I like school. That’s probably clear from reading my blog. I enjoy learning about science and mathematics, and the school system is one that I’ve learned to navigate with ease. Sure, I sometimes have complaints and suggestions for improvements, but on the whole, I enjoy going to school.
That being said, there is one part of school that I think doesn’t prepare us well for life outside of education. It’s about learning how to pick yourself.
What does this mean? Well, think about an activity you enjoy doing, or want to do in the future. Are you doing it right now? If not, what’s holding you back? Many of us in school are hoping that at some point, a person will look at our performance in school and choose us. In the meantime, we just need to work hard and do well at school, because everything else will be taken care of. It’s as if we believe doing our work in school will somehow make people decide to choose us, and then we can finally do the work we want.
Here’s a radical idea though: what if you just started now?
All by yourself, without the blessing of others. What if right now, today, you decided to finally start that project you wanted to do? Instead of waiting to be picked, you decide to do the work the work that’s important to you.
I first heard this idea from writer Seth Godin on his blog (so don’t give me any credit for it, I didn’t come up with it). He writes a lot about doing the work that matters to you and not being afraid to start. When I read this idea, I realized that this has big implications for how we conduct ourselves in school.
Think about it. We go to school, study in order to get good grades, and then hope that we will get the right internship, or be accepted into a prestigious school, or get to work on a big science experiment. Throughout our education, the lesson is clear: work hard, and wait to get picked. The system of applying for grants, scholarships, and schools reinforces this message over and over. Work hard, give us proof, and then wait to be picked.
This was an important realization for me, because a lot of the things I want to do with my life aren’t things that I need to wait to be picked. Instead, I can choose to take initiative and start.
My particular interests are in writing here on my blog about science and mathematics, drawing my webcomic Handwaving, and teaching/researching. Those are my main interests, and the great thing is that I don’t need to be picked to do these. I’ve written on my blog now for over two years, and I’ve been drawing consistently for my webcomic for nearly a year. I didn’t wait until someone told me I was good enough to write or draw ideas about science and mathematics. I chose to start on my own. Even for teaching, there’s nothing stopping me from teaching. Sure, I might not be a college professor at the moment, but that doesn’t stop me from taking other opportunities such as with tutoring or giving presentations. I can still do this. I can pick myself.
Of course, I want to be clear that this doesn’t work for all interests. If you want to be a miner, you’re going to need access to a mine. There’s no way around that. But more and more, the kinds of interests that people have are ones that require creativity, but whose investments in terms of cost are minimal. This is particularly true for a lot of science-related activities, such as science communication. We have better access to tools than ever before. If you want to do science communication, all it takes is for you to pick yourself.
It’s such a simple thing, yet I can imagine the immediate retorts.
“But no one is going to hire me to do this work I love!”
“Okay, I want to do science communication, but no one will listen to me.”
These are valid concerns, but they are tangential to my point. Picking yourself isn’t a guarantee that you will be hired by the organization of your dreams or that you will have an audience. Chances are you won’t have anything, at least not at first. But the key is that this doesn’t matter! What’s important is to start. Everything starts from there. When I started writing on my blog, I had nobody reading my work. Two years later and over a hundred thousand words later, I still have almost no one reading my work. Does this mean I haven’t made any progress? Not at all. First, I’ve improved my writing skills just by showing up every day to write. Plus, my work is still there, waiting to be read. It’s not going anywhere. I’m slowly building up my portfolio of work that proves how serious I am about writing and explaining scientific and mathematical ideas. As such, when someone shows up to my blog and sees the huge backlog, they will be more likely to stick with me. I’ve shown that I’m here for the long haul, even if no one is reading quite yet.
I want this to sink in. I’ve shown up week after week for about two years and I have no “results” to show for it. I don’t have a huge following that reads my words, but that’s not what picking yourself means. Picking yourself is about saying, “This work is important enough to me that I will do it even if no one else sees it.” My strategy for this blog is simple. I will keep writing until pure stubbornness sees me through. And then, once I’ve gotten to the point where people read my work, it won’t be a fluke. It will be because I chose myself many years ago, without anyone else.
This is my message to you. If there’s something you want to do, particularly if it has to do with spreading your interest in science and mathematics, please don’t wait for someone else to pick you. I know this is what you’re used to, because it was what I was used to through school. But the truth is that waiting for someone to pick you is a losing strategy. Some may get lucky and be picked, but most won’t be. Instead, my suggestion to you is to think about something you’ve wanted to do for a long time, and just start. Don’t overthink it. Start with the equipment and resources you have. More than anything, don’t expect big results. People won’t care about your work, at least not at first. That’s not part of the deal when you pick yourself. The deal is to decide that this work is important to you, and that you will do it no matter what.
I started thinking about this in the context of where I am in my education. I’m finishing my undergraduate degree and looking to go to graduate school. I can’t help but see a lot of my education as jumping through hoops just to get the chance of doing research. There’s a lot of work involved just to get the opportunity of being picked. I’ve been wondering if there’s another way, or at least a different road I can travel for my other interests (such as writing). This is why I started this blog, and it’s why I chose myself instead of waiting for someone else to do it. I’m still travelling down the academic road (it’s what I want to do), but I’m also making choosing myself in these other areas.
Picking yourself sounds scary, because it means you’re committing yourself to something. You’re taking a stand and saying, “This is important to me.” However, the truth is that it’s so liberating. Instead of worrying about others choosing you, the choice is on you. Yes, this means you won’t suddenly jump to stardom, but the slow burn is likely more sustainable anyway.
Don’t wait for someone else to pick you. It’s not going to happen, and it will just be an exercise in frustration. Take the initiative to pick yourself. It’s worth it.
Learning Without Excitement
It’s not fun to do.
Right now, there’s nothing stopping you from teaching yourself advanced mathematics, a new language, design, the details of theatre, or any other subject. The resources available to you are vast and often, free.
So why don’t we all spend our time learning?
Simple: there’s no point to it.
No, I’m not saying that learning how to code is pointless. Rather, I’m saying that, without any need to code, it’s difficult to muster up the effort required to learn. We just don’t like learning information that we’ll never use, because we feel like we could spend our time learning something we are interested in (and we are right).
If you’ve ever tried to pick up a mathematics textbook and learn on your own, chances are you know that it’s difficult. Unless you’re super interested in the topic, learning the details won’t be engaging. Sooner or later, we will give up.
What’s the solution, then?
I don’t know, but I can tell you that following your interests is a good way to start. The reality is that learning isn’t easy. Even for a subject you enjoy, it’s a challenge. As such, you need to find a reason for learning what you want to do.
Here are a few ideas:
- If you want to write, find a story to tell, an idea to explore, or a journal to fill out. Don’t think of it as “learning how to write”. Think of it as “learning how to tell this story”.
- If you want to code, don’t open a textbook that shows you the basic syntax. Instead, decide on a project that you’re excited about, and learn how to code with this project as a scaffold.
- If you want to draw, don’t mindlessly draw random objects. Yes, that’s a great way to develop the skill of drawing, but it won’t do much to make you motivated. Instead, figure out why you wanted to draw in the first place, and draw the things that make you excited. Maybe this means starting a webcomic, or perhaps making incredibly detailed drawings. Whatever it is, do what you’re excited about first.
My point here isn’t to make you think that the fundamentals and formal training are useless. Rather, it’s simply to make the observation that learning without excitement is difficult. If you try to learn something without the excitement, you will likely quit.
In a way, it’s good to be deluded about the realities of learning a subject. Without this delusion, we wouldn’t even begin.
Heuristics Lead to Rigour
As you learn more ideas in mathematics, it’s easy to start feeling like certain ideas are “below” you. This often comes in the form of saying that ideas are “trivial”, as if they shouldn’t take up any of your time. This can be exacerbated further in mathematics by the idea of rigour. Once we learn that not all proofs are equal, it can be tempting to say, “Okay, I get this proof, but that’s not the whole story. You’re not being fully rigorous here.” We can then get caught in the cycle of thinking of our work as more important than “basic” facts.
One place where I think this plays out is in the use of animations, diagrams, and heuristics to teach a topic. Here, I’m using the term “heuristics” as an umbrella term that encapsulates anything that wouldn’t be called a strict rigourous proof. This might mean using words instead of precise mathematical statements, or diagrams like I mentioned above. The point is that we substitute some of the abstractness in order to get at the core idea.
The problem I have with people who dislike this sort of learning by analogy is that I don’t think heuristics are ever the end goal. Sure, you might use a nice diagram to get an idea of what is happening within the mathematics, but it’s not the whole story. If you ask someone who specializes in making animations or diagrams that illustrate certain mathematical concepts, I doubt they would tell you to stop once you understand their explanation. Instead, they will tell you to dig deeper to understand all of the nuance that’s available. In fact, I suspect that they would insist you do this, because that’s where a deep understanding comes from. It’s not enough to just go at the surface level. To really learn, you have to go deeper.
Think about it from the perspective of the creator. They are attempting to take a complex idea and distill it down into a heuristic that is easier to grasp. It takes a ton of work to simplify ideas. However, when they are done this process, they still understand the inner details that may be hidden. That’s because they have done the difficult work of understanding the whole structure before reshaping it for others. As such, they know both the heuristic and the actual details. This is a powerful place to be.
For the audience though, they begin with the heuristic. That’s not a problem, but I think we should be more upfront with how we encourage people to dive deeper into a topic. Sure, the heuristics can give a person some taste for the idea, but the real fun starts when you go further and see how all the details play out. That might not be evident from the first exposure, but it’s something you can absolutely improve on.
I don’t think there’s anything wrong with using heuristics. Sure, they may not be the full story, but in order to prepare yourself for the full story, you need to learn the basics first. This is where heuristics really shine. They can give you an overview that lets you decide how much further you want to go. However, I think we need to also acknowledge a second function: heuristics can get you engaged with an initial idea more than the pure mathematics probably will.
Heck, we start learning by using heuristics. When we first learn mathematics, we aren’t talking about epsilons and deltas. We don’t begin our mathematics education with, “Let S be a set.” We start with familiar ideas, we build analogies, and we slowly learn how those initial ideas can be expanded and made more rigorous using mathematics.
There’s a place for both rigour and heuristics in mathematics education. Roughly speaking, I would say that you end up getting a “heuristics sandwich”. When first learning, you use heuristics. Then, you get sophisticated enough to understand the full details. Finally, once you’re really comfortable, you fall back to the heuristics because they are easier to reason with. There’s nothing bad with either of them. Rather, it’s useful to have a bit of both.
Knowledge From Repetition
When I’m sitting in class and listening to the professor, I get frustrated when they go over a concept once and then continue on as if everything is clear. I think to myself, “Do they really expect us to understand a concept from just one example?”
And yet, I realize that I do the same thing when I’m working with a student.
This is a bit disconcerting, and it reminds me that teaching is a tricky business. You never want to make things confusing or too brief for your students, but you also can’t read their minds. You don’t know know how well they are absorbing the material. Furthermore, the most difficult part is when a topic seems straightforward to you, but actually isn’t that clear for newcomers.
I’ve studied mathematics and physics for many years now, so a lot of ideas seem clear to me. For example, I can deal with algebraic expressions without batting an eye (as can any student with enough years of experience). I’m not stymied if I see fractions on top of fractions, and I can factor or expand expressions as I see fit.
At this point, there’s nothing “special” about this ability. It’s what I would call basic knowledge (for someone with the years of working with the concepts as I have), but the crucial point is that this isn’t basic to many people. If you’re used to doing multivariable calculus and dealing with derivatives and integrals until the cows come home, it can be difficult to remember that some people don’t have this skill. Even if it’s obvious to you, there’s a good chance it isn’t obvious to others.
I’m not trying to admonish anyone here. Rather, I’m writing this to remind myself that the students I tutor aren’t necessarily able to “soak up” the crux of a concept through one example. I know that I’ve felt like I’m wasting time when I do multiple examples with a student. After all, we went through the computation once. What’s the use in doing basically the same thing over again with slightly different numbers?
It doesn’t seem like a big difference, but I know from experience that it’s what helps you make sense of new concept. Multiple examples are worth the time, even if they seem superfluous for you, the teacher. For the student, it’s exactly what they need.
I don’t know why it took me so long to figure this out. It has been right in front of me for years. That’s one of the good things about being a tutor while I’m still a student. Even though I might be a few years removed from the classes that the students I work with are taking, I still am in their position as a student myself. The class content isn’t the same, but the experience I have as a student is still similar.
I now try to incorporate more examples and problems into the tutoring sessions I hold with students. Instead of assuming that they have a firm grasp after one example, I do multiple with them in order to let them get comfortable with the idea. Is it as efficient as zipping through many concepts with only one example each? No, but I think the extra time is worth it.
When I think back to my own experience, the times I became really comfortable with concepts are when I did many problems. It was at this point that I understood all of the features of a given idea. After that, I didn’t need to go through more examples to understand at a deeper level, but it took a lot of initial work to get to that point.
I like to think of it a bit like building any other skill. Suppose you want to get better at basketball. You can work on your ball handling, your shooting, your passing, and your movement. If you want to improve any of these areas, there’s not much to it in terms of seeing and understanding what’s happening. However, there’s a big difference between seeing what it means to have good shooting skills and actually having a decent shot. For the latter, you need to go through hours of repetition. It’s not that there’s anything “new” that you couldn’t see by watching someone else shoot, but by going through the repetition you get to feel what it’s like to take a good shot. That only comes with repetition, and I think it’s a similar story with learning mathematics.
As educators, I think we need to be mindful of this when working with students. It might seem like one example is enough to get the point across to a student, but I suspect this often isn’t the case. Instead, we need to be better at offering multiple examples to students. If we don’t, we can expect only a vague understanding. Getting a concept across is a strong function of the amount of repetition which is dedicated to it. Do only a few examples and students won’t come away from the experience with a deep understanding. This is true no matter how basic the idea is. Therefore, I would suggest to forget about how “easy” it seems to you. That’s probably a bad barometer. If you get to the point where you feel like doing another example would be a waste of time, do it. That’s probably a good start in correcting our biased perspective.
In some sense, learning is repetition. It’s as simple (and as difficult) as that. Repetition often feels like a waste of time, but I think we need to shift our thinking on this point. The more I think about it, the more I like the idea of repetition and a greater number of examples.
Don’t be afraid of wasting time. Going through a bunch of content but only having students understand the ideas at a shallow level isn’t really better.