AREAS OF COMMONALITY AND DIVERGENCE IN THE LITERATURE
6.1 Category 1 − Considerations of mathematics
In terms of (i) reference to mathematical content, knowledge and/or techniques, there is overwhelming and almost unanimous agreement in the literature that, irrespective of the name used, mathematical literacy, numeracy and/or quantitative literacy involves mathematical content, knowledge and skills. The selection of quotations below illustrate this point:
Numeracy is the ability of a person to make effective use of appropriate
mathematical competencies for successful participation in everyday life, including personal life, at school, at work and in the wider community. It involves
understanding real-life contexts, applying appropriate mathematical competencies, communicating the results of these to others, and critically evaluating
mathematically based statements and results. (Neill, 2001, p. 7)
[Numeracy]: the ability to access, use, interpret and communicate mathematical information and ideas in order to engage in and manage the mathematical demands of a range of situations in adult life. (OECD, 2012a, p. 34)
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 meets the needs of the individual’s life as a constructive, concerned and reflective citizen. (OECD, 2009, p. 84)35 The concept of quantitative literacy is rooted in the connection between mathematics and reason. (Richards, 2001, p. 35)
… my general sense of what quantitative literacy should be: the predilection and ability to make use of various modes of mathematical thought and knowledge to make sense of situations we encounter as we make our way through the world.
(Schoenfeld, 2001, p. 51)
Quantitative Literacy is the ability to identify, understand and use quantitative arguments in everyday contexts. (Hughes-Hallett, 2003, p. 91)
All of the quotations also hint at a further element of common agreement with respect to mathematical literacy, numeracy and/or quantitative literacy. Namely, (ii) reference to the
‘use’ or ‘application’ of mathematical content/knowledge/techniques to solve problems.
There is widespread acknowledgement and agreement that mathematical literacy, numeracy and/or quantitative literacy, involves more than simply mathematical content.
Rather, as suggested by Neill (2001, p. 2), “Numerate behaviour can then be analogously defined as: The standard mathematical tools, especially when used for other than mathematical purposes.” In other words, a key intention in the development of mathematically literate, numerate and/or quantitatively literate behaviour is the ability to use mathematical content, 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)
The test of numeracy, as of any literacy, is whether a person naturally uses appropriate skills in many different contexts. (L. A. Steen, et al., 2001, p. 6)
Quantitative literacy is not about how much mathematics a person knows but about how well it can be used. (Hughes-Hallett, 2003, p. 91)
35 There have been five PISA studies − in 1999, 2003, 2006, 2009 and 2012 (OECD, 1999, 2003, 2006, 2009, 2012b). All of the pre-2012 studies are comprised of three components: Reading, Mathematics (or Mathematical Literacy) and Science. The 2012 study, by contrast, includes two additional components – one on Problem-Solving and the second on Financial Literacy. In each year of implementation, one domain is given the opportunity for revision, and this works on a rotational system. For the Mathematics domain, this reformulation occurred in 2006 and again in 2012 (Stacey, 2012) (note that the 2012 study was under construction at the time of writing of the original draft of this part of the study, and at the time of finalising this study the 2012 results has not yet been released to the public). The importance of this point is that for the Mathematics component of the studies, in most parts the contents − and especially the theoretical framework that underpins the domain – have remained largely unchanged since the inception of the original study in 1999, bar minor tweaking. For this reason, rather than always referencing all four studies, I commonly only reference the most recent study – namely, the one conducted in 2009 (which is identical to the 2006 study), and this reference must be interpreted to be representative of all previous studies. However, in the event that the 2009 document contains information that is different to the 1999 document or to any of the other documents, then in such cases I reference the immediately relevant document and, where necessary, state the difference.
Any attempt at defining ‘mathematical literacy’ faces the problem that it cannot be conceptualised exclusively in terms of mathematical knowledge, because it is about an individual’s capacity to use and apply this knowledge. (Jablonka, 2003, p. 78, emphasis in original text)
An important part of mathematical literacy is using, doing and recognizing mathematics in a variety of situations.
(de Lange, 2003, p. 80)
Thus, literacy in mathematics is about the functionality of the mathematics you have learned at school. (de Lange, 2006, p. 16)
Moreover, the ‘problems’ that are the focus of this application of mathematics are to be based in extra-mathematical and/or real-world contexts and experiences (as opposed to esoteric mathematical situations). Mathematical literacy, numeracy and/or quantitative literacy, thus, embody a relationship between mathematical content and knowledge and extra-mathematical contexts, situations and problems:
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)
… being numerate involves more than just knowing mathematics. It implies that to organise their lives as individuals, as workers, and as citizens, adults need to feel confident of their own mathematical capacities and be able to make effective decisions in mathematical situations in real life. (Van Groenestijn, 2003, p. 230) Numerate behavior is observed when people manage a situation or solve a problem in a real context ... (Gal, van Groenestijn, Manly, Schmitt, & Tout, 2005, p. 152)
An important commonality in the above descriptions of numeracy is the presence of mathematical elements in real situations, and the notion that these can be used or addressed by a person in a goal-oriented way, dependent on the needs and interests of the individual within the given context (home, community, workplace, etc.), as well as on his or her dispositions. (Gal et al., 2005, p. 151)
… mathematical literacy refers to the competency to handle situations in work, leisure, home and the public domain which involves what mathematicians would consider ‘mathematical’ competencies. (Christiansen, 2007, p. 92)
Numerate behavior involves managing a situation or solving a problem in a real context, by responding to mathematical content/information/ideas represented in multiple ways. (OECD, 2012a, p. 34)
While there is agreement in the various conceptions of mathematical literacy, numeracy and/or quantitatively literacy on the relationship of the mathematical terrain to the real- world, such agreement is lacking with respect to opinions on the scope, location and specific contexts in which such real-world problems are to be explored. This particular area of divergence is discussed in more detail in the pages below in the section that deals with Arena of application (c.f. page 68 below).
Consideration of the (iii) strands of mathematical content emphasised in the literature is the next sub-category for discussion. Where reference to categories or strands of content is made36, different content groupings or strands are employed and emphasised by different authors. Steen (1990), as an early proponent of quantitative literacy, identifies and emphasises six content strands: Quantity, Dimension, Pattern, Shape, Uncertainty, and Change. In later work, L. A. Steen, et al. (2001, pp. 15-17) modifies his original thinking to focus instead on ‘Skills of Quantitative Literacy’ rather than specific content strands, and includes as part of these skills the categories of Arithmetic, Data, use of Computers, Modeling, Statistics, Chance, and Reasoning. Importantly, while these categories define some content to be included in the development of quantitative literacy, they also extend beyond simple content categories to include mathematical and logical skills required for solving problems based in context (L. A. Steen, et al., 2001, p. 17).
Following on from the early work of Steen, the OECD-PISA assessment frameworks highlight four content categories, comprising the labels Quantity, Change and Relationship, Space and Shape, and Uncertainty (OECD, 2009, pp. 93-104)37. The ALL and PIAAC frameworks38 (Gal et al., 2005; PIAAC Numeracy Expert Group, 2009) place emphasis on the strands of Quantity and Number; Dimension and Shape; Patterns, Functions and Relationships; Change; [these previous two strands are combined into a single strand of Patterns, Relationships and Change in PIAAC] and Data and Chance.
Closer to home, the curriculum statement document for the South African subject-matter domain of Mathematical Literacy (DoE, 2003a) categorises the curriculum for the subject according to the four strands39 Numbers and Operations in Context, Patterns and
36 Explicit reference to specific strands of mathematical content is more prevalent in studies, reports and frameworks that have been written to explicitly define a structure for pedagogic and/or assessment practice associated with mathematically literate, numerate and/or quantitatively literate behaviour. For example:
the OECD-PISA assessment frameworks (1999, 2003, 2006, 2009, 2012b);
the International Adult Literacy Survey (ALL) (Gal et al., 2005);
the NCSALL Report titled The Components of Numeracy (Ginsburg et al., 2006);
Neill’s (2001) description of components of numeracy for the New Zealand Curriculum;
and the NCED publication Mathematics and Democracy (L. A. Steen, et al., 2001) that outlines the components of quantitative literacy from an American perspective.
There is, however, a multitude of literature that provide alternative focus on unpacking expressions and descriptions of behaviour associated with conceptions of mathematical literacy, numeracy and/or quantitative literacy rather than identifying specific categories of content and arenas of application through and within which such behaviour is to be developed. Pugalee (1999), Jablonka (2003) and Frankenstein (2009a) are examples of authors who contribute to this body of literature.
37 The 1999 PISA framework (OECD, 1999, pp. 47-50) uses different categories to subsequent PISA frameworks, with emphasis on Chance, Change and Growth, Space and Shape, Quantitative Reasoning, Uncertainty, and Dependency and Relationships.
38 ALL stands for Adult Literacy and Lifeskills Survey and was developed as an international study to measure the numeracy and literacy levels of adults. The ALL study is the successor of the IALS study (International Adult Literacy Survey) which represented the world’s first large-scale, comparative assessment of adult literacy (see (Organization for Economic Co-operation and Development (OECD) &
Statistics Canada, 2000) and (Kirsch, Jungblut, & Mosenthal, 1998)). The latest successor to the ALL study is the Programme for the International Assessment of Adult Competencies (PIAAC), developed in 2009 and currently in implementation phase. See Gal and Tout (2012) for a comprehensive discussion of the history of the ALL study and of the key theoretical components of the study.
39 The curriculum document uses the word ‘Learning Outcome’ rather than strand and explains the meaning of a Learning Outcome as follows:
A learning Outcome is a statement of an intended result of learning and teaching. It describes the knowledge, skills and values that learners should acquire by the end of the Further Education and Training band. (DoE, 2003a, p. 7)
Importantly, each Learning Outcome is an overarching category that comprises not only content but also contexts and competencies that are to be developed in each grade.
Relationships, Space, Shape and Measurement, and Data Handling. The more recent Curriculum and Assessment Policy Statement40 (DBE, 2011a) for the subject makes use of the five topics of Numbers and calculations with numbers, Patterns, relationships and representations, Measurement, Maps, plans and other representations of the physical world, Data Handling, and Probability.
Irrespective of the titles used for the different content strands, the following commonalities emerge:
emphasis on number concepts, including number formats (e.g. percentages ratios, decimals, fractions), and rules and techniques for calculations involving numbers;
emphasis on relationships between quantities and representations of those relationships (e.g. in tables, graphs and equations);
emphasis on how different quantities change in relation to each other and ways of measuring and representing that change;
emphasis on concepts related to measurement, including physical measurement and calculations involving measured values (e.g. area calculations);
emphasis on 2-D and 3-D space, including visualisation of 2-and 3-D shapes, calculations for such shapes (e.g. volume calculations), and visualisation for other 2- and 3-D object like maps and plans;
emphasis on working with statistical data and the use of statistical tools (e.g. tables, graphs, measures) to interpret and make sense of such data;
emphasis on the notion of chance (likelihood or probability).
Another observation is necessary, in that irrespective of the title used to describe the particular organisation of the content through which the development of mathematically literate, numerate and/or quantitatively literate behaviour is to be explored, this organisation is distinctly different from the traditional content strands of Algebra, Geometry, Trigonometry, and Calculus found in historical mathematics classroom.
Rather, conceptions of mathematical literacy, numeracy and/or quantitative literacy deliberately organise content through reference to ‘big ideas’ (OECD, 1999) (Gal et al., 2005), ‘overarching ideas’ (OECD, 2003, 2006, 2009), or ‘phenomenological categories’
(de Lange, 2003, 2006) that contain not only reference to mathematical content but also to contexts of application, problem situations, and skills required for solving problems in such contexts and situations. ‘Content’ in mathematical literacy, numeracy and/or quantitative literacy is, thus, not only mathematical content; rather, it implies a whole spectrum of components – including mathematical, contextual, and competencies (OECD, 2009, p. 90) − required for solving problems based in real-life situations.
Both de Lange (2003, p. 78) and the OECD-PISA documents (OECD, 2003, p. 34) offer a motivation for this move:
Mathematical concepts, structures, and ideas have been invented as tools to
organize the phenomena in the natural, social and mental worlds. In the real world, the phenomena that lend themselves to mathematical treatment do not come organized as they are in the school curriculum structures. Rarely do real-life
problems arise in ways and contexts that allow their understanding and solutions to be achieved through an application of knowledge from a single content strand.
40 A detailed discussion of differences and shifts between the NCS and the CAPS structures is provided at a later point in this part of the study (c.f. sub-section 8.3.2 starting on page 122 below).
The use of phenomenological categories “encompassing set[s] of phenomena and concepts that make sense together and may be encountered across a multitude of quite different situations” (de Lange, 2006, p. 21; OECD, 2006, p. 83) is a deliberate attempt to make provision for an approach to solving real-world problems that requires integrated use of a range of mathematical components as well as consideration of other phenomena.
In other words, the use of ‘big ideas’ or overarching categories allows for the content component of an activity to be organised in terms of the phenomena to be described by that content:
PISA therefore identifies mathematical content by listing a small set of overarching ideas that represent broad categories of real-world phenomena through which opportunities to explore and use mathematics arise in our interactions with the world. (OECD, 2009, p. 91)
In summary, the use of phenomenological categories allows for a prioritising of not only mathematical but also contextual and competency related considerations in interactions with real-world phenomenon.
Descriptions of the (iv) scope of mathematical content41 appropriate for mathematical literacy, numeracy and/or quantitative literacy is an area of significant divergence in the literature. Some authors suggest that only basic or elementary mathematics is required for sense-making practices of problems situated in real-world scenarios. For example, Christiansen (2007) suggests that,
Thus, all of these examples illustrate how mathematics beyond simple arithmetic is not really central to performance in everyday situations, because whatever little mathematics is used, it is subordinated to the principles of the activity. (p. 97) Steen (1999, 2003b; 2001) follows suit, emphasising the need for elementary and/or basic mathematical content and skills – such as arithmetic, percentages, ratios, simple algebra, measurement, estimation, logic, data analysis, and geometric reasoning − over abstract concepts. Others who offer similar opinion include the ALL framework (Gal et al., 2005),
41 In identifying differential emphasis on different forms or scope of mathematical content in the
literature, I run the risk of significantly undermining how different authors identify the complex interplay between mathematical content and real-world problem situations. For example, to say that Steen
prioritises a notion of numeracy or quantitative literacy that comprises only basic mathematical content in no way gives credence to the complex interaction between such content and the intricacies of real-world problem situations that Steen conceptualises and verbalises as being part of numerate and/or
quantitatively literate behaviour. As eloquently summarised by L. A. Steen, et al. (2001),
Typical numeracy challenges involve real data and uncertain procedures but require primarily elementary mathematics. In contrast, typical school mathematics problems involve simplified numbers and straightforward procedures but require sophisticated abstract concepts. (p. 6) While not intending to underscore the intricate relationship between content and application, the differential emphasis placed by different authors on the scope of mathematical content required in the development of mathematically literate, numerate and/or quantitatively literate behaviour suggests variation in the way in which these authors view the role and scope of such behaviour. For example, an author who views trigonometry as an essential component of interactions with the real-world clearly has a different ‘mathematical gaze’ than the author who suggests that only basic arithmetic is required. This distinction has implications for the extent to which authors prioritise the development of mathematical knowledge as a component of mathematically literate, numerate and/or quantitatively literate behaviour, as well as the scope of the real-world that they deem appropriate for consideration in the development of such behaviour.
Hughes-Hallett (2003), Richardson and McCallum (2003), and Howe (2003, p. 185) – who suggests as a requirement “comfort with numbers”.
Other authors suggest that application in real-world situations exists on a hierarchy, with some types of application requiring relatively elementary mathematical content and others requiring more complex and abstract content. Neill (2001, p. 11), for example, suggests that “Numeracy has a hierarchy of levels ranging from the understanding and use of a few basic ideas of number through to complex mathematical applications.” De Lange (2003, p. 81) argues similarly for a distinction to be made between basic and advanced levels of mathematical literacy. The basic level is perceived as a requirement for all learners up to a particular age at school and irrespective of their future career ambitions, and the advanced level defined by the requirements for participation in post- school social, economic and workplace community of practices. The OECD-PISA frameworks (OECD, 1999, 2003, 2006, 2009) also adopt a wider view of the scope of mathematical content relevant for the development of mathematically literate behaviour.
This is particularly evident in some of the examples of problem situations or types shown in the frameworks, which require the use of esoteric mathematical concepts such as trigonometry and geometric reasoning. This allowance for a wider scope of mathematical content beyond elementary concepts stems from a key area of focus of the study to measure “ability to pose, formulate, solve, and interpret problems using mathematics within a variety of situations or contexts”, but where ‘contexts’ refers to both purely mathematical situations as well as real-world situations (OECD, 2009, p. 85).
Worth noting is the general lack of clarification provided of terms like ‘basic’,
‘elementary’, ‘advanced’ and ‘abstract’ in descriptions of the scope of mathematical content. This lack of clarity gives rise to a host of questions and uncertainties, such as:
what constitutes ‘basic’ or ‘elementary’ mathematics?; and when does basic or elementary mathematics stop being such and turn instead into ‘advanced’ mathematics?;
what constitutes non-abstract mathematics, and how are abstract and non-abstract mathematical contents to be differentiated? Added to this is the level of generality (or lack of specificity) used to identify the scope of real-world situations to which this mathematics can be applied. General phrases such as ‘the world’, ‘everyday contexts’,
‘the real-world’, ‘the individual’s life’, ‘personal life’, ‘the workplace’, and ‘society’, are commonly branded as possible arenas of application of the mathematical content. Yet, within each of these arenas there is tremendous variation in the possible problem scenarios and in the level of abstraction and complexity of the mathematical content required to model and/or make sense of such situations. The point is simply this: there is no certainty or agreement over what constitutes appropriate and sufficient mathematical content for the development of mathematically literate behaviour, of the limit of such content, or of the scope of suitable real-world contexts.
Instead, different authors promote different priorities and scope of mathematical content depending on the particular conception of legitimate knowledge and associated
behaviour encapsulated and/or promoted in that literature and the particular use-value envisioned for that behaviour in a particular society. Being mathematically literate in South Africa, and the mathematical content and contexts of application appropriate to the development of such behaviour, is very different to being mathematically literate in any other country. It is, perhaps, in the light of considerations such as this that Ewell (2001) makes the following statement: