Identification and Analysis of Conceptual Difficulties

Study Case Identification and Analysis of Conceptual Difficulties of Electrical Engineering Students in Calculus Courses Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract This study was prompted by the low academic achievement of Electrical Engineering students in Calculus courses. This course is a fundamental foundation for students to understand various mathematical concepts that form the basis of Electrical Engineering. One of the main factors contributing to low learning outcomes is an internal factor, namely students' difficulties in solving problems related to Calculus material. This study aims to: (1) describe the types of difficulties experienced by Electrical Engineering students in solving Calculus problems; and (2) identify internal factors that contribute to these learning difficulties. This study uses a descriptive qualitative approach with research subjects consisting of first-semester students in class A at the University of Cincinnati in the 2022/2023 academic year. The instruments used were Calculus course assignments and interview guidelines. The results showed that: (1) students' difficulties in solving Calculus problems included difficulties in applying mathematical concepts and principles; and (2) the causes were dominated by internal factors related to individual psychological and cognitive conditions. Keywords: Difficulty Analysis, Calculus, Electrical Engineering INTRODUCTION Mathematics, as a basic science, plays an important role in human life. As a universal science, mathematics is the main foundation for the development of modern technology, supports various disciplines, and encourages logical and analytical thinking skills in humans. A lack of mastery of basic mathematical concepts will hinder abilities at subsequent levels, which ultimately leads to difficulties in understanding and MICHAEL KURNIAWAN 1 Study Case applying advanced concepts [1]. Therefore, mathematics education needs to be provided continuously from elementary, secondary, to higher education levels. There are two main factors that influence students' ability to master mathematics, namely internal and external factors. External factors include perceptions of mathematics, the ability to understand symbols, and mastery of basic concepts [2]. Meanwhile, internal factors include mental readiness, self-confidence, motivation to learn, and perseverance in facing academic difficulties. These internal factors are often the main cause of students' failure to understand mathematics material, as seen from the various mistakes made in solving problems. Students who experience learning difficulties usually show various symptoms, including: (a) low academic achievement; (b) learning outcomes that are not proportional to the effort made; (c) delays in completing assignments compared to classmates; (d) indifference towards the learning process and academic results; (e) deviant behavior such as not attending classes, arriving late, not completing assignments, or disturbing classmates; and (f) emotional symptoms such as being easily irritated, depressed, or angry for no apparent reason [3]. Based on initial observations, it was found that many Electrical Engineering students received low grades in Calculus. This is a cause for concern, given that Calculus is a very important starting point for mastering advanced mathematical concepts that form the basis of engineering learning. The low learning outcomes are evident from the many mistakes made by students, such as misunderstanding concepts, incorrect mathematical calculations, inability to apply concepts in real contexts, and inappropriate problem-solving strategies. Therefore, special attention is needed from lecturers to improve students' mathematical skills, especially in Calculus, which serves as the foundation for Electrical Engineering. This study focuses on analyzing the types of difficulties students encounter in solving problems related to Differential Equations in Calculus courses. This analysis is expected to help lecturers understand the sources of students' difficulties in applying mathematical concepts and principles, so that learning strategies can be adjusted to minimize errors and improve student learning outcomes. MICHAEL KURNIAWAN 2 Study Case METHOD This research is a qualitative study with a descriptive approach. According to Moleong in [1], qualitative research aims to understand phenomena experienced by research subjects—such as behavior, perceptions, motivations, and actions—holistically and presented descriptively through words and language. The main objective of this approach is to provide a comprehensive and in-depth description of an event or phenomenon in order to gain new insights [2]. The research subjects were first-semester students in the Electrical Engineering Program, Class A, at the University of Cincinnati who had taken Calculus courses. The objective is to identify the types of errors made by students in solving Calculus problems, particularly on the topic of Differential Equations. The sampling technique used is purposive sampling, which is the selection of participants based on specific considerations. The five students who made the most errors were selected as research subjects. Data collection was conducted directly through tests, interviews, and observations of student behavior during the learning process. The research instruments consisted of: 1. Test instruments, in the form of essay questions related to Calculus material. 2. Non-test instruments, in the form of interview guidelines containing questions about students' learning experiences and strategies in solving problems. Data analysis was conducted qualitatively based on the stages of data reduction, data presentation, and conclusion drawing with verification [3]. The test data was analyzed to identify the types of errors made by students, then classified based on the difficulty in applying mathematical concepts and principles. Meanwhile, the interview data was analyzed descriptively to explore the causes of learning difficulties experienced by students during the learning process. RESULTS AND DISCUSSION The research data was obtained from five students who were selected as research subjects. These five students were the participants with the highest error rates in completing Calculus questions. The scores obtained from 10 questions are summarized in Table 1. MICHAEL KURNIAWAN 3 Study Case Table 1. Calculus Course Assignment Scores of Electrical Engineering Students Based on the table above, the lowest score was 28 points and the highest score was 65 points out of a maximum score of 100. These results indicate that the students' understanding of Calculus concepts is still relatively low. Of the five questions related to Differential Equations, question number 3 was the most difficult with the lowest average score. Analysis of Student 1's Errors Student 1 scored 32 points out of a maximum of 100 points, indicating a low level of mastery of the material. The most significant errors occurred in questions 4 and 5. Figure 1. Student 1's Error in Question 4 Student 1 did not understand how to solve differential equations in fractional form, indicating a weakness in basic concepts. MICHAEL KURNIAWAN 4 Study Case Figure 2. Student 1's Error on Question Number 5 The student failed to apply the principles of Differential Equations, indicating a lack of conceptual understanding during the learning process. Analysis of Student 2's Errors Student 2 obtained the highest score, namely 65 points. However, they still failed to solve question number 4. Figure 3. Student 2's mistake on question number 4 The mistake is similar to that of Student 1, namely not understanding the method for solving fractional differential equations. Analysis of Student 3's mistakes Student 3 received the lowest score, namely 28 points. Figure 4. Student 3's Error on Question Number 1 The student almost answered correctly but made a minor conceptual error. MICHAEL KURNIAWAN 5 Study Case Figure 5. Student 3's Error on Question Number 2 The appearance of irrelevant expressions such as “11x” indicates that the student was unable to identify the required mathematical principles. Analysis of Student 4's Errors Student 4 scored 60 points and also made an error on question number 4. The main error was the same as Student 2's, namely not understanding the form of fractions in Differential Equations. Analysis of Student 5's Errors Student 5 scored 30 points out of 100. Of the five questions, only three were answered. Figure 6. Student 5's Errors on Question Number 3 The student demonstrated a partial understanding of the concept of Differential Equations, but procedural errors caused the loss of most points. In addition to the test results, interviews were conducted to identify the factors causing the errors. The percentage of the analysis results shows: 1. Lack of understanding of the concept of Differential Equations – 23% 2. Forgetting differential algebraic operations – 37% 3. Difficulty applying formulas even after learning them – 40% These factors all originate from internal aspects of the students. MICHAEL KURNIAWAN 6 Study Case DISCUSSION Data analysis shows that the main difficulty students have in solving Calculus problems, especially Differential Equations, is related to their weak mastery of basic mathematical concepts and principles. As a result, most students fail to achieve the minimum passing score of 60 points out of 100. These results support the view [4] that learning difficulties are a condition in which students perform below the established standard due to obstacles in understanding the material. Based on interviews, several main causes were found: 1. Not paying attention during class (23%) 2. Not reviewing the material after class (30%) 3. Forgetting algebraic operations (22%) 4. Not being able to apply formulas even after learning them (25%) These factors indicate the dominance of internal factors that are non-intellectual in nature, such as motivation and discipline in learning. In line with the opinion [5], differences in individual conditions such as enthusiasm, focus, and comprehension affect a person's level of learning difficulties. Therefore, learning strategies are needed that not only strengthen conceptual understanding but also improve internal factors such as motivation, self-confidence, and a positive attitude towards mathematics. In addition, external support such as additional guidance or peer tutoring can also help students achieve better learning outcomes at the University of Cincinnati. CONCLUSION Based on the results of the research and analysis, the learning difficulties of Electrical Engineering students in solving Calculus problems can be categorized into two main groups: 1. Difficulties in Using Mathematical Concepts These include: • inability to define concepts verbally; • inability to distinguish between examples and non-examples of a concept; and • difficulty comparing and confirming relationships between concepts. MICHAEL KURNIAWAN 7 Study Case 2. Difficulties in Using Mathematical Principles These include: • inability to recognize when a particular principle should be applied; and • errors in applying principles due to weaknesses in mathematical operations. The main factors causing these difficulties originate from internal aspects of the students, including: lack of attention during learning, not studying before exams, forgetting algebraic operations, and difficulty applying formulas to the context of the given problems. REFERENCE MICHAEL KURNIAWAN 8