Reflecting on Skemp - Sept. 6

Initially, I found Skemp's theories to be quite dense and difficult to understand; however, with his use of analogies, much became clear. In particular, the illustration of faux ami using the football analogy was enlightening for how incompatible, and therefore disruptive, these two understandings (relational and instrumental) can be in the mathematics classroom. It also served as a reminder for how useful analogies can be for understanding, and for retaining ideas.

My own experience in math classes was comparable to what Skemp describes in the first mathematical mismatch: I wanted the quick simple way to get the correct answer, while some of my teachers tried to introduce more foundational understanding. After reflecting on Skemp's reasons why some teachers still use instrumental instruction, I was able to unpack my attraction to instrumental understanding as a teenager. Firstly, it was how my math education began, and so the shift in my latter high school years was very uncomfortable. Secondly, I had aspirations for higher education, and in my mind, this required high grades, not deeper understanding. As someone with a good memory, the model of memorizing rules worked quite well, but I saw my more curious classmates, often labeled difficult and disruptive to class dynamics, strain under this instrumental framework. By the time I got to Math 11 and 12 when relational understanding was introduced to me, these students had already accepted that math wasn't "for them" -- the damage had been done. Therefore, I'm curious how relational understanding would allow for more students to continue in math education, while also wondering if this would mean fewer students achieve the A+ level. 

As I've stated, I worked well under what I can now describe as the instrumental understanding framework. And I do believe it has its merits. Like Skemp said, simple rules are often helpful and easy to memorize. Further, when dealing with adolescent brains the instant gratification of rule-computation-right answer is nothing to scoff at. But, in seeing how many serious deficits I had in basic mathematical understanding by the time I reached university, I do agree with Skemp that relational understanding is far more beneficial in the long run. Ideally, I can imagine a classroom scenario that somehow balances the motivation granted by instant gratification with a more complete understanding of the underlying concepts. In my opinion, Skemp focuses too much on what constitutes the term "mathematics" and not enough on concrete solutions. Of course, relational understanding sounds idyllic, but as he alludes to, we will not be teaching math in a vacuum. Our instruction will exist within the context of previous math instructors and provincial curriculums. How can we as individuals grapple with this debate within sometimes restrictive contexts?