[orcid=0000-0003-3204-238X<EMAIL_ADDRESS>]

[orcid=0000-0001-8116-9446<EMAIL_ADDRESS>]

# Ontomathedu Ontology Enrichment Method

Konstantin S. Nikolaev  Federal State Institution of the Federal Research
Center NIISI RAS  Kazan Federal University  Olga A. Nevzorova

###### Abstract

Nowadays, distance learning technologies have become very popular. The recent
pandemic has had a particularly strong impact on the development of distance
education technologies. Kazan Federal University has a distance learning
system based on LMS Moodle. This article describes the structure of the
OntoMathEdu ecosystem aimed at improving the process of teaching school
mathematics courses, and also provides a method for improving the OntoMathEdu
ontology structure based on identifying new connections between contextually
related concepts.

###### keywords:

OntoMathEdu ontologysoftware serviceeducational applicationontology enrichment

##  1 Introduction

The ontological approach is often used to organize training in a more
structured format, as well as to help the student form a full picture of the
current discipline, to more clearly understand the existing patterns and
relationships between concepts in the problem area under consideration. The
basis of such an educational infrastructure can serve as a qualitative
ontology. Currently, there are many training systems that use various
ontologies to organize subject data. For example, Lutoshkina et al. in the
work [1], the ontology of the subject area is used to systematize the
multimedia content of electronic courses (the connections between the concepts
of ontology are used to construct an integral structure of the lesson). In
addition, Stancin et al. [2] provide a detailed overview of the use of
ontologies in education, among which the most interesting for us are
ontologies describing the subject under study and e-learning services. Here
are some examples of such solutions:

1\. LiReWiapproach (Conde et al. [3]). Educational ontology is used to
describe the topics to be mastered by students and the pedagogical relations
between the topics.

2\. MOOC-KG (Dang et al. [4]). To build a knowledge graph, an ontology is used
that models knowledge about online learning resources.

3\. MKMSE (Martinez-Ramirez et al. [5]). Ontology is used to store
mathematical knowledge.

4\. Ontology is used to represent the semantic description of document
resources and the relationships between document resources and other types of
resources (Hai [6]).

5\. CALMS (Erazo-Garzon et al. [7]). Ontologies are used to define concepts
and semantic relations between academic information and conceptual aspects.

6\. Ontology is used as an expression of knowledge to support the ends of the
content of education. (Kubekov et al. [8])

##  2 Structure of the OntoMathEdu ecosystem

The article discusses the structure and main tasks of the infrastructure of
software components built around the ontology of school mathematics
OntoMathEdu [9, 10]. Ontology OntoMathEdu is a knowledge system that describes
a set of concepts, relationships between them and statements within the
framework of a school course in planimetry. Figure 1 shows the composition of
the OntoMathEdu digital ecosystem. The ecosystem consists of the following
components.

Figure 1: Components of the OntoMathEdu ecosystem

  1. 1.

Intelligent digital educational platform for school mathematics. This
component occupies a central place in the OntoMathEdu ecosystem. Its main
purpose is the application of semantic and ontological technologies in
teaching school mathematics.

  2. 2.

Collection of questions. This collection is used as an input data set for
various components of the ecosystem, in particular, for the module "Ontology
Enrichment Services", as well as in the central component "Intelligent Digital
Educational Platform for School Mathematics". The collection is constantly
updated.

  3. 3.

Formula storage. This component contains formulas extracted from school
geometry textbooks. Formulas are presented in various formats (plain text,
LaTeX, OpenMath). We plan to use this repository as an auxiliary data set for
building test tasks and other information components.

  4. 4.

Digital educational resources. This component combines all auxiliary data
sources located on the Internet (for example, open educational databases,
resources for the construction of geometric shapes, etc.)

  5. 5.

OntoMathEdu ontology. The OntoMathEdu ontology is a reflection of the level of
knowledge corresponding to the level of school mathematics. It serves as the
main repository of concepts and their relationships that are used by other
services.

  6. 6.

Ontology enrichment service. This component includes a set of methods that are
used to clarify the relationships between the concepts of ontologies and
improve the horizontal connectivity of ontologies.

  7. 7.

Test generator. This component is used to automatically create new test tasks
based on the analysis of the structure and concepts of existing tasks.

  8. 8.

Search by semantic formulas. A search module that performs semantic search of
mathematical texts present in the ecosystem. As a result of the search, it
returns a set of texts containing the requested concept and offers a list of
related concepts for further navigation through the text database.

  9. 9.

Recommendation system. This module helps the user to study various sections of
the training course, providing an individual learning trajectory.

##  3 Ontology enrichment service

The ontology OntoMathEdu is currently in the active stage, and contains more
than 900 concepts and more than 20 relationships. The task of introducing new
relations between concepts is urgent. We have developed a method for
introducing new relations between concepts based on the use of a collection of
questions. To find new connections between concepts, we use a collection of
test questions on school geometry. Our main idea is as follows. Each question
contains some basic concepts of ontology, about which the question is asked.
We assume that the answer to this question can be found with the help of
ontology, in which there is a chain of concepts of limited length connecting
the original concepts. In addition, the search for such connections allows us
to evaluate the structural properties of the ontology, namely the horizontal
connectivity of the resource.

The source of the texts is a collection of school test tasks on geometry with
the markup of nominal groups. These nominal groups will act as candidates to
search for the corresponding concept in the OntoMathEdu ontology. A noun group
of the first type (Noun phrase, NP) is understood as a syntactic construction
in which there is a vertex (noun) and dependent words. An example of NP is a
"finite set of points", where the real "set" is the vertex of a given phrase
with two dependent words. A nominal group of the second type (Prepositional
phrase, PP) is a phrase in which the main word is a preposition. An example of
PP is the phrase "up to two characters after the dot", where the preposition
"to" is the vertex of the nominal group.

Then, for each pair of concepts, all existing relationships are searched, and
the search is performed twice: in the first case, only hierarchical
relationships (rdfs:subClassOf and ome:hasChild) are used as relationships
between concepts, and in the second case, all possible relationships are used.
We call the first type of relationship as a "hierarchical relationship", and
the second as a "complete relationship". The presence of an optimal
relationship is an event in which the shortest complete link is shorter than
the hierarchical one. This connection is optimal, because any pair of concepts
is somehow connected by a hierarchical connection (we can always climb from
the first concept to the node shared with the second concept and descend to
the second concept), but this connection will not be the shortest if there are
direct connections between the concepts under consideration.

The resulting set of shortest links with an indication of optimality is sent
for analysis to a group of experts for manual updating of the ontology. It can
also be used as auxiliary information when checking test tasks.

Figure 2: Block diagram of the ontology enrichment method

Figure 2 shows a block diagram of the task of enriching ontology through
method names. The following sections will describe the main stages of the
algorithm for finding optimal connections between concepts in the texts of
questions with the name of the method. Here are some comments on Figure 2.

  1. 1.

Extraction of NP and PP. At this stage, we need to select a set of NP and PP
in the texts of questions and text variants of answers (this means that we do
not consider answer options for computational problems and for problems with
the choice of an answer presented as a number or a set of characters that are
not a dictionary unit). Thanks to the already preliminary markup of the text
by the nominal groups NP (TERM1 tag) and PP (TERM2 tag), this stage is reduced
to highlighting the contents of these tags using the standard xml library for
Python 3.

  2. 2.

Tokenization and normalization of phrases from nominal groups (NP and PP):
TokenizeAndLemmatizeTexts() function. Tokenization of phrases received at the
last stage is performed using the nltk library. Normalization of the resulting
set of tokens is carried out using the pymorphy2 library, which has the
necessary functionality to bring words to a normal form. For example, for the
word "triangles", the normal form is "triangle".

  3. 3.

Tokenization and normalization of ontology concepts:
TokenizeAndLemmatizeOntology() function. Tokenization of phrases received at
the last stage is performed using the nltk library. Tokens are normalized
using the pymorphy2 library.

  4. 4.

Match search: Match() function. Next, we need to find similar sequences of
tokens in the question text and names of ontological concepts, using a
modified Jacquard measure to work with a sequence of strings. The sequence
element in our case is a lemma from the phrases NP or PP. However, since the
words in the compared sequences may not completely match, we also apply the
Jacquard measure when searching for common words between sequences. In this
case, the elements of the sequence will be individual letters.

When performing this method, we will introduce two thresholds: a lower
accuracy threshold at which two words are considered the same, and a lower
accuracy threshold at which two sequences of words are considered similar.
Their values were obtained experimentally on a specific set of questions.

  5. 5.

Search for connections through hierarchical relations: the
SearchHierarchicalConnections() function. In this method, we look for
connections between pairs of concepts located in the same question, using only
hierarchical relationships (rdfs:subClassOf and ome:hasChild) as relationships
between neighboring concepts. This type of connection is guaranteed to be
present between any two concepts in the ontology, but it is not necessarily
optimal. We select the shortest connection from the resulting set of this
method.

  6. 6.

Search for connections through any relations: Search All Connections(). In
this method, we look for connections between pairs of concepts located in the
same question, using all types of relations as relations between neighboring
concepts. We select the shortest connection from the resulting set of this
method.

  7. 7.

Comparison of shortest distances with Previous methods: CompareDistances()
function. In this method, we check a link using all types of links, which is
shorter than the corresponding link using only hierarchical links. If such a
connection is found, then the optimal connection between a pair of concepts is
confirmed. Table 3 shows examples of optimal connections found in ontology,
indicating the lengths of these connections.

##  4 Conclusion

This article examines the structure of the OntoMathEdu ecosystem, designed to
support the process of personalized teaching of school mathematics, and also
provides an algorithm for enriching ontology, based on the contextual
proximity of concepts in geometry tasks, which contributes to improving the
horizontal connectivity of ontology.

Future work is related to the implementation of the personalization component
in all implemented test generators, as well as the overall development of
these components. In addition, it is planned to develop a component for the
automatic analysis of a detailed response by drawing up a framework (scheme)
of the solution, and filtering tasks by types of these frameworks.

Table 1: Examples of optimal connections found Source theorem |  First concept |  Second concept |  Link length
---|---|---|---
If the diagonals of a given quadrilateral are perpendicular, then this quadrilateral has: 1) equal middle lines, 2)<perpendicular> <middle lines> 3) equal opposite angles |  Perpendicular |  Triangle middle line |  3
If a circle can be described around a quadrilateral, then 1) the center of the circle is equidistant from the sides of the quadrilateral 2) the bisectors of the angles of the quadrilateral intersect at one point 3) the sum of <opposite angles> is equal to <right angle> |  Opposite corners of a quadrilateral |  Right angle |  4
<Segments> connecting the midpoints of the sides of any quadrilateral and the segment connecting the midpoints of <diagonals> |  The middle of the segment |  Diagonal of the polygon |  3
The sum of two axial symmetries with intersecting axes can be replaced by <rotation> around the <point> at which the ax-es intersect at an angle equal to the angle between the axes |  Point |  Rotation angle |  4

###### Acknowledgements.

The article was prepared within the framework of the government task of the
Federal State Institution of the Federal Research Center NIISI RAS for
2022-2024 (topic No. 0580-2022-0014 (FNEF-2022-0014)).

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