Sorry Brett but it seems you have been caught up in a web of terminology.

The student’s response to question 1 is absolutely valid. The distinction between equality and equivalence is not significant to the procedural computation of 5x3. The student is entitled to employ an equivalent representation of the given calculation.

It is also appropriate (and even important) to present the commutativity property of real numbers, since that is the domain of focus for students. The conceptual discussion of equivalence, and examples of non-commutative sets, is a worthwhile digression. It should not, however, distract from the axiomatic truth that order of multiplication is irrelevant to the numerical outcome.

The student’s response to question 2 only betrays the convention of rows/columns as the format for arrays. They clearly grasp the underlying concept.

The bottom line is that this student clearly gets it — it appears they can perform calculations with some fluency using valid methods. The rigid prescription of conceptual models does little to enhance students’ procedural or conceptual knowledge.

Don’t lose the forest for the trees.