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> Incorrect. In the example I gave one merely needed to understand multiplication, addition, and the distributive property. And you, who are observably skilled

> Incorrect. In the example I gave one merely needed to understand multiplication, addition, and the distributive property. And you, who are observably skilled in the art of mathematics, failed to follow a grade school level formula. So much for ambiguous syntax not being an impediment to understanding even when all concepts involved are comprehended. So which notation would you propose that would allow me to understand the formula without knowing those concepts? > It's baffling to me that anyone would attempt to argue that ambiguous syntax is not an issue for explaining mathematics. Please understand that the following unflattering comparison is intended merely for clarity and not to insult. To me it feels like trying to argue with a flat earther that the world is round. The absurdity makes argumentation virtually impossible. I am not saying it's not "an issue", I am saying it's not the issue. Ambiguous notation is a problem, yes, but when our language is ambiguous too I don't think it's a problem you can fully solve. And definitely one that will be solved by removing notation. > I'm saying the mathematics guild is stubbornly ignoring notational advances that further reduces cognitive load. Such as? Because I still haven't seen those advances. > Then how do we explain the complete lack of measurable progress in the average American secondary school student's mathematical ability? It appears to me that regardless of how much effort has been expended, little has come of it. I'd guess that has more to do with local policies and resources than any effort of any mathematician. It seems weird to evaluate the work of a global community based on a measurements of schools in a specific country.

User23 5y ago
First I want to say that I really appreciate your responses. You're obviously writing in good faith and I'm finding this interesting. Thank you for helping me clarify my own understanding. > So which notation would you propose that would allow me to understand the formula without knowing those concepts? We agree on the importance of understanding concepts. I showed the difficulty for one who does understand the concepts. Surely a student who is learning them and thus by definition doesn’t know them would have even greater difficulties. I propose that a notational convention that forgoes the invisible multiplication operator and that distinguishes between ordering operations and function application rather than using parentheses for both would be considerably more clear. Such as, for example, the one Dijkstra adopted. I’m confident that you would have easily understood either of the following: sin.(s + i + n) Or s*i*n*(s + i + n) A student who doesn't understand one or more of the necessary concepts will at least be able to see from the syntax that different concepts are being expressed. I don’t think 1-3 extra glyphs is too high a price to pay for that reduction in ambiguity. There is still ambiguity of course, such as are the expressions in R or C? I like type declarations for that reason. By the way I do agree with you that no syntactic improvement can completely eliminate ambiguity. I just think we should still do our level best to avoid introducing it gratuitously. Once the student has achieved proficiency with the requisite concepts using a more sensible notation, supposing they have an interest in further study, they will then need to learn the customary notation on account of the great and valuable body of work that uses it. I think that’s acceptable though since learning new syntax will always be a part of learning math and as you say the real challenge is mastering concepts and not syntax. > It seems weird to evaluate the work of a global community based on a measurements of schools in a specific country. You make a good point here. It would be better to make a global comparison using something like PISA math scores and seeing if didactic innovation has resulted in any improvement in the places it has been implemented. I’m not aware of any such improvement, but the world is a big place so that’s hardly evidence that there isn’t any.
- gjulianm 5y ago
  > Surely a student who is learning them and thus by definition doesn’t know them would have even greater difficulties But a student who is learning them will not start by looking at the equation, but at the concept, and that concept will disambiguate between sin() and s·i·n. The example you use is interesting. Yes, that would remove ambiguity between sin() and sin. But how would that notation evolve? If in most instances it's clear when you have function application and when it's multiplication, people will stop writing it. Same with the multiplication operator. Not to mention that notation also introduces ambiguity, because the dot is part of the written language, so you'll have instances where it isn't clear whether the dot means "apply function" and when it means "sentence stop". Even assuming that the notation stays and doesn't devolve to something that's faster to write and to read, what did we actually achieve? We wouldn't have removed the complexity of learning trigonometric functions. You wouldn't stop a student from doing sin.(a + b) = sin.a + sin.b, for example, or trying to use the same formulas for sin and cos. My point is that while you will have some instances where notation could be improved (and naming too, for example closed and open sets are confusing because they are not inverse properties) because most of the time what's difficult is the concept itself, notation is like an extra step after having gone up four flights of stairs.
  - User23 5y ago
    > But a student who is learning them will not start by looking at the equation, but at the concept, and that concept will disambiguate between sin() and s·i·n. Some symbolic (broadly construed, including drawings, vocalizations, etc) representation is going to be required to communicate any concept. Why not use one that's minimally ambiguous? I get your point that we can hope the student will not struggle too much with the meaning of sin on the day the day the sine function is being taught. Even so, concepts once introduced usually appear elsewhere. We shouldn't be surprised if after learning both the sine and the invisible multiplication operator, the student might be confused as to whether or not some string is a sequence of multiplications or a function name. This isn't just hypothetical either, I've been in enough math classes to see the students of ordinary intelligence struggle with this. > The example you use is interesting. Yes, that would remove ambiguity between sin() and sin. But how would that notation evolve? If in most instances it's clear when you have function application and when it's multiplication, people will stop writing it. Same with the multiplication operator. I consider using invisible operators to be generally unwise. I wouldn't consider adding ambiguity to save a key or pen stroke a wise trade-off. I'm aware that many mathematicians do, and all I can say to that is that it baffles me. In humility I'm willing to allow that they know something I don't, so perhaps my bafflement is a personal defect. Even so I don't think I want to repair it. In my own work I appreciate the clarity too much. And since that work is just reasoning about programs I want to write and amounts to personal notes and not something I have any interest in publishing, it doesn't much matter to anyone else what notation I use. > Not to mention that notation also introduces ambiguity, because the dot is part of the written language, so you'll have instances where it isn't clear whether the dot means "apply function" and when it means "sentence stop". LaTeX and other comparable typesetting software adequately solve for this. As for manuscript, there are also ways to indicate whether a portion thereof is a formula or explanatory text. What I really think is important though isn't the choice of the glyph "." but avoiding using the same symbol for two completely unrelated concepts like precedence and function application. > Even assuming that the notation stays and doesn't devolve to something that's faster to write and to read, what did we actually achieve? We wouldn't have removed the complexity of learning trigonometric functions. You wouldn't stop a student from doing sin.(a + b) = sin.a + sin.b, for example, or trying to use the same formulas for sin and cos. Reading speed is by chunk and not character count. I challenge the notion that f(x+y) is faster to read than f.(x+y). As for being slower to write, I doubt that any mathematics beyond the most basic arithmetic are constrained by typing speed. I accept that there may be a stronger argument for some kind of shorthand in manuscript, but I still doubt the savings are worth it. As an aside I find this pleasantly parseable: sin.(a + b) = 1/csc.(a + b) although a student who failed to recognize that application binds more strongly than division might suffer. That mathematics deals with parsed expressions and not substrings is certainly a vital concept. > My point is that while you will have some instances where notation could be improved (and naming too, for example closed and open sets are confusing because they are not inverse properties) because most of the time what's difficult is the concept itself, notation is like an extra step after having gone up four flights of stairs. I think a better analogy is that it's like going up a set of stairs where the occasional step is false and drops into a pit. Once one gets used to the pits one can navigate the stairs virtually as well as if they weren't there, but that's hardly an argument in favor of booby trapping the stairs. It certainly will make things considerably harder for first time climbers. Nevertheless, I continue to agree that learning concepts is the more challenging and interesting part of mathematics. I also welcome improvements in clarifying concepts. Sadly, making a complicated concept easier to understand is a much greater challenge than making an ambiguous and muddled syntax unambiguous and clear. My preference is that we pursue both, because they're complementary.

User23 5y ago

First I want to say that I really appreciate your responses. You're obviously writing in good faith and I'm finding this interesting. Thank you for helping me clarify my own understanding. > So which notation would you propose that would allow me to understand the formula without knowing those concepts? We agree on the importance of understanding concepts. I showed the difficulty for one who does understand the concepts. Surely a student who is learning them and thus by definition doesn’t know them would have even greater difficulties. I propose that a notational convention that forgoes the invisible multiplication operator and that distinguishes between ordering operations and function application rather than using parentheses for both would be considerably more clear. Such as, for example, the one Dijkstra adopted. I’m confident that you would have easily understood either of the following: sin.(s + i + n) Or s*i*n*(s + i + n) A student who doesn't understand one or more of the necessary concepts will at least be able to see from the syntax that different concepts are being expressed. I don’t think 1-3 extra glyphs is too high a price to pay for that reduction in ambiguity. There is still ambiguity of course, such as are the expressions in R or C? I like type declarations for that reason. By the way I do agree with you that no syntactic improvement can completely eliminate ambiguity. I just think we should still do our level best to avoid introducing it gratuitously. Once the student has achieved proficiency with the requisite concepts using a more sensible notation, supposing they have an interest in further study, they will then need to learn the customary notation on account of the great and valuable body of work that uses it. I think that’s acceptable though since learning new syntax will always be a part of learning math and as you say the real challenge is mastering concepts and not syntax. > It seems weird to evaluate the work of a global community based on a measurements of schools in a specific country. You make a good point here. It would be better to make a global comparison using something like PISA math scores and seeing if didactic innovation has resulted in any improvement in the places it has been implemented. I’m not aware of any such improvement, but the world is a big place so that’s hardly evidence that there isn’t any.

gjulianm 5y ago
> Surely a student who is learning them and thus by definition doesn’t know them would have even greater difficulties But a student who is learning them will not start by looking at the equation, but at the concept, and that concept will disambiguate between sin() and s·i·n. The example you use is interesting. Yes, that would remove ambiguity between sin() and sin. But how would that notation evolve? If in most instances it's clear when you have function application and when it's multiplication, people will stop writing it. Same with the multiplication operator. Not to mention that notation also introduces ambiguity, because the dot is part of the written language, so you'll have instances where it isn't clear whether the dot means "apply function" and when it means "sentence stop". Even assuming that the notation stays and doesn't devolve to something that's faster to write and to read, what did we actually achieve? We wouldn't have removed the complexity of learning trigonometric functions. You wouldn't stop a student from doing sin.(a + b) = sin.a + sin.b, for example, or trying to use the same formulas for sin and cos. My point is that while you will have some instances where notation could be improved (and naming too, for example closed and open sets are confusing because they are not inverse properties) because most of the time what's difficult is the concept itself, notation is like an extra step after having gone up four flights of stairs.
- User23 5y ago
  > But a student who is learning them will not start by looking at the equation, but at the concept, and that concept will disambiguate between sin() and s·i·n. Some symbolic (broadly construed, including drawings, vocalizations, etc) representation is going to be required to communicate any concept. Why not use one that's minimally ambiguous? I get your point that we can hope the student will not struggle too much with the meaning of sin on the day the day the sine function is being taught. Even so, concepts once introduced usually appear elsewhere. We shouldn't be surprised if after learning both the sine and the invisible multiplication operator, the student might be confused as to whether or not some string is a sequence of multiplications or a function name. This isn't just hypothetical either, I've been in enough math classes to see the students of ordinary intelligence struggle with this. > The example you use is interesting. Yes, that would remove ambiguity between sin() and sin. But how would that notation evolve? If in most instances it's clear when you have function application and when it's multiplication, people will stop writing it. Same with the multiplication operator. I consider using invisible operators to be generally unwise. I wouldn't consider adding ambiguity to save a key or pen stroke a wise trade-off. I'm aware that many mathematicians do, and all I can say to that is that it baffles me. In humility I'm willing to allow that they know something I don't, so perhaps my bafflement is a personal defect. Even so I don't think I want to repair it. In my own work I appreciate the clarity too much. And since that work is just reasoning about programs I want to write and amounts to personal notes and not something I have any interest in publishing, it doesn't much matter to anyone else what notation I use. > Not to mention that notation also introduces ambiguity, because the dot is part of the written language, so you'll have instances where it isn't clear whether the dot means "apply function" and when it means "sentence stop". LaTeX and other comparable typesetting software adequately solve for this. As for manuscript, there are also ways to indicate whether a portion thereof is a formula or explanatory text. What I really think is important though isn't the choice of the glyph "." but avoiding using the same symbol for two completely unrelated concepts like precedence and function application. > Even assuming that the notation stays and doesn't devolve to something that's faster to write and to read, what did we actually achieve? We wouldn't have removed the complexity of learning trigonometric functions. You wouldn't stop a student from doing sin.(a + b) = sin.a + sin.b, for example, or trying to use the same formulas for sin and cos. Reading speed is by chunk and not character count. I challenge the notion that f(x+y) is faster to read than f.(x+y). As for being slower to write, I doubt that any mathematics beyond the most basic arithmetic are constrained by typing speed. I accept that there may be a stronger argument for some kind of shorthand in manuscript, but I still doubt the savings are worth it. As an aside I find this pleasantly parseable: sin.(a + b) = 1/csc.(a + b) although a student who failed to recognize that application binds more strongly than division might suffer. That mathematics deals with parsed expressions and not substrings is certainly a vital concept. > My point is that while you will have some instances where notation could be improved (and naming too, for example closed and open sets are confusing because they are not inverse properties) because most of the time what's difficult is the concept itself, notation is like an extra step after having gone up four flights of stairs. I think a better analogy is that it's like going up a set of stairs where the occasional step is false and drops into a pit. Once one gets used to the pits one can navigate the stairs virtually as well as if they weren't there, but that's hardly an argument in favor of booby trapping the stairs. It certainly will make things considerably harder for first time climbers. Nevertheless, I continue to agree that learning concepts is the more challenging and interesting part of mathematics. I also welcome improvements in clarifying concepts. Sadly, making a complicated concept easier to understand is a much greater challenge than making an ambiguous and muddled syntax unambiguous and clear. My preference is that we pursue both, because they're complementary.