Section (v) The Contrast and the Gap Between the Mechanistic and the Conceptual Approach to the Notion of Derivative

An Interpretive Account: The Analysis

The Mechanistic As Opposed To The Conceptual Approach To The Notion Of Derivative. Camille appears absent-minded in the beginning of the session: for instance she interferes in order to modify Eleanor's suggestion for B6i (from n=2 to n>2) and follows her correction up with a proof for n=1. Moreover in B6ii, whereas Eleanor has given the answer (n=2) before they start the proof, Camille interrupts the tutor in order to ask how they knew 'it was n=2 straightaway'. The tutor's wry answer ('because Eleanor knows the answer and I know the answer!') overshadows the possibility that Camille's comment is not simply a sign of absent-mindedness but a genuine expression of discomfort with discovering the answer intuitively and then proving it formally. The tutor, in order to confirm that Camille is following the discussion, asks for the reason they are looking at this limit. Camille's immediate responses indicate that she is following the argument of the discussion.

That Camille has conceptual preoccupations becomes evident in C1. The tutor's response, T1, is rather unbalanced in its emphasis: the tutor does give the mathematically correct answer that the derivative might exist and not necessarily be continuous, but she also unnecessarily emphasises that, in this case, the derivative is continuous. Later she adds the necessary condition 'as long as we make sure that this exponent is at least one' but at that moment her statement sounded absolute. C2 then is a sign of the following misunderstanding:

The tutor's response with regard to the particular function causes Camille to repeat her question explicitly this time. Camille's responses subsequently demonstrate that she knows definitions and calculational methods but it is the tutor who highlights the next logical step in the proof for B6ii of the question: it is now necessary they prove that for n=1 the derivative at zero does not exist. This idea appears to be elusive to the students who, having found the right answers in B6ii and B6iii, did not complete the proof formally since they did not cover all the cases. The scene will be repeated a bit later in B6iii.

Another noteworthy occurrence in B6ii is Eleanor's claim that cos1/x²®¥ as x®0 (the limit does not exist). The tutor corrects Eleanor, and proceeds without any comment, but it would have been interesting to explore the students' fusing the perceptions of tending-to-infinity and not-tending-to-a-limit. Interestingly this conceptual puzzlement does not seem to affect the student's performance who, with ease and precision, suggests the method for the next part of B6 (calculate the derivative and then find its limit at zero). Both students calculate the derivative impeccably.

The next piece of evidence that, despite evident technical proficiency, their conceptual understanding is not at all at the same level, comes from Camille who calculates the derivative of xⁿcos1/x² efficiently. In the end she adds '"n=2'. Since B6iii follows B6ii, in which the existence of the derivative at zero has been proved "n=2, Camille extends this claim to the existence of the derivative for any x. This is not the case since the derivative exists "nÎÀ for x¹0. Camille's reaction is a sign that there is a gap between a conceptual approach to the notion of derivative (determining the value of the derivative point by point by calculating the limit of the ratio) and the mechanics of the algebra of limits/derivatives (here for instance (fg)'=f'g+fg'). It is again frustrating that the tutor does not point out the connection between the two approaches and leaves the two concept images of derivative dissociated from each other.

For not very visible reasons, Camille's response to the tutor's next question (under which condition for x, her calculation of the derivative is correct ) is x>0, whereas the only value of x that must be excluded from the derivative of f_n is zero. However, that Camille has understood the mechanics of the question, is clear from her eloquent 'the question is to find for which n to have the limit of this derivative to be zero' and her answer and explanation to the question (n=4). Then again, in a comment that sounds distracting from the impression of clarity and eloquence that Camille gives, she asks what would happen if x>1 — ignoring that the limits involved in the process are all taken when x®0. Here it is possible that Camille's concept image of the domain of f_n being Â-{0} (x can be anything other than zero; therefore it can be >1) overshadows her concept image of the limiting process which requires x to be taken as close to zero as possible. Even though x can be greater than one, the values of f_n for x>1 are irrelevant to the question. Another instance of this possible conflict between the point-wise/static and the procedural/ time-dependent perception of the variable in a function that I encountered in these tutorials is the following: when discussing the limit of a sequence s_n (lim_n®¥s_n), some students ask what the limit is if n®0. In this case wondering about the values of s_n as n®0 is not simply irrelevant, as in the case above, but is also evidence of an incomplete image of the domain of the sequence s_n (nÎÀ).

Subsequently, as commented in an earlier paragraph, the tutor reminds to the students that, despite having found that the derivative is continuous at zero for n=4, they still have to exclude the cases for n=2 and n=3.

Then Eleanor claims that the second part of f'₂ (namely (2sin1/x²)/x) tends to infinity. It is likely that Eleanor has been applying here the argument that was used repeatedly in today's tutorial. That is: Likewise,

x_ncos1/x²®0, because |cos1/x²|<1 and x_n®0, as x®0

(1/x)sin1/x²®¥, because |sin1/x²|<1 and x_n®¥, as x®0

which is again a mechanistic and inaccurate transfer of argument. Camille appears to have a more concrete image of the situation at this moment. She notes about (1/x)sin1/x²: 'But it's not quite vast'. In her unconventional choice of words, I think, lies a conception of limf(x)=¥ as f becoming 'vast', taking very large values.

Before closing I would like to highlight, on a socioaffective note, Camille's persistent attention-grabbing interventions. In this last case Eleanor, whose shyness is reflected in her mumbling, has suggested that (1/x)sin1/x²®¥. The tutor hasn't heard that and asks for a repetition. Camille, who has heard what Eleanor said, comments upon Eleanor's claim. The tutor, annoyed, bursts with T9. Camille then repeats what Eleanor said (most students when confronted by the tutor in this manner would be quiet and let Eleanor repeat). Camille's persistence has repeatedly caused friction between her and the tutor who seems to prefer more submissive students.

Finally I only briefly point out the tutor's metaphor on well-behaved (continuous) and badly-behaved (discontinuous or tending to infinity) functions. She says about f'₂: 'the first part is 'well-behaved and you cannot quite cancel the bad behaviour of' the second part'. And she adds:' If they both behave badly they might somehow cancel out the bad behaviour...but that's not gonna happen here'.