[signal11]

> ... 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.

[jmilloy]

That's not part of the definition. That's a separate property.

I'm not sure we should care about that in elementary school, so the point is not to defend the teacher but only that you can't use the definition as an argument against the teacher.

[chris_wot]

No, I'm afraid you've not given a complete definition of multiplication. You need to also show that multiplication is commutative, which is indeed a property of multiplication but MUST be included in the definition.

At the child's level (primary age child, NOT high-school) then it is unnecessary to introduce the distributive property. But you honestly have to make the associative property very, very clear of the child will potentially have problems down the track!

(Edit: brain fart - I said associative when I meant commutative. Oops!)

[jhanschoo]

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?

[chris_wot]

As I've said, that's a misuse of the repeated addition technique.

[orless]

But 5 + 5 + 5 WAS repeated addition!

[jmilloy]

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.

[jasode]

>define $EQUIVALENCE [...] and then mention $PROPERTY

When you keep pointing back to "a x b = b+b...+b", as The Definition without including the properties, it means you're mixing up the orthography[0] of multiplication with the real underlying idea of multiplication.

A math definition includes that all properties must simultaneously be true. It's the limitations of writing (orthography[0]) that we state things one thing before the other. The phrase "and then" used as a sequential condition is not applicable. Instead, if all properties are true, you thus have the definition.

Here's another "definition"[1] that states the summation in reverse order: "In simple algebra, multiplication is the process of calculating the result when a number a is taken b times."

e.g. "when a number 5 is taken 3 times" ... which is the repeated addition the child carried out.

That wikipedia stated multiplication as "a x b = b+b...+b" while Wolfram MathWorld stated it as "a is taken b times." is a difference in orthography and not definition. Unfortunately, you're working backward from an arbitrary orthography and judging the child to be wrong.

[0]https://en.wikipedia.org/wiki/Orthography

[1]https://books.google.com/books?id=aFDWuZZslUUC&pg=PA1974&lpg...

The contents of the Weisstein book was also used in Wolfram MathWorld:

[2]http://mathworld.wolfram.com/Multiplication.html

[signal11]

I'd just like to add a point to jasode's excellent point about getting hung up about orthography, which is that please don't define math using English. It's a terrible thing to do -- for example, the en-us 5x3 = 5 times 3 = 3+3+3+3+3 fails for Spanish speakers. For another, English itself is not very "standard" - some variants of British English would actually read 5x3 as "5, 3 times". Math exists outside of human language and teaching kids should adapt to this reality.

[jeremysmyth]

+1 data point: In my anglophone school system we say 5x3 as "5 multiplied by 3", not "5 times 3". These semantics lead children (and me) to think 5 + 5 + 5 instead of 3 + 3 + 3 + 3 + 3.

[chris_wot]

What are your sources?