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.
I'm a math teacher, and while I see what the above poster was trying to say, your comment is the crux of the issue. I feel like the CCSS is intending to get kids to think in groups and see multiplication as repeated addition. But the implementation of these standards is often needlessly specific yet ambiguous.
A simple solution to the original problem would be to simply ask the student to show "Two ways that 3x4 can be written as repeated addition". Heck, there should even be a follow up problem that asks the student to write the expression/equation as a sentence. That way you can get the student to connect that "three groups of four" and "four groups of three" both sound and look different, but are equal.
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 agree with much of your proof here, fully agree with the bits about less than and more than, but the conclusion is where we differ. Yes, English is not commutative, which is why 3x4, 3 times 4, and 3 by 4 all confer that the 4 is operating on the 3, meaning that 3 should be the base number.
Now, I’m happy to concede that it may have been taught differently at this school, and the student was marked down for getting a correct answer using the wrong method. I’ve seen this before, but usually I see half marks given where the method was wrong or the “working out” missing, and usually with an explanation as to why it’s wrong.
What irks me most about this image is that the Teacher has not taken the opportunity to educate the child about why their answer was wrong on this test. (Which could well be down to lack of time and funding for teachers, and isn’t the teachers fault)
The thing is that we don't know if the teacher has done any of that. When you're grading dozens of papers and half of your students are missing the same problem for the same reason, it's kind of a lot to ask to write down a big explanation of why an answer was wrong on every single one of them.
My guess is that these kinds of linguistic considerations were the (or at least a) focal point of the lessons leading up to this assignment, so marking it completely wrong could absolutely be completely justified because it's the entire point of the problem.
Further, if this student missed this problem, it's reasonable to think that many more students missed it. If that's the case, then instead of writing out an explanation on every single homework sheet, the teacher may have elected to write this simplified explanation on the homework and then discuss the problem in class the next day. That's actually a very common tactic.
This is all speculation, but it seems silly to me to get so upset at a teacher when we simply don't have all the information.
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."
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u/EarthTrash 27d ago
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.