Precalculus Course Research Articles

Pre-service mathematics teachers using Geogebra to learn about instantaneous rate of change

This study is based on an exploratory teaching experiment, integrating the use of GeoGebra, carried out with Brazilian pre-service mathematics teachers that were attending a Pre-Calculus course. We focus on the contributions of GeoGebra to the learning that the students evidence about the ‘instantaneous rate of change’, in this teaching experiment context. The results show that students benefited from the exploration of tasks using GeoGebra, evidencing significant learning in terms of the instantaneous rate of change, favoured by its multiple dynamic resources, which promote visualization, connections between different representations, and processes of mathematical reasoning associated with this concept. These elements constitute a model that shows the contributions of this software in the Calculus’ learning process. In view of these results, the realization of exploratory tasks using GeoGebra is highlighted as a didactic proposal with potential for teaching Calculus in pre-service mathematics teachers courses, as this experience may encourage them and support their future teaching practices.

How Precalculus Course Coordination Can Impact Students’ Academic Performance

This study aims to measure the impact of course coordination and instructor support to implement a research-based curriculum on students’ academic performance. Our results confirm that a careful curriculum design in addition to a dedicated course coordination can have a positive impact on students’ learning. A reflective teaching philosophy and opportunity to engage in critical conversations about teaching and learning promoted through course coordination can influence classroom practice and result in improved student outcomes. We recommend that departments should recognize the importance of course coordination and encourage faculty to work closely towards the common aim of delivering the best teaching practices.

The MSU SEMINAL Project: Incorporating Principles of Culturally Responsive Teaching in a Pre-Calculus Course

In this paper, we provide a report of preliminary findings of the Morgan State University (MSU) Student Engagement in Mathematics through an Institutional Network for Active Learning (SEMINAL) project and its focus on culturally responsive teaching (CRT) as an active learning framework for pre-calculus instruction. The paper concludes with a summary of wins and challenges faced by MSU’s Mathematics Department. Lessons learned by MSU SEMINAL will help inform other departments on implementing CRT in a pre-calculus course.

Building and Sustaining Success in Precalculus: A Multi-Pronged Approach

Nationwide, many students fail to complete the key mathematics courses that are required for most STEM majors, including Precalculus. This paper describes the rationale, implementation, and impact of the redesign of Precalculus at one regional west coast institution. Prior to the redesign, pass rates in Precalculus were modest (75.8% of all students, calendar years 2009–2014). The redesigned course emphasizes active learning including group work based on conceptual questions and utilizes standards-based grading with a supporting emphasis on growth mindset. In the first three semesters of implementation, 88% of students passed the redesigned course. While the authors focus on the redesign of a Precalculus course, it is our hope that instructors of other core mathematics courses for STEM majors may find information of value in this work.

Uma prática pedagógica no ensino de funções utilizando o GeoGebra em um curso de pré-cálculo híbrido

Este trabalho tem por objetivo apresentar uma proposta de prática pedagógica que foi desenvolvida em um Curso de Pré- Cálculo Híbrido na Universidade Federal do Rio Grande- FURG. Para embasar essa proposta realizamos uma pesquisa nos anais das edições XI e XII do Encontro Nacional de Educação Matemática- ENEM. Optamos pela construção de um mapa teórico para analisar as temáticas que emergiram nas pesquisas sobre práticas educativas de Cálculo Diferencial e Integral (CDI). Para elaboração do mapa foram considerados os artigos referentes aos relatos de experiência e às comunicações científicas que continham a palavra CÁLCULO no título, que retratasse uma atividade pedagógica na área de CDI. No processo de análise dos resumos emergiram duas categorias, cujos temas versaram sobre tecnologias digitais no ensino de Cálculo e resolução de problemas e a história da Matemática como estratégia para ensinar/aprender Cálculo. Após o mapeamento, elaboramos uma atividade utilizando o software GeoGebra online no ensino de funções afim e quadrática para mostrar a ação dos coeficientes em cada uma das funções estudadas. Concluímos que os estudantes conseguiram visualizar com mais dinamismo quando o aplicativo foi utilizado.

Developing Problem-Solving Skills in Active Learning Pre-Calculus Courses

ABSTRACT Historically, math education at the high school and introductory college levels has focused on computational skills. With the advancement of computational technologies, problemsolving and other higher-order thinking skills should become the focal point. This article discusses ways in which the University of Minnesota has integrated problem-solving as an important component in pre-calculus courses.

The Calculus Mastery Exam: A Report on the Use of Gateway-Inspired Assessment Tools at Liberal Arts Colleges

ABSTRACT Gateway testing is an important pedagogic tool employed by many university mathematics departments in calculus and precalculus courses. With a goal of ensuring that students attain needed basic skills in courses with a conceptual “reform” orientation, these tests provide an efficient means of assessing a large volume of student work, while at the same time motivating students to engage in the needed skills-based practice. In this article, we describe our experience over the past decade in utilizing a version of this idea, which we call Mastery Exams, in the context of small liberal arts institutions. We discuss students' experience of autonomous mastery goal attainment inherent to our approach in the context of Self Determination Theory applied by educational psychology researchers in a number of higher education settings. We describe the implementation of these exams in the small-class-size setting and present data from a small study carried out at Washington & Jefferson College in 2012. We found that students in the Mastery Exam group showed significant gains in a pre/post-test of nonroutine calculus tasks compared to the non Mastery Exam group.

Active Learning in an Undergraduate Precalculus Course: Insights from a Course Redesign

Addressed toward course coordinators and departmental leaders interested in redesigning an undergraduate mathematics course, we highlight the steps our team at the University of Maryland, College Park has taken in bringing active learning to our Precalculus course. We provide examples of ways in which we have coordinated active learning across redesigned sections, as well as preliminary results of the effects of active learning on student success. Finally, given the broad set of experiences and backgrounds of our team members, we argue for the unique insight our redesign offers in restructuring a gatekeeper mathematics course to be focused on active learning.

The Reasons and Ways to Implement Service-Learning in a Precalculus Class

Service-learning provides a versatile pedagogical platform that can improve the academic performance of participating students and open space for a discussion of issues of social justice within a mathematics classroom. We describe our Precalculus course with a service-learning component, in which university students regularly tutor algebra prerequisites to students in the community. We provide interested readers with an insight into our instructor experience, thus helping them bring service-learning to a traditional mathematics course. We quantify the resulting gains in the content knowledge of student-tutors and a potential shift in their beliefs about mathematical problem solving. Reflective diaries provide a view of students' mathematical and personal growth. We hope that these manifest benefits may encourage mathematics departments to implement service-learning as part of their academic curricula.

How the Amount and Spacing of Retrieval Practice Affect the Short- and Long-Term Retention of Mathematics Knowledge

Retrieving information from memory increases the likelihood the information will be remembered later. The strategic use of retrieval to enhance memory is known as retrieval practice. Teachers can exert considerable control over students’ retrieval practice, dictating when and how much students practice. Laboratory research has shown that retention benefits from increasing the amount of practice (i.e., the number of times information is retrieved) and from spacing practice out over time. Although retrieval practice is a prominent part of the learning experience in certain educational domains, such as mathematics, relatively little research has examined how retention of actual classroom content is affected by increasing the amount and spacing of retrieval practice. Here, we implemented a complete within-subjects crossing of practice amount (baseline versus increased) and practice spacing (baseline versus increased) in a precalculus course for engineering students. Practice consisted of answering quiz questions. We assessed retention of precalculus knowledge at two educationally relevant time points: the end of the precalculus course (within-semester) and the beginning of a calculus course 4 weeks later (across-semester). Within-semester retention benefited significantly from practicing more and from spacing out practice, although some evidence suggested that the effect of amount of practice was less robust than the effect of spacing. Across-semester retention benefited exclusively from increasing spacing. Given that retaining precalculus knowledge across semesters is crucial for success in higher-level mathematics, these findings support increasing spacing in real-world mathematics education. We discuss how our findings fit within the larger literature on the memory-enhancing effects of retrieval practice.

Can Math Competency Predict Success in Nursing School?

Nursing research into math competency and the impact it has on success in nursing school has produced mixed results. This retrospective comparative study investigated whether math coursework predicted program completion in an associate degree nursing program. Review of student transcripts (n=473) found that how a student performed in calculus and precalculus courses, and a student’s overall total math GPA was associated with program completion. Grades in algebra & trigonometry or statistics courses had no effect on graduation rates.

Competency-Based Learning for Motivation and Academic Performance in a Pre-calculus Course

This research evaluated the effect of the Competency-Based Learning strategy supported by the Web on academic performance and motivation variables, in a Pre-calculus distance course. Moodle was used as the management platform for the course. The participants in the study were students of recent entry to the university in the careers of Engineering in Teleinformatics and Engineering in Works and Services at the University Center of the South Coast of The University of Guadalajara. The Motivated Strategies for Learning Questionnaire (MSLQ) was applied at the beginning and at the end of the instruction to evaluate the significance of the Competency-Based Learning (CBL). Two groups were formed, the experimental group composed of students who participated under the Competency-Based Learning strategy, and the control group, who completed the pre-calculus course in a face-to-face fashion. The data provided in the present study was analyzed using the Statistical Analysis System (SAS), version 9.4. t-tests were performed for independent and paired samples. Parametric Analysis of Variance (ANOVA) was used to evaluate the significance of the academic performance variables and motivation of the students submitted to the CBL strategy. The Motivation variable was evaluated through a pre-test and a post-test. The results showed that the students who participated under the instructional strategy of CBL in the Pre-calculus course obtained higher academic performance than those obtained by their peers in the traditional group. The students who took the course under the CBL showed the same levels of intrinsic motivation throughout the course. The target levels of extrinsic orientation of the students who participated in the experimental group suffered changes in their levels. The results of the analysis supported the hypothesis and showed that the experimental group reached levels of intrinsic motivation superior to those achieved by the participants in the control group.

Flipping the precalculus classroom

ABSTRACTThe flipped classroom is becoming a popular course structure in many academic disciplines, but particularly in STEM disciplines, including mathematics. Considerable research has addressed potential advantages and challenges of teaching a flipped course, as well as examining students' attitudes towards the flipped classroom. Studies on students' academic performance in a flipped classroom remain relatively scarce, and have shown mixed results. This paper reports on a study using a flipped classroom design, combined with interleaved practice, in a 5-credit precalculus course at a regional 4-year university. I evaluated the students' performance on the final exam compared to a similarly sized random sample of students from non-flipped sections of the same course. This evaluation was done by a single-blind student research assistant, to reduce researcher bias. I also tracked students success in the calculus sequence. Since grades in subsequent courses were determined by other instructors, they should also be free of researcher bias. I found the students in the flipped sections performed better on the final, with the difference being statistically significant. I found that the percentage of students completing subsequent calculus courses with a or better was not significantly different between the students coming from the flipped and the non-flipped sections of precalculus.

Increasing Interpersonal Interactions in an Online Course: Does Increased Instructor E-mail Activity and a Voluntary In-Person Meeting Time Facilitate Student Learning?

Distance learning is expanding rapidly in universities. While theoretical and qualitative literature stress the critical role of effective interpersonal interactions in motivating students and facilitating learning in online environments, quantitative evidence on the benefits of increased interpersonal interactions on student learning outcomes is limited. This study examines the effect of providing a voluntary meeting time and increasing instructor e-mail activity on student grades in a fully online Pre-Calculus course at a public university. Student selection into courses was minimal since students only had access to one condition at a time. We further use a propensity score matching strategy to address demographic variations in student characteristics across cohorts. Our results indicate that the increased interpersonal interaction opportunities increased final exam scores by 0.22 standard deviations and improved passing rates by 19 percentage points. The Rosenbaum’s sensitivity analysis indicates that it is unlikely that these results are due to omitted variable bias.

Diagnostic Effects of an Early Mastery Activity in College Algebra and Precalculus

The purpose of this study was to investigate implementation of an early intervention mastery activity during the first two weeks of college algebra and precalculus courses at a large U.S. public university. Statistical modeling of (N = 935) students’ performance in the courses, including a logistic regression model of pass/fail course achievement with students’ high school rank, ACT Mathematics scores, and performance on the intervention as explanatory variables, suggested significant independent differences in course performance across performance levels on the early mastery activity. An evaluation of diagnostic validity for the model yielded a 19% false negative rate (predicted to fail the course, but passed) and a 7% false positive rate (students predicted to pass the course, but failed), suggesting the early mastery activity, when combined with admissions indicators of mathematics readiness, may be useful in better identifying students at risk of failing their first university mathematics course. This strategy, which also yields information for focused intervention efforts, is currently being explored through a campus-wide advising tool at the research site.

The Ambiguity of the Inverse Secant

Defining inverse trigonometric functions involves choosing ranges for the functions. The choices made for the inverse sine, cosine, tangent, and cotangent functions follow generally accepted conventions. However, different authors make different choices when defining y = arcsec x and y = arccsc x for negative x. I first discovered that the definitions of these functions were not a settled convention when I found an alternate definition in Schaum's (Ayers and Mendelson 2012) and Anton's (1995) books. The more commonly used definition is simpler and results in a function more easily evaluated and for that reason is preferable when introducing the inverse trigonometric functions in an algebra or precalculus course. As we shall see, though, the alternate definition of the inverse secant function has many advantages when we move on to calculus. Since we have a choice in our definitions, we should choose what makes the most sense in context.

Incentivizing with Bonus in a College Statistics Course

Many studies have argued the negative effects of external rewards on internal motivation while others assert that external motivation does not necessarily undermine intrinsic motivation. At a private university, students were given the option to earn bonus points for achieving mastery in the online homework systems associated with Statistics and Pre-Calculus courses. The results showed a significant difference in online homework grades and final exam scores, dependent upon when the incentive was given. The findings of this research suggest that college students thrive when incentivized. When compared to the students who were not incentivized, the incentivized group had a statistically significantly higher mean for both online homework scores and final exam scores. Many of the incentivized students chose to take the opportunity to earn the bonus points to increase the final semester grade, which apparently also helped to increase the content knowledge necessary for the final exam.

Notes on Teaching Precalculus to a Blind Student in a College Precalculus Course

Notes on Teaching Precalculus to a Blind Student in a College Precalculus Course

Limiting resources to build classroom community in a flipped pre-service teacher course for pre-service middle school teachers

Students are at risk of struggling academically and socially in the first two semesters following a transfer to a university setting. Helping transfer students in this critical period is complicated by a large variance in students' prerequisite knowledge. A flipped classroom with a large group work component may help students form cohort bonds; however, it is difficult to get groups of students comfortable enough to engage in such discussions. The purpose of this mixed method case study was to investigate how limiting groups' resources to a single smartpen and notebook helped students form a strong classroom community that integrates all students. Sixteen pre-service middle school teachers from seven different transfer institutions were enrolled in a flipped precalculus course using smartpens to record their activities. Findings indicated that the smartpens helped students build stronger cohort bonds and encouraged students to confront their own misconceptions about the material.

Precalculus teachers’ perspectives on using graphing calculators: an example from one curriculum

ABSTRACTGraphing calculators are hand-held technological tools currently used in mathematics classrooms. Teachers’ perspectives on using graphing calculators are important in terms of exploring what teachers think about using such technology in advanced mathematics courses, particularly precalculus courses. A descriptive intrinsic case study was conducted to analyse the perspectives of 11 teachers using graphing calculators with potential Computer Algebra System (CAS) capability while teaching Functions, Statistics, and Trigonometry, a precalculus course for 11th-grade students developed by the University of Chicago School Mathematics Project. Data were collected from multiple sources as part of a curriculum evaluation study conducted during the 2007–2008 school year. Although all teachers were using the same curriculum that integrated CAS into the instructional materials, teachers had mixed views about the technology. Graphing calculator features were used much more than CAS features, with many teachers concerned about the use of CAS because of pressures from external assessments. In addition, several teachers found it overwhelming to learn a new technology at the same time they were learning a new curriculum. The results have implications for curriculum developers and others working with teachers to update curriculum and the use of advanced technologies simultaneously.