Isolating approaches: How middle-division physics students coordinate forms and representations in complex algebra

Mathematical reasoning with algebraic and geometric representations is essential for success in upper-division and graduate-level physics courses. Complex algebra requires student to fluently move between algebraic and geometric representations. By designing a task for middle-division physics students to translate a geometric representation to several algebraic representations, we identified students’ facilities and difficulties with complex number algebra. Students were successful when they isolated approaches, which was characterized by students’ use of circle trigonometry, triangle trigonometry, the Pythagorean theorem, and the square of the norm approaches. When students were not successful, they inappropriately mixed approaches to the problem, including failing to distinguish between triangle and circle trigonometry.

Figure 1

Interview prompt given to students in two parts as the third question in the protocol developed for Fall 2015. Part (a) was given immediately prior to and separately from part (b). Both parts were given using the same diagram and equation.

Figure 2

The angle $\varphi$ is defined to be counterclockwise from the positive real axis and is associated with a circle trigonometry approach. The angle $\alpha$ is provided in the interview prompt as the angle from the negative imaginary axis and is used in the triangle trigonometry approach.

Figure 3

Codes are applied to students’ interviews in groupings according to episodes where neither the interviewer nor student referred to the goals in the prompt. Tables 2 and 3 include descriptions of the codes. Correct steps are in black, and incorrect steps are in red. Resulting algebraic representations are given below the box by either rectangular ($R$), polar ($P$), or exponential ($E$) form and by parameters provided by the prompt. Question marks indicate there is no corresponding code for significant parts of solving. Blank boxes indicate no relevant complex algebra manipulations were completed. Red boxes and slashes indicate that coded steps were blended in an incorrect combination. Full transcripts and corresponding coding can be found in Ref. [20]. *Interviewer provided information to the student.