The framing of this study is grounded in Piaget’s cognitive constructivism, particularly the adaptation processes of assimilation and accommodation. Piaget’s (1970) theory provided a lens for understanding the emergence of learners’ errors and misconceptions by emphasising that new knowledge is constructed in relation to existing cognitive structures. Learners often draw on prior knowledge that may be incomplete or only partially applicable to a new context, and this prior knowledge influences how they interpret and respond to mathematical tasks (Smith, DiSessa & Roschelle 1994). Such interpretations can offer valuable insight into the underlying conceptual challenges learners face and, consequently, highlight the instructional importance of attending to errors and misconceptions.

This theory posits that learners build advanced knowledge by connecting it to their existing knowledge (Smith et al. 1994). Smith et al. (1994) observed that learners often have incomplete understandings shaped by their prior knowledge, which can provide valuable insights for deeper comprehension. This perspective highlights the importance of errors and misconceptions in informing topic-sensitive instructional approaches.

Woolfolk (2010) explained assimilation as the process of integrating new knowledge into an existing knowledge structure. Accommodation occurs when learners encounter challenges that prompt them to reorganise or modify their cognitive structures (Smith et al. 1994). Errors and misconceptions are therefore argued to manifest when these processes are misaligned. Smith et al. (1994) suggested that when learners assimilate new knowledge without accommodating it, overgeneralisations based on prior correct knowledge may arise, leading to errors and misconceptions.

Netti et al. (2016) highlighted the importance of studying student errors in the context of mathematical proofs through the concepts of assimilation and accommodation. Their research investigates how difficulties in constructing mathematical proofs stem from these cognitive processes. They identify incomplete or unrelated schemas as primary causes of misconceptions and invalid proofs. Based on think-aloud protocols with university students engaged in proof tasks, their study reveals that insufficient assimilation hinders the integration of problem components. Inadequate accommodation hinders the necessary cognitive adjustments required during problem-solving, leading to fragmented reasoning (Netti et al. 2016). Although Netti et al. (2016) focused on university students, their use of assimilation and accommodation to study proof construction offers a valuable lens for framing errors within the context of PrR tasks in this study.

To operationalise errors within this cognitive constructivist perspective, this study employs Newman’s (1977) Error Analysis (NEA) model, which categorises errors into five stages: reading (decoding symbols), comprehension (understanding problem requirements), transformation (selecting appropriate models), process skills (executing correct procedures) and encoding (articulating solutions). In this study, the NEA model is conceptualised within the theoretical constructs of assimilation and accommodation to identify learners’ errors and misconceptions as follows:

By integrating NEA model with the processes of assimilation and accommodation, this tool positions errors not as isolated mistakes but as indicators of schema development in engaging with PrR tasks. In this study, the tool was used to analyse learners’ written responses and transcripts from semi-structured interviews. This approach provides a coherent framework linking learners’ cognition to error patterns in trigonometric tasks that embed tenets of PrR.

To enhance credibility, member checking was conducted by sharing summarised interview findings with the 10 interviewed learners and their written responses to confirm interpretations, aligning with Lincoln and Guba (1985) trustworthiness criteria. For written response analysis, inter-rater reliability was supplemented by researcher triangulation, in which preliminary error categorisations were discussed with a third researcher to mitigate bias. Reflexivity was maintained by acknowledging the researchers’ positions as mathematics education researchers, which may influence interpretations of learners’ errors and misconceptions. A reflexive journal was kept to document coding decisions and potential biases. The worksheet and interview questions were aligned with Thompson et al.’s (2012) framework of PrR tasks and Van de Walle, Karp and Bay-Williams’ (2016) open-ended questioning guide, respectively, and analysed using the NEA model, ensuring theoretical consistency.

Ethical clearance to conduct this study was obtained from the Walter Sisulu University Faculty of Education Research Ethics Committee (FEDREC) (No. 2024/FEDREC/2219). Informed consent was secured from learners, parents and school authorities. Anonymity was maintained using pseudonyms (e.g. S1, S2). For learners, additional measures included conducting interviews in isiXhosa to ensure accessibility and cultural sensitivity, with translations verified for accuracy.

The analysis of learners’ written responses revealed 155 errors. In some instances, learners’ written responses displayed more than one type of error. Table 2 provides a summary of the frequencies and percentages of errors that were identified according to the categories of the NEA model.

The data in Table 2 show that process-skill errors are common, as they account for 50% (77 errors) of the errors identified in learners’ written work. This is followed by comprehension errors, which constitute 30% (47 errors), and transformation, at 14%. Encoding errors were infrequent and accounted for only 6%, whereas none of the errors were reading errors out of all errors identified in learners’ written responses. Errors were frequently observed in tasks that involved conjectures and informal arguments, such as the task depicted in Figure 2, which involves a soccer player kicking a ball, and the task requiring connections between multiple representations shown in Figure 3. In contrast, tasks that demanded formal reasoning, such as proving trigonometric identities (Figure 1), had fewer errors compared to those that required informal reasoning (e.g. conjecturing and providing non-proof arguments).

Process-skill errors involve procedural inaccuracies that may result from a lack of comprehension of the task requirements (Newman 1977). Sometimes these errors occur despite the task being comprehended. Figure 1 shows Learner S1’s written response to the task that requires the enactment of formal deductive proofs. Learner S1’s written response shows a process-skill error that stems from a right-angled triangle that is sketched in an incorrect quadrant.

Moreover, Learner S1 demonstrated process-skill errors while solving a trigonometric identity sinθtanθ=−35. This type of error arises from the difficulties S1 faced when using the ratio −35 to sketch a right-angled triangle in the first quadrant. Furthermore, the learner struggled to isolate sin θ by dividing both sides of the equation by 5, highlighting persistent procedural inaccuracies. The learner seemed not to comprehend the specifics of the task, as the equation 5 sin θ-4 = 0 and the restriction 90° < θ < 270° were not used to prove sinθtanθ=−35. Instead, the learner manipulated sinθtanθ=−35 procedurally without considering the specifics of the task, particularly the given equation and the restriction. This indicates that the learner’s work consists of a comprehension error.

Learner S1 furthermore reflected on and interpreted the errors displayed in the written response in an interview with the researcher (I), as can be seen in Table 3 - Transcript excerpt.

The transcript from the interview reveals that Learner S1 was able to identify key aspects of the task. The learner, in line 2, acknowledges the significance of sinθtanθ and 90° < θ < 270°, demonstrating an initial awareness. The learner overlooks the given equation 5 sin θ – 4 = 0. This is corroborated by the learners’ written response that manipulated sinθtanθ=−35 and used the ratio as a guide to sketch the right-angled triangle, neglecting 5 sin θ – 4 = 0. The lack of comprehension of the specifics of the task and the relevant aspects needed to find a possible solution is reflected in the learner’s response in Line 4:

This response from Learner S1 overlooks the effects of the information provided in sketching the triangle in the correct quadrant, resulting in a solution that is out of context. This perspective is further reinforced when the learner said:

In this response, the learner’s conception of sketching the right-angled triangle in an incorrect quadrant is linked with procedural inaccuracies evident in the written response, in which the learner assigns incorrect signs of the ratio to the quadrant after struggling with algebraic manipulation (Figure 1). This underlying misconception is illuminated by the learner’s over-reliance on a non-reflective approach to procedures and neglect of the task’s contextual requirements. The learner confirms this observation by saying:

This suggests that the learner believes arbitrary numerical substitutions can be made without regard for the task’s contextual requirements. The errors captured in S1’s response and interview reflect some of the common underlying misconceptions observed in this study. Learners did not understand the specifics of the task, including the application of 5 sin θ – 4 = 0 and the restriction 90° < θ < 270°, in assisting to sketch a right-angled triangle in the correct quadrant, to prove sinθtanθ=−35.

Transformation errors occur when learners struggle to represent mathematical information and to select relevant operations required to find a solution to the task at hand (Newman 1977). Comprehension errors occur when learners can read but fail to grasp the meaning of the information, impeding their capacity to understand the context of the task (Newman 1977). Figure 2 presents a task that requires learners to conjecture and provide proofs.

In Figure 2, learners were required to determine whether a soccer player would score by kicking the ball at a 22° angle and to justify their answer. This task was designed to prompt learners’ ability to engage in conjecturing, leveraging the provided background information. Beyond initial conjecture, learners were encouraged to offer an explanation to support their mathematical claim. Figure 3 shows Learner S2’s written response to the task (Figure 2), reflecting comprehension and transformation errors.

Learner S2 used cos θ to find the length y, which is erroneously identified as the height. This is evident from the learner’s written response: ‘Yes, because the height is greater than 4’ (Learner 2), with y calculated as approximately 10.34 m. This suggests a challenge in comprehending the interrelationship among angles, distance and height within the context of the given task. In a right-angled triangle in which θ represents the angle of elevation, cos θ is defined as cosθ=AdjacentHypotenuse. This further suggests a lack of understanding that the height should be derived from the opposite side using either tan θ or sin θ. Consequently, the application of cos θ to ascertain the vertical height constitutes an error in transforming the geometric problem into an appropriate mathematical equation. For instance, the use of tanθ=height(h)11, where h is the vertical height, can be used to guide the conclusion of whether the soccer player will score or not, using the given height of the pole (4 m) as a reference.

Figure 4 presents a task in which learners advise the soccer player on the best angle(s) to kick the ball for scoring, following the task described above. The learner’s written work responds to the second question in Figure 2.

The learner’s (S3) written response in Figure 4 demonstrates a comprehension error with the requirements of the task. The learner provides procedures and concludes that ‘No. the soccer player will not score because the angle 23.69°’ (Learner 3). This response has been identified as demonstrating both comprehension and transformation errors, reflecting challenges in both interpreting the task and formulating an appropriate mathematical model.

The comprehension error in the learner’s work is demonstrated by a challenge to grasp the task’s core instruction, which is to ‘advise the soccer player on the angle(s) at which she/he should kick the ball to stand a chance of scoring’. Rather than proposing a conjectural angle based on the provided information, the learner misinterprets the prompt as requiring a formal calculation, leading to an irrelevant conclusion.

The conversation between the interviewer (I) and the learner (S2) offers insight into the learner’s perspective on the errors identified in their written responses to the two tasks related to the soccer ball scenario. The errors, categorised as comprehension and transformation errors, reveal a misunderstanding of the requirements of the tasks (Questions 1 and 2 in Figure 2). Table 4 - Transcript excerpt illuminates Learner S2’s reflections during the interview.

Learner S2’s responses highlight a comprehension error characterised by a superficial engagement with the task’s key aspects. When asked to identify helpful aspects (Question 1), S2 selects ‘angle’ and ‘goal’, suggesting an initial recognition of critical components of the task. However, the reason provided in response to the interviewer’s question:

This learner’s (S2) reflection suggests a limited understanding of how the variables angle, distance and height interrelate within the context of the problem. This perspective reveals that S2 views the angle as a prompt for applying procedural trigonometric ratios, rather than as a variable to be analysed in relation to distance and height to address the scoring outcome. This misalignment is reflected in the tasks illustrated in Figure 3 and Figure 4, both of which pose a challenge to comprehend the tasks’ requirements to conjecture by proposing an angle (s) that is likely to provide a pathway to scoring.

The transformation error is similarly illuminated by S2’s procedural focus, as seen in the response of the interview:

This statement suggests that S2 perceives problem-solving as a procedural application of trigonometric ratios without considering their appropriate contextual application. S2’s reliance on the application of formulas without relating the 4-m height to the 11-m distance hinders the construction of a suitable solution. The learner’s (S2) perspective, as evident in the interview, reveals a dependence on procedural knowledge that lacks conceptual understanding. This aligns with the errors observed in the written responses in Figure 3 and Figure 4.

The learner’s written response, as depicted in Figure 5, reveals an encoding error that underscores a critical challenge in effectively communicating a mathematical solution despite a procedurally sound approach. The task presents an opportunity for learners to engage in formal deductive reasoning by requiring them to explain the trigonometric identity tanθ=sinθcosθ.

The learner initiated this process appropriately by recalling the identity and subsequently expanding it through the substitution of sinθ=oppositehypotenuse and cosθ=adjacenthypotenuse. This initial engagement reflects the task’s potential to foster reasoning by encouraging the learner to connect conceptual definitions to a derived identity. However, the error manifests in the final answer, in which the learner incorrectly states tanθ=oppositehypotenuse deviating from the established definition of tanθ=oppositeadjacent. This misstep is classified as an encoding error, as it highlights the learner’s difficulty in accurately articulating the solution despite executing the preceding procedural steps correctly. The discrepancy arises not from a lack of comprehension or transformation but from an inability to present the conclusion in alignment with the derived logic.

The absence of the concluding synthesis suggests a broader challenge, in which learners struggle to encapsulate the reasoning process into a relevant final answer. This deficiency formed a noted trend in learners’ difficulties with articulating solutions, particularly in tasks that require deductive reasoning.

Based on the results reported in this study, learners exhibited a variety of errors primarily related to process skills, followed by comprehension, transformation and minor encoding errors.

Process-skill errors often arose from challenges in executing trigonometric procedures fluently. For instance, Learner S1 (Figure 1) demonstrated challenges in proving sinθtanθ=−35. The learner assigned the ratio −35 to the first quadrant, resulting in misplacing the right-angled triangle. Furthermore, the learner struggled to isolate sin θ. These challenges point to persistent difficulties in manipulating trigonometric relationships, especially when learners must coordinate algebraic techniques with trigonometric reasoning.

These errors, emanating from the learner’s challenge to prove sinθtanθ=−35, highlight tensions in the learners’ adaptation to trigonometric processes. S1’s response suggests that prior algebraic schemas, particularly the treatment of ratios as fractional comparisons between quantities learned in earlier grades, shaped the learner’s interpretation of the task. Instead of accommodating the new conceptual structure of trigonometric ratios, which depend on both lengths and angles within specific quadrants, the learner assimilated the task into an existing algebraic schema. This led to overgeneralised strategies, such as treating trigonometric ratios as algebraic variables and assuming that ratios remain invariant across quadrants. Such assimilation without adequate accommodation restricts the learners’ ability to interpret trigonometric ratios relationally and to recognise how quadrant-based sign conventions define the behaviour of trigonometric functions.

These findings are consistent with research conducted both locally (Chake et al. 2025; Chigonga 2016) and internationally (Dhungana et al. 2023; Fahrudin, Mardiyana & Pramudya 2019), which similarly report learners’ difficulties with procedural fluency in trigonometry. However, in contrast to the dominance of process-skill errors found in this study, Adebayo (2023) observed that comprehension errors were most prevalent. Together, these studies indicate that secondary school learners commonly struggle when required to move beyond memorised procedures to the abstract manipulation of trigonometric concepts.

The learner’s written response in Figure 1 further revealed a comprehension error. The written response indicates that learners demonstrated difficulties and deficiencies in comprehending the specifics of the task, particularly the restriction 90° < θ < 270° and the equation 5 sin θ – 4=0. Learners struggled to connect these specific concepts to sketching a right-angled triangle in the correct quadrant.

According to the findings, Learner S1 sketched a right-angled triangle in the first quadrant with a positive adjacent side, despite the task’s given specifics. During the interview, the learner stated, ‘Trigonometric ratios always remain the same, no matter which quadrant’ (Learner 1). This indicates that the learner has assimilated the task into an invariant ratio schema and is resisting adaptation to the unit-circle representation, corroborating the written response given in Figure 1. This error arises from a lack of integration across multiple representations, which can often present challenges for learners (Duval 2006).

Maknun et al. (2022) found similar results, indicating that learners face difficulties when dealing with trigonometric tasks that require multiple representations. This suggests that a clearer explanation of the connections between the restrictions and the trigonometric ratios across different quadrants of the Cartesian plane is essential for helping learners to overcome these challenges.

Transformation errors (14%) indicate that learners also struggled to translate contextual information into appropriate mathematical models. For example, a common misconception involved the selection of cos θ instead of tan θ in modelling the trajectory of a soccer ball. Such errors suggest that learners struggle to apply real-world contexts to trigonometric relationships. Similarly, Obeng et al. (2024) and Fauzi, Sagita and Wicaksono (2022) report that transformation errors often reflect learners’ inability to connect conceptual understanding with modelling practices. In this study, these challenges also limited learners’ ability to formulate conjectures, an essential aspect of PrR that scaffolds the construction of a reasoning trajectory, which could lead to a proof model for the task in Figure 2. Although the task (see Figure 2) encouraged exploration through conjecturing and constructing non-proof arguments, learners’ default to procedures (see the learner’s written response in Figure 4) limited this potential. However, this assisted in illuminating the challenges learners encountered in their construction of trigonometric reasoning.

Although encoding errors accounted for only 6% of the total, they highlight an important concern: some learners were able to carry out the procedural steps correctly but were unable to present a coherent final expression. This suggests a notable gap in learners’ ability to synthesise their reasoning and validate the conclusions they reached. Arhin and Hokor (2021) reported similar findings, observing that many learners’ difficulties emerged at the transformation, processing and encoding stages. In this study, such challenges indicate that even when procedural fluency is achieved, learners may still struggle to consolidate and communicate their mathematical reasoning, particularly in tasks that require evaluating the logical structure and consistency of arguments.

The distribution of errors also varied by task type. Learners made more error in reasoning tasks involving multiple representations and verbal contexts compared to formal procedural PrR tasks. This supports the view that PrR in trigonometry imposes a heavier cognitive load, especially when students must simultaneously coordinate symbolic, visual and verbal information (ed. Bieda 2022; Weingarden, Buchbinder & Liu 2022). The relatively few errors in formal PrR tasks that require deductive reasoning suggest that learners are more comfortable following algorithmic steps than navigating explorative multirepresentational reasoning. This reaffirms the need for instructional strategies that move beyond procedural fluency to conceptual engagement. Importantly, PrR tasks mitigate this by promoting the identification of patterns, conjecturing and argument evaluation. These tenets of PrR tasks foster a shift from passive computation to active exploration of mathematical claims (Thompson et al. 2012).

Three metacognitive strands that revealed error patterns were observed from the learners’ interviews: superficial formula-based reasoning, invariant trigonometric ratio misconception and limited reflection.

The first strand, superficial formula-driven reasoning, emerged as the most dominant perceptual pattern stemming from process-skill error. This was evident in comments such as Learner S2’s: ‘I chose angle because trigonometry always uses angles… when there is an angle, I just need to use the trigonometry formula, and the answer will come out’ (Learner 2). The interviews consistently showed that learners relied on isolated keywords such as ‘angle’ or ‘goal’, which activated memorised formulas rather than reasoning about the contextual demands of the task. As the earlier findings highlighted, some learners treated the presence of an angle as justification for invoking familiar trigonometric ratios, even when the conceptual structure of the situation rendered the formula inappropriate. This strand aligns with what Star (2005) terms a formula-driven view of mathematics, in which correctness is equated with executing procedures rather than selecting or justifying a mathematical model. The example of a learner seen in the findings, dividing height by distance to ‘find the angle’, demonstrates this kind of superficial assimilation, in which the procedure overrides the need to interrogate conceptual appropriateness.

The second strand, the invariant-ratio misconception, generated both comprehension and process-skill errors. Learners assumed that trigonometric ratios remain constant across all quadrants, which led to systematic sign errors and misunderstandings of angle restrictions. For example, Learner S1’s belief that ratios are quadrant-independent hindered the accommodation of quadrant-specific variation, an idea fundamental to understanding the unit circle. The interviews further revealed that learners seldom used visual representations, such as sketching right-angled triangles on the Cartesian plane, which could have supported the reconciliation of symbolic expressions with geometric constraints. These findings align with Duval’s (2006) argument that mathematical understanding depends on coordinating multiple representational registers. Similarly, Büttner and Erath (2023) showed that learners construct trigonometric meaning through transformations between such registers, particularly when tasks deliberately prompt them to reason across representations. The feature of prompting learners to reason between representational registers is inherent in PrR tasks, which facilitates the communication of logical reasoning (Weingarden & Buchbinder 2022), allowing error patterns to be mapped accordingly.

The third strand, limited reflection, had a strong influence on learners’ reasoning. Many learners equated computation with correctness, as seen in S1’s comment: ‘There is nothing wrong; I can use any number I find’ (Learner 1, Line 8 in Table 3). Interviews revealed that learners rarely assessed whether their problem-solving approaches were effective or if their conclusions were logically sound. One learner (S2) expressed this mindset succinctly: ‘…I got an answer, so it must be right’ (Learner 2). This perspective reduces mathematical activity to symbolic manipulation, rather than assessing arguments or assumptions. This error is defined as encoding errors in which learners fail to ensure their final answers correspond with their initial reasoning. Interviews support Mukuka and Tatira’s (2025) view that learners need structured opportunities to read, critique and reflect on their written solutions. The data suggest that learners often simplify exploratory tasks into procedural routines, which diminishes their reflective potential.

Taken together, these three strands illuminate why learners continue to display persistent error patterns despite exposure to tasks designed to develop reasoning and proof. The interview results indicate a need for explicit instructional interventions that include structured prompts. These instructions can consist of tasks that draw on multiple representational registers and the tenets of PrR to help learners interrogate their assumptions. Such opportunities, according to Büttner and Erath (2023) and Thompson et al. (2012), facilitate reflective reasoning. This practice can support learners in accommodating new trigonometric principles, rather than simply assimilating them into familiar procedural habits.

The study focused on a single school, which limits generality. However, this limited focus aligns with the aims of the study’s emphasis on insightful interrogation of data that are characterised by a case study (Budiyanto et al. 2019).

During the preparation of this work, the authors utilised GROK version 4 and QuillBot for language editing (grammar and adhering to the word count) and textual formatting. The content was reviewed and edited by the authors, who take full responsibility for its accuracy and completeness.