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The problem is not that the notation exists and that there are multiple ways to say the same thing, it's that students are forced to memorize all of it for its

by j2kun 10y ago

The problem is not that the notation exists and that there are multiple ways to say the same thing, it's that students are forced to memorize all of it for its own sake. In fact, there aren't just "three" ways to describe 2 * 2 * 2=8, there are infinitely many. Because 2 * 2 * 2=8 shows up in so many different contexts, and notation in each of those contexts highlights a different (hopefully useful) feature for that context, you'll never be able to have "just" one way to say a thing. You can have your favorite, sure, but all notational preferences are aesthetic. FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how "mirroring" the triangle changes the operations at each corner, and whether they actually connect that to the underlying arithmetic operations is just as tenuous as with the classical notation. You see the video maker say, as an afterthought, that once the students are fluent in this beautiful notation they can go about understanding why the it works, but the same problem as before! They're memorizing arbitrary symbol shuffling, maybe reducing cognitive load but also introducing random extra facts along the way like parallel resistance (which algebra students care about that, again?), and the connection between the true idea and the work they're doing is thin.

deleted 10y ago
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noobermin 10y ago
>FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how "mirroring" the triangle changes the operations at each corner, and whether they actually connect that to the underlying arithmetic operations is just as tenuous as with the classical notation. Yes, this times 10^100. One thing to remember is that all notation is yes, tradition, but that tradition is somewhat subject to natural selection, which I'd argue is selection for ease of use. This is for example why we use symbolic notation in the first place, because it is more convenient once it's been learned. I'm not sure how to deal with this. Perhaps one way is to teach the long, tedious way, just do a bazillion multiplications and then say, hey, exponentiation makes it much easier to write! That's usually how we get students to appreciate short-cuts in the first place. Teaching them the short-cut without at least giving them a taste of the internals is usually a recipe for bad abstractions disconnected from the internals.
mnemonicsloth 10y ago
I agree with everything you say here, except for one remark, which, I'm sorry, is the silliest thing I've read all day: > all notational preferences are aesthetic Try multiplying Roman numerals sometime. Or read up on ancient Egyptian fractions. Or learn group theory without Cayley diagrams. Or do algebra on equations written in prose -- prose! -- as was typical everywhere for a thousand years before the Renaissance. And what good are tensors without indices? Or matrices: do matrices have a first and second index, or do they have rows and columns? Yes, matrix algebra with indices one and two is just as true, but that's the wrong observation. It takes what we already know for granted. In fact what we already know is the destination, and the point of departure is what you look at, or stare at, until you understand matrix algebra. How does the brain turn markings on paper into something like abstract truth? Nobody knows, but it's silly to say the markings don't matter. The brain is biology, and in biology everything matters to everything else.
- j2kun 10y ago
  The key word here is preference, which is subjective by definition. Just because certain notations are considered outdated does not mean there is no situation in which they can have merit. To wit, my preferred method for understanding tensors is without indices, and matrix notation is primarily useful for computers, not humans (in that regard I prefer the coordinate-free perspective). In fact, the insistence that linear algebra must be understood entirely using matrices and rows and columns is a red flag in my book! I'm not saying that notation is irrelevant, I'm saying it's a matter of perspective. Of course there are notational breakthroughs throughout history. But what makes a breakthrough depends on what problems you're trying to solve, which is largely why computer scientists like indices and algebraic geometers like commutative diagrams. Each is as unwelcome in the wrong domain as roman numerals are in algebra.

by j2kun 10y ago

deleted 10y ago

[deleted]

noobermin 10y ago

>FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how "mirroring" the triangle changes the operations at each corner, and whether they actually connect that to the underlying arithmetic operations is just as tenuous as with the classical notation. Yes, this times 10^100. One thing to remember is that all notation is yes, tradition, but that tradition is somewhat subject to natural selection, which I'd argue is selection for ease of use. This is for example why we use symbolic notation in the first place, because it is more convenient once it's been learned. I'm not sure how to deal with this. Perhaps one way is to teach the long, tedious way, just do a bazillion multiplications and then say, hey, exponentiation makes it much easier to write! That's usually how we get students to appreciate short-cuts in the first place. Teaching them the short-cut without at least giving them a taste of the internals is usually a recipe for bad abstractions disconnected from the internals.

mnemonicsloth 10y ago

I agree with everything you say here, except for one remark, which, I'm sorry, is the silliest thing I've read all day: > all notational preferences are aesthetic Try multiplying Roman numerals sometime. Or read up on ancient Egyptian fractions. Or learn group theory without Cayley diagrams. Or do algebra on equations written in prose -- prose! -- as was typical everywhere for a thousand years before the Renaissance. And what good are tensors without indices? Or matrices: do matrices have a first and second index, or do they have rows and columns? Yes, matrix algebra with indices one and two is just as true, but that's the wrong observation. It takes what we already know for granted. In fact what we already know is the destination, and the point of departure is what you look at, or stare at, until you understand matrix algebra. How does the brain turn markings on paper into something like abstract truth? Nobody knows, but it's silly to say the markings don't matter. The brain is biology, and in biology everything matters to everything else.

j2kun 10y ago
The key word here is preference, which is subjective by definition. Just because certain notations are considered outdated does not mean there is no situation in which they can have merit. To wit, my preferred method for understanding tensors is without indices, and matrix notation is primarily useful for computers, not humans (in that regard I prefer the coordinate-free perspective). In fact, the insistence that linear algebra must be understood entirely using matrices and rows and columns is a red flag in my book! I'm not saying that notation is irrelevant, I'm saying it's a matter of perspective. Of course there are notational breakthroughs throughout history. But what makes a breakthrough depends on what problems you're trying to solve, which is largely why computer scientists like indices and algebraic geometers like commutative diagrams. Each is as unwelcome in the wrong domain as roman numerals are in algebra.