My initial reaction to this is much the same as Rawb's. It seems like an odd thing to say, unless it's meant in a general way. The East and the West both have their own unique traditions of Marian devotion. They certainly can differ quite a bit in form, but isn't that evidence of the widespread Marian devotion found in the early Church, rather than suggesting that one form came from the other? Particularly when the earliest existent Marian hymns tend to date from around the same time, regardless of the church in question (c. 4th century, which is when the "Magnificat", quoting directly the Vulgate which was produced around that time, is said have been written). The practice of including "Theotokias" (Marian hymns) in every service (as you will find them in the Orthodox Church, anyway) has been dated to the 5th century, so obviously the hymns must've existed and been in use in some fashion before that.
So, certainly there is no straight line from the East to the West in terms of form. Most Latins, then and now, wouldn't have known Greek, let alone also Syriac (St. Ephrem's Marian hymns are a treasure of the whole Church -- read what preeminent Syriac scholar Sebastian Brock has to say about them, if you are interested), Coptic, Ethiopic/Ge'ez, etc., which are also carriers of their own ancient Marian traditions (HH Pope Shenouda III points out in his book "St. Mary in the Orthodox Concept" that the 13th century emperor Zara Yaqoub ordered that every church in the entire country have an altar dedicated to St. Mary, and that in the Ethiopian tradition all heretics are referred to as "enemies of St. Mary"), so we must say that if they did receive their Marian devotion from the Greek-speaking East, it would've been pretty much as Rawb wrote: as part of receiving everything else from the Hellenized East, as well (cf. the switch from Greek to Latin in Western liturgies happened under Pope Victor I, 189-199, though it wouldn't become the norm throughout the West until -- surprise, surprise -- c. 4th century).
If were you, I would ask the professor for clarification as to what exactly he means, because it doesn't really make sense to say that Marian devotion developed in the East and then was transmitted to the West, as something considered separately from the transmission of Christianity itself. :shrug: