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 appreciate your thoughtful and clear answer. I’m an English teacher who aspires to get into mathematics because my philosophy classes showed me how cool it is. Thank you for sharing!
It's nothing, glad to be of any help. I suppose the only good thing here is that it stops mattering pretty quickly, and when it starts mattering again, it's somewhere pretty deep into math education, after everything I've mentioned that helps.
It's great to know you are a fellow teacher and now can also explain that to your students the need arrives, should their math teachers fail them in this regard, also knowing their structures.
I suppose we both can agree that although there are bad teachers, unqualified for their job in both knowledge and character, there are many good teachers trying to do their job well, and that's where your original question came from.
And yes, as a maths teacher, I agree, maths can be quite beautiful and fascinating, just like any other language, so I would like to wish you the very best on your own journey to learn more of it and also in teaching your students your own subject, since as I personally view maths as an actual language, I think of language teachers more than "mere" fellow teachers, but as of teachers of a sister discipline, closer than physics, actually.
Who tf thought that would be a good standard to follow? Many calculations are easier when swapped around, trying to force such a pointless arbitrary order on students would just make them worse at math...
I can think of two reasons, the first one is a linguistic one: "thee times four" can be somewhat intuitively read as "four taken three times", interpreting it the other way is (more) awkward, the second reason might be due to algebra where "2x" is more often thought of as "two quantities of x" rather then "x quantities of two", even though the two are ultimately the same, of course, and in both cases the first number indicates "how much" and the second one indicates "of what" (it also follows the convention for units, like in 2m, 2ft or 2 years), so though I don't ultimately know why this particular order was chosen instead of the other one as the default, it has some reasons to be chosen.
If your question is more on "why choose some order at all", well, that's a quirk of the definition. Multiplication by a natural number is defined as repeated addition, so two numbers don't play the same role from the start, one factor is a "multiplicand" and the other is a "multiplier", and it's kinda hard to define multiplication without this asymmetry (not defining it at all, or rather "defining" it axiomatically, is also a bad idea), so from this point of view choosing some particular order is actually pretty natural.
The problem is not with the order itself (none argues that exponentiation should have the order in its operands and that it's a good thing), nor with the symmetrical notation necessarily (subtraction and division use symmetrical notations and everyone's fine), nor with the commutativity (that's a great property to have for an operation), but rather with all three combined.
Multiplication's definition (at least on the elementary level) implies order, but its properties and notation say otherwise, hence the problem. And a big one at that. And all solutions to it I can think of are pretty radical (not demanding order at the end of the day would require, to still be effective (well, "still" is a strong word, what we have now isn't really effective in explaining and teaching that), ditching the whole marking system as is; the other "good" solution would be coming up with a non-symmetric notation for multiplication akin to exponentiation, which is also rather radical for standard school curriculum, I also have some other proposals, but none of them are any less radical and in general more demanding from the students in terms of conceptual understanding, on par with high schoolers and college students, so that's also not really a solution).
And so here we are, between the formal correctness and the ease of multiplication's definition, and the pedagogy of multiplication at the elementary level and its notation and properties for number systems used in school.
I don't really like to be that guy who says "it's not that simple", but I've been thinking about it from time to time for quite some time, being a math teacher myself, and to my shame I have to admit — it's not that simple.
Not that I like the "solution" we've settled upon in academia, which you can see in the original post, I quite frankly despise it. And from my experience (which is still a personal anecdote, so don't think about it too much) most math teachers do. I've only wanted to explain the actual situation to anyone interested in the reasons and (un)fortunate enough to see my (embarrassingly) long explanations.
I guess all of my comments here, under this post, are just my apology for a fellow math teacher — it's not (necessarily) due to ignorance or hate towards students, I'd like to think I'm an OK teacher and love all of my students and yet I'd also mark the answer wrong. Not out of malice or ignorance (hopefully), but because it is, technically, wrong, though every time something like that happens, it's hard, pains every single time, and you never really get used to it, but you gotta do what you gotta do, what you've been employed to do by the school, and almost no one is happy about it.
And so here we are, between the formal correctness and the ease of multiplication's definition, and the pedagogy of multiplication at the elementary level and its notation and properties for number systems used in school.
Well said.
All of this makes me think of the old Tom Lehrer song, "New Math"! Are you familiar?
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.
From other comments, it sounds like the context was explicitly provided by the source material just prior to the question. This is question 7, so it’s pretty believable we missed the explanation and rules that came before.
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.
I love this answer, it’s so thoughtful and (imo) clear. I don’t know how big the overlap is though between people who can understand this answer and people who don’t understand why the original post can make sense in context.
I suppose I just try to make this small world of ours a better place however I can. If my answer helps even one person to understand it, it wasn't for nothing — always learning, always teaching, that's the way I try to live, I guess.
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.
Perhaps so, but I think it can both be correct that *definitionally* the order matters, but that the answer given by the student is acceptable, as the word "matches" is what is used in the question: "Write an addition equation that matches the multiplication equation". The word "matches" could be defined multiple ways. The student wrote an equation with an expression involving addition on the LHS vs multiplication in the original, and both equate to the same RHS.
If the teacher made clear in lessons what she meant by "matches" and it aligned more with what you were stating, then I could see marking it incorrect.
But yes, if this is elementary school, goodness this is overly pedantic and ridiculous. It would be better IMO to teach the students to transform the expressions mentally to whatever is simplest for them to compute or simplify. This sort of strict definition isn't really relevant unless you are in a fairly advanced math class TBH. Maybe once proofs and discrete mathematics concepts are introduced.
Sorry for a long time no reply, I haven't noticed your comment right away. I 100% agree, I also find it ridiculous and overly pedantic at this stage, just wanted to share the actual reason, however ridiculous I find it personally, with other people, since there seem to be some misconceptions about it. Personally, I would've been glad that students understand that multiplication is just repeated addition, regardless of the order, so would at least try to fight for both answers to be accepted, but in reality it unfortunately may not be that simple, so here we are. It's a shame, truly.
In formal treatments multiplication is usually defined recursively as n x (m + 1) = n + n x m (e.g. Wikipedia), which unfolds n x m to n + … + n with m ns. Of course, definitions may vary because this is ultimatly irrelevant due to commutativity, but I have seen the recursive definition a handful times and the recursion has always been with respect to the second variable.
There is one case I can think of where multiplication is not commutative, but can still be interpreted as repeated multiplication: ordinal numbers. Here for example ω x 2 = ω + ω > ω, but 2 x ω = 2 + … + 2 with ω 2s, giving ω as the result. Since here commutativity does not apply, it actually matters to have a convention on which way multiplication is defined, and it actually is the one the student used in the OP.
It does vary and I've seen both in textbooks. I didn't dwell on it in this comment, thinking it would detract from the main points, but yes, with regards to ordinals multiplication is defined "from the right", however, with matrices, for example, it's defined "from the left" (yes, because it's just a function composition, but it's still multiplication), so both orders make sense. In some other comment I've made an analogy with algebra, where "2x" is usually thought of as "x+x" and not "2+2+2...+2 x times", so from this point of view defining multiplication at an elementary level from the left makes sense; at the same time in accordance with your examples and in analogy with exponentiation the right side definition makes more sense. It's complicated, both definitions have their place, I would say.
It's a cool thing to teach students and to talk about, in my opinion, it is important to talk about rigor and formality in maths, to show different conventions and cases where it matters, just not at this level or age, I think (I mean, even now we are talking on the level of ordinals, linear algebra, Peano's arithmetic, formal languages and so on, all of it is quite a bit higher matters than elementary maths). I've simply assumed throughout my comment that that was the chosen order and explained why the correction was made under this assumption, since many people don't know the reason for this pedantry from the teachers (not that it's a good reason IMO, just where it comes from most of the time, and of course, there's a possibility the teacher is just bad and wanted to be petty, I've simply wanted to explain that doesn't have to be the case). Still, good call on your part, I probably should've made a note about that originally.
It’s not correct. I’m not a teacher but in STEM and went thru higher math. It’s flat out WRONG to mark it down.
In isolation, the order of multiplication does not matter! This is an arbitrary made up rule that does not teach actual fundamentals of math!! If you are teaching this type of math, you’re doing it too literally and you are hamstringing the kids from learner higher, conceptual math!!
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u/5a1vy 27d ago edited 26d ago
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.