Miriam Janíková & Zuzana Pátíková
miriam.janikova@stuba.sk; patikova@utb.cz
Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava & Faculty of Applied Informatics, Tomas Bata University in Zlín
Abstract
This paper explores the automated generation of mathematical problems with a predefined level of difficulty, specifically in the context of finding local extrema of polynomial and rational functions in one or two variables. Using the Moodle platform with the STACK extension, which integrates the Maxima computer algebra system, we demonstrate how technology can enhance the teaching and learning process in higher mathematics. By automating problem generation, educators can efficiently create diverse exercises tailored to students‘ needs while maintaining uniform difficulty.
Our approach focuses on systematically generating problems to ensure pre-defined complexity. For polynomial functions of one variable, tasks are generated in reverse—starting with desired solutions (e.g., integer or rational coordinates) and constructing corresponding polynomials. For rational functions, additional constraints are imposed on the generated parameters to ensure that stationary points have integer coordinates, keeping the tasks of the same difficulty. To achieve this, we classify rational functions based on the structure of their stationary points and describe specific conditions for parameter generation, ensuring that the resulting functions meet the desired criteria and appropriate difficulty levels.
Further, the methodology extends to functions of two variables, where extrema are found by solving systems of equations including partial derivatives. The problem generation process is designed to provide simplified solutions, such as linear or quadratic systems with rational or integer coordinates. This structured approach allows educators to create assignments that are adaptable to different skill levels and learning objectives by controlling the structure of resulting stationary points.
Finally, our experience with the Moodle and STACK system aligns with broader research findings, demonstrating the benefits of automated tools in mathematics education. Feedback from students, particularly in comparison to commercial platforms like Techambition, confirms that interactive and instant-feedback-based practice significantly improves engagement and comprehension. The study highlights the potential of such tools in making mathematics learning more efficient, interactive, and student-centered.
Keywords
Digital Assessment, Automatically Generated tasks, Moodle, STACK, Local Extrema