On Instrumental and Relational Understanding

I would like to add one more argument in favour of teaching relational mathematics over instrumental mathematics: instrumental mathematics justifies the modern student's common complaint that they need not learn the subject when they have access to a computer. If they are being taught only to follow strict procedures to arrive at an answer, then they are correct in believing that a computer can do this much more quickly and reliably. To stay relevant, we need to teach students more than what can be done by Wolfram Alpha - we need to be teaching relational understanding.

It worries me, though, that my own experience of math in school, over thirty years after this article was published, was still more focused on the instrumental than the relational (and I consider myself lucky to have had some excellent math teachers). I was motivated enough to seek out relational understanding on my own, but in retrospect it's not surprising that so many students complained about not being able to understand math when they were not being taught to understand it so much as to do it. As far as I have seen, the math concepts Skemp suggests as ways to promote relational learning - sets, mappings, equivalence - have not been introduced into our curricula. Maybe this is because of another of his points - math curricula are already overstuffed, and we need to trim away some of the content if we are to teach any content effectively. Since this problem hasn't been solved in the last 30 years, it's certainly very difficult, and teachers and curriculum designers will need to work closely together to make any progress.