Enhancing First-year Engineering Students' Trigonometry Learning Experience

The College of Engineering at our institute has been part of the NSF sponsored consortium of A National Model for Engineering Mathematics Education for the last 6 years and saw significant retention improvement in all three majors: Civil Engineering, Electrical & Computer Engineering, and Mechanical Engineering programs. Students in those three majors came from a diversified high school math background, ranging from AP calculus to basic Algebra. Students with weak math background have one thing in common: they all struggle with trigonometry, a key engineering skill for success in all three majors. To equip students with necessary trig skills in our first engineering math course taught by engineering professors, we implemented a three step approach in our class: (1) Made a connection between the classroom trigonometry calculations with the robotic welding operations on the automobile assembly line, so that students can actually see real life applications of the mathematic model of a two-link robot they learned in class . This is largely done by showing class the automobile assembly process videos followed by math model analysis of the robot arm movement. This approach draws attention from all students, particularly the mechanical engineering inclined students. (2) Added a new NAO robot (an autonomous, programmable humanoid robot developed by Aldebaran Robotics, a French robotics company) based trigonometry experiment to provide students hands-on experience of interaction with a humanoid robot. During the experiment, students specify joint angles (or hand location coordinates for inverse kinematics) via a user-friendly computer interface, watch the robot move its arm accordingly, and then hear the robot report verbally the final location coordinates of its hand (or joint angles for inverse kinematics). Students also create MATLAB function and script files to cross-check and validate the measurements. All students loved to play with the NAO robot, especially the electrical and computer engineering inclined students. (3) Developing a new surveying experiment to further enhance the trigonometry learning experience. This particular experiment aims to enhance understanding trigonometry applications in civil engineering and construction management field. The students use two equipment set-ups: Leveling and Theodolite. They use leveler to measure slopes and theodolite to measure angles in the horizontal and vertical planes to calculate the building height and/or width. Page 26659.2 Students enthusiastically embraced the new approach with active classroom participation. The student performance data also showed improvement related to the trigonometry skills. Our next step is to expand this approach to other areas where students showed weakness. 1. The issue with trigonometry and the work done at our institution Trigonometry can be easily one of the least favorable math topics among high school and college freshman students due to its complexity and the numerous formulas the students need to memorize. The issue is compounded by the lack of adequate connection between classroom learning and the engineering applications in the real world. This not only causes anxiety among part of the student body but also ill-prepared them for subsequent courses requiring such knowledge. A student can barely pass calculus I and still be lacking some critical skills in handling trigonometry problems in statics and circuits [1] . The engineering application of trigonometry is everywhere, such as in automobile engine crankshaft design, in robotic arm movement programming for assembly line operation, and in land survey result calculation. To promote the study of trigonometry and to acquaint the students with the use and practical application of trigonometry in the surveying profession, California Professional Land Surveyors Association organizes the Trig-Star [2] annual high school mathematics competition based on the practical application of Trigonometry. At our institution, to enhance our first year engineering math education, we joined several years ago a consortium led by Klingbeil [3-4] to equip the engineering freshmen with the required math skills during their first semester by the engineering faculty. All incoming freshmen are required to take this course, which also has a lab component where students have a chance to build circuits, play with air track, operate oscilloscope, and learn to program with MATLAB. This NSF funded project contributed to the significant improvement on student retention rate in our college over the past several years [5] . In the meantime, we also noticed that the student performances among different subjects are not balanced. This is typically not an issue for students getting B or better who have a good understanding on all subjects. However, a barely passing grade of C does not provide the detailed information on their understanding of the subject required to be successful in completing the subsequent courses. For example, a student performing poorly in trigonometry and complex analysis but did reasonably well in the rest of engineering mathematics may still be ill-prepared for Circuit Theory. Among all subjects, trigonometry stands out as a subject many students struggle with. To improve the situation, we implemented a three-step approach: 1. Made a connection between the classroom trigonometry calculations with the robotic welding operations on the automobile assembly line so that they can actually see how the two-link robot classroom math model in real life applications. P ge 26659.3 2. Added a new NAO robot based trigonometry experiment to provide students hands-on experience of interaction with a humanoid robot. 3. Developing a new surveying experiment to further enhance the trigonometry learning experience. To see the impact of this approach, we analyzed student exam performance in trigonometry before and after the implementation of the first two steps. The exam data showed noticeable improvement on student’s trigonometry skills. We will implement the last step next year and report its result in future publications. 2. Introducing trigonometry with exciting real life applications Introduction to Engineering Mathematics is the first math course each engineering student takes during their first semester in our engineering program. Content-wise, it contains pre-calculus and calculus. Most of our students had different topics and ranges of high school math courses, however, most of them are familiar with the topics in pre-calculus. Therefore, the instructors treated those topics as review sessions with an engineering application spin and progress rather quickly, having trigonometry started early as 3 rd week of class. Our adopted textbook introduced trigonometry with the following example of one and two link robot. It is obviously hard to make a direct connection between the example and the actual robot in most students’ mind. Figure 1. Two link robot model Page 26659.4 To get students excited about the subject of trigonometry, we introduced two video clips at the beginning of the class, first with a Star Wars movie clip showing Padmé Amidala and Anakin Skywalker’s adventure in droid factory on alien planet Geonosis where the battle droids being assembled by robots on an assembly line. We then show another video of modern day automobile assembly line where the car body parts being welded together by robots on the assembly line. During the process, we emphasize the connection among the three items and conveyed a clear message to each student: trigonometry is an exciting subject and you will use it in your future engineering career. Figure 2. Two link robot model used in auto assembly line Students received this introduction enthusiastically based on the amount of questions and interaction. From instructor point of view, the introduction provided a convenient reference point in subsequent lecturing on all trigonometry topics. For example, when the Law of Sine and Law of Cosine were discussed, instead of describing it as a method of calculation involving triangle, the instructor would say that in order for the robot to deliver the weld at the precise location on the car body in an auto assembly line, the automation engineer needs to know the angles of the robot arms so he or she can program the robot movement to accomplish the welding task on the auto assembly line. Most of students, especially those mechanical engineering inclined students, who were car enthusiasts and repaired cars themselves, easily made the connection and received this approach positively. 3. A new NAO robot based trigonometry experiment A new lab component which employed a humanoid robot was developed in the summer of 2013 and was integrated into the existing lab project on trigonometry. The current lab project runs for two lab sessions. In the first session, students focus on taking angle-versus-length measurements with a sun-dial-like instrument and calipers. The simple Plexiglas “sun-dial”, shown in Figure 3, P ge 26659.5 simulates a two-link planar robotic arm similar to that shown in Figure 1. Given an angle, students dial it onto the instrument, then measure the x and y lengths; or vice versa. They also create MATLAB function and script files to cross-check and validate the measurements. In session two, a computer-controlled humanoid robot called NAO replaces the “sun-dial”. A NAO T-14 torso model is shown in Figure 4. The T14 model has fourteen Degrees of Freedom including two for the head, five for each arm and one (open/close) for each hand. The robots were purchased with a grant provided by the W. M. Keck Foundation. Figure 3. The Sun-dial like instrument Figure 4. The NAO T-14 model This part of the lab project was focused on the forward and inverse kinematics of NAO’s left arm. In order to resemble the configuration of the “sun-dial”, we only allowed the shoulder roll and elbow roll angles of NAO’s left arm to be varied but