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.


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.


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.


 PISA 2015

The following are headline trends from 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’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’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.


[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.


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.

Why I think we should leave the EU

I voted to remain in the EU on 23 June 2016. If there was another referendum I would vote to leave. I have two main reasons for this. The first is that I understand more about macroeconomics. This is through reading in the period subsequent to the referendum. Classical economics has failed to make accurate predictions about the national, regional and global economics. While heterodox economics, like Modern Money Theory or Modern Monetary Theory (MMT), post-Keynesian post-Marxist has been much more effective in predicting the 2008 crisis. Part of the reason for this is that it does not assume that people’s decisions are rational, it considers the role of power and hegemonous groups.

Central to MMT is idea that a nation state, with its own sovereign currency, cannot become insolvent, it is solely responsible for creating its own currency and for spending that currency. There are other provisos, like for instance, a nation’s currency must not be pegged to another nation’s currency. This is the problem with the Euro and for those countries that have this currency. None of which are able to use fiscal policy to ensure that their economy works for their own people. In fact the European Central Bank (ECB) imposes austerity, it limits the extent to which each country can invest, through deficit spending. This is why Modern Money Theorists like Steve Keen and Bill Mitchell are eurosceptic. They predict that at some stage the eurozone will come apart, with considerable political and potential social upheaval. In the end, the UK is best out of this close economic union, even though it is not part of the Eurozone. According to Bill Mitchell, it is not brexit that is the problem it is austerity. That’s why those of the left should stop rerunning debates about the legitimacy of the referendum and focus  attention on exposing  the Tory government on its horrendous record on managing the economy. It’s not just their record on managing the economy, its the corruption also, they have abused their power by lining the pockets of their backers by giving them access to running state services for profit and through favourable taxation

My second main issue with the EU is its response the fascist coup in the US. The Donald Trump administration decieved the US voting poblic and persuaded them to put it into power. The US now has a dangerous far-right authoritarian and racist government. There are many who see the EU as an allience to counter the rise of fascism. I have little faith they will do this. The EU has failed to address the rise of the far right within its own borders, it is institutionally powerless. The reason for this is that inspite of being formed to ensure peace between European nations, its development as an economic block has turned neoliberal. The EU, by having the Euro, runs an economic system that minimises state intervention, encourages outsourcing and deregulation. It forces other nations to do the same. The EU will primarily protect its favoured form of political economy. And what we have seen with neoliberalism of the last 30 years, is rising inequality, unemployment or underemployement and stagnating wages, except for the wealthy. This is fodder for the far right, the same conditions that led to Trump in the US. The EU is wedded to neoliberalism, it will defend this and the Euro ahead of anything else, there are too many vested interests to counter any demise. And while the EU is a neoliberal institution, it is part of the problem and not a solution.

Two illustrations: A Labour MEP holds up a sign ‘He is lying’ while British fascist Nigel Farage speaks in the European Parliament. The EU leaders gather in Malta and issue a joint criticism of Trump. Both of these are platitudinous. The antidote to fascism (based roughly on Arendt) is through the public sphere. In Europe this would translate to radical economic and democratic changes, an end to neoliberalism; universal basic income and/or job guarantees; improvements in civic, institutional and workplace democracy; and progressive taxation. Fascism is overcome by empowering people to overcome it. I do not see any of this happening in the UK or Europe. Platitudes and indifference will have dangerous consequences.

Education, policy and pedagogy: It’s the political economy stupid!

At the heart of all the main issues in education at the moment is economics. In fact economics in education has become of increasing importance and is a growing field in itself. Analysis of data to evaluate education policy has been valuable in understanding how schools perform and the achievement and a progress of different types of students, for example.

This approach is in the tradition of classical economics. Underpinning classical economics is the idea that people make rational decisions within markets. This leads to econometric models that can be used to predict the behaviour of markets and the behaviour of the economy as a whole. In education, for example, it leads to predictions about earnings following participation in school-based programmes or interventions, the study of various subjects or attendance at Higher Education.

Classical economics sets its boundaries at the edges of the economic system. It does not concern itself with the political dimensions of economics, apart from say, informing policy makers on resource allocation. This rests on its fundamental principle of rational behaviour.

If we step outside classical economics, we can still see the distributions of wealth and power that classical economists observe, but we can also begin to see the forces that create these systems. It is not simply rationality with, as Adam Smith observed, an invisible hand ensuring that all would be fair in a freemarket society. Karl Marx’s critical analysis of capitalism in the three volumes of Das Kapital showed that the freemarket does not lead to a fair or equipatable distribution of wealth. It necessarily leads to the accumulation of capital. As a consequence there is an exploited working class. And hence economy is necessarily political.

One can be forgiven for thinking that in state education political economy can be ignored. The reason we think like that is that since the end of the Second World War and until recently, we have had no reason to think differently. But now we must. I shall explain by dividing the period between 1945 and the present into three economic phases.

The first phase is from 1945 until 1970. The post-war period saw considerable government spending on health and education and sat alongside a  freemarket economy. Education was grant-funded through local authorities. This investment was seen as a benefit to society as a whole. However, from the late sixties until the 1970s, things changed. The economic context changed and public education economics had to change in response. This leads to our second economic period between 1970 and 2008.

In the 1970s, there was a change from a mixed economy with public sector spending alongside a freemarket economy to neoliberalism, where there was much greater emphasis on the freemarket. During the early seventies there was a crisis in capitalism, in the UK this was characterised by inflation, decreasing company profits and increasing wage demands by the work force. In the end the unions lost, their pay was controlled and businesses were able to maintain their profits to some degree. The worst outcome was that the economic crisis was erroneously blamed on the unions, inefficient nationalised companies and a supposedly bloated public sector. When Margaret Thatcher was elected in 1979, she famously took on the unions, began privatising nationalised companies and reducing the size of the public sector. Underpinning this was the belief privatisation and marketisation was the best way to run our public services. The invisible hand would do its job. Neoliberalism was the political economy that continued until 2008. It was adopted by New Labour in 1997 and was supplemented by an increase in public spending. Although much capital spending in schools was from private capital. For large businesses, neoliberalism created opportunities to profit from public-sector services and subsidised by the state.

The preoccupation through this period of neoliberalism has been on the deficit in public sector finances. That is the difference between tax revenues and government spending. The preoccupation with eliminating deficit spending and an attempt to return a surplus in public finances has the effect of reducing private sector surpluses (I explain this in more depth here). In other words private sector borrowing has to increase, households become more indebted, house prices inflate. This creates demand in the economy (consumers are debt spending) and the banks profit. In 2008 this whole sorry pile of private debt was found to be overvalued and the big banks had to be bailed out by the state. Once again capitalism is in crisis. But the financial crisis of 2008 was a symptom of underlying problems brought about by neoliberalism itself.

The neoliberal period of unregulated freemarket capitalism has resulted in increased wealth inequality, while the richest 10 per cent or so, have got richer the rest have got poorer or are carrying considerable debt. Inequality in society is indicative of a divided and unhealthy society. Wealth and income inequality leads to democratic inequality, where the wealthy are in a position to influence government much more than the less well off. It also leads to health and education inequalities. Furthermore, it leads to a less productive society since there is less investment in workers and their development.

We find ourselves in period of post-capitalism or post-neoliberalism, the collapse of centrist politics is indicative of this also. No longer is the status quo working for a large proportion of society, this is evident in the election of an unequivocally anti-austerity leader of the opposition, and more dramatically the referendum result that will ultimately lead us out of the EU. This was the precipitation of an anti-establishment and anti-status quo vote.

In terms of the character of education, the three economic periods (public sector, neoliberalism and post-capitalism) have shaped schools and pedagogy in particular ways. During the public-sector period (1945 – 1970) practices and organisations were emergent, but drew heavily on the approaches used in traditional establishments, like for example, the grammar school. In an attempt to address diverse social needs and with new ideas developing in the fledgling field of education research, there were attempts to address individual needs using student-centred practices. However, the mainstay of educational practice drew on traditional teacher-centred practices, because it is much easier to prepare for and to manage classrooms.

The neoliberal period (1970 – 2008) can be characterised by increasing accountability, increasing managerialism and perfomativity. The emphasis on accountability means that teachers are expected to ensure students achieve targets and expectations in terms of progress and examination results. There is increased surveillance and attempts to identify effective practice in terms of progress and attainment. In the 2000s this extended to a prescription of classroom practice and pedagogy. While practice remains largely traditional, there are elements of progressive student-centred teaching, but on the whole the latter, apart from among enthusiasts, was superficial. The importance of the social aspects of learning, such as discussion and dialogue, the importance of affect and motivation and the recognition of constructivist learning were recognised and mandated in official views of pedagogy. However, given the demands placed on teachers and the intensity, as a result of being held increasingly accountable for students’ results, these elements were only really implemented in a performative way, to please observers and inspectors rather than placing them at the heart of education.

The post-capitalism period (2008 – present) continues a neoliberal theme, but it does not hide the crisis beneath. Since 2010 the Coalition government and the  Conservative government from 2015, have extended the privatision projects brought in by previous governments. Academies are effectively outsourced education service providers to the state. There is increased emphasis on quantifiable outcomes to monitor the quality of the service provided by schools. In an attempt to make the educational commodity more clearly defined the student-centred aspects of pedagogy have been abandoned and even vilified. The emphasis has been increasingly on narrowly defined definitions of knowledge and the reduction of learning to a process of memorisation of increasingly complex facts. The crisis beneath this, within the post-capitalist school, is the overall reduction in teachers’ pay and conditions, longer working hours, excessive workloads and deprofessionalisation. The recruitment and retention of teachers is increasingly challenging. There are also deep concerns about the impact of intense school experiences on children’s mental health and wellbeing.

Economically, we move into a post-capitalist post-neoliberal world in which economic, technological, social and political forces are undermining existing approaches. Yet the government continues to press ahead with a privatised and marketised approach to education. What we need to do is develop schools and educational practices to respond to community needs in a more holistic way and to draw on contemporary understanding of learning in terms of culture, socialisation and cognitive development. We cannot return to public sector nationalisation of state education, but we must reduce the managerialism and hierarchical structures of schools and academy chains and improve the working conditions and professionalism of teachers. They can be mutualised as community co-operatives, to devolve decision making and to collaborate with communities. This is an antidote to the corporate managerialism of the neoliberal period. While schools cannot mitigate for wealth inequality, they can connect with local communities and help develop confidence and build social and cultural capital. Austerity (deficit reduction) is a political choice and not consistent with the post-capitalist period we find ourselves in (I elaborate on this in a previous post here).

The driving force in state education is political economy and by considering economic and political forces, not only can we better understand policy, practice and pedagogy, we can better design schools and learning to respond to the political economy in which we live.

The politics of mental illness: from R D Laing, The Frankfurt School to Mark Fisher and Capitalist Realism

I recall reading R D Laing’s The Politics of Experience and The Bird of Paradise almost 25 years ago. Laing was a radical psychiatrist, part of the anti-psychiatry movement of the 1960s. Anti-psychiatry viewed a patient’s illness not just as the patient’s but as part of sick society. His was a bold attempt to get inside the minds of those with a mental illness and to recognise the politics of the experience of mental illness.

There is no such ‘condition’ as ‘schizophrenia’, but the label is a social fact and the social fact a political event. The political event, occurring in the civic order of society, imposes definitions and consequences on the labelled person. (Laing, 1967, p. 100)

Much more recently  I read The Grand Hotel Abyss: The Lives of the Frankfurt School by Stuart Jeffries. The Frankfurt School, from shortly after the beginning of the twentieth century until the latter part of the century, fused Marx’s political economy with Freud’s psychoanalysis. Their critical theory offered an analysis of the development of popular culture and its impact on individual psyche. The twentieth century was the century of mass communication, broadcasting and consumerism. What Theodor Adorno, a key thinker in the Frankfurt School, recognised was that the potent combination of mass communication and consumerism was used to not only suppress any revolutionary zeal of the proletariat, but also to enhance capitalism by creating consumers, pandering to base needs and creating superficial and relocated desires leading to consumerism. Adam Curtis’s, The Century of the Self, presents a stunning visual representation of the effects of the acquisition of psychoanalysis by advertisers and their capacity to use this to control our behaviour. It translates human alienation that arises from subjugation and subordination to capitalism to a desire for consumption of unnecessary products. Capitalism becomes an imperial power in the mind of the individual.

The Grand Hotel Abyss is a fine read, it takes you through the lives of individuals involved in the Frankfurt School as they navigate through the latter parts of the First World War, the Wiemar republic, the Third Reich, exile to California and the liberation movements in the US and Europe in the 1960s.

I was just finishing reading it, when I heard about the death of Mark Fisher. Mark Fisher was a critic, theorist and activist. I was alerted to his short book (90 or so pages) Capitalist Realism: Is There No Alternative? by people I follow on Twitter. Intrigued, I downloaded it to my e-reader. It was within the first few pages that I found that Capitalist Realism is both extension and critique of ideas of the Frankfurt School. I can probably best explain what I mean by saying that the Frankfurt School is located in modernity, while Fisher takes something of a postmodern turn. Modernity was a dominant identifiable cultural philosophical movement emerging in con text of mass production and mass consumption. It is concerned with structures, overarching theory and in many ways mechanistic explanations of the relationships between phenomena and experiences.

Postmodernity is the paradigm shift. As society becomes fragmented, communications and broadcasts fragment to provide individualised experience, as liberalism becomes a dominant political ideal, as communities become diverse and heterogeneous. Thought and experience become fragmented. Philosophy cannot rely on the more monolithic modernistic structures. Lyotard, in the The Postmodern Condition: A Report on Knowledge, heralded the end of grand narratives.

What Fisher does in such a concise and powerful way is weave together Marxist political economy and psychoanalysis while acknowledging the postmodern fragmentation, contradictions and ironies. This provides a powerful critique of mental health as a deeply political and politicised experience. In modernity our desires were controlled and manipulated. The postmodern condition is so much more insidious, sure our desires are controlled, but by images and narratives that we create for ourselves, that we construct from the narratives that are presented to us through the media and reinforced by neoliberal structures and organisations which discipline and normalise our actions through performativity and targets.

We are never away from these personalised constructs, it is our own thinking that disciplines and punishes and keeps us alienated from direct and real experience. We persistently live in a fictive world created by capitalist media. When our mental and physical health is under threat the external narratives that we internalise start to unravel. We become politically active. But more often than not we punish and discipline ourselves because we no longer think or behave ‘normally’. More often than not our mental health conditions are medicalised, we are subdued by chemicals and our senses and reactions are dulled until we submit to the reformation of a personal fiction that is within the limits of normality. That we are restored to being an individualised component in a self-governing capitalist system.

An important point Fisher makes is that the postmodern mental control and self-disciplining experience is with us 24/7. In modernity, we would go to work do as we were told and then go home. For Fisher we are constantly self-regulating and self-directed. Work and production continue because we are under the impression that we are autonomous. We are not.

Currently we are seeing an adjustment to world orders and authority. It is as if this shake up of old truths, a crisis of capitalism and a collapse of postmodern fragmentary narratives, open things up so we can see what Lacan called the Real. There are great dangers as well as great possibilities in the future. R D Laing, The Frankfurt School and Mark Fisher have left us with some important insights from which we can proceed. Importantly we should recognise the politics of mental health.

I understand that Mark Fisher died with little money and intestate. There is a campaign by Mark’s friends to raise some money so that Mark’s wife and child can have a little time to grieve and come to terms with his tragic death.

Privatising the universities: the real agenda of the Higher Education and Research Bill

Much like the privatisation of state schools, which I have written about previously, the Coalition Government between 2010 and 2015, and the Conservative Government from 2015, have accelerated the marketisation of Higher Education. A process that began under the previous Conservative Governments under John Major and Margaret Thatcher and continued by New Labour between 1997 and 2010.

You see, neoliberalism has been a dominant political economy since the 1970s. Neoliberalism is a special kind of freemarket capitalism, that extends beyond the traditional domains of capitalism, the factory for example, to what were previously nationalised or public services, the Post Office and Royal Mail, British Rail and domestic energy supply. The expressed benefits of privatising and marketising these services was that competition would lead to greater efficiency, better value and a smaller state.

Neoliberalism has been such a successful manifestation of capitalism (for some), effectively a state subsidised private sector businesses with, by-and-large, something of a monopoly, the private sector has been creeping into to many other public services. The Multi Academy Trusts that run schools are private companies, prisons too are outsourced to private companies like G4S. There was even an attempt to sell off Land Registry lately. Now we are starting to see similar happen with universities.

It is not obvious, because it is not always easy to see the emergence of private from the public. That is, it is not always clear when an organisation is based on private or public capital. The Multi Academy Trusts are private limited companies with a charitable status, and do not make a profit in theory, though many of them generate surpluses. They do not exist in a genuine freemarket, they often have a local monopoly and they are regulated by the state. But for all intents and purposes their quasi-private market status works in the same way as any corporation underpinned by private capital. They are largely autonomous from the state, and state and local democracy, they have customers (parents and children) who have a certain amount of market choice and they provide a service that is codified and quantified by public assessment.

One of Karl Marx’s most important observations was that capitalists, or those that have access to private capital, are obliged to accumulate more capital. By capital he means money, goods or means of production that are engaged in the process of production and sale of goods and services. Accumulation is not simply down to the vice of greed, it is due to the existence of the market, if the capitalist does not accumulate capital at the same rate as the competition they will fail. Capital obliges capital accumulation. Capital, is not just about accumulating money it is more often than not the expansion and replication of existing means of production.

Now, we have observed this in schools over the last few years, large academy chains have grown very quickly and in some cases (see for example Ark Schools) have expanded into overseas projects and spin-offs in this country.

In universities the marketisation and privatisation process began in the 1990s. Ron Dearing was commissioned by the Secretary of State for Education, Gillian Shephard, in the Major government in 1996, to report on Higher Education and consider how it might develop in the subsequent 20 years. It was hear that Dearing proposed that students should pay 25 per cent of their tuition fees. In 1998, Secretary of State for Education, David Blunkett, in Tony Blair’s New Labour Government, introduced student loans with means-tested tuition fees. Student grants of £1710 was replaced by income-contingent student loans.

This is ground zero on the march toward privatisaton and marketisation of universities. Blunkett effectively created a market by making students customers. This changes the dynamic, subtly, students are buyers and have choice where they spend their money.

This was strengthened in 2005 when New Labour gave universities the right to charge tuition fees of up to £3000 per year. When the coalition government came to power in 2010 this was increased to £9000 per year. Where Higher Education had been paid for through government spending it was now being paid for through private debt. It was argued this promoted widening participation in universites, while minimising public spending. This argument is disingenuous in my opinion, it disguises the forces of neoliberalism that underpin the marketisation of Higher Education (you can see the economic argument for this in a previous blog).

This brings me to the Higher Education and Research Bill which, if it passes through parliament, will further cement privatisation in Higher Education. If public and private is considered as a continuum, this is turning the dial up. Firstly, the introduction of private challenger universities will introduce greater competition into the Higher Education ‘market’. Some argue this is a good thing: shake things up a bit, put complacent universities on their toes. Fine, but Higher Education is so much more than simply purchasing knowledge and skills. It is a time for young people to explore and innovate, for students to question themselves and the world. If it is reduced to buying a ‘commodity’, it undermines so much of this. Furthermore, the introduction of the Teaching Excellence Framework (TEF) serves to commidify this experience, the metrics will lead to a race to optimise on narrow outcomes. Worse still, universities will be competing on the outcomes of parochial measures. It will contort Higher Education from providing rich, stimulating and provoking experiences to the acquisition of defined knowledge and skills. Within the TEF metrics courses will score more positively if they are linked to employment. This has the potential to undermine subjects that do not have a direct link with the world of work. These course may disappear, or be reconfigured for employability. This is at a time when we need graduates to develop creative and divergent thinking informed by the broadest study of diverse disciplines.

Of course, I have to admit, this is likely to have little impact on so-called elite universities such as the one I work in. Certainly not to the extent that it will have in many of the newer universities, where it is going to be a much tougher competitive environment. My previous university committed itself  to neoliberalism, it became observably more managerial and hierarchical, as well as expanding by opening campuses in other countries.

I concede that there are things that could be done to improve Higher Education and the quality of teaching and learning in it. But privatisation is the wrong answer.


A reaction to the College of Teaching’s announcement of the planned availability of academic journals for teachers

I was pleased to hear that teachers in England would have access to research literature.  And I was pleased also that the most charming Dr Vincent Lien who had campaigned for access to journals had been acknowledged in this.

My first question was, which journals and how many? But I will assume that access will be to a range of international education research journals and perhaps some subject specific journals.

When I was teaching and growing an interest in research, I wanted access to education research but academic journals were,  by and large, paywalled. Frustrating. Because I also became aware of the tidy profits that the small number of publishers made. In the 2000s though, you could get access to a 10-credit Open University course for around £100 and this would give you access to just about every academic journal for six months or so. Indeed 30-credit and 60-credit courses were affordable too and for a few hundred pounds you could get access for 12 to 18 months. That has all changed now, since the Coalition Government brought in its economic policy to rebalance the wealth of the 99 per cent to the coffers of the 1 per cent. OU courses are now at least double what they were.

This is something of a diversion, but nonetheless important, since my engagement with academic research when I was a teacher was just that ‘engagement’, making sense of theory and practice across a range of disciplines: sociology, psychology, anthropology etc. How could I make sense of my practice in the context of school, policy, the community and in my classroom? How do I understand learning as a social cognitive, biological and cultural phenomena? How could I develop what I did? What are the methodologies and methods for such inquiry?

I was engaging with ideas, theory, concepts as well as results and findings. I was engaged in a process of scholarship. It was time consuming, I gave a lot of my own time to it, as I have the habit of blurring my interests with my work which seems to have been with me always.

This is my concern. In the last ten years teachers’ workload and the intensity with which they work has increased considerable. Accountability has become increasingly pervasive and poor performance is increasingly treated punitaively, though the notion of ‘poor’ is spurious. It has lead to a system of performativity, in which prescribed practices are imposed in pedagogy and assessment. It is difficult for teachers to find the time to engage in research in a scholarly way, to reflect on ideas and concepts in relation to their own practice.

So it worries me that since Michael Gove’s education reforms, teachers are framed as consumers of research, that research provides definitive answers about practice and about the effectiveness of different approaches. In a post last Christmas I wrote about the Book of Intervention, a satirical presentation of the Education Endowment Fund’s (EEF) Toolkit. It was inspired by a visit to a general practitioner who, when I described my symptoms – a heavy cold, as I recall – perused a large volume of listed pharmaceuticals and then presented me with a prescription: medicine that is the result of a randomised control trial. The Michael Gove vision for education research, prompted by Ben Goldacre’s Bad Science, was that educational research should be ‘scientific’ and experimental. The EEF was established to do scientific educational research using randomised control trials. While educational researchers follow the scholarly principle of beginning their inquiry with a research question, the EEF begins with, in dogmatic fashion, a prescribed methodology.

Of course, this dovetails sweetly with Gove, Dominic Cummings and Policy Exchange’s freemarket neoliberal plan for state education. The codification of interventions and pedagogy, scientifically derived, reconstitute education as a quantifiable service and thus makes outsourcing and privatisation so much easier.

On the other hand, teaching is a complex undertaking, complex relationships and complex constructs of learning and behaviour. In spite of whether we know some aspect of teaching is good or bad, or whether we reduce education to a process of memorisation of facts, the complexity remains. Teachers cannot always replicate the practices that were identified in experimental studies, least those practices may not always be appropriate to the unique situation they find themselves. Constantly teachers are faced with varying experiences and interactions that calls upon their professional judgement.

It is my view that teachers should not be seen as consumers of research but as scholars in their own right, where they engage with research and use theory and knowledge to develop their own thinking in relation to teaching and learning. The latter is time consuming and requires more than referencing the EEF Toolkit, it is necessary to read mulitiple sources reflect and discuss with colleagues and academics.

So while I welcome the news that teachers will have access to journals, I think the Chartered College needs to be reminded that this and previous governments have set our schools on a neolberal course, one that has taken time and autonomy away from classroom teachers. It is the outsourcing and privatisation we should be opposing and we should be fighting together to ensure that teachers’ pay and conditions are adequate enough to permit them to be scholars in their own right and not simply consumers of research or worse still that teachers become deliverers of an experimentally-defined centralised curriculum.


There is plenty of money to spend on schools: a Modern Money Theory perspective

It is common and widely accepted that the UK’s finances are in a dire situation. The national debt has grown from £700 billion in 2010 and is set to reach almost £2 trillion in 2020. While the deficit (the difference between government spending and taxation) has been reduced from roughly £100 bn (2010) to £40 bn (2016), there is a commitment to turning the deficit into a surplus. As a consequence many people are willing to accept that we must tighten our belts, make savings and do what we can to avoid the out-of-control debt spiralling further. We do, after all, want to avoid passing on this debt to our children and grandchildren. So it goes.

According to the Institute of Fiscal studies, school spending is being cut by 8 per cent in real terms. Perhaps more. Over the last few weeks I have heard colleagues in the university as well as teachers, headteachers and parents acknowledge that this is necessary given the economic situation and prospects I have outlined above. We have to cut back on spending, we have to reduce the debt, we have to make sacrifices. We have to find low cost and efficient ways of educating children more cheaply. No frills, no expansive (or expensive) learning; the learning of facts in classrooms with austere compliance. For teachers, no professional development, longer hours and real-terms pay cuts.

After all, it all makes complete sense. It may be a bitter pill to swallow, but if we, as individuals, households or as businesses over spend we would end up with an unmanageable debt and we would go bust. It makes sense that the same applies to government spending. If the UK government overspends, day-to-day, we will, as a nation, go bankrupt.

Now here’s the thing, government spending is not an aggregated version of household spending. In our personal lives our spending is independent of our income. We are, for all intents and purposes, free to spend what we like; whether we spend less or more than our income. Clearly, if we spend more we will go into debt and if we spend less (which is almost an impossibility in the Watson household) we save, but fundamentally income and expenditure are independent of one another. Which is where there is big difference with government finances or what we might call macroeconomics. This is probably the most important thing to understand in getting a better grip of how a nation’s economy works.

The difference between you and the UK treasury is that you do not issue your own currency, the UK does. For every transaction with that currency, whether you are paying for schools or buying an ice cream (see Richard Hammond), there is a buyer and a seller. Because at any one point in time there is fixed amount of currency, for everything spent there must be something sold. Across the whole economy income must equal expenditure. There is at the heart of macroeconomics a conservation of the total amount of currency, because it is only the government that has the legal power to create or destroy currency.

Think of it like this, the government creates currency, the pound. If you work in the public sector you are paid in pounds, with which you buy things from the private sector. People working in the private sector get paid in the currency and also buy things. Currency is circulating around the economy (of course, households and businesses can save money buy spending less than they earn but I will come to this later). The mechanism by which the circulation of currency is controlled is through taxation, we have to return a proportion of our earnings through paying tax in that currency. But importantly the government has to create currency and spend it before there is anything to tax. Tax should not, therefore, be seen as a revenue source, but as a means of regulating the amount of currency in circulation. The source of a government’s capacity to spend is through the creation of its own currency. It is therefore recognised that governments, like the UK and US for example, cannot go bust because of their power to create currency and that they buy things in that currency.

These are the fundamental ideas in Modern Monetary (or Money) Theory (MMT) which draws on the ideas of both Keynes and Marx to consider macroeconomics in an alternative way. The starting point is the idea that within a nation with a sovereign currency all incomes must equal expenditure. A macroeconomic view of this income and expenditure balance is usually considered by breaking it up into different sectors. The private sector, the public sector and in trade with the rest of the world. With this we get the following simple equation:

Private sector surplus or deficit + Public sector surplus or deficit + Exports/Imports = 0

The private sector includes both businesses and households, it is desirable that this is in surplus, since the private sector should have reserves to cope with changes. The public sector surplus or deficit is the difference between tax revenues and government spending. It is the figure that governments and media like to bandy around and create such alarm with. But you can see that if imports are high i.e. currency is leaving the UK, and the private sector is running in surplus then we must have a deficit. And in fact, deficit is a normal way of operating the economy and should not be used as an indication of over spending.

What about the national debt? This not really a debt as such, it is equivalent to the accumulation of surpluses in the private sector. The government issues bonds to buy back the reserves created by the banks as a result of the deposit of private sector surpluses. In the UK this is mostly from large businesses and wealthy individuals since quite a lot of ordinary working folk are in debt.

So if deficit and debt do not constrain government spending can we just create money and be done? It is true to some degree, but according to MMT the constraint on spending is inflation. In other words if the government increases spending too quickly demand outstrips supply and prices go up. A government has to increase spending cautiously. At present, however, inflation is not a problem, it is very low and we can afford to increase public spending without worrying too much about inflation.

But an increase in spending would stimulate the economy, increasing economic activity and therefore growing the economy. One issue is to make sure that spending does not result in the accumulation of wealth by large companies and wealthy individuals. This is what has happened as a result of quantitative easing after the financial crisis and it is why the national ‘debt’ has grown.  As this is equivalent to the accumulations in the private sector. QE swelled the coffers of the rich. In order to make sure spending is more fairly distributed we need to consider things like universal basic income, progressive taxation and debt jubilees (paying off household debt).

I have taken you on a whistle-stop tour of MMT, there is much more to read and understand – which is what I continue to do. A good visual account can be found here. I have been reading Modern Monetary Theory and Practice: An Introductory Text by William Mitchell, L. Randall Wray and Martin Watts, which I recommend. The are blogs by Bill Mitchell, New Economics Perspectives. This lecture by L Randall-Wray is an excellent introduction too.

I conclude by outlining or restating — emphasising even — the implications for spending on schools. Debt and deficit are not the barrier to adequate spending on our schools. As the sixth wealthiest nation in the world we can afford to properly fund our schools. There really is no excuse. What is not clear is whether the government are economically illiterate/incompetent or have some other agenda i.e. the privatisation of schools. If you read The Price of Inequality by Joseph Stiglitz or Post-Capitalism:  A Guide to Our Futures by Paul Mason, they suggest the latter is almost certainly true. Capitalism is in a state of crisis, ensuring sustained profits is difficult and therefore companies are wanting to move into areas where governments can support revenue and profits, like transport, health and education. So-called neoliberalism. Expounding the belief that a nation’s economics is analogous to a household or business serves this: reduce the deficit and debt through outsourcing, markets and efficiencies. It seems most likely that government is being influenced by the self-serving who are defending the capacity of the wealthy and large business to accumulate capital.

It is important that we in education ask questions about the economic models that we are presented with. That we do no acquiesce in a state of ‘oh dearism’ and resignation. That we don’t find ourselves trying to mitigate for government cuts by overworking and burning out. It is important that we educate ourselves, challenge the government and act in solidarity to oppose.

I am happy to engage in discussion about the ideas I have presented here.

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.


[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.


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.
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.

The variation in teachers’ pay in large Multi Academy Trusts

Following my analysis in previous blogs of the variation in teachers’ pay in England, I now look at the difference between pay in the larger academy chains and Multi Academy Trusts (MATs). I used data from the National Statistics School workforce in England (November 2015) data again and identified which Trust schools were part of. I look particularly at the larger groups in both primary and secondary. I make comparisons with maintained schools and academies in general. I would add that this analysis is preliminary, but is consistent with the analysis in my previous blog posts.


The following chart shows mean salaries in primary schools in selected large academy chains. Data labels indicate the number of primary schools in the chain. The maintained and academies and free schools bars are the average by type for the whole state sector.


The following table includes conditional formatting comparing mean salary with maintained primary schools.meansalaryprimarybytrusttable2015


The following chart shows mean salaries in secondary schools in selected large academy chains. Data labels indicate the number of secondary schools in the chain. The maintained and academies and free schools bars are the average by type for the whole state sector.


The following table includes conditional formatting comparing mean salary with maintained secondary schools.meansalarysecondarytrusttable2015


It is notable that as with my previous analysis the mean salaries in academies and free schools is less than it is in maintained schools. This analysis shows that this is true for both primary and secondary schools. It is also important to note that MATs who appear to have higher than average pay are likely to have more schools in London. This is certainly true of the Harris Federation where the average pay is influenced by London weighting. However, it has to be acknowledged that the average inner London pay is higher than maintained schools. In my next post I will look at the differences in London and regional pay more closely.


Thanks to JL @dutaut who observed that AET have 67 schools but only 32 primaries and 30 secondaries: the missing ‘five’ are special schools. Where there are discrepancies like this the schools not included are special schools or all-through schools.

My data can be viewed here.



Why do teachers get paid more in maintained schools? – part 2

I have completed some further analysis using the underlying data from the National Statistics School workforce in England (November 2015) data. I have looked at the differences in teachers’ pay in secondary schools between Local Authority (LA) maintained schools and academies. The question raised in my previous blog was: why do teachers get paid more in maintained Schools than they do in academies?

Today’s analysis drew on the underlying data, where the mean full-time equivalent (FTE) pay for each school is presented. Questions where raised in respones to my previous blog about whether the differences was a result of different academy types i.e. converter academy [1] or sponsor led [2], or whether there was some effect owing to higher salaries in London, for example. My analysis here suggests not and it also supports the analysis I presented in my previous blog. It seems that if you are a teacher you are better off working in a maintained school.

The following chart summarises the difference between average pay in converter academies [1], sponsor-led academies[2], free schools and LA maintained schools.


The mean pay in maintained schools is over £700 greater than in academy converter schools and just over £1000 greater than sponsor-led academies.

Now to look at the differences in pay between the different types of schools types, in relation to London weighting, outer London weighting, London fringe pay and regional pay.

School type, mean FTE pay. London and regional weightings

With the exception of outer London and London fringe, teachers are paid more in maintained schools. In these area pay is higher in the small number of free schools, maintained school pay is comparable to pay in converter academies. Consistently the pay in sponsored academies is less than maintained schools.


I have not yet determined why this is from the data. However, my previous blogs on privatisation would suggest that when a service moves out of the public sector there is a natural downard pressure on teachers’ pay and conditions. Perhaps we are seeing this here.


[1] Converter academies are successful schools that have chosen to convert to academies in order to benefit from the increased autonomy academy status brings. They were introduced in 2010 as part of the Coalition government’s plan to broaden the academy programme and eventually enable all schools to become academies.

[2] Sponsored academies are usually set up to replace under-performing schools with the aim of improving educational standards and raising the aspirations of, and career prospects for, pupils from all backgrounds including the most disadvantaged.

Sponsors are responsible for establishing the Academy trust, the governing body and the appointment of the head teacher. They come from a wide variety of backgrounds including businesses, faith communities, universities and individual philanthropists. Outstanding schools and academies may now also become sponsors themselves in order to help less able schools to improve.
Sponsors no longer have to make a financial contribution, or establish or support an endowment fund, as in the past. However, the Government has said any financial contribution made “at their own discretion” would be welcomed as it would provide opportunities for pupils that are not supported through government funding.