Designing the lecture

• Make the lecture have consistent flow. First-year students will have trouble skipping from one topic to another. If they need to know A, B, and C to understand the example, then lay out A, B, and C before starting the example.
• Know the key point you are trying to convey in the lecture, and the most practical and fundamental skills corresponding to that key point.
• Prepare your lecture well, and be ready to answer likely questions.
• Define the topic of the lesson precisely, and set a goal for what you want to explain and teach.
• Rigorous proofs are not always necessary. Simple explanations giving an intuition about the theorem might be more effective than providing all the technical details.
• Have a complete understanding of the material you present.
• Plan examples that illustrate the methods.
• Try to choose great examples, not merely good examples.
• Give examples that are appropriate to the topic you are trying to teach. Focus on typical cases, not pathological cases.
• Plan examples that demonstrate possible pitfalls in the method and how to avoid them.
• Plan specific questions to ask the students during the lecture.
• Remind students of material taught in previous lectures.
• Sacrifice rigor for clarity. It's OK to lie a little, if it conveys an idea more clearly than the truth would.
• Include motivation when possible, and applications when possible. Try to have the applications suit the audience.
• Teach students to do something. Don't just say mathematics in front of them.
• Consider the proper balance between theory (abstract) and examples (concrete).
• Design the lecture for the typical student, not the ideal student or the brightest student.

• Write clearly and with large enough letters, especially labels on graphs and superscripts/subscripts. Make graphs large.
• Plan the global use of the board space: how to divide it up in "panels", what order to use them in, and so on.
• Write more than just equations on the board: write section headings, hypotheses, connective prose, and so on. Use complete sentences.
• Don't overuse abbreviations. Never use an abbreviation for the sole purpose of saving time.
• Be aware that you might be blocking what you have written.
• Practice writing on a blackboard a little - it's different from writing on paper.
• Write out all the steps of a calculation. (At the very least, say "This is a calculation you know how to do using the so-and-so method; the answer is so-and-so.")
• Practice drawing complicated graphs. Practice drawing simple graphs perfectly. Prepare a transparency (and perhaps a handout as well) if a graph is too complicated to draw but necessary for the exposition.
• Write down everything important in your lecture, even if you say it many times.
• Triple-check what you write on the board. Students will often not notice, or feel confident enough to correct, your mistakes.
• If you do make an error, make a big deal about correcting it so that everyone fixes their notes. Cross out, rather than erase, what is wrong.
• Know the room you teach in. Understand which parts of the board (extreme side panels, very bottom) are not visible from all the seats in the room. Learn how to best use any moving chalkboards your room has.
• Use "white space" and subject headings on the board to delineate topics and main points, just like a textbook does.
• Keep additions to previously written sections to a minimum. Don't re-use something you've previously written if the students are going to have to write it out again.

• Speak loudly and clearly. Remember that more than half your class will not be native English speakers.
• Avoid talking in a monotone.
• Use the high-volume end of the voice to emphasize, rather than the low-volume end to de-emphasize.
• Speak out loud everything (both words and mathematics) that you write on the board.
• Pay attention to the pronunciation of technical terms.
• Consider using "dead time" (while writing long sentences, while erasing, while walking across the room) to say things that could help them build intuition or connect different topics together, or even simply reinforce math they already know.

• Consider various methods of generating student response and feedback throughout the course.
• Spend time fully facing the audience.
• Pay attention to the students' reactions. You will see when they are lost, or perhaps when you have made a mistake.
• Make eye contact with all areas of the classroom. Don't favor one side.
• Ask questions regularly to see whether students understand.
• When you want students to answer, give them time to answer. Don't train them to wait for you answer yourself.
• Ask easy questions to keep the students interested, involved, and confident.
• Be respectful to students.
• When soliciting questions, refer to a specific example or proof to spur their thoughts. Ask them sincerely whether they have questions; don't use off-putting phrases such as "No questions, right?" or "Everything's clear?"
• Let the students know you are talking to them, not just talking in front of them.
• Don't be afraid to do something non-essential to recapture students' attention. Tell a joke, mention a historical anecdote, do a magic trick, explain the etymology of the piece of terminlogy you introduced....
• Listen carefully to the student's exact answer. If they're right, try to use the form they used, perhaps indicating if a better form exists. If they're wrong, often you have a great chance to correct a misapprehension.

• Don't be nervous; just feel free to say what you have prepared for the class. Get rid of physical behaviors that will show if you are nervous, such as fidgeting, shuffling your feet, touching your face, and so on.
• Consider starting class with an introduction, motivation, or explanation of the place of the new topic in the general scheme of the class. Not only does it give them a sense of why they should listen, it also gives them a chance to settle down and quiet down.
• When in the classroom, the criterion for whether or not to do something should be "Does this help the students?", rather than "Does this make it easier for me?".
• Don't go so fast that your point is obscured.
• Repeat important points out loud.
• Use notation that is appropriate to the audience and consistent with the textbook. Avoid confusing notation like variables x and X or triple subscripts.
• Your attitude is important. The students can tell whether you are confident and whether you are enthusiastic.
• Practice makes perfect. It takes time and effort to develop good teaching habits.
• Limit the number of important ideas included in one sentence (one paragraph, one example, ...).
• Be realistic about time. Cramming something into too short a time does not help them learn. Give the students some time to think about what you're saying.
• Make sure you're having fun!