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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)

#51

Post by Garlef »

AzzieElbab wrote: post #31
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

That is exactly how I explained multiplication to my 5 years old, right after getting him into programming by playing with profunctor optics

I learned adding from counting disjoint unions of dots.

I learned multiplication from counting squares of dots.

But agreed: I learned about exponentiation first via repeated multiplication.

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

#52

Post by contravariant »

raverbashing wrote: post #32

I'll tell you what's wrong: the mentality of the author of this text Multiplication has started as repeated addition. That's where the idea came from. > It is as if there were two types of addition: regular, random, “wild” addition and the specially-bred variety of addition to which we give the name multiplication. Nobody. Literally nobody said that Of course, you'll need to forget a bit the idea of repeated multipli…

Now that's an interesting point, did it start as repeated addition?

It is just as conceivable that people were faced with a problem like 'each person needs 2 apples, we have 5 people, so we need 10 apples'?

In this case repeated addition is a perfectly fine algorithm to calculate the product but the product itself is not defined as repeated addition, it's the solution to a particular type of problem.

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

#53

Post by Garlef »

threatofrain wrote: post #43
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)?

No (At least not yet: open research question). But neither can "repeated multiplication". How would you multiply 2 with itself "a half" times?

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

#54

Post by yakubin »

prionassembly wrote: post #49
yakubin wrote: post #37

Earlier quoted context omitted.

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...

Number theorists work with integers, not naturals. And they too will agree with me, as they mostly work with polynomials, which show exactly that.

I also didn't say natural numbers are training wheels, just this metaphor, which to use HN lingo: doesn't scale.

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

#55

Post by eigenket »

prionassembly wrote: post #50
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.

It looks exactly like scalar multiplication when your matrices are diagonal.

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

#56

Post by gugagore »

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.…

Algorithms are not finite. E.g. Newton's method.

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

#57

Post by eigenket »

Garlef wrote: post #48
eigenket wrote: post #28

Earlier quoted context omitted.

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.

Those keywords all appeared in the context of set theory long before category theory was thought of.

Even ignoring that as a historical accident (we invented set theory before category theory), those ideas are all natural, meaningful and interesting to examine in the context of sets even if one has never heard of a category.

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

#58

Post by prionassembly »

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…

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.

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

#59

Post by whizzter »

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…

Having been teaching programming I can see an immediate problem with it, whilst it works great as a starter tying it too hard in the mind will require more un-doing when it comes to more advanced concepts where it isn't an appropriate level.

On the other hand knowing the relation is a good hint at how to explain exponentiation.

In the end teachers should be aware of what will be needed at higher grades to use tools such as it as a starting point but not ingrain it too much and try to push students onto thinking of multiplication as it's own functional primitive once they get the hang of basic tables and start moving on to long multiplication.

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

#60

Post by harperlee »

vishnugupta wrote: post #21
threatofrain wrote: post #3

The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?

> but is 2 cm × 3 cm the same thing as 2 × 3? This made me think too, in particular the teacher's example. So; here's what I think it should be calculated. 3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm). What you are doing above is adding two objects each of who are 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm" which then becomes "6cm". However 3 cm + 3 cm != 2 cm X 3 cm as the author stat…

Another way to say it is that 2cm * 3cm is shorthand for (2)(1cm)(i) * (3)(1cm)(j). We are complecting 3 semantics: measure and unit (which compose a module, in vector terms) and “meaning” (dimension, in vector terms).

This is why in high school physics you are taught to always write down the units of each number: typing them helps avoid a big amount of errors.

Note that “typing” and “vector terms” are not essential, just helps me express it with the HN crowd easily.

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