[Maths-Education] The resource for the course

[Stuart Rowlands]

Anne wrote:

> >A very sharp distinction is being made here - are these intended to be
> >mutually exclusive categories?
> 
> When the only high stakes exam was at 16 I could imagine there could 
> be some merging of these categories.  Sadly, now there is testing and 
> preparation for testing and mock testing and 'springboard' and 
> 'catchup' and 'boosting', not to mention selection tests for 
> selective schools, 'optional' (oh yeah?) testing and progress 
> testing, and all kinds of other words for not allowing learners to 
> reflect, dwell and otherwise bask and swim around in mathematics ... 
> and a growing underclass of children for whom none of this is helping 
> them learn maths .... I am inclined to the view that the only 
> humanistic and sane choice is to take the second category and stick 
> to it for all learners.
> 
> Anne W.

Well put!

For me, this all goes back to Skemp's distinction between relational 
understanding (understanding the relations between concepts that 
constitutes the body of knowledge) and instrumental understanding 
(memorising formulae and rule of thumb proceedures so as to pass 
examinations). Anne's original point that developing understandings 
about mathematics make passing tests a lot easier is similar to 
Skemp's argument that developing a relational understanding of a 
topic may take a longer time than memorising appropriate techniques, 
but, paradoxically, is also shorter in the long run. Developing a 
relational understanding develops the cognitive structures necessary 
to handle unfamiliar situations.

The present culture of tests, tests and more tests seems to 
presuppose the behaviorist notion that surviving these tests develops 
the learning of mathematics -  hence the structure of textbooks 
(''The Resource for the course''!). Unfortunately, the textbook 
example supplied by John Monaghan is not new. Some of the modular 
A-level textbooks commissioned by various exam boards encourage the 
'learning' of a chapter so as to answer the corresponding question on 
the corresponding modular exam paper ('stimulus-response'). While 
this may encourage more students to take up A-level mathematics 
(that somehow mathematics becomes more 'accessible' this way), it 
also stifles a conceptual (qualitative, relational) understanding of 
the subject. Coursework becomes a poor excuse to redress the balance.

My fear, however, is that content (mathematics as a body of 
knowledge) will be seen to be the culprit for this current state of 
affairs - the remedy being an emphasis on process ('mathematisation') 
at the expense of content. This is a real fear given that many 
educators would regard the teaching/learning of mathematics as a body 
of knowledge to be 'absolutist'.

If (all) students are to be inducted into a mature cultural 
conversation (in this case, mathematics), then a 'sideward' 
exploration is necessary as well as a 'forward' assimilation of the 
subject matter. This may require a reduction in content, but it 
certainly requires a reduction in the battery of tests.

Modular mathematics and loads of tests may have been seen as a 
solution to making mathematics more accessible, but this will only 
serve to appease the exterior motivation to gain a paper 
qualification and may actually stifle any intrinsic motivation to 
learn mathematics. I'm not against examinations per se, but I see the 
teaching/learning of mathematics for its own sake as the real 
challenge facing us.

Stuart