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

The iniquity of gap filling in mathematics teaching

What are we reducing mathematics teaching to? What are we reducing the experience of learning mathematics to?

The overemphasis on testing and the impact of ‘knowledge’ evangelism is reducing mathematics teaching to a shallow empty routine of teach, test, identifying the missing ‘knowledge’ and then giving pupils more practice.

Simply put, it is finding and filling gaps.

To someone not familiar with learning theory or with experience of teaching mathematics, this can sound logical and even sensible. A rigorous programme which efficiently leads to progress toward mathematical proficiency. And it is as if what went previously, in mathematics teaching and learning, was shambolic and ad hoc.

But gap filling is a superficial characterisation of mathematics learning. It assumes that mathematics is learnt as a set of knowledges and skills that are, themselves, discrete elements. Teach, test, analyse results and re-teach. Some refer to it as formative assessment. But to paraphrase Dylan Wiliam, if your assessment-for-learning (AfL, or formative assessment) practices involve conditionally-formatted spreadsheets, then it isn’t AfL.

AfL involves the diagnostic assessment of learners’ understanding in order teach responsively.

Mathematics is built upon an interconnected network of concepts and processes. These underpinnings also have cultural, political and historical origins. Mathematics cannot be defined as a set of abstracted and freestanding ideas and methods.

In learning mathematics, pupils need experiences and direct teaching that allows them to understand what mathematics is, its significance in their lives and to develop the confidence to use it in novel situations. Focussing on methods and factual elements does not achieve this.

Let me offer an example.

The English National Curriculum has included, since its inception in 1988, a requirement that students need to learn how to find the terms in a sequence by using a term-to-term rule and then a position-to-term rule.

The following linear sequence includes term values and position values. The general position is ‘n’ and an expression for the nth term can be found.

position1234n
term591317

In last Thursday’s Faculty session, I was discussing this with the secondary mathematics PGCE1PGCE = Postgraduate Certificate of Education trainees. The question I posed was: what is the purpose, in the curriculum, of position-to-position rules and term-to-position rules? The trainees suggested that it is important so that students understand sequences and patterns and as an introduction to algebra. These answers represent, respectively, a conceptual and procedural view of learning. Understanding and knowing how to.

But even deeper than this, the underlying purpose is concerned with a fundamental human capacity to identify patterns and use these to make predictions. Mathematics is often considered as the science of patterns. It is through the identification of patterns that we are able to apply inductive reasoning. This is a key part of human survival, as a social being. Through patterns we can make predictions about season, weather and the planning and production of resources.

This capacity can put us on the edge of madness. Some psychologists suggest that conditions such as patternicity and apophenia  are cognitive malfunctions where an individual identifies patterns in what is effectively noise . This has been related to conditions such as schizophrenia. In my view, patternicity is a natural human condition, the difficulty comes in knowing how to examine the validity of a proposed pattern. This is where the investigation of sequences in the mathematics classroom becomes important in learning essential aspects of being human.

The curriculum example, when you really start to dig, is so much more than learning a procedure i.e. learning how to find the next term and how to find an algorithm that links the position value to the term. There are profound philosophical and psychological aspects within this simple concept.

I conjectured in the PGCE session that the original intention and value of this in the curriculum had this in mind. However, its presence has also been shaped by classroom practice. The investigation of linear sequences is an enjoyable activity. One of my favourite resources New York Cop and Other Investigations  is now sadly out of print, but there are a number of other resources available from my colleagues at NRICH. There are some examples of sequence investigations here.

These kinds of investigations provide students with the experience of considering patterns in concrete form, allowing them the chance to make conjectures and predictions, as position-to-position rules and to position-to-term rules. The danger is that this investigative experience is reduced to a process and the exploration of patterns is reduced to a simple and largely meaningless procedure .

Bruner’s  conceptualisation of learning in terms of:

  • Enactive representation (action-based)
  • Iconic representation (image-based)
  • Symbollic representation (language-based)

In this conceptualisation, learners move back and forth between concrete and enactive representations, which may be physical or computer-based manipulatives, moving toward image representations and on to symbollic, language-based and abstract manipulation. Bruner presents a mechanism through which learners move from the real world to the cognitive world.

Returning to how sequences are taught in the classroom.

position1234n
term591317

At primary school pupils might start finding the next terms in sequences. Linear sequences naturally relate to multiplication tables. In the above example, the position-to-position rule is add 4. Primary pupils will be able to identify this and predict further terms.

In secondary school, in year seven, eight and nine, position-to-rules are introduced or what is often referred to as the nth term. It also represents an advance in mathematical reasoning from identifying next terms to understanding how an algorithm can be used to generate the term in a sequence from its position.

In this example the nth term can be found using the expression 4n+1. Understanding how to find and use this expression is more of a challenge than the position-to-position rule, since it requires multistage reasoning in the construction of an algorithm. Pupils need to have secure conceptual understanding of multiplicative reasoning and the properties of operations in mathematics i.e multiplication, division, addition and subtraction. If they have difficulty in finding an expression for the nth term, it is highly likely that they are not secure in the underlying concepts. Furthermore, if it is an algebraic expression that is sought, difficulties and misconceptions in algebra can cause further problems in undertaking what appears to be a fairly simple method. In high quality teaching, we would be looking at pupils’ work in order to diagnose the problems and identify tasks to address these issues.

The gap filling approach relies on teaching more procedurally. I had a well-refined ‘system’ through which I could demonstrate an efficient routine for generating an expression for finding the nth term. Most maths teachers have something similar. Find the difference between terms, this tells you the coefficient of n. In the example I have used it is 4. But 4n would give 4, 4, 8, 12, 16 … We have to add 1 to get  5, 9, 13, 17… . So the expression is 4n+1.

You would think with efficient teaching and practice this is easy. It isn’t, for many pupils this is not straightforward. They will, with support, produce correct answers, but after time or in another context, they struggle. It is as if they have never learnt how to do it. And not just for the reasons I have set out above. Self-theories and mindsets  also have an impact on the extent to which pupils retain procedural learning. I am not going to go into this here but Mark Dawes and I have written about in this paper.

The gap-filling approach does reveal that pupils have not retained procedural knowledge. But all too often the result is to treat the symptom and not diagnose the underlying source of the problem. Research shows us that long-term procedural retention in mathematics is a result of conceptual difficulties and not an issue of retention. Yet, too often the gap-filling approach encourages teachers to repeat the same teaching procedures rather than analyse and diagnose the underlying reasons for non-retention. This means that we need to include more investigative work into mathematics learning. Not because we want learners to ‘discover’ expressions and algebra but in order that they can make deeper conceptual connections.

References

Bandura, A. (1997). Self-efficacy: The exercise of control. W.H. Freeman.
Bell, S., Brown, P., & Buckley, S. (1989). New York Cop and Other Investigations: A Resource Book for Mathematics Teachers. CUP.
Bruner, J. S. (1966). Toward a theory of instruction. Beknap Press of Harvard University.
Dweck, C. S. (1999). Self-theories: their role in motivation, personality, and development. Psychology Press.
Hewitt, D. (1992). Train spotters’ paradise. Mathematics Teaching, 140. http://nrich.maths.org/content/id/7768/Train%20Spotters.pdf
Shermer, M. (2008). Patternicity. Scientific American, 299(6), 48–48. https://doi.org/10.1038/scientificamerican1208-48

Teaching reasoning in mathematics…but what do we mean by reasoning?

It’s an ongoing issue, one on which I frequently cite Robert Recorde’s sixteenth century observation that students must not only learn by rote but also by reason. Recorde’s The Ground of the Artes is one of the earliest mathematics textbooks in English, it is in the form of a dialogue between teacher and pupil (Master and Scholar) and explains the processes of addition, subtraction, multiplication and division. The following dialogue from the chapter on subtraction reveals Recorde’s thoughts on teaching and learning. The discussion of learning by rote and reason is still relevant today.

Master. So may you if you have marked what I have taught you. But because thys thynge (as all other) must be learned by often practice, I wil propounde here ii examples to you, whiche if you often doo practice, you shall be rype and perfect to subtract any other summe lightly…

Scholar. Sir I thanke you, but I thynke I might the better doo it, if you show me the workinge of it.

M. Yea but you must prove yourself to do som thynges that you were never taught, or els you shall not be able to doo any more than you were taught, and were rather to learne by rote (as they cal it) than by reason [my emphasis] (The Ground of Artes, sig.F, i, v cited in Howson 2008, p. 20).

Yet reason has vexed philosophers, psychologists and teachers alike. What do we mean by reason? And how might it be taught? That is, if it can be taught at all.

Here, I intend to address the former, what do we mean by reason? But in the context of learning mathematics. And to do this, I want to draw on the work of Philip Johnson-Laird, who has researched, extensively, the psychology of reasoning since the 1970s. He presents a comprehensive overview of his work in his book, How We Reason (2006). In spite of Johnson-Laird’s groundbreaking and illuminating work, very few of his ideas have been made use of in mathematics classrooms. I want to present an overview of his central argument and offer a suggestion about how that might be applied in the classroom.

A definition of reasoning

For Johnson-Laird goal-directed[1] thinking can be deterministic or nondeterministic. Deterministic thinking is where each step of the thinking process is based on a current state, like in a computer or when we carry out mental arithmetic. Johnson-Laird considers nondeterministic thinking as processes where we explore worlds of possibility. This leads to Johnson-Laird’s definition of reasoning (which he uses interchangeably with the word inference):

A set of processes that construct and evaluate implications among sets of propositions.

For example, pupils are often asked to compare the magnitude of two or more fractions, like in the example below. The propositions are that they are either the same or one or the other is greater. The reasoning process involves evaluating the possibilities. It involves constructs and representations that permit this evaluation process. fraction

In the following section I explain how Johnson-Laird explains the reasoning process.

Reasoning processes: dual processing theory

Dual processing distinguishes between rapid intuitive inferences and slower deliberative reasoning. This is also connected to subconscious and conscious reasoning, subconsciously we make rapid assessments and judgements and consciously we carefully and logically evaluate. Drawing on this model Johnson-Laird demonstrates how reasoning and inference involve mental processes which are carried out on mental representations. Some of which is subconscious and intuitive and some conscious and rational. The mental representations he refers to as mental models. To illustrate this, consider the following problem:

The cup is to the right of the plate.

The spoon is to the left of the plate.

What’s the relation between the cup and the spoon?

Unconsciously we produce a mental model based on our understanding of the premises and from this we draw a conclusion. In some situations we reason unconsciously and rely on intuition and may not be aware of the premises. In conscious reasoning we become at least aware of the premises as well as the conclusion we draw.

This process can be applied to the fraction example we can make intuitive judgements based on simple insights into the situation, or we can consciously evaluate, using mental models to reason the correct solution that 3/8 is the greater.

In many situations, where we are experts particularly, for example as an experienced teacher, we can quickly form impressions of a situation and act without thinking.

When logic and intuition conflict

One of Johnson-Laird’s most important contributions is the recognition that reasoning based on logic is less frequent than reasoning based on unconscious reasoning and heuristics, where we draw on long term memory and use rules of thumbs to draw conclusions. When we use logic and conscious reasoning it is demanding and can present us with contradictions with our everyday reasoning. Take the following example.

We’re all prejudiced against prejudiced people.

Anne is prejudiced against Beth.

So, does it follow that Chuck is prejudiced against Di?

Intuitively we say no, because nothing has been said about Chuck or Di. However, if we follow through the argument, Anne is prejudiced against Beth, so Anne is a prejudiced person and it follows that we are all prejudiced against her, so Di is prejudiced against her. Because Di is a prejudiced person and we are all prejudiced against Di, Chuck is prejudiced against Di.

This illustrates the limited capacity of working memory, it is difficult to hold all the information and make correct inferences. As Johnson-Laird says “Our reasoning is limited in power” (Johnson-Laird, 2009, p. 74).

The self regulation of the management of reasoning is important: knowing when and when not to consciously reason about the situations we meet. Self regulation takes time and experience to develop and this is just one justification for reasoning being part of the mathematics curriculum. It also relates reasoning to emotion and motivation as I explain next.

Emotions and reasoning

Many problems in our lives are concerned with emotions and affective states e.g. pain, stress, fatigue and anxiety. Johnson-Laird suggests that we read our emotions based on mental models of ourselves or self-theories. This has a role in how we reason, we construct mental models of the premises we confront and draw conclusions based on self-theories and our mental models of the world. It is also important to recognise that emotions have an important role in motivating us to act. Johnson-Laird recognises that any theory of reasoning must also account for affect and emotion. There is an interesting and useful link here with the work on self-theories and mindsets by Carol Dweck and on self-efficacy by Albert Bandura. Both of whom use constructs within social psychology that are analogous to Johnson-Laird’s mental models.

The implications for learning mathematics is that it is necessary for teachers to recognise the interconnection of emotion, motivation, confidence and reasoning.

Mental models and reasoning

The cornerstone of Johnson-Laird’s theory of reasoning is a recognition that we don’t simply think about possibilities, we represent possibilities as mental models. The manipulation of mental models leads us to draw conclusions and decide how to act. In learning mathematics students have to become familiar with mathematics as a series of abstract, but interconnected models and processes, and the axiomatic principles that define the rules with which we can manipulate the models. In learning mathematics we have to learn the abstract models that are the basis of mathematics[2] and processes that set the parameters within which these abstractions can be transformed. In the fraction example above, students must know how to mentally represent fractions and the rules that dictate how they can be manipulated in order to derive a conclusion.

The process of teaching and learning of mathematics requires teachers to engage with students’ mental models as articulated through the argumentation. Teachers have to interpret, diagnose and guide the development of more sophisticated mental models of mathematics. This is often referred to as diagnostic teaching in the context of misconceptions (Swan, 2001; Bell, 1993).

To conclude I summarise some implications.

Issues in the teaching of reasoning in mathematics

  • Students need to be more aware of what reasoning means – they need to develop more sophisticated understanding of how we reason: dual processing, the impact of emotions and self-theories (see, for example, Dweck’s Mindsets).
  • The development of reasoning skills involves having increasingly sophisticated models of mathematics and its concepts. The developmental process involves the diagnosis of existing mental models and supporting the development of new schema.
  • Problem solving has an important role in developing reasoning; in testing possibilities and conjectures; in allowing students to explore and develop their own mental mathematical models; in developing argumentation and justification; and building confidence in reasoning.
  • Students need to experiencing work on logic problems (like the ones above) to understand logic, but also to recognise the differences between logic and intuition.

Notes

[1] Goal-directed thinking refers specifically to thinking that has a purpose, there is an intention to achieve a result. This contrasts with musing or day dreaming, for example.

[2] It is worth considering here how abstract representations are equivalent to Johnson-Laird’s mental models. This also relates to Lakoff and Núñez’s (2000) notion of embodiment. This relates action and behaviour, mental models and metaphor.

References

Bandura, A. (1997). Self-efficacy: The exercise of control. New York: W.H. Freeman.
Bell, A. W. (1993). Some experiments in diagnostic teaching. Educational Studies in Mathematics, 24(1), 115–137. https://doi.org/10.1007/BF01273297
Dweck, C. S. (1999). Self-theories: their role in motivation, personality, and development. Philadelphia, PA: Psychology Press.
Howson, A. G. (1982). A history of mathematics education in England. Cambridge: Cambridge University Press.
Johnson-Laird, P. N. (1983). Mental models: towards a cognitive science of language, inference, and consciousness. Cambridge: Cambridge University Press.
Johnson-Laird, P. N. (2009). How we reason. Oxford: Oxford University Press.
Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: how the embodied mind brings mathematics into being. New York: Basic Books.
Swan, M. (2001). Dealing with misconceptions in mathematics. In P. Gates (Ed.), Issues in mathematics teaching. London: RoutledgeFalmer.