The inevitability of long term goals is that you will face moments where it seems like the task you’ve set for yourself is too large, and that you’ll never accomplish what you want. These slumps happen all the time, and I’ve found that they usually occur – perhaps somewhat ironically – after a stretch of good progress. What once was novel and fun to work on now seems to be a lot of work for only marginal gains.
It’s at this point that you need two things. First, you need a decent amount of discipline to keep chugging along and doing the work that is needed. For myself, this means getting up early to run even when I feel like doing nothing (particularly as the weather starts to morph into winter conditions). It also means carving out a chunk of time each week to write, even though I’m often loaded with homework that needs to be completed. Finding the discipline to do these things when it seems like you’re not even making progress is what will get you to that next level of progress.
The second thing you need is the ability to change what you do in order to give yourself that mental switch to get back to feeling good about your passion. While training requires some sort of routine (I would never be able to run around 125 km per week without one), it’s dangerous to get too complacent. Doing so makes your chosen craft to seem like work and just another thing to get done in the day. Getting in this mindset can bring you a lot of frustration and ultimately kill your passion for what you do. Therefore, finding a way to infuse some sort of novelty into your craft is only a good thing. For myself, that’s doing different workouts and finding new trails to run. I’m not doing anything stupid just because it’s new, but I try to bring variety into my training because it keeps me loving the sport.
For many of us, the wish is to keep on doing our craft for a good portion of our lives. Consequently, our goal should be to do things that keep our love for our craft going. Conversely, we should avoid the things that will make us hate doing our favourite work, mainly overloading ourselves. If we want to keep chugging along with our work, it’s essential to work through the slumps and persevere.
I find it amusing that I could probably impress my family at the dinner table by using terminology from my mathematics classes. If I used words like logarithmic, multivariable calculus, hypersurfaces, tangent planes, and linear approximations, I could get them to think that what I’m doing is pretty advanced stuff that they wouldn’t even be able to wrap their heads around.
But that’s not the case. Many of these terms can be explained much more simply. Tangent planes have an easy geometric visualization that is easy to understand, and the linear approximation is essentially the tangent plane! Even those two terms sound complex but are really the same and not that difficult to understand.
Mathematics is good at defining concepts rigorously, but the terminology used in mathematics can make the discipline seem very confusing, even when it isn’t. In this sense, there’s a sort of “masking” going on. The terms can seem more difficult than the concepts themselves.
This is why it’s difficult to communicate mathematics with people outside of the discipline. A lot of the concepts can make sense, but the terminology is difficult if one does not have a solid background in mathematics. And since explaining a concept requires terminology, those outside of the discipline tend to find the whole thing just too complicated.
For some subjects within mathematics, it’s possible to sidestep this issue through the use of visuals. If the concept you’re trying to explain has any sort of geometric intuition, it’s usually enough to show them the geometric situation to gain an understanding of what is going on. Unfortunately, this isn’t possible with many subjects, such as linear algebra or anything involving more than three dimensions. At this point, the mathematics becomes the sole anchor point for a concept, and terminology is important.
Terminology in mathematics is usually precise, offering little wiggle room. However, the language that is used is often inaccessible to the casual observer outside of mathematics. This is a shame, since many mathematical ideas seem complicated from their names, yet are downright simple when explained. As such, there’s a potential to get more people interested in mathematics by breaking down these language barriers. Sure, it’s not necessarily rigorous, but the truth is that those outside of mathematics won’t really care about the more subtle points of a definition. Instead, simply introducing them to the subject is more beneficial in the long run.
Let’s try not to mask mathematics from the public with a veil of complex terminology. Instead, let’s try to move it away whenever someone asks a question concerning mathematics. Hopefully, we can then get many more people interested in mathematical concepts.
This Isn’t The Forefront
During primary, secondary, and CÉGEP, mathematics is pretty similar in its style. Essentially, the idea is to give students the tools and skills necessary to be able to solve various problems (both in pure mathematics and applications such as physics or chemistry). There is a large focus on formulas and strategies to solve problems.
However, there’s one key distinction between this education and the “real world” of working within these disciplines: the problems for the former have solutions, while the ones for the latter are waiting for solutions. This is important, because it means the skills and work ethic that got you to the discipline throughout school won’t necessarily work now.
It’s a reminder that we cannot stay complacent about our progress. Yes, it’s incredibly important that we learn and retain the knowledge from our education, but there is a large difference between struggling with a question and being able to peak at a few steps of the solution to help you get on the right track and banging your head against the wall because you’re breaking new ground and you find yourself stuck.
Remember, the forefront of a discipline is much different than the path to it.
If there is one thing in particular I dislike about many tests in science and mathematics, it’s how we link the ability to do well on tests with the ability to do science.
The biggest problem I have with this link is that the things we test are not the things we actually care about as scientists, mathematicians, or educators. A great example is the terminology question. This is a question that basically requires a student to know a definition without needing to understand what is important about the concept. Instead, they can “fake it” by knowing the correct definition. These questions don’t actually aid in understanding concepts, they just make sure you know what the proper terms are.
I saw a question like this the other day where I was tutoring a student. It asked, “Write down the above equation in functional form.” The equation was something like $ 3y -4x = 0 $, and I couldn’t immediately understand what the question was. I knew that it involved manipulating the equation, but I didn’t even know what functional form was. Thinking back on it now, I imagine it must be an equation in the form of $y=mx+b$, but I still thought it was strange at the time.
What is the actual value of the question? To me, it only seems like a way of saying, “Write the above equation for y in terms of x.” That makes more sense to me, and it’s only a few words more than “functional form”, yet it is more descriptive. Additionally, I fear it slots the learning of equations and algebra into “archetypes”, which isn’t productive.
Another issue is that science and mathematics (and perhaps other fields, but I am not as familiar with them so I do not know) have serious naming issues. If you were to imagine the body of knowledge science has uncovered as a book in which anyone who has something new to contribute writes their own page and inserts it into the book, you wouldn’t be far off. Science is oftentimes a jumble of confusing terms, with a lot of historical baggage.
For example, I remember learning in my astrophysics class about the different classification of stars. I learnt that we had a whole system of stars: O, B, A, F, G, K, M. At first, I wondered why on earth the classification wasn’t alphabetical, since it classified the brightness of stars. However, I soon found out that the reality was the the original classifications weren’t particularly good, so the names were kept and the order was switched around in order to have them increasing.
Therefore, what could have been an easy process to remember the classifications became a mnemonic: Oh Be A Fine Girl/Guy, Kiss Me. While funny, it isn’t exactly intuitive to learn this classification, and this is something we were indeed testing on. My question then becomes: does being able to remember classifications well on a test really solidify our understanding of which stars are brighter? I’d argue probably not.
This isn’t the only example, either. If you’ve ever been privy to a science or mathematics class, you’ll quickly learn that equations and phenomena aren’t necessarily described in ways that are entirely useful for a person hearing the name. Instead, the concepts will usually carry the name of the person who discovered, invented, or came up with the concept. This is a way for many scientists and mathematicians to stay “immortal” in time. After all, few physics students are ever going to forget who Newton is, just like few biology students will forget about Darwin, or how mathematics students will not forget about Leibniz, Newton, Pythagoras, or many of the French mathematicians. We don’t know these people because we are all fans of the history of science (though some of us are). Really, it’s due to constant barrage of hearing these names in concepts in classes that solidifies these people in our minds.
Unfortunately, this isn’t a great recipe to actually knowing what these concepts means. Saying “Newton’s Laws of Motion” only gives me one real piece of information: the laws will be about motion. But these could have easily been referred to as the three general laws of motion. Except you will almost never hear that. Instead, you’ll hear, “Newton’s Laws”, because it’s a succinct way to describe them. And there’s nothing wrong with describing concepts in concise terms. The problem to me occurs when students get tested on knowing these things, where the names have almost no use.
I’m reminded of some words I’ve heard from Richard Feynman, who described an encounter he had with a kid who criticized his father because he never really explained the names of birds to Richard when they went on walks. However, Feynman shoots back, saying that his father taught him about the birds instead of just what the names were, and that this was much more valuable and interesting. Indeed, this is the sort of thing I imagine. Not that we shouldn’t understand what things are called, but that it’s always more important to know the concept than just the name, and we should be tested accordingly. Personally, I know that it would knock off a lot of stress from remembering how concepts worked, because I wouldn’t have to link a name to a concept. Instead, I could just explain the concept. “Explain Newton’s Second Law” would be asked as “How does motion relate to forces?” The latter question is much more descriptive than simply giving the name of a scientist.
I hope that a shift comes in mathematics and science soon, bringing questions that dig deeper instead of looking at surface-level detail about names.