PART III A Synthesis of the Findings in the Area of Foundational Analysis

PART III A Synthesis of the Findings in the Area of Foundational Analysis. Indications for the Cross-Topical Synthesis in Chapter 10

In this chapter the students' first experiences of mathematical formalism were explored in a series of Episodes from the first weeks of observation. As noted in the Interlude, the concept that emerged as paradigmatically problematic in these first weeks was the notion of supremum/infimum [1, 4, 6, 7]. Below I list a number of the novices' problematic perceptions mentioned in the Episodes:

• a set contains its inf,

• ignoring the second condition of the definition of supremum (Approximation Lemma),

• perplexity with the alternation of the terms 'greatest', 'least', 'upper' and 'lower' in the definitions of sup and inf,

• a number smaller than the supremum of a set, must necessarily be in the set.

Given the epistemological relevance of sup and inf to the notion of limit, these difficulties can be also seen as a prelude to their difficulties with limits and continuity explored in Chapter 7, whether this limit is a finite number or infinity [5]. Also, since the notions of sup and inf refer to sets, inevitably the students' set-theoretical perceptions were implicit in their finding suprema and infima as well as of Ç and È as operators between sets [6]. In the same context, and because a large part of the process of finding suprema and infima involves the manipulation of algebraic inequalities (and in general the arithmetical handling of algebraic quantities), the students' concept image of sup and inf, as well as their action related to these concepts, was also influenced by their algebraic skills as bequeathed by school mathematics. Finally in terms of the students' handling of the new mathematical formalism and logic, quantified logical propositions proved major obstacles; also their handling of the newly formalised proving technique of Mathematical Induction also revealed some of their attitudes towards mathematical proof.

In particular the notions of sup and inf were discussed in problems related to finding the sup and inf of various given sets [4, 6, 7], or to the Archimedean Property [1, 8] the proof of which involves the Completeness Axiom which in turn contains the notion of supremum. Formulating and proving the Archimedean Property revealed the students' difficulty in the encounter with mathematical formalism both in terms of the reasoning used as well as the new forms of expression [also in 6]:

• the former (reasoning) was revealed in their weakness to co-ordinate a negated quantified logical proposition with the definition of sup in order to provide a proof by contradiction of the Archimedean Property [1, 5ii]; or in their difficulty [7] when confronted with the multiplicity of options in the course of a proof and with the need to co-ordinate a variety of information in order to pick an effective option; also [8] in their difficulty to articulate adequately a justification of some steps in a proof.

• the latter (semantics) was revealed in attempts to mimic the b/b writing technique of the lectures by avoiding ordinary language and introducing quantifiers and set-theoretical language [1].

The novices are not at ease with the assumptions they are allowed to make when engaged in proving fundamental statements [2]: that is to distinguish, formulate and prove a universally quantified statement, even though they demonstrate an adequate initial, intuitive grasp of the proof. Specifically, the students do not appear ready to choose arbitrary numbers, establishing thus the universality of their proof, and maintain this arbitrariness through the proof with consistency. So, in a sense, questions, such as CD2.1 in [2] in which it is not clarified to the students, for instance, what statements regarding the real numbers can be assumed, emerge as problematic. As seen [2] in the students' over- and under- reaction to the requirements for rigour, students seem to be vulnerable to issues related to assumed knowledge: in other words they have been sensitised to the increased requirements of rigour in the new course but then abandoned to clarify these requirements on their own.

This sensitisation reflects to a large extent that the novices are in need of, and engaged in, a cognitive shift from unrigorous to rigorous forms of mathematical thinking. Some evidence:

• their highly metamathematical discussions of

- the meaning of £ in the triangle inequality (as a sign that is used in the triangle inequality not to denote inequality, but to denote a variety of options for equality) [3] and

- the Base Clause of Mathematical Induction [3] (the debate of whether the Base Clause should be stated for n=1 or n=2)

• their ambivalence on what knowledge can be assumed and how assuming knowledge is compatible with the demands for axiomatic rigour made by the lecturers and tutors in the beginning of the course [5].

The juxtaposition of the above with the epistemologically founded responses of the tutor, highlight a strong characteristic of mathematical expertise that for the time the students are missing: how mathematical experience empowers hindsight, reinforces a more fruitful use of intuition and secures the embeddedness of mathematical knowledge. Incidentally I mention here the slightly more alarming evidence[8] of a perception of the Base Clause of Mathematical Induction and the n=k step as perfunctory (only the k+1 step of the proof was deemed important by the student).

Interestingly this juxtaposition between expert and novice approaches is not as clear-cut as possibly expected [6]: different — but mathematically equivalent — counterexamples seemed to carry different degrees of conviction both for novices and the expert. This was discussed in terms of the deeply subjective and vulnerable character of mathematical cognition.

Sensitisation to the requirements for rigour does not always imply that the novices are willing to attempt formalisation [1, 5ii]. On the contrary [4, 5i], in most cases difficulty in formalising leads to reluctance, avoidance and preference for concrete, intuitive arguments. Some evidence [4]: a student abstaining from using the Completeness Axiom from a proof because he knew the Axiom in terms of suprema and he did not realise that a symmetrical statement holds in terms of infima; or [5i] formalisation is not thought of as necessary when a proposition is perceived as obviously true. So, while some students [5] have conceptualised the necessity to be formal and struggle with the materialisation of this conceptualisation, others are still engaged in the vicious circle of assuming in their proofs what is to them intuitively obvious or what they are actually being asked to prove.

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