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Is it grammatically correct to say “I teach myself subject”?

Context: my girlfriend is Italian, and remarked that “I teach myself” has no meaningful equivalent in Italian — ordinarily they would use “to learn” to say “I’m learning”, where the learning is happening under one’s own effort, and not being taught by a second party. In English, I’d often say “I teach myself”.

She asked whether I’d use “I teach myself” in formal English and I genuinely don’t know. Is this considered grammatically correct?

It is formally correct grammar, following only accepted rules of standard English. However it is not idiomatic; the circumstances would have to be very particular to say that.

The more idiomatic/natural way of saying what you mean by 'I teach myself' is:

I am teaching myself X.

The present simple 'I teach' is just not that common as is in English, whereas the present progressive is so much more common.

As to the choice of 'to learn' or 'to teach oneself' (not the issue of grammar but the issue of lexical meaning), you could just as well say 'I am learning' in English also. But the two are synonymous.

That said, just for fun I checked Google NGrams to see the relative frequency of the three, "am learning", "am teaching myself", and "I teach myself". "Am learning is by far more popular, but it is interesting that "I teach myself" is more common than "am teaching myself". I think this is because, after looking at the examples, "I teach myself" appears (very naturally) in many non-declarative sentences, such as "How do I teach myself", which is a very different context and so sort of doesn't count.

What this means to me is that "I teach myself" or the more natural "I am teaching myself" might be used, but more likely just say it like in Italian "I am learning".

It's fine.

Lots of verbs in English take a reflexive pronoun like "myself." To teach is one of them. In the case of to teach, it takes two objects: one is the person being taught, and the other is the subject or skill being taught (see Oxford Dictionaries entry "teach" subdefinition 1.1 for the double object use). The reflexive reflects the possibility of someone (like "Lisa") teaching herself something through practice, reading, and other resources.

FluentU, a blog on English language usage, highlights the use of the verb to teach with the reflexive pronoun. It focuses on the present progressive form of the verb:

I am teaching myself. 

You are teaching yourself.

Lisa is teaching herself. / She is teaching herself.
Frank is teaching himself. / He is teaching himself.

We are teaching ourselves.

You are teaching yourselves.

They are teaching themselves.

Why the progressive? This is because teaching tends to be a repetitive action over a period of time, which the progressive or continuous aspect expresses. So "I'm teaching myself (subject)" would be totally fine too.

Still, the simple present also expresses a general habit and can be used. In this sense,

I teach myself (subject)

is understood fine in many contexts. In fact, "I teach myself" is twice as common in a Google Books search than "I'm teaching myself" or "I am teaching myself." General Google results replicate the pattern. I suggest consulting the book results or more general web results to see how the expressions are used in context.

The simple answer is that it is correct. But the real question isn't about English grammar, but about the differences between Italian and English grammar, to see why we would express things differently.

I can see two reasons you might not be able to say this or why if you did, it might not have the required meaning.

Firstly, reflexive verbs are much more common in Italian than in English. They are explained here. In addition there is no exact equivalent of "myself", so they would just say mi lavo "I wash me" where we would say "I wash myself" or, more usually, use the verb intransitively: "I wash". Because they are so common, usually corresponding to intransitive verbs in English, or sometimes to a passive construction (mi chiamo "I am called"), you would miss the sense you get in English of emphasising the use of transitive verb, and that you are both the subject and the object. Rather, they are usually used for things you normally do by yourself (mi rado "I shave") without actually thinking about doing something to yourself.

You will have noticed I have not used the verb "to teach". This is because of the second problem:

In English, teach can take two objects:

I teach children chess

and although one of these is considered an indirect object as shown by the fact that we can say

I teach chess to the children

we usually just allow the two objects with no preposition. You can't do this in Italian. You always teach the subject a "to" the people, as shown in Collins Italian Dictionary

to teach sb sth or teach sth to sb insegnare qc a qn

This means that you simply cannot teach yourself, unless you are the subject being taught.

Thus we see that it is quite reasonable to have to express the same thing in different ways in different languages.