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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)

#271

Post by smolder »

pdonis wrote: post #199
zamadatix wrote: post #177

Earlier quoted context omitted.

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.

They don't need this information when learning that 3 groups of 5 fruit are 15 fruits. It's not in the natural order of understanding principles.

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

#273

Post by jacobolus »

lhorie wrote: post #211
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…

> teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..." My kids are learning multiplication. I honestly don't feel like they're being lied to, or confused about what they are being taught. Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). It's certainly not how I learned (I had…

> Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`).

This is not a "crutch", beyond the extent to which a decimal place-value system is a crutch. I would instead call it a broader and more fluent view of the number system.

Describing e.g. 18 = 2·10 – 2·1 instead of 1·10 + 8·1 is a perfectly valid alternative representation which happens to often be more convenient when multiplying.

Either way when we multiply we break each multiplicand into a sum, multiply the components from each combinatorially, and then add the results together.

18·6 = (1·10 + 8)6 = 10(1·6) + 8·6 = 60 + 48 = 108

vs.

18·6 = (2·10 – 2)6 = 10(2·6) – 2·6 = 120 – 12 = 108

Regularly discussing the alternative ways to represent a number and choosing the most convenient for the current goal builds what is called "number sense": fluency with the place-value system, basic properties of integers, relationships between numbers, base ten, and in a broader way facility with manipulating data structures.

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

#274

Post by jimhefferon »

iio8999 wrote: post #184
dragontamer wrote: post #176

Earlier quoted context omitted.

pi(3 + 5) == 3pi + 5pi == (Pi + Pi + Pi) + (Pi + Pi + Pi + Pi + Pi) == 8-pi. Am I missing something here? -------- This "multiplication is repeated addition through the distributed property" thing works on freaking __matricies__. They don't even have to be numbers or even related. Pi * ([ 1 0 ; + [ 1 0 ; 0 1 ] 2 1 ]) Pi * [ 1 0 ; + Pi * [ 1 0 ; 0 1 ] 2 1 ] [ Pi 0 ; + [ Pi 0 ; 0 Pi ] 2Pi Pi ] [ 2Pi 0 ; 2Pi 2Pi ] You c…

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?

You extend to the rationals in the natural way, and then use continuity to define what happens for irrationals. (Of course, in a discrete context that doesn't apply. But the article's author is a grade school teacher.)

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

#275

Post by wcarey »

pdonis wrote: post #199
zamadatix wrote: post #177

Earlier quoted context omitted.

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.

How would you teach physics without lying to children?

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

#276

Post by smolder »

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…

Refusing to answer curious students questions isn't helpful. Giving them a hint and telling them you'll get back to it works fine. They're smart enough to understand they aren't having to wait because they're incapable of understanding, but because other things need teaching first. I know I did. The good teachers were encouraging of the curiosity while back-burnering something, others responded less patiently with things like "we're not there yet!", which made them seem like bad teachers to me, even back to elementary.

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

#277

Post by mnl »

dragontamer wrote: post #268
mnl wrote: post #239

Earlier quoted context omitted.

Without the distributive property you have two magmas in general for the same set that don't see each other. I'm not sure what your point is. When you have two binary operations you need some sort of distributive property to build a structure. All this is trivial, if you consider a ring, you get a·0=0 as a property, if your starting point is the Peano axioms for the arithmetic of natural numbers that's one of them, f…

> I'd like to see how "repeated addition" works in polynomial rings. Consider the following polynomial: x0 * b^0 + x1 * b^1 + x2 * b^2 ... xn * b^n, where "n" goes to both positive infinity and negative infinity. When "b = 10" and when "x" can be numbers from [0-9], we have the so called base-10 set of real numbers, do we not? IIRC, if b = sqrt(-1) * 10, we then have the set of complex numbers (a non-intuitive result…

A polynomial is a polynomial, a decimal representation of a real number is a decimal representation of a real number, and your representation of complex numbers has funny properties once you begin exponentiating that.

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].

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

#278

Post by bitwize »

In mathematics, sometimes a wrong approach can be used as a teaching tool until you get the concept, at which the correct definition may be disclosed. Example: derivative calculus is easier to teach with infinitesimals, even though the derivative is correctly defined in terms of limits.

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

#279

Post by bawolff »

> Is that really how we want our students to think? Multiplication is not a mere sub-species of addition. Multiplication is its own animal, an independent operation.

Wtf, why not? Why wouldn't you want people to draw connections between different parts of math. The whole point of math is to find the patterns and interconnections.

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

#280

Post by dragontamer »

mnl wrote: post #277
dragontamer wrote: post #268

Earlier quoted context omitted.

> I'd like to see how "repeated addition" works in polynomial rings. Consider the following polynomial: x0 * b^0 + x1 * b^1 + x2 * b^2 ... xn * b^n, where "n" goes to both positive infinity and negative infinity. When "b = 10" and when "x" can be numbers from [0-9], we have the so called base-10 set of real numbers, do we not? IIRC, if b = sqrt(-1) * 10, we then have the set of complex numbers (a non-intuitive result…

A polynomial is a polynomial, a decimal representation of a real number is a decimal representation of a real number, and your representation of complex numbers has funny properties once you begin exponentiating that. 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].

> 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, ALL REAL NUMBERS ARE POLYNOMIALS with a base of 10.

Or to put it another way: when x == 10, your polynomial of (1 + x + x^2) * (1-x^3) == 111 * (-999). That is to say: real numbers are simply polynomials where "x" has been defined to be a particular number, instead of an abstract entity. We call that number the radix-base.

If you instead defined the base to be x = 16 (hexadecimal numbers), you'd get 111 * (-FFF), which you'll find will satisfy similar properties. Now leave x-undefined (since it could be 10 or 16), and what do you get?

Polynomial math. Or so called "Carry-less multiplication" (https://en.wikipedia.org/wiki/Carry-less_product). We don't have a ring yet though: we still need to perform a modulus on all those polynomials to return to a proper ring (and if the modulus is irreducable, we have a Galois field). But we can already see how polynomials and the Reals are so closely related.

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