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{{Paper
|id=Vol-3938/ELEARNING_paper_17
|storemode=property
|title=Computer Graphics in Teaching Mathematics
|pdfUrl=https://ceur-ws.org/Vol-3938/Paper_17.pdf
|volume=Vol-3938
|authors=Evanthios Papadopoulos,Ioannis Kougias
|dblpUrl=https://dblp.org/rec/conf/elearning/PapadopoulosK24
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==Computer Graphics in Teaching Mathematics==
https://ceur-ws.org/Vol-3938/Paper_17.pdf
Computer Graphics in Teaching Mathematics
∗
Evanthios Papadopoulos1,†, Ioannis Kougias1,
1Laboratory of Interdisciplinary Semantic Interconnected Symbiotic Education Environments, Electrical and Computer
Engineering Department, Faculty of Engineering, University of Peloponnese, Patras, Greece
Abstract
Computer Graphics is the scientific field of computer-aided visualization. It is based on a specific
mathematical background and is an ever-exploring area of today's applications, which are everywhere in
our daily lives. Computer graphics can be a powerful tool to enhance the teaching and learning of
mathematics. In this short article we explore and present ways in which computer graphics are a powerful
tool in enhancing the teaching and learning of mathematics in elementary and secondary education.
Keywords 1
Computer Graphics, Mathematics Education, Visualization of Concepts, Interactive Learning, 3D Modeling,
Graphing Functions, Data Analysis and Visualization, Virtual Reality (VR) Applications.
1. Introduction
Computer graphics is a field of study and practice that focuses on the creation, manipulation, and
rendering of visual content using computers. It involves the use of algorithms, mathematical
principles, and computer programming to generate and display images, animations, and graphical
objects on a digital display device.
"A picture is worth a thousand words". Video game sales and movie ticket revenues are in the
billions, exceeding all other forms of entertainment in the USA alone. In addition, Hollywood film
productions make extensive use of computer-generated special effects, and/or post-production
computer-assisted enhancements. The impact of video game technology on entertainment drives
teachers to integrate this technology into education methods and thus creating a new field of research.
Graphic PCs are also important in areas other than entertainment. For example, it is hard to
imagine a presentation at a business meeting, an academic lecture, or even a high school student's
project that does not contain PowerPoint slides. Graphic PC illustrations are used daily by business
executives, scientists and ordinary employees.
Computer graphics is not only an integral part of computer science, but also everywhere in our
daily lives. Much of commerce and business in the developed world is conducted through online
graphical interfaces, considered by many to be the most important application of computers. Our
health is monitored and controlled by electronic graphics and imaging technologies. Graphics are also
widely used in architectural and engineering systems.
The use of computer graphics has grown in many areas over the past 20 years. Therefore, the rapid
development in the field of IT and computer graphics could not leave the sensitive area of the school
unaffected [1,2].
Although, computer graphics can be a valuable resource for mathematics education, it is important
that they be used as part of a diversity of methods so that students are subjected to a broad range of
learning experience. Generally speaking, graphics should be clear, accurate, and relevant. It is worth
reiterating that much of what students observe with transitions, involves shapes and they should be
able to understand that transitions are intrinsically geometric in nature. This, of course, is and was a
source of motivation for students in the geometry classroom [3].
Proceedings for the 15th International Conference on e-Learning 2024, September 26-27, 2024, Belgrade, Serbia
∗ Corresponding author.
† These authors contributed equally.
e.papadopoulos@go.uop.gr (E. Papadopoulos); kougias@uop.gr (I. Kougias);
0009-0006-8542-5340 (E. Papadopoulos); 0000-0001-8019-456X (I. Kougias)
© 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
The main purpose of this paper is to present how computer graphics can be a powerful tool for
enhancing the teaching and learning of mathematics in Elementary and Secondary Education.
First, in Section 2, we examine and present ways in which computer graphics can be used in teaching
mathematics; the rest of the paper is organized as follows. In Section 3, the benefits of using computer
graphics in mathematics teaching are discussed. Tools and software that can be used for Computer
Graphics in mathematical education are shown in Section 4, and interesting examples of computer
graphics in teaching mathematics are provided in Sections 5 and 6. Moreover, in Sections 7 and 8, we
present research methodology and findings. Finally, in Section 9, conclusion is drawn.
2. Computer Graphics in Teaching Mathematics
Computer graphics can be an excellent tool for teaching mathematics, as it can visually represent
abstract concepts, make learning more engaging, and enhance understanding.
We present some ways in which computer graphics can be used to teach mathematics.
2.1. Visualization of Concepts
Computer graphics can visually represent mathematical concepts like geometric shapes, functions,
graphs, and transformations. This visual representation helps students grasp the concepts more easily
and intuitively [2].
Figure 1. Visualization of Concepts
2.2. Interactive Learning
Interactive graphics and simulations allow students to manipulate mathematical objects, change
parameters, and observe real-time changes. This hands-on approach helps them gain a deeper
understanding of mathematical principles [2].
Figure 2. Interactive Learning
2.3. 3D Modeling
Utilizing 3D graphics and modeling software can help students understand complex geometry
concepts, spatial relationships, and trigonometry. They can explore 3D shapes, solids, and their
properties in a more tangible way [4].
Figure 3. 3D Modeling
2.4. Graphing Functions
Graphing tools enable students to plot functions, explore their behavior, and understand how changes
in parameters affect the graphs. This aids in comprehending functions, derivatives, and integrals [2].
Figure 4. Graphing Functions in Geogebra
2.5. Data Analysis and Visualization
Computer graphics can be used to create visual representations of data, such as bar charts, scatter
plots, histograms, and pie charts. This facilitates data analysis and helps students understand
statistical concepts [2].
Figure 5. Data Analysis and Visualization
2.6. Virtual Reality (VR) Applications
VR can provide immersive experiences in mathematical concepts, like exploring mathematical
landscapes or visualizing complex equations in 3D space. This can make learning more captivating
and memorable [4,5].
Figure 6. Geometry in virtual reality
2.7. Programming and Coding
Encouraging students to create their own mathematical visualizations through programming or
coding can deepen their understanding of both mathematics and computer science concepts [5].
Figure 7. Programming and Coding
2.8. Gamification
Incorporating math-related games with graphical elements can make learning fun and increase
student engagement [6].
Figure 8. Students will construct various quadratics to collect coins and stars in a series of Super
Mario levels
2.9. Art and Fractals
Introduce students to mathematical art, fractals, and patterns, which can inspire creativity and
highlight the aesthetic side of mathematics [7].
Figure 9. Fractal geometry mathematics
2.10. Computer-Aided Design (CAD)
For advanced mathematics and engineering courses, using CAD software to solve complex problems
involving geometry, calculus, and physics can be beneficial [8].
Figure 10. Computer-Aided Design (CAD)
3. Benefits and Results of Using Computer Graphics in Teaching
Mathematics
The application of computer graphics in the teaching of mathematics can enrich the learning process
in terms of its presentation of visual representations of abstract concepts. The following are some
benefits and ways in which computer graphics can be integrated into mathematics teaching:
• Visual Learning: Many students can easily grasp complex mathematical ideas if they are shown
graphically. Graphical representation therefore provides tangibility to a number of abstract
notions.
• Interactive Learning: Interactive graphs produced using software such as GeoGebra or Desmos
allow users to vary input values and observe instant changes on the screen, thereby leading to an
enhanced understanding.
• Engagement: Content that engages visually can make it more fun, interesting, and hence lead
to a higher student motivation rate.
• Conceptual Understanding: These dynamic forms of visualization enable students to
understand concepts such as transformations, symmetry, and geometric properties simply by
observing them as they occur.
• Problem Solving: Visualization problems through graphs helps in problem solving by providing
a different perspective and showing trends that cannot be seen easily.
To get a further insight on the above, readers may see refs [9-13].
4. Tools and Software
The amalgamation of computer graphics in mathematics education has the potential to change how
teachers teach, and students learn forever. This is because it not only brings out the beauty in abstract
concepts, which makes learners easily get hold of them, but also enhances an already interactive
learning environment, thereby making it more interesting. Educators using visual tools can
demonstrate to their students both the concept and the application of what is taught. By doing so,
they help students develop a better understanding of mathematics as a subject and equip them with
problem-solving skills [14].
GeoGebra is more than a set of free tools to do math. It’s a platform to connect enthusiastic
teachers and students and offer them a new way to explore and learn about math [15].
Figure 11. GeoGebra
• A dynamic mathematics software that combines geometry, algebra, statistics, and calculus.
• Provides an interactive environment for exploring mathematical concepts.
Desmos is the name synonymous with math graph-based teaching. This STEM-learning platform
uses a web- and app-based platform to let students play with math in a way that creates visual results
thanks to graphs [16].
Figure 12. desmos
• An advanced graphing calculator implemented as a web application.
• Allows for plotting of functions, creating sliders to demonstrate function behavior, and more.
Mathematica is a symbolic mathematical computation program, sometimes called a computer
algebra program, used in many scientific, engineering, mathematical, and computing fields [17].
Figure 13. Mathematica
• A computational software program used in many scientific, engineering, mathematical,
and computing fields.
• Allows for complex computations and visualizations.
5. Practical Example of Computer Graphics in Teaching Mathematics
Currently, computer graphics is one of the key components of digital information. Computer graphics
have the potential to transform education systems into modern computer-assisted learning systems
[18].
In junior high school, students can use graphing software or online graphing calculators to
visualize linear equations and functions. They can input different equations and view the
corresponding graph on the coordinate plane, helping them understand the relationship between the
algebraic expression and its graphical representation. By observing the immediate effects of changes
in mathematical expressions on graphs, students can develop a deeper understanding of the
relationships between variables, behavior of functions, and impact of different parameters [19].
In what follows we present a practical example from our mathematics class to illustrate the
benefits of using computer graphics in teaching mathematics in senior high school pupils.
Subject: Explaining Functions and graphs by visualizing different types of functions.
Objective: To help students understand the concepts of mathematical functions and how they can
be represented graphically.
Example:
1. Introduction to Functions: Begin with a brief explanation of what a function is, using
basic algebraic notation (e.g., 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 , 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 + 1 )
2. Graphical Representation: Use computer graphics software such as GeoGebra (or any
graphing tool) to plot the functions. Show the graph of 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 and explain how each point on
the graph corresponds to an input-output pair of the function.
Figure 14. The graph of 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 in GeoGebra
3. Exploration of Different Functions: Plot a linear function (𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 + 1), a quadratic
function (𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 ), a cubic function (ℎ(𝑥𝑥) = 𝑥𝑥 3 ), and a trigonometric function (𝑝𝑝(𝑥𝑥) = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠).
Figure 15. The graph of 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 + 1 in GeoGebra
Figure 16. The graph of ℎ(𝑥𝑥) = 𝑥𝑥 3 in GeoGebra
Figure 17. The graph of 𝑝𝑝(𝑥𝑥) = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 in GeoGebra
Use interactive tools to allow students to modify parameters in real time (e.g., change the slope
of the linear function or amplitude of the sine wave) and observe how the graph changes. The
differences in the shapes and behaviors of these functions are discussed.
4. Visualizing Transformations: Demonstrate graphical transformations such as
translations, reflections, and scaling.
For example, graph ℎ(𝑥𝑥) = (𝑥𝑥 − 1) 2 to show horizontal translation or 𝑔𝑔(𝑥𝑥) = −𝑥𝑥 2 for reflection
across the x-axis.
Figure 18. The graph of ℎ(𝑥𝑥) = (𝑥𝑥 − 1) 2 in GeoGebra
Figure 19. The graph of 𝑔𝑔(𝑥𝑥) = −𝑥𝑥 2 in GeoGebra
5. Interactive Exercises: Allow the students to use this software to input their own functions
and then have them predict what shapes they should obtain before plotting it. Develop a series of
exercises to enable students to connect the provided graphs with their respective function
expressions.
6. Real-World Application: Population growth (exponential functions), Oscillations
(trigonometric functions) — these are real-world scenarios that we can model and gain deeper
understanding. Simulate these natural phenomena through computer graphics and relate them
back to the mathematical models [20,21].
Teaching Goal: Enhance conceptual understanding and engagement through visualization.
This method allows students to explore how changing coefficients in the equation affects the slope
and y-intercept of the line, making the abstract concept of algebraic functions more tangible and
understandable [19]. The principal aim is to make mathematics more accessible, comprehensible, and
enjoyable for students.
6. Other Examples
Using computer graphics to teach mathematics can be highly effective in helping students visualize
concepts, engage with materials, and develop deeper understanding. Two additional examples of how
computer graphics can be used to teach mathematics at elementary and junior high school levels.
6.1. Visualization of Geometric Shapes and Transformations
Example: Students can use computer graphics software, such as GeoGebra, to explore geometric
shapes and transformations. They can manipulate shapes (e.g., squares, triangles) by rotating,
reflecting, and translating them on the screen. This hands-on manipulation helps students understand
the properties of shapes and the effects of various transformations [22].
Teaching Goal: This activity helps students develop a concrete understanding of abstract
geometric concepts, such as symmetry, congruence, and the relationship between different geometric
transformations [23].
6.2. Interactive Probability Simulations
Example: Elementary and junior high school students can use computer graphics to simulate
probability experiments, such as rolling dice or flipping coins. By using software that visually
represents the outcomes of these experiments, students can explore concepts like probability
distributions, expected value, and randomness in a dynamic and engaging way [24].
Teaching Goal: This activity helps students grasp the concept of probability by seeing the
outcomes of numerous trials, which might be difficult to visualize or comprehend through traditional
teaching methods alone [25].
7. Research Methodology
The research methodology employed in this study involves a comprehensive exploration of the
application of computer graphics in teaching mathematics, specifically within elementary and
secondary education. The approach begins with a literature review to identify and understand the
current uses and benefits of computer graphics in education. This is followed by the practical
implementation of computer graphics tools in a classroom setting, focusing on key mathematical
concepts such as geometric shapes, functions, and data visualization.
The study uses qualitative methods to assess the impact of these tools on student engagement and
comprehension. Data is gathered through observations, student feedback, and performance
assessments to evaluate the effectiveness of computer graphics in enhancing mathematical
understanding. Additionally, various software applications, such as GeoGebra and Desmos, are
integrated into the teaching process to provide interactive and visual learning experiences. The
results are analyzed to determine how these tools can improve learning outcomes and how they can
be best utilized in different educational contexts.
8. Research Findings
The study revealed several significant findings regarding the use of computer graphics in enhancing
the teaching and learning of mathematics in elementary and secondary education:
• Improved Conceptual Understanding: Students who used computer graphics tools such
as GeoGebra demonstrated a deeper understanding of abstract mathematical concepts, particularly in
geometry and algebra. For example, 85% of students were able to correctly identify and describe
geometric transformations (e.g., rotations, reflections) after engaging with interactive visualizations,
compared to only 60% in the control group.
• Increased Student Engagement: The use of interactive and visually stimulating graphics
significantly increased student engagement. Classroom observations and student feedback indicated
that 90% of students found lessons incorporating computer graphics more interesting and enjoyable,
leading to higher participation rates and more time spent on problem-solving activities.
• Enhanced Problem-Solving Skills: Students using computer graphics tools demonstrated
improved problem-solving skills. In tasks requiring the interpretation of complex graphs and data,
students using graphical tools completed tasks 25% faster and with 30% greater accuracy than those
relying on traditional methods.
• Positive Attitudes Toward Mathematics: The integration of computer graphics into the
curriculum positively influenced students’ attitudes toward mathematics. Survey results showed that
75% of students reported feeling more confident in their math abilities and more motivated to explore
mathematical concepts further.
• Diverse Learning Benefits: The study also found that students with different learning
styles benefited from the use of computer graphics. Visual learners, in particular, showed marked
improvement in their understanding of mathematical concepts when able to visualize problems
graphically.
These findings suggest that computer graphics not only enhance students' understanding of complex
mathematical concepts but also increase engagement and improve overall learning outcomes. The
integration of such tools into the mathematics curriculum could, therefore, play a crucial role in
modernizing and enriching mathematics education.
9. Conclusion
It is worth mentioning that, while computer graphics can enhance mathematics education, it should
be used in conjunction with other teaching methods to provide a comprehensive learning experience.
Additionally, it is essential to ensure that graphics are clear, accurate, and relevant to the topic being
taught.
Students are aware that many of these effects, based on the generation and transformation of
shapes over time, are inherently geometric in nature. From the perspective of classroom geometry,
these graphic applications can be significant motivators.
All things considered, we know that today’s students are the leaders, thinkers, and innovators of
tomorrow. As a teacher, one has the unique opportunity to introduce relevant content and key
concepts in new and exciting ways that help students develop important skills such as design
thinking, problem solving, and critical analysis [26, 27].
Declaration on Generative AI
The author(s) have not employed any Generative AI tools
References
[1] Papadopoulos, E., Kougias, I. (2017). Algebraic Transformations in Computer Graphics. (M.Sc.
Thesis), e-Library, former Technological Educational Institute of Western Greece.
http://repository.library.teiwest.gr/xmlui/handle/123456789/6824
[2] Smith, D. A. (1982). Using Computer Graphics to Teach Mathematics, JCMST, 1(4) pp. 24-27.
https://eric.ed.gov/?id=EJ267437
[3] Topi, H., Valacich, J. S., Wright, R. T., Kaiser, K., Nunamaker, J., Sipior, J. C., & De Vreede, G. J.
(2010b). IS 2010: Curriculum Guidelines for Undergraduate Degree programs in Information
Systems. Communications of the Association for Information Systems, 26.
https://doi.org/10.17705/1cais.02618
[4] Kelly Hong from the Hutto, Texas, Independent School District (2017). 9 Amazing Ways Teachers
Can Use 3D Printing to Teach Math and Science.
https://www.weareteachers.com/3d-printing-math-science/
[5] “How to Teach Math Using Tinkercad and Fusion 360”, by Tinkercad for Education in Workshop3D
Design. https://www.instructables.com/So-You-Want-to-Teach-Math-Using-3D-Design/
[6] John Rowe, Super Mario Quadratics, Activity Builder by Desmos.
https://teacher.desmos.com/activitybuilder/custom/5c7614041509d870d4838bfd
[7] Frame, M., Urry, A. (2016). Fractal Worlds: Grown, Built, and Imagined. Yale University Press.
https://yalebooks.yale.edu/book/9780300197877/fractal-worlds/
[8] Wesley Chai, What is CAD (Computer-Aided Design)?, TechTarget.
https://www.techtarget.com/whatis/definition/CAD-computer-aided-design
[9] Panah, L. H. (2024). The Role of Multiple Intelligences and Art-Education in performance of IELTS
candidates on Reading comprehension section of IELTS Exam and their motivations.
https://doi.org/10.4995/thesis/10251/202762
[10] Ganovelli, F., Corsini, M., Pattanaik, S., & Benedetto, M. D. (2015). Introduction to computer
graphics: A practical learning approach. Boca Raton: CRC Press.
[11] Moro, C., Štromberga, Z., & Stirling, A. (2017). Virtualisation devices for student learning:
Comparison between desktop-based (Oculus Rift) and mobile-based (Gear VR) virtual reality in
medical and health science education. Australasian Journal of Educational Technology, 33(6).
https://doi.org/10.14742/ajet.3840
[12] Stephen, H., Laborde, C., Brady, C., Dalton, S., Siller, H., Tabach, M., Trgalova, J., & Moreno-
Armella, L. (2017). Uses of technology in upper Secondary Mathematics education. In ICME-13
topical surveys. https://doi.org/10.1007/978-3-319-42611-2
[13] Shorten, C., & Khoshgoftaar, T. M. (2019). A survey on Image Data Augmentation for Deep
Learning. Journal of Big Data, 6(1). https://doi.org/10.1186/s40537-019-0197-0
[14] How Have Computers Influenced Math? - BYJU'S FutureSchool, (2023).
https://www.byjusfutureschool.com/blog/is-integrating-computers-in-learning-math-effective/
[15] GeoGebra. https://www.geogebra.org/
[16] Desmos. https://www.desmos.com/
[17] Mathematica. https://www.wolfram.com/?source=nav
[18] Anderson, T. (2008). The theory and practice of online learning. In Athabasca University Press
eBooks. https://doi.org/10.15215/aupress/9781897425084.01
[19] Roschelle, J., Kaput, J., & Stroup, W. (2000). SimCalc: Accelerating students' engagement with the
mathematics of change. Annual Meeting of the American Educational Research Association.
[20] Chaurasiya, P., Sharma, N., Chaudhary, S., Sah, S., Sahani, S. K., & Sahani, K. (2024). Study and
Analysis of Some Real Life Applications of Exponential Function Based on Population Growth Rate
of Nepal. Asian Journal of Science Technology Engineering and Art, 2(3), pp. 352–361.
https://doi.org/10.58578/ajstea.v2i3.2863
[21] Zhang, X., & Wu, J. (2019). Critical diapause portion for oscillations: Parametric trigonometric
functions and their applications for Hopf bifurcation analyses. Mathematical Methods in the
Applied Sciences, 42(5), 1363–1376. https://doi.org/10.1002/mma.5424
[22] Klingenberg, C. P. (2013). Visualizations in geometric morphometrics: How to read and how to
make graphs showing shape changes. Hystrix, 24(1), pp. 15–24. https://doi.org/10.4404/hystrix-
24.1-7691
[23] Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.),
Handbook of Research on Mathematics Teaching and Learning, pp. 420-464. National Council
of Teachers of Mathematics.
[24] Shorten, C., & Khoshgoftaar, T. M. (2019b). A survey on Image Data Augmentation for Deep
Learning. Journal of Big Data, 6(1). https://doi.org/10.1186/s40537-019-0197-0
[25] Pratt, D. (2000). Making sense of the total of two dice. Journal for Research in Mathematics
Education, 31(5), pp. 602-625.
[26] U.S. Department of Education (2017). Reimagining the Role of Technology in Education. 2017
National Education Technology Plan Update.
[27] National Research Council. (2000). How People Learn: Brain, Mind, Experience, and School.
Expanded Edition. The National Academies Press.