THIS STUDY AS THE DEVELOPMENT OF A ‘LANGUAGE OF DESCRIPTION’
2.2 The claims and concepts of the language of description
The discussion above highlighted macro-level descriptions and differences between internal and external languages of description. The discussion below now shifts to a more micro-level analysis by focussing on the specific components of an internal language of description.
As was mentioned above, Jablonka and Bergsten (2010, p. 27) argue that a theory comprises a system of theoretical entities, basic principles and a methodology that links
4 See Part 3 starting on page 132 for a detailed discussion of Dowling’s theory and page 167 for a discussion of the relevance of his work to the structure and contents of the internal language of description developed in this study.
5 See Part 5 starting on page 253 for a discussion of the external language of description for this study.
For a discussion of the primary methodology (of textual analysis) and associated methods (draw from the field of semiotics) employed in this study in the analysis of empirical resources, see Part 6 starting on page 360.
the theoretical entities or principles to a specific empirical activity or field. Palm (2009, p. 6), in drawing on the work of Niss (2007, p. 1308), argues in a similar way that a theory is a system of interrelated concepts and claims. The concepts comprise an organised network, linked through a hierarchy, and commonly positioned in a research framework (which can be theoretical, practical or conceptual). The claims – or theoretical entities (Jablonka & Bergsten, 2010, p. 27) – of a theory refer to a domain or class of domains consisting of objects, processes, situations and phenomena: the claims comprise a
‘theoretical manifesto’ (Dowling, 1994, p. 125) of hypotheses, assumptions or axioms about the domain or class of domains which are taken as fundamental, or statements about the domain which draw from or are based on the fundamental claims. These statements often evolve through application of the theory to a specific empirical space as the theory is modified to ensure an effective and comprehensive reading of the space. Thus, a theory can comprise a set of claims about an object and a framework that facilitates analysis of that object.
2.2.1 Claim
In contrast to Dowling’s argument that mathematical and everyday practices are incommensurate, my language of description presents the claim (i.e. theoretical proposition) in the form of a hypothesis that: mathematics is useful and empowering for making sense of real-world contexts and/or problems encountered in real-world settings.
This claim, however, requires several qualifications.
Firstly, the claim is only valid if the motivation and focus of the problem-solving process involving the use of mathematics in contextual situations is for the sense-making of the contextual situations or problems encountered in those situations and not for the learning of mathematical content (which is the focal point of Dowling’s theorising). As such, where the domain of Dowling’s theory is mathematical knowledge as employed within the site of high school Mathematics, the domain of my language of description is the structure of knowledge associated with a conception of mathematically literate behaviour that is separated from or positioned outside of the domain of scientific mathematics and where engagement with mathematical and contextual structures are (supposedly) equally valued and prioritised. And the above claim for the language of description developed in this study is only valid in this ‘external-to-scientific-mathematics’ domain.6 Importantly, the stated claim is also grounded on the assumption that since a primary intention in the language of description is on the use of mathematics as a tool for making sense of real- world situations and not on the learning of mathematical content, a necessary level of mathematical competency is already in place. As a consequence, a format for the subject- matter domain of Mathematical Literacy that is aligned to the structure of knowledge outlined in the language of description presented in this study7 should not have as a primary or ultimate goal the learning of mathematical content or the development of mathematical competency, although this may occur during the process of preparation for life.
6 In Part 5 in this study I refer to the knowledge domain of mathematical literacy as a ‘blended’ domain – namely, as the blending of knowledge and practices associated with both mathematical and contextual structures. See Chapter 17 and sub-section 17.2.1 on page 272 for a discussion of this point.
7 Or, for that matter, any course that aims to develop mathematically literate behaviour, but where this process takes place outside of the domain of scientific mathematics or separated from the teaching of formal mathematical content.
Secondly, building and extending on from the previous qualification is the recognition and acceptance that although mathematics is useful for making sense of and/or for solving everyday real-world problems, mathematics is not enough. Rather, mathematical solutions and models are limited, and there are commonly a variety of extra-mathematical factors which influence the decisions that people make in solving problems in real-world situations. In other words, mathematics is viewed as just one of many tools that can be employed in developing an understanding of a context and associated problems. As such, effective pedagogic practices associated with the a format of the subject Mathematical Literacy that is aligned to the structure of knowledge outlined in the language of description developed in this study must give recognition, exposure, and credence to informal and non-mathematical techniques, structures and considerations which affect and reflect the reality of participation in real-world problem scenarios.
Thirdly, for mathematics to be useful as a tool for the sense-making of real-world situations and problems encountered in those situations, there is an inherent expectation that the person solving the problems is able to apply the mathematical content in a variety of real-world contexts and for a variety of problems. This includes the use of integrated content and skills, developing appropriate models, and, crucially, the ability to identify which techniques and content are appropriate for use in a particular setting and the limitations of the mathematical solution and/or model. As such, the usefulness of mathematics for making sense of the real-world is reliant on the ability of the practitioner to mathematise and model (albeit, in reference to the second qualification above, with an understanding of the limitations of the mathematical solution and/or model).
The final qualification, which relates to the claim and to the previous three qualifications, involves the issue of a ‘critical gaze’. Namely, the usefulness of mathematical content and techniques for modelling and sense-making of everyday situations cannot end at the level of ‘sense-making’. Rather, sense-making must be accompanied by a critical gaze – a level of ‘reflective knowing’ (Skovsmose, 1992, 1994b) − that offers a critique of existing structures and recognition of alternative approaches to problem-solving scenarios. This critical gaze, taken together with the combination of calculation and sense-making techniques, provides the means for developing a more comprehensive and critical understanding of the structure of participation in contextual environments and of possible alternative forms of participation in those environments.
2.2.2 Concepts
The claim (theoretical proposition) of the internal theoretical language is accompanied by a conceptual framework (c.f. F. Lester, 2005, pp. 458-460)8 that identifies different
8 F. Lester (2005, pp. 458-460) defines a research framework as “a basic structure of the ideas
(i.e. abstractions and relationships) that serve as the basis for a phenomenon that is to be investigated.”, and distinguishes three different frameworks: theoretical, practical and conceptual. The framework that comprises the components of the internal language of description developed in this study falls within the description of a conceptual framework: “A conceptual framework is an argument including different points of view and culminating in a series of reasons for adopting some points – i.e. some ideas or concepts – and not others.” (Eisenhart, 1991, p. 209). Importantly, there are two key elements of
conceptual frameworks which are particularly characteristic of the developed internal language presented in this study. Firstly, a conceptual framework is built on an array of previous and current research from various sources, rather than on a single theory (F. Lester, 2005, p. 460). Secondly, the framework need not be limited to drawing on theories; rather, the local knowledge of practitioners who participate in activities in the terrain under investigation can also be used to inform the questions raised by the researcher and the structure and contents of the framework (Eisenhart, 1991, p. 209).
concepts, the collective and integrative of which are seen to characterise a particular structure of knowledge and participation for the subject Mathematical Literacy that prioritise a life-preparedness orientation. These concepts are borne out of analysis of various perspectives of the components/traits/behaviours/knowledge associated with mathematically literate behaviour. As such, the conceptual framework that characterises the language of description (and the various concepts that make up that conceptual framework) reflects a conglomerate of views and opinions. This conglomerate has have been translated into a language for describing the structure of knowledge needed in the knowledge domain of mathematical literacy (and associated forms of participation in the subject-matter domain of Mathematical Literacy) to facilitate a life-preparedness orientation for the subject.
The conceptual framework envisioned for the theoretical language of description for the structure of knowledge in the knowledge domain of mathematical literacy comprises five interrelated components:
Contextual domain of reconstituted real-world contexts;
Everyday domain of practice;
Mathematical Competency domain of practice;
Modelling domain of practice;
A domainof practice involving Reasoning and Reflection on both contextual and mathematical elements.
Each of these domains of practice is discussed and theorised in detail in Part 4 of the study (c.f. Chapter 14 and section 14.4 on page 194 below).