Here's my response to this:
https://medium.com/@highsource/the-only-reason-for-this-answ...
The only reason for this answer to be marked as “wrong” is the teacher saying “You did not apply the strategy I’ve tought EXACTLY as I tought it. You have dared to understand the idea and acted on your understanding INSTEAD OF mechanically applying the actions I told you to.”
Most of your argument does not have a stand.
You’re quoting “the definition of multiplication” which says “adding as many copies of one of them as the value of another one” and use this as basis to argument that 3+3+3+3+3 would have been correct and 5+5+5 is wrong. But even this definition does not say “the first one” and “the second one”. It says “one of them” and “another one”. So 3+3+3+3+3 is just as correct as 5+5+5. Period.
You’re quiting definitions of equal “being the same in quantity, size, degree, or value” and “ equivalent” as “equal in value, amount, function, or meaning”. First point here: the task said nothing about “equivalence”. It just said “solve 5x3”, applying the repetitive addition strategy. So it absolutely does not matter if 5+5+5 is equivalent to 3+3+3+3+3 or not.
Next point, you say that 5+5+5 is equal to 3+3+3+3+3 but not equivalent. If you explicitly add some trivia like banana bundles then you can somehow argument that there is some difference in amount, function or meaning. But only if you explicitly add these details. In the original task, there are no such details so there is no way you can show difference in amount, function or meaning.
I agree that using a commutative property before it was itroduced would have been wrong. But the child here did not use the commutative property! Absolutely not. The child applied the repetitive addition strategy, just (obviously) not at the EXACT convention that the teacher taught. This has nothing to do with multiplication being commutative at this point.
The whole point of this answer being wrong is for the teacher to enforce application of the taught rules or strategies EXACTLY how they are taught. There is no sensible reason for the repetitive addition strategy to be applied as 5+5+5 instead of 3+3+3+3+3. Only the convention and “do as I said”.
Whether “do as I taught” is a good thing or a bad thing really depends. For some children it is really important that they follow the teacher mechanically, repeating exactly what they were told. This way they are at least guaranteed to manage the basic mechanical tasks. So the teacher is more or less guaranteed to have some borderline success with them.
But many children understand things on a much deeper level from the very beginning. They understand the sense and the reason and the logic of math much deeper than the basic mechanics. And once they understand the internals like the absolut truth of 5+5+5 and 3+3+3+3+3 giving the same result, it becomes illogical that one answer is right and the other one is wrong due to “you have not done this EXACTLY as I have tought you”. You see, math is the absolute truth, so if your conventions and enforcements contradict that, these conventions and enforcements are simply wrong. Yes, maybe you first have to do “wrong” for the better good later on, but don’t pretend you’re right.
Finally, you bring the point of “Respect the teachers” because they are “ they are qualified experts on child education”.
Oh, my, I don’t even know where to begin.
There are really different kinds of teachers, some doing great jobs and some, well, not-so-great. Of course you have to respect them as you would respect any other human being.
But this does not mean that teachers or teaching programs are infallible. Respect does not mean they are always right, because, you know, “they are qualified experts” and that they can’t be criticized.
I had around 12 or 15 different math classes in the university and really different kinds of professors. Most of them respected the thinking and understanding above all. They did not care if I did a proof exactly as they taught it— or came up with something original (which was, admittedly, mostly, because I skipped the lection). But there were also some which insisted on exactly the same proofs and even notations as they once wrote on the whiteboard. Reasoning: it was harder for them to check the correctness of the proof if it was not in their exact notation!
Should I have respected this? I did not and I have brought a few cases to the higher university commissions and had all of the wrongful evaluations dismissed.
You point to the dangers of children later on not understanding matrix operations or “equals” vs. == vs. ===. For me much more dangerous is teaching mechanics and punishing for misunderstanding. I have never ever saw a student or a programmer who had troubles with matrix operations, vector multiplication, or === in JavaScript because of they’ve grasped the commutative property of multiplication for numbers too early.
But I have seen a lot of people thinking and working mechanically with once-learned mindsets which they are afraid or uncabale of leaving. I am afraid, this is exactly the mindset which is enforced by “5+5+5 is wrong because this is not how I taught it”.
Let me tell you this. If my kid would have brought this from school, I’d explain him that 3+3+3+3+3 is just as valid as 5+5+5. But I would have also point out that sometimes it’s not just math that you learn in the math lesson. That you also learn social skills — like that the teacher expects you to be conformant to his or her rules. You have to be able to recognize this in this person. You have to be not just clever enough to understand that 5+5+5 is the right answer. You have to be clever enough to see that there is the other correct answer, 3+3+3+3+3, and that the teacher probably expects that one instead.