I recently had a chance to do one of my favorite (and my students' least favorite) things: talk about words in math class. Math words. I also had the opportunity to use one of my favorite math-teacher-type resources: a dictionary. I don't mean the glossary out of a math book, or a page from Wolfram MathWorld, or any one of the approximately 10.5 million web results (as of this writing) that the Google spits out when prompted with "math" + "dictionary." I'm talking about a good, old fashioned English dictionary, one of three left in my room by the previous English-teaching occupant: Webster's Ninth New Collegiate, circa 1989.
Let me bracket this entire discussion by saying that I in no way endorse the use of English dictionaries for math definitions. They're horrible. Even more horrible, if that's possible, than math texts. For instance, I die a tiny little death every time I see the phrase "straight line." But where dictionaries really shine, and where textbooks are almost wholly silent, is in etymology. So the other day, instead of doing an activity titled (really) Investigation: An Asym...What?, I thought it would be much more interesting for us to do a lexicographical investigation into the roots of the asymptote.
Asymptotes get a bad rap. They are, I think, right up there with logarithms in the category of High School Math Concepts Most Likely to Turn Your Stomach as an Adult Reflecting Upon High School Math Concepts. They're kind of abstract, philosophically suspect, and limited to one or two sections in the average textbook. Not exactly auspicious.
And then of course there's the name. It sounds ridiculous. Even as math teachers, we think of it primarily as a punchline. I'll bet that you, dear reader, can currently name at least two math blogs, twitter handles, math team names, lessons, or activities that play on "asymptote." But, like most inside jokes, it's funny to us insiders primarily because it's confounding to everybody else, e.g. our students. In that light, An Asym...What? is supremely unhelpful. After all, shouldn't we be making math things (including words) seem less mysterious? So, in that spirit, wherefore "asymptote?"
Let us begin with a bird. A Greek bird. Upon that bird you will likely find a wing [pteron]. Upon that wing you might find feathers, which are wont to fall [piptein] from time to time. With such concepts established, you might then want to consider, in a particularly lovely and poetic way, what happens when two things come together [sym]. In fact, you might say they "fall together," which is to say that they meet [sympiptein]. You might then inflect that infinitive to capture the idea of meeting [symptotos]. And, naturally, since you have a prefix that denotes without or not [a-], you can succinctly capture the idea of not meeting [asymptotos].
By inspecting the language with the same scientific curiosity that we loose upon the mathematics, we can demystify it. In fact, "asymptote" is an incredibly simple and apt description of precisely the feature we hope our students understand! Plus, as a bonus, that's a freaking cool story.
- Apothem [Gr. something laid down + away from]
- Complement [L. to fill up]
- Perimeter [Gr. to measure + around]
- Recursive [L. to run + back]
- Tangent [O.E. to touch gently]; That's just so beautiful and intuitive that I can't stand it.
- Vertex [L. summit,]; derived from to turn. How great is that, Calc teachers!
If you want a visual mnemonic device for asymptote, here's a video I made: