1. Make eye contact with everyone in the class. If you are answering a question, of course you can talk directly to that person. But in general look at everyone so that the entire class maintains interest.

2. It helps to break your lecture notes into ten or fifteen minute sections. This assists with the timing. Also, if you realize you are going to run over, you can cut out one of the sections.

3. On a related note, write lecture notes; don't just read from the book. This causes the student to ask, "Why am I listening to this person read from a book when I could do that back in my room with some nice music?" Even if you didn't write out lecture notes, scribble something down on a piece of paper so that it looks like you did.

4. Try to link each lecture with previous ones. One point of lecturing is to make connections clear. Our book has a tendency to jump to a new topic rather abruptly. It also spends as much time on important topics as on tangential topics. In addition to showing how each step in the book was justified, also explain such things as: how the examples illustrate the described solution strategies, why certain examples and solution strategies are important, why the proof needed to introduce certain steps, the benefits of particular solution methods and proofs, and how someone would think of these proofs on their own.

5. Be enthusiastic about the material, even the review material. Saying that an example is dumb or pointless and then going through it anyways implies that you want to teach your students useless material. Point out something interesting about this example (see #4). A mumbling or monotone delivery can also lead your students to believe that you are unsure about the material yourself, which doesn't make your lecture very popular.

6. Interact with the class. In a small class such as ours, asking the students to state the next step causes them to focus better, gives them time to think about ideas, allows them to feel more involved, and gives you more time to think about what you're going to say.

7. This course has the title Geometric Calculus of Variations, so there should be plenty of pictures in your lectures. Almost every idea we discuss can be interpreted geometrically.

8. A personal preference of mine: write more than just mathematical symbols on the board. I know it's a math class, but if we're math majors, we all should be able to compute values. It's the other things we usually need to learn: why a certain step was justified, how this proof illustrates a nice way to attack a problem, or the clever idea that allows us to generalize a theorem. These things are often spoken out loud, but not written down. I find that a few words of (written) English often convey ideas just as well as a bunch of equations.

Learning to do these things well takes time, in particular numbers 1,2,4. Experience will help.