Finishing up on reading the article, I can't help to have an impression that this article is a well-constructed retelling of the same idea "the more you learn the better". The idea is inherently correct, but the article bringing up the new definition of relational and instrumental understanding offers very little new ideas on top of "learn more". The author used an analogy of drawing the schema of a city mentally as an example of relational understanding; however, in the eyes of a city planner, such drawing is merely an instrumental understanding of the city. To a city planner, the author does not understand why hotel would be allocated at a certain place, why one place would be closer to another, he can now draft routes but did not learn the logic behind why these buildings were issued permits to be built at their places. To sewage planners, the city planner's understanding of city plan would also be instrumental on a certain level. Sewage and power grids had to be planned due to geological or other factors, and these capacities in turn limited building permits, in their eyes city planner doesn't understand that. In their eyes city planner and author's understanding are all instrumental, just at different levels. One may argue that the author only wanted to draft routes and his understanding was sufficiently relational for his purpose, then such understanding is equivalent to the instrumental understanding defined by the city planner. The definition of relational understanding is now merely a deeper layer of instrumental understanding in a more knowledgeable system.
Taking the concept back to math as a tool, the example of one student doing division wrong because he extended methods from multiplication to facilitate the definition was very poor. The student did not fully grasp the definition of fractions application within multiplication's boundaries, this shows a lack of definition and precise description in teaching. If the student operates division within a complete strict definition, doing correct steps and get correct answers, the understanding could easily be defined as within instrumental understanding. Digging similar mathematical definitions or methods through all the layers, we can eventually reach to the bottom of the layers and discuss why define 1+1 equals 2, and discuss elemental definitions in number theory. Similarly, the mismatch in unit in area calculation is another lack of definition. In physics, dividing 10 newton by 2 square meters gives 5 pascals, math as a tool in such physics calculations abides by the definition and is a correct application. The terms relational understanding and instrumental understanding has no fundamental distinctions, relational can simply be defined as n-th layer of instrumental understanding in author' examples. For different subjects like statistics, physics, or chemistry they realistically require different levels of "instrumental understanding" in different topics, and at certain level the "instrumental understanding" is redefined as "relational" according to author's definition. The article is a convoluted retelling of the importance of building solid foundations with new terms, and to a certain level, a dangerous idea that implies learning up to a certain level is enough, instead of searching for more.
This is a bold and critical response. I liked how you challenged Skemp’s framework directly, especially with your city planner analogy — it was a creative way to argue that relational is just deeper instrumental. Your questioning of Skemp’s classroom examples also showed strong critical engagement. To push it even further, you could connect this critique back to teaching practice — what would it mean for how you teach if relational and instrumental are just layers of the same process? Still, this is sharp, original thinking — excellent work.
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