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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)

#161

Post by pdonis »

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 to teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct...", and that a different approach is possible. One can argue about the pros and cons of each approach, but I don't think it is helpful to just dismiss one side of the argument as "pedantic falsehoods".

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

#162

Post by maweki »

I think what the author is missing is the word "scalar". I do understand the sentiment that in general multiplication is not repeated addition. But multiplication with a scalar is.

In the usual curriculum you do the unit analysis as a separate step, as it's usually in the physics lessons where units are used. In math it's unitless.

Of course, we should start teaching immediately that you can't add apples and pears. Maybe we should just do this explicitly there, that 5 times 3 apples is fine, as 5 is scalar. Explain that immediately and then add the unit analysis in physics.

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

#163

Post by michaelwu »

OlleTO wrote: post #153

Proof: The derivative of x^2 is x. x^2 = x * x = x + x ... + x (x times) Therefore, d/dx(x^2) = d/dx(x * x) = d/dx (x + x + ... + x (x times)) = d/dx(x) + d/dx(x) + ... + d/dx(x) (x times) = 1 + 1 + ... + 1 (x times) = x

It's actually 2x...

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

#164

Post by pdonis »

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

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 properties of addition.

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

#165

Post by daveFNbuck »

swiley wrote: post #145
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…

Anyone that teaches this way should find something else to do. Memorizing a formula is easy but if you want to do well later you must be able to complete the square.

I never learned to complete the square, and I managed to do well enough later to get a degree in math.

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

#166

Post by dragontamer »

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.

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 form of math you can every think of. A and b could be matrices, vectors, polynomials, prime numbers, integers, rational, complex, real, or Galois polynomials over a weird modulus ring thingy inside of a vector inside of a matrix. Doesn't matter, addition and multiplication are and always defined in that manner.

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

#167

Post by jschwartzi »

munchbunny wrote: post #125
spacedcowboy wrote: post #104

Earlier quoted context omitted.

I have a PhD in physics and more maths qualifications than I can shake a stick at; to me, multiplication is repeated addition. I’m not sure what the teacher is trying to do here, but I do think the outcome of what they’re trying to do is far more complicated than the simple “multiplication is repeated addition”. I also happen to have an 8-year-old going through third grade right now, and when we were talking through…

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'…

I think the heart of the issue is whether it's more useful to teach children how to multiply two abstract numbers together as a kind of "mathematical procedure" that they need to memorize, or whether it's more useful to teach children that if they measure two sides of a square with a measuring tape, they can "multiply" the measurement and that the result is now in "square inches" rather than regular inches. And the schism is that some people believe that procedural memorization is useful because after 20+ years of education they've gotten through the good part, and other people believe that the procedural memorization does kids a disservice by divorcing mathematical thinking from the concrete world entirely.

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

#168

Post by redshirtrob »

eru wrote: post #92
sudosysgen wrote: post #73

Earlier quoted context omitted.

We did that derivation in class, actually! I was so lucky to have a great math teacher in high school.

That's not the norm?

I don't think so. I was on the Calculus track in high school so we derived it in...Pre-Calculus.

Prior to that the quadratic formula was something that seemed to be handed down from on high. We used it in Algebra II and maybe even before that, but I had no idea where it came from.

It was a mind-opening experience when we derived it in class one day. Our teacher didn't ruin the surprise. She just said, let's complete the square on a general quadratic equation. And there it was. The quadratic formula!

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

#169

Post by DanielMcLaury »

yakubin wrote: post #29
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. Only for rational numbers. Doesn't work for real and complex numbers. > It's even a useful thing to do, how else would you define multiplication? Axiomatically, not algorithmically.

Multiplication of real and complex numbers is typically defined by starting from repeated addition, extending this notion to rationals, and then extending that notion to reals by taking limits.

How exactly are you going to present multiplication of real numbers axiomatically without essentially including an axiom that bootstraps everything from repeated addition?

I suppose you can try defining the reals as "the unique complete ordered field" or the complex numbers as "the unique algebraically closed field of characteristic zero with cardinality c," but I don't think either of those are pedagogically useful to someone who is still learning what multiplication is.

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

#170

Post by pdonis »

maweki wrote: post #162

I think what the author is missing is the word "scalar". I do understand the sentiment that in general multiplication is not repeated addition. But multiplication with a scalar is. In the usual curriculum you do the unit analysis as a separate step, as it's usually in the physics lessons where units are used. In math it's unitless. Of course, we should start teaching immediately that you can't add apples and pears. M…

> I do understand the sentiment that in general multiplication is not repeated addition. But multiplication with a scalar is.

"Scalar" is too broad; irrational numbers are scalars, but multiplication by an irrational number is not repeated addition.

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