Algebra plays a central role in secondary school mathematics and functions as a gateway to higher-order mathematical thinking. Proficiency in algebra is widely recognised as essential for students’ success in advanced mathematical topics, including functions, equations, and calculus (Kieran, 2016). Despite its importance, a substantial body of research has consistently shown that students encounter persistent difficulties in learning algebraic operations. These difficulties include simplifying expressions, distinguishing between constants and variables, and correctly applying fundamental algebraic properties ((Chamundeswari, 2014); (Muchoko et al., 2019); (Demonty et al., 2018)). Such difficulties are often rooted in the overgeneralisation of arithmetic rules, superficial interpretations of symbolic representations, and a reliance on procedural strategies rather than conceptual reasoning ((Booth et al., 2016); (Welder, 2012)). As a result, many students struggle to move beyond arithmetic reasoning and develop a structurally coherent understanding of algebra.

In mathematics education research, these persistent difficulties are commonly conceptualised as epistemological obstacles, which arise when students’ prior knowledge becomes inadequate or misleading in new mathematical contexts ((Herscovics, 2018); (Subroto & Suryadi, 2018-11)). Epistemological obstacles differ from simple errors because they are embedded in learners’ ways of thinking and reasoning, rather than in momentary lapses or miscalculations. Recent studies further indicate that epistemological obstacles in algebra continue to impede students’ progression from arithmetic reasoning to algebraic thinking, with long-term consequences for their mathematical development and learning trajectories (Utami & Prabawanto, 2023-10). These obstacles therefore reflect deeper issues related to how mathematical knowledge is constructed, justified, and applied by learners, particularly in relation to variables, equality, and symbolic manipulation across different algebraic situations and problem contexts.

Although numerous studies have investigated students’ algebraic errors and misconceptions and proposed instructional responses to address them ((Fauziah et al., 2023); (Utami & Prabawanto, 2023-10); (Wilujeng & Alvarez, 2025)), much of the existing research remains largely descriptive. In many cases, students’ difficulties are catalogued as incorrect answers or procedural failures, without a systematic analysis of the epistemic structure underlying their mathematical activity. Consequently, the relationship between students’ observable techniques and the mathematical rationales that legitimise those techniques often remains insufficiently explored ((Kabadaş & Mumcu, 2024); (Dassa et al., 2024-12)). This descriptive focus limits the explanatory power of prior studies and constrains efforts to understand why certain difficulties persist across tasks and contexts. These limitations highlight the need for an analytical framework that links students’ actions on algebraic tasks to the theoretical foundations of mathematical knowledge.

The Anthropological Theory of the Didactic (ATD) offers such an analytical framework through the concept of praxeology, which conceptualises mathematical activity as an organised system of tasks (T), techniques (τ), technologies (θ), and theories (Θ) (Gascón, 2024). Praxeological analysis enables researchers to examine not only what students do when solving mathematical problems but also how their techniques are justified, stabilised, or remain fragile. Through this lens, students’ mathematical activity can be analysed in terms of both practical performance and epistemic justification. While praxeological approaches have been widely applied to the analysis of textbooks, curricula, and institutional mathematical practices ((Hochmuth & Peters, 2021); (Utami et al., 2022); (Panjaitan et al., 2025); (Agustito et al., 2025)), relatively few studies have employed praxeology to analyse students’ empirical responses to algebraic tasks, particularly in relation to epistemological obstacles.

Research within the tradition of Didactical Design Research (DDR) has emphasised the importance of identifying learning obstacles as a foundation for instructional improvement and didactical decision-making ((Ruli et al., 2019); (Supriadi, 2019); (Rohimatunnisa et al., 2025)). DDR highlights the pedagogical significance of analysing obstacles prior to designing instructional interventions. In the present study, however, DDR is not adopted as a full methodological framework involving iterative design experiments, classroom enactments, or metapedidactical reflection. Instead, its conceptual orientation serves as a theoretical inspiration, informing the analytical focus on epistemological obstacles as phenomena with didactic relevance. In this sense, DDR provides a background perspective for interpreting the instructional implications of praxeological findings rather than functioning as the primary research methodology ((Pauji et al., 2023); (Fardian et al., 2025)).

Against this background, the present study is a praxeological analysis of junior secondary students’ epistemological obstacles in learning algebraic operations, grounded in empirical data from students’ written work and interviews. By mapping students’ difficulties onto the components of praxeology, this study seeks to provide a systematic account of how epistemological obstacles are manifested in students’ algebraic activity. This approach moves beyond surface-level error identification by examining the relationships between tasks, techniques, technologies, and theories within students’ reasoning. Through this contribution, the study aims to enrich mathematics education research by offering a theoretically grounded and analytically precise understanding of students’ algebra learning difficulties and their underlying epistemic structures.

The findings of this study are presented through a praxeological analysis of students’ responses to four tasks on algebraic operations. Each response was analytically reconstructed into tasks (T), techniques (τ), technologies (θ), and theories (Θ) in accordance with the Anthropological Theory of the Didactic. In this study, techniques refer to the observable procedures used by students, technologies denote the explanations or justifications articulated by students to legitimate their techniques, and theories represent the mathematical norms or structures that are expected to support such justifications but may be implicit, fragile, or absent. This reconstruction enabled a systematic identification of epistemological obstacles by revealing disconnections between what students could do and how they justified their actions. The praxeological profile of Student 1 is presented in Table 1.

The praxeological analysis of Student 1 reveals a clear contrast between stable procedural techniques and fragile or absent epistemic justification. In Problem 1, the student successfully expanded the expression and combined like terms to obtain the correct result . This indicates procedural fluency at the level of technique (τ). However, the student’s justification remained procedural. It was not grounded in an explicit articulation of the distributive property as a general mathematical principle, suggesting a limited technological foundation (θ). More pronounced epistemological obstacles emerged in tasks requiring justification and relational reasoning. In Problem 2, the student did not attempt to prove the identity and expressed uncertainty about how to proceed, indicating that algebraic properties were not internalized as tools for validation. In Problem 3, the substitution of Reflects a fragile conception of variables as generalized quantities and an absence of the theoretical norm underlying algebraic equality. Finally, the omission of the contextual task (Problem 4) suggests difficulty in transferring symbolic techniques to contextual modeling. Overall, Student 1’s praxeology is characterized by stable techniques in routine manipulation but epistemological obstacles arising from the disconnection between techniques, justifications, and underlying theoretical structures.

The interview data further illuminate the praxeological structure underlying Student 1’s written responses as presented in Table 2. In the simplification task (Problem 1), the student articulated a procedural justification focused on “multiplying the numbers outside the brackets” and combining like terms. This explanation reflects a stable technique (τ) supported by a procedural form

of technology (θ), but without explicit reference to algebraic properties as general principles. As such, the student’s success in routine manipulation was not accompanied by a fully articulated theoretical justification. In contrast, the interview responses to Problems 2 and 3 reveal more pronounced epistemological obstacles. The student’s uncertainty about how to begin proving the indicates that algebraic properties were not internalized as tools for justification. In the equality task, the substitution of a variable with a fixed number (“I thought s was like a number”) reflects a fragile conception of variables as generalized quantities and an absence of the theoretical norm that underpins algebraic equality. Finally, the lack of response to the contextual problem suggests difficulty in mobilizing symbolic techniques beyond routine procedural contexts. Taken together, the interview data corroborate the praxeological analysis by showing that Student 1’s difficulties arise not from a lack of procedural skill, but from weak or absent connections between techniques, justifications, and underlying theoretical structures.

The praxeological analysis of Student 2 reveals a more articulated epistemic structure than that of Student 1, particularly in tasks involving algebraic justification as presented in Table 3. In Problems 1 and 2, the student demonstrated stable techniques (τ) supported by explicit technological discourse (θ). Notably, in proving the identity The student explicitly referred to distributive, commutative, and associative properties, indicating that these properties functioned as tools for justification rather than merely procedural rules. This suggests the emergence of an underlying theoretical awareness (Θ) in routine proof contexts. However, this theoretical activation was not consistently mobilized across tasks. In the equality problem, the student simplified both sides independently without attempting to establish equivalence, revealing a fragile conception of equality as a relational structure. Here, the technique was procedurally correct, but the absence of a relational theoretical norm prevented the student from resolving the discrepancy. By contrast, in the contextual problem, the student successfully translated the situation into algebraic expressions and manipulated them meaningfully, indicating coherence between technique, justification, and context. Overall, Student 2’s praxeology is characterized by strong procedural and emerging theoretical resources, with epistemological obstacles localized primarily in the structural interpretation of algebraic equality.

The interview data further clarify the praxeological structure underlying Student 2’s written responses as presented in Table 4. In Problems 1 and 2, the student demonstrated stable techniques (τ) supported by explicit technological discourse (θ). Notably, when proving the identity , the student explicitly invoked the distributive, commutative, and associative properties, indicating that these properties functioned as tools for justification rather than as implicit procedural rules. This suggests that, in routine proof contexts, the underlying theoretical norms of elementary algebra (Θ) were partially activated and operationalized. However, this theoretical activation was not consistently mobilized across all tasks. In the equality problem, the student simplified both sides of the equation independently but did not attempt to establish their equivalence, treating the equality as two separate computations. This reveals a localized epistemological obstacle in the relational interpretation of equality, despite otherwise coherent techniques and justifications. By contrast, in the contextual problem, the student successfully translated the situation into algebraic expressions and manipulated them meaningfully, demonstrating alignment between technique, justification, and context. Overall, Student 2’s praxeology is characterized by strong procedural techniques and explicit technologies, with epistemological obstacles emerging specifically where relational theoretical norms, such as equality as equivalence, are required but not activated.

The praxeological analysis of Student 3’s work reveals stable procedural techniques accompanied by fragile epistemic support as presented in Table 5. In Problems 1 and 2, the student successfully expanded expressions and regrouped terms to obtain correct results, indicating fluency at the level of technique (τ) in routine symbolic manipulation. However, the accompanying explanations remained procedural. They did not explicitly articulate algebraic properties as general principles for justification, suggesting that the technological level (θ) was limited and the underlying theoretical norms (Θ) were only implicitly activated. More pronounced epistemological obstacles emerged in tasks requiring relational reasoning and contextual consistency. In the equality task, the student simplified only the left-hand side of the expression and explicitly expressed uncertainty about how to determine equivalence, indicating the absence of a relational conception of equality as a theoretical norm. Similarly, in the contextual problem, although the student attempted to model the situation symbolically, an incorrect handling of subtraction led to an inaccurate result. This suggests difficulty in maintaining coherence between symbolic techniques and contextual interpretation. Overall, Student 3’s praxeology is characterized by effective techniques in routine tasks but epistemological obstacles arising from weak connections between techniques, justifications, and theoretical structures. To further clarify the nature of these obstacles, a follow-up interview was conducted. The interview data provided insight into the students’ procedural reasoning, limited justifications, and expressed uncertainties, thereby substantiating the praxeological interpretation of fragile technologies and absent theoretical norms. Selected excerpts from the interview are presented in Table 6.

The praxeological analysis of Student 3 reveals stable procedural techniques accompanied by fragile epistemic justification. In Problems 1 and 2, the student successfully expanded expressions and regrouped terms to obtain correct results, indicating fluency at the level of technique (τ). However, the accompanying technologies were limited to procedural explanations (“expanding” and “adding”) and made no explicit reference to algebraic properties as general principles for justification. As a result, the underlying theoretical norms (Θ) were only partially activated and remained implicit. More pronounced epistemological obstacles emerged in tasks requiring relational reasoning and contextual consistency. In the equality task, the student simplified only one side of the equation and explicitly expressed uncertainty about how to determine whether the two sides were equivalent, indicating the absence of a relational conception of equality. Similarly, in the contextual problem, although the student attempted to model the situation symbolically, an incorrect handling of subtraction led to an inaccurate result. These difficulties suggest that while Student 3’s techniques

function effectively in routine symbolic manipulation, they are not consistently supported by robust technologies and theoretical structures when tasks demand justification or transfer across contexts. This disconnection between technique, justification, and theory characterizes the epistemological obstacles identified in Student 3’s praxeology. To capture these patterns systematically, Student 4’s work was reconstructed into tasks, techniques, technologies, and theories. The details of this praxeological analysis are presented in Table 7.

The praxeological analysis of Student 4’s work reveals stable procedural techniques in routine manipulation, alongside significant epistemological obstacles in tasks that require justification and structural reasoning. In the simplification task (Problem 1), the student expanded expressions and combined like terms correctly, indicating fluency at the level of technique (τ). However, the accompanying explanation remained procedural and did not explicitly articulate algebraic properties as general principles, suggesting limited technological support (θ) and only implicit activation of theoretical norms (Θ). More pronounced epistemological obstacles emerged in proof-related tasks. In Problem 2, the student did not attempt to prove the identity , indicating uncertainty in using algebraic properties as tools for justification. In the equality task (Problem 3),

the student substituted the variable with a fixed number, revealing a fragile conception of variables as generalized quantities and an absence of the relational meaning of equality as a theoretical norm. By contrast, in the contextual problem (Problem 4), the student successfully modeled the situation symbolically and obtained the correct result, indicating coherence between technique and context. Overall, Student 4’s praxeology is characterized by effective techniques in routine and contextual tasks, but epistemological obstacles arise from weak or absent theoretical structures in proof-based and relational-reasoning contexts. To further elucidate these obstacles, a follow-up interview was conducted. The interview data provided insight into the students’ procedural reasoning, variable interpretation, and expressed uncertainties, thereby substantiating the praxeological interpretation of fragile technologies and the absence of theoretical norms. Selected excerpts from the interview are presented in Table 8.

The interview data provide further insight into the praxeological structure underlying Student 4’s written responses. In the simplification task (Problem 1), the student articulated a procedural justification focused on multiplying each term inside the parentheses and combining like terms. This explanation indicates stable techniques (τ) supported by a procedural form of technology (θ), while the underlying algebraic properties remained implicit rather than explicitly articulated as general principles (Θ). More pronounced epistemological obstacles emerged in tasks requiring justification and structural reasoning. In Problem 2, the student did not attempt to prove the identity and explicitly expressed uncertainty about how to begin, indicating that algebraic properties were not internalized as tools for justification. In the equality task (Problem 3), the student substituted a variable with a fixed number, reflecting a fragile conception of variables as generalized quantities and an absence of equality's relational meaning as a theoretical norm. By contrast, in the contextual problem (Problem 4), the student successfully modeled the situation symbolically and obtained the correct result, demonstrating coherence between technique and context. Overall, the interview corroborates the praxeological analysis by showing that Student 4’s epistemological obstacles do not stem from a lack of procedural skill, but from weak or absent connections between techniques, justifications, and underlying theoretical structures. Following this analysis, Student 5's responses were examined to further explore patterns of procedural fluency and epistemological obstacles across cases as presented in Table 9.

The praxeological analysis of Student 5’s work reveals a relatively articulated epistemic structure that nonetheless remains fragile across tasks. In Problems 1 and 2, the student demonstrated stable procedural techniques (τ) supported by procedural justifications (θ), successfully simplifying expressions and proving the identity . In these

contexts, algebraic properties appeared to function as emerging tools for justification, indicating partial activation of the underlying theoretical norms (Θ). However, this theoretical activation was not consistently mobilized. In the equality task, the student simplified both sides of the expression independently but did not attempt to establish equivalence, revealing a limited relational conception of equality. Similarly, in the contextual problem, although the student represented the situation symbolically, an error in handling subtraction led to an incorrect result. These difficulties suggest that while Student 5 possesses strong procedural skills and emerging theoretical awareness in routine tasks, epistemological obstacles arise when coordination between techniques, justifications, and theoretical structures is required across contexts. Thus, Student 5’s praxeology illustrates how partial theoretical activation may coexist with persistent epistemological obstacles in algebraic reasoning. To further examine the student’s reasoning and justifications, a follow-up interview was conducted. The interview data provide additional insight into how algebraic techniques were explained, how properties were invoked, and where epistemological obstacles became apparent. Selected excerpts from the interview are presented in Table 10.

The interview with Student 5 provides important insights into the relationship between procedural techniques and epistemic justification in algebraic reasoning. In Problem 1, the student articulated a clear sequence of operations, describing how each fraction was expanded distributively and how like terms were subsequently combined. This explanation reflects stable procedural techniques (τ) supported by operational justifications (θ), indicating fluency in routine symbolic manipulation. In Problem 2, the student explained the proof of the identity by expanding, adding, and regrouping terms. Although algebraic properties such as distributivity and regrouping were effectively used, these properties functioned primarily as procedural resources rather than as explicitly theorized principles. This suggests an emerging but localized activation of the underlying theoretical norms (Θ), limited to familiar identity tasks. However, in Problem 3, the student simplified both sides of the equation independently without establishing equivalence. The student’s explanation indicates that equality was interpreted as a comparison between two final expressions rather than as a relational statement that requires justification. This reveals an epistemological obstacle in coordinating techniques and justifications within a relational conception of equality. Similarly, in the contextual problem, although the student constructed an algebraic representation of the situation, an error in handling subtraction led to an incorrect result. This suggests difficulty in consistently transferring symbolic procedures to applied contexts. Overall, the interview corroborates the praxeological analysis by showing that Student 5 possesses strong procedural competence and partial theoretical awareness but encounters epistemological obstacles when coordinating techniques, justifications, and theoretical structures across different types of algebraic tasks.

This study examined students’ algebraic activity through the praxeological lens of the Anthropological Theory of the Didactic (ATD), distinguishing tasks (T), techniques (τ), technologies (θ), and theories (Θ) across four types of algebraic problems: routine simplification, identity proof, equality between forms, and contextual modeling. While praxeological analysis has been widely applied to textbooks, instructional practices, and curriculum design ((Putra & Aljarrah, 2021); (Dewi & Juandi, 2025-08); (Llanos & Otero, 2024-07)), fewer studies have employed it to systematically investigate students’ discourse as evidence for the presence or absence of technological and theoretical justification. The findings of this study reveal a consistent pattern in which procedural competence coexists with epistemological fragility, particularly in tasks requiring justification, relational reasoning, and generality.

Across students, routine simplification tasks exhibited strong procedural fluency. Most participants successfully expanded expressions and combined like terms to reach the correct simplified form, indicating that the required techniques (τ) were well established. However, students’ explanations were predominantly procedural narratives rather than explicit justifications. Statements such as “I multiplied each term inside the brackets” describe what was done but do not articulate why the step is mathematically valid. In praxeological terms, such utterances do not fully function as technologies (θ), because they fail to invoke the underlying theoretical norms (Θ), such as the distributive property as a general algebraic law. This finding suggests that mastery of technique can mask an absence of theoretical grounding, a phenomenon that may remain invisible when analysis focuses solely on final answers.

The epistemological gap becomes more pronounced in identity and equality tasks. Several students either avoided attempting the identity proof or limited their work to simplifying one side of the equation. Even when both sides were simplified, students often failed to establish their equivalence explicitly. This pattern indicates that the equals sign is often interpreted operationally—as an instruction to calculate, rather than relationally, as a statement asserting equivalence between two expressions. The absence of explicit reference to properties such as commutativity, associativity, or distributivity as justificatory tools reflects a missing Θ concerning equality as a relation preserved under transformation. These findings align with (Bosch, 2015-07) observation that epistemological obstacles in school algebra often arise when students are required to move from procedural manipulation to structural reasoning.

Interestingly, performance on contextual problems reveals a different configuration. Some students who struggled with proof and formal equality were nevertheless able to model contextual situations appropriately and carry out relevant calculations. This suggests that contextual sense-making and formal symbolic reasoning may develop as partially independent praxeologies. Prior studies have similarly reported that students can succeed in word-problem modeling despite difficulties with formal algebraic structures ((Ningrum et al., 2019-04); (Powell & Fuchs, 2014)). From a cognitive perspective, (Nathan et al., 1992) argue that contextual problems engage semantic and situational reasoning processes that differ from those required for formal symbolic manipulation. In the present study, however, contextual success was often accompanied by sign errors or inconsistent operations, indicating that semantic understanding alone is insufficient without stable theoretical norms governing algebraic manipulation.

Three interrelated explanations account for these patterns. First, classroom practices may emphasize procedural fluency over justificatory discourse, providing limited opportunities for students to articulate why algebraic steps are valid. This imbalance reflects the longstanding tension between procedural and conceptual knowledge in algebra ((Hiebert & Lefevre, 2013); (Kieran, 2013)) and is consistent with findings that justification does not emerge spontaneously without explicit instructional support ((Simon & Blume, 1996); (Lannin, 2005)). Second, students’ conception of variables often collapses into a specific numerical value, undermining the generality required for algebraic proof and identity reasoning. Such misconceptions have been widely documented as barriers to algebraic generalization ((Martinez & Castro Superfine, 2012); (Žanko et al., 2019)). Third, the semantic load of contextual problems may overload working memory, increasing the likelihood of sign errors even when students possess relevant procedural skills ((Kieran, 2013); (Wladis, 2019-01)).

From a didactical design perspective inspired by DDR principles, these findings carry important implications. Tasks that foreground relational equality, requiring symmetric transformations on both sides of an equation, are needed to counter the operational interpretation of the equals sign ((Harbour et al., 2016); (Jones et al., 2012)). Instruction should also explicitly surface technologies (θ) by prompting students to name the properties that justify their actions and to reflect on the validity of each step (Ayala-Altamirano & Molina, 2021). Furthermore, variable-as-general-number conceptions can be strengthened through tasks that contrast “true-for-all” and “true-for-some” statements, helping students distinguish algebraic generality from arithmetic instantiation (Malisani & Spagnolo, 2009). Finally, contextual transfer can be supported by representational tools that explicitly track quantities and operations, reducing cognitive load and improving sign consistency (Booth et al., 2015).

A key strength of the praxeological approach lies in its capacity to analytically separate technique from technology, thereby making visible situations in which students can perform algebraic steps (τ) without accessing the theoretical norms (Θ) that would legitimate those steps. By restricting the identification of Θ to what can reasonably be inferred from student discourse, this study avoids overattributing theoretical understanding and offers a more conservative, discourse-grounded account of epistemological obstacles ((Winsløw, 2007); (Zakiah et al., 2025)). Nevertheless, the study is limited by its small sample size and task set, which constrains generalisability (Marek & Laumann, 2025). In addition, interview prompts may have shaped the forms of justification students provided, introducing potential interpretive bias ((Dooren, 2025-08); (Lammers et al., 2013)). The absence of systematic analysis of classroom interaction further limits ecological triangulation ((Abrahamson & Sánchez-García, 2016); (Ma & Norwich, 2007)). The praxeological analyses indicate that procedural competence in algebra can coexist with significant epistemological obstacles in proof, equality reasoning, and contextual transfer. Addressing these obstacles points to the need for instructional designs that foreground relational equality, explicit justification, and stable conceptions of variables, thereby supporting students’ progression from procedural fluency toward structural and theoretical understanding.

This study contributes to the praxeology literature within ATD by demonstrating that epistemological obstacles in algebra are not limited to the absence of techniques or technologies, but may also take the form of technologies that remain epistemically weak, that is, student explanations that narrate procedures without invoking the theoretical norms that justify them. While previous praxeological studies have primarily examined teachers’ praxeologies and agency in instructional design (e.g., Mensah et al., 2024; Mensah, 2025), the present study extends this line of work by providing an empirically grounded account of students’ praxeological configurations, highlighting systematic misalignments between τ, θ, and Θ. In particular, the findings reveal a persistent tension between contextual modeling praxeologies and formal algebraic praxeologies, suggesting that success in one domain does not guarantee access to the theoretical structures required in the other. Methodologically, the study advances ATD-based analysis by adopting a conservative inferential stance toward Θ, coding theoretical elements only when supported by student discourse rather than analyst assumptions. This approach strengthens the explanatory power of praxeological analysis and clarifies how epistemological obstacles can be identified with greater precision in studies of student mathematical activity. The praxeological patterns summarised in Table 11 demonstrate how epistemological obstacles in students’ algebraic activity can be systematically located in the misalignment between techniques (τ), technologies (θ), and theories (Θ), thereby strengthening the explanatory potential of ATD in empirical studies of student learning.

This study addressed its objective of examining students’ epistemological obstacles in algebraic operations by applying a praxeological framework that distinguishes techniques (τ), technologies (θ), and theories (Θ). The findings show that students’ difficulties are not primarily rooted in a lack of procedural skill, but in persistent misalignments between what students do and how they justify their actions. In particular, the analysis reveals that procedural fluency in routine simplification can coexist with fragile or absent justificatory discourse, especially in tasks involving proof, relational interpretations of equality, variable generalization, and contextual transfer. These results clarify the nature of students’ epistemological obstacles by demonstrating that the correctness of procedures alone does not guarantee access to the theoretical norms underlying algebraic reasoning. From a didactical perspective, the study points to several concrete implications for teaching and curriculum design. Algebra instruction should move beyond an emphasis on obtaining correct simplified forms toward systematically engaging students in justifying transformations, establishing equivalence relationally, and articulating the properties that legitimate algebraic steps. Teachers are encouraged to design tasks that require symmetric manipulation of both sides of an equation, explicit use of algebraic properties as warrants for reasoning, and sustained work with variables as generalized quantities rather than as fixed numerical values. In addition, contextual problem-solving should be supported through representational scaffolds that help students track quantities and maintain sign consistency, thereby strengthening the coordination between symbolic manipulation and situational meaning. Conceptually, the articulation between praxeological analysis and principles drawn from Didactical Design Research (DDR) proved valuable for mapping students’ epistemological obstacles and identifying directions for potential instructional design. In this study, DDR functions as a didactic horizon rather than a comprehensive methodological framework, supporting the interpretation of praxeological findings without extending to the design of experiments. While the study's scope is limited by a small sample size, a restricted task set, and the absence of broader classroom interaction data, the results offer theoretically grounded insights with clear didactic relevance. Future research should therefore involve larger and more diverse samples, adopt longitudinal designs, and incorporate systematic analyses of classroom discourse to examine further how students’ praxeological configurations develop from procedural fluency toward stable structural and theoretical understanding in algebra.

The authors would like to thank the participating school, teachers, and students for their cooperation and valuable contributions to this study. Appreciation is also extended to colleagues who provided constructive feedback during the development of this manuscript. The authors gratefully acknowledge the institutional support provided by the Rector of Universitas Majalengka through an internal research grant.