Abstract
Background: This study explored Grade 10 learners’ errors and their perceptions of the errors they made in solving trigonometric proof-related reasoning (PrR) tasks.
Aim: This embedded case study follows a qualitative approach to examine learners’ errors in written responses to trigonometric PrR tasks and their perceptions to illuminate the misconceptions that often hinder the development of PrR skills.
Setting: A secondary school in the rural area of Nkozo in the Eastern Cape province, South Africa.
Methods: A custom-designed test was administered to 40 Grade 10 learners. Purposive sampling was used to select 30 learners’ scripts for analysis of written responses. Following this analysis, 10 learners were purposively selected for semi-structured interviews. Guided by cognitive constructivism adaptation processes, the study employed Newman’s Error Analysis Model as an analytical tool.
Results: Process-skill (50%) errors were dominant, followed by comprehension errors (30%), transformation errors (14%), encoding errors (6%) and reading errors (0%). Learners encountered challenges with incorrect formula application, misinterpretation of angle constraints and difficulty coordinating multiple representations. The written responses indicated a tendency to rely on procedures with minimal understanding, defaulting to surface-level strategies and superficial reasoning even when tasks required conjecturing and evaluating mathematical claims. The interviews supported these findings, revealing limited metacognitive awareness and misconceptions about quadrant-based trigonometric ratios that are considered invariant, which hindered effective reasoning.
Conclusion: The study highlights the significance of instructional strategies that foster conceptual understanding, multiple representational fluency and reflective reasoning.
Contribution: This research deepens the existing understanding of common error patterns in trigonometry and learner perceptions. It offers insights for developing PrR skills in trigonometry.
Keywords: errors; misconceptions; proof-related tasks; trigonometry; learner perceptions.
Introduction
The teaching and learning of trigonometry remain challenging across diverse contexts. In South Africa, studies by Ngcobo, Madonsela and Brijlall (2019), Chake, Willie and Phage (2025) demonstrate that teachers struggle to support learners’ understanding of trigonometric concepts. Comparable challenges have been reported in Ghana (Owusu et al. 2025) and in rural South American contexts (Sayster 2023), indicating a broader pattern in which learners struggle to engage meaningfully with trigonometry. A key contributor to these challenges is the subject’s abstract nature, which compounds the difficulties learners face when attempting to interpret, manipulate and justify trigonometric relationships (Dhungana, Pant & Dahal 2023).
Within rural South African contexts, these challenges are particularly pronounced. Studies in the Eastern Cape, including those by Tunzana, Mukuka and Tatira (2025) and by Mukuka and Tatira (2025), illuminated a range of persistent errors and misconceptions among both school learners and preservice teachers. Although situated in different educational levels, the studies converge in showing that unresolved misconceptions in trigonometry are frequently carried forward and become entrenched in later learning. These misconceptions often give rise to systematic errors, reinforced by learners’ inclination to regard any recalled or loosely related trigonometric statement as mathematically worthy (Star 2005). Similar patterns are evident in broader high school settings, in which transformation and processing errors continue to hinder learners’ problem-solving in trigonometry (Obeng et al. 2024).
Other studies, such as those by Gardee and Brodie (2021), had attributed learners’ lack of understanding and underperformance in trigonometry to errors and misconceptions that arise during the learning process. Ahuja (2018), Arhin and Hokor (2021) and Gardee and Brodie (2021) suggested that it is essential to shed light on these errors and misconceptions, as they provide opportunities for achieving a better understanding of how learners construct the meaning of mathematical concepts. Consequently, errors and misconceptions can be viewed as diagnostic measures that inform the design of context-specific instruction and interventions tailored to learners’ immediate learning needs.
Several studies (ed. Bieda 2022; Brodie 2014; Chigonga 2016; Luneta & Makonye 2010; Matindike & Makonye 2023; Tunzana et al. 2025) have indicated the central role of learners’ errors and misconceptions in teaching and learning. These studies offer a deeper understanding of the challenges and potential intervention measures that can inform teacher practice. For instance, Tunzana et al. (2025) investigated Grade 11 learners’ understanding of trigonometry in a rural Eastern Cape school. Their findings reveal that learners often use sine and cosine without considering context and struggle to recognise trigonometric function behaviours across quadrants, frequently making errors in algebraic manipulations. The study identified effective intervention strategies, including individualised instruction, small-group tutorials and audio-visual aids, which enhanced conceptual understanding. It also highlighted that persistent errors and misconceptions often stem from foundational gaps in earlier grades.
Mutodi, Mosimege and Maphutha (2023) similarly argued that learners’ errors arise from underdeveloped primary conceptions of trigonometric ideas. Considering this, this study focuses on errors made by Grade 10 learners, in which trigonometry is formally introduced according to the Curriculum Assessment Policy Statement (CAPS). The examination of these errors provides insight into learners’ construction of trigonometric meaning during the formative stage.
Errors and misconceptions are related but distinct. Riccomini (2005) distinguished unsystematic errors, which are occasional and unintended, from systematic errors, which recur and signal conceptual gaps that may be masked by incorrect answers. Luneta and Makonye (2010) described errors as observable deviations from correctness in written or verbal work. Misconceptions are less visible cognitive distortions that can even produce correct answers and are harder to remedy through instructional means. Guided by these distinctions, this study analysed learners’ written responses and conducted interviews to identify their errors and probe the misconceptions underlying their trigonometric reasoning.
Trigonometry in CAPS integrates algebraic, geometric and graphical relationships through key functions (Department of Basic Education 2011). Its multiple representational registers demand translation and connection, which learners often find challenging (Büttner & Erath 2023; Duval 2006; Maknun, Rosjanuardi & Jupri 2022), leading to errors in these transitions (ed. Bieda 2022). This positions trigonometry as a relevant topic of inquiry because of these complexities.
A substantial body of research highlights the diagnostic value of learners’ errors and misconceptions in understanding their difficulties with abstract mathematical concepts (Ahuja 2018; Gardee & Brodie 2021; Matindike & Makonye 2023; Owusu et al. 2025; Sayster 2023). These studies consistently demonstrate that errors are an integral part of learning and can foster mathematical identity (Gardee & Brodie 2021) and knowledge refinement.
Understanding learners’ misconceptions is therefore essential for effective teaching. As Luneta (2025) argued, such knowledge forms part of the ‘zone of mathematical competence’, a key component of teachers’ knowledge that underpins the development of appropriate instructional approaches. Despite the established importance of identifying and interpreting learners’ errors, limited attention has been paid specifically to the errors exhibited by Grade 10 learners when working with trigonometric proof-related reasoning (PrR) tasks. This gap underscores the need for a focused investigation into the nature of these errors and their instructional implications.
Proof-related reasoning tasks provide learners with opportunities to engage in processes central to mathematical thinking, including identifying patterns, formulating conjectures, constructing proofs and evaluating arguments (Thompson, Senk & Johnson 2012). Because these tasks require both informal explanation and formal justification, they offer a productive terrain for making learners’ reasoning visible (Weingarden & Buchbinder 2023).
This study shows that such tasks (PrR) not only prompt learners to articulate their thinking but also expose persistent errors and deeply held misconceptions that may remain hidden in routine procedural tasks. Furthermore, this research addresses the gap in studying errors in the context of PrR by examining the errors made by Grade 10 learners in trigonometric PrR tasks. Guided by the two research questions outlined in the following, the study identifies error patterns that emerge from learners’ written responses to trigonometric PrR tasks in a rural school context. It also explores how learners interpret their own errors to inform contextually responsive instructional strategies.
In this way, the study aims to contribute to the literature on trigonometric learning difficulties and teacher instructional practice. The study further demonstrates the diagnostic value of PrR tasks in revealing learners’ errors and misconceptions. The following research questions guided the study:
- What errors do learners make when solving trigonometric PrR tasks?
- How do learners perceive and interpret the errors they made while solving trigonometric PrR tasks?
Framing of the study
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:
- Reading and comprehension errors often arise from incomplete assimilation, in which learners struggle to integrate elements of a problem into existing schemas, resulting in a shallow understanding of the task.
- Transformation errors indicate insufficient accommodation, in which learners fail to adjust existing schemas to fit new contexts, such as choosing an inappropriate trigonometric ratio.
- Process-skill errors reflect an imbalance between assimilation and accommodation when procedures learnt by rote are over-assimilated without adjusting to contextual demands.
- Encoding errors represent a final accommodation failure, in which learners follow correct steps but fail to synthesise coherent conclusions.
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.
Research methods and design
The study employed a qualitative approach to analyse errors and learners’ perceptions in trigonometric PrR tasks. Data from learners’ written responses and semi-structured interviews in the context of an embedded case study design were analysed using NEA. Embedded case study research is characterised by a case with multiple units of analysis (Budiyanto, Prananto & Tan 2019). This enables the researcher to gain insights from both a broad context and specific components within the study. In this study, we focused on learners’ written responses to tasks that incorporated tenets of PrR, which provided a broad overview of error patterns.
Semi-structured interviews with the learners provided a valuable component of the data, offering insights into their perceptions and interpretations of the errors they made. This combination of methods allowed for a detailed analysis of the errors and misconceptions that learners encountered in trigonometry at a school in a rural area of the Eastern Cape. The interviews were conducted in isiXhosa to maximise learner comfort and expressiveness. All interviews were audio recorded, transcribed and translated into English by a professional translator. A second linguist verified the translations to ensure accuracy and fidelity to learners’ intended meanings.
A custom design test guided by Thompson et al.’s (2012) framework for PrR was administered to 40 Grade 10 learners at a rural school in the Eastern Cape. The test consisted of 10 test items, each comprising tasks that bear PrR opportunities. Sampling proceeded in two stages, outlined in Box 1.
Analysis proceeded in two complementary phases. Firstly, learners’ written responses were examined to identify recurring errors. Secondly, transcripts from semi-structured interviews were analysed deductively using the NEA model as seen in Table 1, which provided an organising framework for identifying themes related to learners’ approaches to reasoning, justification and proof. Themes arising from interview data were then triangulated with patterns evident in the written responses. Table 1 presents the NEA model, including types of errors, descriptions and instances for each type of error.
| TABLE 1: Analytical tool of error analysis. |
Trustworthiness measures
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 considerations
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.
Results
The written responses of learners were analysed deductively using NEA. This approach enabled the researchers to identify distinct categories of errors, thereby revealing trends and patterns in the errors made by learners when tackling various PrR tasks. This analysis sought to address the first research question (RQ1): What errors do learners make when solving trigonometric PrR tasks? To gain insights into the underlying misconceptions of learners, task-based interviews were conducted. The themes that emerged from these interviews were subsequently identified using the NEA model, which facilitated an exploration of learners’ metacognitive reflections and their subjective interpretations of the errors they made. This analysis addressed the second research question (RQ2): How do learners perceive and interpret the errors they made while solving trigonometric PrR tasks? The study identified a variety of errors, primarily related to process skills, comprehension and transformation. Encoding errors were found but occurred infrequently among learners’ written responses (Table 1). Additionally, interviews with learners revealed three main trends: reasoning that relies on the use of formulae, the belief in invariant trigonometric ratios across quadrants and a lack of metacognitive reflection on the procedures and answers provided. To elaborate on these findings, a descriptive overview of learners’ errors in their written responses is provided. This is supported by learners’ perceptions and interpretations of the errors they made.
Errors identified in learners’ written responses
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.
| TABLE 2: Frequencies and percentages of errors. |
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).
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FIGURE 1: Learner S1’s response with process skill and comprehension errors. |
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FIGURE 2: Task requiring conjecturing and enactment of proof. |
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FIGURE 3: Learner S2’s written response identified as comprehension and transformation error. |
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Process skill and comprehension errors
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  . This type of error arises from the difficulties S1 faced when using the ratio  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  . Instead, the learner manipulated  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.
| TABLE 3: Transcript excerpt S1’s reflection and interpretation of errors. |
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  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  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:
‘It’s just a story; it doesn’t change whether the answer is correct or not’. (Learner 1, Line 4 in Table 3)
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:
‘It does not change because all trigonometric ratios always remain the same, no matter which quadrant’. (Learner 1, Line 6 in Table 3)
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:
‘There is nothing wrong. I can use any number I find’. (Learner 1, Line 8 in Table 3)
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  .
Transformation and comprehension error
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  . 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  , 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.
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FIGURE 4: Comprehension and transformation error from Learner S3’s written response. |
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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.
| TABLE 4: Transcript excerpt S2’s reflection and interpretation of errors. |
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:
‘I chose angle because trigonometry always uses angles. If I know the angle, I will use the formula that I know’. (Learner 2, Line 4 in Table 4)
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:
‘Yes, because when there is an angle, I just need to use the trigonometry formula, and the answer will come out’. (Learner 2, Line 6 in Table 4)
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.
Encoding errors
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  .
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FIGURE 5: Encoding errors from the learner’s (S4) response. |
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The learner initiated this process appropriately by recalling the identity and subsequently expanding it through the substitution of  and  . 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  deviating from the established definition of  . 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.
Discussion of results
The primary objective of this study was to identify errors made by Grade 10 learners in their written responses to trigonometric PrR tasks. The study furthermore aimed to illuminate learners’ perceptions regarding the errors they made to uncover misconceptions. This was done with the hope of uncovering errors and misconceptions that are pertinent to PrR to assist in context-sensitive instructional responses.
Learners’ errors from written responses
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  . The learner assigned the ratio  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  , 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).
Learners’ perceptions and interpretations of errors from interviews
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.
Conclusion
This study examined the errors made by Grade 10 learners in trigonometric PrR tasks and their perceptions at a rural school in the Eastern Cape. Findings identified process-skill errors (50%) as dominant, followed by comprehension (30%), transformation (14%), encoding errors (6%) and reading errors (0%). These findings reveal persistent procedural and conceptual challenges, particularly with quadrant-based sign conventions and the coordination of multiple representations.
Learners’ overreliance on memorised formulas, coupled with limited metacognitive reflection, constrained the exploratory potential of the tasks and contributed to the error patterns observed. Importantly, the tenets of PrR tasks played a diagnostic role. These tasks required learners to engage in core reasoning activities such as identifying patterns, forming conjectures, offering non-proof arguments, evaluating mathematical claims and providing formal proofs. They created spaces wherein deep-seated misconceptions became visible.
The demand to justify, explain and critique mathematical statements exposed not only procedural fragility but also learners’ difficulties in coordinating representations, selecting appropriate models and reflecting on the logical coherence of their arguments. Through these reasoning-focused activities, learners’ errors and misconceptions about trigonometric ratios, quadrant-based sign behaviour and representational constraints were illuminated and captured systematically in this study.
The study recommends that teacher instruction should foster learners’ abilities to reason flexibly, reflect on their solution processes and justify their steps. Rather than focusing exclusively on procedural practice, teaching should incorporate tasks that require learners to explain, critique and construct mathematical arguments. This suggests that teacher professional development programmes should focus on equipping both prospective and in-service teachers with the skills needed to select and use tasks that promote PrR tasks. This approach will aid in identifying learners’ deep-seated misconceptions and enable teachers to customise their instructional design to address the specific learning requirements of their learners.
The study also demonstrated the importance of using tasks that provoke learners’ reasoning. By engaging learners in activities such as explaining patterns and making conjectures, the study highlights the value of these tasks (PrR). They allow learners to communicate their thinking both informally and formally, creating fertile ground for uncovering learners’ errors and misconceptions.
Limitations
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).
Artificial Intelligence (AI) assisted technology
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.
Acknowledgements
The authors would like to express their gratitude to all participants who shared their insights and information, making this research possible. Their contributions were invaluable to the success of this study. We extend special appreciation to the assistant researchers who diligently participated in data analysis, ensuring inter-rater reliability and upholding the quality criteria of the research process.
This article is based on research originally conducted as partial fulfilment of Yamkela Msi’s Honours degree in Mathematics Education titled ‘Errors Amid Proof-related reasoning tasks in trigonometry: An analysis of Grade 10 learners’ written responses’, submitted to the Faculty of Education, Department of Mathematics, Sciences and Technology Education, Walter Sisulu University, in 2024. The thesis was supervised by Matome E. Matsepe. The thesis was reworked, revised, and adapted into a journal article for publication. The original thesis is available upon request.
Competing interests
The author reported that they received funding from the National Research Foundation (NRF), which may be affected by the research reported in the enclosed publication. The authors have disclosed those interests fully and have implemented an approved plan for managing any potential conflicts arising from their involvement. The terms of these funding arrangements have been reviewed and approved by the affiliated university in accordance with its policy on objectivity in research.
CRediT authorship contribution
Yamkela Msi: Conceptualisation, Formal analysis, Investigation, Methodology. Matome E. Matsepe: Conceptualisation, Supervision, Writing – original draft, Writing – review & editing. Kate Sehowa: Conceptualisation, Methodology. All authors reviewed the article, contributed to the discussion of results, approved the final version for submission and publication, and take responsibility for the integrity of its findings.
Funding information
The National Research Foundation (NRF) provided funding for the research project that led to the current article.
Data availability
The authors confirm that the data supporting this study and its findings are available within the article and its listed references, or from the corresponding author, Matome E. Matsepe, upon request.
Disclaimer
The views and opinions expressed in this article are those of the authors and are the product of professional research. It does not necessarily reflect the official policy or position of any affiliated institution, funder, agency or the publisher. The authors are responsible for this article’s findings and content.
References
Adebayo, R., 2023, ‘An analysis of Grade 11 learners’ errors in trigonometric function graphs’, Master’s thesis, University of Pretoria, Pretoria, viewed 04 May 2025, from https://bit.ly/47BsY0K.
Ahuja, A., 2018, ‘Errors as learning opportunities: Cases from mathematics teaching learning’, in V. Kapur & S. Ghose (eds.), Dynamic learning spaces in education, pp. 125–140, Springer, Singapore.
Arhin, J. & Hokor, E., 2021, ‘Analysis of high school students’ errors in solving trigonometry problems’, Journal of Mathematics and Science Teacher 1(1), em003. https://doi.org/10.29333/mathsciteacher/11076
Bieda, K.M. (ed.), 2022, Conceptions and consequences of mathematical argumentation, justification, and proof, Springer, Cham.
Brodie, K., 2014, ‘Learning about learner errors in professional learning communities’, Educational Studies in Mathematics 85(2), 221–239. https://doi.org/10.1007/s10649-013-9507-1
Budiyanto, C., Prananto, A. & Tan, F.T., 2019, ‘Designing embedded case study research approach in educational research’, International Journal of Pedagogy and Teacher Education (IJPTE) 3(1), 1–18. https://doi.org/10.20961/ijpte.v3i1.15002
Büttner, M. & Erath, K., 2023, ‘The role of representations in developing a conceptual understanding of trigonometry’, in Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13), Alfréd Rényi Institute of Mathematics, ERME, Budapest, Hungary, July 10–14, pp. 723–730.
Chake, K.G., Willie, T. & Phage, P.I., 2025, ‘Teaching trigonometry in Alfred Nzo West Education District: Educator perspective and challenges’, Journal of Inquiry Based Activities 15, 61–82, viewed 10 March 2025, from https://www.ated.info.tr/index.php/ated/article/view/240.
Chigonga, B., 2016, Learners’ errors when solving trigonometric equations and suggested interventions from Grade 12 mathematics teachers, pp. 163–176, International Society for Technology in Education, Washington, DC.
Department of Basic Education, 2011, Curriculum and Assessment Policy Statement (CAPS): Mathematics Grades 10–12, Government Printing Works, Pretoria.
Dhungana, S., Pant, B.P. & Dahal, N., 2023, ‘Students’ experience in learning trigonometry in high school mathematics: A phenomenological study’, Mathematics Teaching Research Journal 15(4), 184–201, viewed 08 May 2025, from https://eric.ed.gov/?id=EJ1409234.
Duval, R., 2006, ‘A cognitive analysis of problems of comprehension in a learning of mathematics’, Educational Studies in Mathematics 61(1), 103–131. https://doi.org/10.1007/s10649-006-0400-z
Fahrudin, D., Mardiyana, M. & Pramudya, I., 2019, ‘Profile of students’ errors in trigonometry equations’, Journal of Physics: Conference Series 1188, 012044. https://doi.org/10.1088/1742-6596/1188/1/012044
Fauzi, N.I., Sagita, L. & Wicaksono, B., 2022, ‘Analysis of student errors in solving trigonometry problems based on the Newman procedure’, Kalamatika: Jurnal Pendidikan Matematika 7(1), 1–14. https://doi.org/10.22236/KALAMATIKA.vol7no1.2022pp1-14
Gardee, A. & Brodie, K., 2021, ‘A framework for learners’ mathematical identities: A critical realist perspective’, Didactica Mathematicae 43, 5–25. https://doi.org/10.14708/dm.v43i1.7114
Lincoln, Y.S. & Guba, E.G., 1985, Naturalistic inquiry, Sage, Beverly Hills, CA.
Luneta, K., 2025, ‘Mathematics teachers’ zone of mathematical competence’, African Journal of Teacher Education and Development 4(1), a140. https://doi.org/10.4102/ajoted.v4i1.14
Luneta, K. & Makonye, J.P., 2010, ‘Learner errors and misconceptions in elementary analysis: A case study of a Grade 12 class in South Africa’, Acta Academica 42(3), 33–56.
Maknun, C.L., Rosjanuardi, R. & Jupri, A., 2022, ‘Epistemological obstacle in learning trigonometry’, Mathematics Teaching Research Journal 14(2), 5–25.
Matindike, F. & Makonye, J.P., 2023, ‘An APOS analysis of Grade 11 learners’ errors and misconceptions under hyperbolic functions: A case study at a rural high school in Limpopo Province in South Africa’, SAGE Open 13(3), 1–14. https://doi.org/10.1177/21582440231190348
Mukuka, A. & Tatira, B., 2025, ‘Analysis of pre-service mathematics teachers’ proof comprehension through Toulmin’s argumentation model’, Journal on Mathematics Education 16(1), 111–130. https://doi.org/10.22342/jme.v16i1.pp111-130
Mutodi, P., Mosimege, M. & Maphutha, K., 2023, ‘Learners’ misconceptions and errors when solving inequalities’, Mathematics Education Journal 7(2), 136–158. https://doi.org/10.22219/mej.v7i2.27675
Netti, S., Nusantara, T., Subanji, S., Abadyo, A. & Anwar, L., 2016, ‘The failure to construct proof based on assimilation and accommodation framework from Piaget’, International Education Studies 9(12), 12–22. https://doi.org/10.5539/ies.v9n12p12
Newman, A.M., 1977, ‘Analysis of sixth-grade pupils’ errors on written mathematical tasks’, Victorian Institute for Educational Research Bulletin 31, 31–43.
Ngcobo, A.Z., Madonsela, S.P. & Brijlall, D., 2019, ‘The teaching and learning of trigonometry’, The Independent Journal of Teaching and Learning 14(2), 72–91. https://doi.org/10.10520/EJC-1d6fb8bc84
Obeng, B.A., Banson, G.M., Owusu, E. & Owusu, R., 2024, ‘Analysis of senior high school students’ errors in solving trigonometry’, Cogent Education 11(1), 2385119. https://doi.org/10.1080/2331186X.2024.2385119
Owusu, E., Banson, G.M., Lotey, E.K., Owusu, R., Ofosua, B. & Arthur, Y.D., 2025, ‘Students’ academic struggles in solving trigonometry problems: An investigative study on students’ programs of study’, Cogent Education 12(1), 1–19. https://doi.org/10.1080/2331186X.2025.2563166
Phillips, M.D., Basson, J., Botha, C. & Odendaal, J., 2016, Mind Action series: Mathematics. Grade 10, 1st edn., All Copy Publishers, Bellville.
Piaget, J. 1970, ‘Piaget’s theory’, in P.H. Mussen (ed.), Carmichael’s manual of child psychology, transl. G. Gellerier & J. Langer, 3rd edn., vol. 1, pp. 703–732, Wiley, New York.
Riccomini, P.J., 2005, ‘Identification and remediation of systematic error patterns in subtraction’, Learning Disability Quarterly 28(3), 233–242. https://doi.org/10.2307/1593661
Sayster, A., 2023, High-school students’ productive struggles during the simplification of trigonometrical expressions and the proving of trigonometrical identities, Faculty of Science, Department of Mathematics and Applied Mathematics, viewed 08 May 2025, from http://hdl.handle.net/11427/38539.
Smith III, J.P., DiSessa, A.A. & Roschelle, J., 1994, ‘Misconceptions reconceived: A constructivist analysis of knowledge in transition’, Journal of the Learning Sciences 3(2), 115–163. https://doi.org/10.1207/s15327809jls0302_1
Star, J.R., 2005, ‘Reconceptualizing procedural knowledge’, Journal for Research in Mathematics Education 36(5), 404–411. https://doi.org/10.2307/30034943
Thompson, D.R., Senk, S.L. & Johnson, G.J., 2012, ‘Opportunities to learn reasoning and proof in high school mathematics textbooks’, Journal for Research in Mathematics Education 43(3), 253–295. https://doi.org/10.5951/jresematheduc.43.3.0253
Tunzana, V., Mukuka, A. & Tatira, B., 2025, ‘Exploring Grade 11 learners’ understanding of trigonometric equations: An evaluation of a remedial teaching intervention’, Mathematics Teaching Research Journal 17(2), 80–103.
Van de Walle, J.A., Karp, K.S. & Bay-Williams, J.M., 2016, Elementary and middle school mathematics, Pearson Education, London.
Weingarden, M., Buchbinder, O. & Liu, J., 2022, ‘Opportunities for reasoning-and-proving in mathematical tasks: A discursive perspective’, in A.E. Lischka, E.B. Dyer, R.S. Jones, M. Strayer & S. Drown (eds.), Proceedings of the 44th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, pp. 121–129, Middle Tennessee State University, Nashville, TN.
Weingarden, M. & Buchbinder, O., 2022, ‘A Sociocultural Perspective on Beginning Teachers Enacting Reasoning and Proving Practices’, in Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education, January.
Weingarden, M. & Buchbinder, O., 2023, ‘Teacher learning to teach mathematics via reasoning and proving: A discursive analysis of lesson plans modifications’, in I.M. Christiansen (ed.), Frontiers in Education, Vol. 8, p. 1154531, Frontiers Media SA.
Woolfolk, A., 2010, Educational psychology, Pearson Education, Upper Saddle River, NJ.
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