The contribution of standardized extended caution indices to the prediction of performance in college mathematics

Abstract

This study examined the relationships between standardized extended caution indices (ECI2(,z) & ECI4(,z)), mathematical ability, and course performance (precalculus or calculus) to determine whether or not: (a) ECI(,z)s improve the prediction of performance; and, (b) the relationships between mathematics ability and course performance indicate test bias. ECI2(,z) and ECI4(,z) were computed for a representative group of freshmen entering Texas A&M University in the fall of 1983 and attending a mid-August new student conference (N = 710 {algebra}; N = 536 {trigonometry}). Course performance was determined by converting first semester precalculus (n = 270) or calculus (n = 143) semester point totals to standardized scores. Three examinations from the Texas A&M University Precalculus Skills Test, parallel Algebra Forms A & B and Trigonometry, were administered. Historically, students have not prepared for the examinations (Green, 1982). Due to the extremely high correlations between ECI2(,z) and ECI4(,z) (.985 {n = 708} for algebra and .992 {n = 531} for trigonometry), the individual-based index (ECI4(,zi)) was used in subsequent analyses. The sequential sums of squares F tests for a linear model with mathematics abilities, standardized extended caution indices, and the respective interactions yielded the following results for precalculus students. Algebra ability (F = 26.92), trigonometry ability (F = 7.72), ECI4(,z) for algebra (F = 5.14), and ECI4(,z) for trigonometry (F = 5.43) were significant main effects at the .05 level of significance (n = 209). For calculus students, algebra (F = 45.46) and trigonometry (F = 6.40) were significant at the 0.05 level (n = 138). An examination of test bias for aberrant (top one-third) and consistent (bottom two-thirds) groups found that regression slopes were significantly different for the precalculus groups on trigonometry (F = 2.031, p = .044). A Johnson-Neyman simultaneous region of significance at the .05 level showed that differential prediction was restricted to students with better than average trigonometry abilities (0.097 logits or above). Trigonometry test scores for aberrantly responding, more able, precalculus students would be systematically underpredicted by major group or common regression. Since ECI4(,z) was not significantly correlated with trigonometry ability for precalculus students (r = -.056, n = 223), these results indicate that the individual-based standardized caution index should be used to supplement test score information for tests that require highly specific solution strategies (e.g., trigonometry tests) when the students are assessed long after exposure to the test's content and when students do not prepare for the examination.