I don't understand how the assessment of the major can also serve as an assessment of general education. Aren't we assessing very different competencies?
To a great extent, yes, what we expect students to learn from the major program is different from what we expect them to learn from general education. However, we can find ways to assess general education learning outcomes within the major by choosing to assess those learning outcomes that are of most value to the major. For example, do we want students to be able to write well within the students' disciplines. If so, we can use the writing projects that they do for the major to assess student writing ability.
But, what has that to do with general education? Well, it can help inform our general education writing program. If students still have major weaknesses in their writing, even after they are in the major, perhaps the writing program can be modified to address those weaknesses. Or the faculty may decide upon a different solution or solutions. Some institutions, for example, have implemented a junior or senior level writing course for each major. Some have devised "writing intensive" courses within the majors. Some have writing-across-the curriculum or writing-across the disciplines workshops for faculty who want to improve students' abilities to write. Although such solutions are not part of the general education program, the assessment of a general education competency (in this case, writing) has led to the program changes.
NOTE: it may not be worthwhile trying to assess all general education competencies in all the majors, for some will be more pertinent to the major than others.
Please see the PowerPoint presentation on "alternative assessment" for ways that assessment can be implemented within the major.