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Background: I am a junior university staff member, and I will teach an introductory course in linear algebra for the first time in a few months.

I see that many teachers devote time to writing lecture notes. To me, this looks odd. Whatever I can write will not be half as good as a well-written textbook. so my first choice would be looking for a good book, recommend it to the students, and stick with it as much as possible for my lectures. "Everyone writes their own notes" looks like a model in which there is a lot of needless duplication of work.

Why do people write detailed lecture notes for basic courses? What is the advantage with respect to following closely a textbook?

Are the two approaches really alternative, or am I misunderstanding the role of lecture notes? Should I do both?

  • 21
    Lecture notes can be an opportunity for the instructor to organize and clarify the material in his or her own mind. – Jim Conant Sep 18 '14 at 16:10
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    Whatever I can write will not be half as good as a well-written textbook. — [citation needed] – JeffE Sep 19 '14 at 3:01
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    If a teacher were to teach nothing more than what is in a textbook, why would anyone bother taking the class? it would be much cheaper to just buy the book. – user2813274 Sep 19 '14 at 3:32
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    @JeffE That statement must hold true at least for the majority of the lecturers, by definition of "well-written textbook" (and since I am teaching this course for the first time, I rate to be in the lower half, when it comes to teaching skills). Of course, though, Dunning–Kruger applies. – Federico Poloni Sep 19 '14 at 6:29
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    Of course, though, Dunning–Kruger applies. — So does Impostor Syndrome. – JeffE Sep 19 '14 at 17:58

12 Answers 12

37

As someone who's tried to do both, there are some very valid reasons to prepare "formal" lecture notes.

The primary reason why you'd want to create your own notes is that for many courses, a single good text is not available, and as a result, the instructor has to cobble together material from a number of different sources to produce a coherent set of lecture notes—or recommend that students work with multiple source texts. (Given the out-of-control nature of textbook prices, the latter alternative is unlikely to work out well.)

If you have a single-text class, it may not be necessary to provide students an additional set of notes, provided your lectures stick to the main text material. However, if you bring in alternative or additional topics into your lectures, you may want to include notes for those topics, and refer students to the textbook for places where you follow the "standard" outline.

  • I would imagine that you can benefit from the lecture notes. When teaching the same class over the years all you need to do to prepare for a lecture is to read your notes. – afaust Sep 18 '14 at 16:09
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    @afaust That is a good point, but preparing quick notes for my own use is very different from putting together release-quality material that I can give to the students. – Federico Poloni Sep 18 '14 at 16:53
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    @FedericoPoloni For some people there might not be that much of a difference though. Especially after many years of teaching a class and tweaking/adding relevant material to the notes. That's probably how the first drafts of a lot of textbooks started. – Roger Fan Sep 18 '14 at 19:48
32

As you are teaching linear algebra, I will use it as an example. Gilbert Strang teaches/taught linear algebra at MIT. He also wrote the textbook Introduction to Linear Algebra. You might expect the course to follow along perfectly with the textbook. If you look at the syllabus from when Strang was teaching the course or now what you will see is there are significant deviations. It goes section: 3.6, 8.2, 4.1-4.4, 8.5, 5.2-5.3, 6.1-6.2, 6.6, 8.3, 6.3.

If the author of the textbook cannot even happily follow his own ordering of the topic, it is not surprising that many teachers feel the need to create their own notes that go in the order and cover the material in the depth that they want.

  • 1
    I have taken a closer look at Strang's ILA and this answer, and I must say I now disagree with your example. If you check your list of numbers, you'll see that the only significant deviation is the position of the sections of Chapter 8, which are interspersed with the rest of the material, essentially in the same order as the book. Chapter 8 contains applications of linear algebra, so essentially motivated examples. So it seems to me that the structure of the course stayed basically the same, apart from these examples. – Federico Poloni Sep 28 '14 at 9:43
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    @FedericoPoloni but chapter 8 is not inserted in a ordered way. Further, even removing the chapter 8 insertions, chapter 6 is taught out of order (6.2, 6.6, 6.3), and at the end of the course chapter 10 is taught before chapter 9. I also think the fact that he chose to write the book with an applications chapter, bought taught it interspersed is the issue that people face when choosing textbooks and designing a syllabus. – StrongBad Sep 28 '14 at 10:33
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I have written lecture notes for a couple of courses, and in one of those cases I assigned a textbook as well. Here are some of my reasons.

  • For a course in which I have some flexibility as to the content, I may find that no single textbook includes all the material I want to teach. Asking students to buy three or four books is rather obnoxious, especially if for some of them I will only be referring to a few pages. When I write lecture notes, I can include exactly the material that I want to include in the course.

  • Even when I am generally following a textbook, I often find places where I want to go off on a tangent, or discuss additional related material, or maybe just cover the same material with a different treatment. I feel like me teaching a class should add value beyond the student reading the textbook, and one way to do that is to tell students how I personally think about and understand the material in question. It can be helpful for the student to have that in writing. In part this is probably arrogance, but I really do feel I have insights to offer that are not contained, or not as well expressed, in even the most "well-written" textbook.

  • If I am going to need fairly detailed and precise notes to lecture, I might as well type them - they'll be neater, easier to read, and I can refer to them next time I teach the course. If I'm going to that trouble, I might as well make them available to the students.

  • I find that writing lecture notes for an audience other than myself is a really effective way to teach myself material, and understand it at a deeper level. It very often leads me to new insights on something that I thought I understood.

  • Written lecture notes that are posted on a website can be helpful to anyone in the world, not just the students in my course. I've been able to answer questions on MathOverflow and Math.SE by pointing people at my lecture notes.

  • For high-level courses (especially graduate topics courses), there may not be any textbook on the relevant material - I am assembling it from the research literature. But in order to use material from a research paper in a course, I usually have to rewrite a lot of it - filling in background and omitted details, and so on. So it becomes lecture notes.

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    Indeed, and especially points 3 and 4. I would be very nervous of teaching anything, at any level, where I hadn't done point 4; and having done that, I might as well do point 3, too. – Norman Gray Feb 24 '15 at 10:27
13

Here are some other advantages from my point of view:

  • Writing lecture notes enables you to communicate to the students exactly what is examinable. Most textbooks are bloated with irrelevant material. See any first year textbook in economics or statistics, for example. That thing doesn't need to have 500 pages and colour pictures and a $150 price tag.

  • Writing lecture notes gives you valuable practice at academic writing and presenting your thoughts in a coherent manner. When you first try to write papers, it is tempting to try to intimidate the reader. Writing lecture notes helps to break this habit.

  • Good lecture notes might be used or cited by others if you put them on your website. This can be a good way of advertising yourself.

  • If you happen to teach the course again, going through the pain of writing lecture notes the first time you teach it will make it much easier to teach it in subsequent years, because you will be teaching from your own notes and you know exactly what you are doing.

  • Students will be very happy if they find a mistake in your notes! So you get an army of free proofreaders.

  • In an ideal world, your notes might be so good that you can one day publish them in book form. This has never happened to me.

10

Students often prefer lecture notes, for several reasons. They don't have to pay the cost of a textbook, and they don't have to worry about keeping a textbook in good condition for resale. This means that they are much more likely to be willing to make notes, highlights, underlinings etc on the notes themselves during the lectures. Providing printed lecture notes and letting them make notes on them creates a good balance: students are not frantically copying down everything without having time to think; nor are they likely to slip into a passive listening mode.

Furthermore, having a PDF of the notes available for download is useful for those students (and lecturers) who prefer notes in electronic format for searching etc.

7

I have no experience with preparing a whole course, so I will leave this aspect to others.

The vast majority, if not all, textbooks originated from lecture notes, usually from professors who held that course repeatedly and refined their notes over the years (often this is stated in the introduction of the textbook). Now, you might argue that these textbooks already exist¹ and there is no point in repeating this process as somebody already has done the work.

However, if everybody acted this way, there would not be any didactical improvement or adaption to current needs – which is very important for the future of science, as otherwise each new generation of scientists would take longer to reach the ever progressing forefront of knowledge. Just take a look at very old linear-algebra books: They are much more focussed on calculating stuff (which has become less important nowadays), have a more clumsy notation and leave out certain concepts that are considered relevant today. If some people in the past had not chosen to create their own lecture notes, which eventually turned into textbooks, you would probably have been learning linear algebra as taught in these books.

On the other hand, if everybody wrote textbooks, we would drown in them.

So, going only from this point of view: If you have a textbook you are perfectly happy with, stick to it. If, however, you have a didactical style and approach different from all existing textbooks, you might as well try it – and perhaps write a new textbook one day and advance the didactics of linear algebra.


¹ and in the case of linear algebra: tons of them

5

I think many people write their own lecture notes because they want to present the subject as it is living in their own mind, not as someone else presents it. You can really only ever try to convey your own perspective, and even in mathematics, this can be significantly different from anyone else's.

As a very small example, if you look at most linear algebra texts at how the formula for multiplying matrices is presented, you will find one of two tactics:

  1. It is just a definition, and you had better get used to it

  2. Matrix multiplication is defined the way it is so it corresponds to composition of linear functions. The proof of this is a computation which may involve a few too many summation signs for beginning students to follow fully.

My own perspective on the issue is the following:

a. I introduce matrices as record keeping devices for linear maps: the columns tell you where the basis vectors go

b. I spend time thinking about covectors, (i.e. a matrix which is just a row), and how applying a row vector to a vector is the same as taking a dot product with the transpose of the covector.

c. Realize that for a matrix M, M_{ij} = e_j^\top M e_i, since by definition Me_i is the i^{th} row, and e_j^\top of a vector just selects the j^{th} column.

d. So to find the (AB)_{ij} we just need to compute e_j^\top AB e_i = (e_j^\top A)(B e_i), which is the j^{th} row of A dotted with the i^{th} column of B. This is the standard formula, but it has been "chunked" in such a way that it makes it understandable (at least to me!).

This sequence a - d really represents thinking about a matrix as representing a bilinear form, and it is through this lens that the formula for matrix multiplication makes the most sense to me. You do not have to mention this to the students at this stage to make the sequence a-d understandable and memorable.

I find that this kind of thing occurs constantly. When I read a textbook, I usually find that I have no idea what is going on, and I have to develop some sort of narrative structure which makes sense of it. This becomes my understanding of the material. If I am teaching something, I must teach my perspective. So I often end up writing lecture notes.

3

I want to address just one point from the question. You write

Whatever I can write will not be half as good as a well-written textbook.

and I have to say that while I can't write a better presentation than the text on everything or even many thing, its a rare semester course where there isn't some sections of the course for which I can prepare a better presentation for the students in front of me than the writer has.

If I use Halliday and Resnick, I can present a terser and more insightful version of one section here and another one there. If instead I use Feynman, I may have to address some sections in more detail for the students at my small state school than the great man used for his at CalTech.


And as the other have said the exercise of preparing notes refreshes and clarifies the material.

I'm a lazy man, and whenever the presentation in the book seems to be appropriate to my students (or nearly so), I simple re-express the books methods with only minor changes in emphasis, and adjustment to agree with the demos that I am actually going to do rather than the ones that I would want if I had the prep-room of my dreams.

3

When I was a student, I hated professors not choosing a textbook and only giving (very often badly written) lecture notes. I think following a single textbook, suggesting optionally one or two more, plus some extra material here and there, is very convenient for the students since a good book is coherent, tested, well written and edited, with useful pictures, and often evolved through several years of lectures. You might lose some diversity, but you gain simplicity and coherence (and a more pleasant formatting). The best students will anyway look for additional texts on their own, either during the course or afterwards. Of course additional or very specific or innovative material can require lecture notes. In my experience, lecture notes are often good for the teacher; not always for the student.

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    You'd probably hate professors choosing bad textbooks even more. (And that happens rather often, as lecturers tend to choose textbooks for their readability to them, rater than for their readability to the student; I think there are certain books which are taught from chiefly for their table-of-contents.) – darij grinberg Sep 21 '14 at 22:20
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I've taught many segments of the basic business curriculum. I did not write up lectures in advance. I used the textbook as an outline, and I delivered an extemporaneous interactive discussion of the entire material in the text. This involved lots of definitions, examples, and diagrams of relationships (all of which I would require the students to recall and/or recreate on exams). I merely kept track of each section's progress, and informed them that they were responsible for everything in the book (as much as I tried to ensure I would cover all exam material, I didn't want to guarantee it, and I did want them to read the books they had paid so much for.)

This style worked well for me, but it seems not to be the tack most professors take.

There would have been benefits to writing up notes for lectures. It would

  • force me to review all of the material more carefully prior to the lecture
  • provide an outline for material I knew would be justifiably testable

The downsides were that it would :

  • increase time required for preparation of materials
  • likely reduce the material covered, because it was my goal to cover the entire text (which we did successfully)

I actually even did find time to cover extra material outside the text. For example, we covered Michael Porter's Five Forces in the Marketing course (which I expected them to recreate for the final, having reviewed it nearly half a dozen times in class).

Perhaps you could do the same. It may depend on the materials you have to teach with (especially the textbook.) But you need to review your expectations with your department chair. It may be that they require a textbook that is impossible to follow as the order for your class, for whatever reason. It may be that you'll need to take notes to ensure that you cover any prerequisite materials, and in the correct order, because you'll be teaching linear algebra.

  • Why "likely reduce the material covered"? – Federico Poloni Sep 19 '14 at 6:19
  • Unless I outline the entire text, the text will be more complete than my outline. – Aaron Hall Sep 19 '14 at 12:05
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This answer only involves the field of Computer Science, so other fields may not have the same benefits.

I wrote my own beginning C.S. course (programming) lecture notes (link). The benefits that I found are:

  • Not requiring a textbook. The textbooks in C.S. are very large and there are quite a few of them, so students/readers may not know which one to pick.
  • Not requiring a computer. Students/readers may not have current access to a computer (or any access), so they need to be viable for them too.
  • Plenty of examples. Success in programming is possible only through many examples, and not much of the theoretical aspects of programming. Therefore, my notes have many examples to get the concept across.
  • Learning the material from a different perspective than the one you originally had.
0

As a student I loved lecturers who made formal notes.

  • It contained everything we needed to know
  • No buying expensive books or waiting for the library
  • No flipping chapters between multiple text books

And it was easier to understand, since it matched his teaching style and you could ask him to explain (since he wrote it).

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