I will start with a few things that I enjoyed about reading the text. I really liked how Skemp started with introducing the idea of faux amis. Two faux amis I found very interesting in math, while difficult to explain to student, are "general" and "always". Also, I found the music class analogy which Skemp made very interesting. It makes me think about how in junior level of math class, almost all kids seems to be capable of doing those arithmetic. But once "x" and "y" are introduced, to some students, the same operations somehow seemed like a new set of operations to learn.
An idea that I always kind of had in mind is that, often mathematical insight comes after performing mathematical methodology repeatedly. Because, this is the case for me. But as I entered university, I learned from my study buddy that, some math learners really struggle on solving a problem unless they have understood the methodology offered relationally, which is totally not the case for me. I was so good at imitate whatever was done in an example on the textbook and get the right answer without understanding any principle underlying. After realizing that some people need principles before methodologies, I start to believe that both instrumental and relational explanation are needed in teaching. I recognize the value of "instrumental teaching" as described by Skemp, because as a prevalent idea, "math is a tool". When students ask me why they learn "pre-calculus", well, I answer, "because calculus is very useful in many discipline, so you might want to prepare for it". But at the same time, mathematical insight is, almost self-evidently, what we really appreciate and eager to share with others. And of course as Skemp has pointed out, it benefits students in long-term. Combined with all the situational factors Skemp has outlined, I think a mix of instrumental and relational teaching is the direction that I would like to go for.
This is a thoughtful and personal reflection. I liked how you picked up on Skemp’s faux amis and connected it to your own classroom challenges with words like “general” and “always.” Your comparison between your own learning style and your study buddy’s was insightful — it showed why both instrumental and relational approaches have value. To take it further, you could push on whether there are times when one approach actually gets in the way of the other. Still, this is clear and reflective writing — nicely done.
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