In a recent speech the UK’s prime minister, Rishi Sunak, once again raised the profile of mathematics on the basis that ‘our children’s jobs will require more analytical skills than ever before’. Such pronouncements are ubiquitous from political leaders who see education as a solution to an economic problem; or as David Cameron once described it: ‘the best inoculation against unemployment’. The latest policy approach to this problem is Teaching for Mastery (TfM) – a widely enacted, and government endorsed, programme of professional development led by the National Centre for Excellence in Teaching Mathematics.
TfM provides teachers with structure and guidance, and the positive message that all children can succeed at mathematics. Its approach focuses on teaching classes together rather than grouping by ability, and on honing down mathematical ideas so that teaching is focused on
the most important conceptual knowledge and understanding that pupils need as they progress … [which] provide a coherent, linked framework to support pupils’ mastery of the primary mathematics curriculum.
DfE, 2020, p. 5
This government guidance asserts the kind of management of learning mathematics that is de rigueur in modern classrooms, where teachers’ primary responsibility is to demonstrate their control of pupils’ progress. However, in our recent article in the Curriculum Journal we question whether such a confident, objective, linear view of learning mathematics makes sense.
To evaluate this question, we used Anna Sfard’s (1991) seminal paper on the psychological basis of mathematical development. Her argument is that mathematical ideas are not simply objects but are understood simultaneously as concepts and processes: 2 + 3 both makes 5 and is 5. Nor are these linear. You don’t ‘learn addition’ and then ‘learn multiplication’; you learn something of addition and learn more about it as you put it to use in multiplication. Thus, mathematics certainly develops, but it requires learners to hold on to, make use of, and, crucially, be comfortable with ‘partial understanding’ at each stage. As the famous mathematician John von Neumann is reputed to have said: ‘you don’t understand mathematics, you just get used to it.’
‘If mathematics isn’t as the Teaching for Mastery curriculum suggests, what then sustains teachers’ belief in linear, step-by-step understanding?’
Our paper asks: if mathematics isn’t as the TfM curriculum suggests, what then sustains teachers’ belief in linear, step-by-step understanding? Our answer, drawn from Thomas Popkewitz, is that the translation of the discipline of mathematics to a pedagogic strategy has followed not the path of mathematical development but that of child development, as understood by cognitive social-psychology. Does this matter? If such psychological development leads to all pupils successfully achieving school outcomes, then is it not the best way forward? Two problems arise. First, there is little evidence of any closure of the attainment ‘gap’, so long the Holy Grail of England’s education politics. Second, we have a national curriculum in which
pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. [But] … those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.
DfE, 2013, p. 99
‘Grasping’ concepts ‘sufficiently fluently’ is a psychological construction used by teachers to demonstrate their apparent control over learning; it is not a reflection of how mathematical ideas are likely to develop. In our ongoing research we are starting to explore how this plays out – and seeing that, while teachers are enthused by TfM (a good thing), far from being taught together as a class, pupils are being regrouped into those who are ‘fluent’ and those who are not, with only the former getting access to rich and sophisticated problem-solving challenges – which, of course, are the fundamental heart of the ‘analytical skills’ that Sunak calls for.
This post is based on the article ‘The policy and practice of mathematics mastery: The effects of neoliberalism and neoconservatism on curriculum reform’ by Nick Pratt and Julie Alderton, published in the Curriculum Journal.
Department for Education [DfE]. (2013). The national curriculum in England: Key stages 1 and 2 framework document. https://www.gov.uk/government/publications/national-curriculum-in-england-primary-curriculum
Department for Education [DfE]. (2020). Mathematics guidance: Key stages 1 and 2: Non-statutory guidance for the national curriculum in England. https://www.gov.uk/government/publications/teaching-mathematics-in-primary-schools
Popkewitz, T. S. (1987). The formation of school subjects and the political context of schooling. In T. S. Popkewitz (Ed.), The formation of school subjects: The struggle for creating an American institution (1st ed., pp. 1–24). Falmer Press.
Pratt, N., & Alderton, J. (2022). The policy and practice of mathematics mastery: The effects of neoliberalism and neoconservatism on curriculum reform, Curriculum Journal. Advance online publication. https://doi.org/10.1002/curj.202
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715