Our sir teaches Mathematics . . . . . . English.

Answer: Option B

1. Option A: across
Across means from one side to the other.
It does not fit the context of teaching two subjects.

2. Option B: besides
Besides means in addition to.
This is the correct option because it indicates that English is another subject taught in addition to Mathematics.

3. Option C: beside
Beside means next to or by the side of.
It does not fit the context of teaching subjects.

4. Option D: both
Both is a pronoun indicating two things.
However, it requires a conjunction like and to make the sentence grammatically correct (e.g., "Our sir teaches both Mathematics and English").
Since the question is looking for a preposition, this option is incorrect.

Therefore, the best option is B: besides, as it accurately conveys that the teacher teaches English in addition to Mathematics.