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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)

#301

Post by bawolff »

I feel like people here have a weird definition of "truth". Platonism aside, there is no outside ground truth on what "addition" or "multiplication" is. It is an abstract symbol defined solely by what it does to numbers. As long as your definition describes its behaviour, your definition is no more or less right than any other definition.

If you want to define multiplication as repeated addition (and addition as repeated invocation of the succesor function), by all means do so. Its not wrong.

Sure you could say it doesn't work for complex numbers, but usually when you introduce complex numbers you say something like, multiplication works as normal except i×i=-1. I hardly see that as a problem with the original definition of multiplication. Its more about the definition of i than the definition of multiplication.

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

#302

Post by pdonis »

allturtles wrote: post #217
pdonis wrote: post #205

Earlier quoted context omitted.

I think this viewpoint is pernicious. A child doesn't have to be "smart" in order to deserve being told the truth. Or I could turn your remark around: what makes you, the adult, think you are so much smarter than the child that you can correctly judge what lies are OK to tell them? Are all adults really that smart? (Are any of us?)

An abstraction/simplification/shorthand is not a lie. People don't say "multiplication is repeated addition" because they are trying to hide the truth for some selfish reason. It's a pedagogical strategy to help people learn a new abstraction by analogy to an old one. These kind of crutches are a necessity, you can't introduce all the complexity of the world to someone all at once. This applies to every subject - sci…

> An abstraction/simplification/shorthand is not a lie.

"Multiplication is repeated addition" is not "an abstraction/simplification/shorthand". Doing that for multiplication would be saying something like "multiplication is a distinct primitive operation, but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now." Is that really so hard?

> Is it a "lie" to teach kids just learning chess that queens are worth more than any other piece and you should always protect your queen, even though there are advanced situations when it makes sense to sacrifice your queen for no immediate material gain?

If you tell them everything you just said, no, you're not lying. But if you just tell them "always protect your queen", without explaining anything about why and without saying that there are some advanced situations where you might break this rule, yes, you're lying. It only takes a couple of sentences to add that extra information. Again, is that really so hard?

> In this specific case, introducing the ideas of "operations" and "counting numbers" into the picture muddies the waters, most kids who are just learning multiplication won't have any idea what you mean by those concepts.

Um, what? We're assuming they already know about addition of whole numbers. So it is simple to tell them "this addition thing that you learned, that's an example of an operation", and "those whole number thingies that you learned how to add, those are numbers". Once more, is that really so hard?

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

#303

Post by pdonis »

SilasX wrote: post #288
pdonis wrote: post #284

Earlier quoted context omitted.

> I don't think it's right to call it a lie It's not a lie to tell them "this is a simplified model that doesn't include everything, but you'll be able to add more complexities to it later". But that's not what "multiplication is repeated addition" says. You would say something like: "repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated case…

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 later turn out to be wrong.

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

#304

Post by im3w1l »

pdonis wrote: post #197
dash2 wrote: post #175

Earlier quoted context omitted.

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.

You can do it like this:

First define a*b for integers by addition.

Define the the rational numbers as ordered pairs of integers where the last number may not be 0. Consider the rational number (a,b) to be equivalent to (c, d) if a*d = b*c

(Exercise for the reader: Prove that this is an equivalence relation)

Now define (a, b) + (c, d) = (ad + bc, bd), (a, b) * (c, d) = (ac, bd)

(Exercise for the reader: Prove that the subset of rationals (n, 1) behaves just like the integers)

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

#305

Post by bawolff »

jmull wrote: post #183

To me the "So What’s the Problem?" section has a lot of irrelevant stuff. The problem with "multiplication is repeated addition" is that the concept breaks down once you move on from integers. 2 X 3: "add 2 together, three times" - works well ½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore. I can see why teachers who aren't that comfortable with m…

> ½ X ½:

Or just re-arrange the terms to be (1*1)÷(2*2). No need to add half, half a time, just understand that multiplication distributes over fractions.

That said i think the concept would make sense to most school children as a way of extending the intuitive concept of multiplication of natural numbers.

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

#306

Post by roenxi »

djur wrote: post #294
pdonis wrote: post #282

Earlier quoted context omitted.

Um, by not lying to them? Do you have a specific example of where you think you need to lie to teach physics?

Explaining the structure of an atom without having to explain atomic orbitals and standing waves. Explaining classical mechanics without including a bunch of caveats about relativistic speeds.

"When they performed [this experiment] at [experimental accuracy] they observed [that result] which makes them think [atomic orbitals and standing waves]."

Someone who thinks they understand physics without considering experimental accuracy doesn't understand physics.

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

#308

Post by roenxi »

majormajor wrote: post #213
pdonis wrote: post #199

Earlier quoted context omitted.

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.

The continuing series of qualifications "it's kinda like this, for what you're talking about" gets pretty rough there, even when explaining, say, Kubernetes to adults! It can be a confidence crusher. There's also the flip side response which is always asking a lot of questions about "well then what are the other sorts of numbers" and eventually getting shut down "we're not talking about that now" which comes back to…

> It can be a confidence crusher.

If confidence gets priority over truth in explanations, then society will churn out people who are confidently wrong. This is a bad idea even if everybody does that and even if it is the traditional approach.

If people were honest that they don't know something then the world at large would be a lot nicer to live in.

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

#309

Post by jacobolus »

drdec wrote: post #287
shadowofneptune wrote: post #135

Earlier quoted context omitted.

The way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.

The point of the Geometry course in that curriculum is to teach the student how to do proofs. The Geometry itself is secondary.

Unfortunately this is typically done in a counterproductive way.

See http://jwilson.coe.uga.edu/TiMER/Schoenfeld%20(1988)%20Good%...

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

#310

Post by perl4ever »

pdonis wrote: post #302
allturtles wrote: post #217

Earlier quoted context omitted.

An abstraction/simplification/shorthand is not a lie. People don't say "multiplication is repeated addition" because they are trying to hide the truth for some selfish reason. It's a pedagogical strategy to help people learn a new abstraction by analogy to an old one. These kind of crutches are a necessity, you can't introduce all the complexity of the world to someone all at once. This applies to every subject - sci…

> An abstraction/simplification/shorthand is not a lie. "Multiplication is repeated addition" is not "an abstraction/simplification/shorthand". Doing that for multiplication would be saying something like "multiplication is a distinct primitive operation, but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now." Is that really so hard? > Is it a "lie" to teach kids just l…

> but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now

I think it's an arbitrary perspective, whether you treat the whole number case as primary or the generalization as primary.

People may prefer to consider the extended definition more "real", but I think the argument for going the other way is that usually the original limited form of something is more likely agreed upon by most people, whereas the generalization can be done in multiple ways which may owe something to history and culture, or context.

I feel like math is fundamentally different than physics, where the more advanced theory is objectively closer to correct. With math, it's more of an arbitrary aesthetic or social judgment. Nothing ever stops you from generalizing anything even more than anyone did yet, right?

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