A robust grasp of electromagnetic (EM) plane waves is helpful for studying advanced Optics topics such as reflection, interference, and the wavelength dependence of refractive index.  Although there have been many studies of students' understanding of waves, only a small fraction have been dedicated to plane waves, which are particularly challenging both conceptually and mathematically.  In this study, nine upper-level science/engineering majors were prompted to explain how a provided mathematical representation of an EM plane wave dictated something conceptually planar.  An unexpected initial finding is that students' initial responses always went outside the realm of the prompt: they either focused on aspects of the mathematics that were unrelated to planes, or they relied upon outside information to make the connection to planes.  This revealed difficulty in blending the conceptual and mathematical representations.