AREAS OF COMMONALITY AND DIVERGENCE IN THE LITERATURE
6.3 Category 3 - Arena of application
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.
The OECD-PISA frameworks (OECD, 1999, 2003, 2006, 2009) are an obvious exception to this, where both ‘intra-mathematical’ and ‘extra-mathematical’49 tasks are deemed appropriate.
However, the ‘real-world’ is a big place and, so, what scope of this world is envisioned to be appropriate for the development of mathematically literate, numerate and/or quantitatively literate behaviour? Analysis of the sub-categories of Location and Context identified by Neill (2001) provide some insight in this regard and indicate areas of commonality within the literature. In particular, the (i) locations are largely organised into four overarching categories: (1) ‘personal life’ and/or the ‘home’; (2) the ‘workplace’;
(3) the ‘community’; and (4) ‘society’.50 The rationale behind these categories is, seemingly, for an outward expansion of world-view from possibly familiar locations (issues relating to personal life) to likely less familiar locations (national and global issue).
In contrast to the consistent reference to common locations, references to specific (ii) contexts are considerably more varied and it is more difficult to identify categories or trends within references to contexts. This is to be expected, especially if one takes seriously de Lange (2003, pp. 87-88) contention that mathematical literacy means different things to different people in different cultures and, so, needs to be ‘culturally attuned’ to the needs of a specific population. This implies that the specific problems relevant to the location of, for example, the ‘home’ differ and vary from one country, population and/or community to another depending on the needs of that group at a particular point in time. For example, personal finance in Zimbabwe – and the contexts that are deemed suitable and relevant to exploring this issue − may mean something completely different to what would be constituted as relevant personal finance contexts in New Zealand.
A related issue with respect to the types of contexts deemed appropriate is the level of
‘authenticity’ of such contexts. Namely, the extent to which the contexts genuinely and realistically represent the structure of participation that is considered to be legitimate in a particular real-life scenario (as legitimated and endorsed by those who participate in the context on a regular basis). The level of authenticity is also reflected in the accompanying structures of knowledge and forms of communication that are similarly considered legitimate and appropriate for use in that context. And, an equally important issue is the
49 “If a task refers only to mathematical objects, symbols or structures, and makes no reference to matters outside the mathematical world, the context of the task is considered as intra-mathematical, and the task will be classified as belonging to a ‘scientific’ situation type. … More typically, problems encountered in the day-to-day experience of the student are not stated in explicit mathematical terms. They refer to real- world objects. These tasks are called ‘extra-mathematical’, and the student must translate these problem contexts into a mathematical form.” (OECD, 2003, p. 33)
50 Common alternative terms to those listed here include: ‘individual’s life’, ‘private life’, ‘family life’,
‘everyday life’, and ‘day-to-day life’; ‘educational life’ and ‘occupational life’; ‘public life’, ‘citizenship’,
‘national issues’ and ‘global issues’.
Clearly there is the potential for each of these categories to comprise sub-categories. For example, the category of ‘personal life’ could include the sub-categories of ‘school’, ‘personal finance’, ‘personal health’, and so on.
Again, the OECD-PISA frameworks are an exception here, referring to the consistent situation types of personal, educational or occupational, public but also including the additional and lesser emphasised domain of ‘scientific’.
The ALL and PIAAC frameworks (Gal et al., 2005; PIAAC Numeracy Expert Group, 2009) and the work by Ginsburg et al. (2006) also offer a different perspective, with both including the additional location of
‘further learning’. A possible reason for this is that these two studies deal primarily with adult numeracy rather than with school-based numeracy
potential implication for problems that do not exhibit a particular level of authenticity.
The work of du Feu (2001) provides important insight into this issue.
Du Feu (2001, p. 2) argues that mathematical questions involving the use of or reference to contexts can be classified into one of five categories: (i) context-free: this would contain purely esoteric mathematical contents with no reference to a real-world situation;
(ii) real: these are genuine or real contexts with real names, messy numbers and data (e.g. an unmodified cell phone bill); (iii) cleaned: “These are essentially real contexts, but where the mathematical model has been simplified in order to make the question accessible to the user or possible in the time constraints of examinations.” (du Feu, 2001, p. 2) (e.g. a cell phone bill that a teacher has (re)developed and simplified from an original
‘real’ resource); (iv) parable: these are fictitious contexts, using fictitious names and people, and where it is obvious that the situation is not real. Such problems are often written in the form “A person has to …” and the intention of such problems is to teach a particular mathematical concept or to make a specific point (hence the name ‘parable’);
(v) contrived: these are contexts that are constructed to fit a particular mathematical concept – irrespective of how appropriate or reflective this is of how the situation actually works in real-life. These types of contexts offer the pretence of reality (through the inclusion of real names, photographs, and other localising resources) by presenting the contexts (and problem-solving activities related to those contexts) as reflective of the structure of participation that is legitimated and endorsed in the real-world.
Having identified these different types of contexts, du Feu (2001, p. 4) goes on to problematise the usage of both parable and, particularly, contrived contexts: “I do not think that contrived examples have any place in mathematics testing or textbooks”.
Furthermore, learners who employ real-world techniques and considerations in
contrived contexts (by believing that the contexts are realistic) rather than mathematical considerations and calculations are likely to be penalised for their non-mathematical techniques. This sends a strong message that what happens in the mathematics
classroom and in the real-world are completely divorced from each other (du Feu, 2001, p. 3), which, in turn, has implications for the extent to which learners are able to
successfully employ appropriate techniques in solving real real-life problems beyond school:
If the mathematics teaching does not differentiate between real and imaginary, students are likely to suspend belief and not try to use mathematics when they encounter real problems later on. (du Feu, 2001, p. 3)
As regards the usage of the other three types of contexts, du Feu (2001, p. 4) argues that all three are appropriate but that a balanced approach must be employed.
Alongside du Feu’s concerns regarding the use of contrived contexts lies another issue for consideration. Namely, that allowance for a predominance of context-free, cleaned and parable contexts signifies a prioritising of mathematical considerations and knowledge over real-world forms of participation and authentic sense-making of real- world problems. Context-free problems clearly have a mathematical prioritising agenda, as do parable contexts with an explicitly mathematical bias.
As regards cleaned contexts, an important caveat is necessary. As mentioned previously, the level of authenticity of a context refers to the extent to which a context employed within the setting of a learning process or task accurately reflects the reality of how the context is experienced by participants who engage in that context in their daily lives.
However, a call for a high level of authenticity must also be accompanied by acknowledgement and recognition that any context must, inevitably, undergo some degree of recontextualisation when it is incorporated into a learning process or system, together with some form of selection of contextual elements worthy of exploration and other elements that are appropriate to be ignored. It would be naïve of me to suggest that it is possible to study a real-life setting in all aspects and to take into consideration every possible influence and permutation which may determine functioning in that context. Rather, my call for a high degree of authenticity can be rephrased as an appeal to focus attention on how a person might think, act and respond in a particular context and how that context may be experienced by participants in the real-world. Of lesser importance is the extent to which the context illustrates the utility of mathematics or how it is experienced from a reconstructed and mathematically biased perspective. That said, if contexts are cleaned to reduce the complexity of the contextual elements in order to make the mathematical elements more prominent or to ensure that mathematical engagement with the context yields mathematically manageable and sensible results, then for me this signifies a prioritising of the mathematical terrain over the contextual terrain. The result is a consequent reduction in the degree of life-preparedness
facilitated. Importantly, this is in no way intended to imply that context-free, parable or cleaned contexts must be avoided. Rather, I simply want to emphasise the point that if there is an intention to promote a life-preparedness agenda where an interest in
contextual sense-making practices and contextually appropriate forms of knowledge and participation are prioritised over mathematical structures, then the level of authenticity of the contexts referenced is of critical importance. If, however, the dominant priority involves mathematical considerations and the development of mathematical knowledge, then the usage of different types of contexts − other than contrived (for reasons
discussed above) − becomes appropriate. It is also not my intention to deny the utility and suitability of cleaned contexts in the development of behaviour driven by a contextually dominant agenda, but, rather, to caution that such contexts must be used alongside (and not as a replacement of) authentic contexts and problem situations. This ensures that an experience of the ‘messiness’ and complexity of real-world participation is included in the learning process, hereby facilitating access to a heightened degree of life-preparedness.
In light of both du Feu’s concerns and the issue raised above, it becomes important to determine the extent to which ‘authenticity’ is prioritised in the literature, since this provides further insight into whether mathematical or contextual or life-preparedness agendas predominate. As has already been mentioned, the OECD-PISA frameworks (OECD, 1999, 2003, 2006, 2009, 2012b) include the possibility of both intra-and-extra mathematical problem situations, which suggests the possibility of the inclusion of context-free, parable, cleaned and real contexts within the PISA assessment items. These frameworks exhibit an explicitly mathematical agenda, with a primary focus on assessing the extent to which learners are able to use specific mathematical content, knowledge and skills to solve problems that have some connection to reality. As the following quotation
suggests, it is the mathematical component and not the context that is of primary concerns in these assessments:51
The problem derives its quality not primarily from its closeness to the real world but from the fact that it is mathematically interesting and calls on competencies that are related to mathematical literacy. (OECD, 2009, p. 93, my emphasis)
The ALL (and PIAAC) framework by contrasts, places explicit emphasis on the use of tasks with a high degree of realism and highlights this issue as a key area of distinction to the OECD-PISA framework: “PISA puts only partial emphasis on the realism of tasks.”
(Gal et al., 2005, p. 149). A justification for this distinctive emphasis on realism is also provided:
The philosophy behind the design of mathematical assessments for PISA, GED, and similar assessments is based on assumptions about what it means to “know math” or “be able to do math” in a schooling context; hence, the assessment design assumes that it is legitimate to use a certain degree of formalization of math
symbols and to present contrived math tasks. This assumption does not fit the assessment of skills of adults who may have been out of school for many years.52 (Gal et al., 2005, p. 149)
Unfortunately, no explanation of the term ‘realistic’ is provided in the ALL framework and, so, it is not possible to determine whether realistic problems include only ‘real’
contexts or also ‘cleaned’ contexts. However, the examples of ‘realistic’ resources provided in the framework certainly suggest a deliberate emphasis on the prioritising of authentic contexts and the sense-making of legitimate and appropriate forms of participation in those contexts – and, certainly, to a greater extent than is the case in the PISA frameworks. It is worth pointing out that this emphasis on engagement with authentic contexts is accompanied in the PIAAC framework with a cautionary note on two fronts. Firstly, recognition is given of the complexity of engagement with authentic
51 It is, perhaps, worth mentioning that one of the primary areas for consideration in the revision of the 2012 OECD-PISA framework and associated test items is a move towards a reduction in both contextual elements and associated text, and greater explicitness of the mathematical components of each problem- task situation. This call has come in the wake of protests raised by certain participating countries who have questioned whether past PISA task items have provided an adequate indication of the mathematical ability of participating learners due to the inclusion and possible interference of text and contextual elements which have, potentially, served to restrict and inhibit demonstration of mathematical problem- solving abilities. Such protests have argued that past PISA items have relied too heavily on language literacy as a requirement for managing mathematically driven tasks. It is also in line these similar concerns that calls were made for the title ‘Mathematical Literacy’ (as employed in the framework to describe the mathematical domain of the study) to be modified in the 2012 study to a title that no longer contains reference to the word ‘literacy’. This, however, has not come to fruition (c.f. OECD, 2012b). See Stacey (2012) for a discussion of these and other issues pertaining to the 2012 OECD-PISA study. All other comments described here were outlined in Stacey’s presentation at the 2012 ICME conference in Seoul, Korea.
52 This quotation draws attention to an important point, namely that the PISA study is concerned the domain of school based knowledge while the ALL study is concerned with the domain of post-school adult numeracy. So, while the PISA study is driven by a goal for the assessment of a particular form of mathematical ability, the ALLS study is driven by an alternative goal for measuring the extent to which adults exhibit a form of numerate behaviour in their day-to-day lives. It is this difference in goals that prompts different requirements with respect to the situations and contexts that are constituted as valid and/or appropriate for exploration. It is also worth noticing that a conception of the knowledge domain of mathematical literacy that promotes a life-preparedness orientation is aligned closer to the conception of appropriate mathematically literate behaviour and associated knowledge described in adult numeracy frameworks (such as the ALL study) than to that described in the PISA (and other mathematically oriented) frameworks.
contexts in the context of formal large-scale assessment practices, and of the necessary sacrifice of particular elements of the contextual environment to facilitate increased accessibility of the context and prioritisation of key skills or concepts (PIAAC Numeracy Expert Group, 2009, p. 26). Secondly, the cultural basis of the notion of authenticity is highlighted: “The desire to retain authenticity, however, may at times be at odds with the need to establish cultural appropriateness of tasks and stimuli and reduce context effects.”
(PIAAC Numeracy Expert Group, 2009, pp. 30-31).
Moving beyond assessment frameworks, a reading of the other literature reveals far less explicit specificity of the types of contexts and the level of authenticity of contexts through which mathematically literate, numerate and/or quantitatively literate behaviour is perceived to develop. Most authors are simply content to use the terms ‘real’, ‘real- world’, ‘real-life’, ‘everyday’, ‘realistic’, or ‘authentic’, without explanation of the precise meanings of these terms or whether they allow for the inclusion of contrived, parable-like and cleaned contexts, or whether such contexts are restricted purely to real contexts. Pugalee et al. (2002, p. 303), for example, stress that “Authentic tasks are a critical tool in developing the level of mathematical understanding and conceptualizing indicative of mathematical literacy.”, but fail to explain precisely what is meant by the term ‘authentic’ in relation to the level of ‘realness’ of the task situation and whether cleaned, modified or constructed tasks are included in this description. In the majority of the literature on mathematical literacy, numeracy and/or quantitative literacy there is, seemingly, an assumption that the reader understands the scope of reality intended and signified in or through the usage of the words ‘realistic’, ‘authentic’ and ‘real-world’.
However, as Ginsburg et al. (2006, p. 7) problematize, “realistic is not real”, and it is primarily through real contexts – with messy numbers, multiple solutions, complex variables and extraneous factors − that genuine understanding of the structure of knowledge and legitimate participation in a context is achieved. Realistic contexts, on the other hand, are designed to resemble real situations, but the design or modification of the context is suited to the promotion of the learning of a particular (commonly mathematical) concept (Ginsburg et al., 2006, p. 7). It is this light that It is this light that Ginsburg et al.
(2006) suggest that,
The contrast between decontextualized, abstracted mathematics (e.g., “What is 23
× 13”) and highly contextualized mathematics (e.g., “When can you retire and how do you know?”) might be best described as a continuum from abstract to real, with
“realistic” somewhere in the middle. (p. 7)
Wiggins (2003) is the only other author in the literature read who also emphasises the importance of authentic contexts, but, confusingly for this discussion, uses the words
‘authentic’ and ‘realistic’ interchangeably. For Wiggins (2003),
How should we define “realistic”? An assessment task, problem, or project is realistic if it is faithful to how mathematics is actually practiced when real people are challenged by problems involving numeracy. The task(s) must reflect the ways in which a person’s knowledge and abilities are tested in real-world situations. (p.
127)
In other words, if a task or teaching situation involves a realistic context, then adequate engagement with the task or situation must entail engagement with the forms of knowledge, participation, communication and decision-making practices employed by people who operate in those situations in the real-world on a daily basis. This does not mean that alternative forms of participation (including mathematical forms) cannot be