From the point of view of (mostly) a student (full disclosure: a physics student!) I vastly prefer to have handed out notes.

I have three undergrad degrees: math, physics, and astronomy (each from different departments with distinct styles), and am now a physics grad student, so I've seen a number of styles coming from each of these areas. And I have to say, one of the absolute worst styles is math without any handouts (beaten only by the style where the professor exclusively tells stories or does not show up to classes!).

As Terry Tao says below (or, more likely, above, if you have this sorted by votes!) it can be very frustrating to spend hours being confused because of a simple mistake in the lecture, or a simple transcription mistake in my notes. But more than this, one of the (IMHO) major failings in most of my undergrad math classes was that they focused largely on discussing proofs and technical details, and not at all on *why* anything was like it was.

So, I could, e.g., come out of an analysis lecture being able to prove the uncountability of the reals with Cantor's diagonalization argument, but, since we went through it so fast and covering every detail, not having any understanding of why it worked, what the significant parts of the proof were, or why it is important. Even though in retrospect there was nothing tricky or difficult at all about this argument, I remember having a great deal of trouble understanding this at the time.

However, in the (few) math classes I had that handed out notes with technical details carefully spelled out in them, and in class focused on discussing why things were like they were, in developing some intuition, and trying to modify assumptions and see how things changes, I had a **greatly** improved understanding (and ability to correctly do the homework and exams!).

It has always seemed far easier for me to go through the technical details myself, taking my time, than to go through the qualitative arguments myself in a field with which I am not familiar.

In fact, I found that these kinds of more qualitative lectures actually significantly increased my ability to carefully prove things, since I was able to better understand how the steps in a proof were linked together. As opposed to understanding the technical details in each step, which are useful, but only useful to make a step careful after you are sure the step makes sense!

In the classes I've taught, and had enough control over that I have been able to do like I wanted (which, admittedly, is only one class, a baby-quantum field theory for undergrads class!) I typed up notes for my students, and found that the students who put extra time into understanding the class did substantially better than in classes where I have not been able to do this.

Students who did not put extra time outside of class into studying did not do any better, but claimed to like the notes better.

So I've felt that it's a good idea to hand out notes, since, at the very least, it will reward the students who are working hard and putting extra time into the class. Those students are the ones we want to continue in our fields, after all!

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