Section (viii) Leading Didactical Style as a Potential Propagator of Passive Learning. Resisting the Contingency of Multiple Answers to a Mathematical Question
Context: See
Extract 8.8Structure: The group of students featured in this section are particularly non-participant. Also the tutor uses closed questioning extensively. One of the implications is that the students' responses, even when correct, sound unnatural and unconvincing. In the following, the students appear rather inarticulate and the tutor leading.
The Episode: A Factual Account. See
Extract 8.8An Interpretive Account: The Analysis
Throughout the observation and recording period I found it rather difficult to distinguish between cause and effect in the cases where the tutor used very leading, closed questioning and the students were passive and reluctant participants in these minimal dialectics. My impression however is that generally it is the tutors who set the tone and style of the sessions. It is an exceptional case (some of these cases are reported in this and other chapters) when a student, keen on dynamic dialectics, forces the tutor to adapt.
Here the students are not keen participants. Their responses to the tutor's closed questions sound more like a product of enforcement than conviction. In the following, I present the effects of this approach within the context of question B3.
How Patricia was led to realise the redundancy in her thinking. Patricia's inarticulate answers
• 'you can deduce the one from the other one' and
• 'can we just say that because E' is linearly independent...?'
led me to believe that, even though the tutor prompted her to admit the redundancy of proving both conditions, she does not actually know why. In fact the second answer cited above might suggest Patricia believes that linear combinations of linearly independent vectors are linearly independent vectors.
The Success of a More Cognitively-Friendly Approach. In the second part of the question the tutor's strategy slightly changes. She appears as if she is trying to clarify first the students' knowledge of what a matrix of a map is-and-does and then ask them to apply this refreshed knowledge to finding the matrix in question. This proves more successful than her earlier strategy even though the students again rather mechanically transfer the application of the idea from bases E and F to bases E' and F. The contrast between their inarticulate utterances
• 'Te equals f. All the elements in it...'
• 'that's the matrix of the transformation'.
• 'The coefficients of the sums...f1 plus f2...'
and the fact that eventually the students dictate the necessary calculations for the evaluation of Tei is spectacular. The students appear to become more easily better executioners of algorithmic plans than interpreters of the role and significance of the concepts they are introduced to.
Complications Resulting From Multiplicity of Answers. In B3 it is Te'3=0 with respect to the matrix given in the question. So while calculating Te'1 and Te'2 has led to finding vectors f'1 and f'2 for basis F' of W, Te'3 has not. Zero cannot be in a set of linearly independent vectors. The two students have tackled this complication in different ways. Beth decided that since Te'3=0 and we cannot have zero, the basis will contain only two elements (B1). Cary on the other hand, convinced that a basis of W must necessarily have three elements, decided to keep zero in the basis (C1).
With the tutor's prompting, Cary notes that, if we include zero in the basis, then our set of vectors ceases to be linearly independent (C2). The tutor wants the students to understand that they have to find f'3 replacing zero with another vector, not necessarily via the same method that led them to find f'1 and f'2. Beth however seems at ease with the idea that since the process determined these two acceptable vectors only, the dimension is 2 (B2). Finally through B3 to B8 she is led to change her mind into the multiplicity of possibilities for f'3 cannot be zero.
In the above, the students appear to resist the idea of determining f'3 in a non-unique way. f'3 can be any element of W as long as it is linearly independent of f'1 and f'2. This potential pluralism of answers, while a common mathematical occurrence (examples: there is often more than one solution in a differential equation; there is an infinite number of solutions in an under-defined system of simultaneous equations) seems to be alien to their mathematical experiences and hence they resist it. The expectation of singular answers to mathematical problems, patronisingly designed to present mathematics as a polished, clear-cut deterministic activity, is here revealed to be a detrimental, die-hard habit. I note that the rest of the students in this college, that were confronted with the possibility of various choices for f'3, responded in similar ways.
Conclusion: In the above, a case was explored where the novices seem to resist the potential pluralism of answers to a mathematical problem. Moreover in the process a gap was observed between the students' calculating skills and their ability to reason articulately about their mathematical actions. In terms of teaching style the highly leading and predetermined closed questioning was illustrated as inadequate and unnatural, whereas a more cognitively friendly exploratory style was proved relatively more successful. It is likely that such a teaching approach deepens the above mentioned gap between algorithmic and conceptual understanding. Both passivity and reluctance to participate as well as resistance to the potential of mathematical pluralism reflect a deeply unadventurous and conservative learning style.