Section (viii) An Example of the Tension Between Novice and Expert Approaches to Mathematical Reasoning: The Need to Learn How to Achieve Mathematical Resonance by Creatively Co-ordinating and Manipulating Relevant Knowledge
Context: See
Extract 9.8Structure: This Extract follows Extract 7.8. Cathy presents her proof and the tutor suggests an alternative. The two proofs reflect some of the differences between a novice and an expert approach to mathematical reasoning.
The Episode: A Factual Account. See
Extract 9.8An Interpretive Account: The Analysis
I think that it bears some significance that Cathy is a bit reluctant to present her solution in the beginning of Extract 9.8; given that Extract 9.8 follows the Extract 7.8, it is possible that she begins to suspect that, though correct, her approach is not exactly up to the standards of elegance and resonance with the material she has been taught recently. This was the crucial point made in Episode 7.8. Possibly under the influence of the discussion there Cathy makes the rather aesthetic comment ' you may not like it' which can be seen as a sign of a developing taste for a certain mode of reasoning.
Similarly to Episode 7.8 Cathy resorts to a solid arithmetical handling (to prove that integers a and b are equal, it is sufficient to prove that a=b and b=a). The tutor on the other hand employs a theorem and invests the arithmetical relationship given in the question (hcf(o(x),o(y))=1) with its group-theoretical meaning (<x>
Ç<y>={e}).Again, the juxtaposition highlights the differences between an expert and a novice approach. There are some redundancies in Cathy's way as well as some unclarified points — not necessarily visible in the abbreviated version in Extract 9.8. Most important though maybe her starting to conceptualise the need, beyond the merely aesthetic, for an embedded, contextualised mode of reasoning and for an organically connected argumentation, in a way which will turn the coherence and connectedness of mathematical theories to her benefit. In other words via this juxtaposition of approaches, she might begin to learn about the benefits of mathematical expertise.
Conclusion: In the above, a novice manages with minimum principles (mostly arithmetical), albeit not in an impeccable formal way, to reach the conclusion and complete a proof. This approach is juxtaposed with the tutor's proof which is well embedded in the mathematical context of the course (lecture content, problem sheet material, textbook approaches) and thus shorter and more relevant. Therefore as an example of the differences between novice and expert approaches to mathematical reasoning, this Episode is evidence to a novice characteristic: that, even when undoubtedly imaginative and skilful, novices have not acquired yet the skill of association and co-ordination of relevant knowledge which equips mathematical reasoning with the power of resonance.