This article deeply resonated with me. I have always felt that I had two different experiences learning math, and that the math I studied in university varied drastically from the math I had learned in high school. I knew that it didn’t necessarily have to do with the content, but rather how I was learning and my engagement with the material. Reading this article, it became clear to me that I had experienced an instrumental learning of mathematics in high school and a relational understanding of mathematics in university. Something that prior to this I would have called under the names “learning math” and “doing math”. This brings me to the first thing that made me stop when reading the article. Skemp says “I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’”. This gave me pause because I had felt the difference between these two subjects, and I wholeheartedly agreed with him. I stopped again when he wrote “This difference is so marked that even relational mathematicians often use instrumental thinking.”. I stopped because I felt ‘called out’ by this, I do think that I am a relational mathematician who uses instrumental thinking. The most obvious example of this to me was when I was writing my thesis. I had to go through every line of mathematics with a fine-tooth comb and re-teach myself why everything worked the way that it did. Of course, when I had initially written it, I didn’t fully understand every step that I was taking, instead I was relying on my instrumental understanding of certain concepts. So again, in this idea I agree with Skemp.
Another thing that stopped me in the article was when Skemp wrote “… since an imaginary opponent who thinks differently from oneself is a good device for making clearer to oneself why one does think this way.” It made me stop because it allowed me to see the necessity of discourse amongst differently thinking people. Being in an echo chamber is nice – it’s validating, it’s easy, and it can be exciting to be surrounded by like-minded individuals. I think that I fall into echo chambers easily because they are comfortable. It is uncomfortable however, to have your ideas challenged, pushed, and prodded. This idea that it is good for one’s own growth to push back against your own ideas is not new to me, but something that I need to be reminded of, to keep pushing into the uncomfortable dialogues and re-evaluating my own ideas and values.
This is a rich and reflective response. I like how you connected Skemp’s distinction to your own journey from high school to university and even to your thesis-writing — that made the theory feel lived. Picking up on his point about needing an “imaginary opponent” was smart, and linking it to echo chambers and growth shows you’re applying his ideas beyond math. To sharpen it further, you might have pushed on where Skemp’s view could be challenged, but as it stands this is deep, honest engagement. Excellent work.
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