Preface Welcome to the sixth volum e of Research in Collegiate Mathematics Education (RCME VI). Th e presen t volume , lik e th e previou s volume s i n thi s series , reflect s the importance of research in mathematics education a t the collegiate level. I n 1994 the first volum e of RCME appeared , an d volume V was published i n 2003 thus, we are commemoratin g mor e tha n a decad e o f RCME wit h volum e VI . I n RCME. I, Ed Dubinsky , Ala n Schoenfeld , an d Ji m Kapu t stated : We hope to serve two audiences. RCME i s for researcher s i n col- legiate mathematic s education , an d fo r th e wide r communit y o f mathematicians wh o ma y b e intereste d i n thes e issue s bot h fo r fundamental intellectua l reason s an d becaus e o f application s t o their instruction , (p . vii) Since tha t first volume , RCME ha s continue d th e ai m o f servin g thes e tw o audi - ences, providin g a bridg e o f communicatio n betwee n tw o academi c communities , both o f which ar e intereste d i n improvin g th e teachin g an d learnin g o f mathemat - ics at th e colleg e level. Researc h i n collegiat e mathematic s educatio n i s a comple x task that require s the study of phenomena from a variety of theoretical perspective s and methodologies, with special attention give n to the implementation o f didactical situations in the classroom. Tha t i s why, since its beginning, RCME ha s publishe d studies base d o n differen t theoretica l approache s t o th e problem s o f learnin g an d teaching mathematic s a t th e colleg e level . Mathematics educatio n i s a relatively ne w discipline and the research traditio n at the college level is even more recent. I n their article about research on undergrad - uate teachin g an d learnin g i n RCME. Ill, Anni e an d Joh n Selde n cite d Debora h Ball's cal l t o action : "W e nee d t o kno w mor e abou t th e kind s o f mathematica l understanding tha t matte r i n teaching , ho w t o hel p teacher s develo p thos e under - standings, an d ho w t o hel p teacher s lear n mathematic s in , an d from , thei r dail y practice." Pro m it s beginnings , RCME ha s addresse d thi s nee d an d ha s becom e an importan t foru m fo r th e publicatio n o f studie s relate d t o th e issue s o f learnin g and teachin g a t th e colleg e level. Jus t a s importantly, i t i s a venue for ne w ways of dealing with teachin g an d learnin g topic s in collegiat e mathematic s practice . Overview o f thi s Volum e The eight papers in this volume come from researcher s in six different countries : the Unite d States , Spain , Denmark , Canada , France , an d th e Unite d Kingdom . They dea l wit h five genera l themes : calculu s an d rea l analysis , comple x analysis , linear algebra, proofs, and the transition to graduate work in mathematics. Th e first four studies , comin g fro m thre e differen t countrie s an d fou r differen t approaches , address curriculu m an d th e learnin g an d teachin g o f calculu s an d rea l analysis . Vll

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