standardized tests

As if the SAT didn't already have a less-than-stellar reputation in terms of racial bias and the possible inability to truly measure a student's cognitive abilities, now a story about an expensive blunder is once again making the rounds on social media. Back in 1982, one math question on the test was completely impossible to answer on the multiple-choice Scantron. How was that possible? Because the correct answer hadn't even been listed.

Classic SAT test. Photo by Nguyen Dang Hoang Nhu on Unsplash

Here was the question: Picture two circles, a large one marked B and a smaller one next to it with an arrow, marked A. "In the figure above, the radius of Circle A is 1/3 the radius of Circle B. Starting from the position shown in the figure, Circle A rolls around Circle B. At the end of how many revolutions of Circle A will the center of the circle first reach its starting point?" Is it A, 3/2; B, three; C, six; D, 9/2; or E, nine?

On the Veritasium YouTube page, they explain that if you were to look at the problem logically, you'd conclude the answer was B, three. Because the circumference of a circle is 2πr, and the radius of Circle B is three times that of Circle A, "logically it should take three full rotations of Circle A to roll around." However, that answer is wrong.

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In Jack Murtagh's piece "The SAT Problem that Everyone Got Wrong" for Scientific American, he conveys it all came down to the Coin Rotation Paradox (take note of this if you want to sound super intelligent on your next date or job interview).

You can try this yourself. Murtagh writes, "Here's how the paradox works: Place two quarters flat on a table so that they are touching. Holding one coin stationary on the table, roll the other quarter around it, keeping edge contact between the two without slipping. When the moving quarter returns to its starting location, how many full rotations has it made?"

Again, most test takers assumed that the answer was three. But "in fact, Circle A makes four rotations on its trip—again, exactly one more rotation than intuition expects. The paradox was so far from the test writers’ awareness that four wasn’t offered as an option among the possible answers, so even the most astute students were forced to submit a wrong response."

Why in fact was this the case? On the Scientific American YouTube page, it's explained again: "If you replace the larger circle with a straight line of the same length, then the smaller circle would indeed make three rotations. Somehow the circular path creates an extra rotation. And to see why, just imagine rotating a circle around a single point. There are two sources of rotation here. One from rolling along a path—and the longer the path is, the more rotations. And another from revolving around an object, which creates one extra rotation, no matter its size."

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Okay, one more try. Here, it's relayed in terms of actual astrophysics: "This general principle extends far beyond a mathematical fun fact. In fact, it's essential in astronomy for accurate timekeeping. When we count 365 days going by in a year—365.24, to be precise—we say we're just counting how many rotations the Earth makes in one orbit around the Sun. But it's not that simple. All this counting is done from the perspective of you on Earth. To an external observer, they'll see the Earth do one extra rotation to account for its circular path around the Sun. So while we count 365.24 days in a year, they count 366.24 days in a year."

What might be equally interesting is that out of 300,000 SAT test-takers who got that question at the time, only three wrote in to the College Board to challenge the answer. Ultimately, they had to fix the test, which cost them over $100,000. (In 1982, that's at least ten Happy Meals.)

The comment section on YouTube was buzzing.

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This person suggests following your gut, even if that does mean challenging a professor or other authority figure: "In college, I took a poetry class and once had an answer marked wrong on a test. Confident in my response, I reached out to the poet themselves, who affirmed I was right and even communicated this to my professor. Despite not being a fan of poetry, that moment made me quite proud!"

Another person commented on the reasoning behind the paradox itself: "That part about the circle rotating around the triangle was mind-blowing. You instantly understand why it's not the same if the circle rolls on a flat line or rolls on a curved line."

And for this person, it brought peace of mind: "This was the one SAT I took, and I remember the question that didn't have a correct answer, and it wasn't until today that I understood the right answer. I can die happy now."