Paper presented at the British Educational Research Association Annual Conference, The Queen's University of Belfast, Northern Ireland, 27th - 30th August 1998
The perspectives of this symposium has several strands: a concern about performance in mathematics; problems of understanding mathematics in 'context'; and how mathematics can be used in science and technology (design and technology in England and Wales).
In the UK and elsewhere, politicians and others react to poor international comparisons such as given in TIMSS. Employers and colleges complain about the decline in standards of the mathematics of school graduates. This leads to concerns about what and how mathematics is taught. The UK government wants to raise standards through a subject-based national curriculum linked to an assessment system, and the focus has been on the subject of mathematics rather than mathematics elsewhere in the curriculum. This is certainly the focus in recent moves to improve numeracy in primary schools. There is, however, a commitment to the application of mathematics in other subjects, but in general this is rather weak, with vague statements that provide no guidance on how this should be tackled.
There is a well recognised contrast between the poor performance of individuals in using arithmetic skills in school mathematics problems compared to that on problems in everyday life that involve the same skills (e.g. Lave, 1988). In mathematics lessons the focus is on the mathematical concept or procedure, often devoid of context. When contextual features are added to problems in mathematics in school, the task for students is to extract the mathematics and then solve the mathematics problem. This contrasts with everyday life, where the mathematics is a tool to solve a problem. This problem solving is active, flexible and inventive; problems are transformed and abandoned, an approach prohibited in most mathematics classrooms. Mathematics is taught in an abstract fashion in part to enable pupils to develop generalised tools they can use in a variety of contexts. This approach assumes that the mathematics is learned in a decontextualised fashion. The notion that mathematics learned in a mathematics classroom is in fact decontextualised and can be transferred to any context is problematic and is rejected by those who argue for situated learning (Lave and Wenger, 1991; Lave, 1996; Boaler, 1998). The idea of 'situatedness' means that context is crucially interwoven with knowledge, and that 'decontextualised knowledge', so much a feature of educational institutions, is an unhelpful idea. Lave and Wenger reject the existence of such knowledge, arguing that schools themselves, for example, are very specific contexts. So, they argue, abstract knowledge as found in mathematics classrooms and concrete knowledge are not distinct, but the former is simply disconnected from the activity of the community of practice. (The issue of what constitutes the 'community of practice' we will come to shortly.)
The mathematics classroom can therefore be seen as one context for learning. Saljo and Wyndhamn (1996) have shown how a change of context, even within school, affects the performance of students solving 'identical' problems associated with interpreting postage rates for packages. In mathematics lessons students tended to use mathematical operations that were inappropriate, operations that they would not use in say a social studies lessons; in such lessons they dealt with the interpretation in a way more reflective of everyday life, with considerably more success. Mathematical ideas in everyday life, or the science lesson, are yet other contexts for learning mathematics, and it is likely that the learning will be different.
Real Mathematics Education, developed at the Freudenthal Institute in the Netherlands, is one response to such problems of context. This approach takes seriously the context, i.e. the particularities of the situation, alongside the invariants of the mathematics that could be used to model aspects of this situation. Where mathematics is the subject or, to use the earlier terminology, where the community of practice that pupils are being introduced to is that of mathematicians, this approach is acceptable because there may be less concern about learning to use the mathematics. The aim is to encourage students to mathematise about the world, with the focus on mathematical thinking as an end in itself. In theory there should be no conflict because engineers or scientists need to mathematise about the world they are respectively changing or investigating; this is the essence of using mathematics for them. However, there is evidence that knowledge, even of this theoretical kind, is intimately associated with the situations within which it is used, what Gott (1989) calls device knowledge. Thus different communities of practice may view the knowledge of mathematics differently. An example is given by Bissell and Dillon (1993), where electronic engineers look upon an electronic circuit, not in terms of the differential equation that models its behaviour, but in terms of the effects on the function of some of the circuit's components (which of course can be traced to elements of the differential equation). Engineers will use ways of thinking that model more closely how the circuit behaves (e.g. Nyquist diagrams) rather than the 'abstract' differential equation, although of course these diagrams are derived from such equations (Dillon, 1994). Thus the different ends and the different contexts of use seem to result in a different conception of the mathematics.
Thus mathematicians, scientists and technologists, as communities of practice, are not evidently represented in school classrooms, rather what we have is the 'community of schooled adults', to use a term coined by Lave and Wenger (1991). Teachers are not usually practising engineers, scientists or mathematicians. They are part of communities of technology, science and mathematics teachers. Although each community is distinct sin some way that reflects their 'parent' community, there are aspects that concern schooling that are different from the parent differences. Such communities represent school versions of those that are found outside schools, and the activities, problems and knowledge will not correspond exactly with the communities that they apparently hold as their reference points. For example, science teachers are concerned with the assessment of pupils carrying out practical investigations and technology teachers with assessing made products. Each of these activities produces approaches and methods that do not reflect the work of professional scientists and technologists (e.g. the need to have individual assessment means that technological activity becomes an individual activity in school compared to a group activity in industry and criteria for success are hence different). Nevertheless they will have features of the 'parent' communities that may differentiate them. What is required is a way of reflecting on these community differences in the classroom situation and to disentangle it from those that relate to school and classroom cultures.
Cobb et al (1997) have illustrated how classrooms can be seen as microcultures, which contain a variety of levels of norms. They see these norms as a hierarchy (or series of layers) with the highest being social norms of the classroom, for example, the place of discussion in the classroom. Such a feature would be constant over a variety of activities etc.; a primary school teacher might see discussion as a feature of all her teaching, as a part of her view of how children learn and their need to learn how to collaborate. Thus students would need to explain and justify solutions, make sense of others' justifications etc. They see these norms as common across discussions that take place in a variety of subjects. At a lower level are sociomathematical norms, that are peculiar to how a teacher might deal with mathematical solutions. For example, the need to seek alternative solutions to mathematical problems, and what counts as an alternative solution etc. These norms change across subjects; in technology different criteria will be used to judge alternative solutions (indeed the nature of solutions are different). At the lowest level are classroom mathematical practices. These will be detailed practices that develop as the students learn. Cobb et al (1997) give the example of students starting the year counting in 'ones', and it was an acceptable practice to explain word numbers and numerals in these terms. Later in the school year, they had developed an understanding of 'tens', counting in tens and ones became the acceptable mathematical practice in the classroom. There was no need to explain this practice of explaining. Their analysis, although applied to elementary (primary) mathematics lessons, ignores the fact that the 'classroom' is also a situation in which other norms and practices related to sociomathematical norms and mathematical practices, associated with other subjects, take place. Equally, if the secondary school is considered, students move from one classroom to the other where differences in norms and practices will occur at all the three levels.
Such views of mathematics in the various situations of the mathematics, science and technology classrooms, can help to throw light on some of the studies we represent in this symposium. However, although our research was informed by the situated perspective, we did not employ the kind of data collection and hence analysis that Cobb and his colleagues discuss. Nevertheless we have observed mathematics in science and mathematics lessons in primary schools (Spence in this symposium). This idea of different norms and practices could provide a basis for future work.
Concern within schools about these links raises issues about the use of mathematics in the context of different subjects. Science and technology educators at school and college level are concerned about the need for such links, but the increased separation of the subjects in the UK national curriculum make this difficult. These educators see mathematics as integral to science and technology, yet there is little analysis of the use of mathematics and how students are educated in its use in these subjects.
Research at the Open University on issues in co-ordinating learning across subjects in primary and secondary schools uses mathematics, science and technology lessons as a source of data for understanding how students might use their mathematics in 'everyday' contexts. When students design and make products (e.g. a storage device) or carry out science investigative work, they use mathematical ideas that they have or will encounter in mathematics lessons. This occurs naturally, uncued and purposefully, as it does in everyday life. There are claims of the opportunities these subjects offer for the development of mathematical understanding (e.g. Burghes, Price and Twyford, 1996), but little empirical work. Where evidence does exist in the USA, for example, it derives from trying to change mathematics classrooms to include technological activity such as design (Middle School Mathematics through Applications Project).
The funded studies reported in this symposium investigate the mathematics used in technology classrooms in primary and secondary schools, and in science investigations in a primary school. This enables a contrast of the use of mathematics in both science and technology and the two types of school.
If classrooms can be seen as microcultures, how are mathematical ideas treated differently in science and technology classrooms from that in mathematics classrooms and what are the reasons for these differences? The assumption behind such questions is that there is some evident treatment of mathematics (its concepts and procedures) by the teachers of science and technology, and we present evidence that shows this assumption is incorrect. Indeed what we see is the idea of transparency (Lave and Wenger, 1991); the tools, both cognitive and physical make visible and invisible the ideas that are embedded within. We see examples of this in the design and technology classroom, where tools and procedures associated with their use have mathematical ideas embedded in them. In science the conceptual tools also have embedded mathematical ideas; for example laws that indicate relationships between science concepts (force and acceleration) are essentially mathematical relationships, but these relationships may feature less in explanations of science teachers.
We have already drawn attention to the fact that the national curriculum sees links between mathematics, science and technology, but recognised that these are unhelpful. What, at the school level is done to encourage such links, given this lack of curriculum specification at national level? More important, it is assumed that these unspecified links are unproblematic to make, either in the nature of the mathematical concepts and topics covered, or the purposes to which the mathematics is put.
If the links are poorly addressed or problematic, what can be done to ensure that student learning is improved, in all three types of 'classrooms' (bearing in mind that these are classroom microcultures rather than physical entities)? We suggest that at the very least an explicit recognition of the mathematical opportunities along with some strategies to deal with them are needed. Such strategies need to recognise the different purposes in each of the three classrooms, and the different ways in which the curriculum is dealt with in the school (Murphy, 1998).
Finally, these approaches have to be sensitive to the different phases of schooling. We started our research in primary schools with some view that perhaps the situation there would be different from that in secondary schools, where the departmental and subject separation was reflected in the way mathematics is viewed in each of the areas. However, it appears that in primary schools the situation is no better, albeit that the teacher in question for the three subjects (mathematics, science and technology) is usually the same person. This would allow classroom norms to be the same, but the sociomathematical norms and mathematical practices may vary as the teacher deals with the three subjects. While the similarities between the situation in primary and secondary schools may be so in terms of the present approaches, the potential to change may be better in the primary school where the single teacher is able to consider the needs in the three areas. The departmental structure of the secondary school may have many more barriers to such change. Whatever the conditions of change, nothing will be achieved as long as the use of mathematics in the three subjects is treated as unproblematic, which, as indicated earlier, seems not to be the case.
Bissell, C. C. & Dillon, C. R. (1993) 'Back to the backs of envelopes', The Times Higher Education Supplement, 10 September, p. 16.
Burghes, D., Price, N. and Twyford, J. (1996) 'The interface between mathematics and design and technology in secondary schools', The Curriculum Journal, 7 (1), 33-50.
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Lave, J. (1996) 'The practice of learning'. In Chaiklin and Lave (1996, pp. 3-32).
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Murphy, P. (1998) 'Sums and more sums? - numeracy and the curriculum.' (editorial) Primary Science Review, 53, 2-3.
Saljo, R. and Wyndhamn, J. (1996) 'Solving everyday problems in the formal setting: an empirical study of the school as context for thought'. In Chaiklin and Lave (1996, pp. 327-342).
This document was added to the Education-line database 02 October 1998