Attempting Problems

Flipping to the back of the book.

One of the most important things you can do when learning a new subject is attempting it with your best effort. While this sounds simple, so many people think they are too good for this step and skip it.

I was reminded of this at the beginning of last semester. One of my professors said, “When you read the chapters in the book, do the examples that are there. It’s extremely important that you do. It might be tempting to go through and read the solution of a problem and then say, ‘Oh yes, that makes sense.’ However, doing this won’t help you out in the long run.”

I couldn’t have put it better myself. This is exactly the kind of issue I run into all the time, and I know that it can affect others just as much. We have this idea that reading implies learning, but there are many ways to read a textbook and learn next to nothing about solving problems. This includes skipping through the exercises and examples, preferring to just read the solutions. This is good up until the point that you are faced with a question on your own, such as on a test. At that point, you may find that the problems are not so easy.

It’s simple to say, “Oh yes, this solution makes sense. I get it.” It’s much more difficult to think of what that solution is on your own. But this is what doing the examples in a textbook is for! They are there to allow you to think of the solution without a helping hand. The solutions are for after you’ve done this step.

This is so important, because I see people all the time take ten seconds to read a problem before saying, “I don’t know what to do.” Well, of course you don’t know what to do! You’ve barely had time to think about it at all.

The purpose of doing all these exercises is to give you a sense of what works and what doesn’t. As a consequence, you start to gain an eye for what a solution might look like. When I tell the students I tutor how I solved the problem they are struggling with, they look up at me in disbelief, as if they would never be able to do what I did. But they’re wrong. I didn’t do anything special. In fact, I did something I’ve done a bunch of times. That doesn’t mean I was as good when I was in the same year as them. It just means I’ve been at it longer and have put the work in to make those problems trivial.

It seems like a waste of time, but the real key to learning and succeeding in your studies is to attempt to do problems on your own, without referring to your text. I’m not suggesting you memorize every single formula in physics. Rather, I’m talking about learning to come up with your own strategies to solve problems. From there, it’s just a lot of practice and refinement. If you prefer to skip the work and read the solutions, this skill won’t be developed. You’ll always be looking for the “trick” to solve problems, instead of applying techniques. Therefore, it’s in your best interest to work through as many problems as you can. It will pay off in your future.

Permission to Learn

When you were in elementary school, did you get a say in what you learned? How about in secondary school? Odds are, you were given a set of classes, and your time at school looked something like this:

You're going to study this or you won't be graduating.

We all had these instances when we wondered, “Why am I learning this?” We were then either told that we had no choice, or that that there were benefits to learning that material that we couldn’t see right now. One of these must have seemed reasonable, because you continued on.

We then get to university, where the classes are more varied, yet we still have to take certain ones. Hopefully by this point you like the program that you’re in, so this doesn’t seem as bad.

My question to you is this. Have you ever decided to learn something despite it not being a part of a class? Did you ever decide to learn something just because you were curious?

If the answer is “no”, I encourage you to ask yourself why you haven’t. I’m sure that there must have been something that piqued your interest at one point. Why didn’t you follow up on it?

The point I want to get at here is that, now more than ever, we don’t need permission to learn something new. Sure, what you’re curious about might not be on the test in class. It might be unrelated. But you’re curious, aren’t you? That’s enough, and it’s why you need to learn on your own.

This doesn’t tend to be an instinct, because our education system doesn’t prepare us for this. From the get-go, we’re told what to study. Forget about everything else, it’s not as important to you. Focus on what you need to know for the test.

That’s fine advice if you want to get good marks, but it doesn’t help you grow. We have so many resources available now that you can learn pretty much anything you want. Why not use that opportunity to move away from the curriculum you got from school and learn something that interests you?

We can all use the reminder. We don’t have to be in a class to learn something new. Giving yourself permission to learn something new is a powerful first step to growing much more than you thought possible.

Motivating New Tools

Multiple roads lead to the answer.

One of the great things about mathematics is that there tend to be multiple ways to solve a problem. This implies two things. First, there is no “correct” way to finding a solution. As long as the approach is logically sound and produces the answer, it is a good method to solve the problem. Second, multiple methods imply a fastest method. This method might take less work, be more obvious, or even be simpler. Of course, keep in mind that this is a subjective measure of “simple”. One method that might be simple or obvious to one student could seem bizarre to another. It’s a matter of perspective, yet the fact remains that a given student will find one approach to a problem easier than another.

If one approach is “easier” than another, it lends itself well to developing new mathematics. In particular, a lot of mathematics that is taught in school (in particular, secondary school) is there because it solves a historical problem. (Whether this historical problem had any practical use is a different question!) In other words, a lot of the tools of mathematics are taught to students because they solved a problem.

In post-secondary education, the standard example would be of integral calculus. How much area is under a given curve, and how does this relate to the derivative of that function? These kinds of questions are answered with the tools of calculus, and have applications all over science.

The problem, from the student’s point of view, is that this isn’t how the new tools tend to be presented. Instead, they are given the new tools without the prior motivation, which can make learning these new tools seem pointless. This is even more poignant when students are learning techniques to solve problems they have already learned how to solve. I can hear the question: Why do I need to learn about a new way to solve a problem I know how to solve?

This question isn’t answered as often as it should be. Remember, before a student gets into pure mathematics (where the emphasis is more on proofs and results), mathematics is computational in nature. This means the student is learning how to use new tools to solve problems. While “real-world” examples are given, they don’t tend to capture the heart of why these tools are useful.

Here’s an example where the motivation for the tools is good. Students in physics start by learning the kinematic equations of motion, which hold when the acceleration is constant (such as the case of gravity on the surface of the Earth). Once students learn calculus, they are able to tackle problems in which the acceleration continuously changes. Armed with these tools, they can now address complex problems that their old tools were not equipped to solve. The use of calculus is clear to a physics student. It’s not that it let’s you solve equations of motion. It’s that calculus is able to model the continuous change that is present in a lot of our world. That’s the point of calculus, in a nutshell. How can we quantify change, even when it isn’t constant? Physics students get to experience this sort of motivation.

On the other hand, a lot of students don’t get to see the motivation underlying their tools. Instead, they are introduced to a new rule or concept, and then are assigned practice in order to get better at applying the rule. There’s little mention as to why these rules are as they are, or why these tools are important.

I want to note here that I’m not advocating for motivating mathematics with “real-world” applications. I don’t care if the mathematics that one learns has any inherent use, because that’s not the point (at least, not the whole point) of mathematics. What I am interested in is the question, “What made a mathematician so inclined to invent this mathematical tool?” That’s an interesting question that isn’t dependent on an external application.

Teachers need to find these reasons and explain it to the students. It will achieve two important functions. First, it will be easier for students to answer conceptual questions, because they will know why this tool was needed. By knowing the “why”, students associate mathematical tools with a specific purpose. The second function is that students will be more on board with a new method. Have you ever tried teaching a new method of solving a problem and got a lot of resistance from students who prefer the older method? That’s not an accident. Either both methods are equally good (in which case the students have a point about not wanting to learn a new method), or else the new method is better. But, you aren’t teaching it in a way that convinces them to come on board with you. The latter part here is crucial. If the new method is truly better, you need to show that to the students. I always think back to addition and multiplication. If you have a bunch of objects you need to count, we all now know that it’s so much easier to arrange them into a rectangular array so that we can multiply them. We would much rather arrange 800 objects into 40 rows of 20 and perform (40)(20)=800 instead of counting each object individually. The “tool” of using arrays with multiplication just blows addition out of the water! You need to replicate this experience with your students for all topics.

If you feel like your students aren’t understanding why a new mathematical tool is so much better than an older one, I would argue that it wasn’t presented in a way that made this fact clear. Sure, in hindsight it might be clear, but by the time students have this benefit of hindsight, they will have long-completed the class. In order to get them to understand how a certain tool solves a problem the “best”, presenting the historical need for this tool is a great way to go. It’s not that teaching things in a historical manner is the best way to teach, but history does give us some guidance as to what problems can motivate a new tool. Use this as guidance when you’re planning your new lesson, and chances are the students will be both more on board with you, and will have a better conceptual idea of why they need to learn this new tool.

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.