Commas make a lot of people bonkers. Don't feel bad.
The comma is required, but the colon is incorrect, as is the hyphen in social studies. I would also use a period instead of semi-colon, and would say things are founded ON, not BY, and I don't think you need "to learn." Lastly, those subjects ARE education, so it seems odd to say that you're founding education on something that is education.
You can also just say "... an array of subjects, including math, reading, science, social studies, and writing."
So my version would be:
While in placement, I have observed Mrs. McCauley teaching an array of subjects, including math, reading, science, social studies and writing. These subjects are critical for students because they are the basis on which all future education is founded.