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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)

#41

Post by parenthesis »

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…

> 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.

That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square.

In fact, I have a high school textbook that makes that derivation.

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

#42

Post by D13Fd »

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 and all of the more advanced concepts the author (and the related articles) discuss can be taught later.

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

#43

Post by threatofrain »

Garlef wrote: post #13

My favourite way of thinking about these operations comes from category theory: * addition: the cardinality of the disjoint union of two sets * multiplication: the cardinality of the direct product of two sets * exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a

Can this perspective accommodate fractional exponents, like 2^(0.5)?

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

#44

Post by rafaelturk »

As software developers we can defenetly point the many flaws in this article:

1. IS A CLICKBAIT

2. Course you can decopose multiplications in a series of repeated additions.

3. Extra points for the flawed argument of adding different, well you can't because they are different.

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

#45

Post by Garlef »

nicky0 wrote: post #25
Garlef wrote: post #18

Earlier quoted context omitted.

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...

But talking about category theory makes you sound smarter.

I'm personally more scared of set theory.

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

#46

Post by EGreg »

A few years ago, I was teaching a class and decided to record a short series on the fundamentals of mathematics, for complete "beginners" - whether they be kids or adults.

This is the first video in the series Thinking Mathematically (after the introductory video, "Why think Mathematically?") which I put on YouTube under a channel of the same name. It proceeds through the sets of numbers, N -> Z -> Q -> R -> C and yes it's for beginners. Would love some feedback:

https://www.youtube.com/watch?v=Wd1lzqR3xW0

The other videos you'll find on that channel are:

  1. Why think mathematically?
  2. Numbers and Algebra
  3. Sets and Infinity 
  4. Logic and Probability
Here is the channel with all the videos. They might be useful to share with people who you want to understand these concepts from the ground up:

https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA

(PS: I am generalizing this approach in our upcoming app, https://teaching.app)

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

#48

Post by Garlef »

eigenket wrote: post #28
Garlef wrote: post #18

Earlier quoted context omitted.

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...

Yeah but your comment didn't use the concept of a "category" at all, whereas you used "set" three times. Its definitely more natural to say these definitions come from set theory.

To me, "disjoint union", "direct product" and "set of maps" (as a special case of "internal hom") are the central keywords in what I wrote: Concepts from category theory.

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

#49

Post by prionassembly »

yakubin wrote: post #37
WallWextra wrote: post #33

Earlier quoted context omitted.

If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.

Not algorithms. There will be infinite addition involved, and algorithms are finite. Thinking of multiplication as repeated addition also won't explain anything about it. It's a separate operation. Deal with it. For similar reasons, you can't calculate x-th power of a number, when x is irrational, by decomposing it into exponentiation and roots. This metaphor is just training wheels. At some point you should lose it.…

Multiplication on the naturals is repeated addition.

You might see the natural numbers as training wheels for higher mathematics, but number theorists might disagree...

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

#50

Post by prionassembly »

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…

The better line of thought here might be "A*B is not multiplication, it's function composition".

It literally looks nothing like multiplication when carried out manually.

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