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.
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
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).
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).
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:
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).
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.
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