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.

Why Can't We Give An Answer to 0/0?

Student: What’s the answer to 0/0?

Teacher: It’s undefined.

Student: Then why can’t we define it?

Teacher: Because that’s just how division works.

The exchange above is one that happens often when students start learning division. It’s a simple question, arising since we can divide by any other number, so why not this one? Unfortunately, the answers given to this question don’t attack the consequences of defining 0/0, but explains it as something given by authority.

I often like to remind students of a very important lesson when they start wondering about the definitions and restrictions that come up. We get to make up the definitions used in mathematics. We aren’t forced to use a certain definition for a concept if we don’t want to! Mathematics is about defining concepts and building from them, but we get to choose those starting points.

One specific example where things can get controversial is when we start considering an expression like 0/0. In this case, what is the answer?

As the teacher said above, we leave this expression as undefined. In other words, we say that the statement just doesn’t make sense, and we move on. However, if the definitions are up to us, why can’t we define these to take on certain values? What’s the harm in that?

This is a good question, and answering it will give us insight into the notion of divisibility. What does it mean for a number to “divide” another? If you ponder this question, you will find that we can divide b by a if we can write b=ac, where c is another element. Note that I’m using the word “element” here because we don’t necessarily have to be working with integers, though that is the setting which is most familiar.

There’s another property that we would like, even though you may have implicity assumed this. It’s that if we can indeed write b=ac, the element c is unique. If we take the example of 20/5, we know that the only way to write this is 20=(5)(4). Four is the unique number that we get when performing the division.

So what about zero? If we want to find an answer to 0/0, we need to find a number such that 0=0a. What kind of numbers fit the bill? One works, since (0)(1)=0, but two works as well. Actually, you will quite quickly realize that all numbers work. That’s because you’re multiplying by zero, which “collapses” every single number to zero when you multiply them together.

Now we’re faced with a bit of a dilemma. Which number should we choose to be the answer? Remember, we have all the freedom in the world to create our own definition of things! Let’s say we choose 0/0=5. Then, what happens if we consider (0/0)(0/0)? On the one hand, we know that each term in the parentheses is 5, so we should get a result of 25. However, if we do the multiplication in our usual way, we also get that (0/0)(0/0)=(0/0)=5. As such, we would conclude that 5=25. This is clearly not a very good number system, since any time I owe you twenty-five dollars, I’ll only give you back five. We know that those two numbers should be different, so it’s a bit of a concern when we manage to say that they are equal to each other.

The lesson we learn from this is that, while we do have the freedom to define concepts however we wish, we also want to be consistent. If we can transform five into twenty-five, it turns out that we can make basically any number we want equal to five. This is not a system in which the usual arithmetic rules apply. That’s not inherently bad, but we need to ask ourselves if the tradeoffs are worth it. Is it a good idea to have 0/0 being equal to any number we want as soon as we say it’s equal to a specific number, or is it better to leave it as undefined? As a community, mathematicians have obviously chosen the path of not defining terms like 0/0, and it’s precisely for this reason. Our common sense goes out the window once we start allowing these kinds of expressions.

In fact, you may have seen “pseudo-proofs” that 1=2, and these proofs rely on the fact that they are sneaking in a “divide by zero” operation at some point. Of course, these “proofs” won’t mention that, but that is the trick that is being played.

While it is easy to define these expressions to take on a certain value, there’s a reason why this isn’t done. We aren’t just lazy and don’t want to divide by zero. It’s that zero is a fundamentally different sort of number, and dividing by it doesn’t give us a unique answer.

However, it is much more fascinating to delve into why we don’t divide by zero, rather than simply forcing you to memorize this in class, don’t you think?

The Limits of Life

Note: I received a copy of this book as an ARC from NetGalley. It comes out next week. I tried to stick to the concepts he describes, but I’m sorry if you see any bias.

Diagram of the limits of life.

If you ever read a piece describing evolution or biology, chances are you will read something about how life admits infinite variety. Words like boundless and limitless get thrown around quite a bit. These make for nice narratives, but the truth is that life isn’t quite so free to do what it wants.

To be clear, I’m not saying that life isn’t diverse or that evolution is wrong. What I’m trying to get across is that one needs to be careful about what “infinite variety” means.

If you’re talking about the fact that the exact colour of one’s eyes can take on a gradient of values, then sure, there’s a functionally infinite amount of choice. (Eye colour isn’t anything special, it’s just an example.) In that sense, there’s an infinite amount of variety to be found in nature.

However, this doesn’t imply that life can take any form. Biology is beholden to something, and that something is the collection of equations of physics.

Evolution cannot just roam free and explore any form of life it wants. Evolution must comply with the equations and principles from physics. This immediately implies that life has limits, which means life cannot have infinite variety.

In his new book, The Equations of Life, Charles Cockell explores how physics can inform biological questions. It’s a great book that takes simple physical principles and shows how they have consequences for life. He links equations to the qualitative aspects of different life forms in a way that is easy to understand and shows how physics and biology do work together.

An analogy that I found useful was how he explains that the variety of life is more like a zoo than an infinite expanse. While there is room for a lot of variety, life is contained within a sharp perimeter, given by physical principles.

He writes:

No evolutionary roll of the dice can overcome a lack of a solvent within which to do biochemistry or the energetic extremes of high temperatures. The details, the temperature sensitivity of this and that protein, may well modify the exact transition between the living and the dead, particularly for individual life forms, maybe for life as a whole. But in broad scope, life’s boundaries, the insuperable laws of physics, establish a solid wall that bounds us all together.

One particular characteristic he explores is the notion of temperature. What kind of temperature limits are imposed on life? Is there an environment that is just too cold or too hot? This turns out to be an interesting question where the nature of atomic bonds in molecules plays a large part.

For high temperatures, there is a point in which the temperature of the environment will break the bonds of the atoms that make up the living organism. For carbon-carbon bonds, this limit is about 450 degrees Celsius. After that, the bond breaks. In other words, no more living organism. The important part isn’t the particular value. Rather, it’s the fact that we know we can’t have carbon-based life at a temperature of 1000 degrees Celsius. It just won’t work. There are similar constraints at the high end of temperature for other atoms as well. There’s no getting around them because they are physical principles that apply to everything.

There’s a similar story for the lower end of the range. As Cockell writes, the issue here is that molecules don’t move fast when it’s cold. As a result, any sort of radiation can kill an organism because it will not be able to repair itself fast enough. Imagine you’re building a house and someone is removing bricks faster than you’re putting them on. Even though you keep on working, eventually there is no house left.

Cockell tackles similar ranges in the book, such as pH level. During these excursions, his point isn’t to derive a value and say, “There we go! Life can’t pass this barrier.” Rather, it’s to acknowledge that there does exist a point in which life cannot cross a certain barrier. These barriers can’t be jumped over by clever tricks from evolution. They are a fact of our universe.

This was one of the key lessons I got from The Equations of Life. As a physics student, I found myself interested in the relationship between physics and other sciences. Cockell does a great job of illuminating that relationship. Even if you’re not brushed up on your classical physics, this is a book that will reel you in. Plus, he has a wealth of citations at the end, if you ever want to explore the topics in more detail!