The process of learning with another person is a tricky thing. When you’re on your own, it’s fine to ask as many dumb questions as you can come up with. After all, nobody is going to judge you, since you can simply look up the answer in a book or online. As such, learning by yourself is safe (though it can be slow).
On the other hand, I’ve come to realize how vulnerable you can feel while learning with another person. One person holds the knowledge of a subject (the teacher), while the other is hoping to gain knowledge (the student). However, there’s the added complication that the student doesn’t want to appear stupid in front of the teacher. One reason might be that the student respects the teacher, but a more general reason is that we don’t want to appear less intelligent to anyone. This is only exacerbated when you’re in a large class. Even though this worry is mostly unfounded, we stress about how others will perceive us.
As a student, I’ve felt this firsthand. Even though I work hard and understand a small amount in my field, I can’t help but think that others might judge me if I start exposing the limits of my intellect. It’s a useless worry because the groups I’m a part of are quite supportive, but this doesn’t lessen the anxiety I feel.
This has made me reflect on the kinds of learning environments we create. Are we thinking about how safe the environment is? Sure, we want to have teachers that are knowledgeable, but I would argue that those who can create a safe environment for learning are much more valuable.
There’s one particular aspect of the learning process that I think encapsulates how students think of a teacher: how teachers ask questions. What kind of tone do they use for the question? Do they make it seem like this is an inquisition, or is it a question to guide the student along? Is the question at an appropriate level? Perhaps most importantly, what kind of reaction does the teacher have when a student gives an answer that is partially (or totally) wrong?
These seem like questions which can be answered by instinct, but I think that’s leaving a huge opportunity on the table. If you’re not thinking about the way you ask students questions, you’re probably not doing your best work.
I’ve had a variety of teachers in my life, and some were better than others at asking questions. It could be nerve-wracking at times to answer a question, because you knew that the teacher would shred your answer if you were wrong. I’ve also had classes in which no one would answer questions. I would argue that this is a sign of not creating a safe environment.1
Flipping the script
As a tutor, I get the opportunity to see this play out from the other side of the equation. When I work with students, I try to ask them questions in order to figure out if they understand the topic we’re covering. What I’ve realized is that I haven’t approached this with the idea of creating a safe learning environment.
Each time I ask a question, I don’t care if they get it wrong. I don’t judge them or think less of them if they can’t answer a question. However, what I understand now is that this doesn’t matter to them. To them, they want to avoid looking “bad” in front of me anyway. As such, if I probe this area of weakness, it’s quite painful to accept that they don’t fully understand it. I’m basing this off of my own experience as a student, but I don’t think it’s a huge leap to say that others feel the same.
When I’m the student, I know that the teacher has my best interests in mind (for the most part). This isn’t a huge comfort though, and I still try to not expose any weakness. As such, I shouldn’t be surprised if students feel the same way when they work with me.
I’m fully aware that I need to ask questions in order to help them learn. It’s inevitable that they will get some of these questions wrong. However, I’m realizing now that perhaps a crucial first step is to make them comfortable with me. In essence, they need to not only intellectually know that they are in a safe learning environment, but feel it too. I don’t have answers for this yet, but I’m hoping to learn more as I continue teaching.
I suspect that the length of time I work with a student helps foster this feeling of safety. If I’ve worked with a student for two years, there’s a good chance they will be comfortable with me compared to a student who only just began with me. That being said, I’m interested in learning new ways to foster this feeling, because I think it can really help.
If you’re a teacher of any kind, please think about how you’re structuring your sessions to encourage safe learning. It might sound like an odd thing, but having students be unafraid to try an answer that they are unsure about is a fantastic thing. Getting to that point though doesn’t come for free, and requires some work. By prioritizing a safe learning environment, you’re sending the message to students that the questions you ask aren’t designed to expose their weaknesses to every other student, but are there to help them grow.
The more we think about how we ask our students questions, the better we can connect with them. And isn’t that what we want, in the end?
Of course, I’ve also had many other classes in which the class (including myself) was just lethargic or lazy, and didn’t want to answer questions. ↩
The Right Time
Do you hate mathematics? Have you found that the rules you have to follow seem to make no sense? No matter what you do, it never feels obvious, like the teacher says it should. Perhaps you can follow your teacher through a problem, but you know that if you ever had to do the problem on your own, there’s no way you could do it.
For someone that identifies with the above paragraph, I want to let you in on a secret. I feel the same way at least once a week, in terms of things not making sense. And I’m in the midst of completing a degree in mathematics.
The reason is simple: learning mathematics isn’t easy, and it comes in different levels. The latter is a crucial distinction to understand, because it can be the difference between finding mathematics engaging and creative versus being just a bunch of symbols.
There’s a reason that students have so many mathematics classes during elementary and secondary school. It’s because there’s a lot of material to cover, and one has to have a certain level of mathematical maturity in order to make sense of it all. Imagine if you had to learn about trigonometry when you were in the fifth grade. Sure, maybe you could learn how to manipulate a few symbols, but I doubt most would understand how trigonometry works. This is because students in the fifth grade aren’t ready to tackle those kinds of topics yet. As such, it would be a mistake to teach them trigonometry at that time1.
This example generalizes to any topic within mathematics. Mathematics is a process of layering new abstraction and tools to understand objects. Expecting students to understand topics without grasping the layers underneath is a recipe for confusion. Not only that, but it frustrates students. If you ask many of them, I know they want to learn. It’s just that some concepts make no sense to them, and they don’t have the time to go back and master the previous layers during school. This leads to poor results and lack of desire to do mathematics. Should we be surprised by their low enthusiasm?
After my first year in the physics program, I did research for the university in the summer. The field I worked in was alternative theories of gravity. These are theories that modify the usual recipe of general relativity in order to explain different features of our universe that we observe (particularly on the large-scale).
To understand this area of research, it should come as no surprise that a knowledge of general relativity is a must. To understand general relativity, you need to be comfortable with tensor equations, which are an abstraction of the idea of vectors and matrices. Notice that these don’t give you any insight into how these alternative theories of gravity work. They’re just the pre-requisites that allow one to comprehend what’s going on.
Coming into the summer, I hadn’t taken any course that helped with this knowledge. I didn’t take a course in special relativity, electromagnetism, Lagrangian mechanics, or any mathematics course that introduced tensors. This meant I had to learn everything on my own, from scratch.
There are some who are able to learn things on their own with ease. I’m not one of those people. It was a struggle to wade through the concepts. Not only was I learning new physics, I was also learning new mathematics. It was difficult, because I had no prior tools to work from. By the end of the summer, I would say that I had an “operational” understanding of the subject. I could do some computations, but I didn’t have a holistic understanding of the subject by any means.
I was in the same scenario as the one I described at the beginning. Things were confusing, and it was difficult to see the “whole” picture. This made it frustrating because I could more or less follow an argument, but I couldn’t see why it began as it did. In essence, I wasn’t approaching the subject at the right time.
Fast forward a year later, and things make a lot more sense. I’ve taken courses in special relativity and Lagrangian mechanics, which has made my understanding of general relativity much better. Just going over the same notes I wrote last year is so much easier. It’s remarkable how much difference a year makes, once you’ve taken the right courses.
My recommendation is this. If you find yourself struggling with a subject, ask yourself if you’re comfortable with the material that has come before. Chances are, you will find that it’s the surrounding details that make the current discussion difficult.
Everyone is capable of learning something new. However, you need to keep in mind the timing. If you’re trying to understand concepts that are more advanced than you are ready for, you’re bound to feel frustrated and confused. That’s not a sign that mathematics is not your thing2. It’s a sign that you’re not prepared.
What can you do about it? The first step is to be honest that you’re in over your head. This can be difficult, particularly if you’re in school and are supposed to know the material that was covered in previous years. However, the most important thing is understanding. There’s no shame in admitting that you don’t quite grasp a subject that you’ve already “finished”. In fact, it will help you out in the long term.
The second step is to go back within the subject until you can explain the concepts to someone else. That’s your new “foundation point”. You will build up your knowledge from there. It doesn’t matter how far back you go. Keep going until the concepts are obvious to you. Once you’ve found that point, revisit the next concepts one by one, until each one seems evident to you. This will take some time, but it will ensure that you don’t have a shaky foundation.
I can’t deny this is time-consuming. Worse, you will have to juggle doing this while also working through your present subject in school. But, the idea here is to get back to a place where the learning feels like the “next logical step”. If learning is frustrating, we are unlikely to continue. That frustration is often a symptom of learning new things without mastering previous material, so a great way to proceed is to go back back and strengthen the fundamentals.
It’s all about tackling new subjects at the right time. If you do that, learning will be much more rewarding.
Note that I’m not saying that young students can’t earn advance material. Instead, we should be aware of the prior knowledge that students need in order to tackle certain topics. ↩
I’ve concentrated on mathematics and physics here, but this concept applies in general. You want to make sure that you don’t try to take something on without the proper preparation. ↩
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.