her answer of 2/8. Of course, most teachers — indeed, most reasonably well-‐educated adults — would know that the answer of 2/8 is erroneous. Knowing this is not enough for teachers, however. The question that the teacher has to deal with next — and it is a big question — is what to do about it.
You might like to think about what you would do in this situation. I’d encourage you to not just think about it in your head, but to try your response/explanation on someone [we actually did this in the presentation, in small groups]. What are the words you need to say? What are the materials you will use? How will you address the fact that the student’s materials seem to give 2/8 as the answer? What is the critical mathematical issue that underlies the problem here? Should a grade 5 teacher be able to deal with this scenario? After all, it’s “just” Grade 5 maths!
I should point out that responding to this student IS a hard thing. You need to have a profound understanding (cf. Ma, 1999) of the fact that fractions are determined by a relationship to a “whole”. Given that some aspects of teaching are “hard”, and that there are constraints in teacher education courses that mean that not every teaching issue that might arise can be addressed, how do we know that someone is “safe to practise” as a Grade 5 teacher? The idea of “safe to practise” has been around for some time in the medical, health, and veterinary professions.
The emphasis in “safe to practise” for health, seems to be on the idea of “competence”. In some sense, this seems to be a base-‐level requirement: in order to practise safely, you must have at least this level of functionality. Let’s take a look at another example from (maths) teaching. The next couple of slides are taken from extracts of a video-‐taped lesson on the teaching of Pythagoras’ Theorem to a Grade 8 class in the United Kingdom. At the end of what appears to be a very engaging, interactive, multi-‐faceted lesson, the students seem able to use Pythagoras’ theorem to find the unknown side-‐length in right-‐angled triangles. On closer inspection, though, there are some critical issues. The first slide highlights one: the teacher writes up a bunch of numbers and symbols (6, 8, +, x, √, =) and challenges students produce an answer of 10, hoping that they find √(6x6 + 8x8). Surprisingly, a few students manage to do this, but although the teacher connects this to the 6-‐8-‐10 right-‐angled triangle, the scope of the discovery in relationship to the whole family of different right-‐angled triangles does not appear to be investigated at all. The second slide highlights a second issue, more trivial but still important: there is no right angle marked on the golf-‐course dog-‐leg and yet the expectation is to “do Pythagoras”.
There is a tension here. The students certainly learned how to “use” Pythagoras — and so in this sense the lesson is effective — but there were many critical mathematical and conceptual things that appeared to be absent, to the point that I am not sure that the students learned all the mathematically significant things that they needed to, especially about the actual discipline of mathematics itself (and not just the specifics of Pythagoras’ theorem). Is this teacher “safe to practise”? Is he “competent”? His colleague teacher, in reviewing the lesson, certainly thought he was.
[Important caveat: the video shows neither the entirety of the lesson nor any of the following lesson, so it may be that some of my concerns are assuaged, but there are other issues which cannot be so dismissed.]
At the very least, just as for medicine, our teacher education programs should ensure that graduates are “safe to practise” — that they are competent in the classroom, with no egregious errors/problems in their teaching. But competent in what? As in medicine, the areas of competence for teaching are many and varied.
The challenge, though, is that mere competence may not be enough. Our Grade 5 teacher in the fractions example may well know how to teach fractions using good materials, but may struggle to find the insight to resolve the student’s interpretation of fraction addition with the resulting wrong answer; our Pythagoras’ theorem teacher has great teaching strategies for helping students to learn what to do, but he failed to highlight what Pythagoras’ theorem means as a theorem that applies to all right-‐angled triangles and not to other triangles. In the classroom on the next slide, where students are learning about probability and having a lesson on sample space, there is a range of pedagogical, mathematical, and social issues occurring. Some of these are routine — in the sense that a teacher who is “safe to practise” should have “stock standard” strategies and explanations and approaches that will allow him or her to address them. Other issues, like the fact that Jess’ approach to constructing the sample space is different from the teacher’s (highlighted below) may not be so easily addressed. The teacher may require some insight, in the moment after Jess has shown her work, in order to evaluate and weigh up her work, figure out what she was thinking, ascertain its mathematical correctness, perhaps decide if it can be adapted (if it isn’t correct), perhaps decide if it can be generalised (if it is correct, and so might it be used in other circumstances), and so on. This is more than being “safe to practise”; this is something of a “knack” for being able to see what is going on, understand the student, and know what to do next. Those teachers who design really nice activities for students — the activities that engage students, but hit the nail on the head with regard to developing understanding of concepts as well — also have such a knack.
This additional, higher level of functioning has also been recognised in the medical profession, in the idea of “clinical acumen” or “clinical judgement”. (Note that the parallels with medicine aren’t perfect, since teaching goes beyond the diagnosis and treatment of problems.)
Clinical acumen is most evident when dealing with “contingency”, by which we mean the unexpected, the different, the non-‐routine. This parallels the work of Rowland et al. (e.g., 2005), and the idea of “contingency” as a component of the “Knowledge Quartet”. Can we come up with a parallel concept for teaching?
Here is my attempt.
It is really important to appreciate that this created solution comes out of teacher’s existing knowledge, notably of content and of pedagogy, and that there is much that we know — from research into education — about learning and teaching already. This knowledge provides the foundation, but there are times during teaching when the teacher has to construct something that he or she didn’t know about before: a way to resolve a student’s misconceptions, a task to assess understanding, an activity that will reinforce conceptual understanding, a real-‐world example of a certain phenomenon, and so on.
In particular, this definition acknowledges the dynamic nature of knowledge development in teaching. Some knowledge is acquired through teacher training, professional learning, mentoring, and the like; other aspects may be constructed by the teacher in response to a need. The trick is — as in medicine — finding out whether or not that acumen is effective. Since all my examples so far have been mathematical, an area in which I believe I am “safe to practise” and where I hope I have some pedagogical acumen, I thought it might be salutary to take me “out of area”. Would I be safe to practise as an English teacher? Am I likely to have pedagogical acumen for this area? I have done no teacher training in English but I would say I have high-‐level literacy [spot the grammatical errors in this document — aaggghhh!!] and am moderately well-‐read. Could I teach a secondary English lesson, perhaps using Harry Potter?
Pedagogical acumen is the practical reasoning employed when a teacher interprets the full context of the educational situation (as best it can be seen) and draws on existing knowledge to create an effective response.
[This scenario is “potentially scary” both for me attempting to be an English teacher and for any poor students upon whom I might be inflicted!] What are some of the things I might be called upon to teach?
I think I can deal with apostrophes. I know the rules; I even think I have an analogy that helps with deciding what to do in the “one hour’s time” case. But do I have a way of teaching about apostrophes beyond dictatorially laying down an interminable list of
rules? [Yes, I know sentences should not begin with “But”!] In this case, my content knowledge may be okay, but maybe my pedagogical content knowledge is lacking. The question about teaching a lesson on the wizarding/muggle world parallels is trickier. Here I suspect my content knowledge would be stretched (e.g., I can’t talk about “metaphor” and “simile” fluently), and, again, pedagogical content knowledge is lacking too (e.g., I am struggling to think of a good starter activity that would get students thinking about the parallels, so I know enough to know that I need a good starter activity, but I am not sure I know enough to actually construct one). Perhaps I might just be able to devise something, but it would be a “safe to practise” something at best; I doubt that I have the pedagogical acumen — in part because my foundations are lacking anyway — to devise something of the more inspirational kind (my efforts would, perhaps, be like a lesson taught by a Lockhart or a Hagrid, rather than a McGonagall). As for teaching about caricature … I think I could do some of it — so, perhaps, I am “safe to practise” — but, again, the real challenge would be how to inspire students with the idea and to run with it, and to tease it out to build deep conceptual understanding of it. How could I become better as an English teacher? I think some professional learning could help me become “safer to practise” but can I learn to develop pedagogical acumen? If I want my pre-‐service teachers to become the ones who can address the fractions problem, realise what’s missing in the Pythagoras lesson, and identify Jess’s sample space, then what to I need to do as teacher educator to help them get there?
This brings us back to the teachers who are teaching out of area. As a potential English teacher, I felt a little hamstrung by my lack of content knowledge and my lack of PCK. I need both as foundations if I am to develop pedagogical acumen, I think.
I certainly think it is important that we aim for more than just “safe to practise” in our teachers. It is good to have a doctor who can treat a broken wrist, but it is even better if the doctor can realise that the wrist break arose from balance difficulties and that such problems are symptoms of a certain disease. If we want to meet the needs of students in an effective rather than merely functional way, then it is pedagogical acumen that we need.
References General Medical Council (UK). (2013). The meaning of fitness to practise. Retrieved from http://www.gmc-‐uk.org/guidance/21721.asp Keates, H. (2012). “Teaching Veterinary Anaesthesia: Are we there yet?”. Presentation at the conference on “Building a Culture of Evidence-‐based Practice in Teacher Preparation for Mathematics Teaching” (CEMENT) (7-‐8 June 2012). Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates. Montgomery, K. (2006). How doctors think: Clinical judgment and the practice of medicine. Oxford, UK: Oxford University Press. Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255-‐281.