MATHEMATICAL ACTIVITIES
10.1 The incommensurability of academic and everyday knowledge .1 Academic and everyday activities as fundamentally different domains
147
CHAPTER 10
THE RELATIONSHIP BETWEEN ACADEMIC AND
148 The proposition that I wish to develop is that the division of labour constitutes two distinctive modes of social relations. These modes generate, respectively, academic and everyday practices and knowledges which are, thereby, mutually
incommensurable. (p. 209)
Inherent in the quotation above is the argument that the academic and the everyday constitute distinct regions of practice and fundamentally different domains of practice (Dowling, 2009b, p. 8), and that each domain of practice is a separate system that is comprised of different actors, social relations, structures, practices, and criteria for legitimate participation and communication (Dowling, 1998, p. 24). Knowledge within each of these domains is, therefore, differently acquired and employed (Ensor & Galant, 2005, p. 286). As suggested by Wheelahan (2010, p. 76), while the purpose of academic disciplines is to produce knowledge about the objects that they study, everyday knowledge is developed through the strategies and techniques that people employ while they seek to deal with issues and solve problems that are important to them in their everyday lives.
Importantly, given the distinctive nature of these different domains of practice, transfer of knowledge, practice and language from one domain to another is not a linear process.
Rather, knowledge, practice or language must be ‘recontextualised’ as it moves from one domain of practice to another (Dowling, 1998, p. 24; Ensor & Galant, 2005, p. 286). As P. Dowling (2008) argues,
For some time, now, I (and I’m certainly not alone in this) have been arguing that the meanings of utterances and other actions do not carry over between different contexts and that what defines a context as such is the nature of the alliance and/or opposition in respect of which an utterance (or re-utterance) or action (or re-action) stands as a tactic. So utterances and actions are recycled within contexts −
sometimes achieving status as slogans − and between contexts as resources for different, often quite different, purposes; the result is a recontextualising of the source utterance or action. (p. 1)
This opinion is also shared by both Walkerdine and Ensor & Galant:
I argue that this may be the same signifier [i.e. the use of the word ‘more’] as in the practices of the home, but it is not the same sign. … I argue further that such signifiers are made to signify when united with a signified within a particular practice, from which they take their meaning. Such practices are discursively regulated with the participants positioned in particular ways. The production of mathematical signs within practices renders them at once both socially and historically specific. (Walkerdine, 1990, p. 53)
It [the paper] has gone on to suggest that we should understand the relationship between different sites of practice not as ‘transfer’ but as recontextualisation, a process which delocates, relocates and reconfigures forms of knowledge in terms of the social imperatives, identities and internal structuring of different sites. (Ensor
& Galant, 2005, p. 297).
Dowling characterises the distinctive nature of academic and everyday knowledge and practices using the concept of ‘Discursive Saturation’ and links this concept to Basil Bernstein’s concepts of ‘Vertical and Horizontal discourses’. These concepts are now discussed.
149 10.1.1.1 Discursive Saturation
Discursive Saturation refers to the extent to which an activity – and, hence, the structure of legitimate and endorsed participation in that activity − is regulated by language, most commonly evidenced in the pedagogic texts used in the teaching and learning process:
for example, curriculum documents, textbooks, and assessments. Activities in which the structure of legitimate participation is highly regulated by language use are considered to have strong discursive saturation (DS+). Academic disciplines, such as high school mathematics, would be classified as exhibiting DS+. This is because the language and associated word/vocabulary signifiers (or what Dowling refers to as symbolic resources) employed in such disciplines is often highly specialised and generalisable. Furthermore, an understanding of the language (and symbolic resources) is often a pre-determining factor for effective, successful and legitimate and/or endorsed participation and communication in the discipline.(Dowling, 1995b, p. 213; 1998, pp. 32, 103 & 138).
Activities in which legitimate and endorsed participation is weakly regulated by
language use are considered to have low discursive saturation (DS−). Everyday practices would be classified as exhibiting DS− as the language is largely unspecialised and localised and often not generalisable beyond the context in which the language is employed. Furthermore, in everyday practices an understanding of generalisable specialised language is often not pre-determinant for successful and/or endorsed participation in that practice (Dowling, 1995b, p. 213; 1998, pp. 32, 103 & 138).
Dowling (1998) summarises as follows:
Practices exhibiting high discursive saturation are associated with a degree of context-independence or generalization; practices exhibiting low discursive saturation are associated with comparative context-dependency or localization.
Mathematics is clearly a case of high discursive saturation, a practice which is highly organised at the level of discourse and so produces generalized utterances.
… Domestic and manual practices are examples of low discursive saturation, because they are not generally highly organised at the level of discourse and so they produce localized utterances. (pp. 103-104)
Importantly, Dowling (1994, p. 128; 1998, p. 138) argues further that a crucial distinction between activities exhibiting high and low discursive saturation is the extent to which the regulative and evaluative principles are recognizable within the discourse of the practice.
As summarised by Jablonka and Bergsten (2010, p. 40), “‘Discursive Saturation’ is a dimension that describes the extent to which a practice … has explicit principles of regulation.” Activities exhibiting high discursive saturation generate descriptions that draw explicitly and overtly on the principles regulating the activity, primarily through engagement with generalising and specialising strategies (Dowling, 1998, p. 138). It is for this reason that participation in such activities provides direct access to the regulative and evaluative principles of the activity and, hence, to apprenticeship in the activity. It is also for this reason that when everyday contexts or problems (characterised by low discursive saturation) are introduced into an academic/mathematical discipline (characterised by high discursive saturation), the everyday context is subordinated to the discourse, principles and criteria of legitimate participation of the high discursive saturation activity. As Dowling (1998, p. 138) suggests, “Indeed, such subordination is to a large extent necessary, because of the relative inflexibility of the grammar of the recontextualizing esoteric domain.”
150 Activities exhibiting low discursive saturation, on the other hand, contain language that is highly localised and dependent on the immediate context in which the language is developed or employed, and characterised primarily by localising and fragmenting strategies (Dowling, 1998, p. 139). As such, descriptions of the regulative and evaluative principles of the activity are only made implicitly in such activities. It is for this reason that Dowling argues that participation in practices that exhibit low discursive saturation restricts or inhibits access to the regulative principles of the activity and, hence, position participants as dependents or objects in the activity.
10.1.1.2 Vertical and Horizontal Discourses
Dowling’s theorising on the concept of discursive saturation, by his own admission, relates directly to Basil Bernstein’s (1999) notion of vertical and horizontal discourses (Dowling, 1995b, pp. 219-222; 1996, p. 408; Ensor & Galant, 2005, p. 289). Detailed analysis and discussion of these concepts is beyond the scope of this immediate discussion and I am assuming that given the prevalence of these concepts in education research the researcher is already familiar with the concepts. However, a detailed discussion of these concepts is provided on the supplementary materials supplied for download here and/or on a CD attached to the book version of this thesis.102 The discussion here, then, is focused particularly on the relevance of these concepts to Dowling’s argument regarding the incommensurability of academic and everyday practices is necessary.
The distinction offered by Bernstein (1999) between vertical and horizontal discourses, between hierarchical and horizontal knowledge structures, and between knowledge structures with strong and weak grammars, is of direct relevance to this study in the following way. The to-be-developed language of description of a structure of knowledge for the domain of mathematical literacy that prioritises a life-preparedness orientation is situated as a horizontal knowledge structure exhibiting relatively weak grammar within a vertical discourse. Furthermore, this language of description promotes the view that a link from this horizontal structure to the authentic everyday world (i.e. to a horizontal discourse) is both possible and desirable for enhancing understanding of the world, and for expanding and developing the ‘mathematical literacy gaze’ of the students who participate in the subject. Mathematics, on the other hand, which forms the principle site of analysis for Dowling, is situated as a horizontal knowledge structure exhibiting relatively strong grammar within a vertical discourse. And one of Dowling’s key arguments is that attempts at links between this structure and the everyday world result in the imposition of a mathematical gaze and the mythologising of a virtual reality.
Bernstein’s conception of discourses, knowledge structures and grammars, thus, provides a structure for distinguishing components of the language of description of knowledge in the subject-matter domain of Mathematical Literacy to the structure of knowledge in other school-based academic disciplines (such as mathematics). The positioning of the language of description for the structure of knowledge of the domain of mathematical literacy as a particular discourse and knowledge structure is illustrated in Figure 22 below.
102 The discussion in the additional materials dealt in detail with the characteristics of vertical and horizontal discourses, the differentiation between horizontal and vertical knowledge structures, and the concepts of verticality and grammaticality.
151
Figure 22: Positioning the developed language of description as a particular form of discourse and knowledge structure
For Dowling, the difference in the structure of the activities that give rise to horizontal (everyday) and vertical (academic) discourses and the categorisation of horizontal and vertical discourses as comprising DS+ and DS− practices respectively, posits academic and everyday practices as fundamentally different activities. Movement between these activities is, then, not possible without a radical recontextualisation of the principles and/or structures of one of the activities. Furthermore, every practice and region of practice contain different role players, different alliances between the role players in that practice, and different use of language and discursive resources in the practice. As such, moving from one practice to another does not simply involve transferring skills and concepts from one setting to another. Rather, such movement involves a completely new negotiation of social relations and alliances, the development of a new form of endorsed communication, and a recontextualisation of the particular meaning of language, structures, and principles that define the practice and the structure of legitimate participation in the practice (Dowling, 2010b, p. 5).
This is confirmed by Bernstein (1999) who argues,
As part of the move to make specialised knowledges more accessible to the young, segments of horizontal discourse are recontextualised and inserted in the contents of school subjects. However, such recontextualisation does not necessarily lead to more effective acquisition … A segmental competence, or segmental literacy, acquired through horizontal discourse, may not be activated in its official
recontextualising as part of a vertical discourse, for space, time, disposition, social relation and relevance have all changed. (p. 169)
Strong Grammars (e.g. Mathematics)
Weak Grammars (e.g. History) Strong
Grammars
DISCOURSES
(comprising different forms of knowledge)
Vertical Discourse
(abstract / theoretical / generalisable knowledge)
Horizontal Discourse (everyday / common sense /
localized knowledge)
Hierarchical knowledge structures
(e.g. natural science)
Horizontal knowledge structures (e.g. mathematics; social sciences; economics; humanities)
Location of the
Knowledge domain of mathematical literacy described in the internal language
(to be presented in this study) Linked to authentic everyday contexts and
problems
Location of
esoteric domain mathematics
152 All of the above, suggests Dowling, points to a mythologizing. Namely, “The notion that schooling can have direct relevance to the everyday, which is to say, non-academic world is thus revealed as itself mythical.” (Dowling, 1995b, p. 221).
10.1.2 The subordination of the everyday to the academic
The second part to Dowling’s argument concerning the incommensurability of academic and everyday practices relates to what Dowling (1998, p. 24) terms the ‘principle of recontextualisation’. Namely,
… insofar as an activity can be empirically described as exhibiting a particular structure of social relations, then this structure will tend to subordinate to its own principles any practice that is recruited from another activity. (Dowling, 1998, p.
24)
Key to Dowling’s argument is the contention that in interactions between different activities, one activity tends to cast a ‘gaze’ over the other activity and to recontextualise the activity according to its own principles, structures, discursive characteristics and knowledge forms, hereby ignoring crucial principles, structures and knowledge forms of the recontextualised activity that make the two activities unique and different (Dowling, 1998, p. 121). Following this line of thinking, in interactions between academic and everyday practices it is commonly the academic practice that is privileged over the everyday, with an ‘academic gaze’ cast over the everyday practice such that the everyday is recontextualised according to academic principles: “the everyday setting is, after all, no more than a token. The Esoteric Domain can always be prioritised.” (Dowling, 1995b, p. 221). The result is that the image of the everyday practice that is presented within the academic practice is only a mythical impression. Namely, the everyday practice is presented not in the way it would actually occur but, rather, in a way that the academic thinks is might or should occur – and so, running becomes about speed, distance and time;
painting about surface area; and cooking about ratios and conversions. As Dowling (1998, p. 32) suggests, “There is, of necessity, a dislocation between these academic contexts and the context of evaluation of the practices which are mythologised.” And the consequence of this involves “presenting distorted practices to students who may not have had the chance to participate in the relevant activity and so know better.” (Dowling, 2010b, p. 5)
10.1.3 Summary (of the discussion on the incommensurability of academic and everyday knowledge and practices)
In this section of the chapter I have presented a discussion of Dowling’s view on the incommensurability of academic and everyday activities in reference to two arguments.
The first purports academic and everyday activities as fundamentally different domains of practice in terms of social relations, actors, language, discursive resources, structures and knowledge forms. This was done in reference to Dowling’s concept of discursive saturation and Bernstein’s conceptions of horizontal and vertical discourses. The second argument presents the notion of a ‘gaze’ and suggests that when everyday activities are incorporated into academic activities, the everyday activity is subordinated to the ‘gaze’
(i.e. principles, structures, contents, language, discursive characteristics, routines or techniques, knowledge forms) of the academic activity. The consequence is that the everyday activity presented within the academic activity is a mythologizing of the
153 activity rather than an accurate or realistic representation of the structure of legitimate participation and communication in the activity. Taken together, these two arguments suggest that interactions between academic and everyday activities are unequally yoked and that the result of such interactions is always the loss of something in one or other of the activities, most commonly in the everyday activities.
The discussion in this section of the chapter has focused on establishing – at a general level − the distinctive nature of the structure of knowledge and communication (discourse) associated with academic and non-academic or everyday practices and forms of participation. In section 10.2 below, the discussion now shifts to focus on a more specific and directed site of academic activity – namely, on the relationship between mathematical knowledge (as a form of academic knowledge) and non-mathematical (everyday) knowledge, practices and forms of communication and participation.