How might a teacher go about cheating? There are any number of possibilities, from the brazen to the sophisticated. A fifth-grade student in Oakland recently came home from school and gaily told her mother that her super-nice teacher had written the answers to the state exam right there on the chalkboard. Such instances are certainly rare, for placing your fate in the hands of thirty prepubescent witnesses doesn’t seem like a risk that even the worst teacher would take. (The Oakland teacher was duly fired.) There are more subtle ways to inflate students’ scores. A teacher can simply give students extra time to complete the test. If she obtains a copy of the exam early—that is, illegitimately—she can prepare them for specific questions. More broadly, she can “teach to the test,” basing her lesson plans on questions from past years’ exams, which isn’t considered cheating but certainly violates the spirit of the test. Since these tests all have multiple-choice answers, with no penalty for wrong guesses, a teacher might instruct her students to randomly fill in every blank as the clock is winding down, perhaps inserting a long string of Bs or an alternating pattern of Bs and Cs. She might even fill in the blanks for them after they’ve left the room.

But if a teacher really wanted to cheat—and make it worth her while—she might collect her students’ answer sheets and, in the hour or so before turning them in to be read by an electronic scanner, erase the wrong answers and fill in correct ones. (And you always thought that no. 2 pencil was for the children to change their answers.) If this kind of teacher cheating is truly going on, how might it be detected?

To catch a cheater, it helps to think like one. If you were willing to erase your students’ wrong answers and fill in correct ones, you probably wouldn’t want to change too many wrong answers. That would clearly be a tip-off. You probably wouldn’t even want to change answers on every student’s test—another tip-off. Nor, in all likelihood, would you have enough time, because the answer sheets are turned in soon after the test is over. So what you might do is select a string of eight or ten consecutive questions and fill in the correct answers for, say, one-half or two-thirds of your students. You could easily memorize a short pattern of correct answers, and it would be a lot faster to erase and change that pattern than to go through each student’s answer sheet individually. You might even think to focus your activity toward the end of the test, where the questions tend to be harder than the earlier questions. In that way, you’d be most likely to substitute correct answers for wrong ones.

If economics is a science primarily concerned with incentives, it is also—fortunately—a science with statistical tools to measure how people respond to those incentives. All you need are some data.

In this case, the Chicago Public School system obliged. It made available a database of the test answers for every CPS student from third grade through seventh grade from 1993 to 2000. This amounts to roughly 30,000 students per grade per year, more than 700,000 sets of test answers, and nearly 100 million individual answers. The data, organized by classroom, included each student’s question-by-question answer strings for reading and math tests. (The actual paper answer sheets were not included; they were habitually shredded soon after a test.) The data also included some information about each teacher and demographic information for every student, as well as his or her past and future test scores—which would prove a key element in detecting the teacher cheating.

Now it was time to construct an algorithm that could tease some conclusions from this mass of data. What might a cheating teacher’s classroom look like?

The first thing to search for would be unusual answer patterns in a given classroom: blocks of identical answers, for instance, especially among the harder questions. If ten very bright students (as indicated by past and future test scores) gave correct answers to the exam’s first five questions (typically the easiest ones), such an identical block shouldn’t be considered suspicious. But if ten poor students gave correct answers to the last five questions on the exam (the hardest ones), that’s worth looking into. Another red flag would be a strange pattern within any one student’s exam—such as getting the hard questions right while missing the easy ones—especially when measured against the thousands of students in other classrooms who scored similarly on the same test. Furthermore, the algorithm would seek out a classroom full of students who performed far better than their past scores would have predicted and who then went on to score significantly lower the following year. A dramatic one-year spike in test scores might initially be attributed to a good teacher; but with a dramatic fall to follow, there’s a strong likelihood that the spike was brought about by artificial means.

Consider now the answer strings from the students in two sixth-grade Chicago classrooms who took the identical math test. Each horizontal row represents one student’s answers. The letter a, b, c, or d indicates a correct answer; a number indicates a wrong answer, with 1 corresponding to a, 2 corresponding to b, and so on. A zero represents an answer that was left blank. One of these classrooms almost certainly had a cheating teacher and the other did not. Try to tell the difference—although be forewarned that it’s not easy with the naked eye.

Classroom A

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Classroom B

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If you guessed that classroom A was the cheating classroom, congratulations. Here again are the answer strings from classroom A, now reordered by a computer that has been asked to apply the cheating algorithm and seek out suspicious patterns.