PART III A Synthesis of the Findings in the Area of Linear Algebra. Indications for the Cross-Topical Synthesis in Chapter 10
In this chapter the students' first experiences of Linear Algebra concepts were explored in a series of Episodes from the first and the beginning of the second term of observation. As noted in the Interlude, the concept that emerged as paradigmatically problematic in the area of Linear Algebra was the concept of span /spanning set which was explored from a linguistic and a geometric/visual perspective: the novices [
1] often seem to miss the grammatical link between the terms spanning set and span — which reflects their cause-and-effect conceptual link. This linguistic deficiency mirrors and partly determines the novices' restricted understanding of these new concepts. Specifically, their understanding seems to be also influenced by their responses to the requirements of generalisation. Within the same mathematical task [1],• some students learned to reproduce the strategy, suggested by the tutor, of generating <v1, v2> from <v1>, when v2Ï<v1>, and <v1, v2, v3> from <v1, v2>, when v3Ï<v1, v2>. Some did not seem to accommodate the acculturating process of learning and applying the rule as readily as others.
• one of the students seemed to perceive <v1, v2> as <v1>+<v2>. This implies that the student resists the idea of a construction larger than its parts and it also resonates with findings in other contexts, for instance functions, on the persistence of linearity in the novices' thought processes.
So the students appear as if they respond to the tutor's stimuli for generalisation with various degrees of readiness. On what seemed initially to be a paradox, among the observed students, the only one who attempted to generalise with some facility, was the one who complained about the difficulty of generalisation. Consciousness of difficulty was then conjectured as enhancing the possibility of overcoming difficulty.
Tutors and students also seem to perceive routine-ness and simplicity of task differently: for instance [
6], when looking for the 'usual' basis for P3(Â), the students either failed to respond or made severely decontextualised suggestions (for instance factorisation). Moreover the students unanimously appeared weak in interpreting functional information (e.g. in the 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). These differences in perception make it necessary to reconsider what is traditionally thought of as a simple task. Especially when at the same time that the students appear incapable to perform a 'simple' task, they are capable of remarkable extensions and generalisations.In their attempts to understand the vectorial context, the novices seemed to adhere strongly to the metaphor of the plane which they frequently interpreted rather literally: they use strictly geometrical language regarding, for instance, vector addition. This attitude was vividly illustrated in a students' suggestion [
1] to add vectors by 'dropping perpendiculars', that is by using a Cartesian orthogonal system, instead of using the Parallelogram Rule: as a result, two different aspects of the geometrical metaphor of the plane (Cartesian orthogonal system, Parallelogram Rule) interfered with each other and with the novice's understanding of the construction of <v1, v2>. Other illustrations of this attitude include the students' body language as well as their orientation on the plane [1] (minus for left, plus for right).Novices tend to adapt metaphors in an excessively literal manner mainly because the suggested metaphor — the plane in this case [
1, 2, 3] — has been associated with convenient and familiar algorithms. The likelihood of this explanation is reinforced by the students' tendency to apply their algorithmic competence even within contexts they do not have a good grasp of: e.g.[2] applying the Subspace Test on a subset of ÂÂ whose contents they are not aware of. Soon this action-in-void brings their conceptual difficulties to the surface: difficulties with the zero element of ÂÂ and confusion of f with f(x) [2].Some students however try to avoid this meaningless action, when trapped in a context they are not aware of, and attempt to accommodate the new concept in what they already know [
2]: for instance, while struggling with seeing the zero function as the zero element of ÂÂ, a student attempted a meaningful construction of ÂÂ. In the process it is revealed how her concept image of a vector on the plane (a directed line segment) interfered with the notion of vector as an element of a vector space; also her weakness in perceiving function as an object-element of a set.Similar struggles with constructing a meaning for the newly introduced concept of spanning set (the difficulty with the generating aspects of the spanning process seems to characterise these struggles) mostly consisted of efforts to employ familiar metaphors, for instance from the study of functions. In one instance [
3] — characteristic of the tendencies among the novices — the student exaggerated the tutor's persistent recommendation to use the metaphor of the plane and appeared to believe that every co-ordinate system is a spanning set, including polar co-ordinates. In the process she also appeared to see basis as the smallest spanning set and, as her consistent use of the article 'the' for spanning sets indicates, she seemed to believe that a spanning set and a basis are unique. This contradicts her image of basis as the smallest spanning set which implies the existence of more than one spanning sets. The two images seemed to co-exist independently. Finally the student wondered about the utility of spanning sets. This reflects an inclination to search for a purpose in the introduction of new concepts whose usefulness is not readily visible to the novices.Apart from wondering about the utility of the new concepts the students also seem to acquire a variety of explicit or implicit images related to the concepts that co-exist, or exist despite of, the concept definition. Examples:
• the elements of a basis of a subspace do not necessarily belong to the subspace [
• a basis of a vector space can be reduced to a basis of a subspace [
5].Along with these sometimes inadequate or contradictory concept images, the students' inadequate knowledge [
7] of necessary definitions (kerT, ImT) and the linguistic and graphical inconsistencies associated with these concepts also becomes an obstacle to building up meaning for the new concepts as well as proving arguments. Trying to work within ÂÂ, while thinking of the concept of function disconnected from relevant notions such as domain and range [2], is an example of the role played by these obstacles.In the above, the novices appear to respond in a variety of ways to the introduction of new concepts: by regressing to the adoption of familiar metaphors (substituting thus the power of the abstraction in the new concepts with the convenience of a familiar context); by concentrating on a competent if narrow minded execution of algorithmic tasks; by engaging in a struggle for a meaningful construction of the new concept. For instance while discussing the new concept of linear mapping some students adhered to an application of the definition, even when they could not give an example of the concept; others tried to critically embed the new concept in their previous knowledge (for instance, by asking whether particular examples of mappings they know about fit in the definition of linearity). In understanding the notion of linear mapping the novices' familiarity with the metaphor of the two- dimensional plane seemed to play a positive role. This specificity of context however seemed to have a more controversial effect in cases like finding the zero element of a vector space, where the novices associate the zero vector with the number zero. So, in this case, specificity of context impedes the novice's understanding of the zero vector as, for instance, a function (also in [
3]) or a matrix.The students also appear to be in difficulty to express formally an intuitively grasped idea regarding spanning sets [
1, 4, 7, 8], even in the cases where they appear alerted to the necessity of a formal argument [7]. One of the students' linguistic unease [1] — because English is her second language — was illustrated as a metaphor for the difficulties engendered by mathematics as a symbolic and formal language too. The extreme formalistic nature of some of the new notation, for instance the unnecessarily convoluted use of Kronecker's Delta in [4] also seems to obstruct the development of a student's problem-solving thinking. In this case, the difference between the expert's facility with formalism and the novice's unease is so obvious that while the expert and the novices may agree [4] about the method of approaching a problem, they differ in terms of the implementation of the approach: as a result, their interaction evolves into an initiation process during which the students, with variable ease, become familiar with the new notational tools of mathematical formalism. Hence their learning becomes a specific struggle for accommodating to this new tool — whose appearance maybe intimidating — the vivid intuitive ideas they have about the solution of the problem.In the above, the students' learning is described in terms of their interaction with an expert. In this interaction, the efficiency of the different ways of mathematical persuasion varies [
5]: Socratic closed questioning, in which unproved theorems were used tacitly, appeared to be less convincing for the novices than Refutation by Counterexample, in which the examples vividly and concretely pointed out the argument. In the latter case the strength of the students' conviction as well as their imaginative practising of generalisation were exemplified [5]. At the same time the logic of implication and of proof by contradiction are reasoning techniques that the novices appear to struggle to master [7].
The students' interaction with their tutor was in most cases described as an enculturation process in which the tutor organises the students' intuitive ideas in a demystifying manner and in which the novice seems to learn about particular proofs as well as about thinking in formalistic terms. In some cases [7] the interaction between tutor and students is both cognitively and affectively quite intense and in a manner which highlights the inseparability of cognition from the affective factors under whose influence it takes place.