In Medieval thought and teaching theory, the seven liberal arts are
Rhetoric,
Logic, and
Grammar (the Trivium), and
Music,
Arithmetic,
Geometry and
Astronomy (the Quadrivium), being considered beginning and intermediate courses of study respectively.
Philosophy was considered to be the source of all learning, and a fitting introduction to
Theology, the crown and end-point to medieval studies.
Surprising though it may seem to some contemporary readers, most of the beginning material was taught using pagan authors, who were acknowleged as such: the first book usually given to read was the Distiches of Cato, a selection of Roman moral maxims, which were considered the best introduction to Latin vocabulary. These were learned by repeating after the teacher (the European sphere of influence didn't discover until the late 20th century that it's easier to learn by repeating with the teacher, as had been done for centuries in Asia). Rhetoric, which meant "composition" (not "lying" as it does now) also included the basics of civil law and the proper forms of debate, and was taught from Cicero and example. Logic, which also included the proper definition of words, was partially taught from Aristotle's Categories and On Interpretation, and partially from a forgotten book called the Isagoges of Porphyry, an author who ironically enough, wrote a notable book called "Against the Christians"! Grammar, meaning Latin grammar was taught from an author called Donatus, which is still pretty good if you want to learn Latin.
Arithmetic included number theory, and was as involved with studying primes, squares, triangles, and other geometrical figures in the numerical world as the basics of addition and subtraction -- due to the cumbersome nature of the Roman number system and the lack of any cheap material to write on (remember that parchment is leather) most calculation was done on a "sand table" or a form of abacus in which the counters were laid on parallel lines drawn on any convenient flat surface. Music involved mainly the theory of modes, scales, harmonics, and the like, and was taught with the help of a movable-bridged instrument called a monochord. Geometry was pretty much the way it is now, and focussed on proofs. Astronomy was from Ptolemy, geocentric, of course, and included navigation and astrological data as well -- while earthly maps, which had been getting more accurate in the late Roman period, degenerated into the T-O pattern star maps were more-or-less accurate.
With these, and a little more, in the early Middle Ages, you could start your own college, and many did just that!