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

Abstract

In thisstudy,Iarticulatehowaprospectivesecondarymathematicsteacher reconstructs complexnumbersuponthesetofrealnumbersinthecontextofthe solution setsofquadraticequations.Previousresearchhasindicatedthatonceasked the meaningof x and y in theCartesianformofacomplexnumberwhichisformally de ned as x + iy where x and y are realnumbers,bothstudentsandteacherswere able tostatethat x and y are realnumbers,yetconsideredthemseparatelyrather than beingcomponentsofasingleentity.Thus,thequestionarisesastowhat x and y refer toalgebraicallyandgeometrically;why x and y havetoberealnumbersand what itmeanstobeanelementofthesetofcomplexnumbers.Thisstudyexplicates a prospectivesecondarymathematicsteacher'sanswerstothesequestionsthroughthe articulation oftheparticipant'squantitativereasoningbyconsideringSfard's(1991) theory onthedualnatureofthemathematicalconceptions.Withthisaccount,Iintend to contributetomathematicseducationbyprovidingevidenceonhowthedevelopment of theelementsofcomplexnumbers,whichisthroughshrinking/stretchingofthe distance(s) betweentherootsandthex-coordinateofthevertexofanyquadratic functions' graph,a ordsconceptualizinganycomplexnumberasasingleentityina well-de nedsetratherthanonlyanalgebraicprescriptionofcertainoperations.As the resultoftheinstructionalsequenceinthisstudy,theparticipantpresentsthiswell- de ned setasthesetconsistingoftherootsofquadraticequationswithrealcoe cients.