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

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

#331

Post by tsian2 » Wed, Mar 03, 2021, 10:32 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…

As someone with an almost solely intuitive understanding of mathematics, I still find the "repeated addition" idea useful when it comes to fractions. If 4 * 4 is 4 repeated 4 times (4 + 4 + 4 + 4). Then 1/2 * 1/2 is a half repeated half times, or a half of a half, which is a quarter. The numbers get hard to work with but the intuitive idea is still there.

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

#332

Post by pdonis » Wed, Mar 03, 2021, 10:32 PM UTC

kortilla wrote: post #324

roenxi wrote: post #308

Earlier quoted context omitted.

> 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.

The confidence is in the ability to learn a topic, not confidence about the knowledge.

It seems to me that the way to build people's confidence in their ability to learn is to allow them to learn. That means not making oversimplified statements because you don't think they're "ready" for more details. It means giving them the details, and letting them decide when they've had enough for now. (Of course one's time, say in a classroom, will be limited, so at some point one has to say "we don't have time to go into this further in class now"--but that's still letting them know that there are more details, and they can dig into them further on their own.)

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

#333

Post by thaumasiotes » Wed, Mar 03, 2021, 10:35 PM UTC

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.

Are you kidding? This is the exact opposite of the truth; the nature of multiplication as repeated addition is the entire reason why multiplying by 0 gives the additive identity. It's exactly the same as how exponentiating by 0 gives the multiplicative identity, since exponentiation is just repeated multiplication. And this is so fundamental that 1 is frequently referred to by this property, as "the empty product".

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

#334

Post by alentist » Wed, Mar 03, 2021, 10:36 PM UTC

aaplok wrote: post #289

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 is literally repeated addition, as in "taking a number multiple times". It is in the name. The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation…

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.

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

#335

Post by pdonis » Wed, Mar 03, 2021, 10:36 PM UTC

perl4ever wrote: post #310

pdonis wrote: post #302

Earlier quoted context omitted.

> 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…

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

Axiomatically, I think it can go either way. But one still has to recognize, as you do, that there are more cases than just the whole number case, and that what works for the whole number case might not work for other cases.

> the generalization can be done in multiple ways

There are certainly cases of this, but I don't think the case under discussion is one of them. There is only one generalization of the whole numbers under discussion here, the one from whole numbers to rationals to reals (and on to complex numbers if you want to take it that far, and still further on to matrices for some people in this discussion). There aren't multiple ways to do that: the rationals, reals, and complex numbers are all unique sets.

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

#336

Post by dragontamer » Wed, Mar 03, 2021, 10:37 PM UTC

mnl wrote: post #315

dragontamer wrote: post #280

Earlier quoted context omitted.

> I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x]. Is that not just (1 * (1-x^3) + x * (1-x^3) + x^2 * (1-x^3)) ?? The polynomial itself gives us the means at which we logically split up the multiplication into component parts. Just as 3.14 * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3, when we move onto polynomials, we do the same exact thing. EDIT: remember, AL…

Precisely, you're not repeating p(x) q(x)-times, you've used that p(x) is a linear combination of monomials and then the distributive property of Z[x]. Now, you could argue that this is exactly a way to "add repeatedly", but at some point pushing analogies stops being helpful to your students.

> Now, you could argue that this is exactly a way to "add repeatedly", but at some point pushing analogies stops being helpful to your students.

That's not what this blogpost is arguing about. This blogpost is arguing that "Multiplication is Repeated Addition" is unhelpful at the elementary school level and stops being true at some point.

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My argument is otherwise. "Multiplication is Repeated Addition" is clearly helpful in grade school. Almost everybody I know has learned Multiplication through that method.

Secondly: I cannot think of a single instance where its not true. Yes, I've had to use linear-combinations to extend it out to polynomials, but clearly the property holds even in polynomial-land.

Its not useful to teach multiplication of polynomials with "Repeated Addition". But the advice is "not wrong", in fact, polynomial multiplication continues to see many similarities with Real and Complex multiplication. Especially if we consider a "Basis" to be analogs to the thing that's repeatedly-added.

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

#337

Post by macspoofing » Wed, Mar 03, 2021, 10:38 PM UTC

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…

>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 - because it isn't helpful as you incorporate fractions, and negative numbers.

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

#338

Post by thaumasiotes » Wed, Mar 03, 2021, 10:38 PM UTC

dragontamer wrote: post #198

mnl wrote: post #193

Earlier quoted context omitted.

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 (mo…

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

You don't think it's easier to say "n·5 = n·(5+0) = n·5 + n·0"?

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

#339

Post by alentist » Wed, Mar 03, 2021, 10:39 PM UTC

rustybolt wrote: post #27

These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication? This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you…

> You can definitely see multiplication as repeated addition.

Doesn't work in every context where multiplication is defined.

> It's even a useful thing to do, how else would you define multiplication?

Perhaps something along the lines of: A multiplication is an endomorphism on an additive semigroup.

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

#340

Post by JackFr » Wed, Mar 03, 2021, 10:39 PM UTC

pdonis wrote: post #282

wcarey wrote: post #275

Earlier quoted context omitted.

How would you teach physics without lying to children?

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

We lie when when we teach calculus by relying on hand-wavy 'limit' proofs, without grounding students in the foundational aspects of point set topology.

It's much more productive pedagogically to get an intuition for slope and area than it is to get an intuition for compactness and the infinite intersection of open sets, but slope and area are A LIE.