Section (v) Preliminary Conceptions of Limit and Infinite Largeness. The Two-Step Battle Between Intuition and Formalisation: Conceptualising and Materialising the Necessity for Formal Proof
Context: See
Extract 6.5i and iiStructure: In Kelle's Extract the tutor criticises the student's written approach to CD2.4 and then they discuss the proof of parts i and ii. In Jack's and Andrew's Extract the tutor is disappointed with their treatment of CD2.4ii. The students subsequently attempt unsuccessfully to defend Andrew's approach and the tutor presents two alternatives for the proof. The latter sparks off a metamathematical discussion.
The Episode: A Factual Account. See
Extract 6.5i (Kelle) and Extract 6.5ii (Jack and Andrew)An Interpretive Account: The Analysis
Kelle's Problematic Handling of the e-Definition of Limsn. (On Extract 6.5i) Kelle in the beginning of the Episode is rather uncomfortable with the tutor's rephrasing of CD2.4i in terms of limits, sequences and convergence. Despite his implicit request to stay within a familiar lexical territory, during the incident the tutor occasionally returns to this terminology. However Kelle's major difficulty seems to be the manipulation of the
e-definition of limit.K1 and K2 are hesitant and confused attempts to reproduce the proposition in CD2.4i and Kelle sounds baffled with the quantifiers for n and
e. K1, K2 are followed by the tutor's objection (T1) and her pictorial exposition on what CD2.4i is actually about. K3 is then received by the tutor as progress (T3) and she proceeds with elaborating on the nature of e. K4 is evidence of how undecided Kelle is about the nature of e: e can be any number, e is a number we can choose, e is a number we must find. T4 then is an attempt to establish a connection between Kelle's intuitive knowledge that qn® 0 and e as a tool to formally express this knowledge. K5 and K6 indicate that this connection has not been established. Kelle is convinced that qn® 0 and he does not hesitate to use this to-him-established fact (K6) in his vague attempt to answer the tutor's formal question about proving it. When the tutor intervenes in order to interrupt the vicious circle of his thinking and completes the proof, he listens and concludes in a dramatically expressive frustrated tone 'It's so obvious that it goes to nought'. In part ii, prompted by the tutor about h>1 implying 1/h<1, Kelle vaguely suggests 'inverting' which the tutor interprets as applying CD2.4i on CD2.4ii putting q=1/h.As an observer I am not convinced that Kelle has been persuaded of the necessity or has learned how to present his intuitive ideas formally, namely via the
e-definition of limit. In Chapter 7 (Calculus), ample evidence is provided of the hardship such learning entails. The purpose for presenting this Extract in this Chapter (on the Foundations of Analysis) is to begin drawing a picture of how Analysis as a tool for quantifying and manipulating relations appears as a very difficult language for the learner from the very start. Moreover the difficulty of the language seems to be worsened by the novices' reluctance to accept its necessity and utility.The Success of Interactive Dynamics and a Noble Failure. (On Extract 6.5ii) Andrew's intention is to prove that S has no upper bound by assuming it has and then reaching contradiction. Unfortunately this intention is not clearly stated by Andrew and it takes several interventions by the tutor to clarify it (up to and including Andrew's writing on the b/b). Then Andrew's thinking collapses:
h has been given as a real number >1. Andrew thinks that this allows him to take h as a natural number n+1 and write hn as (n+1)n. In the latter not only h loses its generality (from an arbitrary real number it becomes a natural number) but it also acquires an illegitimate double meaning:• first, (remember: S was defined as the set of all powers of
• second, n is defined by Andrew as a specific natural number used to substitute
h with n+1.Andrew's confusion is recognised by the tutor and Jack who laughs. Andrew resigns but, as his next question to the tutor indicates, in his mind the idea of contradiction still lingers. The tutor also is still open about investigating the idea of approaching the proof by contradiction and Jack also tries to defend it. Jack actually tries to pin down the contradiction by constructing a number,
hm, which he suggests then they try and prove it's the largest number in S. So, in a sense, Jack tries to develop Andrew's idea a step further, or in the tutor's earlier words 'to justify his belief by showing some strategy': Andrew has suggested assuming that S has an upper bound and Jack suggests a number which they can try and prove it's an upper bound. He then hopes that on the way they shall reach contradiction. Jack's suggestion is dubious in two ways:• first, even if hm is proved not to be the largest number of S, it is not implied that any other element of S is not its largest number, and,
• second, as the tutor notes, if m is an upper bound of S, it is not then necessary that hm is the largest number of S.
So in the first, Jack appears to be unaware of the necessity to prove that S has no largest number in general — and not to choose one number, for instance hm, and then prove it is not the largest number of S; and in the second he has unjustifiably assumed that, if m is an upper bound of S, then hm is the largest number in S.
Andrew's subsequent suggestion that hm is accessible but not attainable sounds like a finitistic attempt to express the arbitrary largeness of hn and is perhaps his first digression from the idea of contradiction. He thus rephrases the sought-for result to 'trying to prove that hm is the largest number in S'. Andrew's 'you can actually... there' is an expression of his intuitive idea of the arbitrary largeness of hm but this is all too vague for the tutor to accept as plausible arguments.
In the above (apart from the students' interesting conceptions of infinite largeness and apart from Andrew's confused semantics in his attempt to construct a contradiction for CD2.4i) what looms as intriguing is the dynamics of the didactic interaction. Andrew's strength of belief in his proof by contradiction gradually diminishes but is not annihilated. Jack laughs with Andrew's flawed argument but tries to take the idea of contradiction a step further. Despite the final failure, the novices here exhibit a tendency to act on refuted arguments and modify them until they become adequate. In contrast to other Episodes in this Chapter, where the inadequacy of the novices' arguments has been pointed out to them, here, given a forum to discuss these arguments, they reach this understanding in a mutually critical and argumentative procedure. Subsequently, when the tutor decides to pursue no further the idea of contradiction and suggests two different proofs for CD2.4ii, the students, possibly having the urgency of their dynamic action removed, resume their traditional, passive position and only participate when asked closed questions.
The students' interaction with the tutor becomes again more dynamic in the presentation of the tutor's second proof for CD2.4ii with Jack's suggestion to 'expand' (1+z)n. This suggestion comes to conflict with what the tutor had in mind (to use the inequality (1+z)n>1+nz proved in the previous week) and he insists on trying to elicit this from the students. With Andrew's distracting suggestion ('factorisation') the tutor, possibly in fear of losing completely the attention achieved so far in the group, retreats and accepts Jack's suggestion. In the end he reminds them of the inequality they could have used instead of the binomial expansion. So in this case the dynamic interaction between the tutor and the students has resulted in a redirection of the proving procedure towards the students' more basic choice than the tutor's (that is, Jack's suggestion to use the Binomial Theorem — with which he is already quite familiar — as opposed to the more recently introduced inequality suggested by the tutor).
A Comprehensive Dialogue on the Novices' Major Metamathematical Trouble. The concluding part of Extract 6.5ii is a dialogue that consolidates in the most illustrative way the students' difficulty with distinguishing between assumed knowledge and knowledge to be proved. Because it captures what emerges as one of the themes underlying my discourse on the problematic aspects of the novices' encounter with mathematical abstraction at the beginning of their studies, it will not be further discussed here but it will be incorporated in a later, more theoretical discussion of the issue.
Conclusion: In the above, a discussion of a novice's problematic handling of the e-definition of limit revealed his reluctance and difficulty in formalising what he thinks as an obviously true proposition. Moreover, and in the context of discussing the students' conceptions of infinite largeness, their ambivalence on what knowledge can be assumed and how assuming knowledge is compatible with the demands for axiomatic rigour made by the lecturers and tutors in the beginning of the course were touched upon. From a didactical point of view, dynamic interaction among the tutor and the students proved a fruitful way of refuting the students' flawed approaches. On the other hand, exposition seemed rather inevitable for the presentation of correct proofs.
Apart from the common mathematical content, what links the two Extracts is the contrast between the informal, intuitive and confused approach of the student in Extract 6.5i and the more willingly formal approaches of the students in Extract 6.5ii. In none of the Extracts do the students achieve fully a formal presentation, but what differentiates the first student from the other two is that the latter seem to have conceptualised the necessity to be formal and struggle with the materialisation of this conceptualisation, whereas the former is still engaged in the vicious circle of assuming in his proofs what is to him intuitively obvious and what he is actually being asked to prove. The different states of mind of the students in these Extracts imply their different cognitive needs. This diversity implies the necessity for a didactical flexibility with regard to the accommodation of cognitive needs.