Section (vi) Looking for the 'Usual' Basis of P3(Â): Decontextualised Knowledge and the Ambiguous Nature of 1

Context: See Extract 8.6

Structure: In the following, the strictly tutor-led discussion is interspersed with the students' individual or collective responses that reflect a wide variety of their difficulties with the topic as well as evidence of some novice reasoning attitudes.

The Episode: A Factual Account. See Extract 8.6

An Interpretive Account: The Analysis

In the following I focus on:

• Patricia's and Beth's different responses to the almost identical teaching stimuli regarding the discovery of the basis {1, x, x2, x3} for P3(Â),

• the unanimous interpretation of T(1) as 2 instead of 1, and

• the case of two students' imaginative suggestions for extensions of the knowledge discussed in the tutorial.

The Iteration of an Elicitation Process and its Different Outcomes. Both Patricia and Beth have difficulties with reconstructing the 'usual' basis for P3(Â). Responding to the difficulty the tutor prompts them with a recommendation to think again 'what p(x) looks like'. In both tutorials, the request for a general expression of p(x) yields

p(x) = ax3+bx2+cx+d.

Patricia then 'sees the point' and notices (P2) that p(x) is a linear combination of x3, x2, x and d ('the constant'). With further prompting she sees d as d-times-one. I note here that the students' difficulty in 'seeing' 1 as x0 results in the phenomenon elaborated in the next part of this section (on T(1)).

Beth however seems to be in more difficulty than Patricia; it takes her longer to find the 'usual' basis. Her case seems to be more complicated anyway: at least Patricia, in her first attempt (P1) to find a basis for P3(Â), suggested polynomials. Beth starts from 'matrices' as the tutor calls (1, 0, 0, 0) etc.. It is likely that Beth was misled by the tutor's persistent use of the term 'usual basis' (not specifying for which vector space) and suggested the 'usual basis' for Â4. However she suggests

p(x) = ax3+bx2+cx+d.

as the general form of a polynomial in Â4. Subsequently B1 is muttered and it is likely that Beth was trying to utter 'xs, where s=1, 2, 3'. The tutor rather impatiently interrupts her in order to remind her of what they are looking for: 'simple polynomials that p(x) is made out of'. B2 and B3 illustrate Beth's interpretation of the tutor's 'simplicity': she suggests linear polynomials, or in other words polynomials of degree one. Beth actually refers to the decomposition (factorisation) of a polynomial of the third degree to the product of three linear ones. In that sense B2 and B3 reflect how decontextualised Beth's thinking is and how the tutor's responses to her difficulty have failed to address the essential reasons behind this difficulty: Beth does not seem to have realised so far the rationale behind looking for a basis; or what is a basis and what kind of expression for the elements of a vector space it provides.

The tutor subsequently explains that it is not factorisation they are looking for but the construction of p(x) as a linear combination of four polynomials. The tutor, presumably thinking that Beth may have a problem with the notion of linear combination, asks her what a linear combination is. B4 shows that Beth knows what a linear combination is — actually we already know that she knows from her earlier reply (p(x) = ax3+bx2+cx+d). So Beth's difficulty in seeing the 'usual basis' behind this linear combination cannot be attributed to the absence of prerequisite knowledge: there is evidence that Beth knows what a linear combination is and she also knows that she needs to break p(x) into 'simple' polynomials. The fact that she suggests factorisation may reflect the possibility that she has not solidly consummated the idea that the operation between the polynomials in P3(Â) is addition. So her difficulty to see the basis can be attributed to her lack of the ability to piece together these items of prerequisite knowledge in a meaningful way and in resonance with the needs of the problem she is asked to solve.

Finally Beth does see the correspondence between

ad1+bd2+cd3+dd4

and

ax3+bx2+cx+d

and correctly gives the 'usual basis' {1, x, x2, x3} for P3(Â). This however is not necessarily a fact which guarantees that she finally succeeded in co-ordinating the necessary pieces of knowledge. She yielded the answer intended by the tutor but it is questionable (there is no further evidence in Extract 8.6) whether she resolved the mystery of why her previous suggestion (factorisation) was not accepted.

The unanimous interpretation of T(1) as 2. Most of the students in this tutorial when asked to calculate T(1), where T(p(x))=p(x+1), reply '2'. As explained in B6, the students appear to be seeing T as an action according to which 'we are adding one'. Actually in her elliptic expression (B6) Beth captures the essence of her and her peers' difficulty: what is missing from her sentence is the object on which the action of the verb is suggesting: add one to what? Underlying the students' response is seeing 1 as the number on which to apply the action suggested by 'adding one' and not as

p(x) = 0x3+0x2+0x1+1x0 = 1

which therefore would help them see T(1) as

T(p(x)) = p(x+1) = 0(x+1)3+0(x+1)2+0(x+1)1+1(x+1)0 = 1

Abidul's '1+x' is probably closer to the right answer: she seems to have thought:

I have to make x into x+1

I have no x, I have 1

I'll do this once, or leave it the same so x+1

If this is at all close to her thinking, then at least she has noticed the absence of x which the other students have not. In any case the students appear weak in interpreting functional information: they understand the action required by the rule of T but lack the crucial understanding of the objects which this action is to be applied on. The fragility of this understanding becomes evident in cases where the nature of these objects is ambiguous, such as 1 (a number? a polynomial?). In this case, it seems that what seems to be a routine algorithmic procedure (the application of T) is not routine at all: what this raises is the issue that notions like 'routine' and 'simplicity' are far from being unquestionably clear; especially when used in relation to the novices' learning and understanding.

Finally, I briefly note Cleo's and Camille's imaginative extensions of the discussion in this tutorial: Cleo hypothesises about another basis (1, 1+x, (1+x)2, (1+x)3) and Camille reverses the question tackled in this tutorial (find the matrix of a mapping T given a basis) to: given a matrix, a diagonal one, find suitable bases for the vector spaces between which T is defined. I also note that when the tutor and Camille go into the details of how Camille's suggestion can be realised, Cleo, unable to follow, stays behind. This differentiation raises several issues related to teaching; from a learning point of view however what is significant here is that the students reach the end of the tutorial with a capacity for extensions and generalisations.

Conclusion: In the above, the tutor's use of closed questioning results in the elicitation of responses from the students that reflect their difficulties with new algebraic concepts. Also algorithmic procedures, deemed by the tutor to be routine and simple, turn out to be problematic for the novices. While, for instance, looking for the 'usual' basis for P3(Â) the students either fail to respond to the task or make suggestions that are severely decontextualised and not in resonance with the needs of the task. Moreover the students unanimously appear weak in interpreting functional information (interpretation of T(1) as 2). The fragility of their understanding seems to depend seriously on the ambiguity of some mathematical objects (1, for instance, as a number or as the constant function of value one). In contrast to this difficulty with 'simple' tasks, the novices at times appear capable of remarkable extensions and generalisations. This difficulty with 'simplicity' raises the issue of whether closed questioning allows an exploration (and service) of the students' actual cognitive needs or is simply a rigid implementation of the tutor's predetermined agenda.

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