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.

 

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 nMEHe 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.

 

 

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

Ahn, H. S., Usher, E. L., Butz, A., & Bong, M. (2016). Cultural differences in the understanding of modelling and feedback as sources of self‐efficacy information. British Journal of Educational Psychology, 86(1), 112–136.
Bandura, A. (1997). Self-efficacy: The exercise of control. W.H. Freeman.
Pajares, F. (1999). Self-efficacy, motivation constructs, and mathematics performance of entering middle school students. Contemporary Educational Psychology, 24(2), 124–139. https://doi.org/10.1006/ceps.1998.0991
Pajares, F., & Miller, M. D. (1994). Role of self-efficacy and self-concept beliefs in mathematical problem solving: A path analysis. Journal of Educational Psychology, 86(2), 193–203. https://doi.org/10.1037/0022-0663.86.2.193

Malcolm Swan – a few memories about my PhD supervisor

Malcolm was a kind-hearted supervisor. His real passion was designing mathematical tasks. I was lucky enough to become his PhD student at the Shell Centre in the University of Nottingham in 2010. The Shell Centre provided me with funding to evaluate the impact of the Bowland Professional Development materials on secondary mathematics teachers’ beliefs and practices. The materials were designed by Malcolm based on his years of experience designing tasks and classroom materials. They are superb.

Malcolm and I found that we had a spare couple of days in San Francisco in 2012, having been working with Alan Schoenfeld at UC Berkeley. We had a great time, Malcolm was always engaging and gentle company. Here we are fooling around on a cable car.

What I could never really figure out is how Malcolm could understand how students and teachers would think and act when working on the activities he designed. But he seemed to know. I know he did lots of careful observation and would refine his designs as a result. But he had something extra, some extra bit of magical imagination. It was like that of any creative, an artist, poet or writer, he had an imagined world, a very sophisticated one. When you engage with a Malcolm task you are entering his world. It is a wonderful world.

You don’t just venture alone into Malcolm’s world, he entices you to go as a group. His tasks are wonderfully infectious. Even before I met him I was using the Improving Learning in Mathematics (Standards Units) materials with low-attaining learners who had lost a lot of confidence in mathematics. They couldn’t help but argue and think together. I remember smiling at two or three year 10 girls, who initially refused to suffer the indignity of doing maths while in detention, but within a short time they were furiously debating the meaning of negative numbers and operations. A testament to the power of Malcolm’s task design.

We didn’t always agree during my time as a student. Sometimes it could be downright frustrating. Malcolm had his ideas and I had mine. But we got through. Malcolm was always patient. We realised we were never going to agree on how teachers’ beliefs worked and how they influenced what teachers did in the classroom. But through this it made me make sure I knew my stuff. It made me a better academic.

It was only in the last year or so, while training new mathematics teachers, that I really realised what a profound influence Malcolm had on my thinking about mathematics education. I stress to trainees the importance of tasks in assessment. That is real assessment, diagnostic assessment. Using tasks so you can see and understand the deep concepts and processes that learners struggle with or master. None of your gap filling rubbish.

Malcolm will be missed. Gone way too soon. But in my practice as a teacher educator and researcher Malcolm is with me always. Farewell Malcolm, I’m sure there’s a corner of heaven really busy with a card sort of yours right now.

One of my favourites from the Standards Units or Improving Learning in Mathematics

International Comparisons in Mathematics Education – Mathematics Education Masters Seminar 8 February 2016, Faculty of Education, University of Cambridge

I have changed this seminar from previous years, this year I have taken a political economy turn. This means that I want to look at international comparisons in mathematics education (in relation to school mathematics) not just on the basis of the comparison of teaching and learning but in the context of political economy, particularly globalization. Generally, discussions about international comparative research begin with comparisons of the performance of students in different nation states, this is often followed with consideration of the differences in societies, politics and culture in the compared jurisdictions and then there are questions about the validity of such studies and whether the comparisons are fair.

When we think about international comparisons in mathematics education we tend to think about those studies that are based on assessment, like for example PISA. International comparisons are, however, not limited to assessment-based surveys, comparative studies have been undertaken on teaching and learning, based on observing and comparing classroom practices in different countries.

International comparative education is an aspect of, indeed a phenomena of, globalization. So here I want to promote a discussion not only about the nature of this kind of research, what it can tell us and what it can’t, but what is its role in the context globalization, and how does globalization play a part in education?

I begin with an explanation of the idea of globalization in economic, political and cultural terms. I follow this with a summary of the key lines of international comparative research in education. Finally, I present a critique beginning at the level of the different types of study and finally in relation to globalization.

Globalization

This is a term that describes the a variety of economic, political, cultural, ideological and environmental processes. This contested idea first appeared in the 1940s, though really became widespread in the 1990s.

Globalization is a set of social processes that lead to the social condition of globality, through the growing consciousness of global connectivity (Steger, 2013, p. 1).

It is a contested idea:

There is no consensus on exactly what processes constitute globalization, but common themes include the creation of networks, expansion of social relations, and the acceleration of social exchange (Steger, 2013, p. 1).

The economic, political and cultural dimensions of globalization:

The economic dimension of globalization

The Bretton Woods conference held at the end of the Second World War established binding rules about economic activity. Currency was fixed to the gold value of the US dollar. The International Monetary Fund (IMF) was created. Systems of trade were developed through the General Agreement on Tarriffs and Trade (GATT) in 1947, which was later to become the World Trade Organisation (WTO) in 1995.

Between 1945 and the early 1970s, nations’ economies worked through interventionism (government spending) with a freemarket economy. This resulted in growth and full employment in the US and UK, for example.

By the late 1960s this system was not working and led to high inflation in the UK and similar economic problems in US. It was assumed that interventionism of Keynsian economics was no longer working. This gave way to neoliberalism, typified by the policies of Margaret Thatcher in the UK and US President Ronald Reagan. Neoliberalism is based on the belief that the freemarket is the ultimate economic arbiter, that the big state and state intervention hampers this. This includes the replacement of public services by outsourced and private providers and the privatisation of nationalised industries (Harvey, 2011).

Global trade increased from $57 billion in 1947 to an astonishing $14.9 trillion in 2010  (Steger, 2013). While millions were lifted out of poverty, levels of inequality within developed nations were increasing. For example in the US, between 2002 and 2007, the top 1 per cent seized 65 per cent of the national income growth (Stiglitz, 2012, p. 3). A similar pattern can be found in other developed nations under neoliberalism.

The global financial crisis in 2008 revealed that neoliberalism had fatal problems with its reliance on private debt and consumer spending as the basis of economic growth. The level of low-grade private debt in the US, resulted in the insolvency of banks which had a knock on effect around the world.

The political dimension of globalization

The political dimension of globalization goes beyond the nation-state. It sees the rise of super national organisations like the United Nations and the European Union. These, arguably, threaten the role of nation-states.

actrade_9780199662661_graphic_016-full

The nation-state in a globalizing world Source: Jan Aart Scholte, ‘The globalization of world politics’, in John Baylis and Steve Smith (eds.), The Globalization of World Politics, 2nd edn. (Oxford University Press, 2001), p. 22.

The Organisation for Economic Cooperation and Development (OECD) is one such international organisation. Founded in 1960, it has 35 member countries, its express aim is to stimulate economic progress and world trade.

The cultural dimension of globalization

This involves the spread of culture in a globalized world. Critics claim that culture can become overly homogenized, with dominance of powerful nations like the US. This has an impact on language, day-to-day life as well as arts and culture. There are clearly aspects of globalized culture that have an impact on education.

A brief history of comparative education

In 1959, the International Association for the Evaluation of Educational Achievement (IEA) piloted a conceptual and methodological framework for large-scale international studies (Owens, 2013). These developed into a formalised First International Mathematics Study in 1964 (FIMS), a second between 1982 and 1983 (SIMS) and a third, Trends in International Mathematics and Science Study in 1995 (TIMSS). These were designed to measure students’ problem solving skills. This developed through the 1980s and 1990s with expansions from maths and science to other subject areas. Since 1995, TIMSS has monitored trends in mathematics and science achievement every four years, at the fourth and eighth grades. TIMSS 2015 is the sixth such assessment, providing 20 years of trends. TIMSS 2015 can be found here.

The Programme for International Student Assessment (PISA) sponsored by the Organisation for Economic Cooperation and Development (OECD) was introduced in 2000. PISA is a triennial international survey which aims to evaluate education systems worldwide by testing the skills and knowledge of 15-year-old students. In 2015 over half a million students, representing 28 million 15-year-olds in 72 countries and economies, took the internationally agreed two-hour test. Students were assessed in science, mathematics, reading, collaborative problem solving and financial literacy.

The result of the 2015 survey are here.

Classroom research

While there have been the survey and assessment based approaches to comparative education research. Comparative studies have been undertaken of classroom practice.

TIMSS video study

The Third International Mathematics and Science Study (TIMSS) 1999 Video Study is a follow-up and expansion of the TIMSS 1995 Video Study of mathematics teaching. The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom (Stigler & Hiebert, 1999) compares practices in eighth-grade classrooms in Germany, Japan and the United States. It is a seminal work that highlighted cultural scripts in teaching as well as drawing attention to practices in South East Asia, including notably, lesson study. In my view it was this work initiated pedagogy-envy in the UK and to some degree the US. It was both the character of practice and results from assessment-based comparisons that created this phenomena. But I will return to this.

Larger and more ambitious than the first, the 1999 TIMSS Video Study investigated eighth-grade science as well as mathematics, expanded the number of countries from three to seven, and included more countries with relatively high achievement on TIMSS assessments in comparison to the United States. The TIMSS video study involved videotaping and analyzing teaching practices in more than one thousand classrooms.

Learners’ perspective study (LPS)

This built on the ideas and methodology of the TIMSS video study. Although a centralised methodology was developed at the University of Melbourne, by David Clarke and collaborators, data collection and analysis was undertaken by local teams in each jurisdiction. It examines the patterns of participation in competently-taught eighth grade mathematics classrooms in sixteen countries in a more integrated and comprehensive fashion than has been attempted in previous international studies. Research teams now participating in the Learners’ Perspective study are based in universities in Australia, China, the Czech Republic, Germany, Israel, Japan, Korea, New Zealand, Norway, The Philippines, Portugal, Singapore, South Africa, Sweden, the United Kingdom and the USA.

The results of the Learner’s Perspective Study are reported in a Book Series, published by Sense Publishers . The first three volumes are: Mathematics Classrooms in Twelve Countries: The Insider’s PerspectiveMaking Connections: Comparing Mathematics Classrooms Around the World and Mathematical Tasks in Classrooms around the world. 

I reviewed the fifth volume, Algebra Teaching Around the World (Watson, 2016).

You can also find a discussion on cultural practices and scripts here. I am currently working on a paper with my colleagues, Lizzie Kimber and Louis Major looking at the role of cultural scripts and habitus in teaching.

What do comparative assessments tell us?

TIMSS 2015

The mathematics content can be found here. Achievement results are summarised using item response theory (ITR) scaling, with most achievements scores in the range 300 to 700.

The following charts present headline results from TIMSS 2015. The aim is to show comparative trends in achievement.

math-trends-in-mathematics-achievement-grade-8-trendgraphs

 PISA 2015

The following are headline trends from PISA 2015

pisa-2015

Critique of comparative studies – from Askew et al. (2010)

Askew et al. (2010) present an analysis of the findings of assessment-based international comparisons:

  •  Findings from repeated TIMSS and PISA studies add to our knowledge of changes over time, but these international studies are limited by their lack of longitudinal data that examines learning through tracking the same pupils over several years of schooling (see Theme 2: What rankings tell us, page 18).
  • Not all high attaining countries have closed the attainment gap between pupils from differing socio-economic backgrounds (see Theme 7: Attainment gaps, page 28).
  • Findings from TIMSS suggest the match between curriculum content and the TIMSS test items matters more than teaching in explaining international differences, although the quality of teaching still has a significant effect on mathematical learning (see Theme 1: Impact of teaching, page 16).

And the following are specific observations.

Finland

Finland’s pupils have been considered high performers in mathematics given their success in recent PISA studies. Finland ranked first in 2003 (although the Canadian province of Ontario was the highest scoring) and second after Hong Kong in 2006. This success was a surprise both in Finland and elsewhere (Pehkonen, Ahtee, and Lavonen, 2007). Efforts to understand this achievement have been hampered by a limited research base. The Finnish education system consists of comprehensive school education at both primary and lower secondary levels. Children start school at the age of seven and there are nine years of compulsory schooling. All types of education in Finland are free and well supported.

Singapore

Singapore’s educational structure comprises six years of primary, four years of secondary and two years pre-university. Only the first four years of primary follow a common curriculum: pupils follow one of two ‘orientation’ curricula in the last two years of primary, one of these being a reduced curriculum at a slower pace. There is a leaving exam at the end of primary: some pupils take a different exam if they have followed the ‘reduced’ curriculum. There are three courses at secondary school: around 60% of pupils follow an ‘express course’ leading to a GCE O-level in four years, 25% a ‘Normal’ (academic) course leading to O-level in five years (or an N-level in four years) and 15% in a ‘normal’ (technical) course leading to N-level. Between 20% and 25% of pupils continue to university. While the curriculum is centrally mandated and there is high-stakes assessment, schools have flexibility over the implementation of the curriculum. Since the 1990s there is no longer a single state-mandated textbook, with commercial publishers producing textbooks in an open market. A five-fold curriculum framework emphasises attitudes and meta-cognition as well as skills, concepts and processes. Compared to its near neighbours, Singapore’s pupils do report more enjoyment of mathematics.

 How valid are international studies?

It is not our intention to reiterate the arguments pointing out the difficulties and flaws in studies of international comparisons of mathematics education. For example, there may be considerable differences in the extent to which schools and students feel the tests are important. In PISA 2006 the comparison of first round school participation rates between Finland (100%) and the United Kingdom (76%) is telling. Perhaps more striking is the oft-quoted anecdote from TIMSS 1995 of Korean students marching into the examination hall behind the national flag. Others provide further cogent arguments into the shortcomings of TIMSS and PISA (see for example Brown, 1998; Goldstein, 2004).

The role of international comparisons

The purpose of the OECD for example is international economic activity and trade. However well meaning, the intentions of the organisation in the methodology and administration of PISA, education is framed in terms of trade and economics. Given the OECD’s role in globalization, it is necessary to question what impact it has on global education. It is fundamentally committed to economic globalization and questions have to be asked about its impact on equality. Capitalism, in particularly neoliberalism, increases competition by creating markets in public services[1]. Within this system, there are winners and losers and hence inequality is advanced. On the other hand, PISA and TIMSS are useful in looking at trends in educational performance within individual countries. Assuming, that is, the measures represent ‘quality’ in education. This, though, is a major assumption.

Comparisons of practices in different countries are valuable in understanding pedagogy and practice within different cultures. There is a danger that, taken with assessment-based comparisons, aspects of practice are copied, with an assumption of causality.  England has been invested in policy borrowing, and at times, cherry picking aspects from so-called high-performing jurisdictions. However, this overlooks the complexity of the systems being borrowed from and naive assumptions about causality.

As the global and economic landscape changes rapidly, as capitalism stumbles through another crisis, it is necessary to rethink the role of international comparisons. Greater attention needs to be given to social justice and the environment as opposed to a preoccupation with growth which leads to and exacerbates inequality.

We need to pay more attention to the local, within a globally connected world. Development needs to attend to geographical and cultural locality, it is through this that local communities are empowered. We also need to think about what a mathematics education might look like. Is it in the kinds of assessments used in PISA or TIMSS, and does it look like the kinds of practices we see in video studies across the world? I am not sure that it does, but that is for another time.

Finally, as I have argued elsewhere, we need to consider the driving forces of political economy. We thought we had reached the end of history in this respect and our attention to this had subsided. The global financial crisis has restarted history, we have to consider the globalized forces that impact on what we do in education.

Notes

[1] Since writing this I discovered the following article on the impact of PISA in Europe.

Grek, S. (2009). Governing by numbers: the PISA ‘effect’ in Europe. Journal of Education Policy, 24(1), 23–37. https://doi.org/10.1080/02680930802412669https://doi.org/10.1080/02680930802412669.

References

Askew, M., Hodgen, J., Hossain, S., & Bretscher, N. (2010). Values and variables: Mathematics education in high-performing countries. London: Nuffield Foudation.

Harvey, D.(2011). A brief history of neoliberalism (Reprinted). Oxford: Oxford Univ. Press.

Owens, T. L. (2013). Thinking beyond league tables: a review of key PISA research questions. In H.-D. Meyer, A. Benavot, & D. Phillips (Eds.), PISA, power, and policy: the emergence of global educational governance (pp. 27–49). Oxford: Symposium Books.

Steger, M. (2013). Globalization: A Very Short Introduction. Oxford University Press.

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.

Stiglitz, J. (2012). The price of inequality. London ; New York: Penguin Books.

Watson, S. (2016). Algebra teaching around the world. Research in Mathematics Education18(2), 211–214.