A Novel Method for Teaching Geometry Concepts using Ontologies

This research explores using ontology to enhance geometry education, addressing students’ difficulties in understanding the subject. Traditional teaching methods do not effectively utilize computer capabilities. The study examines how an ontology, encompassing the Hellenic Curriculum for second-grade Junior High School and additional materials, can improve teaching methods. Developed with student input, the ontology was applied over a school year. Results indicate that this approach facilitates better understanding and application of geometric concepts, offering a structured framework, organizing concepts, and creating personalized, interactive learning experiences for both students and teachers.

Introduction

The study of Geometry in junior high school is a fundamental aspect of mathematics education, yet students often struggle to grasp its abstract concepts and their interconnectedness. This challenge highlights the need for innovative teaching tools that enhance comprehension and engagement. While technological advancements in education, particularly Information and Communication Technologies (ICT), have shown promise in fostering interest and improving outcomes in subjects like Geometry [1], [2], there remains a gap in the application of structured knowledge systems, such as Ontologies, to support learning. Current practices largely focus on tools like blogs, wikis, and interactive applications, but the potential of machine-supported systems for knowledge organization and reuse, such as Semantic Web Ontologies, is underexplored in mathematics education [3].

This paper presents the development of an Ontology specifically designed for junior high school Geometry. It aims to facilitate the visualization of relationships between geometric shapes, their properties, and interconnections. Also, to provide a constantly updated pool of examples and exercises to support effective learning. And even more importantly to encourage constructivist learning by enabling students to actively engage with mathematical concepts through hypothesis creation, argument development, and collaborative exploration [4].

This approach states that knowledge involves acquiring new understanding, behaviors, skills, values, and preferences [5]. Unlike traditional methods that emphasize on problem-solving techniques, this study focuses on helping students comprehend geometrical terms and concepts through inquiry-based and constructivist methods. Teachers, on the other hand, acting as facilitators, can leverage Ontologies to analyze teaching materials, improve learning outcomes, and foster knowledge exploration beyond behavioral models [6].

Ontologies have strongly demonstrated their value as cognitive tools in e-learning, enabling the representation of complex relationships, fostering knowledge sharing, and supporting semantic web applications. As they provide intuitive assistance, they promote active student engagement, self-directed learning, and the construction of knowledge driven by curiosity rather than reward [7]. Furthermore, Ontologies assist in educational assessment by revealing hidden conceptual connections and axioms within a given domain. The latest makes them particularly suitable for structuring and enriching the Geometry curriculum [7].

This study draws on fundamental principles of Geometry and demonstrates how they can be integrated into an Ontology that aligns with the junior high school curriculum. The approach also considers the broader cognitive and technological benefits of Ontologies, including their role in organizing, visualizing, and navigating knowledge [7]. In doing so, this research contributes to addressing the underrepresentation of geometrics’ mathematical semantics on the Web of Data and does highlights the potential of Ontologies to transform Geometry education.

The remainder of this paper is organized as follows: First, we discuss the use and benefits of Ontologies in education, providing several examples of implemented Ontologies. Next, we describe the background of the students involved in the specific implementation detailed in this paper. We then outline the steps taken to develop the Geometrical concepts included in the Ontology. Following this, we examine the technological foundation of the Ontology, including the tools, programming language, database, and knowledge organization system used. We proceed with an evaluation of this work and conclude with a discussion of its limitations and potential future directions.

Materials and Methods

Ontologies and their Use in Education

Since an Ontology is an educational domain of knowledge that describes the rich network of relationships between the concepts it comprises [8], there are several Ontologies in the education sector that support the teaching process. As proposed by [9] the use of Ontologies for teaching the C programming language to undergraduate students. The Ontology of [10] provides the LUISA (Learning content management system Using Innovative Semantic Web services Architecture) framework which includes a Learning Object (LO) annotation tool, extending a Learning Object Metadata Repository Ontology. The [11] examine in detail approaches used in an Intelligent Web Teacher for knowledge representation and management through Ontologies. This intelligence creates personalized learning paths based on the individual user preferences including the present knowledge state and the preferred didactic approach. An Ontology proposed by [12] presents a software framework for eLearning systems that are Ontology driven, based on an Ontology for designing competency-based learning and knowledge management applications. A different use of Ontologies, not focused on the learning itself but on the mapping of curricula of EU countries to resources and competencies, is covered in [13]. The approach of [14] is quite interesting, as they applied seven Ontologies of various aspects on the aquatic environment used to support educational contents annotation and retrieval while Semantic web services offered a vast amount of knowledge and information to the AquaRing European Commission funded project.

Another application of Ontologies in education is in content management. Ontologies can be used to describe the domain knowledge of educational resources and provide a common vocabulary for indexing and searching.

An Ontology could also serve as a repository of information or a database of domain knowledge and empirical experiences (strategies, common mistakes and techniques to use) [15]. The integrated use of the GEOGEBRA software in the Ontology can make the learning process even more active, allowing for an interaction between teachers and students [16]. This leads to the proposition that when students use Ontologies, they increase their skills and knowledge and help them to develop further, while at the same time allowing them to combine resources and develop their progress along with the expectations of their teachers.

It has been well-documented that Ontologies, which fall in the general area of ICT tools, can provide students with an interactive learning process student. It has been shown that they have a positive effect on learning geometrical shapes even when used in the first grade and kindergarten levels, using the background of Realistic Mathematics Education’ (RME) for Geometry concepts theory [17]. It has been also shown that students possess a positive critical thinking disposition in general but have a weakly positive inclination to truth-seeking and systematicity. With more time spent in the digital learning environment, the critical thinking disposition gets stronger, especially in the dimension of analyticity and systematicity [18].

The visual representation of an Ontology can significantly assists mathematics teachers in understanding key concepts in a class, particularly in Geometry, where grasping the relationships between geometric shapes and properties is challenging without a clear framework. An Ontology serves as a visual tool that clarifies these connections, enhancing Geometry teaching, especially for teachers with limited mathematical expertise. Integrating Ontologies into the Geometry curriculum can improve teaching effectiveness. Our goal is to create an efficient environment that enhances teaching and helps students deepen their understanding of geometric theory in junior high school.

Geometric Theory Covered by the Circle’s Ontology

In this work, our primary purpose is to give instructors and students with an efficient and effective environment for the teaching of geometry during the second year of junior high school by utilizing the idea of ontologies. There is no requirement that the pupils have any prior knowledge of the theory being taught. They advance their knowledge as an Ontology is being built up. At the end of this process, the entire theory as mentioned in the relevant grade curriculum instructions will be covered thoroughly. The provision of the theoretical part is accompanied with the provision of a variety of examples and exercises which are relevant to the theory embedded within the Ontology environment and come as an integral part of it.

The basic geometric concepts of the relevant covered theory are presented in the order of appearance in the steps taken to build the Ontology. More specifically, since at this level many basic concepts of the geometric theory have as a key point the Circle, over the core concept of the Circle other concepts are built up using ideas from the concept of the Ontology. These concepts include the chord and its arc; The maximum length chord which is the diameter; The inscribed and central angles created initially by the radii of the circle, dividing it into arcs that have these angles; and the regular polygons created in conjunction with the epicenters, inscribed angles and their arcs. We will include in our Ontology the role of the diameter in the creation of the right angle and the right triangle. Finally, based on the concept of the Circle, the theory covered extends to that of the surface areas, trigonometry and the volume of a cylinder, thus covering three-dimensional shape objects as well.

Basic Circle’s Concepts

The starting point for the concept of the Circle was based on the concepts of the center and its radius since the center and the radius provide the geometric definition of the Circle. Other simple concepts such as the diameter, the arc and the cord complete the elementary concepts of the Circle. Then, the concepts of the inscribed and of the central angles were introduced and properly connected. After giving the students, a typical example of an inscribed and a central angle over a common arc, the relation connecting these two angles was made clear (Fig 1). More specifically, the relationship between the two types of angles, the inscribed and the centralis demonstrated as in Fig. 1, where the cursor slides over the horizontal line, sets the degrees of central angle, changing its arc and the degrees of the inscribed angle. The ratio between the degrees of inscribed to central angle is displayed on the XY plot which reveals that the central angle is twice than the inscribed over the same arc.

Fig. 1. Inscribed and central angle connected.

At the end, all the basic concepts were connected through their properties and characteristics as it is thoroughly described in the authors previous publications [1], [19]. This is that an inscribed angle based on the arc of a central angle is half of the size of the central angle, and there is no limitation on which point of the periphery the two legs of the inscribed angle are crossing to create the vertex.

The starting basic concepts of the Ontology are shown in Fig. 2, where B stands for the broader concept and N for the narrower concept of the relationship which are explained later on the Technological Background of the Implementation of the Circle’s Ontology chapter. In Ontology, the terms “broader” and “narrower” are utilized to denote the hierarchical relationships that exist between concepts. A broader concept is one that encompasses a wider range of sub-concepts or specific instances, while a narrower concept refers to a more specific subset of the broader concept. This taxonomy is particularly valuable for organizing and categorizing knowledge in a methodical and coherent manner, thereby facilitating comprehension and navigation of intricate information by both humans and machines. For example, the concept of a circle is broader than that of circle radius, as it incorporates various sub-concepts within its domain. Such hierarchical relationships are instrumental in constructing a framework for comprehending the relationships between concepts and the world around us, as exemplified by the discipline of Geometry in this specific case.

Fig. 2. Basic concepts of the Circle.

In order to be more specific a starting concept, for example, is the center of the Circle and the radius. Geometrically speaking, O(K, ρ) is how a circle is defined where K is the center and ρ is the radius. The arc and the chord are the first next concepts to come, focusing on the circle itself only (Fig. 2).

Regular Polygons

Regular Polygons were constructed with the assistance of the Circle. Six equal central angles that define equilateral triangles fit easily within a Circle. The vertices of these triangles, except for those on the center of the Circle, when connected with a straight line, form a Regular Polygon. The internal angle of the Regular Polygon can be calculated because it is a central angle of the Circle. This calculation is based on the number of the polygon’s sides. Moreover, the angle of the polygon between two adjacent sides can be calculated. This angle is also an inscribed angle of the Circle (Fig. 3).

Fig. 3. Exploring polygons.

The isosceles and equilateral triangles and the square are three particular cases of Regular Polygons. Inverting the relation between the Circle and the Regular Polygon, the concept of the circumcircle is also unveiled. These four last concepts, along with the basic concept of a Regular Polygon, are added on the Ontology of the Circle (the corresponding part on Fig. 11).

Right Triangles

As in the case of the Regular Polygons, we took advantage of the concepts of the inscribed and of the central angle; in the Right Triangles we used an inscribed angle to attach that concept to the Circle’s Ontology. We created an inscribed angle of a semicircle arc and the diameter of the Circle to be its chord. The shape was a Right Triangle. The cord of this angle was the hypotenuse of the Right Triangle. Its properties were related to the diameter of the Circle. The midline of the hypotenuse was related directly to the radius of the Circle.

The most important and most known theorem of Geometry, the Pythagorean theorem, emerged from the concept of Regular Polygons; specifically, from the concept of the square (Fig. 4). Thus, the Pythagorean Theorem through the notions of Regular Polygons becomes part of the Ontology of the Circle [20].

Fig. 4. Square and right triangles in a circle.

The Right Triangles have been now added to the Circle’s Ontology, followed by the Pythagorean Theorem, the hypotenuse and its midline (the corresponding part on Fig. 11).

Circle Circumference and Surface Area

In order to evaluate the surface area of the Circle and to add it to the Ontology as a new concept, we used the concept of the Regular Polygons. Starting with a simple polygon, such as a square, and gradually increasing the number of the angles of the polygon, when the number of the angles increased, the surface area of an orthogonal parallelogram equaled the surface area of the Circle (Fig. 5).

Fig. 5. Approximated surface of Circle, in GEOGEBRA environment.

Apart from the full surface area of the Circle, a sector from the cyclic disc was also explored as a concept and added to the collection of the concepts of the Ontology. In this exploration process, other concepts related to the Circle were added, like the length of an arc and the circumference (Fig. 6) of the Circle [21].

Fig. 6. Circumference of Circle, in GEOGEBRA environment.

An overview of the Circle’s Ontology up to this point is displayed in the corresponding part on Fig. 11.

Trigonometry

The basic trigonometric equation sin²x + cos²x = 1 is a concept that connects to the Ontology of the Circle. The relation originates from an inscribed angle of the Circle of a radius of 1. A vertical line from the one side of the arc of this angle, to the other side of the angle created a Right Triangle (Fig. 7).

Fig. 7. Inscribed angle to create a right triangle for trigonometry.

When the Pythagorean theorem was applied, the mathematic field of trigonometry began to unfold. The basic functions of sin (x) and cos (x) were also explored [22] (Fig. 8).

Fig. 8. Inscribed angle to create a Right Triangle for Trigonometry in Geogebra enviroment.

Contributing to the Ontology of the Circle (the corresponding part on Fig. 11).

Cylinder

The final concept to be added to the Ontology of the Circle was the cylinder. The vessel to embed the new concepts was the Circle itself and an orthogonal parallelogram having its longest side length equal to the length of the periphery of the Circle (Fig. 9).

Fig. 9. Circle and Parallelogram in vertical position.

The surface area of the bent around the circle parallelogram and the volume created were the concepts of the stereometric object that were added to the Ontology [23] (Fig. 10).

Fig. 10. Cylinder’s volume created by bending an orthogonal parallelogram (Fig. 12) around the circumference of a circle in GEOGEBRA environment.

At this point, all the required concepts by the curriculum for the 2^nd year junior high school Geometry, were covered by the Circle’s Ontology (Fig. 11).

Fig. 11. Circle’s Ontology includes Cylinder’s concepts.

The Technological Background of the Implementation of the Circle’s Ontology

As presented in the previous section, the Ontology under implementation covers the Circle and other geometrical concepts that are closely related to the circle. The curriculum of the second year junior high school covers the subjects of central and inscribed angles, Regular Polygons, the Pythagorean Theorem and other introductory trigonometry concepts. All these concepts are included in our Ontology along with the surface of the Circle or a sector of it and other derived terms. In this section, we present all the steps that were taken to build the Ontology of the Circle.

For our research we had to choose a programming language to write, a database to store data and a knowledge organization system. The Ontologies are based on the Resource Descriptive Framework (RDF). In order to represent the data of the Ontology using the RDF, the Turtle [26] language was chosen. This choice was based on the fact that it would be a simple and effective approach to create the RDF triples. The Turtle language has the ability to represent such triples in a plain text form using summaries (abbreviations). Thus, an individual can easily read the resulting text and it is equally easy to define the RDF triples.

To represent the Ontology of the Circle and its concepts in a visual environment like a graph, we used the GraphDB database [24], one of the few triple stores that can perform semantic inferencing at scale, allowing users to derive new semantic facts from existing ones. GraphDB can manage massive loads, queries and inferences in real time. It is also able to display the whole graph of the Ontology or just a part of interest along with its related concepts. When all the concepts have been defined in the Turtle file, they are accessible through the GraphDB environment.

Finally, the need for a knowledge organization system was covered by SKOS (Simple Knowledge Organization System) [25]. SKOS provides a standard way to represent knowledge organization systems using the RDF (Resource Description Framework). Encoding this information in RDF allows the information to be passed between computer applications in an interoperable way. Using RDF also allows knowledge organization systems to be used in distributed, decentralized metadata applications. Decentralized metadata is becoming a typical scenario, where service providers add value to metadata harvested from multiple sources.

The Turtle file needed to build the Ontology of the Circle should contain the concepts of interest. The task to be conducted was to connect these concepts to each other. Based on the theories in Geometry, a Circle is defined and described with the following concepts: center, radius, diameter, arc, chord, inscribed angle, central angle. However, the concepts of Regular Polygons and Right Triangles can be connected to the concept of the Circle, which leads to the conclusion that the concepts of the Circle, the Right Triangle and the Regular Polygon belong in the broader (represented as B in Figs. 2 and 11) concept of Schema, an element needed to be represented in the Ontology. On the other hand, elements related to these concepts belong to the narrower concept. In the SKOS way of expressing concepts, we named that broader Schema as Shape,

•

For it to include the concepts of Circle, Triangle and Regular Polygon. For the Schema definition, these three concepts are connected with the narrower concept, as:

•

•

•

The properties are described in a step-by-step method declaration, starting with the Circle, to fully construct the Ontology by working on each notion separately:

• preferable label➜ “”

• a definition of the concept➜ “”

• an example➜ “”

• a visual example created in GeoGebra➜  https://ggbm.at/asdew;

Declaring other concepts related in narrower concept of Circle:

•

•

•

•

•

•

•

Related concepts to the Circle through other concepts:

•

•

•

•

•

•

•

The above process continues for all the concepts that were included in the Ontology.

Discusion

This Ontology was built step-by-step with the participation of the first experimental group of students of 2nd grade junior high, during a whole school year period. The following year the Ontology was completed in its entirety and implemented. It was time for in class practicing and evaluation with a second experimental group of students of 2nd grade junior high. This experimental group consisted of two different classes of 2nd grade. One class would follow the Ontology approach in order to take advantage and evaluate its effectiveness to improve the teaching process. The other class would follow the standard teaching process as a control group and they would be taught using the classical methods they were used to, without any assistance of any additional software at all. Those two classes were chosen because during the previous year—in 1st grade—had the same teacher and the same measured characteristics. The two classes had almost the same synthesis, same number of students and same proportion between boys and girls. There were no criteria set on choosing those two classes, neither based on their cognitive levels nor on other characteristics like their interest in Mathematics or the level of their participation during the lessons. No student of either class moved from one group to the other. It is clear that all students in both classes shared the same knowledge from the first grade because both classes had the same mathematics teacher. The total number of students was 54.

For the evaluation process of the Ontology contribution to be precise and objective, their single teacher on every lesson during the year of 1st and 2nd grade, had a preprinted form to mark specific characteristics to be measured (Fig. 12). These characteristics compared between the two classes of the second experimental group, would display the Ontology’s effect, if any.

Fig. 12. Evaluation form for Ontology teaching method.

The results from both classes—experimental and control—has been evaluated both for the technological contribution as well as for the pedagogical effects. Three key dimensions were selected to be assessed regarding the impact that the Ontology based teaching method had on students: their attitude during the new teaching process, their cognitive e level change and their achievements. The assessed key indicators connected to the attitude dimension were the level of focusing during the lesson, the number of questions asked, the expressed interest, the participation of each student and the eagerness for seeking answers. The key indicators expressing the cognitive dimension were the level of understanding, the ability to combine the knowledge, and the quality of questions (a true question or a reminder). Finally, as key indicators for their achievements were all the exams (scheduled or not), the tests (not scheduled short time exams) and the given homework.

The first class of students was using the schools’ computer lab where GraphDB holding the Ontology, was providing the knowledge background to be explored. Alongside the Ontology, there was GEOGEBRA to assist with any recalculations and live transformation of graphics. The students within class were divided into small groups. These groups were free to choose their organization. Either one group member was explaining to the rest of the team or a group was exchanging ideas with another group. Compared to the classic teaching process, the new subject that the students were to be taught, i.e., the inscribed angles, was simply announced to the students. The students were given plenty of time to explore what the GraphDB could do for them. As the current students are very familiar with ICT technologies, the Ontology in GraphDB environment was merely asking to be discovered. The students firstly searched for the geometrical concepts. After a glare on the strict mathematical definition, they switched to the relations and properties of the inscribed angle. The central angle was now associated with the inscribed angle and the students picked an example out of the ones available to see how it worked. Almost all of the students looked over a solved exercise, aiming to become familiar with how a problem of that kind is expressed in words and what is the standard process to follow in order to solve relevant problems. The students were very enthusiastic with their new experience and the following teaching process became more of an open discussion than a sterilized classroom. The feeling at the end was excitement, not only for the students but also for the teacher involved in that experiment. The conclusions of that first concept, were communicated to all participants in the building of the Ontology. The initial building team was even happier than the teachers with these results. The Ontology in GraphDB was constantly gaining more and more as a process, as the educational subjects were being taught under this different way of teaching. Moreover, students from other classes expressed their interest in meeting with this “different” approach. The enthusiasm was kept high during the year and the results summed up at the end, depicted the added value and the overall gains.

We present the level of improvement based on the key indicators mentioned previously, measured exactly in the same way for the same period of time for both classes. The results of the key dimensions and their indicators (variables) are shown in Table I.

Table I. Measuring Improvements

Dimension	Variable	Experimental class	Control class	Improvement (%)	Average improvement (%)
Attitude	Focusing	7.3	8.9	21.9	41.0
	Questions asked	5.4	7.8	44.4
	Interest	6.8	9.2	35.3
	Participation	7.1	9.6	35.2
	Seeking answers	4.7	7.9	68.1
Cognitive	Understanding	6.8	9.4	38.2	67.8
	Combine knowledge	4.2	7.3	73.8
	Quality of questions	3.5	6.7	91.4
Achievements	Exams	7.4	9.6	29.7	25.8
	Tests	7.1	9.5	33.8
	Homework	8.6	9.8	14.0
Total number of students		27	27

We see that the students were doing well with exams, tests and their homework from the very beginning. Although there is a positive change in all variables of the Achievement dimension, which is still worth mentioning, what is even more important is the positive change in the other two dimensions, i.e., in Attitude and Cognitive.

The students’ interest was growing within the course timeline, as they were becoming more focused during the lessons and more interested in the cognitive area. Moreover, an increasing engagement has been recorded, but mostly this engagement involved the seeking of answers related to the teaching theme. It has also been recorded that there existed a change in their attitude, reflected in the Attitude dimension, showing that they were more self-motivated than ever and leaving time for the teacher to teach, since they were noticeably more self-disciplined. It was paramount that the students’ quest for answers was boosting their focusing and participation.

As a result of the attitude improvement (Table I), their level of understanding also improved because they were acting like professionally engaged researchers. A real change in the Cognitive level is the big change when looking at the ways they managed to combine knowledge. It seemed as this was a process either easier to perform or already known! The big change was captured by the quality of questions that they were imposing. A big change happened when from reminding questions or asking non relevant ones, they were now asking questions so close to the point.

From the teacher’s part, it is quite amazing how the collaborative level has increased in an effortless way. This captured change supported not only the teaching process but also the entire educational process. It is obvious that the new environment made them happier and more eager to gain knowledge, more self-motivated and with greater receptiveness.

Apart from the improvement of the students’ knowledge level, the teachers realized that the flow of teaching became easier and smoother, allowing them to devote themselves even more in their primary scope, teaching new material and covering the required curriculum. The students were also pushing their teachers to increase the difficulty of the everyday challenges that they were given in the form of homework or in class exercises, setting the level higher than the teachers originally anticipated. This general attitude of students and teachers taking part in this experiment mobilised other classes that were not participating, making them eager and willing to start along. The same effect was observed over teachers of other subjects like physics and chemistry teachers, who were imposing questions like if the same experiment could be applied on their field and how difficult that would be. The enthusiasm of these positives results was spread all over the school.

Conclusions

This study on a Geometry Ontology faces some limitations. It focused on second-grade junior high students, chosen because they were just beginning Geometry and learning computer skills. However, challenges arose in integrating Algebra concepts, and limited technology access meant students could not each use a tablet or PC for individual experimentation. The COVID-19 pandemic also restricted the study’s scope. The small sample size and single public school setting limit generalizability, and results may vary across different regions and school types.

Despite these limitations, the Ontology effectively supported students’ understanding of geometric concepts, reducing reliance on rote memorization and enhancing engagement through group projects and practical applications. Teachers valued the Ontology’s support for comprehensive topic discussions, and its success sparked interest in developing similar Ontologies for other subjects. Extending this Geometry Ontology throughout all middle school years could deepen students’ understanding and offer a solid foundation for higher-level studies, contributing valuable educational insights for complex subjects.

Building on these findings, future research could address the limitations by expanding the study to a more diverse student population, including different grade levels, school types, and regions. Increasing the sample size and incorporating a broader range of technological resources, such as providing individual devices for all students, would further enhance the potential of the Ontology for personalized learning. Exploring the impact of the Ontology in a combined or fully online learning environment, particularly post-pandemic, could offer new insights into how digital tools can support education in varying contexts. Ultimately, these adjustments could lead to a more comprehensive understanding of how Ontologies can reshape math education across various educational settings and stages of learning.

For future research, incorporating long-term assessments will be crucial to evaluate the sustained impact of the Geometry Ontology on students’ learning outcomes. While this study provided valuable insights into the Ontology’s effectiveness in the short term, a more extended evaluation period would help determine how well the knowledge gained is retained over time and applied to more advanced mathematical concepts. Long-term assessments could involve periodic testing, follow-up surveys with both students and teachers, and monitoring students’ performance in subsequent grades. This would provide a deeper understanding of how the Ontology contributes to long-term retention, conceptual development, and the transfer of learning across different mathematical domains. Moreover, such assessments would help identify areas for improvement in the Ontology’s design and functionality, ensuring it remains a valuable tool in the evolving educational landscape.