Memory Aids

In secondary mathematics, students are allowed to prepare their own memory aid for an exam. It must be handwritten and can only cover one piece of paper. Usually, this memory aid is used to write formulas or challenging examples of different concepts so that one isn’t lost when writing an exam.

As a tutor, I have some experience advising others on their memory aids (as well as my own time being a secondary student, of course). I’ve seen some memory aids that were nearly blank and some that were filled to the brim. Both approaches can work, and it entirely depends on the person.

However, one thing I’ve seen time and time again is a tendency for one to write down the formula as provided by a formal definition. This is an important note, because a formal definition can be much more confusing to a student than the rigorous definition. And, as the students I’ve tutored tend to have difficulties in the subject of mathematics, these definitions can make little sense.

I remember an interaction I had with a specific student, who was having a trouble with a certain problem. I looked on his memory aid, and pointed out the exact concept to him. He looked at the writing as if written in a foreign language, and I could plainly tell he didn’t understand what was there.

Then it clicked for me.

“Do you understand the formula and definition you wrote down?” I asked, and he shook his head.

Now, I’m not trying to make fun of my student, but I’m trying to illustrate a broad point. Memory aids have turned into a piece of paper in which we write down formulas that we don’t even know what their functions are! My student simply wrote down the exact formulas from a generic copy of a textbook. Of course he wouldn’t be able to understand it when he needed to use the memory aid, he just copied a textbook.

With a memory aid, the goal should be to not forget any formulas. What I mean by this is that it’s nice to not lose points on a test because you’ve forgotten the exact composition of the quadratic formula. However, a memory aid should not be a used as a crash-course on a topic. It can be done, but the trouble is better spent trying to understand the topic before an exam. Once you’re in an exam, the memory aid should help remind you of the composition of formulas, not how to do a problem (though I have been in situations in which the example I wrote on my memory aid is indeed a test question). This comes down to the fact that mathematics and science is about solving problems, not memorizing formulas.

I’ve written a lot about memory aids, formulas, and memorization here already, but it’s a great topic to think about. As it stands, we send somewhat confusing signals to students: sometimes it is appropriate to have a memory aid, and sometimes it is not.

The truth, of course, is that the kind of knowledge we want to teach students shouldn’t be something we can place on a memory aid. For mathematics and most of the sciences, a memory aid can be completely appropriate, yet not give away the contents of an exam. To make sure this does not happen, there’s no problem to giving students pre-made memory aids to put them all on an even playing field.

What I want to avoid are situations in which students blindly copy formulas onto their memory aids without knowing what they actually mean. Whenever possible, write down formulas in ways that you understand, and it will save you from that terrible feeling in an exam where you do not understand what you wrote on your memory aid at all.

Electric Shock

As people who are working very hard to improve themselves in their passion, it’s tempting to get caught in a loop where we do everything that we should be doing for maximum performance. This is particularly prevalent for people who work on these pursuits as side passions, where the activity is important to them but does not constitute a living.

Because we work so hard at what we do, we walk a fine line between pushing too hard and not realizing our full potential. Let’s face it: the amount of time we invest in our craft can be both physically and mentally exhausting. Therefore, we must be extra careful to avoid burnout and hating what we do.

The strategy I work with goes like this. When I find myself in a situation where I feel like it’s taking an enormous amount of effort to put my usual work with, I don’t keep on pushing. I’ve made this mistake enough that I know it doesn’t work, and actually makes you feel worse. Instead, I back off immediately, not wanting to even approach an instance where I take a full week off. If I back off early, I am able to jump back into training much more quickly.

My mantra is patience. I need to be patient with myself, content with slow growth than trying to do everything as perfect as possible. I also need to be patient with my dips in motivation, accepting them and backing off instead of continuing forward. I treat these dips in motivations like electric shocks, stopping as soon as I start to fall into a state of low motivation. I’ve built up enough of a base that one day won’t make a training cycle, and it’s better that I take an extra day off than push on when fatigued and beat down.

If you want to keep on cultivating your passion, be mindful of those dips in motivation, and be wise enough to take a step back if things aren’t going well. Don’t push through and make it worse in the long term.

Science Education and Self Learning

There seems to be a resurgence in the idea that students can learn without being constricted to a classroom, that they should be free to pursue learning anything that makes them curious. The rationale is that a curious student will love what he or she is learning, instead of slowly being told that they must learn it.

This sentiment has echos for me on my thoughts about running. If I was forced to go outside and run a lot at a young age, I don’t know if I would be enjoying running as I do today. More likely, I think, would be that I’d dread running because it would be another chore to complete, something else to get done.

Contrast that with my attitude towards running now, and there’s a stark difference. I couldn’t be more in love with running than I am. It’s one of the most fulfilling activities in my life. Yet, I can very well imagine what would happen if I was told to run, instead of discovering it on my own.

Going back to education and science in particular, it’s easy to see how this situation could occur. Traditional education follows a model based on a curriculum. In essence, there are people who’ve decided that these classes are important and vital to being a productive member of society, making them mandatory.

This can be viewed in two ways. On the one hand, it guarantees that students are exposed to science. There’s no obligation for others to show a young student what science is. Instead, it’s baked right into school, making it an efficient way to expose students to science.

On the other hand, it can be a nightmare when a student is badly exposed to science. Science can be like water: when it’s good quality, it demands to be drunk. When it’s dirty and polluted, it can make you sick and turn you away. Science is much the same.

The early years are the most important. This is where the bedrock is placed. Without a solid foundation of science, it can be difficult to have a student want to pursue science later. After all, as science gets more intriguing and wonderful, it also gets more complex. Therefore, it’s imperative that a student has the foundation necessary to handle the increased complexities. In science, amazing and complex form a positive linear relationship.

But what happens when a student does not enjoy science as a kid? Simply put, he or she will move on to something else. The student will pursue a different path.

Is that bad? Of course not. Students should be free to pursue any sort of path that they are excited about. If there’s one thing that I would wish upon any student, it is that he or she finds a subject that absolutely fascinates them, and to then explore it. Whether that be in the visual or liberal arts, machinery and hands-on careers, athletics, business, education, or as a scientist, every option is good. As long the student enjoys the subject, that should be more than enough.

But what I’m more interested in is if science itself has let down some students? And, unfortunately, I think the answer is yes.

Because of the mandatory science classes, there’s a huge responsibility for science teachers to educate students in a way that will stimulate their curiosity. If not, they risk putting the potential scientific future of the student in jeopardy.

The point I’m trying to make is that we mustn’t feel the need to force students into science, but we should equally try to avoid making them feel lost within science.

This starts by making science exciting and cool at the beginning. Let’s be honest: students aren’t going to be excited by seeing a bunch of mathematical equations. However, they will be more interested by seeing a live demonstration, then understanding how that situation can be expressed using mathematics. The mathematics aren’t just given. There’s context and meaning associated with them.

Furthermore, at the youngest of ages, science should be about observation and curiosity. Getting students to ask questions about the world is the first step towards getting them to describe it. Once they see that their questions can be posed and used in an experiment, the hook of science is there.

As it stands, I feel like we classify students too soon. From a young age, we repeat certain messages to them. “You’re more of an arts student” or “You’re a nerd, so you fit well in the sciences”. These are presumptions that we make, labeling students to the point that they begin to believe those labels. Once that happens, we have failed them. If there’s one thing that is strong in all of us, it is our belief of who we are. Whether you believe you’re a scientist, an artist, a writer, or any other label, those beliefs aren’t easily changed. Therefore, someone that does not feel like science is a welcoming path for them will remember. Once the door seems shut, there will be little effort to investigate again to confirm that it’s shut.

This is a sad reality. To change it, we need to push back against labels. If we want more people (and more diversity) in science, we need to encourage systems that promote science in an accessible way. We need teachers that are willing to educate students in a way that science seems amazing, not boring and needlessly difficult. Yes, science can be complex and the mathematics required to make sense of many concepts can be tricky, but as long as the joy of scientific discovery is there, the student will not mind putting in the effort to learn complex mathematics.

Another thing that irks me is when, as mentioned above, people say that students shouldn’t have structured courses or that tests aren’t useful to the students at all. When I hear this line of thinking, there is usually another that goes with it: business and entrepreneurship. It appears to me that nearly every call for students learning on their own has some sort of roots in business. The idea is that students should be learning through building things. They should be working on something tangible to them, and this will be a learning experience in and of itself.

I take issue with these arguments, because they are exclusionary. The truth is that each subject or domain has its own ways of transferring information. It’s very possible that hands-on learning and public iteration are key components of teaching business students to be entrepreneurs. However, I don’t subscribe to the idea that everyone has to build something. Or at least, not in the way that it will become a future source of revenue.

In science, entrepreneurship isn’t particularly needed. Yes, one does do research on a topic, which is kind of like building a product. However, the problem is that science has such a deep background that it is essential to know a part of that background if one wants to successfully research scientific topics. Once again, this does not necessarily need to be taught through the traditional education system, but it does need to be taught.

Furthermore, the traditional model of education is a good fit for science education. The reason is that education places a structure on students. When they begin, there’s a path forged before them, inviting them along. At the end, we can be confident that the students acquired the knowledge that we wanted them to have. I’ve heard people discuss how traditional education is industrial and produces “replaceable” people. I don’t think there’s quite enough nuance in that statement. If we think about it, we want students to have the same scientific background. Not because they will be replaceable, but because they will be well-suited to jump into any scientific domain they wish.

It goes back to building a strong foundation. If we don’t provide students with a solid scientific foundation during their education, we put them at a disadvantage once they have free reign on their scientific careers.

Instead of making students forge their own path during school, the curriculum is designed in order to ensure a smooth transition from subject to subject. Unlike other disciplines, it is difficult (not impossible, but difficult) to learn science in a non-linear fashion. Sure, there are small parts that can change, but generally there is a set progression. After all, it wouldn’t make sense to have students learning about vectors before they’ve even learned the fundamentals of algebra. The latter almost needs to be taught first, or else there’s a good chance the student won’t thoroughly understand what is being taught.

Which brings me back to the starting point, about students teaching themselves. In theory, it’s a valid idea. Students can indeed learn science from resources other than educational institutions. However, there are two big drawbacks. For one, it is difficult to get the feedback and progression that is essential to learning science at a reasonable pace (which is of interest to everyone). Additionally, a specific reason for science lies with lab sessions. In science, labs are a key part of getting a complete understanding of scientific phenomena. In the eyes of many students, they are somewhat tedious and boring. Still, they are important to convey tangible links between theory and what actually happens in our world.

The trouble is that the equipment used in labs is expensive. And I don’t just mean a little expensive. The instruments are incredibly expensive. Whether a student is using volatile chemicals, a $3000 scale, or very fine laser equipment, science labs use equipment that is terribly expensive. For an educational institution, it’s a reasonable expense. However, I highly doubt many self-learners would want to spend the money required to do these labs. Consequently, these students would miss a fundamental part of their scientific education due to a lack of lab equipment. This is an argument that I find is often forgotten in discussions about self-learning. Yes, there is a wealth of resources online, but there’s no replacement to having lab equipment and conducting an experiment.

Education is more free and more readily available than ever. Whether you want to learn a new language, an art skill, or complicated physics, the information is out there for those who want to search for it. Even better, a good portion of it is free, and often high quality. Therefore, it may make sense (on its face) to pursue self-learning.

However, the drawbacks are much worse than the advantages. While you do indeed get to learn about topics and subjects that interest you, the responsibility is usually on you to learn the content. There aren’t exams to mark your achievement. Perhaps that’s a good thing for you, but there is no escaping the reality that traditional education is structured. Whether that is good or bad can be debated, but having a certain amount of structure will indeed help the student progress in a way that is logical and helpful to them.

Furthermore, science education is a particular section of education that needs to be examined with care. It’s easy to say that all the information is out there in the vastness of the internet, but it’s quite another to find it and absorb it. With so many different methods to teaching science, it’s critical that there is some sort of consistency. Without it, the learning process becomes just as much of translating from one teacher to another than the learning itself.

If we want to have people enjoy their science education, creating a progression is essential to enjoyment. Sure, it may be fun to talk about explosions and space and the wonders of life, but it’s even more wondrous when the student understands the structures and mathematics behind it. That can only happen by creating that foundation on which a student can build off of.

Information is free and everywhere online now, but there’s no true replacement to taking a structured science course that includes lab sessions, structured progression, and teachers that can really make the difference between a love for science and an abandoned domain.


If you want to be a good hockey player, gymnast, swimmer, basketball player, tennis player, badminton player, or runner, what do you need to do?

You need to be a good athlete.

First and foremost, being able to move athletically is key. Before you do good at a particular sport, being able to run, jump, and throw is key. Are you able to move with grace and fluidity, or are you an awkward mess of limbs when you move? Can you move within all planes of motion, not just one? These are questions that are required to be answered if you want to be good at any sport.

Being good at any sport is more than just the sport-specific skills. Having the right base is just as crucial.

Have you ever wondered why the same people seem to be the dominant ones in a gym class, no matter what sport is being played?

People seem surprised or impressed when they see that I’m able to play competently at badminton, tennis, squash, hockey, basketball, soccer, and more, while my actual primary sport is none of those, but running.

However, if you think about it, there’s nothing that surprising about it. I can do well in all of these sports because I’m a good athlete before anything else. I’ve established a good base, and this allows me to try many different sports and be good at them. From running, I’ve established a strong amount of endurance, and I’ve worked at also being quick with my feet. Combine speed with endurance, and you’ve got a recipe for a person being able to enter many different sports. Furthermore, if you add a good sense of proprioception into the mix, you’ve created a player that can be very dangerous in many sports.

This is all to say that when you establish the base, you’ve given yourself an advantage in any sport you do. Therefore, being good at running, jumping, throwing, and keeping your balance can be the greatest investment in your time you’ve ever made.

Before sport-specific skills, building that foundational base is key.