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> We agree that multiplication makes things bigger, right? Huh!? I have luckily never heard anyone try to explain multiplication that way before. It's a horrib

by feoren 5y ago

> We agree that multiplication makes things bigger, right? Huh!? I have luckily never heard anyone try to explain multiplication that way before. It's a horrible mental model for multiplication! > Imaginary numbers let us rotate numbers. Kind of ... it's useful to visualize multiplication by imaginary numbers as analogous to rotation, but this makes it sound like imaginary numbers are useful because they let us "rotate numbers", like that's something we always wanted to do. This is neither the essence of nor the impetus behind imaginary numbers. > The number e is a little machine that grows as fast as it can No no no! We happen to use e as an exponential base a lot because it's convenient, but it's exponentiation, not e, that grows quickly! f(x) = 4^x grows faster than e^x does; e^-x shrinks; e^0 stays where it is ... this is a horrible intuition to have! > The Pythagorean Theorem explains how all shapes behave (not just triangles) It works on other shapes because you draw triangles in those shapes ... These are awful, awful intuitions about math. We should not take advice about how to learn math from someone who so deeply struggles with basic concepts about it! Edit: this article is so bad that I just have to add another example: > Euler’s Formula makes a circular path. But you just said that e is a little machine that grows as fast as it can! How can it just spin in a circle!? His own intuitions are completely inconsistent! Besides, Euler's Formula has no free variables ...

twojacobtwo 5y ago
You seem to be specifically looking to find a frame in which the author is incorrect, which is basically a sign of the problem he's getting at in the article - the inability to allow ideas to be reduced to 'cartoonish' principles in order to facilitate a different (more engaging?) method of learning math, at the expense of being "technically correct". All your points seem to be making the same type of argument, so I'll just address the first one from my perspective, with the author's points in mind. > Huh!? I have luckily never heard anyone try to explain multiplication that way before. It's a horrible mental model for multiplication! The model definitely worked for me when I first encountered it and it worked for many of my early friends (a couple of whom are now accredited mathematicians). The other cases surrounding multiplication revealed themselves naturally, as I encountered multiplication by 0, multiplication by negatives, etc. (exceptions/cases noted by the author). The basic model gave me a way to think about how I could put the tool (multiplication, in this case) to use without being overwhelming and my self-driven discovery of the unexpected parts made me much more interested in what else I might be missing than if someone had given me a full, technically correct definition from the get-go. The key is that this is specifically aimed at those who are learning, not those who are 'learned'.
- feoren 5y ago
  > You seem to be specifically looking to find a frame in which the author is incorrect Actually I read the article because I love the idea, and agreed with everything he said up until halfway through, when it completely veered off into nonsense-land with awful examples. I have no idea why you've decided you know my motivation for criticizing the article. I'm criticizing it because it is bad! > The model definitely worked for me when I first encountered it and it worked for many of my early friends. Unfortunately it doesn't sound like either of us really have data on this, your anecdotes notwithstanding. But I have to admit I really wonder whether you're (a) remembering your elementary-school mental state correctly, and (b) correct about your friends' mental states as well. After all, you claimed to know my motivation for writing my comment, so maybe you're not as much of a mind-reader as you think. And it's really hard to remember the way you thought about something N decades ago! Maybe you're thinking of multiplication as being "repeated addition"? Great! That's a good cartoon of multiplication. That's a foothold into the concept that doesn't lead you astray. The reason it's a good cartoon is that it is correct -- technically correct -- for a sensible subset of numbers, and it extends naturally to other concepts. Multiplying 7 by 0.5 really is analogous to "adding 0.5 sevens together"; multiplying 7 by -2 is analogous to "adding -2 sevens together". And it's useful if you've been needing to do lots of addition and you're tired of hitting the + button on your calculator over and over. It's immediately useful because addition is useful, and it makes repeated addition easier! But "it makes things bigger" gives you none of this power -- none whatsoever. It just gives you a weird, bad mental image of it inflating a number into a bigger number ... somehow. It's not correct even for the Whole Numbers -- multiplying by 1 does not make anything bigger! You don't even need 0 for it to be wrong! And if you multiply by 0.5 or 17lbs or i, there's absolutely no way of stretching "making it bigger" into an analogy that's at all useful. It's just bad, bad, bad, bad. Completely different from "multiplication is repeated addition"! The rule "it makes things bigger, except when it doesn't" is exactly what people hate about math. A weird, opaque non-definition that isn't even true most of the time.
- patrick451 5y ago
  For a time, I too shared the idea that multiplication makes things bigger. I don't remember if it was taught this way, or that was just what I inferred on my own. In any case, it was an absolutely terrible way to think about it. Sure, it was fine until fractions. But unwinding that poor mental model was a jarring, painful event. I remember to this day banging my head against the figurative wall, crying literal tears trying to understand how multiplication by a fraction made something smaller.

by feoren 5y ago

twojacobtwo 5y ago

You seem to be specifically looking to find a frame in which the author is incorrect, which is basically a sign of the problem he's getting at in the article - the inability to allow ideas to be reduced to 'cartoonish' principles in order to facilitate a different (more engaging?) method of learning math, at the expense of being "technically correct". All your points seem to be making the same type of argument, so I'll just address the first one from my perspective, with the author's points in mind. > Huh!? I have luckily never heard anyone try to explain multiplication that way before. It's a horrible mental model for multiplication! The model definitely worked for me when I first encountered it and it worked for many of my early friends (a couple of whom are now accredited mathematicians). The other cases surrounding multiplication revealed themselves naturally, as I encountered multiplication by 0, multiplication by negatives, etc. (exceptions/cases noted by the author). The basic model gave me a way to think about how I could put the tool (multiplication, in this case) to use without being overwhelming and my self-driven discovery of the unexpected parts made me much more interested in what else I might be missing than if someone had given me a full, technically correct definition from the get-go. The key is that this is specifically aimed at those who are learning, not those who are 'learned'.

feoren 5y ago
> You seem to be specifically looking to find a frame in which the author is incorrect Actually I read the article because I love the idea, and agreed with everything he said up until halfway through, when it completely veered off into nonsense-land with awful examples. I have no idea why you've decided you know my motivation for criticizing the article. I'm criticizing it because it is bad! > The model definitely worked for me when I first encountered it and it worked for many of my early friends. Unfortunately it doesn't sound like either of us really have data on this, your anecdotes notwithstanding. But I have to admit I really wonder whether you're (a) remembering your elementary-school mental state correctly, and (b) correct about your friends' mental states as well. After all, you claimed to know my motivation for writing my comment, so maybe you're not as much of a mind-reader as you think. And it's really hard to remember the way you thought about something N decades ago! Maybe you're thinking of multiplication as being "repeated addition"? Great! That's a good cartoon of multiplication. That's a foothold into the concept that doesn't lead you astray. The reason it's a good cartoon is that it is correct -- technically correct -- for a sensible subset of numbers, and it extends naturally to other concepts. Multiplying 7 by 0.5 really is analogous to "adding 0.5 sevens together"; multiplying 7 by -2 is analogous to "adding -2 sevens together". And it's useful if you've been needing to do lots of addition and you're tired of hitting the + button on your calculator over and over. It's immediately useful because addition is useful, and it makes repeated addition easier! But "it makes things bigger" gives you none of this power -- none whatsoever. It just gives you a weird, bad mental image of it inflating a number into a bigger number ... somehow. It's not correct even for the Whole Numbers -- multiplying by 1 does not make anything bigger! You don't even need 0 for it to be wrong! And if you multiply by 0.5 or 17lbs or i, there's absolutely no way of stretching "making it bigger" into an analogy that's at all useful. It's just bad, bad, bad, bad. Completely different from "multiplication is repeated addition"! The rule "it makes things bigger, except when it doesn't" is exactly what people hate about math. A weird, opaque non-definition that isn't even true most of the time.
patrick451 5y ago
For a time, I too shared the idea that multiplication makes things bigger. I don't remember if it was taught this way, or that was just what I inferred on my own. In any case, it was an absolutely terrible way to think about it. Sure, it was fine until fractions. But unwinding that poor mental model was a jarring, painful event. I remember to this day banging my head against the figurative wall, crying literal tears trying to understand how multiplication by a fraction made something smaller.