Building On Top of Each Other

I once had a mathematics teacher who would say something that bugged me: what we’re doing is easy. I am barely being hyperbolic when I say that this teacher would say this for every single concept we learned. Therefore, I couldn’t help but think that surely not everything could be this easy.

The saying would take on an ironic meaning to me, because even for the most difficult concepts we would learn in the class, my teacher still saw it as easy. I would frequently turn to my friend and exclaim, “How can it all be easy?”

He had no answer for me, and I thought that there wouldn’t be one. However, after finishing the course and reflecting on my experience, I can start to see what my teacher was talking about. It’s not that the concepts are super easy to understand. Instead, it’s that the leap from what we previously knew to what we learned is not huge, so it shouldn’t be too difficult to understand the new material.

Here is a concrete example. When I was in secondary school, I learned about algebra and functions. At the end of secondary school, I learned about more complicated functions and curves (quadratic, hyperbolic, and ellipses), plus I was introduced to the notion of vectors.

In the beginning of CÉGEP, I learned how to manipulate vectors by addition, subtraction, dot products, and cross products. This was mainly seen through the lens of physics, but also through my mathematics courses.

Fast forward to my final mathematics course (Calculus III), and I began to learn how to put these ideas together. Suddenly, vectors weren’t static anymore. They were affected by parameters such as time or angles, meaning they became curves.

Now, imagine trying to show this to a student who doesn’t already have a good grasp on vectors or basic functions. It would be basically impossible. The leap from knowing nothing about vectors to working with vector-valued functions is too drastic. However, if one progresses from simple functions and vectors to these more advanced topics, it’s more of a transition than a leap. The topic that proceeds makes sense given the previous material that was seen.

For myself in that class, it meant that once I learned the polar coordinate system and graphing in three dimensions, everything we did was both in Cartesian and polar coordinates. Taking derivatives or integrals was done with both systems because it was the obvious next step to take.

I can now say that I understand what my teacher meant by the material being easy. It’s not that it would be easy to anyone, but that it should be easy for us, given our progression (and since I had the same teacher for all my mathematics classes, I know they had a good idea for the progression).

The implication of this statement though is that the fundamentals are so important. If one wants to make the next logical transition to a new concept, the previous concept must be understood. By rewinding the clock all the way back, one arrives at the absolute beginning of learning the first important concept. If this concept isn’t understood well, then it can have consequences down the line in terms of why a newer concept doesn’t make sense.

Let’s face it: school isn’t made for us to fail. It’s designed in a way such that a student can succeed. Therefore, the progression should be appropriate, and the responsibility is on the student to understand the fundamentals before moving on. Yes, this can be difficult when one is in a class that seems to be moving on despite you not understanding, but that’s when either the teacher or the student needs to step up and take a moment to review. If not, a student is just pushing their problems down the road.

If you feel like you’re making a huge jump in your learning that you don’t understand, it’s likely that the fundamentals you learned aren’t completely absorbed. Strengthen them, and those leaps will become baby steps forward.

The Skill of Spotting

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If I had to pick one skill that I’d say helps me out in science and mathematics the most, it would be the skill of “spotting”. Simply put, it’s how good one is at figuring out the essence of a question. Personally, it’s a skill that I cherish, because it allows me to do so well during my exams. Rarely do I read a question and go blank. Instead, I’ll immediately get to work, going through the steps needed to solve the problem.

Of course, there are really two ways to solve a problem: the “make it up as you go along” strategy, and the “spotting” strategy. The former can work, but it’s likely a longer method and can result in a lot of dead ends or useless work. However, the “spotting” strategy allows one to know exactly what the steps are to solve the problem.

For example, if I’m given a problem on mathematical induction, I know the steps I need to do. First, I write the base case out to show that it is indeed true, and then I do the calculation with $k=1$ and $k=2$, just to show the pattern seems to hold. Next, I write down the equation with k, and then I try to solve it by induction using k+1 in conjunction with my base case and assumed truth of the statement with k.

This is my “format” for solving induction. What you can notice is that there isn’t any numbers or specific examples in the above case. It’s just a template for solving problems involving mathematical induction. By “spotting” the problem, I can just apply this template and be confident that I will get the correct answer.

This is why the “spotting” strategy is so good. It takes the thinking out of the present moment by referencing a template for a specific kind of problem.

So how do you cultivate this skill? The best answer is that you have to do a lot of problems. By practicing over and over, you will begin to sort out problems into various categories. You’ll start recognizing exactly what you have to do when you get a problem with acceleration, velocity, and position, or when trying to find a tangent plane to a surface. The values and functions might be different, but the approach is the same. Slowly, you will amass an archive of examples that you can reference in your mind when faced with a new problem.


When I enter a test now, I always try to spot. If I’ve prepared well for the exam, this shouldn’t be a problem. After all, the content of the exam is stuff I’ve seen, so there’s no reason I shouldn’t recognize it. That’s not to say that a certain question won’t be difficult, but the odds are with me that I’ll be capable of spotting a problem.

This calms me down immeasurably during a test. Since I’m hyper-focused on getting a great grade, it’s a relief to me when I know exactly what I have to do to solve a question, and all that’s left is for me to go through the motions. Basically, I practice a lot so that I can offload the work on the test to my vast archive of previous examples in order to solve the question.

My advice to you is this: if you want to have an easier time during tests, learn to spot similar questions. It will save you the mental energy of always figuring out from scratch what you should do. It’s like using formulas. After you’ve been exposed to the proof, there’s no need to prove the formula you use on every question you answer. You just use it. Likewise, don’t waste time reformulating the same strategy over and over again to solve a question. Use the one that worked, and just apply it.

It will save you a lot of time and make tests go that much more smoothly.

Slumps

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.

Masking

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.