**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.