Andrew's effort to reproduce a formalistic presentation is evident in the right-hand cells of the above table. He is introducing names for each one the numbers or elements in A1-A3 and tries to define them with precision (example: his definition of p, q and x in the first cell). He is also interrupting the flow of his thinking in order to discuss notational issues or terminology (proper factor and notation for order of an element in A1). He is trying to invest his approach with respectability by stressing that the idea used in this proof was also used in GRF5.1 (he refers to the lateralisation of cases in Extract 9.3).

Jack and the tutor seem to be gradually frustrated by Andrew's chaotic monologue and his equally unclear effort to formalise on the b/b. As the tutor said in the beginning, Andrew seemed to have a generally correct intuitive grasp of the argument in his writing but his presentation was confused. In A1 another element comes through: Andrew indeed is trying to reach contradiction by pointing at the existence of more than one proper subgroup but he seems to ignore that there is already one proper subgroup of G, {e}, and that he only needs to construct one more. In A1 Andrew sounds as if he is anxiously trying to point at two proper subgroups and because he has assumed n, the order of G, to be the product of two numbers p and q, he is struggling to mould these two proper subgroups so that they are of order p and q. In a sense he is the victim of his own notation because p and q are the notation he used to express the hypothesis that n is not a prime: if n had been assumed to be 36, for instance, then p and q could have been 6 and 6, or 3 and 12, 4 and 9, 2 and 18. Any of these combinations give the product of 36, but isn't it arbitrary to choose one of them and then try and find elements in the group of that order?

So Andrew has introduced quite heavy-handed notation without necessarily having control over it. Coupled with his muddled monologue in A1, he causes confusion to Jack whose two interventions ('can p equal q' and 'not happy with the logic behind that') perhaps do not address the main drawback of Andrew's presentation but reflect Jack's "gut reaction" of discomfort with it.

Things worsen in A2 where n acquires a second role, as a divisor of pq which is now, in Andrew's words the order of G. Andrew insists on trying to construct a proper subgroup of order p (again arbitrarily he chooses to pursue an element of order p and not of q). Again he tries to support his approach by comparing it to the already accepted approach used in GRF5.6. Jack's and the tutor's patience seems to be almost exhausted. I also note that in the tutor's response there is evidence that the tutor still thinks that n in Andrew's words is the order of G; Andrew has not made it explicit — it is very likely he hasn't even noticed- that he changed the role of n.

A3 is evidence of how Andrew's disillusionment begins to collapse. His intuitive belief is fortunately still strong: 'I must be able to choose something!' he exclaims. With 'what am I doing?' he seems to realise that his formalising effort has failed and has also damaged the clarity of his thinking. By turning his back to the b/b, he disregards what he did so far and with 'so, if n is the order of G and n is not a prime then for a factor of G there must be a proper subgroup of G of order equal to that factor' regresses to ordinary language but in an illuminating way. Even though he does not truly explain in depth where the contradiction lies, the strategy of the proof is there, transparent and consummate.

Andrew's abandoning the effort to express formally can be seen as a regression to more "primitive" forms of expression: it shows that Andrew as a novice has not been enculturated into the conventions of mathematical formalism. He seems to pursue an enculturation, possibly more keenly than other students in these tutorials, but he hasn't quite mastered it yet. His writing on the b/b is a clumsy imitation of textbook writing and this clumsiness almost puts Andrew's idea into the risk of being perceived as using the converse of Lagrange's Theorem. Also his monologues are at times incoherent and possibly reflect the lack of clarity in his mind about the formalistic approach he wishes to espouse.

I finally note that Andrew's shifts of approach, whether seen as a regaining of confidence in ordinary logic and language, or seen as a regression to a familiar but not quite formally acceptable way of mathematical expression, only take place because of the tolerant learning environment created by the tutor. Whether Andrew took a step towards mathematical maturity in Extract 9.5, or he conservatively abandoned his effort to formalise and regressed to the convenience and familiarity of ordinary language, the incident has been illuminating (possibly for him too) as to the tensions that tantalise the novices' decision making with regard to their choice of expression.