Response to Skemp

    While reading Richard Skemp's article, "Relational Understanding and Instrumental Understanding", I was immediately struck by two things. First of all, that there is indeed an important distinction between relational and instrumental understanding, especially with regards to teaching/learning mathematics. And secondly, that it's perhaps a bit of a false-dichotomy-situation to believe "understanding" falls neatly into one of these two categories. In my mind, math can certainly be taught more "relationally" or more "instrumentally." It's tempting then to think that these two types of understanding exist on a spectrum. But even then, it's not difficult to imagine how one might simultaneously possess both types of understanding of a given topic. I think what's closer to the truth, is that there are degrees of both types of understanding.

    I imagine a human brain in a jar, hooked up to two gauges: one measuring instrumental thinking, and the other measuring relational thinking. As the brain gets fed different math problems, I think you would see the values of each gauge change, independently of each other. I believe we're all engaged with these two different "styles" of understanding to varying degrees, at different times, to suit our needs. I think part of the beauty of mathematics, is the sense that you can always push the boundary of relational understanding. Even something as simple as arithmetic can be generalized to sets and mapping rules, which can be further generalized to "basic" axioms. It's definitely not always desirable to engage in the most generalized level of relational thinking. But on the other hand, if you are seriously only interested in instrumental thinking, you might as well give up and let computers do all of your thinking for you (mostly joking).

    I believe an effective educator necessarily appreciates both "types" of mathematics. It's obvious that math encompasses both the relational underpinnings of it's topics, as well as all of its instrumental applications. I definitely tend to agree with Skemp and his view that there needs to be more of a focus on relational mathematics in schools. And I do really appreciate his justifications for teaching instrumental maths also. I've found that a lot students do tend towards the picture painted by Skemp, that of a student looking for quick and easy rules to pass their exams - very practical. The included passages from Sir Hermann Bondi especially resonated with me, and I think a lot of the satisfaction that you would expect a human brain to get out of studying maths is tragically blunted by an overemphasis on teaching the subject instrumentally. One of my personal goals as an educator is to pass along my passion for the mathematics. But to do that, you do need to make the subject matter appealing and accessible to your audience. I often see the instrumental approach to teaching as the "hook," which can then be expounded upon by all of the juicy relational ideas. Again, I think that's an oversimplification. There are no easy rules or 100% accurate "theoretical formulations," but I do believe that a truly excellent teacher is able to balance these two styles of teaching/understanding to suit the needs of their students.