Using Newman's error analysis model to analyze the errors of senior high school students in solving circle theorem problems

Introduction

Mathematics is a fundamental discipline that explores abstract structures, patterns, and relationships, serving as the language of science and technology (Ye et al., 2023). It is divided into various branches, including arithmetic, algebra, geometry, calculus, and statistics, each providing tools for analyzing and solving diverse real-world problems. Beyond its practical applications, mathematics is a pursuit of logic and reasoning, enabling the formulation and proof of theories that underlie natural phenomena and human innovation (Hendriana et al., 2022). It plays a pivotal role in fields such as physics, engineering, economics, computer science, and medicine, offering precise methodologies for modeling, predicting, and optimizing systems. The discipline fosters critical thinking, creativity, and problem-solving skills, forming the backbone of advancements in artificial intelligence, cryptography, and data analysis. As a universal language, mathematics bridges cultures and epochs, contributing to the understanding of the universe, from the smallest particles to the vastness of space (Youvan, 2024).

Many perceive mathematics as a challenging but essential discipline that develops critical thinking and problem-solving skills. However, some students view it as abstract, irrelevant to real life, or excessively difficult, which often leads to math anxiety and reduced interest. For instance, a study by Gjicali and Lipnevich (2021) revealed that students’ attitudes toward mathematics are shaped by their past experiences, social influences, and perceived utility of the subject. Positive experiences with supportive teachers and practical applications tend to foster a favorable perception of mathematics. Conversely, negative experiences, such as repeated failure or overly abstract teaching methods, can create a negative mindset toward the subject (Yeager and Dweck, 2020). In addition, students’ cultural and educational contexts also play a role; in some cultures, mathematics is highly valued and associated with success, whereas in others, it may not hold the same significance (Acharya et al., 2021). Research highlights the importance of teaching approaches that connect mathematical concepts to real-world contexts, which can enhance students’ appreciation and interest in the subject (Sujatha and Vinayakan, 2023).

Circle theorems, which include properties like the angle subtended by a diameter being a right angle and the angles in the same segment being equal, are frequently seen as abstract and challenging to grasp without proper visualization tools. Studies suggest that integrating dynamic geometry software (DGS) like GeoGebra enhances students’ understanding by allowing them to manipulate and visualize the relationships dynamically (Osypova and Tatochenko, 2021; Zhang et al., 2023; Ziatdinov and Valles, 2022). Furthermore, research highlights the role of instructional strategies in shaping perceptions, as teacher-centered approaches often fail to address students’ conceptual gaps. By contrast, inquiry-based learning and real-life applications have been found to improve attitudes and foster deeper understanding (Wen et al., 2023). A study by Mifetu (2023) noted that students who engage in practical problem-solving tasks involving circle theorems are more likely to perceive the topic as relevant and manageable, reducing anxiety associated with abstract mathematical proofs.

The examiners’ reports for the West African Senior School Certificate Examination (WASSCE) from 2021 to 2023 reveal valuable insights into students’ performances on circle theorem questions. According to the reports, many students struggled with applying circle geometry theorems effectively. In 2021, approximately 65% of students were able to solve circle theorem questions, but many did so with notable errors in their calculations or reasoning. Common mistakes included incorrect use of theorems, especially the inscribed angle theorem, and poor handling of geometrical properties like tangents and secants. The error rate in solving these problems was approximately in the range of 30%–40%. In 2022, there was a slight improvement, with approximately 70% of students attempting circle theorem questions, but similar issues persisted, particularly in visualizing and applying theorems to complex problems. The error rate for this group remained approximately 35%. By 2023, the number of students solving circle theorem problems increased to nearly 75%, although errors persisted, primarily because of a lack of understanding of the underlying geometric principles. The error rate, however, showed a slight decrease to approximately 30%, indicating that some students were refining their understanding of the concepts, likely because of better instructional strategies or additional practice.

Newman's Error Hierarchical Model (NEHM) is a cognitive framework that helps in identifying and understanding errors students make when solving mathematical problems, particularly in geometry and circle theorems. The model categorizes errors into five levels: (1) Reading errors (REs), (2) Comprehension errors (CEs), (3) Transformation errors (TEs), (4) Process skill errors (PSEs), and (5) Encoding errors (EEs). In applying this model to circle theorem problems, NEHM can assist educators in diagnosing specific types of mistakes students make at each stage of problem-solving. For instance, a reading error might occur when a student misinterprets a diagram or a written instruction, while a comprehension error arises if the student does not fully understand the principles of circle theorems, such as the relationship between angles and chords. Transformation errors occur when students misunderstand the language or symbols of the problem, such as misinterpreting the given geometric figures or not recognizing key terms in the problem statement. In the context of circle theorems, transformation errors might involve confusion between the properties of tangents, chords, or angles in a circle. The fourth level, processing errors, is related to the application of mathematical principles and operations. Students may fail to apply the correct theorem (e.g., the angle subtended by a chord at the center is twice the angle subtended at the circumference) or misapply relationships like those between tangents and radii. At this stage, common mistakes in solving circle-related problems could involve incorrect angle calculations or misinterpreting geometric relationships between intersecting lines and arcs. Finally, the fifth level, called encoding errors, arises during the representation of the solution, which includes issues like miswriting the final expression or overlooking steps when transitioning from a geometric diagram to symbolic notation.

Newman's Error Analysis (NEA) framework has been employed in many elementary and secondary school mathematics contexts, although its use has mostly concentrated on subjects other than complex mathematical ideas. Obeng et al. (2024) used the NEA framework to study the mistakes of senior high school students when answering trigonometric problems. In a similar vein, Triliana and Asih (2019) used the NEA method to examine students’ mistakes in tasks involving probability. Singh et al. (2010) used the Newman approach to examine the errors made by Primary Four students in written mathematics assignments in different research studies carried out in Malaysia. In addition, Irianti et al. (2024) examined students’ challenges in answering applied mathematics questions using Newman's five mistake types. The majority of current research has focused on topics like algebraic reasoning, arithmetic operations, and solving geometry problems (e.g., Jannah et al., 2024; Zamzam and Patricia, 2018).

Thus, existing studies have documented students’ difficulties in circle geometry, including difficulties with conceptual understanding, geometric relationships, and the interpretation and application of circle properties. For example, Ndlovu and Mji (2012) examined misconceptions in deductive geometric proof using a circle geometry task and identified difficulties in students’ understanding of geometric arguments. Similar studies have investigated high school learners’ challenges in solving circle geometry problems and used Newman's Error Analysis to identify difficulties associated with students’ understanding and application of geometric concepts (Brijlall and Abakah, 2022; Aidoo-Bervell, 2021; Kania et al., 2024). More recently, Apuri et al. (2026) applied Newman's Error Analysis to students’ errors in solving circle problems and reported prominent comprehension, transformation, and process skill errors. These studies indicate that students’ difficulties in circle geometry can occur at different stages of mathematical problem-solving.

Despite these contributions, the types of errors high school students commit when solving circle theorem–related problems have not been extensively examined (Apuri et al., 2026). This study therefore extends existing work by applying Newman's Error Hierarchical Model to written responses to circle theorem problems and complementing the error classification with semi-structured interviews to identify error types and investigate the perceived underlying causes of errors from a student’s perspective. This approach provides evidence not only on the types and frequency of errors occurring across Newman's categories but also on the factors that interviewed students perceived as contributing to the difficulties evident in their solutions.

Literature review

Geometry is widely recognized as an area of mathematics in which students experience difficulties, particularly when solving problems that require them to interpret diagrams, recognize relationships among geometric properties, and coordinate several pieces of information. These difficulties are evident in circle geometry, where successful problem-solving requires more than recalling isolated theorems. Mosia et al. (2023), for example, examined Grade 12 learners’ difficulties in Euclidean geometry and found weaknesses in the conceptualization of geometric properties, visualization, circle theorems, and geometry terminology. Their findings suggest that students’ difficulties may arise from both their understanding of geometric concepts and their ability to interpret and use geometric information during problem-solving. Similarly, Riastuti et al. (2017) reported that errors in geometry were associated with students’ spatial abilities, with students with weaker spatial abilities experiencing particular difficulty in visualizing geometric relationships. These findings indicate that errors in circle theorem problems may reflect difficulties at different points in the problem-solving process rather than a single lack of knowledge of the relevant theorems.

Research focusing more specifically on circle geometry has also identified conceptual and procedural difficulties among secondary school students. Mosia et al. (2023) found that learners often struggled with circle theorem concepts when examination questions required a higher level of geometric reasoning than the level at which many learners were operating. Makhubele et al. (2014), using Van Hiele's theory of geometric thinking, similarly reported difficulties involving the application of geometric rules, conceptual understanding, and problem-solving among secondary school learners. In the Ghanaian context, Ansong et al. (2021) identified several factors associated with students’ difficulties in solving circle theorem problems, including limited instructional resources, inadequate preparation, weak foundational knowledge, and classroom conditions. Bosson-Amedenu (2017) also identified circle theorems as a particularly challenging area for Ghanaian senior high school students. Collectively, these studies suggest that students’ difficulties with circle theorems can involve conceptual knowledge, visualization, geometric reasoning, and instructional experiences. Nevertheless, these studies do not all provide a detailed classification of the specific errors that occur while students work through individual problem-solving stages.

Several frameworks could be used to examine students’ difficulties in solving geometry problems, but their analytical emphases differ. Van Hiele's model, for example, is useful for examining the level of geometric thinking underlying students’ reasoning (Bonyah and Larbi, 2021), but it focuses primarily on the development and sophistication of geometric understanding rather than on identifying the specific point at which an error occurs during problem-solving. Similarly, Polya's problem-solving framework provides a useful account of broad processes such as understanding the problem, devising a plan, carrying out the plan, and looking back (Pradana, 2024; Wahab et al., 2024), but it does not provide a similarly detailed classification of errors within these processes. Newman's Error Analysis Model (NEAM) is particularly appropriate for the present study because it provides a structured way of locating errors across successive stages of mathematical problem-solving, namely reading, comprehension, transformation, process skills, and encoding. This focus corresponds directly to the study's objective of identifying the types of errors students make in their written solutions (Obeng et al., 2024; Owusu et al., 2026). Newman (1977) proposed these five stages as a means of examining the processes through which learners work when solving mathematical problems. The framework has subsequently been applied across several mathematical topics, including algebra, trigonometry, statistics and probability, functions and graphs, fractions and ratios, measurement, and geometry. Wardhani and Argaswar (2022), for instance, used the model to examine students’ errors in solving trigonometry word problems and identified difficulties across the different stages of the problem-solving process. Other studies have applied Newman's framework specifically to geometry. Riastuti et al. (2017) used Newman's analysis alongside students’ spatial intelligence to examine geometry errors, while Zamzam and Patricia (2018) found that transformation and process skill errors occurred frequently among students solving geometry problems. Likewise, Putri and Nurlaelah (2022) identified reading, comprehension, transformation, process skill errors, and encoding errors in students’ solutions to surface-area problems involving polyhedra.

Evidence specifically concerning circle theorem problems further strengthens the relevance of this approach. Usman and Hussaini (2017) reported that transformation and process skill difficulties were prominent when Newman's framework was used to examine students’ errors in mathematics, although their study was not focused on circle theorems. More directly, Apuri et al. (2026) examined errors made by 258 Ghanaian senior high school students when solving circle theorem problems using Newman's Error Analysis. They found that comprehension, transformation, process skill errors, and encoding errors were common and further identified student-related factors, teacher-related factors, and misconceptions as sources associated with these difficulties. The study therefore provides evidence that errors in circle theorem problem-solving can occur at multiple stages and may be associated with both learner and instructional conditions. Its focus on Ghanaian senior high school students also provides a useful basis for examining the issue within the same educational context. These studies demonstrate the usefulness of Newman's framework for moving beyond whether an answer is correct or incorrect to identifying where difficulties arise during the solution process.

Figure 1 shows five different levels of errors: RE, CE, TE, PSE, and EE, which are arranged in a hierarchical structure.

Reading errors occur when students make mistakes in interpreting the text of a problem before attempting to solve it, often due to difficulties in understanding the language or symbols used, as noted by Newman (1977), who emphasized that comprehension of the problem text is the critical first step in problem-solving. Comprehension errors arise when students misinterpret the relationships or properties within a problem, such as misunderstanding terms like “tangent,” “chord,” or “central angle” in circle theorems, leading to incorrect application of relevant theorems or formulas. Transformation errors occur when students struggle to convert verbal or visual information into mathematical expressions, failing to represent the problem in an algebraic or quantitative form necessary for solution. Process skill errors, on the other hand, emerge during the procedural phase of problem-solving, where students may understand the problem but make mistakes in execution, such as misapplying formulas, making computational errors, or mishandling mathematical procedures. Finally, encoding errors happen when students inaccurately record or represent their mathematical work, resulting in incorrect final answers despite correctly understanding and processing earlier steps in the problem-solving process.

Methodology

Design

This study adopted a mixed-methods research design to provide a comprehensive understanding of students’ errors in solving circle theorem problems. The quantitative component enabled the researcher to determine the frequency and patterns of errors committed by students, while the qualitative component helped explain the underlying reasons and thought processes behind these errors. The integration of both approaches allowed for a deeper and more balanced analysis, making the findings more reliable and informative.

Participants of the study

The study was conducted at a selected senior high technical school in Ghana, where mathematics is a compulsory subject for all students. The study population consisted of all 100 Form Two students enrolled in the school during the 2024/2025 academic year. Form Two students were selected because they had completed instruction on circle theorems and were therefore in a position to demonstrate their understanding and the difficulties they encountered when solving circle theorem problems. This school was selected because it was accessible to the researchers, permission was granted to conduct the study, and instruction on circle theorems had been completed before data collection began. Although the study was conducted in only one school, this setting allowed the researchers to examine students’ errors within their actual learning environment. The findings are therefore interpreted within the context of the participating school rather than generalized to all Senior High School students in Ghana.

A sample of 80 students was selected from the population of 100 using Yamane's (1967) sample size formula with a 0.05 margin of error. The average age of the participants was approximately 17 years. A simple random sampling technique was used to select the 80 participants. Specifically, 100 pieces of paper were prepared, with 80 labeled “yes” and 20 labeled “no.” Each student was asked to select one piece of paper randomly without replacement. Students who selected “yes” were included in the study, whereas those who selected “no” were excluded. This procedure gave each student an equal probability of selection and reduced the possibility of selection bias within the school.

To obtain qualitative evidence on the difficulties reflected in students’ written solutions, five students were purposively selected for individual interviews after the circle theorem achievement test (CTAT) was scored. Selection was based primarily on students’ low overall performance on the CTAT because these students provided information-rich cases for exploring the difficulties associated with the errors identified in the written responses. The interview component was intended to complement, rather than statistically represent, the quantitative error patterns. The five students were therefore selected as cases for obtaining explanatory accounts of the observed errors rather than as a maximum-variation sample of the entire cohort. Where students had the same score at the selection threshold, the researchers considered the error patterns evident in their written responses and selected students whose solutions contained clear examples of the error categories identified in the study. This approach ensured that the interviews were directly connected to the written error analysis. The qualitative findings are consequently interpreted as participants’ accounts of the difficulties they experienced and are not presented as representative of all students in the sample.

Data collection tool(s), validity, and reliability

The study used two main instruments for data collection: the CTAT (Supplementary Appendix A) and a semi-structured interview guide. Both instruments were developed with reference to NEHM, which provided the framework for identifying and examining students’ errors in solving circle theorem problems. The CTAT consisted of six open-ended questions developed from a combination of past examination questions and researcher-designed items. The questions were intended to assess students’ understanding of circle theorem concepts, their ability to apply the theorems correctly, and the errors evident in their problem-solving processes.

To establish content validity, the CTAT was reviewed by three experienced Senior High School mathematics teachers who assessed the clarity, relevance, and appropriateness of the items in relation to the study objectives. Their comments and recommendations were incorporated into the final version of the instrument. A pilot test was subsequently conducted with 30 students from a school that was not included in the main study. The pilot data were analyzed using SPSS version 23, and the CTAT achieved a split-half reliability coefficient of 0.802, indicating good internal consistency.

The CTAT was administered by the researchers during normal school hours in the participants’ classrooms under uniform testing conditions. Before the test began, students were informed about the purpose of the study and assured that their responses would be treated confidentially and used only for research purposes. They were instructed to work independently and show all relevant steps in their solutions so that their reasoning and error patterns could be examined. After the CTAT had been scored and the students’ written responses had been examined, five students who obtained the lowest scores were purposively selected for individual interviews. Low performance on the CTAT was the sole selection criterion for the interview sample. The interviews were intended to obtain more detailed explanations of the difficulties and errors identified in these students’ written solutions. The interviews were conducted individually in a quiet room within the school and lasted between 20 and 30 min. An interview guide was used to explore how the students interpreted the questions, the strategies they used to solve the problems, and their perceived reasons for the errors evident in their written responses. Follow-up questions were asked where clarification was required.

The interview questions were reviewed by experts in Mathematics Education and research methodology to assess their clarity, relevance, and alignment with the study objectives. Their suggestions were considered when refining the interview guide before data collection. The guide was also pilot-tested with five students who had characteristics similar to those of the study participants to identify and address potentially unclear questions. Consistency during the interviews was maintained by using the same core questions and procedures with all five participants. The trustworthiness of the qualitative data was addressed through consideration of credibility, dependability, confirmability, and transferability.

Data analysis

The quantitative data obtained from the CTAT were coded and analyzed using Microsoft Excel. Frequencies and percentages were computed to describe the number and proportion of students who responded correctly, incorrectly, or did not attempt each test item. The identified errors were then classified according to the five categories of NEHM: reading, comprehension, transformation, processing skills, and encoding. These five categories served as a priori codes for the analysis of students’ written solutions. The frequency of each error category across the test items was calculated and ranked to determine the most frequently occurring types of errors.

To obtain qualitative evidence on the difficulties reflected in students’ written solutions, five students were purposively selected for individual interviews after the CTAT was scored. Selection was based primarily on students’ low overall performance on the CTAT because these students provided information-rich cases for exploring the difficulties evident in their written responses. The five students were not selected to correspond to the five Newman error categories; rather, they were selected as low-performing cases whose written solutions contained errors relevant to the quantitative analysis. These explanations were coded inductively because the specific contributing factors were allowed to emerge from students’ accounts rather than being imposed in advance by the NEHM framework. The interview component was intended to complement the written error analysis by providing students’ accounts of difficulties associated with the errors identified in their solutions. Where students had the same score at the selection threshold, the researchers considered the error patterns evident in their written responses and selected students whose solutions contained clear examples of the error categories identified in the study. The qualitative findings are therefore presented as illustrative accounts of participants’ perceived difficulties and are not treated as representative explanations of each Newman error category.

Error-coding protocol

Because the classification of students’ errors involved categorical judgments, procedures were implemented to promote consistency in coding. To achieve this, the written responses were coded deductively using an adapted Newman Error Analysis Model. To ensure consistency in classifying errors, a conceptualization and operational coding rubric was developed to specify the criteria and observable indicators for each error category (see Table 1). The NEHM-based coding scheme was evaluated through an inter-rater reliability check using Cohen's kappa, which is widely recommended for assessing agreement in categorical data (Donner et al., 2023). Here, the five error categories specified by Newman's Error Hierarchical Model were established as a priori coding categories. Two expert raters independently coded a sample of students’ scripts using the NEHM coding guide developed for this study. The coded data were then analyzed in SPSS, yielding a Cohen's kappa coefficient of 0.85. This value is above the commonly accepted threshold of 0.70 for educational research, indicating a high level of agreement between the raters and confirming that the coding scheme was reliable and consistently applied. Following the reliability assessment, the raters discussed discrepancies and reached an agreement on the classification of the relevant responses before the remaining scripts were coded. This procedure helped maintain consistency in the application of the predetermined NEHM categories throughout the analysis.

Table 1. Operationalization and indicators of Newman error types in students’ written responses.

Source: Adapted from Apuri et al. (2026).

Findings

Students’ achievement test data

This study focused on the errors students committed when solving circle theorem–related problems. Figure 2 presents the frequency of students who attempted each test item and responded correctly, responded incorrectly, or did not attempt the item.

Reference to Figure 2 indicates that, for test item 1, 37.5% (N = 30) of the students answered correctly, 55% (N = 44) provided incorrect responses, and 7.5% (N = 6) did not attend to the question. Interestingly, test item 2 recorded the lowest correct response rate, with only 2.5% (N = 2) answering correctly, 72.5% (N = 58) answering incorrectly, and 25% (N = 20) not attempting it. Test item 3 was correctly answered by 26.25% (N = 21) of the respondents, while 56.25% (N = 45) gave an incorrect answer, and 17.5% (N = 14) refused to answer the question. Furthermore, test item 4 received correct responses from 31.25% (N = 25) of the respondents and incorrect responses from 58.75% (N = 47), while 10% (N = 8) did not attempt it. Test item 5 had the highest correct response rate, with 77.5% (N = 62) answering correctly, 21.25% (N = 17) answering incorrectly, and 1.25% (N = 1) not attempting it. Finally, concerning test item 6, 35% (N = 28) of students gave a correct answer, 48.75% (N = 39) failed to provide a correct answer to the question, and 16.25% (N = 13) did not attempt the question. Drawing from the results, test item 2 was the most challenging for students, while test item 5 was the most successfully answered.

Error types committed by students in solving circle theorem problems

Upon critically analyzing the responses given by students through the lens of Newman's (1977) framework, we found that students were committing certain types of errors that revealed both gaps in conceptual understanding and weaknesses in procedural execution related to circle theorems. These included reading, comprehension, transformation, process skills, and encoding. The types and frequencies of the 259 coded errors identified across the six test items are summarized in Table 2.

Table 2. Error types committed by students and their ranks.

As exhibited in Table 2, emerging as the most common, making up 39.38% (N = 102) of all errors, are transformation errors. These often result from the misapplication of theorems, implying that although some students could sometimes recall some theorems, they struggled to apply them correctly in the given context. Process skill errors were the second most frequent at 35.14% (N = 91), indicating challenges in executing the necessary steps even when the right approach is identified. Recorded at 15.83% (N = 41) with reference to Table 2 are comprehension errors, which reflected difficulties in interpreting the problem and understanding what is required. Reading errors were less frequent at 6.56% (N = 17), while encoding errors were not seen that much, representing only 3.09% (N = 8). In total, we observed 259 errors across all test items attempted, highlighting that the most significant errors, as far as circle theorem problem-solving is concerned, lie in the correct application of theorems and carrying out problem-solving processes.

Example of students’ errors and perceived contributing factors in students’ errors in circle theorem problem-solving

The analysis of students’ written responses identified five types of errors based on Newman's Error Hierarchical Model: reading, comprehension, transformation, processing skills, and encoding. Findings from the interview session, comprising five purposively selected students, were used to obtain insight into the difficulties associated with these errors. The findings presented below therefore describe the errors identified in the written responses and the perceived contributing factors reported by the interviewed students. The interview accounts are used to contextualize and explain the error patterns rather than to establish causal relationships for the entire sample.

Reading error

The analysis we conducted with regard to the responses students gave indicated that reading errors were often associated with how students approached the questions, rather than a complete absence of knowledge. By this, we mean not that students do not have knowledge about the question posed, but they misread the questions, leaving behind critical details needed to answer the questions. The written responses suggested that some students overlooked critical details in the questions. The interview excerpt presented below provides further insight into possible factors underlying this difficulty for one of the interviewed students. These difficulties were reflected in some students’ incorrect solutions, where critical details in the questions were overlooked or misinterpreted. Figure 3 shows an example of students’ reading errors.

Reading errors were identified across several test items. With reference to Figure 3, Q1, the student used the point V, where XZ and WY intersect, as the center of the circle, misinterpreting the “point of intersection” as the “center,” even though the point O was stated clearly as the center of the circle. As a result, ∠YVZ was viewed as an isosceles triangle and further equated ∠VZY to ∠VYZ, which does not hold per the question given. The student therefore arrived at an incorrect solution. For Q3, the student's solution demonstrated competence in recalling and correctly applying relevant theorems. However, although the question expected the student to compute the “reflex” of ∠QOS, the student, due to a lack of attention and hastily engaging with the posed question, unfortunately skipped or overlooked the main focus of the question, which, in turn, led the student to provide a wrong answer. The interview provided further insight into the factors that the student perceived as contributing to this type of error.

*RSC*: “You managed to answer most of the circle theorem questions, but you still lost some marks. How do you feel after seeing your score on the test?”

*STD18*: “Hmm, Sir, honestly, I don’t feel too good about it.”

*RSC*: “I understand. What do you think caused those errors?”

*STD18*: “I think it’s because I didn’t take my time to read the questions well; I was afraid of time running out.”

*RSC:* “That makes sense, so sorry about that. Aside from this, is there anything you think caused such errors?”

*STD18:* “Yes. I sometimes get confused with some of the terminology used in the questions. So, when I am not sure of the meaning, I guess.”

The interview excerpt provides an individual account of factors that may have contributed to the reading error observed in the student's written response. The student attributed the mistake to rushing through the questions because of concern about time, which affected careful reading. The student also reported difficulty with some mathematical terminology and indicated that uncertainty about the meaning of such terms sometimes led to guessing. These comments provide contextual insight into the particular student's reading error but should not be taken as evidence that the same factors were experienced by all interview participants.

Comprehension error

Following closely, students being able to read a given problem concerning NEHM amounts to comprehension. Unlike reading errors, which involve difficulties in accurately reading or recognizing information in the problem, comprehension errors reflect difficulties in making sense of the relationships and concepts presented in the problem. A student may be able to read the question accurately but still struggle to make sense of what is being asked or how to apply a given theorem correctly. Figure 4 shows examples of students’ comprehension errors.

Depicted in Figure 4 are examples of students’ responses involving comprehension errors. Referencing the student response for Q2 as shown in the above figure, the student equated let ∠CRS=∠RPC, which is accepted because both/CR/ and/CP/ are radii and ΔCPR is an isosceles triangle. However, in the next step, the student summed the angles in ΔCPR and equated them to zero (0) as the final answer, which is nowhere related to what the question expected. In the solution, the student wrote, “Since they all subtend from the circumference”. To the student, ∠PCR subtends on the circumference, which is not so, suggesting a lack of conceptual understanding. The response suggests that the student had difficulty determining how to proceed with the problem. Likewise, according to the response to Q6 shown in Figure 4, the student viewed the computed ∠PSR as 1/2∠POR(204∘) to get 102, which in the first place was wrong. The student further summed the angles in the quadrilateral PORS, including “x”, and later got the value for “x” as zero (0). This response suggests that the student had difficulty understanding the relevant concepts and relationships required to solve the problem. The interview excerpt below provides the student's account of factors that may have contributed to this difficulty.

RSC: “You did your best, but it was not enough to get the question right. What do you think made you and some of your friends make these errors?”

STD44: “Ok, Sir, the issue is that I do not understand circle theorems. I think most of us don't understand the theorems well”

*RSC:* “So, what do you think made the understanding of the circle theorems difficult?”

STD44: “For me, I will say the way it was taught. Our teacher wrote the theorems on the board and solved a few examples with us, and asked us to apply and solve some. That day I couldn't do it, and till now I don't understand. I think my friends too.”

RSC: “So sorry for that. Now, apart from the way it is taught, is there any other reason why you think students make mistakes?”

STD44: “I don't normally try solving circle theorems. Like most of my friends, I don't solve questions like this; we solve the other ones, like the ones we understand.”

The interview excerpt provides an individual account in which the student attributed the comprehension difficulty to limited understanding of circle theorems and the way the topic had been taught. The student also reported limited personal practice with circle theorem problems and suggested that this reduced opportunities to encounter different problem types. For this student, limited practice may therefore have contributed to difficulties in recognizing when and how particular theorems should be applied. These interview responses are interpreted as the student's perception of the factors contributing to the observed difficulty rather than as evidence of a common experience among all participants.

Transformation error

Transformation errors, being the third level according to NEHM, occur in a situation where students incorrectly interpret or manipulate a mathematical problem from one form into another. Problem-solving in circle theorems most of the time requires students to interpret a given diagram or problem and then transform it into a mathematical form, usually as equations, expressions, or angle relationships, before proceeding with finding the solution. Unlike comprehension errors, transformation errors usually involve difficulty in applying the correct geometric transformation to the given figures. An example is shown in Figure 5.

Shown in Figure 5 are examples of students’ transformation errors. Drawing from the response a student gave to Q4 suggests that the student had an idea about how to approach the posed question. The student started well by correctly solving for the angle (∠OYW), crucial for proceeding with the rest of the solution. However, the Pythagorean theorem was applied in the wrong context by the student, as /YW/ was unknown. As such, there was no way that approach would lead to the correct answer. These error types were frequently observed in the students’ written responses. For the response given to Q2 below Q4 in Figure 5, the student established many relationships, all of which were not essential to finding the expression for q∘. Even though no values were assigned to the angles in the diagram, the student went on to get 60∘ as the value for q∘. The majority of students were observed not attempting Q2, while several of those who attempted it were unable to transform the information presented in the diagram into a workable mathematical equation or expression. The interview excerpt below provides an individual account of a factor that the student associated with this type of difficulty.

*RSC:* “When you were solving the circle theorem question, what did you find most difficult?”

STD8: “I think the main problem was trying to picture how the angles are related inside the circle. I couldn't see which angle was supposed to equal which.”

*RSC:* “So would you say the way the diagram was drawn affected your understanding and how to approach the question?”

STD8: “Yes, a lot. If the circle has too many lines or markings, I get confused about which lines or points to use, even though I know some of the theorems. I sometimes mix them up.”

The interview excerpt from STD8 suggests that difficulties with visualization and spatial reasoning may have contributed to the transformation error observed in this student's written response. The student reported difficulty determining how angles were related within a complex diagram and indicated that multiple lines or markings could cause confusion when deciding which geometric elements were relevant. This account is consistent with the transformation difficulty evident in the student's written response, particularly the difficulty in identifying the relevant relationships represented in the diagram. The observation here is therefore presented as an interpretation of this participant's account rather than as a claim about all five interview participants.

Processing skill error

Processing skill errors in geometry, in particular, circle theorem problem-solving occur not because of students’ inability to transform a given diagram or information into a workable mathematical equation or expression but arise when students encounter challenges in carrying out the actual steps of solving a problem, even when they have some conceptual understanding of the problem. Errors of this nature most of the time show up when students are required to connect and make use of multiple mathematical operations or reasoning steps to successfully solve a given problem. A student may be able to recall a theorem or concept but may stumble when integrating geometric reasoning with calculations or following the logical sequence of steps needed to reach an answer. Therefore, here, the problem lies not in a lack of knowledge but in the challenge of managing the chain of processes involved. Below are examples of students’ processing skill errors depicted in Figure 6.

Processing skill errors were frequently observed in the students’ written responses; however, referring to what has been shown in Figure 6, it is interesting that the student at the end had the final answer to be correct (38∘) for Q5, but the process was not mathematically acceptable. An error in the solution process started from where the student computed the value for ∠ACO as 1/2∠ACB, having computed the value for ∠ACB as 1/2∠AOB=70∘. As a result, the student had a value of 35∘ for ∠ACO. From the diagram and according to the information given, ΔAOC is an isosceles triangle; thus, /AO/ = /OC/. As such, ∠ACO is supposed to equal the value given for ∠OAC, which is 32∘ and not 35∘. The error here, as a result, rendered the rest of the solution process incorrect, even though they demonstrated some level of circle theorem understanding, illustrating how a learner may remember the correct theorem or idea yet struggle to integrate geometric reasoning with calculations or to maintain the logical sequence of steps required for the solution. In addition, the response given by a student to Q4 indicates that, even though the student started well, the right trigonometry ratio was not used. The student correctly computed the value for ∠OYW. As such, /OW/ as extended by the student, became the opposite side of the calculated angle. Yet, the student used it as the adjacent side, leading to using the cosine ratio (“CAH”) instead of the sine ratio (“SOH”). Following this, the right approach to the required answer was distorted. The interview excerpt below provides the individual student's account of a factor that may have contributed to this type of error.

RSC: “When you were solving the circle theorem problem earlier, I noticed in your solutions that you identified the correct theorem at first and started well. Yet, you were not able to reach the final answer correctly. Can you tell me why that happened?”

STD73: “Yes, I remembered the theorem about the angle at the centre being twice the angle at the circumference. But after that, I wasn't sure how to continue. I needed to find some angles and then add and subtract some angles before I continue. Sir, I think the process is long.”

RSC: “So, would you say the problem is more about going through many steps in solving the question”?

*STD73:* “Hmm, yes, I will say yes because I could recall the theorem, but then combining it with the other processes while also thinking about the diagram most often makes me feel like too many things at once, and I got confused.”

RSC: “That sounds like it placed a lot of demand on you. Do you think consistently solving these kinds of problems would help?”

STD73: “Yes, I think so. Circle theorem questions for me are confusing, so I don't normally solve questions under it. When I solve and get lost, I stop. But I think if someone helps me to practice more, it will help.”

In this interview, STD73 attributed the processing difficulty to the number of reasoning steps required to solve the problem. The student explained that recalling a theorem, applying it to the diagram, and simultaneously carrying out angle manipulations and other calculations became difficult to manage. The student also indicated that limited practice with circle theorem problems contributed to the tendency to stop when difficulties were encountered. This account suggests that, for this participant, the number of interconnected reasoning steps and limited practice may have contributed to the observed processing skill error.

Encoding error

Unlike conceptual or procedural errors, encoding errors, according to NEHM, are not about misunderstanding the problem or applying the wrong method. Instead, they arise when students have correctly carried out the necessary processes but fail to state their final answer accurately. This may involve ignoring instructions such as rounding to a specified decimal place, giving results in the wrong unit, poor handwriting, or leaving answers incomplete. Thus, encoding errors highlight the gap between knowing how to solve a problem and formally expressing the solution as expected in mathematics assessments. Figure 7 is an illustration of students’ encoding errors.

The responses shown in Figure 7 indicate that the errors leading to loss of marks in these cases occurred at the stage of stating the final answer rather than in the preceding conceptual or procedural work. For Q2, the student went through the right process but did not round the final answer to two decimal places as the question required them to do. In a similar vein, the final answer for Q1 was not stated correctly because of poor handwriting, confusing “X” with “Y”. The handwriting also made it difficult to determine whether the student had written “∠XZX ” or “∠XZY ” The interview excerpt below provides the student's account of what contributed to the encoding error.

RSC: “I noticed you solved the question correctly, but you still lost some marks at the end. What do you think might have caused that?”

STD31: “I think I did not pay close attention. I rushed through the last part, so I didn't notice that I wrote the answer without leaving it in two decimal places.”

*RSC:* “So, would you say it was more about not giving enough attention rather than not knowing how to solve it?”

STD31: “Yes, exactly. I understood the steps, but I wasn't careful enough when writing the final answer.”

The interview excerpt suggests that STD31 attributed the encoding error to inattention during the final stage of the solution. The student reported rushing through the final part of the question and consequently overlooking the instruction to express the answer to two decimal places. This account suggests that, for this student, insufficient attention to the final response contributed to the observed encoding error. The written responses also showed that unclear handwriting contributed to encoding difficulties in some cases.

Discussion

Our study explored the different types of errors that high school students made when solving circle theorem problems and the perceived factors associated with these errors. The analysis was guided by NEHM, which classifies student errors into five main types: reading, comprehension, transformation, process skill, and encoding. From the results, transformation errors were the most frequent, accounting for 39.38% (N = 102) of all identified errors, followed by processing skill errors at 35.14% (N = 91). Although the written responses indicated that some students could recall and apply basic circle theorem concepts, the errors also revealed difficulties with the deeper conceptual and procedural demands of problem-solving.

The high occurrence of transformation errors (39.38%) suggests that many students struggled to move from visual representations to algebraic or symbolic forms. This aligns with the findings of Nathan et al. (2017), who also observed that students often fail to link geometric ideas with their symbolic representations. In this study, the transformation difficulties were associated with students’ difficulty in identifying and visualizing how geometric elements relate to one another before expressing these relationships mathematically. The interview account from STD8 similarly indicated difficulty in mentally representing relationships among angles and other elements within complex diagrams, suggesting that visualization and spatial reasoning may have contributed to the transformation difficulties observed in the written responses. This supports the work of Lowrie et al. (2019) and İbili et al. (2020), who both noted that students’ spatial visualization skills play a key role in geometry learning. We suggest that from STD8's interview account, there is the possibility that instructional experiences that provide limited opportunities for visual exploration and application of theorems may be associated with students’ transformation difficulties. The participant's account suggested that difficulty interpreting complex diagrams and identifying relevant geometric relationships may become more pronounced when students have limited opportunities to engage with varied problem situations. These findings suggest the need for instructional opportunities that support students in interpreting diagrams and explicitly connecting visual information with mathematical relationships.

Another notable issue was the high rate of process skill errors (35.14%). These errors appeared when students correctly identified the appropriate theorem but made mistakes in carrying out the necessary steps or calculations. Similar observations were made by Lithner (2017) and Hansen (2022), who found that difficulties in following sequential reasoning steps frequently led to incomplete or incorrect solutions. The interview account from STD73 indicated that having to recall a theorem, apply it to the diagram, and simultaneously carry out angle manipulations and other calculations was difficult to manage. In other words, processing difficulty was attributed to the number of reasoning steps required to solve the problem. These findings suggest that regular practice, scaffolding, and guided problem-solving activities may support students in managing the multiple reasoning steps involved in circle theorem problem-solving and developing greater fluency in geometric reasoning.

Although comprehension errors (15.83) were less frequent than transformation and process skill errors, they indicated difficulties in making sense of mathematical relationships and problem requirements. These occurred when students misunderstood question requirements or failed to grasp the meaning of key terms. The account provided by STD44 identified the way circle theorems had been taught and limited personal practice with circle theorem problems as factors that the participant perceived as contributing to the observed comprehension difficulty. This finding supports studies by Wienecke et al. (2025) and Dohaney et al. (2015), who argued that instruction dominated by note-taking and teacher explanation tends to weaken students’ comprehension and retention of geometric concepts. STD44 also reported limited engagement with circle theorem problems outside the examples provided in class and indicated that this limited practice made it difficult to recognize when and how particular theorems should be applied. The account suggests that greater opportunities to engage independently with varied circle theorem problems and to discuss the meaning of geometric relationships may support students’ conceptual understanding.

Reading errors (6.56%), although less frequent, still highlighted issues related to mathematical literacy and attention to detail. Some students misread diagrams, overlooked crucial terms such as “center” or “point of intersection,” or skimmed through instructions too quickly. In the interview with STD18, the participant attributed the observed reading difficulty to rushing because of concern about time and to difficulty interpreting some mathematical terminology. This observation is consistent with findings by Sylvia et al. (2023) and Torppa et al. (2020), who linked difficulties in interpreting mathematical problems with reading-related challenges and affective factors such as test anxiety. The participant's account suggests that supporting students in attending carefully to problem statements and interpreting mathematical vocabulary may help reduce errors at the reading stage

Last, encoding errors (3.09%), although the least frequent, still reflected lapses in precision and presentation. These occurred when students wrote unclear final answers, ignored rounding instructions, or used incorrect units. Although these mistakes may appear minor, the written responses indicated that some occurred at the final stage of presenting the solution rather than during the preceding conceptual or procedural work. Similar observations were discussed by Efklides (2011) and Owusu et al. (2026), who linked self-monitoring and metacognitive regulation with students’ ability to monitor and evaluate their mathematical problem-solving processes. The interview account suggested that inattentiveness during the final stage of solution writing contributed to this type of error for the selected participant. Such findings point to the importance of encouraging students to review their solutions and check whether their final responses satisfy the requirements of the question.

Conclusion

This study examined the errors of senior high school students in solving circle theorem problems using NEHM, which categorizes errors into reading, comprehension, transformation, processing skills, and encoding. The findings showed that transformation errors were the most frequent, followed by processing skills, comprehension, reading, and encoding errors. The written responses indicated that students experienced difficulties in translating geometric information into mathematical relationships and in carrying out the multiple reasoning steps required to obtain correct solutions. These patterns point to the importance of conceptual understanding, diagram interpretation, and procedural coordination in circle theorem problem-solving. The interview accounts provided further contextual insight into these difficulties. Individual participants associated their observed errors with factors such as difficulty interpreting mathematical terminology, limited practice with circle theorem problems, challenges in visualizing relationships within diagrams, difficulty managing multiple reasoning steps, and inattention when presenting final answers. These accounts represent the perceptions of the interviewed students and should not be interpreted as established causes or as evidence of experiences shared by the wider student population. The study therefore suggests that NEHM can provide a useful framework for diagnosing the stages at which errors occur in circle theorem problem-solving and for identifying areas that may require instructional attention. Such diagnostic information may assist teachers in designing learning activities that provide greater opportunities for interpreting diagrams, connecting geometric relationships, practicing varied problem types, and monitoring solution processes.

Limitations and suggestions for further studies

Although this study provides valuable insights into the errors senior high school students make when solving circle theorem problems, it has some limitations. The study was conducted in only one senior high technical school and involved a relatively small sample of Form Two students; therefore, the findings should be interpreted within the context of the study and should not be generalized to a single school. In addition, the study focused solely on circle theorem problems using Newman's Error Analysis Model, which means the results may not reflect students’ error patterns in other areas of mathematics. The qualitative component also has limitations. Only five students were interviewed, and the participants were selected based on their low overall performance on the CTAT rather than through maximum-variation sampling across the wider student group. Although their written responses and interview accounts provided useful contextual information about the difficulties identified in the quantitative analysis, the small qualitative sample limited the range of perspectives available for interpreting the reasons associated with the different Newman error categories. The interview findings should therefore be understood as illustrative accounts of the selected students’ perceived difficulties rather than as a comprehensive explanation of the factors underlying each type of error. Future studies could involve a larger and more diverse qualitative sample, with participants selected to capture variation in performance and error patterns across the Newman categories. Future research should also include participants from different schools and regions to improve the generalizability of the findings. It would also be worthwhile to investigate students’ errors in other mathematics topics and examine the effectiveness of instructional approaches such as GeoGebra, inquiry-based learning, and other visualization strategies in reducing students’ errors and improving their conceptual understanding and problem-solving skills.

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Supplementary material

The Supplementary Material for this article can be found online at: [https://www.frontiersin.org/articles/10.3389/feduc.2026.1875952/full#supplementary-material](https://www.frontiersin.org/articles/10.3389/feduc.2026.1875952 /full#supplementary-material)

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Footnotes

Edited by: Lim Hooi Lian, University of Science Malaysia, Malaysia

Reviewed by: Adriel Roman, Laguna State Polytechnic University, Philippines

Ibnu Rafi, National Sun Yat-sen University, Taiwan