The previous chapter provided a multifaceted review of the consulted and reviewed literature in the sphere of learners’ mathematical discourses and functions. The researcher concurs with the assertion by Walliman (2011) in highlighting the role and value of the literature review process in the development of a theoretical or conceptual framework. The latter author asserts that: “ The [literature] review can be used to show where you have gained inspiration to develop your ideas – and that does not just have to be only from academic sources. It should also demonstrate that you have a good understanding of the current conceptual frameworks in your subject, and that you can take a stance in placing your work within these” (Walliman, 2011, p. 57). The current chapter then centralises the inter-relatedness of mathematical discourse, mathematical functions and the learning of functions within the conceptual parameters or frameworks of commognition and social learning as the foundational guiding theories in the study in conjunction with the relevant philosophical assumptions (Knobloch, 2010).
Research studies are mainly based on some specific paradigms, frameworks or perspectives which necessarily establish the ‘boundaries’ for scientific investigation (Ramenyi & Bannister, 2013). Theoretical frameworks themselves are abstract in their nature, but systematically present generalisations which explain the association between and among a phenomenon’s variables. In this study, mathematical discourse, mathematical functions and the learning of functions are posited as inter-related variables of commognition and social learning as specific theoretical frameworks. The latter coheres with the perspectives of authors such as Knobloch (2010), Ramenyi and Bannister (2013) and Saunders, Lewis, and Thornhill (2012); who emphasise that a theory is fundamentally a systematically and symbolically organised representation of perspectives pertaining to the reality of ideas, concepts or phenomena that are of interest to the researcher. For the purpose of this study, the theoretical framework or perspectives (paradigms) provided a context within which the key principles are defined and related to the practical domain of the study, in terms of which the research problem could also be conceptually relevant to the study’s adopted philosophical assumptions (Kumar, 2012). Philosophical assumptions are not peripheral to the researcher’s own system of values, and mainly refer to the basic principles or paradigms (philosophically rooted points of view) which are subject to application with no need for proof or verification (Knobloch, 2010). In this regard, the assumptions guide the particular philosophical approach or “stance” adopted by the study in the observation of phenomena or critical research variables that are the subject of both observation and investigation (Marshall & Rossman, 2011). Additionally, assumptions (basic philosophical/ intellectual abstract ideas or concepts) could be ontological (assumptions based
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on the nature (“being”) of reality); epistemological (nature and construction of knowledge and its reality); methodological (the means by which knowledge of reality is acquired); axiological (extent to which detailed attention has been allocated to the various components and aspects of the whole study); and rhetorical (the researcher’s art or capacity of written and oral persuasiveness). In this study, a hybrid approach was adopted in terms of the degree of applicability of different assumptions to different aspects of the research process.
3.2 Definition of Key Concepts
The definition of the key concepts is closely related with the conceptual and theoretical context and parameters of this study, since it is on the basis of such definitions that clarity and logic are accorded to the very thematically concepts which are necessarily embedded in the learners’ mathematical discourse and functions, as well as their social learning (Narasimhan, 2009). As a result of the thematic coherence and logic-seeking orientation of the definitions of the below- cited key concepts, their alphabetic sequence does not necessarily signify any order of prioritisation or appearance in the study. The key concepts identified in this study are: asymptote, the Discourse Profile of the Hyperbola and Exponential Function (DPHEF), functions discourse, hyperbola, and mathematical discourse.
3.2.1 Asymptote
The term ‘asymptote’ derives from the Greek word literally meaning, “Not falling together” (Kupstov, 2001). Mathematically, it means that there were no points of intersection in a graph (Mpofu & Pournara, 2018). However, it is now common knowledge that the horizontal asymptotes sometimes intersect with the curve. Information about a curve is conveyed in an asymptote. In sketching functions, one of the important steps is to determine the asymptote. The mathematical definition of the asymptote, used by the community of mathematicians, differs from the school definition. The school definition is context-specific, as opposed to the purely mathematical definition. Throughout this study, the researcher has attempted to discuss the difference between the definitions of the asymptotes, and then showing the difference between the asymptote and the removable discontinuity.
An asymptote of a curve is a line constructed such that the distance between the curve and the line approaches zero as one or both x and y tends to infinity (Kidron, 2011). There are three types of asymptotes, the vertical, horizontal and oblique asymptotes. The horizontal asymptote may or may not intersect with the curve. The vertical asymptote does not intersect with the graph. An oblique asymptote is in the form of y = mx + c. Participants of this study only did functions with only the horizontal and vertical asymptotes that either coincide or are parallel with the axes. Often participants’ definition of an asymptote was closely related to what they (participants) had seen in their learning.
39 3.2.2 Commognition
Communication is a mechanism for the facilitation and execution of mathematical discourse by combining and linking communication with cognition (Sfard (2008). In commognition, learners participate in a mathematical discourse by both talking and thinking. “Thinking is regarded as communication with oneself” (Sfard, 2012). Thinking is expressed in verbal or written form. In this regard, participation in mathematical discourse is seen in action, and development is seen in change of discourse (Sfard, 2012). The more fundamental aim of discussion is to enable learners to think and talk like mathematicians. In the event that the conversation in the discourse remains the same, then development has not taken place. Since learning is a sub-set of development, the focus is not on the change in the learner, but the change in the discourse. Development in the discourse is manifested by the learners’ use of new rules in the mathematical discourse (Sfard, 2012). When learners use new rules, their communication changes. Therefore, learners would have developed in their commognition and mathematical discourse in the event that they are able to make a difference between forms of translation and show this change in their communication (Vyncke, 2012).
3.2.3 DPHEF (Discourse Profile of the Hyperbola and Exponential Function)
The DPHEF is an analytical tool used to analyse the mathematical discourse of the learners who participated in this study. The DPHEF was developed by the researcher from the original Discourse Arithmetic profile used by Ben-Yahuda et al (2005). The DPHEF was designed to identify and distinguish between the superficial and dispositional differences in learners’ mathematical discourse. As an analytical tool, the DPHEF can be used for any mathematical function, and is not restricted to the hyperbola and the exponential function only. From the perspective (rather than assumption) of the researcher, the DPHEF it is envisaged that the DPHEF will contribute towards learners’ knowledge, perceptions, and experiences in the learning of asymptotes of the hyperbola and exponential functions.
The DPHEF has two categories of word use, the mathematical and the colloquial. The mathematical use of words is defined as a form of communication about which all members of the community of mathematicians have the same interpretation (Tachie & Chireshe, 2013). An immediate example is: An increasing exponential function has a base that is more than one. The colloquial use of words relates to a mixture of mathematical and non-mathematical words. An example could be: An increasing exponential function is the one that goes up. The use of words is ambiguous and is prone to different interpretation by different communities of people. 3.2.4 Functions discourse
The functions discourse is grounded on the realisation and conceptualisation of Algebra as a branch of Mathematics (Caspi & Sfard, 2012; Nachlieli & Tabach, 2012). The words and symbols used for functions distinguish the functions discourse from other discourses. A
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function can be represented in four various ways in commognitive terms. A function can be represented verbally by using words to describe a phenomenon; algebraically by using symbols to communicate; numerically by using ordered pairs mostly expressed in tables; and graphically by either sketching or drawing a graph. In this study, the participants mostly opted for graphical and algebraic representations of the functions. Figure 3.1 below illustrates the various developmental stages of a functions discourse, as well as the developmental linkage to routines.
Figure 3.1 Development of the functions discourse
Source: Researcher’s own initiative derived from the review of literature
Figure 3.1 above illustrates the centrality of words pertaining to the development of the functions discourse. These words include asymptote, axes, intercept, to name just a few. These words usually characterise visual mediators in the form of graphs, tables and formulae. Having had exposure to, and knowledge of these words, learners are then able to generalise based on their interpretation of the words and their attendant visual mediators. All three tenets of the functions discourse (word use, visual mediators and endorsed narratives) collectively determine whether learners function at the level of ritualised or exploratory routines. Generally, all learners begin from ritualised routines, and some proceed to exploratory routines while others take some time to move out of the ritualised routines. Competence in the hyperbola functions discourse comes after learners have had competence in each of the sub-discourses on inverse proportion, graphs, symbolic expressions and equations, and tables of values.