Section (iv) A Novice's Struggle for a Meaningful Interpretation of the Definitions of Centraliser and Conjugacy Class: Request For Examples and For a Teleological Rationale Behind the Definitions

An Interpretive Account: The Analysis

The Episode is triggered by C1, Connie's frustrated declaration of non-understanding. I note the emphasis she has put on 'really' as well as her phrasing: 'what they are': this is a purely existential statement which deviates considerably from the standard approach, of the lectures for instance, which is to familiarise with a new concept through its definition. The tutor responds with the definition of C(x) and C2 is a comment on the definition. I note that in C2 as well as in C7 Connie's use of the words 'swap back' and 'sends it back' possibly reflect an action-dominated concept image of the group operation. In xy=yx, 'swapping back' is the result of the commutativity of the particular x and y. And in C7 the inverse x^-1 is 'sending x back to itself'. This latter verbalisation can perhaps be associated with the action perceived aspect of the inverse function (if f sends x to f(x), then f^-1 sends f(x) back to x, so f^-1 is a way of coming 'back' to x).

With regard to the definition of centraliser the symmetry of the expression xy=yx seems to engender the false impression that C(x) and C(y) are the same. I think Connie here is confused with the quantifiers behind x and y: in the definition of C(x), x is fixed and y runs through G. Some of the y, the ones that commute with x, are elements on C(x). The tutor (T1) illustrates that C(y) is the set of all the elements that commute with y, whereas C(x) is the set of all the elements of the group that commute with x. In a sense the verbal presentation of the definition of C(x) seems to be less prone to the engendering of the false impression because it puts the emphasis on the fixed nature of x and the variable nature of y.

Giving the definition of a centraliser however does not satisfy Connie who in C3 emphatically requests some examples 'so that I can understand'. It is a rare occasion in these tutorials that the novices turn up with such eloquent and clear cut declarations of what helps them understand. In this case, Connie makes a statement about the value of examples in her constructing a meaning for a new concept. The tutor's example of an Abelian group is slightly disappointing because it does not enrich the concept image of a centraliser as a new entity; it possibly trivialises it. With the second example however Connie, who also corrects the tutor's claim that C((12)) is {e} and suggests (12)ÎC((12)), participates in the construction of C((12)) and requests no further explanations on the concept of centraliser. It may be reasonable to assume that at least temporarily she is content with her newly acquired image of the concept.

C4 and C5 reflect Connie's preoccupation with the role of g in

C_x = { gxg^-1, gÎG}

I note that in both C4 and C5 it is not clear whether Connie is talking about C_g or C_x. The tutor seems to shift from one to the other adapting each time to Connie's words but never pointing at his shift explicitly. His clearest statement is the process-oriented T3 in which he defines C_x by suggesting a way to construct it: run through all gÎG and construct gxg^-1. The set of gxg^-1 is C_x.

C6 is evidence of the impact that the quasi-algorithmic T3 had: Connie realises that the construction of C_x is an x-centred action and also that running through gÎG may generate more than one conjugate for x. I note that Connie's confusion with the concept of conjugate seems to be underlain by the same confusion with regard to the notion of fixed and variable element as in the definition of centraliser. 'how do you choose g', 'is it for every g' in C4 and C5 as well as C6 possibly reflect an interplay in her mind between the fixed and the variable, between the single- and the multi-valued. In C6 she seems to have accepted the idea of the multi-valuedness of the conjugates of x: 'x has several conjugates'.

Symmetrically to the definition of centraliser, the tutor proceeds with examples (one trivial, one not) — this time without Connie asking for them. Unfortunately using examples here is not as successful as with the notion of centraliser, because Connie finds the calculations for C(12) too complicated. Still concentrating on trying to understand what a conjugate is, she abandons the example and returns to a more theoretical exploration of the concept (C7 and C8). C7 reveals that underlying her questioning about conjugates is her confusing them with inverses. To her the raison-d'-être of an inverse is that it 'sends x back to itself' (see comments above). However despite the presence of an inverse in the definition of conjugate she does not see what a conjugate actually does, what it is for and in what sense it differs from an inverse. In C8 she explicitly demands a justification for the introduction of conjugates. C8 encapsulates, like C3, another learning problem that the novices' seem to be preoccupied with in various mathematical contexts: the absence of a teleological rationale in the introduction of most new concepts.

The tutor's reaction (he notes that she is 'leaping ahead') to her request for justifying the purpose of introducing conjugates is typical of the dominant approach in university mathematics according to which concepts are introduced arbitrarily and gradually begin to make sense as organic parts of the mathematical discourse. Here, for instance, conjugates will begin to make sense when normal subgroups are introduced. For the tutor, Connie is 'leaping ahead' because defining a concept logically precedes justifying it. Psychologically however things are probably different: a large part of Connie's preoccupation with the new concept seems to be constructing a meaning of conjugate that encompasses a reason for its existence and not necessarily a fully-blown expression of the definition. In this sense, Connie does not 'leap ahead' at all: she is simply struggling to imbue the new concept with a meaningful, action-oriented interpretation and, for her, this meaning-imbuing process is an essential part of her understanding of the concept.