When I was in high school, I tutored students in lower level math. One time, I sat down to help a freshman who had scored around a 30/100 on a basic trigonometry exam covering the definition of sine, cosine, and tangent. I told the student I thought I could help and asked to see their exam. My eye was immediately drawn to something they had scribbled along the top of the page:
Everyone will immediately recognize this as a horrible misspelling of the acronym SOHCAHTOA, which every student in the US is taught to remember basic trig relations. From this, it is pretty clear why they roughly got one out of three problems right. Nevermind that her definition would mean that for any angle, sine = cosine = tangent. The student was trying to apply a gimmicky rule without the slightest understanding of the physical problem the math describes. If they could have spelled it correctly, they likely could have made an A with zero actual understanding of basic trig. But wait, it gets worse!
I see this all the time at Georgia Tech. On any given semester during the first lecture of my statics course, I could give a quiz with two questions:
- What is the dot product of two vectors A and B?
- Compute the magnitude of the projection of A onto B.
For the first problem, 9 out of 10 would say “A B cosine theta”. For the second problem, probably 1 out of a 100 could do it, even though it’s just a dot product. They can give you the equation of the dot product but they have no idea what it functionally means or accomplishes. When I was an undergrad I was much the same way. In all honesty, it wasn’t until I was well into graduate school that a lot of the mathematical pieces started falling into place in my mind. It was then that I really started to understand how beautiful and powerful even basic calculus is.
Feynman encountered something similar during a sabbatical in Brazil, namely that no science was being taught there, just memorization. From his essay, O Americano, Outra Vez!:
Then I gave the analogy of a Greek scholar who loves the Greek language, who knows that in his own country there aren’t many children studying Greek. But he comes to another country, where he is delighted to find everybody studying Greek–even the smaller kids in the elementary schools. He goes to the examination of a student who is coming to get his degree in Greek, and asks him, “What were Socrates’ ideas on the relationship between Truth and Beauty?”–and the student can’t answer. Then he asks the student, “What did Socrates say to Plato in the Third Symposium?” the student lights up and goes, “Brrrrrrrrr-up”–he tells you everything, word for word, that Socrates said, in beautiful Greek.
But what Socrates was talking about in the Third Symposium was the relationship between Truth and Beauty!
What this Greek scholar discovers is, the students in another country learn Greek by first learning to pronounce the letters, then the words, and then sentences and paragraphs. They can recite, word for word, what Socrates siad, without realizing that those Greek words actually mean something.
Feynman’s solution was to incorporate the experience of nature in all teaching. Experiments, applications, actually doing things, will help provide vision past equations into the fundamental nature of things.
I have tried to do this in various ways in my statics course, mainly focusing on case studies of structural collapse. I think collapsed buildings are interesting and exciting to my students; definitely one of the few ways I’ve discovered to get them thinking about a physical problem. But still I’m left pondering how I can do a better job translating between the classroom to the real world.