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

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

#151

Post by prionassembly »

magicalhippo wrote: post #132
prionassembly wrote: post #58

Earlier quoted context omitted.

You can teach (some intuition for) multiplication without numbers by multiplying lengths to get areas. Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows. Let the school teacher press the algorithms on him, teach him your hacker's sense of wonder.

> Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows. How do you do that without adding up the number of seats in the rows or columns (ie repeated addition)?

Using a calculator.

Not a troll: the ability to compute products and the idea that (horizontal length) times (vertical length) = (area) are different things.

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

#152

Post by jschwartzi »

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 kind of addition then it all breaks down and you can't reason about much at all with those tools. In fact you can't even get over the hurdle of doing dimensional analysis in chemistry or physics in the first place. It doesn't make any sense. And you'll have been set up to fail by your teachers.

They're not saying "we should be teaching kindergarteners using Euclid" they're suggesting that focusing on the process of doing multiplication hides a lot of very useful insights about why someone would do multiplication in the first place.

The biggest problem with math as it was taught to me in grade school is that it was focused on rote memorization of a procedure. Which is fine if you need to calculate how long to cut 2x4s or stair stringers but it doesn't help you at all if you need to understand why that works the way it does. And god help you if nobody ever shows you the exact method of calculation for those two things in the first place. You're set up to fail if you only focus on process.

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

#154

Post by magicalhippo »

prionassembly wrote: post #151
magicalhippo wrote: post #132

Earlier quoted context omitted.

> Then show e.g. how the number of seats in a theater is the number of seats in a row * the number of rows. How do you do that without adding up the number of seats in the rows or columns (ie repeated addition)?

Using a calculator. Not a troll: the ability to compute products and the idea that (horizontal length) times (vertical length) = (area) are different things.

So teaching multiplication ends up as "whatever the calculator does when I press this button"?

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

#155

Post by johndoe42377 »

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

[dead]

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

#156

Post by IdiocyInAction »

sudosysgen wrote: post #73
parenthesis wrote: post #41

Earlier quoted context omitted.

> 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.

We did that derivation in class, actually! I was so lucky to have a great math teacher in high school.

Yeah, we got it as bonus homework, to derive it ourselves.

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

#157

Post by eternalban »

spacedcowboy wrote: post #134
munchbunny wrote: post #125

Earlier quoted context omitted.

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'…

Your last two paragraphs are, I believe, the crux of my own argument. Sure, matrices aren’t even commutative, and complex numbers have their own quirks because of i ^2 == -1, but these concepts build on the earlier axioms the kid has learnt. Our entire education system is built on “lies-to-children”, and as you progress they point out that what you comfortably believed was a gross simplification. This is no different…

Speaking of “lies to children”, and since you are a physicist, it was only in my last year studying EE that Feynman’s QED came my way and just imagine my surprise on finding that photons do not travel in a straight line.

(I do agree with your pedagogical position.)

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

#158

Post by eigenket »

prionassembly wrote: post #143
eigenket wrote: post #67

Earlier quoted context omitted.

We're talking about matrix/matrix multiplication here, not matrix/vector, right?

I was talking about matrix/matrix composition (rather than multiplication) at first. Then I talked about matrix application . A matrix is a function. Not all functions can be represented by matrices (although all smooth functions can be represented by a Taylor series that sums over matrices (and "the sum C=A+B" really means "the function C such that Cx = Ax+Bx for all x in the range of both A and B"))

I rather dislike this way of saying things. A matrix is a matrix, its a table of numbers, and one can usefully define operations like addition and multiplication on them.

A linear map is a linear map, it maps between vector spaces and it obeys some nice axioms. You can define addition and composition as operations on them.

It is a quite interesting and non-trivial theorem that if you fix a particular choice of basis then you get "for free" a bijection between linear maps and matrices. The bijection between matrices and linear maps is completely dependent on the basis you choose, however, and there certainly isn't a canonical way to choose the basis.

Often it is natural to change basis to make it easier solve some particular problem, and then the matrix that represents a particular linear map will change, but the properties of the linear map won't change (for example its rank, kernel, eigenvectors/values etc).

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

#159

Post by scatters »

shawnz wrote: post #112
scatters wrote: post #99

Earlier quoted context omitted.

> how else would you define multiplication? In Church arithmetic, multiplication is defined as function composition; that is, the dot operator `.` in Haskell-like languages. Addition is considerably more complex; it is defined as `lift (.)`. Exponentiation is even simpler; it is function application, so the `id` function - except that we write exponentiation backwards, so actually `flip id`.

Interesting, what about tetration then?

Good question: you can define m↑↑n as:

    \m n -> n (exp m) 1'
where `exp` is Church exponentiation, `flip id`, and `1'` is the Church numeral for 1, that is, `id`. According to pointfree.io this expands to:

    flip flip id . flip id . flip id
Which isn't particularly illuminating. Pentation isn't much worse, though:

    flip flip id . flip id . flip flip id . flip id . flip id

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

#160

Post by jmull »

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…

It's just that matrix multiplication is not the same thing as scaler multiplication. I think you need to understand that no matter how you define or think of scaler multiplication.
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