### Participation

When I was in secondary school, I really hated going to French class (I still wouldn’t particularly enjoy it). It wasn’t that the teacher was horrible or anything. Instead, it was simply because I didn’t like taking language classes other than English. I really excelled in English, and the huge chasm in my abilities in French versus English weren’t something I liked being reminded of. In English, not only was I well read and could write, I could also speak well. Conversely, speaking was my weakest link in French. It’s frustrating being able to know all the words that you want to say in your head (and even being able to think them), but not be able to actually express them. As such, I didn’t participate in class at all, preferring to only listen.

The result of this was that I came close to getting a horrible mark in the class for never participating. Since I didn’t want that, I changed my habits about halfway through the term and participated as needed. I even found that participating made me more engaged in the class, rather than just waiting for it to end.

The thing I see over and over again in a science or mathematics class now that I am out of secondary school and into higher education is the lack of participation in class. And I know it’s not because the students aren’t listening or are bored. Rather, no one participates because we are scared to be wrong. Almost every time a professor asks a question in my classes, they are met with silence, because we never want to answer a question and be wrong. I’m definitely a part of this group, and I struggle with this all the time.

Rationally, there’s no better place to be wrong than when giving answers to teachers. If the teacher is doing their job correctly, a mistake in an answer is an opportunity to expand someone’s mind and teach them a way to think. Done wrong, it makes a student feel unwelcome to share their answers again, most likely closing the door on future participation. I feel this all the time in class, and it’s why I don’t answer all that often. However, I’d almost wager that I am one of the most frequent contributors, showing just how infrequent we participate in class.

Being on the other end of the situation, I know that it’s not a fun feeling when no one wants to answer the question you pose in class. As such, I struggle not answering in class. After all, it makes no difference if I get something wrong in class. Who cares if others think I’m an idiot for asking the question? Who cares if it’s obvious to everyone else, but not me?

Personally, I’ve always struggled with this to some degree. I’m not actually shy, but I’m always so obsessed with being “good” at subjects that I don’t want to appear like I’m a novice. Therefore, I try to hide my lack of knowledge by doing the “rough work” at home, so I can appear to be knowledgeable during class. The truth, of course, is that the class is there for me to learn, not to know everything before we start.

This is why I encourage participation in mathematics and science class. We need to stop being worried about what others will think of our hypotheses and ideas. Yes, people will judge them, but who really cares? On an intellectual level, I know that asking questions and giving answers in class (even if I’m not sure about them) is a way to help me, not hinder me. As such, it’s one of my personal goals to work on, and I hope you can reflect on this as well. Do you participate in school, even when you aren’t sure about your answer? Do you ask questions, no matter how dumb? If the answer is “no”, I encourage you to work on this with me.

Participation in class is a key component to learning. Do what most people aren’t, and use it.

### Transforming

If I gave a problem to one of my friends who aren’t in physics or mathematics, they’d probably say that it’s way too complicated for them to solve. What’s amazing to me though, is how they are so often wrong about that assumption. Truthfully, many of the problems that I tackle at school (not actual scientific problems) are relatively easy and just require transforming the problem. What I mean by this is that our first line of attack for a new situation is to try and transform it into an old situation that we know how to do.

For example, when I first learned about Lagrange multipliers this year, we started solving a problem using the method that came into our minds first. This involved solving for a variable in one function and then substituting it into the other.

Only at that point did my professor show us the motivation behind using Lagrange multipliers. In the particular problem we were dealing with, we were trying to find the maximize the sum of three numbers while ensuring their product was a certain value. I learnt how Lagrange multipliers preserve the symmetry in a situation and made it relatively more straightforward and systematic to solve.

At the same time, we didn’t just jump into this new method. We started by transforming it into a manner that we could solved and worked from there. After that, we were shown how the alternative method was generally easier to work with. The crucial point is that I didn’t start from absolutely nothing. The Lagrange method was shown after the “regular” method we had used, making it easier to follow.

Remember: a lot of the “complicated” things you will learn are just variations on a theme, adjusted slightly for your present situation.

### Relating

One of the most difficult things for me to do when I am learning is to make the conceptual leap from one idea to the next. Often, I’m not confused about how to do a problem. Rather, I’m stuck on an idea that preceded it which I am not fully on board with.

This is the reality that I’m frequently confronted with when I tutor students. Most of the time, the actual application of an equation to a problem isn’t an issue. Instead, the trouble comes from some conceptual piece of the puzzle that simply isn’t clicking into place. Usually, this comes from the teacher not fully explaining a concept, which then makes the student confused about how to jump from one idea to another. It’s also often a simple thing that manifests itself in the form of the student not being able to do any of the problem.

An example of a gap left in explanations to me is how students have to learn about various functions and what they look like. The list includes the following functions:

– Constant: $y = c$

– Linear or First-Degree: $y = mx + b$

– Quadratic or Second-Degree: $y = ax^2+bx+c$

– Inverse: $y = \frac{k}{x}$

– Exponential: $y = ab^x$

Ignoring the fact that the exponential function is thrown in there completely randomly, the other ones are given a bunch of different names. However, what is missing from this list is how the functions relate to one another. If we remover the random exponential function that was thrown in, a much more intuitive pattern can emerge from this list: each function is a successive power of $x$.

When I first showed this to the student I was tutoring, they didn’t even know what I was talking about when I used the idea of a negative exponent. But after walking through some examples with them, it was much clearer.

Here’s my modified list:

• Inverse function (negative one degree): $y=kx^{-1}$
• Constant function (zero degree): $y=cx^0=c$ (Ignoring the case of $x=0$)
• Linear Function (first degree): $y=mx^1+b$
• Quadratic function (second degree): $y=ax^2+bx+c$

If you look carefully, there are two important things I did here. First, I listed them in a logical order. They are ordered from the smallest power of $x$ to the largest power. This makes the functions easy to recognize. Count the powers and you’re good to go.

Second, I explicitly showed extra work in this list. I showed the powers of one and zero even though the final answer does not necessarily show them. The reason is simple: the students are learning and this is an easy foothold to take when learning about functions. When you’re suddenly bombarded with all these possibilities for functions, it can be difficult to keep them all straight in your mind. This way, you can easily see how the powers of $x$ are in each equation.

This is a simple point, but it’s something that is so important to get right in the beginning of teaching a concept. The longer this goes unnoticed, the more trouble the student will have since they won’t ever feel as if they have a comfortable foothold into the concept. Think about walking on ice. You’d feel a lot safer if you were wearing crampons or skates as opposed to summer shoes. In the same way, our goal is to give students these conceptual footholds, and the best way is usually through relating with something they already know.

### Cementing

A common thread I see between many young students who don’t seem to “get” mathematics is that they aren’t told to look at the way they are taught mathematics as only that: a way. Unfortunately, the impression that is made on them is that mathematics is a strict set of rules that cannot be broken and must be followed every moment.

Imagine instead I was learning the English language, and that I wanted to express the emotion of anger. When I ask this, the teacher tells me, “You can say that you’re ‘angry’.”

Here’s my question: wouldn’t it be absurd if I went along my whole life using that one word to express how I was feeling? Instead of using the variety of synonyms that the English language provides in order to bring more nuance to my state of mind, I’d be repeating the same word over and over again, because I know that it will generally express what I want. In essence, it’s a “safe” option. It then gets cemented in my brain through repeated use, and I am stuck in this cycle forever.

I hope you agree that this wouldn’t be a very good thing to do to a student. Making them only see one way of looking at the world and not using any other words to describe anger robs them of expressing themselves, since it can never fully capture the particular shade of anger in that moment. However, we are doing the exact same thing to our students in mathematics!

Consider one of the earliest functions that are exposed to students: $y = f(x)$. This is taught to be a function of the independent variable $x$. After this is taught, example upon example is given using this form. However, students are then taught that functions don’t have to look like $f(x)$, but can be in the form of $f(y)$ as well, with an independent variable $y$. This isn’t necessarily a problem, but from the countless examples and repetition in the previous form, using this other method becomes confusing. Then, to make matters worse, they are taught that one can often switch between these forms by manipulating the variables.

As one can imagine, this confuses many people. How can $y$ turn from a dependent variable to an independent variable by just arranging the equation in a different manner? When do we have to use $f(x)$, and when should we use $f(y)$? These questions may be obvious to us, the tutors and teachers, but it isn’t so obvious to the students. I’ve seen it myself.

In my eyes, the solution is to make mathematics more dynamic. Instead of introducing the variables $x$ and $y$ as the de facto standard for the rest of their mathematical lives, show them that the variables we use are just their for us, the people doing the mathematics. It doesn’t really matter if something is called $x$, $y$, or $lambda$, they’re just symbols. From there, I believe students will have more confidence in using variables that aren’t your traditional $x$ and $y$. The way in which I try to help this with the students I tutor is to use the terms “horizontal” and “vertical” axis instead of the $x$ or $y$ axis. My hope is that this will get them out of their standard way of mathematics and make them realize that a lot of the things you do in mathematics is more out of convention than of need.

Obviously, the end goal is to get students using mainly just $x$ and $y$ for a lot of mathematics. However, the point is that these are just the traditional ways of doing things. There’s no requirement that the y-axis be the vertical axis. As such, I’d much rather change the names of variables and get them to think in ways that aren’t the usual, just to keep them aware that all of mathematics does not revolve around $x$ and $y$.