A model for developing the multiplication of complex numbers

This study investigated a prospective mathematics teacher's development of the multiplication of complex numbers in Cartesian, vector and polar forms during an instructional sequence involving quantitative reasoning. The teaching experiment methodology was used for the design of the instructional sequence and the conduct of the study. For this, upon completion of the articulation of the understanding of the multiplication of complex numbers by the researcher, a written pre-assessment, also used as the written post-assessment, was given to 28 prospective mathematics teachers. One hour long pre-clinical interviews were conducted with two of the participants. Based on the analysis of the pre-interviews, one of the participants was selected. Teaching experiments included 4 teaching sessions upon completion of which a written post-assessment and also a post-clinical interview was conducted. Results showed that in contrast to the data from the pre-interview, the prospective mathematics teacher was able to develop the multiplication of complex numbers in Cartesian, vector and polar forms in a quantitative structure in relation to each other. Particularly, results showed that she was able to explain the reasoning behind the polar form pointing to one of the complex numbers as operand vector acting on the other one and justify the relationship between the multiplication of complex numbers in three forms. Results also pointed to some difficulties the participant had encountered while developing the reasoning behind the polar form of the multiplication of complex numbers. Results in juxtaposition to each other suggested a model for developing the multiplication of complex numbers quantitatively.