This classic think piece from Richard Skemp, despite being now almost 40 years old, still gets a great deal of attention amongst mathematics educators. I'd somehow missed it in my own preparation, which I find surprising how much I've spent studying the math wars. Skemp describes two perspectives on understanding mathematics, one which he calls relational and the other he describes as instrumental. Relational understanding is related to what we might think of a "deeper" understanding, something that reflects how and why mathematics works and is applied. Instrumental understanding relates to those reliable and typically efficient procedures we apply to produce mathematically correct answers. The part of Skemp's article that really sticks out for me is his suggestion that "mathematics" might be used too broadly: "I used to think that maths teachers were all teaching the same subject, some doing it better than others. I now believe that there are two effectively different subjects being taught under the same name, 'mathematics'" (p. 91)
Today +Chris Robinson, +Joshua Fisher, +Nat Banting, +Nik Doran, and I (pictured left-to-right along the bottom) met via Google+ Hangout to discuss Skemp's article. We discussed examples of each kind of understanding, whether one is a subset or prerequisite to the other, and the various influences that lead us to emphasize one over the other, such as curriculum and assessment.
(Some research needs to be summarized, while some needs to be expanded on. That's my explanation for why it will likely take you longer to watch the video than to read the original article.)
I've been in discussions like this before and I always find them to be fascinating. Still, it seems difficult to really get at the root: What is mathematics, and how can and should our beliefs about mathematics change? There are also strong implications for curriculum design and learning theory, as finding the right balance in our approach to both kinds of understanding should lead to better student outcomes.
Skemp, R. R. (1976/2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95. Originally published in Mathematics Teaching. Retrieved from http://www.jstor.org/stable/41182357