Blog Post #1: Response to Skemp

Richard Skemp presents arguments for and against relational understanding and instrumental understanding. He admits that neither is ubiquitously preferable over the other but presents the merits of each. It seems that Skemp feels that, when possible, cultivating a relational understanding of a concept is preferable to cultivating an instrumental understanding.

One idea that I found interesting was that relational understanding can be more efficient than instrumental understanding. In college, I rarely memorized formulas, instead I tried to make sure that I could replicate the derivation of the formula from first principles. This was efficient in terms of how many equations I spent time trying to remember but it was not efficient in that I would very often spend time (on exams even) re-deriving equations that I needed. So, I would argue that efficiency needs to be evaluated in the domains of both memory and time in order to properly compare two strategies.

Another spot that caught my attention was the idea that math could be taught primarily for its aesthetics rather than for utility, as is done with music. I think that this would be a fascinating experiment, however, I am skeptical that it would find widespread appeal due to high barriers to entry and the effort required compared to other aesthetic mediums.

My interest in this topic is to try to better understand what the actual difference is between relational understanding and instrumental understanding. Skemp suggests that relational understanding is 'knowing what to do and why'. However, this condition is flawed because it is unclear what it means to know 'why'. If a student says "I know that to add two numbers together I can imagine pouring two glasses of water into a bowl and measuring the combined amount." This seems to demonstrate an understanding of addition, however, can it really be said that the student knows 'why'? Does the student need to be able to articulate something like "what is meant by addition is the combination of two collections of quantifiable like items"? Do they need to appreciate that their imagined physical procedure can never be done perfectly in practice? Consider another student that explains "We do this problem in way X. Why? Because the that is the way the teacher did it and they know better." The student was able to answer 'why', what makes their answer inadequate for establishing relational knowledge? It seems to me that the question of whether or not somebody knows why something is a particular way is quite difficult to answer. However, I think that the obvious alternative explanation that Skemp did not address is that relational understanding is established through relations. We can safely say that the student described understands addition in relation to the combination of quantities of fluids. I think that remembering this perspective is important because it ultimately emphasizes the feeling of creating a web of knowledge as we grow our relational understanding of mathematics. It also helps explain how relational understanding. In this picture instrumental knowledge is much more clearly distinct.