It is always an interesting experience when I find myself reading a formalization of an idea that has been floating around in my head for some time, but that I never got around to properly researching. Skemp's conception of 2 types of mathematics, largely differentiated by motivation and process, made me stop because it is something that I noticed early on in my post-secondary studies. In particular, it was through discussions with people in applied fields, such as engineering, that I learned how many people value mathematics simply as a tool and do not engage with the deeper, more artistic elements of the field.
I also stopped when Skemp first introduced his definitions of "relational" vs. "instrumental" understanding because my thoughts immediately went to the current discourse in the math world around AI proofs. Sparked by OpenAI's ethically questionable brute forcing of the Navier-Stokes smoothness problem, the STEM community has been abuzz the last week with discussions incredibly similar to the arguments made in Skemp's paper. All this drama has culminated in Terence Tao penning a letter signed by 25 Fields Medalists titled "A Severe Misalignment of AI in Mathematics" in which he details the dangers and travesty of pursuing mathematics simply for the result, and not for the process (and what is learned along the way). This coming exactly 50 years after Skemp wrote "Relational Understanding and Instrumental Understanding" is quite amazing and speaks to how much Skemp's paper is still relevant.
Finally, when Skemp was discussing the attributes and prevalence of instrumental understanding I stopped because 3 big things came to mind:
1. The fields that incentivize this form of understanding (engineering, economics, programming, etc.) are some of the most socially "valuable" fields because they create tangibly useful things in people's lives (I'm pretending the type of money economists work with is tangible for the sake of the argument). There are many consequences of this, some obvious and some less so.
2. Having worked in Quebec the last 4 years helping students pass the grade 10 standardized math exam there, I was ostensibly forced to teach instrumental understanding due to the structures of their education system. This realization has helped me put words to an element of my dislike of standardized exams that was mostly just intuitive up to this point.
3. Most teachers who lay the foundational mathematical knowledge in children on which everything is built only have an instrumental understanding of math themselves, so how can they be expected to teach relational understanding? This serves to deepen my belief that there should be math specialists in elementary schools.
By this point it's probably pretty clear where I stand on the issue raised by Skemp, but just to make it abundantly clear:
Relational Understanding > Instrumental Understanding
Sure I see the value of instrumental understanding as a stepping stone in certain contexts, but ultimately if you only know how to swing a hammer, eventually you're going to break something and not know how to fix it.
P.S. Sorry this wasn't very brief; I really liked this paper.
So fascinating to connect this paper with the AI-generated proofs we are seeing now, and the huge discussions this is generating! And perhaps this AI speeding-up of proofs is really just the natural consequence, when our society tacitly teaches that getting a correct result quickly is all that matters. (Is that even true in 'instrumentalized' uses of mathematics in engineering, economics, etc.?) Standardized exams, and teachers who don't really understand the mathematics in depth -- also very important themes worth exploring in our classes! I'm so glad you enjoyed this article.
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