As a student and someone who tutors others in science and mathematics, I’ve been able to get a lot of experience on both the teaching and the learning side of education. It has given me a better appreciation of the difficulty of our job as teachers trying to get students to understand. In particular, I’ve learned that being deliberate in my explanations is important if I want students to get what I’m explaining. Sure, I can have them fend for themselves, but the consequence is that they can get confused and frustrated for no good reason.

With a few taps, we can compare ourselves with thousands of other people. The comparisons can be anything we can think of. If there’s some kind of performance metric associated with your activity, you can bet there are places to compare results. It’s the nature of things. Humans like competition and comparison, so we build places where these comparisons are easy to find.

As a student, I’m used to diving right into the technical details of a topic. I don’t mind working through a wall of algebra, because that’s what I’m used to. If I wanted to describe how I learn in my classes, it would be: mathematics first, “high-level” understanding second. This isn’t a bad thing. I don’t mind going through the details first. Sure, I might not know how the concept relates to other ideas immediately, but I can learn that later.

When you first start solving a problem in mathematics, the goal is often to find a way to express the problem as some sort of differential equation. During this initial search, you don’t care how the equation looks. It’s more important to get it written down so that you can proceed.