4 ms·

It is interesting to see kids invent their own perfectly adequate, if peculiar, notation. In the first example with the quadrilaterals, the answers were all rig

It is interesting to see kids invent their own perfectly adequate, if peculiar, notation. In the first example with the quadrilaterals, the answers were all right. The math is not wrong, just written differently.

gohrt 11y ago
That gave me mixed feelings. On the one hand, the teacher did an admirable job (for a teacher) of seeing the correct thinking behind the "wrong" notation. On the other hand, the teacher still accused the student of making a mistake.
- Natsu 11y ago
  If nobody else understands your right answer, it may not be worth much. At this level, it doesn't matter at all--anyone can see based on the constraints what they're doing and understand them. But this may not always be the case. For example, when they get to intervals and suddenly need to properly discriminate between intervals like (1,2], (1,2), [1,2) and [1,2]. I might ding that by just one point--if you use non-standard notation, you must properly define it. You won't always be dealing with cases where your idiosyncrasies have obvious answers and it's better to form good habits that allow you to communicate clearly with others. There may be many good ways to depart from convention and I'd hesitate to stop anyone from doing that, but I think that if defining your departure from convention is too burdensome to be worthwhile, then you do not have a compelling case to depart from it in the first place.
- Gibbon1 11y ago
  If I were a teacher and I saw a student come up with a peculiar but correct notation, I'd be ecstatic.
  - jacobolus 11y ago
    Indeed, developing new notation should be not only allowed, not only encouraged, but even required as an assignment at various points during mathematical training, at every age level. Also to be encouraged is conscientious analysis and critique of existing notations, algorithms, and explanations. Society advances in two ways: in small incremental steps, and in large leaps. When we stick to existing notations and concepts we can make incremental improvements. But the large leaps often come from creating better notations and better conceptual understandings, which can turn substantial amounts of knowledge and logical thinking into obvious consequences of an organized and intuitive system.
    - daniel-levin 11y ago
      Agreed. Feynman diagrams are a great example of this phenomenon. They might be 'pictures', but they are still mathematical notation.
userbinator 11y ago
I'd say "the notation x/y means a fraction, and cannot be used for coordinates. Likewise, (x, y) means a coordinate and cannot be used for fractions. They are related but should not be conflated."
- gizmo686 11y ago
  That makes it sound way to absolute. You should say something more like "please write coordinates like (x,y)". Incidentally, in some contexts, (x,y) is how we write fractions (they are just an equivalence class based on an ordered pair of integers). As long as I am on the subject, (x,y) also means the inner product of x and one. And I had one teacher for whom (x,y) means f(x,y) where f would change more or less every section based on convience. On second thought, maybe we should be a little stricter in teaching students that math notation is rigid. If we raise a generation of mathematicians believing it, then they might clean up the current mess of notation we have today.
  - alextgordon 11y ago
    If you want to make a mathematician sweat, just ask them whether ((a, b), c) is the same as (a, b, c).
    - mjcohen 11y ago
      The answer is "What does '(a, b)' mean?"
    - Someone 11y ago
      As I once said after someone, after a discussion on some trivial mathematical argument that he didn't grasp, said "OK, but one plus one equals two, right?": "it depends".
    - asgard1024 11y ago
      Yes, up to isomorphism.

gohrt 11y ago

That gave me mixed feelings. On the one hand, the teacher did an admirable job (for a teacher) of seeing the correct thinking behind the "wrong" notation. On the other hand, the teacher still accused the student of making a mistake.

Natsu 11y ago
If nobody else understands your right answer, it may not be worth much. At this level, it doesn't matter at all--anyone can see based on the constraints what they're doing and understand them. But this may not always be the case. For example, when they get to intervals and suddenly need to properly discriminate between intervals like (1,2], (1,2), [1,2) and [1,2]. I might ding that by just one point--if you use non-standard notation, you must properly define it. You won't always be dealing with cases where your idiosyncrasies have obvious answers and it's better to form good habits that allow you to communicate clearly with others. There may be many good ways to depart from convention and I'd hesitate to stop anyone from doing that, but I think that if defining your departure from convention is too burdensome to be worthwhile, then you do not have a compelling case to depart from it in the first place.
Gibbon1 11y ago
If I were a teacher and I saw a student come up with a peculiar but correct notation, I'd be ecstatic.
- jacobolus 11y ago
  Indeed, developing new notation should be not only allowed, not only encouraged, but even required as an assignment at various points during mathematical training, at every age level. Also to be encouraged is conscientious analysis and critique of existing notations, algorithms, and explanations. Society advances in two ways: in small incremental steps, and in large leaps. When we stick to existing notations and concepts we can make incremental improvements. But the large leaps often come from creating better notations and better conceptual understandings, which can turn substantial amounts of knowledge and logical thinking into obvious consequences of an organized and intuitive system.
  - daniel-levin 11y ago
    Agreed. Feynman diagrams are a great example of this phenomenon. They might be 'pictures', but they are still mathematical notation.

userbinator 11y ago

I'd say "the notation x/y means a fraction, and cannot be used for coordinates. Likewise, (x, y) means a coordinate and cannot be used for fractions. They are related but should not be conflated."

gizmo686 11y ago
That makes it sound way to absolute. You should say something more like "please write coordinates like (x,y)". Incidentally, in some contexts, (x,y) is how we write fractions (they are just an equivalence class based on an ordered pair of integers). As long as I am on the subject, (x,y) also means the inner product of x and one. And I had one teacher for whom (x,y) means f(x,y) where f would change more or less every section based on convience. On second thought, maybe we should be a little stricter in teaching students that math notation is rigid. If we raise a generation of mathematicians believing it, then they might clean up the current mess of notation we have today.
- alextgordon 11y ago
  If you want to make a mathematician sweat, just ask them whether ((a, b), c) is the same as (a, b, c).
  - mjcohen 11y ago
    The answer is "What does '(a, b)' mean?"
  - Someone 11y ago
    As I once said after someone, after a discussion on some trivial mathematical argument that he didn't grasp, said "OK, but one plus one equals two, right?": "it depends".
  - asgard1024 11y ago
    Yes, up to isomorphism.