Teachers’ knowledge of different forms of complex numbers through quantitative reasoning

This study investigated teachers' conceptualizations of different forms of complex numbers in a professional development study (PD) conducted with classroom teaching experiments focusing on quantitative reasoning. We report from five secondary school mathematics teachers' post-interview data and the post-written sessions upon completion of the PD. Results showed that all the participants could relate the formal definition of complex numbers with the roots of quadratic equations both algebraically and geometrically. Participants could also explain the Cartesian and polar form relationship using vectors with connections to the roots of quadratic equations. They further explained the Euler form by pointing out that the polar form of any complex number on a circle determines a function of Θ from R to C. Albeit making connections, results also pointed to some teachers' difficulties in certain points. Altogether, results suggest that quantitative reasoning might lay a foundation for connecting different forms of complex numbers. We discuss implications for curriculum, teaching, and teacher education.