The two different notions of understanding confused me. While relational understanding was intuitive enough, instrumental understanding, or the author's articulation of it, was rather confusing. Once he brought up the example of the circle, diameter, and why the formula for the circumference is as it is, I understood the concept of instrumental understanding much better.
I loved his recollection of the teacher who "became suspicious" that his students were not fully understanding relationally what concepts were going into calculating area. The "test" of having different dimensions was a clever and decisive method to check whether his students were understanding instrumentally or relationally. It makes me wonder how I will check to see if my students are understanding relationally, or if I will even notice that they aren't understanding relationally.
The notion of a mismatched teacher with his text is an interesting one. Having an instrumental teacher in general seems like the wrong hire, but the resulting confusion of pairing him with a relationally centered textbook emphasizes the mismatch. It reminds me of many "by the book", needlessly strict teachers that I had in grade school.
I agree with his stance. Relational understanding is better than instrumental understanding. Instrumental understanding is just memorization and calculation. Computers can do that for us. Relational understanding is conceptual intuitiveness. It breeds creativity to branch out into other problems. And further, it strikes motivation and determination to tackle those problems. That, a computer cannot do.
Fascinating discussion, Ian!
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