Richard H. Escobales, Jr.
May / June 19989
Richard H. Escobales, Jr., has served on the mathematics faculty of Canisius College (Buffalo, N.Y.) since 1973. His research interests lie in the differential geometry of foliations, but he has also been concerned for a long time about the mathematical preparation of high school students. [He thanks Professor Hung-Hsi Wu of Berkeley for asking him to write this note and for his encouragement.]
Significant changes are occurring in New York State. The “Mathematics A Regents Assessment” (Math A Test) and the “Mathematics B Regents Assessment” (Math B Test) will eventually replace the yearly exams given in the current three-year Regents curriculum in high school mathematics, Courses I, II, and III. This sequence itself replaced the venerable Ninth, Tenth, and Eleventh Year Mathematics (Math 9, 10, 11). The latter no-nonsense curriculum was substantially the one that insulated New York students from the worst excesses of the “new math” of the 1960s. Math 9, 10, and 11 represented New York State’s one-time commitment to excellence in high school mathematics. Some college professors of mathematics I know believe that with some updating, this curriculum could well serve students of the twenty-first century.
Under the new dispensation, the Math A Test is the only exam in mathematics that a high school student must pass to receive a Regents diploma. Those wanting an Advanced Regents diploma will also be expected to pass the Math B Test. The requirement to pass Math A will be imposed on essentially all high school students. For weaker students, this requirement represents a step up. Students who graduate now with a local diploma have only to complete a Regents Competency Test in mathematics. Designed to cover two years of high school mathematics, the Math A Test will, in a more realistic view, cover about one and one-half years of high school mathematics.
But in raising the bar for weaker students, the new programseems to be lowering that bar for better ones. And there is a serious problem of whether most or all of the harder topics will be back-loaded to the Math B Test or ignored entirely. “Will some able students refuse to take the harder Math B, since now they can get a Regents diploma without it?” worries the Math Committee.
The Math Committee, which is advisory to New York’s commissioner of education and of which I am a member, has recommended that there be no proofs in Euclidean or analytic geometry on the Math A Test, nor will questions involving the use of the quadratic formula appear on that test. Since there were some word problems on the pilot exams for the Math A Test (a pleasant surprise), it appears likely that a few such problems will appear on future tests. The Math Committee wants the Math A Test to require algebraic thinking and not just arithmetic thinking. But I would also like high school graduates to know the quadratic formula and have at least some experience with deductive reasoning in a geometric setting.
The pilot exams for the Math A Test included a generous list of formulas, most of which should be internalized by high school graduates. Whether these formulas will be provided for students on future Math A Tests is uncertain. But one is encouraged by the report in the Rochester Democrat and Chronicle of 1 September 1997 that New York’s education commissioner “said he will probably strike the formula page.” The formulas given on the Math A pilots included those for the areas of a rectangle, square, circle, and triangle, the formula for the slope of a line, and the point-slope and slope-intercept forms of a line’s equation.
The word theorem does not occur in the vague mathematicsstandard (another “standards” disaster) of New York’s 1996 “Learning Standards for Mathematics, Science, and Technology” (MST), which the Math A Test roughly reflects. Miraculously, theorem makes its cameo appearance as one of the formulas on the Math A Pilot: “Pythagorean Theorem c2 = a2 + b2.”
New York will publish guidelines so that teachers will be able to prepare students for Math A and Math B tests. Will these guidelines even approach the mathematical quality present in the Math 9, 10, and 11 syllabi? That is unlikely, but stay tuned. I believe that tests should be based on a carefully designed, well articulated, published curriculum. The tests should be subordinated to the curriculum, not vice versa. And so my answer to the question posed in the title of this article is an emphatic no.
The author dedicates this essay to the memory of Thomas E. Flemming, S.J., of Canisius College and Professor Chich-Han Sah of SUNY at Stony Brook.
The final reaction to the question “Should High-Stakes Tests Drive Mathematics Curriculum and Instruction?” begins on page 15.