Section (ii) A Novice's Inquiry on the Concept of Equivalence Class and of Coset: Bestowing Meaning Through Ambivalent Uses of Geometrical Metaphors

An Interpretive Account: The Analysis

I note that the boxed parts of the discussion are the three highlights of the analysis.

Literal Interpretation of a Drawing I: Equivalence Classes as Straight Lines. Fig.2a is the tutor's visual representation of f, a mapping of G on itself. In order to avoid a diagram in which G and Imf would be separated (a misleading idea since ImfÍG), the tutor prefers to represent an element a of the domain as a dot and its equivalence class (generally defined as the set of elements in the domain that are mapped on the same value as a) as a line segment. This metaphorical representation however seems to escape Camille who interprets fig.2a literally and wonders (C1) why equivalence classes are straight lines. T1 and fig.2b are attempts to set the record straight and emphasise the metaphorical nature of the representation. Soon, however, Camille seems to repeat analogous interpretations with regard to the notion of coset (C10 onwards).

C1 and C2 reveal Camille's preoccupation with the notion of an equivalence class which extends later, more intensely, to the relevant notion of a coset. C3-C5 however reflect her puzzlement with the main aspect of the isomorphism theorem (there is an isomorphism between the elements of a group and their equivalence classes). I note that, in this Chapter, the students are repeatedly unsettled by the idea of

(for instance Extract 9.6). Here C3 is a sign of this confusion. Camille quotes the lecturer's definition of g (g(e_a)=f(a)) and then claims 'we don't know what g is'. It seems that commonly used phrasing such as 'define a correspondence between the elements of a group and their equivalence classes' is not perceived by the novice as a clear establishment of a function; or in C3 Camille does not see the = sign as a sign of definition but as a sign of equality. Her confusion then is the outcome of knowing what lies on the right hand of the equality and not knowing what lies on its left.

Moreover Camille's confusion can be justified on another basis: the tutor's expression 'a correspondence between the elements of a group and their equivalence classes' is equivalent but not identical to the lecturer's g(e_a)=f(a). What the lecturer seems to have said is the following:

This is more specific than the tutor's expression and there is not much to guarantee that a novice necessarily ought to see that the two expressions in essence coincide. The tutor's subsequent explanations clarify the definition of g, as well as its properties, but it is noticeable that it is due to Camille's persistence that these clarifications are finally being uttered.

Literal Interpretation of a Drawing II: Cosets as Squares. A Multi-Faceted Tentative Construction of a Meaningful Image of the Concept of Coset. The discussion of the correspondence between the elements of a group and their equivalence classes evokes in Camille a query on another correspondence: 'the 1-1 correspondence between the conjugates of x and x'. Remarkably Camille demonstrates precise knowledge of the relevant definitions (centraliser, conjugate) as well as a relation between the two concepts. I note that, unlike Camille, most of the students in these tutorials at this stage were incapable of reproducing definitions of even simpler group-theoretical constructs mentioned in the lectures.

However Camille in her demonstration of knowledge has not used the term coset at all. The term occurs for the first time in the tutor's words and captures Camille's attention. Subsequently and in the rest of the Episode it seems that the notion of coset constitutes a large part of her preoccupation : C6-C12 seem to be persistent, multiple attempts to imbue it with some meaning. C6 comes through as a surprisingly philosophical and abstract question which raises a very fundamental existential issue with regard to the notion of coset: what is surprising about C6 is that it comes in the middle of the tutor's describing a quite sophisticated construction (establishing a correspondence between the cosets of the centraliser and the conjugates of an element x in a group) and shifts the conversation from the strictly and specifically mathematical (represented by the tutor) to the metamathematical. Camille has been attentively listening to the tutor's demonstration of the construction and has given the very strong impression that, throughout, she has been processing the dense information provided by the tutor. C6 however illustrates that this processing must have been motivated mostly by the desire to construct a representation of coset — visual, 'material' — than consume the tutor's argument. From then on, as said earlier, C6-C12 is a series of successive attempts at interpreting the concept of coset.

C6 is a nearly platonistic enquiry on the nature of cosets as objects, as entities. Camille's entities in C6 do not necessarily act or interact. In C7 the questioning of the nature of these objects takes the form of an exploration of their raison-d'-être (very similar to Connie's enquiry on conjugates in Extract 9.4). C8 is a dissection of a coset which equates a coset with how it comes into existence. I note that so far T4-T6 do not seem to have a direct impact on the genesis of Camille's ideas of what a coset is. C9 is a geometrical interpretation of C8 derived from the notion (and notation for) transformations, and in particular translations. The tutor carefully tunes in (T7) but Camille accelerates her tentative condensation of her conception of coset in a geometrical image in a questionable way: C10 (in parallel with C1) illustrate how the line between a metaphorical and a literal interpretation of a picture is thin and severely disguised under the heavy weight impact of visual imagery. The tutor is surprised and alarmed (T8) by Camille's intention to 'apply [this idea] on squares'. C11 is evidence that Camille is too preoccupied with her image construction to be influenced by T8 and she furthers the interpretation of her fig.2c in a less controversial but highly ambivalent way. T9 is one more effort on the tutor's side to tune in and transform the student's images from within. Surprisingly then Camille turns in a shift to a more abstract property of cosets in which however the geometrical jargon ('size' in C12) is maintained. The tutor (T10) has completely adopted Camille's metaphor and contributes another observation on cosets.

Finally Camille ceases the effort to interpret further the notion of coset once she acquires an image of cosets that is satisfying and clear to her. That Camille is content with what she has acquired can be assumed on the basis of the evidence, given during observation, that this student does not bring a conversation to an end until she acquires a satisfactory (to her) understanding. The issue that C6-C12 raise is whether the quality of the acquired perception of a coset — via a multiplicity of metaphors and visual representations — justifies Camille's eventual sense of content. Given that the tutor cautiously surrenders in adopting Camille's metaphor but does not cross-check whether the intended (by the tutor) and the acquired (by Camille) image of a coset coincide, the questions raised by this issue ought to remain open.