In this chapter the students' first experiences of fundamental group-theoretical concepts were explored in a series of Episodes from the second term of observation. The novices here appeared as especially lacking in basic knowledge which resulted in their lack of understanding of an argument for a proof or the proposition of a theorem [7]. As paradigmatically problematic in this sense emerged the concept of coset. So, for example within the context of Lagrange's Theorem the students' perceptions of certain set-theoretical properties (distinct and disjoint sets, Ç and È) seemed to interfere with their understanding of cosets.

While constructing cosets the students also appeared to be in difficulty with the abstract nature of the operation between elements of a group [1]: interferences from the properties of numerical operations were observed. In fear of reinforcing these interferences, the tutors discourage the students from using metaphorical expressions such as divided by. Somehow paradoxically, however, they encourage them to say multiply with the inverse which does not rule out, I think, the possibility of numerical interferences. Similarly problematic turned out to be the linguistic metaphors such as times and powers of — also used sometimes unclearly interchangeably by the students — with regard to cyclic groups [1].

Linguistic condensation of meaning also causes difficulties, for instance [1] in the context of the concept of order of an element of a group, which is an abbreviation for the order of the group generated by an element. Notationally the abbreviation is similar: |g| is the commonly used notation instead of |<g>|. Moreover another aspect of the very problematic encounter with the notion of |g| seems to be the static and operational duality of the concept: |g| is the number of elements in <g> and, at the same time, the number of times the power of g has to be taken in order to cover all the elements of <g>. After |g|, the powers of g in <g> start repeating themselves. So, in a sense, order of an element is a notion containing both information about a static characteristic of <g> (its cardinality) and information about a way to construct <g> (take the power of g, |g| times). This type of duality is commonly seen as a source of cognitive strain for novices (functions, limits) and it is likely that order of an element is not an exception.

As in Chapters 6-8, the novices are primarily engaged in a meaning bestowing process [2, 4] with regard to the newly introduced concepts: they inquire about the raison-d'-être of the concepts. Examples:

The novices also appeared to be in difficulty in conceptualising a mapping between elements of a group and sets of elements of the group [2, 6] (two characteristic examples of such mappings involving the concepts of conjugate, centraliser and equivalence class and in the context of the First Isomorphism Theorem for Groups were presented). As in previous occasions in these tutorials, where the notion of sets of sets and of mappings between sets whose elements are gradually departing further from the simple arithmetical correspondences the students are familiar with from school mathematics, the increasing degree of abstraction as well as the students' problematic perceptions of the notions of domain and range of these mappings rendered their understanding an extremely complex process.

Within the abstract context of Group Theory a contrast between the novice and the expert approaches to abstract mathematical reasoning was observed: so the expert's logical, succinct and benefiting from hindsight argumentation was contrasted with the novice's more-naturalistically-born-out-of-the-proof argumentation [3]. In one instance [8], a novice managed with minimum principles (mostly arithmetical), albeit not in an impeccable formal way, to reach the conclusion and complete a proof. Her approach was juxtaposed with the tutor's proof which was well embedded in the mathematical context of the course (lecture content, problem sheet material, textbook approaches) and thus was shorter and more relevant. Typically most novices have not acquired yet the skill of association and co-ordination of relevant knowledge which equips mathematical reasoning with the power of resonance. Also the contrast between expert and novice approaches has an enculturating function: in the process the novices are clarifying the modes of formal expression at which they are expected to aim.

The students' participation in this enculturating effort is not successful when it is based on a superficial imitation of formalising practices. The instance [5] of a novice's attempt at a formalistic presentation of his argument — which fails possibly because of his tendency to imitate textbook writing indecipherably and of his inconsistencies in the introduction of new notation — is characteristic: unable to continue, he switched to ordinary language, regressing thus to a form of expression with which he felt his argument would be articulated in more clarity and precision.

Learning emerged here as inseparable from the social forces under the influence of which it takes place — here tutoring: the instance [7] of a novice's viciously circular struggle for the construction of a meaning for the new concept of coset was presented as a characteristic example. This instance was described as a succession of mutual misunderstandings between the student and the tutor who repeated identically the argument for the proof of the question on which the discussion was based. The sources of the novice's difficulties were thus not largely disclosed but the circular dialectics finally — and after the student's persistence — seemed to spiral down to an exploration of gradually more basic knowledge relating to the question and in particular to the construction of cosets (for which the tutor employed a variety of devices: cosets as parcels; cosets as translates). This spiral journey coupled with the persistence of ineffective ideas in the student's mind suggest the complexity of the didactical interaction and also, possibly, the need for an emphasis on constructive learning processes (that is processes that cautiously build on solid previous knowledge or allow revisiting and reconstructing previous knowledge with facility). A section of Chapter 10 — the cross-topical synthesis of the observations on the novice mathematician's cognition presented in Chapters 6-9 — is allocated to the brief elaboration on this and other didactical observations.

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