The difference between MA and EdM can be found on the TC FAQ website. I don't believe, by their words, that you must have graduate credits to start the EdM program. The length of the programs do differ, though.
How do the Master of Arts (M.A.) and the Master of Education (Ed.M.) degrees differ?
The Master of Arts requires the completion of a minimum of 30 points. The Master of Education requires a minimum of 60 points to complete the degree.
The Ed.M. is an advanced Master's degree and applicants will often have professional experience before applying.
How long will it take to complete a degree at TC?
Length of time to complete a degree varies by program, degree, credit requirement, and whether or not students are pursuing their degrees on a part-time or full-time basis and have field work requirements. In general, with a full-time course load, a 32 credit M.A. program can be completed in 1 calendar year and the 60 credit Ed.M. in 2 calendar years. The length of time to complete a doctoral program can range from 4-7 years. Factors determining length of study of a doctoral program include: the number of previous graduate credits transferred into the program and the nature of student's research and dissertation.
Here is the link: http://www.tc.columbia.edu/admissions/index.asp?Id=Info&Info=FAQ I applied for the EdM, too. So, there shouldn't be any worries.