Form of knowledge of the discourse of engaging with LMT

I have discussed how selections into what should be considered mathematics education curriculum, and so the discourse of engaging with LMT is as a result of a recontextualising principle. Therefore, it is necessary to understand the form of knowledge of the discourse of engaging with LMT using Bernstein’s (2000) description of the internal principles of discourses, in terms of their construction and social base. Understanding the forms of knowledge of discourses is part of Bernstein’s (op cit) later work since his earlier work, as already discussed, was concerned with different principles of pedagogic transmission/acquisition, and the contexts that generate them and the implications for change. More specifically, he was concerned with the pedagogic communication the device makes possible with focus on who gets access to what in terms of forms of knowledge made available within the context of education.

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Bernstein (2000) distinguishes two forms of knowledge, which are Vertical discourse and Horizontal discourse where the former is the written form concerned with symbolic mastery; and the latter is the oral form concerned with practical mastery. These forms are referred to by Bourdieu as “the function to which they give rise, one form creating symbolic, and the other practical mastery” (Bernstein, 2000, p. 155). In the educational field, he describes Vertical discourse as “school(ed) knowledge” and Horizontal discourse “as everyday common sense knowledge or ‘official’ and ‘local’ knowledge” (Bernstein, 2000, p. 156). These two forms of knowledge have different structural features and are organised differently although at school level, segments of Horizontal discourse could be fused into the Vertical discourse to enhance the access of school knowledge. Such fusion could provide learners with possible ways of solving problems in their everyday world if they arise. This means “Vertical discourses are reduced to a set of strategies to become resources for allegedly improving the effectiveness of the repertoires made available in Horizontal discourse” (Bernstein, 2000, p. 169).

Horizontal discourse as a ‘form of knowledge’ is referred to as ‘common’ because it is easily accessible to everyone, it is applicable to all, and has a common history pertaining to people’s common life experiences (Bernstein, 2000). Moreover, its features are that “it is likely to be oral, local, context dependent and specific, tacit, multi-layered and contradictory across but not within contexts” (Bernstein, 2000, p. 157). Most of them all, it is “segmentally organised” in terms of “the sites of realization of this discourse” (Bernstein, 2000, p. 157). These segments are differentiated with some being more important than others. Circulation of knowledge within Horizontal discourse is made possible if forms of recontextualising are implicit and that the organising principles are not that systematic. The distributive rules within Horizontal discourse regulate “the circulation of knowledge, behaviour and expectations according to status/position” (Bernstein 2000, p. 157).

Horizontal discourse is further described as consisting of “a set of strategies which are local, segmentally organised, context specific and dependant, for maximising encounters with persons and habitats” (Bernstein 2000, p.157). To describe how this set of strategies plays out, Bernstein introduces two terms, which are repertoire and reservoir. Repertoire entails “the set of strategies any one individual possesses and their analogic potential for contextual transfer” while reservoir entails “the total set of strategies possessed by all

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members of this community” (Bernstein, 2000, p. 158). Based on these descriptions, he explains that members of a particular community could have a common focus but each bringing with them different repertoires to bear on it. The differences between repertoires are a function of experiences members of a community encounter in terms of contexts, activities and their associated issues.

Bernstein (2000) says that the regulation on the relation between reservoir and repertoire depends on the structuring of social relationships. That is, if members of a community are isolated or excluded from each other, then the social base in which the reservoir or repertoire should develop is weakened. This results in a reduction in effectiveness because circulation or exchange of strategies is restricted. Moreover, restriction to circulation or exchange of strategies specialises, classify and privatise knowledge. If there is a huge reduction in isolation or exclusion between members of a community, then the social base pertaining to “circulation of strategies, of procedures and their exchange” is enhanced (Bernstein, 2000, p. 158). This suggests an expansion of repertoire and reservoir and the effectiveness that is brought about because of the circulation and exchange of strategies.

Therefore, Bernstein (2000, p. 158) argues that “structuring of the social relationships generate the forms of discourses but the discourse in turn is structuring a form of consciousness, its contextual mode of orientation and realisation, and motivates forms of social solidarity”. This suggests that acquisition is segmentally structured, and pedagogic practice vary with segment and across social groups or class. Bernstein (op cit) also says that common place for segment pedagogy is in the family, among peers or local community, and transmission of pedagogy is “by modelling, by showing, or by explicit modes”. Moreover, in a segmental pedagogy acquisition of competence is through repeating; pedagogy is exhaustive in the context of its enactment; and acquiring common competence is important and competition is inevitable among peers. For example, pedagogies for acquiring competences in counting change are distinct from those required for acquiring competences in addressing different individuals. As already stated, an individual operating in a Horizontal discourse need not necessarily have one strategy to bear on a particular context. This is because of the implicitness of forms of recontextualising, and unsystematic organising principles.

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The oppositional ‘form of knowledge’ of a Horizontal discourse is a Vertical discourse. Circulation of knowledge within a Vertical discourse takes the form of strong distributive rules regulating access, transmission and evaluation (Bernstein 2000). He therefore states that circulation is made possible if forms of recontextualising affecting distribution in terms of time, space and actors are made explicit. That is recontextualisation and evaluation ought to be explicit, and influenced by strong distributive rules. A Vertical discourse is described in two ways. The one description is that it is about Hierarchical Knowledge Structures consisting of “coherent, explicit and systematically principled structure, hierarchically organised as in the sciences” (Bernstein 2000, p. 157). That is, Hierarchical Knowledge Structures are produced by integrating codes at the level of meanings, and their users appear “to be motivated towards greater and greater integrating propositions, operating at more and more abstract levels” (Bernstein 2000, p. 161). A sign of development is recognised if there is progress in terms of theory in that it has become more general and integrating than the previous one, that is, the oppositions within this knowledge structure are between theories.

The other description is that a Vertical discourse is about Horizontal Knowledge Structures which take “the form of a series of specialised languages with specialised modes of interrogation and specialized criteria for the production and circulation of texts as in the social sciences and humanities” (Bernstein, 2000, p. 157). He states that Horizontal Knowledge Structures are produced by collection or serial codes concerned with the ‘integration of language’ and the ‘accumulation of languages’. Bernstein’s (op cit) argument is that these languages are contextual in that they are specific to a particular Horizontal Knowledge Structure and therefore not translatable to others because assumptions made for each are usually different and in most cases opposing. Moreover, criteria for legitimate text for each language are different in terms of what is considered as evidence and legitimate questions or a legitimate problem. Bernstein goes on to say that just as each language is specialised and excluding, so are its users. A sign of development is then recognized if a new language is introduced, hence “a fresh perspective, a new set of questions, a new set of connections, and an apparently new problematic, and most importantly a new set of speakers” (Bernstein, 2000, p. 162). In contrast to Hierarchical knowledge structures, the oppositions within this knowledge structure are between languages.

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Bernstein (2000) further makes a distinction within Horizontal Knowledge Structures between languages which have theoretical concepts or discursive resources that could be used to interrogate the empirical field to generate descriptions or relations by applying rigorous restrictions, and those languages which do not have such powers. The former description is referred to as having strong grammars while the latter is referred to as having weak grammars. Examples of strong grammars include Mathematics, Logic, Economics, Linguistics, and parts of Psychology, although Mathematics and Logic might not necessarily address an empirical phenomenon but have “a set of discrete languages for particular problems” and possess “the strongest grammars” (Bernstein, 2000, p. 163). Moreover, examples of weak grammars include Sociology, Social Anthropology, and cultural studies.

Acquisition in the Hierarchical Knowledge Structure, according to Bernstein (2000) is by correct usage of the theories, for example, those developed in Physics since as a strong grammar the theories clearly outline what it is. There will be no worry of what one is speaking or writing in terms of language since there is explicitness in what Physics entails. What is required here is mastery of procedures of investigation and instruments of observation to enhance understanding of the theory. A shift from one theory to another is an indication that there is “an extension of its explanatory/descriptive powers” rather than “a break in the language” (Bernstein, 2000, p. 163). Moreover, weak grammars generate problems of acquisition in that acquirers might not necessarily recognise what they are speaking or writing.

Since Horizontal Knowledge Structure consist of a range of languages, for any one transmission to occur, selections and what is privileged in terms of recontextualisation is inevitable (Bernstein, 2000). This requires taking into consideration the social base of the recontextualising principle in terms of “whose perspective is it? How is it generated and legitimated” (Bernstein, 2000, p. 164)? Bernstein (2000, p. 164) says that “the dominant perspective within any transmission could depend on power relations among the teachers, or of pressure from groups of acquirers, or indirect or direct pressures from the market or state”. That is, the Horizontal Knowledge Structure to be acquired is constructed by a perspective which becomes a principle of the recontextualisation. This at the level of the acquirer translates into possessing the recognition and realization rules of what could be considered ‘truth’ and this would influence how the acquirer reads, evaluates and creates

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texts, hence acquisition of a ‘gaze’. Bernstein’s argument is that if acquisition of the specialised language is to be made possible, oral transmission and experience as a result of a social interactional relationship with those who poses the ‘gaze’ is necessary.

A “deeper resemblance” is recognised between Horizontal Knowledge Structure weak grammar and Horizontal discourse that is characterised as a specialised practice responsible for satisfying the material requirements of its segments (Bernstein, 2000, p. 165). Common features between these two forms of knowledge include aspects that they are horizontally organised, serial, segmented, and their contents are volatile. Volatility in the case of a horizontal discourse “refers to the referents of this discourse”. Moreover, volatility in the case of Horizontal Knowledge Structure with a weak grammar modality “refers to additions and omissions of the specialised language of a particular Horizontal Knowledge Structure” (Bernstein 2000, p. 165). Acquisition is also implicit and contextual with reference to the segmentation of a Horizontal discourse and specialised languages of a Horizontal Knowledge Structure weak grammar.

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Figure 6: A map of discourses and knowledge structures by Bernstein

(2000, p. 168)

Figure 6 provides a synopsis of Bernstein’s (2000) descriptions of forms of knowledge with reference to discourses and knowledge structures as already described. Having provided Bernstein’s (2000) descriptions of forms of knowledge, I can now be in a position to describe the form of knowledge of the discourse of engaging with LMT. I would like to state here that the discourse of engaging with LMT is a Vertical discourse of the form Horizontal Knowledge Structure with modality of a weak grammar, hence resembling a Horizontal discourse. I say so because mathematics education courses of which the discourse of engaging with LMT is part of is a Vertical discourse of the form Horizontal Knowledge Structure because the production and circulation of texts is

Discourse Vertical within Power Relations between Horizontal within Hierarchical knowledge structures Horizontal knowledge structures Reservoir D.R Repertoires Grammars D.R strong weak Transmission Explicit Tacit (crafts) D.R = Distributive Rules

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dependent on specialised languages. This includes modes of interrogation and criteria. Moreover, the fields of specialised mathematics education research or mathematics education from which the recontextualisation is possible for the constitution of mathematics education courses has seen the development of new languages. Some of these specialised languages in these fields could have a strong or weak grammar.

As for strong grammar, the fields have seen the introduction of concepts which have been used to interrogate the empirical field, hence the expansion of their conceptualisation. For example, as illuminated in Chapters 2 and 3 of this thesis Shulman’s (1986, 1987) seminal work on knowledge base for teaching has seen the development of concepts such as PCK and SMK. There is also an invitation for researchers to engage with these concepts so that they could be further elaborated as in of themselves were not conclusive. For example, Ball et al. (2008) in their effort to respond to Shulman’s invitation developed knowledge domains to illuminate their conception of MKT. This has seen SMK to include CCK, SCK, and knowledge on the horizon while PCK includes KCT, KCS, and curricular knowledge. This development of a language of description is contextual in that the concern is about knowledge base for teaching, hence MKT.

Since the discourse of engaging with LMT is part of PCK, it follows that it is a Horizontal Knowledge Structure with modality of weak grammar whose transmission and acquisition could either be explicit or implicit. Drawing from the literature reviewed in Chapter 3 of this thesis, the specialized language that entails the discourse of engaging with LMT includes three categories, namely, developing in learners both instrumental and relational understanding which in a grounded way has been described as mathematical reasoning; focusing on learners’ errors; and creating an environment where teacher can listen to learners. With specific focus on learners’ errors, the specialised language is on the nature of error and the strategies for carrying out error analysis.

In my study, as already stated, one of my concerns is what teacher-educators select and privilege in terms of recontextualising the discourse of engaging with LMT for transmission to occur, that is, from the perspective of teacher-educators. Therefore, what each teacher-educator recontextualises from the fields of specialised mathematics education research, specialised mathematics education, and other discourses could vary. In Chapter 6 of this thesis, I am going to show how these selections and privileging resonate with categories of the discourse of engaging with LMT established in the

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literature, and their positions in them. Since teacher-educators’ responsibility is to teach their student-teachers what and how to teach school mathematics, choices of what they select and privilege as the discourse of engaging with LMT depends on their knowledge and understanding of what teaching and learning school mathematics entails. This is based on the assumption that each teacher-educator’s experience is unique, hence implicit transmission. Therefore, if for each teacher-educator, aspects of what entails the categories of LMT are diverse, the expectation is that the messages their student-teachers receive will also be diverse although contextual with reference to what is in focus.

It will be interesting, in Chapter 7 of this thesis, to see what and how of student-teachers’ recontextualisation, which at the level of the acquirer is about their recognition and realisation rules of the discourse of engaging with LMT, hence their participation. Student-teachers’ recognition and realisation rules could have an influence on their reading, evaluating, and creating texts. Of interest also is how this resonates with their teacher-educators’ selections and privileging of the discourse of engaging with LMT. This is as a result of a social interactional relationship with their teacher-educators that happen to possess the gaze to enable acquisition of what entails the specialised language of the discourse of engaging with LMT. Moreover, other aspects such as student-teachers’ own experiences that are contextual could also be factored into the social interactional relationship. Recontextualising is inevitable as shown by Ensor (2001) when she followed the pre-service teachers into their classrooms to establish how what they had learned in the mathematics education courses was recontextualised into their classrooms. My interest in the study is to establish the nature of this recontextualising in relation to the discourse of engaging with LMT.

Exploring and explaining student-teachers’ recontextualising (that is their recognition and realisation rules) is also extended to when the focus is on a selection of common algebraic errors reported in the field of mathematics education research. In Chapters 8 and 9 of this thesis, I establish possible reservoirs of how the sources of errors for each scenario (3 scenarios for each Chapter) could be explained from the specialised field of mathematics education research. For each scenario, I also establish possible student-teachers’ shared repertoires in terms of the task of carrying out error analysis, and their positions in them. Focus here is on what and how of student-teachers’ recognition of the errors. This suggests student-teachers’ descriptions of the errors in terms of their recognition and

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realization rules of what entails the discourse of engaging with LMT. In doing so, I also establish the relationship between the possible reservoirs and student-teachers’ shared repertoires with reference to descriptions of the errors in the scenarios. Conclusions are then made about the nature of a social interactional relationship between teacher- educators and their student-teachers pertaining to the discourse of engaging with LMT.