Modeling and Evaluation of the Mathematical Educational Ontology

Abstract

In this paper, we discuss the current stage of development of the educational mathematical ontology OntoMathEdu, firstly presented by us at INTED 2019 and CICM 2019. This ontology is intended to be used as 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 concepts, covering secondary school mathematics curriculum. The linguistic layer provides linguistic grounding for these concepts, and the foundation ontology layer provides them with meta-ontological annotations. Our current work is dedicated to development of prerequisite relationships of the OntoMathEdu ontology. We introduce these relationships by manual arrangement of the concepts of OntoMathEdu by educational levels. After that, we conduct preliminary evaluation of the ontology. The ontology will be used as a foundation of the new digital educational platform of Kazan Federal University.

Keywords

• Prerequisite
• Ontology
• Mathematical education
• OntoMathEdu

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Organization of knowledge for educational purposes requires complementing logical relations between concepts with 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. Prerequisite relationships are used in such tasks as automatic reading list generation [ 1 ], curriculum planning [ 2, 3 ], evaluation of educational resources [ 4 ] and prediction of academic performance [ 5 ].

While manual annotation of prerequisite relationships by expert is a timeconsuming, it is still the most effective approach and can complement automatic approaches for extraction of these relationships from collections of technical documents ________________________________________________________________________ Copyright © 2020 for this paper by its authors.

Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). [ 6 ], MOOC courses [ 7 ], dependencies among university courses [ 8 ], learning paths of students [ 9 ], Wikipedia [ 10, 11 ] and Linked Open Data [ 12 ].

This work is dedicated to development of prerequisite relationships of the educational mathematical ontology OntoMathEdu [ 13 ]. These relationships are introduced by manual arrangement of the concepts by educational levels.

The main contributions of this paper are two-fold: (i) developing prerequisite relationships of the OntoMathEdu ontology; (ii) preliminary evaluation of this ontology.

The rest of the paper is organized as follows: In Section 2 we describe the OntoMathEdu ontology. In Section 3 we introduce educational levels of OntoMathEdu. And in Section 4 we conduct a preliminary evaluation of the ontology. 2

OntoMathEdu description

In this section, we describe OntoMathEdu, a new educational mathematical ontology [ 13 ]. This ontology is intended to be used as a Linked Open Data hub for mathematical education, a linguistic resource for intelligent mathematical language processing and an end-user reference educational database.

OntoMathEdu 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 description of concept contains its name in English, Russian and Tatar, axioms, and relations with other concepts. Additionally, the concepts have been semi-automatically interlinked with DBpedia [ 14 ] on the basis of the approach proposed in [ 15 ].

Fig 1 represents an example of the Diameter of a circle concept in the WebProtégé editor.

The concepts are organized in two main hierarchies: the hierarchy of objects and the hierarchy of reified relationships (also there are three temporary hierarchies that will be dissolved). Fig. 2 represents the top-level hierarchies and the top-level concepts of the hierarchy of objects.

Fig 3 represents a fragment of the hierarchy of objects, containing the Diagonal of a trapezoid concept and its parents. There are four paths from this concept to the top concept Object, including the following: Diagonal of a trapezoid → Diagonal of a quadrilateral → Diagonal of a polygon → Line segment of a polygon → Line segment → Curve → Geometric figure on the Plane → Object. There are two meta-ontological types of the concepts: kinds and roles.

A kind is a concept that is rigid and ontologically independent [ 16 ]. So, for example, the Triangle concept is a kind, because any triangle is always a triangle, regardless of its relationship with other figures. Fig. 4 represents an example of a kind concept (namely, the Triangle concept).

A role is a concept that is anti-rigid and ontologically dependent [ 16 ]. An object can be an instance of a role class only by virtue of its relationship with another object. So, for example, the Side of a triangle concepts is a role, since a line segment is a side of a triangle not by itself, but only in relation to a certain triangle. Fig 5 represents an example of one of the role concepts, namely the Side of a triangle concept. Each instance of this concept is related to an instance of the Triangle kind concept by the relation of ontological dependence.

Properties of concepts are defined by the axioms, expressed by the formalism of description logics. For example, the description of the Triangle concept at Fig 4 contains axioms, stating that any instance of this concept determined by 1 side and 2 angles, or by 2 sides and 3 points.

Relations between concepts are represented in the ontology in a reified form, i.e. as ontological concepts, not as ontological properties. Thus, the relationships between concepts are first-order entities, and can be a subject of a statement. All instances of a relationship are linked to its participants by object properties.

Fig 6 represent an example of a reified relationship concept, namely, Mutual arrangement between a circumscribed triangle and an inscribed circle. Each instance of this concept is linked to its participants, namely to an instance of the Circumscribed triangle role concept and an instance of the Inscribed circle role concept.

The linguistic layer contains multilingual lexicons under development, providing linguistic grounding for the concepts from the domain ontology layer.

A lexicon consists in (a) lexical entries, denoting mathematical concepts; (b) forms of lexical entries; (c) syntactic trees of multi-word lexical entries, (d) and syntactic frames. A syntactic frame contains a subcategorization model for a particular lexical entry and its mapping to parameters of a corresponding math concept Fig 7 represents an example of the “Riemann integral of f over x from a to b” lexical entry, where the “from a” dependent constituent expresses the lower limit of integration, “to b” express the upper limit, and “of f” express the integrated function. The lexicons are expressed in terms of Lemon [ 17 ], LexInfo, OLiA [ 18 ] and PREMON [ 19 ] ontologies. According to the project, the lexicons will be interlinked with the external lexical resources from the Linguistic Linked Open Data (LLOD) cloud [ 20 ], first of all in English [ 21, 22 ], Russian [ 23 ] and Tatar [ 24 ].

The foundation ontology layer provides the concepts with meta-ontological annotations, defined by the foundation ontology UFO [ 16 ]. 3

Educational Levels

In addition to universal statements about mathematical concepts, the ontology contains the statements that are linked to special concepts named viewpoints. The following types of points of view are currently being developed: ─ Definitions. From different points of view, the same concept can be defined differently. These different definitions can determine some concepts through different systems of other concepts. ─ Educational levels. To implement the principle of consistency and continuity in teaching concepts in the field of geometry, we introduced the notion of educational level and applied this to the presentation of ontology concepts.

Let us consider consistency in the study of the Triangle topic. This topic is studied in grades 7–9, including grades with advanced math program.

Table 1 presents the first level of studying definitions of the Triangle concept in a grade 7 (this is basic level). This level includes four stages of studying this topic in grade 7. At the second level (in a grade 8), the Triangle concept is expanded by the two new concepts (Inscribed triangle and Subscribed triangle). At the third level (in advanced course), other types of triangles defined in the ontology are also considered.

This means the possibility of a parallel study of these pairs of concepts that can be arranged in any sequence and it will be better to study these concepts simultaneously by comparing their properties. The second level includes concepts studied in grades 78. The third level includes concepts studied in grades 8–9 and in grades with advanced math program and also the concepts of previous levels. To take into account the methodological features of teaching mathematics, it is necessary to determine object properties in the OntoMathEdu ontology, which we shall conditionally name didactic relations.

In the current version of the OntoMathEdu ontology the following didactic relations are defined: 1. The Studied simultaneously relation connects the concepts that should be studied together, for example, the Line and Ray concepts; 2. The Studied later relation (the inverse relation of the Studied earlier). For example, the Isosceles triangle concept is studied later than the Acute triangle concept. The Studied later relation as well as its inverse relation, are transitive, therefore we can build the sequences of the Studied later relations, which form a certain sequence of concepts in learning; 3. The Concept-level relation determines the relevance of the concept to the educational level, for example, the concept Triangle is connected by the Concept-level relation with a stage 1 of the first educational level (see Table 1). The Concept-level relation is used as a criterion for building a learning sequence of concepts. 4

Analysis of the OntoMathEdu ontology

In this section, we report the results of a preliminary evaluation of the OntoMathEdu ontology.

The structural properties of this ontology were analyzed using the analytical software tools of the OntoIntegrator system [ 25 ]. The OntoIntegrator system is a development tool focused on the tasks of automatic text processing using various ontological models. The main functional capabilities of this system are: ─ designing ontological models of arbitrary structure with wide data visualization capabilities; ─ development of scientific applications related to text processing; ─ natural language processing based on ontological and linguistic models.

The analytical tools of the OntoIntegrator system allow us to explore various structural properties of ontologies. When using these tools for the analysis of the OntoMathEdu ontology, quantitative and qualitative results were obtained that made it possible to identify some structural features, as well as to identify specific steps for improving the ontology.

In total, 776 concepts, 5 hierarchies, 2338 text inputs of concepts, 836 classsubclass relations were defined in the OntoMathEdu ontology.

The Fig. 8 represents a diagram of the distribution of objects by subclasses in the Object hierarchy, here 1 is the Assertion subclass, 2 is the Geometric figure on a plane subclass, 3 is the Task subclass, 4 is the Tool for measuring or drawing geometry shapes subclass, 5 is the Method subclass, 6 is the Undetectable concepts of plane geometry subclass. As already noted, the OntoMathEdu ontology was built manually based on school textbooks. The general names were used to denote the names of important concepts (problems, theorems, methods, etc.). Below the results of linguistic analysis of the names of ontological concepts were carried out. Fig. 9 shows the frequency distribution of concept names by the number of words in their names. The most frequent classes are two- and three-words concept names which are related to the main objects of the subject area. More longer names (more than 5 words) actually refer to the formulations of standard problems and theorems of plane geometry. Thus, a feature of the OntoMathEdu ontology is not only the systematization of elementary geometry objects, but also the systematization of typical problems, theorems, and drawing methods, which is important for application in the education.

Examples of concept names are given in the Table 2.

When developing an ontology for education, it would be useful to have data about the significance of concepts in the training course. Data on the frequency of concepts in the textbooks by Sharygin [ 26 ] and Atanasyan [ 27 ], the relationships of highfrequency concepts (the contextual environment of high-frequency concepts) contributes to the identification of the most important concepts of academic discipline. Subsequent ranking concepts in terms of their significance may be useful for testing. High-frequency concepts (with frequency of occurrence) for two school geometry textbooks are given in the Table 3 and the Table 4, and low-frequency concepts are given in the Table 5 and the Table 6.

The linguistic-statistical analysis of ontology concepts showed that the OntoMathEdu ontology not only contains a systematization of the main objects of the subject area, but also includes a taxonomy of the main typical problems studied in the school geometry course. The latter circumstance makes this resource especially useful for use in education. Frequency analysis of educational texts allowed to identify the most important concepts of ontology, which can subsequently be used in ranking ontological concepts in the process of studying geometry. 5

Conclusion

In this paper, we describe educational levels of the OntoMathEdu ontology, and conduct its preliminary evaluation.

The ontology will be used as a foundation of a new digital educational platform under development at Kazan Federal University

This work was funded by RFBR, projects #19-29-14084, and by the Government Program of Competitive Development of Kazan Federal University.

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