*Bits, ink, particles, and words.*

When I do nearly any assignment for homework, I’ll make a rough copy of my work before writing the copy that I will hand in. I do it for a practical reason: while it may take extra time to write my work twice, the truth is that I often take a lot of detours on my first try tackling a problem. I go down dead-ends, make little mistakes here and there that need to be corrected, and generally do a lot of messy work. Once I get the correct answer, I can tidy all my work up in order to make my final copy as concise as possible.

When I work with younger students in subjects like mathematics or physics, it doesn’t take much to impress them with my ability to quickly see through a problem and calculate things that would take them minutes. Just like any other student at my level, we often skip the use of calculators because it’s easier to just focus on the work we are doing and do the arithmetic in our head. The most prominent example of this, however, is in algebra.

Even though mathematics is one of the subjects I enjoy studying, it’s not always easy. Nor does everything always make sense to me. One of these things was the notation for an integral.

If there’s one thing I’ve learned about mathematics during my many years at school, it’s that having a solid foundational understanding of the main components can go a long way towards learning new subjects within mathematics. Unfortunately, this is what is often lacking for students, and it can have the knock-on effect of making later concepts more difficult to grasp.