The point isn't "matrix multiplication != scalar multiplication". The point is "the process of evaluating a reducible expression inherently discards information about the original expression", which is a fact about evalution rather than any specific operator. The fact that "the information discarded is of little consequence to the compressed result" is a quirk specific to scalar multiplication. Thus, the commutative property distracts from explicitly modeling the "AST" so that the student understands what multiplication represents under the hood, beyond the rote memorization of scalar multiplication tables.
Perhaps an analogous situation would be: Suppose a teacher wanted to introduce the notion of limits to a calculus curriculum. "That makes zero sense. The only things a student needs to know are the shortcuts for each parent function, e.g. that (d/dx x^2) reduces to (2x) via handwavey magic." But what if an engineer needs to integrate over an arbitrary curve? Can students solve the problem without being comfortable with Riemann Sums? Maybe 1st-year calc students should rederive the shortcuts from scratch? "Except we're talking about a math course, not an engineering course."
> In particular this bit regarding multiplier/multiplicand makes zero sense to me.
> Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.
It's bizarre to simias (and you, I assume) because y'all can't imagine performing the operation without thunking. (Don't get me wrong, I think "repeated addition" is the best method. I'm just attempting to explain the opposite perspective so that it feels less bizarre.)
Perhaps an analogous situation would be: Suppose a teacher wanted to introduce the notion of limits to a calculus curriculum. "That makes zero sense. The only things a student needs to know are the shortcuts for each parent function, e.g. that (d/dx x^2) reduces to (2x) via handwavey magic." But what if an engineer needs to integrate over an arbitrary curve? Can students solve the problem without being comfortable with Riemann Sums? Maybe 1st-year calc students should rederive the shortcuts from scratch? "Except we're talking about a math course, not an engineering course."
> In particular this bit regarding multiplier/multiplicand makes zero sense to me.
> Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.
It's bizarre to simias (and you, I assume) because y'all can't imagine performing the operation without thunking. (Don't get me wrong, I think "repeated addition" is the best method. I'm just attempting to explain the opposite perspective so that it feels less bizarre.)