"Girl, 18, More Math-Ignorant than Governor"

July 08, 2004

James Taranto's Best of the Web got our attention with a little item highlighting some sadly typical behavior: a student not only not knowing, but not knowing how much she doesn't know.

WFTV reports that Luana Marques, a newly graduated 18-year-old, tried to stump Florida Governor Jeb Bush with a geometry problem which she and her friends remembered being on the FCAT:

"What are the angles on a three-four-five-triangle?"

The governor gave a steely grin and then stalled a bit. "The angles would be . . . if I was going to guess . . . Three-four-five. Three-four-five. I don't know, 125, 90 and whatever remains on 180?"

Marques had an answer, although it wasn't the right one: "It's 30-60-90."

The correct answer was 90 degrees, 53.1 degrees and 36.9 degrees, said Michelle Taylor, a graduate student in mathematics at the University of Florida, when told about the governor's pop quiz.

Thank goodness the WFTV reporter knew to ask someone who knew.

We have three problems with this story:

1. This problem doesn't appear on the FCAT. We have no doubt that there was some question involving either a 3-4-5 right triangle or a 30-60-90 right triangle, but not both. We will publicly eat a copy of the FCAT if anyone can prove us wrong.
   
   
2. Anyone familiar with either 3-4-5 or 30-60-90 right triangles would have seen the student's mistake. The 3-4-5 right triangle is the classic shape to use when learning the Pythagorean Theorem, for when simplifying a² + b² = c² there are no messy radicals left over. (This is, of course, assuming that students were taught to do things like factor numbers and find square roots--both skills found in middle school and earlier.)
   
   And the 30-60-90 right triangle is also a classic shape, easily derived from bisecting an equilateral triangle--which means that the short leg and the hypotenuse are always in a 1 : 2 ratio. (Plug those two numbers into the Pythagorean Theorem and you find the third side to be √3.) Once a student learns 30-60-90 triangles she should immediately recognize the ratio 1 : √3 : 2 (and its multiples).
   
   The bottom line? If a student has learned either of these triangles, they cannot be confused with each other.
   
   
3. Although neither the Governor nor the student knew the correct answer, the Governor was the only one who admitted it. Now this is the really appalling aspect of this tale. No one expects grownups in non-scientific fields to remember high-school geometry, so the Governor's ignorance is understandable. But here is not only one student, but her friends as well, who are confoundingly ignorant on basic facts from 10th grade geometry, while being convinced they know their stuff.

This reminds us of an international study of math ability, where after taking the test students were asked to grade their own confidence in their math abilities. The high schoolers from America bombed the test, significantly outperformed by the Singapore high schoolers, who fairly aced it.

But the Americans outperformed the Singapore students in the affective arena, judging their own math skills to be most excellent. Meanwhile, the students who actually knew the most math correctly recognized that there was so much more advanced math they didn't know, thus they were much more modest in describing their own abilities.

This situation is perfectly described by the title of Charles Sykes' book, Dumbing Down our Kids: Why American Children Feel Good About Themselves But Can't Read, Write, or Add.

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So how do you find the angles in a 3-4-5 triangle? The first is easy, since 3² + 4² = 5² then it must be a right triangle, so one angle is 90°. If we call the larger acute angle "Theta" and we remember from the Princess of Trigonometry SohCahToa that the Sine of an angle is the ratio of the opposite side to the hypotenuse, then we can write:

Sin(Theta) = 4 / 5

Solving for Theta we get: Sin^-1(4 / 5) = Theta, or about 53.1°, and because all the angles of a triangle sum to 180° then the other acute angle is 36.9°.

Any questions?

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Update: Kimberly Swygert over at Number Two Pencil has more commentary on this tale, and her commenters are the most excellent ones (except for one who wrote, "You're all wrong" while being wrong herself--oops!) in this discussion.