Section (ii) The Novel Notion of Continuity: Proof By Definition Or With the Algebra of Limits

Section (ii) The Novel Notion of Continuity: Proof By Definition Or With the Algebra of Limits. A Battle of Ambivalent Preferences and the Cognitive Effect of a Hidden Agenda

Context: See Extract 7.2

Structure: In the following, the tutor leads Kelle through a proof for CD5.1i by, first, graphing intx and, then, by definition. Subsequently, for the rest of the question, she changes her strategy by suggesting using the algebra of limits. Kelle's unease with this change — as well as several difficulties with the novel notion of continuity — then come to the surface.

The Episode: A Factual Account. See Extract 7.2

An Interpretive Account: The Analysis

Throughout the Episode the tutor seems to recommend quite uniformly a strategy for tackling problems with limits:

make a drawing

find out about the limit

prove the limit formally.

In these tutorials this recommendation is very common. The students usually receive it with hesitation: in a variety of other cases they are asked not to rely on pictures because pictures are inaccurate. Also guessing a limit is a rather vague, non-specific suggestion. As a result this type of recommendation seems to be adding a mystifying veil to the already obscure notion of limit. In this Episode however the use of pictures is not as problematic as the tutor's ambiguous and unjustified alternation between different approaches to the third step of the above procedure: proving the limit. I note that ambiguous and unjustified here are not used as criticisms of the tutor's mathematical handling of CD5.1: her alternation between different approaches springs from her long mathematical experience according to which it is sometimes convenient to determine the limit by using the definition, or side limits, or the algebra of limits. But she does not justify this alternation of approaches to Kelle from whose perspective this ambiguity maybe interpreted as the tutor contradicting herself.

Kelle in the beginning of the Episode complies with the recommendation to look at the drawing and notices the discontinuity at the integers. The tutor accepts the observation and transforms it into a specific question:

look for lim_x®x₀f(x) when x₀ is or is not an integer.

For the latter case, Kelle switches to d -e mode and tries in vain (K1) to reproduce the formal definition of limit. With the tutor's prompting he realises that he first has to define f(x), then f(x₀) and then look for the limit. Looking at fig.2a makes him lose generality and reply on the basis of looking at the interval (0,1) instead of (n, n+1) twice. K2 then reflects Kelle's preoccupation with whether

Proving that f is discontinuous at the integers

is equivalent to

Proving that f is continuous except at the integers,

which is not elaborated upon by the tutor who proceeds with proving the other case for the limit. Consistent with her declaration of strategy she insists on using the definition (as opposed to Kelle's suggestion for side-limits, probably evoked by fig.2a). Her reason is a preference for avoiding 'quoting big theorems' and again she recommends looking at the drawing.

Kelle complies and contributes to the construction of the drawing for CD5.1ii. Maintaining her control, the tutor outlines the available strategies: formal definition or the algebra of limits. Kelle, influenced by his tutor's expressed preference so far, suggests using the definition but then she announces they will use contradiction and the algebra of limits. Kelle complies again and tries to construct an argument by contradiction but, as indicated earlier by K1, he is in trouble with articulating formal propositions and he fails (he vaguely starts by not negating the proposition). The tutor corrects his faltering logic and states the argument for the contradiction. He adopts the argument and, still influenced by his tutor's expressed preference for the formal definition of limit, suggests its use. The tutor reminds him of her newly expressed preference for the algebra of limits and takes over.

K3-K5 is evidence that Kelle, not confidently and far from independently or efficiently, begins to be able to reproduce the tutor's strategy as exhibited in CD5.1i-iv.

The Essential Role of 'Pictures'. The tutor repeatedly invites Kelle to use 'pictures' and 'intuition' in order to understand the limiting process. Like other tutors she encourages the novices to engage in a 'guessing' intuitive activity before proceeding with formal reasoning. It is very likely that Kelle's inability to start CD5.1 lies in his not using this recommendation which is the only technical (but ambiguous) advice he has received with regard to the exploration of limits. It is remarkable that, once given the 'picture' of intx, Kelle immediately observes its points of continuity and discontinuity. I note here that his two mistakes (that f(x)=0 and that x lies between 0 and 1) can be attributed to the fact that he was looking at the specific interval (0,1) and not at the general (n, n+1) suggested by the tutor. This underlines I think the novice's internal fixedness to specificity and it addresses the tension between the tutor's pressure for statements of generality and the student's literal interpretation of the information contained in the graph. It also raises the issue of the role that powerful but constraining representations exert on thinking.

The Problematic Handling of Formalism: Definitions and Logic. In this Chapter the students often try to reproduce the formal definitions of limits and continuity they have been given in the lectures. K1 is a very typical example of a novice's attempt to verbalise a formal statement which is disconnected from his concept image and is in striking contrast with Kelle's immediate grasp of the limits involved in CD5.1i while observing the graph of intx. So in K1

• he initially omits the universality of the definition of limit "e,

• he confuses the necessity of finding a d with finding an e,

but

• he reproduces f(x)-l<e recalling then that the inequality must hold "e.

By recalling the "e at the final stage, he misses the link between d and e, namely that introducing d and e is a way of quantifying the limiting process: as x approaches x₀, f(x) approaches f(x₀).

Similar difficulty with a formal expression is exhibited later when Kelle tries to formulate the argument for the contradiction in CD5.1ii. However the student demonstrates a remarkable caution for detail when he wonders whether proving that f is discontinuous at the integers is equivalent to proving that f is continuous except at the integers: this is a quite fine linguistic distinction to be made by a novice who has just seemed to be incapable of providing an adequate formal definition of continuity. Assuming that making this linguistic distinction reflects a clarity of his global vision of the proof's structure then this clarity is in contrast with his

struggling and rather failing attempts to use the d-e definition of limit, and,

his inability to negate a formal proposition in order to construct a proof by contradiction.

This contrast, as evidence in this Chapter repeatedly suggests, is characteristic of Kelle's novice behaviour.

First Principles Versus Theorem Quoting: a Battle of Ambivalent Preferences. In the Episode the tutor is seen as altering her preferred strategy for finding the limits in CD5.1. What the tutor may tacitly aspires to convey is the need for flexibility to adapt to a mathematical question's specific needs and to master the use of multiple approaches. What I think Kelle has received though is very ambiguous instructions in which the rules of how to choose an approach (First Principles/Definitions or Theorem Quoting) remain unarticulated. Therefore it is likely that the process of alternating among the various approaches remains mystified for Kelle. There is no actual contradiction in preferring FP to TQ in one case and TQ to FP in another; it is an entitlement and a virtue of the mathematician to master the skill of optimal preference. But it is a didactical action that induces a rather uncreative uncertainty and confusion. Kelle in other words is cognitively a victim of his tutor's hidden agenda.

Changing the strategy to using the algebra of limits and switching to the use of proof by contradiction seems to be a more powerful shift than Kelle can handle and the discussion degenerates into exposition by the tutor for both CD5.1ii and iv. Interaction is re-established for CD5.1iii but despite Kelle's participation in K3-K5 due to the tutor's exercise of firm control and inclination to exposition it is very difficult to evaluate the learning outcome and confirm that Kelle would have been able to carry further his observation that intsinx is discontinuous at the points where sinx = n and transform it into a proof.

Conclusion: In the above, the novice's thinking with regard to the newly introduced concept of continuity and the formal definition of limit seemed to be influenced by the following three elements:

• the role of visualisation: the use of pictures in guessing limits; pictures as specific representations potentially opposing generality,

• the role of formal notation and logic as a vehicle for mathematical arguments: the difficulty of evoking the formal definition of limit; the difficulty of constructing a logical statement of an argument for a proof by contradiction; preoccupation with the use of the term not-continuous as equivalent to discontinuous.,

• the juxtaposition of proof by theorem quoting and proof by first principles: the effect of a not overtly justified alternation using the definition of limit and using the algebra of limits.

Didactically, the dialectics of very closed questioning and of constant reclaiming of control on the part of the tutor have proved unhelpful in supporting the student's conceptual development with regard to the concept of limit. Also, methodologically, the tutor's style has been interfering with the researcher's attempt to understand the cognitive state of the student's thinking.

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