In this article we argue that in South Africa the current format of legitimised participation and practice in the examination papers for Mathematical Literacy restricts successful apprenticeship in the discipline of scientific mathematics and limits empowered preparation for real-world functioning. The currency of the subject, then, is brought into question. We further argue that the positioning of the subject as a compulsory alternative to Mathematics and the differential distribution of these two subjects to differing groups of learners facilitates the (re)production and sustainment of educational disadvantage. We draw on Dowling’s theoretical constructs of differing domains of mathematical practice and positions and focus analysis on a collection of nationally set exemplar Grade 12 examination papers to identify legitimised forms of participation in the subject. We conclude by arguing for a reconceptualised structure of knowledge and participation in Mathematical Literacy and make preliminary recommendations in this regard.

The secondary school subject Mathematical Literacy^{1}

We contend that current criticism is grounded in three main concerns relating to the structure, status and practices of the subject. Firstly, the ML examinations are perceived to be considerably less demanding than those in Mathematics – and the high pass rate of 87.1% in the subject compared to 59.1% in Mathematics in the 2013 academic year provides some validation for this concern. This state of affairs is seen to contribute to the exodus of increasing numbers of learners from Mathematics to ML (enrolment in Mathematics has decreased significantly since the introduction of ML – from 60.1% in 2006 to 42.7% in 2013) (Department of Basic Education [DBE],

At a general level this article builds on the work of others such as Christiansen (

The following structure applies in the article. In the next section we provide an overview of aspects of Dowling’s (

A key issue we seek to highlight in this article is that despite curriculum intentions for the prioritisation of contextual sense-making practices, participation in ML is endorsed and evaluated primarily according to mathematical structures and mathematically legitimised forms of practice. The work of Dowling (

Importantly for the contents of this article – focused as it is on the empirical terrain of the subject ML – a key aspect of Dowling’s work involves analysis of the relationship between mathematical and extra-mathematical knowledge, contents, discourse and practices. A central argument in this regard is that academic (generally) and mathematical (specifically) activities are incommensurate with everyday activities and that academic mathematical knowledge cannot be used as a theory for facilitating adequate or appropriate understanding of everyday practices. For Dowling, exclusive or predominant participation in particular forms of contextualised mathematics practices inhibits mathematical understanding and affords only a limited degree of life-preparation (1995a, p. 9, 1995b, p. 209). Dowling argues further that the consequence of this is particularly experienced in the schooling system where mathematics focusing on relevance is commonly made available to learners who are deemed to have lower mathematical ability (many of whom are located in predominantly working-class environments) whilst abstract mathematics is made available to supposedly higher ability learners (many of whom are located in better resourced schools situated in middle-class environments). The ‘weaker’ learners from poorer socio-economic environments are, thus, exposed to a form of mathematics that is limiting, both in terms of mathematical and real-world understanding and also in terms of future study and career opportunity. It is in relation to this situation that emphasis on relevance in mathematics is deemed to facilitate the production and sustainment of a degree of educational difference and disadvantage (Dowling,

The discussion in this section of the article is specifically concerned with what Dowling (^{2}

Dowling (

Domains of mathematical practice.

The esoteric domain of mathematical practice is characterised by a high degree of institutionalised practice (I+) and comprises explicitly specialised, generalisable and abstracted mathematical contents, contexts, principles, symbols and statements: ‘the esoteric domain comprises the specialised forms of expression and content which are unambiguously mathematics’ (Dowling,

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)

However, school mathematics contains more than just this highly specialised non-negotiable domain of practice. Rather, pedagogic practice facilitates the casting of a gaze beyond the esoteric domain to establish links between this domain and the extra-mathematical world: ‘The practice [

The expressive domain of practice is also constituted through the imposition of a mathematical gaze from the esoteric domain on the terrain of the extra-mathematical and represents an alternative form of recontextualisation than in the public domain. In this domain, non-mathematical modes of expression (I−) are appropriated for use within explicitly intra-mathematical contexts and are employed to give expression to specialised mathematical contents (I+) (Dowling,

As with the public and expressive domains of practice, the descriptive domain is a further form of esoteric domain gaze recontextualisation. In this domain, specialised mathematical modes of expression (I+) are employed to model non-specialised contents and/or extra-mathematical contexts (I−) (Dowling,

Importantly, the generative, regulative and evaluative esoteric domain principles that define the recontextualisation process and, consequently, the structure of legitimate participation in these domains, cannot be fully realised in practices that remain exclusively in these domains and which do not make a deliberate move into the esoteric domain and towards a degree of abstraction and generalisation:

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 [

Furthermore, the identified domains of practice are not mutually exclusive in the sense that engagement with public domain contents precludes engagement with esoteric domain contents. Rather, and as is discussed in more detail below, for Dowling the development of mathematical knowledge and, particularly, successful apprenticeship in the discipline of mathematics, are facilitated through traversal of the entire terrain.

The activity of Mathematics also constitutes positions in relation to how knowledge and available practices are distributed to participants in the activity (Dowling,

By contrast, participants in the Apprenticeship position engage in the practices of the activity with the intention, at some point in the future, of becoming 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,

The Dependent position is a subordinated position to the Apprentice in respect to the Subject. This position is occupied by participants who are unable to access (or are denied access to) the regulating principles of an activity, commonly through the interference of extra-mathematical elements that obscure these principles. In such instances, participants are dependent on the Subject to make visible and explicit the regulating principles according to which any mathematisation processes of non-mathematical elements have been conducted. This position is particularly characteristic of practices that remain primarily within the expressive or descriptive domains of mathematical practice, where the inclusion of non-mathematical expression and contents can serve to inhibit access to the regulating esoteric mathematical principles that structure a problem. Participants in the Dependent position are not construed as potential future Subjects (as with the Apprentice position). Consequently, the final career outcome of such participants is less certain: the Apprentice will become the Subject, but the only certainty for the Dependent is their reliance on the Subject to mediate the practices of the activity (Dowling,

Importantly, participants in the Dependent position may be fully aware that they are operating outside of the public domain and that encountered problems are mathematical in nature, but are thereby reliant on a Subject of the activity to make visible and accessible the underlying generative and regulating esoteric domain knowledge, principles and practices. Not so with the Objectified position. This position occurs primarily in relation to public domain practices characterised by the recontextualisation of real-world practices according to the principles of the esoteric domain – via an imposed mathematical gaze by either the Subject or another party on an extra-mathematical context. When practices are recontextualised in this way, participants are invited to recognise themselves in the problems, as though the problems are their own and relate to and have relevance to their lives: learners are invited to become objects in the problems (Dowling,

In establishing an explicit connection between the differing domains of practice and positions in these domains, consideration must be given to the conditions under which successful apprenticeship in mathematics is to be achieved. Ensor and Galant (

The reverse is true of practices that remain in the public domain and, to a lesser extent, the expressive and descriptive domains. Participants exposed exclusively to public domain practices do not gain direct access to the esoteric mathematical principles underpinning the practices since these are obscured and overshadowed by the interference of weakly institutionalised contents and modes of expression. Such participants are more likely to be positioned as Dependents or Objects: the mythologising of the public domain practices as valid representations of reality and the objectification of participants in the problem-solving process render the participants dependent on the Subject of the activity to make explicit the underlying (mathematical) regulating principles and the criteria according to which mathematisation processes have been conducted (by either the Subject or by another party) in the generation of public domain contents (Dowling,

However, this does not mean that the teaching of mathematics should confine itself only to the esoteric domain. Rather, potential subjects for an activity are 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,

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. (p. 27)

As such and in summary, 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,

According to Dowling (

Examinations in ML are characterised by two examination papers that are differentiated according to cognitive demand. Paper 1, classified as a basic skills paper, is focused on the assessment of proficiency of basic skills and knowledge of both mathematical and contextual contents; it comprises questions posed primarily at the two lowest levels of the four-level assessment taxonomy. Paper 2, by contrast, characterised as an applications paper, is focused on assessment of the ability to engage with both mathematical and non-mathematical techniques and considerations in contextual problem-solving processes. This paper comprises questions posed primarily at the two highest levels of the assessment taxonomy (DBE,

Crucially, the CAPS curriculum document prioritises as a primary goal in the subject, engagement with authentic contexts and resources that bear a high degree of resemblance to real-world practices (as opposed to contrived, mathematised or fictitious contexts) and a focus on the development of an enhanced understanding of these contexts (as opposed to a dominant emphasis on the development of mathematical knowledge) (DBE,

Consider the question extract shown in

Remodelled version of a Paper 1 examination question.

The first thing to notice is that the questions are pre-empted by and based on a contextual scenario; this is a common strategy and occurrence in the examinations as well as in pedagogic practices in the subject. Although our analysis is focused primarily on the type of practice prioritised in engagement with these scenarios, it is worth noting that many of the contextual scenarios employed draw on deliberately constructed fictitious situations and resources. For example, although the scenario of electricity costs is realistic in South Africa, there are no such systems as Cheep-Cheep and Bright-Sparks (and no such telephone system as the Scamtho 250 cited in the examination paper) and the tariff structures associated with electricity or telephone contracts are seldom as simple (or as simply presented) as portrayed here. In the context of the examinations, employed contextual scenarios are largely deliberately constructed to facilitate evaluation of particular mathematical and calculation-based processes.

With respect to the questions developed for engagement with this contextual scenario, Question 1.2.1 is characterised by the usage of, primarily, non-specialised references to everyday forms of expression, with no explicit signification given on the structure of the institutionalised mathematical content required for answering the question (and it is only through the inclusion of the vocabulary signifier calculate that an indication is given of a requirement for a form of mathematical engagement with the scenario). In this question, then, participants are led to believe that this is an actual real-world scenario and that they are engaging with the scenario in a way that reflects real-world practice. As a result, this question is categorised as reflecting a form of mythologised practice associated with the public domain. Question 1.2.2 (a), by contrast, makes reference to a resource involving largely non-specialised contents (namely, electricity costs), but employs a specialised mode of expression through reference to missing variables that have been imposed on the unspecialised context. As such, this question and the resource required for the successful completion of the question are categorised as reflecting a form of practice associated with the descriptive domain. A similar classification applies to Question 1.2.2 (b), where a specialised mathematical mode of expression (i.e. a graph) is referenced for use in relation to an extra-mathematical context and unspecialised contents.

The information in

Categorisation of the exemplar examination questions according to Dowling’s (

Variables | Public | Expressive | Descriptive | Esoteric |
---|---|---|---|---|

Count | 32 | 0 | 26 | 0 |

% count | 55.2 | 0.0 | 44.8 | 0.0 |

Mark totals | 72 | 0 | 78 | 0 |

% marks | 48.0 | 0.0 | 52.0 | 0.0 |

Count | 22 | 0 | 18 | 0 |

% count | 55.0 | 0.0 | 45.0 | 0.0 |

Mark totals | 82 | 0 | 68 | 0 |

% marks | 54.7 | 0.0 | 45.3 | 0.0 |

Count | 54 | 0 | 44 | 0 |

% count | 55.1 | 0.0 | 44.9 | 0.0 |

Mark totals | 154 | 0 | 146 | 0 |

% marks | 51.3 | 0.0 | 48.7 | 0.0 |

The complete absence of both esoteric and expressive domain contents in the examinations is immediately noticeable from the information shown in ^{4}

This situation is exacerbated by the fact that the esoteric mathematical generative and evaluative principles that regulate the criteria for successful and legitimate participation in the subject are rendered hidden and inaccessible to the participants through a dominant focus on questions in the examinations that require engagement with public and descriptive domain practices. In other words, despite the complete exclusion of esoteric domain practices, the authors of the examination papers deliberately prioritise engagement with forms of practice that reflect varying degrees of esoteric domain recontextualisation. In this regard, the number of questions associated with public domain practices dominate throughout both examination papers (although more marks are allocated to descriptive domain practices in the Paper 1 examination) and instances of objectification are commonplace: ‘1.4 Write down another reason, excluding the profit, why the committee decided to use venue ABC.’; ‘3.1.4 (c) Justify Megan’s claim that the price of a 9-year-old pre-owned Smart car could be worth R50 000.’ (DBE, ^{5}

The prioritisation of public and descriptive domain practices to the complete exclusion of esoteric domain practices ensures that learners in the subject are positioned as dependents in the learning process, commonly objectified by the problem-solving scenarios, but seldom (if ever) given direct and explicit access to the specialised esoteric principles that define and regulate the structure of legitimate participation in these scenarios. Learners in the subject are continuously required to engage mathematical principles in contextual problem-solving situations, but seldom (if ever) exposed to processes involving generalisation and abstraction of these principles. Learners in the subject are consistently required to engage in mathematised problem-solving scenarios, but seldom (if ever) empowered to engage in the mathematisation processes. Instead, learners are reliant on their teachers to uncover and make explicit the mathematically structured principles (decided on by others) according to which legitimate and endorsed participation in the problem-solving processes are defined and evaluated. Time and time again, learners are exposed to mathematised forms of contextual situations that bear only limited resemblance to reality and, yet, are presented as opportunities for empowered real-world functioning. All of the above indexes not only a limited and limiting form of participation in the discipline of mathematics, but also stunted preparation for enhanced real-world functionality.

However, as noted in the introduction, the pass rate in ML is significantly higher than that in Mathematics. This begs the question that if ML is not affording access to mathematics nor to life preparation, then what is the high pass rate indicative of? We contend that it is the distinction between the two examination papers and, specifically, the presence of the basic skills paper (which only assesses questions posed at the two lowest levels of the taxonomy of cognitive demand) that is contributing to the significantly high pass rate in the subject. In this regard, the high pass rate is indicative only of the ability of the learners to engage in simplistic and low-level numeracy-type calculations – hence the criticism that ML is an easier qualification than Mathematics and offers less opportunity for career recruitment.

As a final observation, it is worth mentioning that the dominance of public and descriptive domain practices in the examinations highlights a degree of inconsistency with the intention of the CAPS curriculum and with the statement of intention and philosophy for the subject espoused in that curriculum. The CAPS curriculum, both through stated intention (see DBE,

Since it is the contents and the traits of practices associated with the esoteric domain contents that are privileged for ‘recruiting to careers’ (Dowling,

Given the identified problematic structure of existing mathematically legitimised forms of participation in the subject, we suggest that an alternative conception of the structure of legitimate knowledge and participation is necessary. Although it is still in development, we posit tentatively at this stage that this revised knowledge structure is dominated by a life-preparedness orientation (Venkat,

At the time of final edits to this article the official end-of-year 2014 Grade 12 Mathematical Literacy examinations (DBE,

This article was written whilst M.N. was a fellow in the Sasol Inzalo fellowship programme and would not have been possible without the generous support, financial and other, of the Sasol Inzalo Foundation.

The authors declare that we have no financial or personal relationships that might have inappropriately influenced us in writing this article.

M.N. (University of Nottingham) authored the article based on research conducted for his doctoral dissertation and was responsible for all alterations made during the review process. I.M.C. (University of KwaZulu-Natal) was the doctoral supervisor and provided substantial feedback on the component of the doctoral dissertation on which this article is based and also on the contents of this article.

In South Africa, participation in the subject Mathematics (comprising scientific mathematics contents) is compulsory up to the end of Grade 9. At the beginning of Grade 10, learners are required to choose between participation in either Mathematics or Mathematical Literacy. Mathematics is characterised by the study of scientific, abstract or esoteric mathematical contents; Mathematical Literacy, on the other hand, is characterised by engagement with everyday problem-solving situations and the utility of mathematics in those situations. A detailed discussion of the intention and philosophy of the subject, together with analysis of the existing body of literature on the subject, is provided in (North,

Dowling also constitutes

The decision to focus analysis on a set of

Question 3.1 references a table of seemingly authentic statistical data (together with a source reference for the data); Question 5.1.5 references a tax bracket table; Question 2.1 includes information and a photograph of the SALT telescope in Sutherland; Question 2.1.3 shows a cross-sectional representation of the telescope (which an Internet search revealed is an authentic resource sourced from the website for the South African Astronomical Observatory (see

See (Cooper & Dunne,