Let's Read: In Praise of Lectures by T. W. Körner

Alright, let's read! I'm on a roll of venting against lectures.

In Praise of Lectures (2004), Thomas William Körner

The essay is off to a strong start that lifted my mood.

For these students attendance at lectures has a magical rather than a real significance. They attend lectures regularly (religiously, as one might say) taking care to sit as far from the lecturer as possible (it is not good to attract the attention of little understood but powerful forces) and take complete notes. Some lecturers give out the notes at such speed (often aided by the technological equivalent of the Tibetan prayer wheel — an overhead projector) that the congregation is fully occupied but most fail in this task... 

The notes of the lecture are then kept untouched until the holidays or, more usually, the week before the exams when they are carefully highlighted with day-glow yellow pens (a process known as revision). When more than 50% of the notes have been highlighted, revision is said to be complete, the magical power of the notes is exhausted and they are carefully placed in a file never to be consulted again.

The invention of the Xerox machine [or Internet] means that only one student need attend each lecture, the remainder being freed ... Nor would this student need to take very extensive notes since everything done in the lecture is better done in the textbooks.

But then the essay took a sudden turn and started praising lectures.

Why, if lectures have all the disadvantages that I have shown, do they persist in going to them? The surprising answer is that many mathematicians find it easier to learn from lectures than from books. In my opinion there are several interlinked reasons for this.

the reasons are almost all without force. They only apply to some lectures and some books.

(1) A lecture presents the mathematics as a growing thing and not as a timeless snapshot. We learn more by watching a house being built than by inspecting it afterwards.

Some books give lots of historical perspective. some lectures don't. It's not intrinsic to either forms of teaching.

(2) As I said above, the mathematics of lecture is composed in real time. If the mathematics is hard the lecturer and, therefore, her audience are compelled to go slowly but they can speed past the easy parts. In a book the mathematics, whether hard or easy, slips by at the the same steady pace.

Absolute bullshit. A book often do skip over the easy parts by things like "it's obvious", and spend a lot of ink on the parts that are hard. The preface, and the starts of chapters and sections, often have quick summaries as to what's intended in the parts that follows, what is important, what is not, how to read, and what to skip.

(3) ... When a lecture is going well they can seize the moment to push the audience just a little further than they could normally expect to go. A book can not respond to our moods.

This is true. Great!

Though most of the lecturers I attended have no effective response to the mood.
(and personally, no amount of lecturing could teach me, so even if they could stop and repeat it can't help me.)

(4) The author of a book can seldom resist the temptation to add just one extra point... The lecturer is forced by the lecture format to concentrate on the essentials

Again, it's not intrinsic to either forms of teaching: Just write the book so that the essential points are summarized at the beginnings and ends of chapters, and the inessential parts are marked off as starred.

(5) In a book the author is on her best behaviour; remarks which go down well in lectures look flat on the printed page. A lecturer can say ‘This is boring but necessary’ or ‘It took me three days to work this out’ in a way an author cannot.

Bullshit. Authors can and do. I write like that! Am I not an author?

So basically, Körner gave five reasons why lectures are better than books. 4 turned out to be without force, since those "advantages" are not intrinsic to lectures or books. A good book would avoid the problems and gain the advantages, and a shoddy lecture would have the problems and lose the advantages.

And there's a crucial advantage of books over lectures: lectures are not choosable unlike books. If you took a lecture with a bad lecturer, that's it, you are stuck unless you quit. If you got a bad book, change! Good books are just easier and easier to come by now, thanks to printing technology and Internet. Good lecturers are as rare as ever.

Mathematics textbooks show us how mathematicians write mathematics (admittedly an important skill to acquire) but lectures show us how mathematicians do mathematics.

That's a nice and noble thing, but first, that's not true for any lectures I took, and second, why not tutorials? Won't they work much better if the students can do math under guidance, instead of watching the professor stand up and perform?

It'd be like learning driving by watching the teacher drive for 50 minutes, and only then, maybe, drive yourself. Much better if you just start driving after the barest instructions, fail, get taught more, drive better, get taught more, etc. In other words, a tutorial, or a discussion-problem-solve problem-discussion loop, not a lecture.

And the fact remains: some math textbooks really teach you how to do math. I can think of a few, like The Cauchy-Schwarz Master Class, J. Michael Steele, generatingfunctionology, Herbert S. Wilf, A View from the Top, Alex Iosevich.

And you know how I learned to do math? I just work through them and justify everything to my own satisfaction, catching lots of mistakes as I go. That is better than going to any lectures and taught me how to DO math.

The author then proceeds to give two analogous situations where a lecture-like way of studying is best for instruction:

1. Science Awakening (1950), Bartel Leendert van der Waerden: It is talking about how Greek geometry was taught by live lectures, and how it was better than reading the books, and how it ended up causing the downfall:

An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasise essentials and point out how the proof was found. All of this disappears in the written formulation of the strictly classical style. The proofs are logically sound... but one fails to see the guiding line of thought.

... as soon as some external cause brought about an interruption in the oral tradition, and only books remained it became extremely difficult to assimilate the work of the great precursors and next to impossible to pass beyond it.

 2. A parable of "touring a new town"

Suppose that that you visit a large town and you wish to learn how to get around. One way of learning is to go by foot on a guided tour which includes the main landmarks... But, without that first tour given by a native, you would find it very hard to learn your way about town. Lectures by themselves can not give you a full understanding of a piece of mathematics but, without lectures to get you started, it is very hard to gain that full understanding.

The comparison with "Greek geometry" and "touring a new town" are deceptive. The author picked them precisely for their physicalness, completely not justifying how could other math subjects be just as physical as those and thus justify using lectures for them.

Touring a new town is really something that requires a full-bodied experience, but that's not true for algebra, or even geometry. You can be stuck in a prison cell, awaiting execution, and still do mathematics, while you can't go hiking without, actually, like, go and hike.

I am here thinking about the story of Éamon de Valera, revolutionary, and later president of Ireland, who sought solace in quaternion algebra while awaiting execution after a failed revolution:

While awaiting his execution in Kilmainham gaol after the Easter Rising in 1916, he scratched into the wall of his cell the fundamental equation for quaternion multiplication, $i^2 = j^2 = k^2 =ijk = -1$, a poignant echo of Hamilton’s famous flash of insight at Broome Bridge, on Dublin’s Royal Canal, in 1843.

And think about this in another way, the story about Greek geometry is seen as a cautionary tale against lecturing. Because they relied on oral tradition/lecturing, and didn't bother with writing good books, once the chain of orality is broken it's all over. What they should have done is to try to develop better notations instead of relying on oral traditions, so that even without oral instruction, the knowledge could carry on.

In my view students should treat lectures not as a note taking exercise but as a dialogue between themselves and the lecturer. They should try to follow the argument as it emerges and not just take it down blindly.

A "dialogue", really?? A lecture, if viewed as a dialogue, is autistic. The knowledgeable side of the dialogue speaks at length about whatever they know, and the other side sits and listens and rarely breaks the flow. And autistic dialogues are commonly regarded as ineffective and boring.

The author then proceeds to deal with some questions which are pretty much what I agree with so I'm not going to bother describing.

And we finally reach the ending, where the author hastily reach the conclusion:

‘O King’ replied Euclid ‘in Egypt there are royal roads and roads for the common people, but there are no royal roads in geometry.’ Mathematics is hard, there are no easy ways to understanding but the lecture, properly used, is the easiest way that I know.

Absolute unsubstantiated bullshit, especially after devoting the first 2 pages arguing convincingly the badness of lectures, followed by 3 pages of unconvincing arguments about how good lectures could be.