A purposive sampling technique was employed to select participants for this study, which was conducted at a private university located in Majalengka, West Java, Indonesia. Initially, three first-year preservice mathematics teachers enrolled in a Basic Mathematics course within the Bachelor of Education program in mathematics were selected. To enhance the depth and validity of the findings, the sample size was expanded to six to ensure greater diversity and triangulation of findings. Participants were selected based on the following refined criteria:

The expanded participant pool allowed for greater variability in experiences and cognitive patterns, strengthening the transferability of the study’s conclusions while maintaining a focus on in-depth exploration.

A multi-method data collection strategy was implemented, grounded in APOS theory. Instructional activities followed the activities class, discussion, and exercises learning cycle, designed to foster deep engagement and active learning according to (Mutambara & Tsakeni, 2022). The data collection methods included:

These methods were designed to ensure triangulation and validity. They capture cognitive obstacles from multiple perspectives while aligning with the APOS framework.

The four stages of learning a mathematical concept, as proposed in APOS theory, are based on the foundational work of Dubinsky (1997), which provides a clear framework for understanding the genetic decomposition of complex numbers. Building upon this, the researchers developed the following genetic decomposition for numbers, which serves as a tool for analyzing the cognitive processes involved in mastering such concepts.

Thematic analysis in this study followed a structured coding protocol grounded in the APOS theory (Dubinsky & Wilson, 2013), aimed at identifying students' cognitive constructions and obstacles in learning algebraic concepts. The analysis was conducted in four sequential stages:

Raw data-including interview transcripts, written responses, and classroom observation notes-were segmented into components corresponding to the APOS framework: Action, Process, Object, and Schema. This step enabled researchers to map cognitive responses to specific theoretical constructs.

Within each APOS component, instances of misconceptions, incomplete mental constructions, and procedural errors were identified. These obstacles were coded and categorized to reveal areas where participants struggled in forming coherent mathematical understanding.

A detailed cross-case analysis was performed to compare cognitive patterns across the six participants. This involved examining the consistency and variation in how each participant navigated through the APOS stages. The analysis revealed recurring cognitive difficulties, especially in the Action and Process stages, such as rigid reliance on algorithms, difficulty transitioning from procedures to mental processes, and weak abstraction of algebraic structures. However, distinct context-specific challenges were also identified. For instance, participants with stronger procedural fluency sometimes lacked representational flexibility, while others demonstrated conceptual insight but struggled with symbolic manipulation. These comparisons provided insight into both shared learning trajectories and individualized obstacles, allowing for a nuanced understanding of pre-service teachers' cognitive development.

Participants’ learning trajectories were analyzed to trace the development and transformation of introductory number and algebraic concepts through the lens of APOS theory. This genetic decomposition illuminated how participants constructed mental actions, progressed to internal processes, and eventually formed structured schemas.

This analysis served as a theoretical model to interpret how participants overcame (or failed to overcome) specific cognitive barriers.Table 1 presents the refined genetic decomposition aligned with basic algebraic concepts and the APOS framework.

To ensure reliability and credibility, the coding scheme and interpretive themes were reviewed in peer debriefing sessions with two independent researchers. Discrepancies were discussed and resolved through consensus. Additionally, triangulation of data sources-including interviews, worksheets, classroom observations, and reflective journals-enhanced the trustworthiness of the findings.

Ethical approval for the study was granted by the institutional review board, ensuring compliance with established ethical research standards. Informed consent was obtained from all participants, and their confidentiality was rigorously safeguarded. Pseudonyms were employed throughout the study, and all audio and video recordings were securely stored on encrypted devices. Participants were kept informed about the study's progress at regular intervals, promoting transparency and fostering trust.

Many participants experienced difficulty simplifying logarithmic expressions in the logarithm and matrix questions. Most of the answers showed confusion in applying the basic rules of logarithms or in connecting the concept of logarithms with matrix operations. When asked to determine the determinant of the logarithm matrix, many participants could not solve the problem correctly. Their answers were more procedural, without a deep understanding of the basic principles of logarithms and how these concepts are applied in a matrix context (Ndlovu & Brijlall, 2015). This suggests that

most participants were operating at the action stage in APOS theory, where they focused more on the mechanical steps than understanding the underlying concepts.

Similar findings have been documented in recent studies. For instance, (Aksu et al., 2022) found that pre-service mathematics teachers often rely on memorised procedures without demonstrating conceptual understanding, especially in algebraic contexts. Likewise, (Mustafa & Derya, 2016) reported that many university students could carry out symbolic manipulation correctly while failing to interpret the underlying mathematical structures, indicating a predominance of action-level reasoning. These results underscore the persistent gap between procedural fluency and conceptual understanding among prospective mathematics teachers, particularly during the early phases of cognitive development in algebraic thinking.

Figure 1 shows the problem involves multiplying two matrices containing logarithmic expressions, which requires not only algebraic manipulation but also a strong conceptual understanding of the properties of logarithms and matrix operations. In the action stage, students apply basic procedural steps such as substituting logarithm values ​​and performing basic matrix operations. However, errors in calculations or misapplication of logarithm properties often indicate a reliance on memorised procedures rather than an understanding of the underlying concepts. In the process stage, students move from performing individual operations to understanding the relationships between matrix elements. However, challenges may arise due to misunderstandings regarding the rules of logarithms or the procedure for matrix multiplication, highlighting the need for a deeper understanding of the interrelationships of these operations. Thus, the Object and Schema stages cannot be performed.

The following is a discussion that highlights the cognitive challenges faced by students to provide a clearer picture of the difficulties they experience in understanding mathematical proofs and how teachers help them overcome these obstacles;

Teacher: Before we proceed, let us start with this simple problem. How do we simplify log a(xy)? Can anyone explain?

Student 1: For me, I divide loga(x) and loga(y).

Teacher: Ah, that is a common mistake. Remember, the correct property is log a(xy)= log a(x) + log a(y). So we cannot divide them. Can anyone show us the correct way?

Student 2: So, we add log a(x) and log a(y)? That sounds reasonable, but what if we have to multiply in logarithms?

Teacher: Yes, that is right. Remember this as a rule that applies when we work with logarithms. Now, what if we try to calculate the determinant of this matrix?

Student 3: I am confused, sir. Should we calculate these logarithms one by one first or go straight to the matrix operation?

Teacher: Let's simplify the elements of the logarithm. Let's do the steps one by one.

Student 1: And then?

Teacher: After we simplify, the matrix becomes easier to calculate. Does anyone have difficulty with simplifying logarithms?

Student 1: I am still a bit confused, ma'am, about how to visualise the relationship between logarithms and these matrix operations. Why do we need to convert logarithms to ordinary numbers?

Teacher: That is a great question! It is important because we want to simplify the calculations to focus on the bigger steps. Remember, logarithms help us express numbers in a form that's easier to process in some cases.

Research on mathematical errors among students reveals common challenges in matrix operations, algebraic calculations, and problem-solving. Factors contributing to these errors include insufficient practice, lack of attention to problem details, and poor retention of mathematical concepts. Addressing these challenges requires a focus on strengthening conceptual understanding and providing diverse problem-solving opportunities for students.

Here is a discussion that illustrates the cognitive obstacles students face at the Process stage of APOS theory, along with how teachers can help them overcome these obstacles to improve their understanding of mathematical proofs:

Teacher: Now, how do you calculate the determinant of the simplified matrix?

Student 2: I tried using the determinant formula, which is ad−bc, but I am afraid I will make a mistake when simplifying.

Teacher: Okay, let us see together. What if we use the elements that we simplified earlier?

Student 3: Adi, we have a matrix, and now we calculate its determinant.

Student 1: The determinant is 2(ad−bc). I am a little hesitant to calculate it

Teacher: Good! Is anyone having trouble with this step? What happens if we use logarithms in a more complicated context?

Student 2: Do the properties of logarithms still apply, like log a(xn)=n log a(x)? I often get confused about when to apply them.

Teacher: Yes, they are important, and sometimes they can be confusing. The key is understanding when we can manipulate exponents or logarithms in that way. Remember, whenever we see an exponential or logarithmic form, we can try to exploit these properties.

Research on mathematical abstraction and connections reveals challenges students face in advanced topics like matrix algebra and logarithms. Students also have difficulty generalising concepts and connecting logarithms with other applications, such as converting to exponential form. These issues stem from various obstacles, including cognitive, genetic, psychological, and epistemological obstacles. To address these challenges, interventions like reinforcing prerequisite material, providing scaffolding, and using convergent questioning are recommended. Additionally, fostering connections between different representations, part-whole relationships, and mathematical procedures is crucial for problem-solving success. Improving instructional methods based on understanding students' mental constructions can help overcome mathematical abstraction and connection difficulties.

Here is a discussion that illustrates the cognitive obstacles students face at the Object stage of APOS theory, along with how teachers can help them overcome these obstacles to improve their understanding of mathematical proofs:

Teacher: At this point, we want to look at logarithms and matrices not as separate elements but as an integrated object. What happens if we change the elements of this matrix to exponential form? Does the determinant remain the same?

Student 1: Would we get the same result for the determinant? I do not think I understand the relationship between logarithms and exponentials.

Teacher: That is a good question. Let us try to understand that logarithms are the inverse of exponentials, so changing the form can affect the way we view the matrix. What if we try to visualise this matrix in a graphical context?

Student 2: I am still confused about how logarithms affect this matrix operation. Why do we need to know how these elements work together?”

Teacher: Logarithms give us a way to understand changes in a more structured form. By seeing these elements as part of a larger object, we can integrate their properties more easily. This will help us understand more concepts in the future.”

The APOS theory provides a framework for understanding students' conceptualization of mathematical concepts. While APOS theory has been influential in mathematics education research, it is not universally accepted as a Kuhnian paradigm and may be better described as a Lakatosian research program.

Here is a discussion that illustrates the cognitive obstacles students face and how teachers help them overcome these obstacles, mainly focusing on the conceptual integration at the Schema stage:

Teacher: Look at the relationship between logarithms and matrices in a broader context. How can we apply these concepts to solve other problems, such as systems of exponential equations?

Student 3: Can we relate logarithms and matrices to solve systems of exponential equations? I am still not sure how these concepts are related.”

Teacher: That is right! We can use the properties of logarithms and matrix operations to solve such systems. Imagine connecting these two concepts. What patterns do you notice when you compare the determinant results with different logarithm elements?

Student 2: I am beginning to see how these previously separate concepts can be related. However, I am still having trouble finding the right pattern.”

Teacher: This is an important step in forming our mental schema. The more you connect these concepts, the easier it will be for you to recognise patterns and use them to solve other problems.

This discussion helps students address the cognitive obstacles they encounter in understanding logarithms and matrices. Obstacles such as confusion in applying the properties of logarithms, concerns in understanding matrix operations and difficulties in seeing relationships between mathematical concepts can be addressed stepwise from a more mechanical to a more integrated understanding. The APOS-based approach allows students to build more profound and flexible understandings, helping them overcome cognitive obstacles and preparing them for more complex mathematical challenges.

Group discussions provided further insight into the cognitive obstacles that participants faced. Based on the students' work, the APOS theoretical framework can effectively analyse cognitive obstacles in mathematical proof. The given problem requires solving a system of linear equations to determine the parameters a, b, and c in the quadratic equation y=ax 2 +bx+c. This process demands a conceptual understanding of algebraic structures, procedural fluency in manipulating equations, and the ability to transition from an operational to an abstract comprehension of quadratic functions. Many students encounter difficulties at the action and process stages, where they focus on procedural execution without a deeper grasp of the underlying mathematical relationships. Challenges in transitioning to the object and schema stages often manifest in errors related to equation manipulation, misinterpretation of system constraints, and the inability to generalise results. Addressing these cognitive obstacles through targeted instructional strategies can enhance students' ability to construct mathematical proofs and develop a robust conceptual understanding of quadratic equations.

Figure 2 presents the written response of a participant solving a system of linear equations to determine the coefficients a, b, and c in the quadratic function y=ax²+bx+c. The problem requires substituting given coordinate points into the general quadratic equation, forming a system of

equations, and solving for the unknowns. Using the APOS framework, cognitive obstacles in the participant's problem-solving process can be analysed across different stages.

In the action stage, students demonstrate the initial ability to substitute point values (x, y) into a quadratic equation. Although this step is mechanical, some students tend to make errors in basic algebraic operations, such as incorrect addition or elimination of equations. A common obstacle at this stage is difficulty understanding the initial steps relevant to the problem, such as determining the relationship between the given points and the coefficients a, b, and c. In the process stage, the systematic elimination of variables indicates an emerging grasp of equation-solving techniques. However, minor computational errors or inconsistencies may highlight difficulties in comprehending relationships between equations.

Moving to the object stage, the participant arrives at specific values for a, b, and c, demonstrating an ability to conceptualise the quadratic function as a structured mathematical entity. However, any inconsistencies in their final solution could suggest gaps in recognising broader algebraic implications. At the schema stage, a fully developed understanding would require the participant to recognise alternative solution methods, generalise their approach to different quadratic problems, and validate the correctness of their results. This analysis underscores common cognitive obstacles in solving quadratic equations and highlights the need for instructional interventions that enhance both procedural fluency and conceptual comprehension.

The following is a conversation that highlights the cognitive obstacles that students face during the stages of solving a quadratic equation system problem, which illustrates cognitive obstacles at the Action Stage according to APOS theory:

Teacher: We will solve the system of equations to find the values of aa, bb, and cc in the quadratic equation y=ax 2 +bx+c. For example, we are given points such as (x1,y1), (x2,y2), and (x3,y3). Try to substitute the values of these points into the quadratic equation.

Student 1: Okay, I substitute the values ​​of x1 and y1 into the equation. I get y1=ax12+bx1+c. Then I continue for the other points.

Teacher: Good. However, pay attention to whether the substitution results are correct. Are you sure there are no addition or multiplication errors in basic algebra?

Student 2: I feel a little confused about the negative signs in the equation. Why are the results not as expected?

Teacher: Pay attention again to each step and the negative signs. At this stage, it is important to be careful with each algebraic operation you do.

Research on students' difficulties with quadratic equations reveals common errors across educational contexts. These include mishandling signs, computational errors in introductory algebra, and challenges in applying equation-solving procedures. Students struggle with factorization, using the quadratic formula, and distinguishing between solving and simplifying equations. Performance patterns are similar across well-resourced and poorly-resourced schools, suggesting systemic issues in curriculum design. To address these challenges, studies recommend immediate feedback, whole-class discussions of errors, and targeted instructional strategies.

Here is a conversation that highlights the cognitive obstacles students face at the Process stage of solving quadratic equation systems, illustrating the obstacles according to APOS theory at the Process stage and the need for reinforcing understanding:

Teacher: Next, let us try the elimination process. From the system of equations you have, try to eliminate one of the variables to reduce the number of equations. What do you get after the first step?"

Student 3: I tried to eliminate c, but the result was not what I thought. There was something wrong with my steps.

Teacher: Did you notice how the negative sign affected this elimination? Sometimes, mistakes happen because we are not consistent in handling these signs.

Student 1: I struggle to subtract two equations with negative signs. I often get confused about whether I need to change the sign.

Teacher: This is indeed a common mistake at this stage. When eliminating, be sure to double-check each step and sign and ensure that the logic of elimination remains consistent. Try to visualise these steps to make them clearer.

Research suggests that students face challenges in mastering systems of linear equations (SLEs) and related algebraic concepts. Students may struggle to validate solutions at the object stage or understand that elimination results must satisfy all original equations. Some students achieve only an action-level understanding of unique solutions but struggle with non-unique solutions requiring process and object conceptions.

Interestingly, algebraic expertise involves retraining the visual system to perceive hierarchical structure in equations, with object-based attention playing a role in parsing expressions. Mathematical modelling tasks can help characterise students' learning routes and identify difficulties, particularly in validating results. These findings highlight the complexity of learning algebraic concepts and suggest potential strategies for improving instruction.

Here is a conversation highlighting the cognitive obstacles students face during the stages of solving quadratic equation systems problems, specifically focusing on the object stage of APOS theory and the process of forming a mathematical object:

Teacher: At this stage, we need to look at the elimination result as a more complete object. Does this elimination result satisfy all the equations you started with?

Student 2: I feel confused because I only did the algebraic part. However, I am not sure if my elimination result is consistent with the whole equation.

Teacher: This is a common problem. At this stage, you should start to see the system of equations and the elimination result as an integrated object, not just separate parts. Try to double-check each solution you find, and make sure that it satisfies all the original equations.

Student 3: I sometimes get caught up in algebraic manipulations and ignore whether the result fits the whole equation. This makes me doubt my final solution.

Research on students' understanding of quadratic equations reveals several challenges. Students often struggle to integrate their knowledge of systems of equations, algebraic operations, and parameter relationships within the quadratic context. The lack of flexibility in thinking is evident in students' inability to connect algebraic and geometric concepts, such as relating solutions to parabolas passing through specific points. Small algebraic errors frequently impact final results, reflecting weak mental schema structures. To address these issues, researchers suggest allocating more time for developing conceptual understanding and procedural fluency, incorporating figural pattern generalisation to stimulate visual-algebraic reasoning coordination, and explicitly considering mental constructions in the learning process.

The following conversation highlights the cognitive obstacles students face during the stages of solving quadratic equation systems problems, specifically focusing on the Schema stage of APOS theory and the integration of concepts at this stage:

Teacher: At this stage, we should be able to see how the relationship between the parameters aa, bb, and cc affects the form of the quadratic equation. Try to relate your results to its geometric form, which is the parabola that passes through these points.

Student 1: I find it difficult to relate the algebraic result to the form of the parabola. How do I know that the values ​​of a, b, and c affect the form of the parabola?

Teacher: This is a very common challenge at this stage. Try to visualise the parabola using a graph. What happens if the value of a is greater or less? How does that affect the slope of the parabola?

Student 2: I feel confused because I only do the algebraic process without understanding how the results change the graphical form. I cannot yet connect the numbers to the graph that is formed.

Teacher: At this stage, you should be able to relate the algebraic result to the graphical form of the quadratic equation. This is important for developing a broader understanding because the results you get apply not only to the numbers but also to the geometric form.

This conversation illustrates how students encounter cognitive obstacles at each problemsolving stage according to the APOS theory. Students struggle to comprehend and correctly execute fundamental algebraic procedures at the action stage, often making errors in basic manipulations. Moving to the process stage, they begin to construct an understanding but continue to face challenges in algebraic manipulation, particularly in handling negative signs and maintaining procedural accuracy. At the object stage, difficulties arise in perceiving the system of equations as a structured mathematical entity rather than a sequence of isolated steps. In the schema stage, students cannot connect their algebraic solutions to geometric representations, specifically the parabolic nature of quadratic equations.

This analysis highlights the necessity of targeted learning interventions to facilitate students' progression through the APOS theory more effectively. For the Action stage, instructional strategies should emphasise a deep understanding of fundamental algebraic properties through structured exercises. During the process stage, students must be encouraged to reflect on their problemsolving steps and develop skills to verify the consistency of their results. To address challenges in the Object and Schema stages, incorporating visual representations-such as parabolic graphscan enhance students' comprehension of the relationships between coefficients, equations, and their graphical interpretations. These findings further support the need for an APOS-based pedagogical approach that ensures seamless transitions between cognitive stages, ultimately mitigating mathematical proof and problem-solving obstacles.

The analysis of students' work in solving systems of linear equations to determine the parameters a, b, and c in the quadratic equation y=ax²+bx+c reveals distinct cognitive obstacles at each stage of the APOS framework. Research on pre-service mathematics teachers' understanding of complex mathematical concepts has identified various cognitive barriers within the APOS framework. 2016Ndlovu and Brijlall () found that many pre-service teachers remain at the Action and Process stages when learning determinant concepts, often demonstrating procedural knowledge without a deep conceptual understanding. Similar challenges are observed in logarithmic differentiation, where misconceptions persist among learners at the Action and early Process stages (Asghary et al., 2023). Additionally, difficulties in solving logarithmic equations, especially in proofs, highlight significant gaps in foundational knowledge of logarithmic concepts and rules (Okoye-Ogbalu & Nnadozie, 2024). These findings underscore the need for mathematics teacher education programs to emphasise both procedural fluency and conceptual understanding, as addressing misconceptions and enhancing competency will better prepare pre-service teachers for effective instruction.

Students face significant challenges when working with logarithms, often exhibiting errors rooted in both conceptual and procedural misunderstandings. Common mistakes include technical miscalculations, misapplication of algebraic rules, and difficulty applying logarithmic properties correctly ((Bardini et al., 2004);(Rafi & Retnawati, 2018)). Rather than relying on rote memorisation and intuition, which often lead to incorrect reasoning, students should be encouraged to develop a solid understanding of logarithmic definitions ((Dintarini, 2018);(Tatira & Mukuka, 2024)). Conceptual errors are the most prevalent among error types, followed by procedural and technical mistakes ((Angraini et al., 2024);(Yodiatmana & Kartini, 2022)). These difficulties arise from weak foundational knowledge, leading to misconceptions such as treating "log" as a variable or failing to recognise structural relationships between different logarithmic expressions ((Campo-Meneses et al., 2021);(Yodiatmana & Kartini, 2022)). While students perform adequately on routine calculations, they struggle with problems requiring higher-order thinking skills (Kania & Kusumah, 2025).

Findings from this study corroborate the patterns reported in prior research, particularly regarding students' dominance at the action stage in APOS theory, where reliance on mechanical procedures often overshadows conceptual understanding. Similar to the findings of (Bardini et al., 2004) and (Yodiatmana & Kartini, 2022), our participants frequently misapplied logarithmic rules, revealing a lack of structural comprehension. Moreover, while previous studies identified isolated errors in procedural or technical steps, this study extends those insights by illustrating how such errors cluster within specific APOS stages, offering a more detailed mapping of cognitive obstacles in learning logarithms. Thus, consistent with (Dintarini, 2018) and (Tatira & Mukuka, 2024) our results emphasize the importance of strengthening conceptual foundations, especially during the transition from process to object stage. This deeper alignment between our findings and the literature reinforces the relevance of APOS theory as a lens to analyze and address the layered nature of algebraic difficulties among pre-service mathematics teachers.

At the Action stage, students primarily engage in the mechanical execution of algebraic procedures, such as substituting given coordinate points (x,y) into quadratic equations. This stage involves fundamental algebraic operations, yet students frequently make errors in basic manipulations, such as handling negative signs, addition, and subtraction. These errors indicate incomplete mastery of algebraic skills, which are essential before progressing to more complex problem-solving tasks. Consequently, the Action stage tends to emphasize mechanical execution without integrating the conceptual reasoning behind each step ((Langi’ et al., 2023);(Trigueros et al., 2024)). In contrast, the Process stage marks a shift in students' understanding-from performing individual operations to comprehending relationships between mathematical elements, such as those found in matrix theory. At this stage, students begin to recognize patterns in logarithmic simplifications and execute matrix multiplication with greater accuracy ((Feriyanto & Putri, 2020);(Kania et al., 2023)). Despite this progress, misconceptions regarding logarithmic properties or matrix multiplication rules often persist, underscoring the need for a deeper, more nuanced understanding of these operations' interrelations. Moreover, during the Process stage, students begin to recognize relationships within systems of linear equations and apply methods such as elimination or substitution. A significant challenge at this stage is managing procedural complexity. Many students rely on memorized steps without fully comprehending the underlying logic, leading to errors when eliminating variables or processing signs. Additionally, students often fail to verify the consistency of intermediate results with the original equations, revealing a fragile conceptual grasp of the system. These findings suggest that instructional strategies should encourage students to critically engage with each step of their solution process, emphasizing the verification of results as key to strengthening their conceptual understanding.

At the Object stage, students are expected to view the system of equations and the elimination process as an integrated mathematical structure. However, obstacles at this stage include the lack of solution validation. While students may successfully determine the values of aaa, bbb, and ccc, they often neglect to check whether these values satisfy all original equations. This oversight suggests that students perceive problem-solving as a sequence of isolated steps, hindering their ability to generalize and apply strategies in different contexts. Reflective problem-solving tasks that encourage students to check their solutions against the entire system of equations are essential at this stage. Students must understand these concepts structurally (Abu-Hilal et al., 2013) and move beyond individual calculations (Padmanabhan et al., 2013). Inconsistencies in determinant calculations or misapplied logarithmic transformations may indicate gaps in their understanding of determinant formulas and logarithmic properties, suggesting the need for further conceptual development.

At the Schema stage, students should synthesize their understanding across multiple representations, linking algebraic solutions to their geometric interpretations (e.g., as a parabola). However, many students struggle to make this connection, demonstrating a limited ability to generalize their findings beyond the immediate problem. This indicates that current instructional strategies may not adequately emphasize the conceptual coherence between algebraic manipulations and their geometric significance. Integrating visual aids, such as graphs or geometric representations, could significantly enhance students' ability to make these conceptual links. At this stage, (Oktaç et al., 2022) stated that students should develop the ability to generalize their approach to broader contexts, such as solving matrix equations involving logarithmic expressions in various applications. Learners should not only understand individual mathematical objects but also integrate them and apply their understanding to solve complex problems across different areas of mathematics.

These findings highlight the necessity for an APOS-based pedagogical approach that systematically addresses cognitive obstacles through structured learning interventions. Progressing from Action to Process requires reinforcing algebraic fluency with exercises that emphasize both procedural accuracy and conceptual understanding. Advancing from Process to Object requires reflective tasks that encourage students to critically analyze their solution steps, verify results, and understand the mathematical structures behind their work. Educators should incorporate visual representations (e.g., graphing quadratic functions) and contextual applications that help students internalize the interconnected relationships between algebraic expressions and their geometric interpretations, facilitating progression from Object to Schema.

Group discussion plays a strategic role in addressing cognitive obstacles encountered by students, particularly in algebraic learning involving logarithms, matrices, and quadratic functions. Research on the APOS framework supports the idea that group discussions significantly enhance students' progression through each stage of mathematical understanding (Hlangwani & Dhlamini, 2024)(Tuktamyshov & Gorskaya, 2024).

At the Action stage, students often rely on mechanical procedures without understanding the conceptual foundations of the mathematics involved. Group discussions help students identify and correct misconceptions ((Saleemad et al., 2022); (Wester, 2021)), such as errors in applying logarithmic properties, by creating a space where they can articulate their reasoning and receive peer feedback. Such collaborative settings not only highlight individual mistakes but also encourage deeper engagement with the conceptual relationships underlying the procedures. As a result, students move beyond rote memorization, developing a more meaningful understanding of the material ((Andrews et al., 2020) ; (Schwarz et al., 2022)). These discussions foster critical thinking and create opportunities for students to explore the rationale behind their actions, thus promoting the development of procedural fluency alongside conceptual awareness.

As students progress to the process stage, they begin to form connections between procedural steps and underlying theories. Group discussions offer a reflective space where students can critically evaluate their reasoning and refine their approaches. This collaborative validation process helps students recognize how logarithmic and matrix properties influence their solutions within broader mathematical contexts ((Campo-Meneses et al., 2021) ; (Kolar & Hodnik, 2021) ). By discussing the interactions between different mathematical operations, such as the properties of logarithms in the context of matrix transformations, students gain insight into how various mathematical elements integrate. This transition from procedural fluency to conceptual understanding is particularly enhanced by group interaction, which allows for the sharing of diverse perspectives on solving problems ((Faridayanti et al., 2025); (Lee et al., 2024)).

At the object stage, students are expected to perceive mathematical concepts as integrated entities. Group discussions facilitate this by providing opportunities for students to explore the interaction between concepts (Ballard et al., 2023)(Oppong et al., 2024) such as logarithms, matrices, and quadratic functions. For instance, students can analyze how logarithmic determinants behave within matrix structures, deepening their understanding of these interconnections and recognizing their role in more complex problem-solving scenarios ((Junarti et al., 2022);(Kania et al., 2024)). By discussing how concepts interrelate, students develop a more holistic understanding of mathematical structures, enabling them to see the 'big picture' rather than isolated concepts.

Finally, at the schema stage, students must integrate their acquired knowledge into a comprehensive framework that connects logarithms, matrices, and quadratic functions across various applications. Group discussions expose students to multiple problem-solving approaches, encouraging them to explore alternative solutions and overcome conceptual obstacles (Moru & Mathunya, 2022). The collaborative nature of these discussions fosters higher-order cognitive development, enabling students to apply their knowledge in more abstract or applied contexts, such as solving real-world problems (Osman et al., 2016) or exploring advanced mathematical topics (Yuanita et al., 2023). By presenting and defending their approaches in a group setting, students can refine their thinking, challenge their assumptions, and extend their problem-solving strategies to new and unfamiliar contexts. This not only enhances their understanding of the material but also helps develop the cognitive flexibility needed for abstract mathematical reasoning (Ningrum et al., 2020)(Richland & Simms, 2015).

In sum, group discussions offer a powerful tool for overcoming cognitive obstacles at each stage of the APOS framework. By fostering reflection, validation, and integration of mathematical concepts, these discussions help students progress from isolated procedural skills to a deep, interconnected understanding of mathematical ideas. Educators should therefore incorporate collaborative learning opportunities into their teaching practices, as these can significantly enhance students' ability to solve complex mathematical problems and apply their knowledge in diverse contexts.