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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)

#121

Post by eigenket »

fifilura wrote: post #75
D13Fd wrote: post #42

I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is repeated addition. For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases a…

Lego is a great tool for this. I also taught prime numbers by handing out (round) 1x1 pieces and told him to figure out which ones could be made into squares. The teacher was surprised when - in the middle of a more convoluted explanation for primes - he just said "primes are numbers that can't be arranged as rectangles".

*rectangles (hopefully)

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

#122

Post by spacedcowboy »

bidirectional wrote: post #114
horsawlarway wrote: post #109

Earlier quoted context omitted.

pi + pi + pi + (.141592... of pi) ~= 9.8696 You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.

Pi is definitely not a fractional number... I don't think it's cheating at all, multiplication on the naturals is repeated addition, that's not the case for the reals.

Interesting how computers (which only understand ‘1’ and ‘0’ can do multiplication of reals, then. Unless you’re intel, of course... (no, I will never let it go :)

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

#123

Post by masswerk »

horsawlarway wrote: post #109
midjji wrote: post #101

Earlier quoted context omitted.

Not defending the article, but how would you compute: \pi*\pi using repeated addition?

pi + pi + pi + (.141592... of pi) ~= 9.8696 You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.

However, pi is already a ratio…

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

#124

Post by Cerium »

bidirectional wrote: post #114
horsawlarway wrote: post #109

Earlier quoted context omitted.

pi + pi + pi + (.141592... of pi) ~= 9.8696 You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.

Pi is definitely not a fractional number... I don't think it's cheating at all, multiplication on the naturals is repeated addition, that's not the case for the reals.

Can we not consider that the algorithm taught for multiplication of real numbers is repeated multiplication of natural numbers which can be seen as repeated addition of natural numbers so we could define an addition only algorithm for multiplication of real numbers.

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

#125

Post by munchbunny »

spacedcowboy wrote: post #104
Agingcoder wrote: post #9

I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…

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's not particularly useful to talk about those other ways to think about multiplication until you actually need to.

It's too easy once you've mastered the concepts to forget how beginners look at them and struggle to understand them. I think the author isn't remembering what it's like to try to understand multiplication as a new concept - I certainly can't remember.

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

#127

Post by spekcular »

Garlef wrote: post #18
Rerarom wrote: post #15

Earlier quoted context omitted.

This is just set theory, not category theory.

Neither is more foundational than the other. So I think the use of "just" is not justified here. https://mathoverflow.net/questions/360578/category-theory-an...

First, that's not what your link says. (See, e.g., the comments to the first answer.)

Second, I find that claim highly dubious. To do metamathematics and talk about the relative strength of various axiomatic systems, you need to talk about large cardinals. So that, at least to me, seems like a good argument that set theory is foundational to mathematics in a way that category theory is not, since the latter has no non-eliminable place in the (study of) contemporary foundations of mathematics.

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

#128

Post by horsawlarway »

bidirectional wrote: post #114
horsawlarway wrote: post #109

Earlier quoted context omitted.

pi + pi + pi + (.141592... of pi) ~= 9.8696 You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.

Pi is definitely not a fractional number... I don't think it's cheating at all, multiplication on the naturals is repeated addition, that's not the case for the reals.

Sure it is, as long as you're willing to repeat in increments of real numbers.

And that's my point, basically - By the time we're discussing real numbers, we need multiplication as an operator, because we're discussing ratios already.

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

#129

Post by gugagore »

yakubin wrote: post #108
gugagore wrote: post #94

Earlier quoted context omitted.

If you use a termination condition that has to do with convergence of iterates instead of a number of iterations (often the case), then you generally don't know beforehand the length of the finite sequences. Maybe you know a bound, but in general you might not even have that. In an important sense, it only becomes a finite algorithm. It isn't one. You cannot write the finite sequence of instructions down. It's got lo…

> To your point about approximations vs not, if you have an algorithm that, for any desired approximation accuracy can compute the square root to that accuracy in a finite number of steps, then that process is as much "the square root" as anything involving the real numbers. Not really, since approximations, no matter how accurate, don't preserve algebraic properties. You only get to know what it's bigger/smaller tha…

I think I understand what you mean, so let me dial back "anything involving the real numbers".

If you are representing or thinking of "sqrt(2)" as "the positive solution to x^2 = 2", then you preserve algebraic properties. But you generally (correct me if I'm wrong) don't get to know whether it's bigger or smaller than something else of the form "the _choose_uniquely_ solution to _some_equation_" unless you rely on an argument where you invoke approximations.

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

#130

Post by jeff-davis »

meepmorp wrote: post #117
eru wrote: post #92

Earlier quoted context omitted.

That's not the norm?

Oh god, no. Not at all, especially in the US.

Are you sure? My teacher did that.

It doesn't take very long, and I can definitely see someone forgetting all about it.

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