Section (vii) A Frustrating Vicious Circle of a Novice's Struggle to Construct a Meaning of a Coset

An Interpretive Account: The Analysis

First I note that the structure of this session is circular and repetitive. I also note that the tutor is patient and compassionate of Connie's repeated folding back to the same questions. In fact he could probably be deemed partly responsible for the repetitive circularity of the session: when there is evidence that Connie does not satisfactorily understand his argument for GRF7.3ii, he merely repeats the argument — almost invariably. It takes two of Connie's requests for clarity (C1 and her disappointed sighing shortly after that) to realise that he has to 'go into it in detail'; namely to re-address his arguments by referring back to basic definitions and concepts such as the coset. Throughout the session it is not clear that he realises the degree to which Connie's difficulty with the argument in GRF7.3ii can be attributed to her unrequited struggle for the construction of a meaning for coset. My main aim here is to give an account of this struggle.

Connie's Struggle for a Meaningful Conceptualisation of Cosets. This is a session on Connie's very problematic confrontation of the new concept of cosets. The evidence of her difficulty is ample but in a way it is also not very revealing: hints at potential sources of Connie's difficulty are not really suggested in Extract 9.7 but I think, maybe a bit paradoxically, this situation-of-non-disclosure renders this piece of data quite powerful. In its elusiveness this is a piece of data which reflects a very common cognitive situation: the cognising subject feeling unease and requesting enlightenment from the tutor. Unable to articulate the problematic aspects of her cognition of a new concept, the student's request comes through as vague and imprecise. In turn the tutor, being himself unaware of the sources of the problem, keeps providing feedback that seems to frustrate the student's attempt to make sense of the new concept. The outcome is a cognitive situation trapped in a vicious circle of mutual misunderstanding. The following presentation illustrates this vicious circle.

Before proceeding with the account of Connie's attempts at making sense of cosets, I briefly mention three incidents that are, peripherally, part of the picture of her difficulty:

In the following, I present Connie's meaning-making attempts.

C1 is an imprecise interpretation of the requested proposition in GRF7.3ii. Despite the signs of vagueness and confusion in Connie's words the tutor presents his argument for the proof of GRF7.3ii (from now on referred to as The Argument). Connie is then evidently confused with G/K and G-K and the tutor repeats The Argument. Only when Connie sighs with disappointment he suggests 'getting into more detail' (a sign that the tutor is perhaps more sensitive to affective as opposed to cognitive signs of perplexity).

His back-to-basics trip is short though (it includes a vivid metaphor of cosets as parcels produced from multiplying the elements of a subgroup with elements of the group) and, when he returns to The Argument, Connie (C4 and C5) folds back to redefining the basic concepts involved in The Argument. I note here that, in these two utterances, Connie appears concerned about the idea of K being a subgroup AND a coset. She listens to the tutor's explanation that K is both a subgroup and a coset, but generally cosets are not subgroups because they do not contain the identity element of the group. She then returns to The Argument and disappointingly observes that 'the coset of K in G is K' which is a sign that the tutor's explanations have not been entirely received and also that Connie thinks in terms of one coset only (maybe similarly to her thinking about conjugates in Extract 9.4).

The tutor returns to the presentation of The Argument and talks briefly about a slightly more general case than [G:K]=2, [G:K]=3. The latter, the case for [G:K]=3, seems to attract Connie's attention and a bit later she appears repeatedly asking about this case.

I note that the tutor's drawing (see fig.7) seems to enhance Connie's concept image of cosets (C7) even though her use of the term 'subgroups' instead of 'cosets' may not be an entirely coincidental slip of the tongue. The tutor corrects 'subgroups' to 'cosets' and also notes — with regard to 'half' — that her phrasing applies to 'finite situations'.

He then responds to Connie's queries about the case for [G:K]=3 with an explanation of how the cosets would look like in that case. He also juxtaposes the cases [G:K]=3 and [G:K]=2 by returning to The Argument. Similarly to her question about K and G-K, Connie asks whether the cosets g₁K and g₂K in the [G:K]=3 case are also subgroups (C11: I note her use of the vague word 'things'. She still hasn't sorted out what they are. No progress seems to have been made.)

The tutor's subsequent explanations include a second metaphor for cosets (the first one was cosets-as-parcels): 'translates of subgroup by a group element which is not the identity element'. Given the tutor's picture that lies on the paper in front of her and his latest metaphor, Connie continues her inquiry (C12). Connie is striving for a rediscovery of the concept, in her own terms, in the order of her own thinking. In this quest some elements of ambition can be traced: she has been playing around with numbers 2 and 3 and then she is asking about the case where the 'index' (meaning the order' of the group) is a prime (C14 and C15). So she seems to attempt a generalisation.

C18 is disappointing and ambiguous: she hasn't realised that for every subgroup there are plenty of cosets. So to her, one subgroup means one coset. It is true that if G is the subgroup then there is only one coset and it is G itself but Connie sounds very muddled and it is not very likely that she can distinguish this case. The rather frustrating circularity in her mind becomes evident when she asks again about the case [G:K]=3 (C20). C22 may be evidence that her impasse lies in her weakness to understand why left cosets and right cosets aren't always the same. Maybe she is interpreting sameness as uniting-to-give-the-same-whole, namely G (C23).

C24 and C25 illustrate that Connie is still struggling with understanding the operation according to which a coset gK is constructed (multiply an element gÎG with all the elements in a subgroup K of G; the products of these multiplications are the elements of gK). C26 and C27 then is a return to The Argument. C28 is then a brief, but possibly not very reliable, reassurance that Connie is satisfied with her understanding of the Argument.