Research i n Collegiat e Mathematics Education . V I

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fl CBMS Conference Boar d of the Mathematical Sciences

Issues in Mathematics Education Volume 1 3

Research i n Collegiat e Mathematics Education . V I

Fernando Hit t Guershon Hare l

Annie Selde n Editors

Shandy Hauk , Production Editor

:jSi!iAr America n Mathematica l Societ y ^ " ^ Providence, Rhod e Islan d £,

in cooperatio n wit h % \ „, „ Mathematica l Associatio n o f America " a ^ ™ ^

Washington, D . C .

http://dx.doi.org/10.1090/cbmath/013

E D I T O R I A L C O M M I T T E E

John Dosse y Solomon Friedber g

Glenda Lappa n W. Jame s Lewi s

2000 Mathematics Subject Classification. P r i m a r y 00-XX , 97-XX .

ISBN-13: 978-0-8218-4243- 0 ISBN-10: 0-8218-4243- 9

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Contents

Preface vi i Fernando Hitt, Guershon Havel, and Annie Selden

An Imag e o f Calculu s Reform : Students ' Experience s of Harvar d Calculu s

Jon R. Star and John P. Smith III

Effects o f Concept-Base d Instructio n o n Calculu s Students ' Acquisition o f Conceptua l Understandin g and Procedura l Skil l 2 7

Kelly K. Chappell

Constructing a Concep t Imag e o f Convergenc e o f Sequence s in th e va n Hiel e Framewor k 6 1

Maria Angeles Navarro and Pedro Perez Carreras

Developing an d Assessin g Specifi c Competencie s in a Firs t Cours e o n Rea l Analysi s 9 9

Niels Gr0nb(Ek and Carl Winsl0w

Introductory Comple x Analysi s a t Tw o Britis h Columbi a Universities: Th e Firs t Wee k - Comple x Number s 13 9

Peter Danenhower

Using Geometr y t o Teac h an d Lear n Linea r Algebr a 17 1 Ghislaine Gueudet-Chartier

VI CONTENTS

Investigating an d Teachin g th e Processe s Use d to Construc t Proof s 19 7

Keith Weber

The Transitio n t o Independen t Graduat e Studie s 23 3 in Mathematic s

Janet Duffin and Adrian Simpson

Preface

Welcome to the sixth volum e o f 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 educatio n a t the collegiate level. I n 199 4 the firs t volum e of RCME appeared , an d volum e 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 whic h ar e intereste d i n improvin g th e teachin g an d learnin g o f mathemat -ics a t 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 specia l attention give n to the implementation o f didactica l situations i n 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 th e research traditio n at the college level is even more recent. I n their articl e about researc h 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 fo r ne w ways of dealing wit h teachin g an d learnin g topic s i n 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

viii P R E F A C E

The firs t pape r i s a n analysi s o f calculu s refor m a s implemente d a t a larg e U.S . university. Th e second , als o conducte d a t a U.S . university , addresse s students ' acquisition o f conceptua l understandin g an d procedura l skill s i n calculus . Th e third, conducte d a t a Spanis h university , deal s wit h students ' constructio n o f a viable concep t imag e linke d t o th e forma l definitio n o f convergenc e fo r sequences . The fourth articl e involves the development an d assessmen t o f student competencie s in a firs t cours e i n rea l analysi s a t a Danis h university . Th e fift h pape r focuse s on th e teachin g an d learnin g o f comple x analysi s a t tw o Canadia n universitie s and use s APO S theor y t o conside r difficultie s student s experienc e whe n shiftin g from on e representatio n o f a proble m t o another . Th e sixt h study , complete d a t a French university , concentrate s o n th e teachin g an d learnin g o f linea r algebr a an d considers whethe r a n emphasi s o n th e constructio n o f geometrica l intuitio n aid s students' understanding s i n generalizin g linea r algebr a concept s fro m R 2 an d M 3

to M n. Th e sevent h pape r present s a n instructiona l approach , carrie d ou t wit h U.S. undergraduates , fo r improvin g students ' performance s i n constructin g grou p theory proofs . Finally , there is a study of mathematics Ph.D . students i n the Unite d Kingdom tha t describe s severa l differen t style s o f learnin g i n mathematic s an d identifies obstacle s tha t student s migh t experienc e whe n adoptin g a particular one .

The Curriculu m an d th e Learnin g an d Teachin g o f Calculus an d Rea l Analysi s

This volum e begin s wit h a stud y b y Jo n Sta r an d Joh n Smit h II I abou t cal -culus refor m a t a larg e U.S . university : "A n Imag e o f Calculu s Reform. " I n thi s paper, Sta r an d Smit h investigat e th e implementatio n o f a calculu s cours e base d on th e Harvar d Consortiu m Calculus . Th e author s explor e wha t happene d whe n the facult y decide d t o transition fro m a large lecture format t o cooperative learnin g in smal l groups . The y documen t ho w th e instructor s taugh t an d ho w th e student s learned. Th e author s conside r th e mathematica l performanc e o f ninetee n student s from severa l points of view, including interviews and student journals. A main claim that emerge s fro m thei r stud y i s tha t "th e 'taught ' curriculu m di d no t diffe r muc h from traditiona l practic e i n calculus. " On e proble m tha t become s apparen t whe n teaching mathematic s i n a cooperativ e learnin g environmen t i s that th e instructo r needs t o b e awar e o f th e importan t rol e tha t eac h membe r o f a smal l grou p ca n play i n th e learnin g process . Also , th e instructo r ofte n mus t pa y attentio n t o th e positions taken by each group and lea d the class to a general consensus - somethin g that i s very difficul t t o arriv e a t i n every session fo r ever y concept . I n addition , th e instructor need s t o develo p a certai n dispositio n t o teac h wit h cooperativ e smal l groups an d shoul d b e prepare d t o resolv e difficultie s tha t migh t aris e durin g thes e sessions. A n instructo r wh o i s unfamilia r wit h thi s metho d woul d probabl y expe -rience difficultie s durin g it s implementatio n and , a s a consequence , affec t students ' performances. Eve n whe n a n instructo r ha s develope d som e experienc e wit h th e use o f cooperativ e smal l groups , thi s doe s no t guarante e th e complet e succes s o f students. Thi s ma y b e s o particularly i f they ar e participatin g i n thi s environmen t for th e firs t time , a s was the cas e i n thi s study .

The secon d paper , writte n b y Kell y Chappel l an d title d "Effect s o f Concept -Based Instructio n o n Calculu s Students ' Acquisitio n o f Conceptua l Understand -ing an d Procedura l Skill, " compare s studen t achievemen t i n two teaching formats . Chappell compares concept-based an d procedure-based instructiona l environments ,

PREFACE IX

and document s students ' calculu s performanc e i n each . Researc h usin g suc h com -parisons i s wel l established . Richar d Skem p (1978 ) i n hi s articl e "Relationa l Un -derstanding an d Instrumenta l Understanding, " pointe d ou t th e importanc e o f an -alyzing th e developmen t o f tw o kind s o f understandin g durin g th e constructio n o f mathematical concepts , referrin g t o the m a s relationa l an d instrumenta l under -standing. Similarly , Hieber t an d Lefevr e (1986 ) i n thei r chapte r "Conceptua l an d Procedural Knowledge, " argue d tha t

Mathematical knowledge , in its fullest sense , includes significant , fundamental relationship s betwee n conceptua l an d procedura l knowledge. Student s ar e no t full y competen t i n mathematic s i f either kin d o f knowledg e i s deficien t o r i f the y bot h hav e bee n acquired bu t remai n separat e entities . Whe n concept s an d pro -cedures ar e no t connected , student s ma y hav e a goo d intuitiv e feel fo r mathematic s bu t no t solv e th e problems , o r the y ma y generate answer s bu t no t understan d wha t the y ar e doing , (p. 9)

In thi s volume , Chappel l focuse s o n these connection s betwee n procedur e an d con -cept i n calculu s learnin g situate d i n technologica l environment s (graphin g calcu -lators an d computer-base d algebr a systems) . Sh e provide s additiona l evidenc e a t the post-secondar y leve l tha t concept-base d instructio n ca n effectivel y foste r th e development o f studen t understandin g withou t sacrificin g skil l proficiency .

In the third pape r "Constructin g a Concept Imag e of Convergence o f Sequences in th e va n Hiel e Framework, " Mari a Angele s Navarr o an d Pedr o Pere z Carrera s discuss students ' difficultie s i n understanding th e limi t concept . The y propos e first teaching thi s concep t fro m a visua l approac h usin g particula r software , the n i n a formal way . Navarr o an d Carrera s fram e thei r stud y i n terms o f concept imag e an d concept definition . A s Tal l an d Vinne r (1981 ) originall y stated ,

We shal l us e th e ter m concept image t o describ e th e tota l cog -nitive structur e tha t i s associate d wit h th e concept , which in -cludes al l th e menta l picture s an d associate d propertie s an d processes. I t i s buil t u p ove r th e year s throug h experience s o f all kinds , changing a s th e individua l meet s ne w stimul i an d ma -tures. (p.152 )

In contrast, a concept definitio n i s the conventional linguistic formulation o f a math-ematical objec t - it s mathematica l if-and-onl y i f definition . Navarr o an d Carrera s tried t o promot e th e developmen t o f a ric h concep t imag e tha t woul d provid e stu -dents wit h a soli d backgroun d fo r constructin g th e concep t definition . I n orde r to understan d students ' performances , the y foun d i t usefu l t o interpre t students ' productions usin g a van Hiel e mode l o f learning. A s the author s poin t out , th e va n Hiele mode l wa s initiall y applie d t o geometri c concept s a t mor e elementar y level s of education ; bu t i n thi s stud y the y appl y i t i n highe r education . Th e va n Hiel e model asserts that th e learne r move s sequentially throug h five levels of understand -ing (se e Teppo , 1991 ) an d provide s a descriptio n o f th e five phases throug h whic h one can hel p student s move , fro m on e leve l to th e next . Th e level s are : Basi c leve l (or Leve l 0) , Visualization ; Leve l 1 , Analysis ; Leve l 2 , Informa l deduction ; Leve l 3, Deduction ; Leve l 4 , Rigor . Mainl y use d t o promot e understandin g i n geometry , this mode l wa s modifie d b y Navarr o an d Carrera s t o examin e learnin g o f th e con -cept o f convergence . I n thei r visua l approac h wit h computers , th e author s use d a somewhat peculia r notation ; instea d o f th e usua l notatio n fo r a sequenc e (n , sn),

X PREFACE

they use d (1/n , sn). Th e authors ' experimenta l result s sho w tha t thi s notatio n did no t imped e th e constructio n o f a forma l definition . Thei r stud y le d the m t o a characterization o f a methodology fo r introducin g th e notio n o f convergence :

• I t shoul d allo w the introductio n o f the concep t withou t requirin g th e stu -dent t o hav e an y kin d o f logica l o r operationa l skill .

• I t shoul d conve y th e essenc e o f th e processe s o f reasonin g o f a n infinit e kind.

• A n identifiabl e hierarch y o f level s o f reasoning mus t b e present . • I t mus t prepar e th e studen t fo r th e "jump " t o th e forma l definition .

In the fourt h paper , Niel s Gr0nbask an d Car l Winsl0w dea l with the importan t issue o f developin g an d assessin g specifi c studen t competencie s i n a first cours e in rea l analysis . Th e articl e demonstrates , i n a practica l way , ho w t o implemen t a series of activities to develop students' understandin g an d asses s student competen -cies. Th e author s followe d th e Frenc h theoretica l structur e o f didacti c engineerin g (see, for example , Artigue , 1988 , 1990) . Didacti c engineering appeare d i n Prance i n the 1980 s as a response t o understandin g th e complexit y o f classroom phenomena . It i s about th e relationships between research an d actio n in the educational system . This researc h methodolog y take s int o accoun t th e theor y o f didactica l situation s in mathematic s o f Gu y Broussea u (1997) . I t consist s of : (a ) a n epistemologica l analysis o f th e mathematica l conten t i n th e stud y (som e author s als o conside r i t important t o analyz e th e historica l developmen t o f th e particula r mathematica l concept), (b ) a n analysi s o f students ' conception s abou t tha t concept , (c ) a n a priori analysi s o f th e didactica l variable s tha t ma y pla y a significan t rol e i n th e study, (d ) th e actua l teachin g experimen t alon g wit h a posteriori analysi s thereof , and (e ) a proces s o f validation . Usin g thi s methodology , Gr0nbas k an d Winsl0 w report o n a priori analyse s o f the curriculum , th e student s i n the sens e o f learnin g in th e curriculum , an d teachin g an d assessing , i n tha t order . I n thei r a posteriori analyses, the y follo w th e revers e order . Finally , th e author s reflec t o n thei r first cycle o f usin g th e didactica l engineerin g approac h t o revis e bot h th e curriculu m and teachin g o f rea l analysis .

Teaching an d Learnin g Comple x Analysi s

Peter Danenhower' s article , "Introductor y Comple x Analysi s a t Tw o Britis h Columbia Universities, " document s som e of the problems students fac e when learn -ing complex analysis . Ver y little previous research has been conducted i n this area . Danenhower complete d a large research stud y and , i n this article , he focuses o n th e introductory topi c o f comple x number s an d thei r representations . W e kno w tha t any representatio n o f a mathematica l objec t onl y partiall y represent s tha t object . Yet, w e expec t student s t o appl y a concep t flexibly i n a variet y o f mathematica l situations, shiftin g fro m on e representatio n t o anothe r a s appropriate . Thus , i t i s important t o understan d ho w student s articulat e representation s t o arriv e a t th e construction o f a mathematica l objec t an d t o understan d ho w the y dea l wit h sev -eral representation s o f th e sam e mathematica l object . Danenhower' s theoretica l approach i s based o n th e Action-Process-Object-Schem a (APOS ) framewor k (see , for example , Asial a e t al. , 1996) . Danenhowe r focuse s o n the depictio n o f problem s when shiftin g fro m on e representatio n t o anothe r an d describe s differen t level s o f understanding i n accordanc e wit h th e APO S framework .

PREFACE XI

Teaching an d Learnin g Linea r Algebr a

The pape r "Usin g Geometr y t o Teac h an d Lear n Linea r Algebra, " writte n b y Ghislaine Gueudet-Chartier , consider s geometrica l intuitio n a s a n ai d t o learnin g linear algebra . Thi s mixed-method s quantitativ e an d qualitativ e stud y ha s a theo-retical framework base d o n Fischbein's work , Intuition in Science and Mathematics (1987). T o buil d a definitio n o f geometrica l intuition , Gueudet-Char t ier use s Fis -chbein's definitio n o f model : " A syste m B represent s a mode l o f syste m A if , o n the basi s of a certain isomorphism , a description o r a solution produce d i n terms of A ma y b e reflecte d consistentl y i n term s o f B an d vic e versa. " I n th e first par t o f her article , th e autho r define s th e meanin g o f geometrica l intuitio n a s "th e us e o f models stemmin g fro m a geometry. " After a revie w o f th e historica l developmen t of linea r algebra , sh e discusse s response s t o a questionnair e abou t geometr y an d linear algebr a give n t o 2 8 Frenc h universit y linea r algebr a teacher s an d present s an examinatio n o f geometr y an d drawin g us e i n linea r algebr a texts . Gueudet -Chartier continue s b y reportin g o n th e result s o f followin g on e universit y linea r algebra cours e an d interviewin g th e teache r an d eigh t students . Th e teache r be -lieved that b y developing a geometrical model when working in R2 an d R 3, student s would generaliz e mor e o r les s easil y t o R n. Indeed , i n th e interview , th e teache r said: "Fo r quadratic forms , al l the phenomena alread y happen i n three-dimensiona l spaces. I t i s necessar y t o understan d ho w t o mov e fro m 2 to 3 . Afte r that , ther e is nothin g new. " Contrar y t o thi s teacher' s belief , th e author' s analysi s reveal s problems tha t student s fac e whe n passin g fro m R 2 an d R 3 t o R n .

Teaching an d Learnin g Proof s

Many researcher s hav e addresse d th e them e o f mathematica l proof , bu t i t i s always interestin g becaus e o f it s complexity . Th e pape r b y Keit h Weber , "Investi -gating and Teachin g the Processe s Used to Construc t Proofs, " consider s this them e in relatio n t o understandin g grou p homomorphisms . Hi s revie w o f th e literatur e asserts thre e cause s o f studen t difficultie s wit h proof . Th e first caus e i s tha t stu -dents ofte n posses s a n inaccurat e conceptio n o f wha t constitute s a mathematica l proof. Th e secon d caus e i s that student s ma y lac k a n understandin g o f a theore m or a concep t an d systematicall y misappl y it . However , eve n i f students hav e accu -rate conception s o f proo f an d th e abilit y t o deriv e logica l inferences , a s necessar y conditions fo r proof-writin g competence , thes e skill s alon e ar e no t sufficient . Th e third caus e i s tha t student s wh o canno t construc t proof s ofte n d o no t hav e effec -tive strategie s fo r doin g so . I n conductin g tw o studies , Webe r addresse d thi s thir d cause o f students ' difficultie s wit h proof s i n th e contex t o f group homomorphisms . He describe s a prescriptiv e procedur e fo r constructin g proof s tha t h e teste d wit h students an d analyze s th e results . Th e hig h percentag e o f student s constructin g valid proof s support s hi s claim tha t learnin g a powerful procedur e t o prov e certai n kinds of statements abou t grou p homomorphisms improve d students ' performance .

Learning Graduat e Mathematic s

The pape r b y Jane t Duffi n an d Adria n Simpson , "Th e Transitio n t o Indepen -dent Graduat e Studie s i n Mathematics, " document s change s i n students ' learnin g styles whe n goin g fro m undergraduat e t o graduat e study . Th e author s conducte d a qualitative stud y i n the Unite d Kingdom , interviewin g thirtee n Ph.D . students i n

Xl l PREFACE

mathematics. The y describ e thei r methodolog y a s a "conversatio n wit h a purpose" to explor e thes e students ' way s o f learning ne w area s o f mathematics . Buildin g o n their literatur e review , th e author s develo p a n expande d versio n o f thei r previou s theory o f learning styles and explor e the changes i n those styles that thei r graduat e student participant s experienced . The y identif y fou r categorie s o f learnin g styles : (a) Natural, wher e a student see s new knowledge connected to old; (b ) Alien, wher e a studen t construct s ne w knowledg e tha t i s separated fro m th e old , bu t tha t con -tinues t o exis t a t graduat e leve l i n a radicall y differen t for m aki n t o mathematica l formalism; (c ) Coherence, wher e a studen t construct s ne w knowledg e fro m old , with a separatio n o f new knowledg e fro m old , bu t wher e th e ne w knowledg e need s to hav e a clea r interna l structure ; and , (d ) Flexible, wher e a studen t construct s a n ability t o adap t he r o r hi s learnin g styl e t o th e perceive d valu e t o b e gaine d fro m the learning . Duffi n an d Simpso n stat e that ,

The transitio n t o independen t graduat e stud y i n mathematic s thus seem s to b e stabl e onl y fo r natura l learners . Alie n learner s predominately respon d i n tw o ways : b y alterin g thei r learnin g style towards a natural stanc e by more clearly seeking analogica l links o r b y developin g a learning styl e (whic h w e had no t previ -ously see n a t othe r levels ) i n which the y retai n th e separatenes s of a ne w are a o f mathematics , bu t see k interna l coherenc e an d structure fo r tha t ne w area .

Finally, th e editors express deep gratitude an d appreciatio n t o Cath y Kesse l fo r her professiona l wor k and the specia l attention sh e gave to the pas t issue s of RCME for which she served as Managing Editor . Cath y did her best t o strengthen authors ' contributions t o th e tw o mos t recentl y publishe d volumes , an d w e than k he r fo r that wonderfu l work . W e ar e als o gratefu l t o Shand y Hauk , wh o ha s generousl y given he r tim e t o tak e o n a simila r rol e a s Productio n Edito r fo r thi s volume . W e wish to acknowledg e al l those wh o have given o f their tim e an d expertis e t o revie w manuscripts fo r thi s an d previou s RCME volumes . Th e productio n o f this volum e has bee n greatl y enhance d du e t o thei r significan t contributions .

Fernando Hit t Guershon Hare l

Annie Selde n

References

Artigue, M. (1988) . Ingenieri e didactique . Recherches en Didactique des Mathe-matiques, 9(3), 281-308 .

Artigue, M . (1990) . Epistemologi e e t didactique . Recherches en Didactique des Mathematiques, 10(2, 3) , 241-286 .

Asiala, M. , Brow n A. , DeVries , D . J. , Dubinsky , E. , Mathews , D. , & Thomas , K. (1996) . A framewor k fo r researc h an d curriculu m developmen t i n un -dergraduate mathematic s education . I n J . Kaput , A . H. Schoenfeld , & E. Dubinsky (Eds.) , Research in collegiate mathematics education. II. (pp . 1-32). Providence , RI : American Mathematica l Society .

Brousseau, G . (1997) . Theory of didactical situations in mathematics. (N . Bal -acheff, M . Cooper , R . Sutherland , & V. Warfield , Eds . an d Trans. ) Dor -drecht, Th e Netherlands : Kluwer .

PREFACE X l l l

Dubinsky, E. , Schoenfeld , A . H. , & Kaput J . (1994) . Preface . Research in col-legiate mathematics education. I. (pp . vii-xi). Providence , RI : America n Mathematical Society .

Fischbein, E . (1987) . Intuition in science and mathematics: An educational approach. Boston : D . Reidel .

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Teppo, A . (1991) . Va n Hiel e level s o f geometri c though t revisited . Mathematics Teacher, 84, 210-221 .

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Research i n Collegiat e Mathematic s Educatio n

Editorial Polic y

The paper s publishe d i n thes e volume s wil l serv e bot h pur e an d applie d pur -poses, contributin g t o the field of research i n collegiate mathematic s educatio n an d informing th e direc t improvemen t o f post-secondary mathematic s instruction . Th e dual purpose s impl y dua l bu t overlappin g audience s an d article s wil l var y i n thei r relationship t o thes e purposes . Th e bes t papers , however , wil l interes t bot h audi -ences an d serv e bot h purposes .

Content. W e invit e paper s reportin g o n researc h tha t addresse s an y an d al l aspects of collegiate mathematics education . Researc h may focus on learning within particular mathematica l domains . I t may be concerned with more general cognitiv e processes such a s problem solving , skil l acquisition, conceptua l development , math -ematical creativity , cognitiv e styles , etc . Researc h report s ma y dea l wit h issue s associated wit h variation s i n teachin g methods , classroo m o r laborator y contexts , or discours e patterns . Mor e broadly , researc h ma y b e concerne d wit h institutiona l arrangements intende d t o suppor t learnin g an d teaching , e.g . curriculu m design , assessment practices , o r strategie s fo r facult y development .

Method. W e expec t an d encourag e a broa d spectru m o f researc h method s ranging fro m traditiona l statistically-oriente d studie s o f populations , o r eve n sur -veys, t o clos e studie s o f individuals , bot h shor t an d lon g term . Empirica l studie s may well be supplemented b y historical, ethnographic, o r theoretical analyse s focus-ing directly on the educational matter a t hand. Theoretica l analyses may illuminat e or otherwis e organiz e empiricall y base d wor k b y th e autho r o r tha t o f others , o r perhaps give specific directio n to future work . I n al l cases, we expect tha t publishe d work will acknowledge an d buil d upon tha t o f others—not necessaril y t o agree with or accep t others ' work , bu t t o take tha t wor k int o accoun t a s par t o f the proces s of building th e integrate d bod y o f reliabl e knowledge , perspective , an d metho d tha t constitutes th e field o f research i n collegiat e mathematic s education .

Review Procedures . Al l papers, includin g invite d submissions , wil l be eval -uated b y a minimum o f three referees , on e of whom wil l be a volume editor . Paper s will be judged o n th e basi s o f their originality , intellectua l quality , readabilit y b y a diverse audience , an d th e exten t t o which they serv e the pure an d applie d purpose s identified earlier .

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Submissions. Paper s o f an y reasonabl e lengt h wil l b e considered , bu t th e likelihood o f acceptanc e wil l b e smalle r fo r ver y larg e manuscripts . Manuscript s should hav e citation s an d bibliographie s accordin g t o th e forma t o f th e America n Psychological Association as described in the fifth edition of the Publication Manual of the American Psychological Association.

Note tha t th e RCME volume s ar e produce d fo r electroni c submissio n t o th e AMS. Accepte d manuscript s shoul d b e prepared usin g AMS-I^T^X an d th e CBM S author package s availabl e fro m th e AM S we b site , www.ams.org . Illustration s should als o b e prepare d i n a for m suitabl e fo r electroni c submissio n (namely , en -capsulated postscrip t files). Additiona l informatio n fo r final formattin g will b e provided upo n acceptanc e o f a work fo r publication .

For further information , se e the Issues in Mathematics Educatio n sectio n of the Conference Boar d o f the Mathematica l Science s we b sit e a t www.cbmsweb.org .

Correspondence. Befor e submittin g a manuscript , sen d a n abstrac t t o on e of the curren t editor s o f RCME:

Fernando Hit t Dere k Holto n Pa t Thompso n ferhitt@yahoo.com dholton@maths.otago.ac.n z pat.thompson@asu.ed u

Subsequent correspondenc e ma y b e wit h th e productio n edito r o r with th e volum e editor wh o ha s bee n assigne d primar y responsibilit y fo r decision s regardin g th e manuscript.

Titles i n Thi s Serie s

13 Fernand o Hitt , Guersho n Harel , an d Anni e Selden , Editors , Researc h i n collegiat e

mathematics education . VI , 200 6

12 Anni e Selden , E d Dubinsky , Guersho n Harel , an d Fernand o Hitt , Editors ,

Research i n collegiat e mathematic s education . V , 200 3

11 Conferenc e Boar d o f th e Mathematica l Sciences , Th e mathematica l educatio n o f

teachers, 200 1

10 Solomo n Friedber g e t al. , Teachin g mathematic s i n college s an d universities : Cas e

studies fo r today' s classroom . Availabl e i n studen t an d facult y editions , 200 1

9 Rober t R e y s an d Jerem y Kilpatrick , Editors , On e field , man y paths : U . S . doctora l

programs i n mathematic s education , 200 1

8 E d Dubinsky , Ala n H . Schoenfeld , an d J i m Kaput , Editors , Researc h i n collegiat e

mathematics education . IV , 200 1

7 Ala n H . Schoenfeld , J i m Kaput , an d E d Dubinsky , Editors , Researc h i n collegiat e

mathematics education . Ill , 199 8

6 J i m Kaput , Ala n H . Schoenfeld , an d E d Dubinsky , Editors , Researc h i n collegiat e

mathematics education . II , 199 6

5 Naom i D . Fisher , Harve y B . Keynes , an d Phil i p D . Wagreich , Editors , Changin g

the culture : Mathematic s educatio n i n th e researc h community , 199 5

4 E d Dubinsky , Ala n H . Schoenfeld , an d J i m Kaput , Editors , Researc h i n collegiat e

mathematics education . I , 199 4

3 Naom i D . Fisher , Harve y B . Keynes , an d Phil i p D . Wagreich , Editors ,

Mathematicians an d educatio n refor m 1990-1991 , 199 3

2 Naom i D . Fisher , Harve y B . Keynes , an d Phil i p D . Wagreich , Editors ,

Mathematicians an d educatio n refor m 1989-1990 , 199 1

1 Naom i D . Fisher , Harve y B . Keynes , an d Phil i p D . Wagreich , Editors , Mathematicians an d educatio n reform : Proceeding s o f th e Jul y 6-8 , 198 8 workshop , 199 0