Grade 12 rural teachers’ Technological Pedagogical Content Knowledge and challenges while using GeoGebra to teach Euclidean geometry

Introduction: The integration of dynamic mathematics software like GeoGebra in recent years into classroom teaching has gained international attention for its potential to enable learners’ conceptual understanding in geometry. This study explores the challenges teachers face in integrating GeoGebra into Grade 12 Euclidean geometry lessons, addressing significant gaps in the literature regarding technology use in rural educational settings.

Methods: Qualitative data from semi-structured interviews, classroom observations, and Video-Stimulated Recall Interviews with five teachers reveal that inadequate training in Technological Knowledge (TK) limits teachers’ ability to effectively employ GeoGebra, resulting in issues such as inaccuracies in geometric representations and reduced learner engagement.

Results: Observations indicate variation in classroom environments where infrastructural support directly impacts the effectiveness of technology integration in mathematics teaching. The study further highlights effective pedagogical strategies, as evidenced by teachers’ proactive use of GeoGebra during the observed lessons. Participants express a strong desire for more structured professional development to improve their technology integration skills, emphasizing the critical need for comprehensive training programs.

Discussion: Ultimately, this research contributes to international literature by highlighting the necessity of robust pedagogical and technological foundations for teachers, thereby advocating for ongoing professional development that equips teachers with the skills required to integrate technology effectively in geometry education.

Introduction

Euclidean Geometry presents significant challenges for learners, contributing to its removal from the South African math curriculum in 2006 (Machisi, 2021; Bansilal and Ubah, 2019; Tachie and Otto, 2021). Despite these concerns, the Department of Basic Education positions Euclidean Geometry as vital for developing critical and logical thinking skills, accounting for 33% of Grade 12 mathematics assessments (Department of Basic Education [DBE], 2011). The formative learning process emphasizes the acquisition of knowledge and skills related to spatial recognition and problem-solving, which are foundational for advanced geometry studies (Naidoo and Kapofu, 2020). Marange and Tatira (2023), p. 1 states that, “the teaching of Euclidean geometry is characterized by ineffective instructional methods used by in-service teachers as well as the low proficiency levels by learners.” This being the case, it is assumed that using GeoGebra in teaching the topic can transform teachers’ pedagogical approaches from teacher-centric approaches to explorative and dialogical approaches, which in turn are believed to play a critical role in facilitating learners’ conceptual understanding (Machisi, 2021). Of concern for the current study is that there is dearth of research that has explored how teachers within rural schools, particularly within the South African context, employ GeoGebra during geometry instruction. Accordingly, the current study explored rural Grade 12 teachers’ technological and pedagogical content knowledge as well as associated challenges during Euclidean geometry lessons.

GeoGebra has been widely recognized for its potential to enhance teaching and learning in mathematics, especially in geometry. Teachers generally perceive GeoGebra positively, appreciating its ability to support visual learning, foster learner engagement, and facilitate conceptual understanding of abstract mathematical ideas (Zakaria and Lee, 2012; Wassie and Zergaw, 2019). The persistent underachievement in Euclidean Geometry has prompted calls for integrating technological tools such as GeoGebra to enhance teaching effectiveness (Bansilal and Ubah, 2019; Naidoo and Kapofu, 2020). Despite the evidence supporting technology use, many South African mathematics teachers continue to rely on traditional methods, such as chalkboard instruction, which limits effective conceptual development (Mokotjo, 2020). Many teachers experience technophobia, defined as an irrational fear of technology, which hinders their willingness to adopt innovative teaching methods (Lam, 2016; Solihin, 2021). This reliance on outdated pedagogical practices detracts from the learning experience, as drawing complex geometric figures on a chalkboard can be time-consuming and hinder instructional clarity (Tachie and Otto, 2021; Machisi, 2021; Sibiya, 2020). Thus, further research is necessary to explore teachers’ perceptions of integrating technology and their pedagogical choices in the classroom. The research questions we address in this paper are:

- How do Grade 12 mathematics teachers integrate GeoGebra during Euclidean Geometry lessons?

- What challenges do teachers in rural classrooms face when integrating GeoGebra?

Literature review

Understanding GeoGebra and its utilization in Euclidean geometry

GeoGebra is an open-source software that serves as an effective pedagogical tool for teaching and learning mathematics. It seamlessly integrates algebra, geometry, and calculus within a Dynamic Geometry Software (DGS) environment, combining features of both DGS and Computer Algebra Systems (CAS). As shown in Figure 1, GeoGebra offers synchronized visualization and manipulation of mathematical objects through its algebra and geometry windows (Jelatu and Ardana, 2018). The software facilitates the demonstration of geometric relationships, visualization of various representations, and supports mathematical investigations. Additionally, it allows teachers to create teaching materials and promotes collaboration and communication in the representation of mathematical ideas. GeoGebra’s “drag mode” functionality enables direct manipulation of geometric constructions, facilitating the exploration of mathematical conjectures and proofs (Uwurukundo et al., 2022).

FIGURE 1

Figure 1. GeoGebra window interface (Authors’ own construction).

The study by Jelatu and Ardana (2018) demonstrated the positive effects of integrating GeoGebra, reinforcing the potential benefits of using learning software to teach Euclidean geometry concepts (Nzaramyimana et al., 2021). Similarly, Uwurukundo et al. (2022) advocate for the use of GeoGebra in mathematics education, noting its favorable impact on learners’ attitudes and experiences in three-dimensional geometry. These findings highlight GeoGebra’s role in promoting active learning, improving mathematics performance, and increasing learner interest in the subject. This highlights the importance of investigating teachers’ perceptions and methods of integrating GeoGebra in teaching Euclidean geometry, particularly in Limpopo, where research in this area is lacking.

The impact of GeoGebra in teaching Euclidean geometry

Since the launch of GeoGebra, numerous studies have explored its effectiveness in teaching mathematical concepts, particularly Euclidean geometry (Bayaga et al., 2019; Singh, 2018; Khalil et al., 2017). For instance, Zengin (2018) found that integrating GeoGebra into math education in Turkey effectively assists both learners and teachers in visualizing and exploring geometric concepts. Putri et al. (2021) emphasized that GeoGebra enables learners to compare hand-drawn geometric objects with digital ones, facilitating deeper evaluations of their work. Similarly, Bayaga et al. (2019) noted that learners can easily manipulate variables using GeoGebra’s dynamic features, enhancing their understanding of geometric properties. Horzum and Ünlü (2017) further posited that GeoGebra enhances instruction by visualizing content and providing additional exemplification.

In Indonesia, Triwahyuningtyas et al. (2019) confirmed that GeoGebra’s classic application aids learners in grasping the formation of geometric spaces through instant visualization. This aligns with findings from Mudaly and Fletcher’s (2019) study in KwaZulu-Natal, South Africa, which demonstrated that using GeoGebra on iPads significantly improved learners’ comprehension of linear functions and fostered positive attitudes toward the application. Additionally, Jelatu and Ardana (2018) showed that GeoGebra-assisted approaches led to a better understanding of geometry concepts compared to traditional teaching methods. Collectively, these studies highlight GeoGebra’s potential to enhance mathematical learning experiences across various educational contexts.

Furthermore, Wassie and Zergaw (2019) argued that GeoGebra’s impact as a critical educational resource requires a well-designed corresponding curriculum that ensures technological integration in teaching and studying GeoGebra’s inputs and additional values in increasing learners’ high achievement. In a similar research study, Seloraji and Eu (2017) argued that GeoGebra integrated teaching was found to improve learners’ geometric skills in Malaysia. In Zimbabwe, the study conducted by Mukamba and Makamure (2020) revealed that the research classes backed up Vygotsky’s social learning hypothesis were effective because there were interactions amongst the learners and knowledge was socially constructed. As a result, it was further proposed that with the help of GeoGebra, learners could be scaffolded easily in their knowledge of geometric transformations using the visualization features in GeoGebra. This resonates with previous studies such as Uwurukundo et al. (2022), Bhagat and Chang (2015) who argue that GeoGebra facilitates content visualization and understanding through exploration, leading to improved learner attitudes and performance in geometry. Of importance to note is that the software does not remove the role of the teacher in the learning and teaching processes. However, the GeoGebra’s interactive features should be used by teachers and learners, allowing learners to manipulate geometric objects, experiment with different situations, and observe the effects of changes in real-time and make generalizations about the behavior of mathematical objects.

Theoretical framework

The Technological Pedagogical Content Knowledge (TPACK) framework, developed by Koehler and Mishra (2009), extends Shulman’s notion of Pedagogical Content Knowledge (PCK) by incorporating the role of technology in teaching and learning. TPACK identifies the essential knowledge domains that teachers must possess to effectively combine technology with pedagogy and content. The framework emphasizes the integration of these facets to enhance educational practices (Koehler et al., 2011). TPACK comprizes three primary domains as depicted in Figure 2: Content Knowledge (CK), Technological Knowledge (TK), and Pedagogical Knowledge (PK), with each domain influencing the others, leading to secondary knowledge areas such as Technological Pedagogical Knowledge (TPK), Technological Content Knowledge (TCK), and PCK (Koehler et al., 2011). Context is pivotal within TPACK, as teaching does not occur in isolation; various contextual factors, including resources and teachers’ technological proficiency, significantly shape pedagogical strategies.

FIGURE 2

Figure 2. Technological Pedagogical Content Knowledge (TPACK) framework [adapted from Koehler and Mishra (2009), p. 63].

The integration of GeoGebra into Grade 12 Euclidean geometry teaching illustrates how teachers’ CK influences their use of technology, facilitating more effective pedagogical choices. Furthermore, teachers’ ability to effectively use GeoGebra is informed by their TPK, highlighting the importance of aligning technology with pedagogical approaches tailored to curriculum objectives (Ammade et al., 2020). The interconnection of TCK shows how technological tools enhance and reshape the teaching of content, encouraging deeper learner engagement. An exploration of these dynamics within the TPACK framework reveals the nuances of teachers’ perceptions and strategies for integrating GeoGebra, providing a framework for understanding their pedagogical decision-making processes. Ultimately, this comprehensive examination highlights the necessity for a balanced integration of CK, TK, PK, TPK, PCK, and TCK, creating meaningful learning experiences that foster mathematical understanding while allowing learners to navigate the complexities of geometric reasoning through innovative technological engagement.

In the current study, CK allows enables the exploration of how teachers make informed pedagogical decisions pertaining to how to best leverage GeoGebra tools to make geometric principles available to the learners and facilitate visualization of the concepts, allowing learners to meaningfully explore geometric concepts (Cox and Graham, 2009; Mishra and Koehler, 2006). PK allows us to gain insight into how teachers adapt their teaching approaches to cater for the diverse learning needs, ensuring that the integration of GeoGebra enables learners’ epistemological access to Euclidean geometry concepts (Santos and Castro, 2020; Godoy et al., 2021). In addition, TK allows for the examination of the participating teachers’ proficiency in integrating GeoGebra into the teaching of Euclidean geometry in Mopani West District classrooms.

Using TPK, we explored how the participating teachers adapted their teaching approaches to make Euclidean geometry lessons available for the learners, and how their integration of GeoGebra facilitated and/or constrained effective teaching of the concepts (Godoy et al., 2021; Santos and Castro, 2020). PCK allows us to gain insight into how teachers incorporated GeoGebra to facilitate learners’ learning of Euclidean geometry while still ensuring that they demonstrate high levels of content knowledge in the topic (Lai et al., 2022; Kim, 2024). Furthermore, TCK pertains to teachers’ technological knowledge essential to effectively utilize the GeoGebra software during Euclidean geometry lessons, especially encompassing teachers’ familiarity with the different features, tools, and applications, navigating the algebra window and the geometric and graphics window as depicted earlier in Figure 1.

Materials and methods

Purposeful sampling, a non-probability technique, was utilized to select schools and teachers likely to provide valuable insights (Creswell and Creswell, 2017). Teachers were chosen based on their proficiency in teaching Grade 12 Euclidean geometry and their experience with GeoGebra within the Maruleng region of Limpopo Province, South Africa. The selection criteria included teaching Grade 12 mathematics and prior use of GeoGebra. This study involves five Grade 12 mathematics teachers with expertise in mathematics learning area and teaching experiences ranging from 4 to 10 years to enrich the findings. To effectively address the research questions, qualitative data collection was guided by the interpretive paradigm (Mbhiza, 2024). According to Yin (2009), case study research is particularly suited for investigating contemporary issues through multiple data sources. In alignment with this approach, the current study utilized semi-structured interviews, non-participatory classroom observations, and Video-Stimulated Recall Interviews (VSRIs) to ensure methodological triangulation and enhance the credibility of findings (Creswell and Creswell, 2017).

Unstructured classroom observations allowed for naturalistic data collection, capturing teachers’ integration of GeoGebra without researcher interference, thereby improving dependability of the findings. VSRIs enabled reflective dialogue as teachers reviewed video recordings of their own lessons, promoting self-awareness and professional growth (Voithofer and Nelson, 2021; Mbhiza, 2019). Semi-structured interviews provided nuanced insights into teachers’ perceptions, challenges, and pedagogical strategies related to GeoGebra use (Mbhiza, 2021). Each teacher participated in two interview sessions, with all interactions audio-recorded and transcribed verbatim. The transcribed data were subjected to thematic analysis, allowing for the identification of recurring patterns and themes that illuminate the complexities of teachers’ pedagogical decisions (Catalano and Creswell, 2013). Triangulation across the three data sources strengthened the validity of the findings by cross-verifying evidence and capturing multiple dimensions of the teaching experience. To enhance the trustworthiness of this study, multiple strategies were employed across four key criteria of qualitative rigor: credibility, dependability, transferability, and confirmability. To ensure credibility, member checking was conducted by sharing interview summaries with the participating teachers for validation. Triangulation across interviews, classroom observations, and VSRIs further reinforced credibility. In addition to this, an audit trail documenting decisions throughout data collection and analysis was maintained and coding verification was achieved through independent review of a subset of transcripts to ensure consistency in theme development (Creswell and Creswell, 2017).

In the current study, we also provide the thick descriptions of the school settings to enable readers to assess applicability to other contexts. Reflexivity was practiced through researcher memos and reflective journaling, acknowledging positionality and potential biases. All interpretations were grounded in participants’ accounts and corroborated by multiple data sources that we employed in this study.

Data analysis and findings

The analysis and findings section identifies the teachers’ technological and pedagogical content knowledge as well as the associated challenges they experience in integrating GeoGebra in teaching Euclidean geometry. The first section focuses on the GeoGebra integration challenges that teachers face that were identified from semi-structured interviews. The second section represents the classroom settings and how such settings facilitated and/or constrained teachers’ effective integration of GeoGebra during teaching. To explore and understand the participating teachers’ ways of integrating GeoGebra in their lessons, we present five episodes, each representing the participating teachers’ individual lessons as depicted in Table 1.

TABLE 1

Table 1. Different cases and selected teaching episodes.

Challenges identified from semi-structured interviews

The integration of GeoGebra into Grade 12 Euclidean geometry presents several challenges for Mr. Sam, Mr. Rathete, Mr. Marutha, Ms. Masa, and Mr. Thapedi. Insights from semi-structured interviews reveal the complexities they encounter in effectively utilizing this technology. These challenges can be examined through the Technological Knowledge (TK) component of the Technological Pedagogical Content Knowledge (TPACK) framework, which illustrates the importance of teachers’ understanding of educational technology and its application in teaching Euclidean Geometry.

Mr. Sam’s experience highlights the challenges stemming from inadequate training in GeoGebra. He remarked, “I went to a workshop once, where I was introduced to it the first time, but there was a lot of software at the same time, so even though I use it, I am not good at it.” This reflects a gap in his Technological Knowledge (TK) as he struggles to navigate the software due to insufficient focused training (Tachie and Otto, 2021). His reliance on self-directed learning through online tutorials shows an effort to bridge this gap but emphasizes the need for structured professional development to enhance his proficiency. Without a solid foundation in TK, Mr. Sam may struggle to make informed pedagogical decisions regarding technology integration. Similarly, Mr. Rathete shared, “The only time when we… were trying to introduce us to it… was just like a beginner course.” His limited exposure to GeoGebra reflects a broader issue of inadequate teacher training for effective technology use. This lack of TK constrains his ability to facilitate interactive explorations and visualizations of geometric concepts, essential for improving learners’ understanding (Tachie and Otto, 2021). Ultimately, the absence of comprehensive training hampers his ability to adapt to new technological tools, negatively affecting his teaching effectiveness. Mr. Marutha’s approach to integrating GeoGebra equally highlights the challenges faced by teachers who lack formal training. He mentioned,

I was not trained. I just heard about it from mathematics curriculum advisers, and I started using it. While my usage of it can help learners visualize the diagrams in geometry, you should see them, I am struggling, and this can make learners to miss important maths content from the bad representations you see.

His reliance on self-learning through downloaded tutorials shows a proactive attitude but highlights a significant challenge: the independent development of TK can result in inconsistencies in understanding and applying the software. This shows the necessity of providing structured training to equip teachers with the skills needed for effective technology integration, thereby enhancing learners’ access to geometric concepts.

In addition, Ms. Masa acknowledges her need for more training, stating,

“I would say yes, and I would say no. I was trained, but the training wasn’t that comprehensive, it was just the basics really which now as I use the software, I realize that I need more knowledge about navigating the software… I still need more training on it because it is a great tool for teaching learners geometry, but without the skill as a teacher, we seem to be going nowhere slowly”.

This situation highlights a common challenge teachers face when integrating technology without sufficient support. This illustrates a deficiency in TK because the lack training relating to the integration of the technological tool leads to frustration and hampers her ability to fully utilize GeoGebra, as she stated, “the training wasn’t that comprehensive” and “I realize that I need more knowledge.” Research (Marange and Tatira, 2023) shows that a strong foundation in TK is vital for teachers to make informed pedagogical decisions that enhance learner engagement and understanding, as reflected in her comment, “we seem to be going nowhere slowly”. Mr. Thapedi also notes his desire for more intensive training, stating, “Yes, at the time I was at the varsity level, we were taught how to use GeoGebra, though it was not that intensive.” His recognition of the need for further training highlights the importance of ongoing professional development in improving teachers’ TK. Without adequate training and in turn lack of TK, teachers may struggle to effectively integrate GeoGebra, limiting their ability to facilitate interactive learning experiences that foster conceptual understanding (Yildiz et al., 2017).

Across teachers, patterns emerged that TK challenges limited teachers’ effective integration of GeoGebra during teaching. The challenges these teachers encounter in integrating GeoGebra into Grade 12 Euclidean Geometry highlight the importance of Technological Knowledge within the TPACK framework (Tachie and Otto, 2021). Their experiences indicate that insufficient training and support results in TK challenges, which then severely impede their ability to effectively use technology, despite positive attitudes toward GeoGebra integration. Participants suggest that targeted professional development is crucial for overcoming these challenges and enhancing teachers’ TK, which would ultimately improve GeoGebra integration and mathematics instruction in rural classrooms.

Classroom settings and teachers’ pedagogical actions¹

This section examines the strategies each teacher employed to integrate GeoGebra into their Euclidean geometry lessons, alongside their perceptions of using the software in mathematics teaching. For each participating teacher, we selected one lesson to demonstrate their pedagogical actions while integrating GeoGebra during lessons on Euclidean Geometry, forming five individual episodes for the paper. That is, for each participating teacher, we begin by presenting the classroom setting for each teacher, to contextualize the inherent factors that are at play in shaping teachers’ technological pedagogical content knowledge.

The case of Mr. Sam’s teaching

Mr. Sam’s classroom lacked the structure necessary for effective integration of GeoGebra, creating significant challenges in teaching Euclidean geometry. He was required to transport a portable whiteboard and overhead projector, rather than using fixed equipment, which complicated the teaching process and limited interactive learning opportunities (see Figure 3). This setup not only impeded GeoGebra’s utilization but also restricted collaborative activities that could enhance learners’ understanding of geometric concepts. The need to manage portable teaching tools illustrates broader issues in rural educational settings, where inadequate infrastructure limits innovative teaching practices. Effective technology integration demands a supportive environment, which is often absent in rural classrooms. Moreover, the disparity in digital literacy between urban and rural teachers exacerbates this challenge; Chen (2024) notes that urban pre-service teachers are generally more adaptable to technology-integrated teaching approaches, such as the flipped classroom model, compared to their rural peers.

FIGURE 3

Figure 3. Mr. Sam’s classroom setting.

In view of the above classroom setting, in this section we present and analyze two episodes selected from Mr. Sam’s lesson, to explore and understand how the teacher integrated GoeoGebra during the lessons.

Episode 1

To begin the lesson, Mr. Sam handed his learners a question paper with a geometric problem he planned to illustrate using GeoGebra. He explained that he would draw the corresponding diagram on the software. As he created the diagram, he analyzed it, explaining the geometric properties while measuring lengths and angles. Initially, the learners seemed disengaged until they observed the drawing process. Although Mr. Sam’s diagram closely matched the intended representations, he struggled with angle measurements. For example, he intended to depict angle E as 106°, but instead recorded it as 121.13°, ultimately suggesting that the learners accept the latter value (Figure 4).

FIGURE 4

Figure 4. Picture of the geometric diagram drawn by Mr. Sam.

This moment captured our attention, raising questions about how Mr. Sam would address his challenges during the lesson. Notably, his struggle aligns with comments made during a semi-structured interview, where he revealed a lack of adequate training on integrating GeoGebra into his teaching practices. This deficiency likely contributes to his difficulties in accurately constructing angle measurements. Mr. Sam’s challenges reflect gaps in his TK, particularly in using GeoGebra effectively to represent mathematical concepts (Widodo, 2023). His reliance on self-taught skills restricts his ability to make informed pedagogical decisions that could enhance learner engagement and understanding, thereby highlighting the teacher’s TPK since her lack of TK affected the effectiveness of his pedagogical choices and actions during the lesson. The initial disengagement of learners, which shifted upon seeing the diagram drawn, demonstrates the need for improved pedagogical strategies that utilize technology to create interactive learning environments (Saðlam-Kaya, 2019). Mishra and Koehler (2006) emphasize that effective technology integration requires a robust understanding of how technology can enhance content delivery and improve learning experiences.

Figure 4 shows the diagram created by Mr. Sam, intended for solving unknown angles and lengths in his geometry lesson. However, the diagram lacks precision, indicating his struggle with accurately drawing geometric representations. This further reflects TK challenges because the teacher demonstrated lack of skills to utilize GeoGebra to produce accurate mathematical diagrams. To supplement it, he discussed the properties of cyclic quadrilaterals, noting that “the four sides of the quadrilateral are chords of the circle, the sum of the opposite interior angles equals 180°, and the exterior angle is equal to the opposite interior angle,” demonstrating correct CK. He then instructed learners to use GeoGebra to show that exterior angle G equals opposite interior angle B, encouraging them to adjust the drawing as needed. Unfortunately, when trying to manipulate line DG to achieve an angle of 112°, he recorded an angle of 110° instead, reinforcing our interpretation about his lack of TK. To keep learners engaged, he suggested considering 112° and 110° as equivalent, revealing a potential loss of confidence in his lesson plan. He lamented, “It is a pity this diagram was not drawn according to the scale, so we won’t be able to prove other properties.”

During a Video-Stimulated Recall Interview (VSRl), Mr. Sam explained his choice to use labels from the questions rather than those on the diagram, saying, “Oh, eish, I did not notice that as I constructed the diagram; you know I want to help learners understand the concepts, but the technology failed me.” This illustrates that a lack of adequate TK can undermine the teaching effectiveness, even when the teacher demonstrates good CK. Mr. Sam then moved on to prove that angles E and E were equal, referencing the tangent-chord theorem. After finding them initially unequal using GeoGebra, he remarked, “Our tangent is not tangent; let me adjust it.” Following the adjustment, he successfully demonstrated the equality of the angles. He then asked, “What is the size of angle O₁?” This exchange showcases the integration of PCK and TPCK in his teaching approach.

Mr. Sam: Which theorem are we going to use?

Learners: Angles at the center are equal to twice the angle at the circumference.

Mr. Sam: (guiding learners through the calculation), Good, for us to get that when to measure angle B = 110.6 times two, use your calculator to get the answer.

Learners: (The learners calculated the angle as 221.2° and verbalized the answer).

Mr. Sam: The angle that we are getting is the angle inside, and the point is the angle of revolution. Let’s subtract 221.2° from 360.

Mr. Sam’s challenges with GeoGebra, despite his knowledge of Euclidean geometry, reveals the necessity of robust training in TK within the TPCK framework. His difficulties in accurately representing geometric concepts negatively impacted learner engagement, illustrating that a solid foundation in TK is essential for facilitating effective learning experiences. Research shows that strong TK allows teachers to make informed pedagogical choices that enhance interactive explorations and visualizations, thereby improving learners’ conceptual understanding (Wassie and Zergaw, 2019). In Mr. Sam’s case, the deficiency in TK related to the utilization of GeoGebra diminished his confidence and teaching effectiveness, highlighting the critical need for comprehensive professional development in technology integration for teachers.

The case of Mr. Thapedi’s teaching

According to Putri et al. (2021), effective classroom design should incorporate natural light, flexible spaces, comfortable furniture, strategic use of color, and technology that supports learning. Mr. Thapedi’s classroom exemplifies this ideal setup, as it is well-suited for the integration of GeoGebra (see Figures 5, 6) Equipped with a whiteboard and a projector, he utilized these technological tools to enhance the learning experience and facilitate learners’ visualization of geometric concepts.

FIGURE 5

Figure 5. Mr. Thapedi’s classroom setup.

FIGURE 6

Figure 6. Shows all angles in a triangle.

Episode 2

Mr. Thapedi began his lesson by engaging learners in an exploration of the properties of polygons, specifically focusing on triangles. He effectively utilized GeoGebra to illustrate that a triangle is a polygon with three sides, demonstrating his strong Content Knowledge (CK) of geometry. As he constructed a triangle using the software (Figure 6), he highlighted that “the sum of its angles equals 180 degrees …always, when we add the angles inside the triangle, we always get 180,” demonstrating a strong CK by linking the geometric concept to a visual representation, emphasizing the mathematical convention that the learners should at all times observe that the three interior angles in any triangle always add up to make a total of 180 degrees. That is, the teacher articulated the triangle sum theorem and utilized GeoGebra to bring it into focus for the learners. This integration of technology into his teaching reflects his strong TK, as he adeptly navigated GeoGebra’s features to enhance learners’ understanding of the theorem.

Mr. Thapedi actively encouraged his learners to engage with GeoGebra, facilitating their exploration of geometric figures and properties. This approach not only fosters learner autonomy but also promotes collaboration as learners work together on constructing and analyzing geometric shapes. His integration of GeoGebra demonstrates a strong foundation in both TPK and PCK, effectively combining his geometry content expertise with suitable pedagogical and technological strategies to enhance learning. Research shows that GeoGebra improves learners’ mathematical connections and encourages cooperative learning, resulting in greater engagement and understanding of geometric concepts (Septian, 2022; Bayaga et al., 2019). By allowing learners to interact with the software, Mr. Thapedi cultivated an environment that promotes exploration and discovery, which is vital for developing conceptual understanding in mathematics, particularly in Euclidean geometry (Putri et al., 2021).

His ability to navigate GeoGebra while guiding learners demonstrates a good TPK, as he effectively integrated technology with pedagogical strategies to enhance the learning experience (Marange and Tatira, 2023). Mr. Thapedi effectively demonstrated his ability to utilize GeoGebra software to illustrate the properties of an isosceles triangle, where he confirmed that “the triangle has two equal sides measuring 12.58 cm.” He further explained that “the angles opposite the equal sides are also equal,” reinforcing his strong CK of geometric principles. By extending the base side of the triangle to create an exterior angle (Figure 7), he investigated and emphasized the property, stating that “the exterior angle of a triangle is equal to the sum of the two opposite interior angles.” This integration of GeoGebra into his teaching also reflects a strong foundation in PCK, as Mr. Thapedi skilfully combined his understanding of the content with effective pedagogical strategies to facilitate learning. His enthusiasm for integrating GeoGebra was evident, and he transitioned smoothly from discussing triangles to exploring circles to further investigate the properties of polygons, specifically the characteristics of cyclic quadrilaterals as depicted in Figure 8, further revealing his strong TK.

FIGURE 7

Figure 7. Illustration of an exterior triangle.

FIGURE 8

Figure 8. The illustration of a cyclic quadrilateral.

During the demonstration of a cyclic quadrilateral using GeoGebra, Mr. Thapedi effectively labeled the points, sides, and angles, highlighting his proficiency with the software and reinforcing his robust TCK. His confidence in using GeoGebra not only enhances his teaching but also encouraged collaborative learning as learners explored geometric properties together. Previous research reveals that dynamic geometry software like GeoGebra can significantly improve learners’ understanding of geometric concepts and foster a more interactive learning environment (Su et al., 2022; Puig et al., 2022).

Mr. Thapedi successfully demonstrated the properties of a cyclic quadrilateral, highlighting several key characteristics:

1. All vertices must lie on the circumference of the circle.

2. The opposite angles are supplementary.

3. The sum of all angles equals 360 degrees.

4. The exterior angle of the quadrilateral is equal to the sum of the two opposite interior angles.

Figure 9 illustrates Mr. Thapedi’s depiction that the angle at the center of the circle is twice the angle at the circumference. To explore these properties, he constructed two radii from the center to distinct points on the circumference, forming a triangle. He labeled the central angle as α = 121.24° and the circumference angle as β = 60.16°, concluding that the central angle is twice the circumference angle. However, the central angle was not precisely double the circumference angle, thereby failing to visually demonstrate “the angle at the center is twice the angle at the circumference when both are subtended by the same arc.” During the Video-Stimulated Recall Interview (VSRI), when questioned about the discrepancy, Mr. Thapedi expressed uncertainty: “I’m not sure if I’m not doing the correct thing or if the software is not measuring accurately based on how I’ve drawn my lines since they were not drawn to scale.” Mr. Thapedi’s difficulties with GeoGebra alongside his VSRI statement reveal that strong CK combined with inadequate TCK can limit learners’ access to mathematical understanding due to inaccurate representations (Puig et al., 2022). Additionally, his lack of reflexivity during the lesson regarding angle accuracy may stem from limited PCK, which is vital for verifying mathematical concepts during instruction (Machisi, 2021). Mr. Thapedi then constructed a chord from points B to C (Figure 10) in an attempt to prove that triangle ABC is isosceles.

FIGURE 9

Figure 9. The angle at the center is twice the angle at the circumference.

FIGURE 10

Figure 10. Illustration of properties of an isosceles triangle in a cyclic quadrilateral.

He then measured angles C and B, recording values of C = 28.94° and B = 29.83°. He attempted to convince his learners that the two base angles of the isosceles triangle were equal, even though line segment BC was not accurately drawn, indicating a TK challenge. Figure 11 illustrates the concept that the opposite sides of an isosceles triangle are equal. Mr. Thapedi also measured the sides, finding BC = 6.91 cm and BE = 6.81 cm, and concluded that BC = BE, reinforcing the property that the opposite sides of an isosceles triangle are equal.

FIGURE 11

Figure 11. Illustration of two opposite sides of an isosceles being equal.

One learner asked, “Why are the sides not equal if we are using mathematical software? Isn’t the software supposed to provide accurate measurements?” Mr. Thapedi replied, “Since we draw manually, it does not guarantee accuracy; therefore, if you encounter such a situation, you should round off to the nearest whole number, and those numbers should not be far apart from each other.” This exchange illustrates Mr. Thapedi’s need for stronger TCK, as his inaccuracies and the learner’s inquiry expose a gap in ensuring that technology accurately reflects the mathematical principles being taught. This aligns with Marange and Tatira’s (2023) assertion that effective technology integration in mathematics education requires teachers to have a solid foundation in TCK to use software tools for precise representations. Additionally, the potential for misperceptions due to inaccuracies in the drawn figures indicates Mr. Thapedi’s meager TPK. By effectively integrating technology with pedagogical strategies, he could create a more conducive learning environment that reduces learner misperceptions, especially given his strong CK. Mr. Thapedi also extended chord EF to construct the exterior angle of a cyclic quadrilateral (Figure 12).

FIGURE 12

Figure 12. Picture of informal assessment drawn by Mr. Thapedi on GeoGebra.

Consider the extract below concerning the Figure 12.

1. Mr. Thapedi: What can you say about the exterior angle of a cyclic quad?

2. Learner: An exterior angle of a cyclic quad is equal to the opposite interior angle of a cyclic quad.

3. Mr. Thapedi, kindly come and show us.

4. Learner: eish, sir, eh

This dialogue illustrates a holistic teaching approach that goes beyond traditional methods by incorporating communication to enhance learner engagement (Scott et al., 2011). It also suggests that the learner may lack the confidence to effectively integrate GeoGebra into their understanding of the theorem, as indicated by their reactions of “eish and eh.” Mr. Thapedi measured the exterior angle of the cyclic quadrilateral at 61.63°, while the opposite interior angle was recorded as β = 60.61°. According to the theorem, these angles should be equal; the 1.02° discrepancy raises concerns about potential misperceptions among learners. In addition, Mr. Thapedi measured the sides of an isosceles triangle, finding BC = 6.91 cm and BE = 6.81 cm, thus reinforcing that BC ≠ BE and highlighting the importance of TCK alongside high CK. To conclude the lesson, he conducted a post-lesson assessment to evaluate whether the learners achieved the lesson’s objectives, as shown in Figure 13.

FIGURE 13

Figure 13. Picture of post-lesson assessment on Microsoft Word.

Mr Thapedi drew the very same question on GeoGebra (Figure 13), to demonstrate to the learners how the construction is done using the software.

The case of Ms. Masa’s teaching

Figure 14 depicts the technology setting in Ms. Masa’s classroom in which the projector screen was mounted on the wall. The classroom environment was indeed favorable for the effective utilization of GeoGebra, as spatial arrangements facilitated learners’ ability to view the monitor clearly. Specifically, learners were able to see the whiteboard without difficulty, and the monitor’s visibility was enhanced by adequate lighting, which improved the clarity of the displayed content. The design of the classroom, including the arrangement of desks and the positioning of the monitor, contributed positively to the learning experience. Research (Liu et al., 2020) indicates that an organized spatial layout can significantly enhance visibility and engagement in educational settings.

FIGURE 14

Figure 14. Learner at the board/screen.

Furthermore, appropriate lighting conditions are critical for maximizing visibility and ensuring that all learners can effectively interact with digital tools such as GeoGebra (Liu et al., 2018). The combination of these factors not only supports the visibility of instructional materials but also fosters a conducive learning environment that encourages active participation and collaboration among learners (Chakraborty and Yadav, 2022).

Episode 3

In her lesson, Ms. Masa aimed to prove two key theorems using GeoGebra software: first, that the angle formed between a tangent and a chord at the point of contact is equal to the angle subtended by the chord in the alternate segment; and second, that angles at the circumference subtended by the same arc or chord are equal. During VSRI, Ms. Masa was asked why she chose to focus exclusively on these two theorems. She explained,

The learners find the tan-chord theorem difficult to prove and often confuse the angles in question. In contrast, when it comes to the theorem of angles in the same segment, learners often confuse angles subtended at the center or opposite segment with those relevant to the theorem. I wanted to emphasize that the angles we are referring to are on the circle’s circumference and in the same segment.

Ms. Masa’s statement reflects her awareness of her learners’ challenges and her commitment to correcting misperceptions, showcasing her strong PCK. By concentrating on specific theorems, she aimed to clarify key geometric concepts that often confuse learners, thereby enhancing their understanding of Euclidean geometry. Research (Pamungkas et al., 2023) suggests that effective teaching necessitates a profound grasp of both content and pedagogy, particularly in geometry where misunderstandings frequently occur.

The lesson commenced with Ms. Masa asking learners to identify and describe essential lines in Euclidean geometry, such as tangents, radii, chords, and secants. This focus on terminology is vital, as each mathematical topic has specific language that learners must master to succeed (Kusumah et al., 2020). This illustrates a strong PCK because asking learners to engage in the identification of the mathematical objects acted to transform the content for them, stimulating their recall of the concepts that would be used during the topic. Although she used GeoGebra to illustrate the drawing of circles, chords, and tangents, she primarily relied on traditional teaching methods rather than engaging learners in hands-on activities with the software. An overreliance on traditional teaching methods may indicate a limitation in TPK, as it reflects the teacher’s inability to effectively integrate technological tools with pedagogical strategies to achieve a balanced and conceptually sound instructional approach. This reliance may restrict opportunities for active learning and exploration that GeoGebra offers, which are crucial for developing a deeper understanding of geometric concepts (Topuz and Birgin, 2020).

As Ms. Masa measured the angles formed by the chords and the tangent, she recorded angles D = 60° and B1 = 60.67°. When learners noticed the discrepancy and asked why the angles were not equal, she explained that the lines were not drawn to scale, which could lead to measurement inaccuracies, thereby revealing the shortfall in her TK. This highlights the importance of precision when using GeoGebra, designed to provide accurate geometric relationships. Ms. Masa’s confidence in demonstrating that angles at the circumference subtended by the same arc are equal reflects her strong CK, while her accurate measurements of angles I and K (I = K = 47.56° and J = L = 27.82°) further confirm her understanding of the material.

Ms. Masa was overjoyed because she had finally determined that angle I was equal to angle K. She concluded by noting that angles in the same segment are equal, thus demonstrating her good CK and TCK.

The case of Mr. Rathete’s teaching

Mr. Rathete’s classroom was not adequately equipped for effective technology integration. Initially planning to project his lesson onto a white sheet, he switched to a small television due to spatial constraints (Figure 15). The overcrowded classroom required both the laptop and TV to be placed on a desk, obstructing visibility for some learners. This arrangement hindered the learning experience and posed connectivity challenges for Mr. Rathete, consuming valuable teaching time. Despite these issues, he remained calm and continued to set up without displaying frustration.

FIGURE 15

Figure 15. Mr. Rathete’s classroom setup.

The integration of technology in education enhances learner engagement and outcomes when implemented effectively. Research shows that appropriate technology use can significantly boost learner attention and interaction (Tachie and Otto, 2021). Successful integration depends on proficient classroom management skills and a supportive environment; without them, motivation and communication may suffer (Andrei, 2016). Mr. Rathete’s experience signifies the challenges tied to inadequate classroom conditions, emphasizing the necessity for teacher preparedness and nurturing environments that promote effective technology use. The relationship between classroom management, teacher confidence, and technology setup is vital for creating an engaging learning atmosphere. Notably, despite the suboptimal setup in Mr. Rathete’s classroom, learners had access to their own tablets during lessons (Figure 16).

FIGURE 16

Figure 16. Learners with tablets.

During interviews, the teacher mentioned that his learners are in the position of tablets; therefore, his learners were connected to the network and able to use GeoGebra during the lesson. The South African Department of Basic Education recognizes that digital literacies are essential to teachers and learners and that digital competencies prepare learners for the world of work in the information age. In the case of Mr. Rathete, his learners got tablets from the Department of Basic Education, to facilitate effective integration of technology during mathematics learning.

Episode 4

At the onset of the lesson, Mr. Rathete commenced with an introduction to the fundamental concepts of Euclidean geometry. He began by demonstrating the properties of straight lines and angles, specifically illustrating how angles formed on straight lines are supplementary (Figure 17), illustrating his strong TK. To facilitate understanding, he engaged the learners by asking them to interpret the theorem he presented, while he used his arms to visually represent the concept of supplementary angles. This interactive approach appeared to resonate well with the learners, who responded positively to the digital learning environment he had created. In addition, Mr. Rathete made a point to remind the learners about the properties of triangles, reinforcing their prior knowledge in this area.

FIGURE 17

Figure 17. Illustrations of all angles on the straight line add up to 180°.

The teaching of Euclidean geometry, particularly in South Africa, faces numerous challenges, including a lack of adequate training for teachers and insufficient understanding of mathematical concepts among learners (Tachie and Otto, 2021). Rabaza and Hamilton (2022) indicate that effective teaching strategies, such as the use of worked-out examples using technology, can significantly enhance teachers’ performance in mathematics. Mr. Rathete effectively illustrated that the exterior angle of a triangle is equal to the sum of the two opposite interior angles (Figure 18). He continued to review Grade 9 geometry content to ensure that learners had a solid foundation of the concepts. This aligns with research indicating that teachers must connect previous and current lessons to prevent learners from perceiving mathematics as a series of disconnected events, which demonstrates a solid and comprehensive PCK as it opened up opportunities to gain awareness of learners’ prior knowledge (Picado-Alfaro et al., 2022).

FIGURE 18

Figure 18. Illustration of exterior angle triangle.

During the Visual Structured Reflection Interviews (VSIRs), Mr. Rathete was asked to clarify the objectives of these lessons, seeking clarity on what learners were expected to have achieved by the lesson’s conclusion. Mr. Rathete explained that the lesson aimed to present Euclidean geometry concepts using GeoGebra. He said:

The purpose of this lesson is to project Euclidean geometry content lesson using GeoGebra. Remember, Euclidean geometry is all about shapes, lines, and angles and how they interact with each other so that you can conclude conjectures. Therefore, I wanted to visualize these lessons so that learners could see what was meant by the geometrical statements and theorems. It is not easy to teach these chapters using the traditional teaching method. I cannot draw lines and angles accurately, and this chapter needs a high level of accuracy so that you will be able to prove theorems.

In this statement, Mr. Rathete emphasized that Euclidean geometry focuses on shapes, lines, and angles, and their interactions, which allow for the formulation of conjectures. To facilitate understanding, he aimed to visualize these concepts so that learners could grasp the meaning behind various geometric statements and theorems. He also acknowledged the challenges of teaching these topics through traditional methods, noting that accurately drawing lines and angles is crucial for proving theorems.

Our observations indicate that Mr. Rathete has a strong understanding of Euclidean geometry. He emphasized the importance of mastering content, asserting that a lack of knowledge would hinder learners’ ability to demonstrate geometric theorems. During the lesson, he expertly summarized Grade 9 geometry content before advancing to Grade 12 Euclidean geometry, demonstrating theorems such as the relationships between angles and the tangent-chord theorem. Mr. Rathete showcased his proficiency with GeoGebra, with the lesson running smoothly and without technical issues. He prepares lessons using GeoGebra offline and imports them into Microsoft Word for presentation. In a Video-Stimulated Interview (VSRI), he explained,

“Ma’am, working with GeoGebra can sometimes frustrate you. Anything can happen, and the learner will think you didn’t prepare, which can astonish you. So I prefer to prepare the lesson in my spare time and project it simply to avoid distractions that waste time.”

This statement demonstrates the importance of lesson preparation in mitigating potential challenges, demonstrating his PCK to support effective TPCK. To continue the lesson, Mr. Rathete distributed a handout (Figure 19) and instructed one learner to construct the diagram using GeoGebra while the others used their tablets. The resulting diagram is illustrated in Figure 20, reflecting Mr. Rathete’s effectiveness in teaching GeoGebra skills. Although the learners constructed the diagram well, they struggled to measure angles as precisely as on paper, prompting Mr. Rathete to provide assistance. This support exemplifies Vygotsky’s Zone of Proximal Development, where the teacher bridges the gap between learners’ current abilities and their potential knowledge (Vygotsky, 1987).

FIGURE 19

Figure 19. Handout of the activity.

FIGURE 20

Figure 20. Diagram drawn by a learner.

In TPCK, this pedagogical move demonstrates the teachers’ thorough PCK, allowing learners to explore mathematical concepts and steps in to facilitate their learning when faced with challenges to use GeoGebra (Tachie and Otto, 2021). During the VSRI, Mr. Rathete was asked to elaborate on the rationale behind asking a learner to draw and measure the angles. He stated that, “GeoGebra promotes a learner-centered strategy and cooperative learning. I did this to assess their level of understanding.” Mr. Rathete’s effective use of GeoGebra, combined with his planned and thoughtful teaching strategies, fosters a conducive learning environment where learners can develop a deeper understanding of geometric concepts. His emphasis on learner-cantered approaches and the use of visual aids significantly contribute to enhancing learners’ engagement and comprehension in geometry.

The case of Mr. Marutha’s teaching

Mr. Marutha’s lessons took place in a laboratory environment, although it was not specified whether it was designated for mathematics, science, or computer studies. The overhead projector was mounted on a stable ceiling stand, ensuring optimal alignment with the display sheet intended for lesson projection (Figure 21). Mr. Marutha utilized a laptop connected to the projector via an HDMI cable, which also offered wireless connectivity options. The power supply was reliable, and the lighting system was adjustable, allowing for a balance between brightness and dimness to enhance the visibility of the projected content while ensuring adequate illumination for learners to take notes.

FIGURE 21

Figure 21. Mr. Marutha’s classroom setting.

The classroom accommodated approximately 30 learners and was characterized by its spaciousness, cleanliness, and well-furnished layout, which facilitated Mr. Marutha’s movement without disrupting the learning environment. The following section presents an episode extracted from Mr. Marutha’s first observed lesson.

Episode 5

In the first observed lesson, Mr. Marutha initiated the session by prompting learners to distinguish between chords and tangents. The learners articulated the differences effectively, demonstrating their understanding of these geometric concepts. Mr. Marutha then encouraged one learner to articulate the tangent-chord theorem, which was correctly stated as “the angle between the tangent and the chord is equal to the angle subtended by the chord in the alternate segment,” thus demonstrating a good CK. This foundational understanding is crucial as it sets the stage for deeper exploration of geometric relationships. To visualize these concepts, Mr. Marutha illustrated a circle with a centre labeled “O” and a tangent line (Figure 22).

FIGURE 22

Figure 22. Radius is perpendicular to the chord at 90°.

During VSRI, Mr. Marutha was asked about the lesson’s objectives, to which he responded that the aim was to help learners visualize the tangent-chord theorem, thereby enhancing their comprehension of the theorem. Consider his statement: “The aim of this learners is to virtualize lesson themed with tan-chord theorem, hoping that they will understand the theorem better.” This aligns with the pedagogical approach that emphasizes visualization in learning geometry, as supported by various studies indicating that dynamic geometry software like GeoGebra can significantly improve learners’ understanding of geometric concepts (Jelatu and Ardana, 2018; Bayaga et al., 2019). Specifically, when probed further about his choice of focusing on the tangent-chord theorem, Mr. Marutha explained that many of his learners struggle with this theorem which illustrates his strong PCK because he was aware of his students’ learning difficulties and diverse learning needs. He expressed hope that utilizing GeoGebra would allow learners to explore the theorem interactively, thereby enhancing their understanding and application of the concept. He said:

Ma’am from my teaching experience most of my learners find it difficult to understand Tan-chord. I hope if I give them a chance to explore it through GeoGebra it can enhance their understanding and they will be able to apply the theorem since GeoGebra is there to improve our conceptual understanding.

Continuing the lesson, Mr. Marutha drew a radius perpendicular to the tangent to illustrate the relationship between these two lines (Figure 23). His TK with GeoGebra was evident, as he effectively utilized its features to demonstrate that the angle formed by two perpendicular lines is 90 degrees. The classroom atmosphere was notably positive, as learners were engaged and pleased to confirm the validity of this geometric principle. The integration of GeoGebra in the lesson not only aided in visualizing mathematical concepts but also enhanced learner engagement and understanding, revealing the teacher’s positive TPK. This observation resonate with findings from previous studies that when the teacher’s TPK is high, there is a high chance for enhanced learner engagement during the lessons (Bayaga et al., 2019; Nzaramyimana et al., 2021).

FIGURE 23

Figure 23. Illustration of two tangents from the same points are equal.

Mr. Marutha introduced the theorem stating that two tangents from the same external point are equal, a fundamental concept in geometry that requires thorough planning and solid CK. His ability to engage learners through recitation demonstrates effective PCK that encourages active participation, which is crucial in mathematics teaching (Bobis et al., 2016). To deepen understanding, he extended the discussion to prove that a tangent is perpendicular to the intersected chord. He prompted learners to recall how to determine if two lines are perpendicular, and they correctly responded that they check for a 90° angle. This interaction showcases Mr. Marutha’s strong CK and his PCK and TCK in eliciting prior knowledge, essential for constructing new understanding (Koehler et al., 2013).

His confidence in using GeoGebra was evident as he demonstrated constructing a line from the circle’s center to the midpoint of chord GH, measuring it at 0.9 units and calculating the midpoint using GeoGebra’s calculator feature, highlighting his developed TK (Mokotjo, 2020).

Furthermore, the exchange between Mr. Marutha and his learners further illustrates the effective use of questioning to promote critical thinking. Consider this exchange:

Mr. Marutha: What is the value of angle I

Learners: I is equals to 90 degrees.

Mr. Marutha: now we will check if indeed it is 90° …

In this exchange, Mr. Marutha asked for the value of angle I, and the learners confidently responded with 90 degrees. Mr. Marutha then measured angle I, finding it to be 89.93 degrees, which he used to demonstrate the concept of approximation in mathematics. His candid admission during the VSRI about not always achieving exact values with GeoGebra reflects a growth mindset and a recognition of the challenges inherent in integrating technology into mathematics teaching. He expressed a desire for further training in GeoGebra, that he requires confidence and competence in using technology effectively in the classroom. He said:

I don’t know, that is the challenge I sometimes face, sometimes I get exact values sometimes I get values that I close to the exact answer. Remember I have never been trained to use GeoGebra, I need help with this soft.

He further carried on with the lesson by proving that Δ AGI and Δ AHI are similar, supported by the AAA statement. He measured angle G = 41, 64°; I1 = 89.95°, I2 = 89.95°, H = 41, 64° and he asked learners to calculate the third angle A, they recited that angle A = 48,41. He then concluded that Δ AGI and Δ AHI are similar, demonstrating a good CK. Mr. Marutha was engaging his learners while integrating GeoGebra, demonstrating a high TPK and PCK in his teaching (Koehler et al., 2013). From our observation, it was clearly shown that Mr. Marutha did not struggle with the GeoGebra features to draw and measure throughout the lesson, which demonstrated his TK in integrating GeoGebra in teaching geometry.

Discussion

The findings of this study reveal critical insights into the challenges teachers encounter when integrating GeoGebra into their Grade 12 Euclidean geometry lessons. Teachers such as Mr. Sam and Mr. Thapedi face significant obstacles stemming from inadequate training in Technological Knowledge (TK), which restricts their ability to effectively use the software. This aligns with the research conducted by Mokotjo and Mokhele-Makgalwa (2021), which emphasizes the need for enhanced professional development to improve teachers’ confidence and competence in integrating technology into their teaching. Furthermore, Mr. Marutha’s proactive use of GeoGebra in visualizing and elucidating geometric concepts demonstrates the potential positive impacts of adequate training and structural support. As highlighted by (Widianti, 2023), effective technology integration can lead to improved learner engagement and understanding, particularly when teachers are well-prepared to leverage these tools.

Additionally, the variability in classroom settings and resources across these case studies emphasizes the importance of a supportive and well-designed learning environment for successful technology integration. In Mr. Rathete’s case, the poor classroom setup impeded both visibility and learner interaction, illustrating how infrastructure issues can limit innovative teaching practices. Conversely, Mr. Marutha’s well-equipped laboratory classroom supported effective teaching through enhanced visibility and engagement. This mirrors the findings from studies which emphasize that organized spatial layouts can significantly improve learning outcomes. The integration of GeoGebra, when coupled with appropriate pedagogical strategies and a conducive environment, fosters active participation and collaborative learning among learners, as shown in the positive dynamics observed in Mr. Thapedi’s classroom.

Across teachers, these patterns emerged: Technological Knowledge (TK) was the most significant constraint within the TPACK framework, with all participants acknowledging gaps in their ability to navigate GeoGebra effectively. Limited or superficial training resulted in inconsistent application of the software, leading to inaccuracies in geometric constructions and reliance on approximations during lessons. While teachers demonstrated strong CK and, in some cases, sound PCK, their TPK varied widely. Those with better-equipped classrooms and prior exposure to GeoGebra integrated it more confidently, fostering interactive and learner-centered environments. Conversely, teachers with inadequate infrastructure or minimal training tended to revert to traditional methods, limiting opportunities for active engagement as well as inaccurate representation of mathematical objects. Reflexive comments during Video-Stimulated Recall Interviews revealed a shared recognition that targeted professional development and infrastructural support are essential for strengthening TK and achieving meaningful technology integration in rural mathematics classrooms.

This study contributes to international literature by filling identified gaps regarding the practical challenges of technology integration in mathematics education, particularly within the rural South African context. By highlighting the critical importance of TK and effective pedagogical strategies in the successful use of GeoGebra, the findings align with scholars like Yildiz et al. (2017), Marange and Tatira (2023) who advocate for comprehensive training and support systems for teachers. This study illuminates the urgent need for targeted professional development programs that equip teachers with both the technical and pedagogical skills necessary to foster a deeper understanding of mathematical concepts among learners, thereby enhancing teaching practices globally.

Study limitations

While this research provides valuable insights into teachers’ integration of GeoGebra in rural South African classrooms, several limitations should be acknowledged. Firstly, the study involved only five Grade 12 mathematics teachers. Although appropriate for an in-depth qualitative case study, this limits the generalizability of findings to the broader population of rural teachers. Secondly, substantial portion of the data comes from VSRIs where teachers reflect on their own challenges and practices. Such self-reported data may be influenced by social desirability, limited self-awareness, or reluctance to disclose difficulties, introducing potential bias.

Conclusion

This study examined the challenges and pedagogical strategies associated with integrating GeoGebra into Grade 12 Euclidean Geometry lessons. The findings reveal that inadequate training in Technological Knowledge (TK) significantly hinders teachers’ ability to effectively utilize GeoGebra, resulting in issues such as inaccuracies in representations and reduced learner engagement. The expressed desire for more comprehensive professional development among teachers unearths the necessity for targeted training that enhances proficiency with educational technology. The current study also highlights the critical role of classroom environments in facilitating successful technology integration. A well-equipped classroom can enhance teaching effectiveness and student interaction, whereas suboptimal setups can limit engagement and mathematical communication. This contrast emphasizes the need for supportive infrastructure to accommodate innovative teaching practices, particularly in rural settings where resources may be constrained. Ultimately, this study addresses gaps in the existing literature concerning the challenges of integrating technology in mathematics education, especially within the South African context. The findings advocate for a robust framework for professional development that equips teachers with the skills necessary to seamlessly incorporate technology into their teaching. As teachers continue to navigate the complexities of technology in the classroom, cultivating a strong foundation in both pedagogical and technological knowledge will be essential for fostering effective and engaging learning experiences in geometry.

Based on the study’s findings, the Department of Basic Education (DBE) and provincial education authorities should prioritize structured and sustained professional development programs focused on GeoGebra integration. Current one-off workshops are insufficient as iterated by the teachers; instead, multi-phase training that includes hands-on practice, follow-up mentoring, and classroom-based support is needed. Additionally, policies should address infrastructural disparities by ensuring reliable access to projectors, computers, and internet connectivity in rural schools. Dedicated funding streams for technology integration in mathematics education, particularly in under-resourced regions, would help bridge the digital divide.

In addition, district advisors should embed GeoGebra training within pre-service and in-service programs, emphasizing not only technical skills but also pedagogical strategies for leveraging dynamic geometry software to enhance conceptual understanding. Collaborative learning communities, such as professional learning circles, can support teachers in sharing best practices and troubleshooting challenges. District advisors should also conduct classroom-based coaching to help teachers translate training into effective practice.

Further studies should explore the longitudinal impact of GeoGebra integration on both teacher development and learner outcomes. Specifically, research should examine whether sustained professional development improves teachers’ TPACK over time and how this translates into measurable gains in learners’ performance. Comparative studies between rural and urban contexts could illuminate systemic inequities. Finally, investigating learner engagement and attitudes toward technology-mediated geometry teaching could offer insights into optimizing classroom practices.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics statement

The study was ethically approved by the University of South Africa and the Limpopo Department of Education. The studies were conducted in accordance with the local legislation and institutional requirements. Written informed consent for participation in this study was provided by the participants’ legal guardians/next of kin. Written informed consent was obtained from the individual(s), and minor(s)’ legal guardian/next of kin, for the publication of any potentially identifiable images or data included in this article.

Author contributions

MM: Funding acquisition, Methodology, Writing – original draft, Writing – review & editing, Conceptualization, Formal analysis. HM: Formal analysis, Methodology, Writing – original draft, Writing – review & editing, Funding acquisition, Supervision.

Funding

The author(s) declared that no financial support was received for this work and/or its publication.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was used in the creation of this manuscript. Generative AI was used to ensure consistency in the use of American English style.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

1. ^It is important to note that the quality of the images in this section demonstrates the inherent infrastructural challenges within the observed rural classrooms, including the visibility of the diagrams during the lessons.

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