Tuesday, March 1, 2011

The (Vampire) Tale of Calculus

The thing I love most about teaching calculus is that it seems to be the first course where it's easy to tell the story of the math. All the other courses leading into it seem to be a random collection of skills that are only loosely based on one another. For example, Algebra II (at least in our district) hops around from quadratics to exponentials to trigonometry, probability, systems of equations, and conics without much to tie them together.

Calculus flows very nicely, though.

We begin with our student adventurer moving to a new state and having to make new friends with abstract reasoning and rigorous work. Students find themselves walking along curves with limits only sometimes meeting their friends at an appointed spot and sometimes even walking towards a place that doesn't exist. The once familiar functions are treated as trivial, new functions are scarier and uglier than students thought possible making their much hated trig functions seem easy in comparison.

These infinitely (or infinitesmally) long walks along curves lead them to an otherworldly being: the derivative. At first the derivative is completely scary and unreasonable. "What does f(x+h) even mean?!" It's right there at the edge of their understanding and is beckoning them to come play, but has large wolverine-like teeth and is intimidating.

When students finally open up to the derivative's definition after walking the curves and even sometimes leaving the safety of the curves for tangents, they begin to see the true nature of derivatives. They discover the power rule. They are comfortable around the new creature, even when some of the scarier parts show up like the chain rule or even the quotient rule comes with a haunting melody to soothe its use.

Students finally let themselves fall in love with derivatives (which scarcely 2 months before none of them would've dreamed might exist) when it leaves them for a bit. Natural classroom forces tell students to leave the derivatives on the table while we go explore a new idea...

Their old friend, areas, come to visit. Brooding students want to return to derivatives as they begin to feel strange feelings towards Riemann sums, but we have left the derivative behind. Slightly different paths along accumulating functions where they step along thin rectangular blocks or trapezoidal strips lead them to a deeper appreciation of what area really means.

When students are finally beginning to forget about derivatives and are growing accustomed to loving the one they're with, the Fundamental Theorem of Calculus appears and reveals that these two supernatural beings are not only related, but at war with one another in an ancient struggle between infinities and zeros.

On the one hand, it is comforting to see the old derivative friend reappear. On the other, it is awkward to pair the old love with the new friend. The rest of the year is spent coming to terms with how the two relate and can become friends themselves.

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