I disagree.

For your main argument seems to be that there is a particular skill related to higher mathematics that should be partly achieved by students in a very young age.

The problem with this is that there is another equally important skill which is contrary to the one you seem to favour here:

For an algebraist a function is a set-theoretic object and therefore something extensional rather than intentional (so thing rather than meaning related: Math as a science of structure rather than of rewriting devices).

In the case of multiplication that is the set {((0,0),0),((0,1),0), ((1,0),0), ((1,2),2), ((2,1),2),…}

In fact what the kid was likely to think about was not the a commutative statement about the representation of multiplication, which was given by the teacher, but just another representation.

Teachers representation:

1 x n=n

(m+1) x n=m x n + n

Students representation:

m * 1=m

m * (n+1)=m * n + m

Both representations satisfy a commutativity statement but are very different in your understand of “meaning”. But why favour one over the other? The answer is: Do not! For from a mathematical perspective that is not important, important is what they represent and that one gives some correct way (not THE way) of explaining their solutions.

That the student did what s/he did shows that s/he got multiplication as a set-theoretical object. For the student did not just wrote 15 without any explanation, s/he gave an argument by stating 5+5+5. Not following the rule and stating another equivalently correct rule (by writing 5+5+5) is evoking Gödel’s incompleteness statements. Which might be interpreted as that the creative mind of a human cannot be imitated by a rewriting device. The student did a grate job here… discouraging her/him with such a marking is horrible.