Section (iii) LimS , S lim and the Right to Exchange Limits. The Superiority of Proof Via First Principles and the Convention of Foundational Rigour
Context: See
Extract 7.3Structure: In the following, the tutor refutes the students' choice of strategy in CD7.1 on historicoepistemological grounds and addresses the question of principles involved in assuming statements that have not previously been proved in the course. The students have correctly used a theorem which, however, will not be proved until next term. They discuss the limited validity of the theorem and whether they are entitled to use it.
The Episode: A Factual Account. See
Extract 7.3An Interpretive Account: The Analysis
The Student's Foundational Preoccupations and the Tutor's Preference for Proofs by First Principles. It is intriguing that despite the imperfect inductive proofs both students presented in CD7.1, they seem to have been theoretically alerted to the dangers of 'assuming any of the foundations', 'going sort of backwards' (Andrew) and about 'the idea of assuming limits' (Jack is not happy about that). The tutor elaborates upon these preoccupations by demonstrating a proof for CD7.1 which is clearly based on first principles. In fact the proof via the binomial theorem is an almost arithmetical one and it employs mostly school mathematical knowledge. Thus at a metacognitive level the tutor's preference of the proof-by-binomial-theorem over the proof-by-induction signifies a tentative, much needed bridging of the gap between elementary and advanced mathematical thinking. However I note here that, in the tutor's words, his preference for the proof via the binomial theorem is grounded on historicoepistemological, and rather than didactical reasons.
The tutor's preference for proof by first principles in CD7.1 may look as if it is largely a matter of aesthetics. It is the discussion of CD7.2 which makes it more obvious that he is seriously concerned with the implications of the students' practice of assuming unproved knowledge. There is evidence in this Episode that within the course lie conflicts with regard to the knowledge to be assumed and with regard to the varying demands for rigour. The students 'instinctively' use a theorem on the exchange of double limits which holds only under some conditions, without either stating clearly the theorem, or proving it. Formally speaking this approach is unacceptable in a foundational Continuity-Differentiability course in which Analysis is built upon a few, selected first principles. At the same time, in the course on Analytical-Numerical Methods and Differential Equations, the technique of unjustifiedly exchanging limits in order to simplify calculations is very common and one that the students are encouraged to employ. The borderline of the above distinction is for the novices rather blurred and ambiguity often results in treating questions, such as CD7.2, unrigorously. The tutor's statement 'It's the job of this course [Cont-Diff] to question' the conditions of the theorem and his promise that the theorem will be 'reinstated next term with all the full rigour that mathematics is capable of' is a tentative clarification that tries to resolve the ambiguity. The tutor here tries to draw clear boundaries between the different domains of mathematical activity in the course and specify the rules of the game of rigour within each one of these domains. In other words he tries to articulate, more explicitly than most of the other tutors participating in this study, the terms of the didactical contract by stating what is expected of the students in the various domains of their mathematical activity.
The Role of Conventions: the Right to Use the Limited Validity of a Theorem and the Right to Make Unjustified Statements. The students' unconditional exchange of double limits provides evidence for another metamathematical observation with regard to their attitude towards counterexamples: at first both students are convinced that exchanging double limits is 'not necessarily' acceptable. To formalise this conviction the tutor suggests constructing a counterexample. Jack immediately responds with a not very successful but reasonable attempt. Andrew sounds more confused with the act of looking for a counterexample than the example itself. Finally both students appear sceptical when presented with one:
• Andrew, who in previous Episodes has been uncomfortable with the role of a counterexample as disproving totally the truth of a proposition, wonders whether the tutor's argument is 'useful': it ruins their proof; also, in the particular case of CD7.2, the limits can be exchanged.
• Jack's explanation on why he used an unproved theorem is even more unconventional: 'you can not-justify yourself on paper as long as you can justify yourself in a tutorial'. What Jack does not realise is the basic convention of an introductory course that, on the b/b, on paper or in a tutorial, arguments are expected to be self-contained and not rely on tacit explanations. Jack's comment actually reflects the common practice in published mathematical research according to which theorems, that appear in journals for instance, contain statements, the proof of which is merely hinted at, or briefly outlined, by the author. I note however that it is questionable whether Jack is knowledgeable enough to adopt this practice successfully.
In both cases the students doubt the value of the tutor's effort, perhaps because they see it as a bureaucratic hindrance to the unfolding of their argument: whether the theorem has been proved in the course or not, it can actually be proved and that makes their proof for CD7.2 valid. The tutor realises the undermining power of the students' thoughts and presents them with a less questionable proof, disentangled from matters of principle and from conventions.
Conclusion: In the above, evidence was given of the tension between novice and expert practices. In particular:
• the expert appeared capable of distinguishing between different approaches to a proof relevant to the notion of derivative and of expressing a preference for a proof via first principles (via using the binomial theorem), as opposed to a proof by mathematical induction, on historicoepistemological grounds.
• the novices appeared alerted to the dangers of not clarifying the foundations of their proofs but in practice they turned out to be confused and weak in doing so. Especially one student expressed concern with 'assuming limits'.
• the novices generalised the finite case of lim(a+b+c)=lima+limb+limc into limS=Slim which is actually a double limit, thus disregarding unconsciously the conditions for the exchange of limits.
• the novices acted with suspicion and hesitation towards the tutor's request to consider the limited validity of the theorem via a counterexample:
- one of the students objected to considering the cases where the theorem does not hold since it holds in the particular case of the Episode (CD7.2)
- the other student negotiated his right to use elliptic arguments (a common professional practice).
• the expert appears capable of distinguishing between expected practices in different domains: for instance, in applied mathematics, retrospective use of unproved results is allowed as opposed to foundational courses where it is not.
The conflict can be seen as an enculturation process during which the novices learn about certain conditions of the didactical and epistemological contract in common mathematical activity: in the interaction it is revealed that they have been unconsciously assuming the validity of exchanging double limits, using thus a theorem which is yet unproved in the course. What the novices seem to be learning here is that they need to exercise control over their mathematical reasoning in order to avoid subconscious and unjustified decisions and that this skill to control is a characteristic of appropriate mathematical behaviour.