Controversial Question in New York State Algebra 2 and Trigonometry Regents Exam

Controversial Question in New York State Algebra 2 and Trigonometry Regents Exam

They say that there is no such thing as a stupid question. New York State mathematics teachers whose students took the Regents Exam in Algebra 2 and Trigonometry last month (June 2011) are likely to disagree. The test contained a controversial question that asked students to find the inverse of a non-invertible function. Here’s the problem in question:

The problem was in the 2-point, or short answer free response, portion of the exam, testing the learning standard that demands students “determine the inverse of a function and use composition to justify the result.” (A2.A.45) The wording of the question strongly implies that the inverse of the function does indeed exist. However, since the function given is not one-to-one, there is no inverse. Teachers got loud, complaining to representatives of the Board of Regents, the group that writes, edits, and distributes the exam. The Board responded with a memo called, “Scoring Clarification for Teachers,” which acknowledged several ways that students could interpret the question and demonstrate their understanding of invertibility of functions.

Was the response satisfactory? The Board's memo cites “variations in the use of [inverse] notation throughout New York State,” which seems to evade blame for a lousy question. A prominent math teacher blogger responded on his blog, “How could the test-makers not be aware of ‘variations in notation’? Also, notice how there is an asymmetric justification burden on a kid claiming (correctly) that the inverse does not exist.” A lousy question shakes the faith that teachers and students have in the standardized test as a valid assessment of student understanding. For instance, the same blogger concluded, “I have no confidence in New York State’s ability to create a good test of mathematics, at any level.”

It is my sincere hope that this controversy and the appearance of a misleading question will lead to both (a) more opportunities to explore the meaning of invertible functions and one-to-one functions, demanding students to be more savvy test-takers; and (b) increased scrutiny and more careful construction of New York’s Regents exams. In short, as educators, better instruction and better assessment should be our smart answer to this, or any, stupid question.