Abstract
Although the problems of obtaining educational information in e-learning have been successfully resolved, the testing of obtained knowledge still requires a solution. Principles of knowledge assessment by means of tests in mathematics subjects administered in e-learning environment have been developed. An algorithm of component analysis and formula expression indexing for automatic analysis is suggested. This approach allows diversifying the types of questions for consequent verification of knowledge in computer systems. Modification of the formula editor provides the creation of templates for formula entering and the use of indexing algorithm of the expression elements. This makes it possible to avoid ambiguity of responses entered by students. Tasks with computer verification of an analytical symbolic expression can be applied to automate the verification of skills and knowledge in natural sciences, in particular, physics, mathematics, by appropriate modification of the template.
 Development of information technologies led to the development of new models of teaching and learning, especially in the field of mathematical sciences. It is shown that during experimental laboratory work in engineering subjects especially effective are those activities that combine the use of knowledge and skills acquired during the study of other related disciplines. It is shown that research based on interdisciplinary interaction has great potential.
 In addition, the application and advantages of expert systems in open education to improve the teaching-learning process are discussed. It is proposed to use expert systems in the designing of laboratory works and tests. It is also shown how expert systems should be used to optimize the process of creating adaptive type laboratory works and to formulate individual test tasks. The application of the expert system is illustrated on the example of Discrete Mathematics.
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