Use Scratch Paper
Date Written: June 20, 2024; Last Modified: June 20, 2024If you tracked each student’s eyes throughout the duration of an exam, then the objects that are stared at the most (after the exam papers, of course) are the ceiling, floor, and windows. When proctoring, I notice that a lot of students have eyes that wander away from the exam paper while deep in thought (not a bad thing), and I do this a lot as well. It’s a natural movement to aid in thinking and processing complex ideas before putting them onto a piece of paper.
However, I found that it’s much easier to process and sort through arguments and ideas once they’re on a piece of scratch paper. You don’t have to write things out in much detail at all, but just having a few symbols jotted down frees up a lot of working memory to think about other things more efficiently.
For instance, consider the following example:
Problem 1.
Using the \(\epsilon \)-\(\delta \) definition of continuity, prove that the function \(f(x) = 3x^2+6x+1\) is continuous at \(x=7\).
It is extremely difficult for me to just write out a solution in one go, especially if you try to do things in your head. It’s possible to work out the difference of squares stuff completely mentally, but I wouldn’t trust myself to do it quickly or correctly.
Rather, it’s best (in my opinion) to jot down \[f(x)-f(y) = 3(x-y)(x+y)+6(x-y).\] From there, I can either compute mentally or work out on paper that a feasible choice of \(\delta \) is \(\frac{\epsilon }{48+3 \epsilon }\).
I’ve seen several exams where technical problems are attempted several times over due to tight computations such as the above. Use scratch paper! It will speed things up and make things easier.