This work is joint with Jason Fulman from USC. It is known, by Zelevinsky for example, that if $\chi$ is any real-valued irreducible complex character of $GL(n,q)$, then $\chi$ is the character of a real representation, that is, the Frobenius-Schur indicator of $\chi$ is 1. It follows that the sum of the degrees of these real-valued characters of $GL(n,q)$ is equal to the number of elements in the group which square to the identity, which we can count. On the other hand, we may use symmetric functions to obtain a generating function for the degree sum for real-valued characters. We use this generating function, along with classical $q$-series identites to obtain a new combinatorial proof that all real-valued characters of $GL(n,q)$ have indicator 1. In the case of the finite unitary group $U(n,q)$, the Frobenius-Schur indicators of its characters in general are unknown. We compute a generating function for the sum of the real character degrees for this group, again using symmetric function theory, and also by applying the results for $GL(n,q)$ and a change of variables $q \mapsto -q$. We obtain a generating function for the sum of the degrees of real-valued characters of $U(n,q)$ which have indicator 1, and one for the sum of those with indicator -1. We are then able to expand these generating functions using symmetric function identities due to Ole Waarnar, and obtain interesting combinatorial expressions for these sums.