So in school, I learnt about addition, multiplication, and exponentiation, just like most decent students. I did fairly well at math in high-school, but mostly by remembering what each operation did. It was certainly pointed out that there was some higher-order relationship between these operations, but our teachers did not go there.
Of course, the notation for each of these three common operations did not share a common schema, so it was impossible (at least for me) to deduce what comes after exponentiation, in this sequence of operations, just by looking for patterns in the notation of these operations.
A year or two years ago, I was introduced to the notion of tetration, the hyperoperator sequence, and Knuth's up-arrow notation, which indicated to me where humanity had gone about documenting the bigger picture on this matter.
I have much study to do here, and much more to learn, if I am to understand this properly.
Point of cognitive efficiency/ethics/pedagogy: I think we should unify the notation for the hyperoperations, so that the nature of the relationship between addition, multiplication, exponentiation, tetration, etc. becomes more obvious to users of mathematical language.
Update: Addition and Multiplication are commutative, but Exponentiation is not. Is the latter categorically different? More thought required.
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