How students process equations in solving quantitative synthesis problems? Role of mathematical complexity in students’ mathematical performance

We examine students’ mathematical performance on quantitative “synthesis problems” with varying mathematical complexity. Synthesis problems are tasks comprising multiple concepts typically taught in different chapters. Mathematical performance refers to the formulation, combination, and simplification of equations. Generally speaking, formulation and combination of equations require conceptual reasoning; simplification of equations requires manipulation of equations as computational tools. Mathematical complexity is operationally defined by the number and the type of equations to be manipulated concurrently due to the number of unknowns in each equation. We use two types of synthesis problems, namely, sequential and simultaneous tasks. Sequential synthesis tasks require a chronological application of pertinent concepts, and simultaneous synthesis tasks require a concurrent application of the pertinent concepts. A total of 179 physics major students from a second year mechanics course participated in the study. Data were collected from written tasks and individual interviews. Results show that mathematical complexity negatively influences the students’ mathematical performance on both types of synthesis problems. However, for the sequential synthesis tasks, it interferes only with the students’ simplification of equations. For the simultaneous synthesis tasks, mathematical complexity additionally impedes the students’ formulation and combination of equations. Several reasons may explain this difference, including the students’ different approaches to the two types of synthesis problems, cognitive load, and the variation of mathematical complexity within each synthesis type.

Figure 1

Two types of synthesis problem: (a) sequential and (b) simultaneous.

Figure 2

The three versions of the sequential synthesis problems.

Figure 3

The four essential equations for tackling the sequential synthesis problems.

Figure 4

The two versions of the simultaneous synthesis problem.

Figure 5

The three essential equations for solving the simultaneous synthesis problems.

Figure 6

Mathematical performance on synthesis tasks with varying mathematical complexity: (a) sequential and (b) simultaneous synthesis task.

Figure 7

Number of students formulating all the fundamental equations, combining the formulated equations and simplifying the combined equations for the sequential and simultaneous synthesis tasks with varying mathematical complexity.

Figure 8

Examples of interview responses for the three versions of the sequential synthesis task.

Figure 9

Examples of interviews and written responses for the complex and simple simultaneous synthesis problems.

Figure 10

Interview responses for students failing to combine the formulated equations, and those failing to simplify the combined equations for the sequential synthesis tasks.

Figure 11

Written solutions of those students who failed to combine their formulated equations for the (a) simple, (b) intermediate, and (c) complex sequential synthesis tasks.

Figure 12

Written solutions of those students who failed to simplify their combined equations for the (a) simple, (b) intermediate, and (c) complex sequential synthesis tasks.

Figure 13

Written solutions of those students combining the formulated equations and simplifying the combined equations for the (a) simple, and (b) complex simultaneous synthesis tasks.