Advice on Lecturing

Presentation

• Polls as to whether the audience understands something may be helpful, but bear in mind that people are rarely willing to admit that they do not understand, so those who do not raise their hands are probably confused.
• Students are usually reluctant to ask for clarification, or even to correct typos. Encourage them to do so, and thank them when they do. Asking if there are any questions, particularly at the end of a difficult proof, can be helpful.
• Use blackboards in an intelligent order, raising each after you have finished writing on it.
• Try not to rush at the end of lectures, as this can result in insufficient explanation.
• Do not use powerpoint / pre-written slides.
• Write on the board everything the students are to copy down. Additional jottings should be clearly distinguished.
• It is better to lecture with as little reference to notes as possible. This forces you to work things out as you go along and so puts you more on a level with your audience; you will naturally slow down and explain in more detail the difficult sections. However only do this if you are confident of being able to remember everything correctly.
• At the start of each lecture give a brief statement of the position reached so far. In theory students should read through and understand their notes after each lecture, but in practice few do.
• Once you have written a line of a proof, never erase and replace symbols in the line as part of the deduction, as this is very confusing for students copying down notes. Instead write another line with the new symbols.

Global layout

• At the start of the first lecture give a brief course outline and list of chapter headings. Give a list of recommended books with explanations as to how each is good.
• Number all theorems and examples (and if you wish definitions) with a single numbering system. Then refer back to earlier results you are using (at least until they become standard) by explicit number. Also refer to questions on the examples sheets by their numbers. Phrases such as "by earlier examples" are entirely unhelpful.
• Give numbered chapter headings and (normally) subheadings. There is no need to number sub-sub-headings. Then theorem numbering should use the chapter number, but not the sub-chapter number, as a prefix. E.g. in chapter 4 section 3 theorems / examples might be numbered 4.12, 4.13, 4.14..... They would not be numbered 4.3.1, 4.3.2, 4.3.3

Local layout

• The usual headings are: "Definition", "Theorem/Lemma/Proposition/Corollary", "Application/Example", "Remark"
• Most of the text should be under one of these headings, although sometimes a general discussion is helpful, particularly at the beginning of chapters.
• Define significant terms separately under "Definition" headings rather than in the statements or proofs of theorems that use them.
• Underline words that are being defined
• In general a result that is merely used to prove subsequent theorems will be a lemma: a significant result will be a theorem: and a consequence of a theorem will be a corollary. For example Urysohn's Lemma is a theorem, not a lemma.
• An application is a use of a theorem that is significant in its own right. An example is merely to aid understanding, and so can be largely ignored by students once they have worked out what is going on.
• Where possible state theorems as "Let [conditions] then [results]". Avoid convoluted phrasing such as "Let [conditions] and suppose that [more conditions]. Then [results] provided [even more conditions]". Other acceptable expressions are: "Let [conditions] then [one thing] iff [another thing]" and "Let [conditions] then the following are equivalent: [item 1], [item 2], ...".
• If there are several results in a theorem, each with different preconditions, then write something like "Let [general conditions]: 1. If [specific conditions 1] then [results 1]. 2. If [specific conditions 2] then [results 2]".
• Sketches of proofs or partial proofs should be labelled as such at the start

Logical flow

• Modularise as far as possible. It should be possible to omit any or all proofs when reading the notes and still have them make sense. If you wish to refer back to a result from part way through a previous proof then instead pull it out as a lemma.
• If possible each line of the notes should be understandable and proved by the preceding lines, and should not rely on anything later. A linear flow of reasoning is easier to follow, especially when revising.
• Prove all necessary lemmas before a theorem itself. Do not conclude proofs with "we are done by the following lemma..."
• Where possible, prove theorems immediately after stating them, and explore the implications later.
• Give self-contained proofs for each theorem, rather than giving a general discussion and then stating a theorem which was proved in the discussion.
• If you are using an unusual result from another course, or from nowhere at all, then write it out clearly and state explicitly that it will not be proved. You may wish to give a reference to a text book in which it is proved. The same applies to proofs or parts of proofs that are left to the examples sheets. You should only outsource in this way proofs that the students will be readily able to do, and the whole concept calls for caution.
• For complex chapters it can be helpful to draw a diagram with a dot for each lemma / theorem and arrows between them to indicate dependance. This will help you to understand the flow of reasoning and hence decide on an appropriate ordering.

Examples sheets

• Questions should increase in difficulty from straightforward checks of understanding to the very difficult (which should be starred)
• Around 12 questions, or slightly more for higher years, is optimal
• Avoid adding large numbers of questions whose method of solution has little to do with the course, as they are likely to distract from questions that really do matter.
• Try to target questions at common areas of misunderstanding.
• Release examples sheets, especially the first one, as early as possible

Misc

• Draw attention to difficult sections. Never attempt to skate over them.
• Avoid re-defining the same word in subsequent chapters. If you are compelled to do so by standard naming conventions then take extreme care to distinguish the two definitions.
• Try to use notation that will minimise the risk of different symbols being confused with each other. Even a confusion that is obviously ridiculous to the lecturer may be made by a student who has little understanding.
• A great responsibility rests upon the lecturer to ensure that their proofs, especially difficult ones, are correct.
• Explain the meanings of results, and why you are doing things in a particular way. A mass of logically valid but unmotivated work is very difficult to follow.
• Printed notes have some uses, particularly for courses that have a lot of algebraic manipulation. However, they often discourage students from coming to lectures or from taking notes in them. Students will always be able to borrow notes from friends so printed notes are not necessary to ensure everyone ends up with access to the material. Notes giving only definitions and statements of theorems without proofs may be a good compromise.

Meta

• The rate at which attendance falls throughout the term (after the first few lectures) is a good indication of the quality of the lecturing.
• Suggestions to the lecturer about his lecturing style, for example the request to speak more slowly, are only made rarely and when there is general consensus; so they should be taken very seriously.