I spent the last eight years observing teachers in mathematics classrooms, trying to work out the relationship between their thought and action, and before that I spent eight years in the classroom myself. Where, I gave some thought to what I was doing the classroom.
It is trite to say that the classroom is a complex place, but it is no less true to say it. Learning is perplexing. Mathematics education in a state school is mystifying. It is no surprise then that theorists in education have all but given up theorising it all, but prefer to take a partial view – to look at aspects, elements or cases within the totality.
A popular explanation in mathematics education research is that teachers’ actions and behaviours are underpinned by their knowledge and beliefs. It follows then, that to change teachers’ behaviours in the classroom, one might deploy professional development that is designed to develop the teachers’ knowledge and that changes beliefs. There are many things wrong with this approach, not least is that in many cases it doesn’t lead to lasting changes. Fundamentally, it is the treatment of the teacher as deficient with little value is given to teacher autonomy and agency. There are theoretical problems too. Theory is based on knowledge and beliefs have many associations with constructivist perspectives on the psychology of learning. Knowledge and beliefs, in relation to teachers’ thoughts and actions, suggest that the teacher constructs, mentally, a guide fraction and then they follow it. What this disregards, is the effect of in-the-moment responses and decisions by the teacher. A teacher’s thinking is much more dynamic than the constructivist view might suggest.
While critical now, it was against this theoretical backdrop that I begin my research into mathematics teacher professional learning in 2010 as part of my PhD research.
Mathematics teachers’ beliefs were the preferred explanation of my funders and supervisors of teachers’ thinking and their classroom practices. It was also their preferred explanation of how teachers learn new practices and approaches. I had an extended period where I was critically engaged with research and theory around teachers’ beliefs.
While the popular account of mathematics teachers’ actions was based on their knowledge and beliefs, there were competing views coming from a ‘social’ perspective on learning. In this teacher learning involves a process of becoming socialised into a ‘community of practice’. It is an indoctrination into practices and ‘ways of doing things’ – adopting the principles, language and ideas of the mathematics teaching profession, especially as it is in the locality. ‘Change’ or teacher learning must involve some change in the community to permit the individual teacher to change.
As I began to collect data, I felt that the ‘constructivist’ (that based on knowledge and beliefs) and the sociocultural both were valid but partial explanations of what was happening. The research literature appeared to show that the constructivist and sociocultural views of teacher learning were mostly in an ideological conflictual impasse.
My classroom observations revealed another aspect of professional action, which where non-cognitive factors such as motivation and confidence. These appeared to have a considerable impact on the way in which teachers taught, whether they would implement ambitious teaching approaches as opposed to whether they would stick more resolutely to the orthodox teacher explanation, followed by student practice. Ambitious teaching (Stylianides & Stylianides, 2014) is where greater mathematical authority and authorship (Povey & Burton, 1999) is given to the students. It is more demanding on teachers since the lesson becomes less predictable, the teacher devolves control. And while this can offer a positive learning experience for students, they can actually experience what it is like to be a mathematician, it can also substantially increase the level of anxiety in the classroom. This also makes the teacher anxious. This increased anxiety can encourage the teacher to return to well-established routines, routines like traditional chalk-‘n’-talk followed by student practice.
Earlier this year I presented a paper at the Congress of the European Society for Research in Utrecht. In this paper, I revisit the research into teacher thinking, or particularly, teacher decision making and the nature of the choices they make in the classroom (Watson, 2019). Based on my research (I have actually spent about four years looking at one teacher do one lesson and his reflections on his thinking during the lesson), I believed that the character of the lesson was heavily influenced by the momentary decisions that teachers make. They constantly have a choice to follow well-established routines or to open the learning up and give more mathematical author/ship/ity to the students.
The research into classroom decision-making revealed that a primary aim of the teacher was to maintain the ‘flow’ of the lesson (Clark & Peterson, 1986), that is to maintain it as a socially smooth-running experience. If you imagine a middle-class dinner party with a degree of formality, there are a number of social routines and passages of discourse that fill the time without creating an awkward situation in which someone might feel ‘uncomfortable’. In such a situation the level of discomfort might lead to an unpredictable or ‘controversial’ response. The ‘smooth running’ of the dinner is destroyed (I don’t say that this is a good or bad thing, least to say that such things are the inspiration for Mike Leigh e.g. Abigail’s Party).
While the teacher in a mathematics classroom might have less interest in middle class aspirations as the basis for wanting to maintain flow and smooth runningness in their class, there is a similar motive for affective containment – for staying in comfort zones.
And I am not the only one to deploy the analogy of dinner. Stigler and Hiebert, in their video study of practice in the USA, Germany and Japan, observed a culturally-specific ‘script’ in the mathematics lessons they observed. They suggested that the routines in mathematics classrooms were culturally embedded and that they were smooth running because teachers and students all knew the parameters of the script that they were expected to follow.
Family dinner is a cultural activity. Cultural activities are represented in cultural scripts, generalized knowledge about an event that resides in the heads of participants. These scripts guide behavior and also tell participants what to expect. Within a culture, these scripts are widely shared, and therefore they are hard to see. Family dinner is such a familiar activity that it sounds strange to point out all its customary features. We rarely think about how it might be different from what it is. On the other hand, we certainly would notice if a feature were violated; we’d be surprised, for example, to be offered a menu at a family dinner, or to be presented with a check at the end of the meal (Stigler & Hiebert, 1999, Kindle locations 1098-1103).
In my recent work on teacher decision making, I have created an integrated model of teacher decision making which incorporates cognitive psychology and social psychology: it reflects the cognitive, affective, social and cultural aspects of human action. I sketch this out in a little more detail in the conference paper I mentioned earlier (Watson, 2019), but to summarise the key ideas around teacher decision making in the classroom: decisions begin with the senses. The teacher observes a class’s and individuals’ behaviours. The teacher continues to implement their lesson plan (a mental model or script of the lesson) until there is something that draws their attention, it might be a student having difficulty with the activities or tasks or some other behaviour that is raising the level of anxiety in the classroom. The effect of this is that the teacher’s attention turns to the phenomena and the teacher’s level of anxiety might increase. All this is taking place unconsciously using the autonomic nervous system (the limbic system). It might be that the teacher responds unconsciously, there might be a routine or ‘script’ in the teacher’s memory that they might deploy because it is a fairly routine situation to deal with. An experienced teacher does not need to do lot of conscious deliberation over the situations they meet, they have experienced many similar patterns of behaviour and are able to use this embedded knowledge to respond without thinking. This is a useful thing in demanding situations, since conscious reasoning is demanding on the body’s resources. Yet, there are situations in which the teacher might meet a difficult situation in which they have to think more deeply about a possible course of action. And while meditation on an issue is often of value, in fast-moving and demanding environments like the maths classroom, it is an indulgence that has limited opportunity to be enjoyed. The teacher is very much relying on culturally embedded scripts and pre-thought routines to guide their actions in the lesson.
The cybernetics of teacher decision making
I want to examine teacher decision making using cybernetics. Because, I think it will tell us more about the classroom environment rather than just focussing on individuals. I am going to treat the mathematics classroom (or any classroom) as a dynamic system. This deemphasises the individuals in the classroom and incorporates all objects and matter. We therefore have a complex dynamic system, within which there are other complex dynamic systems i.e. the teacher and the individual students. You will note that I am not treating them as ‘black boxes’ but as dynamic systems that co-exist.
A surviving dynamic system
The classroom as a part of an institution, as part of an education system, must endure as system. It has to be contained and ‘productive’ whatever that might mean in this context. If it ends up out of control at least it is time limited (and I have had some classes that have gone out of control and observed classes that have been close to degenerating into an out-of-control state). The state of being out-of-control ends with the end of the lesson. The condition that the individuals leave the class might have an effect on other classes, but the instability of the system has ended with the buzzer or bell. Stafford Beer points out that institutions and organisations have to be surviving dynamic systems, they have to adapt to their contexts and internal and external perturbations in order to remain stable.
A central law in the stability of dynamic systems is Ashby’s Law of Requisite Variety. This tells us that the only way in which variety can be absorbed by a dynamic system is through matching it with the system’s variety. This is to say that whatever the number of possible states of the environment or the context, the only way a dynamic system can maintain stability is by having a matching number of possible states. It is not always possible to design systems so that they have enough variety to counter their environment or context’s variety. A mathematics classroom is a complex context and to control or attenuate the variety, the system is regulated by introducing rules and practices. The teacher provides the regulatory function by applying and enforcing rules and controlling behaviour. The ‘variety’ in respect to the individual and collective students is attenuated to match the variety available, not only in the class but also in the school. Schools have limited resources and limited flexibility, so there is a great need for students to conform in order that they do not exceed the variety available in the school and create instability.
If students are not from school-oriented backgrounds then the level of variety increases a further few notches and the school with its finite resources and organisational inflexibility must introduce further regulation. However, in many cases though, this scope for regulation is not possible and the law of requisite variety is not met and you get a ‘troubled’ school.
While regulation has the effect of maintaining the stability of the classroom it has an effect on the learning that is taking place in the lesson. Part of the regulation process leads to a ‘traditional’ approach to learning, the teacher explanation followed by student practice. All this is inhibiting variety to keep the classroom ‘stable’.
There is dissonance here, a tension or a conflict; regulation of variety to match the limited variety of the school and the education system and other hand this regulation has an impact on the learning process. Let us think here of individuals as dynamic systems engaged in learning a complex subject like mathematics. The curriculum is determinate, it is a body of knowledge and practices, but represents a regulated version of what mathematics is as a dynamic system. The mathematics curriculum is determinate while mathematics is an indeterminate dynamic system.
In cybernetic terms the learning of a dynamic system is developing adaptability: to develop the capacity to survive amongst complexity and unknowability. Yet, the attenuation that takes place in the classroom, in the school and in the education system does not provide an environment in which students can develop and use ‘variety’. As a society we tend to ignore the indeterminacy and accept the assumption that learning must be determinate and that the society we live in is determinate. Effectively, our education system is attenuative of variety, which is the process of social reproduction that Marxists refer to.
The mathematics classroom as ontological theatre
But I am drawing myself into a cybernetic analysis of the education system – something that I don’t quite want to do quite yet. I just remark that the education system is significant in the work of the teacher as a dynamic system. But where I need to get back to presently is the ontological theatre of the mathematics classroom.
Ontological theatre is a term used by Andrew Pickering in the opening of his book, The Cybernetic Brain – a book that tells the story of the British Cyberneticians.
Cybernetics presents a view of the world as ‘theatre’. These are performances, rather than Enlightenment representations. The philosophical basis of cybernetics is ontological, it is performance that creates a reality, that gives the world form. This is weird if one thinks of it in terms of entities. External objects ‘exist’, they are not formed through performance, they are already there. But don’t think of entities, don’t think of the world as the object of our thought, think how it is brought into being by being a product of the formation and interaction of dynamic systems. This is not agents bringing the world into being, but about dynamic systems interacting with agency as an ‘output’ of the processes. That is not to say we don’t have control i.e. free will. Our free will is the capacity to assert our adaptability and not, as it is often considered to be us asserting ourselves on the future. No! We can’t do that.
An ontological theater […] a vision of the world in which fluid and dynamic entities evolve together in a decentred fashion, exploring each other’s properties in a performative back-and-forth dance of agency (Pickering, 2010, p. 106).
This is from the chapter on Ross Ashby, we see the suggestion that ‘entities’ are dynamic systems in equilibrium in a complex and unknowable environment.
In order to consider the ontological theatre of the classroom, we have to dig deeper and think about what we mean by thinking (and learning) in cybernetic terms. You will see some links not just now but in what I have already written that there are some shared concerns that are raised by the new materialists and even the object-oriented ontologists.
Clark, C. M., & Peterson, P. L. (1986). Teachers’ thought processes. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd ed., pp. 255–296). New York: Macmillan.
Pickering, A. (2010). The cybernetic brain: sketches of another future. Chicago ; London: University of Chicago Press.
Povey, H., & Burton, L. (1999). Learners as authors in the mathematics classroom. In L. Burton (Ed.), Learning mathematics: from hierarchies to networks (pp. 232–245). London: Falmer.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: best ideas from the world’s teachers for improving education in the classroom. New York: Free Press.
Stylianides, G. J., & Stylianides, A. J. (2014). The role of instructional engineering in reducing the uncertainties of ambitious teaching. Cognition and Instruction, 32(4), 374–415. https://doi.org/10.1080/07370008.2014.948682
Watson, S. (2019). Revisiting teacher decision making in the mathematics classroom: a multidisciplinary approach. Presented at the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11), Utrecht University.