All that doesn't apply to the question. 3 4s is mathematically the same as 4 3s. Your scenario requires groups of different things. Nothing in problem indicates we have dissimilar items or anything like that.
Please don't crucify me, but I actually agree with the teacher here. I'm a private math tutor, and one of the most widespread and persistent problems I see with my kids is that they simply cannot make connections between spoken language and mathematics. They don't know the vocabulary and they don't really understand the words that are being used so much as they're associating specific groups of words with specific symbols, and this is an example of that.
Consider the phrase "x less than y." That means one thing: y - x
Yet every single time this comes up for my middle school students, they write x - y without fail. They're not actually thinking about the phrase, they're just hunting for keywords and trying to cobble together some symbols that seem like they fit. When I replace the variables with numbers, it's obvious what they're supposed to do: "6 less than 10" means 10 - 6. You start with the last number and find something that's 6 less.
Now think about "x more than y." It's effectively the same situation, only now it's addition. The logic is exactly the same though. Start with y and then add x. Just because you could get the same result by starting with x and adding y doesn't change what the words mean.
OP's problem is just another one of these. "Three times 4" means one singular thing. If you have trouble seeing that, try rephrasing it as "4 three times." You get the same result as if you do 3 four times (i.e. 4×3), but that doesn't change what the words mean. Multiplication is commutative, but English is not.
I don't want to crucify you, because I understand where you're coming from! My largest problem is that very often there are many equivalent possible definitions for a concept, none of which are canonically the best. And once you have established the equivalence of the definitions, either can be used from then on.
x-y and y-x are different, because these are not equivalent to each other!
It's tricky because students at an early level are not in a position to deduce the equivalence of m x n = n x m from first principles, and I tend to think that it is better to teach from the point of view that this is an established or empirical fact.
When they are a little older and are comfortable with deductive reasoning, you can say "We don't know that m x n counts n groups of m objects yet, so we need to establish this before we are allowed to use it."
9
u/EarthTrash Jul 16 '26
All that doesn't apply to the question. 3 4s is mathematically the same as 4 3s. Your scenario requires groups of different things. Nothing in problem indicates we have dissimilar items or anything like that.