14 December 2016

Imagine sitting in a lecture about the development of perspective in art. The lecturer presents some illustrative examples of art from various time periods, starting with works that represent objects as sized independent of their distance, and then going through examples of one point projection, two point projection, and so on. If you were taking notes on this lecture, what would you be writing down?

You'd probably note the main points like how we define one point projection, two point projection, et cetera. Perhaps you'd write down a reference to a work of art that demonstrates these ideas to look at it later. What you wouldn't do is look at an example picture and then try to sketch a copy of it in your notebook while the lecturer is talking.

Now let's think about math class. Unlike what we expect to happen in an art lecture, a common strategy for note-taking is to copy down everything that's written on the blackboard. The effect, which will be unsurprising to some people, is that those students aren't able to really listen to the surrounding verbal exposition, and come out of the lecture only really "hearing" the things that were written down.

On one of my first days at MIT, I was in an introduction to topology class taught by James Munkres. He straight up told the class to take no notes during his lectures. If we wanted notes, they were in the book. The lectures were meant to give additional context that doesn't necessarily appear in the explicit material, like historical viewpoints or approaches that were tried but turned out to be worse or not working at all.

Munkres's class was certainly special, because he had clearly been teaching this course for so long that almost the entire semester was scripted. Not only was the material presented in class the same as in the book, but the actual order and style of presentation was the same as well. In fact, if you wanted to know what the next subject would be, all you'd have to do is turn the page.

Not every class will have that kind of parallelism between the lectures and the reading material. I am absolutely sympathetic to the idea that people want to have reading material along with the verbal lectures to learn from. However, I've seen plenty of people closely examining their notes over and over again trying to find where they're misunderstanding, when what actually happened was that they copied the blackboard down incorrectly. I believe that the approach of copying down the blackboard is unreliable, and furthermore gets the solution almost entirely backwards.

The purpose of a lecture is to implant a little seed into your mind. That seed isn't yet the understanding that you're hoping for. The real learning comes from working through problems, where you see how that little seed's ideas sprout into multiple methods that you incorporate into solutions. In reality, I think the best way to get written reference material is to write out full solutions for homework problems. But if the process of listening to a lecture followed by working out some problems makes an emotional connection, then you might even find that the written material is completely unneeded.

I'm sure you can find plenty of people who swear by taking notes, and probably even some who say that copying down the blackboard is the best way for them to learn. However, people are usually horrible at judging their own understanding, and it's important to think about where these claims are coming from.

So let's imagine a math student Jane. Jane is used to taking notes during classes. When her professor shows the solution to an example problem on the board, she's making a note of each step involved. When she reads a homework problem, she might think, "Ah! We saw a problem like this in class!" and proceed to find the relevant section of the notes to use as a guide.

Then Jane reads on the internet that taking notes is harmful! Wow, that's a surprise, since the notes are so often useful to guide her along during homework sessions. But okay, Jane is willing to test out her assumptions and tries going through a lecture without taking notes.

Homework time rolls around and Jane is looking at the first problem not knowing what to do. "If only I had taken notes, I could just be looking at them right now," she thinks. She stretches her memory and recalls a couple of the intermediate statements that were said in class, which eventually leads her to cobble together a solution after a few hours that she thinks is correct, but isn't quite sure. On the second problem, she's completely lost and doesn't even understand what it's asking.

The decision to not take notes in class made the homework session considerably more time consuming and difficult, so Jane comes to the conclusion that taking notes works for her, even if it might not work for other people.

At the end of this story, Jane will be one of those people who says that taking notes helps her learn math. But she never actually measured her understanding of the subject. She measured her ability to do the homework in the days immediately following the lecture. The true test of understanding comes well after that, and even well after the final exam. The first test of understanding will come in the next semester, when she's taking a class that assumes knowledge of everything that she's learning now.

If we look at the effect that the notes had on Jane's study habits, you might begin to see how they can be detrimental. That homework problem that Jane didn't understand at all? She might have been able to write down a solution if she had notes to use as a guide, but what's preventing her from becoming a mathematical version of a Chinese Room?

Perhaps this fictional story convinced you, and perhaps it didn't. I'm not going to claim I have proof that notes are detrimental to every student, or even on average. I don't know about any research in that area. But if you want to try out not taking notes for yourself, here are my recommendations for how to do it.

  • During lecture, try to find the main new ideas being presented. If something is just algebraic manipulation, trust yourself to be able to do that on the homework if you need to.
  • If the course doesn't have written materials available, do write down definitions. Keep these very short. Most definitions are only a single sentence. If you're writing more than that you're probably writing something that's not included in the definition.
  • Be ready to struggle with the homework. Being stuck on a problem for hours is extremely common for mathematicians, even though it doesn't always seem that way. On one of my problem sets at MIT I was stuck near the end of a solution for around ten hours before realizing that it could be finished in a reasonably simple manner.
  • When you get your homework back, make sure you have a complete and correct solution. If it's the one you turned in, great. If the teacher posts homework solutions, read through and keep that. Those solutions are now your notes.
  • When exam time comes, go through those homework problems as study materials. If you end up getting stuck on one of those problems again, chances are it'll be in the same place you got stuck the first time, and your mind will connect the dots.