How Deliberate Are Your Explanations?

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.

The Whole Ride

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.

Due to this, whenever we find an activity we like, there will always be a push to compare your results. How good are you? Are you better than these top performers? Where do you fall in the hierarchy? These are questions that can be answered quickly due to our love of performance metrics.

This can lead to people moving away from the reason they began the activity, and instead becoming motivated by the performance metrics themselves. They might not even want to, but peers will push them toward it. To peers, this will seem like encouragement. They could say things like, “You’re so good. You just have to go for the bigger stage!” On its face, this sounds like advice from people who want to help. However, what it often accomplishes is a feeling in the person that they must go for bigger things with respect to the activity they do. After all, if they spend so much time doing it, shouldn’t they try and make something come of it?

This is where people fall down the slippery slope of moving away from doing what they enjoy to doing it because they feel obligated to. There’s a big difference between doing an activity on your own terms versus doing it with plenty of people watching, in the chase for external results.

I’m not saying the latter is bad. Rather, I want to highlight that the motivations behind an activity are different in both cases, and it’s important to be able to distinguish between them. The trouble comes when a person starts out by doing an activity because they love it, but slowly transitions into doing it because they feel like they have to do it. It’s a terrible feeling, and the primary cause is due to the chase of external results in order to compare against others.

The truth is that you’re on your own journey. No one else is with you for all of it. This is important, because it means that you should be happy with what you spend your time doing. The external results might be fun, but they won’t carry you through day after day. They’re too brief. Instead, you need something else that makes you do the work. One necessary condition would be doing something you enjoy. If you don’t enjoy what you’re doing and are only doing it because you feel obligated, you will be miserable. You might enjoy those brief moments of comparison where you can get external recognition, but the bulk of your life is not like this. It’s a steady march, day after day.

The goal should be to seek something that satisfies you during your whole journey. This is much more difficult. If you strip everything away, and never get recognized, would you still continue doing what you do?

The answer to this question says it all.

One-Sentence Summary

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.

The unfortunate thing is that this “high-level” understanding doesn’t usually come from teachers explaining ideas to you. Instead, I’ve found that it comes from observing how ideas are explained in class, and putting these observations together over a long period of time.

I’ll start with a simple example. In secondary school mathematics, one of the ideas is that of factoring quadratic expressions. Different methods are taught, including grouping, completing the square, and “filling in the blanks” by looking at the various coefficients. When you’re learning these techniques, they all seem different. You might even be told that you need to use a certain method for a certain kind of expression you see. As such, you follow what the teacher says. It’s only much later on that you realize that these different methods were essentially doing the same thing. They look different at first, but that’s only at the surface. Once you realize this, you also understand that calling them different “methods” is not helpful. Unfortunately, this moment of realization comes much later when you have completed the class.

This isn’t just an issue in mathematics. As a physics undergraduate, I’ve taken many physics classes over the years. Some of these classes seem distinct, but once you take a step back, you can’t escape the fact that they are all about the physical world. This means there are many connections between classes, waiting to be discovered.

If you have good physics professors, they won’t hesitate to point these out. That’s because understanding these connections can give you a better picture of the concept you’re studying. It removes the details and let’s you worry about the heart of the matter. While this might not help you calculate the electric field at a certain point, it will give you the ability to understand a subject at its broadest.

Grasping the conceptual

I’ve hinted at this before on the blog, but I’ve noticed that while I learn a lot of the technical details at school, I haven’t done a good job of grasping the overarching principles behind certain subjects. I think this conceptual understanding is quite important, and yet, due in part to the school system not rewarding this type of knowledge, I haven’t focused on it. This is something I want to work on, because it allows one to move fluidly within a conversation without having to pull up the technical details.

I’ve been thinking about ways this can be done. One method that I think holds promise is the idea of a “one-sentence summary”. It’s exactly what it sounds like. With any given topic, what is the one sentence that can describe its essence? Of course, I’m also barring myself from turning into a punctuation master and creating a run-on sentence that lasts forever. Instead, I’m thinking of something short and sweet, preferably memorable. The key here isn’t to find the perfect sentence, but one that is good enough.

Will it encapsulate everything about the subject? If the subject has any breadth, of course not. But that’s also not the point. Instead, I want to capture the bare minimum of the subject. If someone heard my sentence, would they agree with it? Perhaps they have other ideas that could take precedence over this one, but at least they agree that the one I gave is important.

In order to illustrate this, I’ll give you a few examples of one-sentence summaries that I’ve come up with.

  • Entropy: A count of how many micro configurations give rise to the same macro properties.
  • Metric: A tool to let you measure distances and angles in some geometry.
  • Statistical mechanics: What properties appear when a lot of smaller parts form a system?

As you can see, these aren’t perfect. I’m sure that people with more expertise and experience with these ideas than I have can give better one-sentence summaries. However, the goal here isn’t to write the absolute best sentence for any topic. Rather, it’s to write the best sentence that I can think of at this moment in time. There’s no use in trying to achieve some absolute. What might be even more interesting is to update your one-sentence summaries whenever you think you have a better understanding, and keep the older ones. What you will end up with is a chronological record of your thoughts on a subject. It will let you see how your understanding has changed over the years.

These are just preliminary ideas to get you started. Adapt this as you see fit. The key is to try and gain some high-level understanding of a subject. The method itself doesn’t matter. I know that I’ve spent so much time in the weeds of subjects that it’s worth trying to take a broader perspective.

I want to be clear: I’m not looking to replace my usual method of learning. The details are important, and I don’t want to minimize that. If you want to learn about science or mathematics in particular, sooner or later you have to dig into those details and work with them. There’s no escaping that. Conceptual understanding alone won’t make you pass a test that deals with calculations.

That being said, I think we too often eschew conceptual understanding, as if it’s not as good as “the real thing” of calculation. I think this undersells the value of knowing a subject at a high-level. In my mind, they are both important, but distinct. If you just focus on the details, it will take a long time for the high-level understanding to come. One-sentence summaries could be a tool that can help you, but I don’t care if you use my method. Please, just find something that works for you. It’s so great to feel at ease with a concept to the point that you can talk about it at a high-level without worrying about the details (or that you’re butchering them).

Familiar Forms

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.

However, once you do have an equation, the first step is not to try and solve it. That’s a rookie mistake. Instead, the question you should be asking yourself is, “Can I put this equation into a form I recognize?” Asking yourself this question can save a ton of time in solving a problem. After all, if you can recognize the form of the equation, then you know the answer without doing any more work.

This might seem like an edge case that never happens in practice, but that’s not true. In particular, mathematicians have studied the solutions of many ordinary and partial differential equations, and know the answers. Therefore, if you’re working with a differential equation (which is almost always the case in physics), you might be able to save yourself a lot of time if you recognize the form of the equation.

For example, any student in physics who has taken more than a few courses will recognize the differential equation representing simple or damped harmonic motion. Physics students come across it all the time. This equation comes up when considering swinging pendulums, motion of a spring, electrical circuits, stability of circular orbits, and even in the Schrödinger equation. It’s what you might call a pervasive equation.

I can guarantee that professors don’t go over the solution to this equation after perhaps the first semester of physics. The reason is that students learn how to solve this differential equation, so there’s no need to go through all the work again and again. Instead, they identify the equation, and then give the solution.

However, it might not always be obvious that an equation satisfies the differential equation for harmonic motion. If there are a bunch of constants littered everywhere in the equation (due to the physical situation), it can be difficult to see the underlying equation. How do we deal with this so that we can try and identify an equation?

The trick is to change variables and bundle up constants together as much as you can. If your equation has constants littered everywhere, see if you can divide the constants out so that you have less in total. In the same vein, if you can see a simple change of variables that will allow you to “absorb” some of your constants in the differential equation, that can also help in simplifying the equation.

The goal here is to try to make your equation as generic as possible. That’s often the best way to compare it to the known equations in mathematics which have solutions. When you look at solutions to differential equations, they won’t be given in terms of parameters like the mass of a particle. The constants will be generic. Therefore, it’s often in one’s best interest to “clean up” a differential equation as soon as possible in order to make it recognizable.

Remember, there’s nothing wrong with ploughing ahead and solving the equation right off. It can still work. It’s just that the constants present in an equation that are specific to the problem can muddy the waters of the solution. By dividing constants out and changing variables, the equation will shed its “particular” qualities, showing only the essence underneath. Then, one can save time by identifying it with a known differential equation.

The point when solving a physical problem isn’t to go through all of the mathematical detail for no reason. If a solution is already known, there’s no point to ignore that. Use the fact that you can recognize solutions to speed up your problem solving. In the end, it’s the physical solution itself that matters.