Only Numbers and Algebra

Learning mathematics in school and doing mathematics in general are not the same thing.

This might seem obvious, but I worry a lot about students that don’t have a chance to realize this before they are turned off from mathematics forever. The reason is that the message which is sent to students throughout their years in elementary and secondary school is that mathematics is all about numbers, but this is false.

Sure, a lot of mathematics involves numbers, but it’s a mistake to make the leap that mathematics is all about numbers1. Mathematics is a way of thinking. There’s so much more than equations and formulas within mathematics.

Students get a quick glimpse of this when they are young. During their first years being exposed to mathematics, they learn about shapes and patterns. There’s no algebra involved. Their sense of numbers is only beginning to get sharpened, so the curriculum is focused on other areas of mathematics. Notice how they are still learning without needing to transform everything into equations.

Fast-forward a few years, and the focus has shifted. Now, students are getting used to doing arithmetic, and applying this knowledge to algebra. From here on out, most of the emphasis is on getting students to be proficient with equations. Students learn about probability and statistics, graphing equations, solving quadratics, doing word problems, solving trigonometric relations, exploring vectors, and thinking about geometry. However, in all of these subjects, the emphasis is almost always on using algebra to solve problems. Even in the case of geometry, the use of symmetry is often substituted for brute-force equations. And what do students learn equations are for? Plugging in numbers to get an answer out. The theme of a problem becomes finding the right equation that will spit out the desired answer.

I’m not saying that we should ban equations from ever being used in class. They are a great tool to get a handle on the essence of a problem. But this total emphasis on algebra shoves aside other areas of mathematics. Areas like graph theory and it’s parent field, discrete mathematics, are quite accessible to students at the secondary level, and do away with a lot of algebra2. The point isn’t to do away with algebra, but to look at some subjects which don’t emphasize its use.

What if the student didn’t have to touch an equation, unless absolutely necessary?

I think this change would help some students see that mathematics isn’t just about “finding the number” to solve a problem. Rather, it’s a method of thinking about how we can boil down the essence of a question or a problem into something we can manipulate. This is not limited to the areas I highlighted above. Heck, I can even see a project where students learn how to use equations in order to create mathematical art. Even this is different, because it removes the emphasis on equations (though they are still there!) and more toward creation.

Students tend to have strong opinions on mathematics, which is informed by their experience in secondary school. I’m hoping that we can do more to remove the idea of mathematics being all about numbers and equations. It’s so much more, and I think diversifying our offering to students is key to changing that mindset that some students come away with. For some, this is the mindset they will carry through for the rest of their lives. Even if they don’t study mathematics anymore, I want them to leave with a fair view of the subject, not one informed only by solving endless equations.

  1. In fact, I would argue that students don’t even get to encounter the mathematics of numbers, which is the study of number theory. This isn’t part of the curriculum (at least where I’m from). 

  2. I also realize that we do use a lot of numbers in discrete mathematics. But it’s of a slightly different type. Instead of formulas, we often compute quantities like permutations and combinations which can be visualized and don’t require as many equations. 

Not Necessary

In mathematics, the terms “necessary” and “sufficient” have technical meanings. These terms come about when looking at two statements P and Q. If we say that P is sufficient for Q, then that means if P is true, Q automatically has to be true (P implies Q). On the other hand, if P is only necessary for Q, having P be true doesn’t mean Q has to be true (but the other way works, so Q implies P). If we have the P is both necessary and sufficient for Q, that means having one gives us the other for free. They are tied together and are inseparable.

The reason I bring this up is because I know some students have a tendency to “go overboard” in their quest for good grades. They will do anything to get the best grade, and anything less than perfect is unacceptable. This leads them to working more than everyone else, risking burnout so that they can get the best grades. If you’re a student reading this, you might recognize yourself in these words. I know I do.

The issue is that we’ve turned grades into a “necessary” condition to being considered “good” in school. This is dangerous, because grades are not under your total control. Yes, your work is what gets graded, but it’s your teacher who makes the final decision. Even in subjects where grading schemes are more rigid, like mathematics and science, your teacher is the one assigning your grades. As such, they are external factors that can’t be fully controlled through hard work.

Yet, we convince ourselves that we can control them. We turn the goal of perfect grades into a necessary condition to being a success. When this inevitably doesn’t work out, we then feel like failures. Worse, we might resolve to work even more, just to make sure it doesn’t happen again. This pushes us further into the cycle, creating an unattainable goal.

We need to stop looking at great grades as a necessary condition to being a good student. Sure, we need to work hard and give our best, but we should view that as a sufficient condition to being a good student. We can always give our best effort. That’s something we can control, since it’s an internal choice. Unlike getting perfect grades, it is realistic to say that we will do our best in every assignment and test we do. We have to learn that this is enough.

The beauty of this strategy is that we get to decide if we are a success. Did you do your best and worked hard for that last assignment? If so, then you can look at yourself as a successful student. Notice how this isn’t contingent on getting good grades. Instead, it’s about keeping your focus on aspects of your learning that you can control.

Remember, trying to control external factors in your life is a hopeless pursuit. It may work from time to time, but overall it is a losing strategy. Instead, apply your effort to things which you can control, like your effort. You will find that it makes you much more relaxed, leading to better work.

Memorization in Education

Educators have unrealistic views about memorization.

If you’re reading this as a student, think about any time a teacher spoke about their thoughts on memorization. Unless the point of the class was to memorize certain facts, I’m guessing the teacher did not love the idea of memorization. In fact, for those who teach subjects such as physics or mathematics, they might have gone on a rant about how memorization is a terrible thing to do and one should focus on understanding the material. They might tell you that it’s not a problem if you forget a small detail in a formula. You know the concepts, so you should be capable of working out the details if you ever forget them.

If this describes a teacher you’ve had, they weren’t being fair at all.

Before you start thinking that I see myself as apart from this issue, don’t worry, I’m just as culpable. I’ve given reasons quite close to what I wrote above to students, even though on reflection I know it’s not true.

It’s a seductive message. If you know your core concepts, you can work out anything you forget. Therefore, you don’t have to be a “memorization machine”. You can focus on deeper understanding, not on memorizing formulas. It’s the dream of every teacher. It indicates the students are engaged with the material, not on passing. What more could a teacher want?

The problem is that our educational system does not reward deep understanding. Sure, deeper understanding can be good for the student in the long-term, but there is no incentive present in the system. This creates a dilemma for the students. Is it worth investing the struggle and the effort for deeper understanding, or should they only focus on getting good grades?

“But wait,” you might protest. “Getting good grades in school requires the student to have a deep understanding of the subject.”

This argument is incorrect. A student that has a deep understanding of the subject might do well, but it’s by no means necessary for success. From personal experience, I can say that getting good grades in a class did not mean I had a deep understanding of the subject. It just meant I did well on the assignments and the tests, which is what the grade reflects.

This is why most students don’t have deep understanding as their primary objective. They are making a calculation, and coming up with the answer that the most important thing is to get good grades. Since a deep understanding isn’t necessary for success in a class, students won’t seek it out. I’m not saying that they don’t want to have a deeper understanding of a subject. It’s just that they won’t be sad if it doesn’t happen.

The world rewards good grades, so that’s what students have learned to chase. It means that if they don’t understand a concept, looking for a deeper explanation may not be the first thing they try. Instead, a student may opt for memorization, because it’s the most efficient way to achieve their goal. As teachers or tutors, we can’t blame them for that. They’re acting in their own best interest (at least, in the moment).

So what’s the solution? As usual, I have no clue, but I do have some suggestions. First, we need to accept that memorization is how students choose to study in some classes. That’s not going to stop until our grade-crazy system is changed. Instead, we should encourage students to look at our explanations and ask themselves which is easier to remember. If it’s easier for them to memorize the information, than I don’t think we should do anything about it. A deeper understanding of a subject should end up looking “obvious” to students, in the way that memorization is not. After all, a deep understanding creates a sense of cohesion that pure memorization cannot.

The other suggestion is to create questions that don’t fit the cookie-cutter mold. By asking students about the concepts in a different way than usual, they won’t be able to just bring up a formula. Questions that aren’t only plug-and-play will reward students for doing more than memorizing. Plus, it should translate to getting good grades.

It’s a long road from memorization to deeper understanding, and the external incentives aren’t there to encourage students. However, there are long-term benefits of a deeper understanding, so it’s important to convince students that memorizing isn’t the only way forward.

What Counts As Cheating?

This sounds like an easy question, but I think I have a bit of a different perspective on it.

In science and mathematics programs, it’s not too difficult to cheat. I’m not talking about cheating on tests or exams, but on assignments. That’s because most professors don’t write their own problems. Instead, they assign them from textbooks. The upside is obvious: they don’t have to spend time crafting new questions every semester. Unfortunately, the downside is that textbooks questions tend to be posted online. No matter how obscure the book is, there’s a good chance that someone has taken the time to post the answers.

This presents a dilemma for the student. On the one hand, the answers to the assignments are right there, waiting to be looked at. On the other hand, is looking at those answers considered cheating?

I understand those who take the perspective that it is cheating. After all, the student isn’t doing the work themselves. They are outsourcing it to someone else online. Because of this, it should count as cheating. I think this is a good line of reasoning, and it makes a compelling case as to why students who use online answers are cheating.

But, I still disagree with this assessment.

I think I have a more nuanced view of the problem. As in most things in life, the truth is that it depends on the situation. It’s easy to slap a declarative statement on any discussion, but it often hides the complexity of the situation. I think this is the case here, and I want to make an argument for when looking at the answers online shouldn’t be considered cheating.

This is how a typical assignment goes. The teacher gives it to the students, along with a due date. The students then work on it during that period, asking questions if necessary. Most of the questions get answered, but some might not. The assignment is handed in, and is handed back perhaps two weeks later. By that time, the student has forgotten the content of the assignment, and going over corrections isn’t the most compelling thing (especially since there is new homework to do).

But what if instead the student works through the homework, trying all of the questions, and then looks online to check their work? By doing this, they can check their line of reasoning and make the necessary corrections to their argument before handing in the assignment.

I think this method has several benefits. First, it gives the student quick feedback on their work. Instead of waiting weeks before an assignment is handed back, they can check their work as they do it. This keeps the problem fresh in the student’s mind. Plus, it’s more motivating to do corrections during the time you’re working on an assignment than later on when you have other things to do.

Second, checking the answers online encourages deeper thinking about a problem. How so? Suppose you work through a problem, and think that it’s straightforward. You then compare your answer to one online, and you see that there’s a whole lot more to the question than you realized. This makes you stop and say, “Oh wait, I forgot about this particular aspect.” The result is that you go back and figure out what you did wrong. If you never checked your work, you would only see the mistake you made a few weeks later. By that time, you probably won’t put reworking through the problem high on your list of priorities. That’s why it’s worth checking your answers while you write them. It gives you a chance to dig deeper on a problem that you went through too quickly.

Third, checking your answers online helps you avoid losing marks. Yes, I know this is controversial. If you’re getting help online, did you really “earn” it? My answer to this is “yes”. That’s because I’m thinking about what the main purpose of school is supposed to be: to learn. Not to get things on the first try, not to produce the best work on your own, but to become competent in whatever subject you’re studying.

Tell me who you would consider to be “learning” more. On the one hand, you have a student who does their homework, doesn’t check their answers online, and gets decent grades. On the other hand, you have someone who does the questions on their own, then checks online and makes the appropriate changes before submitting their work, and gets great grades. I would say that the person who consults the answers online is doing more work. Not only do they try the problems themselves, they then do the extra work of making sure their answers are good and doing the necessary corrections. By the end of the assignment, I would say this person has thought more about the questions than the person who does the work themselves and submits it as-is.

Sure, the second student consulted the answers online. But in the grand scheme of things, I don’t think this should be a bad thing. The point is that they are invested enough in their understanding (and success) that they took the extra time to look at the answers and fix their own. By the time exams roll around, I would predict that this kind of student has had more experience with the content than the other person. This is why I don’t think shooting for better marks is a bad thing. The marks help you as a student, and it’s not the point of learning anyway.

Like anything, this perspective has some caveats. You can absolutely find yourself in a situation where you forego even trying the assignment because you looked at the answers online. As you can imagine, this isn’t in the spirit of what I meant. If you’re going straight to the answers online, you’re not learning too much (except how to find the answers you need). Likewise, if you finish your problems and “check” online by just copying the answers you find, that’s of limited use as well. The best way to do it is to slowly look at the answers online, and make sure they match what you have. Anytime something is different, don’t keep on reading the answer. Go back to your own work and see if you can spot an error in reasoning.

The reason I think my perspective here is a bit controversial is because some people might complain that you don’t end up “earning” your grades. I can understand that viewpoint. However, if I’m completely honest, I don’t think it matters in the long run. In my eyes, grades are a tool to get you where you want to be in your education. They aren’t really an indicator of learning. It’s unfortunate, but that’s the way it is. Therefore, I don’t put too much stock in my grades, preferring to see if I can explain the concepts I learn.

The ugly truth is that school doesn’t reward you for trying your best and learning through growth. The educational system rewards those who can perform well on the first try, which is why I actually think using the answers you find online isn’t cheating, but a good use of your time. Of course, you need to do this responsibly, but if you do the work first, I don’t think it’s a bad thing to check your work online.