OntoMathEdu Educational Mathematical Ontology: Prerequisites, Educational Levels and Educational Projections

Abstract

The work is dedicated to development of prerequisite relationships of the educational mathematical ontology OntoMathEdu. The concept A is called a prerequisite for the concept B, if a learner must study the concept A before approaching the concept B. OntoMathEdu provides two approaches for defining prerequisite relationships: directly by establishing a relationship between concepts and indirectly by arrangement the concepts by educational levels. Prerequisite relationships and educational projections will be used in developing of digital mathematical educational platform of Kazan Federal University.

Keywords

• Prerequisite
• Ontology
• Mathematical education
• OntoMathEdu

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This work is dedicated to development of prerequisite relationships of the educational mathematical ontology OntoMathEdu [ 1-3 ].

This ontology is intended to be a Linked Open Data hub for mathematical education, a linguistic resource for intelligent mathematical language processing and an end-user reference educational database. The ontology is organized in three layers: a foundational ontology layer, a domain ontology layer and a linguistic layer. The domain ontology layer contains language-independent math concepts from the secondary school mathematics curriculum. The concepts are organized in two hierarchies: a hierarchy of objects (such as Line segment, Triangle, Inscribed polygon, or Pythagorean Theorem) and a hierarchy of reified relationships (such as Relationship between a tangent line and a circle). The linguistic layer contains multilingual lexicons, providing linguistic grounding for the concepts from the domain ontology layer. The foundation ontology layer provides the concepts with meta-ontological annotations. The current version of OntoMathEdu contains 896 concepts from the secondary school Euclidean plane geometry curriculum.

OntoMathEdu is a component of OntoMath digital ecosystem [ 4 ], an ecosystem of ontologies, text analytics tools, and applications for mathematical knowledge management, including semantic search for mathematical formulas [ 5 ] and a recommender system for mathematical papers [ 6 ]. OntoMath, in turn, underlines the Lobachevskii-DML digital mathematical library (https://lobachevskii-dml.ru/) [ 7 ] and a digital educational mathematical platform of Kazan Federal University under development.

For the ontology can be used for educational purposes, the logical relations between concepts must be complemented with the prerequisite ones. The concept A is called a prerequisite for the concept B, if a learner must study the concept A before approaching the concept B. For example, comprehension of the Addition concept is required to grasp the concept of Multiplication, and, more interesting, to grasp the very concept of Function, even though, from the logical point of view the later concept is more fundamental and is used in the definitions of the first two.

Prerequisite relationships are used in such tasks as automatic reading list generation [ 8 ], curriculum planning [ 9, 10 ], evaluation of educational resources [ 11 ] and prediction of academic performance [ 12 ]. 2

Prerequisites, Educational Levels and Educational Projections

In contrast to logical relationships between concepts, prerequisite relationships are not universal and are relativized to particular education systems: given two concepts A and B, the prerequisite relation can hold between them in one education system, but doesn’t hold in another. In particular, for the concept A that is prerequisite of the concept B in one education system, the following options are possible with respect to another education system: ─ A is a prerequisite of B too. For example, Circle is a prerequisite of Circumference in both the Russian and the UK education systems. ─ A is a prerequisite of C, and C is a prerequisite of B. For example, in UK education system, Angle is a prerequisite of Alternate interior angles, while in Russian education system, Angle is a prerequisite of Alternate angles and Alternate angles is prerequisite of Alternate interior angles. ─ B is a prerequisite of A. There aren’t examples of this pattern in the current version of OntoMathEdu, but such pair of concepts can be Set and Function, or Circle and Disk. ─ A isn’t a prerequisite of B, because A and B are learned independently. For example, in the UK education system, Plane motion is a prerequisite of Area of a polygon, while in the Russian education system it isn’t, because Plane motion and Area of a polygon are independent. ─ A isn’t a prerequisite of B, because A or B are not studied at all. For example, the prerequisite relation holds between Angle and Complementary angles concepts in the UK education system, but doesn’t hold in the Russian education system, because the Complementary angles concept is not studied in it.

OntoMathEdu provides two approaches for defining prerequisite relationships: a direct and an indirect ones.

Direct approach. According to the direct approach, a prerequisite relationship is established directly between two concepts.

In order to relativize the relation to an education system, we intend using of “Descriptions and Situations” (D&S) design pattern, based on the top-level ontology DOLCE + DnS Ultralite [ 13-15 ]. However, manual annotation of D&S’s is labourintensive task.

As an alternative, the concepts can be linked by a subpropertis of the has prerequisite object property, corresponding to education systems, namely: has prerequisite according to the Russian education system, has prerequisite according to the UK education system and other.

Indirect approach. According to indirect approach, prerequisite relationships are established by arrangement of the concepts by educational levels.

Educational levels are the successive segments of the curriculum of an education system and roughly correspond to education grades. In the UK education system, the education levels are: Key stage 1 (1st–2nd years of study), Key stage 2 (3rd–6th years of study), Key stage 3 (7th–9th years of study), Kеy stage 4 (10th–11th years of study). In the Russian education system, the education levels are: 7 grade, 8 grade, 8 grade (extended), 9 grade, 9 grade (extended), and Additional program. Every concept can belong to one education level of a given education system.

Just like concepts, educational levels are also related by prerequisite relation. The level L1 is called a prerequisite for the level L2, if a learner must study the content of the level L1 before approaching the content of the level L2. In terms of the direct prerequisite relationships between concepts, a prerequisite relationship between two levels can be interpreted as follows: if the level L1 is called a prerequisite for the level L2, then for every concept, belonging to L2 there is a prerequisite concept, belonging to L1.

Educational projections. Arrangement of concepts to educational levels allows to extract a projection of ontology to an education system (educational projection). An educational projection of the OntoMathEdu ontology to education system S is a fragment of the ontology, containing all the concepts, that belong to educational levels of this education system. For example, the Russian education projection of OntoMathEdu consists in the concepts, belonging to grade 7, grade 8, grade 8 (extended), etc. 3

Conclusion

In this paper, we describe two approaches for defining prerequisite relationships: directly by establishing a relationship between concepts and indirectly by arrangement the concepts by educational levels. Arrangement of concepts by educational levels, in turn, allows to extract a projection of ontology to an education system (educational projection). Prerequisite relationships and educational projections will be used in developing of digital mathematical educational platform of Kazan Federal University.

This work was funded by RFBR, projects #19-29-14084 and #18-47-160007. Contribution of A. Kirillovich was partially funded by the state assignment to the Federal State Institution “Scientific Research Institute for System Analysis of the Russian Academy of Sciences” for scientific research

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