We welcome the opportunity provided by the organisers of this conference to engage with issues in mathematics education from the perspective of the social. For more than 20 years psychology has offered many researchers in mathematics education a scientific paradigm, some illusion of certainty about the individual and about what the problems are concerning learning mathematics, but it has not given adequate answers to why children fail. Sociology, in particular Bernstein's model, gives a very precise description of the pedagogic mechanism through which educational and social inequality is reproduced in schools. We will argue that a systematic reading of Bernstein's theoretical frame, and also of the research done using this frame, can help: to understand the nature and the character of traditionally organised schools; to understand whether initiatives which change the way mathematics is taught, in England and elsewhere, are the appropriate ones when judged with pedagogical and especially with sociological criteria and arguments; and to articulate a place of intervention towards greater equality and access.
I. Introduction - Starting points and basic assumptions
We will speak from the point of view of sociology and its possible contribution to the question mentioned in the title of this paper: why children fail in school mathematics, in particular, and what the field of mathematics education can do about it. We will, mainly, draw on the work of Basil Bernstein (1971, 1990, 1996), which is concerned with showing how the pedagogic text is constructed, distributed, acquired and assessed. We will also consider empirical research studies in mathematics education and related areas that have been influenced by his work, and which, in Davies' (1995) words, will make Bernstein's work relevant to the analysis of the classroom.
As mentioned, in the light of Bernstein's work, to deal with the question of the systematic failure of certain categories of pupils is to engage with the processes through which the pedagogic (mathematical) text is produced, acquired and assessed. The starting points of such an investigation are as follows:
1b. One can talk about constructions of school knowledge at different levels. For example we can talk about the level of curriculum construction (e.g. a national curriculum), the production of textbooks, or the interactions in the mathematics classroom. Bernstein's model deals with them as different expressions of school knowledge.
1c. Independently of the level we are analysing, the three basic elements of school knowledge - what Bernstein calls message systems - are present. These message systems are: content; pedagogy; and evaluation. Therefore, when constructing or analysing school knowledge, all three message systems and their interconnections are at issue.
1d. To be able to analyse, i.e., specify principles of construction of school knowledge, at any level, one can use one common conceptual framework, and one common research tool.
Regarding assessment, Morais & Miranda (1996) studied the extent to which students understand the evaluation criteria, and more specifically, teachers' marking criteria and procedures, that is, the extent to which students have recognition and realisation rules in the assessing context. Cooper, Dunne & Rogers (1997; see also Cooper & Dunne, 1998) explored the national system of testing in Britain, and more specifically the way different items are structured (differentiation in classification and framing values). The results show that certain categories of children do not recognise the context of the question and therefore their answers draw on everyday resources rather than the specialised resources of mathematics. Singh (1993, 1995) examined discourses of computer contexts in primary school classrooms.
Indeed, the empirical research which we have located are amongst the best combination of theoretical and empirical approaches to the question of success and failure of pupils, and exemplify aspects of Bernstein's conceptual framework (see also Davies, 1995b).
The construction of specialised educational discourses, such as mathematics education, science education, etc. involve the recontextualisation of discourses from the field of knowledge production, such as psychology and sociology. We will call this recontextualisation 2. This is a second plane within the field of recontextualization, which affects the construction of school knowledge. An interesting tension can be discerned already here in classifying academics working in specialised educational discourses. Are they producers or recontextualisers of knowledge?)
Mathematics education as an area of study plays an important role in the construction of school mathematics. This happens either directly through its participation in the processes of recontextualization for the construction of school mathematics and school textbooks, or indirectly through the dissemination of research findings in the education community (e.g. through in-service teacher training courses). Therefore, in what follows, we argue that the analysis of the field of mathematics education (of its assumptions and its directions) is an important part in the analysis of school mathematics.
II. The internal structuring of mathematics education discourse
Lamnias & Tsatsaroni (1997a & 1997b; cf Bernstein 1996, ch.9) in attempting to analyse the internal structure of education discourse (the discourse of pedagogy, or of didactics) have argued that there might be a possible relation between specialised education discourses and the type of school practices that prevail or are made available to teachers. More specifically, it has been argued that the structure of the existent education discourse seems to consist of a series of language games that attempt to establish local hegemonies within the discourse and to create pedagogical knowledge and forms of school practice.
One distinguishing feature of these language games is a sharp distinction between theory and practice. This means that each of these language games produces and disseminates, mainly, theory, this way attempting to influence the construction of curricula, textbooks or forms of school practice. Therefore, because of the way each of these languages operates, and because of the power relations between different agents in the field of recontextualization 1 (i.e., between official state agents, textbook producers, academic community) its influence at the level of recontextualization 1 is probably limited or produces many contradictions; though this is a matter of empirical investigation. However, we wish to point out the similarities between the typology to which we refer in the text below, which has been constructed with reference to specialised education discourses and Dowling's (1998) typology which analyses forms of school knowledge in school mathematics textbooks in England)
Thus, when one attempts to analyse the types of school knowledge/school practice that each of the language games in the specialised discourse of education proposes, one comes up with the typology below. The typology and the revealing of the implicit social assumptions of each type of practice were developed by Lamnias & Tsatsaroni (1997b) by making a systematic reading of Bernstein and attempting to assess the usefulness of his theoretical framework in analysing the most general trends of the discourse of pedagogy in Greece and beyond.
Comments on the typology
C - Classification, in Bernstein's framework, refers to relations between categories (e.g. between contents), and defines what counts as valid contents.
F - Framing refers to relations within categories (e.g. pedagogical relations) and defines what counts as valid pedagogical transmissions (teaching and learning)
Cr - Criteria refers to what constitutes valid communications from the point of view of the pupil, i.e, refers to evaluation.
b. Types of pedagogic practice
* Type 1 relates to a mode of education discourse which was once hegemonised by behaviourism, and, now, probably, by neo-behaviourism. It was imposed and is still the predominant type in many traditionally organised schools. It is characterised by strong classifications, strong framing and explicit, formal and concrete criteria of evaluation.
* Type 2 relates to a mode of education discourse that is hegemonised by developmental and other areas of psychology. It is characterised by strong classifications, weak framing and informal criteria of evaluation.
* Type 3 relates to a mode of education discourse which has had some important influences from sociological critiques of the neglect of pupils' everyday knowledge and the hierarchisation between everyday and school forms of knowledge (Young, 1971) influenced by critical sociology. It is characterised by weak classifications, weak framings and as a matter of principle rejects the idea of testing and evaluation. However, as Lamnias & Tsatsaroni (1997b) have argued, and empirical studies have shown (Cooper, 1998), the criteria of evaluation are in fact an implicit feature of this type of practice.
c. Sociological assumptions implicit in the typology of pedagogic practice
Bernstein's theoretical framework, the main concepts of which are shown in the diagram below, connects the macro- with the micro-level of analysis and purports to describe the pedagogical mechanism that is responsible for social and educational inequality.
Using the diagram above, we can make the following points:
III. Pedagogic practice as a methodological choice in the analysis of the school classroom.
In order to appreciate from a methodological point of view the turn to the analysis of pedagogical practice, either at the level of recontextualization 1 or 2, of the research studies mentioned in the preceding sections, we will refer to Bernstein's basic model which attempts to synthesise the macro- with the micro- level of analysis.
Bernstein analyses the traditional school and the traditional pedagogic mechanism and constructs a model based on the pairs of concepts: Power and social control, classification and framing, recognition and realisation rules (see diagram 2). These concepts link the macro- with the micro-level of analysis. At the communicative and interactional level of the school classroom the interrelationship of these concepts is expressed in the construction of the type of pedagogic practice and reveals the social and educational determinants of the processes of knowledge acquisition. The diagram is the outcome of Bernstein's "... attempt to link the societal, institutional, intrapsychic realms and to demonstrate how the microprocesses of schooling relate to complex institutional and societal forces" (Sadovnik, 1995, p.25).
Therefore, with the synthesis of the macro-/micro-level (Bernstein, 1990, ch. 5; Lamnias & Tsatsaroni, 1997a & 1997b), the model succeeds in describing, in a systematic way, the pedagogic mechanism responsible for the reproduction of educational and social inequality. In particular, the specialised communicative context of the classroom is structured with:
At this point we should state, with Ladwig (1997), that: "Scientific sense will suggest that it is wise to guard against over zealously defending a core theoretical framework that may well require fundamental reconsideration.", though it is obvious that we are suggesting to push this framework as far as it will go. Indeed this has been suggested by Bernstein himself (1998), when, in self-criticism, he confesses that he adopts the method of "productive imperfection" in his writings. The conceptual tensions thus generated have to do the following:
In many cases in published texts, talks, interviews Bernstein compares his model to other theories of cultural reproduction (especially Bourdieu & Passeron, 1977) and emphasises that these theories are more concerned with how the external unequal social relations are relayed and legitimated by the education system. They make a diagnosis of its pathology, focusing in particular on the ideological messages of the pedagogic text and not on how the pedagogic text (e.g. school maths as a text) is produced, distributed, acquired and evaluated.
The notion of the (pedagogic) text shows the influence that linguistics had on Bernstein's work and on the model. A series of other terms such as discourse and context, as well as the concern with "a language of description" at the methodological level, confirm these influences. And if the criticism that sociology borrows these terms from linguistics without any attempt to incorporate them in the sociological vocabulary (Chalaby, 1996) cannot hold in Bernstein's case, still the structural linguistics on which his model relies, makes his approach a structural one. This approach is able to construct typologies (of pedagogical practice) but it is a theoretical and empirical question whether it can read the dynamics of classroom practice and the actions of concrete pupils.
One could argue that, for analytic purposes, a distinction could be made between a structural and a textual analysis, which can probably function in a complementary way. This question is indeed important in assessing the conceptual framework and the model that Bernstein's work has developed. That depends a lot on theoretical (and empirical) work that is required to define the notion of discourse, of context, of text, the "subject" and "reading".
Dowling (1998, ch. 6), commenting on the language of description, which relies considerably on Bernstein's work, writes that the typology is a tool for a better, constructive description of social reality. The product, the description, is not facts, and not representations of reality, but "the description produces systematic order". However, the attempt to thematise the presuppositions about language (text, context, discourse, etc.) that are implicit in the model and the typology of pedagogical practices is linked directly to this tendency to produce systematic order (Derrida, 1988), the emphasis more on typologies than on reading and more on subjects regulated by all powerful social, institutional and linguistic structures, rather than on the subject as a fully blown textual category, as fluid as the fluidicity assumed in post-structuralist conceptions of language.
We conclude that, in view of the move to synthesise the macro/micro-level of analysis of educational processes and its important consequences in thematising the implicit assumptions in the constructions of the pedagogic-math text, it might now be the time to deconstruct "The Sociology of Mathematics Education", i.e., the philosophical assumptions about language implicit in it. This would help to open the field of research concerned with the analysis of school practices, though in our view not without an opening, at the same time, to other fields of inquiry, and indeed to an inter-disciplinary research programme.
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