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Children's learning of fractions : a comparison study of user-controlled computer-based learning vs.… Cotter, Dale S. 1990

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C H I L D R E N ' S L E A R N I N G O F F R A C T I O N S A C O M P A R I S O N S T U D Y O F U S E R - C O N T R O L L E D C O M P U T E R - B A S E D L E A R N I N G V S . N O N I N T E R A C T I V E L E A R N I N G E N V I R O N M E N T S B y D a l e S . Cotter B . A . , T h e Un i ve rs i t y o f B r i t i s h C o l u m b i a , 1980 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A R T S in T H E F A C U L T Y O F E D U C A T I O N W e accept this thesis as c o n f o r m i n g to the required standard T h e Un ivers i t y o f B r i t i sh C o l u m b i a September , 1990 © D a l e S . Cotter , 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics and Science Education The University of British Columbia Vancouver, Canada Date October 11, 1990. DE-6 (2/88) Page ii . A B S T R A C T Thesis Supervisor: Dr. Marv Westrom The purpose of this study was to investigate the effectiveness of the software program Visual Fractions in teaching basic fraction concepts and the effect that student control over the construction of fraction diagrams had on their learning. The Visual Fractions program provides a diagram and two fractions in numeric form. The diagram consists of a figure divided into partitions with some of the partitions shaded. One fraction represents the shaded parts of the whole and the other represents the unshaded parts. Students can control the total number of partitions and whether each is shaded. Manipulating the diagram changes the value of the fractions. A Non-interactive (crippled) version of the software was designed to eliminate the user-control aspect of the program. Users of this program could click to generate a new fraction, but had no control over the choice of fraction. The computer randomly generated a new fraction and displayed the corresponding diagram each time. A third treatment, Fraction Flash Cards, was designed to simulate the Noninteractive version of the program, without the computer. The students received Flash Cards containing images of the computer-generated fraction diagrams. The study consisted of a pilot project during which data collection techniques were tested and revised and the main study. Sixty-four subjects were taken from four intact classes of grade four students. The students were randomly assigned to one of the three Treatment Groups or the Control Group. Three different sets of data were collected: a pretest and postest on fractions, structured interviews, and field notes taken by the researcher during the treatment process. Page i i i . In Treatment Group One, students used the Interactive Version of Visual Fractions. Here, students could create fractions at their command. There is evidence to suggest that this type of interactive control is a critical factor in learning (Merrill, 1987). In Treatment Group Two, students used the Noninteractive version of the software. Students could control the rate of observing fractions and fraction diagrams, but not the value of the fraction. Students in Treatment Group Three used the Flash Cards. Motivation appears to strongly affect one's ability to learn and children appear to be highly motivated to use computers. The purpose of this treatment was to control for any achievement gain that may have been due to the novelty of using computers. The four Groups were compared using analysis of variance with repeated measures. Significance at the 0.01 level was found for the tests and the interaction. A study of the interaction showed that there was no significant difference between the gains of the Visual Fractions Noninteractive Group, the Flash Card Group, or the Control Group. However the gain achieved by the Visual Fractions Interactive Group was significant. From this study, it is clear that the Visual Fractions Interactive program which provides students the opportunity to construct fraction diagrams with immediate feedback, is an effective method of teaching fractions. Page iv. Table of Contents Abstract i i Table of Contents i v List of Figures and Tables v i i Acknowledgements v i i i Chapter One: Introduction 1 The P r o b l e m 1 Concrete Manipulat ive A i d s 2 Computer Ass is ted Instruction 2 V i s u a l Fract ions - A User -Cont ro l led Env i ronment 3 Noninteract ive Learn ing Envi ronments 5 Expectat ions of the Study 6 Prob lem Statement 6 Purpose o f the Study 6 N e e d for the Study 7 L imi ta t ions o f the Study 7 Chapter Two: Review of the literature 9 The Deve lopment o f Fract ion Concepts 9 T h e D i f f i cu l t y is Pr imar i l y Conceptual 9 Def in i t ion o f Fractions is C o m p l e x 10 Chi ldren's Construct ion of Mathemat ica l Ideas 13 T h e Connect ion Between Concepts and Procedures 14 The l e a r n i n g Env i ronment is Cr i t i ca l 16 T h e R o l e o f P h y s i c a l M o d e l s 16 Instructional T i m e must be Effect ive 17 T h e R o l e o f Att i tude i n Student's L e a r n i n g 18 Connect ions between Attitude and Ach ievement 19 M a t h Anx ie ty 19 The R o l e o f Mic rocomputers in Educat ion 20 Student Att i tudes towards Mic rocomputers 20 The R o l e o f Intellectual M o d e l s 21 Learner Cont ro l 23 Microcomputers and Mathemat ics Learn ing 25 T h e D e s i g n o f Educat ional Software i s Important 28 Page v. S u m m a r y 29 Chapter Three: Methodology 31 S a m p l i n g 32 T h e P i lo t Study 32 Research Des ign 36 The Research Des ign M o d e l 37 Descr ipt ion o f Data Col lect ion 3 8 The Pretest/Posttest 38 Student Interviews 4 6 Observations o f the Treatment 48 The Treatment 48 V i s u a l Fractions Interactive Group 48 V i s u a l Fractions Noninteract ive G r o u p 49 F l a s h C a r d G r o u p 50 C o n t r o l G r o u p 51 Data Ana lys i s 51 Chapter Four: Results 52 Treatment Session Report 52 V i s u a l Fractions Interactive Treatment 52 V i s u a l Fractions Noninteractive Treatment 53 F lash C a r d Treatment 54 Quantitative Data Col lect ion 56 Pretest/Posttest 56 S u m m a r y Reject ion o f the N u l l Hypothesis 60 Student Interviews 60 S u m m a r y Report o f Interview F ind ings 71 S imi lar i t ies Between Groups 71 Di f ferences Between Groups 72 Chapter Five: Summary, Conclusions and Recommendations 75 D i s c u s s i o n o f Research Results 7 6 Conc lus ions - The V a l u e o f V i s u a l Fractions 77 Recommendat ions for Future Research 78 Page v i . References 79 A p p e n d i x A - Interview Procedure 84 A p p e n d i x B - Parental Consent F o r m 87 A p p e n d i x C - Pretest/Posttest 88 A p p e n d i x D - Sample F lash Card 93 A p p e n d i x E - P i lo t Study Pretest/Posttest 9 4 Page vii . List of Figures and Tables Figure Page F igu re 1.1 E x a m p l e o f Computer Screen Image 4 F igu re 3.1 Example o f Page O n e - P i l o t Study Test 33 F igu re 3.2 M o d e l o f Research Des ign 36 F igu re 3.3 Example Shading Question wi th Large Denominator 39 F igu re 3.4 E x a m p l e N a m i n g Question with Large Denominator 4 0 F igure 3.5 Example Part it ioning Question 41 F igu re 3 .6 Example N a m i n g Quest ion W i t h Unequal Partit ions 42 F igu re 3.7 E x a m p l e Shading Question w i th Equivalent Fractions 43 F igu re 3.8 Example Narn ing Quest ion w i th Equa l Partit ions 4 4 F igu re 4 .1 Compar i son o f M e a n s 59 F igu re 4 .2 Interaction Effects 60 Table Page Tab le 4.1 C o m p a r i s o n o f the F o u r G r o u p s o n the T w o Tests 57 Tab le 4 .2 G r o u p M e a n Scores and Standard Deviat ions 58 Tab le 4.3 Calcu lat ion o f Interaction Effects 59 Page v i i i . Acknowledgments I w o u l d l ike to thank Dr . M a r v W e s t r o m , D r . D o u g O w e n s , D r . Wa l te r B o l d t , K e n Hughes and R o b M a c d u f f fo r their assistance i n the preparation o f this document. I w o u l d also l ike to thank D r . D . Whi t taker for agreeing to be on m y committee. In particular, I a m endebted to Dr . W e s t r o m and R o b M a c d u f f for the p rogramming w o r k they d i d for this study. I w o u l d l ike to thank the administrators w h o a l lowed me to conduct the study at their schools and a l l o f the chi ldren w h o participated i n the project. I w o u l d also l i k e to thank m y husband, M i k e , and m y son, A n t h o n y , and m y mother, Jean Ingram for their support and understanding. F i n a l l y , I w o u l d l i k e to dedicate this thesis to M r s . Mar jo r ie West , without whose support and encouragement this study w o u l d not have been poss ib le . Page 1 Chapter One Introduction The Problem / can't do Math -1 don't understand fractions. T h i s c o m m o n compla in t is shared by m a n y school ch i ld ren and even some adults. T o o often ch i ldren d o not overcome these in i t ia l d i f f icu l t ies and , w i t h conf idence eroded, cont inue to experience further d i f f i cu l ty and frustration i n their study o f mathematics . A c o m m o n f i n d i n g i n m u c h o f the research indicates that these di f f icul t ies are p r imar i l y conceptual (Hector & F randsen , 1985). Ch i ld ren d o not seem to have an adequate understanding o f basic fractions upon w h i c h to base their further study o f f ract ion equivalence and operations. E lementary students tend to operate on a mechanica l or procedural leve l when w o r k i n g w i th fractions rather than a conceptual one (Peck & Jencks 1981). T h e y prefer to apply often- inappropriate rules in a rote fashion without t ry ing to interpret the m e a n i n g o f their answers, rather than t ry ing to f i n d reasonable responses through est imat ion sk i l l s w h i c h represent genuine understanding (Behr , W a c h s m u t h & P o s t , 1985) . Researchers have been studying the p r o b l e m o f chi ldren's learning o f fractions for many years. T h e focus o f debate has cont inual ly shifted between an emphasis on conceptual and procedura l k n o w l e d g e . Current ly , researchers are interested i n t ry ing to i m p r o v e chi ldren's understanding o f f ract ions, but also in the relat ionship between these two types o f learn ing (Hiebert & L e f e v r e , 1986). There are several suggestions in the literature concern ing the issue o f h o w to develop greater understanding o f fractions i n elementary school ch i ld ren . A t what age should chi ldren be f o r m a l l y int roduced to fractions? W h e n should ch i ldren begin their study o f f ract ion Page 2 equiva lence and operations? S h o u l d concrete manipulat ive aids be used and i f so, should they be avai lable for longer periods o f t ime and at the upper elementary grades as w e l l as p r imary? H o w m u c h t ime is needed to develop an adequate concept o f fract ion? A complete understanding o f f ract ion i n v o l v e s several subconcepts (i.e., rat io , d e c i m a l , etc.) . W h e n shou ld each o f these subconcepts be introduced? S h o u l d certain subconcepts such as dec imals be emphas ized over others? If more t ime needs to be spent studying basic fractions concepts, f r o m where w i l l that extra time come? F r o m w h i c h area o f the mathematics c u r r i c u l u m (or other subjects) w i l l this learning t ime be taken? Other relevant issues concern teaching methods, student attitudes, learn ing environment and avai lable resources. These are just a f e w o f the issues w i th w h i c h the literature o n fractions learning i s cont inual ly concerned . Concrete Manipulative Aids T h e literature on the p r o b l e m o f chi ldren's learning o f f ract ions prov ides m a n y recommendat ions for he lp ing ch i ldren to improve their understanding. F o r example , earl ier int roduct ion o f basic f ract ion concepts (i.e., grade three instead o f grade four) , more t ime spent o n basic concepts before m o v i n g o n to the study o f operations and equivalence, more emphasis o n understanding and est imation sk i l l s rather than memor i za t ion o f procedural ru les , and a greater use o f concrete manipulat ive aids, especial ly in the upper elementary grades where these materials are not no rmal l y found. Computer Assisted Instruction M i c r o c o m p u t e r s m a y be an effect ive c lassroom tool for teaching fract ion concepts. E v e n at the kindergarten leve l , m a n y ch i ld ren are us ing computers as a regular part o f their week ly , i f not da i l y routine. E x a c t l y what ro le mic ros p lay i n the c lass room varies f r o m school to s c h o o l and f r o m c lass room to c lassroom. Computers are t yp ica l l y used i n the schools for w o r d process ing , admin is t rat ion , computer assisted inst ruct ion, and for computer Page 3 p r o g r a m m i n g . W o r d Process ing and C . A . I . are the most c o m m o n w a y s that m i c r o s are current ly used i n the c lass room. T h e literature o n the use o f microcomputers i n the teaching o f mathematics inc ludes d iscuss ion o f issues such as what are the best uses o f the computer (i .e., d r i l l and pract ice p rograms, C . A . I . , or p r o g r a m m i n g ? ) , are a l l C . A . I . p rograms educat iona l l y sound?, are some appl icat ions better than others? V i s u a l F r a c t i o n s ^ - A U s e r - C o n t r o l l e d E n v i r o n m e n t A l t h o u g h there is great potential for computer assisted instruction in the teaching o f mathematics sk i l l s and concepts, not a l l software programs are equal ly effect ive i n u t i l i z ing this p o w e r f u l potent ial . W h a t is needed are software programs w h i c h ut i l i ze computer t ime uniquely and ef fect ive ly by p rov id ing learning environments w h i c h cannot be dupl icated w i t h other c l a s s r o o m tools. V i s u a l Fract ions is a M a c i n t o s h computer software p rogram w h i c h a l l o w s students to manipulate f ract ion diagrams and symbols w i t h speed and accuracy . F ract ions are created instantaneously at the chi ld 's c o m m a n d to represent parts o f a w h o l e , equiva lence and operations. T h e most basic part o f the p rogram invo lves on ly one type o f f ract ion d iagram - a vert ical bar d iagram and the corresponding written fraction s y m b o l s . T h e user is able to change the size o f the shaded port ion o f the fract ion d iagram to graphica l l y d isp lay fractions ranging i n size from halves to twentieths. A s the size o f the fraction is changed, the program supplies the correct mathematical s y m b o l . T h e c h i l d is able to see the associat ion between the f ract ion represented by the d iagram and the s y m b o l . T h i s associat ion takes o n mean ing because the c h i l d is the one determining the size o f the fract ion parts. T h e s y m b o l f o r m o f each fract ion is d isp layed i n large numbers beside the graphic representation to help establish the l i nk between concept and f o r m . Because o f the speed and accuracy o f the computer , many fractions can be d isp layed i n a very short time and st i l l t The Visual Fractions program was developed by Dr. M . Westrom and the Computers in Education Research Group at the University of British Columbia. Page 4 retain the interest o f the user. Constant repetit ion o f a mathemat ica l l y -sound v i sua l a id is the p r inc ipa l feature i n this unique approach to learning fractions. A l s o , the concept o f equal s i zed parts shou ld become apparent to the c h i l d because he cannot create unequal size pieces regardless o f hovy he m a y attempt to do so. T h e computer w i l l a lways d i v ide the d iagram into the nearest equa l - s i zed port ions requested. Figure 1.1 - Example Screen - Visual Fractions A l t h o u g h not spec i f ica l ly manipulat ive i n the tradit ional sense, V i s u a l Fract ions is interactive and user -d i rected. It prov ides a fascinat ing and mot ivat ing experience for students i n a c h i l d -cont ro l led , structured- learning environment. A l t h o u g h the computer actual ly draws the f ract ions, it i s the c h i l d w h o creates his o w n knowledge by detenr i in ing the fract ion the Page 5 computer w i l l m a k e . Because o f the int r ins ical ly interesting nature o f the program and because ch i ldren are normal l y h igh ly mot ivated to use computers for learning, learning shou ld take p lace inc identa l ly and without effort. Noninteractive Learning Environments In order to determine whether V i s u a l Fract ions is a v a l i d use o f computer time and student time, the researcher attempted to dupl icate the essential elements o f the learning environment created by the p rog ram i n another setting. T h e researcher designed several sets o f f ract ion R a s h Cards (see appendix D ) w h i c h simulate the type o f in format ion d isp layed by the f ract ion d iagrams in the computer p rogram. T h e students i n the H a s h C a r d G r o u p had an equal amount o f t ime and teacher assistance to explore the cards as the students i n the C o m p u t e r G r o u p had w i t h the V i s u a l Fract ions program. W i t h the cards, students cou ld create fractions by l i f t ing the f laps on the cards to reveal either a fract ion d iagram or the cor responding wr i t ten s y m b o l . These cards were designed to be very s imi la r to the d isp lay i n the V i s u a l F ract ions p rogram, except fo r two cr i t i ca l differences. F i rs t , the c h i l d i n the F l a s h C a r d G r o u p is not rea l ly i n cont ro l o f his o w n learning. In the V i s u a l Fract ions treatment, there is a real sense o f control over the learning situation; ch i ldren feel as though they are m a k i n g the fract ions through their o w n efforts ( i . e . , s l id ing the mouse button back and forth o n the d iagram). Th i s is not quite the same as mere ly l i f t i ng f laps to reveal ready-made f ract ion d iagrams. W i t h the V i s u a l Fract ions p rogram, the fract ion d iagrams do not exist unt i l the c h i l d creates them. There is ev idence to suggest that this type o f interactive cont ro l is a c r i t i ca l factor i n learning ( M e r r i l l , 1987). A l s o , because mot ivat ion appears to strongly affect one's abi l i ty to learn and ch i ldren appear to be h igh ly mot ivated to use computers , educators should consider u t i l i z ing this mot ivat ional factor by encouraging greater use o f computers to teach concepts currently taught by more tradit ional methods. In order to ensure that any posi t ive results gained f r o m the V i s u a l Fract ions treatment are not due to the nove l ty o f us ing computers , the research des ign inc ludes a third G r o u p w h i c h Page 6 w i l l use the computer to v i e w the same flash cards used by the F l a s h C a r d G r o u p . T h e on ly di f ference w i l l be that the cards w i l l be presented randomly on the computer screen. The user m a y contro l the rate at w h i c h the cards are presented but has no other contro l over the learning envi ronment . T h i s type o f experience is s imi lar to the type o f m i n i m a l interactive learning i n v o l v e d i n turning the pages o f a text b o o k or f l i p p i n g through a stack o f fract ion f lash cards. Expectations of the Study It was expected that as a result o f us ing the p rogram for as l i tt le as one s ix ty -minute session, the students i n the V i s u a l Fract ions G r o u p w o u l d have a better understanding o f fractions than those i n either o f the other two exper imental G r o u p s . E a c h G r o u p was g i ven an equivalent amount o f t ime to explore the learning aids. Super ior performance was anticipated both f r o m the results o f the posttest and f r o m several in -depth interv iews wi th students i n each G r o u p . Students i n the V i s u a l Fract ions G r o u p were expected to i m p r o v e substantial ly i n their abi l i ty to recognize the correct wri t ten symbols for fractions and also be able to correct ly shade i n the required number o f fract ion parts when p rov ided w i t h the s y m b o l notat ion. It was assumed that this abi l i ty w i l l represent an i m p r o v e d understanding o f the concept o f f ract ion def ined as part to who le relat ionship o f equal s ized parts. Problem Statement Does learner-control over the construction of fraction diagrams improve learning? Set i n a larger context, there is the abi l i ty o f the computer to prov ide opportunit ies for learn ing exper iences for students. V i s u a l Fract ions is one o f these. Is it an effect ive one? If so, what are it's ef fect ive characterist ics? Page 7 Purpose of the Study T h e purpose o f the study was to examine whether a software p rogram des igned to a l l o w learner -control over the construct ion o f f ract ion diagrams program is effect ive i n he lp ing ch i ld ren deve lop a better understanding o f fractions. In part icular , this study compared the use o f computers w i t h a more tradit ional c lass room a i d , f ract ion f lash cards. T h e study examined whether attitude toward computers had any effect on the abi l i ty to benefit from the program. T h i s was accompl i shed by the inc lus ion o f a T h i r d Treatment G r o u p w h i c h had an opportunity to v i e w fract ion f lash cards on a computer screen but without contro l over the construct ion o f the fraction d iagrams. Need for the Study M i c r o c o m p u t e r s are currently not be ing ut i l i zed to their f u l l potential i n the school system. Computers st i l l represent a reasonably cost ly investment and avai labi l i ty is usual ly l imi ted to between f i f teen and thirty machines per school . In order to der ive the greatest return o n investment, m i c r o s shou ld be used to per fo rm funct ions that cannot be done w i t h other resources. T h i s study examines the use o f a mic rocompute r in a unique w a y . N o other c lass room too l can instantly create f ract ions at the chi ld 's w i l l i n the w a y V i s u a l Fract ions can . T h i s study w i l l examine this unique approach to us ing computers compared w i th a more t yp ica l use o f computers , a d r i l l p rogram. T h e study compared the use o f the computer d r i l l p rogram to tradit ional f lash card materials. C h i l d r e n often d o not have a good conceptual understanding o f fract ions. Researchers have been s tudy ing this p r o b l e m fo r m a n y years. Severa l suggestions have been proposed, but the p r o b l e m st i l l persists. Th i s study examined a re lat ively new too l , the mic rocomputer and determine whether it might be used to help ch i ldren better understand the concept o f fraction. Page 8 Limitations of the Study T h e f o l l o w i n g are l imitat ions to this study: T h i s study used a convenience sample chosen f r o m one elementary schoo l and used t w o intact classes. There was the poss ib i l i t y o f non-representation due to the s o c i o - e c o n o m i c status o f the s c h o o l o r the abi l i ty l eve l o f the t w o classes. There was a lso the danger o f contaminat ion due to student's d iscuss ing the treatment w i t h their classmates i n a different G r o u p . H o w e v e r , because o f the nature o f the learn ing situation (i.e., i t w o u l d be d i f f i cu l t to learn s i m p l y f r o m a verba l descr ipt ion o f the treatment) and the short length o f the study, this contaminat ion was u n l i k e l y to occur . T h e exposure to the p rog ram is l i m i t e d to one one -hour session. It m a y be that this t ime l i m i t i s not suf f ic ient and student's w o u l d have benefitted f r o m a longer exposure. It was assumed i n this study that students answered truthful ly on the tests and i n the interv iews and that they per fo rmed to the best o f their abi l i ty dur ing the treatments. Page 9 Chapter Two Review of the literature The Development of Fraction Concepts T h e study o f f ract ions has a lways been a struggle fo r ch i ld ren . T h e research ident i f ies a number o f poss ib le reasons fo r this d i f f i cu l t y . In part icular , f ract ions are cons idered to be conceptual ly d i f f i cu l t and i n v o l v e complex def ini t ions The Difficulty is Primarily Conceptual A c o m m o n f i n d i n g i n m u c h o f the research suggests that the d i f f icu l t ies ch i ld ren have w i t h f ract ions are p r i m a r i l y conceptual . They go through the motions of operations on fractions but have not been exposed to the kinds of experiences that could provide them with necessary understandings (Hector & Frandsen, 1981). It is ind icated i n the literature that there are s igni f icant p rob lems i n learning and app ly ing concepts related to rat ional numbers (Behr , W a c h s m u t h , Post & L e s h 1984). H o p e and O w e n s (1987) refer to f ract ions as an inherent ly d i f f i cu l t abstraction w h i c h early c iv i l i zat ions went to great lengths to avo id . Hart (1981) states that ch i ld ren are not conf ident i n their use o f fract ions. She says that whenever poss ib le they w i l l app ly w h o l e number concepts to operations o n fract ions, preferr ing a remainder type answer to one w h i c h states a f ract ion. A s a result o f twenty in -depth interv iews w i t h grade s ix ch i ld ren , P e c k and Jencks (1981) report that about 5 5 % o f the students were unable to demonstrate that they possessed a m e a n i n g f u l concept o f f ract ion. 3 5 % o f the students appeared to have a correct concept o f f ract ion but were unable to extend this knowledge to w o r k w i t h operations and equivalence. T h e authors state that fewer than 1 0 % o f a l l student's in terv iewed had acquired an adequate conceptual base to guide them i n their study o f fract ions. T h e twenty students in terv iewed Page 10 are reported to be t yp ica l o f hundreds o f s imi la r interv iews the authors have conducted w i th grade s i x , seven and n ine students. Definition of Fractions is Complex O n e o f the reasons that ch i ldren m a y have ( i i f f iculty w i t h fractions is that it is d i f f i cu l t to def ine what is meant by f ract ion. T h e not ion o f f ract ion i s not an obv ious or easy concept poss ib ly because there are m a n y different interpretations. E a r l y w o r k by Piaget defines f ract ions as a par t -who le concept but deals o n l y w i t h cont inuous quantity tasks such as length and area. Seven subconcepts o f a f ract ion are ident i f ied i n his w o r k . These subconcepts are s u m m a r i z e d by Hiebert and Tonnessen (1978) as the real i zat ion that: 1. T h e w h o l e is subd iv is ib le . 2. A f ract ion i m p l i e s a determinable number o f parts. 3. T h e subd iv i s ion must be exhaust ive. 4. There is a f i x e d re lat ion between the number o f parts and the number o f d iv is ions . 5. T h e parts have a nest ing or h ierarchical character. 6. T h e w h o l e is conserved under subd i v i s ion . 7. A l l parts must be equal . A c c o r d i n g to P iaget , chi ldren's understanding o f f ract ion as a par t -who le re lat ionship deve lops sequential ly through several stages beg inn ing w i t h an underetanding o f d i v i s i o n into t w o equa l parts, then into fourths and f ina l l y d i v i s i o n into thirds, f i f ths and sixths. T h e p r o b l e m o f de f in ing what is meant by f ract ion is further compl i ca ted w h e n one considers the d i f ference between discrete and cont inuous objects. H ieber t and Tonnessen (1978) argue that Piaget's subconcepts are appl icable o n l y fo r cont inuous quantity representations and ch i ld ren do not necessari ly perceive sets as a w h o l e w h i c h must be Page 1 1 d i v i d e d into parts. T h e y state that chi ldren's strategies used to so lve discrete quantity (set/subset) p rob lems are extremely different than the strategies they e m p l o y i n cont inuous quant i ty tasks. H u n t i n g (1983) suggests that ch i ld ren m a y be able to transfer knowledge about discrete quantit ies to cont inuous p r o b l e m tasks but p robab ly not the reverse. H e recommends that ch i ldren receive instruct ion o n fractions i n both cont inuous and discrete f ract ion quantit ies. T h o m a s K i e r e n (1976) has done extensive w o r k i n the area o f rat ional number interpretation. H e has ident i f ied s ix subconstructs w h i c h he c l a i m s are necessary fo r a fu l l y funct iona l f ract ional number construct. They are par t -who le , d e c i m a l , rat io , quotient, measure and operator. K i e r e n states that instruct ion w i t h rat ional numbers is t yp ica l l y based upon a computat iona l construct. Ra t iona l numbers , he says, are v i e w e d as objects o f computat iona l manipu la t ion and extensions o f w h o l e number computat ions. A c c o r d i n g to K i e r e n , l itt le effort has been made to develop i n the c h i l d a broader construct o f rat ional numbers w h i c h w o u l d inc lude a var iety o f experiences i n a l l s ix subconstruct areas. H o p e and O w e n s (1987) are i n agreement w i t h K i e r e n . T h e y state that each o f these subconcepts is important and a c h i l d must have experience i n a l l areas i n order to have a f u l l y funct ional concept o f f ract ion. W h i l e it is important for educators to recognize the var ious interpretations o f f ract ion and address each, m a n y studies are l i m i t e d to the not ion o f f ract ion def ined as a part to who le relat ionship. In order to f i n d out where ch i ld ren have gone w r o n g i n deve lop ing a concept o f f ract ion , it is appropriate to start by t ry ing to understand what ch i ld ren k n o w about the most basic concept o f f ract ion. It is general ly accepted i n the literature that there exists a hierarchy i n the development o f f ract ion concept. N o v i l l i s (1976) for example , has ut i l i zed p ic to r ia l models to represent this hierarchy ranging f r o m a part to w h o l e m o d e l w h i c h she ca l l s reg ion , to a part o f a set m o d e l and f ina l l y to the most d i f f i cu l t m o d e l , the number l ine . Part to w h o l e relat ionships t yp ica l l y receive the most attention i n the fo rmat ive years o f schoo l ing . A c c o r d i n g to N o v i l l i s , this time does not Page 12 appear to be wel l - spent or adequate enough to help ch i ld ren develop a good base l ine on w h i c h to b u i l d a more complete concept o f f ract ion. K i e r e n (1976) has further deve loped his perspect ive o n the def in i t ion o f f ract ions as a part to w h o l e c o m p a r i s o n . H e refers to this d i v i d i n g o f a w h o l e into equa l parts as part i t ion ing. K i e r e n c l a i m s that w i t h i n this subconcept, understanding i s deve lopmenta l i n nature and depends upon the maturat ion o f the c h i l d through var ious age-related stages. Poth ier and S a w a d a (1983) further deve loped K ieren 's theory to ident i fy f i v e leve ls i n the progression o f understanding o f the part i t ion ing process. T h e y c l a i m that each level is distinguished by certain characteristics, procedural behaviors, and partitioning capabilities. T h e f i ve successive stages are master ing: 1. L e v e l 1: S h a r i n g (not ion o f 1/2). 2 . L e v e l 2 : A l g o r i t h m i c h a l v i n g (denominators w i t h powers o f two) . 3 . L e v e l 3 : Evenness (fractions w i t h even denominators) . 4 . L e v e l 4 : Oddness (fractions w i t h o d d denominators) . 5 . L e v e l 5 : C o m p o s i t i o n (fractions w i t h composi te denominators) . C h i l d r e n are not concerned about equal i ty o f parts i n a f ract ion unt i l l eve l three. T h e y are able to d i v ide a d i a g r a m into eight or even sixteen pieces at l eve l two but d o not concern themselves w i t h whether the pieces represent a/a/r share. It takes considerably more t ime and exper ience for ch i ldren to attain leve l 3 reasoning than is often avai lable i n the school sys tem. Research by P e c k and Jencks (1981) c o n f i r m s this theory. W h e n asked to d raw sketches o f one th i rd , one half , one fourth and other s imp le f ract ions, t yp ica l grade six students d rew sketches w h i c h were o n l y vaguely related to the mean ing o f the fract ion s y m b o l s . P e c k and Jencks report that even at the grade s ix l e v e l , most students in terv iewed were unaware that the partit ions they made had to be the same size. C h i l d r e n struggle par t icu lar ly w i t h the not ion o f o d d numbered fract ions (i .e., thirds and f i fths) especia l ly w h e n asked to i l lustrate these fractions us ing c i rcu lar d iagrams. T y p i c a l l y ch i ldren w i l l Page 13 divide a circle in half and then divide one side into half again to illustrate one third. They do not appear to be concerned with the notion of equal sized pieces even when their partitions are grossly uneven. This concept of equality should be firmly acquired by the child before any further instruction is given on more complex fraction concepts (Peck & Jencks, 1981). Children's Construction of Mathematical Ideas Even if one can arrive at a satisfactory definition of what is meant by fraction, it is another task altogether to find out what children understand about rational numbers. Researchers are continually trying to understand how children construct basic mathematical ideas. Hunting (1983) writes: Since basic fraction concepts are the seedbed for many important mathematical ideas-including notions of equivalence, inverse, decimals, probability, ratio, and proportion-the mental mechanisms that support a knowledge of fractions needs to be understood (p. 182). How do we know for example, when a student understands a concept such as one half or two thirds? Lesh, Behr and Post (1987) propose an answer to this question based upon their work from three different National Science Foundation funded projects. These researchers have developed five distinctive types of representation that occur in mathematics learning. They are: 1. Experienced-based scripts (i.e., real world events). 2. Manipulative models. 3. Pictures or diagrams. 4. Spoken languages. 5. Written languages. According to the Lesh et al. (1987), translations among and within these five representations are critical to genuine understanding of mathematical concepts. Thus, if a child really knows what we mean when we say one half, he or she will be able to translate this knowledge in a Page 14 variety of different ways as illustrated by the five different representational systems. He or she should be able to recognize the idea in a variety of representational systems, flexibly manipulate the idea within each representational system and translate the idea from one system to another. Lesh argues that it is in trying to translate an inadequately developed concept from one representational system to another that students begin to have serious difficulty. Another point of view expressed in the literature suggests that children come to school with well-defined concepts including an understanding of fractions (Pothier and Sawada, 1983). Virtually every child they say, understands what it means to divide a candy bar in half. However, this type of understanding may simply be limited to a notion of my share as opposed to any meaningful insight into the idea of equal parts of a whole. In other words, just because a child understands that an object may be divided into parts of two, three or perhaps even four, does not necessarily mean that this is sufficient knowledge for the acquisition of any meaningful concept of fraction. Pothier and Sawada argue that this type of knowledge is merely rote learning as a result of sharing activities and does not involve an understanding of half in a number sense. They state that this is often indicated by the child's liberal use of the term half in expressions like break in half in four pieces and split in half in three pieces. The Connection Between Concepts and Procedures Even if children arrive at school with some niclimentary ideas about fractions, they are often unable to connect this knowledge with the new information they receive about fraction symbols and operations. According to a study by Peck, Jencks, and Chatterley (1980), students have no idea how to relate fractions symbols with their previous experiences and thus have no base for reasoning in problem situations. Hiebert (1984) states that one possible reason for this failure to link understanding with symbols may be that many children fail to establish meaning for the symbols they are taught. Mathematics becomes, for Page 15 many students, an exercise in following the right rules (p. 501) . Students appear to f o l l o w rules ind isc r iminate ly . E v e n w h e n confronted w i t h the unreasonableness o f their answers, ch i ld ren w i l l insist u p o n rule rather than reason. A c c o r d i n g to Hiebert , Children see little or no connection between the understandings they possess about the number system and it's properties and the rules of form they have memorized for operating on symbols (p. 504). P e c k and Jencks (1981) report that w h e n asked to solve f ract ion p rob lems , ch i ld ren typ ica l l y search fo r rules that often have no meaning for them. T h e y seem to bel ieve that any rule will do and even w h e n ch i ld ren m i s a p p l y prev ious ly learned rules, they are often unable to te l l that they have done so. Further P e c k and Jencks state, these ch i ld ren are dependent on external influences to determine the validity of their answers (p. 345) . T h e most l i k e l y reason for this is that ch i ldren are general ly poor at est imation sk i l l s . They do not l i k e to engage i n act iv i t ies w h i c h require practice i n est imation. A c h i l d seems more secure i n a situation w h i c h has a r ight and a w r o n g answer. A c c o r d i n g to a study by B e h r , W a c h s m u t h & Post (1985) , c h i l d r e n w h o scored l o w o n rat ional number order and equiva lence p rob lems s h o w e d uncertain or inaccurate use ( i f any) o f the est imation process. A d d i t i o n a l l y , students whose rat ional number concepts are weak w i l l often confuse their k n o w l e d g e o f w h o l e number operations w i t h that o f f ract ions (Behr , W a c h s m u t h , Post & L e s h , 1984). E s t i m a t i o n sk i l l s are however , accord ing to B e h r ei al., fundamental to the development of a viable concept of rational number (p. 128). T h e di f ference between conceptual (development o f meanings) and procedural (appl ication o f rules) k n o w l e d g e has been the focus o f m u c h debate i n mathematics educat ion for m a n y years and continues to this present day. H iebert and Le fev re (1986) summar ize this debate as one w h i c h has constantly alternated over the years between one focus and the other. In 1 8 9 5 , f o r e x a m p l e , D e w e y argued for i m p r o v e d understanding. In 1922, T h o r n d i k e emphas i zed the learn ing o f sk i l l s i n order to m a x i m i z e retention. B r o w n w e l l turned the Page 16 debate in 1935 back to an argument for increased understanding. In 1960, Bruner reinforced this position until Gagne refocused the argument in 1977 on skill learning. Currently the debate over these two issues has taken on a different perspective. No longer are the two issues considered in isolation of each other. Instead researchers are interested in the connection between these two types of learning. The Learning Environment is Critical One way to help children make the connection between concepts and procedures is to ensure that their learning occurs in an environment most conducive to the assimilation of knowledge. An aspect of learning environments that has been researched and discussed extensively in the literature is the role of physical models. The Role of Physical Models According to Lavetelli (1970), the curriculum should apply Piagetian techniques to the learning environment which would encourage students to be active participants in the process of learning. Concrete materials (especially for young children) she says, are an excellent way to allow for this participation. She states that the teacher should guide learning but not force children to parrot the right answers. Being given the right answers does not convince a child. They need to be convinced by their own actions. Being able to construct one's own knowledge, says Lavatelli, may be a key factor in learning and retention. Simply using concrete materials to teach fractions however, is not the complete solution to the problem. A major focus of the Rational Number Project, a multiuniversity research project funded by the National Science Foundation and conducted over a five year period from 1979 to 1983, was the role of concrete manipulative materials or physical models in facilitating the development of a rational number concept. One of the many findings of the project was that even after extensive instruction on the order and equivalence of fractions, a Page 17 significant number of fourth grade students demonstrated a substantial lack of understanding when asked to apply fraction knowledge to new situations. (Behr, Wachsmuth, Post and Lesh, 1984). The recommendations of these researchers include introducing fractions in the third grade with a heavy emphasis on unit fractions and plenty of repetitive hands on experiences. The emphasis on teaching the meaning of fractions using manipulative aids in a variety of learning experiences has been clearly established in the literature (Carpenter, Coburn, Reys,& Wilson, 1976). Hiebert (1984) points out however, that the concern over teaching understanding using physical models is not as new a perspective as the literature often suggests. He cites Van Engen's work (1949) as an example of someone whose objective was to help children link mathematical symbols with the concrete objects or events that they represent. Indeed, researchers have been calling attention to the need for a more active participation of the child in his own learning for many years and yet the problem with fractions still persists. Gunderson and Gunderson (1957) noted that children showed good understanding of fractions when using manipulative materials. Children they argued, can benefit from systematic instruction in the meaning and use of fractions rather than memorizing rules and generalizations. Thirty three years however, have passed since these recommendations were made and in spite of the continued use of concrete manipulative materials, children still have problems with fractions. The problem appears to be more complex than an initial examination of the issue reveals (Chaffe-Stengel & Noddings, 1982). Instructional Time must be Effective In addition to providing children with concrete manipulative aids, teachers must be prepared to allow students enough time to use the materials effectively and in an appropriate learning environment. Carpenter et al. (1976) recommend a greater focus on initial conceptual work prior to any formal work with algorithms. They point out that merely increasing instruction Page 18 time w i l l not guarantee better achievement results. Instruction time, they say, shou ld be w e l l o rganized i n order to teach the concept. Further, w h e n faced w i t h an already o v e r - c r o w d e d c u r r i c u l u m , most teachers d o not w e l c o m e the recommendat ion to s i m p l y spend more time teaching concepts ( Johanson 1988). A d d i t i o n a l l y , says Johanson, computers are s t i l l re lat ive ly expens ive and their proper use must be just i f ied . H o p e and O w e n s (1987) warn us that ch i ld ren w i l l not deve lop mean ing i n vacuo . A child, they say, can develop meaning only through carefully structured experiences in a context or setting used to represent a chosen subconstruct (p. 29) . T h e purpose o f these p h y s i c a l settings is to p rov ide an env i ronment where ch i ldren can manipulate objects i n order to eventual ly establ ish a l i nk to a s y m b o l i c setting. A c c o r d i n g to H o p e and O w e n s , these learn ing env i ronments must be carefu l ly structured i n order to establish the mean ing o f some mathemat ica l concept. They emphasize the impor tance o f understanding the language used in the p h y s i c a l setting. It is unfortunately often true that the development o f understanding is inh ib i ted not because o f a m i s s i n g conceptual l i nk , but rather a misunderstanding o f the language used to develop the concept. T h e ro le that language p lays i n deve lop ing concepts has rece ived a great deal o f attention i n the l iterature. F o r example , L e s h , B e h r & Post (1987) report that most fourth through eight grade students have a ser iously def ic ient understanding about the models and languages needed to represent, describe and i l lustrate mathematical ideas. In attempting to understand chi ldren's concepts o f f ract ions, researchers must try to deve lop methods to ensure that they and the c h i l d are us ing the same language when ta lk ing about mathematical ideas. F o r e x a m p l e , does the c h i l d understand terms such as d i v i d e , numerator , cut up into equal p ieces, shade i n the f ract ion part, etc.? The Role of Attitude in Student's Learning In addi t ion to p r o v i d i n g an appropriate learning envi ronment , educators must concern themselves w i t h the ro le o f the student's attitude toward that envi ronment . Page 19 Connections between Attitude and Achievement There i s ev idence i n the literature to suggest that a student's attitude toward a learning exper ience m a y affect h is abi l i ty to learn. Extens ive rev iews have been done o f research o n attitude towards mathemat ics (Ha l yadyna , Shaughnessy and Shaughnessy , 1983). M u c h o f this research has been dedicated to f ind ing a posi t ive relat ionship between attitude towards and achievement i n , mathemat ics . ( C h a p m a n , 1984; Corbett , 1984). T h e l i n k has been c lear ly establ ished; anxiety does seem to have an effect upon achievement and c lear ly many y o u n g people ident i fy themselves as suffer ing f r o m this phob ia . H a l a y d y n a et a l . (1983) , p rov ide a g o o d def in i t ion o f what is c o m m o n l y meant by attitude towards mathematics. T h e y def ine it as a general emot iona l d ispos i t ion toward the school subject o f mathematics . T h e y go on to suggest that a pos i t ive attitude toward mathematics is v a l u e d for several reasons, one o f w h i c h is that attitude is often pos i t i ve ly related to achievement. Math Anxiety H a v i n g accepted the connect ion between attitude and achievement, several researchers have tr ied to f i n d w a y s to reduce anxiety and improve achievement. Mathemat ics educators have cont inua l l y been p lagued w i t h the question o f h o w one can foster understanding o f and thereby i m p r o v e achievement i n school mathematics. T h e term Math Anxiety i s not a new one; an abundance o f study has been devoted to this one aspect o f attitude toward mathematics . T h e literature gives several different interpretations to the not ion o f M a t h A n x i e t y . It is often referred to as a state o f be ing anxious about mathematics achievement and abi l i ty . W o r d s l i k e fear, pan ic , d i s l i k e , hate, and mathophobia are used to descr ibe this p h e n o m e n a ( M a r i o n , 1984; Papert , 1980). In another article o n mathematics achievement and attitude product iv i ty , T s a i and W a l b e r g (1983) , report that mathematics attitude is in f luenced by home condi t ions and achievement. Page 20 T h e y suggest that causal i ty m a y be rec iproca l : the more one learns, the higher the attitude, and the higher the attitude, the greater one learns. In conc lus ion , they state that i m p r o v i n g attitude and encouraging greater learning are both important and long - te rm results. T h e ind icat ion i n the literature is that the use o f phys ica l models to faci l i tate understanding is appropriate but not necessari ly effect ive unless c o m b i n e d w i t h a carefu l ly structured learning env i ronment where language barriers are not a p r o b l e m and student attitudes toward the learn ing experience are posi t ive . T h e learning env i ronment must also be constructed to teach a speci f ic concept i n an appropriate t ime f rame. M i c r o c o m p u t e r s have been shown to be an effect ive c lass room tool requi r ing s igni f icant ly less instruction t ime over convent ional c l a s s r o o m techniques ( K u l i k , K u l i k & B a n g e r t - D o w n s , 1 9 8 5 ) . T h e R o l e o f M i c r o c o m p u t e r s i n E d u c a t i o n Just as attitude towards mathematics affects achievement, student attitudes toward computers are important to success i n computer related programs ( L o y d & Gressard , 1984). S t u d e n t A t t i t u d e s t o w a r d s M i c r o c o m p u t e r s In a study des igned to measure attitude towards computers , these authors reported that students o n the who le had a fa i r ly pos i t ive attitude toward computers . D a l t o n and H a n n a f i n (1985) , report that ch i ld ren bel ieve computer -made learning is more pleasant than convent ional instruct ional strategies due to the inf in i te patience and non- judgemental nature o f the computer . T h e students were reported to have deve loped posi t ive attitudes toward the computer w h i c h prov ides consistent feedback and never showed signs o f frustration o r anger . In another study o n student attitudes toward computers , evaluated over a t w o year per iod , the researchers f o u n d that student interaction w i t h computers d i d not result i n less interaction w i t h teachers and classmates inspite o f the c l a i m s o f some cr i t ics o f C . A . I . ( G r i s w o l d , Page 21 1984) . T h e researcher r e c o m m e n d s the use o f computers and C . A . I , to encourage students to recognize their role and gain confidence in controlling the course of their own learning. T h e reason for this recommendat ion i s that student's perceptions o f their responsib i l i ty fo r success and their academic se l f -conf idence were consistently related to C . A . I , after cont ro l l ing fo r the effects o f gender, minor i t y status and achievement. The Role of Intellectual Models T h e w o r k o f S e y m o u r Papert (1980) lends some further insight into the not ion o f learn ing env i ronments , mic rocomputers and ch i ldren cont ro l l ing the course o f their o w n learning. H e proposes a theory o f learn ing based upon the idea that each o f us develops a co l lec t ion o f m o d e l s w h i c h r e m a i n w i t h us throughout our l i ves and to w h i c h w e relate a l l new k n o w l e d g e . In his case, the gearbox o f an automobi le served as a m o d e l w h i c h he appropriated at a very y o u n g age and to w h i c h he later related his ideas about mathematics . T h u s , fo r Papert , mathemat ics was easy but o n l y because o f the conceptual m o d e l o f gears to w h i c h he related it. In fact, Papert contends that anything is easy if you can assimilate it to your collection of models (p v i i ) . T h e idea o f gears i s an excel lent m o d e l to represent mathematical ideas because a c h i l d can relate to the mathematical knowledge as w e l l as the body knowledge not ion o f turning gears. H e says that the c h i l d can in a sense become a gear as he tries to go through the mot ions o f understanding h o w they w o r k . Papert advises educators and researchers to l o o k for ways to create condit ions under w h i c h intel lectual mode ls w i l l take root. W e need to real ize however , that the not ion o f intel lectual models is a personal matter and each person col lects their o w n set o f models as they mature. W h a t serves as an ideal m o d e l fo r one c h i l d m a y do nothing for another c h i l d i f he cannot relate to it. Papert reminds us that the gears m o d e l w o r k e d for h i m because he f e l l i n l o v e w i t h gears. In order to understand h o w educators can facil itate this development o f intel lectual models , it is appropriate to consider h o w Papert's not ion o f intel lectual models can be related to the i d e a o f structured learning environments. Papert argues that the computer i s the idea l Page 22 m e d i u m current ly avai lable to prov ide these learning experiences for ch i ldren. H e argues that computers c a n i n a sense become all things to all people because o f their power to s imulate countless funct ions i n countless ways . Computers can serve as the too l to create gear box m o d e l s o r m i c r o w o r l d s w h i c h w i l l g ive m e a n i n g to concepts as they are presented to o r more accurately stated, deve loped by the c h i l d . It i s important not to confuse Papert's not ion o f m i c r o w o r l d s w i t h what is t radi t ional ly meant by structured learn ing environments. Structured learning envi ronments are designed to teach a speci f ic concept. M i c r o w o r l d s are learning environments des igned to prov ide the in fo rmat ion and tools necessary for a c h i l d to d iscover k n o w l e d g e about a part icular topic . T h e y are envi ronments w h i c h are i n part p rov ided for and i n part created by the c h i l d where o n l y issues o f importance to h i m are re levant Current ly , the onus is o n the c l a s s r o o m teacher to faci l i tate the creation o f a m i c r o w o r l d fo r the student. T h e result m a y be the same as i n a structured learn ing env i ronment , but i n a m i c r o w o r l d , the c h i l d is construct ing his o w n k n o w l e d g e and thus learning should be more mean ing fu l and retention greater. Papert's ideas are strongly l i n k e d to Jean Piaget w h o is k n o w n as the author o f learn ing wi thout cu r r i cu lum. Papert (1980) refers to Piagetian Learning as learning w h i c h occurs w i thout f o r m a l inst ruct ion. H e deviates f r o m Piaget 's theory however , i n that Papert be l ieves that a l though ch i ld ren b u i l d their o w n k n o w l e d g e , they need r i c h tools to d o so. W h e n k n o w l e d g e is s l o w to develop, Piaget w o u l d attribute this to the c o m p l e x i t y o f the concept be ing learned. Papert o n the other hand, bel ieves that an impover i shed learning env i ronment is responsib le for the s lower intel lectual development i n the c h i l d . Thus , as l o n g as the learn ing env i ronment is r i ch i n tools for bu i ld ing k n o w l e d g e , the c h i l d shou ld be left to deve lop his o w n learn ing . Th i s i s not the same th ing as a carefu l ly structured learning env i ronment designed to teach a c h i l d a speci f ic concept. Page 23 Learner Control F o r the past decade, a theory o f inst ruct ional des ign has been proposed by a group o f people under the name o f C o m p o n e n t D e s i g n Theory w h i c h suggests that learn ing , resu l t ing f r o m instruct ion is most eff ic ient and effect ive i f a proper combinat ion o f in fo rmat ion presentation fo rms are used to their greatest advantage. ( M e r r i l l , 1987). E m b e d d e d i n this theory is the p romot ion o f learner contro l as a m e c h a n i s m for adapting to i n d i v i d u a l di f ferences i n learn ing style. Because o f increased technology and avai lab i l i ty , a w i d e var iety o f inst ruct ional techniques are n o w possib le . A c c o r d i n g to M e r r i l l (1987) , instruct ional design theory has not kept pace w i t h the increased capabi l i t ies o f computer hardware and software. A s an extension o f the o r ig ina l inst ruct ional des ign theory, M e r r i l l proposes a New Component Design Theory w h i c h extends the or ig ina l theory to take advantage o f the increased capabi l i t ies o f computers part icular ly i n the areas o f intervention and presentation. T rad i t iona l l y , most instruct ional software was based upon a structured or tutor ial type m o d e l , also k n o w n by the term Branched Program Instructional Model (Reige luth , 1983). In this type o f learn ing situation, the student is presented w i t h some in fo rmat ion (usual ly a screen o f text and/or graphics) , asked a quest ion, p r o v i d e d w i t h an answer and poss ib ly remed ia l mater ia l depending upon his response, and then the cyc le is repeated. M e r r i l l (1987) argues that this type o f learning env i ronment is best i n two c i rcumstances: to act as a secondary re inforcement to a p r imary , more experient ial env i ronment and to correct or help students overcome misconcept ions or misunderstandings after hav ing exp lo red some exper ient ia l env i ronment . H o w e v e r , he says that the computer is m u c h more than just just a tutor and should not be l i m i t e d to this type o f funct ion . One o f the most effect ive instruct ional uses o f the computer , he argues, is that w h i c h a l lows the student to interact d i rect ly w i t h the subject matter. A c c o r d i n g to M e r r i l l (1987) , an exper ient ial m o d e l o f instruct ion is m u c h different f r o m a tutorial m o d e l because the student is able to interact d i rect ly and have contro l over some type o f exper ient ia l representation o f subject matter. H e Page 24 refers to an exper ient ia l representation as a control lable m i c r o w o r l d w h i c h a l lows the student to explore and d i s c o v e r the re lat ionships i n v o l v e d . H e warns us, however , that exp lorat ion is o n l y one type o f transaction and m a y not be suff ic ient to enable the student to learn the necessary procedures or to understand a l l o f the relat ionships i n c l u d e d i n the exper ient ial s imulat ion . H e suggests that students m a y learn more f r o m a p r o g r a m w h i c h inc ludes elements o f demonstrat ion, explanat ion and predict ion as w e l l as components o f exp lorat ion . T h e student, he argues, m a y learn more f r o m some f o r m o f structured transactions than f r o m open ended learner contro l led explorat ion. Researchers are not a l l i n agreement w i t h the idea that learner contro l presents the best learn ing env i ronment . A study by B e l l a n d et a l . (1985) into the nature o f mot ivat ion and learning indicates that sel f -d i rected learning environments m a y not necessari ly be the best situation for a l l people . T h e y argue that computer software programs w h i c h d o not p rov ide any external p r o m p t i n g m a y not be effect ive learning envi ronments for l o w achievers w h o can not keep themselves on task. Students w h o are not self mot ivated fo r example , require some external cont ro l to help them learn. Sof tware developers , they say, need to address this quest ion w h e n des ign ing educat ional programs for a l l types o f learners. R o w l a n d and Stuessy (1988) concur w i t h this f i nd ing . T h e y argue that a l though one o f the greatest aspects o f C . A . I , is it's abi l i ty to p rov ide i n d i v i d u a l i z e d inst ruct ion , it is poss ib le that not everyone is able to benefit f r o m this interactive nature o f some computer programs. T h e i r concern is that some learners are predisposed to a part icular cogni t ive style and are not eas i ly adapted to n e w modes o f instruct ion. In order to test their theory, R o w l a n d and Stuessy conducted a study to determine what effect the mode of C . A . I (tutorial versus s imulat ion (wh ich invo lves learner control)) has o n achievement and understanding o f concept relat ionships. T h e y were also interested i n determining whether cogni t ive learning styles c o u l d be purposely matched to the type o f C . A . I i n order to increase performance. C o g n i t i v e learn ing styles were def ined as hol ist o r serialists and were determined from a Page 25 sel f -administered study preference questionnaire. T h e results o f their study indicate that cogni t ive style does interact w i t h mode of C . A . I , to in f luence student achievement and that learners are more effect ive w h e n matched to the appropriate mode o f C . A . I . Microcomputers and Mathematics Learning M i c r o c o m p u t e r s are already beg inn ing to have an effect o n mathematics education. T h e Nat iona l C o u n c i l o f Teachers o f Mathemat ics ( N C T M ) considered this impact at the 1984 conference. A s igni f icant statement result ing f r o m that conference was that the major in f luence o f technology o n mathematics education is its potential to shift the focus o f instruct ion f r o m an emphasis o n manipulat ive sk i l l s to an emphasis on deve lop ing concepts, re lat ionships , structures and p r o b l e m - s o l v i n g sk i l l s ( N C T M . report, 1981). B e c a u s e o f the int roduct ion o f computers and calculators , the C o u n c i l argues i n their report that it is no longer the most important goa l o f school mathematics that students be prof ic ient in computat ional sk i l l s . T h e C o u n c i l cal ls fo r a change i n focus to take advantage o f emerg ing technology . N o t a l l studies o f microcomputers and achievement report pos i t ive results. T h e results o f a study ( M o r r i s , 1983) undertaken to determine h o w m i c r o - c o m p u t e r activit ies integrate into regular instruct ion and h o w such activit ies affect student achievement and attitude indicate m i x e d results. The des ign o f the study inc luded a pretest, posttest and both an exper imental (one intact class) and a C o n t r o l G r o u p (a second intact c lass) . It was determined by an achievement pre-test analysis us ing an independent t-test, that the two classes were c lose enough that it w a s u n l i k e l y they came f r o m populat ions that were different. T h e study i n v o l v e d a four w e e k unit o n co-ordinate geometry where one class rece ived instruct ion as w e l l as an opportunity to spend t ime at the computer p l a y i n g several games w h i c h requi red the use o f co -ord inate and d i rect ional movement concepts. T h e second class rece ived o n l y instruct ion and no computer t ime. A t the end o f the four week unit , both classes took an achievement test o n the content o f the unit. Students were also asked to respond to an Page 26 attitude scale w h i c h related to computers and their use i n school . T h e achievement post-test analys is p r o v e d to be s igni f icant at the .02 leve l . The authors conc luded that the addi t ion o f computers to regular instruct ion m a y have s igni f icant ly i m p r o v e d achievement. The i r research des ign does not however , take into account that any extra re inforcement , computer o r otherwise, m a y have accounted for the improvement . It m a y s i m p l y be that the E x p e r i m e n t a l G r o u p had lots more opportunity to learn the concepts. T h e results o n the attitude survey were not encouraging . Students were , accord ing to the author, frustrated by the [apparent] limitations of computers, attributing to computers the ability to think and control the world. T h i s study raises several interesting concerns and makes recommendat ions for future studies. F o r example , they ask What specific aspects or characteristics of computers affect the learning of mathematics? (Mor r i s , 1983). In another study o f us ing microcomputers to re inforce ar i thmetic s k i l l s , the results are also not c o n c l u s i v e . Car r ie r , Post , & H e c k (1985) c l a i m that a l though their study prov ides some support for the c l a i m that microcomputers enhance achievement i n schoo l mathematics, further study is warranted especia l ly i n areas other than s i m p l y reinforcement o f arithmetic s k i l l s . T h e major i ty o f the literature o n computer uses i n educat ion however , is general ly favorable and indicates that computers m a y prov ide an effect ive means o f i m p r o v i n g performance i n a var iety o f subjects i n c l u d i n g mathematics (Henderson, L a n d e s m a n , & K a c h u c k , 1985; G r i s w o l d , 1984) . A c c o r d i n g to B u r n s and B o z e m a n (1981) , m a n y studies o n the implementat ion o f computer based instruct ion point to a s ignif icant enhancement o f learning i n mathematics . Jo l i coeur and Berger (1988) report i n their rev iew of the literature that C . A . I , (computer assisted instruction) i s an effect ive means for i m p r o v i n g academic s k i l l , requ i r ing s ign i f icant ly less time than tradit ional c lass room methods. Studies b y K u l i k et a l . (1985) support this pos i t ion . Page 27 A l t h o u g h documented i n the literature that the computer m a y be an effect ive c lass room too l , it is nonetheless true that not a l l software programs currently avai lable are equal ly effect ive (Jol icoeur and Berger , 1988). S o m e are successful and ut i l i ze the computer e f f ic ient ly and others d o not. T h e development o f instruct ional software has c o m e a l o n g w a y i n the past twenty years. O r i g i n a l l y students were not able to interact w i t h the computer to tai lor the learn ing situation to meet their o w n needs. T h e potential o f the computer was not be ing rea l i zed . M o s t software programs avai lable were o f a l inear, d r i l l and pract ice nature. T o d a y , however , most instruct ional programs a l l o w m u c h greater use o f the interactive capabi l i t ies o f the mic rocompute r (Brandon , 1988). R e v i e w s o f the l iterature o n the effects o f currently avai lable instruct ional software have general ly been pos i t ive . Studies typ ica l l y report i m p r o v e d grades, reduced instruct ion time and pos i t i ve attitudes towards computers (B randon , 1988). Other research indicates that C . A . I , improves academic sk i l l s i n s ign i f icant ly less time than tradit ional instruct ional methods ( K u l i k et a l . , 1985). There are some negative f ind ings i n the research concern ing the effect iveness o f C . A . I . . (C la rk , 1985) argues that m u c h o f the research o n the effects o f instruct ional software contains serious des ign f l a w s . C u r r i c u l a r content and method o f instruct ion for e x a m p l e , have not been proper ly cont ro l led in m a n y studies. H e states that instruct ional software is no more effect ive than tradit ional instruction w h e n method and content are appropriately cont ro l led . (C la rk 1983). In fact he says, Media are mere vehicles that deliver instruction but do not influence student achievement any more than the truck that delivers groceries causes changes in our nutrition (p. 445) . A c c o r d i n g to M a n d e l l and M a n d e l l (1989) , cr i t ics o f computer use i n educat ion bel ieve that the current interest i n computers is o n l y temporary and w i l l eventual ly wane just as d i d the interest i n other fo rms of m e d i a as educat ional tools (i .e., te lev i s ion , p r o g r a m m e d - l e a r n i n g books and mov ies ) . A d v o c a t e s o f computers i n educat ion o n the other hand, argue that there is a fundamental dif ference between computers and other fo rms o f m e d i a . Computers a l l o w for active part ic ipat ion and interaction w i t h the Page 28 computer . Further , u n l i k e te lev is ion and mov ies , computers have become pervasive i n the w o r k fo rce . T h e major i ty o f the literature tends to report pos i t ive results i n currently avai lable software. Jo l i ceur and B e r g e r (1988) deve loped a study w h i c h analyzed fractions and spe l l ing software programs to determine i f they were an effect ive teaching m e d i a . Resul ts f r o m these studies indicated that students w h o used the fractions software p r o g r a m learned s igni f icant ly m o r e than students w h o d i d not. S o m e programs were c lear ly superior to others i n the s tudy . The Design of Educational Software is Important In des ign ing educat ional software, a p rogrammer shou ld cons ider several pedagogica l issues. Ease o f use o f the p r o g r a m , for e x a m p l e , i s an important c r i ter ion . Sof tware developers must be carefu l not to des ign programs w h i c h w i l l waste the student's t ime in learn ing h o w to operate the p rogram. L e r o n (1985) thoughtfu l ly warns us that s i m p l y engaging in mathematical activities does not ensure that children will learn the mathematical concepts involved. Jo l i coeur and Berger (1988) suggest that in fo rmat ion process ing methods are important cogni t ive concepts to keep i n m i n d w h e n deve lop ing educat ional software. T h e y say that to produce optimal learning conditions, educational software must provide students with basic conceptual tools that they can use to build and store more complex concepts (p 13 -19) . In add i t ion , they argue, any exercises must encourage students to process in format ion semant ical ly . T h e y state that w h e n not p r o v i d e d w i t h conceptual tools , students mere ly m e m o r i z e n e w in format ion and thus retention can be expected to dec l ine fa i r l y rap id ly . W h a t is needed then, are software programs w h i c h ut i l i ze computer time uniquely and ef fect ive ly by p r o v i d i n g learning environments w h i c h cannot be dup l icated w i t h other c lass room tools. Page 29 Summary C h i l d r e n struggle greatly w i t h the concept o f rat ional number . T h e d i f f i cu l ty ch i ld ren have w i t h fractions is p r imar i l y conceptual . T h e literature identif ies many different interpretations o f f ract ion. C h i l d r e n often h o l d incomplete ideas poss ib ly because there are so m a n y different def in i t ions. There appears to be a deve lopmenta l hierarchy i n h o w ch i ld ren develop concepts i n c l u d i n g those related to fractions. It is important that ch i ldren begin to understand the concept o f f ract ion by understanding the relat ionship o f equal parts o f a who le . Greater emphasis must be p laced o n the idea that f ract ion parts must be equal i n size. C h i l d r e n struggle w i t h this concept part icular ly as it relates to fractions w i t h o d d numbered denominators and c i rcu la r d iagrams (e.g., i l lustrat ing one th i rd o f a c i rc le ) . T h e recommendat ion i n the literature is that educators increase their emphasis o n teaching ' m e a n i n g and understanding o f a l l the different interpretations o f fract ions. C h i l d r e n shou ld be taught to analyze the reasonableness o f their answers us ing est imat ion sk i l l s and to re ly less o n the i n d i s a i m i n a t e use o f procedural rules. A n increased amount o f t ime spent on the top ic , earl ier in t roduct ion , structured learning envi ronments and greater use o f concrete manipu lat i ve aids are a l l suggested as poss ib le solutions to the p r o b l e m ch i ld ren have understanding f ract ion concepts . Educators shou ld be aware o f the ro le o f language w h e n d iscuss ing mathemat ica l ideas w i th ch i ldren . In addit ion to tradit ional concrete manipulat ive mater ials , educators shou ld consider us ing the mic rocomputer as a c lass room tool . T h e current literature indicates that the mic rocompute r has been successful ly used i n m a n y areas o f the c u r r i c u l u m i n c l u d i n g mathematics instruct ion. Teachers should choose software programs that are pedagogica l ly sound and take into considerat ion the i n d i v i d u a l needs o f their students. S o m e students need more p r o m p t i n g and external mot ivat ion than some software programs prov ide . A l s o , issues Page 30 such as ease o f use, t ime o n task, student attitude toward the subject and the learn ing env i ronment and avai lab i l i ty o f equipment must be taken into considerat ion. A possib le learn ing tool fo r teaching fractions w h i c h deserves further explorat ion is the mic rocomputer . Ideal ly , software programs should be designed to teach spec i f ic f ract ion concepts i n a time-efficient, structured learning environment. S u c h a learn ing env i ronment is in t r ins ica l ly mot i va t ing and ut i l izes the power o f the microcomputer i n a unique way . In other words , this envi ronment rel ies o n the unique abi l i t ies o f the mic rocompute r to imp lement it and cannot be dupl icated by other means. It is the hypothesis o f this researcher that V i s u a l Fract ions i s such a program. Page 31 Chapter Three Methodology T h e purpose o f the study was to determine whether V i s u a l Fract ions , a software p r o g r a m designed to teach f ract ion concepts o n the M a c i n t o s h computer , is an effect ive method o f instruct ion. In part icular , the issue o f learner cont ro l over the learn ing env i ronment was e x a m i n e d . T h e study w a s des igned to answer three quest ions: 1. D o ch i ld ren w h o have contro l over their learning, learn more? 2. Is V i s u a l Fract ions an effect ive means o f teaching basic f ract ion concepts? 3. D o e s V i s u a l Fract ions help ch i ldren m o v e toward an understanding o f fractions based upon a concept o f equal parts o f a w h o l e ? T h i s study addressed several issues related to the use o f computers for instruct ional purposes. B e c a u s e o f the f ind ings o f several pr ior research projects, it was assumed that computers are an appropriate method o f instruction i n mathematics educat ion. A rev iew of the literature also revealed that ch i ldren typical ly have a ru le -based understanding o f f ract ions and general ly d o not develop sound basic f ract ion concepts before be ing required to p e r f o r m operations and equiva lence tasks. T h e software p rogram, V i s u a l Fract ions was therefore spec i f ica l l y m o d i f i e d for the purpose o f this study to prov ide a learn ing env i ronment des igned to teach basic f ract ion concepts. T h i s chapter contains a d iscuss ion o f the sample , p i lo t study, research des ign ( instrumentation and treatments) and data analysis techniques. Page 32 Sampling T h e study used a convenience sampl ing o f four classes o f grade four students taken f r o m t w o schools i n B u r n a b y , B r i t i s h C o l u m b i a , (a suburb o f V a n c o u v e r ) . T h e y are referred to as S c h o o l 1 and S c h o o l 2 . There were four groups i n the study. T w o groups were i n v o l v e d us ing computers , the V i s u a l Fract ions Interactive G r o u p and T h e V i s u a l Fract ions Non in te rac t i ve G r o u p . T h e other t w o groups, F l a s h C a r d G r o u p and the C o n t r o l G r o u p , d i d not use computers . T h e computer groups were compr i sed o f students f r o m S c h o o l 1 and the non -computer two groups were f r o m S c h o o l 2 A l l students were r a n d o m l y assigned to the f o u r G r o u p s . The Pilot Study T h e p i lo t study occur red at S c h o o l 1. Three grade three students and four grade four students part ic ipated i n the p i lo t study. These students represented a w i d e range o f abi l i ty levels . A l l o f the ch i ld ren had w o r k e d w i t h M a c i n t o s h computers pr ior to the treatment and were comfor tab le w i t h the mouse po int ing device. T h e or ig ina l pretest (appendix E ) cons isted o f 4 pages (a total o f 12 questions) arranged i n progressive d i f f i cu l t y . Shad ing quest ions c o m p r i s e d the f irst two pages and the last two pages conta ined questions requi r ing n u m e r i c answers. T w o examples were p r o v i d e d for each page o f quest ions. T h e test was p i lo ted o n several ch i ld ren (4 intact classes o f grade 3 and 4 chi ldren) as w e l l as the 7 students i n the p i lo t study. It was determined f r o m the results o f these tests that even grade three students w i t h l i tt le understanding or f o r m a l instruct ion i n fractions c o u l d score re lat ive ly h igh o n the test s i m p l y by c o p y i n g the pattern presented i n the example questions. F o r e x a m p l e , o n page one o f the pretest, the student is asked to shade i n the d iagrams to i l lustrate the requested f ract ion (see f igure 3.1) Students s i m p l y studied the example and transferred what they saw there to the questions asked. Severa l students exp la ined this Page 33 technique to the researcher claiming it was how they answered lots of worksheets which they didn't understand. Figure 3.1 Example of Page One-Pilot Study Pretest SHADE IN THE CORRECT FRACTIONAL PART EXAMPLE A EXAMPLE B 3 4 1 3 tf/Sf/S •/////'< '//•///, 1. 1 I I I 3 It was decided therefore that this test was an inappropriate measure of understanding of fraction concepts and the pretest was redesigned to address these problems. The subsequent test used during the study was focussed more accurately on the examination of fractions and less on worksheet answering skills. Because the original pretest/posttest was used for the pilot study, it is difficult to determine whether these students showed any improved understanding of fractions. There were however, some important discoveries about the program itself and how children reacted to it The three grade three students in the pilot study were extremely enthusiastic and tended to work in a cooperative manner with each other rather than as individual learners. Two were of average ability level but appeared to have no prior knowledge of fractions or at least no formal fraction language to express their ideas when asked prior to the session. The only Page 34 instruct ion g iven to the students at the beg inn ing o f the sessions was a d i rect ive i n the f o r m o f a chal lenge to d iscover h o w the p rogram w o r k e d . T h e students w i l l i n g l y shared their d iscover ies and he lped each other to understand h o w the p rogram w o r k e d . T h e y q u i c k l y f e l l into a pattern o f cha l leng ing each other to make different fractions on the computer . Because they d i d not k n o w the terms one third or two fifths to descr ibe what they were seeing o n the computer screen, they s i m p l y used terms l i k e one out of three and two out of five. T h e y spent a lot o f t ime w o r k i n g w i t h the p r o g r a m button w h i c h hides the numbers f r o m v i e w . T h e y w o u l d c a l l out a f ract ion to each other, try to m a k e it on the screen and then uncover the numbers to see i f they were correct. These gir ls had n o trouble w o r k i n g w i t h fractions w i t h denominators o f twenty and c o u l d easi ly make any f ract ion y o u asked them to after o n l y thirty minutes o f p l a y i n g w i t h the p rogram. T h e th i rd g i r l tended to w o r k more o n her o w n and was somewhat s lower than the other two i n understanding h o w the p rog ram w o r k e d . She had a p r o b l e m staying o n task w i t h almost any o f her requ i red schoo l work , however , her attitude toward schoo l had changed quite cons iderab ly after w o r k i n g w i t h the software p rogram. A c c o r d i n g to this chi ld 's a ide, she had become ext remely enthusiastic about fractions and had designated herself the class expert and tutor! T h i s c h i l d c o u l d not answer any o f the questions o n the pretest, but made remarkable improvement o n the posttest. In l ight o f this chi ld 's n o r m a l l y apathetic approach to her schoo l w o r k , they were amazed at this t ransformation. T h e fou r grade four students w h o part ic ipated i n the study were o f v a r y i n g abi l i ty ranges. T h e y were int roduced to the computer p rogram i n the l ibrary o f the schoo l where there was o n l y one computer avai lab le , thus there was no poss ib i l i t y o f peer interact ion. T h e first student, a g i r l o f h i g h ab i l i t y l e v e l , was very q u i c k to understand h o w the p r o g r a m w o r k e d and concentrated very intently fo r over an hour. She asked n o questions and d i d not m a k e any effort to c o m m u n i c a t e w i t h us dur ing her entire session. A f t e r the hour was over , w e had to stop her and ask her to rewr i te the test fo r us. It was later determined that she was an Page 35 i E S L student and quite shy. H e r score i m p r o v e d s l ight ly o n the posttest, but as it was quite h igh to beg in w i t h , there wasn't a lot o f r o o m for i m p r o v e m e n t T h i s student was quite fascinated w i t h the t w o - w a y rectangle part o f the program and spent m u c h o f her t ime m a k i n g patterns and seeing re lat ionships between the numbers and the pictures. She wasn't as interested i n the button w h i c h h i d the numbers f r o m sight as the grade three gi r ls were. A l t h o u g h she d i d not c o m m u n i c a t e her thoughts, it seemed as though she was t ry ing to rea l ly understand exact l y h o w the p rogram w o r k e d and w h y she had to decrease the numerator before she c o u l d increase the denominator . She m a y have benefitted f r o m a more sophist icated vers ion o f this p rogram. T h e second grade four student was a boy was o f average abi l i ty and p r o m p t l y i n f o r m e d us o f this fact w h e n asked i f he en joyed mathematics. H e d i d not l i k e the chal lenge instructions issued to h i m to s i m p l y play with the computer and try to figure out how the program works. H e rea l l y struggled w i t h the lack o f direct ions and i n complete contrast to the first grade four student, appeared very frustrated and cont inual ly asked questions. A f t e r several attempts at t ry ing to understand the p rogram he i n f o r m e d us that he was f in ished. A t this po int he was s h o w n h o w to h ide the numer ic symbo ls i n order to try and guess w h i c h f ract ion he was creat ing. F r o m this point o n he was able to w o r k m u c h more independently and appeared to enjoy the p rog ram very m u c h . H e had to be stopped after approx imately fo r t y - f i ve minutes because o f l unch hour. T h e th i rd grade four student was o f l ower abi l i ty l eve l i n most subjects i n c l u d i n g mathematics . She per formed extremely w e l l o n the posttest to the amazement o f her teachers. Ex tens ive interv iews w i th this student lead the researcher to bel ieve that she d i d not rea l ly understand fract ions but had somehow learned the system and was able to respond f r o m m e m o r y rather than f r o m real understanding. Page 36 T h e fourth student i n grade four was exposed to the software p rog ram i n the computer lab a long w i t h the three grade three students. T h i s student was not h igh l y mot ivated and d i d not appear to enjoy the p rog ram at a l l . H e had trouble w i th the chal lenge to f i n d out h o w the p r o g r a m w o r k e d . H e seemed to be the type o f student whose learning was not se l f -d i rected at a l l and needed constant teacher intervention to keep h i m o n task. H e asked to be a l l o w e d to return to his c lass after 30 minutes. Perhaps this type o f student w o u l d have benefitted more f r o m the p r o g r a m i f he was g iven some assistance i n the basic operat ion o f the software o r i f he had more exper ience w i th sel f -d i rected learning envi ronments . H e d i d not seem to want to explore o n his o w n and consequently appeared to learn very l itt le Research Design T h e research des ign adopted was a di f ference study us ing a two- factor (treatment and test), quas i -exper imenta l des ign . Factor A consisted o f four Treatment G roups , and Factor B had repeated measures, pretest and posttest. Page 37 T h e R e s e a r c h D e s i g n M o d e l T h e f o l l o w i n g m o d e l (f igure 3.2) i l lustrates the design o f the study: F i g u r e 3 . 2 M o d e l o f R e s e a r c h D e s i g n GROUP N PRETEST POSTTEST Al 1 VISUAL - B l B2 FRACTIONS -INTERACTIVE 22 A2 1 VISUAL FRACTIONS - B l B2 NON-INTERACTIVE 22 A3 1 FRACTION B l B2 FLASH CARDS 9 A4 1 CONTROL 11 B l B2 E a c h o f the three Treatment G r o u p s was required to write a pretest. Three days after the pretest, each G r o u p was exposed to a treatment session o f approx imate ly 20 to 55 minutes i n durat ion. These three G r o u p s then rewrote the same test immediate ly f o l l o w i n g the treatment. T h e C o n t r o l G r o u p wrote the tests i n the m o r n i n g and rewrote the same test i n the afternoon o f the same day , but rece ived n o treatment. T h e four G r o u p s i n the study were: 1 . V i s u a l F ract ions Interactive C o m p u t e r G r o u p . Page 38 2 . V i s u a l Fract ions Noninteract ive C o m p u t e r G r o u p . 3 . F r a c t i o n F l a s h C a r d G r o u p . 4 . C o n t r o l G r o u p . In addi t ion to this quantitative data, there was a qual i tat ive component to this study, i n the f o r m o f student in terv iews. Interviews were conducted pr ior to the pretest and after the posttest. Description of Data Collection Three sets o f data were co l lected : the pretest/posttest o f f ract ion knowledge , student in terv iews and observat ions o f the treatment process. The Pretest/Posttest T h e purpose o f the pretest and posttest ( A p p e n d i x C ) was to m a k e an object ive measure o f student's understanding o f f ract ions. T h i s was done to determine whether exposure to the p r o g r a m , V i s u a l Fract ions , was more effect ive i n deve lop ing a good concept o f f ract ions than a s imi la r amount o f time spent w o r k i n g w i t h a noninteractive vers ion o f the same learn ing env i ronment . T h e o r ig ina l pretest/posttest designed for this study was p i lo ted w i t h over 100 grade 3 and 4 students. It was d iscovered that because o f the re lat ive ly easy questions at the beg inn ing o f the test and the two example questions g iven for each set o f quest ions, students w i t h l i tt le or no understanding o f f ract ions were able to score very w e l l . A s a consequence the pretest w a s redesigned to test fractions rather than worksheet competence. E x a m p l e s were not p rov ided and the student had to read and answer a l l o f the quest ions independent ly us ing their k n o w l e d g e o f fract ions. There were 17 questions on the pretest. The questions were not arranged i n any part icular order accord ing to type or degree o f d i f f i cu l t y . S o m e questions requi red shading, others requi red the student to wr i te numer ic s y m b o l answers. Rectang les , squares and c i r c le d iagrams were used throughout the test. T h e same test w a s administered as the posttest. T h e m a x i m u m poss ib le score o n the test was Page 39 68. E a c h answer w a s rated a n o m i n a l scale score between zero and four based o n h o w c lose l y the response resembled a correct understanding o f fract ions. A n s w e r s were g iven a score o f zero i f no attempt was made to answer the quest ion. A score o f one was g iven i f some effort (mar) was made but there was no obv ious relat ionship between the answer and either the numerator o r denominator o f the fract ion.or i f the attempt to shade appeared to be comple te l y arbitrary. A score o f t w o was assigned i f the answer corresponded i n some w a y w i t h either the numerator or denominator , but not both or i f the shading represented some discernable reasoning process, though not accurate. T h e answer d i d not have to be correct as l o n g as there was some apparent purpose for , or re lat ionship between, the numerator o r denominator g i ven by the c h i l d . F o r example , quest ion 1 o n the test asks the c h i l d What fraction is illustrated by the shaded portion of the diagram ? (a rectangle d i v i d e d into fifteen parts w i t h 7 shaded in) . I f a c h i l d answered 7/12, they w o u l d have scored 2 points . T h e numerator 7 corresponds correct ly , but the denominator is not correct and there is no apparent l o g i c a l reason for it. A score o f three was assigned i f there was a re lat ionship between the answer and the correct numerator and denominator or i f the d r a w i n g was rough l y correct but not as accurate as it c o u l d have been. F o r example , quest ion 16 shows a rectangle w i t h 1/3 shaded .Students w h o answered 1/2 w o u l d have scored three points because there is one shaded part and two unshaded parts .Both the numerator and denominator i n the chi ld 's answer has a l o g i c a l purpose, though not correct . A score o f four was assigned o n l y to correct answers (numerator and denominator) and reasonably accurate efforts at shading. There were three basic types o f questions i n c l u d e d i n the test. These three types o f questions are r a n d o m l y dispersed throughout the test. F o u r types o f quest ions were used: 1 . T h e first type o f quest ion was designed to test f ract ion concept and est imat ion s k i l l . Students t yp i ca l l y w o r k w i t h halves, thirds, fourths and f i f ths w h e n f irst learning about f ract ions. Students w i t h a g o o d understanding o f basic f ract ion concepts Page 40 should be able to transfer knowledge learned about these smal ler numbers to any p r o b l e m w i t h a reasonably larger denominator (e.g. f i fteenths and twentieths). S o m e o f the quest ions requ i red shading , others requi red answers to shaded- in d iagrams. Quest ions i n this group were : 1, 8 , 9 , 10, 14. F igu re 3.3 shows an e x a m p l e o f a shading p r o b l e m w h i c h tested understanding o f a large denominator p r o b l e m (question 14): Figure 3.3 Example Shading Question With Large Denominator F igu re 3.4 shows an example o f a quest ion w i t h a large denominator w h i c h requires the student to name the f ract ion part shaded (question 1): Figure 3.4 Example Naming Question with Large Denominator 1. WHAT FRACTION IS REPRESENTED BY THE SHADED PORTION OF THIS PICTURE? •///, ///, /// •/// •/// '///. •///. •///. ///. •///. •///. ///, ///. ///. "•yrrw-*//* ///s ///* ///* f//* *//* *//* t//s *//* , *//** *//* *//* *//* *//* *//* /// /// ///. ///. ///. ///. ///• ///. ///, ///. ///. •///, •/// 7//, ///• ///. *//* '//* '/**• *//* '//* *//* *//* *//* *//* t//t ///* ***** *//* *//* rrnr, SS/i ///* //Si ///, ///, ///. ///. •///, •///. / / • . ///. •SS/. / / • . •///. ///t ///, SSS4 SS/t ///4 ///t ///, //fi S//4 ///, f/f* sss* ss/* ///* /S/4 /S/4 *sss '/// rs/s *sss '/// '/// */// */// •SS/ '/// '/// */// '/// >/// '/// '/// *sss // •sss. ^///. •///. ^///. '///. *///. *///. *sss. *///. *///. *///. *///> ///. ^/// Page 41 2 . T h e second type o f quest ion was designed to test the idea that f ract ion parts must be o f equa l s ize. A g a i n , some o f the questions requi red shading and others requi red answers to shaded - in d iagrams. Quest ions i n this group were : 3 , 4 , 5 , 11 , 12, 13a, 13b, 15. F igu re 3.5 i l lustrates a typ ica l example o f a shading p r o b l e m o f the second type (question 3) : Figure 3.5 Example Partitioning Question * 3 . SHADE IN 2/3 OF THIS CIRCLE F igu re 3 .6 (question 5) i l lustrates the same type o f quest ion except that here the student was requi red to determine what f ract ion was represented by the shaded por t ion . (Before the treatment, most students thought that this d i a g r a m i l lustrated the f ract ion 1/3). Page 42 F i g u r e 3.6 E x a m p l e N a m i n g Q u e s t i o n W i t h U n e q u a l P a r t i t i o n s *5 WHAT FRACTION IS REPRESENTED BY THE SHADED PORTION OF THIS DIAGRAM? 3. The third type of question was designed to test mastery of basic fraction concepts. These questions were in a sense equivalency tasks and go beyond basic understanding, however, in the context of the test and with the diagrams provided, students without formal instruction in equivalent fractions still should have been able to figure out the answer to these two problems. Questions in this group were: 6 and 7. Figure 3.7 (question 6) illustrates an example of this type of problem. Page 43 Figure 3.7 Example Shading Question with Equivalent Fractions * 7 SHADE IN 1/4 OF THIS RECTANGLE 4 . The fourth type o f p r o b l e m inc luded o n the test i n v o l v e d the abi l i ty to recognize very c o m m o n l y used f ract ions, 1/3 and 1/4. Students were requi red to recognize the shaded- in por t ion and wri te the correct numer ic s y m b o l . These questions were i n c l u d e d to see whether students w h o had l i tt le or no understanding o f fractions (demonstrated by an inabi l i ty to correct ly answer any o f the other questions o n the test) c o u l d st i l l answer these t w o questions correct ly s i m p l y f r o m rote m e m o r y . They are the most c o m m o n l y used fract ion i l lustrations and it is poss ib le that students are able to cor recdy answer these questions wi thout any understanding o f what a f ract ion is . F igure 3.8 (question 16) i l lustrates an example o f this type o f quest ion . Page 44 F i g u r e 3.8 E x a m p l e N a m i n g Q u e s t i o n w i t h E q u a l P a r t i t i o n s *16 WHAT FRACTION IS REPRESENTED BY THE SHADED PORTION OF THIS RECTANGLE? AAA/AA//////////////////•/////////////////•/////////// /////•////////•/////////////////////•//////////////•// V/A/A/AAAT/AA/A/AA/AAA/.A/A/AA/AA/AAA/A/A/A/^/\ E a c h quest ion was constructed w i t h a speci f ic purpose and diagnost ic intent. See A p p e n d i x C fo r i l lustrat ions o f each quest ion. T h e intent o f each question is descr ibed b e l o w : Q u e s t i o n 1: T h i s quest ion was i n c l u d e d to determine whether students c o u l d handle a f ract ion w i th a large denominator . T h e shading is purposely scattered throughout the d i a g r a m in a arbitrary manner . Q u e s t i o n 2: T h i s quest ion was i n c l u d e d to see whether students can wr i te a f rac t ion s y m b o l correct ly fo r a quest ion they should have seen m a n y t imes. It is l i ke l y that even students w i t h l i t t le or no understanding o f fract ions w i l l have been able to answer this quest ion . Q u e s t i o n 3: T h i s quest ion requi red the student to shade i n 2/3 o f a c i r c l e . It w a s a d i f f i cu l t quest ion and was inc luded to see whether the ch i ldren understood that f ract ion parts must be of equal size A N D were able to act on that knowledge by d i v i d i n g the d iagram correctly . Page 45 Q u e s t i o n 4 : T h i s quest ion was del iberate ly des igned to be c o n f u s i n g . A t f i rst g lance it l o o k s l i k e one ha l f o f the d i a g r a m is shaded, w h i c h i n fact it i s . H o w e v e r , the l ine separating the right hand side makes it seem l ike the answer c o u l d be 2/3. T h e student w h o understands that f ract ion parts must be the same size should have answered 1/2 or 2/4, not 2/3. Q u e s t i o n 5 : T h i s quest ion requi red students to real ize f ract ion parts must be equa l and answer 1/4, not 1/3. T h i s is a c o m m o n textbook p r o b l e m . Q u e s t i o n 6 : T h i s quest ion i s d i f f i c u l t fo r c h i l d r e n w h o d o not have a sound concept o f f ract ion and espec ia l l y a k n o w l e d g e o f what the numerator and denominator represent. T h e student had to real ize that a l though the question asked for 1/4, they were to shade i n 2/8 o f the d iag ram. T h i s was a v i s u a l p r o b l e m , not an equiva lency task because the students were p rov ided w i t h a d iagram as w e l l as a numer ic s y m b o l . T h e f ract ion d i a g r a m was d i v i d e d w i t h a broken rather than a s o l i d l ine to help students real ize that the l ines were not absolute. Q u e s t i o n 7 : T h i s was the probab ly the most d i f f i cu l t quest ion o n the test. It requ i red the student to ignore the so l id l ines d i v i d i n g the d i a g r a m into f i f ths and instead to shade i n 3/4. T h i s quest ion requi red a s o l i d understanding o f basic f ract ion concepts i n order to answer it correct ly . Q u e s t i o n s 8 , 9 , 1 0 , a n d 1 4 : These quest ions were des igned to see whether students c o u l d estimate large numbers o f parts and shade i n the correct response. Q u e s t i o n s 11 — 1 3 b : These quest ions a l l tested the f ract ion concept o f equa l parts o f a w h o l e . Q u e s t i o n s 15 a n d 1 6 : These t w o quest ions are des igned to see whether the student w i t h a g o o d concept o f f ract ion w i l l answer 1/4 to the top one and 1/3 to the bot tom one. They Page 46 were del iberately p laced together to m a k e the student th ink at f irst glance that both should be 1/3. Student Interviews T h e purpose o f the student in terv iews was to see h o w m u c h the students k n e w about f ract ions and to see whether there were any di f ferences between the three G r o u p s ' responses after the t reatment It i s often d i f f i cu l t to determine w h y a student answers a quest ion i n a part icular w a y and the interv iews were designed part ly as test o f the pretest/posttest, to determine whether it was an accurate evaluat ion o f student's k n o w l e d g e o f f ract ions. F i v e students were ran d om ly selected f r o m each Treatment G r o u p and in terv iewed twice . T h e first set o f interv iews occured d i rect ly after the pretest and pr io r to the treatment. T h e same students were interv iewed again after the treatment. Students were asked to exp la in what they were d o i n g as they worked . Interviews were aud io recorded. T h e procedure was the same for both interv iews. Students were asked the f o l l o w i n g seven quest ions d u r i n g the in terv iew ( A p p e n d i x A ) . Question 1: Students were asked to demonstrate an ab i l i t y to take a p iece o f paper (8 1/2 by 11 inches) and d i v i d e it into five sevenths. Question 2: Students were asked whether they thought f ract ion parts a l l had to be the same size. T h i s quest ion was repeated at the end o f the interv iew to see whether the interv iew process caused the student to change his t tank ing or i f he retained his o r ig ina l answer throughout the in terv iew process. Students were quest ioned to m a k e sure that they understood the di rect ions o f the test and i n part icular that they k n e w what part o f the fract ion the shaded area o f the d iagram represented. T h e y were asked to identity the shaded and unshaded f ract ion o f the same d iagram. T h e c o m m o n l y used fract ions 1/3 and 2/3 was used for this quest ion to see whether students might have been able to answer this f r o m m e m o r y wi thout real understanding. It was assumed that students w h o c o u l d correct ly answer this Page 47 quest ion but n o others o n the interv iew, were answer ing f r o m m e m o r y rather than f r o m an understanding o f f ract ions. Question 3: Students were asked to e x p l a i n what a f rac t ion w a s to someone f r o m outer space w h o had never heard the w o r d before. Question 4: Students were asked w h i c h was b igger , 8/12 o f a chocolate bar o r one w h o l e chocolate bar? T h i s quest ion was i n c l u d e d to see whether students thought 8/12 was larger than one w h o l e . It w a s also i n c l u d e d to see whether they understood the d i f ference between 8 out o f 12 chocolate bars and 8/12 o f one chocolate bar. Students w h o answered that 8/12 was larger had the quest ion repeated to them changing the w o r d a chocolate bar to one chocolate bar. Question 5: Students were asked to determine f r o m both a d i a g r a m and a numer ic representation, w h i c h was bigger , 2/3 or 4/6. T h e pictures were arranged to m a k e the v i s u a l c lue most obv ious . T h e purpose was to see whether students c o u l d operate f r o m a v i s u a l express ion o f f ract ions and not just numer ic . Question 6: T h i s quest ion asked students to ident i f y w h i c h o f t w o d iagrams represented 4/10 w h e n i n fact both d i d . T h e purpose was to see whether students c o u l d ident i fy the bot tom picture as 4/10, recogn i z ing that f ract ion parts must be o f equal s ize o r whether they w o u l d ignore the size o f the f ract ion parts and answer 1/7. Question 7 : T h e f i n a l quest ion i n the in te rv iew was a lso des igned to test the student's understanding o f f ract ion size. A t f irst glance it appeared to be 1/3, h o w e v e r the student w h o c lear ly understands fract ion parts must be the same size should have rea l i zed that the correct answer was 1/5. Page 48 Observations of the Treatment In addi t ion to the qual i tat ive data p rov ided by the student interv iews, the research des ign for this study i n c l u d e d a report and analysis o f the treatment process. D a t a was co l lec ted i n the f o r m o f notes by the researcher du r ing the treatment sessions fo r each G r o u p . G e n e r a l impress ions , no ise l e v e l , peer interact ions, student comments , frustrations, time o n task, etc were noted and reported fo r each G r o u p . A summary o f this data i s reported i n Chapter f o u r . The Treatment There were fou r G r o u p s , three Treatment G r o u p s and one C o n t r o l G roup .Parenta l consent fo rms (see appendix B ) were taken to the schoo l by the researcher one w e e k pr io r to the exper iment. T h e teacher was responsible fo r the distr ibut ion and co l lec t ion o f the s igned consent forms. Parents were g iven three poss ib i l i t ies : a l l o w i n g their c h i l d to participate i n the a l l phases o f the exper iment , a l l o w i n g their c h i l d to participate i n the experiment, but not be in terv iewed or not a l l o w i n g their c h i l d to participate at a l l . T w o parents d i d not want their c h i l d to participate at a l l and twelve a l l o w e d them to participate, but not be interv iewed. Visual Fractions Interactive Group Students selected for the first computer treatment were g iven an opportunity to explore the interact ive vers ion o f the V i s u a l Fract ions software program.There were o n l y four screen o n the entire p rogram. T h e first screen was s i m p l y a title screen w i t h an arrow button ind icat ing that the user shou ld c l i c k on the arrow to proceed to the second screen. T h e next three screens d isp layed different types o f f ract ion d iagrams - a hor izonta l bar d i a g r a m , a t w o - w a y rectangle d iagram (a square) and a c i rcu lar d iagram. E a c h d iagram was o n a separate screen w h i c h a lso d i sp layed the cor responding wr i t ten f ract ion symbols . T h e user was able to change the size o f the shaded port ion o f the f ract ion d iagram to graphical ly d isp lay fractions r a n g i n g i n s ize f r o m halves to twentieths. A s the s ize o f the f ract ion was changed, the Page 49 program suppl ied the correct mathematical s y m b o l . T h e c h i l d was able to see the association between the f ract ion represented by the d iagram and the s y m b o l . T h e s y m b o l f o r m o f each f ract ion was d i sp layed i n large numbers beside the graphic representation to help establish the l i n k between concept and f o r m . T h e student had the opt ion o f c l i c k i n g o n the numer ic s y m b o l and h i d i n g i t f r o m v i e w . Because o f the speed and accuracy o f the computer , m a n y fract ions c o u l d be d isp layed i n a very short t ime The user c o u l d m o v e between the four screens s i m p l y by c l i c k i n g o n the arrow button located at the bot tom o f each s c r e e n . . T h e Students i n this G r o u p were a l l o w e d one session o f up to an hour i n durat ion.Students were taken out o f their regular c lass room schedule i n t w o groups to use the computers . E a c h student w o r k e d ind i v idua l l y on a M a c i n t o s h computer . T h e researcher exp la ined that the ch i ld 's task w a s to d iscover h o w the p rog ram w o r k e d . T h e researcher remained i n the r o o m d u r i n g the students time o n the computers and answered general questions about the operat ion o f the computer , reso lved hardware and network p rob lems, exp la ined h o w to operate the m o u s e , etc. N o instruct ions were p r o v i d e d to the students o n h o w to operate the p rog ram. T h e researcher d i d not te l l the students h o w the p r o g r a m w o r k e d , but p rov ided pos i t i ve feedback and re inforcement to students as they began to understand h o w the p r o g r a m w o r k e d . T h e researcher encouraged students w h o were h a v i n g d i f f i cu l t y getting started. T h e researcher also made notes o n interesting student comments , questions and behav iors d u r i n g the computer session. Visual Fractions Noninteractive Group In order to ensure that any pos i t ive results gained f r o m the V i s u a l Fract ions treatment were not due to the novel ty o f us ing computers , the research design inc luded a noninteract ive ve rs ion o f the p r o g r a m w h i c h used the computer to v i e w the same f lash cards used by the F l a s h C a r d G r o u p . T h e o n l y d i f ference was that the f ract ion d iagrams were d isp layed Page 50 r a n d o m l y o n the computer screen. T h e user was able to cont ro l the rate at w h i c h the cards were presented but had no other contro l over the learning envi ronment . T h e procedure fo r this G r o u p was the same as the Interactive G r o u p . Instructions were the same and this G r o u p was p r o v i d e d w i t h the same opportunity to spend as l o n g as they wanted up to one hour us ing the software p rogram. Flash Card Group Students selected for the F l a s h C a r d G r o u p had an opportunity to explore a more tradit ional m e t h o d o f inst ruct ion o n fract ions. Students i n this G r o u p were p r o v i d e d w i t h a set o f f ract ion f lash cards designed by the researcher designed to s imulate the noninteractive vers ion o f the V i s u a l Fract ions p rogram. See A p p e n d i x D for an example o f the F l a s h t C a r d s . W i t h the cards, students were able to create fractions by l i f t i n g the f laps o n the cards to reveal either a f ract ion d i a g r a m or the corresponding writ ten s y m b o l . These cards were des igned to be very s imi la r to the d isp lay i n the V i s u a l Fract ions p rogram, except fo r two c r i t i ca l d i f ferences. F i rst , the c h i l d i n the F l a s h C a r d G r o u p was not rea l l y i n cont ro l o f his o w n learn ing . In the V i s u a l Fract ions treatment, there was a real sense o f cont ro l over the learn ing situation; ch i ldren felt as though they were m a k i n g the fractions through their o w n efforts ( i . e . , by s l i d i n g the mouse button back and forth o n the d iagram) T h i s G r o u p had the same amount o f t ime to examine the materials and f igure out h o w they w o r k e d . Students were taken out o f their regular c lass room schedule i n groups o f three or four . T h e researcher remained i n the r o o m dur ing the session and observed the act iv i ty o f the ch i ld ren . Quest ions were answered about the cards and what to d o w i t h them i n a vague and general w a y , usua l l y by respond ing w i t h another quest ion , i .e. , What do you think you could do with the cards? T h e researcher made notes o n student behavior , comments and Page 51 group interactions d u r i n g the sessions. T h e researcher administered the treatment to two groups o f four students and one group o f three students. These students were r a n d o m l y selected f r o m the two grade four classes at S c h o o l 2. A l l students rece ived the treatment o n the same day. C o n t r o l G r o u p T h e C o n t r o l G r o u p d i d not receive any treatment. T h e y wrote the test i n the m o r n i n g w i t h the students selected to be i n the F l a s h C a r d G r o u p and d i d not k n o w ahead o f time w h i c h G r o u p they were i n . T h e y rewrote the same test at the end o f the afternoon o n the same day. D a t a A n a l y s i s T h e data f r o m the pretest and posttest was c o d e d o n a f i ve point scale f r o m zero to four. Responses were assigned a score based upon h o w c lose l y they approx imated a correct understanding o f the quest ion. F o r example , i f the student made no attempt to answer the quest ion , they scored zero. I f they made some effort but there was no apparent l o g i c to their response, they score one. I f there was a l o g i c a l (though not necessari ly correct) reason fo r the denominator or the numerator , they scored two . I f there was a l o g i c a l reason for both the numerator and denominator , they scored three. If the answer was correct , they scored four . These scores were analyzed us ing B M V D P for a repeated measures A N O V A . D a t a co l lected from the in terv iew process was summar i zed and overa l l general izat ions, i n d i v i d u a l and G r o u p responses were reported. T h e researcher's observat ions o f the treatment process are also reported. T h e qual i tat ive data was analyzed by the researcher. Page 52 C h a p t e r F o u r R e s u l t s In this chapter, an analysis o f the f indings f r o m both the quantitative and the qual i tat ive part o f the study are presented. F i rs t is a report o f the three treatment sessions i n c l u d i n g student per formance, attitudes, etc. T h e second part focuses o n the results o f the quantitative data. A repeated measures A N O V A was used w i t h one treatment factor (the four Groups ) and one repeated measures factor (the pretest and posttest). T h e f ina l part o f the chapter presents a report o f the student interv iews i n c l u d i n g general f ind ings , G r o u p s imi lar i t ies and ind i v idua l reports where relevant. T r e a t m e n t S e s s i o n R e p o r t T h e f o l l o w i n g i s a general report o f the procedures o f each o f the G r o u p s i n the study. V i s u a l F r a c t i o n s I n t e r a c t i v e T r e a t m e n t T h e ch i ld ren were exposed to the treatment i n two groups. T h e computer lab was operat ing ef f ic ient ly fo r the first treatment and there were suff ic ient computers for each student to w o r k by themselves . F o r the second group, there were some hardware p rob lems and this caused some interference w i t h the session. These students were l i m i t e d to approx imately 50 minutes rather than the f u l l hour exposure. T h e f o l l o w i n g comments are w i t h regard to the general cond i t ion o f the treatment sessions and apply to both sessions. M o s t o f the students were general ly enthusiastic and cooperat ive. T h e y were s i m p l y issued the chal lenge to discover how the program worked. T h e ch i ld ren seemed confused at first and not certain h o w to go about the assigned task. M a n y o f them ca l led for c la r i f i cat ion and some stubbornly demanded more in format ion . T h e researcher kept c o m m u n i c a t i o n to an Page 53 absolute r run imum. T h e o n l y comments made were designed to encourage the students to keep t ry ing , to p l a y w i t h , and to exp lore the p rogram. W h e n students thought they k n e w what the p rog ram was about, they were keen to share it w i t h the researcher and each other. T h e researcher of fered posi t ive praise for the d iscovery and encouraged any student- init iated peer interaction. M o s t o f the students d i d not want to leave the lab w h e n the hour was up . There was however , a s m a l l group o f students f r o m one c lass were w o r k i n g o n a d inosaur project w h o wanted to go back and f in i sh it . These students said that n o r m a l l y they w o u l d have preferred to stay and w o r k o n computer , but the d inosaur art project was more appeal ing . A l l o f the students i n both sessions expressed a desire to use the p r o g r a m again. Visual Fractions Noninteractive Treatment These students were d i v i d e d into two different groups for treatment. One o f these groups exper ienced s i m i l a r hardware p rob lems as one o f the Interactive G r o u p s , however , the p r o b l e m d i d not interfere w i t h the learn ing process. Students i n the Noninteract ive V i s u a l Fract ions Treatment had a very different reaction pattern to the treatment sessions. They were very keen at f i rst and understood r ight away what was expected o f them. A f t e r approx imate ly f i ve minutes , most o f them felt they understood what the p rog ram was about and h o w to operate it. A f t e r approx imately f i f teen minutes , however , they became bored and terminated the p r o g r a m vo luntar i l y . O n l y four out o f the twenty two students w h o used this v e r s i o n o f the p r o g r a m expressed a desire to use it again . T h e f o l l o w i n g prob lems affected both the Interactive and the Noninteract ive G r o u p s : There were some prob lems w i t h hardware mal funct ions i n the M a c i n t o s h lab. These p rob lems d i d not, however , appear to affect the learn ing process and were dealt w i t h p r o m p t l y . Page 54 One potent ial ly very serious p r o b l e m in the treatment process m a y have been the interference f r o m a d inosaur project w h i c h the students i n one c lass were w o r k i n g o n . It seemed that this unstructured learn ing act iv i ty was not the n o r m for these students and they c lear ly d i d not want to miss out o n this unique opportunity to p lay w i t h p last ic ine and chat w i t h their f r iends in an i n f o r m a l session back i n the c lass room. T h i s probably interfered w i t h their ab i l i t y to concentrate o n and benefit f u l l y f r o m the treatment. H o w e v e r , as students f r o m this c lass were d i v i d e d r a n d o m l y , ha l f into each o f the two computer treatments, this p r o b l e m w o u l d have affected both G r o u p s and not just one. T h u s the results , fo r the two different computer methods w o u l d not have been contaminated i n favour o f one G r o u p or the other because o f this r a n d o m placement o f students equal ly into both G r o u p s . Flash Card Treatment Students i n this G r o u p were d i v i d e d into three groups and o f three or four students each. T h e groups were exposed to the treatment i n a smal l o f f i ce conta in ing one round table on w h i c h a l l the f lash cards were p laced . T h e students i n a l l three groups seemed to be extremely k e e n , h igh ly mot ivated and very cooperative. O n l y one student tr ied to examine the cards independently and he was very q u i c k l y conscr ipted into the cooperat ive activit ies o f h is peers. These students were g iven the same direct ive as the students i n the computer sessions: to examine the materials and try to f igure out h o w they w o r k e d or what they c o u l d be used for . M o s t o f the students expressed a desire to have a set o f the cards i n their c lass room and c o u l d th ink o f several activit ies associated w i t h them. T h e students i n a l l three groups sorted the cards by denominator and ranked them f r o m highest to lowest . A t no point d i d the researcher ever suggest that there was a right w a y to use the cards. S o m e of the students recognized equivalent fractions and started to categorize them accord ing to equ iva lency . N o n e o f the students wanted to leave w h e n their t ime was up . T h e f o l l o w i n g comments pertain to both the F l a s h C a r d and the C o n t r o l G r o u p : Page 55 A l t h o u g h the results do not indicate greater improvement , the students i n the F l a s h C a r d G r o u p had several advantages, some o f w h i c h were external and unavoidable and others w h i c h were in t r ins ica l l y part o f the research des ign . N o n e o f the students i n any the three f lash card groups wanted to leave w h e n the session t ime was up and a l l sa id they rea l ly en joyed p l a y i n g w i t h the cards. It m a y be that the smal l group situation and the pr i v i lege o f be ing chosen over their classmates m a y have contr ibuted to the intense interest i n the learn ing situation. T h e other Treatment G r o u p s d i d not enjoy this same advantage because everyone i n the class (wi th the except ion o f those w h o d i d not have parental permiss ion) part ic ipated i n the study. A l s o , i n the F l a s h C a r d G r o u p , the novel ty o f the cards (they were br ight , c o l o u r f u l , laminated cards w i t h ve l c ro f laps) m a y also have in f luenced the student's behaviour . T h i s h i g h degree o f concentration and enjoyment m a y have been the result o f these unforeseen advantages. Because o f the smal ler sample size, these students were exposed to the treatment i n smal l groups o f three o f four . It was c lear f r o m the researcher's point o f v i e w that these students were more on task and concentrating m u c h more intently that the students i n either o f the other t w o treatments. It is poss ib le that the presence o f the researcher i n a s m a l l r o o m , s i t t ing at one r o u n d table w i t h a handfu l o f students m a y have been responsib le for focus ing their attention o n the task at hand. A d d i t i o n a l l y , these students k n e w that they were chosen ( randomly) to part ic ipate i n the treatment whereas i n the other G r o u p s , everyone w h o returned their permiss ion sl ip was permitted to participate. Th i s exc lus iv i t y m a y also have contr ibuted to their greater enthusiasm. T h e smal l group m a y also have been a contr ibut ing factor i n the peer interact ion w h i c h was c lear ly very intent i n this Treatment G r o u p . In fact, o n l y one student tr ied to examine the cards independently and was eventual ly d rawn into the group act iv i t ies o f h is c lassmates. F i n a l l y , this c lass had just completed a unit on fractions and had a greater in i t ia l understanding o f fractions i n general p r io r to the treatment .This instruct ional advantage was c lear ly evident from the pretest results and it m a y have been Page 56 responsible for the absence o f s ignif icant improvement f r o m the pretest to the posttest i n spite o f the ev ident enthusiasm and concentrat ion on the treatment (i.e., a ceiling effect). Quantitative Data Collection T h e f o l l o w i n g is a general report o f the quantitative data co l lec t ion , consist ing o f a statistical report o f the pretest and posttest results and a summary reject ing the N u l l Hypothes is . Pretest/Posttest " T h e pretest/posttest was administered to a l l 6 4 subjects in the study. There were four G r o u p s in the study, three treatment G r o u p s (two C o m p u t e r Treatment and one F l a s h C a r d G r o u p ) , and the fourth G r o u p w h i c h served as a cont ro l and d i d not rece ive any treatment. T h e same test was used as both the pretest and the posttest. T h e total poss ib le score o n the test was 64. T h e results o f the tests were ana lyzed us ing B M D P 2 V for a two - fac to r ( four G r o u p s by t w o Tests) analys is o f var iance w i t h repeated measures (pretest and posttest) on the second factor . Tab le 4.1 presents a summary table o f statistical s igni f icance for each o f the two factors i n the study and their interact ion. Factor A represents the m a i n effect Groups o f w h i c h there were four . T h e A N O V A indicates that w h e n considered by themselves, there is no s ign i f icant d i f ference between any o f the four G r o u p s i n the study. Factor B represents the m a i n effect Tests o f w h i c h there were two , the pretest and the posttest. T h e same test was administered as the pretest and the posttest. T h e A N O V A indicates that there is a s ignif icant d i f ference between the scores o n the pretest and the posttest at the .01 leve l Page 57 T a b l e 4 . 1 C o m p a r i s o n o f the F o u r G r o u p s o n the T w o Tes ts Source S u m o f Degrees o f M e a n F P Squares F r e e d o m Square A (Groups) 8 7 5 . 5 6 3 2 9 1 . 8 5 1.58 0 . 2 0 4 5 E r r o r 1 1 0 9 . 9 3 6 0 1 8 5 . 1 6 B (Tests) 3 4 6 . 9 7 1 3 6 4 . 9 7 8 . 7 5 0 . 0 0 4 4 A B (Interaction) 4 8 2 . 2 4 3 1 6 0 . 7 4 4 . 0 5 0 . 0 1 0 9 E r r o r 2 3 7 8 . 7 5 60 3 9 . 6 4 T a b l e 4 . 2 shows a c o m p a r i s o n o f the means scores fo r each G r o u p . F i g u r e 4 .1 d isp lays the d i f ference between the mean scores for each G r o u p . The m a x i m u m poss ib le score on the test w a s 6 8 . T h e greatest d i f ference between pretest and posttest means occurs w i t h G r o u p 1, the Interactive V i s u a l Fract ions G r o u p . G r o u p 1 had an a lmost 10 point (15%) increase i n mean scores between the two tests. T h e results indicate that the C o n t r o l G r o u p had an a lmost zero (43.8 - 43.9) increase i n overa l l mean scores. T h e di f ference i n mean scores between the pretest and posttest fo r G r o u p 2 and 3 were both less than three points . T h e pretest score fo r G r o u p 3 was m u c h higher than the other three G r o u p s . The i r posttest score however , was very s i m i l a r to the posttest score for G r o u p 1, ind ica t ing an increase o f less than t w o points . G r o u p 3 (the F l a s h C a r d G r o u p ) and G r o u p 4 (the C o n t r o l G r o u p ) were c o m p r i s e d o f a r a n d o m select ion o f students f r o m two different c lasses, one o f w h i c h had just c o m p l e t e d an intensive introductory unit o n fract ions. T h e other two G r o u p s d i d not have a s imi la r experience that year. It is probable that this addit ional instruct ional t ime accounted for the h igher pretest score for G r o u p 3 . Page 58 T a b l e 4 . 2 G r o u p M e a n S c o r e s a n d S t a n d a r d D e v i a t i o n s G r o u p Pretest S D Posttest S D M e a n s A l 4 1 . 2 0 1 2 . 4 0 . 5 1 . 0 0 9 . 4 8 4 6 . 1 0 A 2 4 1 . 0 0 1 1 . 3 9 4 3 . 5 0 8 . 9 6 4 2 . 3 0 A 3 4 9 . 3 0 7 . 9 5 5 1 . 0 0 7 . 5 9 - 5 0 . 1 0 A 4 4 3 . 8 0 1 1 . 7 9 4 4 . 0 0 1 2 . 6 2 4 3 . 9 0 C o l M e a n s 4 2 . 7 0 4 7 . 2 0 4 5 . 0 0 Note : A l = V i s u a l Fract ions Interactive G r o u p A 2 = V i s u a l Fract ions Noninteract ive G r o u p A 3 = F l a s h C a r d G r o u p A 4 = C o n t r o l G r o u p T h e graph o f the overa l l means (f igure 4.1) , suggests that the s igni f icant factor B (tests) can be exp la ined b y the d i f ference between pretest and posttest scores for G r o u p 1 ( A l ) . T h i s d i f ference between the pretest and posttest scores does not seem to h o l d for the other G r o u p s because the l ines appear to be para l le l fo r these G r o u p s w i t h i n chance. T h e graph suggests that a spec ia l study o f the interact ion effects shou ld be made . Page 59 Figure 4.1 Comparison of Means 55 -r 3 5-1 1 1 1 A1 A2 A3 A4 Visual Visual Flash Card Control Fractions Fractions Group Group Interactive Noninteractive Table 4.3 Calculation of Interaction Effects Inter- cell row col grand action value mean mean + mean . = effect A1B1 41.2 46.1 42.7 + 45.0 -2.6 A1B2 51.0 46.1 47.2 + 45.0 = + 2.7 A2B1 41.0 42.3 42.7 + 45.0 = + 1.0 A2B2 43.5 42.3 47.2 + 45.0 = + 0.5 A3B1 49.3 50.1 42.7 + 45.00 = - 1.2 A3B2 44.0 43.9 47.2 + 45.00 = -2.3 A4B1 43.8 43.9 42.7 + 45.00 = -2.3 A4B2 44.0 43.9 47.2 + 45.00 = - 2.0 F igure 4 .2 depicts a compar ison o f the interact ion effects ( B A ) . W h e n the interaction effects are graphed, the results show that fo r the G r o u p 1, the posttest per formance was re lat ive ly Page 60 superior to the pretest per formance. Th i s superior ity i n performance o n the posttest relat ive to the pretest, however , does not h o l d for the other three Groups . T h e graph shows that the A B effects l i e c lose to zero for the other three G r o u p s and therefore the same results as for the f irst group cannot be substantiated.Although the interact ion effect, A B , i s not s igni f icant at the .01 l e v e l , the probabi l i t y o f error is so c lose to 0 .01 (p = 0 . 0 1 0 9 ) , that it i s reasonable to assume there an interaction effect is w i th G r o u p 1, the V i s u a l Fract ions Interactive G r o u p . F i g u r e 4.2 I n t e r a c t i o n E f f e c t s 3 T I nter-action Effects Fractions Fractions Group Group Interactive Noninteractive S u m m a r y : There was no d i f ference found between the groups. T h e s igni f icant d i f ference between the pretest and the posttest c o u l d not be interpreted because o f the s igni f icant interaction. A n examinat ion o f the interaction effects revealed a s ignif icant difference i n the mean score for Page 61 treatment one, the V i s u a l Fract ions Interactive G r o u p . T h e other groups d i d not change s ign i f icant ly between the pretest and the posttest Student Interviews A p p r o x i m a t e l y 12 parents chose the opt ion o f a l l o w i n g their c h i l d to participate i n the study but not to be interv iewed. M a n y o f the ch i ldren i n the class were eager to participate in the in terv iew process and requested to be chosen. F i v e ch i ld ren f r o m each Treatment G r o u p were r a n d o m l y selected for interv iews f r o m those w h o had been g iven parental permiss ion . C h i l d r e n were taken out o f their regular c lass room routine and in terv iewed pr ivately for approx imate ly 10 to 15 minutes , i n the m o r n i n g pr io r to the computer treatment and immedia te l y after w r i t i n g the pretest. E a c h c h i l d was interv iewed a second time as soon as poss ib le after w r i t i n g the posttest. N o one had to wai t longer than three hours to be in terv iewed after the posttest. Computer treatment Groups. B e c a u s e o f r a n d o m ass ignment o f c h i l d r e n into Treatment G r o u p s , s i x o f the ch i ld ren interv iewed at S c h o o l 1 were ass igned to the Noninteract ive G r o u p and o n l y four were i n the V i s u a l Fract ions Interactive G r o u p . Interviews were conducted at a table i n the corr idor due to l imi ted space avai lable i n the schoo l . There was considerable interference due to noise and other distract ions, however this d i d not appear to affect any o f the chi ldrens' abi l i ty to concentrate o n the questions. M o s t ch i ld ren d i d not appear to m i n d m i s s i n g their c lassroom activ i ty to participate i n the in te rv iew , and appeared to try their best at answer ing the quest ions. H o w e v e r one b o y was not part icular ly interested i n t ry ing to answer the questions and dur ing the second interv iew he c o m p l a i n e d that he was m i s s i n g sharing time. It is poss ib le that he k n e w more than he was w i l l i n g to revea l . Control and Flash Card Groups. A t S c h o o l 2 , none o f the c h i l d r e n i n the C o n t r o l G r o u p were in terv iewed. F o u r ch i ld ren f r o m one class were in te rv iewed and one f r o m the Page 62 other c lass. A g a i n , the uneven representation f r o m the one c lass was due to r a n d o m select ion o f students. A pr ivate o f f i ce was p rov ided for the purpose o f in te rv iew ing students and a l l the students appeared to answer the questions to the best o f their abi l i ty . T h e data f r o m one student i n the Noninteract ive G r o u p and one student f r o m the F l a s h C a r d G r o u p was not usable as second interv iews c o u l d not be obtained for these t w o students because they, were absent d u r i n g the scheduled t ime fo r the second interv iew. T h e interv iew process d i d not appear to s igni f icant ly affect the chi ldrens' learn ing . A t the beg inn ing o f the in terv iew, every c h i l d was asked whether they thought f ract ion parts had to be o f equal s ize. T h i s quest ion was repeated at the end o f the interv iew. In every case, the ch i ld ren d i d not change their o r ig ina l answer to this quest ion dur ing either the first o r second in terv iew. Students w h o d i d change their m i n d , d i d so between the end o f the first in terv iew and p r io r to the second interv iew. It is therefore reasonable to conc lude that students d i d not change their m i n d as a result o f the interv iew process. Question One: The task was to show 517 of a piece of paper by tearing it. Th is quest ion appeared to be quite d i f f i cu l t fo r most students i n grade four. M a n y were not able to attempt and most d i d not try to measure o r m a k e their pieces at a l l s imi lar i n size pr ior to the treatment. A f t e r the treatment sessions, a few students d i d m a k e some effort to measure , but i n most cases this was not very successfu l . Visual Fractions Interactive. Students i n the V i s u a l F rac t ions Interact ive G r o u p responded to this quest ion w i t h a large var iance i n abi l i ty . B e f o r e the treatment, two o f the students in terv iewed w o u l d not even attempt the quest ion. T h e other two were able to d i v ide the paper into seven pieces and give back f i ve o f them, but made no attempt to m a k e the p ieces even rough ly the same size. N o one made any effort to measure their w o r k at a l l . A f t e r the treatment, dur ing the second interv iew, every student at least attempted the quest ion w i t h v a r y i n g degrees o f success. T h i s time two of the students tr ied to measure Page 63 very carefu l l y i n order to have pieces o f the same size. Three students were s t i l l unable to answer the quest ion correct ly , but it was apparent that they at least understood what to do even though they c o u l d not d o i t T h i s task appears to be very d i f f i cu l t fo r students o f this age. N o n i n t e r a c t i v e . Students i n the Non in te rac t i ve G r o u p had a greater i n i t i a l va r iance i n their response, however , each student retained their o r ig ina l answer from the f irst to the second in terv iew. Three o f the f i ve ch i ld ren i n this group appeared to understand what was be ing asked o f them. O f these three, no one tr ied to measure the pieces o f paper or m a k e the fraction parts i n any w a y s imi lar , let alone the same size. O f the r e m a i n i n g t w o ch i ld ren , one student w o u l d not even try the p r o b l e m . T h e other student ended up w i t h eight unequal p ieces o f w h i c h she kept three and returned f i ve . A s igni f icant f m d i n g w i t h this quest ion is that fo r a l l f i ve ch i ld ren i n this group, their in i t ia l approach to the quest ion and f ina l answer remained almost ident ica l from the first to the second interv iew. The treatment d i d not appear to affect their abi l i ty to answer this part icular quest ion at a l l . F l a s h C a r d s . Students i n the F l a s h C a r d G r o u p were s i m i l a r to the Non in te rac t i ve G r o u p in that they had a large var iance in their abi l i ty to answer this question and no one changed their o r ig ina l response. O n e boy measured very carefu l ly and correct ly answered the quest ion. Three others were able to d i v ide the paper into seven pieces, and g ive back f i ve o f them, but none o f them made any attempt to measure or m a k e the pieces equal . T h e f i f th student d i d not use the entire p iece o f paper. H e d i v i d e d part o f it into f i ve unequal pieces and w h e n asked what the left over p iece was , he c a l l e d it two blanks. Page 64 Question 2: Look at the diagram. One part is shaded and two are not. All the parts in this diagram are the same size. y/////////////ft//*//, / / / / / ^ / / / / / / / / / / / / / / ' / / / / / / / / / > / / / / / / / / / / . ' / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / ' / / / / / / / / / / / / / / / / / / / / ' / / / / / / / / / / / / / / / / / / / / ' / / / / / / / / / / / / / / / / / / / / '/////f////'////t/'/f ' / / / / / / / / / / / / / / / / / / / / a) Do fraction parts always have to be the same size? (yes) b) What fraction is represented by the shaded part of this diagram? (1/3) c) What fraction is represented by the unshaded part? (2/3). T h i s set o f quest ions was i n c l u d e d to see whether students understood the not ion o f equal parts o f a w h o l e and also to see whether they understood the language used dur ing the in te rv iew process (i .e., shaded, unshaded, represented etc.) T h e s imple f ract ion 1/3 w h i c h is c o m m o n l y used i n textbooks was chosen to establish whether the c h i l d understood the language used. It was d iscovered that several ch i ld ren d i d not. T h e y spoke i n terms o f one two (meaning 1 o n the top and 2 on the bottom) rather than one third. A l s o , it was apparent that some ch i ldren c o u l d recognize the fract ion 1/3 f r o m rote m e m o r y w i th very l ittle understanding o f f ract ions. T h e y c o u l d for example correct ly respond one third but not k n o w w h i c h number corresponded w i th the numerator and w h i c h the denominator . (Note: the terms numerator and denominator were not actual ly used; instead students were asked, fo r example , what the bot tom number represented i n 1/3.) V i s u a l F r a c t i o n s I n t e r a c t i v e G r o u p . A l l four students i n the V i s u a l F rac t ions Interactive C o m p u t e r G r o u p responded no i n the in i t ia l interv iew to part A o f this question. A f t e r the treatment a l l four students had changed their m inds and answered yes, fraction Page 65 parts do all have to be of the same size at the beg inn ing o f the second interv iew. E v e r y student was able to respond 1/3 and 2/3 to part B and C. Visual Fractions Noninteractive Group. I n i t ia l l y , a l l f i v e students i n this G r o u p responded no and o n l y one student changed her m i n d dur ing the second in terv iew. She said that she changed her m i n d because the computer had a lways d rawn the f ract ion parts the same size. F o u r o f these students were able to recognize 1/3 and 2/3. O n e student d i d not have the verba l language but requested to wr i te h is answer. H e wrote 1/2 and 2/1 and exp la ined these responses as 1 on the top and 2 parts o n the bot tom and v i c e versa. H e d i d not change his response after the treatment. T h e other student w h o c o u l d not recognize these fractions in i t ia l l y was able to recognize both after the treatment. The Flash Card Group. F o u r o f the ch i ld ren i n this G r o u p d i d not th ink f rac t ion parts had to be the same size and one be l ieved they d i d . A f t e r the treatment, no one changed their t m i n d . E v e r y o n e i n this G r o u p was able to answer 1/3 and 2/3 dur ing both in terv iews , except fo r one student w h o responded two into one and one into two. H e used these terms interchangeably and didn't seem to k n o w w h i c h should correspond w i t h the numerator and w h i c h the denominator . H e made no improvement by the second in terv iew. Question 3: Describe what a fraction is to someone from outer space who had never heard the word before. Th is quest ion resulted i n a large var iance i n student responses. S o m e students were not able to answer at a l l , others gave a schoo l context -based or text b o o k type answer, and a f e w descr ibed real l i fe part i t ioning activit ies. Visual Fractions Interactive Group. There were t w o d i f ferent types o f response to this quest ion. T w o o f the ch i ld ren (both f r o m the same class) responded i n a context -based manner (e.g., A fraction is parts of a block and some are shaded). T h e other two ch i ldren Page 66 were able to talk about fractions i n a more abstract manner (e.g. A fraction is a part of something, for example a chocolate bar, that you want to divide into parts and share). A l l o f these ch i ld ren answered this quest ion i n a very s imi lar manner before and after the treatment. O n e c h i l d added the not ion o f equa l - s i zed parts to his def in i t ion o f f ract ion, thus ve rba l i z ing a more sophist icated understanding o f f ract ion after the treatment. V i s u a l F r a c t i o n s N o n i n t e r a c t i v e G r o u p . O f the five c h i l d r e n i n this G r o u p , one was unable to g ive any sort o f answer , t w o answered f r o m the context o f a schoo l -based task ( i .e. , a m a t h quest ion i n v o l v i n g shapes, shad ing i n , numbers , etc.) and t w o were able to answer i n a more general context (i .e., parts o f something) . A f t e r the treatment, the student w h o c o u l d not answer was able to g ive a reasonable def in i t ion (i.e., p ieces that y o u d i v i d e up) and another student shif ted f r o m a context -based idea to a more general understanding o f f ract ions ( i .e. , sp l i t t ing someth ing into G r o u p s ) . T h u s , two o f the five c h i l d r e n i n this G r o u p were able to verba l i ze a more sophist icated understanding o f f ract ions after the treatment. F l a s h C a r d G r o u p . T h i s G r o u p was s i m i l a r to the other two G r o u p s i n that t w o o f the ch i ld ren referred to fract ions as part o f something and the other three responded i n the context o f a mathemat ica l f o r m u l a (i.e., f ract ions are numbers w i th a l ine i n between them and the top number means the shaded part and the bot tom is the total parts.) T h e on ly change i n this G roup 's answers f r o m one in terv iew to the next w a s i n one boy w h o couldn' t p rov ide a def in i t ion d u r i n g the first interv iew and was able by the second in terv iew to say Fractions are Math.... Question Four: Which is bigger, 8112 of a chocolate bar or one whole Chocolate Bar? It seemed from the answers to this quest ion that some ch i ld ren do not d is t inguish between fraction p rob lems o n discrete vs . cont inuous objects. W h e n ch i ld ren responded 8/12, the quest ion was c lear ly repeated emphas i z ing 8/12 of a chocolate bar. F o r example , one g i r l Page 67 appeared to be w o r k i n g f r o m a cont inuous object concept o f f ract ions. Because o f this she be l ieved that 8/12 was bigger that one w h o l e . She saw 8/12 as 8 out o f 12 chocolate bars even though the quest ion was carefu l l y restated as 8/12 o f one chocolate bar. H e r understanding o f this quest ion had not changed by the second in terv iew. It is p robably because o f this type o f th ink ing or ientat ion that she answered one seventh to the quest ion w h i c h asked her to name the f ract ion i n quest ion 6. T h e correct answer was 4/10 but she c o u l d o n l y see 7 separate boxes. She d i d not see that the big box w a s i n fact 4 out o f 10 parts o f one box . A n o t h e r interest ing observat ion about chi ldren's responses to this quest ion w a s the tendency to def ine 8/12 i n terms o f be ing s imi la r i n size to 1/2 or 1/4, f ract ions w i t h w h i c h they were apparently more fami l ia r . Visual Fractions Interactive Group. A t f i rst o n l y t w o o f the fou r c h i l d r e n i n this G r o u p be l ieved that one w h o l e was larger. One student was not sure and the other thought 8/12 was larger. A f t e r the treatment, a l l four ch i ld ren responded that one w h o l e was bigger , and three o f them c o u l d c lear ly exp la in w h y 8/12 was o n l y part o f one w h o l e . Visual Fractions Noninteractive Group. In this G r o u p , three c h i l d r e n b e l i e v e d one w h o l e was larger than 8/12 and two thought 8/12 was larger. A f t e r the treatment, o n l y one c h i l d changed his m i n d and thought that perhaps one who le was larger after a l l . Flash Cards. In this G r o u p a l l f i ve ch i ld ren thought that one w h o l e w a s larger. Three o f these were able to say they thought so because one w h o l e was the w h o l e th ing and 8/12 was just part. One c h i l d referred to 8/12 as be ing like 1/4. N o one changed their m i n d between the two in terv iews . Question Five: Which diagram represents a bigger fraction or are they both the same size? (The top diagram illustrates 3 equal parts with two shaded and the bottom illustrates 6 equal Page 68 parts with 4 shaded. The area of the two diagrams is the same and they are positioned to make it visually obvious that the shaded portion is the same in both cases,) ********* ********** ********* /A/ A/ A/ AA ///////// ******** ******** ***** *** ******** ***** A/X ******** ******** '^ /^//// ******** ******** ' / / / / / / / ^ / / / / / / / / / / / / / / V A ' / / / / / / / / / / / / / / / / / . • *****************4 * *****************' ******** / / / / / / / / / / / / / / . **************. 4*4*4*4**4*4*4*4*4*4*4*4*4*. / / / / / / / / / / / / / / , / / / / / / / / / / / / / / , **************. 4*4*4*4*4*4*4*4*4*4*4*4*4*4*. 4*4*4***4*4*4*4*4*4*4*4*4*. **************. **************. **************. ****************. ****************. * **** ** * ** *. - * * * *******. **************. f******* ******** f******* ******** ******** ******** ******** ******** ******** ******** ******** ******** ******** ******** ******** ******** ********* 4 ********* *** , ********* 4 ********* *** 4 ******** ****** 4 ******** 4 ******* ****** 4^ * * X X -• A/, " ****** ******' ' *** *** ****, *** / ***' *** *** ***** ***** *** - ' *** *** * * ***** ***** ***** **** **** **** **** ****** ***** **4 ********, **4 ********. '4 ********* ******** ' ****** **** ******* ******* ****** ******* **** ** ***** ***** **** ****** *** A l l o f the ch i ld ren w h o be l ieved one fraction was larger than the other chose 4/6 because it had more pieces. E v e n w h e n asked to ignore the numbers and just concentrate o n the d iag ram, m a n y o f these ch i ld ren s t i l l insisted that the d iagram w h i c h had four shaded pieces was larger. These ch i ld ren were not operat ing f r o m a concept o f f ract ion as a representation o f area. C l e a r l y fractions were associated here o n l y w i t h numbers and the larger the numbers , the b igger the fraction. V i s u a l F r a c t i o n s I n t e r a c t i v e G r o u p . T w o o f the c h i l d r e n i n this G r o u p responded i n the f irst in terv iew that 4/6 was bigger (because there were more pieces) and the other two thought they were both the same (one because it was obv ious from l o o k i n g at the picture and the other because o f a mathematical ca lculat ion. A f t e r the treatment, a l l four ch i ldren rea l i zed that 2/3 o f the d iagram was the same as 4/6. T h e two w h o changed their m i n d both responded that they c o u l d remove the line d i v i d i n g the d iagram into sixths and m a k e it just thirds. B o t h ch i ld ren seemed pleased to have d iscovered the s imi lar i ty between 2/3 and 4/6 by themselves . V i s u a l F r a c t i o n s N o n i n t e r a c t i v e G r o u p . O f the five c h i l d r e n i n this G r o u p , o n l y one reported i n the first in terv iew that the fractions were the same size. The others a l l be l ieved Page 69 that more parts meant b igger f ract ion. A f t e r the treatment, o n l y one student changed her m i n d and dec ided that by l o o k i n g at just the picture y o u c o u l d see they were the same size. T h e student w h o o r ig ina l l y be l ieved that the fractions were the same size started to change his m i n d and say that 2/3 was bigger , but after recons ider ing , he c o n f i r m e d that they were rea l ly the same. Flash Card Group. T w o o f the ch i ld ren i n this G r o u p responded that 2/3 was larger because the numbers were smaller . T h e y be l ieved that the smal ler the number , the larger the va lue . One c h i l d exp la ined this was because his teacher to ld them 2 was larger than four thousandths. These ch i ldren had recently completed a unit o n p lace va lue and dec imals and were probably con fus ing the procedural ru les w h i c h they had learned. T w o o f the ch i ldren i n this G r o u p responded that 2/3 was the same as 4/6 and one thought 4/6 was larger because the numbers were bigger. Question Six: Which of these figures represents 4110? (The two diagrams are again placed to demonstrate that the area is the same.) ********* ********* ********* ********* ******** ********* ********* ********, ********* ********* ********* ********* ********* ********* irrr rrrrrx ********. ********, ********. ********. ********, ********, ********, ********. ********. ********. ********. ********. ********, ********* ********. """" ******* ******** ********* *******/, ********* ********* -******** ********* ********* ********* * * ********* ********* ********* ******** f s s s* s s y y f f r f f f > f\ ********* ********* ********* ********* ********* ********* ******** ********. ********* ******** ******** ********* ********. ********, ********* ********* ******** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** ************************************ *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** T h e general response to this quest ion dur ing the first interv iew was that the top picture represented 4/10 and the bo t tom d i a g r a m s h o w e d 1/7. It was not poss ib le to see any l ines d i v i d i n g the shaded section into fourths, thus ch i ld ren tended to treat this as one large p iece rather than four separate ones. Page 70 Visual Fractions Interactive Group. T h i s G r o u p unders tood this ques t ion better before treatment than the other two Groups . Three o f the ch i ld ren were able to ident i fy both pieces as 4/10 i n the first in terv iew and o n l y one said that the top was 4/10 and the bot tom was 1/7. T h i s student changed her m i n d b y the second interv iew and a l l the students i n this G r o u p then rea l i zed that both fractions represented 4/10 o f the d iagram. Visual Fractions Noninteractive Group. O n l y one student i n this G r o u p a n s w e r e d the quest ion correct ly dur ing the first interv iew. T h e others a l l be l ieved that the top picture represented 4/10 and the bot tom was either 1/6 or 1/7. A f t e r the treatment, on ly one student changed her m i n d and rea l i zed both fractions were actual ly 4/10. T h e other students a l l retained their o r i g ina l answer, except fo r one g i r l w h o said the bot tom fract ion was actual ly 1/7 and not 1/6 as she had o r ig ina l l y thought Flash Card Group. F o u r o f the c h i l d r e n i n this G r o u p were able to g ive the correct response to this quest ion dur ing the first interv iew. O n l y one c h i l d thought that the top was 4/10 and the bot tom was 1 into 6. D u r i n g the second interv iew this c h i l d dec ided that the bot tom was actual ly 1 into 7. Page 71 Question 7: look at the Diagram. a) What fraction part is represented by the shaded part of this diagram? (115) b) Do fraction parts all have to be of the same size? ************** f************J ************** ************** ************** ' / / / / / / / / / / / / / ************** ************** ************** ************** ************** ************** f************* ************** ************** ************** ************** These questions were des igned to test whether ch i ld ren w h o be l ieved f ract ion parts a l l had to be equal , c o u l d i n fact apply that knowledge to correct ly answer 1/5. The d i a g r a m is intended to l o o k l i k e the more c o m m o n l y seen fract ion 1/3 w h i c h was used to ask the same questions at the beg inn ing o f the interv iew. T h e quest ion: Do fraction parts have to be equal? was repeated to see whether any o f the students changed their m inds as a result o f the in terv iew process. In no case d i d any student answer one w a y at the beg inn ing o f an in terv iew and change their m i n d by the end. A n y changes i n response a lways occur red f r o m the negative to the pos i t ive and f r o m the f irst to the second interv iew. V i s u a l F r a c t i o n s I n t e r a c t i v e G r o u p . D u r i n g the f i rst i n t e r v i e w , a l l but one o f the students i n this G r o u p be l ieved that f ract ion parts d i d not have to be o f the same size. O n e boy thought m a y b e they d i d , but wouldn ' t say f o r sure. B y the second in te rv iew, a l l four p r o m p t l y answered yes to this quest ion. In the f irst in terv iew, two ch i ld ren answered 1/3 to this quest ion , one answered 1/3 o r maybe 1/5 (he wasn't sure) and o n l y one student gave the correct answer as 1/5. A f t e r the treatment, three students answered 1/5 and o n l y one was not sure. She k n e w it wasn't 1/3 because the p ieces weren't the same s ize , h o w e v e r she was unable to determine what the correct answer shou ld be. Page 72 Visual Fractions Noninteractive Group. A l l but one student i n this G r o u p be l ieved f ract ion parts d o not have to be the same size and n o one i n this G r o u p changed their m i n d after the treatment. In the f irst in terv iew, four ch i ldren i n this G r o u p answered 1/3 and one student responded one two (meaning one as the numerator and t w o as the denominator ) . E v e n after the treatment, none o f these ch i ld ren changed their m i n d s . N o t one o f them was able to g ive the correct response o f 1/5. Flash Card Group. A l l but one student i n this G r o u p thought f rac t ion parts d i d not have to be the same size and none o f these students changed their m i n d by the second interv iew. T h e one student w h o k n e w they had to be equal rea l i zed this before the treatment and d i d not change his m i n d by the second in terv iew. T w o students responded 1/5 and two responded 1/3 d u r i n g the f irst in terv iew. N o n e o f these students altered their answers by the second in terv iew. T h e f i f th student responded one into two du r ing the f irst in terv iew and one into three dur ing the second interv iew. Summary Report of Interview Findings T h e in terv iew results are compared be low accord ing to s imi lar i t ies and dif ferences between g r o u p s . Similarities Between Groups T h e f o l l o w i n g statements appeared to be true for most o f the ch i ldren in terv iewed regardless o f membersh ip i n treatment G r o u p . Ch i ld ren 's understanding o f f ract ions appears to be ru le o r procedure -based. F o r a great m a n y ch i ld ren , statements such as my teacher told us, o r the books show it like this, o r a fraction is school stuff like math ref lect the idea that their l eve l o f understanding is p rocedura l -based rather than conceptual (i.e., they have m e m o r i z e d rules wi thout pocess ing Page 73 any intu i t ive sense o f what a f ract ion is) . C o n s i d e r f o r example , one student's def in i t ion o f f ract ion: A fraction is well say you had four boxes and two of them were shaded and two of them weren't, you would have to put a line and the two shaded ones would go on top and the unshaded ones on the bottom, my teacher said. T h i s procedura l approach d i d not change after the treatment sessions. O b v i o u s l y it w o u l d take considerably more experience for ch i ldren to develop a more independent o r concept -based learn ing approach. A g a i n , this same student's posttest response to the quest ion o f whether f ract ion parts a l l had to be the same size i l lustrates this po int w e l l : Yes they do because when they [the computer] drew the fractions, they were all the same size. A related issue had to d o w i th whether ch i ldren operated f r o m a context -based f ramework or whether they were able to transfer in fo rmat ion learned i n one situation (i.e., the computer screen) to another (i.e., the posttest). It seemed c lear from the interv iews and from ta lk ing in -depth to several students after the posttest that this transfer was often not made even though it was c lear that a conceptual shift i n understanding had occurred. T h i s was especia l ly apparent fo r the question about whether f ract ion parts had to be o f equa l size. T h e treatment d i d not seem to affect the student's abi l i ty to def ine a fraction. Students d i d not change their o r ig ina l answer to this question dur ing the second interv iew. M a n y o f them def ined fractions w i t h a very l i m i t e d , context -based def in i t ion (e.g., its numbers with a line between them. The top number is the shaded part and the bottom number is the number of parts," etc. V e r y f e w students answered this quest ion i n a more general fash ion (e.g. , a fraction is a part of something or equal parts of a whole). Differences Between Groups In add i t ion to these general f ind ings , some speci f ic observat ions can be d r a w n from each o f the three Treatment G roups . Because o f the smal l sample size o f students in terv iewed i n Page 74 each G r o u p (n= 4 o r 5) , i t i s somewhat dangerous to m a k e general izat ions. H o w e v e r , there appeared to be di f ferences between the V i s u a l Fract ions Interactive G r o u p and the other two G r o u p s o n a lmost every interv iew quest ion. S o m e o f these dif ferences were rather dramatic and are reported here. T h e biggest d i f ference that occur red was i n response to quest ion 2 a and quest ion 8. B o t h o f these asked the quest ion Do fraction parts always have to be the same size? A l l four students i n the Interactive G r o u p responded no to this quest ion dur ing the first in terv iew and a l l four changed their m i n d and said yes dur ing the second interv iew. In contrast, the students i n the Noninteract ive G r o u p a l l responded no at f i rst , but on ly one student changed her m i n d and said yes, they did have to be always the same size. S i m i l a r l y , a l l four students i n the F l a s h C a r d G r o u p sa id no at f i rst and no one changed their m i n d by the second interv iew .There was a c lear d i f ference between the Interactive G r o u p and the other two Treatment Groups i n their response to this quest ion. A n o t h e r d i f ference between the Treatment G r o u p s occur red i n response to quest ion 7 w h i c h requi red the students to correct ly ident i fy the shaded port ion o f a rectangle as 1/5 (It was d i v i d e d into three uneven parts and c o u l d be interpreted as thirds to the undiscern ing eye). B e f o r e the treatment, o n l y one student i n the V i s u a l Fract ions Interactive G r o u p was able to cor rect ly answer this quest ion and even he wasn't total ly c o n v i n c e d o f h is answer. A f t e r the treatment, every c h i l d i n this G r o u p correct ly ident i f ied the f ract ion as 1/5. In contrast, none o f the students i n the Noninteract ive G r o u p were able to answer this quest ion after the treatment. A d d i t i o n a l l y , the F l a s h C a r d G r o u p showed no improvement between interv iews. T w o were able to answer correct ly pr ior to treatment and no one else was able to answer 1/5 afterward. A g a i n this shows a clear dif ference between the Interactive treatment and other t w o G r o u p s . Page 75 All of the students found the task of dividing a piece of paper into 5/7 difficult. The differences between Groups was more subtle in this case. However, because of the difficulty of the question and the nature of the change (from no measuring to some attempt at measuring), the difference is worth noting. Before the treatment, none of the students in the Interactive Fraction Group made any effort to measure the paper in order to make their divisions equal and two students would not even attempt the problem. After the treatment, everyone attempted the problem and two students measured very carefully. There was also great variance in ability to answer this problem in the other two Groups; however in contrast to the Interactive Group, none of the students in the other two Groups altered their original effort at solving this task. The only student who tried to measure the paper did so during the first interview as well. An analysis of the findings from both the quantitative and qualitative data revealed that Visual Fractions is an effective method of teaching basic fraction concepts. A discussion of the findings is presented in Chapter Five. Page 76 Chapter Five Summary, Conclusions and Recommendations T h e purpose o f this study was to determine whether learner -contro l over a computer learn ing envi ronment was more effect ive i n teaching basic f ract ion concepts than a noninteract ive vers ion o f the same program. T h e two computer methods o f instruct ion were c o m p a r e d to a F l a s h C a r d treatment, des igned to contro l for the inf luence o f computers o n learning b y s imulat ing the noninteractive computer environment. Data was obtained by means o f a pretest/posttest. In add i t ion , students in terv iews were conducted and results reported. T h e f i n a l source o f data i s i n the f o r m o f a qual i tat ive report o f student's interact ion w i t h the learn ing envi ronment . T h i s chapter presents a summary report o f the f ind ings o f the P i l o t Study and the research results as w e l l as conc lus ions and recommendat ions fo r further study. The Pilot Study The results f r o m the P i l o t Study indicated that the or ig ina l pretest/posttest d i d not accurately report ch i ldrens ' understanding o f f ract ions. Students were able to answer the quest ions s i m p l y by l o o k i n g at the examples and us ing them to f i l l i n the b lanks i n a s imi la r manner . Apparent l y this technique o f answer ing worksheets correct ly without real ly understanding the concepts i n v o l v e d occurs d a r m i n g l y often i n schools . A c c o r d i n g to more than twenty ch i ld ren i n f i ve different schools w h o wrote the o r ig ina l test, this c o p y i n g technique was something w h i c h they e m p l o y e d regular ly because it seemed to w o r k for them. Unfor tunate ly , the learn ing w h i c h seemed to be occur ing had to d o w i t h worksheet wiseness as opposed to increased conceptual understanding. T h i s f i n d i n g has broad impl i ca t ions fo r the w a y i n w h i c h w e teach a l l subjects, not just mathematics . Teachers need to cons ider very care fu l l y what they want students to learn w h e n des ign ing worksheets . Page 77 Discussion of Research Results The sample was taken from the grade four students in four different classes at two elementary schools in Burnaby, British Columbia (a suburb of Vancouver). The sample may not be representative of a random sampling of grade four students and therefore any conclusions that the researcher makes cannot be generalized to the population of grade four students. The treatment process was limited to one one-hour session. Except for the Group which used the noninteractive version of Visual Fractions, it is probable that many of the students would have benefited from a second exposure. These students were clearly bored after fifteen minutes and it is unlikely that they would have learned anything more from a continued or repeated exposure. Students seemed to work better in smaller groups. The students in the pilot study were observed in groups of four and seemed to be more on task more of the time than the students in the larger Group. Individual instruction may not be as beneficial because the peer interaction appeared to be an important part of the learning process. Some of the students in both computer groups were not concentrating as intently as they could have been because of a desire to return to their class and work on a dinosaur project. It is the opinion of the researcher that this activity represented an unusual opportunity for student's to interact in a informal way with one another and was highly desirable. They therefore did not spend as much time at the computer as they otherwise might have done This situation could have posed a serious threat to the study, however, because the same situation occurred with one half the students in each computer Group, the problem was the same for both treatments. Further, it only affected approximately five or six out of twenty two students in each Treatment Group Page 78 There were several issues concern ing the design o f the f lash cards w h i c h m a y have g iven the H a s h C a r d G r o u p an advantage over the other Treatment G r o u p s w h i c h the researcher d i d not cons ider p r io r to the treatment. F o r example , the cards were a l l c o l o u r coded by denominators and the students q u i c k l y p i c k e d up o n this fact and began to sort the cards into groups by co lour . T h e y a lso l i k e d to f i n d cards w i t h equivalent f ract ions and sort them into groups. T h i s abi l i ty to l o o k at several cards at once was unique to this Treatment G r o u p and the students c lear ly en joyed c o m p a r i n g and sort ing the cards. T h i s type o f act iv i ty was not poss ib le i n either o f the two other Groups because on ly one f ract ion at a t ime c o u l d be seen o n the screen. T h u s , the intent ion o f dup l icat ing without a computer , the Noninteract ive vers ion o f the V i s u a l Fract ions P r o g r a m , was not accompl i shed . A l t h o u g h the des ign o f the H a s h Cards was ve ry s imi la r to the Noninteract ive software p rog ram, they p r o v i d e d two more elements w h i c h c lear ly resulted i n a greater enthusiasm and higher leve l o f concentrat ion. It seems therefore that this result lends support to the theory that learner cont ro l is is a c r i t i ca l factor i n learning. T h e students w h o used the H a s h Cards had a greater degree o f cont ro l over their learning environment than they were intended to have and the result was a greater enthusiasm and desire to learn. W i t h o u t the presence o f the poss ib le ceiling effect, this G r o u p m a y w e l l have s igni f icant ly i m p r o v e d their score f r o m pretest to posttest. Conclusions - The Value of Visual Fractions T h e V i s u a l Fract ions G r o u p scored s ign i f icant ly better than the C o n t r o l G r o u p and this makes it c lear that the testing and passage o f t ime were not responsible for their ga in . A d d i t i o n a l l y their ga in scores were s igni f icant ly superior to the F l a s h C a r d and N o n -interactive G r o u p s ind icat ing that it was not the act iv i ty w i t h fractions nor the opportunity to use the computer that caused their h i g h gain . T h e interactive nature and opportunity fo r cont ro l o f fered to the students appear to be the m a i n factors contr ibut ing to their success. Page 79 The results from this study indicated that the interactive version of Visual Fractions is an effective means of teaching basic fraction concepts. The version of the software program used in the study was specifically adapted to test one aspect of learner control. A more sophisticated version of the program exists which allows students to test themselves by selecting a drill button. The program then asks the students to make various fractions and prompts for incorrect efforts. This added feature would probably improve the intrinsic appeal of the program and permit a greater retention of interest and longer time on task, possibly resulting in a greater improvement of understanding of fraction concepts. Additionally, versions of the program exist which allow students to make comparisons between two fractions, (equivalence) and to perform operations tasks. Recommendations for Future Research It is possible that students in both the Interactive Visual Fractions Group and the Flash Card Group may have benefitted from a second treatment session. Also, based on the interactions of the students in both the pilot study and the Flash Card Group, smaller Groups may be preferable and result in greater improvement in understanding. Fraction concepts are difficult to define and even more difficult to measure. Any test designed to evaluate a child's understanding of fractions is bound to be subjective and therefore possibly not an accurate assessment. In-depth interviews appear to reveal more about the concepts children hold than tests do because the researcher has an opportunity to probe and find out why they answered the way they did. A future study of this topic might include more emphasis on the interview process and less on the testing aspect. In addition to experimenting with smaller groups and longer sessions, it would be interesting to expand the study to include the more sophisticated version of Visual Fractions. Page 80 Students could then be tested to see whether their understanding of equivalent fractions and operations (addition, subtraction, etc.) could be improved by exposure to this software. Page 81 References Barron, L., (1979), Mathematical Experiences for the Early Childhood Years. 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V a n E n g e n , H . , (1949) , A n A n a l y s i s o f M e a n i n g i n A r i t h m e t i c , E lementary S c h o o l J o u r n a l 4 9 , 3 2 1 - 3 2 9 ; 3 9 5 - 4 0 0 . Page 86 Appendix A - Interview Procedure There are 7 questions i n this interv iew. T h e t ime approximate fo r each interv iew is 10 - 1 5 minutes . E a c h student shou ld be in terv iewed pr ivate ly . Student responses are to be audiotaped. In addi t ion , the researcher is to take notes o n such act iv i ty as fac ia l express ion, student m a n n e r i s m , etc. . . . that m a y help to c l a r i f y student responses. T h e student shou ld be greeted and have the procedure exp la ined br ief ly . H e should be r e m i n d e d that the purpose is for a univers i ty research project on h o w chi ldren l e a m , that it is not a test and the outcome w i l l not affect h is grades i n any w a y . T h e c h i l d should be encouraged to re lax and g ive the best answers he can to each quest ion. Procedure 1) A s k the student to take a p iece o f paper, tear i t and g ive back 5/7 o f the paper. 2) A s k : 1) D o fraction parts a l l have to be the same size? 2) W h a t fraction i s represented by the shaded port ion o f this d iagram? 3) W h a t fraction is represented by the unshaded port ion? ' • • / / / / / / / / / / / / • / ' / • / / / / • / / / / A / / / / ' / • / / • / / • A / / / / / / / ' / / / / • / / / / / / / / / / / ' / • / / • A / A / / / / / / / / ' / • A / A / / / / / / / / / • • ' • • • • • • • • / / / • / / • / '/•A/•/•Ay/' / / / / / / / //// AA/A • A/A /AAA • A/A ////, • ••A / / / / /••A / / / / /AAA / • / / /A/A / / / / /•/A ///A / / / / / • / / • //A ' / / • / / / / / / / '///•///A//1 '/•//•A//// ' / / / / / • / / / / / • • • • / / • • / '/•A/////// ' / / / / / / / / / / / / / / / / / • / / / • / • / / • • • / \*** ** ****** ////A/ ////A/ ////•A / / / / / / / / / / / / / / / / / / / / / / / / / / / / • / / / / • / / 3) Pretend that I a m from outer space and have never heard the w o r d f ract ion before. C a n y o u descr ibe what a fraction i s fo r me? Page 87 A s k : W h i c h is bigger , 8/12 o f a chocolate bar or one w h o l e chocolate bar? (show quest ion wr i t ten o n a p iece o f paper). A s k : W h i c h d iagram represents a bigger fract ion? / x x x x ' / / / / ' / / / / / A / A T / X X X X / X X X X / X X X X / A X A / < » / / / / / A / A T / A A A / / / / A / / A / A / / • / / / / / • A / / A / / / / / / / / / / / / X X X X X X X X X X , " ' X X X X X X X X X X X X X X X X X X A T A - - - -A / / 1 / / / / / / / / / / / / X X X X X X X X X X X X / / / / / / / / / / / , " / / / / / / / / / / / , X X X X / / / / / / / / / / / / / / / / / / / / / / / / / • / / • / / • • / • A / A / / X X X X X X X * / A / / A A / / X X X / X X / / / / / / / / / / / / / / / / " ' X X X X X X X X X X X X X X / / / / / / / / / / / / / / / / ^ / / / / / / / / / / / / / / / / / / / / / / / • / / / / / / / • / / / / / / , X X X X X X X X X X X X X X X X X X X X X X X / / / / / / / / / / / / / / / / • / / / / / / • / / / / / / / / / / / / / / / / / / / , / / / / / / / / / / / / / / / / / / / / / / / / / / / / y y / / / / / / / / / / / / / / / / / / / / x x x / A / A A / X X X X , / X X X X / / • X X / x x x x , / • X X X / / X X X / x x x x / x x x x / x x x x / x x x x / x x x x / x x x x / X X X A * / x x x x x x X X X X X X X X X , X X X X X X X X X X X X X X X * / X X X X X X X * / X X / X X X X X X X . / X X x x x x x x x xxx xxxxxx , / X X X X * / X X . / X X X X X X X . / A X /////* " " X X / x x x x x x x . xxx • X X X X X x x x x x x x x x x xxxxx , X X X , x x x x xxx, xxx xxx xxx _ x x x x xxxxx* xxxxx* / x x x x x x x * / X X X X X X X / / x x x x x x x * . / x x x x x x x * X * f * X X X X X X X » x x x x x x x * x x x x x x x . X X * X X / X X X X * / x x x x x x A s k : W h i c h o f these pictures represents 4/10? W h a t f ract ion does the other picture represent? IF / rrrrrrm— * / / / / / / / -/ x x x x x x x . / x x x x x x x * *• / / / / / / > / x x x x x x x . / x x x x x x x * / x x x x x x x . / x x x x x x x * / x x x x x x x * / x x x x x x x . / x x x x x x x . / x x x x x x x * * / / / / / / / . / x x x x x x x . / x x x x x x x . / X X X X X X X . fcV/xx/xx,! / x x x x x x x . / x x x x x x x . / x x x x x x x * / x x x x x x x . / x x x x x x x . / x x x x x x x . / x x x x x x x . / x x x x x x x * / x x x x x x x * / x x x x x x x . / x x x x x x x . / x x x x x x x * / x x x x x x x . / x x x x x x x . / x x x x x x x * / x x x x x x x . / x x x x x x x . / x x x x x x x . / x x x x x x x * / x x x x x x x . / x x x x x x x * / x x x x x x x * ' / / / / / / / . / • x x x x x x * / x x x x x x x * / x x x x x x x . / x x x x x x x * / x x x x x x x * / x x x x x x x * / x x x x x x x * / x x x x x x x * * / / / / / / / , _ _ / x x x x x x x , / x x x x x x x , / x x x x x x x * / x x x x x x x , / x x x x x x x , / x x x x x x x , • x x x x x x x , / x x x x x x x * / x x x x x x x , / x x x x x x x , / x x x x x x x , / x x x x x x x , / x x x x x x x , / x x x x x x x , / x x x x x x x , / x x x x x x x , _ffttttSft -* / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / , / x x x x x x x x x x x x x x x x x x x x x x x x x x x x / x x x x x , / X X X X X X X / / / / / / X X X X X X X X X X X X X X X X X A X X X , / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x . / / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x . * * / / / / / / / / / * • / / / / / / / / / / / / * ' / / / / / / / / / / , / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x , / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x , / X / X X X X X x x x x x x x x x x x x x x x x x x x x x x x x x x x / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x . / X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x , / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x . / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x / X X X X X X X X X X X X X X X X X X X X A A X X X X X X X x x x x x / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x / x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x L o o k at this d iag ram. A s k : Page D o fract ion parts a l l have to be the same size? W h a t f ract ion i s represented by the shaded port ion o f this d iagram? VfSffffssfssf/ V'"****** 4* 4*4*ffft ^ f f f f f f f f f f f f t ^ f f f f f f f f f f f f t ^ f f f f f f f f f f f f j M f f f f f f f f f f f f f t ^ f f f f f f f f f f f f t t f f f f f f f f f f f f * t f f / f f f f f f f f f * t f f f f f f f f f f f f * tSffffffffSSf* t f f f f f f f f f f f f * t f f f f f f f f f f f f * t s f f f f f f f f f f f / t f f f f f f / S f f f f * t f f f f f f f f f f f f t f f f f f f f f f f f f f * t f f f f f f f f f f f f * t f f f f f f f f f f f f t t f f f f f f f f f f f f * NAME: APPENDIX C GRADE: AGE: BIRTHDAY: 1. What fraction is represented by the SHADED portion of this picture? ANSWER: 2. What fraction of this square is SHADED? ANSWER: 3. SHADE in 2/3 of this circle. 4. What fraction is represented by the UNSHADED portion of this picture? ANSWER: Page 91 5. What fraction is represented by the SHADED in portion of this circle? ANSWER: 6. SHADE in 1 /4 of this rectangle. 7. SHADE in 3/4 of this rectangle. 1 J Page 92 8. ESTIMATE! SHADE in 7/20 of this diagram. 9. ESTIMATE! SHADE in 7/12 of this circle. 10. Can you SHADE in a DIFFERENT 7/12 of the same circle? Page 93 What fraction is represented by the SHADED portion of this diagram? ************ ///y///////> ************ ************ ************ ************ ************ ************ ************ ************ ************ ************ ************ ANSWER: 2. What fraction of this rectangle is SHADED? ****************** ****************** /'/St* *****St*/**//**' ****************** ******************* ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** ****************** / / / y / y / / / / / / / / / / / / ****************** ****************** / y / / / / / / y / / / / / / / / / ******************* ANSWER: 13. What fraction of this rectangle is SHADED? f7777>^77777777777773T77777777T77777; ***************************** ***********************************. y ***** ***************************** . ***********************************. ***********************************. ***********************************. ***********************************. ***********************************, ***********************************. ************************************ ***********************************. ***********************************. ***********************************, ANSWER: 13. What fraction is represented by the UNSHADED portion of this diagram? •***************************************************. ***************************************************, **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** **************************************************** ANSWER: Page 94 14. ESTIMATE! SHADE in 5/15 of this rectangle. 15. What fraction is represented by the shaded portion of this diagram? ANSWER: 16. What fraction is represented by the SHADED portion of this diagram? Appendix D - Sample Flash Card APPENDIX E - PILOT STUDY PRETEST / POSTEST ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* NAME GRADE BIRTHDAY YEAR 19 MONTH _ DATE Page 97 Shade in the correct fractional part. Example A 1 3 Example B / / / / / / / / f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f ' S f f f f f f t 'SSSSfff* •Sffffff* •SftSfff, 'fffffSS, SSSSfSSS f f f f f f f f SSSSfSSS f f f f f f f f f f f S f f f f SSSSfSSS 3 4 1. 2 7 5. V £ '9 Z Z \ 9 £ £ ' / • / / • / • / / / • / / ] •/•//•A '/S//S''f'tf/A '////////////. '•/V//V//V//V. ' / / / / • / > / • / / / • / . • / / / / / / / / / / / / , • / • / / • / / / • / / / / . V///////////. '/A/••//ATA '////////////., ' / / / / / / / / / / / / A '/•/////•////-•////////////. '/A/v/v • 1 '/ x '•'• /,*| JO £ rrrrrrrrn sssss/ss* //////A/V /•///> /•/A/A/V '////•///* •/////////////////< A/A< //•//•//<'/'#'/'* A//V/A*A///A/A*> •/••/•A*VA*V/A'/A*'/ •/////•/••/•//////* £ JO \ \ \ w \ \ \ \ v SWW \\\N\, g 9[dujox3 y eidujBxa 86 3§*d Page 99 WRITE T H E CORRECT SYMBOL FOR E A C H F R A C T I O N . Example A ******* ******** ******** ******* ****** 1 4 Example B **** • / / / . ' / / / . ***** '/>/• ••//. ' / / / . **** • /-/>•. '//v. '/•/V '/V/. ' / / • . ' • / / • ' / / / . **** ***** 3 5 ******** ******** */******* ******** ******** ******** ******** 2. *****, V*****, Y*****. Y*****> ****** ****** ****** ****** wm ***** ****** ***** ***** 5. 3. T777T7T7T77T. \*************. ************. ************* ************. ************. ************. •************. '************. '************* '************* ************* ************** '************* ************* ************* ********** ********** '********, ********** ********** ********** ********** '******** ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* •********. ********** ********* \>>*>t>>>A ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* 6. ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ******* ***************************************** ****** ****** ******************************************** ******** **************************************************** ***************************************************** *************************************** / / / / / / / / / / / / , / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / y / / / / / / / / / / / / / / / y / / / . **************************************************** **************************************************** *************************************************** *************************************************** *************************************************** *************************************************** *************************************************** y ************************************************** '9 77777T777T>V77T7T777r777777777T7>771 *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** *********************************** /y yy ************* * * * * * * * * y / / / / y / / / / y / y y / y / y / / / / / / y / y / / / / / / / / / / / / / y / / / / *********************************** yy/ / / /yy / / / /y /y /y /y /yyyy/yy/ / /y / /y / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Y777777T7777 ************ ************ ************ ************ ************ ************ ************ ************ ************ ************ ************ ************ *f********** ************ \**********'*\ ******** ********* ********* ***ft/*/, ********* ********* ********* ********* Sf*'**/*'*'*'. ********* st*******. ********* ********* ********* ********* ********* ********* ********* ********* **/******* ********* ********* //St////**/////*'///. ******************* ******************* ******************* ******************* ******************* ******************* ******************. ******************* ******************* ******************* ******************* ******************* ******************* ******************** ******************* ********************** ******************* ******************* ******************* ******************* ******************* ******************* ******************* ******************* ******************* ******************* ******************* ******************* ******************* rrrrrr, ********* ********* ********* ********* ********* ********* ********* S £ ************* ************* ************* ************* ************* ************* ************* ************* ************* ************* '************ ************* \ rrrrrrr. ******* ******* ******, ******* g 8idwBX3 V sidwBxa NOIlOVaJ H3V3 aOJ 109UAS ±338803 3H1 31I8M 001 3 § B d 

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