Systematic study of student understanding of the relationships between the directions of force, velocity, and acceleration in one dimension

We developed an instrument to systematically investigate student conceptual understanding of the relationships between the directions of net force, velocity, and acceleration in one dimension and report on data collected on the final version of the instrument from over 650 students. Unlike previous work, we simultaneously studied all six possible conditional relations between force, velocity, and acceleration in order to obtain a coherent picture of student understanding of the relations between all three concepts. We present a variety of evidence demonstrating the validity and reliability of the instrument. An analysis of student responses from three different course levels revealed three main findings. First, a significant fraction of students chose “partially correct” responses, and from pre- to post-test, many students moved from “misconception” to partially correct responses, or from partially correct to fully correct responses. Second, there were asymmetries in responding to conditional relations. For example, students answered questions of the form “Given the velocity, what can be inferred about the net force?” differently than converse questions “Given the net force, what can be inferred about the velocity?” Third, there was evidence of hierarchies in student responses, suggesting, for example, that understanding the relation between velocity and acceleration is necessary for understanding the relation between velocity and force, but the converse is not true. Finally, we briefly discuss how these findings might be applied to instruction.

Figure 1

Summary of FVA scores. The graph on the left shows a trend of increasing post-test score with increasing class level for four different calculus mechanics courses ranging from first-year introductory to second-year physics majors. The graph on the right shows pre- and post-test scores for the two different honors introductory mechanics courses.

Figure 2

Percentage of students in the Standard Mechanics course who responded using only one model—either correct, cannot-be-zero, cannot-be-opposite, or misconception—for 3, 4, 5, or 6 out of the 6 questions of the same question type—$\overrightarrow{v}\to \overrightarrow{F}$, $\overrightarrow{F}\to \overrightarrow{v}$, or $\overrightarrow{a}\to \overrightarrow{v}$—on the “multiple context” tests. Note that a majority of students used only one model for 5 or 6 out of 6 questions.

Figure 3

Mean student response percentages for all six conditional relation question types and for three course levels. Black: standard calculus mechanics, $N=228$; white: honors for engineers calculus mechanics, $N=86$; gray: second-year physics majors, $N=65$. Error bars are $\pm 1$ standard error.

Figure 4

Within-student, pretest versus post-test response choice percentages for the three lowest scoring question types. Responses were categorized as either Correct, Part. Correct (partially correct response), or Miscon. (misconceptionlike response). The data presented are from the honors Calculus Mechanics course for engineers, $N=230$, and the first-year honors physics majors Calculus Mechanics course, $N=49$. Note that roughly half of the students did not change their answer from pre- to post-test—these students are represented in the three diagonal columns from back left to front right of each graph. Also, note that a little less than half of students improved by moving into or out of a partially correct response, or directly from the misconception response to the fully correct response—these students are represented in the three columns behind and to the right of the diagonal columns—and roughly 10% of students answered less correctly from pre- to post-test—represented in the three columns to the front and left of the diagonal. (A few examples from the honors for engineers $\overrightarrow{v}\to \overrightarrow{F}$ plot: 23% of students responded with a misconception on the pretest and a misconception on the post-test. 15% of students responded with a misconception on the pretest and a Part. Correct, partially correct response, on the post-test. 8% of students responded with a Part. Correct, partially correct response, on the pretest and a correct response on the post-test.)