Teaching math is mostly done w/o context and history, IMHO a lot of math makes much more sense when the original problem is understood, before the level of abstraction is being raised.
Math is a also a language and a notation. Unless one uses math regularly, there is simply not enough practice/repetition to read/speak this notation.
Math is a tower of abstractions, depending on other abstractions. A lot of topics in math depends on people understanding a lot of basic parts, which means if a student just got by with a prior topic, it is near impossible to catch up/understand what is currently being taught. (Compare to other topics: For example, if a student is bad in their Greek history, they get a fresh start when the topic is industrialization in England w/o any penalty.)
Math in the primary and secondary schools is mostly computation, ‘real math’ is only taught to people studying MINT.
tl;dr
we need a better curriculum in the primary/secondary schools
we need more exercises in reading/writing the mathematical notation (sorry, just understanding math is not enough, because understanding doesn’t make one fluent)
at least in my school years, math was not repeated enough.
reading/understanding math is really hard, at the higher levels, understanding 2-3 pages on a textbook per day is an acceptable pace. I guess all the entertainment nowadays makes it not easier to sit still in a room and get math into ones brain
For me the ‘breakthrough’ with math was, simply to accept that at the higher levels we are speaking about symbols (abstractions) that follow certain rules and everything else is derived by pure logic. Just accepting that one is manipulating symbols with rules to get to other symbols and learning the rules, made it click for me. Disclaimer: Was lucky with great math teachers in university, but even in my university there were people who simply could not accept the game of mathematics and were frustrated, because they wanted easy question/answer style formulas in the sense: When you see this, substitute PI with 3.14 and multiply r by r and write down the number that your calculator shows. They never made any effort to understand where PI comes from, where the radius comes from and why it makes sense.
What is insane, is how many people studied computer science but are totally unable to apply mathematics to the problems they try to solve. Supposedly most of them learned relational algebra and discrete mathematics during their studies (and formal languages/complexity theory)… it is like something is missing in their ability to transfer what they learned in the university to basically the same problems where the symbols have different names. That is something I would love to understand.
tl;dr
For me the ‘breakthrough’ with math was, simply to accept that at the higher levels we are speaking about symbols (abstractions) that follow certain rules and everything else is derived by pure logic. Just accepting that one is manipulating symbols with rules to get to other symbols and learning the rules, made it click for me. Disclaimer: Was lucky with great math teachers in university, but even in my university there were people who simply could not accept the game of mathematics and were frustrated, because they wanted easy question/answer style formulas in the sense: When you see this, substitute PI with 3.14 and multiply r by r and write down the number that your calculator shows. They never made any effort to understand where PI comes from, where the radius comes from and why it makes sense.
What is insane, is how many people studied computer science but are totally unable to apply mathematics to the problems they try to solve. Supposedly most of them learned relational algebra and discrete mathematics during their studies (and formal languages/complexity theory)… it is like something is missing in their ability to transfer what they learned in the university to basically the same problems where the symbols have different names. That is something I would love to understand.