Assessment of Mathematical Behaviour: A Social Perspective

Candia Morgan
Institute of Education, University of London, U.K.

In this paper, assumptions underpinning assessment practices in mathematics education are challenged. It is argued that labelling any activity as ‘mathematical’ is an essentially social act in which power relationships among the participants play a crucial role. The assessment of students’ ‘mathematical’ activity relies on their ability to display certain forms of behaviour appropriate to their assessor’s expectations rather than on any necessary underlying ‘understanding’. Some implications for research into assessment practices in mathematics education are discussed.

There has been increasing interest in assessment in mathematics education over the last ten years, including, in particular, attempts to match assessment methods to developments in the curriculum. While recent research into teaching and learning has increasingly turned to detailed consideration of its socially situated nature, however, the activity of assessment in mathematics has not undergone such scrutiny. Although we are all familiar, in principle, with the roles that summative assessment may play in regulating learners and reproducing the workforce, this has played little role in research related to assessment at the level of the classroom or the interactions between teacher-assessors and students. There is still an assumption that in some sense assessment actually does measure something that is clearly defined and exists objectively. In this paper I wish to explore this assumption and to raise some issues for future research related to assessment in mathematics education, taking a social perspective.

What may be called mathematics? The widely discussed question ‘What is mathematics?’ takes for granted the existence of ‘mathematics’ as an independent entity or field of activity and generally sees it as more or less well defined (even if there may be disputes about the definition). It is thus possible to make statements such as ‘Mathematics is . . . rigorous / systematic / abstract / . . . etc.’, to point to various objects and activities in the world and say ‘That is mathematical’, to point to an individual and say ‘She is a mathematician’ or ‘He is doing mathematics’. I wish to problematise this taken-for-grantedness and to consider how and why particular practices may come to be labelled and accepted as mathematical. This is not a purely philosophical, epistemological problem but has practical significance: assessment (with its multitude of forms and purposes) is pervasive in educational practices, having concrete and often significant consequences for the assessee, and what is assessment in mathematics education if not the labelling of an individual or the action of an individual as ‘mathematical’?

The language that I am using should make it clear that I see the question of how we recognise mathematics and mathematical thinking, not as a question about perception of the properties of some independent phenomenon, but as a socio-cultural issue. Any claim that a particular practice is mathematical must be validated, not by reference to some absolute measure, but by acceptance within a community. The questions remain as to what community this is and which participants in this community have the power to accept a practice as mathematical.

Among the least contentious labellings is that of the activities of academic mathematicians. The paper published in the Journal of the London Mathematical Society must be a mathematical text simply because, if it were not, it would not have been published; the scribblings and discussions within university mathematics departments are mathematical activity because that is what happens in such places. Here we may clearly see the role of the community in defining mathematics. The editors, reviewers, department heads and colleagues have the power to include or exclude and, while there may be some areas of contention within the community (witness, for example, the dissension about the status of ‘experimental’ mathematics), while the boundaries of the discipline are by no means permanently fixed, and some members of the community may have more gate-keeping power than others, no-one from outside the community may challenge the definition. Though outsiders may question the value of a particular mathematical activity they may not question the right of the academic mathematics community to call it mathematics.

The identification of mathematics in school is more problematic, perhaps because the community involved is less homogeneous and the location of power within the community is less certain. On the whole the boundaries between activities within the school are clearly marked (strongly classified in Bernstein’s terms). That which takes place within the mathematics classroom and is sanctioned by the mathematics teacher is mathematics. The mathematics teacher holds the power of definition and may ascribe meaning and value to the activity of students by labelling it as mathematical or not mathematical – in other words, the teacher can declare that a student understands or does not understand that which has been sanctioned as mathematics. There are, however, ways in which the teacher’s right to do this may be challenged. Such challenge may come from the academic mathematics community as with, for example, the ‘New Math’ initiative’s attempt to restructure school mathematics to reflect a particular aspect of academic mathematics. More recently in the UK we have seen attacks from the academic community on the use of ‘investigation’ in the school curriculum on the grounds that it involves and even encourages ‘non-mathematical’ ways of thinking (e.g. LMS, 1995; Wells, 1993). At the same time, challenge may come from students, parents or other teachers, usually occurring in response to changes in curriculum or teaching styles. For example, the teacher attempting to introduce discussion or problem solving into the classroom may have to face the question ‘When are we going to do some real maths?’, where ‘real’ mathematics is seen by students and parents to be pages of written exercises. Each attempt to change the school mathematics curriculum is, in effect, a re-negotiation of what is to be recognised as mathematics. The eventual nature of any particular change is a consequence of which group holds the strongest hand in the negotiation.

Within other disciplines, such as engineering or chemistry, practitioners may also see themselves as doing mathematics on many occasions. It is not always certain, however, that what the engineer or chemist identifies as mathematics within their own practice is the same as what the mathematician sees in engineering or chemistry; this may be one factor in the frequent disputes in universities in the UK about how best to teach ‘mathematics’ to engineering or chemistry students (and whether ‘mathematicians’ should do it). A mathematician involved in designing a course for chemistry undergraduates told me of her difficulty in making sense of a chemist colleague’s articulation of his view of ‘the mathematics’ of Fourier transforms. While both were agreed that students needed to know and understand more than just ‘how to do it’, each had a completely different opinion of the fundamental ideas underlying the theory. In this case, the two colleagues were happy eventually to accept the differences and to recognise both approaches as mathematical and equally valid in mathematical terms. Where communication between practitioners in various academic disciplines is more distant, however, I suspect that mathematicians are likely to be less tolerant of alternatives proposed by non-mathematicians and may challenge their right to label their own activity as mathematical.

Outside academia and the school, the situation is different. Here it is not, on the whole, the participants who identify their practice as mathematical but an outsider (psychologist, anthropologist, sociologist, mathematics educator) who places that label upon it. Indeed, participants may even refute such labelling (Cockcroft, 1982). It has been pointed out (by, for example, (Lave, 1988; Nunes, Schliemann, & Carraher, 1993)) that activities in and out of school which bear an apparent resemblance to one another because they both, for example, produce solutions to isomorphic (from the outsider/researcher’s point of view) numerical or spatial problems, may in fact be very different for reasons that are structured by the objectives, conventions and tools available within each practice. How then do they both come to be labelled as mathematics? In these cases, the labelling is likely to be irrelevant to the participants but may have great relevance to the outsider/researcher in that it creates a legitimate object for study and an audience for their findings.

In some cases, particularly among mathematics educators, mathematics has been identified in artifacts rather than in practices, for example, the "frozen mathematics" seen by Gerdes (Gerdes, 1986) or Harris (Harris, 1987) in baskets or knitted socks. Here the labelling tends to be even more contentious as it pushes the boundaries of what may be included as mathematical activity and who may be included in the community of those who do mathematics. In particular, it attempts to include those groups who have traditionally been less powerful and have had less authority (or inclination?) to label their own activity as mathematical. ‘Does a spider do mathematics?’ Does an individual have to be aware that they are doing mathematics before their behaviour can be labelled as mathematical? If doing mathematics is to be defined as participation in a community of practitioners, it would seem that the answer to that ought to be ‘yes’, for what is mathematical behaviour if it is not that which is recognised as such within the community? Where it is the outsider/researcher who labels behaviour as mathematical, the individual is being recruited into a practice that they may have no awareness of and in which their role is essentially that of an object of study rather than a potentially autonomous or powerful participant.

Recognition of mathematical thinking in school Within the school context, I am particularly concerned with how children’s mathematical thinking may be recognised or, to put this in a way more compatible with the discussion above, how children’s thinking or behaviour may come to be recognised as mathematical within the particular community of the school mathematics classroom. Here I turn to my original concern with the nature of assessment – the official recognition of mathematical behaviour in school. Consider the following argument, characterising the common, often hidden, assumptions that underpin assessment practice and much thinking about assessment: Assumption (for the sake of argument): A child ‘has’ some cognitive state that, if the teacher could have access to it, would be labelled ‘understanding’ of some part of what is recognised as mathematics within the classroom.

But: How may this ‘understanding’ be recognised?

Answer: Through observing behaviour of particular kinds, especially semiotic production.

But (a further assumption that, if pressed, most mathematics educators would accept): There is no necessary connection between ‘understanding’ and behaviour.

Assumption (necessary in order to make any claims about the validity and reliability of assessment): A competent participant in the school mathematics discourse who ‘understands’ will, in appropriate circumstances, display the behaviour that will lead to recognition of that understanding by those in positions of authority within the discourse (including in particular the teacher).

This final assumption, of course, raises many further questions about what it means to be a ‘competent participant’, what might determine ‘appropriate circumstances’, etc. Ultimately, can we separate knowledge of the ‘ground rules’ (Edwards & Mercer, 1987) that might be said to constitute competence in the discourse of school mathematics from ‘understanding’ of mathematical concepts?

The learner thus not only needs to ‘understand’ a particular piece of school mathematics but also needs to know the forms of behaviour that will lead to recognition of this and how (and when) to display these forms of behaviour. In other words, she needs to have access to the realisation rules of the practice that will allow her to produce a ‘legitimate text’ (Bernstein, 1996).

Question: Is it possible to display the behaviour in the circumstances that will lead to recognition without ‘having’ the underlying understanding? At various times, practices within mathematics classrooms such as ‘rote learning’ of multiplication tables have been criticised for producing successful behaviour in learners (i.e. correct, prompt responses to particular questions) without developing ‘the underlying concept’. Such disputes are often presented as being about the relative effectiveness of different teaching methods. Alternatively, they may be seen as competition between different discourses with different sets of meanings for what they both call ‘learning’ and different sets of behaviours recognised as signs of mathematical ‘understanding’ or success.

Competition between ‘traditional’ discourses of school mathematics, which may be seen to take ‘answers’ as signs of learning, and ‘reform’ discourses, which tend to be concerned with behaviours that appear to provide evidence of processes (represented in the US by the NCTM Standards and in the UK by the ATM and other advocates of investigation and problem solving), can certainly give rise to apparent anomolies in assessment practices. For example, one teacher, evaluating the work of a student who had presented a generalised algebraic formula as his solution to an investigation, stated:

He needs to explore. There’s something needed before he could generalise. The student was clearly able, at least in the context of this problem, to generalise symbolically. This is, in many mathematical discourses, seen as a sign of high attainment. But, because he failed to provide the teacher with evidence of the expected ‘exploratory’ behaviour, he was judged not to understand what he had done. The ‘need’ – expressed in the language of pedagogy, suggesting a process by which the student could achieve ‘true’ understanding – is actually a need to display a particular type of behaviour that conforms to the expectations of the teacher.

Mathematics teachers and researchers in mathematics education spend a lot of time attempting to identify whether or not children understand or have learnt. In recent years it has increasingly been recognised that this is not a straightforward process and that assessment instruments may be inadequate in a number of different ways (see, for example, (Ridgway & Passey’s, (1993)) discussion of various types of validity), including those arising from the context of the assessment. Cooper’s (Cooper, 1996) ‘Working Model of The Assessment Process in Relation to Culture’ identifies some of the ‘sources of threat’ to the validity of assessment. His immediate concern is with the ways in which the cultural codes and strategies available to different groups of children may lead them to have differential access to the recognition and realisation rules embedded in a system of assessment of mathematical attainment, giving rise to bias in the assessment. Detailed analysis of teachers’ assessment practices in the context of GCSE coursework (Morgan, 1996a; Morgan, 1996c) suggests that there are also differences between the values and strategies of various teachers and assessors that similarly threaten children’s potential recognition as having learnt. For example, one teacher may read a succinct algebraic generalisation as a sign of a child’s high mathematical ability while another teacher, like the one quoted above, may interpret the same text as a lack of evidence of desired problem solving activity.

The concept of valid assessment itself, however, demands some re-examination within a relativist theory of knowledge and a socio-cultural theory of learning (Galbraith, 1993; Morgan, 1996b). It is commonly accepted that ‘context’ (to put it simply) affects people’s performance on apparently similar tasks. This dualist formulation, however, assumes that there is some underlying constant ‘understanding’ or ‘ability’ that belongs to the individual, independent of context, and could be measured if only we could gain access to it in some ‘pure’ way. This assumption, of course, formed the starting point for the characterisation of common thinking about assessment that I outlined above.

But it is possible to see learning as being "not in heads but in the relations between people" (McDermott, 1993). As is often the case, it is illuminating to look at cases in which individuals are identified as failing to understand or to learn. McDermott’s (Hood, McDermott, & Cole, 1980; McDermott, 1993) analysis of the behaviour of a child identified as ‘learning disabled’ suggests that his disability is not his own inherent characteristic but is brought about by the social situations and relationships with other participants which structure opportunities for him to display behaviour that leads to him being labelled ‘disabled’. The ‘failing’ child does not simply fail to learn. In fact she may be being very successful in learning to take up a particular position within a practice and, eventually, within the wider society (Dowling, 1991).


Towards a social perspective on assessment in mathematics education Current thinking on assessment in mathematics education emphasises a concern for a match between curricular objectives and assessment methods. It is focused on reforming traditional assessment practices that have clearly failed to achieve such a match and, moreover, have often been associated with some obvious lack of equity for some groups of students. The arguments that I have presented in this paper, however, lead me to conclude that thinking at the level of objectives and methods fails to address fundamental issues about the ways in which assessment practices work to distinguish between students and the bases upon which such distinctions are made.

In order to take a critical approach to assessment, we need to examine the behaviours that are recognised as signs of mathematical understanding within the school mathematics community and the ways in which the situations that students find themselves in within the classroom enable them to display these behaviours (or fail to enable them). The values ascribed by teachers to various forms of behaviour are not necessarily obvious or universal. Indeed, the ‘same’ behaviour in different circumstances may be valued very differently. For example, the presentation of very carefully drawn diagrams may in some situations be valued highly, while in others it is interpreted as a sign of a lack of ‘real’ mathematical ability (Morgan, 1998). Yet the advice that teachers offer to their students tends to focus on the importance of taking care in drawing diagrams without distinguishing between different situations.

It seems likely that teachers themselves are often unaware of, or unable to articulate precisely, the forms of student behaviour that they will value as evidence of mathematical understanding. There is a fundamental equity issue here that is often ignored. Those students with the linguistic awareness and skills that are generally associated with advantaged, literate backgrounds are more likely to ‘pick up’ the unspoken distinctions and display the valued behaviour in the appropriate situations. Others, from less advantaged backgrounds, are less likely to come to school with these skills; they must, therefore, rely on their teachers to provide them with the necessary awareness of the forms of behaviour that will be valued. The naive guidance (such as ‘draw a diagram’) commonly provided by teachers is not adequate for such a purpose. An important task for teachers and researchers who are concerned with equity in assessment, therefore, must be to investigate assessment practices at a level of detail that can identify which aspects of students’ behaviour are likely to be recognised as mathematical and valued as signs of mathematical understanding. Such investigations would also need to develop a language to describe these valued behaviours – a language that teachers and students can use both to help students to display the behaviours that will lead to success in the assessment process and critically to interrogate the assessment practices themselves.


Bernstein, B. (1996). Pedagogy, Symbolic Control and Identity: Theory, Research, Critique. London: Taylor & Francis.

Cockcroft, W. H. (1982). Mathematics Counts. London: HMSO.

Cooper, B. (1996). Using data from clinical interviews to explore students’ understanding of mathematics test items: Relating Bernstein and Bourdieu on culture to questions of fairness in testing. Paper presented at the Symposium: Investigating Relationships Between Student Learning and Assessment in Primary Schools, American Educational Research Conference, New York.

Dowling, P. (1991). A touch of class: ability, social class and intertext in SMP 11-16. In D. Pimm & E. Love (Eds.), Teaching and Learning School Mathematics. London: Hodder & Stoughton, 137-152.

Edwards, D., & Mercer, N. (1987). Common Knowledge: The Development of Understanding in the Classroom. London: Methuen.

Galbraith, P. (1993). Paradigms, problems and assessment: some ideological implications. In M. Niss (Ed.), Investigations into Assessment in Mathematics Education: An ICMI Study . Dordrecht: Kluwer Academic Publishers, 73-86.

Gerdes, P. (1986). How to recognise hidden geometrical thinking: a contribution to the development of anthropological mathematics. For the Learning of Mathematics, 6(2), 10-12.

Harris, M. (1987). An example of traditional women’s work as a mathematical resource. For the Learning of Mathematics, 7(3), 26-28.

Hood, L., McDermott, R., & Cole, M. (1980). "Let’s try to make it a good day" - Some not so simple ways. Discourse Processes, 3, 155-168.

Lave, J. (1988). Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. Cambridge: Cambridge University Press.

LMS. (1995). Tackling the Mathematics Problem: London Mathematical Society.

McDermott, R. P. (1993). The acquisition of a child by a learning disability. In S. Chaiklin & J. Lave (Eds.), Understanding Practice: Perspectives on Activity and Context . Cambridge: Cambridge University Press, .

Morgan, C. (1996a). Generic expectations and teacher assessment. Paper presented at the Research into Social Perspectives of Mathematics Education, Institute of Education, University of London.

Morgan, C. (1996b). Language and assessment issues in mathematics education. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4). Valencia, 19-26.

Morgan, C. (1996c). Teacher as examiner: The case of mathematics coursework. Assessment in Education, 3(3), 353-375.

Morgan, C. (1998). Writing Mathematically: The Discourse of Investigation. London: Falmer.

Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street Mathematics and School Mathematics. Cambridge: Cambridge University Press.

Ridgway, J., & Passey, D. (1993). An international view of assessment - Through a glass, darkly. In M. Niss (Ed.), Investigations in Assessment in Mathematics Education: An ICMI Study . Dordrecht: Kluwer, 57-72.

Wells, D. (1993). Problem Solving and Investigations. (3rd (enlarged) ed.). Bristol: Rain Press.