Alternative Perspectives

Imagine I had a block of aluminium and I asked you to describe it to me.

Perhaps you start by describing it qualitatively. Maybe it’s a cube of side length 0.10 m. If there’s a light in the room, perhaps you’d note how reflective the cube is, suggesting that it’s some sort of metal. You might even pick it up and note that it isn’t heavy, suggesting one of the lighter elements.

But this isn’t the only way to describe the block. If you took a more “complete” approach, you might do some tests on the block to determine its composition, concluding that it is indeed aluminium. Then, you could mathematically describe (at least, in principle) every particle that makes up the aluminium block.

What I’m trying to illustrate here is that there are different ways of looking at a block of aluminium, and they can both be right in their own domain. Additionally, it can be good to look at the block as a whole “thing”, instead of a bunch of particles. In other scenarios, the opposite may be true.

However, this is a bit too much for the purposes I want to talk about now. Where I get a lot of use out of different ways of looking at something is for a mathematical or physical concept. When learning, a teacher tends to only do one derivation (if they do one at all) for an idea. That’s great, but there’s always a chance that the way the teacher understands the concept is difficult or not in line with the way you think. Consequently, the proof can seem complicated even though it isn’t, just because you’re not comfortable with the tools involved. It’s kind of like always using traditional running shoes while training, and then suddenly being given racing flats for your race. The racing flats will mostly likely be a lot better than your training shoes, but you won’t reap the benefits because they will feel odd to your feet, making it difficult to run fast. Likewise, a teacher may select a method that is the most efficient, but you can’t appreciate the gains because you’ve suddenly been thrown into using methods you aren’t comfortable with. The solution to this is therefore to seek alternative methods to derive said equation, hopefully in a way that you are familiar with.

That is all well and good, but I think there’s use to doing this even when you’ve understood the first way of deriving a result. By looking at different ways to get to the same answer, you’re effectively giving yourself different ways to “look” at the concept. For example, my professor for waves and optics actually went through several different ways of calculating the intensity of a diffraction pattern. Some used algebra, others used vector addition, but the end result was the same.

What this did was give me a firmer grasp of the ideas at hand. I could explain them in different ways and still get to the same answer, which is both a good sign and makes me comfortable with the concept. Obviously, this might not work for every mathematical or physical idea, but it is a good way to help strengthen your understanding of a subject. I’ve said this before, and I’ll hammer it home again: a strong foundation is the most crucial part of learning.

So go build that foundation.

Interconnecting Histories

Through secondary school, I had to take history classes. During them, I learned the history of Canada, and how there were a bunch of important wars, conflicts, and political moves that brought us to where we are today. I took classes on this for four years, and throughout all of it, the theme was the same: war between the French, British, and the native Americans, and how both the French and British flourished while taking over from the native Americans (something that is often only brushed upon in school, which is odd since it really is a place for ripe ethical debate).

What didn’t happen was a discussion on the scientific breakthroughs of the centuries we studied (roughly 1600-1980). We didn’t learn how the ingenious development of electrical power spread out through the world, nor did we learn how the Copernican heliocentric view of the solar system changed the way people looked at Earth with respect to the rest of the cosmos. We didn’t study the engineering breakthroughs of telecommunications, and how they changed the way information is transmitted around the world. We also didn’t study anything about ancient history, by which I refer to the history of life itself. None of this was studied. It was only war, trade, and politics, and for just one small part of the world.

Of course, I realize that there is only so much material that can fit into a course, and that the total history of our species is long. However, I feel like we do ourselves a real disservice to only focus on a couple of points like we do in secondary school. I fear we give students the impression that what is taught is our (local) history, and we fail to give them the bigger picture. Those other connections I mentioned above are definitely important ones, and I feel like at least a couple deserve to be up there with the material that is taught.

When I reflect back on my experience in history classes, I remember a lot of dates, names, and places that had to be memorized for tests. I also know that the idea of the course was to give students an understanding of the big picture through essay questions, but often they didn’t capture how wonderfully exciting history could be. I think learning about a much more global history that incorporates science into the fold would have generated that excitement more than the order of the provinces of Canada joining the official country.

I don’t know if it is realistic to expect the curriculum to be updated to fill this function. Perhaps this sort of thing could take over one of the optional classes in later years. I feel like I’d be able to do something with this idea, because showing the wonders of science (particularly, when you only have to dive into the qualitative aspects), students will be captivated. There’s always more history behind the broad strokes that are made by curriculum history, and I think many interesting science stories lurk behind them, just waiting to be presented.

Personally, I’ve only begun on this journey a bit. I’ve read some history books on science, but not many. I know that there are many more things left for me to learn, and that’s why I’m going to discover them. I want to make sure that other students think about this opportunity as well, and not like they know their history because they passed four history courses in secondary school.

Notes

It’s no secret that if you walk into a classroom at random (but more specifically, a science or mathematics classroom), you will see a lecturer up front, with a bunch of students listening and taking notes. Sometimes, it might even seem like the students are barely paying attention to the words of the teacher, preferring to just copy what is on the board or screen.

I’ve heard many people criticize this aspect of school, saying that educational institutions have transformed learning into a state of copying what’s up front and not even thinking during lectures. I’m sure you’ve also seen the odd student in a classroom who isn’t taking notes. Often, they just don’t care about paying attention in class, but other times, they feel like they learn better from only listening.

I’ve thought about both of these approaches a lot, and I can’t completely come up with a perfect answer to which one I think is better. It’s probably tradition more than anything that made me take notes through each year of school. I still do, in almost every class.

Critics of taking notes might say, “But you can get all the stuff the professor is saying from books! Lectures are for listening and absorbing the way people think.” (I’ve gotten this line of reasoning as I read Richard Dawkins’ An Appetite For Wonder.)

I agree with the general idea, but I think this misses a crucial aspect of what writing notes does, at least for me. When I write notes, I don’t necessarily write everything down from what the teacher does, because some of it isn’t important. What I do write down is usually different laws or equations, and their derivations. To me, this is useful to have. Yes, I could get it all from a textbook, but writing it down in my own notebook allows me to make the information even more dense. I don’t have to go about explaining all the things that are obvious to me. I can jump right ahead to the heart of the matter. This way, I rarely have to flip through a textbook to find what I want, because I have it in my own writing.

The added benefit of doing this is that I can write down explanations that I need while looking at a concept, which others might not find useful. Since I’m writing notes while the lecture goes on, I can give myself little reminders here and there to help me out for later. This may seem trivial, but it can be a big help for when I look at my notes later (which I do, though Dawkins says he never did).

Additionally, I think there’s something to be said about remembering things you’ve written down. I forget things easily, but it seems like just the act of writing them down helps me remember. I often don’t even need to look at the reminders I write for myself because I remember writing them. I think it’s very possible that this same sort of thing happens when writing notes in class. I don’t have research on this, but writing the ideas down seems to help me internalize them, much like actually doing problems is more useful to me than simply skimming through them. Therefore, I think taking notes isn’t quite as bad as it may seem at first glance.

The way I take notes is simple: I write down equations and lines of arguments. I usually end up writing definitions, but only insofar that they help me understand what is next. Crucially, I try to participate often in class, which means I’m not just writing down notes and staying consumed in the act of writing all the time. I try to stay connected to questions being posed, and I also try and think of my own questions.


My advice is this: if you feel that you’re doing fine without taking notes, there’s absolutely no need for you to start. Personally, I find taking notes in class allows me to skip having to go and look at all the equations and theory later while I’m doing homework. I have it all in front of me in my notes, which are usually easier to read at a glance. If you do take notes, keep in mind that you don’t have to write down everything that is said and put up front in class. A lot of it is context, and the key is to give yourself enough context to remember the content of class, but you don’t have to recreate it. Notes are fine, just don’t go overboard.

A Tutor’s Job

As a tutor, I have the responsibility of helping students with their various classes that are difficult for them. I am supposed to work with the student in order to answer their questions. That’s the deal, at least in my view.

I’ve been thinking a lot though about what my actual job needs to be. Am I giving the students a false sense of what I’m actually going to provide? Should I be clear about what I expect I can do? Usually, I assume that since a student has sought me out, what they want is for me to make them better at the subject. I think that’s a fine desire, but I don’t know if I can really do that.

(Just to be clear, the tutoring I’m talking about is with regards to mathematics and science classes. Therefore, think problems, equations, mathematics, and so on. It’s usually less about theoretical concepts and more about actually calculating things.)

I’ve written about this before, but in my mind there are two “kinds” of learning that you’re supposed to do in class. The first level is conceptual, and it basically means you understand the idea of what you’re learning. For example, knowing the concept of the derivative means that you’re comfortable with the idea that it’s the slope of the tangent line to a curve at a specific point. You might also understand that it’s essentially done using a limit of bringing essentially two points on a curve arbitrarily close to each other. Knowing this also means you have an understanding of what a limit is. All of this is on the conceptual level.

The second level is then (for lack of a better word), the “calculating” level. This level involves – you guessed it – actually calculating and doing problems. Returning to our example, this would mean calculating $frac{d}{dx}(arctan(x))$ or $x^3e^{4x}$. It would also mean doing word problems that involve the derivative in some way. While the first level gives you an understanding of what you’re doing, this level gives you and understanding of how you do it.

Both levels are important, but we all know that the one that is actually tested on at school is the second level. It’s great to understand the idea, but if you can’t actually calculate or solve a problem, you’re “punished” by way of failing the test. Therefore, the level that is the most “useful” is the second one.

The fear I have is that students enter tutoring sessions with me expect to become fluent at the concepts they struggle with (which means improving on the second level), while simultaneously only going to one session a week with me for perhaps an hour. Now, I don’t want to say that nothing gets done in an hour, but I can say that most students will not be able to improve their second level by just dedicating one hour a week with a tutor. It’s a great start, but it’s not enough. Why? Because improving the second level requires one thing: practice. And lots of it. Much more than can be done in a single hour.

That’s why students need to work on problems when they aren’t with me. Just like learning a language, they need to practice regularly and frequently. That’s the way they will improve.

So how do I see my job? I see it as an opportunity to strengthen a student’s conceptual understanding (the first level), as well as giving a student the tools to make calculations and problem solving go faster (the second level). This means I like to spend a good chunk of time on hammering home the ideas of what a student is doing (instead of just saying it’s what they need to do and leave it at that). Additionally, I’ll then help a student go through several examples where they attempt the question and I try to give helpful feedback.

What I don’t like doing is simply a bunch of problems where calculations are made and no discussion or conceptual thinking is involved. This means that once I’ve looked at a type of problem with a student, we will probably move on to something else and not do a bunch of the same type of problem. The reason is simply that I want to spend as much time as possible laying the foundation for the student. Actually building it requires time on their own, working at their own pace and doing a lot of examples. I see myself as someone who will point them in the right direction, but won’t hold their hand all the way until their destination. That’s a kind of time investment that both I and the student can’t do, so I rather focus on giving the student a clear picture of what to do. using this system, I think I can give students the best chance possible to succeed, but the key is that they have to buy into the process by practising on their own as well. Without that, I can’t be an effective tutor (and I’ve experienced this difficulty before).


When I first started tutoring, I simply went and didn’t think of how I was going to help the student. I thought that I would answer their questions, and be there to check their work to make sure they are doing things correctly.

Now though, I’ve seen a glimpse of what works and what doesn’t. I realize that I can’t give them the skills of solving problems with ease. At least, not without them investing time outside of our tutoring sessions. Therefore, I’ve come up with a new way to evaluate a potential student who wants tutoring. I think that I’m going to tell them something along these lines:

What I am able to do is help you get a conceptual understanding of the subject, as well as the best ways I know to solve problems. However, what I need from you is a commitment to practicing on your own time. I can show you all the best techniques in the world, but you won’t ever really be comfortable with them unless you practice outside of this time. If you can do that, then I can be your tutor.

What I’m hoping this does is send a signal to those who are serious about improving. I need a person who is ready to work hard and improve. That’s the key, above else, to improving. My job to set them on the right track, and their job is to do the work to improve. I won’t be going into tutoring blind anymore, because I’ve learned that it’s not on me, ultimately, which helps them to succeed. It’s on the student.