Section (iii) Spanning Sets and the Struggle for a Meaningful Metaphor

An Interpretive Account: The Analysis

In the following, I comment upon Connie's efforts to construct a meaning for spanning sets. The tutor presents definitions and insists on drawing examples from the geometrically familiar case of the plane. He also insists on drawing Connie's attention to some formal properties of spanning sets and bases, such as their non-uniqueness. Dwelling on examples from the plane and insisting on exposition regarding formal properties at times distracts from investigating some of Connie's perceptions as shown below in more detail.

Analytical Versus Geometrical Approach: how resisting alternative approaches impedes understanding. As mentioned in the Context, from the beginning of the session, and in consistence with his recommendations in the morning session, the tutor constantly encourages Connie to think of spanning sets and bases in terms of the two-dimensional Cartesian plane; also to adopt a geometrical perspective. Connie however repeatedly resists (C1, C3). With the tutor's second geometric prompt (T3) she seems to begin seeing the span of (1,0) and (1,1) as the whole plane (C4).

The tutor's subsequent theoretical exposition on the notion of basis does not seem to impress Connie. A few minutes into his exposition Connie interrupts in order to draw his attention back to her specific queries on LA5. Underlying her difficulty with the conditions under which <<S>>=<S> (I note also that what does not appear in the transcript above is Connie's slip-of-the-tongue in which she misinterprets ≤ as 'less than' instead of 'a subspace of') is her discomfort with the notion of spanning. So, despite the fact that it was she who insisted on talking specifically about LA5.24, it is she who interrupts the tutor's exposition on the question in order to discuss the notion of a spanning set (C5). Connie here appears to attempt to construct a parallel between spanning sets and mathematical knowledge she is already familiar with, such as the image of a function ('all about the values that thing can take'). The tutor however does not pursue an inquiry into this construct. On the contrary he prefers picking on Connie's phrasing (THE spanning set) and recommends caution: a spanning set of a set is not unique. His comment, even though mathematically valid, seems to be, didactically, a distraction from illuminating further Connie's attempt to construct a meaning for a spanning set.

However, Connie insists on her image-making (C6). She is struggling for a meaningful metaphor, for an embedding of the concept of spanning set in a familiar and operational context. When the tutor defines <S> as the set of all linear combinations, she wonders 'Why are they useful?' (C7). The tutor then presents her with an explanation according to which spanning sets themselves are not interesting, but bases are. Then again Connie interrupts the tutor's theoretical exposition in order to draw him back to the specificities of her query on how LA5.24 and LA5.25 connect. As said earlier, Connie does not seem to be interested in the tutor's theoretical exposition: instead of another lecture, she seems to need to concentrate on the status of these new concepts. Her not very eloquent expression in C8 hints at her beginning to perceive a basis as a spanning set of a minimum size ('the smallest'). The tutor again tries to shift her attention on non-uniqueness and again she seems to ignore his remark ('THE basis' in C9). Her basic concern is still about the concept of a spanning set and a basis (C10) and the tutor then once again militates for a geometrical approach (plane, space). The inappropriateness of doing so is demonstrated in her extension of

So Connie here seems to think that every co-ordinate system is a basis. To the latter he reacts alarmed but he does not explain why this parallel does not work (T4), leaving Connie probably in doubt about her idea that every co-ordinate system is a basis, but not clear about why this is not so. Finally C12 and C13 are rather fuzzy verbalisations of her final interpretations of what a basis is. The tutor accepts them — I think they do not contain anything immediately wrong but are indeed very ambiguous in their generality.

I note here that, at one point, the tutor suggests banning the term spanning set altogether and replace the relevant phrasing with 'a subspace is spanned by a set S'. This alludes to a more generally observed tendency of the novices to disregard the notion of a spanning set as redundant. I recall here that, in Greek for instance, there is no word for spanning set, only for span. Details will follow in the analysis of more material in this chapter.