We all know that to enjoy a game, you must know the rules of the game. Likewise, to appreciate—and even comprehend—your environment, you must understand the rules of nature. Physics is the study of these rules, which show how everything in nature is beautifully interconnected. Physics taught as the rules of nature can be among the most relevant courses in any school, as educationally mainstream as English and history.

**Mathematical need not mean computational**

Physics has the reputation of being overly mathematical, intimidating many students who are otherwise attracted to science. My teaching experience tells me that it’s not mathematics per se but rather computation that intimidates students. That’s an important distinction. Every serious physics course *is* mathematical, containing equations. But it also can be noncomputational. By postponing problem solving until a follow-up course, an introductory, noncomputational physics course can be enjoyed by math whizzes and math weaklings alike.

**Equations guide thinking**

The laws of physics are central to any physics course and are expressed unambiguously in equation form (Figure 1). Although equations have traditionally been used as recipes for problem solving, they provide deeper insight when used as guides to thinking. A physics student can learn to “read” equations as a music student reads notes on a musical score.

Rather than writing Newton’s second law as *F = ma* (force equals mass times acceleration), I strongly suggest *a = F/m*, which is more like Newton expressed it. Then a student can see why a boulder and feather falling without air resistance (free fall) have equal accelerations (Figure 2).

Any topic is better learned when related to what students already know. Students know the relationship between a circle’s diameter and circumference: *C = πD*. In ratio form, they see that whatever the size of a circle, the ratio *C/D* remains constant: *π*. Similarly, the ratio of gravitational force *F* to mass *m* for freely falling objects yields the constant *g*, the acceleration due to gravity.

**Concepts before computation**

When a teacher spends mere seconds on the concepts in an equation and many minutes on number crunching, students get the impression that physics is all about computation. Instead, focus on the concepts in equations and how they connect, with much less number crunching. Concepts first, computation second. Time normally spent on problem solving can be better allocated to an overview of physics. Then all students can enjoy what many of us already know: that physics can be a student’s most delightful course.

**Examine the whole elephant before measuring its tail**

A physics course can concentrate on a few topics in detail or many topics more generally. I prefer the latter—to study mechanics, properties of matter, heat, waves, light, radioactivity, nuclear fission and fusion, with some time devoted to Einstein’s relativity. A broad overview of physics is valuable to students who continue with physics and also to those who don’t.

**The black hole of physics instruction: kinematics**

To cover a wide range of physics I recommend just skimming through kinematics—the study of motion without regard to forces. Kinematics can swallow more class time than any other topic, because it’s a dandy introduction to numerical problem solving. A main reason for limiting time spent on kinematics is that it addresses no laws of physics. None.

**Exaggerating symbol sizes**

The relationship between terms in an equation can be illustrated by changing the sizes of the symbols. For example, when a cannon is fired, the force acting on the cannonball has the same magnitude as the force that makes the cannon recoil. Although the two forces are equal in strength, the resulting accelerations are enormously different. Tweaking the symbols in Newton’s second law illustrates and provides the explanation (Figure 3). Note the relative sizes of the *m*’s and *a*’s.

**Equations identify and connect concepts**

Some teachers complain when students presented with a problem grasp for an equation. I don’t. I encourage it! Hooray for equations serving as a crutch. Equations identify the concepts involved. For example: We know that a rocket fired in deep space gains speed as long as the thrusting force is maintained. Question: For a constant thrust, will the rocket’s acceleration also increase? The equation for Newton’s second law guides our answer by reminding us that acceleration depends not only on applied force but also on mass. Aha! As fuel is burned, the mass m of the rocket decreases. Hence the acceleration as well as the speed of the rocket increase (Figure 4). The equation nicely guides this discussion.

**Distinguishing between closely related concepts**

Equations help to differentiate closely related concepts such as *velocity* and *acceleration*, which are commonly confused. Well-chosen examples help point out the differences between the two. My favorite is asking for the acceleration of a vertically tossed object at the top of its path, such as little Hudson tossed upward by his dad (Figure 5).

Students will likely say the acceleration of Hudson at the top of his path is zero. This answer is wrong because velocity (which is zero there) is confused with acceleration. The equation *a = F/m* guides thinking to the correct answer, *g*. Barring air drag, the acceleration of any projectile is everywhere *g*, whether moving upward, momentarily at rest at the top of its path, or moving downward.

Newton’s second law involves thinking of three concepts at once: *acceleration*, *force*, and *mass*. A lot of us, me included, have difficulty thinking of two ideas at once. But three ideas? Even Galileo didn’t get around to that! So we have to be patient with students who don’t comprehend these connections and distinctions right away.

* Momentum and energ*y

Exaggerated symbols help explain differing magnitudes of concepts in various circumstances. For instance, symbol sizes nicely illustrate how the amount of force varies during the changes in momentum of colliding objects (Figure 6) and with changes in energy (Figure 7).

**Beyond mechanics**

Given a choice, would students want to spend time on kinematics problems or learn why radiation from their smart phones can’t damage human cells? Radiation energy comes in packets, or photons. The photon energy is related to the radiation frequency by *E = hf*, where *h* is Planck’s constant. It’s easy to see that radiation at low frequencies means low energy of each photon (Figure 8). A bit of number checking will show photon energies much too low to disrupt cells in the human body.

**We can’t change only one thing**

The value of equations isn’t limited to the physics classroom. Equations in general remind us that we can never change only one thing: Change the value of a term on one side of an equation, and you correspondingly change the other side. Whenever you change one thing, something else is also changed. Not being able to change only one thing extends way beyond physics, especially to ecology and to situations that are social and even personal.

**Physics in the educational mainstream**

There are many reasons why physics courses aren’t as common as English and history in secondary schools. Physics is avoided by students who are threatened by math and by others who view it as a “killer course” that will lower their GPAs. Some teachers are quite content with their small classes of mathematically talented students who, like them, enjoy problem solving. These courses should remain, for they provide the vital foundation for future engineers and scientists.

But we shouldn’t shut out the many nonmathematical students who see science as “cool” and would love to learn physics “without numbers.” They would welcome a noncomputational course that emphasizes concepts over mathematical skills. To bring more of the general public into science, a noncomputational survey physics course can precede the higher level physics courses and have a place in the educational mainstream. This approach isn’t just good for individuals—it’s good for the country. Basic science knowledge enables people to understand critical issues such as climate change.

When a learner’s first course in physics is a delightful experience, the rigor of a second course will be welcomed. And in your teaching of physics, it’s fun and rewarding to get to photons and rainbows.

*Paul G. Hewitt *(pghewitt@aol.com)* is the author of the popular textbook *Conceptual Physics, 12th edition,* and coauthor with his daughter Leslie Hewitt and nephew John Suchocki of *Conceptual Physical Science, 6th edition,* both published by Pearson Education.*

**On the web**

A video with more on equations as the rules of nature and as guides to thinking, “Hewitt-Drew-it! Physics for Teachers 1,” is at *http://bit.ly/TST-physics*.

**Editor’s Note**

This article was originally published in the March 2017 issue of *The Science Teacher* journal from the National Science Teachers Association (NSTA).

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## One Comment

Unfortunately I don’t have time to respond to this post with the attention it deserves since it has the possibility of doing so much damage to young impressionable teachers. But, I would hope that the NSTA would retract this especially in light of all of the research recently about retention, student understanding, and methodology. We do are students no favors if all we do is treat physics as a light topic that involves no real understanding of the interactions. If in your school this is your way to grow your classes that is fine, but I have taken the exact opposite approach (full disclosure I use heavily modified modeling materials in my school) and seen my numbers rise year after year after year both in terms of enrollment and in terms of understanding. Students seek understanding, and if all you do is serve them the first 10% of a topic then they get say ” I got an A in Physics” while understanding nothing. I will take a deep dive on a few topics over a survey of all topics any day. There is a reason AP Physics B was changed. To do a good job with that many topics takes time, you don’t get to use less time with less material and say you did just as good of a job.