Precalculus Course Research Articles

Systematic integration of a rubric in first-year engineering calculus courses

ABSTRACT For many first-year engineering students, their first calculus or pre-calculus courses can be a substantial barrier for initial success and persistence to subsequent engineering courses. One approach that may mitigate this challenge for some students is to support their development as stronger metacognitive learners – learning how to self-evaluate and improve based on informed feedback. A structured mathematical problem-solving rubric designed to offer feedback on how to develop and communicate mathematical solutions to engineering problems can be a helpful tool to support metacognitive development. After a semester of careful and systematic instruction, practice, and use of such a rubric in engineering calculus courses, we found significant improvement in pre-calculus students' metacognitive awareness. A median split analysis showed that for both calculus and pre-calculus courses, the lower metacognitive half of each class improved significantly in their metacognitive abilities, as was the intent for the rubric. Students reported an overwhelmingly positive perception of the value of the rubric for their own learning, and the subset of calculus I students who improved in their sophistication of rubric use had significantly higher (12.5%) final exam scores compared to those who did not improve.

Spaced recall reduces forgetting of fundamental mathematical concepts in a post high school precalculus course

The retention of fundamental mathematical skills is imperative to provide a foundation on which new skills are developed. Educators often lament about student retention. Cognitive scientists and educators have explored teaching methods that produce learning which endures over time. We wanted to know if using spaced recall quizzes would prevent our students from forgetting fundamental mathematical concepts at a post high school preparatory school where students attend for 1 year preparing to enter the United States Military Academy (USMA). This approach was implemented in a Precalculus course to determine if it would improve students’ long-term retention. Our goal was to identify an effective classroom strategy that led to student recall of fundamental mathematical concepts through the end of the academic year. The concepts that were considered for long-term retention were 12 concepts identified by USMA’s mathematics department as being fundamental for entering students. These concepts are taught during quarter one of the Precalculus with Introduction to Calculus course at the United States Military Academy Preparatory School. It is expected that students will remember the concepts when they take the post-test 6 months later. Our research shows that spaced recall in the form of quizzing had a statistically significant impact on reducing the forgetting of the fundamental concepts while not adversely effecting performance on current instructional concepts. Additionally, these results persisted across multiple sections of the course taught at different times of the day by six instructors with varying teaching styles and years of teaching experience.

Teaching Real-Life Applications of Exponential Functions to Undergraduate Students Majoring in Science

The paper is dedicated to the problem of optimization of teaching real-life applications of exponential functions at undergraduate level (as a part of Pre-Calculus course). The authors’ original method of using real-life problems for teaching exponential functions is presented. Concrete recommendations for increasing effectiveness of teaching exponential functions are given. The paper contains original courseware which was proved to promote deep understanding of the concept of exponential functions in students, as well as their ability to apply this concept to solving real-life problems. All conclusions and recommendations are based on the results of modern research in education and on the authors’ teaching experience. The effectiveness of presented technique was proven in practical teaching at Voorhees University, South Carolina, USA and at Denmark Technical College, South Carolina, USA.

Sustaining Active Learning in Undergraduate Precalculus: Results and Challenges of a Course Redesign

Addressed toward course coordinators and departmental leaders interested in redesigning undergraduate mathematics courses with a focus on active learning, this report provides an update on progress in redesigning the Precalculus course offered by the Department of Mathematics at the University of Maryland, College Park, following initial indicators of success. We begin by discussing our team’s challenges due to the COVID-19 pandemic and our attempts to navigate these challenges. We then provide results related to student achievement, self-efficacy, and STEM intentions, in addition to addressing the remaining goals of the redesign.

Focusing on foundational Calculus ideas to understand the derivative concept via problem-solving tasks that involve the use of a Dynamic Geometry System

The aim of this paper is to review recently calculus curriculum reforms and research studies that document what types of understanding students develop in their precalculus courses. We argue that it is important to characterize what difficulties students experience to solve tasks that include the use of foundational calculus concepts and to look for possible ways for students to develop a way of reasoning to work on problems that involve variational phenomena. Thus, we identified tasks in which calculus students exhibit limited understanding of essential concepts to approach and solve those tasks. The purpose is to illustrate and discuss how the systematic use of a Dynamic Geometry System (DGS) could provide a set of affordances for students to develop ways of thinking to grasp calculus foundational ideas and to study the derivative concept. Here, we relied on Thurston’s seminal work that emphasizes the relevance for learners to identify, connect, and coordinate different dimensions and meanings (intuitive, symbolic, algorithmic, geometric, physical, and formal) to construct, understand, and apply the concept of derivative.

Concepts Most Successfully Learned and Most Challenging to Students in a Redesigned Pre-Calculus Course

Pre-Calculus is considered one of many gate-keeper courses for STEM college students (Viera Jr et al., 2019). To better support students in Pre-Calculus, various college-level institutions have redesigned their Pre-Calculus courses to address learning challenges and inequities faced by students. For instance, Jones Lanaghan (2021) redesigned the Pre-Calculus course to incorporate a standards-based grading system, active learning practices among students, and emphasis on developing a growth mindset. Students in their redesigned Pre-Calculus course passed the class at a higher rate than those enrolled in the traditional Pre-Calculus course, with a difference of 38.3% higher. Additionally, students in the redesigned Pre-Calculus course passed their Calculus I course at a higher rate, with a difference of 3.1% higher. Despite the overall improvement and encouraging outcomes to the course, further learning challenges still need to be addressed. This study seeks to answer the following research questions: 1) What concepts are most successfully learned by students in a redesigned Pre-Calculus course? 2) What concepts are most challenging to students in a redesigned Pre-Calculus course? Data from 59 students enrolled in all sections of a redesigned Pre-Calculus course in Spring 2023 were analyzed quantitatively, using descriptive statistics. Discussion, limitations, and future research for teaching will also be shared.

Revisiting Past Assignments: Do Precalculus Students Learn More from Revising or Discussing Their Work?

In introductory math courses such as precalculus, students often have difficulty with assignments that require explanations or synthesizing knowledge. When students receive low scores on such assignments, instructors often wish to help students understand the material they struggled with. A common strategy is to allow the students to revise and resubmit the assignment. In a study of a precalculus course, I investigate the effects on learning and confidence of two strategies: allowing homework revisions and offering extra points for meeting with the instructor to discuss the assignment. The data indicate that discussions appeared to have the greatest effect on quiz scores and confidence, although both strategies were beneficial when compared to using neither strategy. In addition, this article looks at student perceptions of the strategies.

Current Approaches for Engaging Parents in Students’ Academic Performance: Basis for Parental Engagement Policy

This action research aims to determine the current approaches that the school has employed to enhancing parental engagement on the academic performance of the students in the new normal. The participants were of the selected 35 high school teacher, which contains 21 Junior High School teachers and 14 Senior High School teacher, from Valeriano E Fugoso Memorial High School at Boys Town Complex Parang, Marikina City. Parental engagement at school approaches includes volunteering, conferencing with teachers, attending workshops, involvement at school decision making. Also, the parental engagement in the student’s education at home approaches includes helping on homework, communicating encouragement and educational expectation. Results displayed that these approaches enabled parental engagement. Mostly, optimistic comment information from the parents were stimulated and promoted better insight of their child’s performance in school. A significant effect on the academic performance of the learner in PreCalculus course towards the engagement of the parents, whether at home or at school in the new normal. Similarly, parents became more involved both at home and school activities because of the effect of the several approaches which have improved performance in PreCalculus.

Increasing the Open Education Resources Capacity of Precalculus Courses at York College and Queensborough Community College

This paper presents a case study conducted by two universities seeking to explore Open Education Resources (OER) in their precalculus course. Students not only gained access to their textbook for free on the first day of class, but also Lumen OHM, an online mathematics assessment platform. The majority of students involved in the study were satisfied with the textbook, characterizing it as useful, accessible, and affordable, and agreeing that they would be very likely to register for a future course with an online text like that used in the course. This study also found that when comparing OER to non-OER settings using the final exam scores, for the most part, there was no significant difference observed in the learning outcomes. Based on these findings, the researchers believe that there is no need to continue teaching with costly physical textbooks when students can achieve the same academic level results with OER.

Communication skills enabled in a pre-calculus course using dynamic geometry software

This article is the result of a research project that aimed to characterize the communication skills developed by students who solved problems of seven workshops related to the notions of variation and change within the framework of a pre-calculus laboratory course mediated by interactive mathematical software at a Colombian public university. Data analysis was performed considering the characterization of the skills of interpretation, explanation, and justification, which are fundamental for the communication process of variational thinking. Here we show evidence of the identification of communicative skills manifested by students when solving problems involving the use of the notions of variation, tendency, and approximation.

Conceptual Maps: A Tool for Assessing the Meaningful Learning of Engineering Students in a Pre-Calculus Course

Context: The difficulty that students of Differential and Integral Calculus, in Engineering programs, show when dealing with the concept of function, one of the fundamental concepts of higher Mathematics, with countless applications in courses of the aforementioned programs. Objectives: To evaluate the contribution of a potentially meaningful teaching unit to promote learning of first-degree polynomial, exponential and logarithmic functions, and to evaluate Engineering students’ learning, through conceptual maps. Design: A potentially meaningful teaching unit of concepts related to first-degree polynomial, exponential and logarithmic functions was applied. The research is based on the Meaningful Learning theory. Scenario and participants: Five Pre-calculus students from Engineering programs at a community university in Rio Grande do Sul. Data collection and analysis: Data collection took place through initial and final assessment instruments, in addition to questionnaires and conceptual maps, during the students' participation in an extracurricular activity, with analysis of the conceptual maps based on an adaptation of Cañas and co-workers’ Topological Taxonomy. Results: The results showed that the adopted methodology has the potential to promote meaningful learning in the context of Engineering Education and that conceptual maps are reliable tools for assessing learning. Conclusion: it is possible to claim, that the first-degree polynomial, exponential and logarithmic functions, once worked in a contextualized way, help in the understanding of different concepts and contribute to the occurrence of a meaningful learning, whenever there are learning conditions for such, which requires considering the students' prior knowledge, in addition to their interest in learning.

An On-Campus Approach to Online Mathematics Teaching: A Case Study on a Pre-Calculus Course

<p style="text-align: justify;">Learning in STEM subjects is to a high degree based upon understanding logic, especially in subjects like mathematics. It has always been challenging to preserve the benefits of on-campus teaching and learning while digitalizing the teaching of mathematics. In this article an approach to design for a suitable online pre-calculus course is discussed, that aims to address the challenges. The main focus will be on student active learning in synchronous online environments, technical teaching methods in lectures, and pre-planning of the course. The final exam in the course was held as a closed-book proctored exam on-campus with pen and paper, providing data on comparisons of the final exam scores with the exam from the previous year, in which the entire course was held on-campus. The results indicate a positive effect from the presented design. Also, student surveys indicated high student satisfaction.</p>

The role of leadership in educational innovation: a comparison of two mathematics departments’ initiation, implementation, and sustainment of active learning

Several studies have shown that the use of active learning strategies can help improve student success and persistence in STEM-related fields. Despite this, widespread adoption of active learning strategies is not yet a reality as institutional change can be difficult to enact. Accordingly, it is important to understand how departments in institutions of higher education can initiate and sustain meaningful change. We use interview data collected from two institutions to examine how leaders at two universities contributed to the initiation, implementation, and sustainability of active learning in undergraduate calculus and precalculus courses. At each institution, we spoke to 27 stakeholders involved in changes (including administrators, department chairs, course coordinators, instructors, and students). Our results show that the success of these changes rested on the ability of leaders to stimulate significant cultural shifts within the mathematics department. We use communities of transformation theory and the four-frame model of organization change in STEM departments in order to better understand how leaders enabled such cultural shifts. Our study highlights actions leaders may take to support efforts at improving education by normalizing the use of active learning strategies and provides potential reasons for the efficacy of such actions. These results underscore the importance of establishing flexible, distributed leadership models that attend to the cultural and operational norms of a department. Such results may inform leaders at other institutions looking to improve education in their STEM departments.

Undergraduate Students’ Perceptions of Features of Active Learning Models for Teaching and Learning to Teach Mathematics

The recent push toward active learning – engaging students in the learning process – is meant to benefit students. Yet there is still much to learn about students’ perceptions of this phenomenon. We share results from an interview study of students’ perceptions of features of two active learning models institutionalized at a large doctoral-granting university – a model for teaching foundational mathematics courses and an early field experience model for teaching preservice secondary students to teach mathematics. These models were implemented simultaneously in a single precalculus course. Interviews were conducted with both student populations (i.e., precalculus students and preservice teachers) to understand which in-class features of the models students noticed and identified as beneficial to their learning. Precalculus students identified specific opportunities related to active learning in the undergraduate mathematics teaching model – working in groups on mathematics tasks that engaged students in sensemaking and interacting with their instructor around mathematics. Preservice teachers identified specific opportunities related to three features of the university field experience model – observing a mathematics instructor enacting ambitious instructional practices, planning and teaching a “real” lesson, and observing student thinking and practicing teaching moves during groupwork. We conclude with pedagogical recommendations about particular features of the models that may help mitigate student resistance to active learning.

What in the World Just Happened? Students’ Self-Regulation in Asynchronous PreCalculus During the COVID-19 Pandemic

We describe students’ learning practices in an online asynchronous PreCalculus course during Fall 2020, the first complete semester of distance learning induced by the COVID-19 pandemic. Results were compiled using a thematic analysis of a questionnaire administered to 43 students enrolled in PreCalculus at a university in the Midwest (United States). Students were given opportunities for active learning and various synchronous Q&A sessions, yet they primarily learned through watching videos and reading worked examples, minimizing interactions with the instructor and available tutors. The questionnaire results show that students knew active learning was helpful, but they were unable to curtail unproductive learning practices. The questionnaire also showed that students struggled to stay motivated and keep to a schedule. We conclude that by developing their study techniques and self-regulatory habits, students will be able to take more control over their learning, particularly in asynchronous classes.

A Faculty Learning Community Implements Research-Based Curriculum and Pedagogy to Redesign Precalculus

: This article reports on a reconceptualization of the Precalculus course experience at Florida International University. We discuss the details of the redesign process—unified within a faculty learning community (FLC) model—along with broader change in the course sequence leading up to Calculus. We provide data, including comparison between the treatment and control groups.

Tight connection to success: the college credit plus program

The state mandated College Credit Plus (CCP) program in Ohio has grown significantly for several years. Many general education core courses are being offered to high school students, on-premises or in a partnering university. In mathematics, students can complete pre-calculus courses while in high school. In forming the partnerships with high schools, universities have applied remediation free standards for initial placement and improved efforts to enhance the school classroom learning experience. The partnerships have facilitated a smoother transition from high school to college, and universities have invested additional resources to ensure the success of CCP students. The high school teachers have completed advanced graduate coursework to qualify and become credentialed in the CCP program. The CCP initiative has successfully bridged the gap that existed between K-12 and universities by tightening the collaboration among high school teachers and university faculty. As a result, Kent State University has seen an increase in freshman registration which we posit is due to CCP. Students who follow the CCP program are significantly better prepared for college. Technology has been used to provide a previously inaccessible experience in mathematics for CCP students. Data on the success of the overall CCP program are highlighted in the article.

Logistic Regression in Analyzing the Determinants of University Students' Mathematics Performance

Mathematics provides an effective way of building mental discipline, encourages logical reasoning, and plays a crucial role in understanding the contents of other subjects. Mathematics performance is affected by several factors including grade, social economic status, background and many more. The aim of this study is to investigate the factors which are related to students’ mathematics performance and to identify the significant factors which affect the students’ mathematics performance of MAT133 course. This study was conducted in UiTM Negeri Sembilan, Kuala Pilah campus. 480 students were selected from various programs who had registered Pre-Calculus course (MAT133). This study employed Logistic regression analysis to test the relationship between predictor variables and the dichotomous outcome variable which is categorized as low or high achiever in MAT 133 course. About 55.7% of the students are high achiever who achieved higher than 70% marks in MAT133. The research findings reveal that students’ intake on July (OR = 2.941), male students (OR = 0.315), students who scored A+, A and A- in Modern Mathematics in SPM exam (OR = 0.340), students who had B+ and B- in Additional Mathematics in SPM (OR = 0.512) and assessment marks for MAT133 were found to have a significantly higher possibility to be high-achiever in MAT133. Acquiring strong mathematical background is a priority as a fundamental preparation for students to learn mathematics in higher institution.

A Network Theory Approach to Curriculum Design

In this paper we hypothesize that education, especially at the scale of curriculum, should be treated as a complex system composed of different ideas and concepts which are inherently connected. Therefore, the task of a good teacher lies in elucidating these connections and helping students make their own connections. Such a pedagogy allows students to personalize learning and strive to be ‘creative’ and make meaning out of old ideas. The novel contribution of this work lies in the mathematical approach we undertake to verify our hypothesis. We take the example of a precalculus course curriculum to make our case. We treat textbooks as exemplars of a specific pedagogy and map several texts into networks of isolated (nodes) and interconnected concepts (edges) thereby permitting computations of metrics which have much relevance to the education theorists, teachers and all others involved in the field of education. We contend that network metrics such as average path length, clustering coefficient and degree distribution provide valuable insights to teachers and students about the kind of pedagogy which encourages good teaching and learning.

МЕТОДИКА ПРЕПОДАВАНИЯ КОНЕЧНОЙ МАТЕМАТИКИ В ПЕДАГОГИЧЕСКИХ УНИВЕРСИТЕТАХ В СВЯЗИ С НОВЫМИ ТЕНДЕНЦИЯМИ РАЗВИТИЯ МАТЕМАТИЧЕСКИХ НАУК

There were introduced new methods of the teaching and instruction of the following parts of the Pre-calculus: (1) Binomial Series; (2) Trigonometry; (3) Partial Fractions. The problems introduced in the article for the Pre-Calculus Course in Finite Mathematics was developed by the author. These unabridged problems are developed within the new trends in the evolutions of the novelty of the syllabi in Mathematics due to the development of the Mathematics Sciences / Theory and Applications. These new trends in the Theory and Application of Mathematics Sciences have been added new demands to the newly revised textbooks and corresponding syllabi for the Mathematics Courses taught at the Junior two years Colleges and Pedagogical Universities.These newly developed problems are reflection of the Development of Mathematical and Engineering Sciences to offer great amount of learning conclusion/sequel to those who pursue a bachelor’s degree at the universities of the pedagogical orientation. The problems presented in the article here are developed and restructured in terms of the newly developed techniques to solve the problem in Finite Mathematics and Engineering sciences. Moreover, the techniques offered in the article here are more likely to get utilized in Advanced Engineering Sciences, too, within the content of the problems, which require to obtain finite numerical solutions to the Real Phenomena Natural Problems in Engineering Sciences and Applied Problems in Mathematical Physics.The aim of this present publication is to offer new advanced techniques and instructional strategies to discuss methodology and instructional strategies of the mathematical training of the students at the Pedagogical Universities. Moreover, these new teaching techniques and strategies introduced may be extended to the engineering sciences at the technical universities, too.The results and scientific novelty of the introduced methodology and learning conclusions and sequel of the new knowledge the students at the Pedagogical universities may be benefited from are in the following list of the learning conclusions, presented in the article here. The students of the pedagogical orientation may attain the mastery skills in the following sections of the combinatorics in Finite Mathematics subject:- The n! Combination of n different terms.- Evaluate the expressions with factorials.- Identify that there are -!!( )!nrnr various of combinations of r identical terms in n variations.- Identify and evaluate the combinatorial coefficients from the Binomial Theorem.- Identify and able to build the Pascal’s triangle of the binomial coefficients.- Utilize the Binomial Theorem to expand the binomial formula for any natural powers.- Utilize the Binomial Theorem to obtain the general formula for the n-th term of binomial expansion.- Utilize the Sigma Symbols in the Binomial Theorem for the n-th terms of the binomial expansion. 19- Generate the expansion for the power of the ex, where e is the base of natural logarithmic function y = f(x) = x.Practical significance: the methods of teaching and new teaching strategies offered here in the article alongside with the application of the new trends in the development of mathematical and mathematics education sciences can be useful for prospective and currently practicing teachers of mathematics. Moreover, the materials presented here in this article can be useful for the educational professionals in their professional development plans to improve the quality in education