What’s Wrong with “Multiplication Is Repeated Addition”? (2008) • Hacker News Board

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

#171

Post by pdonis » Wed, Mar 03, 2021, 4:48 PM UTC

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…

> I'm at a loss how you would teach it without explaining that it is repeated addition.

How much will they later have to unlearn when they find out about other kinds of numbers for which multiplication is not repeated addition?

Quite possibly the same approach won't work for all children; some will be better served by telling them multiplication is repeated addition at first, and then later explaining that it isn't always, it only is for whole numbers. Others might be better served by telling them up front that multiplication is a separate operation, but can be "emulated" for whole numbers by repeated addition. (As a child I was in the latter group; I hated it when teachers told me one thing in one grade, and then said "well, what you were taught before isn't actually correct..." in a later grade. How many more times would they change the rules?)

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

#172

Post by [deleted] » Wed, Mar 03, 2021, 4:50 PM UTC

eigenket wrote: post #121

fifilura wrote: post #75

Earlier quoted context omitted.

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)

[deleted]

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

#173

Post by pdonis » Wed, Mar 03, 2021, 4:50 PM UTC

dragontamer wrote: post #166

jschwartzi wrote: post #152

Earlier quoted context omitted.

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…

> In all fields and rings, multiplication and addition are related by the distributive property.

That's a necessary condition for "repeated addition" to work for multiplication, but not a sufficient one. Try plugging in A = pi to the distributive formula and see how well "repeated addition" works.

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

#174

Post by whatgoodisaroad » Wed, Mar 03, 2021, 4:50 PM UTC

I like this article, but I feel like it misses the value of teaching analysis to students. It's true that it's not-quite-right to say multiplication is repeated addition, but it's also a really nice demonstration of the analytical/generalization approach anyone could use to "invent" multiplication on their own.

This same pattern can be seen driving discoveries in math. - Integrals are generalizations of Riemann sums. - Fractional exponents generalize taking square roots. - The gamma function sorta generalizes factorials. It's okay to start with something not-quite-right and explore how it generalizes.

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

#175

Post by dash2 » Wed, Mar 03, 2021, 4:52 PM UTC

pdonis wrote: post #164

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 pro…

Multiplication by rational fractions p/q is just multiplication by whole p, followed by multiplication by 1/q. 1/q is just the number that when multiplied by q gives 1. So you can cash that all out as repeated addition.

Multiplication by irrationals is just an infinite sum of multiplication by rationals, no? x times pi = x times 3, plus x times 1 / 10, plus....

I don't see there's any conceptual issue here. Mathematicians feel free to correct me.

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

#176

Post by dragontamer » Wed, Mar 03, 2021, 4:53 PM UTC

pdonis wrote: post #173

dragontamer wrote: post #166

Earlier quoted context omitted.

> 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…

> In all fields and rings, multiplication and addition are related by the distributive property. That's a necessary condition for "repeated addition" to work for multiplication, but not a sufficient one. Try plugging in A = pi to the distributive formula and see how well "repeated addition" works.

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 can even "break up" the two matricies into its component parts:

   Pi * ( 1 0  + 0 0  + 1 0  + 0 0  + 0 0 )
          0 0    0 1    0 0    2 0    0 1

And you'll still get the correct answer: multiplication is repeated addition. Even with matricies. Even with vectors. Even in Galois Fields. Even with rational numbers. Even with real numbers. Even with Complex numbers. This property holds through all forms of math that I'm aware of.

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

#177

Post by zamadatix » Wed, Mar 03, 2021, 4:54 PM UTC

pdonis wrote: post #161

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…

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.

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

#178

Post by DanielMcLaury » Wed, Mar 03, 2021, 4:54 PM UTC

andrewprock wrote: post #140

bidirectional wrote: post #114

Earlier quoted context omitted.

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.

This is an interesting statement. While it's true, it obscures the fact that pi is defined as a fraction: Circumference/Diameter. That this fraction cannot be represented as a numeric fraction is one of the great insights of early mathematics.

I wouldn't call the fact that pi is irrational an insight of "early mathematics." We knew that sqrt(2) was irrational around 500 BC if not earlier. We didn't know pi was irrational until around the time of the American revolution.

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

#179

Post by sandworm101 » Wed, Mar 03, 2021, 4:55 PM UTC

>> Multiplication requires different units. >> 2 baskets × 3 apples per basket = 6 apples

No it doesn't. In that example the "baskets" and "per baskets" (ie x/baskets) are cancelling each other. 2 baskets x 3 apples/baskets = 2 x 3 apples = 6 apples. Once you remove the unnecessary terms the actual multiplication is still basic repeated addition.

2 hours x 3 miles per hour = 2 x 3 miles = 3miles+3miles = 6 miles.

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

#180

Post by whatgoodisaroad » Wed, Mar 03, 2021, 4:56 PM UTC

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…

Exactly. Probably every discovery in math starts out with something that's only close to the truth (like multiplication-is-repeated-addition) but generalizing it to open up whole new areas.

This part of math is so exciting and compelling, I feel like it should be more of a focus in schools.