Is "whom" in that sentence correct?
She's a mother of one of my students whom I am teaching on Skype
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Yes, it's correct, although most speakers would shorten I am to I'm.
To be sure, break the sentence into its two main parts.
This student is the object of the second sentence. So the relative pronoun required is whom rather than who.
You might simplify you sentence as:
She is the mother of a student whom I teach.
However, your sentence raises several questions.
The first is whether the average English speaker would use the more formal whom (which is steadily losing ground among native English speakers) in the place of the more common who.
The second is whether you are teaching the mother or her child. The sentence does not make this clear.
Thirdly, she is the mother, not a mother of the student,
And finally you are probably teaching the student via or through Skype rather than on it.
Whom is correct in this sentence. You use "whom" when referring to the object of the verb.
She's a mother of one of my students whom I am teaching on Skype.
The classic test is to see if the verb works with "he/she" or "him/her".
In this case the verb is teaching, and you would say "teaching him", not "teaching he". This confirms that it is the object of the verb, and "whom" is correct.
However the sentence would be better as:
She's the mother of one of my students whom I am teaching on Skype.
(unless you deliberately meant to imply the student has two moms, which is possible these days)