MATHEMATICAL LITERACY IN SOUTH AFRICA
8.3 Definitions, statements of intention and curricular agendas for the subject Mathematical Literacy 70
8.3.1 Definitions, statements of purpose and dominant intention(s) and agenda(s) in the NCS conception of the subject Mathematical
8.3.1.3 Category 3 − Arena of application
mathematically literate person will exercise.” However, they also argue that the NCS is not clear on how the three way interplay between content, contexts and competencies/abilities/behaviours must play out in a classroom situation. Furthermore, despite Bowie and Frith’s acknowledgement of the competencies component of the subject and Brombacher’s insistence on the centrality this component in the development of mathematically literate behaviour, neither the NCS nor any other supporting documents provide a definitive list of competencies, competency clusters or broad categories of competencies to be developed. Rather, in most instances it is mathematical content, and in some instances specific contexts, that are foregrounded in the Assessment Standards listed in the NCS (and replicated in supporting documents). Again, this allows for a potentially differential interpretation of the structure of legitimate participation in the subject and key areas of focus in the subject and, possibly, accounts for the varied opinion on the intended purpose of the subject and the various ‘spectrum of pedagogic agendas’ that have resulted.
The emphasis in the South African conception of Mathematical Literacy on a three-way interplay of content-contexts-competencies is consistent with the perspectives of several bodies of international literature.82 However, the fact that Mathematical Literacy curriculum is not explicitly organised around a clearly defined list, cluster or grouping of skills positions the subject differently to many international perspectives. The implication of this in the South African situation is that the lack of clarity over specified skills has resulted in a prioritising of specialised mathematical knowledge, routines and forms of participation and communication over the development of a general set of widely applicable competencies. This absence of specification of competencies in the subject further elaborates a prioritising of mathematically legitimised forms of participation.
country in which a particular conception of mathematical literacy, numeracy and/or quantitative literacy is promoted. With respect to the overarching categories of
locations, however, the locations of everyday life, workplace and national and/or global issues specified in the subject share stark similarity to the types and scope of locations specified within international perspectives.
A further important consideration with respect to contexts relates to the ‘types’ of contexts deemed appropriate for investigation and engagement in the subject. By ‘types’ I am referring not so much to the scope or locations of the contexts (e.g. personal life;
workplace; etc.) as to the level of authenticity of the contexts. Bowie and Frith (2006) provide an appropriate starting point with the concerns that they raise regarding the types of contexts that are appropriate for the teaching of the subject-matter domain Mathematical Literacy:
If the Mathematical Literacy curriculum is to have credibility as a preparation for coping with the kinds of poorly-defined problems that make up the real demands of life and work, then inauthentic “applications” must be avoided. (p. 32)
For them, inauthentic contexts imply “pseudo-contextualisations” (Bowie & Frith, 2006, p. 32), namely contexts which bear no relation or resemblance to reality, but which are presented as though they are representative of real-life. These pseudo-contextualisations constitute the equivalent of du Feu’s (2001) category of ‘contrived’ contexts.
Frith and Prince (2006, p. 53), citing Usiskin (2001), similarly caution against the use of
“contrived ‘real-life’ examples masquerading as ‘reality’”, and argue that the teaching and learning of the subject Mathematical Literacy requires the use of contexts that are real for those involved83 and that require as much in-depth understanding of the contextual components of the context as of the mathematics employed to investigate those contexts.
This is an important observation since it highlights that understanding of mathematical content is not sufficient for legitimate and endorsed participation in contextual sense- making practices; rather, understanding of the real-world terrain is equally important if enhanced sense-making is to occur.
Given this general agreement on the need for authentic and realistic contexts that bear a high degree of resemblance and connection to the structure of real-world practices, and given du Feu’s (2001) concerns regarding the usage of ‘contrived’ contexts (c.f. page 70 above), it begs the question of what stipulations the NCS and supporting documents for Mathematical Literacy provide with respect to appropriate contexts in the teaching of the subject.
83 This is an interesting and important emphasis in that Frith and Prince are arguing that Mathematical Literacy involves socially situated practice and that the types of contexts that are most real and relevant to a group of learners vary from one community to another (Frith & Prince, 2006, p. 53). This raises
questions regarding the viability of using a national examination to assess competency in this subject and the types of contexts that can be included in such an examination without disadvantaging particular groups of learners. And, if an approach is adopted that learners must not be prevented from being able to answer questions in an assessment due to the unfamiliarity of the context and/or the language used in the context – as the statement “Be careful that the context does not interfere with the mathematics and detract from the mathematics” listed in the Subject Assessment Guidelines (DoE, 2008d; 2009c, p. 5) for the subject suggests − then this posits the mathematical and not the contextual components of the subject as the dominant focus of assessment. And for Frith and Prince, this prioritising of mathematical forms of participation negates the type of behaviour that is necessary to develop an in-depth understanding of the contextual situations (see immediately below).
In most of the curriculum documents there is acknowledgement of the need for relevant contexts, specifically relevant to the lives of the learners:
Each context should be relevant to the learners.
(DoE, 2008d; 2009c, p. 5)
Teachers should choose meaningful contexts to embed the content gleaned from the Assessment Standards in clusters across the Learning Outcomes where possible. (DoE, 2005a, p. 13)
Mathematical Literacy, by its very nature, requires that the subject be rooted in the lives of the learners. (DoE, 2003a, p. 42)
However, alongside this emphasis on relevance there is also an explicit recognition of the central role of the mathematical elements of ‘appropriate’ contexts and that such appropriate contexts must be comprised of mathematical components and/or be of a mathematical nature:
It is through engaging learners in situations of a mathematical nature experienced in their lives that the teacher will bring home to learners the usefulness and importance of mathematical ways of thought in solving problems in such
situations. To this end it is very important for the teacher to incorporate local and topical issues into the Learning Programmes that they design. The practices of the local community, the home environment and local industry provide a wealth of relevant contexts to explore. (DoE, 2003a, p. 42, my emphasis)
Many local and international studies have shown the existence of a set of attitudes described as ‘mathsphobia’ in school-going learners and in the population at large.
It is the responsibility of the teacher, in implementing this curriculum, to endeavour to win learners to Mathematics. Real-life contexts which lend themselves to
mathematical ways of thought are ideal for doing this. (DoE, 2003a, p. 43, my emphasis)
The two quotations above suggest that for the authors of the NCS curriculum, a context is only important in as much as they contain mathematical components that can be extracted to illustrate the usefulness or application of mathematics in facilitating sense- making of particular mathematical aspects of the context. This implies that contexts for which formal mathematical techniques do not provide an effective tool for contextual sense-making practices are not considered to be of value or relevance for investigation in the subject. This sentiment is echoed by Julie (2006, pp. 67-68), who argues that the inclusion of qualitative (i.e. non-mathematical and situational techniques) and/or common sense approaches to solving contextualised problems should be discouraged in the teaching of Mathematical Literacy and are ‘antithetical’ to the goals of the Mathematical Literacy.
Recognition is also provided of the possibility of the use of cleaned and context-free contexts or problems in the teaching of Mathematical Literacy:
Teachers, naturally, also have the freedom to use well-designed simulated problems as context. (DoE, 2005a, p. 14)
Question 1 [in the Mathematical Literacy Grade 12 Paper 1 examination] could contain some basic calculations and simple short questions. (DoE, 2008c; 2009c, p.
4)
Despite these references to different sources and/or locations of possible contexts, there is no explicit statement that the contexts must contain a high degree of resemblance or link to reality (i.e. real and/or cleaned contexts) or a warning against the use of fictitious or contrived contexts. It is, perhaps, no wonder, then, why all of the national Grade 12 Mathematical Literacy examinations papers since the inception of the subject contain a predominance of contrived contexts or constructed contexts, with some purely mathematical and context-free or semi-contextualised calculations, and hardly any questions containing what could be classified as ‘real’ contexts. Two examples of the types of context-free and contrived questions that dominate the Mathematical Literacy national examinations are provided in Figure 14 and Figure 15 on the page below.
In fact, my own analysis of the examination papers since 200884 reveals that, apart from street maps and tables of data, none of the examinations contain any other form of authentic, un-cleaned, and/or genuine resources (e.g. adverts, newspaper articles) relating to a real-life scenario. All of the resources employed are contrived or cleaned. For a subject that is supposedly about engaging with and making sense of the real (and complicated and messy) world, this state of affairs presents a particularly narrow and primarily mathematised experience of the world.
In sum, while the curriculum and supporting documents provide explicit statements of a requirement for the mathematical grounding of appropriate contexts, there is no accompanying statement of a requirement for authenticity or link to reality. The consequence is that all national assessments for the subject and, by implication, pedagogic practices in the subject, include a spectrum of context types, ranging from context-free to contrived, but largely absent of authentic and real contexts. Mathematical knowledge and content, and mathematically endorsed forms of participation and communication with or of that knowledge and/or content, are prioritised in all practices involving the teaching, leaning and assessment of Mathematical Literacy in South Africa. And, as is discussed in Part 3 of this study (c.f. Chapter 11 starting on page 167 below), this state of affairs is particularly problematic when viewed through the lens of Dowling’s (1998) concerns and criticisms levelled against the inclusion of primarily Public Domain practices and associated mathematised representations of reality in the teaching of mathematics.
84 See North (2010) for a detailed analysis of the 2008 Mathematical Literacy National Examinations according to content coverage and cognitive demand, and in relation to whether the examinations mirror the purpose and intention of the subject, including the usage of authentic contexts. In answer to the question posed as the title of the paper How mathematically literate are the matriculants of 2008?, a key conclusion presented is that the examinations “do not reflect the underlying intention and purpose of this subject and fail to assess sufficiently the extent to which students can apply mathematical content to solve and make sense of problems encountered in daily life.” (North, 2010, p. 229). This issue will again be investigated and elaborated in Part 7 (see Chapter 25 starting on page 402) during analysis of a set of Grade 12 exemplar examination papers through the lens of the components of the developed theoretical language of description.
Figure 14: Context-free and semi-contextualised questions in the 2010 Mathematical Literacy Grade 12 Paper 1 national examination paper (DBE, 2010a, p. 3)
Figure 15: A contrived context in the 2010 Mathematical Literacy Grade 12 Paper 2 national examination paper (DBE, 2010b, p. 6)
This lack of specificity of the level of authenticity of the contexts to be dealt with is not limited to the South African situation. Rather, and as was discussed previously, much of the international literature on mathematical literacy, numeracy and/or quantitative literacy remains similarly silent on this issue. This lack of specificity allows for a grey-area to develop in which it is deemed appropriate for non-real (i.e. contrived, cleaned and possibly even context-free) problems to be included in teaching and learning aimed at the
development of mathematically literate, numerate and/or quantitatively literate behaviour. And, as the discussion in the previous paragraph has illustrated, this is precisely what has happened in the subject-matter domain of Mathematical Literacy South Africa, with the consequence that contrived and non-real mathematised problems now dominate pedagogic and assessment practices in the subject.
8.3.1.4 Category 4 – Components (and/or features associated with courses,