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.
I’ve written at length about the importance of knowing your audience while teaching, but I want to go through it again because it’s worth thinking about. If you want to produce explanations that people get, you need to adjust your expectations accordingly.
In this essay, I want to explore a few ways in which this appears while teaching. First, we will look at the importance of good examples that are tailored to the audience. Then, we will see how showing our steps is critical. Finally, we will look at how our expectations need to be shifted when working with students who aren’t as experienced in a subject.
Examples are not all “trivial”
When learning a new topic in science or mathematics, examples help illustrate the topic. They give students a foothold into the new subject. I’m someone who loves to get into abstraction, but if I never look at an example, I can get lost in what to actually do when faced with an application of the theory. Examples help us learn past just the theory.
Therefore, if you’re crafting an explanation, you should think about what kind of examples you want to show. What I’ve found though is that I have a tendency of thinking an example is too easy, when really it is easy for me. This skewed perception of what is actually difficult can be a challenge to get around, but if you want students to have examples that are tailored to their level, you want to think about your skewed perspective.
I have to remind myself of this every time I tutor younger students who are learning the basics of algebra. When working through a problem with them, I’m tempted to start giving variable names to everything and only put in numbers at the very end. I do this because I know it helps you avoid making a mistake in the intermediate steps. The problem though is that this can be enough to overload them. It’s sort of like the issue that people writing programs have. Should I make this program solve my specific problem, or should I try to make it more general? I have a tendency to go to the latter, which can make it confusing for students who want to see specific applications.
This is why I try to adjust my teaching accordingly. I catch myself floating towards the clouds of abstraction, and I reel myself in by giving an example I think is too simple. Odds are, that’s just right for them.
After all, I know that an example is only good if a student can understand what’s happening. If it’s so complicated that they need me to solve the whole thing for them, it’s not good. An example should be suitable to their level. This sounds obvious, but it’s not always what happens. Remember, your perspective is skewed towards thinking all of this stuff is easy. Therefore, you need to manually adjust, or else the students you work with won’t learn from you.
Show your steps
This is another crucial point, and it’s even more important if you’re working in a medium which doesn’t involve active communication (i.e. teaching in a classroom). For example, this is important if you’re writing the solution to a problem or are communicating through text.
The reason showing your steps is critical is that students need these steps when they are working through a new kind of problem. If you skip steps, students will find themselves lost from the jump between two lines. Sure, this step might seem obvious, but remember that it’s obvious to you. Your perspective skews everything. That step which is “self-evident” might not be so clear when a student is working through a derivation line by line. I can’t tell you how many times I’ve tried to work through an argument in a textbook, only to find myself stuck on one line that seems to “magically” transform to the next. This just creates confusion, all for a few lines of saved space.
Of course, if you’re in the room and are explaining the concept to the student, you can then fill these explanations in as you go. However, if you don’t take initiative to show more steps, your student might be too shy or intimidated to ask. This does happen, so it’s a good idea to show more steps than less.
One of my professors does this while teaching in class. They might skip some steps from line to line, and they rely on someone asking a question about what happened in order to explain. These are often just algebraic manipulations, but if you string a few of them together, it can get hard to follow. And explanations only come if someone asks. Therefore, the default is that less work is shown, which I’m sure can be confusing to some.
My overarching point here is that you shouldn’t assume too much knowledge on the part of a student when going through steps in a calculation or derivation. I would suggest asking after the session if the student thought there was too much detail. If so, then you can start to dial back. This way, you aren’t forcing them to admit that they can’t follow, which can be difficult.
Of course, this all demands a knowledge of your audience. You need to know the skill level of the students you’re working with. If they are sufficiently advanced, then by all means don’t go through all of the algebra. But I think it would be better to play it safe. After all, you can always drop steps later on.
Shift your expectations
Throughout this whole essay, my constant message has been to adjust your expectations accordingly. You don’t want to assume too much of students, because that can lead to them not following (and in bad situations, not pointing this out). Therefore, it’s a better idea to shift our expectations ahead of time, and ask for feedback later.
This is always a work in progress. You will constantly have to remind yourself of your skewed perspective. If we get too comfortable, we start making students confused. It’s not easy, but it’s necessary. After all, no one said teaching was easy.
If you want people to understand the explanations you give, you have to be willing to do the difficult work of going to their level. Too often I’ve found myself trying to get students to come up to my vantage point. That’s fine, but the issue is that students often can’t make that jump. If they do, they end up confused as to what is happening. And believe me, there’s a large difference between following the steps and following the idea. The latter is what we should always be shooting for, but I’m afraid that it happens a lot less often than we want.
As such, I want this to be a reminder to myself that explanations need to be tailored to the student. If it’s something totally new, don’t skip steps. That will just make things difficult to follow, for no added benefit. I’d much rather take things slow and have them understand what’s happening than go fast and require more explanations later on. It might seem like a time drag at first, but it is worth it.