9.1 ‘Road Map’ and broad overview of Dowling’s language
9.5 Considerations of (successful) apprenticeship in mathematics
In addressing the issue of apprenticeship in the discipline of mathematics, Ensor and Galant (2005) pose and answer the following question:
How do the structuring of mathematical knowledge, and the relationship between different sites of practice, impact on the nature of apprenticeship, that is, on how we induct learners into mastery of mathematics? Such mastery is achieved in our terms when learners … have grasped the ‘generative principles’ (Dowling 1998) of whatever discourse they have been induced into, and are able to produce
appropriate learning performances. (p. 297) Furthermore:
Apprenticeship of students into mathematics, in Dowling’s terms, involves the successful move from Public to Esoteric Domain. Interruption of this trajectory inhibits students’ ability to master mathematics. (Ensor & Galant, 2005, p. 297) In other words, both Ensor & Galant and Dowling (1998, p. 140) contend that apprenticeship in school mathematics and, hence, eventual mastery in the subject, is only possible if participants have access to Esoteric Domain contents. Given that in this domain the content and expression of all messages are explicitly mathematical, exhibiting vertical discourse and texts and language containing high discursive saturation100, students with access to this domain have access to the regulative principles that underpin the content. Here they gain access to generalising strategies that can be applied to a variety
99 Dowling (1998, p. 153) uses the phrase ‘Visual code of presence’ to refer to the extent to which a textual resource invites a reader to identify with the problem (or with the context and/or characters in a problem) and into believing that they are an active participant in solving the problem.
100 See sub-section 10.1.1.1 on page 149 below for an elaborated discussion of the concept of discursive saturation.
145 of problems and contexts (via the imposition of a ‘mathematical gaze’), and here they have the potential to develop understanding of mathematical concepts.
And this, in turn, is only possible if students are exposed to texts that provide access to Esoteric Domain knowledge and the principles underpinning this knowledge. In other words, students only gain access to the position of Apprentice if they are positioned, by the texts that they encounter and by the teachers administering those texts, in an apprenticed voice and to distributing strategies that generalize and specialize mathematical knowledge (Dowling, 1998, p. 149). And, by gaining access to this Esoteric Domain knowledge and the principles that underpin this knowledge, Apprentices are then able to participate in and dictate the criteria for recontextualisation practices through the imposition of a mathematical gaze on everyday practices.
Fragmenting and localising strategies employed in texts have the opposite effect. These strategies restrict access to Esoteric Domain knowledge and, rather, foreground localised, context-dependent techniques that draw on unspecialised knowledge and meanings.
Students exposed to these types of strategies are, thus, restricted to the Public, Expressive and Descriptive Domains of practice and to associated texts and discourse exhibiting dependency or objectified voices. Within these domains the criteria of the imposed recontextualising gaze are invisible and the presence of non-mathematical elements render the mathematical principles underpinning the practice less visible or even invisible. As such, participants who are given access to Public, Expressive and/or Descriptive Domain practices through fragmenting and/or localising strategies are more likely to be positioned as Dependents or Objects.
However, this does not mean that the teaching of mathematics should confine itself only to the Esoteric Domain. Rather, Dowling argues that potential subjects for an activity are always attracted to an activity through the Public Domain: “The public domain is, in this sense, the principal arena in which an activity selects its apprentices.” (Dowling, 1998, p.
149). As such, if no projection is made from the Esoteric Domain to the Public Domain, then no new apprentices will be “hailed” into the activity (Dowling, 1998, p. 141). The move from the Public Domain to the Esoteric Domain is also not a linear process. Rather, the Expressive Domain of practice provides a bridge from the Public to Esoteric Domain.
Conversely, the Descriptive Domain provides a bridge from the Esoteric to the Public Domain:
There is no natural route into the esoteric domain of mathematics … Nor, of course, can mathematics education begin and remain exclusively in the esoteric domain; there has to be a way in and this will always be via the public domain.
Pedagogic action must then construct trajectories that lead into the esoteric domain via the expressive and that lead to the public domain from the esoteric via the descriptive. … in general, in respect of any specialist region of mathematics, the whole of the map should be traversed in one way or another. (Dowling, 2009b, p.
27)
Despite the limitations of practices that remain exclusively within the Expressive, Descriptive and Public Domains, then, these domains are an essential part of apprenticeship in mathematics. Students are attracted to mathematics through the Public Domain; the Expressive Domain provides a bridge from the Public to the Esoteric Domain; and the Descriptive Domain provides a bridge from the Esoteric to the Public Domain. Successful mathematics teaching, thus, involves facilitating a journey that begins in the Public Domain, moves through the Expressive Domain into the Esoteric
146 Domain, and returns to cast a new gaze on everyday problems in the construction of new Public Domain problems through the use of Descriptive Domain practices.
This described space and movement through the space is illustrated in Figure 21 below.
Figure 21: Apprenticeship in mathematics
As summarised by Dowling (1998, p. 141): “In thus establishing an apprenticed position as a limited subjectivity with respect to any region of the esoteric domain, the apprenticed position will have undergone what can, metaphorically, be described as a 180-degree rotation from the public to the esoteric domain.”
Esoteric domain
Public domain Descriptive
domain
Expressive domain
Apprenticeship
‘Mathematical Gaze’
147