It is fine to start a sentence with And. It gives a slight emphasis to the sentence before it and helps make the point more clear. Many authors use and to start a sentence.
As a teacher, I think if it is done in the correct way and not overused, I think it is fine. I would rather the students put a period and start a new sentence than have some drastic run-on sentence.
Traditionalists will not allow and to begin any new sentences. However, many writers use and to start a sentence which gives additional support/information to the one that precedes it. In other words, when and means in addition or and so/thus a writer uses it effectively as a transitional word, not as a co-ordinating conjunction.
A frequently asked question about conjunctions is whether and or but can be used at the beginning of a sentence. This is what R.W. Burchfield has to say about this use of and:
There is a persistent belief that it is improper to begin a sentence with And, but this prohibition has been cheerfully ignored by standard authors from Anglo-Saxon times onwards. An initial And is a useful aid to writers as the narrative continues.
from The New Fowler's Modern English Usage
The same is true with the conjunction but. A sentence beginning with and or but will tend to draw attention to itself and its transitional function. Writers should examine such sentences with two questions in mind: (1) would the sentence and paragraph function just as well without the initial conjunction? (2) should the sentence in question be connected to the previous sentence? If the initial conjunction still seems appropriate, use it.
The short answer is never.
It is certainly too informal for formal, academic or technical writing. It is acceptable practice in informal writing to begin a sentence with a coordinating conjunction. I would think that for a friendly letter or a post on your blog, beginning a sentence with "and" or "but" or "so" would be acceptable because the tone on a blog is much less formal.
As a college instructor, I constantly correct and mark down papers for this "academic tone" error.
Marilynn07, I feel bad for your students. You're punishing them for something that is often used in formal, acedemic, and technical writing. Using "and" at the beginning of a sentence is similar to using, say, "additionally" at the beginning of a sentence. And I don't think you'd have a problem with that.
Also, your sources don't support your post at all.