Section (vii) Transforming 'beautifully literary' Intuitions Into Mathematical Formalism
Context: See
Extract 8.7Structure: This is a proof and refutation session in which the tutor twice pursues the students' suggestion which turn out to be successful. In this way, the tutor's act is to propel the transformation of the students' intuitive ideas into formal mathematical reasoning.
The Episode: A Factual Account. See
Extract 8.7An Interpretive Account: The Analysis
Transforming 'beautifully literary' Intuitions Into Mathematical Formalism. The interaction of Cathy and the tutor — George's intervention is mathematically crucial but the psychologically rich aspect of the interaction involves mainly Cathy — can be deconstructed as a sequence of actions in which
• the novice has the necessary global grasp and intuitive ideas but is unable to organise them to an effective proof
• the expert organises these ideas in a demystifying process during which the novice learns about LA7.35 as well as about thinking in formalistic terms.
This is the theme of this Section and the analysis here aims at illuminating the details of the interaction from this point of view.
The focus of Extract 8.7 is psychological/affective. Cathy demonstrates a lack of confidence in the rigour of her thinking (not contented with her rigour in the beginning / 'solution then might be wrong' / refuses repeatedly to translate her intuitions to proofs / 'is this formal enough?' she says reluctantly later). These words by Cathy make the Extract 8.7 look like an attempt by the tutor to boost her confidence by flexing her cognitive muscles as well as encouraging her emotionally: Cathy does formalise when encouraged and when clearly introduced to the tools of mathematical formalism. The inseparability of cognition from the affective factors under whose influence it takes place here is evident.
The tutor's intentions seem to be to co-ordinate optimally the students' suggestions as well as act as an expert initiator into the codes of the mathematical culture (in particular as far as the mathematical reasoning used is concerned and as far as the rules of propriety that govern mathematical expression: logic and notation).
Expert Initiation Into the Codes of Mathematical Culture: Mathematical Expression and Notation. A priority in the tutor's agenda seems to be reforming Cathy's written expression of her ideas and definitions. Characteristically Extract 8.7 starts with two such attempts for reform: her writing for ImT and ImT2.
For the first, Cathy has used x to denote an element of V as well as ImT on the grounds that both vectors 'come from V'. This can be misinterpreted as x=Tx which is not what she aimed to say. Moreover her expression 'for v in V' can be misinterpreted as the universally quantified expression "vÎV, while she meant $vÎV. The tutor's critique leads her to more acceptable writing introducing her thus to a notational convention of formal mathematical writing.
Similarly, her writing of ImT2 seems to be a symbolic representation of her rather illustrative idea that T2 is 'applying T twice'. With the tutor's suggestion to think of T2 as a transformation of V, her thinking is clarified and her reconceptualisation of T2 leads to an acceptable expression. In this case the tutor intervenes in order, not merely to modify a problematic representation of a good intuitive idea, but also to channel Cathy's perception of T2 to a direction which is more appropriate for the context of LA7.35.
Once the issue of Cathy's problematic symbolic representations of definitions has been settled, the focus of the interaction seems to shift to the transformation of her 'beautifully literary ideas' into mathematically formed argumentation. Cathy's 'ImT2 has less vectors than ImT' is a rather inaccurate and finitist verbalisation of ImT2ÍImT — in fact in the realm of infinite sets cardinality and Í are two issues that cannot be treated adequately with colloquialisms such as 'less than'. For intuitive reasons that possibly relate to the simplicity of the definition of kernel — the elements of a vector space mapped to zero — Cathy finds the proof for kerT£kerT2 'easier to do formally'. As a result, C1 — despite its containing the same 'less than' expression as previously — turns out as a verbalisation of the heart of the argument. The student's question in the end of C1 — 'Is that formal enough?' — illustrates vividly the pressing sense of obligation towards mathematical formalism that the novices begin to feel.
The second time that the interaction between the students and the tutor takes the form of a transformation of intuitive verbalisations into formalism is with C2 and C3. Cathy seems to have grasped the idea that if kerT contains an element other than zero, then the construction of another element is possible which is contained in kerT2 and not kerT, thus contradicting b. It takes, however, the tutor's insight that Cathy's idea is more than an unjustified guess — and thus it is worth pursuing — and George's contribution to transform Cathy's 'it seems true but...' into a proof for bÞa. The tutor at first organises C3: let vÎImTÇkerT, v¹0. Cathy interprets that as Tv=0 and then as the correct but rather circular T2v=0. The students' impasse, with regard to how vÎImT can be exploited, is resolved by the tutor's reminder that vÎImT is an 'existential' statement:
$wÎV such that v=Tw. Having this relational statement available, Cathy moves on swiftly to her next successful verbalisation, C4. The tutor, again in charge of conditioning the novices' expression within the boundaries of mathematical propriety, translates C4. This accelerates their arrival at the conclusion — which also is due to another tutor intervention with the theorem that, if two subspaces intersect trivially, then the dimension of the sum is the sum of the dimensions.I note that, in the above, the students' weakness in dealing dynamically with the definitions of kerT and ImT, despite Cathy's eventually successful writing of their definitions in the first part of the tutorial, reflects a possible gap between their relational and their instrumental understanding of the concepts.
Finally, C5 is another verbalisation of a suggestion for a proof (aÞb). Its relevance to the novices' preferred approaches to mathematical reasoning makes it more appropriate to be mentioned in the following.
Expert Initiation Into the Codes of Mathematical Culture: Mathematical Reasoning and Proof. Unlike other cases, where extremely closed questioning seemed to stifle the novice's possibility of making connections, here the interaction, even though characterised by the tutor's dynamic interventions, is more prone to discovery than elicitation. The fruitful learning outcome of this approach is reflected in the students' success to reproduce the argument used for kerTÍkerT2 in their proof for ImT2ÍImT; also in George's seeing the analogue between the proofs for bÞc and cÞb.
As far as the cyclic process of proving the equivalence of propositions a, b and c (aÞbÞcÞa) — in contrast to evidence from other tutorials where the students found it difficult to come to terms with the logic of this cyclic process — Cathy's (and George's to a lesser extent) difficulties lay elsewhere: in the formalisation of their ideas for instance as explained above. George appeared unsure about where to start in the chain of proofs and how to continue but had a plan even though he did not manage to pursue it. The tutor himself attributes a great deal to an initial intuitive approach: b and c look the same so maybe they are of the same degree of difficulty. He then moves on to organising the information from the Rank and Nullity Theorem in a way that appears relevant to the data of the question.
Initiation into acceptable and viable ways of mathematical reasoning culminates in the part of the tutorial relating to the proof of aÞb. In C5 ('go backwards') and C6 Cathy demonstrates an approach to mathematical arguments widely preferred by novices: contradiction by assuming the negative form of the argument to be proved. The tutor, for the first time in Extract 8.7, intervenes more dramatically with criticising Cathy's suggestion, not as incorrect but as unnecessarily negative and proceeds with his own suggestion. Since the tutor is not explicit about the reasons that make proof by contradiction less acceptable, it is questionable whether Cathy comes to any rationalisation of why her suggestion is not followed.
Finally T1 is strongly graphic evidence that the tutor is determined to intervene dynamically in the students' approach to argumentation: emotionally laden expressions (the impact of the word 'surely' in an appropriately firm tone of voice) do not support or reinforce a mathematical argument which ought to be grounded on proof.
Conclusion: In the above, the novice appeared to have the necessary global and intuitive grasp of a proof but, even though alerted to the requirements of formal expression, she was unable to organise her ideas effectively. The latter was propelled by the tutor in a demystifying process during which the student seemed to learn about the particular proof as well as about thinking in formalistic terms. The interaction between tutor and students was both cognitively and affectively intense. This highlighted the inseparability of cognition from the affective factors. The Episode stands as a metaphor of the novice's initiation into the codes of expression of the mathematical culture by the expert.