Initially Gran Torino appeared to be a senile old man who constantly forgot and asked Izuku's name, often calling him "Toshinori". Even though he was the one who sent Izuku the internship offer, he would fail to recall Izuku's name for more than a moment. In reality, this proves to be a nothing more than a facade for messing with his newfound protege. 

Gran Torino is actually a very intelligent man with incredibly broad perception. He proved to be a far more capable teacher to Izuku in a mere week then All Might did in almost a year. He was able to read Izuku's progress and personality just by watching the U.A. Sports Festival and indirectly taught him the first step to mastering One For All with a simple parallel between Izuku and his favorite dessert. His sharp mind is also evidenced when he was able to accurately predict All For One's strategy based solely on the media covering the aftermath of Hero Killer Stain.