Thinking Errors and Problem-Solving Strategies of Students

Study Case Thinking Errors and Problem-Solving Strategies of Students in Solving Integral Problems in Engineering Education Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract This study aims to analyze the types of errors made by students in solving calculus problems, particularly on the topic of integral application, with reference to problemsolving ability indicators. The method used was qualitative descriptive with 68 participants from the College of Engineering, University of Cincinnati, who had taken Calculus II in the 2024–2025 academic year. The research instruments included problem-solving-based essay tests and interview guidelines to strengthen the analysis results. The analysis was conducted based on Polya's four problem-solving indicators, namely: understanding the problem, planning the solution, implementing the plan, and reevaluating the solution. The results showed that the most significant errors occurred in the stages of understanding the problem (56.32%) and planning the solution (64.28%), which included conceptual, procedural, and calculation errors. At the stage of understanding the problem, students still had difficulty representing the area or volume of a rotating object that had to be found. At the planning stage, most were unable to choose a solution method that was appropriate for the characteristics of the problem. Meanwhile, at the implementation stage, only about 28.74% of students were able to perform the calculations correctly, and at the evaluation stage, only 7.23% were able to effectively check their results. These findings indicate the need for more appropriate learning strategies to strengthen students' problem-solving abilities in calculus courses. Keywords: error analysis, calculus, integral, problem solving MICHAEL KURNIAWAN 1 Study Case INTRODUCTION Problem-solving skills are a very important mathematical competency for students in the field of engineering. This skill needs to be developed because engineering graduates will be faced with complex problems that require analytical and critical thinking skills to find effective solutions [1]. In practice, problem-solving skills require individuals to analyze situations, devise strategies, and solve problems efficiently [2]. In addition to being relevant in an academic context, this skill is also a key asset for facing challenges in an increasingly dynamic and data-driven world of work [3]. In the context of calculus learning, especially in the subject of integrals, problemsolving skills are a fundamental aspect that supports students' ability to apply mathematical concepts to real-life situations. However, various studies in higher education in the United States show that many students still have difficulty understanding and solving integral problems conceptually and procedurally [4]. For example, research conducted by Johnson [5] found that common student errors include misreading questions, transforming concepts, and errors in writing down the final answer, which account for about 50–75% of total errors. Another finding by Smith et al. [6] shows the distribution of errors based on Polya's problem-solving indicators, with errors in the problem-understanding stage reaching 55.89%, solution planning 50%, plan execution 32.36%, and evaluation 44.12%. Meanwhile, research by Anderson & Lee [7] revealed that students often fail to think critically when determining the radius of a rotating object, integration limits, or when compiling partitions in integral problems—which indicates their weak conceptual understanding and analytical abilities. In addition, a study by Brown et al. [8] found common errors in determining integral functions, applying integral rules, and selecting the appropriate method for calculating the volume of rotating objects. Similar results were also reported by Walker [9], who showed that students with high problem-solving abilities were able to go through the planning, monitoring, and evaluation stages well, while those at moderate or low levels often stopped at the planning and monitoring stages without evaluating the results. This confirms the importance of reflective and evaluative abilities in the process of solving mathematical problems. It is also necessary to strengthen mathematical communication, context-based exercises, and the application of problem-based learning (PBL) models to help students MICHAEL KURNIAWAN 2 Study Case think more systematically and critically [10]. Several studies have also found that even students with high abilities still make procedural errors at the re-examination stage, while those with low abilities often make conceptual errors in understanding the content of the question and in carrying out the steps to solve it [11]. Based on these findings, it can be concluded that students' errors in solving calculus problems—especially on the topic of integrals—occur at various stages, ranging from understanding concepts to evaluating results. Therefore, this study aims to identify and describe students' errors in solving integral application problems, such as calculating the area and volume of rotating objects using the disc, ring, and tube skin methods, based on Polya's four problem-solving indicators: (1) understanding the problem, (2) planning the solution, (3) implementing the plan, and (4) checking the solution. The results of this study are expected to form the basis for the development of more effective learning strategies to improve students' problem-solving skills in calculus courses. METHOD This study used a qualitative descriptive method that aimed to reveal and analyze various forms of student errors in solving calculus problems, particularly on the topic of integral application, which includes calculating the area and volume of rotating objects using the disk method, washer method, and shell method, based on problem-solving ability indicators. A qualitative approach was chosen because it allows researchers to examine in depth the students' thought processes and identify errors at each stage of problem solving [1]. The sampling technique used purposive sampling, taking into account variations in student ability levels (high, medium, low). The research sample consisted of 68 students from the College of Engineering, University of Cincinnati—including 32 students from the Mechanical Engineering program, 17 students from Electrical Engineering, and 19 students from Industrial Engineering. Participants were selected based on the suitability of their academic experience in calculus learning [2]. The main research instrument was a problem-solving test based on calculus questions on the application of integrals, consisting of six essay questions. These questions were designed with reference to Polya's four problem-solving indicators [3], namely: (1) understanding the problem, (2) planning the solution, (3) executing the plan, and (4) evaluating the solution. Each question was designed to cover the conceptual and MICHAEL KURNIAWAN 3 Study Case procedural aspects of integral application, thereby enabling error analysis at each stage of the problem-solving process. Before use, the test instruments were validated by expert reviewers, namely lecturers with experience teaching Calculus II, to ensure the suitability of the content and level of difficulty of the questions. In addition to written tests, this study also used semi-structured interview guidelines to obtain in-depth information about the causes of students' errors. The interview guide also underwent a validation process by expert reviewers to ensure that each question was relevant and able to accurately explore the cognitive aspects of the respondents. The research procedure was carried out in several stages, namely: 1. Development of research instruments, including calculus tests, answer sheets, and interview guides. 2. Test administration, followed by the collection and classification of student answer sheets. 3. Preliminary analysis of answers based on problem-solving indicators to identify conceptual, procedural, and calculation errors. 4. In-depth interviews with a number of selected respondents to explore the causes of errors and confirm the analysis results. 5. Data processing includes reduction (selection of relevant errors), data presentation, and systematic conclusion drawing. This analytical approach follows the principle of data triangulation to ensure the validity of findings and reliability of interpretations [4]. Data analysis was conducted using descriptive qualitative methods with the following steps: 1. Classifying student answers based on Polya's problem-solving indicators [3]. 2. Identifying the types of errors that arise, whether conceptual, procedural, or calculation errors [5]. 3. Interpreting the error results based on interview data to gain a comprehensive understanding of students' thinking patterns and obstacles in solving integral problems. RESULTS AND DISCUSSION Research data was obtained through problem-solving tests and in-depth interviews with College of Engineering students in Calculus II courses on the topic of integral application. The data analysis process followed three main stages as proposed by Miles MICHAEL KURNIAWAN 4 Study Case and Huberman [1]: 1. Data reduction, which involved selecting and classifying students' answers based on Polya's four stages of problem solving: understanding the problem, planning the solution, implementing the plan, and checking the solution. 2. Data presentation, in the form of narrative descriptions and tabulation of identified error indicators (see Table 1). 3. Drawing conclusions and verification, which involves identifying recurring error patterns and linking them to interview results to find conceptual and procedural causes. Table 1. Error Indicators Based on Polya's Stages The credibility of the data was strengthened through triangulation techniques (test results and interviews), member checking (confirmation with students regarding the interpretation of errors), and peer debriefing with two fellow lecturers in the field of calculus to ensure the objectivity of the analysis. Understanding the Problem A total of 56.32% of students made mistakes at the problem understanding stage. The errors that arose included: (1) not reading the question carefully, (2) misinterpreting terms and function forms, (3) failing to identify important information, and (4) being unable to draw graphs or curves correctly. These results are in line with the findings [2] which state that many students have difficulty converting questions into formal mathematical form. This indicates that students' mathematical reading comprehension skills are still low and most still rely on memorizing formulas. These findings are in line with study [3] which shows that more than 58% of students fail to understand the core of the problem before developing a solution strategy. MICHAEL KURNIAWAN 5 Study Case Planning the Solution A total of 64.28% of students made mistakes at this stage. The types of mistakes included: (1) choosing the wrong integral method (e.g., between the disc and tube skin methods), (2) confusion in determining the integral limits, and (3) lack of strategic understanding of the question structure. These errors indicate weak strategic reasoning skills in problem solving, where students rely more on procedural habits than situational analysis [4]. Executing the Plan Only 28.74% of students were able to execute the plan correctly, meaning that 71.26% made mistakes at this stage. Common errors include algebraic operation errors, incorrect application of integral rules, and symbolic inaccuracies. Some students understood the strategy but lost accuracy in its execution. This is in line with the findings [5] which show that calculation errors are often careless errors (50%), conceptual errors (60%), and application errors (70%). Rechecking Solutions The evaluation stage is rarely performed by most students. Only 7.23% check their work through substitution or logical evaluation of the results. Thus, 92.77% of students skip the verification step. According to Polya [3], this stage is crucial for detecting early errors. These findings confirm that students with low metacognitive awareness tend not to reflect on their thinking processes [6]. Analysis of Answers Several sample questions are presented to illustrate patterns of student errors. Figure 1. Sample Answer to Question 1 (Determining the Area of a Cross Section) MICHAEL KURNIAWAN 6 Study Case Students were asked to calculate the area under the curve y=4-(x-2)^2 at the interval x=0 to x=4. The analysis shows common errors in the form of: (1) the graph does not match the function, (2) incorrect integral limits, (3) incorrect simplification of the function, and (4) no verification of the results. Figure 2. Example Answer to Question Number 2 (Determining the Volume of a Rotating Object Using the Disc Method) Errors that appear: (1) not understanding the shape of the rotating object, (2) using incorrect integral limits, and (3) incorrectly applying the volume formula. Figure 3. Example Answer to Question Number 3 (Using the Ring Method) Some students chose the correct method but simplified the first function incorrectly, resulting in an error in the integral calculation. MICHAEL KURNIAWAN 7 Study Case Figure 4. Example Answer to Question Number 4 (Tube Skin Method) The main error lies in misinterpreting the function relative to the axis of rotation and not checking the results. Figure 5. Example Answer to Question Number 5 (Cross-sectional Area of a Hollow Pipe) Students failed to correctly describe the area and incorrectly determined the integration limits, indicating weak visualization skills. Student Interviews In-depth interviews revealed four main factors causing student errors: 1. Conceptual-visual difficulties — students did not know how to start drawing function graphs. 2. Low motivation and frequency of practice — studying only before exams. 3. Weak basic concepts — not understanding function simplification and the relationship between variables. MICHAEL KURNIAWAN 8 Study Case 4. Procedural habits without reflection — do not double-check their work. Triangulation between test results and interviews shows high consistency; students' written errors are in line with their admissions in interviews. Thematic Analysis Cross-item results show a general pattern: students who draw graphs (even with errors) tend to have better solution strategies than those who do not draw at all. This shows that visual representation plays an important role as a link between conceptual understanding and procedural application. Poor visual representation skills cause error propagation, namely conceptual errors that lead to procedural errors. Several studies support the use of digital media such as GeoGebra to help improve representation and problem-solving skills in calculus [7]. In addition, students need to be trained in computational thinking—the ability to decompose, recognize patterns, abstract, and think algorithmically—as a systematic approach to solving complex mathematical problems [8]. CONCLUSION Based on the analysis of the problem-solving abilities of College of Engineering students in solving Calculus II problems on the topic of integral application, the following conclusions were obtained: 1. Errors occurred in almost all stages of Polya's problem solving, with the following percentages: understanding the problem (56.32%), planning the solution (64.28%), executing the plan (71.26%), and checking the solution (92.77%). The first two stages were the main determinants of successful problem solving. 2. The types of errors included conceptual errors (misinterpreting the question or symbols), procedural errors (choosing the wrong integral method, calculation errors), and evaluative errors (not verifying the answer). 3. Low problem-solving ability indicates a lack of mastery of systematic, logical, and reflective thinking. 4. Repeated errors indicate that calculus learning still emphasizes procedural aspects over the development of concepts and critical thinking strategies. MICHAEL KURNIAWAN 9 Study Case REFERENCE MICHAEL KURNIAWAN 10