Investigating student reasoning with multivariable expressions in the context of wave propagation  

Student difficulties with multi-variable expressions are well-documented across topics such as electric fields, potential difference, and capacitance. The presented study focuses on exploring these types of difficulties in the context of mechanical wave propagation on water surfaces. We designed a sequence of questions to probe students' thinking about the relationships among wavelength, propagation speed, and frequency as a wave moves between regions of different depths. Our goal is to identify which reasoning elements place the greatest strain on thinking in this context: linking the medium to propagation speed, analyzing frequency when crossing a boundary, or connecting wavelength, speed, and frequency. Because students often rely on intuitively appealing but incorrect ideas (e.g., a wave moving faster in one medium continues to move faster in another), we interpret their reasoning through the Dual Process Theory of Reasoning (DPToR). The results highlight points of tension in students' reasoning about wave propagation.