Education 65.04 (or any other 65.xx number) is a required course for the New York State approved and registered program for provisional state teacher certification in secondary education. It includes both seminar work on science teaching methods and curriculum and also student teaching experience. It is an integrated course and there is one grade given for all work.
Education 67.04 (or any other 67.xx number) is similar to Education 65.04, except that it does not include the student teaching experience (and so carries fewer credits). It is normally taken by experienced teachers who are changing fields from some other area to science education, and by students who have never had a science teaching methods course that is required for matriculation to graduate programs or as a pre-requisite to other work in science education.
Education 613.1 is a graduate-level version of Education 65.04. The same topics are covered, but students are expected to master them in more depth and do more supplementary reading and provide more comprehensive responses to assignments. Normally 613.1 and 613.2 are to be taken in the same semester. 613.1 is the seminar portion of the course, 613.2 is the student teaching experience. Grades for each course are separate.
Education 613.2 is the student teaching experience component of the 613.1/613.2 combination. It is numbered separately to provide for separate grades in the two courses and to satisfy some state requirements for explicit evidence of that the student teaching requirement has been met, e.g. when not taken as part of a state-registered program.