What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
291–300 of 425 posts
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#292Earlier quoted context omitted.
> to distinguish the element being operated on (the LHS) I'm not sure what you mean, they are both operated on, it is a binary operator and commutative, there is literally no difference.
I agree that it's a generally pointless distinctions, although it might be useful in some cases such as number systems where the multiplication isn't commutative, or for a particular implementation where the distinction matters. After all if you were to code a multiplication that was implemented naively as a series of additions it'd be generally much faster todo 2x1000 than 1000x2. To return to TFA I think the author…
Suppose your friend Alice arranges your wedding. You ask her to arrange the lawn chairs in an arrangement of 2 rows and 3 columns. But she misinterprets your request as 3 rows and 2 columns. Oops. Now a particular family can't all sit in a single row without rearranging the chairs.
If all you care about is the total number of chairs, the order of operands is irrelevant. but if you care about the structure, "2 x 3" may encode information that "6" does not.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#293Earlier quoted context omitted.
> 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!…
> At some point you need a `Nat.mul` that's concretely defined
And since your computer can only perform finite operations on numbers with a finite bit size, which means operations on a set of numbers isomorphic to the integers, then yes, repeated addition works fine for your concrete implementation.
But that's not at all the same as saying that multiplication is repeated addition, without any qualification whatsoever. The latter claim is the claim I'm arguing against.
> Unless you're accepting the real line as being axiomatic
Your computer can't manipulate real numbers. It can only implement finite operations on numbers with a finite bit size, which, as above, is a set of numbers isomorphic to the integers. So if you want to work with the real numbers themselves, abstractly, then you can't expect what the computer does to carry over.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#294Earlier quoted context omitted.
How would you teach physics without lying to children?
Um, by not lying to them? Do you have a specific example of where you think you need to lie to teach physics?
Explaining the structure of an atom without having to explain atomic orbitals and standing waves. Explaining classical mechanics without including a bunch of caveats about relativistic speeds.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#295Earlier quoted context omitted.
> By taking 3 times as usual and then another .14 approximately First, "taking 3 .14 times" doesn't make sense, at least not in the elementary school student's understanding of "repeated addition". Second, pi is irrational, as you evidently realize since you say "approximately". There is no way to add 3 pi times. > If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’…
But what is that number? You only described an algorithm that relies on a vague definition of “part to the right”. See, you can’t even write it down (unlike 1 + 3.14 = 4.14), because pi is not really a number, in a sense. It is an infinite calculation that happens to converge between 3.14 and 3.15. You can never get rid of “pi” in your calculations unless it cancels out naturally, so for practical reasons it’s 3.14 a…
> what is that number?
Pi minus 3. If you are saying you can't actually perform that operation, great! That means you are agreeing with me. See below.
> you cannot multiply by pi by repeated addition
Thank you for agreeing with my main point. See below.
> not because the numeric method is wrong.
If someone is going to claim that multiplication is repeated addition, then their definition of multiplication in terms of repeated addition must cover all cases. You are saying it doesn't cover pi, which means it doesn't cover all cases. So the definition is wrong.
> For generalized reasoning, yes. For teaching, not sure.
Why not? What's wrong with telling kids, multiplication in general is a distinct primitive operation, we are teaching you how to multiply whole numbers using repeated addition, but be aware that this method will not generalize to all cases?
> Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that?
A student who has been told what I just described above is not in either of these positions, so you are arguing against a straw man.
> When they learn, they ask themselves why and it’s a rabbit hole.
No, it's natural human curiosity which should be encouraged, not stomped on. At some point they'll either realize that they're up against material they're not yet ready for, and put it aside for later, or they'll end up being a math prodigy. Both outcomes are better ones than them just being told "this is it, don't ask any questions".
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#296Earlier 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.
> Let's not take the word "lie" out of its context
I'm not the one that used that word. If people are going to reference a concept that's been published in a book and uses that word, they should own it and be prepared to justify it. If they can't do that, they should not use the word in the first place.
To be clear, I'm not saying you are one of those people. But the poster I was responding to is.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#297Earlier quoted context omitted.
I never got over being lied to about electron shells and the 2-8-8 "rule".
How were you lied to about that?
In GCSE Chemistry (taught to 15-16ish year olds), we were taught that the first shell contained two electrons, and subsequent shells contained 8 (up to the final, which may contain fewer).
Then, after GCSEs, during A-levels, we were told (by the same teachers, in the same classrooms) to forget this model, and that the situation was actually more complicated, with s, p, d, f orbitals etc.
I realise that this is was a simplification and not necessarily an outright lie, and can understand why they did it this way. But it was the first time I realised I’d been deliberately taught something that was incomplete or inaccurate.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#298Earlier 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.
They don't need this information when learning that 3 groups of 5 fruit are 15 fruits. It's not in the natural order of understanding principles.
> They don't need this information when learning that 3 groups of 5 fruit are 15 fruits.
They also don't need the "information" that multiplication is repeated addition, period, full stop. But that's what others in this discussion appear to be trying to argue for.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#299Earlier quoted context omitted.
Um, by not lying to them? Do you have a specific example of where you think you need to lie to teach physics?
Explaining the structure of an atom without having to explain atomic orbitals and standing waves. Explaining classical mechanics without including a bunch of caveats about relativistic speeds.
> Explaining the structure of an atom without having to explain atomic orbitals and standing waves.
My high school chemistry teacher had no problem explaining this to me, when teaching the periodic table of the elements, without telling any lies and without going into the details of the quantum mechanics involved. The Pauli exclusion principle and a general statement that the details of the quantum mechanics were out of scope for that class was enough.
> Explaining classical mechanics without including a bunch of caveats about relativistic speeds.
My high school physics teacher had no problem explaining classical mechanics including the caveats. The caveats took only a few minutes early in the semester. What's the problem?
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#300I'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 tend to think of multiplication as a transformation. "repeated additional" almost makes it sounds like we're merely telescoping an interval within a single dimension. Whereas "transformation" (to me), evokes more of a Cartesian Product or Quadrature sort of mental image.