Striving for Creativity in Math: The Drake Equation

During astronomy this year, the kids said they wanted to learn about aliens, so my practicum student decided he’d introduce the Drake Equation. He did a great job, but he ran into some really common snags that all teachers deal with. What follows are my notes as an observing teacher, and, of course, a healthy bit of editorializing.

Can I really get through a post without a JLP reference? It’ll make sense in about 800 more words, promise.

When the students said they wanted to learn about aliens, they probably didn’t have this in mind:

[latex size=3]N=R^{*}\cdot f_p\cdot n_e\cdot f_l\cdot f_i\cdot f_c\cdot L[/latex]

That’s the kind of dark humor that takes you over when you get a physics degree. Oh, you’re interested in something? Come to me, learn the dark secrets of the universe, MATHAMTHAMATHMATHAMATHAMATHMATHMATH.

What’s especially funny about the Drake Equation is that mathematically it’s not all that interesting. It’s 8th-grade probability and a little bit of units analysis thrown in to add a some flavor.

What’s depressing is that students often balk at the sheer volume of variables, surely this is a sign that something in math education is wrong. Right? When multiplication scares students; seriously?

Here’s a shamelessly stolen excerpt from our favorite pedia of the wiki variety:


N = the number of civilizations in our galaxy with which communication might be possible (i.e. which are on our current past light cone);


R* = the average rate of star formation per year in our galaxy

fp = the fraction of those stars that have planets

ne = the average number of planets that can potentially support life per star that has planets

f = the fraction of the above that actually go on to develop life at some point

fi = the fraction of the above that actually go on to develop intelligent life

fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space

L = the length of time for which such civilizations release detectable signals into space

Annnnnd, units:

[latex size=2][planets]=[\frac{star}{year}][none][\frac{planets}{star}][none][none][none][year][/latex]

Here’s where things go awry: My practicum student asked the kids to research values for each variable and make an estimate for N. A reasonable request, but not without fault.

You can guess what the kids did…

Student Use of Google as a Classroom Barometer:

They Googled “Drake Equation,” copied down what they found at the first link, and then sat and stared at their meaningless answers for the next 15 minutes as we tried in vain to get them to actually think about what an L of 1,000 or 10,000 might mean.

This pitfall is common. In my humble 5 years of teaching experience, I’ve learned a single hard-and-fast rule about when students abuse search: they do it when they think that you want an answer. They don’t have want for the answer. They don’t have need for the exploration. They want to finish something so that they can be “done.” It makes me insane that school has made this place so easy for students to get to, and it’s a pretty good barometer for whether they’re engaged or not.

This is the classic issue in education: are the students doing work for you, or are they doing it for themselves with a little prodding/help/guidance from you? I still don’t have this down, but I struggle with it daily, and it’s a much deeper question than we realize.

If students are doing work for themselves, will they ever learn about that random content jumbled together into what we pretend to call curricula? Should they? I’m not so sure anymore. I used to think education was the game of tricking young people into learning about things they wouldn’t otherwise care about, but now I’m not so sure.

So, the Drake Equation mostly flops. My practicum student learns a lot, and we move on.

However, I can’t help but feel that we have to revisit this, so here’s how I’m going to do it:

Inquiry With the Drake Equation:

First things first, why is the Drake Equation so long? This begets a socratic dialogue between you and the students. They already want to know about aliens, so use that momentum to your advantage.

Let’s get out the whiteboards. Give them 3 minutes to write down everything that might affect how many aliens there are. Tell them to be crazy, we’ll pare things down later.

Next, circle the room and have them pick out the things that everyone came up with. Have them find one thing to fight about; maybe it’s the term someone put in for the size of the planet. Maybe it’s how likely the civilization was to realize the redundancy of the letter ‘q.’

Let them fight it out.

Put the final expression on the board right next to Drake’s. Let them fight that out.

Finally, pick some numbers as a class and compare the results of your class’ equation with Drake’s.

Call it a day.

A Few Loose Ends:

Here’s a great interactive Drake Equation editor from the BBC.

Finally, there’s something about Wikipedia’s treatment that I don’t get. It says:

N = the number of civilizations in our galaxy with which communication might be possible (i.e. which are on our current past light cone);

This doesn’t make any sense to me. Can someone please explain how the Drake Equation takes into account the probability that the civilization falls within our light cone, or even close enough to exchange real-time communiques?

Obviously the messages move at the speed of light, so there’s a timing issue that relates the probability of proximity, but I feel like the DE, as it stands, doesn’t account for any of that.

Things get interesting when you start to really think about conversing with the green buggers. Sure, it’s cute to estimate how many of them have developed facebook and scrapped the idea in favor of building spaceships, but let’s remember that you can’t just talk to anyone you want. They have to have lived, and sent messages some time in the past in order to cross the interstellar void.

Here’s my amendment, I feel like how I did during Statistical Physics; there’s just something I don’t get:

[latex size=3]P_{communication}=\frac{L}{d} \cdot [\frac{L}{l_{gal}}]^2[/latex]

Where d is the distance between the two civilization in years (assuming speed of light comms) and lgal is the lifetime of the galaxy.

I added the first term to account for the fact that aliens on the other side of the galaxy–living right now–are communication-worthless to us.

The second term represents the probability that the civilizations exist at the same time, and is squared to represent the probability of two-way communications.

So, the DE becomes:

[latex size=3]N=\frac{L}{d}\cdot[\frac{L}{l_{gal}}]^2\cdot R^{*}\cdot f_p\cdot n_e\cdot f_l\cdot f_i\cdot f_c\cdot L[/latex]

Assuming the galaxy has had an even probabiltiy to produce life, the probability that our time lines overlap and are within our light cone severely limit the number of civilizations that we could talk to.

Looks like Picard was right, stellar archaeology is probably a most interesting field.