Investigating student understanding of simple harmonic motion

This study aimed to investigate students’ understanding and develop instructional material on a topic of simple harmonic motion. Participants were 60 students taking a course on vibrations and wave and 46 students taking a course on Physics 2 and 28 students taking a course on Fundamental Physics 2 on the 2nd semester of an academic year 2016. A 16-question conceptual test and tutorial activities had been developed from previous research findings and evaluated by three physics experts in teaching mechanics before using in a real classroom. Data collection included both qualitative and quantitative methods. Item analysis and whole-test analysis were determined from student responses in the conceptual test. As results, most students had misconceptions about restoring force and they had problems connecting mathematical solutions to real motions, especially phase angle. Moreover, they had problems with interpreting mechanical energy from graphs and diagrams of the motion. These results were used to develop effective instructional materials to enhance student abilities in understanding simple harmonic motion in term of multiple representations.


Introduction
To develop an effective physics instruction, information about student dominated misconceptions and difficulties are essential. Students with misconceptions intended to not change their way of thinking after a traditional instruction [1]. Simple harmonic motion (SHM) is an oscillation subjected to a linear elastic restoring force, and is sinusoidal in time and demonstrates a single resonant frequency. Previous studies found that most students have difficulties in relating concepts with graphical representation, connecting the physics of harmonic motion with the mathematics of differential equations [1,2] and understanding fundamental concepts of angular frequency and energy [2][3][4]. This study aimed to investigate student understanding on a topic of simple harmonic motion (SHM) and then to develop instructional materials based on student difficulties.

Participants
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Comparison between pre-and post-test scores
From all four analyses, both groups did better after the instruction. The average post-test scores of both groups slightly increased. From analysis results of pair samples t-test, average normalized gain and effective size, the pre-and post-test scores difference did important to a certain extent. Interestingly, the 3 rd group did significantly better than the 2 nd group. These results suggested that tutorial activities did help students improve their conceptual understanding in SHM.

Student misconceptions and difficulties
Results of student misconceptions were qualitatively analysis from their initial written answers in the tutorial sheets. Here are a few dominated misconceptions.
Periodic motion versus simple harmonic motion. SHM is a special case of periodic motion where the restoring force is linearly dependent on displacement, which is a sinusoidal function of time.
However, most students thought that SHM is a periodic motion. From question 1 in the first tutorial, students answered that the triangle function represents SHM and explained that this restoring force makes an object oscillating around an equilibrium position. Most students did understand this concept better after the instruction from increasing correct responses in question 6. Difficulties with restoring force. Many students did not realize that the restoring force in SHM has to be a linear function with the displacement. On question 2 in the tutorial, they answered incorrectly that the motion with a restoring force 3 F kx = − could be considered SHM. From question 2, students had difficulties with plotting restoring force as a function of displacement. Incorrect intuition relating frequency with amplitude. Many students thought that amplitude depends on frequency or period. This misconception has been reported in previous studies [2][3][4]. They tended to think that some energy has to be used in order to have larger amplitude, so frequency has to be lower [2] or the period has to be larger. On question 5, most students chose incorrect answers (choice d) indicating this misconception. Difficulties with defining an equilibrium position. Both groups also performed poorly on question 7 and 8, which focused on a concept of an equilibrium position. When a mass-spring system is place on an inclined plane, many students did not realize that a new equilibrium position can be located to easily solve an equation of motion. Difficulties with interpreting graphs. Many students had difficulties in relating displacement-time, velocity-time and acceleration-time graph. On question 16, most students could not correctly identify a correct velocity-time graph from given information. Moreover, most students did not fully understand meaning of phase and could not plot a graph with correct phase.

Conclusions
In this study, the SHM-CS and the SHM tutorial were developed and implemented to identify student conceptual understanding. Undergraduate student misconceptions in SHM were identified. The dominated one is "incorrect intuition relating frequency with amplitude," which was found in previous studies [1,2]. Moreover, most students had difficulties in understanding restoring force, representing SHM in a graphical form and in defining an equilibrium position. The first draft of tutorial activities were addressed most of these misconceptions and difficulties. As a result, students did fairly better on the post SHM-CS. Students should have at least 2 hours in working with the tutorial in order to better develop well-rounded understanding in SHM concepts.