Section (iv) Striving for Meaning and Significance: The New Concept of Fourier Series

An Interpretive Account: The Analysis

Striving for Meaning and Significance. Camille's determination to explore the meaning of Fourier Series of a function is clearly stated in the opening statement of the Episode, C1. Subsequently the student seems to have chosen the strategy of clarifying excruciatingly every relevant term/concept/definition. This results in her frequent interruptions of the tutor who is fairly annoyed and sounds as if she is always on the verge of losing her patience. I note here that this is something not fully conveyed by the dry citation of facts in the transcript. Also I note that the tutor's irritation can be attributed to the fact that Camille has asked her about a topic that she is not entirely and readily familiar with. The reason I begin the analysis of Camille's behaviour with these comments on the tutor's behaviour is that the latter directly influences the former. Moreover Camille's reactions are hard to dissect in terms of the cognitive/affective dichotomy so at least in this Episode there is a combined consideration of both emotional and cognitive issues.

For instance, Camille's three interventions, concerning

can be seen at the same time as pedantic reactions AND signs of insecurity. Then comes the discussion of ~ as a sign of ambiguous meaning. Justifiably Camille, as well as other novices in various occasions, confuse the several uses of ~ that vary from course to course. Camille mentions approximation, equivalence/similarity and equality (C2-C4). All are dismissed by the tutor who prefers ~ to mean 'this is the Fourier series of this function' because this allows writing the Fourier Series of a function f down without worrying about whether it converges to f or not.

Camille again expresses mistrust and turns to Frances — her silent peer who is sitting at the same tutorial with Camille — for support with C5. Frances listlessly admits to no confusion. Camille's perplexity becomes more evident in her subsequent 'but it doesn't really converge' which the tutor does not explore. With her wondering about what is a Fourier Series still pending, Camille turns to the exercise from the lectures (fig.4 ). The tutor explains the calculations that have been omitted by the lecturer and Camille's frustration now turns overtly against the lecturer towards whom she is mistrustful. She seems nevertheless to have grasped the notion of possible odd and even extensions of a function. Her next query is again a sign of mistrust and disbelief towards the lecturer: if odd and even extensions are two possibilities why did the lecturer mention that the possibilities are three? A brief examination of the lecture notes leads to the conclusion that the lecturer meant the ordinary Fourier Series as the third possibility. Satisfied for a moment Camille turns to her initial epistemological question on the Fourier Series of a function with C6.

The tutor's subsequent explanations in T5 are characterised by an evasive vagueness culminating in the final statement '...we are sort of skipping things here a bit because you are doing this clearly from the point of view of using it, so you really need to know the minimum to actually do things. Because the actual theory of it gets quite complicated'. The latter illustrates the conflict encountered earlier in the first term of the novices' studies. While in the course on Continuity and Differentiability there is a clear intention to build Analysis carefully and precisely on sound axiomatics, the Analytical and Numerical Methods course is an exercise in the mechanics of differentiation and an application of rules for solving differential equations. The tutor's statement illustrates the double standards that occasionally apply in these introductory courses. In other words, beneath the tutor's words lies an attempt to discredit Camille's inquisitive epistemological approach (highly desired in more rigorous courses like Continuity and Differentiability) as inappropriate for this particular course. This application of double standards becomes problematic since the dichotomy is poorly perceived by the students who justifiably cannot distinguish confidently among the variations of rigour expected from them at the various stages of the course.

In sum, in this Episode a novice's struggle for learning has been seen in the light of both contextual and emotional factors. In particular in this Episode the mathematical context of Analytical and Numerical Methods seems to determine a variety of aspects of Camille's learning, starting from the interpretation of a symbol (~) and up to the degree of detail and rigour required in the definition of new concepts. Camille has not been told about the particular meaning of ~ in the definition of a Fourier Series and she does not realise that all that is expected of her, at this stage, is to learn about the new concept 'from the point of view of using it'. So she engages in an excruciating decomposition of the concept which turns out to be emotionally frustrating since she doesn't share the same concern with her peer in the tutorial or her tutor. Her increasingly persistent inquiry is a result of this frustration and this is why I think that, in this Episode, the cognitive and affective are intertwined and therefore harder to distinguish and dissect.