A 5x3 rectangle is exactly the same as a 3x5 rectangle. It is not a bundle of bananas, silly. It's just a rectangle, regardless of orientation.

This seems like a far-fetched justification. What's more likely?

1. The teacher understands (and is expecting 9 year olds to learn) the pedantic difference between equivalence and equality. Keep in mind, this being elementary school, the teacher likely is a generalist and also teaches reading, science, and social studies.

OR

2. The teacher has a very rigid grading guide that specifies how much credit is given for any given answer/technique, and is simply blowing through 50+ tests at 1AM, applying this standard grading. Ironically, in this scenario, the teacher is just as pointlessly constrained to "following the rules" as the students. He/she may even agree that it's ridiculous to deduct a point.

It's pretty obvious the kid is incorrect because 5*3 is using the sum monoid and overloading notation.

Or at least that would make more sense than arguing the kid is wrong via Javascript and matrices.

If the author of this article instead references the definition of multiplication of natural numbers on wikipedia [1], then the student is correct since $a \times b = a + a + \dots + a$ with that definition.

Without access to this particular teacher's curriculum materials, it's not possible to know for sure what definition is being referenced by the "repeated addition strategy". I'm inclined to assume the teacher knows what they're doing and has graded the work appropriately.

There are many comments on this thread about multiplication being commutative by definition, but this is not quite correct. Following the same definition of multiplication I cited above, it is a theorem that $a \times b = b \times a$. When I teach Abstract Thinking (a sort of introduction to proof writing course for mathematics students), I have the students write proofs for this property of multiplication of natural numbers, and the other familiar properties (cancellation, distribution, etc.). If anyone is interested, I've broken the steps out into worksheets that I give to my students, and you can see them at the link below. [2] [pdf] (Multiplication of natural numbers is section 5.5.)

[1] https://en.wikipedia.org/wiki/Natural_number#Multiplication [2] [pdf] http://billkronholm.com/wp-content/uploads/2015/10/MATH280.p...

I may have bought the explanation if the answer below hadn't been marked incorrect as well, because the axis on an imaginary array are arbitrary.

This post just made my BS-meter explode. How can this make the front page of HN? I don't even know where to start.

It’s more important than ever for students to understand the difference between equal as a result and equivalence in meaning from a young age because it is a fundamental computer science concept (...) Equivalent means not only are they equal, they are also of the same data type. In other words, they mean the same thing.

Except that this point is totally misguided because 5+5+5 and 3+3+3+3+3 are, in fact, the same thing. A member of the set of natural numbers, commonly known as 15, and that you can write as S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(0))))))))))))))) if you have the patience to do so.

In fact, the author uses the == and === operators in JavaScript to illustrate his point, but of course, 5+5+5 === 3+3+3+3+3 resolves to true in Javascript, and in any language under the sun that compares stuff by value.

And then, if you compare by reference, the result of == or similar operators doesn't depend at all on whether you made your integer by adding up three fives or five threes: in Java,

  Integer i1 = new Integer(5+5+5);
  Integer i2 = new Integer(3+3+3+3+3);
  Integer i3 = new Integer(3+3+3+3+3);
  System.out.println(i1==i2 + " and " + i2==i3);

prints false and false, while using equals instead of == would print true and true.

So I don't see how this nonsense would teach kids anything useful about computer science. The only thing it can do is confuse them.

equivalent is defined as, “equal in value, amount, function, or meaning.” In the above problem 5 x 3 is equal to 5 + 5 + 5, but they’re not necessarily equivalent. Equivalence relates to meaning, so it depends on the meaning of multiplication, as the directions indicate.

First of all, the sign in the exam statement is an equal sign, not an equivalence sign. So if 15 is equal to 5x3, what the student wrote is perfectly fine. Also, "solving" a multiplication means finding out its value, which is what he did.

Secondly, you know what also is a fundamental computer science concept? The logical operation "or". The definition of "equivalent" in the blog is reported as “equal in value, amount, function, or meaning.” "Or", not "and". So the definition doesn't say anything about equality in meaning (however you define it) being a necessary condition for equivalence.

Maybe teachers are "experts on child education" but that doesn't exempt them from knowing something about maths if they are supposed to teach them.

This is a load of crap. "Equivalence" has a mathematical meaning and it isn't the bullshit he tries to play it off as. After interpreting an uncited example blurb on Wikipedia as "the definition of multiplication" he goes on to discuss division, a non-commutative operator, how it's bad for kids to use the commutative property of multiplication before the teacher has taught it, how bundles of bananas of varying quantity (i.e. sets of different sizes) are not equivalent, how the JavaScript type system is weird and how you can't simply flip the dimensions of vectors and expect them to be equivalent.

Neither of these things have anything to do with the original problem. The best thing I can think of him doing, as a self-proclaimed "math evangelist", is to shut up about concepts that are obviously beyond his understanding.

I almost stop reading at the first paragraph: He makes a difference between "Equals" and "Equivalency" and then he gives overlapping definitions:

Equal is defined as, “being the same in ...value.” Whereas equivalent is defined as, “equal in value...”

If you can't be precise in what you write yourself...

But the question is asking the student to find a solution not something equivalent, I will agree with what this article says if the question is "Find the equivalent repeated addition form of the following multiplication problem" but it is not.

This is how you ensure that somebody will never like math.

Here's a view of why it might make sense for a teacher to emphasize one way over the other that doesn't focus on "it's the definition and definitions are important":

> The teacher obviously knows (I’m assuming) that 5 + 5 + 5 is the same as 3 + 3 + 3 + 3 + 3.

> So why would one method be preferred over the other?

> Because thinking of 5 x 3 as, literally, “five groups of three” will help them when they learn how to divide. (That’s what the Common Core standard here is getting at.)

from http://www.patheos.com/blogs/friendlyatheist/2015/10/21/why-...

Also notable is the explicit assumption of good faith:

> Let’s assume for a second that this teacher isn’t an idiot. (I know. I know. Bear with me for a minute.)

> What possible explanation could there be for deducting points from this poor child’s exam?

1) When I was learning what "5 times 3" was I don't think I could even pronounce "commutative," let alone understand what it meant. I did know that 3 x 5 was the same as 5 x 3, but 5 / 3 was not 3 / 5 though!

2) Javascript's equality operators can confuse even longtime practitioners and bringing them up muddles the point.

I think the author is trying to say that 5 x 3, 3 x 5, and 30 / 2 are equivalent but not equal because they represent different operations, which he is somehow equating to types in computer science? This is nonsense, all three expressions are obviously the same type, and they are referentially transparent mathematical expressions, so they would be equal not just equivalent.

3) Setting students up for matrix multiplication? Really?

I'd be fully ok with this if the kid in question was actually learning about "Equals Versus Equivalency," but all of my school experience says that's unlikely. I hope I'm wrong.

The answers were wrong. But I would have given the student at least 1/2 point if not a full point with an explanation.

Would probably depend on how how much the concept was presented in class and the books, etc.

Potentially a straw man, but:

5*(1 + 1 + 1) is equivalent. If the student had written (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1), should the teacher have marked it correct? It's essentially the approach in question 2.

I'd argue that in this example, it doesn't demonstrate an understanding of "repeated addition" and at the very least should warrant follow-up by the teacher. The commutative example is more subtle and context would be nice to understand, but if this was a no consequence homework assignment that led to a quick follow-up by the teacher then it seems like a good move.

This whole thing is hilariously polarizing. People seem to project so much stuff on to what's actually shown on the paper that the arguments don't even make sense a lot of the time - either way. Some things I can get only looking at the paper:

The student got a grade of 4/6. There is no evidence from what can be seen on the photo if there are more than two questions or not - however it appears that after #2 there are instructions about showing work, etc. So it's entirely within reason to think that there were only two questions, yet many many people seem to accuse the teacher of being awful for not assigning partial credit. The evidence present on the page is not enough to conclude either way - there are solid arguments and inferences in each direction, yet everyone seems to argue from a "the answer was all right" or "the answer was all wrong" perspective.

Similarly - everyone is focusing just on the "learning to multiply whole numbers" aspect of this. But I presume this assignment was given in the course of a broader teaching curriculum. Perhaps the goal is to get the kids to arrange things in a certain way, because it provides a bit of foundational knowledge for next steps. Some next steps where pushing the "3+3+3+3+3" version of this is a "better" representation:

* algebraic concepts: 2x becomes x+x, etc. I don't know a way to write 2x in terms of 2+...

* fractions: (this is basically the same as above) 4 * 1/2 is 1/2 + 1/2 + 1/2 + 1/2, but again I don't know how to write it in terms of 4+ without requiring a bigger transform of first doing the multiplication then switching the sign.

* Matricies - here is a case of multiplication that isn't commutative

Point being - when teaching sometimes things are left out at first, for the sake of a simple consistent framework to built more concepts upon. Later - those concepts can help to understand additional properties or adjustments to the original facts. It's not required to teach the rules of commutative and associative and so on immediately.

Another funny thing about this is everyone just assumes that the paper as shown represents the totality of output from the teacher. It's entirely possible that after the quiz or assignment, the teacher gave a lesson on why this (perhaps common) 'mistake' is wrong.

I guess I'm rambling off my original point - but I don't really understand how this entire thing is causing so much vitriol and hate and unfounded speculation - other than a bunch of people projecting their own frustrations with some shitty teachers they had in the past.

The author forgets to mention that in mathematics = represents equivalency.

This is why saying 5 of 3 is better than 5 times 3.

Its a trivial mistake; we don't know the context; the kid could (and maybe was) tutored on the difference. All about nothing.

There are a number of issues with this explanation.

Firstly, as we know, multiplication and addition are associative, which means if you ever teach a child that 5 x 3 is different to 3 x 5, you are imparting wrong information.

The issue is that the question asks the child to "use the repeated addition strategy to solve: 5x3". The reason this is a problem is because "repeated addition" is indeed a strategy to teach children the concept that if you take a multiple of some number the. It is like repeatedly adding that number of items, a number of times. It is used as a stepping stone towards fully understanding multiplication,after on, and takes into account that young children think I terms of what they see. So for example, they see that a dog has 4 legs, and if you have 3 dogs then you add the four legs together three times (one for each dog).

Notice that it's madness to teach this as a. "addition strategy", because at that age "strategy" is far too abstract a concept for most children to grasp. The irony is that teachers then attempt to teach using a technique that uses a low level of abstraction, but when they call it the "addition strategy" they have just attempted to teach this technique that is using a more concrete methodology via language that uses concepts that are arguably more abstract than the concept they are attempting to teach!

You can see that the whole point of that technique is missed completely on that exam because of the question being asked. In fact, to have a student demonstrate understanding the I fact the question should be "I have five jars of Jellybeans. Each jar has 3 Jellybeans in them. Show me how you would represent the number of jars times by the number of Jellybeans in each jar, using addition."

You see, the point of the strategy is entirely being missed here. The author protests that the child will get confused because if they rely on the law of association with subtraction and division they will get the answer wrong, and be confused. But that's not what is happening. The child has clearly understood that actually, 3+3+3+3+3 is the same as 5+5+5. In actual fact, the student has shown a clear understanding of multiplication via addition.

If you think that child will be confused, wait till they get to fractions and numbers with decimal places! Because at that point, you can't use addition to explain multiplication and then you need to explain multiplication in terms of scale. There's actually a case to answer that the entire technique of teaching multiplication via addition is fundamentally flawed and it's better to teach in terms of scale anyway. I don't subscribe to that view, but I can see why it might be held.

I have to also take issue with using the definition from what looks like the Cambridge Dictionary's noun definition is that this is NOT the same precise meaning as equivalence in mathematics. In fact, if you were to use first-order logic, then it would be:

iff 5x3=15 then 5+5+5=15

or,

(5x3=15) ≡ (5+5+5=15)

That satisfies the two expressions logical equivalence. So the statement that this is NOT logically equivalent is entirely wrong.

Furthermore, the author has not read the definition on Wikipedia carefully enough. It says:

The multiplication of two whole numbers is equivalent to adding as many copies of one of them, as the value of the other one

The assumption being made here is that Wikipedia is saying that the number that is to be added up multiple times is the leftmost number in the expression, but it does not in fact say this at all. It says to add "as many copies of one of them", which means it could be referring to the left or right hand value in the multiplication expression.

The common core and the techniques used to teach young children are solid. Unfortunately, it looks like the way they have been used and taught to educators is the problem here! The fact that you can see the framework leaking into a test question shows that there is a fundamental flaw in the pedagogy of whoever is teaching that class.

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.

I am surprised to see exclusively negative reactions here. For one, the question wasn't marked wrong so much as partially incorrect (and partially correct). Secondly, the question is not "What is 5x3?" and the test is not about multiplication. It's explicitly about a specific process, which the teacher has presumably taught and which the student unequivocally got wrong.

If you were asked in the wood shop to make a drawer using dovetail joins and you instead make a drawer using a rabbet joins, well you shouldn't get full credit because the task wasn't about producing a drawer but about dovetail joins.

The learning objective may read "I can use control structures to solve tasks involving repetition", but if the question is "Use a for-loop to print the numbers from 1 to 10", and you used a while-loop, well again you would get partial credit but not full credit.

Many of us are computer programmers, and we know that many successful code solutions to the same problem are of equal quality. The worst code I work with is by people who never seem to care what the code looks like as long as it produced a correct result. Maybe they were only ever taught by teachers who would mark this 100% correct because they thought that answers matter and concepts don't.

My wife is a teacher in the NSW education system (Australia) and I've seen her use the rectangle system. However, the rectangle system is used to also show that if you take the same rectangle with the items placed in it in a uniform distribution, the rotate the rectangle and its contents by 90 degrees the the number of items are the same, but the row and column numbers swap around.

If anything the rectangular system shows that multiplication is commutative, which I feel is its real value. Interestingly enough, that isn't ever explained to most teachers so I'm not surprised if it's being misapplied as a technique for learning!

As against what happens out there in the tough world of work.

Paeno axioms do not define multiplication. Multiplication is considered to be part of second-order arithmetic, which includes Paeno arithmetic (which is a first order system) and augments it to form a stronger set of axioms, of which multiplication is one of them.

The definition of multiplication is indeed in the Wikipedia article, but if you reread it then you'll see it doesn't claim to be a Paeno axiom. A better article to read (after reading about Paeno axioms!) is here:

https://en.m.wikipedia.org/wiki/Second-order_arithmetic#Basi...

Exactly. And this is why we use math operators rather than English terms for math operations. If you want to specifically mean 5 times a group of 3, define a new operator for it, like 5 ○ 3. Don't 'overload' the x operator with your own arbitrary meaning.

It's not an analogy.

I was giving a different example of an algorithm than the one denoted by the assignment. "Use the repeated addition strategy..." is a clear call to demonstrate the application of a named procedure.

This is no accident, by the way. This is a deliberate design decision of those who created the common core standards. One of the goals is to teach algorithmic thinking.

It's pretty easy to google about if you can find your way through the reactionary hissy-fit memes.

You're joking, right?

GP read it as: "5 x3", like he would read "copy x3", or "copy, three times". It's a natural way of reading "5x3", though I personally read it as "5x 3".

Because where I come from, we use the English equivalent of "into" rather than "times". "5 into 3" roughly translates to "5, 3 times".

The meta-point here is that English (or any other language) is crap for math, which is why we use mathematical notation. And this bullcrap syllabus is trying to redefine the "x" operator, which gets my goat.

(5, times 3) or (5 times, 3) both work.

I just consider "times" the name of the multiplication operator.

I think you would find we agree much more than we disagree. Though what I find most infuriating is the blanket assumption (with similar level of disconnect) that what is being taught is mindless or confusing with no value, often simply because it's labeled as a "curriculum" or a "learning objective". You're not automatically right because you "experienced similar BS"; instead you have to realize that you, too, are coming into it with a bias and blindness.

What I see is a a bunch of people who can't stand seeing that red -1, maybe because it has been ingrained in them that they have to be perfect. Or maybe it's natural, and no one helped them git rid of that feeling.

It's so important for young students to feel like they understand and will continue to understand, in order for them to then achieve new understanding. I don't know how to write that without sounding like a theorist, but I sincerely believe it to be true. You've got to get rid of that fear of red ink.

There are tons of poor ways to teach, and poor curricula. This teacher could be doing a fine job with this student (and the parent's the ones that don't get it), or could be seriously hindering the child. I certainly wouldn't teach multiplication strategies this way. But it's not clear to me that marking this particular answer as only partially correct is inherently and unquestionably wrong.

I don't really support the pedantry that's going on in the grading, but I'm not sure I agree with you. To the observer looking at a non-moving rectangle, "length" and "width" are not interchangable. In the same way, there are indeed a "first" and "second" by simple definition of the way English/math notation work (in other words, "left" and "right".) The student was asked to use the "repeated addition strategy" and an "array" -- if the algorithm for doing those was taught using a specific order of the operands, the student is technically wrong to swap them. Whether or not it's fair or useful to deem them wrong when they are giving an equal but non-equivalent answer, or whether or not the algorithm should care about order when the underlying mathematical operation is commutative, are other issues.

It certainly is a repeated addition strategy, but is it the repeated addition strategy given to the students?

The definition of the algorithm given to the student may involve language like "take the first number and..."

The steps are the steps.

The objective and obvious difference between the 5 and the 3 is that the 5 is first and the 3 is second. The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. (The bananas and bundles just illustrates an example for why, in another context, being first or second would be important. But on the test, 5 is still first.)

On the other hand, you are invoking a "repeated addition" that the student was never taught. Your repeated addition strategy is "add <one of numbers> together <the other number> times". The taught repeated addition strategy was "add <the second number> together <the first number> times".

The purpose of marking a test is to communicate to the child whether they understand the topic. Marking an almost correct answer wrong (without elaboration) is bad feedback.

The problem that people are raising here isn't how the child feels, it's how the child thinks. And one thing they might think as a result of this answer is "Oh, I guess multiplication isn't the same both ways. I must have been mistaken." and it might take some time for this misunderstanding to clear up.

The act of calculating the area surely does mean they are the same rectangle. Because they can be written down in more than one way is a weakness of the notation; the math is independent of that.

> After all, you aren't really teaching repeated addition, you are just using it as scaffolding to provide an insight into multiplication!

You may be right. This is the interesting part of the discussion, and you've framed it well. I think it can be scaffolding technique also for the application of definitions, the expansion of symbols to their definition. Perhaps there is a better way to say that (or other examples), but the point is that I don't think that the exclusive value in teaching the technique is soon-to-be discarded scaffolding for multiplying numbers.

> the child (and parent!) was annoyed because it made little sense to mark it as wrong

The impact that the -1 has on the child is also interesting. I think it scored 1 out of 2, so it wasn't marked "wrong" so mach as "partially correct". It should be clear to the student that they basically got it right but slightly misapplied the technique, due to the comment, shouldn't it? If it isn't, it's the result of too much focus on the grade and too little focus on the comment.

It seems to me more likely that it's parents and other adults who see this -1 so negatively, and impose that on the kids. I would have been upset as a kid, too, but the sooner someone could have gotten me to be okay with quantitative imperfection, the better.

Yeah, but the problem is: repeated addition is attempting to take something very concrete like I give four children three marbles each, how many marbles does each child have?

You then use that addition technique to have them add up the number of marbles (in essence it's as if you are asking them to count on their hands, which is a valid technique at this level).

But that helps the child understand the concept of addition in a very literal and concrete fashion, because at the age of 4-5 years old (sometimes older), children don't think at a higher level of abstraction. And using symbols to represent multiplication IS a higher level of abstraction.

It seems to me, a non-educator, that the counting technique has value in word problems. But as soon as the child shows they understand the concept, then you introduce the notation (e.g. 5 x 3), explain the numbers can be added up either as five values added up three times, or three values added up five times.

That the test talked about a "strategy" is not really maths, and frankly it seems to be misapplying a solid teaching technique, leading to confusion, anger and a lack of confidence in the child. If that's happening, then I'd suggest the technique is not all that solid and teachers and other educators should seriously consider whether it is causing more harm than good.

P.S. If you have a Bachelors in Mathematics, then surely you can see that there is a fundamental problem if a child is taught that 5x3 is not the same as 3x5?

I seriously doubt this student "only knows how to produce a particular answer for a particular question". Are you claiming he could compute 3 times 5 but not 5 times 3?

On the contrary, I think this student may be showing that he knew the two techniques, and that he is smart enough to pick the easier computation.

But yes, if your goal is to kill any creativity in intelligent should punish kids that deviate from the lines hard.

If a teacher asked "compute 1000 x 1", no sane kid would do "1 plus 1 equals 2; 2 plus 1 equals 3;...; 999 plus one equals 1000".

This teacher would have failed Carl Friedrich Gauss, too, for computing sum(1,100) in seconds.

A student that sees a multiplication sign and reads it as "sets of" is not better off than one who reads it as multiplication.

a x b = b + ... +b is not a strict definition, it's just a convention.

Check the English Wikipedia:

https://en.wikipedia.org/wiki/Multiplication

It's, as you say, 5x3 = 3 + 3 + 3 + 3 +3.

But now check Russian Wikipedia:

https://ru.wikipedia.org/wiki/%D0%A3%D0%BC%D0%BD%D0%BE%D0%B6...

There you'll see 5x3 = 5 + 5 + 5.

So much for the "very well-defined mathematical meaning".

> ... does have a very well-defined mathematical meaning: a x b := b + ... + b.

Please complete the definition. which is that a x b = b x a, so a x b can also be written as a x .. x a. There is nothing special about the order.

Right, for different definitions of "same". I think that the OP attempts to explain that there are different definitions of "same" that are each valid.

I'm pretty sure that the homework was given as part of a course teaching multiplication. Perhaps what was desired was to first have children able to construct products from repeated addition, before teaching them the commutative property?

Well, maybe you can find a source, but I can only find sources that define multiplication as I have and then mention that multiplication of, say, real numbers, is commutative.