Mathematics: Gateway to the Merry Land of Science... So deal with your math anxiety and quit being such a big, damn wuss about it, you effing coward!!!

Photo: From Colby Keller's Big Shoe Diaries blog.
Submitted as part of the Fearless photo project.

If you want to know what the takeaway is for this piece on math anxiety, you don't even have to read past this opening paragraph, I'll give it to you up front: "Fashionable innumeracy" -- the idea that it's perfectly acceptable to be a mathematical moron -- ends here. Right now. From this moment forward, it is no longer okay to say you're bad at math. It's totally fine if you're not a genius, just quit acting like a math-shy half-wit. If you're faced with a math problem on the job, or in any situation, you can say, "Math is a little tricky for me," or, "I'll need a little help with this," or, even better, "I've got to work on my math." But fleeing in terror and declaring, "I'm bad at math!" while expecting to be given a sympathetic pass for your cowardice ends now. I mean it! Continue to shamelessly avoid math and you will be stripped of your citizenship and deported. Maybe not today, but it's something I'm working on. Seriously. I've written letters.

If you'd like to stall a bit before finally facing your fears, do feel free to read on...

Somewhere in America, there's a hairy, leather-clad biker dude who spends his days dealing drugs and doing math. That's "math" not "meth" -- although he might be doing that as well. Throughout his workday, he's probably doing arithmetic, fractions, even some simple algebra. He's probably even calculating odds in his head using basic statistics. And yet, for however long he was in school before he dropped out, he was probably one of those students who said, "Why I gotta study this [EXPLETIVE]? I ain't never gonna use this [EXPLETIVE]!" It's a classic dodge employed by people who claim they'd rather not "waste" time studying math. The truth, of course, is that math scares the [EXPLETIVE] out of them.

Math anxiety is real, but it isn't the same as test anxiety and it must also be distinguished from other more general anxiety disorders. Math anxiety is almost like a straight-up case of failure anxiety, but not quite... and, honestly, you probably don't have it. At least, not a serious case of it. Put a timed math test in front of some people and they'll get the shakes, the sweats, cry a little, then vomit. For most people, it simply isn't that bad, but the flood of feelings math anxiety induces is unique and unmistakable. It's that moment in class when the word problems, the geometric proofs and the almost unbearable pressure that comes with trying to solve for "x" make you feel as if you've just been dropped into a foreign country where you can't read or speak the language and you really, really need to find a bathroom. Or, as Sheila Tobias says in Overcoming Math Anxiety, "The first thing people remember about failing at math is that it felt like sudden death."

Gobbledygook Pt. 1: Geophysics Without Fear

It was another object lesson in science's ability to obfuscate, intimidate and make you scratch your head till it bleeds. While I was researching the post before this one, I came across this passage in Geodynamics: Applications of Continuum Physics to Geological Problems:

"The gravitational potential anomaly [ΔU] due to a shallow, long wavelength isostatic density distribution is proportional to the dipole moment of the density distribution beneath the point of measurement."

For the layperson, this is the very model of a "What the fuck????" gob of indecipherable science babble. Let's break it down [DON'T BE SCARED, IT'S ONLY SCIENCE]:

Gravitational potential -- If you pick up a brick, hold it above the ground, and then let it go, it won't just stay suspended in mid-air. By lifting it up, against the pull of gravity, you gave it a certain amount of stored energy, called potential energy. And since that energy -- which is equal to the energy the brick will have when it falls -- comes from doing work on the brick against the earth's gravitational pull (and don't forget, the brick is pulling on the earth too), we call that energy gravitational potential.

Gravitational potential anomaly -- General science tip: whenever you see a delta ("Δ") in an equation, that usually refers to some kind of change or difference between two things. "ΔU" represents the gravitational potential anomaly and refers to how different the measured gravitational potential is from a standard reference potential. The difference is due to the fact that the ground beneath us does not have a uniform density, so a reference potential is used to get an idea of how different the pull of gravity is at any given location.

Isostatic density distribution -- You can take two columns of earth, for example one that starts at the top of a mountain and one that starts on the ocean floor, and by the time you burrow down and reach the creamy filling (i.e. the big mass of molten lava that all land masses rest on) you'll have two columns of earth that weigh pretty much the same, but you'll find the mountain column will be less dense than the ocean floor column. The principle of isostatic density distribution tells us this will be true for any two columns of earth we might want to consider. The "long wavelength" part just means you're talking about something massive enough to make a dent in the lithosphere (which consists of the earth's crust and the top part of the mantle layer).

Dipole moment of the density distribution -- This one really threw me at first. I knew about dipole moments as they related to electric charges and magnetic fields, but didn't know what they had to do with rocks and dirt. If a gravitational anomaly is detected, that's an indicator the density of the earth in that area has sort of adjusted itself, compensated to achieve normal isostatic density distribution. Think of the dipole moment as a measure of the extent of the self-adjusting that occurred to get the right density distribution. Plus, it's proportional to the gravitational potential anomaly (as the equation at the beginning shows... trust me, that's what it says).

Moho -- Beneath the earth's crust, but before you get to the mantle layer, there is a boundary called the Mohorovičić discontinuity, or Moho for short. It was named after a Croatian seismologist and marks the depth at which there are notable changes in the earth's chemical composition compared to the crust above it. This has nothing to do with anything, I just like the word "Moho." 

So here's what the bizarre, scary passage at the beginning was saying:

Geophysicists, folks who might be looking for ore and petroleum and things underground, start with a standard measure of what gravity is like for the whole planet. But the earth doesn't have a uniform density, so the gravity is actually a little different depending on where on earth you are. This is why, in some places, you weigh a little less and in other places you weigh a little more (there can be other factors, but here we're concerned with the effects of density). The difference in how much you weigh in a particular location compared to the standard measure depends on the nature of the difference in the density of the earth where you happen to be standing. You can take measurements and calculate the difference because there is a mathematical relationship between how the earth's density at your location changes and how the gravity changes in that same place. Different things alter density in different ways, some of which are known. That's why knowing about gravitational potential anomalies is useful: measuring what the gravity is like at a particular location gives us a hint of what might be underground.

There, you see? Was that so hard? Don't we all feel better now?

Moho.

Sources

• Geodynamics: Applications of Continuum Physics to Geological Problems. ©1982, John Wiley & Sons, Inc.
• Geophysical Methods in Geology, 2nd Ed. P.V. Sharma. ©1986, Elsevier Science Publishing Co., Inc.
• U.S. Geological Survey, http://www.usgs.gov.

Proof: Be Not Afraid

"For years, mainstream thinking about math anxiety assumed that people fear math because they are bad at it. However, a growing body of research shows a much more complicated relationship between math ability and anxiety."

From "A real fear: it's more than stage fright..." by Paul Ruffins, published in Diverse Issues in Higher Education, March 8, 2007

Too many people feel distanced from science because of one thing: Math. Or, more precisely, their math anxiety. It's fine to talk about theories and experiments and genius discoveries, but none of it means dick until some math is attached to it; the nasty, intimidating, complicated kind of math with lots of Greek letters and odd symbols and practically no actual numbers. Science is grounded in math, and math has as its foundation THE PROOF.

Even if you take math classes every semester in high school (shockingly, many people don't -- you know who you are), very little time is spent in class specifically discussing the nature of mathematical proofs. Teachers show them to students all the time and expect students to learn them, but while showing proofs is often a big part of math and science classes, teachers often seem to skip over the whole "What is a proof?" thing. Nobody tells you that most of science and pretty much all of math is about proofs. It's an incredibly important subject that educators hope you'll just sort of pick up as you go along.

There are basically three parts to any proof: The hypothesis (the thing we are trying to prove is true), the arguments (the relevant ideas we will stack like bricks to see if they support the truth of the hypothesis) and the conclusion (where we learn whether or not the arguments show the hypothesis to be true or false).

The arguments are things we know to be true because they have already been proven. Bad arguments ruin good proofs. Consider this proof that penguins can fly from Antonella Cupillari's book The Nuts and Bolts of Proofs:

• Penguins are birds.
  
• All birds are able to fly.
  
• Therefore penguins are able to fly.

The hypothesis is false, but, based on an invalid argument, we are led to believe that it's true. Consider an actual news event that recently occurred in the UK:

• A girl died.
  
• The girl had recently been given a vaccine.
  
• Conclusion: The vaccine is fatal.

A minor panic ensued in England because of people employing this proof who never once considered whether or not the argument was valid (i.e. represented a direct causal link between an event that occurred previously and a subsequent event). It was not valid. They were idiots.

Like any other proof, a math proof is a logical progression of steps, but to say that no intuition or insight is involved is wrong. There's a reason some people are better at it than others. It's why we are in awe of geniuses like Russian mathematician Grigory Perelman who solved the Poincaré conjecture which, according to the Clay Math Institute, is one of the millennium's seven most difficult math hypotheses.

It's about understanding that facing real science means facing math, facing proofs and doing it fearlessly. Genius is NOT a prerequisite. I'll prove it.

The Greek Alphabet: Math Without Numbers (kind of)

This is the sort of thing that drives people away from science:

It's a formula for the magnitude of an electric field as a function of time in an unmagnetized plasma. It's math, but you don't see any numbers. There are numbers there, but they're hidden behind the letters and those mysterious symbols. For the layperson, it makes no sense, even when you realize that most of those "mysterious symbols" are just Greek letters.

Blame Diophantus of Alexandria (c. 200 A.D., but reports of the exact dates of his lifespan vary widely). Before this Greek mathematician came along, equations were simply expressed as sentences written out in words. Diophantus is credited as being the first to use symbols to stand in for the numbers that change (the variables) and the numbers that are used repeatedly (the constants) in different equations. For many centuries after Diophantus, lessons in Latin and Greek were a common part of any aspiring scholar's education, and using Greek letters as symbols in mathematical equations was just the tradition and no big deal.

These days, Greek letters are more commonly associated with fraternities and sororities (at least for the less-than 30% of Americans who have college degrees). One reason equations like the one at the beginning of the post seem so intimidating is because they look so freakin' strange, very much like another language. Check that... mathematics is another language. If you're going to learn it, and stop being intimidated by it, it is helpful to recognize the letters of the Greek alphabet:

If the equation is being presented properly, it should be apparent what the author means for each symbol to represent. A mark's meaning can change depending on which branch of science you're dealing with, but I've never seen the character pi represent anything other than 3.14159,blah,blah,blah. There are other freaky non-Greek symbols to learn, but that's for another post. Download this PDF (127 K) containing a Greek alphabet cheat sheet.