Section (vii) The Gap Between the Novice's Advanced Algorithmic Behaviour and Inadequate Conceptual Understanding. An Example from an Application of the Taylor Series
Context: See
Extract 7.7Structure: Under the firm instruction of the tutor, the students execute the calculations involved in the manipulation of the inequalities in B10. At one instance the students suggest a differentiating technique (for finding the maximum of a variable quantity) with which they are familiar from school but the tutor prefers a more basic, algebraic alternative. The proof is completed with very closed questioning.
The Episode: A Factual Account. See
Extract 7.7An Interpretive Account: The Analysis
Part a of B10 requires a reasonable amount of good algorithmic behaviour: it is about the cautious substitution of x3 for f(x) and 2 for n. Part b however requires going slightly beyond this mechanism of substitution and engage in the manipulation of the Taylor expression for g, g' and g''. This qualitative differentiation in the question passes rather unnoticed by the tutor whose words might perpetuate the students' impression that simple algorithmic behaviour is still sufficient for part b. The tutor's surprise that the students have not completed a question for which 'there is nothing much you can do' is a trivialisation with which the confused students possibly would not agree. Beth then appears ready to 'translate' the Taylor expansion in part a for function g in part b without being concerned for the interval or the point around which the expansion is taken.
Hence, for Beth, the Taylor series, a tool for approximation, is reduced to a machine of plugging in values, detached from possible uses or meaning. With the tutor's instruction, she manages to define the interval and the point around which the expansion is taken and to calculate the expansion for g. Subsequently Beth's observations are correct insights which however she cannot prove.
Cary's suggestion to use a differentiation technique in order to identify the maximum of 2/h+h/2 is in classic contrast (in this chapter it has been repeatedly pointed out) with the tutor's preference for an algebraic technique, based on a few arithmetical principles. I note then that her first attempt to find such a technique fails and the tutor returns to Cary's suggestion. Unlike other occasions the tutor seems here to change her preference for a first-principles approach to a theorem-quoting one a bit more easily, probably under the pressing immediacy of Cary's suggestion. In principle, she may prefer a more basic proof but she does not deny the convenience of a practical alternative, even if it is a philosophically less acceptable one.
Finally, Beth again exhibits considerable facility with algebraic manipulations when towards the end of the Episode contributes substantially to the proof that for all x in [0,2] it is possible to choose h with |h|=1 so that x+h is in [0,2].
Conclusion: In the above, evidence was given of the contrast between the students' relative facility with algorithmic behaviour (the action of algebraic manipulations) and more procedural and conceptual understanding of the Taylor Series as a tool for approximation. Also part of the significance of the instance lies in the short juxtaposition between the tutor's philosophical preference for proofs based on first principles and one of the students' suggestion for using a more sophisticated (but informally familiar from school) tool — identifying the maximum of a variable quantity via differentiation.