No. 43 - Summer 1998

© 1998, Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, CA 94112.

Astrophysics for a Ten-Year-Old Mind

Michael Chabin

Try an experiment. Over the next couple of days, ask a random selection of the people you meet to tell you the difference between 1/4 and 1/5. Be sure to ask politely. Many of us have painful associations with random arithmetic problems given out of context. But if you ask nicely and find a broad range of people, you may notice, as we did, that the vast majority of those surveyed don't know the answer.

Radical members of the "Back to Basics" movement in the U.S. might conclude from this experiment that America's schools are failing and what is needed is a return to the kind of mathematics instruction that has made state lotteries so popular. But they would be wrong. Our survey says nothing about the schools. Almost every American fifth-grader can tell you the difference between 1/4 and 1/5.

What the survey does suggest is that most adults never subtract fractions in that form and they have quite understandably forgotten the method.

Now try a different experiment. Show your subjects the equation e = mc2 and ask what it means. Prepare yourself for a pleasant experience: Not only do people know a lot about this arcane piece of physics, but they like talking about it.

The equation is the most famous in science. It can be found in magazines, comic strips, textbooks, graffiti, posters, and science fiction novels. Remember the little bespectacled chick often under the care of the cartoon rooster Foghorn Leghorn? He had a t-shirt proclaiming this seeimingly arcane physics. An astonishing number of people recognize it. Most associate it with Albert Einstein and many will say it has to do with mass and energy. A heartening number of students and teachers will even tell you the equation states that energy is equal to mass times the speed of light squared.

All of which might lead radical members of the education establishment to claim that American science education is not as bad as recent international surveys indicate and that in order to continue such remarkable successes teachers should be given raises and more professional development time. But they would be just as wrong as the "back to basics" advocates. What people know about Einstein's equation, like much of the useful and practical science they know, is not part of the curriculum and was not learned in school.

In fact, though this is by far the most recognized equation in history, it is not recognized as an equation at all. Rather e=mc2 has become a trademark of science in much the same way golden arches are a trademark of McDonalds. One way to see this is to rearrange it to e= c2m and ask people to tell you what that is. Another way is to ask them to use it.

The equation is almost never shown as:

Energy (in joules) = Mass (in kilograms) X The Speed of Light Squared (in meters squared per second squared)

Maybe that is why many of those who proudly tell you "energy is equal to mass times the speed of light squared" will blush if you then say, "Okay, then, so how much energy will one liter of water yield if it is converted entirely into energy?" Clearly, knowing the equation is one thing, using it is for specialists. Accepting this contradiction is not their fault. Much science is taught by memorization.

Still, it is a shame students don't have a chance to use the equation. It is elegant. It is powerful. It lets them answer exotic questions. And, the "back to basics" movements notwithstanding, learning to use it exploits just the kind of mathematics that can be expected of early middle schoolers.

In 1989 the National Council of Teachers of Mathematics (NCTM) released a set of standards which, outside of California, are used to guide curriculum development. Of the standards for grades 5-8, Standard 9 includes understanding the concepts of "variable, expression, and equation." Scientific notation is implicitly part of the standards and is used in examples given, as is the metric system.

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