Euler’s Method with a double step in teaching the numerical solution of differential equations: programming, computation, visualization

Abstract

The article presents the results of a study aimed at developing and testing a methodology for fostering computational thinking in students of pedagogical specialties through the integrated use of programming, numerical analysis, and visualization. The central component of the research is the implementation of Euler’s method with a double step in the Maple environment as a means of studying the Cauchy problem for an ordinary differential equation. The methodology involves calculating an approximate solution with two different step sizes, constructing graphs, estimating the error using Richardson’s rule, and comparing the results with a reference value. Leveraging Maple’s built-in programming language and visualization capabilities, students implemented the algorithm, interpreted the results, and created graphical objects illustrating the dynamics of the approximate solution and its accuracy. The findings indicate that this form of learning organization fosters the development of flexible computational skills, the ability to perform program-based modeling, and the capacity for critical error analysis. An analysis of typical student errors identified key difficulties related to array handling, correct indexing, and interpretation of residual terms. The discussion of results is carried out in comparison with contemporary international approaches, particularly with studies emphasizing the importance of visualization, algorithmic thinking, and independent implementation of computational methods in programming education. The methodology shows potential for scaling within educational programs aimed at preparing specialists in STEM fields and demonstrates the effectiveness of integrating computer algebra systems into courses on mathematical informatics.