I will be a postdoc in the fall and will be teaching my very first classes aimed at graduate students. One will be an intro class, and the other a topics class.
There are of course many differences between undergraduate and graduate courses. I'm interested in knowing how to take these into account when planning for and teaching a graduate -level course.
Of course there are many differences between undergraduates and graduates, including:
Mathematical independence. In a graduate class, one may generally presume a much greater level of mathematical independence and self-motivation than in an undergraduate class.
Mathematical sophistication. Graduate students generally are more knowledgeable than undergraduates and will benefit from more sophisticated explanations. Graduate students are more likely to appreciate it when the instructor follows sideline topics in class for a bit, or unifies topics with ideas from other areas.
Capacity to be challenged. Graduate students have a far larger capacity to be challenged mathematically. One can give mathematical puzzles, and not be surprised to find that students struggled with them for hours and hours over the weekend.
Because of these differences, I often organize my graduate classes in a much different manner. For example, in order to introduce graduate students to the math research experience, I usually require my graduate students to write a term paper (see my account of this here), which I then collect into a "Proceedings of Graduate Set Theory" volume at the end of the semester and have a session devoted to student talks. (With undergraduates, in my experience, this doesn't work as well.) Also, in lectures I am willing to follow tangents in the topics that might come up for a longer time and more thoroughly than I do in my undergraduate classes, since in the undergraduate classes I feel more compelled to stay closer to the established syllabus.
One difference that is often overlooked is the availability (or non-availability) of mathematical vocabulary. When teaching an advanced course, I can talk essentially as I would talk to a colleague (or to myself). In a lower-level course, I have to watch my language more carefully to avoid using terms that are second nature to me (and to other mathematicians) but unknown to the students.
Some years ago, I was assigned to visit the first-year calculus class being taught by a new junior faculty member, in order to assess the quality of his teaching and, if necessary, to make suggestions for improvement. He did a great job of teaching except for one thing: He used the phrase "necessary and sufficient condition" several times (and pronounced it as if it were a single word). My one suggestion for improvement was to point out that a typical calculus student would, at best, understand that phrase only with some reflection (and at worst would not understand it at all) --- so it is necessary (but not sufficient) to bear in mind the vocabulary that the students are familiar with.
The difference between undergraduate and graduate is a matter of degree, not a clear cut. The difference between first and third year students are as large (or larger) than undergraduates and graduates.
If anything, graduate classes are easier, the students take the courses because they are interested, not because they have to take it as part of the curriculum. You can tell people to study or find out stuff by themselves, you don't have to give them everything predigested. But they are harder, because you have to cover more material, and the material isn't readily available, is much less ordered and worked over, cleaned up and has not yet found its "optimal" exposition.
