Note Taking in Math Class

I'm a math undergrad with a real passion for mathematics (enough to try to maintain a blog about it). As of today, I'm taking four math classes, have a gpa of 3.9, and spend a good amount of my time tutoring. Also, I don't take notes.

I'm not sure what place note taking has in a math classroom. The books that I spent more than $100 a piece on (way too much!) cover all the same content that my classmates' notes do. If I forget the definition of the Laplace Transform, then I don't need notes to look back on, I have a book. Failing that, I have Wikipedia, Wolfram's Math World, and Paul's Online Math Notes. If I need to have problem worked out, I have Khan Academy or any other of a variety of YouTube videos. My school, and I imagine every other school in the world, is packed with thousands of books, many of which are about differential equations. If I need help with Laplace Transforms, or any other topic in math, I have a plethora of sources to reference. Why then, on top of all that, should I take notes?

Further, I think taking notes in math class have negative consequences. Sometimes I look around and notice my classmates too absorbed in their note taking to actually be paying attention in class. The professor might add a bit of interesting information verbally, and my classmates are often too busy copying what's on the board to hear it. Also, there is the problem of divided attention: if your attention is being put into your notes, you are not working on comprehending the material. I assume that these students go back over their notes at a later time and try to make sense of the material then, but that has to be very tough when there is no professor to offer insights.

So why do so many students take notes? I suspect it's because of years of programming by high school and elementary school teachers. I also think it's because students have developed skills that are appropriate for other classes (note taking is very valuable in English or history class) and mistakenly believe that those good habits will translate to their math classes. I think all of this negatively impacts math students' education.

Wrong vs. Not Helpful

In solving a problem, students inevitably make mistakes. This is how we learn. However, there is a huge difference between being wrong and being not helpful.

Wrong

Suppose a student is asked to solve the following for x:

Seeing the 2 next to the x, a student may try to divide both sides by 2. This can be done correctly, but let's assume that the distributive rule is momentarily forgotten:

This leads to the incorrect answer of x = 1, whose falsity can be demonstrated by substituting 1 in for x in the original equation (which would give 6 = 10).

Not Helpful

Now imagine that the student takes the same equation, and subtracts 2x from both sides, giving:

This is not wrong, subtracting 2x is not a violation of any mathematical rule, but neither is it helpful. A student who attacks a problem like this may still be in need of assistance, but a different type of assistance from before.

My Point

This distinction is obvious to educators. It is not, however, necessarily obvious to students. And it needs to be made obvious to students. There is a rampant misconception of math that there is a right way and a wrong way of doing math, and if you're not not doing it correctly, you're wrong. (I previously wrote about that here.) Sometimes you're doing something useless that is not necessarily wrong. And that is ok.

I think a good way to get this distinction across is by way of analogy. Consider chess: there is a notable difference between a wrong move (moving a rook diagonally) and a foolish one (exposing yourself to checkmate).