As a physics teacher, I get to play with toys as part of my job. Physics labs give me the chance to dig out classic favorites such as Slinkys and Hot Wheels cars and put them to educational use. Occasionally I get catalogs from laboratory equipment manufacturers full of strange and sterile contraptions, but I find that students learn better when they personally connect the lessons to familiar things, and the more nostalgic the better. And so, where I am able, I make the childhood playground my laboratory. Merry-go-rounds and teeter-totters are wonderful physics tools, and the student’s natural intuition about the equipment becomes the starting point for the lesson.
Occasionally there are misconceptions to overcome, and they are not given up easily, but these collisions between the expected and the actual provide the opportunity for the deepest learning.
One of my favorite lessons is teaching the principles of the pendulum using a common playground swing. Every student is intimately familiar with the thrill of the swing set, but perhaps none of them have given much thought to what physical properties underlie its motion. Most of them hypothesize that it is the mass of the pendulum that determines its timing, that a larger, heavier student will give the swing a slower motion. But a little investigation will reveal that only the length of the pendulum, the distance from the support bar to the seat, determines the time it takes for the swing to complete one back-and-forth cycle.
When we move to the textbook, the students discover this principle borne out in the formula for a pendulum’s period, T = 2π√(L/g) , where T is the time for one swing, L is the length of the string, and 2, π, and g are all fixed constants. Mass is nowhere to be found. Many students are amazed at their misconception regarding mass, and would have a hard time believing that only length affects a pendulum’s period unless they had seen it and experienced it on a swing themselves.
I really enjoy helping them work through this change in their conceptual understanding, for I think it is where real learning takes place. In most physics textbooks, there is a small mention of Galileo being the first scientist to properly describe the properties of the pendulum, and so several years ago I decided to look up his original work on the subject to see what I could find. But to my surprise, what I found was that I was the one who had the misconception.
And so, after years as a physics teacher, I had to revive my role as the student and learn.
In scouring through Galileo’s The Two New Sciences, I figured the best way to find the section discussing pendulums was to scan for the formula mentioned above. But I did not find it. In fact, I did not find any formula, familiar or otherwise. I finally had to do an Internet word search for “pendulum” in order to locate the specific passage, and what I found intrigued and puzzled me. Galileo discussed the properties of a pendulum as an aid to the larger question of music, why certain combinations of tones are more pleasing to the ear than others. The discussion was about the nature of harmony, and the pendulum merely served as an illustration of the properties of vibration, a simpler visual analogy to what occurs with harmony in the ear. Here is what he had to say about the nature of the pendulum:
As to the times of vibration of bodies suspended by threads of different lengths, they bear to each other the same proportion as the square roots of the lengths of the thread…
Instead of a formula, Galileo presents the relationship as a proportion, two ratios set equal to each other. If one combination of length and time are known, a simple proportion will yield the time for a new, different length. With this method, all of the “clutter” of 2, π, and g from the formula cancel out and all that remains are the significant quantities, length and time, and that one goes as the square root of the other. It was elegant and simple, and it struck me as profound because it focused on the relationship between the two quantities rather than on their individual values.The modern formula is the means of quickly computing an individual value, but Galileo’s proportional reasoning was the original means of identifying the root relationship. The formula must have come later. I had always connected formulas with a proper understanding of physics relationships, but that was not how Galileo thought. He did not think through the use of a formula. I wondered if I could discover how Galileo was trained to think, and if I could do the same.
A few months later I found a key that helped help me understand the difference between Galileo’s mindset and my own. I was reading the first few section of the ancient work Introduction to Arithmetic by Nichomachus where he quickly summarizes the four disciplines of the Quadrivium, Arithmetic, Music, Geometry, and Astronomy before embarking on his work explaining arithmetic. There he defines arithmetic as dealing with absolute quantity, and music as dealing with relative quantity. Arithmetic focuses on the properties of individual numbers, such as even or odd and so forth, but music focuses on the properties of numbers in relationship to each other, such as in harmonic ratios. It then struck me that my education in physics had been an education in Arithmetic only, focusing on how to find the individual numerical “answer”. Give me the length of a pendulum and I can compute for you the specific time for one cycle. I was very good at focusing on individual, absolute quantities through the use of equations.
What I lacked was an education in music, a training of the mind to identify and distinguish relationships between absolute quantities and to find the proper harmony among them. Coming from a public school background, I never had much musical education once I decided that I preferred math and science. The departments were separated physically and philosophically in my high school, and the two groups viewed each other with some smoldering hostility.
But it was the musical mindset that gave Galileo insight into the scientific properties of the pendulum. Music was not a menace to science, but a friend. The individual values of length and time were not so much important in themselves as was the relationship between them, and that required a musical form of thought. To my surprise I found that Isaac Newton used this proportional reasoning as well in his development of the laws of motion, as did the astronomer Johannes Kepler and many other early scientists. Their focus was on discovering the proper and orderly relationships between the individual quantities, which was sometimes complicated and not readily apparent, but it is where the true progress and growth occurred. The modern formulas and equations only came afterwards. And although formulas are powerful and tremendously useful as computational tools, they spring only from a proper understanding of the relationships.
I also discovered that this “musical” concept was not an idea that started with Galileo. Many of the ancient philosophers, most notably the Pythagoreans, believed in the hidden mathematical harmony of the natural world. This root assumption that the world was created in such a way as to be both mathematically described and aesthetically pleasing inspired scientists up through the Scientific Revolution. It’s a concept we have abandoned in the chaos of postmodernity, and therefore no wonder why we languish in advancing natural science
This discovery has given me much to think about. It has strengthened my belief that a Classical education, an education that encompasses all seven of the Liberal Arts, is a superior education. If we wish to produce young men and women who are capable of thinking like Galileo and Kepler, advancing the boundaries of mathematics and science, they must be educated in all areas of study, including music as well as arithmetic, and be able to integrate the disciplines together. It has also changed the way I teach physics, emphasizing a relational approach as well as a formulaic one.
But perhaps more importantly, I have been confronted with my preference for myself as an individual over my relationships with others. In Galileo’s analysis, it is the relationship that gives the individual values context and meaning. The relationship between the individual values is more significant than the individual values themselves. I tend to focus on myself as an absolute, autonomous individual. But perhaps it is my relationship to other individuals, such as my students, my family, and ultimately my Creator, that provides my life real context and meaning. When those relationships are properly understood and executed, when there is harmony, perhaps that is when I find my satisfaction in life.