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Function Art: Linking Mathematics, Technology, and Visual Arts

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Title: Function Art: Linking Mathematics, Technology, and Visual Arts
Language: English
Authors: Guillermo Bautista (ORCID 0000-0001-5471-9326), Jemarie Malacapo, Florenda Gallos-Cronberg, Houssam Kasti (ORCID 0000-0001-6573-8370), Noah Dana-Picard (ORCID 0000-0002-1777-3232), Zsolt Lavicza (ORCID 0000-0002-3701-5068)
Source: School Science and Mathematics. 2026 126(3):251-263.
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 13
Publication Date: 2026
Document Type: Journal Articles
Reports - Research
Descriptors: Mathematics Education, Mathematical Concepts, Art Products, Computer Software, Constructivism (Learning), Computer Uses in Education, Visual Arts, Graphs, Animation, Mathematics Skills, Learning Strategies
DOI: 10.1111/ssm.18373
ISSN: 0036-6803
1949-8594
Abstract: This study investigated students' understanding of mathematical functions and strategies to create artwork using GeoGebra. It was framed by the principles of constructionism and examined how students use functions in creating artworks. We gathered data from students' artworks using the Algebra view and the Construction Protocol in the GeoGebra software. We used descriptive statistics to determine the number and types of functions used, whereas we employed thematic analysis to identify patterns and themes in students' artworks and strategies. The study found that students primarily used quadratic functions. Emerging techniques and strategies included grouping functions by artwork parts, using concepts of reflection and symmetry, combining animation and transformation, and integrating functions with other tools. The findings suggest that allowing students to create artwork through graphing can provide valuable insights into their mathematical skills. This study sheds light on the potential of GeoGebra as a tool and teaching aid for assessing students' mathematical abilities, and it offers valuable insights into students' strategies and techniques when creating mathematical art. This study revealed students' mathematical skills, strategies, and conceptual understanding through the creation of function art using GeoGebra.
Abstractor: As Provided
Entry Date: 2026
Accession Number: EJ1507465
Database: ERIC
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  Value: <anid>AN0194205337;ssm01jun.26;2026Jun03.05:17;v2.2.500</anid> <title id="AN0194205337-1">Function Art: Linking Mathematics, Technology, and Visual Arts </title> <p>This study investigated students' understanding of mathematical functions and strategies to create artwork using GeoGebra. It was framed by the principles of constructionism and examined how students use functions in creating artworks. We gathered data from students' artworks using the Algebra view and the Construction Protocol in the GeoGebra software. We used descriptive statistics to determine the number and types of functions used, whereas we employed thematic analysis to identify patterns and themes in students' artworks and strategies. The study found that students primarily used quadratic functions. Emerging techniques and strategies included grouping functions by artwork parts, using concepts of reflection and symmetry, combining animation and transformation, and integrating functions with other tools. The findings suggest that allowing students to create artwork through graphing can provide valuable insights into their mathematical skills. This study sheds light on the potential of GeoGebra as a tool and teaching aid for assessing students' mathematical abilities, and it offers valuable insights into students' strategies and techniques when creating mathematical art. This study revealed students' mathematical skills, strategies, and conceptual understanding through the creation of function art using GeoGebra.</p> <p>Keywords: art mathematics integration; function art; GeoGebra; STEAM education; transformation of functions</p> <p>Many educators believe that teaching science, mathematics, and technology is better complemented by the arts to enhance creativity and innovativeness among students. Research has shown that art integration provides a venue for self‐expression and personal connection (Land [<reflink idref="bib33" id="ref1">33</reflink>]) and enhances cognitive development (Baker [<reflink idref="bib6" id="ref2">6</reflink>]) as well as retention (Watson [<reflink idref="bib44" id="ref3">44</reflink>]). Brezovnik ([<reflink idref="bib14" id="ref4">14</reflink>]) has argued that art integration improves academic achievement. However, in visual arts, this emphasis primarily lies in integrating art with geometry, with fewer studies focusing on mathematical functions. This paper aims to contribute to filling this gap by focusing on how visual arts interact with mathematical functions. It highlights the integration of the transformation of functions into visual art using the GeoGebra application.</p> <hd id="AN0194205337-2">Background of the Study</hd> <p>This paper is based on a 3‐h webinar conducted by the first author. The webinar focused on using GeoGebra to integrate visual art into the learning of mathematical functions by creating function art. Function art refers to art constructions whose components include graphs or segments of graphs of mathematical functions (Bautista Jr., Prodromou, and Lavicza [<reflink idref="bib8" id="ref5">8</reflink>]; Bautista Jr., Prodromou, Raynes, et al. [<reflink idref="bib9" id="ref6">9</reflink>]). Grade 11 students from a Science High School in the Philippines attended this webinar. The emphasis of the webinar was on the transformation of functions, the effects of parameters of functions on their graphs, and functions with restricted domains to create artworks. To give an example of an artwork, the first author used GeoGebra to demonstrate how to draw a company logo using graphs of functions. Aside from graphs, the Circumcircular Arc tool was used for parts of the artwork where functions were difficult or impossible to use. Due to time constraints, the artwork was not completed during the webinar, but the students were encouraged to complete it independently.</p> <p>After the webinar, students were given 1 week to create a GeoGebra applet, an artwork project which we will refer to in this paper as Project 1. After submitting Project 1, a Function Art competition was conducted, and students were given 1 month to complete their artwork. We will refer to this artwork as Project 2. Project 1 was compulsory and was counted as one of the students' performance tasks which would be a part of their overall academic grade. Project 2 was an optional individual project, although students knew that additional marks would be awarded to those who would participate in the competition. From this point on, we refer to Projects 1 and 2 as GeoGebra artworks or simply artworks unless it becomes necessary to distinguish them. Students were encouraged to use functions for both artworks but were not formally required to. It was also emphasized that using functions would be one of the criteria for grading.</p> <p>This study contributes to STEAM education by introducing a novel approach to integrating mathematics and visual art. Unlike previous studies which mainly focused on the integration of geometry and visual art, this study emphasizes the transformation of functions which provides a fresh perspective on using mathematics software in creating art. Grounded in constructionism (Papert [<reflink idref="bib38" id="ref7">38</reflink>]), this study focuses on the students' preferences in using functions and the strategies they used in creating their artwork.</p> <p>The past studies indicate that (a) there is a concern about the lack of solid connection to content topics in many visual art integration studies (Clapp and Jimenez [<reflink idref="bib17" id="ref8">17</reflink>]; LaJevic [<reflink idref="bib32" id="ref9">32</reflink>]), (b) there are no explicit mentions of competencies in the transformation of functions in the Senior High School Philippine curriculum, and nonspecific competencies in the topic in Junior High School (Department of Education [<reflink idref="bib22" id="ref10">22</reflink>]), and (c) empirical studies are scarce in the integration of arts to mathematical functions (Bautista Jr., Prodromou, and Lavicza [<reflink idref="bib8" id="ref11">8</reflink>]). This study contributes to addressing these gaps. The lack of connection to content topics is addressed by explicitly selecting a particular mathematical content to be integrated into students' artwork. In the local context, this study can be a basis for student projects for Filipino mathematics teachers where the students can enjoy the arts and strengthen their knowledge of the transformation of functions. Finally, this study fills the gap in integrating mathematical functions and arts using technology.</p> <hd id="AN0194205337-3">Research Questions</hd> <p>There are many benefits of integrating arts into technology and visual arts (Brezovnik [<reflink idref="bib14" id="ref12">14</reflink>]). However, studies on integrating the use of mathematical functions and technology into visual arts are scarce. This paper aims to answer the following research questions.</p> <p></p> <ulist> <item> Which mathematical functions would students use in creating their artwork with GeoGebra?</item> <p></p> <item> How would the students use mathematical functions in their artwork (strategies), and which mathematical concepts and skills would be highlighted in creating these artworks?</item> </ulist> <p>This paper highlights how arts, technology, and mathematics are deeply integrated with a focus on mathematical concepts. In addition, it could provide a theme for analyzing students' artworks in terms of functions.</p> <p>The study involved Senior High School students from the northern part of the Philippines. A multiple‐case study design was employed to investigate their artwork. To answer the first research question, we tallied the number of functions and function types and applied descriptive statistics to analyze the data. For the second research question, we used thematic analysis to identify patterns in the students' strategies in using mathematical functions. We then conducted an in‐depth analysis of a smaller subset of artworks that best illustrate the range of identified strategies.</p> <hd id="AN0194205337-4">Literature Review</hd> <p>Constructionism posits that learning is best achieved through the active construction of knowledge using meaningful activities (Papert [<reflink idref="bib38" id="ref13">38</reflink>]). It highlights "independent creation, building, invention, and discovery of new concepts by students through interaction" (Levin et al. [<reflink idref="bib34" id="ref14">34</reflink>], 45). This process involves creating artifacts, such as function art, that reflect one's understanding and making connections between new information and existing knowledge.</p> <p>Constructionism aligns closely with the principles underpinning art integration in STEM activities. The John F. Kennedy Center for the Performing Arts defined art integration as "an approach to teaching in which students construct and demonstrate understanding through an art form. Students engage in a creative process which connects an art form and another subject area and meets evolving objectives in both" (Silverstein and Layne [<reflink idref="bib42" id="ref15">42</reflink>], 3). Deasy ([<reflink idref="bib20" id="ref16">20</reflink>]) defined art integration as "the effort to build a set of relationships between learning in the arts and learning in the other skills and subjects of the curriculum" (p. 3). Both definitions highlight learning in both the arts and the subject matter. When appropriately implemented, art integration may result in a deeper understanding of the arts and at least one subject area (Dell'Erba [<reflink idref="bib21" id="ref17">21</reflink>]). In our study, the "Art" integrated in STEM is function art.</p> <p>Constructionism emphasizes learning through the creation of artifacts and models, a principle that aligns with the use of GeoGebra, a dynamic mathematics software. GeoGebra enables students to construct mathematical objects and explore their properties. It can be used to create microworlds (Carvalho et al. [<reflink idref="bib15" id="ref18">15</reflink>]) where students can build their knowledge and skills interactively. For instance, Abar et al. ([<reflink idref="bib1" id="ref19">1</reflink>]) demonstrated how GeoGebra facilitates learning through the construction of mosaics using isometric transformations. Their study highlighted how students engaged in hands‐on exploration and applied mathematical concepts while creating digital artwork. Schmid et al. ([<reflink idref="bib41" id="ref20">41</reflink>]) demonstrated that integrating GeoGebra with augmented reality and virtual reality enhances students' visualization skills. Similarly, Yunianto et al. ([<reflink idref="bib46" id="ref21">46</reflink>]) explored how GeoGebra facilitates computational thinking, promoting a constructionist learning environment where students can explore and build mathematical ideas about polygons inscribed in circles. These studies underscore the effectiveness of GeoGebra in fostering a constructionist environment.</p> <hd id="AN0194205337-5">GeoGebra as a Dynamic Mathematics Software</hd> <p>GeoGebra is a free dynamic mathematics software that combines geometry, algebra, spreadsheets, graphing, statistics, and calculus in one software (Hohenwarter et al. [<reflink idref="bib28" id="ref22">28</reflink>]). It offers the advantage of being supported by multiple operating systems, hardware, and synchronized representations (Bautista Jr. et al. [<reflink idref="bib10" id="ref23">10</reflink>]). In a literature review by Wassie and Zergaw ([<reflink idref="bib43" id="ref24">43</reflink>]), for the past two decades, GeoGebra has been used by teachers and students as a tool in teaching and learning algebra, calculus, 2D and 3D geometry, mathematical functions, transformations, and proofs. Further, studies show that GeoGebra can improve student achievement and motivation (Arbain and Shukor [<reflink idref="bib4" id="ref25">4</reflink>]; Nzaramyimana et al. [<reflink idref="bib37" id="ref26">37</reflink>]), develop students higher‐order thinking skills (Yohannes and Chen [<reflink idref="bib45" id="ref27">45</reflink>]), and help visualize abstract concepts (Dahal et al. [<reflink idref="bib18" id="ref28">18</reflink>]). It also provides teachers with tools to explain mathematical concepts (Kllogjeri and Kllogjeri [<reflink idref="bib30" id="ref29">30</reflink>]) such as the transformation of functions, which is crucial in creating function art.</p> <p>GeoGebra was selected for this study due to its capability to support multidirectional representations. This feature aligns with our objective of integrating mathematical functions into artistic creations. For instance, a student can simply type the equation of a line in the Algebra view, and GeoGebra will plot the graph in real‐time. Conversely, in the Graphics view, students can draw a line using the Line tool, and GeoGebra will automatically derive the corresponding equation. This multidirectional functionality provides students with a more versatile toolkit for creating their artwork.</p> <p>Moreover, GeoGebra's ability to display both graphs and geometric objects, such as polygons, within a single window offers a rich dynamic environment for students. This functionality provides them with more options in creating their artwork, such as allowing a combination of geometric shapes and function graphs. Its environment which includes the Graphics view, the Algebra view, the Construction Protocol (CP) among others, and its features such as grouping objects by type, by construction order, and by layer, allow researchers to investigate the student's artwork from multiple perspectives.</p> <p>Although GeoGebra has traditionally been used for graphing and mathematical problem‐solving, its application in creating artistic tasks is relatively unexplored. In this study, we challenge students to use GeoGebra in an unconventional way—to produce artworks made from functions and geometric objects.</p> <hd id="AN0194205337-6">Arts Integration, STEM, and STEAM</hd> <p>There is a growing interest in arts integration, particularly in Science, Technology, Engineering, and Mathematics (STEM) education. In recent years, the integration of Arts into STEM has become popular among educators and teachers, resulting in the Science, Technology, Engineering, Arts, and Mathematics (STEAM) paradigm. The primary purpose of the "A" in STEAM is to "improve student engagement, creativity, innovation, and problem‐solving skills" (Perignat and Katz‐Buonincontro [<reflink idref="bib39" id="ref30">39</reflink>], 31). However, despite its widespread popularity, STEM and STEAM have broad and sometimes ambiguous definitions (Aguilera and Ortiz‐Revilla [<reflink idref="bib2" id="ref31">2</reflink>]). In addition, the incorporation of Arts within these disciplines has often been marginalized and overlooked (Clapp and Jimenez [<reflink idref="bib17" id="ref32">17</reflink>]). For instance, research by Liu et al. ([<reflink idref="bib35" id="ref33">35</reflink>]) revealed that arts and aesthetics are often treated as inferior to other components within STEAM activities. In this paper, the authors coined terms such as "STEM‐with‐stickers" and "artless‐STEM effects" to describe specific activities that lacked or inadequately integrated art elements. In particular, some "art integration" activities let students decorate their work using colored or electric engraving pens. In these unfavorable cases, arts integration was reduced to mere decorations and superficial aesthetics (LaJevic [<reflink idref="bib32" id="ref34">32</reflink>]).</p> <p>We address the gaps above through function art. In function art, we elevate the role of art by making it the focal point of the student's activity. While creating their artwork, students select appropriate curves by manipulating the parameters of functions, applying what they have learned about the properties of functions. In this activity, students are active creators experimenting with different mathematical functions. This active creation is the cornerstone of constructionism, where learners build knowledge through active engagement.</p> <hd id="AN0194205337-7">Visual Arts and Mathematical Functions</hd> <p>Mathematical function in school curricula covers a wide range of topics and concepts. In the Philippine mathematics curriculum, students are expected to study the different types of functions (e,g. polynomial, trigonometric, exponential, logarithmic) from Grade 8 to Grade 12 (Department of Education [<reflink idref="bib22" id="ref35">22</reflink>]). With these functions, students are expected to study their respective graphs' properties through transformation and other means. However, there is a lack of clear emphasis on the transformation of functions in the Philippine mathematics curriculum. For instance, in the competency with code "M8AL‐If‐3," although the students are expected to "describe the graph of linear equations in terms of its intercepts and slope" (Department of Education [<reflink idref="bib22" id="ref36">22</reflink>], 222), there is no mention of the analysis of the effects of <emph>m</emph> and <emph>b</emph> in <emph>f</emph>(<emph>x</emph>) = <emph>mx</emph> + <emph>b</emph> on its graph. Further, there is no explicit mention of the transformation of functions in the Senior High School curriculum (Department of Education [<reflink idref="bib23" id="ref37">23</reflink>], [<reflink idref="bib24" id="ref38">24</reflink>]).</p> <p>Although numerous studies have explored the integration of visual arts into mathematics, limited research has focused on integrating arts into the teaching of mathematical functions. In the authors' search for related literature from ERIC, JStor, Google Scholar, and Proquest databases, using the keywords function art, function visual art, algebra art, algebra visual art, function algebra art, and function graph art, we found only seven accessible articles on the integration of arts into functions among them only five were on visual arts. We discuss them below.</p> <p>Davis and Joswick ([<reflink idref="bib19" id="ref39">19</reflink>]) conducted one notable study on the integration of arts into functions. In this study, the authors used the concept of "line of sight" and the Perspective Projection Rule (PPR), a technique used by computers to map points on the horizontal plane onto the vertical plane (the computer screen). In the lesson, the students used images on paper and imagined lines that would connect them to the eye of the observer. This approach emphasized the correspondence between the points on the horizontal and the vertical planes, which was then used to introduce the concept of functions, inverse functions, domain and range, and one‐to‐one correspondence.</p> <p>Three studies sharing similarities with this paper were conducted by Miller ([<reflink idref="bib36" id="ref40">36</reflink>]), Beige ([<reflink idref="bib11" id="ref41">11</reflink>]), and Aktümen and Yildiz ([<reflink idref="bib3" id="ref42">3</reflink>]). All these studies involved students using functions to create artwork. In Miller's study, students modified the parameters of a pre‐made applet of a parametric function to create Frieze patterns. In Beige's study, students used algebraic function graphs to create artwork. Students could work with various graphs, use symmetry and reflections, and join function graphs to create artworks by restricting the functions' domains. Both studies are descriptive, and although the authors claimed that students had increased their understanding of functions more profoundly, no empirical evidence was discussed.</p> <p>Aktümen and Yildiz ([<reflink idref="bib3" id="ref43">3</reflink>]) conducted a study on using GeoGebra by pre‐service teachers to create aesthetically pleasing visuals using graphs of functions. The authors reported that the participants could create beautiful patterns using function graphs, deepened their understanding of mathematical concepts, and gained self‐esteem during the process.</p> <p>This paper shares similarities with the three studies above. However, it introduces a new approach by allowing students to use geometric tools in addition to function graphs. This is possible because GeoGebra enables the creation of functions and geometric objects in the same window. This leeway is grounded on building the expectation of success by creating a perception of personal control (Keller [<reflink idref="bib29" id="ref44">29</reflink>], 50), a core tenet of the constructionist approach. The authors believe that limiting the students to use functions in creating their artwork over a short period could lead to frustration. By allowing more flexibility, students may feel empowered and motivated to engage with the task.</p> <hd id="AN0194205337-8">Conceptual Framework</hd> <p>Our unit of analysis in this study is the function and we use Figure 1 as our conceptual framework. The conceptual framework for categorizing functions is represented by a 3‐dimensional coordinate system with each axis representing a different attribute of functions. The <emph>x</emph>‐axis represents grade levels at which functions are taught in the Philippine K‐12 Mathematics Curriculum Guide (PKMCG) (Department of Education [<reflink idref="bib22" id="ref45">22</reflink>]). The <emph>y</emph>‐axis categorizes the appearance of the function as straight‐horizontal (straight1), straight non‐horizontal (straight2) and curved. We distinguish between horizontal and non‐horizontal graphs due to the common misconceptions about constant functions. For example, in some studies, students do not recognize constant functions as functions.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/SSM/01jun26/ssm18373-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="ssm18373-fig-0001.jpg" title="1 The conceptual framework used in this study." /> </p> <p></p> <p>The <emph>z</emph>‐axis encompasses other properties of functions, such as symmetry, periodicity, and asymptotic behaviors. In this paper, the term "symmetry" refers specifically to reflection symmetry. For example, the graphs of polynomial functions with odd degrees greater than or equal to 3 (see poly1 in Figure 1), taught in Grade 10, are nonsymmetric curves and thus are located at the intersection of G10 and curve coordinates on the <emph>xy</emph>‐plane. Conversely, the graphs of polynomial functions with even degrees greater than or equal to 3 are classified as symmetric and placed directly above their corresponding curve on the <emph>z</emph>‐axis. To enhance clarity in locating points within the framework, we have included gridlines on the <emph>xy</emph>‐plane to facilitate identification of their classifications on the <emph>x</emph>‐ and <emph>y</emph>‐axes.</p> <p>The definitions of functions in the PKMCG follow the standard mathematical definitions of functions. For instance, linear functions are functions of the form <emph>f</emph>(<emph>x</emph>) = <emph>mx</emph> + <emph>b</emph> (p. 183) where <emph>m</emph> and <emph>b</emph> are real numbers, and quadratic functions are of the form <emph>f</emph>(<emph>x</emph>) = <emph>ax</emph><sups>2</sups> + <emph>bx</emph> + <emph>c</emph> (p. 196) where <emph>a</emph>, <emph>b</emph>, and <emph>c</emph> are real numbers and <emph>a</emph> is not equal to 0. However, we discarded functions that were constructed geometrically and only included functions that were created using equations. For instance, when plotting a linear graph in GeoGebra, two methods are available: typing the equation directly or using the Line tool. Our analysis deliberately excludes the latter method, because it does not contribute to the comprehension of functions and their mathematical properties.</p> <p>The authors categorized the functions strategically based on the conceptual framework in Figure 1. We separated linear, quadratic, and other polynomial functions, because they are taught separately and in different grade levels and their properties differ. The linear function was also separated from constant functions because of its use in the artworks and the misconceptions surrounding functions. This structured approach enabled a detailed examination of the frequency and types of functions used in the students' artwork, providing insights into their application of mathematical concepts.</p> <hd id="AN0194205337-10">Materials and Methods</hd> <p>The study aimed to determine which functions the students would use in creating artwork and how they would use these functions. It utilized a multiple case–study design and followed the constructionist approach (Papert [<reflink idref="bib38" id="ref46">38</reflink>]). The unit of analysis was functions. We used a parallel mixed methods study with the quantitative descriptive statistics informing the qualitative coding of the functions involved for each figure to demonstrate the critical connection between artistic growth and constructionism's influence on student autonomy.</p> <p>The study involved 123 Grade 11 students from a school in the northern part of the Philippines. The students were from three classes taught by the same teacher in their mathematics subject. The teacher was already using GeoGebra in these classes before the webinar, so most of the students were already familiar with it. Additionally, these classes were selected, because all of them had access to technology. Hence, the selection of participants follows the purposive and convenience sampling techniques.</p> <p>The students attended a 3‐h webinar on function art which covered the basics of GeoGebra, a review of the transformation of functions (including vertical and horizontal dilations) and a demonstration of creating a simple artwork. After the webinar, the students had 1 week to complete their first artwork (Project 1) and 1 month to complete their second artwork (Project 2). For both projects, students used GeoGebra to draw artwork using function graphs and other tools. In Project 1, 52 artworks were submitted, but one was excluded for not being original. Of the remaining submissions, 24 utilized functions. In Project 2, 29 students participated, with 18 incorporating functions. To analyze students' use of functions, we identified the artworks containing mathematical functions and tallied the function types used in each applet. After tallying, we examined the strategies students used in utilizing functions to create artwork.</p> <p>Table 1 shows the different phases of the analysis.</p> <p>1 TABLE Phases and steps in the analysis of students' artworks.</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left">Phase</th><th align="center">Step</th><th align="center">Description</th></tr></thead><tbody valign="top"><tr><td align="left">Quantitative (Phase 1)</td><td align="center">1</td><td align="center">Collected students' artwork (Project 1: 51, Project 2: 29).</td></tr><tr><td align="center">2</td><td align="center">Identified artworks containing mathematical functions (Project 1: 24, Project 2: 18, total: 42)</td></tr><tr><td align="center">3</td><td align="center">Tallied the types and number of functions contained in each artwork in (2)</td></tr><tr><td align="left">Qualitative (Phase 2)</td><td align="center">4</td><td align="center">Selected all artworks with functions for further analysis</td></tr><tr><td align="center">5</td><td align="center">Coded 42 artworks based on visual features (e.g., symmetry, animation)</td></tr><tr><td align="center">6</td><td align="center">Examined the algebraic properties of functions and other objects in 42 artworks; examined animations in both artworks with and without functions</td></tr><tr><td align="left">Integration (Phase 3)</td><td align="center">7</td><td align="center">Synthesized qualitative coding results with function tally data</td></tr><tr><td align="center">8</td><td align="center">Interpreted how mathematical function use related to visual and strategies</td></tr><tr><td align="center">9</td><td align="center">Connected both qualitative and quantitative data sets in the discussion</td></tr></tbody></table> </ephtml> </p> <p>To answer research question 1, two of the authors independently tallied and categorized the functions in students' artworks. Using the categories in Figure 1, we tallied the number of functions per category. First, we examined the Algebra view and used the "Sort by Object" feature of GeoGebra to isolate functions from other objects. We then coded each artwork by counting the number of functions per category and noting which function was present in each artwork. After coding, we checked for any discrepancies and resolved them. As the categories of functions are clearly defined, interrater reliability was not required. Descriptive statistics such as percentage, mean, median, and standard deviation were used to summarize the types of functions students used in their artwork.</p> <p>To answer research question 2, we conducted thematic analysis (Braun and Clarke [<reflink idref="bib13" id="ref47">13</reflink>]) to examine students' strategies in using functions in their artwork. The analysis went through five phases: familiarizing with the data, generating initial codes, searching and reviewing themes, defining and naming themes, and writing the report (Braun and Clarke [<reflink idref="bib13" id="ref48">13</reflink>]).</p> <p>For each artwork, we coded the strategy used by students in utilizing functions. To do this, firstly, we analyzed the visual appearance of the artwork and coded notable features and strategies (e.g., symmetry, animation). Secondly, we used the Algebra view to examine algebraic properties of the curves particularly functions. Thirdly, we examined other mathematical objects and how students linked them to functions (e.g., how a line segment is connected to linear functions). Lastly, we looked further at the Construction Protocol to see the order of construction. Throughout the process, coding and recoding were performed continuously to refine the emerging themes. We linked quantitative data and qualitative data by looking at the function tally and strategies. For example, we paid attention to functions that are commonly used, such as quadratic functions. The final codes were grouped into themes representing student strategies. The themes of strategies that emerged were grouping functions by artwork parts, animating functions, exploiting transformations, and combining functions with other tools.</p> <hd id="AN0194205337-11">Results</hd> <p>This section presents the students' outputs aligned with the research questions. All the students' names in the discussion are pseudonyms.</p> <hd id="AN0194205337-12">Types of Functions Used in Artworks</hd> <p>In Project 1, 24 out of the 51 artworks submitted by the students (47.1%) incorporated mathematical functions in their artwork. For Project 2, although 29 students participated in the competition, only 18 (62.1%) used functions in their submissions. Table 2 shows the statistics on the number of functions used by the students.</p> <p>2 TABLE Statistics on the number of functions.</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left">Statistics</th><th align="center">P1 (<italic>N</italic> = 24)</th><th align="center">P2 (<italic>N</italic> = 18)</th></tr></thead><tbody valign="top"><tr><td align="left">Mean</td><td align="center">11.28</td><td align="center">18.88</td></tr><tr><td align="left">Median</td><td align="center">9</td><td align="center">9</td></tr><tr><td align="left">SD</td><td align="center">8.21</td><td align="center">25.07</td></tr></tbody></table> </ephtml> </p> <p>Twenty‐four artworks used mathematical functions in Project 1, and the number of functions ranged from 1 to 32 functions per artwork. The average number of functions per artwork was 11.28, and the standard deviation was 8.21. On the other hand, for Project 2, there were 18 artworks, and the number of functions ranged from 1 to 99 per artwork. The average number of functions was 18.88, and the standard deviation was 25.07. Although on average, the number of functions increased from Project 1 to Project 2, the standard deviations were large in both Projects, which means that the number of functions used by the students varied widely among artworks.</p> <p>Figure 2 displays the frequency of the specific function types in the artworks. Quadratic functions were the most popular, appearing in 19 artworks, followed by cosine functions in 11 artworks and sine functions in 10 artworks. The same functions were also the most popular in Project 2. In total, 34 artworks used quadratic functions, 27 (79.4%) used vertex form, 6 (17.6%) used standard form, and 1 (3%) used both. Interestingly, many students chose to draw straight‐line segments using the Segment tool instead of typing the equation of linear functions. Seven artworks used linear functions in Project 1 and only one artwork in Project 2.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/SSM/01jun26/ssm18373-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="ssm18373-fig-0002.jpg" title="2 The number of GeoGebra applets (artworks) that use specific types of functions." /> </p> <p></p> <hd id="AN0194205337-14">Student Strategies in Using Functions</hd> <p>Students used four strategies, which emerged during the thematic analysis, in using functions to construct their artwork. They grouped the functions by artwork parts, used symmetry, reflection, translation, animated functions using the GeoGebra slider, and combined functions with other tools. Constructionist ideas of learning by doing and making are embodied in these strategies. They show how students internalized mathematical principles and leveraged technology while creating their artwork. The following details the strategies that they have used in creating their artwork.</p> <hd id="AN0194205337-15">Grouping Functions by Artwork Parts</hd> <p>There were five artworks where functions or geometric objects were grouped by artwork parts. In Figure 3, for example, Ellen used different functions/tools for specific parts of her artwork. For instance, she used linear functions to draw the lines on the neck, and then she used Circumcircular arcs to draw the curves in Naruto's hair, as shown below.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/SSM/01jun26/ssm18373-fig-0003.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="ssm18373-fig-0003.jpg" title="3 Naruto (anime character) artwork that uses linear and quadratic functions." /> </p> <p></p> <p>She continued the same method in Project 2, using quadratic functions to construct the lower parts of the character's hair, the headband, and the shirt. Another student used a similar strategy where she used functions to draw the lower part of the bird's feathers on its tail and the Circumcircular arc tool for the upper parts.</p> <hd id="AN0194205337-17">Exploiting Symmetry, Reflection, and Translation</hd> <p>There were eight symmetrical artworks: six perfectly symmetrical and two partially symmetrical. The partially symmetrical artworks had only some of their components exhibiting symmetry or could be made symmetrical, as seen in Figure 3. For example, the whisker markings on Naruto's cheeks are symmetrical about the <emph>y</emph>‐axis, whereas the nose and hair are not.</p> <p>Quadratic functions were commonly used for both symmetrical and partially symmetrical artworks. Of the 8 artworks, 6 utilized the vertex form, whereas 2 used the standard form. One example of the latter is the Spiderman artwork created by Lira, in which quadratic functions were used to achieve symmetry and reflection, as shown in Figure 4. This piece was one of only two artworks composed entirely of functions.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/SSM/01jun26/ssm18373-fig-0004.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="ssm18373-fig-0004.jpg" title="4 Artwork that uses symmetry of functions." /> </p> <p></p> <p>In her artwork, Lira used linear and quadratic functions to outline Spiderman's head and mask. For quadratic functions, she created symmetric curves that do not intersect the <emph>y</emph>‐axis by generating a curve then reflecting it across the <emph>y</emph>‐axis. To do this, she alternated the sign of <emph>b</emph> in the standard form of quadratic equation <emph>f</emph>(<emph>x</emph>) = <emph>ax</emph><sups>2</sups> + <emph>bx</emph> + <emph>c</emph> and reversed the order and the signs of the domain interval. Case in point, for the lower borders on the eyes, she used the quadratic equations <emph>s</emph><subs>1</subs>(<emph>x</emph>) = 0.52<emph>x</emph><sups>2</sups> – 1.09<emph>x</emph> and <emph>t</emph><subs>1</subs>(<emph>x</emph>) = 0.52<emph>x</emph><sups>2</sups> + 1.09<emph>x</emph> and the domains [0.19, 2.26] and [−2.26, −0.19], respectively (see Figure 4). This strategy was also used in Naruto's whiskers in Figure 3. For symmetric curves that intersect the <emph>y</emph>‐axis, she used the quadratic function of the form <emph>f</emph>(<emph>x</emph>) = <emph>ax</emph><sups>2</sups> + <emph>c</emph> which inherently produce symmetry about the <emph>y</emph>‐axis. This form was also employed in other two symmetric artworks.</p> <p>Although most of the students carefully graphed the symmetric artworks with precision, they did not apply the same level of accuracy to the other objects. In Ryan's artwork in Figure 5, for example, the functions are symmetric about the <emph>y</emph>‐axis, but the circles and squares are not.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/SSM/01jun26/ssm18373-fig-0005.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="ssm18373-fig-0005.jpg" title="5 The crown artwork." /> </p> <p></p> <p>A total of 11 artworks featured translations, with the number of translated functions ranging from 2 to 6. There were eight artworks with vertical translations (e.g., whiskers in Figure 3) and three with horizontal translations (e.g., curves on the crown in Figure 4). In Figure 3, Ellen used the quadratic form <emph>f</emph>(<emph>x</emph>) = <emph>ax</emph><sups>2</sups> + <emph>bx</emph> + <emph>c</emph> and varied <emph>c</emph> to vertically translate the two lower whiskers. In Figure 5, Ryan used the vertex form of the quadratic equation <emph>f</emph>(<emph>x</emph>) = <emph>a</emph>(<emph>x</emph> − <emph>h</emph>)<sups>2</sups> + <emph>k</emph> and adjusted <emph>h</emph> to achieve horizontal translation, subsequently modifying the domains of the translated function to create symmetric curves (functions <emph>f</emph> and <emph>i</emph> serve as examples).</p> <hd id="AN0194205337-20">Animating Functions</hd> <p>Approximately 29% of the artworks (23 out of 80) included animations, with 11 in Project 1 and 12 in Project 2, though most of these animations featured circles, polygons, and line segments. Only three students used function animations to generate motion in their artwork, all of whom employed trigonometric functions. In Figure 6a, Jasmine used the slider tool to translate the sine function horizontally, resulting in an oscillating motion effect. Additionally, she used the slider to expand and shrink the ghost's mouth and moved its eyes back and forth to create a spooky, animated appearance.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/SSM/01jun26/ssm18373-fig-0006.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="ssm18373-fig-0006.jpg" title="6 Artworks that use animations." /> </p> <p></p> <p>In Figure 6b, Maria used parametric equations and the slider tool to change the domain of a sine function, creating a slithering halo effect around the hand of the character to depict the flow of energy. She used the slider <emph>t</emph> to animate a number and used set up the domain interval to [<emph>t</emph>, <emph>t</emph> + 1].</p> <hd id="AN0194205337-22">Combining Functions With Other Tools</hd> <p>Many artworks revealed students' understanding of the concept of the domain of functions. For example, in the crown artwork in Figure 5, Ryan used three quadratic functions instead of two from the first and third circles, so he set the domain of the first function to [−7.34, −5.99], the second function to [−5.99, −3.58] and the third function to [−3.58, 0]. He also did this to the right side of the artwork by reversing the components and the sign of the domain intervals.</p> <p>Some students used a combination of objects to attend to precision and details. In Teresita's artwork shown in Figure 7a, she used a combination of semicircles, semi‐circular arcs, and function graphs to create the Spotify logo.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/SSM/01jun26/ssm18373-fig-0007.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="ssm18373-fig-0007.jpg" title="7 The Spotify logo artwork." /> </p> <p></p> <p>In Figure 7b, the solid curves are graphs of functions with varying thicknesses to show the use of multiple functions. On the other hand, the dotted and dashed curves are geometric objects created with Circular arc, Circumcircular arc, and Semicircle tools in GeoGebra. Despite the variety of tools, Teresita seamlessly integrated them to create fluid curves. However, some students needed help to define the domain of the functions. For instance, one student failed to use the domain effectively, resulting in many overlapping curves in the artwork.</p> <hd id="AN0194205337-24">Learnings From the Construction Protocol</hd> <p>The CP of GeoGebra automatically records the steps taken during the applet's construction, allowing users to replay them using the arrows at the bottom of the CP window. Users can observe the objects being constructed at each step in the Graphics view. Shown in Figure 8 are the first 11 steps in constructing the Spiderman applet created by Lira. The CP window on the right displays the steps, whereas the corresponding curves appear in the Graphics view on the left. Each curve is numbered according to its corresponding step, and the small black circles indicate the junction points of these curves.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/SSM/01jun26/ssm18373-fig-0008.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="ssm18373-fig-0008.jpg" title="8 The first 11 steps of the Spiderman artwork." /> </p> <p></p> <p>As shown in the figure, Lira began the construction by plotting the curve on the top of the head, followed by the curve on the jaw. Then, she plotted the four straight‐line segments (linear functions) as indicated by the numbers 3–6. Subsequently, she alternated the signs of <emph>b</emph> in the quadratic functions of the form <emph>f</emph>(<emph>x</emph>) = <emph>ax</emph><sups>2</sups> + <emph>bx</emph> + <emph>c</emph> and adjusted their domains appropriately to plot symmetric curves about the <emph>y</emph>‐axis. The adjustments are indicated by the construction order of Curves 8 and 9 and then Curves 10 and 11. From then on, the CP revealed that she plotted the curves on one side and then reflected them on the <emph>y</emph>‐axis to create the curve on the opposite side. She consistently used this approach until the artwork was completed. This approach was also used by the creator of the Naruto function art (Figure 3) in creating the symmetric parts of the artwork.</p> <p>Ryan, the Crown function art creator in Figure 5, used a different method. Rather than using the standard form of the quadratic equation, he used the vertex form <emph>f</emph>(<emph>x</emph>) = <emph>a</emph>(<emph>x</emph> − <emph>h</emph>)<sups>2</sups> + <emph>k</emph>, alternated the sign of <emph>h</emph>, and then adjusted the domain appropriately to reflect the curve on the <emph>y</emph>‐axis. The CP revealed that he created all the curves on the right first (from center to rightmost) and then created all the curves on the left (from center to leftmost). While constructing all the curves on one side before reflecting them on the <emph>y</emph>‐axis may be more time‐consuming than Lira's approach, the number of functions (eight in total) made the time difference negligible.</p> <hd id="AN0194205337-26">Discussion</hd> <p>The following discussion interprets the key findings of the study, highlighting their implications for mathematics education and the integration of function art in STEAM learning. Of the 80 artworks submitted in both projects, only 42 (52.5%) utilized functions. The main reason for this relatively low percentage is likely the lack of emphasis on incorporating functions in grading the students' work. Research indicates that Filipino students are highly grade‐conscious (Bernardo [<reflink idref="bib12" id="ref49">12</reflink>]), suggesting that they would likely have created more functions if the teacher had explicitly stated that it would be included in the grading system. Additionally, the user‐friendly nature of GeoGebra may have contributed to this outcome. Unlike other apps, which feature separate graphing and geometry apps, GeoGebra combines both functionalities in the same app, making it quite tempting for students to rely primarily on geometric tools when creating their artwork. This is likely the reason why many students opted to use the Segment tool to draw straight‐line segments rather than graphs of linear functions.</p> <p>The increase in the number of students using functions and the average number of functions per applet from Project 1 to Project 2 suggests a potential improvement in students' familiarity and confidence in utilizing functions to create their artwork. However, despite this progress, the standard deviation indicates a need for more consistency among students in using functions. This inconsistency may be attributed to the considerable freedom given to students, allowing them to choose any number of functions for their artworks. Considering the substantial standard deviation, it may be beneficial to consider imposing a minimum requirement for the number of functions in future project iterations. Additionally, future research could explore how imposing such restrictions might influence students' creativity and problem‐solving approaches in function art.</p> <hd id="AN0194205337-27">Types of Functions Used in Artworks</hd> <p>The results revealed that the quadratic function is the most used function. Exponential and logarithmic functions are the least used even though the students were learning them during the implementation of this study and had learned quadratic functions 2 years before the implementation. This finding suggests a strong preference for quadratic functions.</p> <p>There are two possible reasons for students' preference for quadratic functions. The first is the emphasis on quadratic functions both in the Philippine mathematics curriculum (Department of Education [<reflink idref="bib22" id="ref50">22</reflink>]) and in the actual teaching time of teachers (Department of Education [<reflink idref="bib25" id="ref51">25</reflink>]). The quadratic function has 15 competencies and is allotted 4 weeks in the teacher's work budget. Due to this longer exposure, the students may have internalized the concepts of quadratic functions. On the other hand, exponential and logarithmic functions are less emphasized, with only 1 week of work budget for each.</p> <p>The second reason is that students may have recognized the advantages of quadratic functions such as their ease of manipulation, particularly the vertex form <emph>f</emph>(<emph>x</emph>) = <emph>a</emph>(<emph>x</emph> − <emph>h</emph>)<sups>2</sups> + <emph>k</emph>. For instance, Ryan used this vertex form on the crown artwork, because it is easy to translate horizontally and vertically just by adjusting the values of <emph>h</emph> and <emph>k</emph>. Even though it is possible for him to use an exponential or logarithmic functions in his artwork, he would need two functions for each of the four curves instead of one.</p> <hd id="AN0194205337-28">Student Strategies in Using Functions</hd> <p>The strategies that students used are graphing functions by artwork parts, exploiting reflection, symmetry, and translation, animating functions, and combining functions with other tools. The mathematical concepts and skills that are highly visible in the students' artworks are the application of symmetry, reflection, and domain.</p> <p>For symmetry and reflection, although some students opted for perfectly symmetrical artwork, like the Spiderman illustration in Figure 4, others chose to incorporate symmetry only in specific objects. For instance, in Figure 3, Ellen ensured that Naruto's facial whiskers made of functions were symmetrical by reflecting them on the <emph>y</emph>‐axis. However, she deliberately maintained asymmetry in other elements such as the nose and certain hair parts, possibly for artistic reasons. This indicates her awareness of symmetry and thoughtful consideration of the balance between artistic expression and mathematical principles (Baur and Fellner [<reflink idref="bib7" id="ref52">7</reflink>]; Galbraith and Jones [<reflink idref="bib27" id="ref53">27</reflink>]).</p> <p>Although studies show students have difficulties understanding domain and range (Baker et al. [<reflink idref="bib5" id="ref54">5</reflink>]), some students leveraged the domain of functions to make their work as precise as possible. Teresita even combined the function graph with different GeoGebra tools to draw seamless curves in creating the Spotify logo in Figure 7. For Teresita to draw such seamless curves, she had to examine the coordinates of the endpoints of the semicircles and the arcs and then consider them in restricting the domains of the functions. It is likely that she spent a lot of time and effort on creating them and therefore supports the objective of STEAM in student engagement and problem‐solving skills (Perignat and Katz‐Buonincontro [<reflink idref="bib39" id="ref55">39</reflink>]). In contrast, despite their understanding of the symmetry of functions, some students sometimes ignore precision if it does not affect the visual appearance of the artwork itself. For example, although Lira clearly demonstrated knowledge of the symmetry of functions and domains in her artwork in Figure 4, she did not mind the 0.01 difference in the endpoints of <emph>p</emph><subs>1</subs> and <emph>q</emph><subs>1</subs>.</p> <p>The concept of domain was used strategically in linear and quadratic functions, particularly for symmetry and reflection. For symmetric artworks, the domain [−<emph>d</emph>, <emph>d</emph>] was used for symmetric curves about the <emph>y</emph>‐axis (e.g., Spiderman). The domain pair [<emph>p</emph>, <emph>q</emph>] and [−<emph>q</emph>, −<emph>p</emph>] were used for curves that are reflections of each other across the <emph>y</emph>‐axis (e.g., Crown artwork, Naruto's whiskers).</p> <p>For translation, although the number of artworks with vertical translation outnumbered those with horizontal, and literature suggests that the former is easier to understand than the latter for students (Lage and Gaisman [<reflink idref="bib31" id="ref56">31</reflink>]), we cannot make any claims from this study, because the opportunities for translation depend on the artwork. For example, in Figures 4 and 7, there were many opportunities for vertical translation, but very few for horizontal translation. In contrast, Figure 5 only offers the opportunity for horizontal translation.</p> <p>One of the most exciting animations among the students' outputs is the sinuous, halo‐like energy depicted by the sine function's slithering effect. Although it seems unlikely that the student derived the equation of the curve, as parametric forms are not covered in their class, it was fascinating to witness her skill in manipulating the function's domain to achieve the slithering effect.</p> <p>This study highlights the efficacy of GeoGebra in analyzing students' artwork. The software's simultaneous multiple representations display enabled the authors to thoroughly examine the properties of functions and objects utilized by the students in their artwork. The ability to group objects based on their construction order, type, and dependency further facilitated a comprehensive and detailed analysis of the artwork. Moreover, the CP, listing the step‐by‐step construction of artworks, was a valuable feature for gaining meaningful insights into how the students crafted their artwork.</p> <hd id="AN0194205337-29">Conclusions and Implications</hd> <p>This study explored how integrating mathematical functions with art using GeoGebra can enhance students' understanding of mathematical concepts. By analyzing students' artwork, we identified that the most used mathematical functions were quadratic functions, and the least popular were exponential and logarithmic functions. The study also uncovered various strategies that students employed, such as grouping functions by artwork parts, using symmetry and reflection, animating vertically and horizontally, and combining functions with other tools. These findings highlight the potential of integrating technology, art, and mathematics to foster creative and analytical skills in students.</p> <p>This study provides insights into how students applied mathematical concepts through function art, particularly in their use of transformations, symmetry, and mathematical representations. By analyzing students' function art projects, we identified strategies that reflect their mathematical understanding and decision‐making, contributing to discussions on the role of technology in STEAM education. GeoGebra's multidirectional representation capabilities provide students with the opportunity to transition between algebra and geometry in both directions. By enabling the incorporation of both geometric objects and algebraic graphs in one window, GeoGebra expands students' strategic options, providing a larger repository of tools that can bridge abstract mathematical concepts and artistic expression.</p> <p>Furthermore, GeoGebra promotes interactive learning by enabling students to visualize and manipulate mathematical functions dynamically, fostering deeper engagement and understanding. These contributions demonstrate GeoGebra's potential to enrich mathematics education, encouraging both creativity and conceptual comprehension, and paving the way for future research in the integration of technology and creative arts in education.</p> <p>This study contributes to addressing some gaps in the literature. Firstly, it strengthens the A in STEAM by providing the students with the context to reinforce their understanding of mathematical concepts such as domain, symmetry, reflection, and translation of functions. Secondly, in the local context, it provides a project‐based model teachers can use to address the curriculum's lack of lessons in function transformation. Lastly, the study addresses the lack of a strong connection between art and content topics (Catchen and DeCristofano [<reflink idref="bib16" id="ref57">16</reflink>]) by selecting a specific mathematical content that targets specific mathematical skills students can use to create artwork.</p> <p>The limitation of this study is that it is not generalizable to the general population of Grade 11 students in the Philippines because the sample came from a Science High School, and students have access to technology and the internet all the time. This stands in contrast to reports indicating that 40% of Filipino students lack access to devices and up to 70% have limited access to the internet in rural areas (Flores [<reflink idref="bib26" id="ref58">26</reflink>]; Santos [<reflink idref="bib40" id="ref59">40</reflink>]). Similar studies in Philippine schools and other grade levels and socioeconomic backgrounds could be conducted to deepen the understanding and make a more generalized claim on this aspect. Further studies may require students to use a minimum number of functions to ensure their use and even require artworks where only functions can be used. In addition, the study only examined the data from the artworks created by the students, and no pre‐test or post‐test assessments were conducted. As a result, the direct impact of the artworks on students' learning outcomes could not be determined conclusively. To address this limitation, future studies should incorporate pre‐test and post‐test evaluations to understand better the intervention's effectiveness in enhancing students' knowledge and skills.</p> <p>Integrating mathematical functions and related concepts into artwork using technology holds significant implications for classroom practice and mathematics education. Applying function art highlights the dynamic interplay between arts, mathematics, and technology, enriching the learning experience, nurturing students' interest in both disciplines, and advancing the integration of creative and analytical thinking across academic domains. We encourage educators to emphasize the creative potential of function art, allowing students to explore diverse artistic expressions using technology while deepening their understanding of fundamental mathematical principles.</p> <p>This study underscores the effectiveness of constructionism in STEAM education, showing that students can develop a deeper understanding of mathematical concepts through creative activities. The integration of constructionism into mathematics education provides a powerful means to engage students and foster a holistic learning experience.</p> <hd id="AN0194205337-30">Acknowledgment</hd> <p>Open access funding provided by Johannes Kepler Universitat Linz/KEMÖ.</p> <ref id="AN0194205337-31"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref19" type="bt">1</bibl> <bibtext> Funding: Open access funding provided by Johannes Kepler Universitat Linz/KEMÖ.</bibtext> </blist> </ref> <ref id="AN0194205337-32"> <title> References </title> <blist> <bibtext> Abar, C. A. A. P., M. V. De Almeida, and Z. Lavicza. 2024. "Arts and Mathematics: GeoGebra Focused on Isometric Transformations." Journal of Mathematics and the Arts18, no. 1–2: 47–65. https://doi.org/10.1080/17513472.2024.2365361.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref31" type="bt">2</bibl> <bibtext> Aguilera, D., and J. Ortiz‐Revilla. 2021. 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  Data: Function Art: Linking Mathematics, Technology, and Visual Arts
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  Data: <searchLink fieldCode="AR" term="%22Guillermo+Bautista%22">Guillermo Bautista</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-5471-9326">0000-0001-5471-9326</externalLink>)<br /><searchLink fieldCode="AR" term="%22Jemarie+Malacapo%22">Jemarie Malacapo</searchLink><br /><searchLink fieldCode="AR" term="%22Florenda+Gallos-Cronberg%22">Florenda Gallos-Cronberg</searchLink><br /><searchLink fieldCode="AR" term="%22Houssam+Kasti%22">Houssam Kasti</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-6573-8370">0000-0001-6573-8370</externalLink>)<br /><searchLink fieldCode="AR" term="%22Noah+Dana-Picard%22">Noah Dana-Picard</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-1777-3232">0000-0002-1777-3232</externalLink>)<br /><searchLink fieldCode="AR" term="%22Zsolt+Lavicza%22">Zsolt Lavicza</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-3701-5068">0000-0002-3701-5068</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22School+Science+and+Mathematics%22"><i>School Science and Mathematics</i></searchLink>. 2026 126(3):251-263.
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  Data: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
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  Data: 13
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  Data: 2026
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  Data: Journal Articles<br />Reports - Research
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  Data: <searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Art+Products%22">Art Products</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+Software%22">Computer Software</searchLink><br /><searchLink fieldCode="DE" term="%22Constructivism+%28Learning%29%22">Constructivism (Learning)</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+Uses+in+Education%22">Computer Uses in Education</searchLink><br /><searchLink fieldCode="DE" term="%22Visual+Arts%22">Visual Arts</searchLink><br /><searchLink fieldCode="DE" term="%22Graphs%22">Graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Animation%22">Animation</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Strategies%22">Learning Strategies</searchLink>
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  Data: 10.1111/ssm.18373
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  Data: 0036-6803<br />1949-8594
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  Data: This study investigated students' understanding of mathematical functions and strategies to create artwork using GeoGebra. It was framed by the principles of constructionism and examined how students use functions in creating artworks. We gathered data from students' artworks using the Algebra view and the Construction Protocol in the GeoGebra software. We used descriptive statistics to determine the number and types of functions used, whereas we employed thematic analysis to identify patterns and themes in students' artworks and strategies. The study found that students primarily used quadratic functions. Emerging techniques and strategies included grouping functions by artwork parts, using concepts of reflection and symmetry, combining animation and transformation, and integrating functions with other tools. The findings suggest that allowing students to create artwork through graphing can provide valuable insights into their mathematical skills. This study sheds light on the potential of GeoGebra as a tool and teaching aid for assessing students' mathematical abilities, and it offers valuable insights into students' strategies and techniques when creating mathematical art. This study revealed students' mathematical skills, strategies, and conceptual understanding through the creation of function art using GeoGebra.
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