Section (ii) The Problem of Clarifying What Knowledge Can Be Assumed in a Proof and the Role of Quantifiers in Establishing the Generality of a Proof

Context: See Extract 6.2

Structure: In the following Ben presents his solution. The tutor refutes his proof. George then outlines his proof. The tutor refutes his formal writing and presents a proof. A discussion follows on the role of quantifiers in establishing the generality of a proof.

The Episode: A Factual Account. See Extract 6.2

An Interpretive Account: The Analysis

What is problematic with Ben's approach. As the tutor observes Ben has attempted a direct application of part i in his proof for part ii. In this, he has ignored that the pair of arbitrary numbers used in part i are rational, whereas in part ii they are real. He interprets 'if x is irrational' as 'there exists x which is irrational' (fig.2a), assumes then that a/x and b/x will then be irrational, with a and b being rational, and, via part i, he deduces the existence of s, a rational number between them. So, by the moment he is interrupted by the tutor, Ben

• has ignored the requirement for choosing arbitrarily the two numbers to start with (his a/x and b/x are not arbitrarily chosen real numbers) and,

• he is using the statement that between two irrational numbers there is always a rational number.

Therefore Ben is committing two serious formalistic errors:

A he ignores the requirement for a universal quantification of his proof (for any two real numbers).

B he is using a previously unproved statement either as a misinterpretation of part i or in effect of lack of rigour (that is, in the absence of the realisation that previously unproved statements cannot be assumed in a mathematical proof).

A Note on Cathy's Questions. Cathy remains non-participant during the Episode, possibly because she has not answered the question in her drafts and thus feels that she has no contribution to make. However towards the end of the tutor's presentation she sounds preoccupied with how his proof preserves the universality of a and b. Like Ben (A) Cathy is not comfortable yet with universal quantification. Moreover if Ben (B) under-reacts to the requirements for rigour (by assuming more than allowed), Cathy seems to over-react (by questioning the assumption of factual knowledge, such as the existence of irrational numbers). In both cases, over- and under- reaction constitute evidence of an unease with issues regarding the knowledge one is allowed to assume.

What is problematic with George's approach. George's initial idea to base the proof for real numbers a and b, in part ii, on a generalisation of the available information on rational and irrational numbers is a demonstration of more confident reasoning than Ben. His is a problem of expressing mathematically a general idea (G1) to which he was led by intuition. So,

 

In his mind he possibly wanted to prove:

 

between any two real numbers there is a rational number (1)

In his draft, as the tutor says, he started doing so by supposing:

 

there is a rational number between two real numbers (2)

In the tutorial he reads his draft:

 

as if he had written (1). But he has written (2).

(2) is a fact, which the tutor emphasises by pointing at 0 and 1 as two real numbers between which there is an irrational number. George then realises (G2 and G3) that there is a substantial difference between his intentions and the reflection of those in his writing. Again, the problematic aspect of the student's expression, here in writing, has been one regarding universal quantification of mathematical statements.

Conclusion: In the above, the students are not at ease with the assumptions they are allowed to make when engaged in proving fundamental statements (e.g. CD2.1): they have found it hard to distinguish, formulate and prove a universally quantified statement (they do not appear ready to choose arbitrary numbers, establishing thus the universality of their proof, and maintain this arbitratiness through the proof with consistency). Ample similar evidence on CD2.1 is available from other tutorials too. CD2.1 seems to be of a particularly problematic nature mostly because it is not clarified to the students what statements regarding the real numbers can be assumed. As seen in Cathy's and Ben's 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 raises the didactical question as to how, in this state of uncertainty about the rules, are the novices expected to play successfully the mathematical game of foundational Analysis.

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