Preface Welcome to th e fift h volum e of Research i n Collegiat e Mathematic s Educatio n (RCME V). This and the four previou s volumes serve purposes similar to those of a journal. Eac h present s reader s wit h peer-reviewe d researc h o n question s regardin g the teachin g an d learnin g o f collegiate mathematics . There i s no w a growin g internationa l communit y o f mathematic s educatio n researchers workin g a t th e post-secondar y level , askin g pertinen t questions , gath - ering an d analyzin g data , an d reflectin g upo n a variety o f pedagogical issues . Lik e mathematics educatio n researc h a t othe r levels , potentia l topic s an d viewpoint s are extremel y broa d an d varied . On e ca n tak e a cognitiv e poin t o f view , askin g how individual learner s come to understand specifi c mathematica l concept s such a s limit, o r one can take a more socio-cultura l stance , examining classroo m situation s to uncover ways, such as the structuring o f whole class and smal l group discussions , that ca n facilitat e learning . Dependin g o n the questio n posed , on e migh t decid e t o conduct a qualitative o r quantitativ e study , o r a combination o f both. Researcher s often begi n wit h a theoretical framework— a "lens " throug h whic h t o analyz e thei r data—and buil d o n th e wor k o f others . However , sometime s whe n littl e i s known , an explorator y stud y ma y b e a way o f getting a "fee l fo r th e territory. " Collegiate mathematic s educatio n researc h i s a relativel y youn g field , bu t in - terest i n i t an d it s result s i s growing . Som e thing s ar e known , bu t muc h remain s to b e uncovered . Th e editor s o f RCME ar e please d t o b e par t o f this developmen t as they ai d researcher s i n bringin g suc h studie s befor e a combined audienc e o f col- leagues, bot h fello w researcher s an d mathematician s wit h a n interes t i n teachin g and learning . Overview o f RCME V The volum e begin s wit h a stud y fro m Mexic o o f students ' understanding s o f variable, a concep t tha t pervade s mathematics . I t i s followed b y tw o studie s deal - ing wit h aspect s o f calculu s reform . Th e firs t o f thes e note s tha t calculu s refor m projects hav e use d a variet y o f interactin g innovation s t o achiev e thei r effect s an d tries t o isolat e informatio n o n th e contributio n o f cooperativ e learning , whil e th e second compare s refor m an d traditiona l students ' use s o f calculus i n a subsequen t engineering mechanic s course . Th e nex t articl e report s a study o f Israeli preservic e teachers' intuition s regardin g th e concep t o f actual infinity , usin g Fischbein' s idea s on intuition . Thi s i s followed b y tw o article s tha t concer n APO S theory . Th e firs t summarizes published an d unpublished studie s regarding students' performance an d attitudes i n courses based o n APOS (Action , Process , Object , Schema ) theor y an d the AC E (activities , clas s discussion , exercises ) teachin g cycle . Th e secon d com - pares an d contrast s th e view s o f mathematica l understandin g provide d b y APO S Vll
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