No, you either misunderstood your professor or he was wrong.
It is entirely practical for a router to connect networks of different classes, and indeed this is BY FAR the most common setup for a router. (Even your home adsl/ethernet router will connect a CLASS C to a /30). It is not uncommon in larger organisations for a single organisation to use routers to connect (for example 10.0.0.0/8) a Class A address with (192.168.0.0/16) a Class B address, while offering a.b.c.x - a Class C address through a DMZ for servers.
Moreover, the whole design of IP addresses is to allow breakdown and aggregation of addresses to reduce load on routing tables - and routing tables are typically large lists of directories stored in routers showing how to reach Class A,B,C and subsets of these through the router.