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

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

#251

Post by SAI_Peregrinus » Wed, Mar 03, 2021, 7:18 PM UTC

qsort wrote: post #88

SAI_Peregrinus wrote: post #76

Earlier quoted context omitted.

Multiplication over the reals is commutative. Matrix multiplication of non-square matrices isn't. Multiplication in a Ring isn't necessarily commutative. Other algebraic structures also have non-commutative multiplication. One could argue that these things aren't "multiplication" even if they are "products" since they don't satisfy all the properties of multiplication over the reals. But it is common to call the use…

I really don't get the point you're making. If we're going to pull out random examples, monoids aren't guaranteed to be abelian; strings and concatenation form a monoid that's not abelian. If you have enough mathematical sophistication to conceptualize a non-commutative ring, you're well past the point where naming conventions are even remotely an issue. The original article was contrasting addition and multiplicatio…

My point is only that the commutative property is not inherent to all multiplication operations, so there can be a distinction between the operands. It's not necessarily a useful distinction, and in the usual use of multiplication it's utterly useless and only adds confusion.

But matrix multiplication is taught in high school (and usually promptly forgotten), it's not particularly advanced math.

Personally I'm of the opinion that the terminology is muddled. There's no need to distinguish the operands of a multiplication over any of the usual domains (reals, rationals, integers, etc). And when you reach the point where it does become important there's generally more than one product operation and we should stop calling it multiplication. "Matrix multiplication" is a bad term. You also typically wouldn't name the operands, since as you note there can be more than two!

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

#252

Post by Sharlin » Wed, Mar 03, 2021, 7:21 PM UTC

JosephRedfern wrote: post #186

pdonis wrote: post #161

Earlier quoted context omitted.

> 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…

I never got over being lied to about electron shells and the 2-8-8 "rule".

To be fair, it once was the best model we had, before we figured out quantum mechanics. That brings up an interesting point, though: the way many of these things are taught closely mirrors the history of how we discovered them in the first place. This is definitely not a coincidence, but I'm also not sure whether it is the most effective way to teach.

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

#253

Post by kmill » Wed, Mar 03, 2021, 7:33 PM UTC

chrisandchips wrote: post #150

I strongly believe that helping younger students gain strong intuition for these operators pays dividends towards their later success in maths. I've always run into the following problem: I try to motivate multiplication as repeated addition, which does help with intuition, but then things totally fall apart when we move on from integers into fractional values. 1/2 * 1/2 -> 1/4. Sure you can teach someone to simply m…

I think it's sort of an illusion (though certainly a useful one) that numbers are all part of the same system. Multiplication of natural numbers is defined to be repeated addition, and I'm not really sure how you could define natural number multiplication in any other way. From the natural numbers you can go on to define the integers, rationals, then reals, and each has its own definition of multiplication, though they each depend on the definition of multiplication from the previous system.

There's a standard way of lifting each type of number to the next type, and this lift is compatible with all the basic operations (the lift is a "homomorphism"), so it's easy to pretend that the real numbers (or complex numbers if you want) are the universal system.

So with your example of going to fractional values, you're right, repeated addition falls apart -- but I'd say that's because it's not the definition for multiplication of rational numbers! Multiplying numerators and denominators is the usual definition, but that gives about as much intuition as does the definition for multiplying natural numbers. Sort of "the point" of multiplication of naturals, I think, is that it represents how many things you have if you arrange them in an n by m grid. Rational numbers show up in geometry with similar shapes (scaling), and for a few reasons you'd want multiplication to represent by how much something scales after a composition of scalings; maybe "the point" of rational number multiplication (at least algebraically) is that you can defer dividing until later, i.e. (a/b) * (c/d) is (a*c)/b / d.

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

#254

Post by WalterBright » Wed, Mar 03, 2021, 7:42 PM UTC

I distinctly remember when I realized that the times in "4 times 3" is 3 added 4 times. That was the end of trying to memorize the times tables. If I forgot what "6 times 9" was I just thought "6 times 10 minus 6" and there it was.

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

#255

Post by [deleted] » Wed, Mar 03, 2021, 7:43 PM UTC

CodeMage wrote: post #248

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.

Let's not take the word "lie" out of its context and then get hung up on it. We're not actually lying to children by simplifying the explanation down to what they can comprehend with the tools they have at that stage.

[deleted]

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

#256

Post by WalterBright » Wed, Mar 03, 2021, 7:45 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…

Teaching Newtonian Mechanics to students is also a lie. But I don't believe starting with General Relativity is going to do anything but confuse students.

I was never confused by "3 baskets of 6 apples, how many apples" questions or thought I'd been lied to about addition.

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

#257

Post by kmill » Wed, Mar 03, 2021, 7:48 PM UTC

pdonis wrote: post #171

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 a…

Maybe I've been doing too much computer-assisted proofs recently, but I'm not understanding what all the fuss is. At some point you need a `Nat.mul` that's concretely defined, and probably the only reasonable definition is recursively using repeated addition (and if your natural numbers are in unary, which is theoretically the simplest, then even `Nat.add` is going to be defined recursively by repeated incrementing! like taking a bead from a pile one at a time and putting it into another).

Then you can define an interface with abstract operations like addition and subtraction, then any given number system can try to implement this interface. Naturals, integers, rationals, reals, polynomials, rational functions, complex numbers, and many others can implement at least the semiring interface. But what's important, I think, is that there's no universal multiplication operation -- it's just a word that fills in for whatever is the right multiplication operation at the moment. It's some ad-hoc polymorphism.

Unless you're accepting the real line as being axiomatic (maybe from geometry, where you can define all the basic operations using ruler-and-compass constructions), all the definitions of multiplication for all the above number systems are going to be, eventually, based on the one for the naturals.

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

#258

Post by qsort » Wed, Mar 03, 2021, 7:49 PM UTC

howling wrote: post #227

qsort wrote: post #100

Earlier quoted context omitted.

Well, no, not really. The standard definition of the reals is as the unique nontrivial totally-ordered, Dedekind-complete, Archimedean field up to isomorphism. So what you would really need is a uniqueness proof, with addition and multiplications "provided" by the hypothesis.

And how do you prove that a totally-ordered, Dedekind-complete, Archimedean field does not lead to contradiction besides constructing it explicitly by bootstrapping from natural numbers?

Nothing in the construction requires you to show an algorithm that given x, y \in R allows you to compute x+y and xy. You would make the usual Dedekind construction and show it satisfies the axioms of such a field. (as a matter of fact, no such algorithm exists in full generality!)

It's probably a tomato/tomato kind of thing, but I'm only objecting to the 'algorithm' part of parent's comment.

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

#259

Post by agumonkey » Wed, Mar 03, 2021, 7:52 PM UTC

munchbunny wrote: post #125

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 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'…

so the iterative pattern doesn't really hold when you go into abstract algebra ?

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

#260

Post by rovolo » Wed, Mar 03, 2021, 7:53 PM UTC

amelius wrote: post #244

By the way, why doesn't exponentiation have units in physics? E.g. why don't we ever have something like kg^s?

This is purely a guess, but I think it's because all the exponents are scalar. You get exponents when the derivative of a function is proportional to itself:

    d/dx (f) = k*f

So exponentials will be scalars because the exponential part is scaling the original value.

Example: population growth is exponential.

    d/dt(p) = (ln(2)/r)*p
    p(t) = p0 * 2^(t/r)
    p : population
    p0: population
    t : days
    r : days (time it takes for the population to double)
    t/r : scalar