Bag of Examples

As a student in mathematics and physics, I’m part of two different worlds. On the one hand, proofs and abstraction come from the side of mathematics. On the other hand, physics is where concrete examples and applications are the norm. In physics (at least, within the scope of undergraduate education), we only care about the mathematical tools that we can apply to a given problem.

Of the two camps, I find myself identifying more with the mathematics side. That’s just because I like abstracting past one application and finding the link between a lot of concepts. I realize that we do this in physics as well, but I find myself at home within the mathematical proofs versus the handwaving that often happens in physics (which isn’t always a bad thing!). I’m definitely not a pure mathematician. I still like to link what I’ve learned with the world, so perhaps I fit best into the “applied mathematics” camp.

The reason I bring this up is because I’ve been thinking of the way we introduce examples in mathematics. If you’re in an advanced mathematics class, chances are the professor will first go over some definitions and perhaps a few theorems before giving examples. Even then, the examples only illustrate the idea, so they aren’t the focus of the class. Instead, the focus is on the proofs and how to go from argument to argument.

This makes sense. After all, advanced mathematics tends to involve abstracting past examples and capturing the general case. This means the examples are less important than the underlying characteristics they share. This of course is the strength of mathematics. When you can look past the immediate features of a specific example and see what many examples have in common, you can come up with theorems that apply more generally.

This came to light in my abstract algebra class. There, I learned about a mathematical object called a ring. The specifics aren’t important, but what is important is that a ring is something we are all familiar with. The integers under addition and multiplication form a ring. The real numbers form a ring. If we look at linear algebra, nxn matrices form a ring. The point is that even if you have no idea of what a ring is, you have likely worked within a ring in your mathematical journey.

As such, the underlying structure of a ring was there, beneath your feet, all this time. Instead though, you studied arithmetic and linear algebra as separate concepts. It turns out though that they are both examples of rings. Therefore, by studying the properties of rings, we can capture the core ideas of matrices and integers or real numbers in one go.

That being said, there’s a downside to abstracting to higher and higher levels. Each time you go up in abstraction, you lose certain features. If you climb a mountain, the landscape of trees you see from the summit will look smooth. Descend the mountain though, and you find that there’s a staggering amount of diversity present within the forest. What looked uniform from above can break up into unique parts upon closer inspection.

I think we can make a similar comparison with abstraction and examples. Sure, abstraction is great in the sense that it captures everything we want in one sweep. But the price we pay for it is that we don’t have specific examples in mind when working through the mathematics. This might not sound like a bad thing, but it makes it difficult to apply our knowledge to specific scenarios.

This is something I also learned in my abstract algebra class. As great as it is to study rings, integral domains, ideals, and fields, it’s also important to find examples for these specific objects. Examples clarify definitions and make the abstractions we study easier to visualize. Without them, it’s difficult to attach meaning to our objects of study.

It’s tempting while studying mathematics to jump straight to the abstraction. I’ve been there, and I know the feeling. You want to do it because you figure that knowing the general case will be a lot better than any specific one. However, this ends up not being true at all. In fact, I would venture to say that specific examples provide the footholds necessary to be comfortable with abstraction. When I feel at ease with a concept, it’s usually because I’ve internalized a specific example and I can see how that example stems from the general case.

Unfortunately, advanced mathematics courses often prioritize proofs and abstraction over concrete examples. This is emphasized through the kinds of problems that are assigned and the amount of class time dedicated to examples.

On the one hand, going through a bunch of examples can seen repetitive in class. Furthermore, problems which involve examples tend to be easier and don’t involve proofs. This isn’t a bad thing, but it does mean that the students don’t get to practice abstract thinking as much. I would venture to say that this is a big reason why professors don’t assign these problems as often.

On the other hand, I find that focusing only on the abstractions prevents you from playing with an idea in specific settings. I found the lack of examples in my classes a hindrance when it came to working on problems that dealt with specific examples. It might seem like applying the general knowledge you know to a specific example would be easy, but I can assure you it’s not. Furthermore, it’s once you succeed in applying the general knowledge to a concrete case that the idea becomes familiar. I would argue that knowing abstract knowledge without being able to apply it anywhere is next to useless. You need to have a balance of both the abstract and the concrete to thrive.

This is why I’ve begun thinking about my “bag of examples”. It’s great to know important mathematical results, but if I can’t illustrate them with an example, it becomes difficult to communicate them. Plus, working through a specific example tends to be the easiest way to grasp an idea.

Good examples become fertile ground for experimentation. When you have a concrete example, you can see how the result you’ve proved works in this case. The best examples can even inform the general result.

I know that my bag of examples is very empty at the moment. However, I want to build it up. Knowing mathematical proofs is great, but having a bag of good examples that you can pull out at any time is an under-appreciated asset.

Finally, I don’t want to forget about the related category of counterexamples. These can be just as important as examples, because they remind us that mathematics can be misleading if we look at just a few cases. Counterexamples force us to be more careful in our hasty generalizations and to remember that the final arbiter of truth is through a proof.

As such, my goal for now is to start amassing examples and counterexample stop illustrate various mathematical ideas. I want to find the shining jewel of examples for any idea. It’s great to keep on learning new material, but sometimes it’s worth pausing and building up a bag of examples.

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.