What’s Wrong with “Multiplication Is Repeated Addition”? (2008) • Hacker News Board

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#351

Post by kortilla » Wed, Mar 03, 2021, 10:53 PM UTC

pdonis wrote: post #303

SilasX wrote: post #288

Earlier quoted context omitted.

Yeah, and that's still too much inferential distance to feed to an elementary schooler.

You appear to have a much more impoverished view of kids and their ability to learn than I do. My experience (not to mention my memory of how I was myself as a kid) is that kids grasp the fact that there can be more to a subject than adults are able to teach them at a particular time and place, so they're ok with adults honestly admitting that. But they do not like adults telling them categorical statements that late…

You’re view of children seems to be more based on your memory of high school. That’s a lot different than your behavior as a 6 year old when these simple primitives are being taught.

Little kids don’t care about nuance when they’re still having difficulties with carries in addition. Your ideal world where we first explain children that base10 isn’t the only way to represent numbers and whatever other caveats simply doesn’t exist.

They don’t have the knowledge required yet to even understand the scenarios when “exceptions to the rule” apply.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#352

Post by [deleted] » Wed, Mar 03, 2021, 10:55 PM UTC

thaumasiotes wrote: post #350

dragontamer wrote: post #343

Earlier quoted context omitted.

The (1 + (-1)) thing was very precisely written. All rings have 0 and 1 as elements. 0 is the additive identity. 1 is the multiplicative identity. 0 and 1 are NOT necessarily numbers. In Linear Algebra of 2x2 matricies, 0 is: [ 0 0 0 0 ] And 1 is: [ 1 0 0 1 ] Because A * 1 == A, A * 0 == [0 0; 0 0]... etc. etc. In general, you cannot really assume much more than "0 exists" and "1 exists" when working with Rings (at l…

> As such, the concept of "5" does not necessarily exist in all possible Ring-systems. Don't think of 5 as a quantity; think of it as a variable name. The proof only depends on the concept of addition and an additive identity (and distribution of multiplication over addition, which you're using anyway); no property of 5 appeared. > the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions…

[deleted]

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#353

Post by codethief » Wed, Mar 03, 2021, 10:56 PM UTC

eigenket wrote: post #119

midjji wrote: post #101

Earlier quoted context omitted.

Not defending the article, but how would you compute: \pi*\pi using repeated addition?

You first define multiplication of natural numbers to be repeated addition, then define multiplication of rationals in terms of multiplication and addition of naturals, then define multiplication of reals in terms of multiplication of Cauchy sequences of rationals ;)

Exactly. I've scrolled through hundreds of comments here now and it really is beyond me how the question of whether you can define x·y in terms of addition for x and y being arbitrary reals is even a matter of debate.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#354

Post by [deleted] » Wed, Mar 03, 2021, 10:57 PM UTC

thaumasiotes wrote: post #350

dragontamer wrote: post #343

Earlier quoted context omitted.

The (1 + (-1)) thing was very precisely written. All rings have 0 and 1 as elements. 0 is the additive identity. 1 is the multiplicative identity. 0 and 1 are NOT necessarily numbers. In Linear Algebra of 2x2 matricies, 0 is: [ 0 0 0 0 ] And 1 is: [ 1 0 0 1 ] Because A * 1 == A, A * 0 == [0 0; 0 0]... etc. etc. In general, you cannot really assume much more than "0 exists" and "1 exists" when working with Rings (at l…

> As such, the concept of "5" does not necessarily exist in all possible Ring-systems. Don't think of 5 as a quantity; think of it as a variable name. The proof only depends on the concept of addition and an additive identity (and distribution of multiplication over addition, which you're using anyway); no property of 5 appeared. > the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions…

[deleted]

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#355

Post by dragontamer » Wed, Mar 03, 2021, 10:58 PM UTC

thaumasiotes wrote: post #350

dragontamer wrote: post #343

Earlier quoted context omitted.

The (1 + (-1)) thing was very precisely written. All rings have 0 and 1 as elements. 0 is the additive identity. 1 is the multiplicative identity. 0 and 1 are NOT necessarily numbers. In Linear Algebra of 2x2 matricies, 0 is: [ 0 0 0 0 ] And 1 is: [ 1 0 0 1 ] Because A * 1 == A, A * 0 == [0 0; 0 0]... etc. etc. In general, you cannot really assume much more than "0 exists" and "1 exists" when working with Rings (at l…

> As such, the concept of "5" does not necessarily exist in all possible Ring-systems. Don't think of 5 as a quantity; think of it as a variable name. The proof only depends on the concept of addition and an additive identity (and distribution of multiplication over addition, which you're using anyway); no property of 5 appeared. > the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions…

I see what you're saying now. So to answer your original question...

> You don't think it's easier to say...

No. I disagree, your way of thinking is harder for me to think. :-)

You're correct, but my mind didn't work like yours. But that's the beautiful thing about mathematics: we both are correct. We just had different viewpoints about how things work. Ultimately, it seems like we're both saying the same thing, although we tweaked the formulas to look like the simplest ways for our own brains.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#356

Post by alentist » Wed, Mar 03, 2021, 10:59 PM UTC

codethief wrote: post #349

alentist wrote: post #334

Earlier quoted context omitted.

Define π times e in terms of repeated addition. I wouldn't say multiplication is literally repeated addition. I'd say it reduces to repeated addition when the multiplier is a natural number.

Maybe it is not literally addition but it certainly reduces to addition in all cases: https://news.ycombinator.com/item?id=26329855

Nope. I'll reply there.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#357

Post by NineStarPoint » Wed, Mar 03, 2021, 11:00 PM UTC

chrisandchips wrote: post #150

I strongly believe that helping younger students gain strong intuition for these operators pays dividends towards their later success in maths. I've always run into the following problem: I try to motivate multiplication as repeated addition, which does help with intuition, but then things totally fall apart when we move on from integers into fractional values. 1/2 * 1/2 -> 1/4. Sure you can teach someone to simply m…

Fractional numbers make sense to me as an extension, but it also requires an intuition of division on the same lines.

Take number n and multiply it by number x/y. To do this, you have to split number n into y parts and take x number of them. So to multiply 8 by 3/4, you split 8 into 4 parts (2 + 2 + 2 + 2) and then take 3 of those parts. This ends up being 2 + 2 + 2 = 6.

For multiplying two fractions, you have to extend it to n/m * x/y. Since you can multiply the top and bottom of a fraction by the same number, you can write n/m as ny/my. Then you can have n/my be your “equal part”, and take x of them. So 1/2 * 1/2, you take 2/4 and split it into 1/4 + 1/4 and 1 of them, so the answer is 1/4.

To me at least, this makes sense as an extension of multiplication is repeated addition. It’s when you get to irrationals that it starts to fall apart, and even then the intuitions the above way of thinking led me to have served me well.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#358

Post by thaumasiotes » Wed, Mar 03, 2021, 11:00 PM UTC

dragontamer wrote: post #355

thaumasiotes wrote: post #350

Earlier quoted context omitted.

> As such, the concept of "5" does not necessarily exist in all possible Ring-systems. Don't think of 5 as a quantity; think of it as a variable name. The proof only depends on the concept of addition and an additive identity (and distribution of multiplication over addition, which you're using anyway); no property of 5 appeared. > the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions…

I see what you're saying now. So to answer your original question... > You don't think it's easier to say... No. I disagree, your way of thinking is harder for me to think. :-) You're correct, but my mind didn't work like yours. But that's the beautiful thing about mathematics: we both are correct. We just had different viewpoints about how things work. Ultimately, it seems like we're both saying the same thing, alth…

Fair enough.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#359

Post by pdonis » Wed, Mar 03, 2021, 11:01 PM UTC

SilasX wrote: post #329

pdonis wrote: post #326

Earlier quoted context omitted.

> loading down literally every claim with that long chain of caveats. I have never proposed doing the latter, so you are attacking a straw man. Once it's understood that you're teaching a simplified, approximate model, you don't have to repeat in every sentence that you're teaching a simplified, approximate model. You just have to not say it's "the Truth", without approximation and without qualification.

I'm relying on these examples you gave of how to do it: >"multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with" >"repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated cases that you'll learn about later" If you disagree that that's "long" or would feel that way i…

> If you disagree that that's "long"

I do. Some of the words might be changed, depending on what words have been used to describe the operation of addition and the set of counting numbers. But, as I think I've pointed out elsewhere in this thread, the very fact that the children know about addition and the counting numbers means they know what an operation is ("a thingie like addition") and what a set of numbers is ("a thingie like the counting numbers").

> you seem to reject the idea of giving the one caveat at the beginning of the course

I don't know where you're getting that from. I have already said the contrary--once you've said it, you don't need to repeat in every sentence.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#360

Post by macspoofing » Wed, Mar 03, 2021, 11:05 PM UTC

dragonwriter wrote: post #345

macspoofing wrote: post #337

Earlier quoted context omitted.

>I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods Are you sure it's pedantic? As a matter of practically, neither people, nor computers actually compute multiplication in such a way. Even children, though they may be exposed to the 'multiplication is repeated addition' concept as an introduction to multiplication, are quickly ushered past this and it is never brought up again -…

> As a matter of practically, neither people, nor computers actually compute multiplication in such a way. Even children, though they may be exposed to the 'multiplication is repeated addition' concept as an introduction to multiplication, are quickly ushered past this and it is never brought up again. The standard manual method of multiplication of large numbers relies on leveraging heavily: (1) the fact that my mul…

>nonnegative integers,

Why, prey tell, did you make a distinction between positive and negative integers? Is it that repeated addition starts to fail as conceptual model there?

>the fact that my multiplication is equivalent to repeated addition for nonnegative integers, ..

Your claim seems to be that in some abstract way any algorithmic approach to multiplication relies on the fact that multiplying non-negative integers (there's that distinction again .. nasty stuff those negative integers are for your argument) can be rewritten as a repeated addition ... but that's not how people (not even children learning multiplication) do it. That's not how computers do it either.

So no, algorithmically multiplying numbers is not the same thing as 'using' the concept of 'multiplication is repeated addition' in any sense, colloquial or otherwise. Also it starts feeling like you're begging the question when you claim that.