by lex

Tue – October 14, 2003

Or why it’s important to be technical.

My son takes AP calculus and physics at Torrey Pines High School in North County, San Diego. It’s hard, he sometimes struggles, and I’m very, very proud of him.

For my own part, I was never a scholar in maths. Somewhere around fourth or fifth grade, we were introduced to negative integers. I rebelled. If you had four pencils, and took away five, you’d find that after you’d taken away four, there were no more pencils to take, I reasoned – you’d have to stop. The theoretical, potential fifth pencil my teacher tried to sell me on seemed so much more adult blather, comforting to say aloud perhaps, like the rosary, but ultimately senseless. I was querulous, she was harassed, I was told to get on board. I stayed in the station, awaiting better logic.

My agnosticism persisted all through algebra, a pointless exercise in mental master debation. What on earth were these letters doing in what was ostensibly a mathematical question, and what was a variable, anyway? And what exactly would one do with the area under the curve, in the real world? No one could be bothered to tell me.

It was not until my junior year at the Naval Academy, when we started to do differential equations, that the light came on. Eureka! Drop a wrench from orbit, and over time it would accelerate at a determinable pace, up until the moment when it entered the atmosphere, where friction would impede the rate of acceleration at an increasingly greater rate (based on air density, interpolated over a changing altitude) and that wrench struck someone’s head at a certain velocity, that any of this applied in the real word. By then it was too late, I was too far gone, and an opportunity was lost.

Believing in the gradual perfectibility of man, I ensured that my son took his math studies seriously. “Math is how the world works,” I would tell him solemnly. “It makes the world turn, or if it does not, explains why, and how fast.” I think he bought it. In ninth grade chemistry, he once exclaimed aloud, “it really is all math!”

Mission accomplished. I was at that time the commanding officer of an FA-18 squadron, The underlying work was technical to a degree, and I got by through experience, memorization and rule of thumb. But I could not pull the curtain back, and see the little man manipulating the controls.

My son had a revelation of sorts, a kind of epiphany that freshman year. He asked me to help him with a math problem, that proved utterly beyond my capability. I made phone calls to mathematical friends, we worked it out, but the scales had fallen from his eyes. I think he realized that he had in a sense overtaken me, at age 13. I was not entirely ready for this, but happy nonetheless. My concern? Having told him for years how important math was, I had demonstrated that a certain degree of success was possible without it. Would this call into question everything else I had tried to teach him? Time would tell.

So far, so good. He’s a good man.

But I still wish that someone would have told me how to get at that fifth pencil, in a way that would have captured my imagination.

Nice one, Bill. I empathise totally with Lex on this one, having struggled with Maths, quadratic equations etc at school, yet getting good grades in geometrical and engineering drawing and positively shining in Physics, a science wrapped in mathematics. My son, by his own admission a reluctant scholar, could out-math his sister when he was nine (and she was four years his senior). The mysteries of the human mind.

HD I had the same problem. I think a large part of it is the way it is taught.. I tried to avoid all math classes unless they were of course required; a class in probability was one of them. Found that fascinating as many things you think are “logical” aren’t necessarily so by the laws of probability.

Was going to a computer trade school in San Diego where for some odd reason I had to be versed in trigonometry and calculus.

Had a friend who was a Phd dropout from the University of CA -in math – and for the paltry sum of a daily taco lunch at a nearby Mexican restaurant, he tutored me in these 2 subjects.

He would generally start out with “OK Bill, this is how they teach it but this is how you should think it”.

Within a few weeks I had “gotten it”.

“t was not until my junior year at the Naval Academy, when we started to do differential equations, that the light came on. Eureka! Drop a wrench from orbit, and over time it would accelerate at a determinable pace, up until the moment when it entered the atmosphere, where friction would impede the rate of acceleration at an increasingly greater rate (based on air density, interpolated over a changing altitude) and that wrench struck someone’s head at a certain velocity, that any of this applied in the real word.”

This makes a lot of sense. I think that an intuitive approach to differential equations…ie, focused on how the equation maps the physical system, rather than formal calculus derivations and analytical solutions (which aren’t usually possible for most real problems anyhow) would help a lot of people in understanding and appreciating math.

There is a team at Marshall University which is reviving mechanical analog computers (used for the solution of differential equations in scientific research during the 1930s-1940s, as well as in naval and other military fire-control systems) in the belief that seeing the direct mechanical equivalent of the equations will help in gaining an intuitive understanding of calculus.

Meant to include link on the work at Marshall:

https://www.marshall.edu/math/DAlab.asp

That is a fascinating exercise at Marshall. The psychology of `Gestalt`! I used this from the time I was qualified as an instructor and in my experience it works every time.