A similar case of the above mentioned ambivalent use of terminology is evident in B1 and B2, which illustrate Beth's fragility of expression in general. Her interpretation of <v₁>=Æ when v₁=0 may be linguistically interpreted in terms of thinking that

Beth's Inflexibility in Moving from One Dimension to Another. Looking at Beth's development of ideas during the tutorial (from B1 to B4 and then her contribution in finding the span of {v₁, v₂, v₃}) I note an inflexibility in understanding the construction of spans as a way to describe gradually spaces of a higher dimension. For instance in B3 she seems to understand that <v₁> is the 'line' that carries v₁. When v₂ is added in the picture she also knows that <v₂> is the 'line' that carries v₂ but finds it surprising that <v₁, v₂> is something more-than-its-parts, namely something more than just the two lines together. Possibly Abidul's 'and' in A3 alludes to the same conception. Similarly, when constructing <v₁, v₂, v₃>, she fails to see the possibility of v₃Ï <v₁, v₂>, therefore delaying to see the incoming third dimension, something which all the other students in this tutorial noticed immediately (see next paragraph).

I see Beth's cognitive behaviour here as impeding her transition to generality and in connection with the questionable use of the metaphor of the plane that dominates these tutorials, I have some reservations as to whether Beth is constructing the necessary links between the definition of Span given earlier in the tutorial and the example of the line, the plane and the space.

From the Line, to the Plane and the Space: the Convenience of Learning the Recipe Well. As briefly mentioned in the previous paragraph all students when asked to construct <v₁, v₂, v₃> appear ready to make the distinction between

v₃Ï<v₁, v₂> and

v₃Î<v₁, v₂>.

Given that only Camille made this observation in the earlier cases — and she is the only one who mentions a generalisation to k dimensions in the end —, it seems reasonable to assume that the students learned to make the distinction from watching the tutor doing so. There is no evidence as to whether the necessity of the distinction has been understood or whether the students would make the same distinction if they were given k instead of 3 vectors to span. In any case the tutor's progressive build-up of the span of one, two and then three vectors seems to have yielded the desired outcome: visualise <v₁, v₂, v₃> as the three-dimensional space. In a sense, this is likely to be learning by mimicking, by acquiring a habit for reasoning in a not completely rationalised way; in other words this is likely to be a case of acculturation to the habit of generalisation in advanced mathematical thinking.

The Questionable Use of the Metaphor of the Line, the Plane and the Space. In the following, I cite evidence of the students' use of language in these tutorials that illustrates the strong geometric approach that dominates their thinking about <v₁>, <v₁, v₂> and <v₁, v₂, v₃>. I stress that I find rather alarming the possibility that this geometric metaphor is taken as the literal message. Additionally in these tutorials, with the exception of Camille, no reference is made to a transition to the geometrically-not-so-easily-conceivable bigger-than-3 dimension. In the following I have compiled some brief lexical references from the students' expressions with the purpose to convey their strictly geometrical — as opposed to the intended algebraic — frame of mind:

I note that the students' pointing at, bending towards the drawings of the tutor as well as the use of their hands (for instance P3 and B3) also convey an impression of their close adherence to the metaphor. I also note that the tutor intervenes only in C1, possibly disturbed by the direct reference to school geometry, in order to explicitly distinguish between vectors on the plane and vectors as elements of a vector space. Moreover in P1 Patricia appears to attempt a more abstract description of <v₁, v₂> but the tutor pulls her back to a 'pictorial' description. Patricia probably was in less need of the scaffolding metaphor of the plane and the pulling back can be seen as trying to condition Patricia within the cognitive frame of the tutor's agenda.

A Verbal Proof of <v₁, v₂>=<v₁> when v₂Î<v₁>. A4 is Abidul's idea of why <v₁, v₂>=<v₁> when v₂Î<v₁> and it illustrates the commonly encountered phenomenon in these tutorials that the novices have an adequate argument in their minds but have difficulty in expressing this argument in formal mathematical terms. Cathy's explicit doubt whether this way of expressing herself is 'formal enough' (Extract 8.7) is another example of the tension between ordinary and formal expression and its consequences on learning.

The Case of Camille: When Affective Factors Determine Crucial Aspects of the Learner's Cognitive Behaviour. Camille is one of the students that have left the most striking impressions on me with regard to their learning style. In this Extract as well as others (for instance Extract 8.2) Camille appears as the most confident and determined among her peers in this college. She is the most vivid participant and interferes incessantly demanding or giving explanations. She is not afraid of risking interpretations that may prove unacceptable, she tends to think constructively and comparatively and leaps to brave generalisations. She talks explicitly about her difficulties and rarely allows the change of the subject if she does not feel satisfied with her understanding of it. Occasionally her queries are about mathematical terminology in English because she has been educated in French; also her mathematical background is at times different to that of some of the other students. In the following, I present the evidence of Camille's behaviour as outlined in this paragraph.

The Interference of English as a Second Language in Learning. In this discussion Camille's thinking is interrupted four times in order to clarify the use of a number of mathematical terms: scope (slip of the tongue for 'span'), directions (for 'vectors'), plan (with French pronunciation for the 'plane'), taking projections/ dropping perpendiculars. These interferences, though rather superficial in their influence on Camille's flow of thinking, illustrate how semantic issues determine understanding. I suggest there is a parallel to these interferences from another language — French in this case — with the interferences on the students' learning from the semantics of mathematics as, among everything else, a symbolic language. The issue reappears in Chapter 9 and is dealt with there in greater detail.

Conceptions of Subspace and Span: Signs of a Constructive Mind. Camille is the only one among the students in these tutorials to look critically at the definitions of Span given in the beginning of the Episode. Her observations about 0ÎS and SÍ<S> as well as her reasoning for them are evidence of a mind in the process of constructing a meaning for <S> from trying to understand its constituent parts, its elements and its properties. Uniquely in these tutorials Camille is also addressing the issue of the equivalence of the two definitions of Span and in particular that <S> is the smallest subspace of V that contains S.

This comparative and constructive approach is again evident when Camille intervenes in Cleo's question about <S> being a subset of V (I note here that several times during this tutorial Cleo appears to be in difficulty to keep up with Camille's thinking: her participation seems to be suppressed by Camille's impulsive reactions. Also Cleo, when personally addressed by the tutor, does not give any evidence that she either follows completely the arguments exchanged in the discussion or is willing to participate extensively): Camille extends the notion of subset of V as a set that, provided that it has the properties of a vector space, becomes a subspace. Apart from the impressive amount of confidence that Camille's reaction exudes — she actually replaces the tutor in this instance by giving explanations to Cleo — it is also a demonstration of her concept image of a subspace as a set endowed with some essential features which she explains in an instrumental and illustrative way.

From a methodological point of view, Camille's confident exposure of her ideas makes her cognitive processes more transparent. This is beneficial to the purposes of the study because this transparency reduces the amount of the interpreter's explanatory intervention; that is, it reduces the degree of necessary analytical processing in the extraction of explanations. Unfortunately Camille is among a minority of students participating in these tutorials who approached tutorials in this way from the beginning. Fortunately, during the two terms of observation, more students developed similar behaviour.

An Idiosyncratic Suggestion for Adding Two Vectors on the Plane. Asked about how can a vector on the plane be expressed in terms of two vectors that span the plane, Camille suggests (fig.1e) taking the projections on the two axes and 'measuring the distance' (C3-C5). Even when given some time to think, and given Cleo's weakness or indifference in changing Camille's mind, she insists. It is likely that Camille's suggestion originates in the common way of identifying the co-ordinates of a point on the plane (or the space) with regard to a Cartesian, orthogonal system. If this is true, then it alludes to the persistence of the geometric metaphors in the students' minds as well as the dubious influence of cross-references to their different aspects (here the Parallelogram Rule as a way of adding two vectors on the plane interferes with the Cartesian orthogonal addition of vectors). C6 is Camille's explanation for C3-C5 and it is reasonable to regard it as a reinforcement of this 'orthogonal' explanation.

I note that the change in Camille's mind and the elicitation of the Parallelogram Rule is achieved by the tutor with three reversals in the flow of her teaching:

With the first and the third the tutor helps Camille rid of her misleading ideas and with the second to replace one of them with the desirable one.

Remarkably the tutor exhibits a considerable amount of flexibility in her efforts to modify Camille's ideas by adopting the student's perspective and then questioning it from within. As a result, in the end, the responsibility for Camille's changing mind is successfully transferred to Camille herself. As the evidence suggests, in this and other chapters, most strikingly Chapter 9, this tutor usually employs highly leading techniques of very closed questioning. Therefore the flexibility she exhibits here is rather unconventional. In a sense Camille's unconventional 'orthogonal' conception of vector addition seems to have yielded an equally unconventional (out of character) response by the tutor.

I note that like Patricia in P3, Camille looks at the plane and decides about the sign of the vector: it is minus because it is 'on the left'. Again the relative sense in which representations, such as a drawing, are supposed to be taken is not entirely understood by the novices.

Finally I note that Camille in this and other tutorials is explicitly expressing her fear of anything beyond three dimensions. As with her difficulty to imagine sets of functions 'after 3 it gets confusing'. Paradoxically she is the only one among the students in these tutorials who achieves a reasonable generalisation — notably unprompted by the tutor — in C7. From a cognitive point of view this may not be paradoxical at all: Camille is more capable of achieving generalisation because at first she is the one who is conscious of the difficulties involved in generalising. Once she has become suspicious or aware of the cognitive leap involved in generalising, she can attempt taking the leap. Consciousness in these terms then emerges as an enhancing determinant of cognition.

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