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|title=Enhancing mathematical understanding through dynamic GeoGebra modeling: A holistic educational approach
|pdfUrl=https://ceur-ws.org/Vol-3820/paper114.pdf
|volume=Vol-3820
|authors=Liudmyla I. Bilousova,Liudmyla E. Gryzun,Valentyna V. Pikalova
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==Enhancing mathematical understanding through dynamic GeoGebra modeling: A holistic educational approach==
https://ceur-ws.org/Vol-3820/paper114.pdf
Liudmyla I. Bilousova et al. CEUR Workshop Proceedings 44–55
Enhancing mathematical understanding through dynamic
GeoGebra modeling: A holistic educational approach
Liudmyla I. Bilousova1 , Liudmyla E. Gryzun2 and Valentyna V. Pikalova3
1
Academy of Cognitive and Natural Sciences, 54 Universytetskyi Ave., Kryvyi Rih, 50086, Ukraine
2
Simon Kuznets Kharkiv National University of Economics, 9A Nauky Ave., Kharkiv, 61166, Ukraine
3
National Technical University “Kharkiv Polytechnic Institute”, 2 Kyrpychova Str., Kharkiv, 61002, Ukraine
Abstract
This paper presents an innovative approach to enhancing mathematical understanding through interactive
modeling using GeoGebra software, based on holistic educational principles. The research describes the de-
velopment and implementation of a comprehensive complex of dynamic mathematical models created within
inter-university projects of the Kharkiv GeoGebra Institute. The complex comprises three distinct categories
of models: fundamental mathematical concept visualization, transdisciplinary connections demonstration, and
real-world problem-solving applications. A systematic methodology for model development was implemented,
incorporating theoretical foundations of holistic education and practical considerations for effective visual-
ization. The paper details the technical implementation using GeoGebra tools, discusses specific examples of
models created, and presents the pedagogical framework for their application. Special attention is given to
the development of supporting didactic materials that guide learners through active investigation using the
dynamic models. The research demonstrates how this approach facilitates deeper mathematical understanding by
connecting abstract concepts with practical applications and fostering active learning through visualization and
experimentation. Results indicate that the developed complex of models effectively supports the implementation
of holistic educational principles in mathematics education, particularly in establishing meaningful connections
between mathematical concepts and their real-world applications.
Keywords
dynamic mathematics software, interactive modeling, mathematical visualization, GeoGebra, transdisciplinary
connections, holistic mathematics education
1. Introduction
Contemporary mathematics education faces significant challenges in fostering deep understanding
and meaningful engagement among students. Recent studies by Vlasenko et al. [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11], Lovianova et al. [12, 13], Kramarenko et al. [14], Kramarenko and Kochina [15], Ponomareva
[16], Merzlykin et al. [17], Tarasenkova et al. [18], Achkan et al. [19] highlight persistent gaps in
mathematics education at both secondary and university levels. These challenges manifest in students’
difficulties with abstract concepts and their practical applications, ultimately affecting their overall
mathematical competence.
The core issues in mathematical education, as identified by Bilousova et al. [20] and diSessa et al.
[21], include:
• Students’ struggle with abstract mathematical concept comprehension
• Limited ability to apply mathematical knowledge to practical tasks
• Diminishing interest in mathematics due to perceived complexity
• Failure to recognize mathematics’ role in other disciplines
CoSinE 2024: 11th Illia O. Teplytskyi Workshop on Computer Simulation in Education, co-located with the XVI International
Conference on Mathematics, Science and Technology Education (ICon-MaSTEd 2024), May 15, 2024, Kryvyi Rih, Ukraine
" Lib215@ukr.net (L. I. Bilousova); Lgr2007@ukr.net (L. E. Gryzun); valentyna.pikalova@khpi.edu.ua (V. V. Pikalova)
~ https://kafis.hneu.net/grizun-lyudmila-eduardivna/ (L. E. Gryzun);
http://web.kpi.kharkov.ua/kmmm/uk/o_kafedre_ua/profesorsko-vikladatskij-sklad/pikalova-valentina-valeriyivna/
(V. V. Pikalova)
0000-0002-2364-1885 (L. I. Bilousova); 0000-0002-5274-5624 (L. E. Gryzun); 0000-0002-0773-2947 (V. V. Pikalova)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
44
Liudmyla I. Bilousova et al. CEUR Workshop Proceedings 44–55
A fundamental challenge lies in students’ lack of holistic understanding of mathematics as both
a theoretical framework and a practical tool for solving interdisciplinary problems. As Singh [22]
emphasizes, this disconnect between abstract mathematical concepts and their real-world applications
often results in decreased motivation and engagement.
The holistic educational paradigm offers a promising approach to address these challenges. According
to Miller [23, 24], holistic education emphasizes:
1. Learner autonomy and active participation
2. Integration of knowledge across disciplines
3. Connection between academic concepts and real-world experiences
4. Development of comprehensive understanding through practical application
Computer Dynamic Models (CDM) emerge as powerful tools for implementing holistic education
principles in mathematics teaching. Research by Semenikhina and Drushliak [25] and Alessi [26]
demonstrates that CDMs can effectively:
• Visualize mathematical concepts in real-time
• Enable active exploration of mathematical relationships
• Facilitate understanding of transdisciplinary connections
• Support development of integrated thinking skills
Among available mathematical software, GeoGebra stands out for its comprehensive modeling
capabilities. Study by Kramarenko et al. [27] highlight GeoGebra’s effectiveness in creating interactive
visualizations and supporting mathematical investigation. The software enables seamless integration
of geometric and algebraic representations, facilitating dynamic visualization and manipulation of
mathematical concepts [28].
The Kharkiv GeoGebra Institute, operating within the International GeoGebra Institute network
since 2010, focuses on:
1. Promoting effective implementation of GeoGebra in mathematical education
2. Supporting research in mathematics, physics, and computer science
3. Advancing STEM education through technology integration
4. Fostering international collaboration in mathematical education
This paper presents the results of an inter-university project conducted through the Kharkiv GeoGebra
Institute, focusing on developing a comprehensive complex of dynamic models for holistic mathematics
learning at the university level.
2. Theoretical framework
The development of our GeoGebra model complex is grounded in both theoretical principles and practical
considerations, implemented through a systematic methodology combining theoretical, empirical, and
modeling approaches.
2.1. Methodological foundation
The project’s initial phase established three fundamental requirements for the model complex:
1. Development of diverse model categories:
• Basic mathematical concept visualization
• Transdisciplinary connection demonstration
• Real-world problem-solving applications
2. Implementation of dynamic, interactive elements to support active learning
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Liudmyla I. Bilousova et al. CEUR Workshop Proceedings 44–55
3. Cloud-based accessibility through www.geogebra.org
Following Bevz [29] and diSessa et al. [21], we approached transdisciplinary connections through
three primary dimensions:
• Content integration across disciplines
• Learning activity structure
• Educational process organization
2.2. Theoretical analysis process
The analytical phase involved comprehensive examination of:
1. Core mathematical concepts and their interdisciplinary applications
2. Curriculum content threads [30, 31]
3. Transdisciplinary connection patterns
This analysis revealed key connection chains between mathematics and other disciplines:
• Mathematics – Computer Science
• Mathematics – Physics
• Physics – Mathematics – Biology
• Mathematics – Economics
• Mathematics – Engineering Design
The theoretical framework was further enhanced through semantic analysis using specialized software
tools, including TextAnalyst 2.0, Text Miner 12.1, and Trope 8.4, enabling identification of key learning
elements and their interconnections across disciplines.
Figure 1: The common scheme of the graph, representing their transdisciplinary links with exact learning
elements (LE1...LEn) of subject domains (SD).
This theoretical foundation guided the subsequent development of practical models and their im-
plementation in educational settings. The framework emphasizes the importance of active learning,
visualization, and practical application in mathematical education, aligning with holistic education
principles outlined by Miller et al. [32] and Mahmoudi et al. [33].
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Liudmyla I. Bilousova et al. CEUR Workshop Proceedings 44–55
Figure 2: The example of the graph for selected LEs, representing the transdisciplinary links for the chain:
Physics – Mathematics – Biology (Used below for the transdisciplinary model “Lens”).
3. Results and discussion
3.1. Model development process
The implementation of the theoretical framework resulted in a systematic model development process
comprising several key phases:
3.1.1. Phase 1: Mathematical model construction
For each model, the development process included:
1. Analysis of transdisciplinary concept relationships
2. Definition of mathematical dependencies for visualization
3. Specification of model parameters (fixed and variable)
4. Selection of appropriate graphical elements
5. Identification of relevant applications and problems
6. Development of supporting didactic materials
3.1.2. Phase 2: GeoGebra implementation
The technical implementation utilized various GeoGebra tools [25, 28]:
• Standard geometric tools (Points, Lines, Polygons)
• Computer Algebra System (CAS) components
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Liudmyla I. Bilousova et al. CEUR Workshop Proceedings 44–55
• Dynamic transformation tools
• Action Object and Movement tools
3.1.3. Phase 3: Testing and refinement
The models underwent rigorous testing and improvement cycles to ensure educational effectiveness
and technical reliability.
3.2. Model categories and examples
3.2.1. Category 1: Basic mathematical concepts
These models focus on fundamental concept visualization and understanding. Notable examples include:
Example: Remarkable curves investigation – epicycloids
Chain of transdisciplinary links: Geometry – Algebra – Mechanics
The model demonstrates epicycloid construction and properties, enabling investigation of:
• Relationship between curve lobes and radius ratios
• Position calculations using geometric parameters
• Transformation between epicycloids and hypocycloids
Figure 3: Episodes of the students’cognitive activity with the dynamic model “Remarkable curves investigation
– epicycloids”.
3.2.2. Category 2: Transdisciplinary connections
These models emphasize interdisciplinary relationships, as demonstrated in the following example:
Example: Lens Model
Chain of transdisciplinary links: Physics – Mathematics – Biology
The model illustrates optical principles through mathematical relationships:
• Lens curvature effects on focal points
• Mathematical relationships in image formation
• Geometric properties of light paths
Supporting tasks include:
1. Investigation of mathematical dependencies
2. Analysis of geometric properties
3. Integration with biological systems (human eye)
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Liudmyla I. Bilousova et al. CEUR Workshop Proceedings 44–55
Figure 4: Scheme of the optical system of a human eye.
3.2.3. Category 3: Real-world applications
This category focuses on practical problem-solving, exemplified by:
Example: Fermat-Torricelli points investigation
The model supports various real-world investigations:
1. Construction and property analysis
2. Application to urban planning
3. Resource optimization problems
3.3. Educational impact and implementation
The implementation of these models demonstrates several key advantages:
1. Enhanced visualization: following principles outlined by Kramarenko et al. [34], the models
provide dynamic visualization of abstract concepts.
2. Active learning: as suggested by Tarasenko et al. [35], interactive elements encourage student
engagement and exploration.
3. Practical application: the models bridge theoretical understanding and practical implementation,
supporting findings by Bilousova et al. [20].
4. Cloud integration: cloud-based accessibility aligns with modern educational needs [36, 37].
3.4. Didactic support framework
The developed didactic support materials include:
1. Transdisciplinary connection tasks
• Concept relationship identification
• Cross-disciplinary application exercises
• Integration-focused problems
2. Practical application tasks
• Real-world problem solving
• Industry-specific applications
• Contextual learning activities
3. Investigation guidelines
• Step-by-step exploration procedures
• Parameter manipulation instructions
• Analysis and conclusion frameworks
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Figure 5: Episodes of transdisciplinary tasks solving, operating the model “Lens”.
3.5. Future research directions
Based on our findings, several promising research directions emerge:
• Long-term impact assessment on student understanding
• Development of additional model categories
• Integration with emerging educational technologies
• Extension to other STEM disciplines
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4. Conclusions
This research demonstrates the successful development and implementation of a comprehensive
GeoGebra-based modeling complex for enhancing mathematical education through a holistic approach.
The key findings and contributions can be summarized in several dimensions:
4.1. Model development framework
The research established a systematic approach to creating educational mathematical models, incorpo-
rating:
• Robust theoretical foundations drawing from holistic education principles
• Structured development methodology across three distinct model categories
• Integration of dynamic visualization with practical applications
• Cloud-based deployment for widespread accessibility
4.2. Educational innovation
The developed complex advances mathematical education through:
1. Enhanced visualization: synamic models provide immediate feedback and interactive exploration
opportunities, supporting findings by Kramarenko et al. [27] regarding the effectiveness of visual
learning in mathematics.
2. Transdisciplinary integration: following principles outlined by Gryzun [38], the models success-
fully bridge multiple disciplines, demonstrating mathematics’ role in various fields.
3. Active learning support: interactive elements encourage student engagement and independent
exploration, aligning with Bilousova et al. [20]’s recommendations for effective mathematics
education.
4. Practical application: teal-world problem-solving capabilities address the gap between theoretical
understanding and practical implementation identified by diSessa et al. [21].
4.3. Technological implementation
The research demonstrates successful utilization of GeoGebra’s capabilities through:
• Effective integration of geometric and algebraic representations
• Development of interactive, user-friendly interfaces
• Implementation of cloud-based accessibility
• Creation of scalable and modifiable models
4.4. Pedagogical implications
The research yields significant implications for mathematics education:
1. Enhanced teaching methodology: the model complex provides educators with tools for imple-
menting holistic teaching approaches, supporting findings by Miller [24] regarding effective
mathematical instruction.
2. Student engagement: interactive elements and real-world applications increase student motivation
and understanding, addressing challenges identified by Singh [22].
3. Flexible learning support: cloud-based accessibility enables both classroom and independent
learning, aligning with modern educational needs [39].
4. Comprehensive understanding: the transdisciplinary approach fosters deeper mathematical
comprehension, supporting principles outlined by Mahmoudi et al. [33].
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4.5. Future directions
The research opens several promising avenues for future investigation:
1. long-term impact studies: systematic evaluation of the model complex’s effectiveness in various
educational contexts.
2. Model extension: development of additional model categories and applications for emerging
educational needs.
3. Technology integration: investigation of integration possibilities with new educational technolo-
gies and platforms.
4. Pedagogical framework: further development of supporting didactic materials and teaching
methodologies.
5. Cross-cultural implementation: study of the model complex’s effectiveness in different educational
systems and cultural contexts.
4.6. Final remarks
This research contributes to the advancement of mathematics education by providing a practical
framework for implementing holistic educational principles through dynamic modeling. The developed
complex of GeoGebra models, supported by comprehensive didactic materials, offers a scalable and
effective approach to enhancing mathematical understanding. The success of this implementation
suggests that similar approaches could be valuable across various educational contexts and disciplines.
The findings underscore the importance of combining theoretical rigor with practical application in
mathematics education, demonstrating how technology can bridge this gap effectively. As educational
technology continues to evolve, the principles and methodologies established in this research provide a
foundation for future developments in mathematical education.
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