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

denisegaskins.com

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

#241

Post by beaconstudios »

pdonis wrote: post #203
dragontamer wrote: post #194

Earlier quoted context omitted.

> You can't. Pi is irrational. You just need to break it up infinitely times. We usually call the sequence 31415926...

> You just need to break it up infinitely times. I've responded to this elsewhere in the thread: I don't think adopting increasingly perverse interpretations of "repeated addition" as you try to include more and more numbers is a useful way to teach multiplication.

what they described is identical to long multiplication. That's not perverse, it's how most people multiply numbers.

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

#242

Post by cannabis_sam »

mnl wrote: post #193
dragontamer wrote: post #166

Earlier quoted context omitted.

> Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related. I just finished studying Galois Fields, which LITERALLY redefines multiplication and addition operators to study new forms of math (IE fields in particular) In all fields and rings, multiplication and addition are related by the distributive property. A(b + c) is equal to Ab + Ac, in literally every…

Yes, they are two binary operations and depending on the sets you consider and which properties you impose for those operations you have different algebraic structures. (This used to be taught at school before "modern mathematics" were considered harmful, maybe they were but at least they were correct). The thing is that as you can write m (let it be a positive integer) as m=1+...+1 (m-times), you can write n·m=n·(1+…

> But it's not a good idea to let ourselves get carried away, we still have two binary operations going on. At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards.

Why is this not a good idea?

Can’t we just accept/postulate that the additive identity is different from the multiplicative identity?

And still define a relationship between the addition and multiplication?

Maybe I misunderstand the issue..

(I’m not trying to be pedantic, but my math background has some holes :)

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

#243

Post by cannabis_sam »

jschwartzi wrote: post #152
cjfd wrote: post #105

Ah yes, there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously. You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be def…

Except this isn't a pedantic falsehood. Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related. Understanding ratio is key to understanding a lot of the physical sciences and you can build a really good intuitive understanding of a lot of simple physical concepts if you can just do dimensional analysis. But if you think of multiplication and division as a…

> Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related.

My apologies if this is a stupid question, but when does the intuitive (layman) understanding of multiplication as repeated addition break down (mathematically)?

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

#246

Post by scarmig »

midjji wrote: post #101
Agingcoder wrote: post #9

I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…

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

How would you compute \pi+\pi using addition?

Irrationals break all our intuitions.

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

#247

Post by maxerickson »

EvanAnderson wrote: post #225
maxerickson wrote: post #222

Earlier quoted context omitted.

I suspect that the problem isn't introducing a given model, rather the problem is then not moving on to additional models. Or putting it another way, I doubt that considering repeated addition gets in the way of understanding scaling.

I will say what we did work thru some multiplication of integers as repeated addition and scaling, just to show that the answers "come out the same". I used the scaling by a fraction to show that the repeated addition method doesn't work on all classes of numbers. (We've talked some about the difference between integers and real numbers.) The gist of my statement was something like: Adding repeatedly is a fine method…

I wasn't really trying to analyze your approach, more speculating that getting in front of the teacher probably doesn't matter. It's the insight that multiplication isn't strictly equivalent to addition that matters, and illustrating that they are often equivalent probably doesn't block that insight.

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

#248

Post by CodeMage »

pdonis wrote: post #199
zamadatix wrote: post #177

Earlier quoted context omitted.

The general concept is known as https://en.wikipedia.org/wiki/Lie-to-children and I think you'd be hard pressed to teach/learn/apply anything of significance without this "well actually it's more complicated than that" approach and recognizing how far you need to go with it to aptly do some task.

There's no need to lie to children. Telling children "multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with" is not a lie.

Let's not take the word "lie" out of its context and then get hung up on it. We're not actually lying to children by simplifying the explanation down to what they can comprehend with the tools they have at that stage.

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

#249

Post by SAI_Peregrinus »

eigenket wrote: post #111
marcosdumay wrote: post #106

Earlier quoted context omitted.

Of course, once you decide to iterate over uncountable sets, infinity starts to appear all the time. This isn't the only case of infinite calculations that can't be computed in practice but that mathematics use all the time anyway; and it reflects quite well the fact that multiplying irrational numbers isn't something that one can do practice. There is no problem with it.

I definitely wasn't trying to indicate that there was a problem with this, just pointing out that the path of using the Surreals (or anything else) to give an "algorithmic" description of real addition or multiplication is probably a bad idea.

I wasn't even attempting to give an "algorithmic" description. Just a formulaic one, that depends on the axiom of infinity (and accepting transfinite induction as a valid process). Since even at least one of the usual process for constructing the Reals (Dedekind cuts) needs this I don't feel it's much of a stretch in reasoning. And the Surreals have a nicer recursive formula for multiplication that eventually turns into repeated sums, and they're a strict superset of the Reals, so it does apply there.

If you want an algorithmic (no possible need an infinite number of steps) explicit construction of anything on the Reals you're going to be disappointed. They're an infinite set.

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

#250

Post by kmill »

amelius wrote: post #244

By the way, why doesn't exponentiation have units in physics? E.g. why don't we ever have something like kg^s?

I think it would have to play well with Taylor series -- for example e^s would be 1 + s + s^2/2 + s^3/6 + s^4/24 + ... -- and that would be mixing units. That's not to say it's meaningless, but rather you'd have to accept measurements like 4s + 5s^2, with incompatible units hanging around. (There's something similar in geometric algebra, I think, where expressions contain terms of different dimensions added together.)
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