Section (ii) The Contrast Between Algorithmic Ability and Conceptual and Contextual Understanding: Applying the Subspace Test and Looking for the Zero Element of
ÂÂContext: See
Extract 8.2Structure: Similarly to Extract 8.1, where the students learned the 'recipe' of finding the span of a set, the students here appear good at organising their algorithmic thinking (see first two applications of the Subspace Test in the Context). Unlike these two applications that involved thinking in terms of Mn(
Â), the application presented in this Extract involves thinking in terms of ÂÂ, with which they are less familiar. This lack of familiarity results in the students' unease with handling the question.Note to the Reader: Same as Section (i).
The Episode: A Factual Account. See
Extract 8.2An Interpretive Account: The Analysis
In the following, I comment upon the students' handling of the zero element of ÂÂ as well as Camille's preoccupation with the contents of ÂÂ. I note that the first frictions (though not mentioned in the Context) with regard to the zero element of a vector space appear in the discussion of the first two applications of the Subspace Test: for instance when Abidul uses the term 'nought' for the zero vector of Mn
(Â), the tutor corrects to 'zero' (meaning: the zero matrix). Subsequently Abidul asks whether she can write 'just 0', not the full matrix with zeros everywhere. The tutor says it's fine as long as it is clear what she means.Another element that seems to be setting up the scene for Extract 8.2 is the fact that the students in the beginning of the tutorials do not know the Subspace Test. However during the tutorial they are given the opportunity to apply it several times and by the time they reach the third application (that is this Extract) they appear to be handling it quite well — for instance, all of the students immediately suggest the evaluation of af+bg when the tutor asks for the second condition of the Subspace Test. Similarly to Extract 8.1, where the students in the course of the tutorial learned the 'recipe' of finding <v1, v2, v3>, the students here are quite good at organising their thinking along solidly outlined algorithmic procedures. The problematic element is that, while the previous two applications of the Subspace Test involved thinking in terms of the elements of Mn(
Â), here the vector space in question, ÂÂ, is less familiar. In the following, I present the events from the perspective of this lack of familiarity and highlight the outcoming learning difficulties.Looking for the Zero Element of an Unclearly Defined Vector Space. Some of the students here appeared ready to start applying the Subspace Test on a set, UÍÂÂ, whose contents were not well aware of. As a result of this rather mechanistic choice, they soon impinge upon the identification of the zero element of ÂÂ. Even though some of the students can recall the formal definition of the zero element in a vector space (a+0=0+a=a,
"aÎV) all of them either remain silent or, after persistent prompting, suggest 'stays the same' or 'x', namely the identity function. Even when Eleanor says 'nought' towards the end of the incident, one is tempted to think that she simply means the real number zero — and not the function z(x)=0.The evidence of the students' difficulty to understand, and furthermore to act upon, ÂÂ is even stronger and more elaborate in the tutorial with Camille. I note that, once the hurdle of identifying the zero element of ÂÂ is overcome, the students are still uncomfortable with the contents of ÂÂ — as their frequent confusion of f with f(x) illustrates. Considering separate values of f as elements of ÂÂ is an indication of their difficulty to view f as an object contained in a set of similar objects, as an element of the set ÂÂ. Again the tutorial with Camille provides strong evidence of this difficulty too.
Camille's Preoccupation with the contents of ÂÂ. C1-C8 illustrate Camille's struggle to construct a meaningful interpretation of ÂÂ; C9-C11 her narrow concept image of a vector and the expansion of this image that the understanding of ÂÂ necessitates.
Given that Camille is a learner who is not afraid of expressing risky ideas (see Extract 8.1), C1 is her first attempt to imbue meaning to ÂÂ. C2-C5 are indications that despite the definitions given by the tutor, Camille — probably not having found them helpful — insists on trying to relate  with Â2. Knowing that the set of (x, f(x)) is a subset of Â2, where f:®Â, her thinking gets entangled with f as [a set of] ordered pairs and f as an object-element of a set. In C6 she seems to have realised that  is a set of functions and she explicitly expresses her difficulty in shifting from a set of elements — for instance matrices — to a set of functions.
C7 then leaves an impression of a second-round effort. Now Camille possibly interprets ÂÂ as a 'power' of Â, thus 'containing' Â — in the same sense that 23 'contains' 2? The chain of tentative interpretations of ÂÂ continues with C8 which seems to be a compromising combination of her entanglement with Â2 and her newly-acquired idea that ÂÂ contains functions. In any case C1-C8 is an impressively explicit illustration of a novice's striving for meaningful interpretations of the concepts — and the notation attached — they are being introduced to. Camille's attempts here highlight the failure of the transmissive model as a descriptor of learning. Camille's struggle cannot be described in terms of transmission and reception of knowledge. What the tutor repeats again and again about ÂÂ, is variably perceived by Camille as her unendingly changing interpretations of ÂÂ indicate. Camille seems to be very much alone in this process even though the steps to the various directions she is taking are obviously fed by the information she receives.
Finally, the narrow image that Camille has of a vector (also evident in C6) is revealed in C9-C11 and, in particular, in her attempt to combine the information that
• the zero vector is a function
• this function has a particular property related to denoting it 'zero vector'
in order to deduce that the zero vector of ÂÂ is the zero function. Camille still seems to struggle with understanding how the elements of ÂÂ can be functions. In particular C11 highlights a schism between the notion of vector and the notion of function.
The evidence in this study suggests (for instance Extracts 8.1 and 8.3) that the students have acquired a persistent geometric image of a vector through teaching that focuses on the examples from the line, the plane and the space. Understandably the teachers are repeatedly suggesting to their students to espouse a geometric approach in order to demystify Linear Algebra and especially Vector Spaces which are deemed as too inaccessible and abstract by the learners and to embed it in already familiar contexts. This practice unfortunately proves short-sighted because, within weeks of their instigating a geometric approach, the tutors are forced to generalise their discourse on vector spaces to cases where the geometric metaphor not only does not apply but also creates distracting interferences.
Conclusion: In the above, the students' narrow-minded algorithmic application of the Subspace Test on a subset of ÂÂ (whose contents they were not aware of) was shown as an action-in-void via their difficulties to identify the zero element of ÂÂ. These difficulties were illustrations of a narrow concept image of vector and of a weakness to perceive function as an object-element of a set. Moreover a student's bold interpretations gave evidence of her uniquely individual struggle for a meaningful construction of ÂÂ. As in Section (i), the students' difficulties with an abstract perception of vectors was here partly attributed to teaching that adheres to a large number of examples from the line, the plane and the space. Finally a methodological point similar to Section (i) was also made.