Within One’s Expertise

Because I am a tutor for secondary students, most people assume that I know everything that needs to be known about those classes. However, that could not be further from the truth. In fact, I frequently encounter problems that the students I help have that I cannot answer. It’s not even that I don’t know the answer. Sometimes, it’s just so far back in my mind that I don’t remember what the exact steps are.

When this happens, I don’t beat around the bush and pretend I know the answer. I tell them I have no clue, and that we’ll work it out. The reason I do this is twofold. First, it’s probably true, since a lot of the secondary mathematics is straightforward after doing more advanced mathematics. But the second reason is that I want to show them that I am not a robot that knows all the answers to their questions. I’m a student just like them (even if I become a teacher as well), and I’m constantly learning. I want them to know that I am just like they are, expect a few years further down the line. That’s it. I’m not a mathematical prodigy or someone who can do all the calculations in their head. I’m just there to give them a hand when they need it.

This is why I’m not too scared of straying from my expertise. Yes, I may not have much experience in all the classes in secondary school (in fact, the majority of students I tutor this year are taking a class I’ve never taken), but there’s no harm in learning. Plus, learning this way also helps me become a better teacher in the future.

I’m definitely not advocating for you to proclaim expertise in every area of your life. I’m just saying that if you are willing to put in the hard work to learn, it’s entirely possible to become more knowledgeable at anything. Therefore, don’t restrict yourself to your personal expertise where you are comfortable. Experiment, and try to learn new things.

Discovery

I find it a bit of a mild tragedy that we as students don’t get to feel the joy of discovery while learning new scientific concepts. Classrooms talk about DNA, magnetism, electricity, gravitation, chemistry, and evolution as if they are mundane things. Ideas are introduced, but rarely is there any sort of “discovery” by the students. Instead, the information is clearly meted out in logical sections with almost no flair.

Imagine if you’re reading a book that is typeset in all capitals with very little spacing between lines. No matter how amazing the book is, it will be difficult for a person to get engaged. This is why books aren’t made like that. They are designed carefully so that you don’t experience any drawbacks from the book itself, allowing you to enjoy the words within.

In this same way, I feel like many classes are poorly designed to stimulate student enjoyment. I’m not saying that science and mathematics classes need to be the highlight of every day for each student, but I’m saying there’s probably a slightly more exciting way to deliver information instead of spoon-feeding it to them in class lectures every single day.

I’ve written about this before, but students can be given bits of context to situate themselves in whatever they are learning. Answers to questions that are a bit more deeper, such as “Why did people care about this when it was first discovered?” instead of “Why am I learning this?”, can be helpful to giving students an appreciation of science and how it was developed throughout history. As I look back at my lessons on history, the only time I’ve learnt about scientific history is during brief snippets from my various science teachers, and those were rare. I think if those moments were shared more, I could gain a much deeper appreciation for why a scientific idea is important.

Additionally, most science classes follow the same tried-and-true formula: lecture with notes, homework, test, final exam. Each day is a new lecture, and the students are expected to take notes and absorb the information as it is handed to them. In this scheme, it’s easy to push productivity. A teacher simply needs to write more on the board in a given lecture, while minimizing questions. The problem with this, however, is that the class is reduced to a mere transcription of information. The students aren’t necessarily learning, they’re just absorbing. In my mind, learning happens when one is engaged, and there’s no better engagement than discovering something. After all, that’s the dream of basically every science student. Why shouldn’t we harness this to good use in classrooms?

I envision it working something like this. Students don’t get the full results. They get bits and pieces, and the teacher has to get them to think. What do these two bits of information suggest? What if we look at them in this way? By gently prodding the thought process of students in one direction or another, we can get them to think and deduce in place of just absorbing new information.

As I get older and continue my studies, I fear that I’ve lost many opportunities when I was younger to discover surprising truths about science. I don’t want to be a sponge that soaks up new information. I want to be the detective that takes in new information, thinks about it, and uses hunches to get new results. I believe this can be stimulated by having students discover more things in the classroom. Yes, it isn’t going to help the productivity of the class. But it will give them the tools to think, which is a big part of what science is all about.

On the Boundary

When I first started learning about physics, I thought it was amazing how we had these equations and patterns that emerge in nature to the point that we could actually predict what would happen if we threw an object or slid it on a certain surface. The classes were interesting to me because they allowed us to describe things we actually saw.

At this point, I wasn’t familiar with the notion of classical physics, or that there was anything other about physics that I was learning. I didn’t know there were inherent limitations into what we were learning. I imagined that physics was as intuitively simple as the principles I was learning.

However, the trouble came when taking those intuitive principles and extending them in my imagination. For example, imagine moving a book across your desk. When you push on it with your hand on one side, the whole book starts to slide across the desk. Have you ever wondered why this is?

It might seem like a stupid question, but hang in there with me for a moment. It seems obvious that the whole book will move despite only touch a small portion of it, but let’s extend this experiment. Imagine having a long broomstick that is one kilometre long. When you push it, what happens?

Intuition will give us the same answer: the broomstick moves as a complete “unit”. However, if we extend this already-crazy scenario even further, what happens when the broomstick is 300,000,000 m long?

If we were to continue with our intuition, we would be able to push one end of the stick and move the whole thing. But wait a moment. If pushing the broomstick means effectively moving some sort of “signal” throughout the atoms of the broomstick from one end to the other, you’ve just made a signal move faster than the speed of light! Since this goes against Einstein’s special relativity, something in our scenario is wrong.

As it turns out, the rod would not actually move together in one moment. Instead, it would take time for the signal to be passed from one atom to the other along the entire broomstick. After all, pushing the broomstick only means moving the atoms on one side of the broomstick, and so the atoms have to “communicate” to the adjacent ones that they too should move, all the way through the broomstick. It is much like what you can see if you film somebody letting go of a slinky with a high-speed camera. The top of the slinky falls first, followed by the next slowest ring and so on.

Personally, I find this extremely cool. It’s unexpected: we’re taught that an object in free-fall will, you know, fall. But here the situation seems to defy the usual assumptions (though a closer inspection would show that indeed, we could describe the slinky with tension as it falls).

What I wish I had more of in my classes of classical physics were boundary conditions: situations that showed the limitations of the wonderful equations we learnt. It’s great and all to know the ideal cases, but I think it would have been equally instructive to show where they don’t work. I feel like we often gloss over what doesn’t work in favour of what does, but when learning, the cases where something doesn’t work can be of great help.

When learning an idea, investigate where it’s domain of applicability exists, and what happens when one leaves it.

Only Part of the Picture

I remember when I was in secondary school, my class learned about conjectures in mathematics class. This section was probably one of the most confusing part of my mathematics class because there was nothing definite about it. I found it strange how we went about doing these problems, and I was happy that we didn’t spend too much time on it (a sign that I probably should have spent more time on it).

Just to give a quick explanation of a conjecture, it’s essentially a statement that does not have a proof, but states a certain property or relationship between things. For example, most of us have seen some form of proof for the Pythagorean theorem, involving squares and geometry. However, suppose you never had seen the proof and was told that the square of the sums of two legs of a triangle is the same as the squared length of the hypotenuse, that would be a conjecture.

If you thought it was false, you’d try and find a counterexample. After trying a couple of different triangles, you’d see that they do indeed fit the conjecture. At one point, you may throw your hands up in frustration and say, “All these stupid triangles fit the conjecture!”

You haven’t actually proved anything, but you now have a stronger suspicion that the conjecture is correct. You couldn’t just keep using triangles with different lengths if you wanted to prove this relation. You would have to take a triangle with arbitrary lengths and do some kind of proof to get to the conjecture. Then you could be sure of the conjecture, and it would become a proved statement.

Instead of learning that though, what I learned was that you needed to either find a counterexample (in which case you were then done), or you had to do three examples of the conjecture and show that it does indeed give the correct answer each time. Basically, it’s a glorified version of arithmetic, with only a bit of thinking involved.

I never got why we were doing this, nor how we were supposed to form a strategy for these conjectures. Was I just supposed to pick random numbers and see if they worked or not? Was I supposed to try and think of a counterexample?

What I think would have made much more sense now was if I was shown that conjectures are like “proofs-in-waiting”. They are ideas looking for proofs. Additionally, I don’t get why we were supposed to just examine the conjectures. After all, they are pretty boring on their own. Instead, we should have had to use those conjectures to learn how to construct proofs. It would have been a great example of the power of proving something.

Unfortunately, the closest I’ve come to proving something as an exercise on my own (read: not through the teacher’s notes on the board) was when I learnt how to do mathematical induction. The idea is that if your premises are true, than this is what you get. It was a powerful way to prove things that happen in series (such as repeated matrix multiplication). It gave me a taste of what mathematics can do.

Therefore, I believe we need to do give more effort to show young students the trajectory of what they will do with the tools they are learning now. Maybe it isn’t always necessary to complicate their lives, but I think it’s important to show them that these ideas will be used in the future. It gives them the sense that mathematics isn’t just a series of hurdles to jump through.