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Fermi Problems

It’s a familiar peeve: The public doesn’t understand science or its workings. Society would be stronger and safer if the citizenry could only judge the reliability of climate change studies, the benefits of vaccines, or even the significance of the Higgs boson. 

This plaint sounds both worthy and familiar, but to lament the impoverished state of science literacy is to flog an expired equine. It’s easy to do, but neither novel nor helpful.

Of course, that’s not to say that we shouldn’t try. The teaching and popularization of science are The Lord’s work. But one can easily allow the perfect to become the enemy of the good. Rather than hope for a future in which everyone has a basic understanding of atomic theory or can evaluate the statistical significance of polls, I’m willing to aspire to a more conditional victory.

I would appreciate a populace able to make order-of-magnitude estimates.

This is a superbly useful skill, and one that can be acquired by young people with no more than a bit of practice. Learning the valences of the elements or the functions of cellular organelles, both topics in high school science, require memorization. Estimating the approximate magnitude of things does not.

And mirabile dictu, no personal electronics are required. Indeed, gadgets are a hindrance. Ask a young person “how much does the Earth weigh?” and he’ll pull out his phone and look it up. The number will be just that—a number—arrived at without effort or the slightest insight.

But this is a number that can be approximated in one’s head with no more than middle school geometry, a sense of the approximate weight of a rock or a car, and about a minute’s thought.

The rough-and-ready answer might be wrong by a factor of two or three, but in most cases that will be adequate for whatever is the purpose at hand. 

To scientists, such questions are known as Fermi problems, after the famous physicist who encouraged colleagues to make back-of-the-envelope calculations. Apparently, one such problem posed by Enrico Fermi himself—and reputedly used by Google when interviewing potential employees—is “how many piano tuners are there in Chicago?” 

Answering this requires making reasonable guesses about such things as the fraction of households having pianos, how long it takes to tune them, etc. But anyone can do that. No background in advanced mathematics is required, just the self-confidence to take on the problem.

If we wish the public to be able to make smart choices about such issues as the relative dangers of football versus driving, how long will it take to burn away the entire Amazon basin, or whether it’s safer to inoculate your child or not, we are fabulizing if we think that hearing the answers in a news report will suffice. Just as skills are developed by practice, not by reading, so too will an ability (and a readiness) to make an order-of-magnitude estimate produce understanding that is deep and long-lasting.

Aside from its utility, this skill rewards its practitioners with personal gratification. It’s an everyday demonstration that quantitative knowledge about the world is not simply handed down from on-high. Observation, simplifying assumptions (“let’s approximate a chicken by a sphere…”), and the simplest calculation can get us close to the truth. It is not solely the province of experts with tweed jackets and a wall covered with sheepskins.

School teachers have long tried to promote an interest in science by maintaining that curiosity and logical thinking are characteristics of us all. “Everyone’s a scientist.” But this felicitous idea is generally followed up with course curricula that are warehouses of facts. Being able to make order-of-magnitude computations with nothing more than one’s wits would be far more satisfying, because then you would know, not because someone else told you, but because you worked it out yourself.

We have inadvertently let a device in everyone’s back pocket become the back of the book for any question requiring a quantitative answer. It needn’t be so, and it shouldn’t.