Observed hierarchy of student proficiency with period, frequency, and angular frequency

In the context of a generic harmonic oscillator, we investigated students’ accuracy in determining the period, frequency, and angular frequency from mathematical and graphical representations. In a series of studies including interviews, free response tests, and multiple choice tests developed in an iterative process, we assessed students in both algebra-based and calculus-based, traditionally instructed university-level introductory physics courses. Using the results, we categorized nine skills necessary for proficiency in determining period, frequency, and angular frequency. Overall results reveal that, postinstruction, proficiency is quite low: only about 20%–40% of students mastered most of the nine skills. Next, we used a semiquantitative, intuitive method to investigate the hierarchical structure of the nine skills. We also employed the more formal item tree analysis method to verify this structure and found that the skills form a multilevel, nonlinear hierarchy, with mastery of some skills being prerequisite for mastery in other skills. Finally, we implemented a targeted, 30-min group-work activity to improve proficiency in these skills and found a 1 standard deviation gain in accuracy. Overall, the results suggest that many students currently lack these essential skills, targeted practice may lead to required mastery, and that the observed hierarchical structure in the skills suggests that instruction should especially attend to the skills lower in the hierarchy.

Figure 1

The nine skill categories needed to determine the period, frequency, and angular frequency from graphical and mathematical representations.

Figure 1a

(Continued)

Figure 2

The proportion of correct responses plotted for each of the nine skill categories by each population of students. Not all students saw questions from each of the nine skills.

Figure 3

A simple, informal hierarchy constructed using cross tabulations. Skills on lower levels are prerequisite to the skills on higher levels. Skills on the same level are not necessarily equivalent. Only algebra-based physics students are included in the figure.

Figure 4

Hierarchy of period, frequency, and angular frequency skills for the algebra-based students obtained using item tree analysis and the DAKS package for R.

Figure 5

Distribution of scores for the tutorial and nontutorial groups on the 28 items relevant to the tutorial. The percentages are based on the students within each treatment.

Figure 6

A comparison between the proportion correct for tutorial students vs nontutorial students for each $T$, $f$, $\omega$ skill, and for four unrelated physics questions.