I think you mean when to use an article and when not to, right, not the difference between “THE” and “A”? (Although that’s worth stating too).
In grammar terms (which some Taiwanese seem to like), the rule is that if the noun is a count noun and is singular, you must use an article (A or THE). Otherwise, it depends on the meaning. The only exception is when a singular noun refers to all examples of that noun (e.g. “Man is a social animal”). Mass nouns (“sugar”, for example – anything you have to say “I want some…” instead of “I want a…” before in English) don’t need and indeed can’t take an article in the singular. In the plural they can, because a plural mass noun means all the different kinds of that thing (“The sugars produced in this area are famous…” assuming they make white sugar, brown sugar, and I don’t know what other kinds of sugar.)
Articles are late-acquired by native speaking kids. It seems counterintuitive to expect learners of English to master them just becuase it seems convenient to present them first in the book since they’re short and common. I’m not saying we should give up on it, just that it’s one of those things (like having learners of Chinese acquire the correct usage of “le” in Mandarin) that takes time and sometimes just never arrives at 100% accuracy.
In my technical writing classes, I used to actually have the kids mark each and every noun and check whether it was a singular count noun or not, and then deal with the article accordingly.
Now, the other thing I’d like to have a magic formula for is to get Taiwanese students to write “a” instead of “one”. Chinese uses “one + measure word” where we would use “a” in English (but also where we would truly use “one”) so I guess it’s just one of those things, but for some reason hearing “one” where it should be “a” irritates me in a linguistic sense. Maybe I need some new hobbies.