AREAS OF COMMONALITY AND DIVERGENCE IN THE LITERATURE
6.2 Category 2 − Interplay of content, contexts and/or competencies
As has already been discussed, a defining feature of mathematically literate, numerate and/or quantitatively literate behaviour involves an interplay between mathematical content and the usage of that content in solving problems related to real-life considerations. Brombacher45 (2007, p. 14) makes use of the diagram shown in
45 Although Brombacher’s work stems from a South African perspective and is written with a particular view towards the subject-matter domain of Mathematical Literacy, much of the content of his writing, by his own admission, is informed by international perspectives relating to conceptions of mathematically literate, numerate and/or quantitatively literate behaviour. It is for this reason that I deem it appropriate to cite his work as part of this more general discussion on international perspectives of mathematical literacy, numeracy and/or quantitative literacy.
Figure 8 on the page below to provide clarity on the specific roles of content and context
− and the relationship between these two facets − in the development of mathematically literate, numerate and/or quantitatively literate behaviour, and to illustrate the perspective that
… the attributes of Mathematical Literacy are developed through interplay between content and context. Content enables us to work on finding solutions to problems that are interesting and relevant, while context gives meaning to the mathematical knowledge and skills (content) that we are teaching. (Brombacher, 2007, p. 129) This dual emphasis on the interplay between content and context is a common feature of much of the literature.
Figure 8: Interplay of content and context in Mathematical Literacy
Despite this emphasis on the duality of the content-context relationship, it is my contention that in much of the literature read there is a prioritising of the content
component (i.e. the mathematical terrain) over considerations for genuine sense-making of the contextual terrain and of the structure of legitimate and appropriate knowledge and participation in that terrain. In other words, the contexts that are deemed appropriate are those which allow for a particular type of mathematical exploration of for
exploration of a particular mathematical concept. Primary focus is on how particular mathematical knowledge and techniques can be applied in a situation or context. In other words, on how mathematics can cast a gaze outside of its own domain. By contrast, lesser and often no emphasis is placed on the use of a variety of appropriate tools and techniques, including mathematical and/or non-mathematical (i.e. situational) techniques, to facilitate a deeper understanding of legitimate forms of knowledge and participation in a situation or context. Contextual and/or qualitative considerations which may affect how a person might actually solve a problem in an everyday situation are not given credence, and contextually derived narratives are not considered as reflecting appropriate or valid solution strategies. Instead, mathematical structures comprise the organising principle of the activity and, as such, it is mathematically generated solutions derived through mathematically structured routines that are
endorsed. Packer (2003, p. 36) suggests as much when, in presenting an argument for a canon of empirical mathematical problems that can be used to develop and measure quantitatively literate behaviour, states that,
Mathematics helps us to solve the problems we want to solve
The problems we solve give meaning to the mathematics we use Mathematical
Content
Real-Life Contexts
The challenge is to identify important, frequently encountered problems that cannot be efficiently solved without using mathematics.
As does Van Groenestijn (2003):
Achieving numeracy is a matter of learning how to use mathematics in real life and how to manage mathematical situations. (p. 233, emphasis in original text)
From one definition or expression to the next, time and time again, the organising principle is the use of mathematical knowledge and techniques to solve problems:
The ability to use mathematics to solve problems is a primary goal of becoming mathematically literate. (Pugalee, 1999, p. 3)
A quantitatively literate person is a person who, with understanding, can both read and represent quantitative information arising in his or her everyday life.
(Richardson & McCallum, 2003, p. 99)
Hence, numeracy courses embedded in school programs must focus on problem- solving activities in which students can apply their acquired mathematical insights and skills and learn how to manage such situations. (Van Groenestijn, 2003, p. 233) Quantitative Literacy is the ability to identify, understand and use quantitative arguments in everyday contexts. (Hughes-Hallett, 2003, p. 91)
This is not to suggest that such emphasis on mathematics as the organising principle is not deliberate or that such authors are unknowingly prioritising mathematical structures over the contextual terrain. The focus on mathematics and the prioritising of mathematical knowledge, techniques and skills is explicitly acknowledged. And this is not surprising if one recognises, as has already been suggested above, that for many the development of mathematically literate behaviour is encompassed in the teaching and learning of scientific mathematics and is signified through empowered and enhanced mathematical ability.
Alongside an emphasis on the interplay of mathematical content and real-world contexts is a further call, from some authors and frameworks46, for the central role of the development of skills or competencies – or what the PIAAC framework refers to as
‘enabling processes’ (PIAAC Numeracy Expert Group, 2009, pp. 29-31) – as characteristic of mathematically literate, numerate and/or quantitatively literate behaviour. De Lange (2003, p. 88) is particularly ardent in this respect, arguing that “the desired competencies, not the mathematical content, are the main criteria …”. This perspective is reflected in the OECD-PISA frameworks (OECD, 2003, 2006, 2009, 2012b) where competencies are viewed as the central component that makes it possible for mathematics to be applied and successfully utilised to solve problems encountered in real-world scenarios:
46 For example: Pugalee (1999); Neill (2001); L. A. Steen, et al. (2001); Niss (2003); Ginsburg et al.
(2006); OECD-PISA frameworks (1999, 2003, 2006, 2009); ALL frameworks (Gal et al., 2005) & (Van Groenestijn, 2003); and the SCANS framework (SCANS, 1991) & (Packer, 2003).
While situations or contexts define the real-world problem areas, and overarching ideas reflect the way in which we look at the world with “mathematical glasses”, the competencies are the core of mathematical literacy. Only when certain competencies are available to students will they be in a position to successfully solve given problems. (OECD, 2003, p. 32)
Importantly, in many of the conceptions that emphasise the central role of competencies, these competencies are presented as the component that binds the content and the contexts together. As suggested by Ginsburg et al. (2006, p. 3), the competencies (i.e. the cognitive and affective component of numeracy) are the “processes that enable an individual to solve problems, and thereby, link the content and context.” Brombacher (2007) offers a similar perspective,
the purpose of the content and the contexts in the mathematical literacy classroom is to develop life skills and competencies. These are the competencies that the individual needs to participate in her/his world as a self-managing individual; as a contributing worker; as a life-long learner; and as a critical citizen. (p. 15)
and illustrates the components of this perspective by means of the diagram shown in Figure 9 on the page below.
I interpret this particular conception of the relationship between content, contexts and competencies to suggest that the content and contexts are perceived as tools used to facilitate the development of a set of skills that have application across a wide range of problems and situations. The competencies, and not the contexts or the content, are, thus, the central component in the development of mathematically literate behaviour. De Lange (2003) again echoes a similar sentiment:
… to effectively transfer their knowledge from one area of application to another, students need experience solving problems in many different situations and contexts … . Making competencies a central emphasis facilitates this process:
competencies are independent of the area of application. (p. 80)
Figure 9: Diagrammatic representation of the relationship between content, context and skills
As an illustration, consider a task in which a learner is instructed on the process for drawing a linear graph to represent costs for a pre-paid electricity scenario. From the position of the conception or perspective described above, the ‘linear’ and ‘electricity’
components are secondary to the development of a graph drawing ability. In other words, heightened emphasis is placed on the ability to draw an appropriate graph (irrespective of the shape or name of the graph) to represent a real-life situation (irrespective of the specific nature of the situation). Of lesser importance is the ability to draw linear graphs to represent a pre-paid electricity situation.
A further facet of this emphasis on competencies is the belief that such competencies are universally applicable across a wide range of contexts and problem situations and are not bounded by any particular strand of content or context. As suggested by Howe (2003, p.
185), “This leads me to suspect that there are certain skills that are to some extent context free and that support the ability to deal with quantitative information in a variety of contexts.”
It is beyond the scope of this study to provide a comprehensive list of all competencies emphasised in the literature. However, it may prove useful to identify particular skills that are emphasised consistently, since this provides an indication of the types of skills most commonly associated with mathematically literate, numerate and/or quantitatively literate
behaviour and, as such, adds further insight into understanding how such behaviour is conceptualised. Common skills include:47&48
fluency with mathematical concepts and tools;
conceptual understanding;
problem-solving, application and/or modelling;
reasoning, insight, and/or reflection;
communication – including communication of mathematical ideas through appropriate usage of operators and symbols, as well as critical communication through the offering of opinions, decision-making, and so on; and
attitudes, dispositions, beliefs and/or values.
It is, perhaps, in an attempt to encapsulate this emphasis on such competencies as those listed above that Bass (2003) provides the following wide-ranging definition or expression of quantitative literacy:
QL appears to be some sort of constellation of knowledge, skills, habits of mind, and dispositions that provide the resources and capacity to deal with the
quantitative aspects of understanding, making sense of, participating in, and solving problems in the worlds that we inhabit, for example, the workplace, the demands of responsible citizenship in a democracy, personal concerns, and cultural enrichment. (p. 247)
Does this emphasis on competencies and the positioning of mathematical and contextual components as tools negate the prioritising of mathematics as the organising principle and the imposition of a mathematical gaze over incorporated real-world scenarios? Not necessarily, since many of the skills are still grounded in a mathematical base and with a mathematical bias. For example, consider the skill of ‘communication’. The type of communication that is envisioned involves, primarily, communication appropriate to the domain of mathematics. Namely, the use of appropriate symbols and mathematical notation, the correct layout of calculations and answers, critical interpretation and comparison of mathematical solutions and options, and the generation of narratives to problem scenarios – through the use of mathematical techniques and routines − that are endorsed primarily according to mathematical knowledge and structures. More general communication skills that reflect common everyday communication strategies – such as the ability to write a paragraph, formulate an argument or opinion, or develop a presentation – are downplayed; instead, the skills that are prioritised are those concerned primarily with the way in which mathematical ideas are communicated in a mathematically logical, appropriate and legitimate way.
47The literature consulted in identifying this list of skills includes: (de Lange, 2003, 2006); (Hughes- Hallett, 2001, 2003); (Gal et al., 2005); (Niss, 2003); (Neill, 2001); (OECD, 2009); (Pugalee, 1999);
(Pugalee, Douville, Lock, & Wallace, 2002); (Packer, 2003); (Richards, 2001); (Richardson &
McCallum, 2003); (Schoenfeld, 2001, p. 53); (L. A. Steen, et al., 2001); (Van Groenestijn, 2003).
48 The description by Kilpatrick et al., (2001) of the ‘Strands of Mathematical Proficiency’ provides detailed discussions of many of the skills listed here. Kilpatrick et al., refer to the strands of (i)
‘procedural fluency’ − which reflects the first skill category listed above; (ii) ‘conceptual understanding’;
(iii) ‘strategic competence’ − which reflects in the problem-solving and/or modelling category; ‘adaptive reasoning’ − which reflects in the reasoning, insight, reflection category; and productive disposition – which reflects in the attitudes, dispositions, beliefs and/or values category.
The Strands of Mathematical Proficiency are referenced specifically in relation to the development of mathematical knowledge and there is overlap between these strands and most of the skills identified as essential in the development of mathematical literacy, numeracy and/or quantitative literacy behaviour.
This provides evidence of the dominance of the mathematical terrain and mathematical goals in many conception of mathematical literacy, numeracy and/or quantitative literacy.
It is also important to point out, though, that not all agree with the conceptualisation of a generalisable and widely applicable set of competencies as a core component of mathematically literate, numerate and/or quantitatively literate behaviour. This is particularly true for perspectives that position mathematical literacy, numeracy and/or quantitative literacy as a socially or culturally situated practice. Jablonka (2003), for example, argues that:
The assumption that it makes sense to search for a universalistic applicable cannon of mathematical skills that can be separated from the context of their use is
doubtful from the perspective of the socio-cultural view of mathematics. It is doubtful whether mathematical skills can be separated from the social dimensions of action and from the purposes and goals of the activity in which they are
embedded. … Such a description ignores the interests and values involved in posing and solving particular problems by means of mathematics. (p. 79)
Ewell (2001, p. 37) offers a similar suggestion, first equating quantitative literacy to a type of literacy and then emphasising the socially or culturally situated nature of literacy.
And Frith and Prince (2006, 2009) follow suite, warning against viewing mathematical literacy as a constellation of skills which have application outside of a particular social setting.
These alternative perspectives indicate, once again, the differential emphasis and sometimes lack of consensus over the components and key areas of focus in descriptions of mathematically literate, numerate and/or quantitatively literate behaviour.