Section (vii) The Overwhelming Linguistic and Conceptual Complexity of the Notions of Sup and Inf
Context: See
Extract 6.7Structure: In the following, the tutor and Cornelia complete the proof for CD2.6i that Cornelia could not finish. The discussion is characterised by very closed questioning.
The Episode: A Factual Account. See
Extract 6.7An Interpretive Account: The Analysis
The Tutor's Three No's to Cornelia. Cornelia has had difficulties and finally gave up proving the second property for the infimum of kS. During the tutor's closed questioning presentation of the proof, her three responses are successively met with his disapproval. The tutor's three No's to Cornelia signify three misunderstandings, on her part, that presumably have constituted part of the stumbling block that led to her giving up on the proof in the first place.
• First (C1), Cornelia ignores that k has been given as negative and, when dividing b>ks by k, replies b/k>s, instead of b/k<s. This is a typical algebraic mistake, that at university level is usually attributed to carelessness, which Cornelia instantly corrects once her attention is drawn by the tutor to the sign of k.
• Secondly (C3), when asked what is s (that in the proof denotes supS) she starts her response with 'the greatest'. She is interrupted by an impatient tutor who reminds her that s is the least upper bound for S. As in Extract 6.4 most students appear in a linguistic unease (in parallel with their difficulty with the new concept) with the terminology commonly used for supremum and infimum, that is least upper and greatest lower bound. Similarly to Kelle who at least three times in Extract 6.4 corrects himself interchanging least with greatest and vice versa, Cornelia, if given the chance, might as well have done so. Coming to terms with a new concept and its complex terminology is a demanding task that novices frequently find hard to carry out. I also note here that the alternation of the terms 'greatest' and 'least' in CD2.6 is even more cognitively demanding because of the reversal due to the negative sign of k.
• The two incidents cited above psychologically build up to Cornelia's resignation, signified by C4 and to her subsequent third and essential flawed response to the tutor's closed questions. When asked about what can be said about b/k, a number shown so far to be smaller than the supremum of the set, Cornelia deduces that it 'must be in the set'. In her response there is evidence of a conception regarding the supremum of a set according to which anything less than the supremum is necessarily contained in the set. This is true for intervals like (a,b) where b is the supremum and anything close to b but less than b belongs to the interval. The tutor notes that the set in their proof can be a 'dotty' one, breaking up thus her continuous and dense concept image.
Also her response is an illustration of a major difficulty of mathematical reasoning encountered by novices: confronted with an inequality such as b/k<s, the available next-steps (observations that will lead to the next step of the proof) are numerous. Whether the decision made by the student is a fruitful one relies on how well she has realised the existence of the various options and on how she will associate the data with the goal of the proof. In this case, Cornelia did not seem to have adequate control of her options even though her response seems to indicate that she follows the tutor's train of thought. In this sense, the tutor's suggestion 'following your nose through the definitions' is a recommendation the meaning of which is not as obvious to the novice as to the expert mathematician. The tutor's metaphor however is insightful in another sense: it captures the detective skills required in most mathematical activities, namely that given the clues one is able to make a choice that will turn out to be effective. Cornelia's response in Extract 6.7 exemplifies the cognitive behaviour of a novice who has not mastered these skills yet.
A Note on an Ostensibly Happy-Ending Story. Cornelia, after a succession of flawed responses that have been dismissed and modified by the tutor, completes his last sentence correctly, leaving thus a last impression of successful fulfilment of the tutor's task (which is to help Cornelia understand the proof for CD2.6 i). It is a methodological constraint of this study that no further evidence is provided with regard to whether Cornelia's last utterance, ak<b, is merely a correct calculation, an automatic algebraic reaction to the tutor's a>b/k; or whether it is a meaningful contribution to his proof and one which shows that her flawed initial conceptualisation of the proof has been reformed. As an observer, my impression is that, through the end of the Episode, Cornelia remains overwhelmed by the complexity of the definitions of supremum and infimum and uncertain of her own understanding.
Conclusion: In the above, evidence was given of difficulties embedded in the notion of supremum and infimum of a set, and in particular of the notion of infimum as the greatest lower bound of a set. From algebraic mistakes in the handling of inequalities to the confused use of the new terminology, additionally perplexed by the alternation of the terms 'greatest', 'least', 'upper' and 'lower', these concepts emerge as essentially problematic. A particular conception revealed in the Episode was the student's belief that a number smaller than the supremum of a set, must necessarily be in the set. At the metamathematical level, the novice seemed to be in hardship with confronting the multiplicity of options in the course of a proof and with co-ordinating a variety of information in order to pick an effective option.