In my high school calculus class, many students
find it difficult to remain perfectly attentive throughout each 45 minute
lecture, taking notes on each problem while also engaging in class to answer
questions or perhaps question the teacher. I am, of course, no exception to
this rule.
In what I’ve learned from my 13 years in the
American public education system (I did not, by the way, take any common core
classes), the math education goes something like this: teacher lectures and
does a few example problems, you the student independently do homework, review
answers, take a test with slightly different questions, and then proceed to
forget the majority of whatever you’ve just learned.
A few days ago, my class began delving deeper
into the realm of limits and sequences. At one point in the lecture, my teacher
interrupted to say something like this: all your life, you’ve been taught to do
math, empirically. You may not have understood why to proceed a certain way
when solving a test or understood which formula to use, but you begin learning
some of these reasonings as you graduate from class to class.
Which I guess, makes sense.
But a while ago, someone had recommended to me a
book entitled How Not to Be Wrong: the Power of Mathematical Thinking by
Jordan Ellenberg. I had read the introduction (only, yeah I know, I know) and
wrote a review of the novel’s beginning anecdote, which can be found here.
Several months later, I decide to take a hike
outside of San Bernardino, to a hot spring site in a place called Deep Creek
Canyon.
The hike and subsequent adventure were actually
wonderful: it felt so good to enjoy the absolute, irrevocable silence of the
canyon -- no noise, traffic, or pollution -- so quiet you could only hear the
blood rushing in your ears.
The springs themselves were a delight as well!
After about an hour hike down into a canyon, we waded across a freezing stream
to have lunch on some sun-drenched rocks overlooking the cold river.
Afterwards, we alternated between the just-on-this-side-of-uncomfortable hot
springs and the freezing creek, not unlike Japanese onsen. Someone had set up a
slackline across the freezing river, which we took turns attempting to cross as
well.
Where does the math come in? you might ask.
Well, we used Newton’s Law of Cooling to calculate the rate at which our body
temperature dropped after plunging into the mountaintop snow-cold water. Just
kidding.
Hardly any math was used that day: we relished
in the torpor of a hidden getaway in the middle of the desert, sprawling in the
drowsy hot springs before feeling the adrenaline paramount to a freediver’s
thrill at submerging in our icy creek.
But -- there was a lull in the middle of the day
wherein my companion pulled out a new copy of a Cicero book, and I my iPhone, finally
beginning to read Ellenberg again.
It didn’t feel like math.
Ellenberg explains one of the basic ideas in
calculus: limits. He begins with explaining how early mathematicians found the
number pi -- anecdotal style.
It’s not a strange concept to me -- irrational
numbers. Pi, Euler’s number, and the like. But discovering irrational numbers?
Basic human intuition understands the concept of whole numbers relatively
easily. One toga, two goats, three laurel wreaths, four stuffed grape leaves.
Half of a stuffed grape leaf may be easy to imagine. But 3.1415 grape leaves?
When graphing curves and parabolas in math, if
you zoom in close enough to any curve, it resembles a line. In statistics, this
method of representing data, known as linear regression, often simplifies
trends when viewing data with a microscope -- that is, looking at only small
chunks of data.
The legend Archimedes decided to find the area
of a circle by inscribing other shapes, like squares, pentagons, and octagons,
inside of a circle. The area of the polygon contained with the circle
represents the area of a circle the more sides that polygon has.
For example, the area of an octagon within a
circle better resembles the circle’s area than a square within a that same
circle. Archimedes theorized that a circle was simply a polygon, and its area
could be calculated by calculating the area of a polygon whose numbers of sides
approaches infinity. Here’s a catchy slogan Ellenberg presents: “straight
locally, curved globally.”
Math has this wonderful, complex, and quirky
history seemingly irreconcilable with what I’ve learned in class, at least up
until this year.
This year, I’ve had the pleasure of encountering
several elegant equations, such as this one:
Euler’s equation uses two of the most irrational
numbers, e and pi, as well as the imaginary i and 1 and 0.
It seems counterintuitive to read about math,
but I think the introduction of mathematical ideas through anecdotes -- such as
how Cauchy, a professor who both infuriated and inspired coworkers and students
alike when he taught his freshmen students his own cutting edge take on
calculus when the curriculum nearly the opposite -- makes math more
approachable, less of a tool you use out of necessity and more of a concept you
develop and harness as your own. It’s a living thing with its own history and
archaic notions, an art form if you look at it a certain way as well. I’m sure
many more complicated equations relating to earth and space are equally as
deserving of awe. But for now, I’ll continue reading on.
Until then!
