Students’ conceptual performance on synthesis physics problems with varying mathematical complexity

A body of research on physics problem solving has focused on single-concept problems. In this study we use “synthesis problems” that involve multiple concepts typically taught in different chapters. We use two types of synthesis problems, sequential and simultaneous synthesis tasks. Sequential problems require a consecutive application of fundamental principles, and simultaneous problems require a concurrent application of pertinent concepts. We explore students’ conceptual performance when they solve quantitative synthesis problems with varying mathematical complexity. Conceptual performance refers to the identification, follow-up, and correct application of the pertinent concepts. Mathematical complexity is determined by the type and the number of equations to be manipulated concurrently due to the number of unknowns in each equation. Data were collected from written tasks and individual interviews administered to physics major students ($N=179$) enrolled in a second year mechanics course. The results indicate that mathematical complexity does not impact students’ conceptual performance on the sequential tasks. In contrast, for the simultaneous problems, mathematical complexity negatively influences the students’ conceptual performance. This difference may be explained by the students’ familiarity with and confidence in particular concepts coupled with cognitive load associated with manipulating complex quantitative equations. Another explanation pertains to the type of synthesis problems, either sequential or simultaneous task. The students split the situation presented in the sequential synthesis tasks into segments but treated the situation in the simultaneous synthesis tasks as a single event.

Figure 1

Types of synthesis problems: (a) sequential and (b) simultaneous.

Figure 2

The three versions of the sequential synthesis problems.

Figure 3

The four essential equations for solving the sequential synthesis problems.

Figure 4

Conceptual performance across sequential synthesis tasks with varying mathematical complexity for the (a) spring event and (b) trajectory event.

Figure 5

Overall performance on the three sequential synthesis tasks. Error bars represent standard errors.

Figure 6

Typical interview responses from the three versions of the sequential synthesis task.

Figure 7

Typical students’ physical actions to sequential tasks for the (a) simple, (b) intermediate, and (c) complex version.

Figure 8

Typical student responses for factors cueing energy conservation and projectile motion.

Figure 9

The two versions of the simultaneous synthesis problem.

Figure 10

The three essential equations for solving the simultaneous synthesis problems.

Figure 11

Conceptual performance on simple and complex simultaneous synthesis tasks for (a) energy transfer, (b) translational motion, and (c) rotational motion.

Figure 12

Overall performance on the two simultaneous synthesis tasks. Error bars represent standard errors.

Figure 13

Typical interview responses from the two versions of the simultaneous synthesis task.

Figure 14

Typical student responses for identifying and not identifying the pertinent concepts for the simultaneous synthesis problem.