by Len Cerny

Professional physicists have an understanding of variables and constants that is far more sophisticated than most junior-level physics students, which enables them to do problem-solving more efficiently. These differences are very noticeable when students are asked to deal with challenging problems such as creating an elliptic integral as an “answer” to a problem. In one example, we asked to students in their first junior-level E&M course to find an equation for the electrostatic potential in all space due to a ring of charge. We let students know that their “answer” would be an elliptic integral, which they could put into Maple. Students faced many challenges while solving this problem, and among these challenges was understanding what was variable and what was constant during integration. This was problematic for students at several points in the problem-solving process, including knowing when they were “done” and had a valid answer. Helping students understand constants and variables in many contexts can help students be more successful problem solvers.

There is a joke about a math teacher who writes $x + 2 = 10$ on the board and asks a student what “$x$” is. The student confidently replies, “$3$. I remember, because that's what you said it was yesterday.”

Students often first learn to think about variables as anything represented by a letter instead of a number. They see it simply as an unknown number to be determined through algebraic manipulation. Incoming students in introductory physics courses, given Newton's Universal Law of Gravitation, may say that they can't answer a question like “What happens to gravitational force when the distance between two objects doubles?” because they don't have all the numbers to plug in. They see the formula, not as a set of relationships, but as a set of letters for which one is the unknown, and for which all the other numbers need to be known. For these students, leaving variables in an “answer” to a physics question seems like leaving the problem unsolved.

At the beginning of the upper-division level, there are still some students who don't spontaneously look at an equation and try to understand how the variables relate to each other. They may need to have their attention drawn to what the equation means and what it says about physical phenomena. However, by the start of upper-division courses most students can look at an equation like $F = GmM/r^2$, and see the relationships between the different quantities. They will also probably realize that $G$ is a constant while the other letters represent things that can be varied from situation to situation. However, some juniors, when told to consider the force on a rocket over the course of its launch into orbit from Earth will need time in order to recognize that some of the “variables” are truly variable for this problem while one “variable” (the mass of Earth) is constant for this problem. Students need to be able to see one letter, “$G$”, as a universal constant, one letter, “$M$”, as a constant for this particular problem, and the other letters as representing things that are variable for this problem.

In introductory physics courses students are often trained to go “equation shopping” for the formula that contains the variables that represent the things they know and the thing they are looking for. They are rarely trained to look at a general equation and ask, “For this particular problem, which quantities vary and which are constant?” When professional physicists address a problem involving equations with a sea of symbols, they very rapidly recognize universal constants and then start making meaning of the rest of the variables and thinking about which variables would be constant or varied in different situations. Students frequently see the same sea of symbols and are overwhelmed and don't know how to start breaking the problem into manageable pieces. We have found it is valuable to specifically tell students to think about what is held constant and what is variable.

In introductory physics, it is frequently hard for students to understand how something going constant speed can have a changing velocity and be accelerating. Similarly, formulas such as $F = mv^2/r$ exacerbate student confusion about the difference between a force of constant magnitude and a constant force. We see similar problems in upper-division physics. For example, for finding the magnetic vector potential in all space due to a spinning charged ring of radius $R$, students need to keep track of numerous quantities during integration that have scalar and directional components. When integrating around the ring using $d\phi$, the ring's speed is the same at all places on the ring, but it's velocity varies. Similarly the current as a vector is different at every point on the ring although the magnitude of that current is constant. Students also need to realize as they find $|\Vec r-\Vec r'|$ that the ring's radius, $R$, is constant, but the position vector to a point on the ring, $\Vec r'$, varies around the ring. Also the position vector to an arbitrary point, $\Vec r$, is constant, but the magnitude of the vector between the arbitrary point and a point on the ring, $|\Vec r-\Vec r'|$ varies during integration. Our experience is that most upper-division students will have difficulties with these issues and will make one or more errors with either the current or the position vectors when they first attempt a problem like this. Additional discussion can be found on the page addressing student understanding of vectors and scalars.

Knowing which variables are variable and which are constant is important for realizing what can be pulled out of an integral. At the junior-level, most students have had some practice “pulling things out” of integrals. Students will also have dealt with things such as the integral of $e^{ax}$, where “$a$” is a “variable” in the integrand that is held constant during integration.

Len, The bridge project convinced me that students see far less of even this simple example than we expect. Most intro level college calculus problem use numbers and not parameters at all. Even when they do use parameters, they are rarely asked to consider what happens when the parameter changes. While the process of learning how to do this typically begins in lower-division physics, I suspect there is less there than we expect, also. We should talk to Dedra. Corinne

Len, it may be useful to add the concept of “parameter” into this discussion. Corinne

However, asking students to create an elliptic integral requires greater sophistication than most incoming juniors initially have. In the case of integrating $e^{ax}$, students are familiar with a specific process for solving for this integral in closed form. When being asked to create an elliptic integral that is not solvable in closed form, students are facing several unfamiliar new things.

Students must recognize that for an elliptic integral to be “doable” by programs such as Maple, that if one is integrating in terms of $d\phi$ that the only variable quantities allowed must be in terms of $\phi$. All other “variables” must be held constant during integration. Students will sometimes think they have reached an “answer” when they have an integrand such as $d{\phi}/|\Vec r-\Vec r'|$. When told that Maple would not be able to evaluate this, they think it is some quirk of the program and don't realize that they have a varying quantity, $\Vec r'$, that is not yet defined in terms of the variable for which they are integrating.

When we ask students to find the electric potential in “all space” due to a ring of charge, students are often confused how solving for the potential at a single arbitrary point can accomplish this goal. Students are often tempted to create triple integrals that deal with integrating across all space. Students need to recognize that for the purposes of integrating, a single arbitrary point can be used as a constant point during integration, and then a program like Maple can perform calculations for multiple points to create a visualization of the field for various specific spatial regions.

Understanding what is a variable and what is a constant is not a simple concept that students either “do or don't have.” Instead, there are many different levels of understanding. Students are often asked to apply a progressively more sophisticated level of understanding without being given explicit instructions as to how to accomplish this. Recognizing the many nuances and levels of understanding that we ask of students is a first step in helping them bridge the gap between their understanding as incoming juniors, and the level of understanding they will need as professionals.