9.1 ‘Road Map’ and broad overview of Dowling’s language
9.3 The Structural Level of Dowling’s language of description − Domains of Practice and Positions
9.3.1 Domains of Practice
Dowling argues that knowledge in the context of an activity can be either strongly or weakly associated, classified, or institutionalised with specialised practices.94 In specific relation to school mathematics, knowledge in the mathematics classroom is either strongly or weakly institutionalised according to specialised mathematical principles.
This association with mathematics occurs in relation to both: the content of a message – including not only knowledge and/or skills but also, according to Sethole, et. al., (2006, p. 119) , to the nature of the context – i.e. mathematical or everyday – from which the content is drawn; and the expression of the message – namely, how the message is
94 Dowling originally used the word ‘classification’ to refer to association with mathematical knowledge (1995b, 1998), but later altered this to refer instead to the ‘institutionalisation’ of knowledge with respect to mathematics (2008; Dowling, 2009a, 2009b). This change signifies a break with Bernstein’s notion of classification, where classification refers to the relations between different educational contents, contexts or categories and the degree of boundary maintenance between those categories (Hoadley, 2007; Maton &
Muller, 2007; Singh, 1997). In contrast, Dowling uses the word ‘institutionalisation’ to refer to “the extent to which a practice exhibits an empirical regularity that marks it out as recognisably distinct from other practices (or from a specific other practice).” (2009a, p. 13).
Bernstein presents a general theory with a high level of abstraction and which can be taken to refer to a variety of sites, including the curriculum, the structure of the school, and forms of knowledge. Dowling, by contrast, narrows the theoretical description to focus specifically on the relationship and distinction between empirical practices such as school mathematics, and positions, relationships and forms of knowledge within those practices. Furthermore Bernstein employs the term classification to refer specifically to a boundary between categories (e.g. between Mathematics and another subject-matter discipline) – and the implication of that boundary on power relations. Dowling, in turn, argues that classification strength is not a fixed quality of a subject-matter domain such as mathematics and, rather, that different elements of the domain can be differently classified according to both or either of content and expression (Straehler-Pohl & Gellert, 2013, p. 319).
139 transmitted and the signifiers (including words/method/language) used to transmit the message.
Strong institutionalisation (I+) of expression and content are, thus, characterised by explicit signification of specialised mathematical language, terminology, visual mediators, contents, skills, routines and explicitly mathematical contexts. Weak institutionalisation (I−), by contract, is characterised by restricted signification of these components together with signification of extra-mathematical and/or non-mathematical elements and contents (Dowling, 2009b, p. 15) (Sethole et al., 2006, p. 119).
Differing possible combinations of strong and weak institutionalisation of mode of expression and contents gives rise to four possible ‘domains of practice’ within the terrain of mathematics: Esoteric Domain, Public Domain, Expressive Domain, and Descriptive Domain (see Figure 20).
Adapted from Ensor and Galant (2005, p. 292) and Dowling (1998, p. 135).
Figure 20: Domains of mathematical practice
The Esoteric Domain is the region of a mathematical activity that is strongly associated with specialised mathematical knowledge with respect to both content and expression. In all respects, activity within this domain contains explicit mathematics and the use of abstracted and generalisable mathematical principles, signifiers, routines and narratives:
“the esoteric domain comprises the specialised forms of expression and content which are unambiguously mathematics.” (Dowling, 1994, p. 130). According to Dowling, this is the
140 domain that contains what is seen to be legitimate mathematical content, language, actions and practices (P. Dowling, 2008, p. 4). As such, the mathematical principles that regulate the practices of the activity and against which the practices of the activity – and participation in the activity − are legitimised and endorsed are explicit in this domain (Dowling, 1994, p. 129). Furthermore, Dowling (1998) argues that it is only in this domain that full access to these regulative principles is possible:
Because ambiguity is minimised in the esoteric domain, specialised denotations and connotations are always prioritised. It is, therefore, only within this domain that the principles which regulate the practices of the activity can attain their full attention. The esoteric domain may be regarded as the regulating domain of an activity in relation to its practices. (p. 135)
As such, it is in this domain of practice that students must engage if they are to be successfully apprenticed into the position of master (Dowling uses the word ‘subject’) (1998, p. 140) of the activity (Dowling, 1998, p. 141; Ensor & Galant, 2005, p. 297).
Although the Esoteric Domain contains the ‘non-negotiable’ aspects of school mathematics, school mathematics contains more than just this strictly mathematical component. Rather, teaching and the role of the teacher (‘pedagogic theory’ in Dowling’s terms) (P. Dowling, 2008, p. 3) has an impact on how mathematical knowledge is transmitted, and it often through the teaching component that attempts are made to cast a gaze outside of the Esoteric Domain and to establish links between the Esoteric Domain and the extra-mathematical and/or everyday world: “The practice [mathematics] must also constitute a more weakly institutionalized region in order to permit entry into it; this is the Public Domain.” (Dowling, 2010a, Slide 2, emphasis in original slide). The result of this ‘mathematical gaze’95, as everyday settings are brought into the mathematics classroom and recontextualised according to mathematical principles96, is the development of the Public Domain of school mathematics as a collection of recontextualised and reformulated or ‘mathematised’ problems (P. Dowling, 2008, p. 4).
As Dowling (2010a) suggests,
In the case of school mathematics, the public domain seems to comprise a collection of everyday activities (such as shopping and other domestic practice) that have been re-shaped, re-contextualised to conform to mathematical principles.
(Slide 2)
Practices and problems in the Public Domain are generally weakly institutionalised in terms of both content (I−) and expression (I−) (Dowling, 1998, pp. 135-136). In other words, the problems in this domain appear to be about something other than mathematics, are not overtly mathematical, and, although it may be obvious that there is mathematics in the problem, the focus appears to be on something other than the mathematics (Dowling, 1998, pp. 135-136). Furthermore, despite the fact that the problems in this domain are recontextualised according to mathematical principles, the regulative and evaluative mathematical principles underpinning the activity are not explicit and rather are ‘hidden’ by the everyday and/or extra-mathematical signifiers and elements (Dowling, 1998, pp. 135-136).
95 A more detailed discussion of the notion of the ‘mathematical gaze’ and associated mathematical myths – the Myths of Reference, Participation, and Emancipation – are provided in sub-section 10.2.2 below starting on page 156.
96 It is in relation to this notion that Straehler-Pohl and Gellert (2013, p. 320) state that the key concepts inherent in Dowling’s domain of practices model are gaze and recontextualisation.
141 It is essential to insert an intervening comment here. In using Dowling’s theories, some have misconstrued the locality of this Public Domain and have equated the Public Domain to the everyday or Public Domain knowledge to everyday knowledge (c.f. (Bernstein, 1999, p. 170, Note 1; Hoadley, 2007, p. 683)). Dowling’s usage of the term refers to the space where everyday contexts, problems, practices, and knowledge are recontextualised according to abstract esoteric principles, practices and knowledge. The Public Domain is the space where the everyday represents a virtual reality of life, a mathematised reality of life. The Public Domain is not the everyday; it is a misrepresentation of the everyday.
The remaining two domains – the Expressive and Descriptive Domains – are also the result of the imposition of a ‘mathematical gaze’ from the Esoteric Domain on the everyday world, and both represent an alternative form of recontextualisation than that which occurs in the Public Domain. In the Expressive Domain, non-mathematical language (expression) is embedded and foregrounded within an explicitly mathematical context and is used to signify and give expression to mathematical content (Dowling, 1998, p. 135) – for example, where fractions are equated to pieces of cake and equations to a seesaw, scale or balance. In the Descriptive Domain, the situation is reversed as mathematical language is used to describe non-mathematical content. For Dowling, this is the domain of mathematical modelling as mathematical concepts and tools (e.g. sine curves) are employed in generating descriptions of extra—mathematical concepts (e.g.
wave or tidal motion). Importantly, the regulative and evaluative principles of the Esoteric Domain cannot be fully realised in practices that remain in either of these domains due to the ‘interference’ of the extra-mathematical components: “the esoteric domain must signify differently because of the recruitment of a non-mathematical setting, so that, once again, the principles of the esoteric domain cannot be made fully explicit within [these]
domain[s].” (Dowling, 1998, p. 137).
Having described the various components of each of the domains, what remains is to identify the different uses and usefulness of this model. Ensor and Galant (2005, p. 291
& 293) argue that the model is powerful for three reasons. Firstly, the model provides a tool for analysing the classification of the contents and/or discourse of an activity such as school mathematics. For example, the model could be used to determine the extent to which a particular text (e.g. an exam or a textbook) employed in the activity of school mathematics privileges certain domains of practice over other domains. Secondly, the model provides a framework for discussing the relationship between mathematics and out-of-school practices, and particularly between Esoteric Domain mathematical content and everyday problems and situations (P. Dowling, 2008, p. 4). In this regard, the model illustrates how the Esoteric Domain contains the non-negotiable content, language and mathematical knowledge of school mathematics. The three other domains (Public, Expressive and Descriptive) are, then, the result of differing interactions between a
‘mathematical gaze’ cast from the Esoteric Domain and everyday extra-mathematical practices. The third and final utility of the model is that it illustrates the process for apprenticeship into mathematics. This topic of ‘apprenticeship’ – and, more specifically, of different ‘positions’ that exist within the domains of practice − requires more elaboration and is dealt with in detail in sub-section 9.3.2 below.
In brief summary, in the discussion above it was posited that mathematics as an activity is comprised of different domains of practice, with each domain differentiated in terms of the extent to which knowledge in that domain involves engagement with weakly or strongly institutionalised mathematical contents. However, according to Dowling (1998, p. 131), activities are not neutral entities and, rather, construct positions in the activity in
142 relation to how knowledge and practices are distributed to different participants in the activity and also in different domains of practice. And it is to the topic of ‘positions’ in the activity of mathematics that the discussion now shifts.
9.3.2 Positions
Dowling (1998, p. 140) identifies four main positions within an activity: Subject, Apprentice, Dependent and Object. The Subject is the most dominant position in an activity: this is the position that has mastered the practices and regulative principles of the activity. Every other position is then, to a greater or lesser extent, subordinated to and/or objectified by the Subject position (Dowling, 1998, p. 140).
The Apprentice position: “The activity, in effect, regulates ‘who’ can say or do or mean
‘what’. Clearly, the activity must provide for the generation of new subjects. … the process of subject generation is appropriately referred to as ‘apprenticeship’.” (Dowling, 1998, p. 140). In other words, participants in the Apprenticeship position engage in actions with the intention, at some point in the future, to become potential Subjects of the activity:
Successful apprenticeship to an activity is achieved (metaphorically) upon the completion of a one-hundred-and-eighty-degree rotation of the apprentice who thereby ‘moves’ from ‘outside’ to ‘inside’ the activity and becomes its Subject.
(Dowling, 1998, p. 123)
Importantly, apprenticeship in mathematics involves successful engagement with the Esoteric Domain contents of the discipline, which, in turn, facilitates the capacity for defining mathematical structures and generative principles according to which a mathematical gaze can be cast over everyday practices. In other words, Apprentices are invited to participate in recontextualising activities that facilitate the development of Public Domain activities.
The third position, the Dependent position, is a subordinated position to the Apprentice in respect to the Subject. In this position, the participants are exposed to mathematical and/or mathematised and recontextualised practices, but where the structure and regulating principles of the activity are decided and imposed by the Subject. As a result, participants in this position are not directly exposed and do not have independent access to the regulating principles. Instead, they are ‘dependent’ on the Subject to make visible these regulating principles – since it is the Subject that determined the criteria for and principles of mathematisation and recontextualisation. Participants in the Dependent position are not construed as potential future Subjects and, as a consequence, the final
‘career’ outcome of this position is less certain: the Apprentice becomes the subject, but there is no certainty what the Dependent will become (Dowling, 1998, p. 141). This position is different from the Objectified position (see below) in that a Dependent may be fully aware that they are operating outside of the everyday world and that encountered problems are mathematical in nature, but is still reliant on the Subject of the activity to interpret and make explicit the regulating principles of the activity.
The fourth position is the Objectified position. This position relates primarily to Public Domain practices that have been recontextualised – through a ‘mathematical gaze’ − according to the principles of the Esoteric Domain. When practices are recontextualised in this way, positions must be created within the recontextualised practices; and
143 participants in the learning process are invited to recognise themselves in these positions in the problems, as though the problems are their own and relate directly to their lives:
students are invited to become objects in the problems (Dowling, 1996, p. 402).
Participants positioned as Objects in such recontextualised Public Domain practices have no control or independent access to the regulative principles of the practice and are again reliant on the Subject to make these principles visible and explicit. Furthermore, because the recontextualised practice is only able to reflect a mythologised version of the actual practice and because the mathematical principles are hidden or obscured in the practice, within the Objectified position the students neither learn sufficiently about mathematics or about extra-mathematical contents (Dowling, 1998, p. 141).
The discussion above has identified different positions which characterise practices associated with the activity of mathematics. In the immediate discussion below, the way in which these positions are determined and distributed to different groups of participants in the activity is explored in brief.