Semester Calculus Research Articles

A STUDY OF WOMEN ENGINEERING STUDENTS AND TIME TO COMPLETION OF FIRST-YEAR REQUIRED COURSES AT TEXAS A&M UNIVERSITY

This paper reports fi ndings on gender that were part of a larger study reviewing time to completion of course work that includes the fi rst two semesters of calculus, chemistry, and physics, which are often considered the stumbling points or “barrier courses” to an engineering baccalaureate degree. Texas A&M University terms these courses core body of knowledge (CBK), and statistical analysis was conducted on two cohorts of fi rst-year enrolling engineering students at the institution. Findings indicate that gender is statistically signifi cantly related to completion of CBK with female engineering students completing required courses faster than males at the .01 level (p = 0.008). Statistical signifi cance for gender and ethnicity was found between white male and white female students at the .01 level (p = 0.008). Descriptive analysis indicated that of the fi ve majors studied (chemical, civil, computer, electrical, and mechanical engineering), women completed CBK faster than men, and African American and Hispanic women completed CBK faster than males of the same ethnicity.

Simple thinking using complex math vs. complex thinking using simple math—A study using model eliciting activities to compare students' abilities in standardized tests to their modelling abilities

Traditional mathematics assessments often fail to identify students who can powerfully and effectively apply mathematics to real-world problems, and many students who excel on traditional assessments often struggle to implement their mathematical knowledge in real-world settings (Lesh & Sriraman, 2005a). This study employs multi-tier design-based research methodologies to explore this phenomenon from a models and modeling perspective. At the researcher level, a Model Eliciting Activity MEA) was developed as a means to measure student performance on a complex real-world task. Student performance data on this activity and on traditional pre- and post-tests were collected from approximately 200 students enrolled in a second semester calculus course in the Science and Engineering department of the University of Southern Denmark during the winter of 2005. The researchers then used the student solutions to the MEA to develop tools for capturing and assessing the strengths and weaknesses of the mathematical models present in these solutions. Performance on the MEA, pre- and post-test were then analyzed both quantitatively and qualitatively to identify trends in the subgroups corresponding to those described by lesh and Sriraman.

CAN GOOD QUESTIONS AND PEER DISCUSSION IMPROVE CALCULUS INSTRUCTION?

ABSTRACT Preliminary report of the results of a project1 to introduce peer instruction into a multi-section first semester calculus course taught largely by novice instructors. This paper summarizes the instructional approaches instructors chose to use, and the subsequent results of student performance on common exams throughout the course of the term. 1 Support for the Good Questions project was provided by the National Science Foundation's Course, Curriculum, and Laboratory Improvement Program under grant DUE-0231154. Opinions expressed are those of the authors and not necessarily those of the Foundation.

The Effects of Non‐CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta‐Analysis

Forty‐two studies comparing students with access to graphing calculators during instruction to students who did not have access to graphing calculators during instruction are the subject of this meta‐analysis. The results on the achievement and attitude levels of students are presented. The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.

HOW HIGH? HOW FAST? HOW LONG? MODELING WATER ROCKET FLIGHT WITH CALCULUS

ABSTRACT We describe an easy and fun project using water rockets to demonstrate applications of single variable calculus concepts. We provide procedures and a supplies list for launching and videotaping a water rocket flight to provide the experimental data. Because of factors such as fuel expulsion and wind effects, the water rocket does not follow the parabolic model of a textbook projectile, so instead we develop a one-variable height vs. time polynomial model by interpolating observed data points. We then analyze this model using methods suitable to a first semester calculus course. We include a list of questions and partial solutions for our project in which students use calculus techniques to find quantities not apparent from direct observation. We also include a list of websites and other resources to complement and extend this project.

TRANSITION ISSUES THAT REFORM CALCULUS STUDENTS EXPERIENCE IN TRADITIONAL SECOND SEMESTER CALCULUS

ABSTRACT The present study, consisting of an original study and a replication study, tracks the performance of the following two groups of students in traditionally-taught second semester calculus: students who took reform first semester calculus and students who took traditionally-taught first semester calculus. As evidenced by first semester calculus course grades that were based on common assessments, the reform students were better prepared to advance to second semester calculus. In second semester calculus, the course grades of the two groups of students were not significantly different, but the second semester withdrawal rates were consistently higher for the students who had taken reform first semester calculus. In order to identify and explore the issues that these reform students faced as they transitioned into traditionally-taught second semester calculus, they were interviewed via focus groups. This article documents the major themes of the interviews and illustrates major themes with student quotations.

EFFECTS OF CONCEPT-BASED INSTRUCTION ON STUDENTS' CONCEPTUAL UNDERSTANDING AND PROCEDURAL KNOWLEDGE OF CALCULUS

ABSTRACT An original study, involving 305 college-level calculus students and 8 instructors, and its replication study, conducted at the same university and involving 303 college-level calculus students and 8 instructors, investigated the effects of instructional environment (concept-based vs. procedure-based) on students' conceptual understanding and procedural knowledge of calculus. Multiple achievement measures were used to determine the degree to which students from different instructional environments had mastered the concepts and the procedures inherent to first semester calculus. Achievement measures included two skills examinations designed to evaluate procedural competence and a midterm examination and a final examination both designed to evaluate procedural skills as well as conceptual understanding. The learning environment resulted in no significant differences in the students' abilities to employ procedural skills, as measured by the skills examinations. The students enrolled in the concept-based learning environment scored significantly higher than the students enrolled in the procedure-based learning environment on assessments that measured conceptual understanding as well as procedural skills (p < .001). The results of the replication study are consistent with the results of the original study, increasing the generalizability of the results. These results provide post-secondary level evidence that concept-based instructional programs can effectively foster the development of student understanding without sacrificing skill proficiency.

On the Work to Fill a Water Tank

a second semester calculus class, a problem arose in which one had to compute the work required to fill a cylindrical water tank from a water source below the tank through a filler pipe at the bottom of the tank (Figure 1). See [1, p. 426] for a similar problem. The typical solution (Figure 2) is to divide the water in the tank into (infinitely) thin horizontal slices and compute the work to lift each slice of water into place (as though it was a solid). particular, the work to lift a slice of water of thickness dy to the level y units above the base of the tank is (wn R2 dy)(y + a), where w is the weight density of water. Therefore, the total work required is fh0(wrr R2(y + a)) dy. After we did the problem, a student asked essentially: In computing the work to lift the slice of water, we said that the force was the weight of the slice of water. Isn't it true that as the tank fills up, the force at the top of the filler pipe becomes greater which means that it takes more work to fill slices of the tank as the water level rises? I am sure that he was not the first person to have raised this question. My explanations why eorem 1, we assumed that g( )/x is bound d near 0. This assumption rules , for example, g(x) = -Jx. Th orem 1 is ctu lly a special case o the following e general theorem. The proof s similar to that given before.

Dimensional analysis: an elegant technique for facilitating the teaching of mathematical modelling

Dimension analysis is promoted as a technique that promotes better understanding of the role of units and dimensions in mathematical modelling problems. The authors' student base consists of undergraduate students from the Science and Engineering Faculties who generally have one or two semesters of calculus and some linear algebra as part of their curriculum. Because of ‘In Service Training’ which is an integral part of their education, they have a reasonable understanding of the link between theory and practice in their particular industry, but manipulating mathematical formulae is not necessarily a strong point. Dimensional analysis involves both dimensionless products and linear algebra and, because of the latter, this branch of mathematical modelling was, until recently, beyond the reach of most undergraduates. However, it has been found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates. This note illustrates the concept of dimensional analysis by examining the simple pendulum problem and shows how dimensionless products can lead to the discovery of the connection between the period of the pendulum swing and its length. Dimensional analysis is shown to lead to interesting systems of linear equations to solve, and can point the way to more quantitative analysis, and two student problems are discussed. It is the authors' experience that dimensional analysis broadens a student's viewpoint to include units and dimensions as an integral part of any physical problem. With this approach coupled with a computer algebra systems such as DERIVE, students can concentrate on understanding the model and the modelling process rather than the solution technique. Finally, it has been observed that students find dimensional analysis fun to do.

THE COMPLEAT MODELER APPLICATIONS OF CALCULUS IN MEDICINE AND BIOLOGY AN MATC COURSE AT DARTMOUTH COLLEGE

ABSTRACT This paper describes a successful course in mathematical biology at Dartmouth College. The course targets premedical students and biology majors, rather than mathematics majors, and requires only one semester of calculus as prerequisite. Real world problems form the basis of student work.

A heuristic formula for d/dx [f(x)g(x)

In this note the author suggests a heuristic formula for the derivative of functions of the form f(x)g(x). This formula provides an alternative way to find derivatives of functions of such a form. It has been successfully taught to a number of first semester calculus classes. The author invites calculus instructors to use this formula in their classes.

Success and Diversity: The Transition through First-Year Calculus in the University

With the current trend away from affirmative action programs, it is increasingly important that universities retain the African American and Latino students that they admit. In 1988 the University of Texas at Austin (UT) established the Emerging Scholars Program (ESP) to challenge high-achieving students to excel in calculus and pursue mathematics, science, or engineering (MSE) majors. The university makes a special effort to recruit African Americans, Latinos, and women, but the ESP is open to any student, regardless of race or ethnicity, who is committed to excel in calculus. We analyze the influence of calculus performance on choosing a mathematics, science, or engineering major by focusing on variations by race, ethnicity, and gender and on the role of students' participation in the ESP. Findings suggest that ESP students earn higher calculus grades than non-ESP students and are more likely to enroll in the second semester of calculus. Also, earning higher calculus grades increases the likelihood that...

INTERDISCIPLINARY POPULATION PROJECTS IN A FIRST SEMESTER CALCULUS COURSE

ABSTRACT We discuss how applications from microbiology and sociology can be used to involve students and give a physical context for critical concepts in a first semester calculus course. Using data they collect themselves (from bacteria or yeast grown in a laboratory, for example), students develop elementary models of population growth in interdisciplinary class projects. Using the tools of calculus, demographic software, and available technology (CAS or graphing calculator), students then proceed from “microcosm to macrocosm,” moving from biology to sociology, to address questions of human population projections.

The Geometric Series in Calculus

(1998). The Geometric Series in Calculus. The American Mathematical Monthly: Vol. 105, No. 1, pp. 36-40.

A NEW LOOK AT THE CLASSIC BOX PROBLEM

ABSTRACT The classic box problem has long been an integral part of the typical first semester calculus course. A slight deviation from the standard problem constraints provides an interesting and non-trivial exercise which can be presented to and readily appreciated by students at a Calculus I level. The approach emphasizes the importance of recognizing patterns, evaluating the “ideal” and the “practical” solution, and estimating error.

TWO FAMILIAR THEOREMS: MODIFIED SO THEIR PROOFS ARE COMPREHENSIBLE TO FIRST SEMESTER CALCULUS STUDENTS

ABSTRACT Error estimates for tangent line approximations and for numerical integration are found using special cases of the error formulas for Taylor's Theorem and the Trapezoidal Rule, respectively. Proofs of these theorems rely on a modification of Rolle's Theorem. By recasting these theorems, their proofs are more accessible to first semester calculus students. The advantages are several. First, students are able to estimate errors in approximations. Second, to take full advantage of the results, students need to use a graphing calculator or computer algebra system. Third, students are introduced to two theorems which use the same proof-technique. Fourth, students become familiar with one of calculus' big-name theorems.

SPLINES AND ROLLER COASTERS A CALCULUS PROJECT USING MAPLE

ABSTRACT The use of a computer algebra system has been advocated as a means of enriching the calculus curriculum. In practice it is not always easy to develop projects which are both stimulating and accessible. This paper describes a set of linked projects which involve students in mathematical modeling without a heavy science “front end”, and which combine in such a way as to create a first semester calculus experience that is, in part, project based.

BRINGING TAYLOR POLYNOMIALS TO THE FOREFRONT OF DIFFERENTIAL EQUATIONS

ABSTRACT Taylor polynomials ordinarily are introduced towards the end of second semester calculus. In this note we propose introducing them very early in the course, as an extension of linear approximations, and allowing them to weave various threads of differential calculus into a coherent fabric. Calculators and computers operating on these polynomials allow the interplay of geometrical, numerical and algebraic aspects of problems to reinforce the concepts. An example motivating the need for mathematical theory is also discussed.

INCREASING STUDENTS' CONCEPTUAL UNDERSTANDING OF FIRST SEMESTER CALCULUS THROUGH WRITING

ABSTRACT Since calculus is highly symbolic in nature, students often try to get through calculus by manipulating the symbols without understanding the meaning of such symbols (i.e. having a procedural but not a conceptual understanding of the topics in calculus). A variety of writing assignments were implemented in one section of a traditional first semester calculus course in an attempt to increase the students' understanding of the ideas, procedures, and concepts of calculus. This paper outlines and describes the philosophy, structure, and outcomes of the course and discusses findings and implications. An appendix of sample writing assignments follows.

SUPPORT SYSTEMS IN BEGINNING CALCULUS

ABSTRACT The authors introduced voluntary study sessions, structured broadly on collaborative learning principles, in a first semester calculus course, and analyzed student study patterns. Instructors were provided with detailed guidelines as to how to interact with students. We report on differences which emerged among students, particularly pertaining to prior calculus experience and gender.