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The problem of visualization of mathematical concepts is not new. Mathematicians have been debating various aspects of this problem, using both philosophical foundations and practical considerations, since the 17th century. The authors provided an overview of sources on mathematical visualization. In the genesis of ideas about the role of visual images in mathematics, extreme positions clashed: from absolutization of visualization to complete denial of its necessity. From the authors’ point of view, balanced and comprehensive solutions to the visualization problem are most in line with the truth. Moreover, the development of computer technologies has provided new opportunities and more advanced tools for visualization of mathematical concepts, which partly contributed to the “rehabilitation” of the ideas of mathematical visualization in the eyes of supporters of “pure” mathematics. The authors have identified a sore point of the most heated discussions. This is a visualization of some Calculus concepts that have a high degree of abstraction. The “ε–δ” language is one of such concepts. Using the example of 5 visual images of different cases of the limit of a function at finite and infinite points, the authors conducted a study to find out whether subsequent visualization really helps improve students’ understanding of the definitions of the limit of a function in the “ε–δ” language. The study involved 152 first-year bachelor’s students in the fields of “Management” and “Construction”. The authors analyzed the students’ self-assessment in terms of understanding the aforementioned definitions of the limit of a function. The results of the study show that, according to students - future engineers, visualization helps them better understand the concept of the limit of a function. This hypothesis can be accepted at a significance level of 0.01, that is, with a reliability of 99%. At the same time, the authors did not find sufficiently convincing confirmation of this hypothesis for students - future managers. Both categories of students reported an improvement in their understanding of the definition of the limit of a function when using visualization with animation. The authors identified the most suitable types of definitions of the limit of a function for visualization using Kendall’s coefficient of concordance.

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Visualization Problems in University Mathematics Courses: The Example of the Limit of a Function

  • 会议论文

  • 发表日期：

    2026年

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  • Victor Krasnoshchekov,
  • Natalia Semenova

摘要

The problem of visualization of mathematical concepts is not new. Mathematicians have been debating various aspects of this problem, using both philosophical foundations and practical considerations, since the 17th century. The authors provided an overview of sources on mathematical visualization. In the genesis of ideas about the role of visual images in mathematics, extreme positions clashed: from absolutization of visualization to complete denial of its necessity. From the authors’ point of view, balanced and comprehensive solutions to the visualization problem are most in line with the truth. Moreover, the development of computer technologies has provided new opportunities and more advanced tools for visualization of mathematical concepts, which partly contributed to the “rehabilitation” of the ideas of mathematical visualization in the eyes of supporters of “pure” mathematics. The authors have identified a sore point of the most heated discussions. This is a visualization of some Calculus concepts that have a high degree of abstraction. The “ε–δ” language is one of such concepts. Using the example of 5 visual images of different cases of the limit of a function at finite and infinite points, the authors conducted a study to find out whether subsequent visualization really helps improve students’ understanding of the definitions of the limit of a function in the “ε–δ” language. The study involved 152 first-year bachelor’s students in the fields of “Management” and “Construction”. The authors analyzed the students’ self-assessment in terms of understanding the aforementioned definitions of the limit of a function. The results of the study show that, according to students - future engineers, visualization helps them better understand the concept of the limit of a function. This hypothesis can be accepted at a significance level of 0.01, that is, with a reliability of 99%. At the same time, the authors did not find sufficiently convincing confirmation of this hypothesis for students - future managers. Both categories of students reported an improvement in their understanding of the definition of the limit of a function when using visualization with animation. The authors identified the most suitable types of definitions of the limit of a function for visualization using Kendall’s coefficient of concordance.