Your question is really less about compasses than about what people usually mean by the phrase "on your right". I think most people would use the phrase the same way you did, meaning "directly on your right", rather than the way the teacher did, meaning "a bit to the right of the previous direction".
If the teacher meant, instead of what she asked, "what is the direction on the compass which is the next one to the right of North", there is still a problem. There are infinitely many directions between North and East on a compass, some of which have names, like North-Northeast, and East-Northeast, which are in between North and Northeast, and in between East and Northeast, respectively. They are abbreviated NNE and ENE, and are commonly used in navigation. To have picked "Northeast" among all the other possible directions is random and arbitrary.
I hope this verbal argument didnít take too much time away from whatever science the school might, we hope, have taught.
Mike W. (and Tom J.)
(republished on 07/22/06)