A number of recent studies reported on ways to identify mathematically talented elementary school students for challenging programming (Assouline & Lupkowski, 1992; Lupkowski-Shoplik & Assouline, 1993; Mills & Bamett, 1992). After these studies demonstrated that the Secondary School Admission Test (SSAT) was an effective out-of-level identification instrument for mathematically talented students younger than age 12, it became critical to determine further the instructional needs of the identified students. The next step was to conduct above-level achievement testing to provide a profile of student strengths and weaknesses so that the standard elementary mathematics curriculum could be adapted for them.
Participants in the studies described here took tests designed for students several grades older because grade-level tests are too easy for talented students and their abilities and achievements are not measured accurately. The goal was to use more difficult tests so that the ceiling effect would not be a factor.
Over the years, researchers have noticed that many mathematically talented students perform better on conceptual tasks than computational tasks (personal communication, Julian Stanley, September 1992). The intent of the studies described here was to document that observation.
Study 1: STEP SubjectsChildren in Texas, Iowa, and Pennsylvania whose parents had nominated them for participation in one of several university mathematics programs for talented elementary school students were subjects for Study 1. All students had scored at the 95th percentile or higher on at least one of the mathematics subtests of a standardized, grade-appropriate test. Twelve were third graders, 18 were fourth graders, and 20 were fifth graders. Thirty-three males and 17 females participated.
InstrumentsStudents took Level 3 or Level 4 of the Sequential Tests of Educational Progress Series II (STEP) Basic Concepts and Computation tests. Detailed information about those tests can be found in the handbook (ETS, 1971). The STEP Basic Concepts Level 4 (STEP4CONC) and the Computation Level 4 (STEP4COMP) tests were designed for second semester 3rd graders through first semester 6th graders. For the Basic Concepts Level 3 (STEP3CONC) and the Computation Level 3 (STEP3COMP) tests, second semester 6th graders through first semester 9th graders comprised the norm group (ETS, 1971).
ProcedureChildren took a level of the STEP that was developed for children two or three years older, following Cohn's (1988) rule of thumb: exceptionally able third graders are given the level of the test that is two grades beyond their grade placement, while fourth and fifth graders are given the test that is three grade levels beyond their grade placement. Tests were administered according to standardized procedures.
Results and DiscussionDescriptive statistics for children participating in Study 1 are available from the authors. Percentile ranks for students in all three grades were higher on the STEPCONC than STEPCOMP tests, indicating that students performed at a higher level on conceptual tasks than on computational tasks. To address the question of whether or not students' scores on the two subtests differed significantly, the scaled scores were assigned percentile ranks. These were obtained from national norms tables two years above grade level for 3rd graders and three years above grade level for 4th and 5th graders. Percentiles were then converted to z-scores using a normal distribution table, and z-scores were compared via t-tests (see Table 1). In all cases, the findings were significant.
Table 1. Study 1: T-tests Using Z-scores Derived from Percentile Ranks
Effect sizes were calculated to determine the practical significance of the Basic Conceptsâ€”Computation difference. They were computed by dividing the difference between the two z-score means by the average standard deviation, thus forming a standard score. For example, an effect size of d = .80 would mean that the Basic Concepts mean is .80 of a standard deviation higher that the Computation mean. By convention, an effect size of d = .20 is considered "small," an effect size of d = .50 is "medium," and an effect size of d = .80 is "large" (Cohen, 1977). All of the effect sizes were in the "medium" range. Thus, the findings in Study 1 can be considered not only significant, but also important.
Study 2: CTPII SubjectsChildren in Texas whose parents had nominated them for participation in a talent identification project were subjects in Study 2. All students had scored at the 90th percentile or higher on at least one of the mathematics subtests of a standardized, grade-appropriate test. During the spring of 1992, 209 students participated, including 2 second graders, 83 third graders, 69 fourth graders, 57 fifth graders, and 6 sixth graders. Sixty-four percent of the subjects were male.
InstrumentationStudy 2 participants took the Level 3 or Level 4 Mathematical Concepts and Mathematics Computation subtests from the Comprehensive Testing Program II (CTPII) battery
Level 3 of the CTPII is designed for third through fifth graders; Level 4 for sixth through eighth graders. The CTPII is a nationally standardized assessment grounded in current classroom content designed to measure the achievement of high-ability first through twelfth graders. (See ERB, 1990 and ETS, 1983 for more specific information.)
ProcedureThird graders took the CTPII Level 3 tests and were compared to fifth graders. Fourth and fifth graders took Level 4 tests and were compared to seventh and eighth graders respectively. The CTPII subsections were administered according to standardized procedures.
Results and DiscussionDescriptive statistics for the third, fourth, and fifth grade participants are available from the authors. To determine whether scores on the CTPII Concepts test were significantly higher than scores on the CTPII Computation test, scaled scores were converted to percentile scores (ETS, 1992), percentile scores were converted to z-scores and the z-scores compared via t-tests (see Table 1). Again, students in all three grades performed significantly better on concepts than computation tests. Effect size analysis in Study 2 indicated a "large" effect at grade three and "medium" effects at grades four and five.
Since the Conceptsâ€”Computation difference was largest for third graders, we examined their test items for possible explanations. The third graders frequently missed many items on fractions and decimals. Other missed items included conversion from centimeters to meters, the zero property of multiplication, and interpretation of a remainder. These topics require direct instruction; it would be difficult for a student to solve these types of problems intuitively.
ConclusionsFor students at each grade level using either the STEP or the CTPII, scores on the mathematics concepts tests were significantly higher than scores on the computation tests. Explanations for this observation include: (1) an underlying cognitive construct causes the difference; (2) talented students are bored with the computation tasks they are required to do repeatedly in typical elementary classrooms, develop the habit of doing calculations quickly, and make mistakes; (3) the differential effect of classroom instruction on opportunities to learn concepts and computational skills (especially at the third grade level); or (4) conceptual material is easier for students to understand intuitively and requires less direct instruction than computational material.
Perhaps the latter explanation is the one we should consider most carefully. Talented students may perform relatively poorly on computation tasks because solving computation problems requires more direct instruction, while conceptual tasks are more intuitive. The lesson we can derive from these findings is that the children are ready for further challenge than they receive in school. They have demonstrated that they are ready for instruction at a high level on computation tasks, and they are ready for instruction at an even higher level on conceptual tasks. They would benefit from enrichment in the regular classroom; being grouped with other talented students; fast-paced mathematics programs; or individualized, mentor-paced instruction (see Lupkowski & Assouline, 1992).
Notes^{1} We thank Julian C. Stanley for his helpful comments on an earlier draft of this paper.^{2} Z-scores were derived from percentile ranks comparing subjects to older students. Third graders were compared to fifth graders in the normative group, fourth graders were compared to seventh graders, and the fifth graders were compared to eighth graders (see Cohn's rule of thumb; Cohn, 1988).
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