Concepts for designing knowledge assessment systems in mathematical disciplines under distance education conditions

Abstract

The article presents a novel approach to the design of knowledge assessment systems for students studying the discipline of Differential Calculus. In contrast to most existing sys-tems, which rely on fixed databases of tasks and answers, the proposed solution introduces a dynamic model that automatically generates tasks based on predefined rules using random numbers. Each generated task represents a mathematical function for which the derivative must be calculated. Before presenting the task to the student, the system checks its validity. The answer is submitted not as a numeric value but in the form of a symbolic mathematical expression. This significantly enhances the quality of assessment by enabling evaluation of both correctness and depth of understanding of differentiation methods. Recent advances in symbolic computation, numerical methods, and machine learning have opened new possibili-ties for developing intelligent systems capable of adapting to the individual needs of learners. Existing platforms — such as Moodle, Khan Academy, ALEKS, Proctorio, and ExamSoft — offer various testing frameworks. However, they are primarily based on static question banks, which often limits their flexibility, adaptability, and ability to provide personalized feedback. This paper outlines the limitations of such systems and highlights the necessity for more adap-tive and interactive approaches in mathematical e-learning environments.
The objective of the study is to develop and test an algorithm for task generation, au-tomatic solution computation, comparison with the student's submitted answer, and determi-nation of the answer's correctness. The study also aims to ensure that the method used allows for formula-based responses to be evaluated fairly, even when the same solution is presented in multiple algebraically equivalent forms. The core of the proposed system is built on a RESTful API architecture. This allows tasks to be generated in real-time using randomized parameters and validated programmatically before delivery. The solution also utilizes Math-Jax for formatting and displaying mathematical content in a clear, readable manner within a web interface. An essential component of the system is its capability to interpret symbolic ex-pressions and compare them analytically and numerically. The comparison is done by evalu-ating the difference between the user-submitted derivative and the system-generated solution over a range of values using both symbolic differentiation and numerical finite difference methods.
A specific example is discussed involving the generation of derivative problems from randomly selected elementary functions—polynomial, exponential, logarithmic, and trigono-metric. The structure of such problems is defined by a set of randomly generated parameters, which enables a practically unlimited variety of tasks. The system includes mechanisms to compare student-submitted expressions with the reference solution through evaluation at mul-tiple points, reducing the likelihood of false positives and ensuring objectivity. The findings suggest that this rule-based task generation approach enables a higher degree of individuali-zation and significantly reduces the dependency on static databases. Moreover, assessing symbolic expressions rather than mere numerical values allows the system to better reflect the actual understanding and application of calculus concepts by students. The proposed system was prototyped and visualized through a UML diagram and web interface, demonstrating its practicality and user-centered design. The results of numerical experiments support the vi-ability of the proposed approach. The system's rule-based task generation and symbolic re-sponse evaluation enhance assessment reliability and objectivity. Unlike systems reliant on preloaded question banks, the proposed method provides flexibility, reduces manual work-load, and supports deeper learning. However, the current system is best suited for problems of basic to moderate complexity, as higher-level tasks may require more sophisticated sym-bolic input and parsing mechanisms.