A prospective secondary mathematics teacher's development of the meaning of the cartesian form of complex numbers

In this study, I articulate how a prospective secondary mathematics teacher reconstructs complex numbers upon the set of real numbers in the context of the solution sets of quadratic equations. Previous research has indicated that once asked the meaning of x and y in the Cartesian form of a complex number which is formally defined as x+iy where x and y are real numbers, both students and teachers were able to state that x and y are real numbers, yet considered them separately rather than being components of a single entity. Thus, the question arises as to what x and y refer to algebraically and geometrically; why x and y have to be real numbers and what it means to be an element of the set of complex numbers. This study explicates a prospective secondary mathematics teacher's answers to these questions through the articulation of the participant's quantitative reasoning by considering Sfard's (1991) theory on the dual nature of the mathematical conceptions. With this account, I intend to contribute to mathematics education by providing evidence on how the development of the elements of complex numbers, which is through shrinking/stretching of the distance(s) between the roots and the x-coordinate of the vertex of any quadratic functions' graph, affords conceptualizing any complex number as a single entity in a well-defined set rather than only an algebraic prescription of certain operations. As the result of the instructional sequence in this study, the participant presents this well-defined set as the set consisting of the roots of quadratic equations with real coefficients. This reconstruction of complex numbers regarding the algebraic and geometric aspects of the quadratic equation's roots also explained why quadratic equations have to have conjugate roots once they have at least one complex root.