College Calculus Research Articles

Analyzing tasks assessed in university Calculus exams: a didactic and historical perspective

Analyzing tasks assessed in university Calculus exams: a didactic and historical perspective

Positive Feedback as a Lever to Boost Students' STEM Outcomes.

Although many college students intend to major in Science, Technology, Engineering, and Mathematics (STEM), dropout from these fields is high, especially among members of historically underrepresented groups, such as women and racial-ethnic minorities. We propose a minimal, yet potentially powerful intervention to broaden participation in STEM: giving positive feedback to students in STEM. Studies 1 and 2 found that giving positive feedback is less normative in math (vs. English) courses, and instructors' feedback-giving practices and students' experiences mirror these norms. However, students who received positive (vs. only objective) feedback on introductory-level college calculus exams showed greater belonging and self-efficacy in math, which predicted better STEM outcomes (i.e., increased interest in STEM and higher final math course grades, respectively, Study 3). These findings were especially strong for racial-ethnic minority students. Giving positive (vs. only objective) feedback is thus a potentially transformative tool that boosts student outcomes, especially for underrepresented groups.

Conceptions of The Instruction of Maxima And Minima Problem-Solving to Calculus Students: A Study on University Education

Theoretical Framework: The teaching of calculus is indispensable in many scientific careers, and maxima and minima problems hold a prominent place among its contents. However, students often encounter difficulties solving optimization problems. Objective: Considering this, our study aims to determine the perception of calculus teachers regarding the importance of contextualization and the use of diverse resources when they teach these problems. Method: Our study has a descriptive nature and to achieve our goal, a sample of university calculus professors from the Dominican Republic was analyzed using a Likert-type test. Results and Discussion: The results demonstrate a favorable trend towards contextualization and the integration of resources in teaching maxima and minima problems, although some professors still prefer traditional methods based on textbooks. Research Implications: The findings highlight the importance of exploring contextual teaching strategies, promoting resource diversity, and adopting innovative pedagogical approaches to enhance learning, comprehension, and student motivation.

Best practices for teaching the concept of the derivative: Lessons from experienced calculus instructors

Much research has reported on difficulties exhibited by students when working with the concept of the derivative in first-semester university calculus. In an effort to generate and share a resource on effective instructional practices related to the teaching of the concept of the derivative, a 12-item questionnaire was administered to experienced calculus instructors in the United States. Most of the 10 experts who participated in this study considered providing ample examples and practice problems, using graphing utilities such as Desmos and GeoGebra, and employing problem solving strategies in the teaching of the concept of the derivative to be effective in supporting students learn about the concept of the derivative, among other things. These experts further remarked on representations of the concept of the derivative and real-world contexts they typically use in their teaching of the concept, in addition to providing rationales for the choice of real-world contexts they typically use in their teaching of the concept. Recommendations for instruction are included.

대학생 TBL·PBL 학습 과정에서 팀 내 과업 갈등이 집단효능감에 미치는 영향 연구

Objectives This study utilized a multilevel approach to investigate the influence of task and relationship conflicts among college team members on collective efficacy during university TBL and PBL classes.
 Methods Over a span of 10 weeks, a total of 176 students were organized into 25 teams participating in a uni-versity calculus mathematics course, which was conducted using TBL and PBL methodologies. At the conclusion of the course, a survey was administered. The study employed a cross-level interaction analysis with random slope models.
 Results The study revealed a significant cross-level moderation effect in the relationship between individual-level task conflict and collective efficacy moderated by team-level relationship conflict. In teams characterized by high relationship conflict, there was a positive correlation between team members' task conflict and collective efficacy. Conversely, in teams with low relationship conflict, team members' task conflict had little impact on collective efficacy. Relationship conflict negatively and significantly affected collective efficacy at both individual and team levels.
 Conclusions This study empirically demonstrated that the influence of team members' task conflict on collective efficacy varies depending on the level of team relationship conflict. This underscores the importance of effective communication during periods of relationship conflict in teams. The study discusses implications for TBL and PBL for both scholars and practitioners.

The supportive role of active learning in a calculus course on low precalculus proficiency students

College calculus plays an important role in STEM students’ degree and career aspirations. One of the key factors considered in assessing a student’s ability to be successful in calculus is their proficiency in topics from prior mathematics courses such as algebra and precalculus. This study set out to examine the impact of students’ precalculus proficiency on their achievement in introductory calculus based on their classroom environment. Results from the implementation of the Modeling Practices in Calculus (MPC) model, an innovative, active learning approach, are presented. Using a randomized-controlled trial research design, students were randomly assigned to MPC and traditional, lecture-based calculus sections. The Precalculus Concept Assessment inventory was administered to gauge students’ precalculus proficiency. We found that students exposed to the MPC model were more likely to be successful in their calculus course, even if they began with low precalculus proficiency. Also, students enrolled in the MPC sections saw significant growth in their precalculus proficiency from the beginning to the end of the semester. Additionally, we observed this model providing support for students in key demographics (low proficiency, female, first and second year undergraduates) in terms of the development of their proficiency that they may not receive in traditional classrooms.

The Effects of Online Mathematics Homework on Learning, Attitude, and Efficacy in a College Calculus Class – A Case Study

This case study examines how online mathematics homework impacts student learning, attitude, and efficacy in a college Calculus I class. This research is important because it provides data on student perceptions for how to effectively implement online homework. Participants for this study completed homework using a blend of online and paper formats. We use a mixed methods approach to analyze quantitative survey data and qualitative data collected in focus groups and interviews. Our findings indicate students benefited from using a combination of paper homework and online homework. Recommendations are made for how instructors can utilize the benefits of both approaches to support student learning, attitude, and efficacy.

A Sterling Look at Stirling’s Constant

Summary In a previous article [3] appearing in this journal, existence of the finite positive limit implicit in Stirling’s approximation was proved, and in so doing this limit was connected to the Euler limit . Because it was not done in the original article, exact determination of the constant () by a novel, elementary technique is the aim of this follow-up. For both articles, the arguments are readily accessible to advanced undergraduate students with a year of university calculus in their background.

곡선 아래 넓이에서 리만합의 극한으로

The transition from high school to university mathematics presents significant challenges for understanding definite integrals. In high school, students learn that definite integral is the difference between the two antiderivatives evaluated at the endpoints. However, in university calculus, the definition of the definite integral is based on the limit of Riemann sums. To facilitate students’ transition, we employed commognitive theory to design teaching and learning activities that incorporated the history of definite integrals. Our study aimed to document changes in students’ discourse on definite integrals and analyze the conflicts they encountered in the transition. In the pre-survey, most students believed that the area under a curve was the definition of the definite integral. However, in the post-survey, many students accepted the limit of Riemann sums as the formal definition. Moreover, students demonstrated growth in substantiating narratives on the connection between areas (or the limits of Riemann sums) and antiderivatives. Utilizing the history of mathematics, we provided suitable contexts to discuss explicitly the change in word use of the definite integral. Students’ self-reports indicated that history helped them understand the reasons behind the new definition of definite integral. Additionally, rather than disregarding the high school definition as incorrect or imprecise, students could relearn it in connection with the university discourse. Our study contributes to designing a smooth transition from high school to university mathematics and for mathematics teachers to adapt to the inverse transition.

Presenting a New Model to Support the Secondary-Tertiary Transition to College Calculus

Although the secondary-tertiary transition has been investigated in mathematics education research with different focuses and theoretical approaches, it remains a major issue for students in the transition. With success in a science, technology, engineering, or mathematics (STEM) major at stake, we investigated a novel approach to support the transition from secondary precalculus or calculus to tertiary calculus. Using the Four Component Instructional Design (4C/ID) model and empirical data from the United States (US) nationally representative FICSMath project, we mapped instructional experiences of students in the transition to theoretical components of the 4C/ID model. From exploratory factor analysis (n=6,140), we found six factors that mapped to the 4C/ID model components and created the new Secondary Precalculus Calculus (SPC) 4C/ID model. In this model, the Learning Task Component represents tasks to engage learners in meaningful problem solving; the Support Component grounds instruction in reasoning and understanding; the Procedure Component integrates group work and graphing calculators to connect concepts to procedures; and the Part-Task Component represents instruction to develop automaticity. The SPC 4C/ID model presents a unique support for precalculus and calculus teachers in the quest of teaching for learning and transfer of learning across the transition.

Comparing high-school and college instructors’ beliefs about college calculus preparation

The expectations gap – the difference between the level at which high schools prepare students and the preparation colleges expect – is believed to be accountable for hindering access to, and success in, higher education. It has been largely attributed to misalignment of priorities and lack of communication among policy makers and educators and should inform instruction at both levels. This study compared expectations of college calculus instructors and of instructors who teach calculus-preparatory courses, in college or high school, on what makes for important preparation for the first college calculus course. A total of N = 293 instructors teaching precalculus and calculus at the high school and college levels, in the United States, completed a survey that assessed participants’ beliefs on three major areas related to calculus preparation and success: the critical content prerequisites for calculus, the importance of different calculus teaching and learning elements discussed in the literature, and the ways in which college calculus instructors are willing to accommodate under-prepared students. This study confirms the existence of an expectations gap on a rather narrower issue ‘adequate preparation for college-level calculus', in terms of content and pedagogy. The significance of the finding and its implications for bridging the expectations gap are discussed.

Analysis of Factors Influenced by Changes in Secondary Curriculum and the SAT System on College Mathematics －Based on the case of A university

This study analyzed the correlation between a student's learning background and their academic achievement in college-level calculus. By focusing on <i>A</i> university cases, this study seeks ways to improve the college calculus curriculum according to changes in the curriculum and the College Scholastic Ability Test (SAT) system. In particular, the academic achievement of college calculus was compared and analyzed according to whether calculus was completed in high school, or if calculus was selectively taken in the SAT. Through this study, it was confirmed that the relationship between one's academic achievement and one's learning background in college calculus differs depending on the student's basic academic ability. In other words, there was a difference in the academic achievement of college calculus according to whether students with a certain level of basic academic ability selectively took calculus in the SAT. On the other hand, it was also confirmed that there was a difference in the academic achievement of college calculus for students who lacked basic academic ability depending on whether or not they had completed calculus in high school. Therefore, universities should prepare a curriculum for college calculus according to the educational background of students by conducting a basic academic ability assessment before granting them admission in order to more accurately grasp their particular educational backgrounds.

CALCULUS TEACHER’S COMPETENCIES AS CORRELATES OF STUDENTS’ LEARNING EXPERIENCES

The good qualities of a calculus teacher are an asset in producing great results for students and themselves. This study aimed to investigate the different competencies of a calculus teacher as correlates of students' experiences in learning at Visayas State University, Philippines. The study employed selected secondary data from the existing study literature such as competencies in teaching calculus subjects and different types of learning experiences of students. The study used some descriptive statistical measures to summarize the selected data and employed Spearman rho correlation to predict significant factors (teacher's competencies) that might influence the student's learning experiences. Results showed that teachers' competencies such as "prepared", "knowledgeable", "mastery", "organized", "decent", and "approachable" are highly correlated to the different learning experiences of students at a 1% level of significance. Hence, calculus teachers must be always knowledgeable in a class through rigorous preparation and studying. It is also concluded that teachers must develop a mastery of calculus subjects and apply an organized routine during class discussions to obtain desired academic performance for students. In addition, a well-groomed and professional may be integrated in front of students as well as open-mindedness and good personality to make interactive and stimulating class discussions. It is recommended that calculus teachers may undergo some rigorous training for calculus teaching to become a knowledgeable and competitive educators. Furthermore, teachers who will teach college calculus must be holding at least a master's degree in line with mathematics education or pure mathematics.

Studying while anxious: mathematics anxiety and the avoidance of solving practice problems during exam preparation in college calculus

Studying while anxious: mathematics anxiety and the avoidance of solving practice problems during exam preparation in college calculus

Student attitudes and achievement in active learning calculus

The present paper explores the relationship between attitudes towards mathematics (ATM) and achievement in college calculus in active learning (AL) and lecture-based (LB) classrooms. Previous work on this relationship has mainly been limited to LB instruction, neglecting the impact of innovative approaches such as AL. Less attention has been paid to the roles played in this relationship by gender, year in college, and initial ATM. Results from a sample of 535 undergraduate students enrolled in 9 AL and 9 LB sections are presented. Data included ATMI surveys’ responses, final grades, and demographics. Correlation and multiple regression analyses were conducted. The influence of instruction on students with low ATM was also examined. Gender and year in college were the main demographic variables considered. Achievement in AL was found to be less dependent on initial ATM in terms of correlation. AL showed higher gains in grades than LB, when controlling for ATM and demographic variables. Effect sizes of AL instruction on grades of students with low ATM were larger than those of students with higher ATM. Furthermore, AL courses had a large effect size (d = 0.81) on female students with lower ATM, confirming its role as a gender equalizer.

Disciplinary Literacy, Agency, and Didactical Texts: Findings from a Calculus Textbook Think-Aloud Study

Textbooks have been the subject of research within and across disciplines, but have not yet been widely studied from a disciplinary literacy perspective. Readerly agency is also understudied in disciplinary literacy. The present paper aims to illuminate both of these areas by examining facets of agency that readers demonstrated during think-aloud interviews with a college calculus textbook, thus exploring the disciplinary literacies that are involved in reading this didactical text. Authors have drawn on the recently forwarded theory of “didactical disciplinary literacy,” as well as definitions of agency in both literacy and math fields, to reveal how agency was enacted by readers using the textbook to negotiate unknowns in calculus. Findings show participants exhibiting agency in a range of ways, but not always in direct relationship to themselves as agents within the field of mathematics. This study invites college instructors and secondary teachers to consider ways they might foster didactical disciplinary literacy skills to support student learners in mathematics, and other disciplines.

Contrasting cases of college calculus instructors: their preferences and potential pedagogy in teaching derivative graphs

This study examined college calculus instructors’ preferences in solving two calculus tasks to examine college calculus instructors’ use of important cognitive roots in understanding derivatives of function. Our results showed that only one instructor consistently uses cognitive roots while other instructors either focus on algebraic methods or expressing the constraints of students’ knowledge, demonstrating their orientations and conflicts. These results potentially show what students experience when differentiation of function is taught.

An analysis of the relationship between high-school pre-calculus and university calculus grades

First-year mathematics instructors at universities across North America and the globe have been noticing a decline in the mathematics skills and preparation of their incoming students, who have been failing out of first-year mathematics courses at alarming rates. Though some universities have implemented placement or diagnostic tests to measure students’ preparedness, many still use high school grades as the only indicator of readiness for university mathematics. However, researchers have questioned how effective these high school grades are at predicting success in university mathematics classes due to factors such as mis-aligned teaching methods, curriculum, and grade inflation. This study analyzes the relationship between grade 12 Pre-Calculus grades and first year university Calculus grades at a large Canadian university over the period from 2001–2015 in an attempt to quantify previous research in a large-scale, longitudinal manner. Results show that there is a significant disconnect between these grades, that the disconnect has been growing over time, and that it is is quite significant for students who are not performing well at the Calculus level. Recommendations moving forward include the implementation of placement examination in a wide-scale manner, and increased communication and collaboration between K-12 and university mathematics educators and administrators.

Examining the mathematical autobiographies of undergraduate health science students

Numerous studies affirm the importance of students’ attitudes for mathematical education. This study uses mathematical autobiographical essays to examine the mathematical experiences and attitudes of undergraduate students enrolled in mathematics classes at a Midwestern University in the United States. The essays of 41 students (20 men, 21 women) out of 170 enrolled in College Algebra, Precalculus, and Calculus were analyzed using the categories of (1) confidence in mathematics ability, (2) value of mathematics, and (3) liking of mathematics. This paper provides context and theoretical depth to student reports of their attitudes in these categories. Statements at the intersections of both confidence and liking, and value and liking provide additional insights. First, students who perceive mathematics as valuable are more likely to like it. Second, the vast majority of students talk about confidence and liking together, and with a positive correlation. Further, these students tend to talk about liking or not liking mathematics because of their perceived ability in mathematics. Our study suggests the importance of proactive teaching strategies to improve students’ perceptions of the value, liking of, and their confidence in, mathematics, and also suggests self-efficacy as a potential theoretical basis for this work.

Self-efficacy and achievement emotions as mediators between learning climate and learning persistence in college calculus: A sequential mediation analysis

The current study aims to deepen our understanding of the mechanisms of motivational and emotional profiles in entry-level calculus. Specifically, we tested a multiple mediation model to examine the linking roles of students' self-efficacy and achievement emotions (anxiety and enjoyment) in the relationship between learning climate and learning persistence in entry-level calculus. A sample of 267 college students enrolled in the entry-level calculus classes at a southern urban university took measures of learning climate, self-efficacy, achievement emotions (enjoyment and anxiety), and learning persistence. A multiple mediation path model was conducted to examine the mediating roles of self-efficacy and achievement emotions in linking learning climate and learning persistence in college calculus. The results revealed that self-efficacy played a prominent role in mediating the relationship between learning climate and learning persistence. Moreover, self-efficacy and anxiety serially mediated the relationship between learning climate and learning persistence.