The Eye of the Dragon

We obscure the beauty of mathematics when we jump straight to the end.

Photo by Emanuela Meli on Unsplash

The last few weeks, for our virtual, quarantine edition of Algebra II, we’ve been exploring the unit circle.

I’ve been trying to wrap my mind around how to describe the way I see too many mathematical concepts introduced, with a standard presentation of the unit circle being in my mind the epitome of that concern. But the dragon is so large and untamable that it has been resistant to words.

Enter this month’s issue of NCTM’s “Mathematics Teacher” (2020.05), specifically the lead article, “What’s in a Name? Language Use as a Mirror into Your Teaching Practice” (by Tracy E. Dobie and Miriam Gamoran Sherin).

In this article, the authors discuss how our language as educators can shape our thinking, and offers a few terms from other languages. One of these, my bright shining moment of terminological epiphany, comes from Chinese: 画龙点睛, which refers to drawing the dragon’s eye, and can be translated as adding the finishing touch.

The metaphor is that a picture of a dragon is not fully complete until the artist has added the detail of the eye, which breathes life into the entire image.

The idea behind the phrase in mathematics education is that this is the moment that the teacher brings together a bunch of disparate threads and creates a single, living, coherent picture out of them.

When I think of how I remember the unit circle (with the special angles marked) being presented to me, I remember it as being slapped on my desk, whole and undissected, with the understanding that I simply memorize its nuances.

But this is like drawing the eye of a dragon and demanding that students extrapolate the rest of the creature from that detail. The unit circle diagram is the end point, a culmination, of a bunch of different threads that have to be tied together to make sense of them.

And I fear that many students see the unit circle as they do the leer of a mythical beast ready to breathe fire onto them. Today, during a review, I brought out the unit circle diagram again, and a student who had first learned that image elsewhere complained: “Oh no, here it comes again!”

I love the patterns that are held in the diagram: The symmetry, the way in which a handful of simple rules can let you build it on demand. It should not be a memorization task, it should be a way to behold the beating heart of mathematics.

But it is only so if it is the final touch at the end of a long journey, the bringing together of multiple concepts. If you have not drawn the dragon, you cannot appreciate the beauty of its eye.

My dissection of the diagram will come in a future article (or more) in this space. I’ve deliberately left the diagram itself out of this article because, if you know what I mean, you don’t need it, and if you don’t know what I mean, I don’t want to do what was done to me.

In a larger level, I urge my colleagues to reflect: Are your dragon’s eyes a beautiful culmination, or are they gory, jarring, and decontextualized?

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