Using Math in Physics: 3. Anchor equations

These are important, but they’re only the first steps in beginning to build scientific skills. Unfortunately, much of school and introductory science focuses solely on these elements. Since lists of facts and procedures are well suited to memorization, students often come away with the epistemological misconception that memorization is all there is to science.⁶ 

—building causal stories and extended chains of reasoning where each step sets up the conditions for the next—is rarely a part of introductory science classes.⁷ While introductory biology classes teach a number of multi-step processes, they tend to be learned more as a memorized string of processes rather than what we would describe in physics as a mechanism of chained causality, where each step produces the conditions that lead to the next.⁸ 

—putting lots of knowledge together in a structured and organized way that makes sense—is the heart and soul of building scientific knowledge and learning to use it. Students accustomed to memorizing lots of terms in introductory biology or lots of molecular structures and reactions in introductory chemistry may miss the coherence in physics, both among its different parts and with everyday experience.

In physics, a few critical equations can help synthesize a lot of knowledge about a content topic. I refer to these fundamental equations as anchor equations to emphasize their special role as stable fixed points in organizing knowledge. When I call on an anchor equation in slides or in our text (the NEXUS/Physics wiki⁹), I put the anchor icon shown at the top of this article to help students realize how valuable those equations are and how often we refer back to them.

Identifying anchor equations and learning how to use them can help students make the transition to a more thoughtful use of math in science. They have four specific benefits:

• Anchor equations encourage synthesis and principle-based reasoning.

• Anchor equations help students learn to blend physical concepts with math symbols.

• Anchor equations code conceptual knowledge.

• Anchor equations organize problem-solving knowledge.

Students in introductory physics rarely have experience in powerful principle-based reasoning, and many see physics (and all the sciences) as an incoherent and hard-to-memorize collection of facts and highly localized procedures—“flash-card science.” While both biology and chemistry have powerful organizing principles (for example, evolution in biology and chemical bonding rules in chemistry) and occasionally have problems that use some math, their fundamental statements are qualitative rather than quantitative. Students often have a hard time seeing equations as providing principles to reason with.

In physics, our powerful theoretical frameworks are tied to mathematics and fundamental equations—Newton’s second law, the work-energy theorem, Maxwell’s equations, the Schrödinger equation, Einstein’s equations of general relativity. This kind of mathematics-based organizational knowledge structure is not common in introductory classes in the other sciences. If we want to help students learn synthesizing skills and principle-based reasoning that includes symbolic math, we have to be explicit about it. Identifying anchor equations and encouraging students to start with them (instead of jumping to a memorized special case equation) can help.

We want students to see equations as a way of saying something physical, not just as a way to calculate. Often, textbooks present all of the equations that are useful for calculation in their end-of-chapter summaries. This sends students the message (an “epistemological meta-message”) that these many equations are independent calculational tools. It encourages memorization and discourages them from making the challenging transition to thinking synthetically with math.

One example of calculational vs. anchor equations is kinematics. End-of-chapter summaries often include lots of equations for calculating velocities and distances under a variety of specific conditions. An alternative approach is to avoid the calculationally useful kinematic equations and instead use only the conceptually more natural kinematic definition equations. Students can do all the same calculations, but they start with equations that make conceptual sense, map their knowledge of the physical system into the equation, and only then manipulate the math to get their answer. This can help students learn to build the physics/math blend and build their confidence that they can reason with math, as well as calculating with it.

The basic kinematics concepts are

• In a given time interval the position may change (velocity).

• In a given time interval the velocity may change (acceleration).

The direct mapping of these physical concepts into math symbology is given by the word equations:

1. The average velocity is given by the change in position (How far did you move?) divided by the time interval (How long did it take to do it?).

2. The average acceleration is given by the change in velocity (How much did it change?) divided by the time interval (How long did it take to do it?).

For motion along a line (1D), we write these symbolically as follows:

1. $<v>=ΔxΔtv=dxdt$
2. $<a>=ΔvΔta=dvdt$

where we use the notations “<…>” to indicate the average over an extended time interval and “Δ” to indicate the change in the following quantity. My class has calculus as a prerequisite so I include the definition of instantaneous velocities.¹⁰ 

Once students learn these coding rules,¹¹ these equations come to be seen as natural and sensible definitions and, if a student has managed to construct the blend, these equations will not need to be memorized. If forgotten, they can be easily reconstructed from the conceptual definition. This illustrates the difference between memorizing an equation and “knowing” it, something the students may have little previous experience with.

Two additional equations let students generate all the usual kinematic equations: the definition of change and the evaluation of average in the case of constant rate of change.

The definition of the delta can be seen as making physical sense:

The final value = the initial value + the change in the value

While this looks trivial, it is only so if you are mapping meaning to the symbols. Students who are still learning to do this often write down something else (perhaps with a + instead of a– or in the wrong direction or they ignore the delta entirely or even put in a ½ from the confusion with the average). This happens much more often than you might expect.

The second equation students need is the average when the change is at a uniform rate (easily shown geometrically):

The average value = ½ (the initial value + the final value) (if the quantity is changing at a constant rate)

Students also get this wrong, replacing it by the difference extraordinarily often. When they do this, it’s a clear signal they are relying on memorization and not sense-making (not making the blend). The fact that this is an epistemological error (“What kind of knowledge should I use here?”) rather than a knowledge error can easily be demonstrated by asking them first, “What’s the average of 10 and 12?” and then, “If the velocity is constant, what should the average be?”

Students often hold on very strongly to their incorrect equation, often not having the epistemological resource “If it doesn’t make sense, the equation I used must be wrong.” Rather, they rely on “I can’t be expected to make sense of an equation, so somehow my sense-making intuition must be wrong.” Developing the resource that equations and intuitions must match is one of the most important skills we can help students develop. Sometimes you do have to modify your intuitions, but sometimes you’ve remembered or used the equation incorrectly.

To help students learn to use equations by blending physics and math, rather than just picking an equation with the right variables¹² and plugging in, we need to give problems that are not easily solved by pulling up a single standard equation. I give some examples in the instructional materials below. More are in the supplementary materials.¹³ 

As trained physicists who blend physics and math and can unpack the knowledge coded in an equation, we often use a shorthand that hides the content being called upon and makes an equation look like a computational tool to be memorized. Newton’s second law is a great example of this. It’s the fundamental powerful principle (anchor equation) that underlies all of our understanding of classical motion at scales from the molecular to the galactic, at speeds up to a significant fraction of the speed of light. It’s the principle that organizes all our knowledge of motion. And yet…

Before I realized all this, I had a tendency to just write “F = ma” for Newton’s second law. I sat up and took notice, however, when, in one of my physics for life sciences classes, after my discussion of springs, a student asked, “Professor, what’s the difference between F = kx and F = ma?” All of a sudden a lot of student behavior that never made sense to me fell into place. Why would they use F = ma for each of the different forces in a problem? Why would they tell me acceleration caused forces? And how would they ever forget it? Well, of course one reason is they were not making the blend, just using each equation as a calculational tool. But a second reason was that they were failing to identify one of the two equations as a core principle (and the other as a crude toy model).

That’s when I began to write text for the NEXUS/Physics wiki online textbook¹⁴ called, “Reading the content in …” in which I identified a few anchor equations and wrote a reading for each in which I specified a basic equation and made explicit the conceptual physics that was coded in the equation. One example is the conceptual kinematic equations discussed above. A second is Newton’s second law.

It’s important, at least for a while, to stress the general validity of Newton 2 and to help students realize that is not just an equation to calculate the relation among two variables and a parameter, but that the full form, with diacritical markers (vectors, subscripts, … which students often ignore), specifies a lot of conceptual ideas.

One way to display this is shown in Fig. 1 at the beginning of this article.¹⁵ Each of the detailed ways we choose to write the equation in that figure specifies a bit of conceptual knowledge that students often fail to bring to bear in solving a problem. In my experience working one-on-one with students, it’s often not that they don’t know these things, they just tend not to see when it would be valuable to call on them. Reminding themselves of the full form of the law can help them recall just what aspects of the physical phenomena they need to bring to bear in building their mental blend when using this equation (it’s about an object, it’s about the sum of forces, directions matter, …).

To let them know that I’m serious about using equations this way, I do not accept “F = ma” as a correct form of N2 (they have to put “net” on the force), and I strongly encourage them to write “a = F^net/m” to remind themselves that it’s forces (the sum of them) that cause acceleration and not the other way around.¹⁶ 

Many students study for each week’s quiz as a way of keeping up in the class (exactly what I want them to do!). But many are frustrated because they feel they’ve done all they can—reviewing the previous week’s work, creating long lists of everything we covered—but still wind up doing poorly on the Monday quiz. They come to me for advice on how to prepare. Then I have them where I want them—unsatisfied with memorizing and ready to try something new! From both of the earlier examples (the conceptual kinematic equations and the detailed form of Newton 2), we can see that these equations code conceptual knowledge. But they can also help organize the basic ideas that students have to learn to use the equations in practical problem-solving situations. This approach also helps them in doing their homework when they are not sure how to get started.

If a student approaches a problem by trying to call on the long list of “the things we’ve learned that I have to memorize” at the end of a typical kinematics chapter, they can get confused and lost, especially since the equations can look contradictory if they don’t pay attention to the conditions attached to each.¹⁷ This is especially true if the problem involves multiple objects, time intervals, distances, and velocities.

A more productive approach is to begin with the basic, conceptually reasonable definitions shown in Fig. 3. These equations can guide the approach to a problem, by unpacking them with the supplementary conceptual knowledge shown in the boxes. Each item in Fig. 3 is conceptually straightforward: a one-step translation of a sensible concept into symbol.

Students can then look at each specific object in a complex problem and decide what the initial and final values of each variable are, when they are known and when they are unknown, and set up the equations needed to solve the problem.

When students learn to not just review, to not just make pages and pages of review notes, but rather to take their reviews or class notes and organize them around a few core ideas—often around an anchor equation—they take a major step in transforming the way that they view the knowledge in the class. Their performance on both quizzes and exams improves dramatically, and they develop a new and surprising sense that “there’s not actually that much to learn,” even though there is—but it seems less if it can be easily unpacked from a small set of “things to know.”

To get students to value anchor equations, it’s important to offer problems that are easy to set up using an anchor equation and a few straightforward manipulations rather than ones that are easy to simply put numbers into a memorized equation and calculate.

Here’s an example. This problem is set up so that each step is easily solved using the kinematic anchor equations in conceptual form for average velocity and acceleration. I strongly encourage you to work this through for yourself and not just to read it, keeping the kinematics anchor equations in mind throughout.

Since one of my learning goals is “meta”—to help students learn to understand the character of physics knowledge and not just “do it”—I give an essay question on most homework sets and exams in which students are asked to think about what’s going on and discuss it. Some of the problems that focus students on anchor equations are of this character (see “Another homework problem”). More examples are given in the supplementary materials associated with this paper.¹³ 

In this paper I’ve identified a few equations as anchors: the two kinematics equations, Newton’s second law, and Coulomb’s law. They seem to have different character. The kinematics equations are straightforward definitions that each connect directly to a concept. Coulomb’s law describes one thing—the electric force—and it collects all that force’s observed properties into a single representation. Newton’s second law also collects a lot of observations, but much more universal and general ones. Are these all anchor equations?

Since my criterion for an anchor equation is that it serves to organize a conceptual space and is powerful and productive in helping to get started solving a wide range of complex problems, my answer for this is an unqualified yes.

But how many might you have in a physics course? How do you decide which equations are “anchors”? These questions don’t have simple answers. Since anchor equations are organizers of mental spaces, they depend both on the context —what physics they are helping to organize—and on the individual—what each person has become comfortable with.

Some equations, such as those mentioned above, are likely to be anchor equations in anyone’s version of a physics class that discusses classical motion. If I am discussing the motion of interacting objects, Hooke’s law (F = –kx) is a not very good phenomenological model of a contact interaction and it has limited utility. But if I am studying the behavior of oscillating systems where throughout a linear assumption will be made, it’s definitely an anchor equation. If I’m teaching a unit on viscous fluid flow (something very valuable in a class for life science students), I could rely on a highly general anchor equation, the work-energy theorem, or I could take one of its consequences, the Hagen-Poiseuille equation, as an anchor for the section.

While I definitely recommend teaching the idea of anchor equations and identifying a few central ones that are appropriate for the instructional context you are creating, perhaps the most important thing for students to learn is that equations can serve not just as calculational tools, but also as conceptual organizers. Helping students see that each topic they are learning is a coherent structure with a few powerful and productive core ideas rather than as a huge collection of unrelated facts to memorize is a fundamental epistemological lesson.

Many of the ideas for this series of papers were developed in the context of studying physics learning in a class for life science majors. A number of problems and activities using anchor equations are offered in the supplementary materials to this paper.¹³ A more extensive collection of readings and activities from this project on the topic of anchor equations is available at the Living Physics Portal (search “Making Meaning with Mathematics: Anchor equations”).

This paper relies heavily on my many years of observing students, organized by what has been learned in physics education research and in particular on the resources framework¹⁸ for thinking about student thinking. While I have not been able to find anything that deals directly with the issue of using equations to organize large conceptual spaces in physics, readers who want to go deeper are encouraged to look into some of the research on dynamic epistemologies,⁶ epistemological framing,¹⁹ and the coding of physical meaning in mathematical symbols.^20,21

I would like to thank the members of the UMd PERG over the last two decades for discussion on these issues. I am grateful to Khala Marshall-Watkins for suggesting the term “anchor equation.” I am also grateful to two anonymous reviewers who made many excellent suggestions for improving my manuscript. The work has been supported in part by a grant from the Howard Hughes Medical Institute and NSF grants 1504366 and 1624478.