An Application-Oriented Course to Improve Student Performance in Mathematics Courses

This is an evidence-based practice paper. Performance in pre-calculus and calculus courses has a strong impact on student success, retention, and graduation in any engineering school. One of the important reasons why students perform poorly in these courses is their failure to make the connection between concepts of mathematics, and engineering problems and applications. Without making this connection, students lose interest in their mathematics courses, resulting in high failure rates. This also strongly affects students’ chances to make satisfactory academic progress within their degree, as mathematics and science courses are usually pre-requisites or corequisites to their core engineering courses. This is a serious problem and must be addressed if students’ retention and graduation rates are to be improved. Starting Fall 2016, New Jersey Institute of Technology (NJIT) is offering a new course: Analytical Methods for Engineering Applications (ENGR101). This is an application-oriented course based on the Wright State University model adopted by over 40 colleges and universities. ENGR101 specifically targets students that are ill-prepared in mathematics based on the performance of a mandatory placement exam that all incoming first-time full-time first-year students take. All students placed in either of the two pre-calculus courses would take ENGR 101 in their first or second semester. Throughout the course, students are introduced to engineering problems and applications that rely on concepts of mathematics. Although this course has only been offered three times thus far, preliminary results are very encouraging. We found that students taking ENGR101 mostly did better in their pre-calculus courses than students who did not take ENGR101. An improvement in performance was also seen in their subsequent-semester mathematics courses. We are committed to offer this course for at least three consecutive years to assess the short-term and long-term effect on students’ academic progress, performance in mathematics courses, and retention and graduation rates. This paper includes information about setting up such a course and the challenges that needed to be met. It also presents the results of our analysis thus far, including a comparison of the performance in mathematics courses of the participants against a control group. Introduction Incoming first year students at our mid-size STEM institution should ideally start in calculus I in the mathematics sequence before they can advance to sophomore-level core engineering courses. However, a high percentage of these students are placed into remedial pre-calculus courses, and do not reach calculus I until their second semester, or even their second year. At our institution, for each student who is placed in a calculus-I course, there is another student who is placed in one of the two pre-calculus courses. This distribution has only slightly improved despite a significant increase in the average student profile in terms of SAT/ACT scores and high school GPA. Moreover, a high percentage of those that are placed into calculus fail or withdraw, and are unable to make timely progress in their major. A key detrimental factor contributing to this is that a majority of the incoming first year students are considered to be underprepared in mathematics. Our university is exploring various options to help these students reach calculus I as soon as possible. Pre-calculus summer boot camp is one of programs successfully implemented at our institution [1]. Other initiatives include: 1) developing sample placement tests for students to practice under the same environment as the original test, 2) making a placement calculator for students to input the scores from the practice placement tests to determine their likely mathematics placement, and 3) establishing a strong outreach to educate students about the impact of their mathematics placement on their engineering curriculum and motivating them to do better on the placement test [2]. Students placed into pre-calculus courses also lose their drive to do well in these courses as they find it difficult to establish a connection between mathematics and engineering. Therefore, they struggle to keep up with the coursework. In addition to a loss in motivation, any delay in entry to calculus I or failure in calculus I is almost automatically equivalent to at least one additional semester of stay at college. More often than not, this leads to students switching to nonengineering majors or leaving the university altogether. It is fine for students to switch majors or leave the institution for the right reasons, but it should not happen for a lack of support or for failing to make the connection between mathematics and engineering courses and to see the big picture. In an attempt to solve this problem, our institution decided to offer an “Engineering 101” introductory course loosely based on the Wright State University (WSU) engineering mathematics education model, starting in Fall 2016. WSU has developed a model with National Science Foundation (NSF) funding to increase student retention and motivation. This model is currently being tested at or has fully been adopted in 40+ engineering schools nationwide [3-4]. The idea is to teach mathematics to incoming first-year students using an application-oriented, hands-on introductory course. This course provides an overview of relevant topics in engineering analytical methods from core sophomore-level engineering courses. These topics are reinforced through extensive examples of their use in lab exercises. Topics include algebraic manipulation of engineering equations; use of trigonometry, vectors and complex numbers, sinusoids and harmonic signals, systems of equations and matrices in engineering applications; and basics of differentiation, and integration. The WSU model has been successfully implemented since 2004. At WSU, every department requiring this course saw an increase in first-year retention in 2004-2005, as compared to baseline data averaged over the prior four years. Overall, WSU saw first-year retention increase from 68.0% to 78.3%. In addition to first-year retention, this model has had a significant impact on student performance in calculus at WSU. Of the students ultimately enrolled in calculus I, 89% of those who had formerly taken this course earned a “C” or better, compared to only 60% of those who had not [5]. The goal of this paper is to analyze the effectiveness of ENGR 101, a similar course offered at NJIT. The following sections discuss the mathematics sequence followed at our university, framework of the course and the analysis and results obtained in detail. The Mathematics Sequence As a standard practice in many four-year colleges, NJIT also requires all incoming first-year students to take a mathematics placement examination. The result of the placement examination is used to gauge the student’s background and competency in various mathematics topics and determines the level of mathematics (calculus I or pre-calculus) the student will begin in his/her first semester. For engineering students, this is particular crucial; as the calculus sequence is a prerequisite to many core engineering courses. Any delay in the completion of calculus I would have drastic impact on the student’s time to graduate. Figure 1 shows the mathematics course sequence followed by engineering students. FIGURE 1THE PRE-CALCULUS COURSE SEQUENCE FOR ENGINEERING STUDENTS Students placed in either of pre-calculus courses (MATH108 or MATH110) are required to take ENGR101. Students placed into MATH111 (calculus I) can choose to take ENGR101, if they want to, but very few actually chose to. MATH108 students taking ENGR101 were offered an opportunity to retake the mathematics placement exam at the end of the semester. Student who did well on the placement exam were able to skip MATH110 and jump to MATH111, thereby saving a course. The new placement test scores only counted if the student successfully completed both MATH108 (C or better grade) and ENGR101 (D or better grade). Course Framework The course, ENGR 101, is a 4-credit course meeting for 90 minutes of lecture two times a week, and 90 minutes of recitation and 90 minutes of lab meetings once a week. The total population of students is divided into a group of 80-90 students per lecture and 25-30 students for both recitation and lab. The course has been significantly revised from the original WSU model to cater specifically to students, who are taking pre-calculus and are one to two terms behind the expected starting point. A tentative course outline for both the lecture and lab portion is shown in Table 1 and 2. TABLE 1TENTATIVE COURSE OUTLINE FOR THE LECTURE PORTION Week 1 Application of Algebra in Engineering – Linear Equations and Quadratic Equations Week 2 Application of Trigonometry in Engineering One and Two-Link Planar Robots Week 3 Introduction to Vectors, Free Body Diagrams Week 4 Exam#1, Introduction to Complex Numbers in Engineering Week 5 Sinusoids and Harmonic Signals in Engineering Week 6 Systems of Equations and Matrices in Engineering Week 7 Introduction to Derivatives in Engineering Application of Derivatives in Dynamics Week 8 Exam #2, Application of Derivatives in Electrical Circuits Week 9 Application of Derivatives in Mechanics of Materials Math 108 Math 110 Math 111 Math 112 Pre-calculus Calculus MATH108 – University Mathematics I MATH111 – Calculus I MATH110 – University Mathematics II MATH112 – Calculus II Week 10 Application of Integrals in Engineering Application of Integrals in Statics Week 11 Application of Integrals in Dynamics