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.

Figure 1. The laws of nature are expressed in equations, behind each of which are fascinating stories.

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).

Figure 2. Just as the ratio C/D is the same for all circles, the ratio F/m is the same for all objects in free fall.

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.

Figure 3. The differences in acceleration are due to the different masses.

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.

Figure 4. As fuel is burned to provide thrust, mass decreases and acceleration increases.

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).

Figure 5. Although the velocity of Hudson varies as he is tossed upward, his acceleration is a constant g.

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 energy
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).

Figure 6. The impulse-momentum equation tells us that the magnitude of force in a collision greatly depends on the time during which the change occurs.

Figure 7. Energy conservation tells us that a small force can ideally lift a huge weight.

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.

Figure 8. The low radiation frequency of smart phones means correspondingly low energy for each photon of radiation.

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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