Ploughing through exam scripts today, I am going through the usual mix of the excellent, the average and the truly awful. Here’s a question: How does somebody go through six years of secondary school and one year of college and still commit the most basic of mathematical heresies? I’m talking about an inability to do the basic algebraic manipulations that are an absolute pre-requisite for making any progress in more advanced mathematics, including calculus. And it’s not just ‘weak’ students who make these mistakes, it’s students with high grades in Leaving Cert Maths and lashings of CAO points.
There is something going badly wrong at second level. While educational theorists and sideline commentators talk at length about ‘problem solving skills’ and ‘creativity’, our students are not acquiring the basic tools (and not just in maths) that they will need to develop into graduates who can confidently address real-world problems.
The thing is, lecturers have been talking about this for years and it is now commonplace for the universities and the institutes to establish ‘clinics’ and ‘learning centres’ in an attempt to get our students up to speed. Why is it that as a nation we are happy to have our third level institutions teaching remedial mathematics? How on earth is that ok?
I think this represents a critical failure of our teaching and learning culture at second level. Learning basic maths is really an exercise in pattern recognition. When you see certain expressions, you know, based on your memory (forged by practice), what the next step should be. It’s a bit like the best chess players who can recognise patterns on a board and then draw on their extensive long-term memory bank of patterns to decide their next move.
But many students’ memories seem to be largely empty of automatically accessible mathematical knowledge and they seem to make fundamental errors based on reasoning that is superficially plausible but ultimately wrong. The common error
A/(B+C) = A/B+A/C
is a case in point. It looks sort of reasonable but a student who had really immersed himself in basic mathematics (and learned the rules) would instantly know the correct ‘next move’ to make rather than deducing a move that is just plain wrong.
I fully appreciate that youngsters are growing up in a world of distractions, in a world where we all have difficulty focusing. But there are certain building-blocks that all disciplines have and they must be acquired even if it might be tedious to do so. Modern advances in digital technology have made information retrieval extraordinarily easy, but making sense of that information and using it to address real world problems, needs a scaffold of basic knowledge and skills. And too many of our students just do not have that scaffold.
I worry that those driving pedagogical change, including the design of new curricula for second level, do not agree with this point of view.