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> As I said, there are hundreds of examples on that site. In that article or in the whole site? Because the article just seems like different notation, no grea

> As I said, there are hundreds of examples on that site. In that article or in the whole site? Because the article just seems like different notation, no great improvements. > In many cases for example it's sufficient to know that an operation is associative to perform some manipulation Whenever that's the case, mathematicians have already found a way to use notation to that advantage. See for example the case of groups and rings, where the operations are usually denoted by '+' and '·' (addition and multiplication) even when they might not necessarily be those operations just because it makes operations more intuitive. > What does the following mean? What concepts must be held in mind to understand it? Well, you need to know what the sinus is and what is addition. If you don't have those concepts it doesn't matter that I write sin(s + i + n) or 'The sinus of s plus i plus n' (not to mention that there's ambiguity there too, does 'of' refer to 's' or 's plus i plus n). > On the other hand if one desires to make mathematics more accessible to persons of ordinary intelligence, then reducing the cognitive load by clearly defining the semantics of a formula using an unambigous syntax is certainly necessary. Evidently though that's of little importance to the mathematical guild. Notation is not the issue when explaining mathematics. The mathematical guild has done a ton of effort in improving math education, and you'll be able to find videos and texts that barely have any notation or equations. Notation exists because once you learn the concepts, it reduces the cognitive load of transmitting those concepts.

User23 5y ago
> Well, you need to know what the sinus is and what is addition. If you don't have those concepts it doesn't matter that I write sin(s + i + n) or 'The sinus of s plus i plus n' (not to mention that there's ambiguity there too, does 'of' refer to 's' or 's plus i plus n). 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. > Notation is not the issue when explaining mathematics. The mathematical guild has done a ton of effort in improving math education, and you'll be able to find videos and texts that barely have any notation or equations. 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. > Notation exists because once you learn the concepts, it reduces the cognitive load of transmitting those concepts. I agree. In fact that's germane to my original point. The mathematics guild is using primitive notation compared to what computing scientists discovered in the latter half of the 20th century. I'm not saying that notation is bad or even that the traditional amgbigous syntax isn't an improvement over just writing everything out in some natural language. I'm saying the mathematics guild is stubbornly ignoring notational advances that further reduces cognitive load. > The mathematical guild has done a ton of effort in improving math education 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.
- gjulianm 5y ago
  > 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.

User23 5y ago

> Well, you need to know what the sinus is and what is addition. If you don't have those concepts it doesn't matter that I write sin(s + i + n) or 'The sinus of s plus i plus n' (not to mention that there's ambiguity there too, does 'of' refer to 's' or 's plus i plus n). 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. > Notation is not the issue when explaining mathematics. The mathematical guild has done a ton of effort in improving math education, and you'll be able to find videos and texts that barely have any notation or equations. 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. > Notation exists because once you learn the concepts, it reduces the cognitive load of transmitting those concepts. I agree. In fact that's germane to my original point. The mathematics guild is using primitive notation compared to what computing scientists discovered in the latter half of the 20th century. I'm not saying that notation is bad or even that the traditional amgbigous syntax isn't an improvement over just writing everything out in some natural language. I'm saying the mathematics guild is stubbornly ignoring notational advances that further reduces cognitive load. > The mathematical guild has done a ton of effort in improving math education 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.

gjulianm 5y ago
> 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.