Why do we use his and him but not her and herm?

I just thought about this the other day. In the sentences below, the word "her", takes on two meanings. The words "him" and "his" are needed to convey the same two meanings (same with they, them, and their). The female version is less verbose. I am just curious about this construct.

My wife teaches her math skills. (she teaches the baby math)
My wife demonstrates her math skills. (sh shows off her skills)

My husband teaches him math skills. (he teaches the baby math)
My husband demonstrates his math skills. (he shows off his math skills)

They teach them math skills. (they teach the kids maths skills)
They demonstrate their math skills. (they show off their math skills)