Precalculus Course Research Articles

Learning through teaching: An online setting

A powerful way for students to master a subject is to engage them in teaching it to someone. This is especially true for mastering concepts in mathematics. In a traditional classroom we can give markers to students and send them to the whiteboard to solve problems. In an online course the electronic whiteboard is not quite as friendly. Though recent innovations make it possible to "write mathematics" on the online electronic whiteboard, the process is quite time-consuming. In our online pre-calculus course we supply a student with a PDF file containing the solution to a problem and ask the student to create a YouTube video in which they play the role of instructor. They are required to explain the steps involved in solving the problem and why each particular step "makes sense." We then provide a critique of their video and require them, if needed, to supply a revised video. The revised videos are made available online for the entire class. In our poster session we present three student videos, the corresponding instructor feedback, and the revised videos.

Using a MOOC Format for a Precalculus Course

We describe recent attempts by the University System of Georgia to develop and pilot a system-wide online Precalculus course for credit. The course was based on full online delivery, using an emporium model for learning. We experienced moderate success with the course, with a completion rate of 61–76% compared with the 4–7% for Massive Open Online Courses reported by studies in the literature. Persistence by students, as measured by logins and content viewed, correlated to student success. Based on student surveys, there is a significant disconnect between student perception of their learning and their performance in the course. We experienced several technical challenges with the course platform and responded to student requests for more video support of the text.

Spaced Retrieval Practice Increases College Students’ Short- and Long-Term Retention of Mathematics Knowledge

A major challenge college students face is retaining the knowledge they acquire in their classes, especially in cumulative disciplines such as engineering, where ultimate success depends on long-term retention of foundational content. Cognitive psychologists have recently recommended various techniques educators might use to increase retention. One technique (spaced retrieval practice) involves extending opportunities to retrieve course content beyond a customarily short temporal window following initial learning. Confirming the technique’s utility requires demonstrating that it increases retention in real classroom settings, with commonly encountered educational content, and that gains endure into subsequent semesters. We manipulated spaced versus massed retrieval practice in a precalculus course for engineering students and followed a subset of students who proceeded into a calculus class the following semester. Spacing versus massing was manipulated within- and between-subjects. Within-subjects, students retained spaced content better than massed content in the precalculus course. Between-subjects, students for whom some retrieval practice was spaced, compared to those for whom all practice was massed, performed better on the final exam in the precalculus class and on exams in the calculus class. These findings suggest that spaced retrieval practice can have a meaningful, long-lasting impact on educational outcomes.

Placement Model for First-Time Freshmen in Calculus I (Math 131): University of Northern Colorado

Two approaches, Linear Discriminant Analysis, and Logistic Regression are used and compared to predict success or failure for first-time freshmen in the first calculus course at a medium-sized public, 4-year institution prior to Fall registration. The predictor variables are high school GPA, the number, and GPA’s of college prep mathematics courses taken in grades 9 to 12, ACT math scores, and the score on a calculus readiness test. First-time freshmen who are predicted to fail are advised to take a precalculus course prior to attempting the first calculus course. Using a prediction model for 2012 based on data from 2010 and 2011, 73.9% of students were classified correctly as either passing or failing Calculus I. Of students predicted to fail, 77% did in fact fail which was almost three times as high as the failure rate for students predicted to pass. The study also found that both precollege achievement factors, (specifically high school math GPA), and a placement test (the calculus readiness test) were significant predictors of success in Calculus I.

Gender Differences in Achievement in an Inquiry-Based Learning Precalculus Course

The authors conducted a two-semester quasi-experimental study in which each author taught a traditional lecture-based section of precalculus and a section using an inquiry-based approach called a Modified Moore Method in which the students worked through and presented the course material. A common final exam was used to compare student achievement. The results were compared for the overall population and by each instructor. Gender proved to be an important variable with the females performing significantly better in the Modified Moore Method sections than their counterparts in the traditional sections while there were no significant differences for the males.

Exploring student perceptions, learning outcome and gender differences in a flipped mathematics course

AbstractThe flipped classroom approach has recently gained prominence in education. However, a review of previous studies shows that the relationship associated with gender difference, student perceptions and learning outcomes has still remained unexplored, and there has been little discussion regarding flipped classroom environment. To fill this gap, this study aimed to provide a further study by developing an empirical study, extending perspectives of research for flipped classroom in education. This study was conducted to respond and investigate two main student perceptions in a flipped precalculus course, namely, situational interest and course satisfaction. By the self‐developed perception measures, situational interest contained feeling, value and topic interest (three factors), while course satisfaction contained course design, system quality, course arrangement and online assessment (four factors). To respectively determine factors on final grades, we assessed the predicting power among those factors. Students' feedback and gender differences were also evaluated to provide a holistic profile of this flipped course. Results showed that feelings predict the final grades in males, while course design predicts the final grades in females. Moreover, the result also showed that even if females and males showed different topic interest in this course, they performed equally well. Some suggestions to effectively implement a flipped course were also provided from students' feedback. The implications of the results were provided for instructors' guidance in implementing flipped classroom. Finally, the study concluded that students' perceptions may be considered as motivational strategies in teaching and learning process to involve students in academic activities for improving their grades in flipped course.

Student and Instructor-centered Approaches to Teaching Precalculus

This paper describes the results of a two-semester-long study of the effects of student-centered instruction on Precalculus courses. We also describe our teaching approaches centering around students, which include a mixture of lectures, student presentations, group work, discussion, and guided investigations. Students were taught with either the traditional, and still dominant, lecture-based, instructor-centered teaching methods, or with the collaborative student-centered methods. We found that the average over all students was higher on quizzes when they were taught with student-centered methods. Moreover, the students in the lower quantiles scored significantly higher than their counterparts in the traditional lecture-based classrooms. This suggests that not only are student-centered approaches more effective than instructor-centered approaches, but that they are particularly helpful to those students who are struggling with Precalculus.

A quantitative analysis of the relationship between an online homework system and student achievement in pre-calculus

Supporting student success in entry-level mathematics courses at the undergraduate level has and continues to be a challenge. Recently we have seen an increased reliance on technological supports including software to supplement more traditional in-class instruction. In this paper, we explore the effects on student performance of the use of a computer software program to supplement instruction in an entry-level mathematics course at the undergraduate level, specifically, a pre-calculus course. Relying on data from multiple sections of the course over various semesters, we compare student performance in those classes utilizing the software against those in which it was not used. Quantitative analysis of the data then leads us to conclusions about the effectiveness of the software as well as recommendations for future iterations of the course and others like it.

The impact of taking a college pre-calculus course on students’ college calculus performance

Poor performance on placement exams keeps many US students who pursue a STEM (science, technology, engineering, mathematics) career from enrolling directly in college calculus. Instead, they must take a pre-calculus course that aims to better prepare them for later calculus coursework. In the USA, enrollment in pre-calculus courses in two- and four-year colleges continues to grow, and these courses are well-populated with students who already took pre-calculus in high school. We examine student performance in college calculus, using regression discontinuity to estimate the effects of taking college pre-calculus or not, in a national US sample of 5507 students at 132 institutions. We find that students who take college pre-calculus do not earn higher calculus grades.

English

The Internet is gradually becoming part and parcel of our lives. As a result, online educational technologies are gaining more acceptance in our educational system. This is more so in higher education. However, online assessment is one aspect of online technologies that has taken time to be accepted by educators. This is primarily due to concerns related to reliability, validity and security of online assessment. This paper presents the results of a study that investigates students' perceptions of online homework that was recently introduced in our precalculus courses, as compared to conventional paper and pencil homework. Furthermore, we share some qualitative data we observed that should be useful to all mathematics educators, especially those planning to implement online homework. Some important issues related specifically to MyMathLab, which is the platform we are currently using for online homework, together with the major problems students faced in getting accustomed to the online environment will be discussed.

Enhancing Learning through Visualization - Bringing Data and Equations to Life

We demonstrate how Mathematica, a powerful visualization and computational tool available to all students faculty and staff, is being used to visually interpret data sets and create "manipulations" that allow students to visualize, in real-time, the effect of modifying a parameter. Examples will be drawn from a variety of disciplines, including literature, history, psychology, and mathematics: we will (1) visualize character location in Shakespeare's plays, (2) graphically display details of categorical interconnections in the elements of a set, such as US presidents who were vice presidents, or who were vice presidents and governors, (3) work with a "digital" Rorschach Test. Currently, we use Mathematica as the central tool for collaboration by students in an online precalculus course. Demonstrations that model specific course topics are assigned to groups of students. The groups are tasked with "explaining" the demonstration. For example, a demonstration might display a sine curve with "sliders" controlling the amplitude, frequency, and phase shift. Students can move the sliders, but can't see the values assigned. They must determine what values are being assigned to the constants in order to obtain the specified graphs. Simple demonstrations are easy to make and thousands of open source demonstrations are available at demonstrations.wolfram.com.

Influences of Teaching Approaches and Class Size on Undergraduate Mathematical Learning

An issue for many mathematics departments is the success rate of precalculus students. In an effort to increase the success rate, this quantitative study investigated how class size and teaching approach influenced student achievement and students' attitudes towards learning mathematics. Students' achievement and their attitudes toward learning mathematics were compared across four treatments of a precalculus course. The four treatments were (a) traditional lecture-based structure, (b) traditional lecture-based structure with a reduced class size, (c) instruction that engaged students in problem solving, and (d) instruction that included opportunities for small collaborative groups. The achievement of students engaged in problem-based learning (PBL) was significantly higher than the other treatments. These findings suggest that undergraduates benefit from instruction that encourages reflection on prior knowledge while developing new ideas through problem solving. Surprisingly, students in the PBE treatment did not continue to outperform students in the other treatments in calculus. These findings suggest the need for longitudinal studies that investigate the long-term effect of teaching approach and small class size on student learning and student success in advanced mathematics courses.

The Student Mathematics Portfolio: Value Added to Student Preparation?

This article describes key elements for educators to successfully implement a student mathematics portfolio in an undergraduate mathematics course. This article offers practical advice for implementing a student mathematics portfolio in a freshman precalculus course. We look at the potential value added to student class preparation and compare our approach to traditional classroom assessments of the portfolio. The article provides anecdotal and student self-efficacy evidence that student mathematics portfolios lead to increased student performance in the course.

Using Student-Made Games to Learn Mathematics

First-year university students design and play their own games, including board, computer, and other kinds of games, to learn mathematical concepts and practice procedures for their pre-calculus and calculus courses.

Building an Understanding of Functions: A Series of Activities for Pre-Calculus

Building block toys can be used to illustrate various concepts connected with functions including graphs and rates of change of linear and exponential functions, piecewise functions, and composition of functions. Five brief activities suitable for a pre-calculus course are described.

On Turning to Market Timing: A Binomial Approximation

Utilizing simple binomial modeling techniques, I consider the potential impact of technical analysis on expected returns, expected volatility, and higher moment risk. While limited as a practical evaluation tool, this approach allows for relatively intuitive observations relevant to the ongoing discourse regarding the efficacy of market timing. I demonstrate the impact that timeframe reference can have on the perception of risk, showing that market timing can resemble an option strategy in the short-run while it is more accurately described as an allocation shift away from the risky asset. Absent the presence of serial autocorrelation and over any time horizon, this allocation shift is highly inefficient as it can magnify volatility (measured as variance or standard deviation) relative to a return equivalent buy-and-hold allocation to the risky asset and provides inferior downside protection as well. However, market-timing's shortcomings as a down-side-protection strategy are likely to be masked over shorter observations periods, particularly if the analysis omits an apples-to-apple comparison to a return equivalent buy-and-hold strategy. Serial autocorrelation improves the outlook for market timing in terms of return, volatility and down-side risk, but the level required to reach a risk-adjusted break-even is likely to be higher than has been present in the empirical record. I utilize techniques which are admittedly elementary in order to develop and document the key observations in way that is hopefully more accessible to a specific target audience - students in early statistics / pre-calculus courses with an interest in continued studies in finance and investing and investors with less exposure to more formal conventions of notation. Advanced students/practitioners will find this treatment a bit unorthodox, but hopefully interesting nonetheless.

Using the Scientific Method to Motivate Biology Students to Study Precalculus

During the last two years we have developed a precalculus course customized around biology by using the scientific method as a framework to engage and motivate biology students. Historically, the precalculus and calculus courses required for the Suffolk County Community College biology curriculum were designed using examples from the physical sciences with little direct relation to biology. The biology-related mathematics in the curriculum was limited to that presented in biology courses. Favorable results from the biology department spreadsheet and web page LICIL project to increase math awareness led us to our current NSF CCLI project to develop an alternative approach to teaching mathematics to biology students. We selected the scientific method because it is taught at the beginning of every introductory biology course and is also the first fundamental level at which mathematics and biology are connected. In this article, we describe the scientific method approach to introducing precalculus, the sequence of topics covered in precalculus and the introductory biology course, and the challenges and successes of implementing this new course where mathematics topics are presented from the standpoint of a scientific experiment. Examples of the computer lab component of the course where Excel is used for data organization and analysis are provided along with how the scientific method can be used in precalculus to make predictive models that have direct relevance to the biology curriculum.

What's My Math Course Got to Do with Biology?

At United States Military Academy, a unit on biological modeling applications forms the culminating component of the first semester core mathematics course for freshmen. The course emphasizes the use of problem-solving strategies and modeling to solve complex and ill-defined problems. Topic areas include functions and their shapes, data fitting, elementary matrix and vector algebra, difference equations, and an introduction to differential calculus. The course includes many topics covered in a traditional precalculus course and in many ways resembles what other colleges often refer to as a math for liberal arts course. Each biological “application” is designed to be investigated in two class periods along with some out-of-class reading. Within each application, students have the opportunity to see the mathematics they have learned in action. In this article we describe a module that was not only well-received by our students, but, in our opinion, helped our students make connections between science and mathematics while enhancing and reviewing the mathematical skills they had developed during the semester. This module investigates an application of Leslie matrices to address age-structured population problems. The module includes a group activity that gives students a feel for how the Leslie model might be applied in the “real” world.

David's Understanding of Functions and Periodicity

This is a study of David, a senior enrolled in a high school precalculus course. David's understandings of functions and periodicity was explored, through clinical interviews and contextualized through classroom observations. Although David's precalculus class was traditional, his understanding of periodic functions was unconventional. David engaged in sense making behaviors even though these behaviors were not encouraged or explicitly taught. A careful analysis of his work revealed that David's understandings of functions, function notation, and periodicity were compartmentalized. However, David was able to skirt compartmentalization through flexibility in problem solving, translation between representations, and transfer of mathematical information from one representation to another.

What are We Developing? A Case Study of a College Mathematics Program

Over one‐third of all college mathematics enrollments are in courses considered to be developmental. While such courses have been the subject of a large body of research, one question that seems not to have been studied empirically is the alignment of the content of developmental and college level mathematics courses. This paper gives the results of such a study, conducted at a medium sized public liberal arts university. While the content of the developmental mathematics courses was used subsequently in the Precalculus course, most of the content of the Intermediate Algebra course was not used in other college level mathematics courses. It is hoped that the results of this paper will serve as a catalyst for other institutions to examine carefully and define the mission of both their developmental and college level mathematics courses.