QUANTITATIVELY LITERATE BEHAVIOUR 14 5.1 Justifying the need for a framework
5.2 A framework for identifying areas of commonality and divergence in the literature
5.2.2 Considerations of perceived ‘orientation’: dominant agendas and intentions
5.2.2.1 Dominant agendas in mathematical literacy and numeracy 17
Julie (2006, p. 62) argues that the various definitions on mathematical literacy can be seen to be on a continuum, with mathematical literacy for entry into mathematics on one end of the spectrum and mathematical literacy for critical interaction with mathematical structures and installations in society on the other end. Jablonka (2003, p. 76) illuminates
17 In identifying the different perspectives, Jablonka (2003, p. 76) initially refers specifically to
mathematical literacy and numeracy and not to quantitative literacy. Only at a later point in the discussion does she refer to quantitative literacy. At no point does she explicitly distinguish between the three or even acknowledge a deliberate interchangeable usage of the three terms. Despite this inconsistency, my intention is to use these different perspectives to organise and identity common threads in the literature that refers to mathematical literacy, numeracy and quantitative literacy.
‘Intention’ (conceptions)
refers to ‘why’ a form of behaviour is promoted and the external (macro-level) impetus for the development of that behaviour.
‘Agenda’ (perspectives)
refers to ‘what’ constitutes the dominant organising principle internally (micro-level view) in the development of a form of knowledge or behaviour.
the in-between categories by suggesting that the following groupings can be identified from the multitude of different perspectives of numeracy and mathematical literacy in the literature: mathematical literacy or numeracy is seen as:
(i) basic mathematical competence: i.e. “knowledge and understanding of fundamental mathematical notions” (Jablonka, 2003, p. 76);
(ii) the ability to perform mathematical calculations18 in everyday contexts;
(iii) the ability to develop mathematical models of both simple and complex real-world contexts;
(iv) the ability to understand and evaluate existing mathematical knowledge, and models and structures developed by others which promote a particular value system and/or perspective.
I contend that the movement from perspectives (i) to (iv) is characterised by a shift in prioritisation of the mathematical terrain to the terrain of real-world. While the first perspective has as an explicit goal the development of mathematical knowledge and techniques, the fourth perspective is oriented towards sense-making of real-world environments and the forms of knowledge that facilitate legitimate and endorsed participation and communication in those environments. This movement is also characterised by a heightened degree of critical engagement with contextual components and awareness of the role of mathematics in informing and shaping preferenced interpretations of real-world practices. Perspectives (ii) and (iii) reflect moderated versions of perspectives (i) and (iv), and exhibit differential degrees of prioritisation of the mathematical terrain and the real-world terrain respectively.
With this in mind – and reflecting back on my usage of the term ‘agenda’ to refer to the dominant area of prioritising in the development of a particular form of mathematically literate, numerate and/or quantitatively literate knowledge or behaviour − it proves useful to view the different perspectives offered by Jablonka as constituting a spectrum of
18 Jablonka (2003, p. 76) uses the word ‘basic’ in describing the type of mathematical calculations envisioned in this perspective: “basic computational and geometrical skills in everyday contexts”. I have deliberately excluded this word from the description of this perspective, as a reading of the literature quickly highlights the dissention amongst authors of the meaning of ‘basic’ when referring to mathematical content in the context of mathematical literacy, numeracy and quantitative literacy.
Furthermore, in many cases the mathematical content specified is certainly not basic (in terms of how a non-mathematician might conceptualise basic content) and, rather, involves abstract algebraic, geometric and trigonometric concepts. Exclusion of the word ‘basic’, then, facilitates for the inclusion of a broader range of literature in this category.
agendas19. This spectrum describes the differential relationship between a prioritising of exclusively mathematical goals on the one extreme of the spectrum and the evaluation of real-world contexts and problems on the other.
My own reading of the literature has, however, prompted an expansion of the four perspectives identified by Jablonka. This is achieved through the inclusion of sub- categories for some of the perspectives. This expanded spectrum of agendas is illustrated in Figure 6 below.
Figure 6: ‘Spectrum of Agendas’ in the literature on mathematical literacy, numeracy and quantitative literacy
Some clarification is necessary. Firstly, with respect to Agenda 1 − with a primary agenda for the development of mathematical competence, a distinction is made between Literacy in mathematics and Numeracy. Some conceptions of mathematical literacy, numeracy and/or quantitative literacy equate mathematically literate, numerate and/or quantitatively literate behaviour with literacy, competency and efficiency in the understanding of complex and abstract mathematical concepts. In other words, to be mathematically literate is seen to equate to being able to demonstrate understanding of abstract
19 The phrase is borrowed from the work of Venkatakrishnan and Graven (2007) who identify four different agendas in the teaching of the subject Mathematical Literacy in South Africa. These agendas encapsulate differential pedagogic practices in the teaching of the subject, practices which “traverse across the purpose of contexts and degree of integration of contexts within pedagogic situations.”
(Venkatakrishnan & Graven, 2007, p. 77). The agendas are: (i) content driven; (ii) mainly content driven;
(iii) content and context driven; and (iv) context driven. Hechter (2011) has expanded this line of research through utilisation of the spectrum of agendas framework to identify and classify five different question types used in classroom-based Mathematical Literacy assessment tasks. These include: (i) purely mathematical questions; (ii) mathematical questions where the context is in service to the mathematics;
(iii) dialectical questions where both content and context are prioritised; (iv) contextual questions where the mathematics is treated in service to the context; and (v) purely contextual questions. The first four question types correlate roughly to pedagogic practices in each of the four pedagogic agendas, but Hechter claims that the contextual questions are positioned outside of this framework. It is important to recognise that although there are similarities between the perspectives identified by Jablonka, the spectrum of agendas identified by Venkat and Graven, and the classification of questions offered by Hechter, each of these authors is operating at different levels of analysis. Jablonka is referring to broad notions or understandings of conceptions of mathematical literacy, while Venkat and Graven and Hechter are operating at the level of the classroom – the former in terms of general pedagogic practices and the latter in terms of specific textual assessment practices.
Contextual Sense- Making Practices
(i.e. Critically evaluating/engaging real-
world problems and the use of mathematics in
those problems) Agenda 4
Mathematical Competence
Agenda 1
1 [a]
Literacy in Mathematics
1 [b]
Numeracy
Modelling Agenda 3
Contextually Dominant Goals
Mathematics in Context
Agenda 2
2 [a]
Application
2 [b]
Numeracy-in- Context Mathematically Dominant Goals Critical engagement
with the mathematical terrain
Critical engagement with the terrain of the real-world
Spectrum of Agendas
mathematical knowledge. Numeracy20, by contrast, is seen to refer to the ability to manage calculations involving foundational and elementary mathematical concepts and principles, such as the ability to work with ratio and proportion, equations, and percentages.
Agenda 2 encapsulates those perspectives who promote a primary agenda for the utility of mathematics in extra-mathematical contextual settings21. Here a distinction is made between Application and Numeracy-in-Context. Starting with the Numeracy-in-Context dimension, this perspective promotes the development of skills associated with the use of elementary mathematics in solving contextual tasks encountered in (supposedly)22 everyday contexts: for example, the use of ratio and proportion to determine best buy options.
Elaboration of the characteristics of the Application perspective as a sub-category of Agenda 2 requires a brief caveat. In the context of the spectrum of agendas above (as well as throughout the contents of this larger study), ‘application’ refers to the imposition23 of (all forms of) mathematics – the application of mathematics – to contextual structures.
The direction of movement in an application is from mathematics to a context (W. Blum, et al., 2002, pp. 153-154). As suggested by Stillman (2012, p. 2), “With applications the direction (mathematics → reality) is the focus. ‘Where can I use this particular piece of mathematical knowledge?’ The model is already learnt and built.” This conception of application is to be contrasted with the Modelling perspective that comprises Agenda 3.
The current usage of the term ‘modelling’ in this spectrum (as well as in the larger study), denotes the process involved in moving from a particular problem situation based in an authentic real-life context (as opposed to a mathematised situation) to a reconstruction of that context, and where the reconstruction is commonly grounded in mathematical
20 Some authors would disagree ardently (and perhaps violently) with the way in which I have equated Numeracy to refer to competency with basic mathematical principles, and would instead argue that Numeracy involves significantly more than just the ability to understand basic mathematical concepts and to perform simple calculations (see, for example, (Neill, 2001), (Gal & Tout, 2012) and (Hogan &
Thornton, 2012)). Although I acknowledge these differing perspectives, my own position is that the term
‘Numeracy’ provides a suitable descriptor for engagement with elementary mathematical contents, and that other terms such as ‘Application’ and ‘Modelling’ provide alternative descriptors for different forms and levels of engagement with mathematical contents.
21 Extra-mathematical settings refer to contextual settings that exist in a domain outside of mathematics.
So, a context drawn from a real-life context such as shopping would be included here. Extra-mathematical settings are contrasted with intra-mathematical contextual settings, which refer to settings that exist and develop exclusively within the domain of mathematics. A particular type of exclusively mathematical problem involving only mathematical entities and signifiers (e.g. factorisation) would classify as an intra- mathematical problem. I make this distinction to clarify that the Agenda 2 perspective is characterised by attempts to move mathematics beyond or outside of its own domain.
22 I have used the word ‘supposedly’ here to emphasise, as Dowling (1998) consistently points out, that although many contexts drawn from the real-world into the mathematics classroom have a base in reality, they are quickly mathematised and recontextualised according to mathematical principles, knowledge and structures. The consequence is that these contexts no longer adequately reflect how a person might act, think, or communicate in that context in their daily lives. Such contexts are then ‘advertised’ as everyday contexts, but are in fact nothing more that mythologised representations of reality.
23 Note that the word ‘imposition’ as employed here does not denote a negative connotation and is not to be equated with impressions of colonisation or subordination. Rather, the word has been deliberately employed to emphasise the particular directional flow of mathematics being placed in or on something else, as opposed to something else drawing in mathematics.
structures24. In modelling processes, the direction of movement is from reality to mathematics (Stillman, 2012, p. 2). This reconstruction provides an alternative view25 of the situation and of possible alternative forms of legitimate participation and communication in the situation, and can sometimes facilitate a different or broader understanding of the situation. Importantly, this conception of modelling does not include word problems or other problem situations which involve “nothing more than a ‘dressing up’ of a purely mathematical problem in the words of a segment of the real world.” (W.
Blum, et al., 2002, p. 153). In other words, modelling as conceived of in this immediate discussion (and extended study) involves the accessing of mathematical resources in making sense of authentic real-world situations and not for the promotion of mathematical learning through exposure to mathematised situations (W. Blum, et al., 2002, pp. 153- 154).
The distinction made between applications and modelling in terms of the direction of movement between the real and mathematical worlds is important in that it suggests a significant difference in the ultimate goal of the processes involved. In applications, the goal is to impose a specific mathematical concept on a situation − to mathematise26 the situation. In modelling, by contrast, mathematics is simply a tool for providing an alternative view or understanding of a situation: the mathematised view is not the ultimate goal but rather a means to an end, where the end is a broader, alternative or more in-depth understanding of the structure of participation in a particular real-world setting. In the
24 A Modelling agenda also comprises a key dimension of the internal language of description of the structure of knowledge for the knowledge domain of mathematical literacy to be presented at a later point in the study. The specific components or stages envisioned for this Modelling agenda are discussed in detail as part of the discussion of this modelling dimension of the internal language. See Part 4, Chapter 14 and sub-section 14.4.4 (starting on page 220) for this elaborated discussion.
25 To clarify the meaning of ‘alternative view’ in the way it is referenced here, consider the following example. A mathematical model can be constructed to show a comparison of the monthly costs on two different cell phone contract systems. The model, thus, provides a particular perspective on costing or pricing issues that could be used to facilitate a decision-making process on the most appropriate cell phone choice for an individual with particular needs. However, this is only one perspective or view of the situation and does not account for other issues that might affect choice − such as the colour or features of the phone, the particular financial situation or constraints of the person exploring the different contracts, and so on. In other words, although the model provides a view of the situation from a particular
(mathematically based) perspective and, in so doing, provides access to an alternative understanding of the situation, the model is only one of several possible perspectives which may cast a gaze over the situation and which may be drawn on to facilitate appropriate and legitimate forms of participation in the situation.
26 The tern ‘mathematise’ originally stems from the work of Freudenthal (1968) and was employed to describe the activity of (re)organising reality or even mathematical contents (van den Heuvel-Panhuizen, 2003, p. 11). This latter aspect of mathematisation (of mathematical contents) is referred to as ‘vertical mathematisation’ (c.f. (Treffers, 1987) – cited in (van den Heuvel-Panhuizen, 2003); also, see
sub-section 14.4.4.2 on page 223 below for a more detailed discussion of this concept). The current usage of the term mathematisation in much of the literature on mathematical literacy, numeracy and/or
quantitative literacy, is more commonly associated with activities involving the (re)organisation of reality according to mathematical structures and principles than with the (re)organisation of mathematical contents.
modelling process, the mathematical approach can be ignored for a different perspective;
in the application process, an accurate mathematical approach is the ultimate goal.27 &28 On the extreme right of the spectrum of agendas is positioned Agenda 4. This
perspective promotes a dominant agenda for engagement in contextual sense-making practices. Namely, utilisation of a variety of skills, techniques and knowledge forms to make sense of contextual situations and of forms of appropriate and legitimate
participation in those contexts, and also to analyse existing structures and to question the underlying assumptions (both mathematical and other) that influence the nature of participation in these structures. In this agenda, the primary goal in a problem-solving process is the development of a broader and/or more complete understanding of a contextual situation or the successful completion of a real-world task. Mathematics is seen as simply one of many tools and considerations that may be imported into and utilised in the problem-solving process to facilitate understanding of the context or completion of the task. In this agenda, authentic real-world contextual situations – and appropriate and legitimate forms of participation in those situations – function as the organising principle of the learning process.
Further clarification of the distinction between Agenda’s 3 and 4 is necessary. As envisioned here, the process of modelling (Agenda 3) involves the development of mathematically structured or informed models to represent real-world situations and the interpretation of those models to deepen understanding of possible forms of legitimate participation and engagement in the situations. Although there is clearly a motivation for enhanced engagement with real-world situations in the Modelling agenda, there remains an emphasis on the mathematical structure of the model and the specifics of the mathematical knowledge and techniques employed in the construction of the model.
Agenda 4, by contrast, is concerned more with developing a comprehensive understanding of a particular real-world situation through consideration of existing forms of legitimate knowledge and communication that facilitate endorsed participation in the situation, together with possible alternative forms of communication and legitimised participation facilitated through engagement with mathematical structures in the situation.
Agenda 4 is also concerned with critical evaluation of existing models (mathematical and others) that claim to provide an enhanced view the structure of legitimate knowledge and participation in a situation and with evaluation of the values and perspectives embodied
27 Note that the distinction that I am making here between modelling and application is specific to the discussion of the spectrum of agendas and also to the language of description of the structure of knowledge for the knowledge domain of mathematical literacy to be presented in Part 4. There is every possibility that a real-world practitioner (e.g. an engineer) who applies scientific mathematical principles to solve a problem (e.g. involving the construction of a bridge) would disagree with this distinction.
28 It is inevitable that some people will disagree with this distinction between applications and modelling with respect to the direction of movement from mathematics to reality. In fact, in many of the texts read, the terms application and modelling are used interchangeably – together with ‘problem-solving’ – to refer to the same process of relating mathematics to a real-world situation. I have simply chosen to make an explicit distinction between these practices to emphasise that there are two possible directions of movement and that I am most interested in the one from reality to mathematics – and I am preferencing the word ‘modelling’ to represent this direction of movement.
In reference to ‘problem-solving’ as distinct from application or modelling, I take problem-solving to refer to the solving of problems within a particular domain of knowledge or practice – while both application and modelling involve a movement outside of a domain to another domain. In the realm of esoteric mathematics instruction, problem-solving would refer to the solving of mathematically based problems through the utilisation of appropriate mathematical knowledge and techniques. In the realm of a contextual domain, problem-solving would refer to the solving of a contextual problem through the utilisation of appropriate contextual knowledge and resources (of which mathematical knowledge may be one such resource). In this sense, problem-solving is an integral part of each of the agendas for
mathematical literacy, numeracy and/or quantitative literacy identified on the spectrum.
in the models which preference particular forms of participation. In Agenda 4 questions are raised as to why a particular form of knowledge (mathematical or other) has been used to describe or make sense of a situation, why specific variables in the situation have been included and others excluded, and the implications of these selection or exclusion strategies on the view afforded by the model. In Agenda 4, mathematical decision-making and solution strategies are questioned, alongside acknowledgement of the restricted view that such strategies afford of real-world situations. This does not suggest that critical evaluation of the suitability and viability of constructed models does not form part of the modelling process. Rather, that Agenda 4 prioritises the opportunity for a critical
‘outsiders’ view of a situation, removed from ambitions for representing the world mathematically, and directed towards deepening an understanding of a real-world situation through a deliberate questioning of the models and structures that claim to represent and describe legitimate forms of participation in the situation.
Crucially, however, in as much as the contextual terrain dominates in Agenda 4, contextual sense-making practices are facilitated in part through engagement with mathematical techniques, knowledge and forms of working, and also with modelling processes. In other words, investigation of possible alternative (and mathematised) forms of participation in a contextual environment is only possible if a degree of mathematical understanding is already in place, and mathematised descriptions of segments of real- world practice can only be considered if modelling processes are available. As such, the agenda of Contextual Sense-Making Practices is supported by elements of the Numeracy- in-Context (Agenda 2 [b]) and Modelling (Agenda 3) agendas. Crucially, however, these two latter agendas are subordinated and in service to the dominant agenda for contextual sense-making practices: in short, any mathematics employed is in service to a broader goal for understanding the context and possible forms of legitimate and endorsed participation in the context.
Notice that Agendas 1 and 2 have been grouped under the banner of ‘Mathematically Dominant Goals’ and Agendas 3 and 4 under ‘Contextually Dominant Goals’. These distinctive groupings have been included to emphasise the overarching goals prioritised in the learning processes, the organising principles that dominate and dictate the structure of knowledge and participation, and the type of behaviour that is expected will develop – all in relation to practices aligned to each agenda. As such, a learning process dominated by Agendas 1 or 2 will prioritise the development of mathematical knowledge or the utilisation of mathematical skills as a primary outcome. By contrast, a learning process dominated by Agendas 3 or 4 will prioritise enhanced functioning in real-world settings as a primary goal.
As a final comment, the agendas are not mutually exclusive in the sense that it is highly likely that an individual may operate in more than one agenda in a single instance of practice depending on their needs or objectives. To illustrate, if we shift the agendas to the level of classroom practice, a teacher may position themselves in each of the
agendas during different phases of a lesson depending on what it is they hope to achieve during the lesson. So, they may begin by teaching an un-contextualised mathematical concept. Thereafter they may move towards a contextualised application of the concept through exposure to real-world components. The concept may then be combined with other concepts in the construction of a model to highlight particular elements of the situation and/or to investigate a particular form of participation in the situation. Finally, a critical discussion may ensue regarding the validity of the model for describing the situation. By operating in this way the teacher will have traversed the entire spectrum.
However, there remains a sense in which a particular agenda dominates with respect to