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
8.3.1.2 Category 2 − Interplay of content, context and/or competencies
Venkatakrishnan and Graven (2006, p. 23) point out, positions Mathematical Literacy as a sub-set of Core Mathematics:
Being literate in Mathematics is an essential requirement for the development of the responsible citizen, the contributing worker and the self-managing person.
Being mathematically literate implies an awareness of the manner in which Mathematics is used to format society. It enables astuteness in the user of the products of Mathematics such as hire-purchase agreements and mathematical arguments in the media, hence the inclusion of Mathematical Literacy as a fundamental requirement in the Further Education and Training curriculum. The development of literacy in Mathematics, in the sense outlined here, is also a fundamental responsibility of the Mathematics teacher and other educators. The requirements of the Assessment Standards in Mathematics ensure this. (DoE, 2003b, p. 62)
By implication, learners in the Mathematical Literacy classroom only become mathematically literate; learners in the Core Mathematics classroom become much more than this. Thus, despite all intentions for Mathematical Literacy and Core Mathematics to be different in ‘kind and purpose’, the reality is that − with both subjects promoting mathematical goals − they are primarily different in level and complexity of mathematical knowledge and competence.
This [a section taken from the statement of purpose in the NCS] does indeed give the impression that whatever mathematics will be evoked from working with “real- life problems” will be subordinate to the solving of those problems, as will
consolidation and extension of mathematical skills. (p. 96)
The Subject Assessment Guideline (SAG) documents for Mathematical Literacy77 send a subtly different message, as the rather extensive quotation below illustrates (DoE, 2005b, 2007; 2008d, pp. 7-8):
Learners must be exposed to both mathematical content and real-life contexts to develop these competencies. On the one hand, mathematical content is needed to make sense of real life contexts; on the other hand, contexts determine the content that is needed.
When teaching and assessing Mathematical Literacy, teachers should avoid
teaching and assessing mathematical content in the absence of context. At the same time teachers must also concentrate on identifying in and extracting from the contexts the underlying mathematics or ‘content’.
Assessment in Mathematical Literacy needs to reflect this interplay between content and context. Learners should use mathematical content to solve problems that are contextually based.
Assessment tasks should be contextually based, that is, based in real-life contexts and use real-life data, and should require learners to select and use appropriate mathematical content in order to complete the task. Some assessment tasks might more explicitly give learners the opportunity to demonstrate their ability to ‘solve equations’, ‘plot points on the Cartesian plane’ or ‘calculate statistics such a mean, median and mode for different sets of data’ while other assessment tasks might be less focused on specific mathematical content and rather draw on a range of content to solve a single problem.
Teachers need to design assessment tasks that provide learners with the opportunity to demonstrate both competence with mathematical content and the ability to make sense of real-life, everyday meaningful problems.
A similar message is explicated in the Teacher Guide78 for Mathematical Literacy:
The challenge for you as the teacher is to use situations or contexts to reveal the underlying mathematics while simultaneously using the mathematics to make sense of the situations or contexts. (DoE, 2006, p. 4)
The message contained in these various documents suggest that legitimate participation in the subject must be characterised by a dual emphasis on the development of
77 These documents complement the NCS and focus specifically on the assessment requirements for the subject. The three SAG documents (i.e. 2005, 2007 and 2008) differ with respect to the examinable curriculum stipulated for each grade, specifically in relation to the composition of the ‘Core Assessment Standards’ listed in each of the documents. The 2007 and 2008 documents, thus, refer to a larger portion of the original curriculum contained in the NCS than the 2005 document.
78 The Teacher Guide document was developed to provide teachers with an illustration of the type of pedagogic approach envisioned in the subject. The teacher guide, thus, contains a collection of units, each of which focus on a particular problem or context, and which illustrate how the curriculum can be covered in an integrated way through those problems and contexts.
mathematical knowledge and content together with the ability to apply such content to make sense of contextualised problems. Is this so different from the message contained in the NCS that focus must be on engaging with contexts? In my opinion, the difference is significant. The NCS is suggesting that primary focus must be on making sense of the context (albeit through engagement with mathematical techniques – hence the deliberate choice of the title ‘Mathematical Literacy’ for the subject). The supporting documents, on the other hand, suggest that the mathematically appropriate techniques and solutions are of central concern (often at the near exclusion and subordination of contextual entities and elements). The former privileges the contextual terrain, while the latter privileges the mathematical terrain.
The Learning Programme Guideline document for Mathematical Literacy79 illustrates this distinction more explicitly:
The teaching and learning of Mathematical Literacy should thus provide opportunities to analyse problems and devise ways to work mathematically in solving them. Opportunities to engage mathematically in this way will also assist learners to become astute consumers of the mathematics reflected in the media.
In summary, Mathematical Literacy aims to develop four important abilities 1. The ability to use basic mathematics to solve problems encountered in everyday life and in work situations.
2. The ability to understand information represented in mathematical ways.
3. The ability to engage critically with mathematically based arguments encountered in daily life.
4. The ability to communicate mathematically.
(DoE, 2005a, p. 8)
The prioritising of mathematised forms of participation over contextual sense-making practices in the subject is clearly evident in these statements.
The final ‘nail in the coffin’ for the subordination of contextual sense-making practices to mathematical considerations appears in the Examination Guideline documents which provide the explicit instruction to teachers to “Be careful that the context doesn’t interfere with the mathematics and detract from the mathematics.” (DoE, 2008c; 2009c, p. 5, my emphasis). And, since the purpose of this document is to specify the criteria according to which the Grade 12 national examinations are to be set (and, hence, according to which teachers must model their teaching and internal assessment), this statement will in all
79 The Learning Programme Guideline document provides guidance on the sequencing of learning, teaching and assessment across each of the grades (DoE, 2005a, p. 15). As such, this document provides teachers with an example of a year plan and illustrates how to construct work schedules and lesson plans.
likelihood override the intention in the NCS to prioritise contextual sense-making practices over predominantly mathematical forms of participation.80
This issue of the mixed messages imprinted in and between different curriculum documents is problematised in various local literatures. Bowie and Frith (2006, pp. 31- 32) argue that the Mathematical Literacy curriculum looks too much like scientific mathematics on two fronts. Firstly with regards to the structuring of the curriculum for both Mathematical Literacy and Core Mathematics according to almost identical Learning Outcome categories (broadly: Numbers, Functional Relationships,
Measurement, and data Handling). And, secondly, in terms of the inclusion of esoteric mathematical content (such as trigonometry and transformation geometry) in the Mathematical Literacy curriculum, and the structuring of the assessment standards predominantly in terms of mathematical content. Christiansen (2006) offers a related observation:
… the ML NCS is a political hybrid product. Though it states that “[t]he approach that needs to be adopted in developing Mathematical Literacy is to engage with contexts rather than applying Mathematics already learned to the context” (chapter 3, ‘contexts’), it has an obvious focus on mathematical skills and concepts
throughout. It is using claims of utility to justify itself, yet its content is distinctly mathematical. (p. 10)
And, in a different paper she offers a similar comment:
… the formulation of the assessment standards are clearly written with a focus on learning the skill, with the contexts as illustrators of the use of mathematics, rather than mathematics being used as a tool to solve a specific problem. (Christiansen, 2007, p. 97)
Venkatakrishnan and Graven (2006) offer a similar suggestion:
The examples provided within the Assessment Standards in the curriculum statement, …, do emphasise the use of ‘real’ problems, but the format tends to stress their use as useful ‘vehicles’ upon which mathematical content can then be carried and foregrounded. (p. 20)
They also go on to point out how the inclusion of lists of teachable and assessable mathematical content at the back of the NCS document (in the ‘Content and Contexts’
section (DoE, 2003a, pp. 38-43)) sends a clear message regarding the central role of the mathematical content component of the subject in teaching and assessment practices (Venkatakrishnan & Graven, 2006, p. 20).
80 Through my own interactions with teachers, national examiners, and exam papers, I have borne witness to how this state of affairs is playing out precisely in this way in the teaching, learning and assessment of the subject-matter domain of Mathematical Literacy. A cursory reading of a Grade 12 matriculation examination immediately reveals a dominant emphasis on mathematical content and techniques throughout the question paper. Every question has a distinctly mathematical focus and accurate mathematical knowledge, techniques, solutions and narratives are prioritised over real-world
considerations and over contextual sense-making practices. In short, the examinations assess the extent to which learners can do mathematics in context rather than the extent to which they can engage with a context (and make use of mathematics in this process). This issue will be explored and evidenced in more detail 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 for the knowledge domain of mathematical literacy.
Mthethwa (2009, p. 111) also problematises such ‘deviations’ amongst the curriculum documents, but from a theoretical perspective. He argues that one of the possible consequences of this discrepancy between the NCS and supporting documents is the development of a gap between the pedagogic recontextualising field (i.e. the work of teachers in the classroom) and the official recontextualising field (i.e. the curriculum and other supporting documents), or between the intended curriculum and the implemented curriculum (c.f. Bernstein, 1996).
While Mthethwa is predicting a possible consequence, Graven and Venkatakrishnan (2007) and Venkatakrishnan and Graven (2007) offer evidence of a ‘spectrum of pedagogic agendas’ that have developed in response to this differential emphasis on mathematical forms of participation and contextual sense-making practices between the curriculum and supporting documents.81 The spectrum of pedagogic agendas is comprised of the following agendas − (i) context driven; (ii) content and context driven;
(iii) mainly content driven; and (iv) content driven − and, as suggested by Venkatakrishnan and Graven (2007, p. 77), these varying pedagogic agendas “traverse across the purpose of contexts and the degree of integration of contexts within pedagogic situations.” As such, within each pedagogic agenda there is a specific prioritising (or not) of contextual sense-making practices over the learning of mathematical content and mathematical forms of participation. Venkatakrishnan and Graven (2007, p. 82) argue that the NCS and supporting documents for Mathematical Literacy prioritise, primarily, the second pedagogic agenda – the content and context driven agenda.
Irrespective of which pedagogic agenda predominates in the curriculum, supporting documents, and classroom practice, the presence of a spectrum of pedagogic agendas provides evidence of the differential interpretation of the relationship between content and contexts in the subject (and associated differential emphasis on the legitimation of contextual and mathematical forms of participation). This differential interpretation is spurred, in all likelihood, by the mixed messages emanating from the curriculum and/or supporting documents. This points to varied and inconsistent opinions amongst the various role-players in the subject (including curriculum developers, examiners and teachers) regarding the primary intention, purpose, and required pedagogy for the subject Mathematical Literacy. This also points to varied and inconsistent opinion regarding the form of legitimate participation in the subject, and the structure of
knowledge (including the dominant domain – mathematical or contextual − from which that knowledge is to be drawn) and pedagogic action that is required to facilitate this legitimate participation
Alongside specifying content and contexts, the NCS also makes reference to specific skills (competencies) which learners are expected to develop. Many of these skills are specified within individual assessment standards, but also in the ‘Competence Descriptions’ which appear near the end of the document (DoE, 2003a, pp. 54-65). This emphasis on skills is shared by Brombacher (2007, p. 15) who argues a key characteristic of the subject Mathematical Literacy involves an interplay between mathematical content and real-world contexts that facilitates the development of a set of competencies which are universally applicable across a range of contexts and problem situations. Bowie and Frith (2006, p. 30) offer a similar perspective, arguing that the definition of Mathematical Literacy provided in the NCS makes it clear that the three key elements in the subject include mathematical content, contexts and “the abilities and behaviours that a
81 This ‘spectrum of pedagogic agendas’ was introduced and discussed in Footnote 19 on page 39 above.
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.