Basically, what is happening when your instructor blips the throttle is that he is acutally getting the engine to stop for an instant, since it doesn't like to run backwards and especially if you have mixture/prime "problems" as others have commented on, and on the prop's last breath bumps against the compression swings the other way and restarts in the other direction. I've seen some actually slow the prop down by applying friction on the spinner.

I also prefer a starter but when I forget it, I normally flip clockwise to bump it off of compression and the engine will start in the correct direction. So, you might try flipping in the opposite direction, as counter-intuitive as it may seem.

Good luck!