Section (v) dim(X+Y)=dimX+dimY-dim(XÇ Y) and the Varying Persuasion of Mathematical Arguments

An Interpretive Account: The Analysis

Note: I note here the expression of a general complaint by the students with regard to lecturing. Jack's 'we've never found a basis for anything' (or his stressing that he remembers the theorem because of his reading and not the lectures) is a direct attack against what is perceived by the students as excessive accumulation of theoretical knowledge in lecturing. The tutor's request to return to the specific discussion of the theorem is a common response of tutors in similar situations: the valuable time of tutorials is not to be spent on general criticisms of the course.

Receiving Coldly a Logical but Not Well-Embedded Mathematical Argument. In Part a of Extract 8.5 the tutor elicits from the students the answer to the question (dim(XÇY)=n-2) with very closed questioning. However, despite the fact that the answer is uttered by one of them, the argument is coldly received by the students. The reason, that Jack gives, is that the tutor's argument is 'just based on a formula we haven't proved'. In addition to that, I note that underlying the tutor's argument are two tacit theorems:

The above theorems constitute necessary knowledge for the construction of the argument and in Part a the students do not give the impression that they possess this knowledge. These epistemological deficiencies coupled with the method of Socratic persuasion by very closed questioning result in the students' scepticism.

What follows in Part b is in striking contrast with Part a: the tutor deconstructs their belief that a basis of a vector space can be reduced to a basis for a subspace by removing some vectors from it via an illustrative counterexample.

The differential efficiency of the two arguments in Part a and Part b can be tentatively explained as follows:

In a similar vein, the collective effort to prove the formula used in LA6.29ii turns out to be quite successful even though there is a great deal of leading by the tutor who sets up the scene by refuting the students' belief that a basis for Z can be reduced to a basis for X or Y(part b) and by replacing it with the new technique of extending a basis for a subspace to a basis for the vector space. The students' thinking initially revolves around a direct application of the technique on X until Jack alludes to the possibility of applying the technique on XÇY. Jack's intuition on XÇY is prompted by Andrew who notes that Jack's suggestion to start from a basis for X does not provide good prospects for constructing a basis that contains a basis for Y.

Finally the proof is collectively completed and an imaginative suggestion is made by Jack who wonders whether the formula can be extended to 'more than two subspaces'. The exchange of arguments between Jack and Andrew is an exemplary case of interactive mathematical creativity even though the creations of this interaction in particular are rather tentative. I note that the tutor does not deter the students from experimenting on their idea. On the contrary he suggests the use of a concrete context (²⁷) for the experimentation.

Note: there is a remarkable repetition of the misleading phrasing 'THE basis' by the students which is commented upon by the tutor. Similar evidence is available from other tutorials and the interviews.