jmc

The program must contain exactly $3$ of the $5$ classes that are not English. Therefore, there are $\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$ valid programs if we ignore the mathematics requirement.

Since there are $2$ math classes (Algebra and Geometry), $3$ of the $5$ classes besides English are not math. Therefore, there is only one program that satisfies the English requirement, but does not satisfy the mathematics requirement (the program consisting of English, History, Art, and Latin). It follows that the number of programs satisfying both requirements is $10-1=\boxed{9}.$