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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)

#371

Post by bigger_cheese »

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…

I remember my High School chemistry lesson from the 90's clearly explained we were working with historical models, we talked about nature of the models developments and the experiments that led to them.

We started with the JJ Thomson "plumb pudding model"

Then we learnt about Rutherford and the gold foil experiment which invalidated Thomson's model.

Then we learnt about emission spectrum and the Bohr model which invalidated Rutherford's model.

The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.

Same thing in physics. We started with debate about nature of light and examined things like lumniferous aether and Michaelson Morley experiment, double slit experiment etc. Then Maxwell and electromagnetism leading to Heinrich Hertz and the photoelectric effect, which led to Planck and Einstein.

I certainly never thought I was being lied to. We were treading along the same path as those who had came before.

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

#372

Post by mnl »

dragontamer wrote: post #336
mnl wrote: post #315

Earlier quoted context omitted.

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. ------- My argument is otherwise. "Multiplication is Repeated Addition" is clea…

I think there might be a bit of a problem with quaternions.

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

#373

Post by smolder »

pdonis wrote: post #298
smolder wrote: post #271

Earlier quoted context omitted.

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.

> They don't need this information when learning that 3 groups of 5 fruit are 15 fruits. They also don't need the "information" that multiplication is repeated addition, period, full stop. But that's what others in this discussion appear to be trying to argue for.

I'm really not sure how you could arrive at an understanding of multiplication without first going through the exercise of adding up X groups of Y to yield X * Y. We don't need to declare to children that the precise definition of multiplication is repeated addition. If that's what you're arguing, we agree. If it's that 3*5 shouldn't be explained as 5+5+5, I'm not sure how kids would be expected to learn the subject. We can't transactionally insert comprehensive math knowledge, so it's got to be incomplete along the way.

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

#374

Post by thaumasiotes »

mnl wrote: post #363
thaumasiotes wrote: post #333

Earlier quoted context omitted.

> 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 repea…

Could you sketch a proof starting from some definition of the operation product of integers as "repeated addition" without using the distributive property, which would imply that we already have another binary operation besides the sum? I was thinking about the Peano axioms when I wrote that (hence the working backwards thing). Obviously if you start with a ring, you don't have to impose it, you get that as a propert…

If you want to work up to it conceptually, then I'd say consider the meaning of something like

  5·3 + 2·7
We add 3 five times, and then we add 7 twice. It is then easy to extend this to

  5·3 + 2·7 + 0·4
and say, OK, add 3 five times, and then add 7 twice, and then add 4 zero times. And you get 29.

At that point you notice that when you say 5·3 means "add 3 five times", you forgot to say what you were adding it to. You're adding it to the identity, 0.

And finally we say that starting from 0 and adding something zero times leaves you where you started, at 0, so we can observe that 0·n = 0.

If you formalize that, you'll end up either deriving or postulating the distributive property, depending on what you start with. But it isn't arbitrary; it's not a coincidence that (as I mentioned above) exponentiation by 0 gives the multiplicative identity, and (as I haven't mentioned yet) exponentiation distributes over multiplication the same way multiplication distributes over addition.

(Asymmetry does start creeping in; aggregate exponentiation isn't as nice since exponentiation doesn't have the nice properties that addition and multiplication do.)

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

#375

Post by ac42 »

Many commenters have already said it: the article is probably not helpful when teaching kids. Or is it? I recall having asked myself these questions at school: if lines are infinitely thin, how is it repeated addition that defines surface areas? What is "inside" the variables we use as units?

Measure theory (for your volume) and the structure of dimensional quantities [1] are non-trivial topics, and while probably not giving good answers, I think the article is certainly asking good questions.

[1] https://terrytao.wordpress.com/2012/12/29/a-mathematical-for...

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

#376

Post by tomxor »

Double_Cast wrote: post #292
simias wrote: post #82

Earlier quoted context omitted.

I agree that it's a generally pointless distinctions, although it might be useful in some cases such as number systems where the multiplication isn't commutative, or for a particular implementation where the distinction matters. After all if you were to code a multiplication that was implemented naively as a series of additions it'd be generally much faster todo 2x1000 than 1000x2. To return to TFA I think the author…

Suppose your friend Alice arranges your wedding. You ask her to arrange the lawn chairs in an arrangement of 2 rows and 3 columns. But she misinterprets your request as 3 rows and 2 columns. Oops. Now a particular family can't all sit in a single row without rearranging the chairs. If all you care about is the total number of chairs, the order of operands is irrelevant. but if you care about the structure, "2 x 3" ma…

Except we are talking about scalar multiplication, that was the premise, not matrices.

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

#377

Post by yarg »

yarg wrote: post #307

It's not circular reasoning - it's bootstrapping the rational implementation atop the integer implementation.

The euclidean algorithm can compute the GCD of the factors of the naive rational form, repeated subtraction (down to zero) of the GCD from the denominator and numerator can be used to find them in their reduced forms.

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

#378

Post by selimthegrim »

JosephRedfern wrote: post #297
pdonis wrote: post #286

Earlier quoted context omitted.

How were you lied to about that?

In GCSE Chemistry (taught to 15-16ish year olds), we were taught that the first shell contained two electrons, and subsequent shells contained 8 (up to the final, which may contain fewer). Then, after GCSEs, during A-levels, we were told (by the same teachers, in the same classrooms) to forget this model, and that the situation was actually more complicated, with s, p, d, f orbitals etc. I realise that this is was a…

Wait until you hear about Px, py orbitals and their actual quantum numbers

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

#379

Post by selimthegrim »

cobaltoxide wrote: post #237
sidlls wrote: post #136

Earlier quoted context omitted.

I have got almost exactly the same situation: PhD, physics, forgotten more math than most folks ever learn, etc., and with an 8 year old learning the same level of mathematics as you describe. I think the only difference might be I used "iterated" rather than "repeated" when helping him. Anyone who is just learning multiplication likely lacks the depth of experience necessary to make use of the "correct" jargon and a…

Another physics PhD chiming in here. I have never before noted the difference between "multiplier" and "multiplicand". The whole article has me rolling my eyes. In fact, I would argue that multiplication being associative shows that this distinction is meaningless.

Surely you knew subtrahend and minuend from vector subtraction? Or divisor:dividend?

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

#380

Post by agumonkey »

spacedcowboy wrote: post #325
agumonkey wrote: post #262

Earlier quoted context omitted.

Since you're a physician, do you think that helps for multiplicative relationships in real world laws ? It took me decades .. sadly, to be comfy with handling U = RI formulas. On the algebraic level it's stupid simple, but for real world physics the meaning is more bidirectional coupling of ratios and amplitudes and taps into a different part of my brain.

Physicist, not physician, but really - I haven't used my physics knowledge directly in a few decades now... I've been a software engineer for most of my life :) As for V=IR (I had to google U=RI, maybe U is the more modern version, but it was always V=IR when I were a lad), I don't really have a problem with ratios. When I was learning equations, the simple rule is "do unto one side whatever you do to the other", so…

hum sorry about the physician.. freudian slip

what I meant about the ratio part is that physicists don't care much about the computation they care about whenever you know one piece, you know the other will variate through a third factor in a multiplicative manner.. maybe it makes sense, but it's not at all a structural notion like (x) === iterate(+,n)

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