Abstract:

This paper is dedicated to the development of a new educational mathematical ontology OntoMathEdu. The ontology is a central component of the digital educational platform under development, and is intended for solving such tasks as: automatic knowledge testing; automatic recommendation of educational materials; semantic annotation of educational texts.

Currently, the ontology covers Euclidean plane geometry‎ only.
In developing this ontology, we draw on our experience in developing the ontology of professional mathematics OntoMathPro. At the same time, the ontology of OntoMathEdu is focused on the educational process, and has the following differences from the ontology of professional mathematics:
1. name of concepts;
2. choice of concepts;
3. conceptualization;
4. the presence of didactic relations between concepts;
5. the presence of points of view.

Ontology OntoMathEdu consists of the following modules: a type hierarchy; a hierarchy of reified relationships; a role hierarchy; and a network of points of view.

The basic ontology hierarchy is a type hierarchy. A type is a concept that is rigid and ontologically independent. So, for example, the concept of “Triangle” is a type, because any triangle is always a triangle, regardless of its relationship with other figures. The top level of the type hierarchy consists of the following concepts: “Plane figure”, “Euclidean plane geometry axiom”; “Euclidean plane geometry theorem”; “Euclidean plane geometry problem”; “Unit of measurement”; “Measurement and construction tool”.

Relations between concepts are represented in ontology in the reified form, i.e. as ontological concepts, not as ontological properties. This, the relationships between concepts are first-order entities, and can be a subject of a statement. The top level of the hierarchy of reified relationships consists of the following concepts: “Plane transformation”, “Metric property of a plane figure”, “Comparison relation between plane figures”, “Arrangement of plane figures”.

A role is a concept that is non-rigid and ontologically dependent. An object can be an instance of a role only by virtue of its relationship with another object. So, for example, the concept “vertex of triangle” is a role, since a point is a vertex not by itself, but only in relation to a certain triangle.

In addition to universal statements, an ontology contains statements relativized to particular points of view. Points of view are represented using the “Descriptions and Situations” design pattern, and are based on the top-level ontology DOLCE + DnS Ultralite. Currently, there are the following types of point of view: definitions and educational levels.

The ontology contains the following relationships: the part-whole relationship; “determined by” relationship; the relation of ontological dependence that binds a role concept to its dependee concept; the “theorem-property” relation; the "theorem-criterion" relation; the “is measured by the formula” relation, which relate geometric figures with their metric properties.

The ontology was evaluated in the task of computer-based testing.

Keywords:

Ontology, mathematical education, Geometry, Planimetry, OntoMathEdu.