CBMS Issue s i n Mathematic s Educatio n Volume 12 , 200 3 First-year Undergraduates ' Difficultie s i n Workin g wit h Different Use s o f Variabl e Maria Triguero s an d Soni a Ursin i ABSTRACT. Mathematic s course s give n a t universit y leve l requir e a goo d un - derstanding o f algebrai c concepts . I n particular , i t i s essentia l t o b e abl e t o work wit h differen t use s o f variable . I n thi s articl e w e presen t th e result s o f a study whic h investigate d first-yea r undergraduates ' capabilit y t o interpre t an d use variables a s unknowns , genera l number s an d variable s i n simpl e functiona l relationships. Th e result s ar e base d o n a detaile d analysi s o f th e response s given b y 16 4 first-yea r colleg e student s t o a questionnair e o f 6 5 open-ende d items an d o n dat a obtaine d whe n fou r student s wer e interviewed . Th e result s show th e persistenc e o f misconception s an d approache s characteristi c o f alge - bra beginner s i n secondar y school . Afte r severa l course s student s d o no t see m to hav e a soli d understandin g o f th e concep t o f variable . Evidenc e i s provide d suggesting that students ' response s ar e a reaction t o ke y signs present i n an ex - pression (quadrati c exponent , equa l sign) an d tha t student s sho w a tendenc y to rel y o n an d t o appl y memorize d rule s withou t examinin g thei r pertinenc e to a give n context . 1. Introductio n The developmen t o f algebrai c languag e an d it s us e fo r differen t purpose s re - quires th e developmen t o f th e concep t o f variable a s a singl e multifacete d concep t that include s differen t aspects . Researcher s hav e emphasise d th e differen t use s o f variable (e.g. , Philipp , 1992 Schoenfel d & Arcavi , 1988) . I n particular , Usiski n (1988) stresse d tha t differen t use s o f variabl e ar e relate d t o differen t conception s of algebra, fo r example , generalize d arithmetic , proble m solving , stud y o f relation - ships, stud y o f structures. Thes e differen t conception s o f algebr a an d th e differen t uses of variable very often appea r mixe d togethe r i n the teaching of school algebra . Although student s ar e usuall y taugh t ho w t o us e each , ther e i s n o emphasi s o n the particula r aspect s characterizin g eac h us e tha t migh t hel p student s differenti - ate them . I n particular , universit y student s ar e expecte d t o hav e acquire d a soli d and flexibl e handlin g o f variable an d th e abilit y t o distinguish betwee n it s differen t uses an d t o handl e the m i n a n integrate d manner . Ou r concer n i s t o investigat e this assumption . Eve n thoug h i n th e las t twent y year s muc h attentio n ha s bee n brought t o secondar y students ' conception s o f variabl e (e.g. , Bills , 2001 Booth , We want t o thank th e editors and Dr . Kirk Weller fo r their valuabl e comments an d suggestion s that helpe d t o improv e thi s paper . ©2003 America n Mathematica l Societ y 1 http://dx.doi.org/10.1090/cbmath/012/01
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