[jacobolus]

The biggest problem is that it is very unfriendly to uninitiated newcomers and makes insufficient effort to draw people in. You end up with a culture that is unfortunately insular and has trouble engaging with even engineers and scientists, much less the general public. It’s also not very friendly to people who approach problems in different ways: symbol pushing has been elevated and anyone who has difficulty with symbol pushing (for whatever reason) ends up at least partly excluded.

Students who have a lot of practice/experience by the time they get to be teenagers (often via extra-curricular help and support) are much better prepared than those without that practice. Which is of course not a problem per se, you see the same in any field and it’s great if kids want to learn ahead of their peers. But then the content, curricular design, and pedagogy of mathematics courses leave students with the impression that those differences in preparation are due to innate differences in aptitude (“I suck at math”; “she’s just a math person”; ...), toss less well prepared students into the deep end to sink without enough support, and ultimately chase a huge number of people away who might otherwise find the subject beautiful and interesting, and could meaningfully contribute.

[red75prime]

Well, we can't know that until we find (or won't find) the more effective way of teaching (or a way to do math without "symbol pushing" for that matter).

Until then it will not be wise to break what works (even for a minority of students).

[jacobolus]

Current incentives are set up to make even the most trivial attempts to run against the mainstream definitions and notations extremely difficult.

[simonh]

I don’t think it’s fair to say they are set up to do that. They weren’t conceived with that purpose. It’s just a fact of life that once we’ve invested a huge amount of effort in one set of conventions it’s very costly to change those conventions.

[jacobolus]

I don’t mean that some secret committee got together to “set up” all of the social incentives of the entire school system, university system, textbook industry, scientific publication system, engineering fields, etc.

What I mean is that there are incentives for the people involved in those systems which are extremely difficult to reform, and as long as the current incentives prevail it is all but impossible for anyone to refactor things like basic mathematical notions and notations.

Switching and retraining costs are high, gaps in inter-operability are expensive, and there is almost nobody who will achieve any career advancement through promoting changes to the high school and early undergraduate curriculum.

Mathematicians are generally most interested in pushing on the shiny boundaries of the field rather than trying to clean up the centuries-old material for novices. Teachers have their hands full enough with their students to do much new research in pedagogy. Practitioners in industry have their own problems to solve.