How do you go about writing your own lecture notes for a new course?

Question

In advanced graduate courses, there are often no textbooks on the material being covered, and sometimes no good introductory material at all. Even when there are textbooks, many professors prefer to write their own notes on the subject.

As a fairly new instructor, the idea of creating a course from scratch seems intimidating. Fortunately, I don't have to do it.

My question has two parts:

What are the most important considerations when writing your own lecture notes for a course? What is the motivation for writing your own notes in a course where textbooks are available?

If this is too broad, let me know and I can degeneralize it.

Answer 1

What are the most important considerations when writing your own lecture notes for a course?

I would say that the single most important consideration is to make life easy for yourself. It is easy to fill your time with writing lecture notes but that isn't always the best use of that time. Nevertheless, it can be useful for the students so here are my tips.

1. Make sure you'll be teaching the course several times.

   It's likely that the notes won't come together until after the end of the course, so you should be thinking a bit that you're writing to help the next lot of students. To ensure that this isn't unnecessary work, make sure that the next lot of students are your students as well.

2. Write as you go along.

   You can never plan everything from the start, particularly at the detail needed for notes. Writing as you go along means that you are writing it knowing what the current lot of students need, so your efforts will be concentrated on where they are most needed.

3. Write as little as possible.

   It's tempting to write as much as possible, but as others have already said this is a temptation to be resisted. Write what the students need right now. If you get a moment to add more later, fine, but don't let that be your top priority.

4. Add value.

   If there are textbooks around, but they aren't really usable as a book for this course, think about how to make your notes be the bridge between the course and the book rather than replacing the book. Even a simple glossary can be of immense value, or a "look up list" of where students can find particular topics (useful if the book is rather large compared to what you're teaching).

5. Use a wiki.

   This is a practical tip. Write your notes on a wiki (preferably a mathematically-enabled one). The advantages of this are:

   1. Psychologically, it's easier to write a short, incomplete wiki page than a, say, PDF document. So writing on a wiki means that you get a little bit up quickly and then gradually improve it.

   2. Hyperlinking your notes is an immense added value. Just imagine a theorem in which all the key terms are hyperlinked to their definitions.

   3. Student's can fix typos themselves instead of continually barraging you with emails.

   4. The notes no longer need to be linear. You can add optional extras, such as a page where you go into greater detail for those interested.

What is the motivation for writing your own notes in a course where textbooks are available?

I wrote notes for a course which was a "cut and shut" course. When I took it over, it used three textbooks (one only a chapter) and only one of those was actually any good. Although I found better replacements, I still felt unhappy about asking students to buy three textbooks for one course (as partial mitigation, I found free online textbooks for some parts). Also, things didn't go smoothly between the books. So I wrote supplemental notes to help bridge the gap between the lectures and the books, and in some places to replace the books altogether.

You can find my notes online on my course wiki.

Answer 2

A motivation for writing course notes, when textbooks/monographs already exist, is to omit some things, first. This might seem strange, until one observes that many standard texts which go through several editions often succumb to the natural pressure, amplified by publishers, to become ever-more encyclopedic. The latter has its uses as reference, but perhaps not so much as a guide to the actual threads of a live course. Further, _making_choices_ of what to pay attention to in the course, as opposed to what to ignore (in the course), imparts considerable information to the students.

As positive side-effect of omitting some iconic-but-not-immediately-essential things, sometimes there is room to include newer or novel things that otherwise wouldn't find a spot, or happen not to be included in the standard texts.

Of course, one's own notes can more accurately support and/or complement one's own lectures and plot-line for a course, so one could argue that picking a standard text and just indicating a subset of it for students could suffice. Even if that is the case, people will see all the other stuff and wonder why you're ignoring it. Intentional, or accidental? Some distractions.

And self-generated notes can be made freely available, and available in electronic formats, and...

As to "important considerations", ... certainly facility with an adequate typesetting software system comes first, or it's hopeless. For a first pass through, don't keep everything in one big file, both because it takes longer to do the inevitable frequent re-typesettings, and because it's hard to find things. Use your computer's file system, etc., to organize things. Keep backups. Some form of "version control" can be useful.

About content: especially if you're trying to lighten the burden of encyclopedic texts, don't fall into the (sometimes appealing) trap of "completeness", or of giving all the iconic examples, or perhaps even working out all details of all proofs of all small things. Instead of contriving artificial exercises, if you've given representative prototypes of relevant arguments, subsequent similar arguments can be left to the reader. Of course, one should be honest with the reader, rather than merely lazy, and be honest with oneself about why something's omitted. (Not to mention competent to confirm that it really is just "more of the same" and routine.)

A different sort of point: write at a level and in a style to address the students you actually have, rather than ones you wished you had, or some idealized students. E.g., don't omit arguments or explanations that you know perfectly well most of your students couldn't fill in, don't couch things in terms you know are more sophisticated than your audience, ...

Coming to some questions of taste: the pressures for traditionally published texts to be fairly formal are not entirely helpful to students or mathematics. That is, there is stylistic pressure to be formal, and to follow a certain (quite recent, modern!) style of definition-lemma-proof-lemma-...-theorem-proof... exercises. Candidly, one often discovers that the exercises are cryptic exhibition of the formative examples that motivated the whole business, but the natural causal sequence has been inverted.

Now, yes, a certain amount of that inversion can be helpful, clarifying, but the standard full-blown version is not. (To name-drop) Gelfand was a consistent advocate of giving the key examples so that "definitions and theorems and proofs" become almost afterthoughts. E.g., definitions should be "justified" by examples...

Especially in fairly sophisticated mathematics, terminology is heavy, usage is inconsistent or slightly abused, etc., and trying to "repair" the situation by cranking up the formality is attractive, but often doomed or at least incredibly expensive. Instead, giving the watershed examples, rather than trying to make "rules" or rigidify language, seems to me to more directly address the genuine issues.

And be prepared to defend (on scientific and pedagogical grounds) any deviation from the iconic sources, because it seems that people are quite happy to believe in the authority of iconic authors' writings. Many people in the math business do seem to believe that definitions given in iconic texts are "sacred", too, and are beyond discussion, so don't be surprised when people are inflexible in this regard! Have explanations ready... Yes, some of your energy will be spent defending your choices to students, and this can be made mostly productive, if you are prepared.

Answer 3

What are the most important considerations when writing your own lecture notes for a course?

My aim is to pick a good path through the syllabus.

• It must be logically sound and well-motivated.

• Important examples and algorithms should go near the start.

• Ideally there will be a running example that can illustrate several different ideas.

• It will fit well enough with previous years' problem sheets that most questions can be asked again.

• Most students are going to get lost at some point: I try to postpone this point as far as possible, even if it means some compression of harder material at the end of the course.

I then write outline notes that have complete definitions and statements of theorems, but only a few examples, and usually no proofs. Then in lectures I can spend time on short questions, examples, and proofs. This helps keep the lectures fresh and interesting.

For an example see my Spring 2013 Notes on Error Correcting Codes.

What is the motivation for writing your own notes in a course where textbooks are available?

1. It helps me think about the material. Writing outline notes is time-consuming, but it makes certain that I have thought hard about all of the material.

2. It saves time, in the long run. Having the big picture fixed reduces the amount of preparation needed for each lecture, sometimes to almost nothing. When preparing I am free to concentrate on small scale problems in particular examples / proofs.

3. It makes sure the syllabus is covered in an efficient way. Textbooks usually cover more material than is needed, and there is the problem of finding good books at the right level. If multiple textbooks are required they will probably have competing conventions, which have to be reconciled.

There is also the less respectable reason that, in the UK system at least, students expect to be able to rely solely on their lecture notes and problem sheets issued by the lecturer, and are unprepared to learn from textbooks. Of course this is a bad thing, but it is a long and hard fight to change.

Answer 4

One important thing is to think ahead about the content. Clarify the goals, both in terms of target theorems and of methods. Try to have a nice goal for the end of the course, and to clarify what you want your students to master, to what level of details, and for each point what you want to show and what you want them to do by themselves. Writing a detailed syllabus should makes you do all that.

About how to proceed without being too intimidated: I usually like to write the title of chapters, then write the title of sections, then write the title of subsections, and only then starting to go into the real writing. Of course, the titles may change during the writing, but then at any time you're not writing a whole book, only a subsection of it. Much less intimidating. Writing all titles first helps you knowing where you are headed at any time.

About the motivation to write your own lecture notes when textbook are available is to learn what you will teach. Even if you know it well, writing it makes you understand it much better because you are forced to go into the details very thoroughly. But this motivation is especially important when you choose to teach an advanced course on a topic you do not master yet, which is probably the most efficient way to learn.

Answer 5

Mine started out as a student (who recognizes having a horrible handwriting) got into the habit of transcribing my scribblings on the blackboard to LaTeX. He shared them with me via git, I fixed up his copying mistakes (and my much more frequent glitches on the blackboard), completed derivations, added examples, ... and published the result for the class. It sort of grew from there, adding new material as the course's contents changed from term to term, filling out with ancilliary matter not directly used but which was needed to check coherence, and ocassional interesting sidetracks.

The reason for starting this was that the syllabus was a coherent collection of themes, but no book covering them decently could be found.

It has turned out a very long project, running (on and off) for a few years now, with no end in sight...

I got much help from my students, which get extra credits for pointing out mistakes, streamlining proofs (or directly making them understandable), or suggestions like new examples. Often I gave out extra homework to read and criticize the sections covered in class (or sometimes sidetracks), that helped them learning the material and got me high-quality feedback.

Answer 6

I have built into my course requirements the expectation that students will, on a rotating basis, neatly rewrite/retype their notes and upload them to a class Wiki or discussion forum, and that other students will add comments and/or questions about those notes. This serves multiple purposes:

• It forces graduate students to learn how to produce neatly formatted mathematical formulas;
• It ensures that if a student is absent for a legitimate purpose then he or she is able to find out what was missed;
• It gives me a way to keep track of improvised departures from my plans;
• It gives me an opportunity to add after-the-fact corrections, clarifications, or better examples;
• It gives me some idea of what is being correctly understood or misunderstood by my students;
• and (what is relevant for this question) it yields a reasonably complete rough draft of a set of lecture notes at the end of the semester, which I can then reuse, edit, revise, or put to another purpose.