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ABSTRACT

Article history:

Received XX Month 2024

Revised XX Month 2024

Accepted XX Month 2024

Online first

Published 1 September 2024

Calculus is among the most important branches of mathematics that is widely used in various fields of science and engineering. It explores changes using basic ideas, including integration, differentiation and limitation. The goal of advanced calculus is to take the basic ideas of calculus to a deeper and more complex level. Basic and advanced calculus have a supportive relationship where learning increasingly difficult problems in advanced calculus requires a solid understanding of basic calculus concepts. Therefore, this study attempts to determine whether the assessment results of engineering students for the subjects calculus I (basic) and calculus II (advanced) have a significant relationship with each other. In addition, student performance in these two subjects was descriptively evaluated. The research sample used was a total of 67 Engineering students who took Calculus I (March - August 23 semesters) and Calculus II (October 23 - February 24 semesters). Students' final assessment scores were taken and analysed using SPSS version 2.0 software through descriptive statistics and Person correlation. Results showed that the performance of students who obtained an A grade in Calculus II had increased by 10% from Calculus I. Based on the Person Correlation analysis, there was a strong positive linear relationship between Calculus I and II with a value of r=0.561 (p-value < 0.05). Although this relationship was significant, only 32.6% of the variation in Calculus II can be explained in this model. It suggests that a wide range of other factors, including learning and teaching strategies used by lecturers or by students themselves, can have an impact on students' performance. Since understanding engineering mathematics necessitates a solid calculus foundation, it is hoped that this study will encourage students to work hard at improving their grasp of the fundamentals of the subject.

Keywords:

Basic Calculus

Advanced Calculus

Pearson Correlation

Relationship

Descriptive

DOI:

10.24191/jcrinn.v9i2

# INTRODUCTION Calculus is a fundamental branch of mathematics that provides essential tools for understanding change and motion. It forms the foundation for advanced studies in various scientific and engineering disciplines, making a solid grasp of its concepts crucial for students pursuing these fields. A good understanding of the importance of calculus in a wide range of careers and engineering education is important for students as they make decisions about how much mathematics to take at university (Huang, C. H., 2011). Many students perceive calculus as a challenging subject to master. A strong foundation in algebra, geometry and trigonometry is essential, which may leave students who lack proficiency in these areas to struggle. Calculus also demands higher-order thinking and multi-step problem-solving skills, which can be intimidating. Additionally, many concepts require the ability to visualise curves, slopes and areas under curves, posing a challenge for some learners. Nevertheless, a study conducted in Northern Province, Zambia, by Collins Chama (2023) showed that students have a positive perception and attitude towards calculus. Learning motivation, learning anxiety, mathematical connection ability, problem-solving ability and students' perceptions of teacher competence significantly affect mathematics learning achievement and are highly correlated ( Retnawati, 2022). Calculus 1, often the first course in a calculus sequence, introduces students to the fundamental concepts of limits, derivatives and the basics of integration. Meanwhile, Calculus 2 builds on this foundation, exploring deeper into techniques of integration, series and sequences, polar coordinates and parametric equations. The progression from Calculus 1 to Calculus 2 requires a thorough understanding of the initial concepts to successfully grasp more complex topics. This transition is not merely a step up in difficulty but a continuation of conceptual understanding. Basic Calculus becomes difficult owing to a lack of knowledge and a poor foundation in Basic Mathematics (Domondon, 2022). The principles learned in Calculus 1 are integral to tackling the more advanced topics in Calculus 2. Therefore, a strong foundation in Calculus 1 is often predictive of a student's success in subsequent calculus courses. Performance in first-term calculus consistently impacts success in second-term calculus for students in science, mathematics, and engineering (Sahmbi, 2020). Positive experiences and success in previous mathematics courses can boost confidence and create a more favourable perception of calculus. This paper aims to investigate the relationship between students' performance in Calculus 1 and their subsequent success in Calculus 2. Specifically, it seeks to answer the following questions: How does proficiency in Calculus 1 affect students' ability to understand and apply concepts in Calculus 2? Understanding these relationships can provide insights into how to better prepare students for advanced mathematical studies. ## LITERATURE REVIEW Today, many studies have examined basic mathematics achievement. Basic mathematics is important in learning mathematics and calculus to the next level. Many recent studies (such as Tazkiya, 2023; Ormonoy, 2022) have shown the importance of solving math problems in elementary school. In the end, problem-solving plays an important role in the process of mastering one or another theoretical material studied in the primary grades, develops students' thinking skills and concerns the difficulties facing students when learning mathematics. One of the reasons for the low overall achievement in mathematics is a lack of mathematical problem-solving skills. Several factors contributing to low achievement in mathematics include students' perception of teacher competence, learning anxiety, problem-solving ability, learning motivation and mathematical connection ability (Wawan, 2022). These elements are influenced by student, teacher, curriculum, school, and family considerations (Algani, 2019). In terms of problem-solving, most students struggle with creative thinking abilities, particularly fluency, adaptability and novelty (Yayuk, 2020). Furthermore, the structure and content of textbooks can impede the development of problem-solving skills as they frequently provide limited possibilities for learning through problem-solving (Masina, 2023). For academic success in mathematics education, it is essential to grasp basic mathematical concepts and recognise mathematical errors. According to recent studies, students’ mathematical pathways are significantly shaped by their ability in basic mathematical concepts. According to studies, performance in higher mathematical subjects later on is highly predicted by having a strong basis in geometry, algebraic manipulation and arithmetic (Smith et al., 2021). Furthermore, identifying and fixing arithmetic errors improves metacognitive awareness and greater conceptual comprehension, both of which are crucial elements of mathematical competence (Johnson & Brown, 2020). According to Lee et al. (2019), these results highlight the need to provide focused instruction on fundamental mathematical concepts and develop a culture of error analysis and feedback in mathematics education. Through these techniques, teachers may help students develop into capable mathematicians and resilient problem solvers, which will ultimately improve their academic performance. Transitioning from high school mathematics to university-level calculus is a critical step for students. Proficiency in fundamental courses, including Mathematics and Additional Mathematics, is frequently seen as critical for success in calculus. Numerous studies have discovered that secondary school mathematics performance predicts university calculus achievement. According to research, having a strong foundation in Basic Mathematics (algebra, geometry, and trigonometry) and Additional Mathematics (more advanced topics such as introductory calculus and linear algebra) improves students' ability to understand complex calculus concepts. Several studies from 2019 onwards have explored this relationship in depth. For instance, Johnson (2020) discovered that students who excelled in high school mathematics were more likely to succeed in their first-year calculus courses. The research highlighted the necessity of algebraic manipulation and trigonometric understanding as fundamental calculus skills. Similarly, Lee and Kim (2019) found a significant correlation between high school mathematics performance and university calculus grades, implying that problem-solving skills learned in high school mathematics are directly relevant to university-level calculus problems. In addition to basic mathematics, current research has emphasised the importance of additional mathematics. Smith and Brown (2021) found that students who completed mathematics courses in secondary school performed much better in university calculus than their counterparts who did not. This finding is further supported by Zhang et al. (2022), who discovered that exposure to complex mathematical concepts in high school, such as differential equations and introductory calculus, gave students a head start on university calculus studies. Parker et al. (2021) mentioned that students who excelled in high school algebra and geometry are more ready to face the complicated concepts of university calculus. Their study, which tracked the performance of over 1,000 students, found that early mastery of basic mathematical concepts is the basis for learning higher calculus topics. Meanwhile, Williams and Jackson (2021) studied 800 students and found that those who completed additional mathematics courses in secondary school were more likely to excel in their first-year calculus courses. According to the study, these courses' advanced problem-solving skills and conceptual knowledge can be directly applied to university-level calculus. These results are corroborated by another research by Martinez et al. (2022), which emphasised that students who are exposed to higher-level mathematics in secondary schools, such as calculus, linear algebra, and differential equations, tend to transition to university mathematics more easily. Their research, which included a thorough examination of student performance data, showed that prior knowledge of these complex subjects lessened the cognitive load when revisited in an academic context. In the same vein, Maisurah (2023) stated that most students could accurately answer mathematical questions, particularly those involving addition, subtraction, multiplication and division, while some others still struggle with the basics of mathematics, particularly when it comes to problems involving variables and numbers. Nonetheless, students must work hard to solidify their understanding of mathematics' foundational ideas, particularly those that involve algebra. The complex relationships between Calculus I and Calculus II have been the subject of a recent study, providing insight into pedagogical approaches, curriculum design, student outcomes and the transfer between these courses. McCoy and Weiss (2019) stressed how crucial it is to successfully sequence calculus courses to improve student achievement. They discovered that students' performance in Calculus II is positively impacted by a well-structured curriculum that expands on ideas covered in Calculus I. Similar to this, Johnson et al. (2020) investigated different Calculus I educational strategies and how they affect Calculus II student performance. According to their research, some teaching strategies helped students grasp concepts more deeply and were more equipped to handle the difficulties of integral calculus. Furthermore, Duncan (2019) offered proof that shows how well students learn when Calculus I and Calculus II are taught in the order they are intended. Duncan demonstrated how a well-thought-out curriculum might support students' learning continuity and comprehension of calculus ideas through longitudinal analysis. A longitudinal study by Li and Zhang (2020) evaluating student learning outcomes in Calculus I and Calculus II provided insights into the variables affecting students' performance and advancement in both courses. Additionally, Smith et al. (2020) explored the effects of Calculus I pedagogical practices on Calculus II achievement by students. Their conclusion emphasised the value of technology-enhanced instruction and active learning strategies in raising student achievement and engagement. The transition from Calculus I to Calculus II was the main topic of Lee and Park's (2019) study, which sought to discover the potential and problems facing students during this crucial stage of their mathematical education. Collectively, these studies highlight the importance of the link between Calculus I and Calculus II in influencing students' mathematical understanding and ability. Thus, educators may improve evidence-based practices that enhance student achievement in calculus and beyond by analysing curriculum design, pedagogical approaches, student outcomes and course transitions. In this study, the researcher's objective is to determine whether the performance of Calculus I influences the performance of Calculus II. # METHODOLOGY This study was conducted on engineering students who took Calculus I during the March-August 2023 semester and Calculus II during the October 2023-February 2024 semester. The study sample consisted of 67 students from diploma programs in Electrical, Mechanical and Civil Engineering. The data used in the study were the final assessment scores of the students for Calculus I and Calculus II, with all students having passed both courses. Analysis was carried out using descriptive and inferential statistics with the Statistical Packages for Social Sciences (SPSS) version 20.0. A descriptive study of the student's grades was initially conducted to examine the percentage frequency of students achieving grades A, B and C in both subjects. Therefore, the grades were slightly consolidated to focus solely on grades A, B and C, as shown in Table 1 below: Table 1. Combination of student score grade | | | | | --------- | --------- | --------------------- | | **Marks** | **Grade** | **Grade Combination** | | 100 - 75 | A+, A, A- | A | | 74 - 60 | B+, B, B- | B | | 59 - 50 | C+, C | C | This study proceeded with regression and correlation analysis to examine the relationship between Calculus I and Calculus II. Pearson's correlation was used as both variables were continuous, with the data being normally distributed. Pearson's correlation coefficient, r, was used to measure the strength and direction of the linear relationship between the independent variable (Calculus I) and the dependent variable (Calculus II). The scale for interpreting the strength of the correlation ranged from -1 to 1. A correlation value close to 1 or -1 indicates a perfect relationship, either positive or negative, while a value of 0 indicates no significant linear relationship between the variables. Table 2 below displays the correlation scale based on the r value as defined by Ghazali and Sufean (2016). Table 2. Correlation Value and Relationship | | | | ------------------------------ | ------------------------- | | **Correlation coefficient, r** | **Strength/Relationship** | | 1.00 | Perfect | | 0.7 - 0.99 | Very Strong | | 0.5 -0.69 | Strong | | 0.3 - 0.49 | Moderate | | 0.1 - 0.29 | Weak | | 0.01 - 0.19 | Very Weak | Source: Ghazali & Sufean, 2016 Meanwhile, a linear regression analysis was performed to explain the tendency of Calculus II score changes based on changes in Calculus I scores. A simple linear regression model was employed in this study. First, the coefficient of determination (R²) value was examined to indicate the percentage of variation in the Calculus II score that can be explained by the Calculus I score. Subsequently, the t-test and p-value were assessed to determine if the regression coefficient is statistically significant, with a p-value of less than 0.05 being considered significant. In contrast, the F value was used to examine the fit of the model that resulted significantly from the relationship between these two variables. # RESULT AND DISCUSSION Figure 1 below shows the descriptive results, which are the percentage bar charts for grades in Calculus I and II. The results indicated that the majority of students obtained a grade B in both subjects, with 48% in Calculus I and 45% in Calculus II. This is followed by grade C with 34% and grade A with 18% in Calculus I. In Calculus II, the second highest percentage was grade A at 28%, followed by grade C at 27%. There was also a noted increase in the percentage of grade A students from Calculus I to Calculus II, with a rise of 10%, while grades B and C saw a decrease of 5% and 7%, respectively. The positive increase in grade A suggested that 18% of students were able to master the basics in Calculus I and maintain excellence in Calculus II (28%). However, the percentage of grade C remained relatively high, implying that while student performance in Calculus was satisfactory, they mainly grasped basic concepts and might struggle with more complex problems or advanced applications. Martinez and Bain (2014) interpreted grade C as a satisfactory level of achievement but indicated a possibility for improvement. Students also need support from lecturers and peers to improve their basic understanding for more comprehensive achievement (Brookhart, 2017). ![](667a5c48c9df5_media/media/image1.png) Fig 1. Bar Chart for percentage grades of Calculus I and Calculus II Next, the results of the correlation analysis were examined to determine the strength and direction of the linear relationship between the variables tested, Calculus I and Calculus II, as shown in Table 3. The relationship between the two variables was found significant at a 5% confidence level (p-value \< 0.05). The results indicated that the Pearson correlation coefficient, r = 0.571, with p = 0.00 \< 0.05, explained that there was a significantly strong positive relationship between Calculus I and Calculus II. Statistically, this test demonstrated a strong association between the variables within the context of the study conducted. Table 3. Correlation between Calculus I and Calculus II | | | | | -------------- | ----------------------- | --------------- | | | | **Calculus II** | | **Calculus I** | **Pearson Correlation** | 0.571 | | | **Sig. (1-tailed)** | 0.00 | | | **N** | 67 | Table 4 below shows the results of the regression analysis conducted in this study. It was revealed that R² = 0.326, indicating that 32.6% of the variation in Calculus II scores can be explained by the Calculus I scores. Although this percentage was relatively low, it suggested that Calculus I scores still influenced Calculus II scores. The remaining 67.4% was influenced by other factors not accounted for by this model. Possible factors include the students' individual approaches to learning and their level of mastery of basic Calculus I concepts, which was moderate based on the grades obtained in Calculus I. Table 4. Summary of Simple Linear Regression Analysis | | | | | | ------------------------ | -------------------------- | ----- | -------- | | | **Regression Coefficient** | **t** | **Sig.** | | **Constant** | 32.364 | 5.079 | 0.00 | | **Calculus I** | 0.551 | 5.603 | 0.00 | | F = 31.397 (Sig. = 0.00) | | | | | R² = 0.326 | | | | a.Dependent Variable: Calculus II, b.Predictors: (Constant), Calculus I To further explain the influence of Calculus I on Calculus II, with a significance value of p = 0.00 (\<0.05), indicated that the regression coefficient for Calculus I was statistically significant, and that there was a significant linear relationship between the two variables. The F-value obtained was 31.397 with a significance value of p = 0.00 (\<0.05), indicating that the regression model used was significant overall. Then, the regression model provided a good explanation of the variation in Calculus II scores based on Calculus I scores. Therefore, it can be concluded that an increase in Calculus I scores tends to be associated with an increase in Calculus II scores, and this relationship is statistically significant. # CONCLUSION The results of this study highlighted the critical role of a strong foundation in Calculus I for succeeding in Calculus II. Even though other factors could also contribute to the outcome, the significant increase in the number of students achieving grade A in Calculus II suggested that mastering the basics in Calculus I can lead to higher achievement in more advanced calculus concepts. However, the persistently high percentage of grade C students indicated that many students only achieve a satisfactory level of understanding. These suggest the necessity for additional support. Providing targeted support for students struggling with fundamental concepts in Calculus I can help improve their performance in Calculus II. This support can include tutoring, peer study groups, as well as additional resources such as online tutorials and practice problems. Future studies should investigate other factors that influence Calculus II performance, such as students' study habits, classroom environment and instructor effectiveness, to develop a more comprehensive understanding regarding the determinants of success in calculus courses. In conclusion, the study demonstrated that a solid grasp of Calculus I concepts is crucial for success in Calculus II. Enhancing foundational knowledge and providing robust support systems can significantly improve students' performance and confidence in tackling advanced mathematical problems. 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