WHAT’S IN A NAME? MATHEMATICAL LITERACY, NUMERACY OR QUANTITATIVE LITERACY?
7.2 My privileging of the term ‘mathematical literacy’
My primary reason for privileging the term mathematical literacy over numeracy and/or quantitative literacy is a pragmatic one: in South Africa there is a subject called Mathematical Literacy and, so, it makes sense to use a term that is recognisable and which has a direct correlation to existing curricular and pedagogic practice.
But, how precisely does my conception of mathematical literacy differ from those for numeracy and quantitative literacy (and even mathematical literacy) described in the literature and outlined above? The remainder of this section is devoted to answering this question.
De Lange (2003, p. 75; 2006, p. 13) points out that many of the definitions and expressions of quantitative literacy and numeracy focus primarily on the numerical or quantitative aspects involved in the application of mathematics. This sentiment is echoed
59 The lack of a space in the word ‘criticalmathematical’ is deliberate.
by Niss (2003, p. 215), who argues that variation in interpretations of the term quantitative literacy “is mainly a matter of how narrowly the word ‘quantitative’ is to be understood, vis ẚ vis the involvement of numbers and numerical data.”. L. A. Steen, et al. (2001, p.
6), for example, provides as expression of quantitative literacy as “The capacity to deal effectively with the quantitative aspects of life”; Hughes-Hallett (2003, p. 93) suggests that the “cornerstone of quantitative literacy is the ability to apply quantitative ideas in new or unfamiliar contexts.”; and Ginsburg et al. (2006, p. 1) focus on quantitative aspects in motivating for the need for increased numeracy amongst adults: “As quantitative and technical aspects of life become more important, adults need higher levels of numeracy to function effectively in their roles as workers, parents, and citizens.”
Both de Lange (2003, 2006) and Niss (2003) argue against this narrowing of focus on primarily quantitative aspects and promote, instead, a conception of mathematical literacy (which is the term they both privilege) that encompasses a broader application of mathematical knowledge. This broader application includes the premise that “All kinds of visualizations belong as well to the literacy aspect of mathematics and constitute an absolutely essential component for literacy …” (de Lange, 2003, p. 76), where ‘all kinds of visualizations’ includes reading maps, understanding plans (de Lange, 2003, p. 76), and, presumably, also other visual or spatial resources such as assembly diagrams, instructions for appliances, models, and so on.
In promotion and defence of this argument, de Lange (2006, p. 14) identifies and differentiates four different literacies − (i) numeracy, (ii) quantitative literacy, (iii) spatial literacy, and (iv) mathematical literacy, each of which reflect a particular relationship to different phenomenological categories. De Lange (2006, p. 15) illustrates the relationship between these literacies and their connection to particular phenomenological categories in the diagram shown in Figure 11 below:
Figure 11: De Lange's (2006) ‘Tree structure of mathematical literacy’
As illustrated in the diagram, De Lange views numeracy as reflective of the ability to work with and perform calculations involving numbers and data. Numeracy is positioned as a particular sub-set of quantitative literacy. Quantitative literacy, by contrast, is seen to involve functionality in a range of categories and, so, includes skills and knowledge associated with more than just numbers and data. Spatial literacy is the third literacy and
is conceptualised as prioritising the development of skills associated with engagement with spatial objects, including 2-and 3-dimensional representations such as maps, plans, navigational tools, geometric representations, and so on. The final literacy, Mathematical literacy, is conceptualised as the overarching literacy comprising all of the other literacies (de Lange, 2003, pp. 80-81; 2006, pp. 14-15).
I find de Lange’s distinction between numeracy and quantitative literacy useful and agree with his assertion that mathematical literacy is the overarching category
comprising both numeracy and quantitative literary plus the additional spatial literacy.
However, I find his association of each literacy type to specific phenomenal categories restrictive. This categorisation suggests that each literacy is determined by the types of contexts or phenomenon that a person is exposed to and is able to show functionality or competence in, rather than the skills and knowledge that they are able to demonstrate in solving problems based in real-life contexts. By contrast, I contend that mathematically literate behaviour is characterised by the ability to decide on and employ appropriate strategies to solve any problem irrespective of the context or structure of the problem.
Furthermore, I envision that such behaviour is not defined or bounded by the particular phenomenological categories in which the problems are situated or the specific
mathematical skills employed to solve those problems. A further reservation of de Lange’s ‘Tree structure of mathematical literacy’ schematic relates to my view that there are other literacies which do not appear in this diagram but which are a central component of mathematically literate behaviour and, as such, need to be considered in any conception of mathematical literacy. These additional literacies are elaborated on in the immediate discussion below.
As such, and in contrast to de Lange, I suggest an alternative conceptualisation of the distinction between numeracy and quantitative literacy that does not rely on an association with phenomenal categories. Namely, that numeracy involves competence with basic mathematical content, calculations, techniques and knowledge, not limited specifically to numbers and quantity but encompassing all mathematical contents; while quantitative literacy involves the functional use of such content, calculations, techniques and knowledge in making sense of problems grounded in real-world situations. Numeracy is thus seen as a pre-requisite for quantitative literacy. The third literacy – that of spatial literacy − remains in line with de Lange’s conceptualisation of the term, involving understanding and competence with spatial representations and objects such as maps, plans, 2-and 3-dimesional views of objects, techniques for estimating distances, and so on. Importantly and distinct from de Lange’s conceptualisation, quantitative literacy (and so also numeracy) is seen as a perquisite for spatial literacy: for example, the ability to use a plan effectively relies on the ability to measure accurately and to perform necessary conversions and calculations.
I content that three further literacies need be added to this existing categorisation of literacies that comprise the overarching framework of mathematical literacy. To begin with, given the inherently text (both spoken and written) rich nature of contextually based problem-solving interactions, a key literacy required in contextual sense-making practices is that of text literacy. Text literacy reflects an individual’s capacity for making sense of messages and information conveyed through written and/or oral text. The conception of text literacy as employed in this study is seen to comprise a combination of both prose literacy and document literacy. Given that ‘prose’ refers to a language form that exhibits a grammatical structure, prose literacy involves “the knowledge and skills needed to understand and use information organised in sentence and paragraph formats” (Kirsch et al., 1998, p. 113). In the context of mathematical literacy related practices, this involves
the capacity to interpret and understand information and instructions encountered in textual and language format that specifies facets of the contextual situation under investigation and/or the specific problem-solving requirements required in relation to that situation. In addition to the overwhelming amount of information communicated in prose format, information is also communicated extensively through the use of documents that are organised in ‘matrix structures’ (i.e. with a clearly identifiable row and column structure) (Mosenthal & Kirsch, 1998, p. 641). Document literacy, then, involves the ability to make sense of information presented in and through tables, signs, indexes, lists, schedules, charts, graphs, maps, and forms (Kirsch & Mosenthal, 1988, p. 2). This form of literacy is particular essential in a format of mathematical literacy oriented towards life-preparedness since, as argued by Kirsch et al. (1998, p. 118), while prose literacy is the dominant form of literacy in schools, documents tend to be the principal form of communication in out-of-school settings. Taken together, prose and document literacy facilitate successful interpretation of and engagement with both quantitative and other information presented in textual formats, including financial documents (bills, invoices, tickets, quotations, payslips), newspaper articles, adverts, tables, graphs, timetables, brochures, and so on. Importantly, within the framework of mathematical literacy textual literacy is seen to interact in an intricate and intertwined way with the other literacies such that successful and enhanced engagement with encountered information is facilitated through a combination of both quantitative and textual literacy knowledge and skills. For example, in making sense of financial documents (document literacy), elements of prose literacy and quantitative skills are employed (e.g. understanding of the meaning of the terminology ‘tax’ and checking that the tax value on an invoice has been correctly determined); and in working with maps (spatial literacy), both document literacy (e.g.
making sense of a distance table) and quantitative skills (e.g. estimating travelling times) may be required.
The next literacy for consideration is that of statistical literacy. Statistics pervade real- world practices, and the ability to function effectively in everyday life requires the capacity for critical engagement with encountered statistical information. However, and as illustrated and argued in detail by Gal (2002), statistical literacy is a separate literacy from numeracy (and from quantitative and spatial literacy, and also from the domain of scientific mathematics) and does not develop in a sustainable form through engagement with these literacies. Instead, statistical literacy develops through a complex interaction of knowledge bases (including literacy skills, statistical knowledge, mathematical knowledge, knowledge of context, and skills and knowledge that facilitate critical analysis of statistical information) and dispositional elements (including beliefs, attitudes, and a critical stance). Where the knowledge bases facilitate interpretation and understanding of statistical information, the dispositional elements highlight a necessary and crucial inclination on the part of the individual to active the knowledge bases in the process of critical engagement with statistical contents (Gal, 2002, p. 4). The conception of mathematical literacy adopted in this study, then, shares this perspective that statistical literacy is distinct from numeracy, quantitative literacy and spatial literacy, and, consequently, requires a unique and dedicated site of development. That said, the interconnectedness of these various literacies, together with textual literacy, is again acknowledged and emphasised: successful engagement with statistical information is reliant on an individuals’ ability to interpret and understand presented information (in both prose and document format) and employ a variety of quantitative techniques that facilitate critical analysis.60
60 In similar vein to the conception of statistical literacy offered by Gal (2002), scientific literacy is another form of literacy that engages mathematical knowledge and structures and, yet, which exists outside of the domain of disciplinary mathematics. As with statistical literacy and also with the
The final literacy to be considered as an essential component of mathematical literacy – what I have termed real-world literacy. This component comprises considerations that directly affect functionality in real-world situations − considerations that are often not of a mathematical nature, which commonly override mathematical considerations, which impact on how people think, act and communicate in real-world situations, and which influence decision-making processes. For example, when buying a house there are many considerations other than cost which affect the type of house bought, including, amongst others, the location, features and condition of the house (and even the attitude and personality of the estate agent). These are real-world considerations that are context- specific, and the rules for making sense of these considerations are often only experienced or learned in the specific context in which they are encountered. These considerations work together with the mathematical considerations to allow for informed decision- making practices.
In the context of mathematical literacy related practices, numeracy, quantitative literacy, spatial literacy, statistical literacy and textual literacy all comprise elements of mathematics or have the potential to facilitate mathematical investigation.61 As such, I refer to these as comprising the mathematically informed practices of mathematical literacy. By contrast, real-world literacy exists completely in the domain of the contextual.
The diagram in Figure 12 below – as an adapted version of de Lange’s (2006) ‘Tree structure of mathematical literacy’ schematic − illustrates my view on the relationship between the various literacies that make up the overarching category of mathematical literacy.
conception of mathematical literacy adopted in this study, both context and mathematical knowledge are required to successfully engage in the practices of science (Watson & Callingham, 2003, p. 5), and yet neither of these terrains alone define the structure of legitimate participation in the discipline. Instead, it is through a complex interaction of particular contents, knowledge, and skills drawn from both of these terrains, together with specialised scientific knowledge, and with beliefs, attitudes and a critical
perspective, which facilitates the structure of legitimate participation. See (Holbrook & Rannikmae, 2009, pp. 276-277) for a discussion of different components and attributes associated with scientific literacy, and (Shamos, 1995) for a discussion of different levels of scientific literacy.
61 Note that, in classifying numeracy, quantitative literacy, spatial literacy and document literacy as comprising predominantly ‘mathematically informed practices’, it is not my intention to deny the presence of contextual elements in problem-solving scenarios involving these literacies or to deny the necessity for engagement with and understanding of these contextual elements for successful completion of problem-solving processes. Rather, in the context of the specific domain of mathematical literacy, I see these literacies as comprising both mathematical and contextual elements and, as such, that practices accessed through these literacies facilitate mathematical forms of engagement with problems and resources encountered. In each of these literacies there is potential for engagement with mathematical structures and elements, and for understanding how mathematical forms of participation facilitate a particular type of understanding of a problem scenario. By contrast, real-world literacy places emphasis on understanding contextual (and commonly non-mathematical) factors that affect action and behaviour in a contextual situation.
Figure 12: The literacies that characterise mathematical literacy
As a final comment, it is crucial to emphasis the ‘literacy’ component of the conception of mathematical literacy as described above. The conception of mathematical literacy adopted in this study is perceived to comprise a collection of various literacies such that being mathematically literate is akin to showing proficiency in all of these domains.
Proficiency in this context is not to be equated to a minimal level of engagement or a set of basic skills – to the ability to only ‘read and write’ the contents of the domains. Instead, proficiency is to be equated with full functional competence in the domain, including the ability to interpret and understand the contents of the domain, to engage critically with these contents, and to communicate these contents effectively and in an accessible format.
Proficiency is to be equated with functional engagement – specifically, with the use of the contents of the domain to engage and solve problems, and to challenge and critique existing structures. Being mathematically literate involves the capacity to interact with complex real-world scenarios, to engage confidently with the language and resources employed in these scenarios, to employ statistical, quantitative and spatial tools to investigate these resources, and to communicate opinions and results in a critical way through a variety of mediums.
This, then, concludes the discussion on my privileging of the term mathematical literacy and my views on various literacies that characterise mathematical literacy. In the next chapter (Chapter 8) discussion now shifts to a focus on the South African school subject Mathematical Literacy and the structure, intention, and dominant orientation of the conception of mathematically literate behaviour prioritised in the subject.
Real-World Literacy (Familiarity with and understanding of the components that affect decision-making processes
in real-world contexts)
Mathematical Literacy
Quantitative Literacy
Mathematically informed practice Contextual practice Numeracy
Spatial Literacy
Text Literacy
Statistical Literacy