What is confusing you is the thermodynamic concept of equilibrium. A system at equilibrium has no external source of energy flowing onto it and is perfectly mixed internally. These are not common circumstances in the real world.
In a system that is in equilibrium, a solid and its corresponding liquid can exist at one temperature. Any extra energy added will, when equilibrium is restored, serve to convert the solid to liquid (this takes energy). Since, at equilibrium, we have perfect mixing this will not result in a temperature change. For water it takes a lot of energy to convert ice to liquid water.
Of course, in the real world, nothing is at equilibrium and there is always a flow of energy in or out of the system. And for an isolated system at equilibrium, when energy is added there will be a period where parts of the system will be hotter than others. But, since being at equilibrium means the system will be perfectly mixed, this is temporary and the system will regain equilibrium with a constant temperature but a different mix of water and ice.
So you professor is right if he specifies that the system is in equilibrium but you are right in the real world for most systems. The idea of equilibrium is a useful idea for understanding the limiting behaviour systems if a little unrealistic as a description of the real world.