Peachy, with all due respect to your "Islam teacher" here is the related bit in Quran:
"And say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what (must ordinarily) appear thereof; that they should draw their veils over their bosoms and not display their beauty except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers or their brothers' sons, or their sisters' sons, or their women,..."
Quran 24:30-31
Note that it is directed at all "believing women" and not just to Mohammad's wives. Almost all Islamic scholars agree that a woman's "beauty" includes our hair as well as pretty much all our body.
Here is another:
"O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks (veils) all over their bodies. That will be better, that they should be known so as not to be bothered."
Quran 33:59
One could argue that cloaks/veils here could also be translated as dresses or garments but it would help to remember that the clothing of the time included head pieces, so they did not really have to be specified.
Again, I am not religious at all but had to study quite a bit of the Quran in school and did my own (anti) research to talk back at the teachers On this issue, unfortunately, I have to say that the desire to cover women up like dolmas is pretty clear.