Computational complexity of college math eigenvalue problems

Providing students with suitably complex exercises is crucial to keeping them motivated and leading them to deeper understanding. Making such problems manually takes mathematics teachers´ precious time which can otherwise be used to mentor students. Many software tools and services for automatic generation of math problems are found on the Web, but all of them provide only materials of high-school level or below. In addition, no standardized methods are provided to evaluate and control the computational complexity of generated problems. General concepts of complexity in various forms can be found in standard textbooks. The authors treat a different flavor of complexity, subjective complexity, where complexity is measured by the difficulty that learners feel. The authors proposed a new framework for evaluating computational complexity from learners´ view, aiming to apply our framework to automatic generation of college math problems with controlled computational complexity. To prove the effectiveness of the new concept of complexity from learners´ perspective, we developed some automatic generation tools based on our framework. In this paper, we will present one of the tools that supports automatic generation of eigenvalue problems. Controlling the complexity of eigenvalue problems involves restricting the number of calculation steps, the heights of the involved rationals, and the algebraic number fields appearing in the model solution of the problem. The small-scale experiments showed the relevance of our framework. Our system makes a path to strict quantitative control on various complexity from the learners´ view. Future work includes the application of our methods to other areas such as differential equations, number theory and so on. We also should prove that this system has a positive effect on the engineering education at college level. Therefore, demonstration experiments in classroom will be scheduled in the next step of our research.