To infinitive and gerund can be an object of the verb teach?
I wonder if these sentences are grammatically correct or incorrect and reasons.
1) I teach you to speak English.
2) I teach you speaking English.
3) I teach to speak English to you.
4) I teach speaking English to you.
I really wonder please let me know.
For what I assume is the intended sense, OP's example #1 is the most natural alternative:
1) I teach you [how] to speak English
But the second version doesn't have that meaning:
2) I teach you speaking English.
In principle that could make sense if we interpret it as meaning I speak in English while teaching you [something, not necessarily how you should speak English]. Or at a stretch we could interpret "speaking English" as a noun phrase equivalent to "spoken English", but that's a rather non-standard usage.
Example #3 is simply ungrammatical. There's not much more to say about it.
Example #4 is syntactically "valid", but if the teacher was any good at his job that's probably not how he would phrase it. He'd probably use the format of example #1 anyway, but if he wanted to focus on the actual subject (the spoken language), as opposed to the goal (that you learn to speak English), he's more likely say:
5) I teach you spoken English
(Note that this version more strongly implies I only teach spoken English, not the written form.)
You teach someone to do something.
He is teaching you to speak English.
to here refers to a purpose, a reason.
If you teach something, you can say:
I teach English. I teach sailing. [the activity of sailing a boat] I am teaching them to speak English.
However, we would not say: I am teaching speaking unless it refers to public speaking, the activity.
I teach public speaking to business people.
The ing forms in your example refer to the activity of teaching, and there we just use the nouns: I teach math or I teach Chinese.
If you want to say to teach someone spoken as opposed to written English, you can say:
I teach English conversation classes.
