It all kicked off on Twitter after I posted a journal article

Did I see that coming? Well, possibly, but I didn’t consciously set out to provoke such a Twitter response when I posted a link to my most recent academic publication on social media. Within a few hours of my article, New Right 2.0: Teacher populism on social media in England, being published by the British Educational Research Journal (BERJ) on Friday 24^th July, the article was receiving unprecedented attention on Twitter. Unprecedented, not only for me, but for BERJ and for an academic publication on education research more generally.

Colleagues and friends contacted me over the weekend to ask me if I was OK. It seems that for many of my associates, the response to my BERJ article was predominantly hostile. A ‘pile on’ as it is frequently referred to.

A screenshot of a cell phone

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It wasn’t so one-sided, however, I was receiving at least as much support through other communication channels as I was facing robust criticism on social media.

The article itself considers how Twitter – and specifically ‘#EduTwitter’ as is my research focus – can be productive and collaborative, but it can frequently become divisive and angry. The educational schism that my paper considers is between the Trads and the Progs. The Trads or traditionalists are a consequence, I argue in the paper, of three factors: the New Right, the coalition of social conservatives and economic liberals that emerged in the 1950s in the UK and US as a reaction to post-war social democracy, Keynesianism and the welfare state; the erosion of state-sector teachers’ working conditions over the last twenty years; and as a result of effects of social media. Trads advocate for robust discipline in the classroom, educational practices that are orientated toward memorisation and for research evidence based on ‘scientific’ research methods. The political positioning of the Trads is characteristically populist, the unheeded teacher against a progressive elite. I coin the term ‘micropopulism’ to distinguish this niche populist tendency. The Progs emerged as a less coherent and less organised reaction to the Trads’ social media presence.

It was pointed out that while much of the reaction to my article denied the existence of Trad micropopulism, the actual Twitter reaction to the article provided demonstrable real-time evidence of the phenomenon and the main argument of the paper: that social media is divisive and can amplify populism in unproductive ways.

The reaction to my article did feature a populist attack on institutions – the academy (i.e. higher education institutions), the British Education Research Association (the professional association for which BERJ is the flagship academic journal) and for peer review.

In the reaction, I am characterised as a ‘gatekeeper’ for the progressive elite that exists in the academy and that has been central to the power that has foisted unscientific progressive education approaches on teachers. There were further important observations in the reaction to my article. I was robustly challenged as characterising Trads as right wing. In fact, at no point during the paper do I make such a suggestion. I do argue that there is a relationship between new right think tanks and Trad micropopulism on social media, but I have never believed that Trads’ primary political associations or voting have been for the Conservative party. What I do find interesting is those self-identifying leftist teachers should be so enthusiastic about the reforms of a new right politician such as Michael Gove. The apparent benefit of Gove’s curriculum reforms seemingly outweighs the transfer of millions of pounds worth of public assets to private interests as part of the ramping up of school academisation since 2010 by the Coalition and subsequent Conservative governments.

Cybernetic decision making in the classroom

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).

Awkward.

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.

Reference

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.

Apparently there are too many PhD students

There have been some conversations in the University, I understand, that there are too many graduate students competing for too few academic jobs. There was some discussion also that we should reduce the number of graduate students. While the first statement might be true, I take issue with the second.

Globally, there might be finite resources and funding for academic work. Certainly in England, I suspect (I am not going to look at the data just now) investment in academic work has probably diminished over the last 40 years. If it has not diminished, then the source of that funding has increasingly come from private sources – whether that be applied research for industry and business or debt-funded undergraduate study. Higher education, in England, is a competitive market. This, I believe, is the source of the pressure. Whether that be the result of tightened funding or consequence of the business/corporate market language is immaterial. The issue is, then, the question of whether there is too much demand from people to do scholarly work. Should we be placing a limit on access to research degrees?

I think not.

As each moment passes, each day, as each year, decade or century passes, we create for ourselves a more complex world – a more complicated world. Our capacity for sophistication holds no bounds. Yet, we also create for ourselves considerable problems. The Enlightenment held for us so much promise. With our minds, we had an unlimited capacity to develop technology and prove ourselves masters of nature. The Enlightenment also gave us the belief that we would be able to solve rationally, moral conundrums. However, we have been repeatedly humbled by nature. If we think about the twentieth century, humanity experienced the most violent century in history. The horror and the destruction were way beyond the experience of being violently consumed by a predator. This was violence on a man-made industrial scale and was not designed with quick dispatch in mind. It was constructed withe cruel and horrific vision.

We do need scholarship – active/activist scholarship – that can help us address the complex problems that humanity faces. These problems while they exist in the chaos of nature are the product of human reason. There is something intuitive about nature’s chaos, as living beings, we can cope with the unknowable and the uncertain. I was saying to the trainee mathematics teachers on Friday, each of us as individuals, has a surface with an almost infinite area. The contact between each of us and our environments is infinite – or approaching infinity, to be more mathematically precise. There is an infinite exchange of data. If we were to remove our cerebrum – the part of the brain responsible for rational thought – then using our limbic brain we could continue to live our lives. We can cope – and we have to cope – without the power of reason, because there is simply too much to reason about.

Antonio Damasio’s book Descartes’ Error begins with the story of Phineas P. Gage, who suffered a life changing accident while at work in the summer of 1848. Gage was a 25-year-old construction foreman. He was working in the construction of the railroad in Vermont. As they blasted through rock to allow the railroad to proceed on a straight course, Gage was setting charge. At four-thirty on a hot afternoon, he put powder and a fuse in a hole. He was distracted momentarily and began tamping down the charge before the man helping him had had chance to cover it with sand. Gage was tamping down the powder directly with an iron bar. The iron bar as it struck the rock caused a spark. The explosion is considerable. The iron bar enters Gage’s left cheek pierces the base of his skull goes through the front of his brain and exits from the top of his head at high velocity. The iron rod apparently was found more than a hundred feet away, covered in blood and brain matter.

What was surprising was that Gage was not killed instantly. And despite serious damage to his brain, he recovered and lived for another 11 years. Of course, the accident resulted in dramatic changes to his personality, Phineas Gage was no longer able to respond to people in a measured way, and within the norms of politeness. However, he did live and Gage’s horrific accident demonstrates how much we rely on our limbic brain – or indeed how little we need our cerebrum.

Rationality in the contemporary university is so heavily influenced by Enlightenment, philosophy. I was only this afternoon listening to Terry Eagleton’s Luxembourg lecture from 2013 in which he talks about culture wars: in the post-Enlightenment, a position of privilege was given to science and there was a devaluation of the humanities. We turned our attention to rationality and treated the arts and humanities as frivolous and valueless. Now our science and our economics (and indeed the condition of contemporary societies) have led us back to a point at which we must critique the Enlightenment. We have created one big stubborn humanity-sized knot, a global scale conundrum of rationality. Our belief and thought, or the belief in the power of thought and rationality, has left us with one big mess. We face global problems with the environment, inequality, poverty and an unprecedented scale of human movement. Rationality is not going to be enough to solve it. Universities in their present form are not going to solve it, and scholars thinking in the way they do not going to solve it. We need the affective, intuitive narrative dimensions of the arts and humanities. We do need critical and embodied scholarship. Scholarship that has the boldness to go beyond the Enlightenment and go beyond Descartes’ Error. Embodied scholarship does not simply take place in the ivory tower it has to be out in the real world amongst people and amongst practice – day-to-day practice as Lefebvre stressed to us.

I know, that some of the most important work I do as a teacher educator, is with professional practitioners in public services. They experience, and they feel every day practice, they feel and experience the impact of our institutions and our policy on many individuals who are powerless. They are engaged in theory and practice. One is not privileged above the other. They must have the experience of doing pure research, but with the framing and experience of the everyday and of practice.

Or, might I be a farmer-scholar? I could spend part of the week working at growing food for me, my family and the community and for the rest the week. I could engage with work at a more theoretical level in relation to what I do now or concerned with the growing food.

The answer then to the excess of research students, is not that we have too many people wanting to be academics, it is that we have to reframe academia and what academic work actually is. To do this we have to think beyond the Enlightenment.

Memory and emotion

In my research into teachers’ beliefs, I often to return the idea of episodic memory which Nespor (1987) takes from Abelson’s (1979) paper on the differences between knowledge and beliefs systems.

Abelson suggests that information in knowledge systems is stored primarily in semantic networks, while belief systems are composed mainly of ‘episodically’-stored material derived from personal experience or from cultural or institutional sources of knowledge transmission (e.g., folklore).
Broadly speaking, semantically-stored knowledge is thought to be broken down or ‘decomposed’ into its logical constituents (abstract semantic categories —principles, propositional structures, or whatever) and organized in terms of semantic lists or associative networks. Episodic memory, by contrast, is organized in terms of personal experiences, episodes or events (Nespor, 1987, p. 320).

Nespor goes on to explain (drawing on Spiro, 1982) the association between affect, emotion and episodic memory:

… mood and emotion are stored as analogue representations of the experiential states associated with bodies of propositional knowledge. They function as a form of background coloration to content representation, the nature of which ‘corresponds to the nature of the felt experience’. When events are associated with a single or dominant experiential quality, their cognitive representation will have a relatively homogeneous coloration and one can speak of the event as having a ‘signature feeling’ (Nespor, 1987, p. 323).

Spiro argues that the ‘coloration’ provides a mechanism by which we can quickly associate events in front of us with similar ‘feelings’ in long-term memory. It allows us not be concerned with content and detail but with the overall affective character of the experience in memory and the events we bear witness to in the real world. This is similar to Johnson-Laird’s communication theory of emotions, that emotions are a primitive form of reason, that can result in culturally and voluntarily compiled responses (Johnson-Laird, 2006).

My purpose here is primarily to do some further scholarship on memory and emotion, to substantiate ore even challenge my initial understanding as set out above. Why is this important or why could this be important? As Reisberg points out:

The study of emotional memories provides a fabulous opportunity to explore the biological basis for memory formation, building both on what we already know about the biological processes relevant to memory, and what we know about the biological concomitants of emotion. The study of emotional memory also is crucial if we are going to understand autobiographical memory… (Reisberg, 2006, p. 15).

Emotional memories appear to be long-lasting and are more accurate than non-emotional or emotionally neutral memories. Emotional memories are important to us because they make us pay attention. Biologically, there is strong evidence of an important role for the amygdala (Reisberg, 2006). Studies of individuals with amygdala damage continue to find emotional images arousing, which suggests the amygdala does not have a role in attention, but in the way people consolidate emotional memories (ibid.). Reisberg steers to a conclusion that emotions play an important part in arousal (there is a considerable body of research arguing just this), but there is a significant psychological and cognitive role for emotion in the way we interpret and make meaning. Making meaning and sense making resonate with other contexts and approaches, see, for example Weick (1995) on the importance of how individuals make sense of themselves in organisations. Bruner (1986, 1990) makes much of meaning and narrative (cf episodic and autobiographical memory): the drive to make meaning is a strong intrinsic motivation.

Neuroimaging provides evidence for the memory enhancing effect of emotion, where there is combined activity involving the amygdala (the emotion-based system) and in the hippocampus and associate medial temporal lobe (memory-based system). Moreover, imaging shows that similar mechanisms take place during coding and retrieval (Dolcos, LaBar, & Cabeza, 2006).

The importance of this multidisciplinary work on emotion and memory, is the emotion, as a subjective account of affect (Massumi, 2002), is embodied, material, sensory and somatic. Emotion helps us make meaning, yet the tradition of humanism and Enlightenment rationality privileges the purely cognitive – the pure reason. Where I started with episodic memories with signature feelings has been enhanced, broadened and substantiated. Central to the human condition and ir/rationality is an embodied and affective experience.

References

Abelson, R. P. (1979). Differences between belief and knowledge systems. Cognitive Science, 3(4), 355–366. https://doi.org/10.1207/s15516709cog0304_4

Bruner, J. S. (1986). Actual minds, possible worlds. Cambridge, Mass.: Harvard University Press.

Bruner, J. S. (1990). Acts of meaning. Cambridge, Mass: Harvard University Press.

Dolcos, F., LaBar, K. S., & Cabeza, R. S. (2006). The memory enhancing effect of emotion: functional neuroimaging evidence. In B. Uttl, N. Ohta, & A. L. Siegenthaler (Eds.), Memory and emotion: interdisciplinary perspectives (pp. 107–133). Malden, MA ; Oxford: Blackwell Pub.

Johnson-Laird, P. N. (2006). How we reason. Oxford, New York: Oxford University Press.

Massumi, B. (2002). Parables for the virtual: movement, affect, sensation. (S. Fish & F. Jameson, Eds.). Durham, NC: Duke University Press. https://doi.org/10.1215/9780822383574

Nespor, J. (1987). The role of beliefs in the practice of teaching. Journal of Curriculum Studies, 19, 317–328. https://doi.org/10.1080/0022027870190403

Reisberg, D. (2006). Memory for emotional episodes: the strength and limits of arousal-based accounts. In B. Uttl, N. Ohta, & A. L. Siegenthaler (Eds.), Memory and emotion: interdisciplinary perspectives (pp. 15–36). Malden, MA ; Oxford: Blackwell Pub.

Spiro, R. J. (1982). Subjectivity and memory. Advances in Psychology, 9, 29–34.

Weick, K. E. (1995). Sensemaking in organizations. SAGE.

Multiplication – the privilege of mathematical thinking

I love John Mason. It is always a pleasure to listen to him as he takes you with him through his exploration of mathematical thinking and learning: “sit there and close your eyes and imagine a number line…” He takes you on a journey of ideas, connections and new understandings of the relationships between concepts and ideas in mathematics.

This evening we explored multiplication in the Faculty of Education.

But it is not the wonderful session that I want to talk about. It is my theme of not Mathematics Education (nME). A kind of meta- hyper- mathematics education. I mentioned to John that I was interested in nME. He looked puzzled, but not dismissive, John is always interested in thinking. I talked about how we had been through a period of relative stability in mathematics education research (I was talking about neo-liberalism). It is the liberalism that it is important in mathematics education research, it allows the freedom of thought and builds on the constructivism following Piaget and Vygotsky: constructing worlds of meaning and mathematical imaginaries as part of the process of learning (and doing) mathematics. It is the neo– in neo-liberalism that has contributed to deepening inequality in the last forty years.

Neo-liberalism, while it indulges some in this kind of constructivist thinking, it is for those, primarily, who have the time and luxury to indulge. If you want a sense of the mathematical indulgence that is associated with social class read G H Hardy’s Mathematician’s Apology. It is an apology for the fact that his position, wealth and privilege gave him access to think about pure mathematics. Wonderful things ensue, of course – the contribution of pure mathematics is without any doubt. Hardy explains how the pursuit of mathematics for its own sake and without purpose often leads to useful applications. It is the pursuit for no particular purpose that makes pure mathematics productive. But it is, in the context of liberal economics with its implicit utilitarianism, limited to a selected elite.

“Ah!” You say, “mathematics is meritocratic, it is blind to socio-economic status, class or even background.”

Well, no it isn’t, the fact that some children from disadvantaged backgrounds get to study mathematics at top universities insufficient to support this claim. Disadvantage children who progress to study mathematics in leading universities generally have a combination of talent, some luck and often or not a great deal of support. Sadly, there is often an unspoken appeal to competition or even social Darwinism: surely it is a fitting way to select the best. Probably not: the top universities’ mathematics departments are by-and-large filled with students who are from middle class or privileged backgrounds.

Let me explain why this (and I can go back to John Mason’s talk for this). Clear your minds – imagine a number line. Now imagine that number line is an elastic band. Stretch it out to three times its length, on what number would the original ‘4’ be. This is the basis of mathematics learning – of rich deep and agile mathematical thinking in which we explore concepts and relationships.

Now imagine that you are 13 years-old, you have one parent. They may be in precarious, low paid work, they may be struggling with their mental health because of debts. They may be struggling with alcohol, they might be worrying about paying the rent or getting evicted. You might live on a road where families face all sorts of difficulties in work and in keeping a roof over their heads. A community working and living precariously. You might have been pushed out of the shiny academy because you are distracted and can’t follow the strict and daunting behaviour policy. Your school is facing problems because there are lots of kids like you facing challenges, the teachers are tired and stressed. They haven’t got the patience for the kind of stuff John is doing. They love it, they love what he does. But they are so so tired. Even if they can, there is lots going on in your head, even your loving parent can’t shield you from their own or even the community’s anxiety and deepening sense of hopelessness.

Now tell me how you are going to shut all this out – this noise – and imagine your number line, even if you have a patient, thoughtful, energetic teacher. How do you stand a real chance? You don’t, it is a lottery for those from disadvantaged backgrounds.

Just before John began his talk, he mentioned the work he had been doing with Cambridge Maths, an initiative run by Cambridge Assessment to develop curriculum. There is no doubt that they are doing wonderful things. I reminded John of Cambridge Assessment’s primary purpose as an arm of the University of Cambridge, in a political and economic climate where the University can’t rely on public funding. Cambridge Assessment is about making money and it follows that Cambridge Maths will have to contribute at some stage. John agreed but argued that any opportunity to develop mathematics education must be taken. He was about to start his wonderful talk and I couldn’t make the following and my final point.

If we really want to make mathematics universal and allow all to indulge in the rich thinking that the study of mathematics promotes, then we have to – we must – start to think critically about it. That is ‘critically’ in the sense of what is driving the agenda: things that are not Mathematics Education – things like political economy. We cannot (must not) put mathematics education in a bubble insulated from political economy. Neo-liberalism fabricates and manufactures consent for economic scarcity (reducing public sector deficits). The consequence is that mathematics education research and development necessarily has to rely on markets and private finance. It is not any-port-in-a-storm to sustain research and development projects; by not resisting we are complicit in the political economy of neoliberalism. If we want universal access to mathematical thinking and a mathematics education for all, then we need to fight for public investment in research and education. We need to campaign against the meanness of economic policy that has marginalised so many and left them without the basic quality of life that creates barriers to the wonderful mathematical journeys that John Mason takes us on.

The heritability of intelligence

The field of behavioural genetics attempts to identify aspects of human behaviour that are heritable. This line of research can be traced back to the nineteenth-century researcher, Francis Galton (1822-1911). Galton was a cousin of Charles Darwin and was inspired by Darwin’s Origin of Species (1859), this led him to investigate heredity. In particular, he was interested in the heritability of ability and intelligence. In 1869 he published Hereditary Genius. In this work, he set out his method of historiometry, where he examined the achievements of relatives of eminent men. He observed that amongst more distant relatives there were fewer eminent people and he concluded that ability is heritable. But Galton did acknowledge the limitations of his research and he anticipated the study of twins as an improvement. Twin studies were an important part of behavioural genetics from the 1920s. I come to this research shortly. But casting aside any doubt about his own research, Galton believed in improving the genetics of human society. He coined the term eugenics and talked about race improvement.

I think that stern compulsion ought to be exerted to prevent the free propagation of the stock of those who are seriously afflicted by lunacy, feeble-mindedness, habitual criminality, and pauperism, but that it is quite different from compulsory marriage .

Galton’s critics were understandably concerned with the idea of selective breeding in order to improve the human race. For Galton, these criticisms appeared to be an overreaction. After all, he said, he was not looking to manufacture compulsory unions, it was simply a matter of restraining “ill-omened marriages” (ibid.). Importantly, it appeared to Galton that it was science and mathematics that had led to the derivation of the facts about heritability. And that we must, for the sake of society, ensure that democracy is “composed of able citizens” (ibid.) and that we must be aware of “the true state of things”. This is regardless of the fact that his own methods were inconclusive.

This is a prime example of how Enlightenment thinking can lead to folly, where blind faith in science – and the scientific method – leads to dangerous conclusions and unethical consequences. Social science is not a science, it is political and a moral philosophy. It can draw on studies based on the scientific method, but we are in error to believe that social science, such as educational research, is a science. Social science does not lead to facts, it leads to the potential to make individual and collective judgements that are informed by theory and evidence. Decisions and judgements leading to evidential claims are based on power and moral choice not on absolute truth.

Undoubtedly Galton was an accomplished individual, he is generally characterised as a polymath. His contributions are startling and impressive. Galton developed the idea of regression to the mean and standard deviation. He also pioneered the use of questionnaires. What he seems to have been unaware of, was the real causes of the conditions of society. While Galton assumed that inequality in society was natural selection – the cream rising to the top – Karl Marx was explaining the existence of poverty as a result of the failings of liberal economics. The free market kept the rich rich and the poor poor and exploited. The conditions of the poor ensured that they were starved, overworked, poorly housed and consequently they were wretched examples of humanity. While Darwin’s natural selection takes place over thousands of generations, the conditions of the working poor in Victorian Britain had developed within a few generations. There is nothing natural about what Galton observed of lunacy, feeble-mindedness, habitual criminality and pauperism. These things were entirely man made.

Galton – and it is pretty much unforgivable – gave those who benefitted from the economic status quo scientific ‘facts’ to justify and explain the tracts of squalor and depravity across industrialised Britain. It was, they could say, just a matter of heredity. That the well off are well off because of their genetic superiority and the poor are that way because of their inferiority. But it becomes more sinister. There were programmes of sterilisation in some European countries and some states in America in the early 1900s. Adolf Hilter was inspired by eugenics; consequently, the Nazis killed thousands of disabled people in the 1930s. The Holocaust was the ultimate in racial cleansing with the gassing of millions of Jews during the Second World War.

Perhaps there is a case for eugenics: Toby Young. Perhaps his father, Michael Young, should have been made aware of the possible consequences of an “ill-omened marriage”. We could have avoided a retrograde step such as Toby Young. But we are all wise after the fact. And as a matter of principle, as you will no doubt have gathered, I am opposed to eugenics. But Young Junior, with his characteristically ill-informed gobshitery, argues for ‘progressive eugenics’ . Young rehashes many of Galton’s original arguments for eugenics with little smatterings of evidence, partial readings and partial understandings. Blah, the best people have the best IQs, blah. I am suddenly struck by the immensity of Galton; he was mistaken and the consequence of his work was the death of millions, but he was no second-rate right-wing establishment bum licker, he was an original thinker. What Young tries to do is to input into his ‘bold’ progressive eugenics some fresh thinking – poor people should be allowed access to genetic manipulation to improve their babies when the technology comes available. Eugenics remains abhorrent and an unacceptable form of social engineering, even if we do prefix it with ‘progressive’ and the state funds designer baby programmes to those on benefits and low incomes.

The father of modern genetics, Gregor Mendel (1822-1884), identified physical characteristics of pea plants that were heritable: plant height, the shape of the peapod, seed shape and colour, and flower position and colour. While biologists at the time believed that inherited traits were blended, Mendel’s experiments showed that there were dominant and recessive traits that are passed from parent to offspring. It was not until the beginning of the twentieth century that the significance of Mendel’s work was recognised. Heritable information is carried in genes which come in pairs and offspring inherit one gene from each parent. Genes are made from DNA (deoxyribonucleic acid). A gene is a length of DNA that codes for a specific protein. A DNA molecule can make copies of itself and it carries information for creating proteins. One gene will code for the protein insulin, which has an important role in controlling the amount of sugar in the blood; human beings have 20,000 to 25,000 genes¹http://www2.le.ac.uk/projects/vgec/highereducation/topics/dnageneschromosomes. In humans, observable heritable traits include height, hair colour, earlobe attachment, tongue rolling, dimples, handedness, freckles, curly hair, red/green colour blindness and hairline shape, for example. DNA codes for proteins that result in the development of these characteristics. Behavioural genetics assumes that if physical characteristics are inherited, then our inherited hardware and architecture can lead to the inheritance of higher-order characteristics such as intelligence and personality. But to what extent is our behaviour, our successes and failures, attributable to our environment and upbringing. Or is it already hard-wired into us genetically? Are we preloaded with certain capacities that can predict our outcomes?

Galton anticipated the use of twin studies to identify the genetic basis of psychological traits such as IQ and personality. The methodology exploits the fact that identical twins share the same genes and that non-identical twins share half their genes. Differences in the behaviours of identical and non-identical twins can be used to estimate the proportions of their behaviour that are inherited. Krapohl et al. , claim that academic achievement is a result of just over 60 per cent heritable characteristics. This, in turn, is a result of heritable intelligence, self-efficacy and personality. The study is based on a classic twin study involving 6,653 pairs of twins in the UK and using GCSE²General Certificate of Secondary Education. The examinations taken at the end of compulsory schooling in the UK at age 16. results. The assumption is that the similarities in the performance of identical twins are entirely genetic since identical twins have the same genes and they have been brought up in the same environment. Underpinning this assumption is the belief that the environments that identical and non-identical twins develop in are similar: each twin in both groups has a similar experience of the environment. This is referred to as the equivalent environment assumption. Yet, it is recognised that identical twins are generally treated in the same way as they grow up, much more so than non-identical twins. Therefore, identical twins’ behaviours may not be attributable to their genes, but to the way in which they were brought up and not treated as two individuals. The equivalent environment assumption is therefore unjustified: at best it means that findings from twin studies overestimate the heritability of psychological aspects, at worst it invalidates these claims altogether .

To counter this, behavioural geneticists have used studies of identical twins that had been separated at birth and who have subsequently been brought up in different environments . However, Joseph argues that the comparisons between identical twins raised together and those raised apart still do not justify the equivalent environment assumption. He identifies the following similarities between identical twins raised together and those raised apart:

They are exactly the same age (birth cohort).
They are always the same sex.
They are almost always the same ethnicity.
Their appearance is strikingly similar (which will elicit more similar treatment from the social environment).
They usually are raised in the same socioeconomic class.
They usually are raised in the same culture.
They shared the same prenatal environment.
Most studied pairs spent a certain amount of time together in the same family environment, were aware of each other’s existence when studied, and often had regular contact over long periods of time .

The study of identical twins raised apart has not provided a valid defence of the equivalent environment assumption. More recently a defence of behavioural genetics has come from genome-wide association studies . GWAS involves scanning markers across the complete sets of DNA, or genomes, of many people to find genetic variations associated with particular behaviours³https://www.genome.gov/20019523/genomewide-association-studies-fact-sheet/. Effectively, this is a hunt for genes or sets of genes that lead to particular behaviours, intelligence or personality. However, it has not been possible to identify the sets of genes that contribute to intelligence and academic achievement. Krapohl et al. found that sets of genes (genome polygenic scores, GPS⁴http://www.nature.com/mp/journal/v20/n1/fig_tab/mp2014105b2.html) explained ~2% of educational achievement. And according to Joseph, this kind of molecular genetic research, is a result of a mistaken belief that the twin studies provide unequivocal evidence that genetic factors contribute to observed variation in behaviours; gene-finding exercises are unlikely to yield results .

Intelligence is a social construct, we judge intelligence based on observed behaviours. In parallel with the work of behavioural geneticists, psychologists have attempted to distil a measure of intelligence⁵An excellent summary of the history intelligence tests and IQ measures can be found here https://logicalincrementalism.wordpress.com/2017/07/22/evolved-minds-and-education-intelligence/. Yet, measures such as intelligence quotient (IQ) are measures of how good participants are at the IQ tests. IQ is confounded by socioeconomic status, the tests have a class bias and reflect the social and cultural capital of particular groups. Yet, I acknowledge also that intelligence involves cognitive processes as well as being a social construct. Intelligence has the following features (but not limited to these): perception and recognition, reasoning, the creative use of existing knowledge, strategic and tactical planning, and the capacity to act and adjust actions as information is updated and the context changes. Our intelligence, our reasoning and the decisions we make are sensitive to emotions and affect. Confidence, self-concepts and motivations have a profound effect on our attainments and will have an effect on any assessment of our intelligence. In addition, our physiological and affective states also have an impact on how intelligently we act. Intelligence is a complex psychosocial construct, it is unsurprising therefore that it continually eludes behavioural geneticists.

Philip Johnson-Laird, a cognitive psychologist and philosopher, explains human reasoning (the core of human intelligence) using mental models . He draws on a dual processing model of human reasoning. One type of reasoning is entirely conscious and can draw on logical analysis to derive solutions and construct action. Although, as Johnson-Laird points out, people are quite bad at logic and there are many situations where there is insufficient information to permit an entirely logical analysis. Information is missing, assumptions have to be made, gaps have to be filled and judgements made using synthetic mental models. This draws on past experience and relies on matching the situation at hand to situations and strategies. The second type of reasoning is almost entirely intuitive and subconscious. Rational conscious thought is demanding and consumes resources, it appears to be hard-wired into humanity to limit rational thought, probably because of the resources it requires and also to allow sensory resources to be available should something unusual come our way, such as a predator. Much of the time, in our day-to-day lives, and in many of the things we do routinely we use subconscious reasoning. In this we rely on shared cultural patterns of behaviours and shared mental models; we are at ease in our communities and families and have a sense of how others will act and respond in these contexts without the constant demand for conscious thought.

The complexity of human reasoning and intelligence is irreducible to a genetic marker, but genetics dictates the format of our central nervous system and our neurophysiology. Intelligence is the ability to reason and act effectively in different situations. Thus, a person with the quality of intelligence would be judged by an observer to have negotiated an obstacle, problem or context, efficiently and effectively. There has to be some degree of challenge in the problem, some complexity or novelty that requires reasoning. The challenge faced cannot be solved using a routine, method or algorithm. In other words, the distinction between a robot and an intelligent being is that the being has the capacity to use creative reasoning processes to solve problems. A robot or artificial intelligence is reliant on routines and algorithms to negotiate the situations it meets. Intelligence is primarily dictated by the way in which we learn to use our ‘hardware’. Experiences, relationships and the contexts in which we learn and how we learn really define how intelligent we become. I want to use the analogy of cinema.

Edison’s patented invention, the Kinetoscope, was introduced in 1891, the Lumière brothers’ first projection of films to a paying audience took place in 1895. Films create an illusion of continuous movement by passing a series of images in front of a light source enabling the images to be projected on a screen. The moving image as a form of collective entertainment spread in the form of photographic images printed on a semi-transparent celluloid base cut into strips 35 mm wide. This was devised by Henry M. Reichenbach for George Eastman in 1889 . Through the twentieth century, cinema technology evolved with the introduction of sound and colour. More recently cinema has used digital technology and computer-generated images. The hardware and technology have evolved, improving the quality of production. Cinema, as an art form, is dependent on the technology but relies on human creativity, reasoning and culture to create a narrative, a visual and auditory effect, a spectacle and artistic form. From the point we had the technology to project a feature film a century ago, the artistic content has not advanced significantly. Clearly, film makers have more sophisticated technology, but the limitation of the art form has been human creativity, society and culture. For example, according to the IMDb⁶http://www.imdb.com/ website, D W Griffith’s 1916 film Intolerance is rated 8 out of 10, while Kathryn Bigelow’s The Hurt Locker (2009) is rated similarly. Clearly, there is a considerable amount of high-quality contemporary output, much more than there was at the time of the release of Intolerance. However, while cinematic hardware has advanced dramatically the quality of the artistic output has progressed at a steadier rate.

Our genetics provide us with the hardware, analogous to cinematic hardware: our brain, central nervous system is like the projection equipment and the film. Human intelligence is like the film content, constructed and devised based on free will, knowledge and in congress with culture and society. Genetics and heredity are important, but only in giving us the hardware. It is our experience of society, knowledge, culture and ourselves that allow us to develop intelligence. Or, indeed a poverty of these things does not permit the development of intelligence.

George Orwell in his anthropological account of the British working class in the 1930s provides a unique insight into the conditions of society and how it impacts on working people living in poverty. He reflects on the intelligence of his boarding house landlord and landlady, the Brookers. Orwell, observes first hand how conditions and political economy crush intelligence and reason.

The most dreadful thing about people like the Brookers is the way they say the same things over and over again. It gives you the feeling that they are nor real people at all, but a kind of ghost for ever rehearsing the same futile rigmarole…But it is no use saying that people like the Brookers are just disgusting and trying to put them out of mind. For they exist in tens and hundreds of thousands; they are one of the characteristic by-products of the modern world. You cannot disregard them if you accept the civilisation that produced them. For this is part of what industrialism has done for us .

Thirty years previously Robert Tressell, observed a similar impact on the working class: conditions of poverty, limited opportunity, repetitive drudgery and exploitation that lead to the apparent absence of intelligence . Like Orwell, Tressell attributes conditions to political economy and the political choices of those who hold power and wealth.

When we consider intelligence we have to look at society, culture and political economy and not at genetics.

References

A clarification of the meaning of ‘self-efficacy’

This post is a response the Andrew Davis’s healthy scepticism about the concept of self-efficacy. This was in response to a recent post:

http://stevenwatson.co.uk/2017/06/recent-research-in-cultural-differences-in-the-development-of-mathematics-self-efficacy/

So what I intend to do here is clarify its origins and meaning drawing on the work of Albert Bandura .

What happens when a pupil strongly believes that they will be successful at a mathematics task when in fact they have no chance of success?

— Andrew John Davis (@ded6ajd) June 28, 2017

Andrew’s first question suggests that self-efficacy is equivalent to a pupil believing they will be successful in a test. That the pupil’s ‘beliefs’ may be inaccurate. I responded by saying that this illustrates the difference between confidence and self-efficacy. On this Bandura says the following:

It should be noted that the construct of self-efficacy differs from the colloquial term “confidence.” Confidence is a nondescript term that refers to strength of belief but does not necessarily specify what the certainty is about. I can be supremely confident that I will fail at an endeavor. Perceived self-efficacy refers to belief in one’s agentive capabilities, that one can produce given levels of attainment. A self-efficacy assessment, therefore, includes both an affirmation of a capability level and the strength of that belief. Confidence is a catchword rather than a construct embedded in a theoretical system. Advances in a field are best achieved by constructs that fully reflect the phenomena of interest and are rooted in a theory that specifies their determinants, mediating processes, and multiple effects. Theory-based constructs pay dividends in understanding and operational guidance. The terms used to characterize personal agency, therefore, represent more than merely lexical preferences .

This makes an important point about the meaning of self-efficacy – “it is a construct embedded in a theoretical system” in contrast with confidence as a “colloquial term” which refers to the strength of belief without necessarily identifying the nature of the task. However, this was not quite the point that Andrew was making. The question he raises is, should mathematics self-efficacy be defined as the true belief an individual has in their capacity to solve mathematics problems?

So why isn't mathematics self-efficacy defined as the *true* belief an individual has in their capacity to solve mathematics problems?

— Andrew John Davis (@ded6ajd) June 28, 2017

To respond to this, to address the distinction between the true belief an individual has in their capacity to solve mathematics problems and mathematics self-efficacy. Bandura defines self-efficacy as follows:

Percieved slef-efficacy refers to beliefs in one’s capabilities to organise and execute the courses of action required to produce given attainments

On the face of it, given Bandura’s definition of self-efficacy, they would appear to be equivalent. But there are subtle differences. Andrew’s definition refers to outcomes alone, whereas Bandura refers to beliefs about personal capability. This is a subtle difference but important and probably best elucidated by looking at the underlying theory.

Origins of self-efficacy: agency and control

The problem that Bandura is addressing in the introduction of self-efficacy is concerned with human agency and control. Agency is concerned with the power, knowledge and disposition an individual has in exercising the right to chose the way to act. Control is related to this, it is the motivation and drive a person has to have agency in their lives.

The striving for control over life circumstances permeates almost everything people do throughout the life course because it provides innumerable personal and social benefits. Uncertainty in important matters is highly unsettling .

Approaching this from social science disciplines other than psychology this might not seem such a big deal. Sociology readily constructs agency and control, it is implicit within the field to consider the impact of the social word on personal freedom. Similarly in anthropology where the effects of culture and society and a key part of theory in this discipline. Yet in psychology, agency, in reference to the social world, is given little attention. B. F. Skinner, for example, considered the individual as having limited agency, behaviours are responses to environmental responses. In behaviourism, there is an absence of ‘self’ or ‘control’.

Another important idea underpinning self-efficacy is the notion of ‘intentionality’: people generate courses of action to suit given purposes .

Intentionality and agency raise the fundamental question of how people actuate the cerebral processes that characterize the exercise of agency and lead to the realization of particular intentions .

So we can see in this how the concept of self-efficacy is different from a self-assessment of how well the individual will perform. Self-efficacy is concerned with agentic action and the construction of a course of action in response to situations.

…the making of choices is aided by reflective thought, through which self-influence is largely exercised. People exert some influence over what they do by the alternatives they consider .

The multidimensionality of self-efficacy

Self-efficacy represents the deliberative processes preceding action. It involves the consideration of alternatives courses of action and decisions about how to proceed. Going beyond this, Bandura locates self-efficacy as a self-concept as a self-assessment. But “self-efficacy beliefs are not simply inert predictors of future performance” . What distinguishes self-efficacy from the ‘inert’ predictor or perceived capability is the multidimensionality of self-efficacy belief systems.

Efficacy beliefs should be measured in terms of particularized judgements of capability that may vary across realms of activity, under different levels of task demands within a given activity demand, and under different situational circumstances .

In attempting to measure self-efficacy, we are not simply asking whether the individual is going to be successful in a particular activity. It requires careful assessment using gradations of task demands within the domain of concern. It also requires a clear definition of the domain of activity and a careful conceptual analysis of the aspects, knowledge, skills and dispositions required.

I hope, I have illustrated here some the key differences between perceived capability in respect to performance in a particular context and the multidimensional multi-faceted concept of self-efficacy. When measured appropriately, self-efficacy is a strong predictor of performance but is not a specifically an assessment of personal capability.

References

Recent research in cultural differences in the development of mathematics self-efficacy

Self-efficacy is a conceptualisation of self-belief, developed by Albert Bandura . It is the belief an individual has in their capacity to be successful in a domain. It is a self-assessment of skills, knowledge and dispositions in a context. It is domain specific in that self-efficacy is contextualised, with demanding but related sets of challenges. The activities cannot be so trivial that the action required is relatively routine or straightforward. We are talking about problem solving in contexts, where there are complex decisions which may have multiple solutions and multiple means by which outcomes might be achieved.

Mathematics self-efficacy is the belief an individual has in their capacity to solve mathematics problems. It is a belief that they can achieve a level of success when undertaking maths-related work. Mathematics self-efficacy correlates with mathematics performance. Mathematics self-efficacy is an important predictor of mathematics performance .

According to Bandura, there are four sources of self-efficacy: enactive mastery experience, vicarious experience, verbal persuasion and physiological and affective states. I will explain each of these in turn:

Enactive mastery experience

Self-efficacy is developed through experience, through working on and solving problems; if we we are to limit our discussion to mathematics self-efficacy. This can be easily understood from our own experience, if we practice and get positive results, i.e. we are successful, then we become more confident. However, Bandura, takes a more profound view of success, a broader view, and allows the possibility of acquisition of self-efficacy even when we fail.

Mathematical self-efficacy is developed not just as a consequence of getting questions right or simply by finding solutions to problems. Self-efficacy is developed through reference to the strategy that we took in solving problems. Effectively, we assess the the approach we took and how it led to the outcome. In developing self-efficacy, we do not assess the outcome in absence of the method we used. This explains why, even though our final result might be wrong, we can develop self-efficacy. The essence is in being be able to connect our actions to the outcome and understand, rationally, how that led to the result.

Vicarious experience

A second but weaker source of self-efficacy is through vicarious experience. We can develop self-efficacy by observing others carry out activities. If the modelled behaviour is self-efficacious then it can provides a source of self-efficacy for the observer. This is especially true if the observer identifies with the individual modelling the behaviour. If, as observer, we see ourselves as similarly, having similar capacities and potentialities, then we are likely to improve our self-efficacy by observing them model actions, and in mathematics, by modelling mathematical problem solving.

If the observer perceives the person modelling the action as considerably different – they might feel that they are more intelligent or more able – then it is less likely that the observer will develop self-efficacy vicariously.

Verbal pursuasion

A third, but still weaker source of self-efficacy, is verbal pursuasion. We can use encouragement to persuade learners of their capacity to be successful. If the encouragement is misplaced and we try and persuade learners that they will be successful and they ultimately fail, there is the possibility that self-efficacy will be undermined. Encouragement must be based on accurate assessment of the individual’s capabilities and potential. Furthermore, if the more knowledgeable other is not trusted by the learner, then it is unlikely that self-efficacy will be developed.

Physiological and affective states

Illness, tiredness and stress all undermine self-efficacy. It is important that learners are challenged and are set challenging objectives. But if the demands becomes overwhelming, then there can be negative effects. Equally, if a learner is unwell or if there are external stressors then self-efficacy is undermined and there will be a noticeable effect on mathematical performance.

Cultural differences in the effects of vicarious experience and verbal pursuasion

consider cultural differences in the extent to which the social sources of self-efficacy impact on self-efficacy and mathematical performance overall. It has previously been suggested that social effects are different in cultures that are predominantly individualist, like the US and Western Europe, to cultures that are collectivist, as in South East Asia.

undertook a quantitative study in the US, the Philippines and in Korea.

The important results are as follows:

Mathematics self-efficacy has a significant positive correlation with students’ mathematics achievement. This is consistent with previous research, both theoretical and empirical. It also provides evidence that this relationship is independent of culture.
Mathematics anxiety is negatively correlated with mathematics self-efficacy. Again this is an expected result and previous research has suggested this also. In more vernacular terms it means that the more confident a learner is in mathematics the less anxious they are.
Students in individualistic cultures report stronger mathematics self-efficacy compared with collectivist cultures. This is often in spite of superior performance by learners in collectivist cultures. This could be because people in collectivist cultures refrain from higher ratings because of a cultural desire to express humility.
Vicarious sources of self-efficacy tend to be from teachers and verbal pursuasion comes from family and peers.

Concluding remarks

This research confirms the importance of self-efficacy in mathematics learning. It challenges the view that mathematics learning should be predominantly rote learning and practice. Problem solving is necessary to develop self-efficacy. Learners need chance to explore and examine non-routine problems. Importantly they need to develop the capacity to assess the strategies they use, themselves and with the support of teachers and peers.

Not only does self-efficacy correlate with mathematics performance, it is also important in respect to mathematics anxiety. The more self-efficacious the individual the less anxious they become.

Finally, although this research considers self-efficacy sources in different cultures, it draws attention to the importance of social sources and within what contexts this might develop. The research shows vicarious sources tend to be from the teacher, verbal pursuasion comes from family and peers. This is important in understanding multiple social roles in learning mathematics.

References

This changes everything

I was feeling numb at five minutes to ten last Thursday. I had been campaigning intensely for the Labour party – both professionally and in a personal capacity – for months. It came up on Twitter, the mainstream media were saying that exit polls predicted a hung parliament. And while the Conservative party were predicted to be the largest party, the result for me marked a major change in British politics. It was going to be an exciting night.

So it turned out. As the results came in through the night it was clear that Labour had increased its share of the vote from April polls of about 25 per cent to 40 per cent in the General Election. This was unprecedented.

What is so significant, is the election result demonstrates strong support for a radically different economic and social policy. Radically different from the consensus that had existed between the major parties since the 1970s.

Keynes is back baby. The manufactured consent around a liberal/ neoliberal political economy which focuses on controlling public-sector spending and facilitating wealth creation has been shaken to its core. Particularly because it was the cause of the 2008 Global Financial Crisis and it prompted the austerity approach adopted by the Coalition government and Tory governments from 2010 to 2017. Neoliberalism and austerity has undermined public services and exacerbated inequality.

When I say Keynes is back, I mean that we stop the purblind view of the importance of wealth creators, but begin to look again at the role of government spending in creating demand. Wealth creators cannot attract wealth unless ordinary people have sufficient money to purchase things in the economy.

Government can increase that wealth through redistribution (e.g. progressive taxation), increased investment in the economy (e.g. through infrastructure, health and education) and more robust regulation of the financial sector (addressing exploitation of private debt). Since the UK government has a sovereign currency it can use its capacity to spend, tax and regulate to rebalance the economy.

Keynes is back, but it’s been upgraded by contemporary economists. I have written about it in the following posts:

http://stevenwatson.co.uk/2017/01/mmt-school-spending/

http://stevenwatson.co.uk/2017/02/spendandtax/

The consequence for teachers, educators and academics is that we have to start thinking differently. We have to think about what education might look like in a post-neoliberal world. Some of my thoughts are in the following post:

http://stevenwatson.co.uk/2017/05/laboureducationge2017/

Since the Labour Party’s positive manifesto has been welcomed by the country, we must now go further and think about how we transform our education system. Transform from a marketised, privatised and commodified system into a democratic system that serves communities and the nation in an inclusive way. Paying attention to social justice, peace, environment, community cohesion and individual and collective intellectual development. A system that must effectively serve people more and serve less those that run and control it.

Exciting times, I look forward to the debate.

Upon the rejection of a research article, nightmare and hope

I have a number of research strands going on at the moment. There is my research into mathematics teachers’ professional development – this goes back to my interests as a teacher and head of department and was followed up with my PhD research at the University of Nottingham. I also have an interest in learning processes in mathematics education, particular around school students engaged in rich tasks and problem solving. This relates, also, to my experiences as a teacher and again was something I looked at obliquely in my PhD research. The professional development I evaluated was to support teachers in implementing approaches that promoted the learning of problem-solving skills.

And then of course – those of you who have been reading my blogposts will know – I have got increasingly interested in political economy and public spending. This a result of my professional development research. I recognised, as part of this research, there are significant constraints and limitations on teachers in having access to good quality professional development. I followed the money, and power, and identified the source of these limitations. You really only have to look at Marx and Keynes to begin to comprehend the basis of decisions about the funding of the public sector. It is not based on a rationality of equity.

I am not going to mention my work on geodemographics here with Tim Mullen-Furness. That’s for another day.

While my research has grown to be diverse, I look up and down my inquiry trail. For a number of reasons, I find myself looking deeply, again, into professional development research particularly in respect to mathematics teachers. In part, this is prompted by the death of my PhD supervisor, Malcolm Swan. Sadly gone too soon. But also by the unexpected rejection of a research article. I submitted a theoretical/ empirical paper to a journal early in 2016. It came back in late summer with the ‘accept subject to major revisions’ tab. I duly revised and wrote a report about my changes and resubmitted. This was last September.

The night before last (these things always seem to come late at night and I always foolishly look at them) I received an email from the editor rejecting the paper. I had expected the reviewers to judge my paper based on the original reviews. But they had looked at it afresh. And rejected the blighter.

‘They have bloody well moved the goal posts’ I thought to myself angrily, as I laid a wake with insomnia. Insomnia directly related to my decision to look at and contemplate the email from the editor. In those dark hours, one can grow irrational. I do. I always have. I enter a dark terrain, like a bad acid trip. I began to consider that this single event may have a catastrophic effect on my progression from probation to tenure. Foolish and irrational, I know, over such a relatively small setback.

But it has focussed my mind on the overall purpose of my research and the direction in which it is going. While I have been merrily skipping on, on to new ground, it has taken me back. It has made me review my core interest. That of professional learning.

I need to thank my resplendent colleague Rupert Higham for his generous mentoring yesterday morning. He has inspired and encouraged me, as has done in the past, to steer my course as I feel appropriate. I must follow my water. He helped me make sense of myself.

In spite of the journal editor and reviewers’ final response to my piece. I recognise that I have been trying to bung my theoretical act onto an empirical stage. I am not anti empirical, its just that I am a thinker and schemer. Those dark terrains, the bad acid trips are the dark side of my imagination. The positive side of my imagination, the hope and vision that my overdosed imagination has given me has always outweighed the negative. As I have got older I can manage and ride out the extreme imagined fear knowing that experiences and people (and a good night’s sleep) can restore my positive frame.

The experience of this, in the last couple of days and the shocking events in Manchester, have, oddly, resulted in me being buoyant today. There are so many challenges in the world, on an unthinkable scale. But today I see my place, the way in which I can contribute, the way in which I can use my imagination to see a better world and contribute to some solutions.

↑ 1	http://www2.le.ac.uk/projects/vgec/highereducation/topics/dnageneschromosomes
↑ 2	General Certificate of Secondary Education. The examinations taken at the end of compulsory schooling in the UK at age 16.
↑ 3	https://www.genome.gov/20019523/genomewide-association-studies-fact-sheet/
↑ 4	http://www.nature.com/mp/journal/v20/n1/fig_tab/mp2014105b2.html
↑ 5	An excellent summary of the history intelligence tests and IQ measures can be found here https://logicalincrementalism.wordpress.com/2017/07/22/evolved-minds-and-education-intelligence/
↑ 6	http://www.imdb.com/