Section (viii) The Contrast Between Novice and Expert Approaches to Mathematical Reasoning. The Example of a Convergent Series
Context: See Extract 7.8
Structure: In the following, the students present, or talk about, their evaluation of four infinite sums. Finitist attitudes come to surface as well as the contrast between the expert tutor's sophisticated and embedded approach to the evaluation and the novices' arithmeticised and decontextualised technique.
The Episode: A Factual Account. See Extract 7.8
An Interpretive Account: The Analysis
The Novices' Finitist Attitudes Towards Infinite Sums. In this Episode, Cliff's and Cathy's attitude towards infinite sums is, in brief, to treat them as finite sums. The students subsequently apply a wide range of arithmetical operations on these finitised infinite sums:
• Cliff 'splits up' å 1/r(r+k) as å (1/kr - 1/k(r+k)).
• Cathy 'breaks' the (-¥)-(+¥) sum in two: (-¥)-0 and 0-(+¥). Then she removes || and calculates the two infinite sums.
• Cathy on å r2/3r: r2=r2-1+1=(r-1)(r+1)+1 and breaks the infinite sum accordingly. Since å 1/3r=1/2, she turns to calculating å (r-1)(r+1)/3r which she rewrites as the sum of its term at zero plus the sum from 1 to ¥. Breaking the infinite sum once more leads her to obtaining 1/3å r2/3r on the right hand side of the equation. Finally by calculating å 2r/3r+1 she reaches the result 3/2.
The students' treatment of the infinite sums, which are limits, as finite quantities is illustrative of the students' attitude towards å and the ease with which they use the notion of rearrangement. Didactically, the danger of the overextended use of the 'right to rearrange' can be proved to the novices via exposure to the large number of cases where it does not hold. As seen in cases like continuity and differentiability, the novices' impression that infinite sums can be broken, rearranged etc. reflect their finitist views of infinity. It also reflects culturally and epistemologically embedded conceptions, or primary mathematical intuitions, about certain mathematical properties, such as the differentiability of all continuous functions, that permeate through the history of mathematics. Teaching, which is oriented towards the overcoming of these epistemological obstacles, can influence the novice's mathematical growth away from these conceptions. On the contrary, the novices' constant and biased exposure to sums that can be broken and rearranged, such as the ones in this Episode, is likely to result in the perpetuation of these conceptions.
The Contrast Between the Expert's Embedded and Sophisticated Approach and the Novice's Decontextualised Technique. Cathy's way of evaluating the sum in SS7.1iv is a refreshing, back-to-arithmetical basics approach. It is not terribly elegant (a few of her 'moves' are repetitive and circular such as writing r2 as r2-1+1, moving 1/3 inside and outside the å several times, etc.) but it is pragmatic and straightforward. It has the feel of handy arithmetic and does show skill and imaginative capacity. I note however that only ostensibly Cathy's solution is basic and arithmetical (the only piece of previous knowledge she explicitly employs is that å1/3r=1/2). This is a deceptive appearance since, behind Cathy's rearrangements, lies the theory that makes them possible. What Cathy seems to be doing here is unconsciously reducing infinity to the finite rules of a game she knows well, namely manipulating algebraic quantities.
On the other hand the tutor's approach is a formal and elegant shortcut in resonance with the material the students have been taught at lectures and the techniques they will need. It is, in other words, a contextualised choice of technique which is generalisable to a large number of infinite sums. It has the benefit of hindsight and of globality. It shows an expert handling, an informed awareness of the facilities available to the craftsman (åxr=1/1-x, letting f(x) be 1/1-x, calculating f' and f'' and noting that f'' can be written in terms of f and f') as opposed to Cathy's decontextualised, hence slightly primitive approach.
None of the above is meant to diminish Cathy's efficient approach which (the dangers of naive rearrangement of the terms in a series aside) yields the correct answer. It only aims at highlighting the inclination of the novice to resort to familiar (here: handling of algebraic expressions) modes of operating at the expense of adopting new, potentially more contextualised and efficient ones.
Conclusion: In the above, the novices' inclination to treat infinite sums as finite quantities was demonstrated and attributed to deeply embedded epistemological beliefs and to the novices' biased exposure to infinite sums that can be harmlessly evaluated with finite techniques. Moreover two approaches to the evaluation of an infinite sum were juxtaposed:
• the novice's basic and arithmetical finitist one, and
• the expert's contextualised, concise and sophisticated and, possibly generalisable to a number of cases, one.
The novice's attitude was attributed to a habitual regression to familiar modes of thinking (manipulation of algebraic quantities) despite the novel experiences of alternative, newer techniques.