Abstract:

In this paper, we present the results obtained from applying an instrument to analyse how high school and college students use the intuitive notion of continuity to visualize and solve problems of geometric-topological nature in the plane and space. The instrument consists of four activities and considers several mathematical concepts such as orientation, deformation, and construction of topological manifolds based on triangles, rectangles, Möbius strip, and a twisted strip. Additionally, certain social aspects are also taken into account, such as academic level and gender, among others. The design of the instrument is inspired by the topological-algebraic demonstrations of certain classical theorems, such as the fundamental theorem of algebra and some theorems of embedding and immersion of real projective spaces.
To engage students in the solution, we introduced the ant “Guita” who has the problem of getting her food. There is a red zone that Guita has to avoid to reach her food. The instrument was applied in three different circumstances: without the support of didactic material and with two types of didactic material, which allows us to study the correlation of continuous solutions to a problem and the use of three-dimensional objects.
In the activity related to triangles, a lower percentage of solutions in three-dimensional space is observed compared to the other activities for which the percentage of solutions in three-dimensional space is statistically the same. For the activities involving triangles and rectangles, a difference in the use of two or three dimension spaces is perceived depending on the type of instrument version provided, which makes evident the need to develop didactic material that aids in a better understanding of continuity and geometric-topological objects in three-dimensional space. Our statistical analysis is essentially based on the application of hypothesis tests, confidence intervals, and correlation.
A noteworthy observation reveals that the prevailing trend among participants is to tackle the problems by employing continuous paths within the original space or through diverse deformations and constructions applied to it.
The study was conducted in various high schools and colleges in the state of Guanajuato, Mexico, with the participation of 175 students.

Keywords:

Continuity, dimension, orientation, Möbius strip, manifolds, correlation.