Skemp Reflection

Relational vs. Instrumental Understanding

After reading Skemp’s article, I am convinced of the general superiority of relational teaching in mathematics instruction. That being said, I believe there are some areas where an instrumental approach can have certain benefits.
Skemp initially describes instrumental knowledge as “rules without reasons.”
The first effect this statement had on me was to remind me of the vast majority of my experience as a math student. Virtually all of my teachers throughout high school taught mathematics in a very rigid, mechanical way, with little thought to deeper analysis or explication.
The second quotation which stood out to me was Skemp’s statement that some students just want “some kind of rule for getting the answer.”
What this reminded me of was some of the experiences I have had as an instructor at Sylvan. Many of my students have had learning difficulties in mathematics. The simple ‘rule to get the answer’ method is helpful for them, as it helps improve their confidence by making them feel successful. Unfortunately, I believe this approach also promotes intellectual laziness and the tendency to look for an ‘easy button’ whenever possible.
“… instrumental understanding […] usually involves a multiplicity of rules rather than fewer principles of more general application.”
Upon reading this, I thought of the analogy of a tool box. Instrumental teaching requires you to carry around a giant box, with many specialized tools. Conversely, an instructional method which promotes relational understanding lets you carry around a lighter box with fewer, more efficient tools.
Writing as a Devil’s Advocate in favour of instrumental instruction, Skemp writes: “They will at least acquire proficiency in a number of mathem-atical techniques which will be of use to them in other subjects …”
Upon initial inspection, I thought this point had some validity. However, after further reflection, I decided that it would be insulting to mathematicians everywhere. This would have us consider math as a mere tool for accomplishing tasks, rather than an independent discipline worthy of study in its own right.
The final quotation which struck me most was that relational instruction is “easier to remember.”
I agree wholeheartedly with this. Throughout my entire undergraduate career, I would always loathe to memorize formulae. Instead, I would learn to derive them, and pore over proofs in my notes until I fully understood how and why they worked. The memorization would then occur almost by accident.