Today, we teach calculus to people as early as secondary school. I read that Oxford was granted a charter in 1248, at which point it was awarding specific "degrees" in various subjects. However, Calculus was not even invented until the 1700s.
Given that today, calculus is considered to be the foundation for higher education in mathematics, what would one study in the 1200s if they were pursuing a math degree?
University students in the 13th century would not pursue a degree in math. Students would be taught the trivium first followed by the quadrivium with all classes and coursework conducted in Latin. The only advanced degrees were theology, law, and medicine with the former being considered the most prestigious.
The University of Cambridge has a nifty little summary of the medieval university.
From my understanding, in medieval universities one would study the Trivium and the Quadrivium.
Trivium = Grammar, Logic, Rhetoric
Quadrivium = arithmetic, geometry, music, astronomy
These were mandatory and composed the seven liberal arts.
After the Quadrivium, the student could specialize further.
I know this only partially answers your question, but I hope it helps.