> 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.
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.