Section (viii) The Difficulty of Realising and Justifying the Steps in a Proof and an Application of the Archimedean Property
Context: See
Extract 6.8Structure: In the following, Jack's proof is modified by the tutor. The tutor then presents a proof for CD3.3ii.
The Episode: A Factual Account. See
Extract 6.8An Interpretive Account: The Analysis
CD3.3 is a problem sheet question mostly relevant to Accumulation Points but its first two parts are still embedded in the context of Foundational Analysis. Therefore, in a sense, CD3.3 illuminates the rites of passage from a study of the foundations of Analysis to a more topological territory, such as the introduction of the concept of Accumulation Point. For this reason it provides a demonstration of how the learning behaviours of the novices are standardly carried on from fundamental Analysis to Topology. A significant part of this behaviour is the students' lack of clarity and precision when presenting otherwise deftly grasped ideas: Jack's presentation of his proof for CD3.3i offers some evidence for that.
The Inability to Justify a Correct but Complex Step in a Proof. Jack appears hesitant in justifying x-1<n=x, which is however approved by the tutor. He eventually achieves a justification of the inequality with the tutor's two prompts. I note here that the second prompt is more leading than the first: it leads Jack to think of n as one natural number among those greater than x-1. As a result of seeing n as one of the numbers >x-1, Jack is perhaps forced to see what has made him choose n and to realise that actually n is the least of these numbers. With Andrew's question about why they need to consider n as the least of these numbers, Jack is further reinforced to articulate the reasons for his choice ('you can't take any natural number bigger than that. You have to take the next one along'). So while in writing he did not feel the pressure for justification, in the dialectic process with Andrew and the tutor (as in Extract 6.1) he is forced to abandon his tendency for tacit-ness.
As discussed in earlier analysis, these tutorials seem to be considered by the tutor as an enculturation process for the novices into mathematical literacy and articulate expression. So his objections to Jack's proof and his demands for clarification are possibly more of a pedagogical character than strictly mathematical. In the same vein Jack's redundant introduction of
e, denoting the already given in the question by fracx, is also criticised by the tutor.The tutor also seems to think of himself as a mediator between the complex presentation of the mathematical content in the problem sheets and the novices. Again as previously observed in this chapter, one of the sources of the novices' difficulties with Foundational Analysis seems to be the problematic role of quantifiers. As a result, and in view of the heavily phrased proposition in CD3.3ii, the tutor suggests a rephrasing. I note however that despite the students' difficulty with the heavily quantified expression, Jack sees through the statement that 'the whole thing is about number bases for natural numbers' and has applied Mathematical Induction correctly to prove it. So, in a sense, what the tutor assumed would be a difficult question has been adequately handled by at least one student, Jack. This raises the didactical issue of the accuracy of the tutors' prophecies. I note however that his general observation that quantifiers are mathematical tools of particular difficulty for the novices resonates with the findings of this study.
Incidentally I also note Jack's remark on the tutor's suggestion for a 'quicker way' than Jack's Mathematical Induction in CD3.3ii. Jack asks: 'Do you go straight to the general case?'. What Jack seems to be thinking is that the Base Clause of Mathematical Induction and the Inductive Hypothesis are perfunctory and that there may be ways that the crucial 'general case' of Mathematical Induction would stand on its own even in the absence of the previous two steps. This and other conceptions of Mathematical Induction in the context of Foundational Analysis will be discussed globally in Part III.
Conclusion: In the above, and in the context of a problem sheet question related to an application of the Archimedean Property, the discussion gives rise to a student's difficulty to articulate adequately a justification of some steps in his proof. Gradually he is led to the realisation of the choices he had made in a dialectic process with his tutor and fellow student. Moreover some evidence was provided of a student's conception of Mathematical Induction according to which the first two steps of the inductive proof may be seen as perfunctory.