AREAS OF COMMONALITY AND DIVERGENCE IN THE LITERATURE
6.5 Category 5 – Dominant orientations in international perspectives of mathematical literacy, numeracy and/or quantitative literacy
In Chapter 5 of this part of the study, I introduced a framework of possible orientations associated with particular conceptions of mathematically literate, numerate and/or quantitatively literate behaviour, as informed by the dimensions of the Agenda(s) and Intentions(s) promoted within the different conceptions (c.f. page 48 above). As described above, these agendas and intentions represent respectively:
Agenda → The dominant perspective or agenda in a body of literature in terms of the extent to which the literature prioritises mathematical knowledge and techniques or contextual sense-making practices (and associated legitimate forms of knowledge participation and communication in those contexts) as the dominant organising principle of an activity.
Intention → The explicit or implicit external impetus for which a particular conception of mathematical literacy, numeracy and/or quantitative literacy is directed
− for example, for more effective participation in the workplace, or for critical citizenship, and so on.
The collective of the identified agendas and intentions were then reflected in a framework shown in Figure 7 on page 48 above.
Employing this framework to the literature read for this study facilitates comparison of the dominant orientations within the various international perspectives on mathematical literacy, numeracy and/or quantitative literacy, and also comparison to the dominant orientation in the South African subject-matter domain of Mathematical Literacy53. Although it is not feasible, within these pages, to provide an explicit description of my reasoning for the specific categorisation of every piece of literature read in the framework, an illustration is necessary. For purposes of this illustration, consider the OECD-PISA frameworks (OECD, 1999, 2003, 2006, 2009). As regards the dominant Agenda(s) in the framework, as has been illustrated at various points in the discussion in the pages above, these frameworks contain an explicit prioritising of mathematical structures over real- world forms of knowledge and participation in the solving of problems. Some examples of this include:
although real-life contexts may be employed, allowance is made for such contexts to be modified to facilitate heightened access to the mathematical components of the problem scenario;
allowance is also made for the inclusion of purely scientific questions and contexts where no extra-mathematical contextual link is included;
the process of mathematising encouraged in these framework is explicitly mathematically grounded and promotes the imposition of a mathematical gaze on real-world structures and the recontextualisation of such structures according to mathematical principles;
despite allowances made for the inclusion of non-contextually based problems, primary focus in the framework is on assessing learners’ ability to perform contextually based calculations (i.e. mathematics in use) and to create and analyse mathematically structured models to represent real-world situations.
All of the above suggests an explicit prioritising of mathematical knowledge, structures and applications over considerations involved in contextual sense-making practices. It is mathematical forms of participation used to develop mathematised forms of understanding that dominate and not an impetus for understanding the real-world (and of how people might think, behave and communicate in that world). This clearly eliminates the framework from being associated with the final life-preparedness agenda dimension in the table. And, although there is some mention of un-contextualised calculations, it is clearly the second and third agendas – namely, the Ability to perform calculation in real- world contexts and Modelling – that dominate in these frameworks. Specifically with respect to Agenda 2, it is the dimension of Applications (Agenda 2 [a]) that dominates and not prioritisation of Numeracy-in-Context (Agenda 2 [b]) type practices. This is evidenced in the expectation in the OECD-PISA frameworks for engagement with scientific and abstract mathematical contents and the use and application of those contents for solving problems encountered in extra-mathematical contexts.
Now consider the dominant Intention espoused within this framework. The following definition of mathematical literacy is provided in the OECD-PISA frameworks:
53 See page 120 below for an elaborated discussion of the comparison of the dominant orientation in the subject-matter domain of Mathematical Literacy to those that characterise international perspectives on mathematical literacy, numeracy and/or quantitative literacy.
Mathematical literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen. (OECD, 2003, p. 24)
This definition suggests that mathematical literacy is seen as something empowering and that possession of the faculty of a mathematical gaze positions the user in a better position to be able to make sense of and cope with the demands of the world. The following quotation illustrates the consequences, from the perspective of the OECD-PISA framework, of not developing this gaze:
Failure to use mathematical notions can result in confused personal decisions, an increased susceptibility to pseudo-sciences, and poorly informed decision-making in professional and public life.54 (OECD, 2003, p. 27)
The OECD-PISA conception of mathematical literacy, thus, falls in line with an intention for the development of Human Capital. Jablonka (2003, p. 81) herself offers the same classification of the OECD-PISA framework, arguing that “PISA is intended to estimate and compare the stock of ‘human capital’.”
Through utilisation of this same approach I have categorised the various literature on mathematical literacy, numeracy and/or quantitative literacy read for this study. This categorisation is shown in Figure 10 on page 82 below.55 From this categorisation, I contend that the largest portion of the literature read embody statements of Intention promoting the development of mathematically literate behaviour as a form of empowerment and for developing and enhancing worth and value for the workplace and the economy: namely, mathematical literacy for the development of Human Capital. A considerably smaller number of articles promote a dominant intention of mathematical literacy for any of the other intentions. This does not mean that such intentions are not present or referenced in some of the literature in which mathematical literacy for Human Capital is prioritised, but, rather, that they do not constitute the dominant intention in such literature. Notice that this dominant intention for the development of Human Capital is also reflected to an exclusive and overwhelming degree in the literature and documents that relate to the South African subject-matter domain of Mathematical Literacy. In South African, participation in the subject is proclaimed as an avenue to enhanced functioning in future career and daily-life practices.
54 My mother-in-law, who achieved a double-G (yes, a ‘GG’!) symbol for mathematics in her matriculation examinations and, yet, who now runs her own business very successfully, would in all likelihood not agree with this statement. She also has no objection to me mentioning her result in this study. She treats it as a point of honour and pride that she achieved the lowest mark ever for mathematics in the history of her school.
55 Note that this figure also includes categorisation of the literature that relates specifically to the South African school subject Mathematical Literacy – even though discussion of this subject has not yet been provided. Detailed discussion of the subject is provided in Chapter 8 below (c.f. page 91), and during that discussion (c.f. page 120) I refer back to this figure to highlight the dominant orientation in the subject in comparison to the dominant orientations that characterise international conceptions of mathematical literacy, numeracy and/or quantitative literacy. In brief, this later discussion highlights that the dominant orientation in the National Curriculum Statement (DoE, 2003a) is characterised by an intention for the development of Human Capital and by agendas for enhancing application and modelling abilities, while in the CAPS curriculum (DBE, 2011a) the agendas shift to a prioritising of modelling and contextual sense-making practices. In contrast to both of these, the national examinations for the subject (under both curriculum structures) promote an almost exclusive focus on numeracy in context, albeit retaining the intention for the development of Human Capital.
As regards dominant Agenda’s promoted in the literature, much of the discussion in the pages above has highlighted different ways and areas of focus in which mathematical components are prioritised over real-world components. It should come as no surprise, then, that my categorisation of the literature reveals that the second and third agenda’s along the spectrum dominate throughout the largest portion of the literature. Namely, focus on mathematical calculations in real-world and/or mathematised contexts – with specific prioritisation of application-related practices (Agenda 2 [a]) rather than engagement with elementary contexts in contextual settings (Agenda 2 [b]) − and the design and analysis of mathematical models to represent real-world situations. And, although these agendas do not promote a prioritising of mathematical considerations at the exclusion of real-world contexts, they certainly point to a prioritising of a mathematical gaze over contextually based problems. Only a limited number of authors stipulate a direct prioritising of a life-preparedness agenda that is accompanied by acknowledgement and consideration of real-world concerns and considerations, and by emphasis on contextual sense-making practices aimed at enhancing understanding of existing and possible alternative forms of knowledge and participation in contextual situations. Interestingly, notice the wide variation in the dominant agenda prioritised in the South African documentation and literature relating to the subject-matter domain of Mathematical Literacy. In contrast to the dominant agenda prioritised in the majority of international literatures read (i.e. for Applications – Agenda 2 [a]), it is the Agenda 2 dimension of Numeracy-in-Context (Agenda 2 [b]) that dominates the South African documentation (most of which reflects government curriculum and assessment documentation). However, there is also some curriculum or assessment documentation for the subject that prioritise agendas of Application (Agenda 2 [a]), Modelling (Agenda 3) and Contextual Sense-Making Practices (Agenda 4). As is discussed later (c.f. page 106 below), this variation in the dominant agenda prioritised in different official documents has resulted in the development of a “spectrum of pedagogic agendas”
(Venkatakrishnan & Graven, 2007) in the subject, each of which comprise different criteria for legitimate participation in the subject and for the structure of knowledge required to facilitate endorsed participation. An elaborated discussion of these and other issues relating specifically to the subject-matter domain of Mathematical Literacy in South Africa is provided in Chapter 8 (starting on page 91) below.
An intervening word of caution is necessary here. I acknowledge that my reading of the literature on mathematical literacy, numeracy and/or quantitative literacy is limited and that a more extensive reading and consequent categorisation of the literature in the framework of agendas and intentions may lead to identification of different trends in the dominant orientations than those that I have alluded to above. I also acknowledge that my selective sourcing of literature may have resulted in a collection of literature which promotes a common perspective and orientation. That said, I have tried to read as widely as possible and to deliberately source literatures that offer differing perspectives on features associated with mathematically literate, numerate and/or quantitatively literate behaviour to facilitate a comprehensive analysis. In sourcing literature I have also chosen to focus particular attention on works that are considered to be pivotal works in the field and, as such, are referenced frequently in discussions and debates (for example, the work of Lynn Arthur Steen). I have also deliberately sourced information on the OECD-PISA and ALL assessment frameworks since these frameworks are used on a global scale and are widely considered to provide an accurate measure of a particular form of mathematically literate behaviour. In light of this, there is a strong case to argue that the dominant orientations (and associated intentions and agendas) that characterise much of the literatures read provide a valid and appropriate reflection of prevailing areas of
prioritising in international (and South African) conceptions of mathematically literate, numerate and/or quantitatively literate behaviour.
A possible and likely reason for this prioritisation of agendas characterised by mathematically structured and legitimised forms of participation stems from the fact that the majority of the conceptions of mathematical literacy, numeracy and/or quantitative literacy do not separate the development of associated behaviour from engagement with scientific mathematics practices or from the domain of mathematics. Instead, the development of mathematically literate, numerate and/or quantitatively literate behaviour is seen to stem through participation in and engagement with all forms of mathematical contents, including scientific contents. In sum, the dominant perspective in much of the international literature is that to be mathematically literate, numerate and/or quantitatively literate means to be literate and/or numerate with mathematics and not with something else other than mathematics. The result of this perspective is that participation in such practices is inevitably, understandably and appropriately legitimated according to mathematical knowledge, structures and forms of communication.56
This, then, brings to an end the analysis and discussion of the forms of behaviour associated with various conceptions of mathematical literacy, numeracy and/or quantitative literacy described in international literatures. Looking ahead, in the next chapter I move to discuss what I perceive to be key distinctions between facets of behaviour associated with the terms mathematical literacy, numeracy and/or quantitative literacy and my reasons for prioritising the term ‘mathematical literacy’. Thereafter (in Chapter 8), I discuss the South African conception of mathematically literate behaviour as encompassed in the subject-matter domain of Mathematical Literacy and compare this conception to the international perspectives described above.
56 Although, it must be said that there is a risk in ‘painting all of these different conceptions with the same brush’ since the basis of legitimisation in different forms of mathematical practice will vary. Namely, the basis of legitimisation for mathematical practices that prioritise modelling will be different from, for example, those that prioritise pure mathematics concepts, or from those that engage applied mathematical principles.
Intention
Mathematical Literacy, Numeracy and/or Quantitative Literacy for:
Agenda Developing Human Capital Cultural
Identity
Environmental Awareness
Social Change
Evaluating Structures
Mathematically dominant goals 1. Mathematical
Competence 1 [a]
Literacy in Mathematics
International (INT) (Niss, 2003); (OECD, 1999, 2003, 2006, 2009) South Africa (SA)
1 [b]
Numeracy
INT SA
2. Mathematics in Context
2 [a]
Application
INT
(Niss, 2003); (OECD, 1999, 2003, 2006, 2009); (L. A. Steen, 1999, 2001a, 2001b, 2003a, 2003b; L. A. Steen, (Ed), 1990; L. A.
Steen, et al., 2001);
SA (DoE, 2003a) 2 [b]
Numeracy-in- Context
INT (Ginsburg et al., 2006); (Neill, 2001) SA
(DoE, 2005b, 2007, 2008c, 2008d, 2009a, 2009b, 2009c) (DBE, 2010a, 2010b, 2012a, 2012b, 2013b, 2013c, 2014b, 2014d)
Contextually dominant goals
3. Modelling
INT
(de Lange, 2003, 2006); (Ewell, 2001); (Ginsburg et al., 2006); (Hughes- Hallett, 2001, 2003); (Niss, 2003); (OECD, 1999, 2003, 2006, 2009);
(Packer, 2003); (Pugalee, 1999; Pugalee et al., 2002); (Richardson &
McCallum, 2003); (Schoenfeld, 2001); (L. A. Steen, 1999, 2001a, 2001b, 2003a, 2003b; L. A. Steen, (Ed), 1990; L. A. Steen, et al., 2001); (Van Groenestijn, 2003); (Venkatakrishnan & Graven, 2007); (Graven &
Venkatakrishnan, 2007)
(D'Ambrosio,
2003) (Fusaro, 1995) (Skovsmose,
1992, 1994a)
(Jablonka, 2003)
SA (Brombacher, 2007);(DoE, 2006); (DBE, 2011a)
(Christiansen, 2006, 2007);
(Julie, 2006) 4. Contextual
Sense-Making Practices
INT (C. Hoyles, Noss, & Pozzi, 2001; C Hoyles, Wolf, Molyneux-
Hodgson, & Kent, 2002) (Fusaro, 1995) (Frankenstein
, 2009b)
(Jablonka, 2003);
SA
(Brombacher, 2007); (DBE, 2011a); (Gal et al., 2005- at a conceptual level); (PIAAC Numeracy Expert Group, 2009 - at a conceptual level)
(Christiansen,
2006, 2007) (Julie, 2007)
Figure 10: Categorisation of the literature on mathematical literacy (including the South African subject Mathematical Literacy), numeracy and/or quantitative literacy according to the dimensions of dominant Agendas and Intention
Prioritisation of Mathematical knowledge and participation
Prioritisation of knowledge / participation that facilitate contextual sense-making practices
SPECTRUM