Part III Didactical Implications: Observations Related to the Teaching of Advanced Mathematics Derived From the Study of the Novice Mathematician's Cognition
In the previous section, the interaction between the students and the tutor was examined as an enculturation process of the novice into the expert's culture. Here I present briefly some didactical observations linked to the teaching of advanced mathematics. I note that these observations were originally made in the Analysis of the Episodes but, for the sake of conciseness, were not mentioned in Parts III of Chapters 6-9. So this section is a brief cross-topical synthesis of the didactical observations made in the study. In the brackets I refer to the Episodes where these observations originate. The aim here is to juxtapose Interactive Dialectics — the materialisation of the novice's induction to the culture of advanced mathematics as an enculturation process — with Exposition — the materialisation of the novice's induction to the culture of advanced mathematics as an acculturation process (as defined in the Interlude)— and suggest that, on the basis of the evidence in this study, the former has generally more didactical potential than the latter.
As elaborated in Parts I and II, a major difficulty of the novice's enculturation into advanced mathematics is the lack of clarity with regard to the increased requirements of rigour in the new course that the novices have to confront — for instance [
6.2] with regard to the knowledge that they are allowed to assume. This raises the didactical question as to how, in this state of uncertainty about the rules, the novices are expected to play successfully the game of advanced mathematical formalism. In some cases [6.4] the tutors prefer to demonstrate these requirements of rigour by presenting exemplars of flawless and rigorous answers — even in cases [6.4] where the student seems to prefer a more conversational style of first refuting his proof and then presenting an acceptable one. In other cases [6.5], dynamic interaction between the tutor and the students proves a fruitful way of refuting the students' flawed approaches but still exposition seems rather inevitable for the presentation of correct proofs. In general this type of dialectics, exemplified in cases [6.2, 6.8] where the tutor and a student discretely scaffolded another student's overcoming of an obstacle in a proof, seem to be a successful means to the enculturating end. In the above, a diversity of tutoring approaches is featured. The diversity of the students' observed cognitive needs implies a need for flexible tutoring approaches that adapt to these needs [9.2].The students were also found to be in difficulty with establishing a creative co-ordination of intuitive and formal practices: very closed questioning and constant reclaiming of control on the part of the tutor seemed to be not very productive as opposed to providing sufficiently detailed explanations that aimed at sustaining the specific students' meaning making process [
7.2, 7.4].The evidence from these tutorials suggests that there are tutors who view interaction with the students at a reflective, meta-topical level as a legitimate part of the tutorial; others view this interaction as a deviation from a carefully predetermined tutoring plan. It is possible that these views entail that the former [
7.3, 7.6] engage in exploring and supporting the students' meaning making processes with more zest than the latter [7.4, 7.5]. I note here that there have been cases in which the tutor engages so substantially in the interactive process that, partly as a result of his own uncertainty — about a particular proof of a statement, not the truth of the statement — he yields almost equal didactical control to the students, who then emerge as dynamic perpetrators of these newly balanced dialectics.Yielding control to the learner seems to be of considerable didactical potential: for instance, when the tutor [
8.1] manages to modify a student's perspective from within, that is by adapting their point of view and challenging it with key prompting questions; or by creating a more debate-friendly learning environment [9.3, 9.5]. Closed questioning on the other hand and highly directive instruction [8.6, 8.10, 9.1] seems to be less efficient and a perpetrator of decontextualised algorithmic behaviour. Directive instruction is mostly based on the expert's prophecies, about, for instance, the simplicity or difficulty of a task [8.6], and these prophecies can prove misleading.A major didactical point, regarding the potential cognitive danger built in the use of geometrical metaphors as a visual aid for the introduction of new and abstract concepts, was made mostly in the context of Linear Algebra (Vector Analysis): some of the students' difficulties with an abstract perception of vectors (beyond the geometric approach) was [
8.2] partly attributed to teaching that focuses on a large number of examples from the line, the plane and the space. However, flexibility on the tutor's part in thinking in the terms of the student's personal metaphors [9.2] seemed to boost a more confident demonstration of thinking on the part of the student. So there seems to be value in the didactical use of metaphorical discourse but only when it does not impede the construction of the intended-by-the-tutor abstract concept image.In the same vein a less biased use of examples — for instance in the context of convergence and divergence of series where the students' overexposure to examples of convergence seemed to encourage finitist attitudes towards infinite sums — was suggested as a way to curtail some of the novices' prejudices, for instance, with regard to the number of convergent series, or continuous functions, or cyclic groups.
The students were found to be largely engaged in an exploration of the raisons-d'-être of newly introduced concepts. To support this exploration, the tutors often decompose the various problematic concepts or theorems (for example: coset or the First Isomorphism Theorem for Groups [
9.6]) into their basic elements. This, even though sometimes based on the tutor's preconceptions of what constitutes the problematic elements of the concept or the theorem, seems to be quite efficient. The students seem to appreciate a new concept when it is launched as a useful apparatus, not as an ideal that exists only because of its definition. One of the reasons that spanning sets emerged as paradigmatically problematic in Chapter 8 was the students' tendency to disregard them and favour instead the concept of basis which is of more obvious utility: a suggestion emerged then to introduce the notion of basis before the notion of a spanning set or even to ban the use of the term spanning set altogether from introductory courses. Research focused on the didactics of particular mathematical areas can substantiate and enrich this type of suggestion.Similar existential inquiries were carried out by the students with regard to some newly introduced theorems: in justifying the importance of these theorems (Lagrange and First Isomorphism Theorem for Groups [9.6]) the tutors often use pragmatic — as opposed to epistemological — arguments. A pragmatic argument was described as an attempt of the tutor to convince the students of the significance of a theorem by repeating that it definitely appears on exam papers because, for instance, it has a 'name attached to it'; an epistemological argument was described as giving an existential rationale to some newly introduced concepts such as justifying the introduction of cosets as a substantial element of studying normality in Group Theory. The latter were suggested as cognitively more powerful, whereas the former were acknowledged as strong motivators.