This study is underpinned by the Pedagogical Content Knowledge (PCK) framework, originally introduced by (Shulman, 1986), to examine the impact of subject-specific supervision on the development of pedagogical skills and content knowledge among mathematics student teachers. PCK is crucial for understanding the intersection of content knowledge and pedagogical expertise, which enhances teaching effectiveness, particularly in highly specific subjects like mathematics (Shulman, 1986). PCK combines knowledge of the subject with teaching techniques that make learning effective and accessible. This framework is especially relevant to mathematics education, where both the depth of content knowledge and the quality of its delivery are essential (Zubaidah et al., 2023). The framework includes several key components, of which three have been adapted to align with the objectives of this study while remaining closely linked to the original elements. The three relevant tenets for this study are.

This includes a comprehensive understanding of mathematics as a subject—its concepts, procedures, and logical underpinnings. According to (Quinio & Cuarto, 2023), deep subject knowledge in mathematics is crucial for teaching as it informs the instructor about common pitfalls and students’ likely misconceptions.

This pertains to the knowledge and skills related to teaching and learning. In mathematics, pedagogical knowledge (PK) involves presenting mathematical concepts understandably and engagingly, adapting instruction to accommodate different learning styles and abilities. (Shulman, 1986) highlights that PK distinguishes an expert teacher from a mere content expert. This includes explaining complex ideas, designing effective learning sequences, and critically assessing student understanding of mathematics.

The ongoing mentorship and feedback characterise effective supervision experienced educators provide novices aimed at enhancing their teaching strategies and content delivery. (Handbook of Research on Critical Thinking and Teacher Education Pedagogy: IGI Global, 2019) noted that supervisors with a strong foundation in PCK can give precise, actionable feedback that is instrumental in developing novice teachers' pedagogical responses to student learning needs. Furthermore, the study by Kamruzzaman (2023) highlights the significance of mentorship in teacher development, emphasising the role of expert guidance in fostering effective teaching practices.

This study specifically examines how supervisors equipped with robust PCK in mathematics influence the pedagogical capabilities of student teachers in South Africa's diverse and challenging ODeL environments. The investigation focuses on evaluating the effects of supervision enriched with Pedagogical Content Knowledge (PCK) on the proficiency of student teachers in presenting mathematical concepts. It employs methods such as classroom observations, evaluations of teaching performance, and feedback discussions to collect insights into the relationship between supervisory practices and teaching effectiveness. Moreover, the study conducts a detailed examination of the teaching results from the student teachers. It qualitatively explores the impact of supervision informed by PCK, emphasising the presence of SSPP versus its absence.

This framework aims to dissect the critical elements of effective mathematics teaching and the role of expert supervision in fostering superior pedagogical outcomes. By rigorously analysing the influence of supervisors' PCK on student teachers, the study hopes to contribute valuable insights into the practices that best support the development of high-quality mathematics instruction. Building on this theoretical foundation, the following literature review evaluates the existing knowledge to offer a broader understanding of how Mathematics teaching practices have been optimised studied in similar contexts.

Optimising mathematics teaching practices in an Open Distance e-learning (ODeL) context, according to (Prates & Matos, 2021), requires an optimal understanding of this educational model's unique challenges and opportunities. The challenges include effective e-learning platform utilisation and emphasising the educator's role in creating tailored materials and fostering interactive and innovative learning experiences to enhance student engagement and assessment. Various studies such as ((Livers, 2012); (Suwidiyanti, 2020)) have evidenced that pairing mathematics student teachers with subject-specific supervisors can significantly enhance instructional skills. The role of subject-specific supervisors is crucial in ensuring quality teaching, particularly in mathematics, where the complexity of the subject matter demands specialized guidance.

Open Distance e-Learning (ODeL) represents a shift from traditional face-to-face education to more flexible and accessible forms of learning, facilitated by digital technology. This shift necessitates a review of supervision approaches to ensure quality teaching and learning outcomes. Supervision in ODeL is multifaceted, involving overseeing instructional practices and providing continuous support and feedback to student teachers. (Sethusha, 2014) conducted a study at UNISA focusing on the challenges experienced by supervisors in an ODeL environment. She found that the geographical dispersion of students posed a significant challenge, as supervisors had to monitor virtual classrooms, review recorded lessons, and provide feedback through digital communication tools. (Sethusha, 2014) emphasized the importance of adapting traditional supervision methods to the ODeL context to ensure effective support for student teachers.

Subject-specific supervision refers to pairing student teachers with supervisors with expertise in the subject area being taught. This approach is particularly beneficial in mathematics education, where specialized knowledge is essential for effective teaching. (Ngara & Simuforosa, 2017) investigated the quality of teaching practice supervision among open-distance students at the Zimbabwe Open University. They found that subject-specific supervisors could provide more targeted feedback and address content-specific challenges more effectively than general supervisors. Their study highlighted the need for supervisors to have in-depth knowledge of the subject matter to offer meaningful guidance and model effective teaching strategies. Additionally, (Ekeh & Ramsaroop, 2022) highlighted that effective supervision in mathematics is essential for addressing issues such as inadequate resources, overcrowded classrooms, and poor pedagogical content knowledge, which are prevalent in many educational settings. Their research underscores the critical role of subject-specific supervisors in overcoming these challenges and enhancing the quality of mathematics education.

Subject-specific supervisors can offer insights into complex mathematical concepts, suggest effective pedagogical approaches, and identify common student misconceptions. (Fatmasari & Suripto, 2018) conducted a study at the Indonesian Open University to evaluate the teaching practice program in a distance learning environment. They found that the involvement of subject-specific supervisors was crucial in helping student teachers develop a deeper understanding of the subject matter and improve their teaching techniques. The supervisors' specialized knowledge enabled them to provide constructive feedback and foster a more rigorous professional development experience for student teachers.

Despite the benefits, there are challenges associated with subject-specific supervision in an ODeL context. One major challenge is the geographical dispersion of students and supervisors, making real-time interaction difficult, as observed by (Musingafi et al., 2019). Their study also highlighted uneven levels of supervisors and mentors expertise.

(Kanapathipillai, 2022) explored innovative supervision approaches utilizing platforms like ZOOM for meetings, email for document sharing, and social media for peer support, proving more effective than traditional methods during the COVID-19 pandemic in open education contexts. He found that advancements in digital communication tools had mitigated some challenges and offered more advantages than onsite supervision. Another challenge is the potential lack of immediate feedback due to the nature of asynchronous learning environments, as (Canals et al., 2020) found in a study conducted on second language learners and teachers. (Jojo, 2022) suggested utilizing detailed rubrics and formative assessment techniques to provide timely and specific feedback on student teachers’ instructional practices, especially relevant to practice teaching. Establishing regular check-ins and feedback sessions can help maintain a continuous support system for student teachers.

Optimizing mathematics teaching practice in an ODeL context through subject-specific supervision holds significant promise for enhancing instructional skills among student teachers. By leveraging the expertise of subject-specific supervisors, student teachers can receive tailored support that addresses content and pedagogical needs. Despite the challenges inherent in ODeL supervision, innovative approaches and technological tools can help bridge the gaps and ensure effective supervision practices, ultimately leading to improved educational outcomes.

This study adopted a qualitative multi-case study design, as recommended by (Mohamed et al., 2024) for inquiries seeking to understand complex teaching and supervision practices within naturalistic settings. Such a design allowed an in-depth exploration of how subject-specific supervision shapes student teachers’ Pedagogical Content Knowledge (PCK), as framed by (Shulman, 1986).

Participants consisted of five student teachers purposively selected from a cohort of 46 within the Mathematics Specialist-Supervisor Pairing Programme (SSPP). Purposive sampling was appropriate because, as (Tongco, 2007) argue, it enables researchers to select individuals who can provide rich, relevant, and information-laden cases. Three participants were studying Intermediate Phase Mathematics, and two were specialising in Natural Sciences and Technology (Grade 9), all of whom displayed initial conceptual or pedagogical difficulties pertinent to PCK development.

Data were collected using a lesson observation schedule, a post-lesson feedback guide, field notes, and reflective supervisory memos. These instruments were explicitly aligned with the PCK components proposed by (Shulman, 1986). To ensure content validity, two mathematics education specialists reviewed the instruments for coherence, alignment, and clarity, following validation procedures recommended by (Nurcahyono et al., 2020).

Classroom observations formed the primary data source, enabling close examination of instructional sequences, students’ responses, and emerging misconceptions, consistent with Nahouli et al. (2023) emphasis on gathering data in real-world contexts. Structured post-lesson feedback discussions supported clarification and elaboration of observed practices. Supplementary artefacts including lesson plans, worksheets, and WhatsApp reflections were collected to provide additional triangulation and contextual grounding.

(Aquilar Solano, 2020) framework guided the establishment of trustworthiness where credibility was enhanced through triangulation of observations, field notes, and reflective discussions. To erify interpretations with participants, researchers conducted member checking during post-lesson feedback sessions. Peer debriefing with fellow SSPP supervisors strengthened interpretive depth, while a maintained audit trail of memos and dated reflections supported dependability and confirmability.

The data were analysed using thematic analysis, selected for its flexibility and capacity to produce rich and complex insights (Fuchs, 2023). Initial themes were guided by the study’s theoretical framework subject-specific knowledge, PK, and supervision and feedback while additional themes, particularly misconceptions, were incorporated as they emerged from the data, consistent with (Schmidt et al., 2020).

Coding began immediately after each post-supervision session, with the first author identifying potential cases and sharing them with co-researchers via WhatsApp for collaborative discussion. These early codes were refined and classified as either theoretically grounded or emergent, and were subsequently organised into cases that informed the final thematic structure.the development of themes.

The researchers obtained ethical clearance from the university where they are employed, and a clearance certificate was issued with the reference number 2021/11/10/90194969/41/AM. (Beckmann, 2017) stresses that obtaining ethical clearance from the university signifies that human participants are protected from harm. Confidentiality and anonymity of all participants were strictly upheld throughout the research process, as acclaimed by (Nii Laryeafio & Ogbewe, 2023) ensuring that any information that could reveal personal and sensitive information was protected.

Student 1 primarily relied on a teacher-centred approach, focusing on direct instruction with minimal student interaction. This method limited opportunities for students to actively engage with the content or participate in discussions. To improve this teaching practice, the supervisor emphasised the importance of reinforcing a solid understanding of mathematical operations, particularly the correct use of brackets in addition. Since brackets usually denote multiplication, they must be correctly separated by a plus sign when used in addition to prevent confusion. The supervisor recommended adopting a more interactive approach, such as having students work in pairs to solve similar problems. This strategy promotes active engagement and enables the teacher to identify and correct any misconceptions in real-time. Shifting away from a teacher-centred approach and integrating these interactive techniques will create a more dynamic and effective learning environment, helping students gain a clearer and more accurate understanding of mathematical concepts.

Student 2 primarily used a teacher-centred approach, focusing more on delivering content than engaging students in the learning process, which limited opportunities for student interaction. The supervisor recommended shifting to a more learner-centred strategy by incorporating techniques like think-pair-share, where students first consider a problem individually and then discuss it with a partner before sharing it with the larger group. This approach encourages students to articulate their understanding and learn collaboratively. Additionally, using visual aids or manipulatives was suggested to make abstract concepts more tangible and engaging for learners.

To further enhance the teaching of the distributive property, it is essential to revisit and solidify your understanding of this concept. Practice applying the distributive property in various contexts to build confidence. Moving away from a lecture-based format, integrating visual aids and hands-on activities can make abstract mathematical concepts more concrete, helping students better grasp and retain the material. This shift to a more interactive and engaging teaching style will foster a deeper understanding and active participation among students.

Student 3 taught Grade 3 mathematics under the topic as "Fractions - Share and group things equally." In this lesson, she taught the division of various geometric shapes such as circles, squares, triangles, etc. Even though the shapes were not drawn accurately, the student successfully illustrated how to divide a circle, square, and other figures. The challenge in this regard was when the student was supposed to divide a triangle into three equal parts, as illustrated in Figure 2.

According to the student, the two triangles illustrated on the right-hand side illustrate the division of a triangle into three equal parts, where the colouring indicates the fraction . This misconception is passed on to the learners, as it can also be seen in the learners’ workbooks. However, since the supervisor in this scenario was a mathematics specialist, he was able to probe the student teacher to understand if this was indeed a misunderstanding, during the post-supervision session. The student confirmed this misconception. Figure 3 shows the triangle that is divided into three equal parts.

Following the supervisory session, the supervisor clarified to the student that in such instances, it is recommended to utilize an equilateral triangle. The student should identify the triangle's centre and subsequently draw lines from the centre to the corners or to the midpoints of the sides in order to divide it into three equal parts.

Student 3's lesson revealed a common misconception in the geometric division of triangles, particularly in accurately dividing them into equal parts. However, the student effectively employed a learner-centred teaching strategy, utilizing charts and encouraging students to work individually, in pairs, and present their answers on the board, which promoted active participation. This was clearly reflected in the learners’ workbooks. To enhance this approach further, the supervisor advised focusing on the precise use of geometric tools and techniques, such as drawing lines from the centre to the vertices to ensure correct division. In addition, to deepen engagement, assigning different roles during group activities, like scribe, presenter, or discussion leader, was recommended to rotate these roles to ensure active participation from all students. This strategy will reinforce understanding through peer teaching and promote a more comprehensive grasp of geometric concepts. Moreover, incorporating models when teaching could significantly enhance students' understanding of fractions and related concepts. Figure 4 represent the involvement of learners by Student 3 and one learner’s workbook indicating that learners actively participated in the lesson.

Student 4 and Student 5 were grouped because their misconceptions were closely related. These students chose to teach "balancing equations" in their lessons, a Natural Sciences and Technology section, both of them in Grade 9. Even though this topic does not fall under mathematics, it applies the knowledge of mathematics. We believe the errors discovered in these lessons are directly linked to the phenomena under investigation and helped strengthen our argument.

According to Student 4, two hydrogen atoms react with one oxygen atom to form water, as represented in Figure 5. Our focus in this regard is on the students' belief that H₂O results from adding one oxygen atom and two hydrogen atoms, as seen in Figure 4, which will become more evident with Student 5. The student understands H+H=H₂; hence, we have H₂O after the chemical reaction, this misunderstanding was confirmed by the student in the post-supervision discussion. The researcher engaged the student in understanding that the chemical reaction should not be confused with addition in mathematics. Moreover, H+H=2H, not H₂, as the student explained it to the students.

The researcher involved in this supervision is a mathematics specialist and could not thoroughly explain the chemical reaction to the student. However, the researcher sought the intervention of a Physical Science teacher, who was also a mathematics teacher, in the absence of the mentor teacher on the day. The Physical Science teacher clarified this chemical reaction and confirmed that it should not be confused with adding terms in mathematics. The in-service teacher explained that there is confusion among students and some teachers who do not fully understand the distinction between atoms and molecules. In its most stable form, oxygen is diatomic (O₂), meaning it exists as a molecule consisting of two oxygen atoms. Hydrogen also exists as a diatomic molecule (H₂). The Physical Science teacher explained further about the chemical reaction and the balancing

of the equations to the student teacher and the supervisor. Figure 6 depicts how Student 5 presented her understanding of the chemical reaction, which is equivalent to the one committed by Student 4.

Student 5 repeated the error that was committed by Student 4 (see Figure 5). This time, clearly confirming the misconception that Student 4 had as displayed in Figure 6: H+H+O=H₂ O. After the lesson, the researcher probed the student to verify if the misconception was the same as that of Student 4. It was discovered that the student believed that adding H and H results in H₂, as we see on the right-hand side of the equation. Empowered by the previous supervision of Student 4 and the explanation from the Physical Science teacher, the researcher managed to clarify to the student that in the equation H₂ + O₂ → H₂O, one molecule of hydrogen gas (H₂) reacts with one molecule of oxygen gas (O₂) to form one molecule of water (H₂O).

Even though Student 4 & 5 displayed similar misconceptions, especially when it comes to the interpretation of the chemical reaction, their teaching approaches were completely different.

Student 4, like Students 1 and 2, primarily relied on a teacher-centred approach with limited student interaction, delivering the lesson with minimal input from the learners. This method significantly reduced opportunities for active engagement, as the student teacher did not ask any questions to involve the learners during the presentation. The supervisor advised that to enhance understanding, particularly in concepts like balancing equations, Student 4 could have posed open-ended questions and encouraged group work. This strategy would have allowed students to hypothesize answers before discussing them as a class, fostering critical thinking and collaboration.

Additionally, the lesson presented a fundamental misunderstanding of chemical reactions, equating them with mathematical addition. To improve, it is crucial for Student 4 to strengthen his understanding of chemical reactions and how they differ from mathematical operations. Collaborating with science educators or mentor teacher could provide valuable insights and help clarify these concepts. Furthermore, shifting towards a more interactive teaching style, incorporating open-ended questions and group discussions, would significantly enhance student engagement and deepen their understanding of the material. By adopting these strategies, Student 4 can create a more dynamic and effective learning environment that promotes critical thinking and active participation.

Student 5 demonstrated strong engagement with her students through a well-executed mix of interactive activities and discussions. She began the lesson by probing the learners' understanding of balancing, using a balancing scale in small groups to illustrate the concept. Student 5 effectively maintained student involvement throughout the lesson by posing questions for individual answers or group discussions and encouraging learners to balance equations in their workbooks.

The supervisor praised this interactive approach but advised further enhancement by introducing more challenging questions to push students' critical thinking and deepen their understanding of chemical equations. To improve further, it is recommended that Student 5 consult with science educators to solidify her understanding of chemical equations, ensuring that the concepts are taught accurately. By continuing to use interactive strategies and incorporating more complex problems, Student 5 can ensure that her students engage with the material and apply the concepts correctly in various contexts, leading to a more effective and comprehensive learning experience.

In Figure 7, the student teacher demonstrated the concept of a balanced equation using a balancing scale to the learners. By employing this model, the student teacher illustrates the fundamental principle that for an equation to be balanced, both sides must be equal in value, similar to how the arms of a scale remain level when the weights on either side are equivalent. This demonstration serves as an intuitive and concrete method to convey the abstract idea of balance in equations, helping to reinforce the mathematical concept that any operation performed on one side must be mirrored on the other to maintain equality. Through this hands-on approach, the student teacher effectively supports learners' understanding of balancing equations by symbolically linking mathematical expressions to physical balance, thereby making the concept more accessible and meaningful. The findings presented in this section are interpreted in the next section by integrating them with the reviewed literature and theoretical framework underpinning this study.

Across the five cases, both Mathematics and Natural Sciences & Technology student teachers demonstrated conceptual and pedagogical gaps that affected the clarity and accuracy of their instruction. A key similarity was that all participants exhibited discipline-specific misconceptions—for example, the misuse of brackets, incorrect application of the distributive property, and misunderstanding of fractional geometry in Mathematics, as well as interpreting chemical equations arithmetically in Natural Sciences. These misconceptions indicated limitations in subject-specific knowledge, a central component of PCK. Another similarity was that all student teachers showed improvement when provided with probing questions, modelling, and targeted subject-specific feedback. This promotess the value of specialist supervision in correcting misconceptions and strengthening pedagogical reasoning.

Differences emerged in the nature of the conceptual errors. Mathematics students tended to struggle with operational and representational concepts (brackets, fractions, geometric interpretation), whereas Natural Sciences students’ challenges centred on conceptual translation, such as distinguishing symbolic representations in chemical equations from arithmetic operations. Additionally, Mathematics student teachers appeared more dependent on procedural steps, while Natural Sciences students struggled more with the conceptual underpinnings of scientific symbols.

These patterns suggest that misconceptions are prevalent across subjects but manifest differently depending on the disciplinary context. Mathematics errors were largely procedural or representational, while Natural Sciences errors were conceptual and symbolic. Despite these differences, the consistent positive response to subject-specific supervisory interventions across all cases reinforces the conclusion that specialist supervisors are better positioned to identify, interpret, and remediate discipline-related misconceptions. Overall, the cross-case analysis highlights that while the subjects differ in content demands, subject-specific supervision enhances both content knowledge and pedagogical competence across STEM domains supporting the effectiveness of the SSPP model in ODeL environments.

The study revealed critical misunderstandings in fundamental mathematical operations among student teachers, particularly concerning the misuse of brackets and the distributive property. Student 1’s incorrect application of brackets as symbols for addition mirrors the findings of (Das, 2020), who emphasizes that such errors can lead to significant computational mistakes and hinder students' progression to more complex mathematical concepts. The SSPP intervention provided corrective feedback that not only addressed the immediate errors but also contributed to a deeper understanding of mathematical notation, a foundational element crucial for students' future success in algebra and arithmetic.

In a similar vein, Student 2's misapplication of the distributive property underscores the importance of multiplicative thinking in mathematics education, as argued by Hurst and Huntley (2020). They contend that a conceptual understanding of multiplication is vital for students to grasp the interconnected nature of mathematical operations. The intervention’s focus on reinforcing these concepts through targeted instruction demonstrates the effectiveness of specialized supervision in mitigating such widespread misconceptions.

Student 3’s lesson on dividing triangles into equal parts revealed what (Maurice-Ventouris et al., 2021) identify as a common visuospatial misunderstanding. The student incorrectly divided the triangle, failing to apply accurate geometric principles. Using diagrams, while powerful for visualising geometric concepts, according to (Saracco, 2023) can lead to errors if not properly understood and applied. The supervisor's feedback emphasised the importance of accurate geometric tool usage, such as drawing lines from the triangle's centre to its vertices, to ensure correct division.