=Paper=
{{Paper
|id=Vol-2543/rpaper26
|storemode=property
|title=Modeling and Evaluation of the Mathematical Educational Ontology
|pdfUrl=https://ceur-ws.org/Vol-2543/rpaper26.pdf
|volume=Vol-2543
|authors=Liliana Shakirova,Marina Falileeva,Alexander Kirillovich,Evgeny Lipachev,Olga Nevzorova,Vladimir Nevzorov
|dblpUrl=https://dblp.org/rec/conf/ssi/ShakirovaFKLNN19
}}
==Modeling and Evaluation of the Mathematical Educational Ontology==
https://ceur-ws.org/Vol-2543/rpaper26.pdf
Modeling and Evaluation of the Mathematical
Educational Ontology
Liliana Shakirova1 [0000-0001-5758-4076], Marina Falileeva1 [0000-0003-2228-7551],
Alexander Kirillovich1 [0000-0001-9680-449X], Evgeny Lipachev1 [0000-0001-7789-2332],
Olga Nevzorova1,2 [0000-0001-8116-9446] and Vladimir Nevzorov3 [0000-0002-1887-5791]
1
Kazan Federal University
2
Institute of Applied Semiotics of Tatarstan Academy of Sciences, Kazan, Russia
3
Kazan National Research Technical University, Kazan, Russia
liliana008@mail.ru, mmwwff@yandex.ru,
{alik.kirillovich, elipachev, onevzoro, nevzorovvn}@gmail.com
Abstract. In this paper, we discuss the current stage of development of the edu-
cational 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 math-
ematical language processing and an end-user reference educational database.
The ontology is organized in three layers: a foundational ontology layer, a do-
main ontology layer and a linguistic layer. The domain ontology layer contains
language-independent concepts, covering secondary school mathematics curric-
ulum. 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 ar-
rangement 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, OntoMath Edu
1 Introduction
Organization of knowledge for educational purposes requires complementing logical
relations between concepts with prerequisite ones. The concept A is called a prerequi-
site 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 time-
consuming, it is still the most effective approach and can complement automatic ap-
proaches 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).
306
[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 educa-
tional 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 rela-
tionships 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 Onto-
MathEdu ontology. In Section 3 we introduce educational levels of OntoMath Edu. 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 mathemati-
cal 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. Addi-
tionally, the concepts have been semi-automatically interlinked with DBpedia [14] on
the basis of the approach proposed in [15].
Fig. 1. Diameter of a circle concept in the WebProtégé editor
307
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 con-
cepts of the hierarchy of objects.
Fig. 2. The hierarchies of concepts
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 seg-
ment → Curve → Geometric figure on the Plane → Object.
308
Fig. 3. The Diagonal of a trapezoid concept in the hierarchy of objects
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 exam-
ple, the Triangle concept is a kind, because any triangle is always a triangle, regard-
less of its relationship with other figures. Fig. 4 represents an example of a kind con-
cept (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 in-
stance of this concept is related to an instance of the Triangle kind concept by the
relation of ontological dependence.
309
Fig. 4. The Triangle kind concept in the Protégé editor
Fig.5. The Side of a triangle role concept in the Protégé editor
310
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 con-
tains 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 ar-
rangement 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.
Fig.6. An example of a reified relationship concept in the Protégé editor
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” ex-
press the upper limit, and “of f” express the integrated function.
311
Fig.7. Syntactic frame for the “Riemann integral of f over x from a to b” lexical entry
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 anno-
tations, defined by the foundation ontology UFO [16].
3 Educational Levels
In addition to universal statements about mathematical concepts, the ontology con-
tains the statements that are linked to special concepts named viewpoints. The follow-
ing types of points of view are currently being developed:
312
─ Definitions. From different points of view, the same concept can be defined differ-
ently. 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.
Table 1. Educational levels for the Triangle topic in the OntoMathEdu ontology
Educational levels Stages of
3rd 2nd 1st studying Concepts
+ + + 1 Triangle
+ + + 2 Acute triangle,
Obtuse triangle,
+ + + 3 Isosceles triangle,
Equilateral triangle
+ + + 4 Right triangle
+ + + 1 Inscribed triangle,
Subscribed triangle
+ + 1 Medial triangle
+ + 2 Orthogonal triangle,
Triangle with vertices at Euler points
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 7-
8. The third level includes concepts studied in grades 8–9 and in grades with ad-
vanced 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.
313
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 trian-
gle 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 ed-
ucational level, for example, the concept Triangle is connected by the Con-
cept-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 se-
quence 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 soft-
ware tools of the OntoIntegrator system [25]. The OntoIntegrator system is a devel-
opment tool focused on the tasks of automatic text processing using various ontologi-
cal 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 struc-
tural properties of ontologies. When using these tools for the analysis of the OntoMa-
thEdu 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 class-
subclass 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 geome-
try shapes subclass, 5 is the Method subclass, 6 is the Undetectable concepts of plane
geometry subclass.
314
Fig. 8. The diagram of the distribution of objects by subclasses in the Object hierarchy
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 distribu-
tion of concept names by the number of words in their names.
Fig. 9. 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 relat-
ed to the main objects of the subject area. More longer names (more than 5 words)
315
actually refer to the formulations of standard problems and theorems of plane geome-
try. Thus, a feature of the OntoMathEdu ontology is not only the systematization of
elementary geometry objects, but also the systematization of typical problems, theo-
rems, and drawing methods, which is important for application in the education.
Examples of concept names are given in the Table 2.
Table 1. Examples of concept names
Length of name Concept name (English translation)
(in words)
1 Astrolabe; Vector; Hyperbola; Hypotenuse; Homothetic
transformation
2 Axiom of congruence; Vertex of a square
3 Semimajor axis of an ellipse; Interior part of an angle
4 Interior part of an angle
Axiom of a zero-vector postponement
5 Tangent line to a circle
Mutual arrangement of points on a line
6 Cutting square into unequal squares
Tangent segment from a point to a circle
7 Theorem about product of segments of intersecting
chords
Axiom of uniqueness of a vector postponement from
given point
8 Rule of finding the coordinates of the product of a vec-
tor by a number;
Axiom about scalar product of a vector into an equal
vector
9 Inversion of angles between straight lines and circles
property
10 Axiom of distributivity of multiplying vector by a real
number related vector addition;
Problem of costructing a triangle given three sides
11 Axiom of distributivity of multiplying vector by a real
number related numbers addition
Theorem about area of a parallelogram with given two
sides and angle between them
12 Rule allow to find the coordinates of sum, difference
and product by a number using coordinates of vectors
Theorem about area of a parallelogram with given side
and the altitude drawn to this side
14 Problem of constructing a triangle given two sides and
the included angle
15 Problem of constructing a triangle given two angles and
the included side
316
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 high-
frequency concepts (the contextual environment of high-frequency concepts) contrib-
utes to the identification of the most important concepts of academic discipline. Sub-
sequent 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.
Table 2. High-frequency concepts in the textbook by Sharygin
Name Count
Point 1595
Line 846
Triangle 793
Circle 765
Angle 632
Line segment 304
Table 3. High-frequency concepts in the textbook by Atanasyan
Name Count
Point 1652
Line 1061
Angle 858
Triangle 848
Line segment 588
Circle 511
Table 4. Low-frequency concepts in the textbook by Sharygin
Name Count
Ellipse 1
Centimetre 1
Trigonometric equality 2
Plane geometry theorem 2
Property of a triangle 2
Adjacent angles 2
317
Table 5. Low-frequency concepts in the textbook by Atanasyan
Name Count
Heptagon 1
Polyline 1
Miter square 2
Roulette 2
Object 2
Perimeter of a rectangle 2
A general assessment of the frequency distribution of ontology concepts is given in
the Table 7.
Table 6. Examples of concept names
Frequency of using Number of concepts Number of concepts
(interval) in in the textbook by in in the textbook by
Sharygin Atanasyan
1000–1620 1 2
500–999 4 4
100–499 10 13
50–99 25 31
10–49 91 111
5 –9 33 42
1 –4 70 61
The linguistic-statistical analysis of ontology concepts showed that the OntoMa-
thEdu 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 ontologi-
cal concepts in the process of studying geometry.
5 Conclusion
In this paper, we describe educational levels of the OntoMathEdu ontology, and con-
duct 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.
318
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