[signal11]

> The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique.

Um, this is sophistry. The question asked for "5 x 3" using repeated addition. The x is a very well defined mathematical operator and "repeated addition" has a very well-defined meaning, and the child has demonstrated it by repeatedly adding 5 three times.

Yes, the child's cardinal sin is he Did Not Do As He Was Taught(tm), but seriously, that's more the teacher's and the school board's problem in my book, not the child's.

[chris_wot]

The way the child is being taught is because the teacher or course administrator has misunderstood the purpose of teaching arithmetic via repeated addition.

Repeated addition is relying on the fact that children see the world in a very concrete way and have not started to understand concepts in a more abstract fashion. Thus you use objects to explain concepts, like: every cat has one tail, I have 3 cats so how many tails are there in total?

You introduce notation in the class, but I can't see how it is valuable to use an abstract expression like 1x3 without a concrete description of the example of cats and tails. After all, you aren't really teaching repeated addition, you are just using it as scaffolding to provide an insight into multiplication!

The fact that the answer given can be shown as wrong has already demonstrated that the child (and parent!) was annoyed because it made little sense to mark it as wrong. It probably caused more harm than good, because now the child questions their understanding of the subject matter, yet ironically they do appear to have grasped the concept!

So at this point, the poor pedagogy of the teacher in misusing the counting technique means that the child starts to doubt themselves unnecessarily, they become locked in to a scaffolding technique that will later need to be discarded anyway. When they hit non-integer rational numbers - numbers with decimal points - they aren't going to be able to add these together, instead they will need to grasp that you can scale down numbers if you multiply any rational number between 0 and 1.

[jmilloy]

> 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.

[chris_wot]

Thanks, we are probably on the same page here :-)

The mark doesn't honestly seem to be the issue here though, at least so far as I can see, but rather that the teacher marked something as wrong when it was right.

[jmilloy]

It doesn't seem like a cardinal sin so much as a small quantitative note that the process was taught a different way that the teacher thinks is important.

Let's suppose one student can follow the procedure when asked but can't actually multiply in application, one student can't follow the procedure correctly but can multiply when needed, and a third can do both. Probably the first student will get questions on this quiz correct but will struggle on much of the rest of the unit, maybe get a low grade or hopefully get the help they need. The second (with the paper shown in the OP), will probably get an high grade because they got partial credit on a silly quiz. The third will get a higher high grade. What's so bad about that?

BTW, appealing to definitions won't work here, because the x does have a very well-defined mathematical meaning: a x b := b + ... + b.

[orless]

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".

[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.

[chris_wot]

What are your sources?