Jeremy Côté

Behind the Equations

In secondary school, students in physics learn about the kinematics equations. These equations describe the motion of objects under a constant acceleration (often gravity). There are several equations, which describe the relationships between acceleration, speed, position, and time. In particular, here is one of the equations:

x(t) = x0 + v0t + at2/2.

This equation lets us find the position at any time t, since the other parameters are known. You might even recognize this as the equation of a parabola.

Students learn about this equation and the others during their first course in physics. They are then encouraged to write down all the equations, and decide on which one to choose based on the parameter that is missing. In the above case, the speed of the object at time t isn’t present.

I once was working with a student, and I could see that they did not see the link between the equations. Wanting to probe this a bit further, I asked, “Isn’t it a bit strange that this exact arrangement of terms describes how the position of an object changes with time? For example, why in the world do we need to have a t2 term in the equation?”

The student agreed that it did seem strange. At the very least, it wasn’t obvious as to why this combination of terms produced the right answer. After all, if you just look at the equation, there’s not much telling you if this is the “right” combination1. So what’s going on here?

Of course, the answer is that the equations do come from somewhere. They come from analyzing Newton’s equation F=ma under the presence of a constant acceleration. But if that’s it, why don’t students learn about this first? After all, they encounter Newton’s equation early on.

The issue is that one needs to use calculus in order to give a satisfactory derivation of the kinematics equations. In secondary school, students don’t have this background of calculus yet, so they cannot follow the steps (even though they are quite simple). As a result, students are presented only with the final answer. The derivation is left out. The implicit message is that the derivation isn’t that important.

The main consequence of this is that students end up seeing these equations as “magic”. In other words, they feel that there’s no way they would have been able to come up with these equations on their own. This is untrue, but the lack of derivation forces them to accept the equations on authority.

I hope you can agree with me that this is not a good situation to be in. Accepting equations on authority is in the opposite spirit of both subjects. This encourages students to memorize equations without understanding where they come from and why they are true. After all, if you know where an equation comes from, you don’t have to worry as much about memorizing it. If you forget it, you can always work it out again.

We are doing a disservice to students by forgetting about working out the details. We are teaching them that it’s more important to know how to use an equation than where it comes from. We are sending the message that equations and formulas are something to be recalled, but the proofs themselves don’t matter.

Yes, it will take more time to present the material. That means there won’t be as much time to do practice problems, and the pace might be slower. But students want to learn about the explanations behind the equations! I’ve found that students are interested in finding the connection between the formulas they have and the concepts they learn. It’s much more satisfying to be able to connect these in one’s mind, and students agree with me.

Therefore, we need to move beyond only presenting formulas. This will require a lot of work to create proofs of equations that students learn. It will also require creativity in the presentation, which is where the expertise of the teacher will come in.

I’m not suggesting that everything needs to be proved. Even in university, some proofs are skipped due to time constraints. But don’t let results stand on their own. If you don’t have time to prove them, give the students the appropriate resources so that they can look at the connections on their own. Do all you can to make sure that students aren’t forced to accept statements and equations out of the blue. If the students don’t have the requisite background, explain that to them. Don’t just say that “it works”. Give some intuition so that their explanation of a concept doesn’t only involve stating the equation.

Our purpose in teaching these concepts should not be about the results. It’s about the links between concepts and the way of thinking that is important. Proofs exemplify these principles, while the formula at the end is just a nice endpoint. The real learning comes through understanding where this equation originates, not the fact that it works.

Anyone can learn to substitute numbers into an equation and get an answer. But why does this equation do the thing you want it to do? Is this form particular, or are there different ones that can be used? Does the equation seem surprising? If so, can a student work from first principles to get back to that equation? These are the questions that should be asked more often in the classroom.

Behind every equation lurks an explanation. Don’t be fooled into thinking the equation itself is the point. Always shine a spotlight behind an equation to illuminate its origin and the reason it works.

  1. The one thing that you could argue is that the dimensions of both sides of the equation are the same. If you look at each term, they all have the dimensions of length, so that matches up. It doesn’t explain why there is a 1/2 in front of a term, though.