You're not looking at it in the right way. The key here is the law of conservation of energy (Thanks Ethereal_Dragon). Energy is never created or destroyed, simply changed from one state to another. So, let's say for example you have a 3.6GHz P4 running at 100% load, that things going to probably produce >100W of heat energy, which will transfer from the processor core to whatever is adjacent to it, be it air (which isn't a very good conductor of heat, so the processor would quickly overheat and either shut itself down or burn out), or a stock aluminium cooler (much better at conducting heat, so takes the heat from the cpu very efficiently), or a zalman crazy copper mega HSF (copper is better at conducting heat than aluminium, so is better again. Silver it the most efficient heat conductor, but it's too expensive to be commercially viable and copper isn't far behind it in the thermal conductivity stakes). No matter what heatsink exists, the CPU STILL produces 100W of heat, and that is transferred away from it into your room!
So, we can look at this whole process in three stages: CPU produces heat - heat transferred to heatsink - heat leaves heatsink and transferred to air inside case.
The Overall CPU temperature is governed by two things:
1. How efficiently the Heatsink (and thermal compound, which is most important, but we won't go into that now!) removes heat from the processor, and
2. How efficiently the heatsink transfers said heat to the air.
The same amount of heat is still being transferred to the air inside your case, which by definition, is also the air inside your room.
What I'm basically saying is this: no matter what heatsink you put on that processor, it witt not affect your room temp because the heat energy is still being trapped in the room. To lower room temps with your PC running you have to either crank the AC up, get a more powerful AC unit, or open a window.
Your teacher's a mule. You will never prove that putting a better HSF on your processor will reduce the amount of heat actually PRODUCED, because it's simply not true!
Once again, the heat is still being produced, just moved away more quickly and efficiently!