For ESL I suggest you say it like the English/British. This is for clarity and there is little chance that you will be misunderstood.
So say "can't" with an "a" as in "argue".
I just came to the US and I'm always confused when people say can't and can
I don't know which word are they saying.
Can't has the t sound at the end, but most American people don't say that t sound. So they just say like Can'
so it's the same thing as can - the opposite word
and the important thing is i don't know to pronounce those 2 words exactly and sometimes I don't understand what they say either.
For ESL I suggest you say it like the English/British. This is for clarity and there is little chance that you will be misunderstood.
So say "can't" with an "a" as in "argue".
Interesting observation. If you ask most Br Eng speakers - probably with the sole exception of people who have studied phonology - 'What's the difference between can and can't?' you'll get the answer 'One's got a t'. But what we really listen for to distinguish the two is the vowel sound. So follow mxreader's advice. Often, when a speaker of Am Eng says 'can't' I have to think twice about what I really heard (and whether one or the other makes more sense in the context - If an American seems to say 'Sorry, I can' I can guess [because of the 'Sorry'] that 'can't is the more likely of the two.
But on the last point you're wrong: there is an audible difference between [kæn] and [kænʖ], just as there is an audible difference in Br Eng between [kæn] and [ka:nʖ]. It's just that the difference, in Br Eng, doesn't depend on your hearing the [ʖ] - which is often an indistinct sound (but it's there, as a spectrograph would show).
b
Even native speakers have to stop someone to say "I'm sorry -- did you just say he can or he cannot?" from time to time. Context will often help, but not always. I agree that it can be very hard to hear the difference sometimes.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
You're not alone.