Cognitive Analysis of Students' Understanding of Integral

Study Case Cognitive Analysis of Students' Understanding of Integral and Differential Concepts in Calculus Learning at the University of Cincinnati Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract Calculus is an important branch of mathematics that plays a broad role in science, economics, and engineering. However, many students still have difficulty understanding the basic concepts of calculus, especially integral and differential material. This study aims to assess the understanding of mathematics education students of these two concepts through case study-based problem solving. The research method used a qualitative approach with case study techniques. Data were obtained through tests and interviews with three randomly selected mathematics education students from the University of Cincinnati class of 2023. The analysis was conducted descriptively through the stages of collection, presentation, and conclusion drawing. The results showed variations in the students' level of understanding in solving integral and differential problems, which were categorized into medium level comprehension and high level comprehension. These findings are expected to form the basis for the development of calculus teaching strategies at the university level to improve students' conceptual understanding. Keywords: calculus, integral, differential, student ability. INTRODUCTION Calculus is one of the main branches of mathematics that covers the concepts of limits, derivatives, and integrals. The term calculus comes from the Latin word calculus, which means “small stone” used for counting [1]. Calculus plays an important role in studying change, just as geometry focuses on shapes and algebra focuses on operations MICHAEL KURNIAWAN 1 Study Case to solve equations [2]. With its wide application in science, economics, and engineering, calculus can solve problems that cannot be solved through basic algebra. The two main branches of calculus—differential and integral—are closely related through the fundamental theorem of calculus [3]. Differential calculus focuses on rates of change, such as the slope of a tangent line, while integral calculus deals with the accumulation of change, such as the area under a curve or volume [4]. In the learning process, students usually study differential calculus before integral calculus, even though historically the concept of integration was discovered first. Derivatives were developed to solve problems such as determining the tangent line of a curve or finding the maximum and minimum values of a given function [5]. A deep understanding of calculus is very important because its applications extend to physics, chemistry, biology, economics, and information technology [6]. The concepts of integrals and differentials enable the calculation of areas under curves and the analysis of changes in function values. Therefore, students are expected to master integral and differential calculations both theoretically and practically in order to be prepared for professional challenges [7]. However, research shows that many students have difficulty understanding the concepts of integrals and differentials in depth, especially when applying them to realworld problem solving [8]. This is due to the abstract nature of calculus and the lack of contextual examples that can be directly applied in everyday life. As a result, calculus is often considered one of the most difficult branches of mathematics at the university level [9]. Previous research by Anderson and Roberts [10] highlights the historical development of calculus and the figures who played a role in its formation. Morrison [11] found that the low level of understanding of calculus among engineering students was due to a lack of interest in learning and weak basic mathematical skills. However, there is still little research that assesses mathematics education students' understanding through case study-based questions, even though this approach can provide a more comprehensive picture of their ability to apply integral and differential concepts. This study aims to fill this gap by evaluating students' ability to solve case-based problems, analyzing their level of understanding, and categorizing them into Low Level Comprehension, Medium Level Comprehension, and High Level Comprehension [12]. MICHAEL KURNIAWAN 2 Study Case METHOD This study used a qualitative approach with a case study method to assess students' understanding of integral and differential calculations. The qualitative approach is descriptive and aims to explore phenomena in depth, with the results presented in narrative form [13]. The case study method is used to observe specific phenomena or problems in context, as well as to find alternative solutions and improve students' critical thinking skills [14]. The research subjects consisted of three students from the mathematics education program at the University of Cincinnati class of 2023 who were selected at random. The first student had high proficiency in calculus and was active in academic activities such as math olympiads. The second student had moderate understanding and good learning motivation, while the third student had low understanding and needed additional guidance in understanding concepts. The research instruments included three case-based questions and interviews designed to measure conceptual understanding and the ability to apply calculus in realworld contexts. Data analysis was conducted using descriptive qualitative methods, evaluating the accuracy of calculations, conceptual understanding, and the relevance of answers to the case studies provided. The analysis stages included data collection, data presentation, and conclusion drawing as described by Johnson [15]. The following are examples of questions given to students: 1. A car moves at a speed of 𝑣(𝑡) = 3𝑡 2 + 6𝑡 − 5 where v(t) is the speed (m/s) and t is the time (seconds). Determine the displacement of the car after traveling for a time of t=5 seconds! 2. A building construction project can be completed in x days with a cost per day of 𝐶(𝑥) = 2𝑥 2 − 600 + 30𝑥 dollars. To minimize the total cost, how many days should the project be completed in? In addition to the written test, interviews were conducted to explore students' understanding of integral and differential concepts. Interview questions about integrals: MICHAEL KURNIAWAN 3 Study Case 1. What do you know about integrals? 2. How do you understand question number one? 3. What do you think the question means? Interview questions about differentials: 1. What do you know about differentials? 2. How do you understand question number two? 3. What do you think the meaning of the question is? RESULTS AND DISCUSSION Based on the results of research on three students from the Department of Mathematics Education at the University of Cincinnati who are taking the History of Mathematics course in the 2023 class, there was variation in the approaches to problem solving. The following presents the results of the analysis of the three respondents' answers to the three questions given. Question 1: A car is moving at a speed of v(t)=3t^2+6t-5, where v(t) is the speed in meters per second and t is the time in seconds. Determine the displacement of the car after traveling for a time of t=5 seconds. Figure 1. Student 1's answer to Question 1 Figure 1 shows how Student 1 solved the question related to integral calculations. Student 1 wrote down the available information well. The first step was to arrange the equation in integral form, then perform the integration process on the given function. After substituting t=5 into the integral result, the displacement was obtained to be 175 meters. This answer is considered correct. To gain a deeper understanding, an interview was conducted with Student 1: Researcher: Do you understand the concepts of integrals and differentials? Student 1: Yes, I understand both. Researcher: What do you know about integrals and differentials? MICHAEL KURNIAWAN 4 Study Case Student 1: Integrals are the opposite of differentials. Researcher: What does the question mean? Student 1: The question asks for the distance traveled in five seconds based on the given velocity function, and I used the concept of integrals to solve it. The interview results show that Student 1 has a good conceptual understanding of integrals, especially in the context of their practical application. He understands that integrals are not just mechanical calculation procedures, but also tools for linking concepts of change (such as velocity) with cumulative quantities (such as distance). This understanding reflects mature abstract and analytical thinking skills, indicating readiness to tackle more complex calculus problems. Figure 2. Student 2’s answer to Question 1 Student 2 was also able to explain the calculation steps well. He wrote down the displacement function as the result of the integral of the velocity function, then calculated the integral value from t=0 to t=5, and obtained a result of 175 meters. Question 2: A school building construction project can be completed in x days, with a cost 2x^2-60x+300 dollars. Determine the completion time of the project so that the total cost is minimum. Figure 3. Student 1's answer to Question 2 MICHAEL KURNIAWAN 5 Study Case Student 1 arranged the cost function into the form f(x), then found its first derivative and equated it with zero to find the minimum point. The result obtained shows that the project should be completed in 150 days. In an interview, Student 1 stated that differentiation is the inverse of integration and is used to determine the minimum value of a function. He demonstrated a good understanding of the application of the concept of derivatives in the context of cost optimization. Figure 4. Student 2’s answer to Question 2 Student 2 multiplied the cost function per day by x to obtain the total cost function, then found its first derivative and determined the critical value by setting f^’ (x)=0. He explained that the optimal value is obtained when the first derivative is zero and the second derivative is positive. This understanding demonstrates mature analytical and conceptual abilities in applying calculus to solve economic or project management problems. Figure 5. Student 3's answer to Question 2 Student 3 created the function and differentiated it, but the final result obtained (±√15) was not in accordance with the context, indicating a conceptual error in interpreting the information in the question. In an interview, Student 3 admitted that he understood the basics of differential theory but had difficulty remembering the application steps systematically. This shows the need to strengthen conceptual understanding and derivative application skills in real contexts. MICHAEL KURNIAWAN 6 Study Case Based on the results of the three students, two levels of understanding of integral and differential material were obtained, as shown in Table 1. Table 1. Level of Understanding of Mathematics Education Students Regarding Integral and Differential Material The results of this analysis show that students can be categorized into two main levels of comprehension: Medium Level and High Level Comprehension, referring to the model described in previous research [1]. Medium Level Comprehension At this level, students understand part of the question's meaning but still have difficulty connecting the information with integral and differential concepts. The steps taken to solve the problem are often incorrect, indicating a need to strengthen conceptual understanding. These findings are in line with previous studies that emphasize the importance of mathematical literacy and critical thinking in solving calculus problems [2–6]. High Level Comprehension Students with a high level of comprehension not only understand the questions conceptually but are also able to relate theory to practical applications, such as optimization and displacement calculations. They demonstrate good analytical skills and are able to identify relevant variables and solution steps. These findings support previous research emphasizing the importance of analytical thinking skills in solving advanced mathematical problems [7–10]. CONCLUSION The results show that most students have a good understanding of integral and differential concepts, but some still need additional guidance. The two main categories of comprehension levels found are Medium Level Comprehension and High Level Comprehension. MICHAEL KURNIAWAN 7 Study Case These findings have important implications for improving the quality of calculus learning at the university level. Curriculum adjustments are needed at the University of Cincinnati to focus more on the practical application of calculus and the development of analytical skills. In addition, evaluation strategies such as providing regular feedback and real-world case-based exercises can help students strengthen their conceptual and applied understanding. Overall, these results are expected to improve students' readiness to face academic and professional challenges in the fields of mathematics education and applied sciences. REFERENCE MICHAEL KURNIAWAN 8 Study Case MICHAEL KURNIAWAN 9