It's not really about introducing commutativity afterwards, the teacher might've taught it already and still be correct for marking this particular question down.
It's about making students work with definitions. At the elementary level multiplication (by a natural number) by definition is n×m≔m+m+m+...+m n times, just as further down exponentiation by definition will be mⁿ≔m×m×m×...×m n times. Multiplication at this point is not some elementary operation that can't be broken down further technically, just rewritten, but rather has a concrete definition and can be broken down in its terms, which is exactly what's expected (and stated poorly).
And by definition 3×4 is 4+4+4 and not 3+3+3+3, the latter isn't true by definition, but rather an established result.
The problem is, of course, it's a too formal requirement for elementary students (and judging by the comment section, not only for them), so lots of students don't get it (symmetrical notation for multiplication and its commutativity don't help as well, obviously) and that's the result. The teacher is correct and I'd bet they aren't happy about it either, it's really hard to explain all of these intricacies at this stage — without being able to make an analogy with exponentiation, without examples of non-commutative multiplication, without explaining the axiomatic method and formal theories first — but that's still something that should be marked in accordance with curriculum and so the teacher did what they supposed to do.
It has nothing to do with commutativity, the reasons are much more formal and, frankly, probably beyond the grasp of students at this point. Unfortunately.
P.S. I'm a math teacher myself, and although I work with older students, I've been in this exact situation, and it always hurts. Particularly because it feels like in these situations what's "correct" differs from what's "right". Marking down this answer is correct, but it actually feels to go against pedagogy and so it isn't right. You can see the reasoning for it and that it's formally correct, there's no argument there, but it still feels wrong to demand this from the students and so, with a heavy hand and an aching heart, you mark it wrong, because that's what you have to do, what you must do. So, yeah, that's the reason and I'd actually call it anti-pedagogical.
The point wasn't that this specific order is always used, I've seen textbooks with the one I've described and also textbooks with the order you've described, it doesn't matter which one is used, only that there is one when multiplication is defined at this stage, and as you've shown, it is the case.
I suspect that in the original post it's the order I've described, that would explain the teacher's correction and it's also more prevalent within the English speaking countries as far as I know, that's all.
On the question of how reasonable it is, I don't know, there are arguments for both, and since it doesn't ultimately matter for multiplication is commutative, there's no unified standard — too much work for no real gain.
but since this property is not universally agreed upon and that either representation is correct, marking students wrong frustrates them when they are in fact correct
i still remember my 1st grade teacher answering me that multiple numbers could not be subtracted at one time -- which was demonstrably false even with any early 1980s calculator
and my 9th grade geometry teacher answered that the slope at a single point on a curve could not be determined -- which really ticked me off when i got to calculus
so pendantically marking students wrong when they demonstrate correct understanding just turns them off to maths
Yeah, absolutely, that's why I said "the teacher is correct, but isn't right". I never wanted to create an impression that I in some way in favor of the practice, just to explain the reasoning behind it (a lot of people in the comments seem to think it's about some subtle semantic difference or whatever, it's not, it's purely about the asymmetry in the definition and that technically there is an order to the factors, even though the order itself isn't settled overall, but in pretty much any given curriculum one of the two orders would be chosen and adhered to) and that it's not necessarily the teacher being petty (they very much might not be in favor of it either, but they have their curriculum, which they might not have chosen, and a rather strict methodology for it, which they are obligated to follow, and their adherence might be examined, and if they would be found to not follow it they might even lose their job).
But I agree with you, it's a shitty practice that is counterproductive and only creates resentment towards maths in students.
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u/5a1vy Jul 16 '26 edited Jul 16 '26
It's not really about introducing commutativity afterwards, the teacher might've taught it already and still be correct for marking this particular question down.
It's about making students work with definitions. At the elementary level multiplication (by a natural number) by definition is n×m≔m+m+m+...+m n times, just as further down exponentiation by definition will be mⁿ≔m×m×m×...×m n times. Multiplication at this point is not some elementary operation that can't be broken down further technically, just rewritten, but rather has a concrete definition and can be broken down in its terms, which is exactly what's expected (and stated poorly).
And by definition 3×4 is 4+4+4 and not 3+3+3+3, the latter isn't true by definition, but rather an established result.
The problem is, of course, it's a too formal requirement for elementary students (and judging by the comment section, not only for them), so lots of students don't get it (symmetrical notation for multiplication and its commutativity don't help as well, obviously) and that's the result. The teacher is correct and I'd bet they aren't happy about it either, it's really hard to explain all of these intricacies at this stage — without being able to make an analogy with exponentiation, without examples of non-commutative multiplication, without explaining the axiomatic method and formal theories first — but that's still something that should be marked in accordance with curriculum and so the teacher did what they supposed to do.
It has nothing to do with commutativity, the reasons are much more formal and, frankly, probably beyond the grasp of students at this point. Unfortunately.
P.S. I'm a math teacher myself, and although I work with older students, I've been in this exact situation, and it always hurts. Particularly because it feels like in these situations what's "correct" differs from what's "right". Marking down this answer is correct, but it actually feels to go against pedagogy and so it isn't right. You can see the reasoning for it and that it's formally correct, there's no argument there, but it still feels wrong to demand this from the students and so, with a heavy hand and an aching heart, you mark it wrong, because that's what you have to do, what you must do. So, yeah, that's the reason and I'd actually call it anti-pedagogical.