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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)

#411

Post by eigenket »

alentist wrote: post #410
eigenket wrote: post #409

Earlier quoted context omitted.

Na, I'm not - I'm explaining how to "compute: \pi*\pi using repeated addition". Step 1 - find a convenient Cauchy sequence that converges to \pi.

> Na, I'm not Yes, you are. Cauchy sequences are a model of the real numbers (which are the Dedekind-complete ordered field). > I'm explaining how to "compute: \pi*\pi using repeated addition" How will you define π times π as a repeated addition, i.e. something of the form x + x + … + x?

There is not one Dedekind-complete ordered field, for every two Dedekind-complete ordered fields there is a unique isomorphism between them. For example the nxn diagonal matrices with real entries with all the entries along the diagonal equal are a Dedekind-complete ordered field.

I don't know why you are conflating the words "compute" and "define", do you understand how these are different words? I was responding to how to "compute: \pi*\pi using repeated addition", this is rather different to defining \pi*\pi as a repeated addition.

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

#412

Post by alentist »

eigenket wrote: post #411
alentist wrote: post #410

Earlier quoted context omitted.

> Na, I'm not Yes, you are. Cauchy sequences are a model of the real numbers (which are the Dedekind-complete ordered field). > I'm explaining how to "compute: \pi*\pi using repeated addition" How will you define π times π as a repeated addition, i.e. something of the form x + x + … + x?

There is not one Dedekind-complete ordered field, for every two Dedekind-complete ordered fields there is a unique isomorphism between them. For example the nxn diagonal matrices with real entries with all the entries along the diagonal equal are a Dedekind-complete ordered field. I don't know why you are conflating the words "compute" and "define", do you understand how these are different words? I was responding to…

> There is not one Dedekind-complete ordered field

Yes there is.

> For example the nxn diagonal matrices with real entries

That's a model, not the theory.

> I don't know why you are conflating the words "compute" and "define"

This whole discussion is about definition.

Your own comment started with "You first define..."

And no, you are not able to compute arbitrary products of arbitrary reals in this way anyway, if by computation you mean a finitary algorithmic process.

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

#413

Post by peterwoerner »

threatofrain wrote: post #267
peterwoerner wrote: post #263

Earlier quoted context omitted.

But you can define multiplication based on abstracting repeated addition. That how we did it my analysis class, although I forget some of the details. So if you say multiplication is repeated addition, it really isn't a lie.

But did your analysis class define addition on the naturals? Then that's an operator for the naturals. And then for integers, rationals, etc.

Yep and how to construct, the naturals from set theory, the integers from the naturals, the rationals from the integers and the reals from the rational. I've forgotten the details at this point, but I remember the conclusions.

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

#414

Post by eigenket »

alentist wrote: post #412
eigenket wrote: post #411

Earlier quoted context omitted.

There is not one Dedekind-complete ordered field, for every two Dedekind-complete ordered fields there is a unique isomorphism between them. For example the nxn diagonal matrices with real entries with all the entries along the diagonal equal are a Dedekind-complete ordered field. I don't know why you are conflating the words "compute" and "define", do you understand how these are different words? I was responding to…

> There is not one Dedekind-complete ordered field Yes there is. > For example the nxn diagonal matrices with real entries That's a model, not the theory. > I don't know why you are conflating the words "compute" and "define" This whole discussion is about definition. Your own comment started with "You first define..." And no, you are not able to compute arbitrary products of arbitrary reals in this way anyway, if by…

Ok, I think your definition of "field" is different to mine. Mine starts "A field is a set F with binary operations + and x" and then goes on to list some properties they have to have, yours seems to be doing something different. If you start with mine you get a whole bunch of different complete ordered fields, but you can easily show they're all isomorphic (e.g. Spivak does this IIRC).

>This whole discussion is about definition

No, it isn't this whole discussion is answering the question about computing pi * pi. Maybe I was slightly sloppy in my first answer to that question, I was only attempting to sketch the method.

I don't really want to compute products of arbitrary reals in that way (or any way), but the method I sketched works for computable reals which is sufficient to cover the case of \pi. By computation I mean something that runs on a Turing machine, but I don't assume finitary (I'm happy for my Turing machine to keep producing more and more bits of precision forever).

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

#415

Post by [deleted] »

agumonkey wrote: post #259
munchbunny wrote: post #125

Earlier quoted context omitted.

I think the useful distinction is that, when you teach multiplication as a mechanical computation (arithmetic) it's useful to talk about it as repeated addition. As you get to negative numbers, rational/irrational numbers, complex numbers, matrices, etc. it becomes more useful to think about multiplication in more abstract ways, among which repeated addition is still often a useful way to look at it. I also think it'…

so the iterative pattern doesn't really hold when you go into abstract algebra ?

[deleted]

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

#416

Post by betterunix2 »

aaplok wrote: post #400
betterunix2 wrote: post #316

Earlier quoted context omitted.

"Multiplication is literally repeated addition" How do you repeat something a negative number of times? What does it mean to multiply two negatives, if multiplication is repeated addition? This is not an overly generalized concept, we teach negative numbers to elementary school students.

I don't know what you mean by "overly". Multiplication by negative numbers is a generalisation of the earlier concept of multiplication by positive numbers. In a sense even multiplying by 1 is a generalisation (at least if you read Euclid he doesn't really define multiplying by 1). I don't claim that we should not teach generalisations to children. What I claim is that of we teach generalisations we at least should b…

That sounds a bit backwards to me. If anything, negative numbers are the generalization (as is 0), and we should have a concept of multiplication that works for such generalizations. Euclid's approach to arithmetic always seemed strained to me, something motivated by a religious view of math where a compass and straightedge were the only tools anyone should need.

Repeated addition is just a technique for multiplication. It is one of many techniques we teach children, and we should leave it at that -- a technique. When we get to multiplication by negative numbers or fractions, we teach children other techniques, and we do not feel any need to try to "define" multiplication in terms of those techniques. Why should we "define" multiplication in terms of repeated addition, and then do circles around ourselves trying to generalize that "definition?" We can just say that multiplication is one operation we can do on numbers; addition is another, and they are related by the distributive law (and it from the distributive law that we can derive the various techniques we use for multiplication).

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

#417

Post by anticensor »

jmull wrote: post #160
y7 wrote: post #39

I think the author's argument becomes more clear when you consider multiplication in rings other than the integers, for example (square) matrices. The product of two matrices A*B does not correspond to repeated addition, and it is not commutative (A*B does not equal B*A in general). I can see that having an engrained belief that multiplication is defined via addition becomes problematic at some point when learning ab…

It's just that matrix multiplication is not the same thing as scaler multiplication. I think you need to understand that no matter how you define or think of scaler multiplication.

Hadamard multiplication of matrices is a scalar multiplication.

https://en.wikipedia.org/wiki/Hadamard_product_(matrices)

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

#418

Post by CodeMage »

pdonis wrote: post #296
CodeMage wrote: post #248

Earlier quoted context omitted.

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.

> Let's not take the word "lie" out of its context I'm not the one that used that word. If people are going to reference a concept that's been published in a book and uses that word, they should own it and be prepared to justify it. If they can't do that, they should not use the word in the first place. To be clear, I'm not saying you are one of those people. But the poster I was responding to is.

I'm not saying you shouldn't use the word. I'm saying you shouldn't ignore its very specific and specialized context. Taking things out of their context often leads to problems, just like in this case.

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

#419

Post by shadowofneptune »

jacobolus wrote: post #309
drdec wrote: post #287

Earlier quoted context omitted.

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%...

Beliefs 1–4 on page 151 are exactly what I thought about geometry until later in college. Thanks for the article, it's fascinating.

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

#420

Post by djur »

pdonis wrote: post #299
djur wrote: post #294

Earlier quoted context omitted.

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.

> Explaining the structure of an atom without having to explain atomic orbitals and standing waves. My high school chemistry teacher had no problem explaining this to me, when teaching the periodic table of the elements, without telling any lies and without going into the details of the quantum mechanics involved. The Pauli exclusion principle and a general statement that the details of the quantum mechanics were out…

What about earlier than high school? I learned about both of those things much earlier.
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