That's a great question, Joan. Art class
is being used as the name of this course, and we don't normally use the definite article with names. That's why I wouldn't say "the Joan," at least not in this context. The names for all basic classes are without the definite article: history, geography, typing, art, ceramics, math,
By the way, in your example art classes
, notice that classes
can be singular or plural, but not art
since it's the descriptive element of this noun adjunct.