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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)

#191

Post by curiousllama »

pdonis wrote: post #161
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…

> 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 I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…

Idk discovering all the lies I believe is one of the most fun parts of learning for me.

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

#192

Post by pdonis »

dragontamer wrote: post #188
iio8999 wrote: post #184

Earlier quoted context omitted.

I think the issue is more about: 3 * pi = pi + pi + pi but how do you represent the other distribution, where 3 is added together pi times?

Break-up pi in whatever means you think is reasonable. 3 * (3 + 0.1 + 0.04 + 0.001 + 0.0005...) Aka: 9.4245... You know, how we've been multiplying 3 * pi for our whole lives. We split pi up into an infinite sum of component numbers (3, 1, 4, 1, 5, 9, 2, 6...) and then combine them together by individually multiplying the parts (3 * 3 + 3 * 0.1 + 3 * 0.04...)

> Break-up pi in whatever means you think is reasonable.

You can't. Pi is irrational.

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

#193

Post by mnl »

dragontamer wrote: post #166
jschwartzi wrote: post #152

Earlier quoted context omitted.

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. 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+...+1), invoke the distributive property for · wrt + and express it as: n·m=n+...+n (m-times), so it looks like "repeated addition" for integers in this case. 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.

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

#194

Post by dragontamer »

pdonis wrote: post #192
dragontamer wrote: post #188

Earlier quoted context omitted.

Break-up pi in whatever means you think is reasonable. 3 * (3 + 0.1 + 0.04 + 0.001 + 0.0005...) Aka: 9.4245... You know, how we've been multiplying 3 * pi for our whole lives. We split pi up into an infinite sum of component numbers (3, 1, 4, 1, 5, 9, 2, 6...) and then combine them together by individually multiplying the parts (3 * 3 + 3 * 0.1 + 3 * 0.04...)

> Break-up pi in whatever means you think is reasonable. You can't. Pi is irrational.

> You can't. Pi is irrational.

You just need to break it up infinitely times. We usually call the sequence 31415926...

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

#195

Post by tarkin2 »

Why not just say:

There is the symbol '*'. And when there's two integers it does one thing, and you can think about it as repeated addition. But when there's something different, it works differently. But now we're just going to concentrate on integers.

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

#196

Post by pdonis »

wruza wrote: post #187
pdonis wrote: post #164

Earlier quoted context omitted.

> You can definitely see multiplication as repeated addition Only if you restrict to whole numbers. But multiplication does not just apply to whole numbers. How would you see multiplication by pi as repeated addition? > how else would you define multiplication? As a separate operation with its own properties. The subject article gives several examples of properties of multiplication that are simply different from pro…

How would you see multiplication by pi as repeated addition? By taking 3 times as usual and then another .14 approximately. E.g. for 2, it is 3+3+28/100, roughly 6.28. (edit: if you’re confused by /100, read it as “two places next to a point”) If you’re asking how to do that exactly , first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:

> By taking 3 times as usual and then another .14 approximately

First, "taking 3 .14 times" doesn't make sense, at least not in the elementary school student's understanding of "repeated addition".

Second, pi is irrational, as you evidently realize since you say "approximately". There is no way to add 3 pi times.

> If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:

Add 1 to the 3 and keep the part to the right of the decimal point the same. Addition is commutative.

Basically, your argument is that multiplication is repeated addition as long as we're willing to adopt increasingly perverse interpretations of "repeated addition" as we expand the scope of the numbers we use. Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?

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

#197

Post by pdonis »

dash2 wrote: post #175
pdonis wrote: post #164

Earlier quoted context omitted.

> You can definitely see multiplication as repeated addition Only if you restrict to whole numbers. But multiplication does not just apply to whole numbers. How would you see multiplication by pi as repeated addition? > how else would you define multiplication? As a separate operation with its own properties. The subject article gives several examples of properties of multiplication that are simply different from pro…

Multiplication by rational fractions p/q is just multiplication by whole p, followed by multiplication by 1/q. 1/q is just the number that when multiplied by q gives 1. So you can cash that all out as repeated addition. Multiplication by irrationals is just an infinite sum of multiplication by rationals, no? x times pi = x times 3, plus x times 1 / 10, plus.... I don't see there's any conceptual issue here. Mathemati…

> multiplication by 1/q

Is not repeated addition. You can't add a number to itself 1/q times. At least, not unless you're willing to adopt increasingly perverse interpretations of "repeated addition" as you try to cover more and more numbers. See my response to wruza upthread.

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

#198

Post by dragontamer »

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+…

> At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards.

n * 0 = n * (1 + (-1)) = n + (-n) = 0.

-----

The 0-element in a Galois Field works identically btw. In GF(5), the 0 element is 5 (5 mod 5 == 0).

n * (5) == n * 0 == n * (1 + (-1)) == n * (1 + 4) == 0 mod5.

For example, if we take "n" == 2, 2 * 5 == 10 mod 5 == 0.

2 * (1 + 4) == 2 + 8 == 2 + 3 (mod 5) == 5 mod 5 == 0. Etc. etc.

-------

This property literally holds in all fields and rings (but not groups).

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

#199

Post by pdonis »

zamadatix wrote: post #177
pdonis wrote: post #161

Earlier quoted context omitted.

> 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 I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…

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.

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

#200

Post by pdonis »

curiousllama wrote: post #191
pdonis wrote: post #161

Earlier quoted context omitted.

> 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 I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…

Idk discovering all the lies I believe is one of the most fun parts of learning for me.

> discovering all the lies I believe

I have no problem with discovering lies I believe because I told them to myself (and I think most of the lies anyone believes are lies they've told themselves). What I have a problem with is discovering that someone else deliberately lied to me. But probably not everyone feels the same way I do about such things.

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