Jack Schneider suggests no:
It is not unreasonable to think that if a teacher with a B average from a good college is sufficient, a teacher with an A average from the Ivy League must be better. Yet that isn’t the case. Consider the content of an average standardized test for seventh-grade math and then check out the senior thesis of a Harvard math major. The two documents are worlds apart. In short, the academic backgrounds of teachers matter; but only up to a point. As research indicates, beyond a certain threshold level of content mastery, there is little return to student achievement. And additional research indicates that higher-prestige colleges do not produce markedly more effective teachers.
On the whole, then, we might conclude not only that American teachers are generally well prepared academically, but also that recruiting a more selective slice of the public into classrooms may be both impractical and unproductive. This, of course, is not to say that teacher training is perfect, or that the pool of teacher candidates is ideal. It does, however, suggest that simple solutions like Teach For America-style recruiting are unlikely to make much of a difference.