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Hmm.. Under the commutative law, 5x3 = 3x5. So it's kinda an extra step 'show your work', but it's not wrong. Does common core not teach commutative, transitive, associative properties, etc?

This way of reading things, for instance, can be used when students learn matrices in multivariable calculus in high school.

Huh? I went to a private school, and was one of the more advanced ones. Taking a combined algebra/Pre-Cal class junior year, we had the option to take AP Calculus, AB or BC. But normal-track students didn't touch calculus, much less do much if anything with matrices. And as far as I can remember, matrix multiplication is the only time when axb =/= bxa. And I don't even remember the different between cross product and dot products.

Hmm.. Under the commutative law, 5x3 = 3x5. So it's kinda an extra step 'show your work', but it's not wrong. Does common core not teach commutative, transitive, associative properties, etc?

They are making a very pedantic distinction without a real difference here. And you can bet they did not mention commutativity of real numbers. This is the intellectual equivalence of creeping legalism being injected into math.

Taking a combined algebra/Pre-Cal class junior year, we had the option to take AP Calculus, AB or BC. But normal-track students didn't touch calculus, much less do much if anything with matrices

For me matrices were first introduced as methods of calculating currents and voltages in linear electric circuits. Later on we got many more use out of them. But it was in the university.

And as far as I can remember, matrix multiplication is the only time when a × b ≠ b × a.

The operator of multiplication can be defined in many ways. There is the matrix product; there is the Hadamard product; there is the Kronecker product; there is the Frobenius product. It would be not particularly wise to "prepare" students for operations with matrices. When they get that far they will be able to explain for themselves why this or that method is used, and what's the difference.

The whole story is sad. Math is famous for many ways to prove something. Here is the page with 114 different proofs of the Pythagorean Theorem. According to those "teachers," 113 of them would be "wrong" - and I'm suspecting that few of them would be able to comprehend some of those proofs.