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

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

#231

Post by EvanAnderson » Wed, Mar 03, 2021, 6:10 PM UTC

matheist wrote: post #218

EvanAnderson wrote: post #210

The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may. I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach mu…

Math PhD here. You're a fantastic dedicated parent and you did no harm. First, as to the merits of your approach --- I think of multiplication as scaling myself. That's totally valid. Second, whether it's even possible to do harm here --- worst case scenario, the metaphor doesn't make sense to your kid and she doesn't use it in her own thinking about mathematics. Everyone has to develop their own intuitions --- like…

I appreciate the reply. I didn't feel too terribly off-base, but seeing the discourse here gave me cause to question.

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

#232

Post by aidenn0 » Wed, Mar 03, 2021, 6:12 PM UTC

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.

If you don't lie to children you confuse and discourage them. I know because my own internal desire for precision often is to the detriment of my pedagogy.

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

#233

Post by hrktb » Wed, Mar 03, 2021, 6:13 PM UTC

SilasX wrote: post #201

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.

Well, that would be true if all children were as smart and analytically adept as you.

It looks like splitting hairs, but I also think it makes a difference to state upfront that it’s an oversimplification.

Kids actually understand the point, and they can decide if they care enough to ask more questions or if it’s good enough for them.

We faced that when teaching divisions. Saying upfront we’d explain falsehoods for the sake of simplicity helped set aside the more difficult questions (infinity, etc.) that came right after. We just say it’s the complicated parts and move on.

The best aspect is they are more receptive to have their mental model broken afterwards, instead of clinging to what you explained as fully true.

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

#234

Post by nathancahill » Wed, Mar 03, 2021, 6:13 PM UTC

loonster wrote: post #182

shadowofneptune wrote: post #135

Earlier quoted context omitted.

The way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.

The best math class I ever had was a drafting class called "Descriptive Geometry". In that college class we used a drafting table to solve math problems. An easy example would be the length of line of the corner seam of a hip roof. Sadly I have forgotten what the more complex problems were. Fantastic class that isn't offered anymore.

Btw, the game Euclidea on iOS is fun for that. You "discover" and prove concepts and progress through higher-orders of geometry proofs.

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

#235

Post by ubasu » Wed, Mar 03, 2021, 6:23 PM UTC

To put it in terms of what the audience here can relate to:

multiplication is an overloaded operation, or in more modern terms, it is doing multiple dispatch, depending on whether the input is a whole number or integer or rational or real or complex

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

#236

Post by dash2 » Wed, Mar 03, 2021, 6:24 PM UTC

pdonis wrote: post #197

dash2 wrote: post #175

Earlier quoted context omitted.

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. Mathemati…

> multiplication by 1/q Is not repeated addition. You can't add a number to itself 1/q times. At least, not unless you're willing to adopt increasingly perverse interpretations of "repeated addition" as you try to cover more and more numbers. See my response to wruza upthread.

Hmm, so just go about it differently: x * p/q with p,q integers is (x*p) / q. The first bit is repeated addition. The second bit means, find the number r such that r * q = x * p. Even if r is not integer, q is integer, so we can try different numbers, add them to themselves q times, and close in on the answer.

So I still think conceptually it's fine to think of it as repeated addition? It might be algorithmically a bad way to do it ("increasingly perverse", taking limits when the number is irrational etc.). I don't know how computers actually implement multiplication - though wikipedia (https://en.wikipedia.org/wiki/Binary_multiplier) says it works via shifts and adds.

Or maybe, it's bad to teach it to kids this way? But then I think we need evidence from educationalists.

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

#237

Post by cobaltoxide » Wed, Mar 03, 2021, 6:31 PM UTC

sidlls wrote: post #136

spacedcowboy wrote: post #104

Earlier quoted context omitted.

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 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.

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

#238

Post by mxfh » Wed, Mar 03, 2021, 6:33 PM UTC

This is more confusing then it helps.

Why mix concepts of Linear Algebra/Physics with basic Arithmetics for no good reason?

The multiplication, that is just repeated addition, is very possible in dimensional analysis and is called scalar multiplication:

The outright worst part is this weird complication:

2 baskets × 3 apples per basket = 6 apples

of 2 x 3 apples = 6 apples.

Just why? This is just a scalar operation.

Staying in strictly mathematical concepts, the operation described in OP are closer to the cross product of orthogonal vectors, then multiplication in my opinion.

[4 0 0] × [0 3 0] = [0 0 12]

https://en.wikipedia.org/wiki/Scalar_multiplication

https://en.wikipedia.org/wiki/Cross_product#Geometric_meanin...

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

#239

Post by mnl » Wed, Mar 03, 2021, 6:35 PM UTC

dragontamer wrote: post #229

mnl wrote: post #204

Earlier quoted context omitted.

Repeated addition that ain't... It boggles my mind that you've been studying Galois theory yet somehow try to reduce the algebraic structures associated to two binary operations to playing with one of them.

The algebraic structures multiplication brings is different from the algebraic structures from addition. That's the point of rings (as opposed to groups). Nonetheless, in a ring (and all fields are rings), multiplication must and always is related to addition, through the distributed property (which I argue, the distributed property IS the mathematical term for "repeated addition"). Without the distributed property,…

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, for the latter seeing it as "repeated addition" makes no sense, for the former, well you have a ring, you have two binary operations, not one, and of course you have some form of distributive property or else you'd be studying this set with just one binary operation at a time.

I'd like to see how "repeated addition" works in polynomial rings.

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

#240

Post by 613style » Wed, Mar 03, 2021, 6:37 PM UTC

jschwartzi wrote: post #152

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…

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…

We can't really talk about what multiplication "is" or "isn't" independently of context. It's an operation on two objects and the context its used in is necessary for defining the operation.

Though it can be helpful to think of multiplication as scaling or rotation in certain contexts, or as repeated addition in others, none of those are a universal truth.

To say it's not related to addition at all is also too broad to be true. They're related in many useful ways that other commenters have pointed out, in addition to the obvious way that in some situations you can define one in terms of the other. Even at the cutting edge, our inability to prove the Goldbach Conjecture might have something to do with not fully understanding the deepest relationships between the operations.