You're kind of right, in a certain sense. The problem is that in relativity you need to be very careful about what you mean when you say "observe".
If you were in a rocket ship traveling away from Earth at half the speed of light, and you looked back at the Earth, then even without time dilation you would see events happening in slow motion because the light takes longer to get to you the further away you are. This is basically the same as the doppler effect. If there were no time dilation, then for every two seconds of your time you would see one second of Earth time. Taking time dilation into account, you see one second of Earth time every 1.73 seconds of your time.
On the return trip (traveling towards Earth at half the speed of light) then if there were no time dilation you would see events speeded up: for every two seconds of your time you would see three seconds happening on Earth. Taking time dilation into account, you see three seconds of Earth time happening in 1.73 seconds of your time.
If you didn't know about relativity, so that you only took the doppler effect into account, it would seem to you that events on Earth were taking place about 16% faster than they ought to. So in that sense you would see time traveling more quickly on Earth.
[Critics are asked to draw the relevant diagrams for themselves before saying that I'm wrong about this; it's surprising, but if you think about it carefully it has to be true. Consider someone on the rocket ship receiving "ticks" from a clock on Earth; over the round trip, you have to receive the right number of ticks for the amount of time that passed on Earth, i.e., you receive more ticks over the round trip from the clock on Earth than from your own clock.]
[Addendum: the key point here, which I overlooked previously, is that the naive observer will do his calculations differently depending on whether he thinks that he is moving or that the Earth is; this both explains the asymmetry in the apparent time dilation and makes it clear why it isn't a good description of reality.]
On the other hand, if you were traveling at half the speed of light in a very big circle around the Earth you would see events on Earth taking place 16% slower than you would expect. So the apparent time dilation (as measured by a naive observer on the rocket ship) varies depending on your relative direction as well as your relative speed.
When doing special relativity properly, we imagine a network of observers all traveling at the same speed but in different places. This can be (somewhat improperly) simplified by supposing that the observer on the rocket ship has some way of seeing what was happening on Earth "right now" without being limited by the speed of light. Technical note: when I say "right now" I mean "right now in the observer's frame of reference".
In this hypothetical case the observer would see things very differently to the naive observer described previously. On both the outgoing trip and the return trip (and also on a circular trip) events on Earth would be seen taking place 16% slower than normal. This is the behaviour your teacher (and John) is describing; an observer on the rocket ship and an observer on Earth both see each other in slow motion. This time dilation depends only on the relative speed, not on the direction of travel, and is real in a sense that the apparent time dilation I described earlier is not.
However, whenever the rocket ship accelerated, something odd would happen.
Suppose you were traveling away from Earth at half the speed of light for a year, took one day to turn around, and then traveled back, also at half the speed of light for a year. As far as you're concerned you would be away for two years. During the first year, on the way out, you would see 10.4 months take place on Earth. During the second year, on the way back, you would also see 10.4 months take place on Earth. During the turnaround, though, in a single day of your time you would see 6.9 months take place on Earth, making a total of 27.7 months. At the end of the trip you've experienced 16% less time than the Earth has, but (from your point of view) it all happened when you accelerated; the rest of the time it was the Earth, not you, that was in slow motion.
From the point of view of an observer on the Earth, mind you, you were in 16% slow motion the entire time; you were in very slow motion while you were accelerating, but because that was only for a short time it didn't contribute significantly to the overall time dilation. This is hard to get your head around, but that's reality for you. :-)
[If you don't happen to have a faster-than-light telescope, you can replace the above with a thought experiment involving a network of observers traveling in tandem with the rocket ship, all going the same speed and accelerating at the same time (in their mutual frame of reference); later on, you collect notes from all the observers recording when (in their frame of reference) they passed the Earth and what Earth date it was at the time.]
