# Letters to my sister, I. What is trigonometry?

The foremost job of any math teacher is to answer the question “Why are we learning this stuff anyway?” It’s an annoying question to hear—why can’t you see the inherent beauty that I do?!—but a good one to ask if you’re burning away your childhood from inside a classroom.

Trigonometry is a hard one to answer for. At its heart is the study of the most basic shapes—circles and triangles—and even more basically, the fundamental narrative concepts of oscillation and cyclicality. Continue reading

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# The first-year dream (On multiplication, pt I)

The other day I was brainlessly strolling through Facebook and shared the following from a page called Brainy Miscellany [1]:

It’s funny if you teach algebra. Or calculus. Or differential equations. Or…

People who teach math call this “the first-year dream”: specifically, it is the hope that along with our multiplicative distributive law ${c(a+b) = ca+cb}$, we have an exponential distributive law, like ${(a+b)^2 = a^2 + b^2}$.

One of my old calculus students, Charles, commented on the post and did what I can always count on him to do: ask “Why?” We took to Facebook messenger and I offered an answer. He asked again, “But why?” Thirty minutes later, we’d gone pretty deep, touching on simple geometry, ring theory axioms, and the philosophy of math. Finally I was able to satisfy Charles with why the first-year dream is only a dream. In fact, we’d be in trouble if it was true.

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