мне кажется, предлог at

да, тут at. и я думаю he will start teach

можно и start to teach и start teaching

спасибо большое

Заменил предлог. Вот ещё справка:

In academic contexts, use at when referring to the institution in general or to a specific one. Note, however, that one would use in for graduate school or college (largely American context). The construction in university is also correct (largely British);
I should point out that in college and in university are rather colloquial constructions, but that's why they're used in everyday situations! In more formal contexts, it would be better to use at college/ at university.
https://english.stackexchange.com/questions/7382/studying-phd-at-the-university-or-studying-phd-in-the-university