The main argument of this study

In the previous chapter, I discussed salient features of mathematical discourse on function objects in learners’ discourse. The question now is; what are the central arguments suggested by the study? Although the sample for my study was too small to allow for generalisations, findings in this study may serve as a basis for making judgments about the properties of the function discourses of the other 26 learners and so I can develop some hypothesis questions for further study. In what follows, I am going to make a number of observations regarding the general properties of learners’ function discourse

160 The analyses of learners’ discourse shed light relating to learners’ abilities to participate38

in the function discourse. This was evident by the presence of all the features of mathematical discourse. In Chapter Two, I indicated that for Sfard (2008), the four features of mathematical discourse are inextricably intertwined. This intertwinement in my study was evident in the fact that in the discussion of each feature, all the other features were present and were interacting with each other.

This study has shown that the learners’ mathematical discourse were error prone and less polished than those of the school mathematical discourse-indicating lack of objectified talk. Learners avoided speaking about formal definition and kept their discourse at the level of routine procedures. The definitions seemed to play no role in learners’ substantiating narratives. Definitions are endorsed narratives (mathematical facts) that describe the objects. This study has shown that learners kept their discourse at a level of processes. This finding led me to think that instruction should reconsider the place of definitions in the process of learning if learners are to be fluent and competent enough in the discourse of mathematics. What this means in practice however is very complex because we know as teachers if we give the learners the formal definition, that may not work. But the whole issue of how we build the words and word use in instruction is very important.

The use of visual triggers was largely demonstrated, but it was cue based, i.e. interpretation of visual mediators was based on their appearance without paying attention to underlying properties of the visual mediators (graph or equation). The visual triggers are prototypic examples that learners were sensitized to thorough previous learning. The data has shown that use of these resulted in errors. This result highlights importance of emphasising on global approach of functions (Even, 1998). This approach addresses important aspects of function discourse, i.e. classification and interpretation.

In light of the above, the learners’ current discourse can be described as passive driven and routine driven. According to Sfard (2008), in order for learners to speak in an objectified

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In this context, ‘participate’ refers to taking part in a mathematical discourse using features of the mathematics discourse. I am not using Sfard’s description of ‘participating in a discourse’ where she associates participating with a metaphor that views learning “as a process of becoming a member of a certain community” (Sfard,1998), as an increasing ability to participate meaningfully in a particular social context.

161 way, their discourse must stand in two legs, that of processes and that of objects. Since the five learners’ function discourse tend to be mainly processual, the implications for instruction is that these learners need to be ushered towards the objectified version of the discourse.

In the analysis, it was evidenced how visual mediators, especially graphs, function as a way of establishing a common focal point and mediational tool. The study has further shown that graphs were the most preferred visual mediator to mediate communication and seemed to have provided learners with the focal point when talking about function objects. However, in order for learners to appreciate the concept of function they need to engage with different forms of function representations. As I mentioned earlier, for learners to be fluent and competent enough on the discourse, mediational diversity should be an additional catalyst for objectification.

The function discourse of the five learners may be described as a combination of colloquial and mathematical discourse, and so, consistent with Sfard’s (2008) assertions that any discourse has colloquial and literate parts. Learners’ discourse indicated a shift from colloquial discourse to a more mathematical discourse. In those contexts learners shifted their routines, word use and narratives to substantiate their narratives. It has been strongly argued that learners’ discourse (combination of colloquial and mathematical discourse) was not linked to errors.

Language of function terminology has played part in learners’ difficulties with functions. And in the analysis it was evident that some of the learners are using words erroneously, and they seem to know what they are talking about. They use hyperbola when they mean parabola because both sound the same. They use the word intersect to mean intercept. The word gradient is used erroneously to identify the coefficient of squared in the quadratic function (y However, the way these words are used resembles that of school mathematics discourse, with some piece of information missing. Words were linked to classification errors. The results of this study substantiate the importance of getting these words clear and we should make a distinction around them. Teachers need to be cognizant of those language specific features of the discourse that seem to hinder learners’ performance. Thus to support meaningful learning of functions, teachers may wish to deliberately capitalize on the existing interplay between learners’ colloquial talk and literate mathematical discourse on functions.

162 The notion of commognition proved useful in an attempt to understand difficulties experienced by learners with the concept of function. In this study a commognitive analysis revealed that, at least some of the difficulties may stem from lack of function terminology, overreliance on visual cues and poor access to endorsed narratives (formal definition). The results of this study have implications for instruction. Teachers need to play a role in helping learners to change the discourse and to develop learners’ mathematical discourse to the level of the experts’ mathematical discourse and the complexity of which should not be undermined.