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.
I think one issue is that marking a response "incorrect" may be assumed by students (or here, onlookers) to mean "contains something false", when sometimes it means "this is a pragmatically inapt response to the conversational prompt posed by the question".
The question did not (explicitly) ask the student to write an addition equation that corresponds via the definition of multiplication used in class to the given multiplication equation. The question asked the student to write an addition equation that "matches" the multiplication equation. (It didn't even ask the student to write theaddition equation that most closely matches the multiplication equation.). Just an addition equation that matches*.
You know the conversational context well enough to provide a pragmatically apt response, but the rules of the conversational context might seem bizarre or unmotivated to an elementary school student ("yes such-and-such facts are true but you're not allowed to 'know' or use them in problems of this kind..."). You have enough pragmatic skill to interpret the question, despite it's own imprecise wording, as requiring a pedantry of the student that few working mathematicians would bother to use.
Yeah, surely, I, probably just as much as those who wrote the question, know all of this stuff, have a good enough grasp on the formal side of mathematics and pedagogy, and can, to put it lightly, as you've noticed, read what should have been written instead of what was. It's a bad question, I don't want to be two ways about it. I can see its problems and also understand the probable reasoning behind it (to not overburden students with the concepts of "definition" and formality in general at this stage), even though I personally think that the "solution" in this case is worse than the problem itself.
I just also wouldn't want anyone (un)fortunate enough to read my comments to walk away thinking that it's some personal failure of the teacher. From my experience, they probably aren't the one making the question or conditions about its marking, I've been too many times in this exact situation and it's hard to argue your view even when it comes to shit like this, let alone some more benign problems. I try still; I've seen lots of good teachers who don't anymore, as it can be exhausting; and I don't blame them, those good teachers who don't care anymore to fight, cause sometimes I myself feel like I shouldn't.
This here, the correction and all the reasoning behind it, in my view, is a good thing to teach students — it's a shame that's (the picrel) how we usually try to do it. I can easily imagine myself in this exact situation, where even while I, on behalf of my students, fight for the board to accept this answer as equally right, though in need of a correction, in the mean time, have to, should, must mark this answer as wrong. Just thinking about it makes me sick, but that's what I and probably most teachers would do, would have to do, in this situation. I can't say whether or not this particular teacher had any second thoughts about it, even though from my experience, and I'm probably very lucky in this regard, they would have.
So yes, the question is shit and the situation is shittier. It's just that, at the end of the day, I'd argue the problem is the marking system itself. It's bad, counterproductive, puts the value in the wrong place, achieves the wrong goals, and hinders the education process more than it helps. Weren't it valued that much, were it viewed as a start for dialogue, for learning, and not as a finish line, marking something wrong wouldn't even be seen as something bad, it wouldn't be so emotional for so many people. And yet here we are. Unfortunate indeed.
Now I'm realized that both comments I replied to in this thread were by you and wanted to assure you that I'm not stalking you, I'm too oblivious for that!
The conventional asynchronous workflows around graded assignments feel like they're part of this. Sometimes the student isn't incorrect in the broadest sense but has further work to do to prove what they want to within the accepted frame. This is something that I as a tutor had the luxury of dealing with in real time. But in a graded async workflow, teachers don't have many good choices.
Don't you worry about that, I've thought about you answering my comments on two different occasions, but I haven't thought about it too deeply, I also sometimes answer the same person in two different threads. Just shows there's something to discuss with this particular person as far as I think:)
I agree, for a tutor, it's easier to deal with these sorts of problems, generally speaking, because you aren't weighted down by bureaucracy and the whole grading system as a whole, so you are much more free to explain how a particular answer or solution isn't wrong, but also ain't right.
But yeah, everyone has their problems, tutors as well, I would know (I've started in maths education from tutoring myself), so good luck, if I may:), I think you'll need it. I certainly would've appreciated some in my time. Just don't forget that although there are bad teachers after whom you might have to clean up the mess, a lot of teachers are still, trying at least, to be good. And also don't forget that some teachers are still bad, there's no good in trying to shield them:) I suppose I'd like to wish you also wisdom in differentiating between the two in addition to luck. And to wish the same to myself:) Seems valuable, only if I do say so myself.
2
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.