Section (i) First Steps of Initiation Into Mathematical Formalism: Meaning and Proof of the Archimedean Property
Context: See
Extract 6.1Structure: After several attempts the students and the tutor agree on a formulation of the ArchPr. With the help of Andrew, Jack completes a proof of the property on the b/b. The tutor critically reviews his writing style and they discuss alternative formulations of the property.
The Episode: A Factual Account. See
Extract 6.1An Interpretive Account: The Analysis
Formulating the ArchPr. The ArchPr says that "xÎÂ $nÎÀ x<n. A1-J1-A2 feature the students' attempts to reconstruct a formulation of the ArchPr and T1-T2 are the tutor's refutations of these attempts. A1 is a partial reconstruction of the ArchPr that maintains its logical structure (the two quantifiers and the inequality) but not its meaning. J1 is an attempt to defend A1 by adding more information on y. A2 is a response to the tutor's objection to J1, that however close to x Jack chooses y, he can choose y even closer: what Andrew says is that in this case he can get even closer to x than the tutor with his new y. The rather meaningless circularity of the students' arguments is dismissed by the tutor who halts the interactive process by asking them to look up an acceptable formulation of the ArchPr. I note here that that same week other students have had difficulty with reconstructing the ArchPr. An illustrative attempt is student Connie's in Extract 6.6: 'The Archimedes' Principle isn't saying that there exists this number >1, bigger than any number?'.
The mathematical meaninglessness of the students' attempts does not however deprive the interaction of its didactical interest as a dynamic interactive process of conjectures and refutations. The dynamics of this process are intriguing because the students are both concentrating on reconstructing the ArchPr as well as to defend themselves and each other to the tutor. J1 initiates the mutually supportive cognitive action of the students and A2 accentuates their complementarity. So in a sense their exchange of words highlights both cognitive (attempt to reconstruct the ArchPr) and socioaffective (support each other and self in order to confront the tutor's objections) aspects of the students' thinking process.
Subsequently the tutor's question about what would be the implication if the ArchPr did not hold is an attempt to highlight its meaning through its negation ($aÎÂ " nÎ À n£ a). Possibly Andrew tries to negate the ArchPr in his mind and, carried away by n=a and ignoring the quantifiers determining n and a, claims that the negation means that the real numbers are bounded above. In this and subsequent chapters the novices' difficulty with negation and quantified statements will be repeatedly mentioned.
Proving the ArchPr: an Interactive Procedure. Jack suggests reaching contradiction by assuming the ArchPr is false. Then À will be bounded above and, since it is non-empty, by the Completeness Axiom it will have a supremum. He then seems unable to continue. Andrew suggests then applying the definition of supremum for e=1. Jack, preoccupied with what he has written on the b/b so far, ignores the suggestion and repeats the definition. The tutor intervenes in order to recommend the use of Andrew's suggestion. In the next few seconds Jack overcomes the impasse he appeared to have reached and completes the proof. Jack accomplishes at first to construct accurately the supposition that the ArchPr is false. This supposition implies that À has a sup. He also reconstructs the definition of supremum but he cannot co-ordinate the two elements (falsity of ArchPr and definition of sup) so that the contradiction emerges. The co-ordination finally takes place in the few seconds of him pausing to think after the tutor insists that he uses Andrew's suggestion. So, in a sense, Andrew and the tutor have discreetly scaffolded Jack's final overcoming of the impasse without heavy-handedly imposing it upon him. With their interventions they have created what appears to be suitable for Jack conditions that helped him find his way out.
Proving the ArchPr: on Jack's writing. The complementarity and mutual support of the two students is also illustrated in Andrew's defending Jack's writing with 'this is what the lecturer wrote'. The problem though is that Jack has been reproducing on his draft and then on the b/b what, according to the tutor, is the lecturer's b/b technique. The tutor encourages the students to modify this abbreviated, semi-colloquial writing style to a more flexible, narrative and descriptively richer writing style. In fact the tutor's critique reflects a contemporary trend in the didactics of advanced mathematics according to which the learners are encouraged to express their ideas in words and not simply in quantified and categorical notation. The reason I present the tutor's critique in detail is that it reflects the use of formalism by the novices which at this stage can be seen as awkward. It also illustrates how the students are urged with lessons like this to a wiser interpretation and manipulation of mathematical formalism.
For instance Jack's use of quantifiers in $ supÀ Î Â is rather problematic: it seems to be a literal translation of 'the supremum of À exists'. As far as the use of ne is concerned, the tutor's comments have a more controversial effect. This is a notation, he claims, that might induce the impression that there exists a function between e and n. I think that it is equally likely to induce the impression simply that n depends on e. Jack seems to be disturbed by this ambiguity as well as Andrew who reckons whether this is a good reason to avoid this notation.
The whole Episode stands as a metaphor for the maturation process which the novices must go through in their first encounters with mathematical formalism. In the above, the tutor is not simply teaching a proof of the ArchPr: he is setting up an initiation into the idiosyncratic precision and rigour of mathematics as
• his critique of Jack's writing,
• his speech on the essence of the ArchPr and how it affects the system within which the Archimedean mathematics works, and,
• his cautionary remark towards the end on the naturality of defining f to be 1/e (and not e to be 1/f)
illustrate. The novices are naturally overwhelmed by the dazzling caution for detail that this new approach to mathematical thinking (and expressing) entails.
Conclusion: In the above, the formulation and proof of the ArchPr stimulated a student-tutor interaction process mostly characterised by the tutor's enculturating interventions with regard to the students' reasoning (handling quantified propositional statements: negation and co-ordination to reach contradiction) and expression. From a didactical point of view, a dialectically successful instance was examined where the tutor and a student discreetly scaffolded another student's overcoming of an obstacle in a proof.