Comparative Effectiveness of Teaching Numerical Methods Using Chebyshev Polynomials and Conventional Techniques

Abstract

Despite the widespread use of Chebyshev polynomials in numerical analysis, evidence of their instructional effectiveness in undergraduate mathematics remains limited. This study compared Chebyshev polynomial-based instruction, conventional teaching, and a combined instructional approach. A quasi-experimental design involved 60 undergraduate mathematics students assigned to three instructional groups. Data were collected using a validated structured questionnaire assessing conceptual understanding, problem-solving skills, learning attitudes, and perceived instructional effectiveness. Maple 18 was used to implement Chebyshev polynomial procedures and conduct numerical simulations. Educational outcomes were analyzed using descriptive statistics and one-way analysis of variance at the .05 significance level. The combined approach received the highest mean rating for conceptual understanding (M = 3.48, SD = 0.58), although differences among the three groups were not statistically significant, F(2, 57) = 0.662, p = .520, η² = .023, indicating a small effect size. The Chebyshev-based approach showed more favorable questionnaire outcomes for problem-solving and student engagement. Numerical experiments demonstrated close agreement between Chebyshev polynomial approximations and benchmark solutions, with negligible numerical errors and high computational accuracy and stability. These findings indicate the computational promise of Chebyshev polynomial methods and suggest potential instructional benefits, while providing limited evidence of differences in conceptual understanding across approaches. The study highlights the potential value of combining computational methods with conventional instruction and recommends considering spectral methods in undergraduate mathematics curricula, supported by professional development for lecturers.