There has been significant recent interest in studying how the number of numerical semigroups of genus $g$ behaves as a function of $g$. Bras-Amorós has shown how to organize the collection of numerical semigroups of genus $g$ into a rooted tree called the ordinarization tree. The ordinarization number of a numerical semigroup $S$ is the length of the path from $S$ back to the root of the tree. We study the problem of counting numerical semigroups of genus $g$ with a fixed ordinarization number $r$. We show how this can be interpreted as a counting problem about integer points in a certain rational polyhedral cone and use ideas from Ehrhart theory to study this problem. We give a formula for the number of numerical semigroups of genus $g$ and ordinarization number $2$, building on the corresponding result of Bras-Amorós for ordinarization number $1$. We show that the ordinarization number of a numerical semigroup generated by two elements is equal to the number of integer points in a certain right triangle with rational vertices. We consider the analogous problem for supersymmetric numerical semigroups with more generators. We also study ordinarization numbers of numerical semigroups generated by an interval.