Never solve math homework problems on the piece of paper you intend to hand in (unless it is a problem that is very, very easy for you).
I used to do my work on the same piece of paper I intended to hand in, and when I encountered a problem I was uncertain about… I froze. I did not dare write anything incorrect on the page I was going to hand in. If I did, I might have to recopy all my work in order to end up with a neat-looking page to hand in. So, I would just skip the problem, telling myself I would ask the teacher about it in class next time. The result was that I greatly slowed my learning how to do that type of problem.
I now know that if I figure out how to do a problem all by myself, I will remember how to solve it for years. But, if I ask someone else how to do a problem I was struggling with, I will probably forget their solution within minutes. So, if my goal is to do well on a final exam a few months from now, I am better off trying to figure the problem out by myself.
And that probably means I will have some, and perhaps many, false starts. Scrap paper gives me the opportunity to doodle, to draw diagrams, to write down the information from the problem in a way that makes sense to me (as opposed to the way the problem was presented), and to make false starts in solving the problem with no negative consequences. I can work big. I can try solving something two or three different ways to verify I get the same answer no matter which approach I take. I can quickly correct mistakes I find in my work without having to erase anything. I can take baby steps in solving a problem if I am uncertain of which direction I should go.
My work on scrap paper is private. Nobody else will see it. Once I think I have solved a problem, I can then work on finding a better / faster / cleaner solution. Only then am I ready to copy the highlights of my work from my scrap paper onto my nice, lined, more expensive notebook paper… checking every step twice for mistakes before I copy it over.
I learned to catch most of my own mistakes this way. I also discovered that by doing the same problem more than once, sometimes three or four times as I searched for the most elegant solution, I figured out answers to many little questions that bugged me about the problem. The “why”, or “how come” sorts of questions. I became faster at working that type of problem. My confidence in my skills grew.
I also caught an embarrassingly large number of “small” mistakes as I copied my work over onto the page I was going to hand in: sign errors, arithmetic errors, errors due to distractions, etc. I learned to read my own work with a questioning eye, looking for all my “usual errors” before I copied a step from scrap paper to notebook paper.
In short, doing all my work on scrap paper improved my confidence, my speed, my enjoyment of the subject, and my grades. It was liberating, empowering, fulfilling… and it saved some money too. And I am not alone… see Killing Trees: The Never-Ending Quest For Scrap Paper.