Through the Minefield
When mathematics makes sense, each piece seems to fit together. There’s no question about how to do things, because it’s all natural. This is what happens when you become good at algebraic manipulations. No matter how hairy the expression, you’re able to deal with it. Double-decker fractions aren’t frightening. Sure, it might be tedious to work through, but it’s doable. It’s sort of like strolling through a path in a meadow. Nothing is blocking you, and the way forward is clear.
Contrast this to when mathematics doesn’t make sense. When this happens, it’s like trying to navigate a minefield. Each step is uncertain. You worry about how one wrong move will ruin everything.
Both experiences happen to everyone. There are times when learning goes smoothly, and times where it is anything but smooth. Of course, some people experience one side of this coin more than others. For those that find mathematics to be a minefield on average, it’s not surprising that they will grow to dislike mathematics. If I had to do something which made no sense to me every single day, it wouldn’t take long before I became frustrating with it, too.
The question then becomes: how do we move the needle so that more people find mathematics to be something they understand?
I want to be clear here that I’m not saying those who enjoy mathematics don’t experience any difficulties. There are times when the concepts are challenging. However, the difference is that they have enough forward momentum that they keep on pushing through the difficulties.
To solve this problem, I think there are two related points that need to be addressed.
Does it make sense?
The first is that mathematics needs to make sense. I would say this is one of the most important aspects of teaching that we don’t emphasize. Mathematics isn’t a bunch of random rules that we need to memorize and apply in just the right way so that the answer is magically spit out. Instead, mathematics is built on a few key rules (axioms) and uses logic to build more complex structure. This means you don’t need to memorize everything. By knowing a few key rules, you can branch out and do a lot more. Furthermore, you should be able to connect what you’re doing back to those axioms.
The way we mess this up is by chopping ideas up into these arbitrary categories that make the concepts seem distinct, even though they aren’t. The most egregious example that comes to mind is the insistence in secondary school to show students how to solve a system of two linear equations by using elimination, substitution, or comparison. These methods are presented to students as different ways to solve an equation, but it’s often not pointed out that these are essentially doing the same thing. Sure, the methods might be slightly different, but there’s not much separating them. After all, comparison is just a special case of substitution, and elimination isn’t that much different either.
What’s the key insight here? Instead of focusing on three arbitrary “methods” to solve a system, the main emphasis should be on the fact that you can do basically anything to an equation, as long as you do it to both sides. That’s the key insight, not these three arbitrary methods.
When we emphasize different methods and forget to mention how they are all similar, students can get confused and think that you’re only allowed to use certain ones depending on the situation. I know this because I’ve worked with students who had this impression. It’s no fault of their own, because these methods are listed as distinct. Why should they expect them to be linked?
At it’s core, mathematics is a discipline that makes sense. Not to me in particular, but to anyone who is willing to sit down and chew through the arguments. It’s not always easy, but the results are accessible. Therefore, instead of emphasizing rules and procedures without talking about the underlying parts, we should focus first on the underlying mechanisms and show how they give rise to our basic rules. My prediction is that this would shift the mindset of students from “mathematics is a bunch of rules that I need to apply in just the right way” to “mathematics is a subject that makes sense if I carefully follow the arguments.”
Taking a small step
The second is that we need students to take small steps. If you’re like me, you’ve tried to work through a problem or a piece of mathematics and became confused when an author suddenly took a large step. The result is that you become disoriented, since the step was too big for your tastes. This can happen to everyone. We all have our own preferred pace in which to tackle a problem. As such, it’s no surprise to me that the “default” step size which is present in textbooks and in teaching practices can be too large for some.
The result is that some students become bewildered and are unable to follow. And since a classroom isn’t often made to suit the needs of one student but of thirty, it means they can be left behind on the mathematical journey. For the student, the only way to catch up is to take what is said as a given and just commit it to memory. Instead of going over each argument in detail and thinking about it, they have to accept it without further investigation. This reinforces the notion that mathematics is a bunch of facts that need to be accepted, instead of a series of reasoned arguments. Who can blame a student for holding this view when they have been left behind?
This is both an easy and a difficult fix. It’s easy in the sense that we know how to help the student. They need to sit down with the material and go through it at their own pace, taking steps that seem reasonable to them. I often catch myself going too fast when working with a student, and when I do I try to slow down, because the explanation isn’t for me. It’s for them. On the other hand, the fix is difficult because teachers can’t give each student this opportunity at all times. The reality is that a classroom is made to serve many students, which means the time each one gets with the teacher is limited.
As a tutor, I get the opportunity to work with the students on a one-on-one manner. This helps, but not everyone has access to a tutor. The best advice I could then give to a student is to see after class if they can go through the arguments that were presented in class. It’s during this time where they can see if the steps taken were acceptable. If not, the student needs to work through the confusion, or else they will be forced to accept the results without understanding them.
Ideally, a student would go through any claim with small enough steps that each one seemed obvious. Sure, that means it might take longer to understand a result, but I would argue it’s preferable to taking the knowledge at face value without understanding the arguments. (Of course, this doesn’t necessarily translate into better grades.)
The unfortunate reality is that I see students who look at mathematics as a minefield, with every step an uncertain one. The reason this happens is because we’ve taught them to value facts over the arguments that link those facts. It’s the links which are so much more important, but since they aren’t emphasized on exams, students don’t internalize them. The result is that a student might get good marks, but this doesn’t mean they understand the mathematics.
My goal as a tutor is to help bridge this link. Instead of getting students to take big, uncertain steps through what looks like a minefield, I want them to take smaller steps through a meadow. At its core, mathematics is understandable. We just need to stop focusing on the results and more on the underlying mechanisms.
Visuals in Mathematics
There’s no doubt that writing is a useful tool. If anything, I’m biased towards writing. I write every day, so I know what it means to use words to craft an explanation. If you can use the right words in the right arrangement, almost everything becomes clear.
That being said, there’s still a difference between writing and communicating. As much as I love writing about physics and mathematics, I realize that using this medium to craft explanations can be problematic. This is why I rarely write pieces with long calculations without using something in addition to words. It’s not that writing is bad. Rather, it’s that writing on its own isn’t great at communicating mathematics and physics concepts.
Thankfully, we have it a lot better today than in the past. If you look at older texts on mathematics or physics, you will see that everything was communicated using words. The end result is that learning required both a desire to understand a new concept and the patience to decode the text. This isn’t great for someone who is having a difficult time with the concept itself. As we know, learning can be difficult, so it should be our priority to craft explanations that lead students to understanding without needless barriers.
Remember, I like writing. I’m not saying we should quit writing explanations about mathematics and physics just because writing isn’t the best medium. I’m suggesting that we should complement our writing with other media.
In particular, consider the under-appreciated diagram. A diagram conveys both words and the relationships between them. If you’re working through a mechanics problem, it’s often helpful to draw a diagram. This lets you see the different constraints on the system and lets you set up the coordinate system. Sure, this could be described in words, but writing it out would be tedious and wouldn’t convey the idea in as simple of a format as a diagram. The best thing about a diagram is that it’s visual, which means you can consume it quickly. You don’t have to parse through a paragraph of text while simultaneously building up the diagram in your mind. Instead, you get the diagram as part of the explanation. This prevents you from building the wrong diagram in your mind, and it’s better at conveying the message than a paragraph.
Here’s another example. Suppose I wanted to convey the fact that the total revenue from an event was the sum of the sales from the three ticket types. I might say that the first type of ticket cost ten dollars, the second type cost fifteen dollars, and the third ticket cost twenty dollars. To find the total revenue, one simply has to multiply the number of the first type of tickets sold by the cost of that ticket, and do the same for the other two ticket types. Taking the sum would give the total revenue.
There’s nothing wrong with writing it out like this. If you’re like me though, the above paragraph is a bit of a mess to follow, with everything being spelled out in words. Instead, we could just label the revenue as R, and the number of tickets sold by each type as a, b, and c. Then, we could skip the confusing paragraph and write:
R = 10a + 15b + 20c.
We were able to compress our long paragraph of explanation into one line that explains what each variable represents, and then an equation giving the relationship. Even better we didn’t lose anything by compressing our explanation. In fact, I think the equation makes it even more clear, since we can imagine how the revenue will change based on the tickets sold. Plus, we have developed methods of manipulating equations using the rules of algebra. As far as I know, we don’t have an “algebra of paragraphs”.
I bring this example up to show that we already avoid using words when we can. I’m not suggesting something novel here. We don’t use equations in mathematics to make the lives of students difficult when they start out. We do it because it makes our lives easier in the long-term. In this same way, I think we should be placing our focus on making mathematics more visual.
It pays to be careful here. When I’m talking about mathematics being “visual”, I don’t think we should only do mathematics by drawing diagrams and sketches. That’s making the pendulum swing too far in one direction. What I would like to see is an emphasis on drawing pictures to accompany an explanation. Keep the algebra and the definitions there if you have to, but don’t stop there. Use visuals to convey an idea whenever you can. Students will thank you for it.
When I read a mathematics textbook, one of the first things given in each chapter is a list of definitions. Because of the nature of mathematics, these definitions are technical. However, what is often forgotten is that definitions tend to stem from some sort of observation. That initial observation should be given to the students. In particular, if the observation is visual, that should be shown. I can’t tell you how many times I’ve read through several definitions only to find myself scratching my head. The times where I was able to understand quickly was when the author included a diagram showing the idea in a visual manner.
If you’re reading about functions and you come across the terms “one-to-one” and “onto”, the definitions can seem cryptic at first. Sure, they are clear, but what do they mean? The best way to give students a visceral feel of what’s happening is by drawing two sets and showing how the elements in one set are mapped to the other. By including this diagram, the idea of a function being “one-to-one” or “onto” becomes clear. It’s not that the definition was inadequate. It’s that definitions can be difficult to parse, whereas a (good) visual leads to near-instant understanding. (Of course, some visuals can be confusing, but I would argue that’s the fault of the author.)
I’ve only given a few examples here, but I hope you have been able to think of others by reading this essay. I think we can sometimes get in the habit of using words instead of visuals because words are easy to type and visuals take longer to prepare. However, my goal here is to convince you that including more visuals in your explanations will make it a lot easier for people to follow you.
It’s not that visuals will automatically transform your explanations into world-class pieces. It’s that visuals will let students absorb your ideas without needing to decode a bunch of text first. I’ve written about this before, but when a student is learning, the best thing they can have is a foothold from where they are to where they need to go. Without that foothold, it’s difficult to get to the next level. The gap becomes too great, and students just get frustrated. Visuals can be those footholds.
How does this change the way I’m going to teach and craft explanations? The biggest change is that I’m not going to let myself slide into writing because it’s comfortable. I love writing, but writing on its own isn’t enough to make a student follow. When I’m trying to learn, the best combination I’ve found is to mix written text with a lot of visuals. This means you get the advantage of being to explain details in depth with words, but you also get the benefit of sprinkling the explanation with visuals. This both breaks up the text (giving the reader natural “break points”) and let’s them ponder over what they’ve read in terms of a visual. I’ve found that this works well for me when I am learning, which is why I will do my best to include more visuals in any exposition piece I write.
At the end of the day, mathematics is about ideas. It also just so happens that we are wired to understand pictures and drawings very well. The saying about a picture being worth a thousand words exists for a reason. Writing is comfortable for me, but it’s not the best tool to use for explaining mathematics and physics concepts. It works well if you use it in conjunction with lots of visuals. But without them, you risk losing readers in a sea of words.
It’s difficult holding a bunch of new information in your head, so make the job easier on the reader by giving them the visuals they need.
One thing I didn’t mention in the main piece was that I am aware that there are some people who have visual impairments. This means using visuals wouldn’t be helpful to them. I’m not sure how to address this, and it’s something I still have to think about. Just because I haven’t had experience with students who are visually impaired doesn’t mean they should be shut out from this discussion.
The second point I wanted to mention is that there’s a whole other wave of mathematics explanations which use animations and movement. I have a lot more thoughts on this, and it will be the subject of a future piece.
Finally, I realize that there’s a certain irony in not including any visuals in this essay. However, my argument is that I’m not explaining a concept in mathematics or physics. Don’t get too mad at me!
The Necessary Details
As a student in science, you’re taught how to understand the details, the gory bits of an argument or a concept. When you learn about Kepler’s law of equal areas being swept out in equal times, you’re not just told that fact, but you prove it. Each part of the argument is explained, and you get a full explanation.
This is great, but the problem is that we don’t get to learn how to explain a concept. We’re given all of the details, but the truth is that they aren’t all useful when explaining the idea to someone else. The other person probably doesn’t care about the technical details. They want the big picture, so focusing on the minutiae doesn’t help them. The result is that they become disinterested.
The details are important, but it depends on the situation. If you’re trying to learn a subject, then sure, look at all of the technical details. However, if you want an overview of an idea, getting into the details isn’t as important.
Achieving this balance is crucial when trying to communicate an idea. What are your goals? Do you want the person to have enough knowledge to explain the concept themselves, or do you want them to understand the idea? You also need to consider what they are looking for. Without aligning these two objectives, attempts to explain science won’t go well.
I’ve noticed this difficulty when trying to describe my work to my family. They have no background in physics, so my explanations can’t be technical. I therefore have to find other ways to explain what I do. Do they understand the details when I’m done explaining? Of course not, but that’s not my goal. My goal is to get them to the point where they could summarize what I do in a few sentences instead of blankly saying, “He studies something in science.”
As I’ve gone through my undergraduate degree, I’ve realized that this is something which isn’t emphasized nearly enough. Perhaps it’s the result of my specific program or the university I attend, but there’s not a lot of emphasis on outreach and explaining what it is we do to a broader audience. Maybe that’s because it doesn’t seem like an “important” part of the job of a scientist, but I wholeheartedly disagree with that sentiment. On a practical level, scientists are mostly funded by government agencies or academic institutions, which means this is public money. As such, I would argue we have an obligation as scientists to explain what we are doing to the public.
On a more philosophical level, I think it’s important to do this because science affects all of our lives. We learn about how the world around us works, and we get to be curious about our place in the universe. Explaining science is a worthy endeavour, and yet science students aren’t prepared to do this. Instead, they focus on working through problem sets.
It all begins with choosing which details are necessary. Of course, we could just say, “If you’re interested in science, look at a textbook!” This might help in terms of disseminating information, but it doesn’t take into account the amount of knowledge the average person has. A standard textbook will likely be too advanced for them, and it’s not designed to inform. It’s designed to give all the details, which is more than a person often wants.
If you’re a science student, my wish is that you take the time to explain what you’re learning to others. In particular, try to explain your research or what you’re learning to those who have little experience in science. By doing this, you will get to practice the art of giving just the right amount of detail. You get the benefit of getting more people to learn about science, and I think you will find that your own knowledge of the subject will become more stable. After all, one of the best ways to internalize a concept is to explain it to others.
Just because we aren’t taught this in university, it doesn’t mean it’s worthless. It might be one of the more important things we do.
It’s game seven of the Stanley Cup Finals, and both teams are about to get on the ice. The camera hovers around them, and you notice that everyone has a complex array of fist bumps, arm movements, and rituals. Some even have smelling salts that they wave in front of them before the match starts. Superstition is rampant, and you roll your eyes as a scientist, knowing that it’s all nonsense.
Except that it isn’t complete nonsense. Sure, there’s no supernatural effect from any of these actions, but they do serve a purpose: convention. By doing the same thing every game, the players are able to calm themselves and focus on the match ahead. By repeating the same actions all the time, it becomes a new normal.
This is hardly a phenomenon of professional hockey, nor is it exclusive to sports. Convention is present in all aspects of our lives, and science is not immune to it. In particular, the field of physics is filled with conventions. You can barely wade into a few pages of a textbook without encountering some (often contrived) convention.
In and of itself, there’s nothing bad with convention in science. It makes things standard so that everyone has an understanding of how things are done. Unfortunately, physics is riddled with historical conventions. The problem with history is that a lot of it was wrong or misguided. As such, we end up having terminology that doesn’t even reflect the actual phenomena, such as cosmic rays, which aren’t photons but are particles.
This can be confusing enough, but there are also historical conventions in terms of notation, even when legibility is reduced. By honouring historical convention, you end up with ridiculous situations like this:
The sane response would be to say, “Alright, let’s get rid of all this bad notation and terminology. Let’s update to the twenty-first century!”
However, there’s an additional problem. You notice these crazy conventions when you’re a student. It’s obvious then, but your professors say that you just have to suck it up and get used to it. Not having much of a choice, you accept it. Fast forward ten years, and you will end up telling your students that they need to get used to convention. After working with the convention for years, you won’t find that there’s anything strange with it, while the young students will find it weird.
Just like that, the cycle continues.
Breaking these historical conventions isn’t easy. It’s a mixture of not wanting to ruin the old notation in the literature and being “used” to the normal way to do things.
I have a different view. We are scientists, which means we shouldn’t be bogged down by historical baggage that has pervaded the field in the past. If a new convention is needed, we should step up to make that change. It might be slow, it might not work, but it’s worth trying. Not for us, but for our students later on.
I know that I won’t be writing those Greek indices on my tensors anytime soon.