### Do You Understand?

When teaching a concept, it’s natural to suppose that the person you are working with isn’t able to follow each statement you make. This is a consequence of the fact that the material is new to them, which means it takes time to understand the material. I’m sure we can all think back to instances in our education where we only partially understood what was going on.

I also predict that the teacher asked the class a common question.

“Do you all understand?”

At first, this seems like a reasonable thing to ask. After all, *shouldn’t* part of your job as a teacher be to make sure that everyone is following the argument? If no one understands the concept, then it’s a waste of both your time and your students’ time. It makes sense to suppose that “checking in” on students to make sure they are following is the right thing to do. If the class answers “yes”, then you can continue on without worrying.

Unfortunately, the reality is not so nice.

In an ideal world, students would make a teacher pause and go over a certain argument again if they couldn’t follow. They would ask questions and the teacher would answer them. Understanding in the class would flourish, and everyone would be happy. But this doesn’t happen. In fact, it rarely even comes *close* to happening. A class might have one or two questions thrown in, but by and large, students don’t ask questions, even if they can’t follow.

Should we blame students for being unwilling to participate in class? I want to argue that instead it is up to us as educators to create learning environments that are more suited to asking questions when something isn’t clear. Furthermore, I want to argue that you should try to *avoid* ever asking the question, “Do you understand?” Instead, there are much better alternatives available to you. I also want to note that the following is applicable to both classroom teaching as well as one-to-one teaching.

## Those three little words

I’ve asked this question so many times that I think I do it almost out of habit when I work with a student. It’s instinctual, because I want to make sure that a student is following what I’m saying. I don’t want to assume that they are following, and then be caught down the road and find out that the argument *wasn’t* clear to them, requiring us to start over. If possible, I think that’s something all teachers strive to avoid. In particular, some students drift off during explanations, so this question also acts as a way to bring them back to the topic.

However, we have to consider this question from the side of the student. What do they think about when they hear the question “Do you understand?” As a student myself, I have two possible reactions. If I do understand, then I don’t have a problem and answer “yes”. If I *don’t* understand, chances are I won’t feel like asking a question is worth it. I tell myself that I will figure it out later on my own. As such, I still answer “yes” to the question. But notice how different these two scenarios are! Despite not understanding, I might still say that I do.

This isn’t the only reason students answer in this way. There are many reasons, depending on the student. Some students will follow the tendency of the group.

A simple scenario:

*If a few students say that they understand, that must have meant the teacher did a good enough job at explaining it. This means I should understand it too (even if I don’t). Therefore, I’ll say “yes” in order to follow the group’s lead. I don’t want to look like I’m dumb or can’t understand, so I’ll hide it.*

A student may *want* to ask a question, but they fear that their question will showcase their ignorance, making them seem stupid to their peers. Depending on the class and the kind of institution one is in, this can have a *big* effect. If the group of students is roughly the same throughout their whole school life, showcasing one’s ignorance to the peers that they will be with every single day might not seem like a great risk to take. Therefore, it seems like a good idea to just say that everything is understood.

This is something that definitely happens, so as teachers, we need to be vigilant for it. Students *won’t* necessarily give truthful responses to their answers. As such, my advice is to not ask the question! Don’t even set them up to be in a situation where a student might be pressured in to saying they understand even if they don’t.

## Rapid response

There’s another angle to this that I think a lot of teachers don’t realize unless it is pointed out to them. While some may argue that their students are being truthful to them when they ask the question (which I think is debatable), **a lot of teachers don’t give ample time to pose this question.**

Have you ever asked your class if they understand the argument that you just went through, while at the same time erasing the board? It makes sense to you to do these tasks simultaneously, but the situation is different from the students’ perspectives. *They* see a teacher who is asking the question in an off-hand way (since you’re barely paying attention and cleaning the board instead), so the students infer that this must have been an easy topic. Therefore, asking for further clarification might seem like an extra hassle that isn’t worth the effort.

Here’s another scenario that I have encountered many times with teachers. After going through a long and detailed argument (where students are grasping on the thread that weaves everything together), the teacher stops writing on the board, turns around, and ask, “Does that all makes sense?”

Meanwhile, the student is just trying to keep pace with the writing on the board. They are still a few lines behind in the argument, and haven’t had enough time to process what just happened, let alone ask themselves if they understand what is going on. They say nothing.

The teacher, seeing no one raise any questions, waits for a moment or two before diving back in, oblivious to the fact that it has barely been two seconds since they addressed the class.

The lesson here is simple. **Give your students ample time to look at what just happened in the class and think about what kind of questions they have.** Yes, that means

*waiting*for a period of longer than three seconds. Yes, that means you might not be able to transition from topic to topic as fast as humanly possible, but the benefit is that the students will feel like they have breathing room in the class. Therefore, if you have built a good learning environment in your classroom, the students will be able to formulate questions for you to answer.

## How to not ask “Do you understand?”

A teacher shouldn’t need such a blunt tool to gauge if their students understand. A teacher who has taken the time get to know their students and make sense of their strengths and weaknesses needs to be able to probe the class for understanding *without* asking that question.

To do this, the best way I know of is to ask conceptual questions that refer to the topics just learned. This way, you’re not able to get away with students saying they understand even if they don’t. The students have to *show* some understanding of what is happening. This is also helpful because it means the students can engage with the material instead of being able to drift off and not pay attention.

How you pose these conceptual questions is up to you, but this alleviates the need to ask students if everything makes sense. As long as they are getting the correct answers to the critical points in your questions, they are demonstrating that they know enough about the new topic.

I know what you’re thinking. This is a great *idea*, but what happens in practice is that there ends up being only one or two students in the class who answer all of the questions. I can’t generalize from these two students to the entire class! Of course, you’re correct. That just means you have to be on the lookout for this pattern. If the same few students are always answering the questions because the rest of the class won’t participate, change up your strategy. Force them to break up into groups and think about these conceptual questions. *Don’t* force them to speak in front of the entire class if they don’t want to, but give each student the opportunity to think about their understanding of the subject in a deeper fashion than a three-second reflection.

I can already imagine another critique of these ideas: they sound great, but there’s already so little time in a class and so much material to cover! Splitting up into groups and discussing would make this even worse.

I hate to break it to you, but that’s the price of admission if you want students to not answer your repetitive question in the affirmative even when they don’t understand. As a teacher, it’s our job to probe their understanding without just asking them. The dynamics of both teaching to a group and teaching to an individual can lead to students telling teachers what they want to hear. I’ve had this happen to me as a tutor. It takes probing a little deeper before they admit they don’t understand. I often kick myself when this happens, because it means the student feels like I will judge them for not understanding a concept. The reality couldn’t be more different. I absolutely *want* to know if they don’t understand, so I can help them get to that point of clarity!

This brings me to my final point. **We need to talk to our students and establish that asking questions is a good thing.** Being unsure about something is great, because it tells you exactly where you have to look to improve your understanding. Asking questions and getting clarification is something I think we all wish we did more of as students. In the short-term, it’s easy to just nod along in class and say that everything makes sense, but if you want to focus on long-term growth, asking questions is so important. That’s why you need to convey this message to your students all the time. What I tell my students now is that they need to be comfortable with stopping me mid-sentence and say, “Jeremy, you’re making no sense.” If I can get them to feel comfortable doing that, then I’ve created a good learning environment for my students.

### Act of Translation

When you’ve understood a subject, there’s a tendency to take parts of the subject for granted. You make more assumptions about how things will work, and why one thing is used versus another. After all, you have spent so much time thinking about the subject that you know it inside and out. This familiarity implies potential sources of confusion don’t even cross your mind because you *know* how wrong they are. In other words, you start to figure that everyone else probably knows the same amount as you do, so certain things aren’t stated.

This happens quite often in teaching. I’ve been on both sides of the equation. The teaching side of the situation is like I wrote above. It’s not that a teacher is ignoring sources of confusion on purpose. It’s that they are so far past that state of confusion that it doesn’t even register. Of *course* that equation works in this way, how else would it go?

As a student, this can be frustrating. Often, the questions I have as a student are ill-posed, and not quite thought out. This is to be expected, since I’m learning the subject! However, it then becomes a challenge when asking a teacher, because I have to try and formulate it in a way that *they* can understand, though I barely understand it myself. This leads to teachers answering questions that weren’t being asked. What’s interesting though is that, as a student, I can understand what one of my classmates is asking, even though the teacher may interpret the question in a different manner. This suggests that there is something to be gained by being in the space between an expert and a beginner. Here, you are able to understand what the beginners are saying, because you are aware of how you had those same difficulties not long in the past. This means that you are in a unique position to help those who are struggling right below your level.

This is part of the reason I am a tutor. I know that it makes for good practice in the art of explanation, because it requires me to relate to the difficulties a student is having. **When they have a question, not only is the answer itself important, but the source of the question is as well.** What kind of perspective did they have on the concept that caused them to ask this kind of question? Does this perspective suggest that they lack some understanding in another related concept, or was it only with this specific concept? These are questions that you need to reflect on when teaching. Remember, you can’t just go and ask them! If you could do that, then you wouldn’t need to be helping them.

If they’re having difficulty, it’s up to *me* to find a way to relate the new concept with something they have already seen. I shouldn’t necessarily seek to convert them to my way of thinking, at least not right off. If a student is having difficulty in some area, what they *don’t* need is to be converted from their way of thinking to my way of thinking. Sometimes, this might be an option if their mental model is off from what the concept, but often only smaller nudges in the right direction are needed. Therefore, the better approach would be to understand *their* way of viewing things, and try to bridge the gap from what they understand to what I understand.

To get a feeling for why trying to completely switch a student’s viewpoint about a concept doesn’t work well, think about *any* subject that you have difficulty in. Then, while working with one particular resource in that subject (say, within a textbook), try and get your question answered by looking at *another* textbook. I bet you will be more confused with the notation and the manner of presentation, which will end with you not getting your question answered. The *better* way to understand a topic you are stuck on is to ask someone who is both familiar with the topic *and* your specific textbook. Then, that person will be able to help you, since they can speak the same language as you.

I’m not saying that students should always stick to one textbook or one resource. Getting information from multiple viewpoints can be helpful. However, trying to use multiple resources while stuck may lead to more problems. At minimum, a student will have to “re-learn” the concepts from the new resource, which takes time.

This is similar to the situation in which a student does not understand a topic and a teacher is trying to parse through that misunderstanding. The teacher has to be aware of the way the student is thinking about the subject, and they have to figure this out indirectly. It’s not an easy job, but it’s the challenge of good teaching (or good tutoring).

Remember, if you’ve taught a subject for a long time, chances are that you’ve acquired blind spots that students can be trapped in. It’s up to you to look around and make sure that you’re understanding the student’s difficulty from *their* point of view. Otherwise, you’re setting yourself up for a conversation in which both student and teacher are speaking past each other.

### Concepts and Mechanics

As someone who teaches, I struggle with striking the right balance between explaining a concept to a student and giving them the steps to *solve* a problem (the mechanics). At first glance, it might seem like these two ideas are the same, but a student experiencing difficulties will often need one without the other.

If a student doesn’t understand a concept, then they will have a difficult time solving a problem. If they don’t understand that Newton’s third law talks about equal and opposite forces, then solving the problem is impossible. Maybe they forgot to include a force in their equations, giving the wrong answer. At the heart of the issue is that, conceptually, they’re missing the details that matter to solving a problem.

On the other hand, there are plenty of students who understand the concepts. Qualitatively, they can explain anything regarding the topic at hand. However, the quantitative aspect can be difficult, even when the concepts are mastered. This is because mathematics isn’t always easy! Even if a student has a rough idea of what they must do to solve a problem, this doesn’t mean they are comfortable with the tools to solve it. I’m familiar with a bunch of physics experiments, but as I’ve seen when going into the laboratory myself, I am *not* adept at carrying them out. Remember, there’s a big difference between understanding and being able to apply knowledge.

To address the former problem, we need to give more explanations. This is when we can review topics seen in class. The most important thing to do is **get the student to explain the concept**. This is at the heart of learning. If you can’t give an explanation of what you read or learned, you haven’t learned anything at all. Often, students are incentivized to put the majority of their attention on following predetermined recipes, while foregoing the explanations of concepts. I know this happens because I do the same thing. It’s difficult to spend your time studying for a test by explaining concepts when you *know* that the test will have little (if any) explanations required.

Despite that, while practicing and getting stuck on problems, it’s a good idea to try and get the student to explain the concept they are struggling with. This will either give you, the teacher, something to grab in order to guide the student to the correct explanation, or the student might even resolve their difficulty by talking about it.

If their difficulty lies in the mathematics themselves, the worst thing to do is start lecturing about the concepts. They know the concepts! Instead, a better move would be to work through a related example with them. That way, the student can see how the pieces move to solve a problem. It often only takes a worked example to give the student the tools they need to continue.

It’s important to note that we often think about addressing the former problem, but not the latter. We don’t want to give the students the answers, and instead we tell them to keep working at it. But that’s the wrong mentality. When first learning a new concept, a student needs the help of examples to ground them. It shows them how the new mathematics or concepts that were developed in theory *apply* to solving problems. It might seem like this should follow through from the theory, but often this doesn’t happen. By hammering home examples of how to use the tools, students can more easily solve problems later on.

Of course, we don’t want students to be *just* equation machines. So emphasize the concepts, but keep in mind that they don’t necessarily translate to the mechanics of solving problems. These are two different skills, and they need to be taught as such if we want students to be adept at both.

### Jumps in Abstraction

If I ask an adult to tell me what *3-5* is, there’s a good chance that they would tell me the answer is 2 without much thought. This kind of arithmetic is simple to us, since we’ve had to do it over and over again through elementary and secondary education. Even if we haven’t used mathematics in a long time, these questions are straightforward.

But it wasn’t always this easy. Remember, we aren’t born with this innate sense of negative numbers. In fact, we wait until the end of elementary school before seeing negative numbers. Before, this, if you ask a young student what *3-5* is, chances are that they will answer “2” (because they figured you meant it the other way), or if you explain that you are indeed talking about *3-5*, they will tell you it isn’t possible.

You and I can both imagine what comes before the number zero. But to the young student, it’s not that they don’t have the imagination. **They don’t even know that this extra richness is there.** This is an important insight, because it signals to us that we get comfortable within our usual mathematical spaces. Consequently, we can become blind to the generalizations that are possible, just like the young student who can’t imagine that there is even such a *thing* as a negative number.

Going up a few levels in education, a lot of secondary school concerns itself with geometry. Students learn about perimeter, then area, and then volume. But the kinds of topics that are covered during these explorations are limited in scope. Students learn about regular solids like cubes, prisms, pyramids, cylinders, cones, and spheres. These are nice, because they allow teachers to combine them to form more complex solids. The task for the student then becomes figuring out how to separate a large, composite object into a bunch of smaller objects and add their corresponding volumes.

This is a great exercise for a student. However, by focusing on these core solids and only dealing with different combinations of them, the students don’t get to see the richness *beyond* those solids. The world isn’t only made up of those solids! We have plenty of other interesting forms that we can find in nature, from a sprawling tree to a curvy egg. These aren’t the simple solids that students are used to. Furthermore, what about the amazing objects that mathematicians have come up with, such as Torricelli’s Trumpet? I can just imagine the interest that would be generated when showing students how this particular object has an infinite surface area, yet somehow has a finite volume! Of course, one would have to work within the constraints of limited calculus knowledge, but I’m certain that this could work.

Sometimes, we have to get out of the thicket of working through particular problems, and figuring out where we are on our mathematical journey. By doing this, students get better at understanding the context that surrounds what they are learning, rather than simply keeping their heads down and working on problem after problem. That strategy may be “productive”, but it will ultimately hamper students’ awareness about the wider mathematical world.

I’m not advocating here for a radical change in how one teaches (that’s a different story). What I’m arguing for here is to give students a broader idea of what the results they see *mean*. It can be as simple as sowing the seeds for deeper connections for students to think about. The goal should be to make sure the students know that there is always more to uncover if they so choose. I absolutely *don’t* want students thinking that they have learned all that exists in mathematics by the time they are done secondary school.

It’s also good to note that this isn’t only important within secondary and elementary schools. This is something that should be done in *all* levels. Throughout this past semester, I’ve been pushed to consider many more mathematical spaces in my abstract algebra class. While many spaces take cues from spaces like the integers, the rational numbers, or the real numbers, there are many other spaces which have similar attributes, and can be studied beside our usual settings. The essence of algebra is preserved, even though we don’t know which explicit space we are talking about. This is both neat and difficult to wrap one’s mind around. It means that you have to step away from the comfort of the familiar spaces and explore new ones. It means opening up your perspective to a vast new world of possibilities.

I want students to get a sense of *that* while studying, to show them that there is always more to learn, if they are so inclined. Let’s do our best to get students out of their comfort zones, and be surprised and delighted every once in a while.