311–320 of 425 posts
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That's not the norm?
I don't think so. I was on the Calculus track in high school so we derived it in...Pre-Calculus. Prior to that the quadratic formula was something that seemed to be handed down from on high. We used it in Algebra II and maybe even before that, but I had no idea where it came from. It was a mind-opening experience when we derived it in class one day. Our teacher didn't ruin the surprise. She just said, let's complete…
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Yeah, and that's still too much inferential distance to feed to an elementary schooler.
You appear to have a much more impoverished view of kids and their ability to learn than I do. My experience (not to mention my memory of how I was myself as a kid) is that kids grasp the fact that there can be more to a subject than adults are able to teach them at a particular time and place, so they're ok with adults honestly admitting that. But they do not like adults telling them categorical statements that late…
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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…
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A polynomial is a polynomial, a decimal representation of a real number is a decimal representation of a real number, and your representation of complex numbers has funny properties once you begin exponentiating that. I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x].
> I'd like to see how you'd show kids that: (1+x+x^2)·(1-x^3) is a "repeated addition", both belong to the ring Z[x]. Is that not just (1 * (1-x^3) + x * (1-x^3) + x^2 * (1-x^3)) ?? The polynomial itself gives us the means at which we logically split up the multiplication into component parts. Just as 3.14 * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3, when we move onto polynomials, we do the same exact thing. EDIT: remember, AL…
Now, you could argue that this is exactly a way to "add repeatedly", but at some point pushing analogies stops being helpful to your students.
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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…
Multiplication is literally repeated addition, as in "taking a number multiple times". It is in the name. The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation…
How do you repeat something a negative number of times? What does it mean to multiply two negatives, if multiplication is repeated addition? This is not an overly generalized concept, we teach negative numbers to elementary school students.
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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…
> Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related. My apologies if this is a stupid question, but when does the intuitive (layman) understanding of multiplication as repeated addition break down (mathematically)?
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…
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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…
What I was mostly responding to in your previous comment is the idea of "emulating" multiplication for whole numbers by repeated addition. What I got from that is that you were thinking of the whole numbers as being inside the real numbers. My point was that to get the real numbers you usually have to start with the whole numbers and build up to the reals (say by Dedekind cuts or Cauchy sequences), and then you can embed the whole numbers inside and pretend that that was where they were the whole time.
I do get that you were giving different pedagogical approaches, and this was just one of them, but I think this one is at least as bad of a misrepresentation as saying "multiplication is repeated addition" without specifying "for whole numbers." That is, unless you're like the Greek geometers and accept a continuum from the start.
(There are other number systems that the integers sit inside, though, that don't themselves sit inside the real or complex numbers, which is a reason I would hesitate to think about whole numbers this way in general.)
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. You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication? This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you…
> 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.
That is one way to derive it, but there are others, including my personal favorite, the resolvents method (which also works for deriving cubic and quartic formulas).