st si

ANEVALUATION OFA PROGRAM OFFERING EXTR1I.

TIME TO HELP STUDENTSACHIEVESUCCESS WITH ACADEMICMATHEMATICS

by

tlGer a l d i neKary Kavanagh,B.6c., B.Ed.

A thesi s sUbmi t t e d in partial fulfilment of the requirementsfor the Degree of Hasterof Education

Depart mentof CurriculumandInstructio:m MemorialUniversityof Newfoundland

March 1991

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Acknowledge ments

The researcherwishesto thank all tho s e who helpedin anyway with thecompl e t i onof this study.

In par ticula r , I wi s h toexpressmyapprec iat i ontoDr.

FrankWolfefor his expe r t assista nceand adv i ce.Without hi s advice and assistance , this st ud y would not have been com- pleted.

I alsowish totha nk the seventeac he r s involvedwiththe teachingof thisprogram who very willinglygaveof theirtime toanswer my questionsconce rning theprog r am. I sincerel y tha nkthesetea ch ersaswell asthete achers involvedwith the regular programwho us ed their cl a s s time to enable th~ ir students to complete the attitudesurveys and objectivete sts . I would especially like totha nk Mrs. Shella Pentonand Mr.

Don Ash who very graciously al l owe d me to vi d e o ta pe their classes.

Finally, Iwo uld like to thank the many studentswhotook the ti me to compl e tethe test andat t i tude survey.

Abs tra ct

The primary purpose of this study was to evaluate a program ot"feringextra time to certain students wh o have a demonstrated difficUlty with academic mathematic s . The int e ntio n of this programis to helpthese students achieve succe sswi th ac ec e n tc mat he ma t ics. As a con sequence , it is hoped thatthe attitudeof these st ude nts toward mathematics ....ould improveif their achievement improved.

Toevaluatethisprogram, several different methodswere undertaken.Allei gh t classesgiven the extra timewere given an attitude surveytwice as a pretE lst and posttest tosee if any change in attitude towards mathematics too k place. As well, the end of year marks wereob t a i ne d for this group of st ud ent s and a comparablegroup who studiedthe same courses but who didnot receive the extra time.In addition, both of these groups were given a 25 it e m test developed by the researcher.Interviewswere conductedwith the seven teachers involved withthe teaching of the classes that had received the extra time. Finally, a videotape was made of two of the eight classesto obtaininformation on how the extratime was used.

The data collected with the attitude surveys was tested using t-tests.The test scores for the two comparable groups was compared using a two-way AnalysisofVa ria nc e (ANOVA).As well, a great deal of qualitative data was collectedon the

i i i

group receiving extratime and this was also reported on.

This studyfocusedon two questions, one dealingwith the achievement of the group arid one dealingwiththeirattitudes towards mathematics. The results indicate a very slight increase in the achievement of these students on the final evaluation. There was no significant difference on the 25 item test. There was also no significant change in tho attitude of these studentsafterthe y received the extra time stUdying mathematics.

Based on the findings of this study, the researcher concluded that evaluating a program is indeed a difficult task. After spending a great deal of time onthis evaluation, I was unable to ascertain whether the program was successful or not. It was recommended that a much more controlled experiment would be necessary to evaluate this type of program.

Iv

TableofContents

Acknowledgements Abstract List of Tables

CHAPTER

INTRODUCTION Rationale for the study The Purpose of the study signif icanceof the Study Constra i n tson th e Study

Page

11 iii viii

I I REVIEW OF RELATED LITERATURE

In t r od u c t i o n

Time: Theori esand Models The Car roll (1 9 6 3 ) Model The Bloom (1976)Model TheWiley-Har n i s ch fege r

(1 974) Model The Bennett (1978) Mode l The ALTMo d e l Allocated Inetruct.Ionel Ti me TimeandAbility to Lear n Ti me andMat hema t i cs

10 10 10 10 12

,.

15

,.

"

24

28

CHAPTER

HI

IV

Attit ude an dAchievement Summary

METHODS AND PROCEDURES

Method

Materials and Instru me nts

PRESENTATJ:ON AND ANALYSIS OF DATA 32 36

3.

3.

43

50

Introduc tion 50

Desc rip tiveData on Group Under StUdy 50

Students 51

Teachers 57

AnalysisofData on Extra Time Group 61 Anal y s i s of Achievement 61 Analysis of At ti tude Data 65

Comparison of Groups 67

Achievemen t 71

Videotape 75

V SUMMARY, CONCLUSION , DISCUSSION,AND RECOMMENDATIONS

Summary Conclus ion Discussion Recommendat ions

vi

77 77

7. .,

.9

Bibligraphy 90 Ap p e ndi c e s

AppendixA - Sand ma nMat hematicsInventory (MAl) 93

Appendix B - StudentQuestio nnaire 102

App en d i x C - Two 2S- I tem Te s t s 105 Appendix D - Ans wer Keys for 25- ItemTests 110

vii

List of Ta bl e s

Page Table 1 TimeAllocated to Mathematics in

NewfoundlandSchoolsPriorto 1982 and Since 1982

Table2 Stu dentscompleti ngtheSandmanMAl Table 3 Enrollmentand Time Allocations to the

Mat hematicscourses

Table" Reliability Coefficie n tsfor Researcher- Made Tests

Ta bl eS PearsonCorrelation of Researcher- Made TestswithCATMScore andJune Ma r k Table6 Reliabil i t iesofthe six Defined Scales

of the MathematicsAttitudeInventory as Determinedby CronbachIs Alpha Coefficient--AllStu de ntsCombi ne d

40

"

47

Table7 Rangeof Scores from Canadian

Achieveme n tTestinMathematics (CATM) 52 Table8 TimeSpent on Homeworkand Mathematics

P..)mewo r k

Ta ble9 Mat hematicsCl asses Mi s s ed During the Schoo l Year

Table10 Source of Help WithMa the ma t i c s If Student Is Experienci ngDifficulty Tabl e11 Achieveme n t of WholeGr oup on June

Examinationsin Mathematics

viii

5J

"

55

62

Table 12 Achievement oi" Students Taking Academic Mathematics 1203

Tabl e 13 Achievement of Students Taking Academic Mathematics 2203

Table 14 Achievementof Group Based on aa-ree»

Researcher-Made Test

Table 15 Comparisonof Pre and PosttestResults of Attitudes Towards Mathematics Table16 Range of Scores from Canadian Achieve-

ment Test inMathematics (Students Receiving six Periods of Mathematics) Table 17 Time Spent on Homework and Mathematics

Homework

Table18 Source of Help With Mathema'cics If Student Is Experiencing Difficulty Ta ble 19 Achievement of Group on June

Examination in Academic Mathematics Table 20 Achievement of Groupon 25-Item

Researcher- MadeTest

Tab l e 21 Analysis of VarianceSummary Table for June Examination Marks Table22 Analysis of Variance Summary Table

for 2S-ItemTest

ix

63

63

65

66

68

70

71

72

73

74

74

CHAPTER I Introduction

Rationaleforth e study

During the last few decades, nt'lc h attention nas been given to research onlearning and achievementasitre r eeee to timeallotments and.t.!.m.gutilization.Educators such as Bloom (1976, 1981), Car r oll (1963), Ge t t i nge r (1979, 198 5 );

Harnisch feger (197 6), wiley (1974 ), and others have cometo believe that allocatingextra instructional time to a t.ude-rt s will result in act>:l..evement gains. Accordingly, since 198 6 several schoolsunder the jurisdiction of the Roman Catholic School Board for st.John's have allocatedextra time to some lower ability students in all effort to help these same students experiencegr e a t e r success with academicma t he ma t i c s. The study reported ne-e was Jeveloped to eval uate this practice. This evaluationwouldprovide informationabout the effectivenessof this approach whichmight lead to suggestions for the improvement ofth e program.

Prior to 1982, Newfoundland and Labrador high school studentscomp l e t e d their high schoolmathematicsprogramover a two year period.Mathematics was taught by uaLnq nine or ten 40-mlnuteclasses during a five or six day cycle.This program providedapproximC:lt~ly60 minutes of instructionper day to mathematics over a six day cycle. rn the reorganizedhigh school program which began in 1982, mathematics courses are

taught with six 40-minuteclasses over a six day cycle for three years.As a consequence, this program provides approxi- mate1y the same total amount of time as in the past or can even result in a slight loss of total time. (See Table 1).

'l'able1.

'l'i me Alloc at e dtoMathe ma tic s in NewfoundlandScho ol sPrior to 1982 an dSince 198 2

Time Allotted for Math- ematics Prior to 1982

Time Allotted for Math- ematics Since 1982

165 days x 60 minutes/day= 9900 minutes/year 9900 minutes/yearYo2 years==

198'JOminutes over 2 years 165 dC'ys x 40 minutes/day:::I

6600 minutes/year 6600 minutes/yearx 3 years=

198 00 minutes over 3 years

This researcher derived the 165 days a year based on the 190 teaching days approved by the government of Newfoundland and Labrador.I deducted the three paid holidays, two days for administrative purposes and 20 days for two sets of formal examinations.The 165 days do not take into account lost time due to storm days, assemblies, and so on. Consequently, mathematics did notgain any extra instructiona ltime over the three year period whenthe reorganized highscnoc'; program was

int roduced.The timeallot tee:. to mathematicsisnow spread out over three ye a r s.

Usually students who enter high scho ol with a poor mathema t ics ba ckgro und have diff i cUl t y wi th high school academic mathematics. Thismay be a result of a la c k of understandingof theprerequisite sk i lls or a lack of effort , or as a result of a combination of factors. However as Stallings (1980)I points out: "zven though high-achieving studentsar e moreinclinedtobeenga g e din academic tas ks ,it is of co ns i d e r abl e importance toall oc ate sUf ficien t time and effo rt toworki ngwith lo w-ac h i evi ngst ude nt sWh oma y notbe so inclined" (pp. 11-12).

In their curriculumguides for mathemat ics, the Depart- ment of Educat ionfor Newfoundland and Labrador mainta inthat

"t he mathematics programs ... ar e designed to provide all students witha sound ba c kg r o und in the basi c mathematical skills necessary...tofunct i on in contemporary society, and for use in the future" (p. 1). The curriculum guides al s o assert that the mathematics programs"recognizethe individual ne eds and abilitiesof studentsand provide a system by which the general objectives may be met in different ways for diff e r e nt students" (p. 2). The philos ophyexpres sedinthe above st atementsseemsto claim that the mathematics cou r -se s of theprovinceare abletome e t thene e ds of"aLd" le arne r s. Even so, some studentsstillexperiencedi f f i c ulty ,whichmay be due to timere s t r a int s on theirmathematicslearning.

A recentTaskForce on Mathematics and ScienceEducation (1989) in Newfoundland and Labrador felt that -scucenee of different aptitudeswillrequiredifferentamountsattimeto lea rn" (p . 220 ). TheTaskFor ce al so suggeststhatmathemat- ic sandsci e nc e are the "subj ect s whichpresentsubstantial difficulti e s for st uden t s , then a clear case exists for allocating as much time as necessary to these SUbj ec ts to ensure that st ude nts suc c e ed" (p . 217) . This would imp l y usingdifferentialamountsof ti me for diff e rentstudents in ord e r to helpallstude ntsexper i e nc e successwithma t he ma t - ic s. One way to receive this extra time, the Task Force Cla ims, would be to use theadditional time thatiscur re nt l y usedfor electiveSUbjects in thecurriculum(suchas typing, physical education, art) for basic academic areas such as mathematicsandscience.

Dur ing the schoo l year 1966-1 98 7 , the Roman Ca t holi c Schoo l BoardforSt.John 's decidedto offeraca de mic naehe- matics to some lower ability st ude nt s for nine ac-nrnuee classesduringthesLx-da y cycle.Thiswouldentail the loss of a one credit course suchas typ i ng, business education, or music.This decision was takenin an effortto helpthesesame st udents expe r i e nc e success with mathematics. The scudentis st udi e d thesame academic course as otherstuden ts (ages 15 and 16) inLevel I mathematics. By taking nine classesto study mathematics, rather than the six recommendedby the governmentcurric ul umguides,these students would becompl e t -

ing only 13 creditsa year as compared to the14 credits also re c o mmen de d by the Departmentof Education. Non ethe l ess , if theywere successful in completingthese 13 creditsa year, they would still be able to meet the graduationrequirements. Thismethod of obtainingext r a timewouldappear to be similar to the one suggested by the Task Force on Mathematics and Sc i e nc e.

Th i s approach was begun in one high schoolin theboard.

Sincethe first year appearedto have been a successful one, the practiceof usingni ne classes instead of theregula rsix for academic mathema ticswas graduallyincreased throughout schools underthe jurisdiction of the Roman Catholic School Board for st.John 's.Thus ,during the school year 1988-1989.

there were eight classes (128students) receivingthisext_~

time .By thistime ,i t becameapparent tha t aneed existedto evaluate the practice of incr easingallocated instructional time ^asit related to the achievement and attitude of these high school students.

The Purposeof the study

Thepurpose of th isstudy wasto evaluat e theeffective- ne s s of a particular mebhodof allocatingextra time tohe l p lower ability studentsachieve in ac a d e mi c ma thematics.This eval uation was carried out to determine if the program was effective . That is, thisstudyat tem pted todet e rmi ne if the method of allocating extra timE" to lowe r abi li ty students

accomplished i ts purpose. More speci fically, the following questio ns were inve s t:ig ao t e d :

1. Are the se st uden ts expe rienci ng success wi th academic mathematics?

Willthe attitude towa r ds mathema t icson thepart of stu d e nt s who are receiving more timeat mat hema tic s change dur i ng theye ar of schooling?

Inadditi on, by comparing theachievement ofthe s e students withstudents of similar ability whodi d not receive the extra time, the researcher attempted to determine ifthe goalsof the program arebeing met. The groups were made comparable accordingto scoresre ceive d onthe Canadian AchievementTest inMa t he ma tic s.

This study could be consideredas"a c tionre s e a r c h ." In the words of Borg (1987 ), "thegoa l of actio n re s e arc his to gather ev i de nc e that can hel p thete a c he r or admin ist r ator make decisions related to thelocal schools" (p.284). One of the goalsof this study has been to assistthe school boa r d in making fut ur e decisio ns abou t continui ng this program to as s i s t some of it s stud e nts.

Significanceof th~

Most ofthe available research on time and achievement has beenfocusedonins t r uc t i on with primary, el e me ntary and jun ior hig h studen ts . There appears to be less re s e a r ch availableon therel a t ionshi pof allocatinginstr uc tiona l ti me

to achi eve ment inmat h ema t ics alllong hig hsc hoo l st ude nts. This is especiallYtruefor Newf o u nd landandLabra dor schools.Ev e n so, mos t of theavailab l e resear c h onti me do esinv o lve the basicsubjec t area sof'ma the ma tic s , reading and language arts. Hi g h school isdifferent from primary andel e me nt a ryschoo ls intha t highscho ols usually allocate fixe d time clas speriods and sUbject teac hers to the va r ious sueje ccs, As a conae - qu ene e, theab i li ty ofahighschoolteacherto alloca teext ra time to mathematicsor anyothe r subjectisrestricted during thenormal schoolday . By co ntrast, at theprimaryor elemen- tary levels , st ude ntsar eusu a ll y taughtall sUbjectsby the same teacher.Th e r e f ore ,the teach e r is ab leto allocateext ra time to anySUbj ect or top ici fhe or shefe els it is ne ed e d. Depen d i ng on the si ze of the school , juni or high scho o l teac he rsma y or Dlaynot be ableto al locateextra t ime.Some junio r highschoolsare simi l a r tohighsc hool swhereas othe r schoolsfunct ion much like th e element ary schools.

The Ta s k Forc e on Ma t hematics and Sci e nc e Edu catio n (1989 ).hasre c o mmended :

•.. that school schedulesandstud ent programs be designed to allowfordi f f e r e n tialtimeallocations inba s i c SUbject areas.Thi s shoul dbe accomplished by ha vi ng studen tsWho ne e d more time ta ke fe we r optionalcou rses, particularly at the intermediate and senior high schoo l leve ls. (Reco mme nda tio n 10. 12,p, 2231

sincethis is the approachalread y begun by the school board, thisstudy may beus e f u l in de t e rmini ng if such a recommenda- tion ca n be accomplished successfully in this particular

constraint son the study

The foll owing can be considered as conse ra Int.s on the study:

1. All students involved in the study came fromone schoolboard withinthe province.

2. The sampleof those studentswhore c e i ved extratime was limited to128 students from theei gh t classes withinthe board'sjurisdi c t i onwho rece ivedthe extratime.

J. The researcher had no control over assignment to classes. Intactclasseshad been previouslyassigne dby the various highschools invol ve dinthe program.

4. The June examination mark in mathematics which servedas one indicationof achievement wasba s e d on diff erent eva l uatio n s withinthefive hi gh schools offering the extra time for mathematics.

5. There s e arch e r hadno controlover the assignment of teachers tothe variouscl a s ses.

6. The researche rhadno control overthe class size or themet hodby whichth e extratimewas allocated .Five of the eight classes involve d were intact classes for the nine periods , whereas three classes had intact classes for six

periodswitha smallernumber of the classal loca t edth e extra threepez-Lods ,

CHAPTERIX A ReviewotRe~ated Literature

Introduo tio n

Thisstudyattempted to evaluatea program of al locating extra. ti meinor de r to help lower abilityst udentsto experi- encesuccessin academic mathematics.In conjunction with this effort,a review of the literature was undertaken todetermine the relationshipbetweentimeand ach ievementespeciallyas it relates to high school mathematics.

Chapter I Iis divided in to five sections. sectionone containsa reviewof theories that link time to achievement.

Section two relates research studies that show a relationship betweenallocatedinstructional time andachievement. studies which show a relationship between time and a student'sability to learn are presentedin sectionthree .section four relates studies involving mathematics achievement specifically to time. The attitude of students as related to achievement is discussedin section five.

Time : Theor ies andModels TheCarroll <1963) model.

Several modelswerefoundwhichlinkedtime and school achievement, one of which was pr opo s e d byCa rro l l (1963). His work ledto much further interest in time as a significant variablefor school learn ing. His "Model of School Le a r ni n g"

11

incorporates timewithfiveothervariables.

1. Ap t i t ud e- -t h e amount or timeast u d e nt wcu Ld need to le a rna task wit h optimal inst r uctio na l conditions.

2. Abilitytounderstand ins t r uc tio n.

3. Quality of instruct lon- -the degr e e to which the te a cher or instructiona l resourc es organizeand present th e task in orderthat the learner can le a r n it as rapidl yand effici ently as possible.

4. Ti meall o wed ferle a r ning- -t heop po r t unit y tolearn.

5. Pers everance--theamount of timethat the learnerIc willin g to spend in learning a taskor concept.

Carroll (196 3 ) recognizedthatthefirs t thr e e variables aredete rmIna nt s of time needed for learningwhereasthe last twomenti onedare determinantsof the time actua lly spentin le a r ni n g. He worked the five va r i a ble s into a mat he ma tica l formula:

(time actuallyspent) degree of learning=f ( timeneeded )

Thisisnot the traditional meaningmathemat.icianshave placed on func tions,but it helped Carroll explainhispoint.

Ca r ro l l (1963 )consideredthe fac t o r s aptitUde, abili ty to understandins t r uc t i o n, and per severanceas beingva r i a bl e s internal tothe learner;whereas, qualityof instruction and opport unitytole arn areexternalcondi tions.He concededtha t thequalityof instructionis an "e l u s i ve " quanti ty wh i l e the timeto lear n isunde r the control ofthe teacher.

12 This stu dy di d not focus on aptitude as de fi n e d by carroll (1963), or perseverance of the students. However, students were selected for this program based on their presumed ability to learnmathematics. The time allowed for learningisthe variable selected for comparison betweenthe groups.

The Bl o o m 11976) mod el.

Bloom (1963) based his model on the work of Carroll (1963). He trans£Clrmed Carrol l's conceptual model into an instructionalmodel known as the "mastery learning model."He feltth a t ifCarroll were correctthenitwouldbe possib leto assist allle arne r s tore ac h high levelsof achievemen t,by increasing the time spentunt il it approachedthe time needed to lear na concept. The main goal of mastery learning is to mani.puLa t.e learni ngvariables to enab le95% ofthe learners to reach mastery on a particularta sk.

Bloom 's (197 6 ) mastery learningmodel is based on th r e e major fact ors. Bloom statedthatthese threefactors are:

1. Student Characteristics: cognitive entry behav- iors- -thebasic prerequisite learning skills that are necess- ary for the new learningtask; and affective entry character- istics- -the motivationthatth e student has or can be moti - vatedto learnata s k.

2. Instruction: the cues,practiceand reinforcement given during the learning pr oc e s s that help meetthe needs of

13 the learners.

3. Learning Ou t c ome s: le v e land typ e of ach ievement , rateof le a rning ,andaf f e c t i ve outco mes .

Bloom (1976) felt if the stude nt ent r y cha rac ter istics and the qualityof instructIon are fittingthen thelearn i ng ou t c o me s wi l l bepos i ti ve.He as sUllied tha t the entry ch a r ac - te r i sticsofthelear ne rand thequal ity of instruct i on ca nbe mod ified to obtai na hi gh erlevel oflearn i ng .He emphas i zed tha t in cre a sing allo c ated time in i tself is not the most efficien t way to improv e achi ev eme nt. All st udents are dif ferentandeach on e br ingsdifferentcharac t er i sticstothe lea r ni n gsit ua tion that de t ermine s howeffic ient l y he/ shewill us e the time allocated to him. Th is chan ge in think in g (whereby it was fe lt that all learne rs co uld reech a hi gh leve lofach i e ve ment ifgiven eno ughtime) ledtonewermodel s whic h pl a ced mor e empha s i son how allocated timecanbe us e d effectivel y.

In an analysis of 27maste r y-lea r n i ng stud ies,Gus key and Gates (1986 ) found that ma s t e ry-l e a r ni ng can res ult in positiv e gains in achie vement . In passi ng , they found the effect s we r e greaterat the elementa ry andjuni orhigh le v e l s than at the se nio r highleve l and thateff ects in lang u ag e arts and soci a l stud i e s we re sl i g htl y hi gh erthan those in mathematic sandsci e nce st ud ies .

WhileBloom (1976) concededthatincreasedalloca tedtime of itself is notthe most ef f ic ient wayto improve achie ve-

14 ment, the main foc us of th is pre s en t study conc e n tra tes increases in allocated time an d how achievement may be affected by this increase. It mu s t be stressed tha t th e reeeercner-ha d no control over the students selected fo r inc lusio n in this program nor any input in t o the kind of instruct ionthey were given.

ThEIl wil ey-Harnischf eger 119 7 " )mo d e l.

The wiley-Harnischfeger(1974)model also considersti me to be a basic ingredient for achievement . They claim tha t achievemen t is determined by two variables: (a) the total time needed by a given pupil to learn a task, and (b ) the tota l ti me the pupil act u allyspends on this task. These var iablesare similar to Carroll 's(1963) model.However,the y claim th at therecan be enormous variations in the tota l alloca tedtime.

Their model distinguishes between different types of classroom even tsandsugges tsdifferent measuresfor inst r uc - tional t ime . They di s tingu ish betwe en the te a c he r' s on- task ti me an d the students' on-t a s k ti me. Furthe r, the i r mode l showshow pupil and teacher~of time inte rac t to influence achievement.

Wiley andKarn i s chfeg e r (1974) alsosaw achievement asa function of ti me but thei rmathematical formulawas .0.r-er Lne- ment of Carrol l's (1963). To them:

ecnIevement'"f (.\iXX)

z

where :

15

w - totalallocated expos ureti me X • per cent active le a r n ing time Y - perce ntusable expo s uretime z • to t alne eded lea rningtime.

ItlIu s t be no t e d thatthi s re s e arc her wasable to study thetotal al locatedtime of the twogroup s being compar ed . Tt.' totalneededti me tole a r n a concept\/ou l d be avery elusive time tode t ermi ne and wouldcertainlyvary fr omstudent to student.

TheBennett 11978) mo d e l .

Amod e l pro posed by Bennett (1978 ) is conside r ed a modif ica ti on of the wU ey-Ha r nischfeger (1974 ) mod e l. He dist i nguis hes between the nominal ascun t;ofti me in school (actua l leng thot the school day and the school ye a r) an d the

act ua l amountof ti meus ed in scn c c t wh e re ext ra ho lida ys.

buildingal terations, and soforth are excluded. He fur t her subdivides thet ime availab l e tolea rn ingby consideringthe timeallocatedtoeach areaof the cu r riculumsuc has mat he- ma tte s, English , scienc e, art, and so on. By de l eting the disruptionsand lackof interest or poorpersi stence of the student,the total act ive learningti me canbedetermine d .

Ac c or d ingtoBennett (19 78),on ly "t heact i ve port i on of thetime as s ignedto^It.task isef fec t ive for learning that task" (p-12 8 ). He fel'ttha t aptitUde andpr ior ecnteve eene, cl ari t yof instructions, task difficulty andpaci ng of the

16 instruction were importan'': to current achievement,Th e r e f or e , he concluded that onlytheti me in which a studentis actually comprehending a task is considered effective for acquiring that tas k. This should determine the total content compre- hendedand have a bearingon the actual achievement.He also felt that feedback is an impo r t a nt characterist ic that influencescomprehension and achievement.

While agreeing withBennett (1978) that active learning time is themost effective, this researcher would have been unableto ascertain this time for the group under study if inde e d it could have been done. In addition. Bennett alst'"

contends tha t aptitude, prior achievement and pacing of instructio n are important for ac h i ev e ment. The students selected to receive the extr a time in mathematics have a record of poor prior achievement in mat hematics. Ifgiving extra time enablesthe teacher to slowtih epace of instruc- tion, then indeed theirachievement may be affected.

TheALTmod e l.

The BeginningTeacher Evaluat ionStudy (BTES), (l9aO) outlined byDenhamand Lieberman (1980), was carried out by i.he Far West LaboratoriesinCa li f orni atoide nt ify tieachLnq activities and otherclassroom conditionsthat would foster learning. The study focused on reading and mathematics instruct ionat the grade two and grade five levels .

Fromth i s study, the concept of academic learn i ngtime

17 (ALT)wasdeveloped. The ALTmodel isba s ed on:

1. Allocated time.

2. Student engagement: time in whi ch a st ud e n t is actively engaged in thetask.

3. Su c ces s rate: tho degree to which a stu dent processes, unde r s t a nds and co r r ectly re s pond s to a le arn i ng task. High ,medium, and low successrateswere determined.

4. Ta s k relevance: the lear n i ng activities of the studentmustbelimi t e d to thecontent categoriesthat areto be cov e r e d on the measure of achievement.

In addition,the BTES (198 0 )investigators id e n ti fie dtwo phases of the instruc tiona l process that they felt woul d resultin increasesofthe ALT.Theseare the"pl a nning" phase and the"i nte r a ction " phase. The planni ngphase includes:

1. Diagnosis: assessing th e pr evi o us knowledge , sk ills, strengths andwe akne s s e s of the student.

2. prescription: decidingcn the appropriategoals, activit ies,groupingsand schedules.

Th eint e r a c t i o n phase inv olves:

1. Presentation: teacher actuallypresentstask or concept to the stUdents.

2. Moni to r ing : te a c he r is ableto determine the student 'sstate ofknowle dg e and skill during task.

3. Feedback: basedon the moni toringof students, teacher determine"whether additional informationor explana- tionis needed.

18 Again, this present study W<lS designed to concent rate allocated time, wit h success being determi nedby a passing grade (a mark greater or equal to 50%)onthe June re port card.

Five theories or modelswere selected for inclusion in thisreviewof theli t e r a tur e,becauseall fivefocus toso me extent on "a llocated time" , which is the main focus of this pre s e nt !;~ud y. Fi nally , sinc e al lof themo de ls identified her e also invo l v e ot her factors which thi s re search er had li t t l e or noco n t r ol over,no particuLarmodelwas selectedas a specific mode l to followforthis study.

Allocated Ins t r u c ti o n a l ;>\!!:!..!:

Alloc atedinstructional ti me can be def inedastheamount of timeuhata teacherhas been assigned to teachacou r s e. In Newfoundland and LabradorI the Department of Education is responsiblefor decidingthe number of daysin a schoolye a r and the number of hoursin ascho ol day,as wellas theamou nt of time to be allocated to a particular SUbj ec t. At the pri mary and el e me nt a ry level, teachers have some freedom within the sch ool day toallocate the amountsof time that theywill use for each SUbject area.At the secondary level, the teachers have little freedom because the government specifies in its Seni or High Sc ho o l Certif i c at i on Handbook (1985) that all two cred itcourses~110-120hours of instructionper year.Si nc e all academic mathematics co urses

19 are two credit courses, no extra time can be obtained during the school day unlesstherewas a changein the high school program.

The allocated time recomme nd ed by the Department of Educatio n is theupper limit.Actually , and oftenfo ranumber of reasons,theamount of time used for instructionwould be less than the 110-120 hours recommended. The Task Force on Mathematicsand scdence (1989) found thatthe time actually usedis consistently lesstha n that allocated.This was tr ue at allle v e l s. Th e Ta s k Forcefound widespreadvariation in time actuallyallocated tomathematicsand science in elemen- taryschools.A number of other researchers (Crocker, undated;

Gettinger, 19841 Powell, 1979; Romberg & Carpen ter, 19 8 6;

Sanford "Evertson, 1983; walberg, 1988) ha ve also found that allocated time varies from school to school and even from classroom to classroom in the same schoo l . As the autho rs suggest, a loss of allocatedtime is a probl'"m no t only in NewfoundlandandLabradorbut in otherprovi ncesandstatesas welL In a study by Duke (1978, as cited in Fredrick s walberg, 1980 ) , high sc hoo l administrators rank skipping class, truancy and lateness as the top th ree causes of instruc tio nal time los t. other factors accounting fo r ti me los t in high school include examinations ,assemblies, storms, furnace problems,teacherstrikes, andso on. As a result of the great deal of time that is lostin school, Wyne and Stuck (1982)have distinguishedbetweenloc a t e dti me and instruction

20 ti me.Th ey de fin e d inst ructio ntime as "the pr op ort ionofti me alloc a t edforaca d e mic ac tiv i ties that 1s act ua l lydevoted to inst ruc t ion"(p, 68) .

The re s e arc h ers of the BTES (1980) studyfoundll.po s i t ive relat ionsh ip betw e e n allocated ti me andstudent achievement in ma th ema t ics and readi ng in grades two and rive. As Fisher et al. (1980 ) re po r t e d on the find ing s of the stud y; "Te ache r s who alloc ate no re time toa particul ar con ten t areaof the cur r i c ul um have st ude nts who achieveat higher leve l s than te a c he r s whoallocateless timeto thatconten tarea "(p.15 ). They concludedas ama j o r findingof the BTESStudythat the amou nt of timeallocated to a particularcurriculum area was posi t i vel y as s oci atedwith st ude nt learn i ng in thatarea.

Even so,no t all research ers areconv inc e dof the effects ofincrea s e d time onachiev emen t. Borg (1980), ina further ana l ys i s of thedatarel ating achievement to allocat ed ti me, found "mostofthe signi fi cant relations h i ps betw e en ach i e ve- ment and all ocate d time are notlarge , accounting for fr om thre e to sixpercen t ofthe re sidual achieveme nt varianc e" (p.

60). Fred ri c k andWalberg (19 80 )claim"timedevot e d tosc hoo l learningappe ars to be amod e stpredictor of ach Levement;" [p, 193). Dempster (1987) appears to agreewiththis andclaims that an inc r ease in allocat e d ti me wil l ha ve no impa ct on le a r n i ng unles s it is accompanied byan increase ofti me on task or engagedtime. Wyne and Stuc k (198 2 ) felt that even tho ug halloc ate dti meha san ef f ect onecr.Ievenen t,"itisnot

21 simply the total qu an tity of timespe ntthat hasthe greatest impact onstudentecnfevemerrt ,butthequal i t y of thequan t i t y oftime spent learning" (p.71).B1a1 (1986) goes sofa r as to claimthatextend ingal l o c ated time mayeven be detrimentalto learn i ng. This couldbe becauseof fat i g u e oc casioned as a result oftheadditional timespent.Bla i furtherclaimsthat :

"If studentschoosenot toput much effortint o their studies, or ch o o se not to con centrate on their st u di es. the amount learned through the simplistic mechanism of additional allocationcouldeither be wasted or prove counterproduc t ive "

(p.40).

In a study of 102 junior high Englishand aa t be na t Ics classes, Sanfordand Ever tson (1983) reportedthat there was a significant re l a t i ons h i p between time use and achievementin the mathematicsclasses butno t in the Englishclasses. For mathematicsclasses, "higherme an classac h i e ve ment gainswere rel ate d to more ti me sp ent inwho le-class instructio n (r=

0.4252) and le s stime spent in seatwo rkIr =-0.41 60 )"[p,166). The r re f ers to the Pearsonr and thefindingswere signifi- cant at p<.05 level.These findingsappear to indicatethat novti me is~makes a difference inachievement gains.The re s e arc he r s felt that the subject matter was important in studying effect ive tea ch ing and ti meut il i zat i on. They found that in the Englis h classes. there wa s a wide variety of instructio na lapproachesandcurricu lumconte nt . Th i s made for a high degree of inc on g r ue nc e betwe e n what was taught in

22 classes dur ingthe year and the end of th e year achievement lIeasure . Sanford an d Evertson did not find th e salle inc o ng ru e nc ein the ..atneaatIcsclass e s .

In a study of allocatedinstructionaltime and ach iev e- mentin highsc h o o l economics,Poindexte r (1985) compa red88 highschoo lst u de nts whorec ei v e d 73 hour s of inst ru c tio nwith 89 adul t high school stud ents who re c e i v e d 60 hours of instr uct iontostud y the same economic s cou r s e wi th the same teacher.Bothgroups of st ud entswe readminist ere d asta ndard- ize dTest of Economi c Li t era c y (For m A for the prete st and Form B forthe post tes t ).She found tha t"the ach i ev emen tgain of adu lt hi gh school st Ud e nts with 60 hours of allo c ated in s t ru ction wa s equal to the achievement of regUl a r high sc h oolallocated13 additional instructional hou r s" (p. 78). Sh e feltthat factors ot herthan allocated timecouldhavehad animpactonthe achievement of students ineconomics.

A study on the te a Chi ng of Basic Reading Skills 1n secondarysch ools reported by Stallings (1980), found that st ude nt reading gains canbe achievedby al locat ing tillle to spe cificreadingactivitiesandtha t the distributionof time canaffect read i ng levels. The findings ofthestudy suggest that spen di ngmoretime on discus s i on and re viewresultsin more student ga in in re ading.

A st ud y by Sc hmi d t (1978 , as cited in Borg, 1980) determinedthe effect of the quantity ofsc hoo ling duringhigh schoolon achievement in six subjectare a s. Hisdata wasdr a wn

23 from anat iona l long itudina l studyof 9192 high schoo l senio rs in 725 scnco t.s. six areas ofthe cu r r icu lum were examined:

mathematics , English , foreign la ngu a g e, fine arts, social studies and science. The st ud y by Schmidt fo und that the quanti ty of schooling received had the greatest effect on achieveme nt in mathematics ,sciencean d English. Mat hema tics was dete rmi nedtobe almos t entirelyle a r ne d in schooland of the six areas stUdied, mathemat ics was the most strongly influencedby potential quantityof schooling. This finding co ntr adictsthefind ingsof Guskeyand Gates (1986).According to Borg (1980), "t he resu ltsof Schmidt 's wor kdemonstrate that the qua ntity of sch ooling a stude nt receives in hig h school doe sha ve a significanteffec t on academicachievement"

(p. 49 ).

Despitethe fact that several researche rshave foundthat allocatedti me Incceaees hadli t t l eor no effect on achieve- ment, most resea rche rshold sacred the view that the time al located for learn ingor thequantity.;:>fins t r uc t i on is sti ll a significantvariabl eaff ectingachi evement (Gett ing e r ,198 5;

Harnischfege r &Wiley, 1976;xarwetc, 1976; Stallings, 1980 ).

According toHus e n (1972), if an adequate amountof time is no t provided or st udents do not spend a sufficient amount of time enga ged inle a rning th e nthedegre e of lea rn ing will be lowe r e d.

\

24 Timeand AbUity to Learn

Allocated time canbe lost during the school year for many reasons. Timeis lost be c au s e of storms, furnac e prob- le ms, assemblies, fu n days, and examinations, toname just a few. This loss of ti me apparently has an effect on stUdent performance.However, the more able studentsappear to be ab le to completesuccessfullythemathematics cour s es.Butslower students who are atadisadvant agewhen theyenterhigh school mayne e d the extratime to be suc cessful with the academic mathemati cscourses.Not allstude nt s lear nat the samerate.

This was one of the mainreasonswhythe sch oo l boardiniti- ated thisprogram fo rthe slower students.It was hop e d that the extra ti me wou ld help the s e same students experience successwithan academic mat hemat icsprogram.

At th ehi g h school level, most SUbjectareas have been allocated a specific amount of time to cover a required course of etudy. Anderson (19B1) refers to theuse of classperiods as a fixed-timecondition. However, he points out tha t "by operating under fixed-timecondit ions,.•. wear e guaranteeing that some stUdentswill learn a great deal, some wi l l learn moderatelywell, and some willnot learn at all " (p. 1). In Newfoundland and Labrador, the Department of Education specifiestha ti l ltwo-credi t courses at thehi g h school level are to be allocated one class periodper day. This may be eno ughtime fo r somestudentsbut it maybeins u ffi c i en t for manyot hers .

25 Thetimeneede dto le a r naco nce pt canvarygreatly from stude nt to ano the r . Decidi ng just how much ti me is requiredto learn a concept is a que s t i on which is at the center of the research on time and le a r ning . According to Walberg(1988 ),answeringthis questio ndependson"wh a t is to bele arn e d , howi tis taug ht,and the student 'saptitude" (p. 83).Gettinger(1984), in an extens ive re v i e w of l iteratu reto de t e rm i n e the individua l differe ncesof stUdentsandthetime neededfo rle a r n i ng, provided some interes tingestimates. For example, she quo teda study in which the fastest students comp.Letied 5000 reading problems ov e r a seven month period usinga computerassistedprogr amwhereas theslow eststud e nts completedon ly1000.Gett i nge r feelstha t "stude nt sdodiffer inthe rat e atwhich the y le ar n school - relatedtasks " (p.23).

She quo tes many studies to support this claim in wh i ch variations in learning time for slow and fast le a r ners vary fr om1.5:1 in one study to a highof 7:1 inanotherst udy.

Fred rick and Walberg (198 0) report tha t Bloom (1976)

"est i mate s thatthe slowest 10 perce nt ofstude nts mayne ed five - to-six ti mesasmuch ti meto le arn asthe most ra p i d10 perce nt, butthey areusuall y not givenit" [p,191). Guskey andGate s (1986), intheirsynthesis of theeffects of mastery le a r n ing. also found studies in which the amount of time needed to learnva ried among fast andslowlearne rs. It was ear l ier not e dthat Bloompro posedtha t byus i nghi s "ma ster y le arni ng" mode l , all le arne r s coul d reach a hig h le v e l of

ae achievemen t it al l ocatedenoughtille . Howe ve r , FUchs and Fuchs (198 6 ) found that aas t e ry le arn in g actually retarded the suc c e s s of hig h achievers in grade one. They fel t that to o muc h allocatedtilllecould slowthei r progre s s. Th e y concluded thatthe low achie v ersrequired more direct ,structured, and elaboratedins t ruct i on.

Maste ry learni ngtechniquescandecrea sethediffe rences betwee nfast and sl owlearners , according to Arl in (1984).

Ge ttinge r and White(197 9)foun d thatthe time ne ed e d tole arn was moreli kelycorrela ted to sch ool ach iev e ment thanwas the 1Qscore of grad e four, five and si xst ude nt s. Arlin (1984) ag.:ees thatdifferences in learn ingab i lit ie sarp :o:eflecte din individual di fferenc es inthe amount otth'lene ed e d to learn ma t e r i a l.Oneway todecrease this gap betweenstude nts is to provide extra ti me especially in areas ....herespecific pre- requisites are requiredbefore procef!di ng tothe ne xt st e p.

Ac a de mi cma thema t icsin the secondary sc hoo l would certainly be a SUbjectrequiringprerequi site ski ll s .

A study by Gettinger (1985) attemptedto eval uatethe exten t to whichallocati ngless timethan theae o unt; needed ....ould affect th e overall achi eve me nt of school related materiaL The study invol ve d 111 four t h and fifth gra de st ud e nts.Thetime ne ede dfor le arning ....asba s edon ebe number of trial s neededto maste rata s kin reading to100% accu racy.

The timeallocated forlea r n ing....as reduced systemat ical lyfor all ch ildren. The study foun d that allocating insufficient

27 learni ngtime had aneg a t i veef f e c t on achievement. Acc o r d in g to Ge t t i nger(1985)."th emost obviousimplica tion fr omthe s e findin gsistha t spending less timetha nne e d e d or allo c at i ng less t ime th an needed doe s affect student achievement " (p. 10 ).This impliestha t slo w le a rne r slI'Iay no t be ableto learn the sa meamountof materi a l in the time as sIgnedtothe faster learn e rs. As a co n s equen c e , all o c at ing ext r a time to slow learners may be one wa y to hel p the mach i eve suc ces s wit h mathematIcs.

Reporting on a study entitled Nat ional Follow Thrqug h Observati on St.u dv 1975, Sta llings (1980) re po rted tha t low ach iev e r s in grade th r e e achieve d more rn mathe matics and reading than highe r achi eve rs whe n the y whe re gi ven an inc r e a s e in time . In a large two-pha s e st udy involvi ng 87 reme d i a lseconda rycreserceee,Stalli ngsfoundthatincreasin g alloc a ted ti me to improve rea d i ng resulted in positivegains if the te a c he r used an interactive on-task approach to inst ruction, but aslightor neg at ive gain was obtainedi fthe teac her al located the ext r a ti me tononintera ctive instr uc - ti on . She fo und that whe n the te a ch ers were supporti ve and gave positive fe edba c k, the st ude nts improved in read i ng achievement. She claimed tha t "low-a ch i e v ing sec ond ary st ude nts.. . pros peredmore andsee me d toneedthis nurturing environment moretha n did those se c o ndaryst ude nt s who were ac h Lev Lnqat a higherlevel" (p . 13) .

Karwe it and Sla v in(1981 ) st ud ied 18pri ma r yandet esen-

28 tary classrooms in rural Ma r y l a n d to see how ti me affected achieveme nt gaine d. They inve s t iga t e d four measures1total sched uledtime, total instruct ional time, total engaged time and engaged rate. Th ey defined "e ng a g e d rate " as the tota l engaged ti me divide d by the total instructional ti me. They fou nd th e strongest relationshipwith achievementoccurred when thetime measures captured thein d i vid u a l ' s engaged tim e.

The i r re sul t s also ind i c ate d that different students wi ll experience different effects of time depending on their positionrelativeto thecl a s s average.Ab ove average students will needless time tomaster the material , whereasstude nts below theaveragemay need moretime.

In summary, from theevidenceof the studies onability.

it wouldappear that the idea of allocatingextra ti me to high schoo l students ..,ith lowe r ability in mathematics could certainly be seen as a posi t ive step toward helping these stud en ts.

Time and Mathematics

Inthissection, studiesinvo lV i ngtime asit relatesto achievement in mathematics we r e inves tiga ted.Very fe w studies could be found concerni ng allocated time and mat hema t ics, especiallyat the highschool le v e l. Accordingto Robitaille (1975) and Ka rwe i t (1964 ), the r e is a need for re s e a r c h in this area.

Walberg (1988) arguesthatmathemati csand sc ience"••.

29 because of th eirhig hly specializedand abstract symbolism, may require the greatestconcentrat ionandperseve rance" (p.

76 ). Welch, Anderson and Harris (1982) . claim that suojeccs (suchas mat hemat ics and fo r eig n languages )that can only be learned in school will show stronger effects fromtime .i.n school than is tr ue for othereubject.e,AccordingtoRomberg and Carpenter (1986) , the researc h on ti meand mathe ma t ics ach ieveme n t cl e a r l y sup port s the fac t : "Wh ile ther e are limitson the amountof timeanycla s s teac her can allocate to mathematics ins t ruc t i o n,thete a c he r who consistentlydevotes le s s time to mathematics instruction than col leagues can expect relativelypoorerstudent achievement" (p.862) .

Husen(1967 )was th e chai rmanofa very largecompa rative studyundertaken byan internationa l pro ject for the evalu- at i onof educati on al achievement(lEA).Thestudy wasa la r ge cooperative cross-nat ional educational research project involving 12 cou ntr i es. The study involved 13-year-old stud e nt s and pre-university students. Many variables were inv e s tig ate d, but forthe purposesof this studythevar i a bl e s of importance were: (A) total hours perweek in sc ho ol: and (b) the hours per week allocated to mathematicsteaching .The stud" found tha t the percentageof time per week devoted to mathematics varied from 11%to 18%among the countries st u d ied. Accordingto Husen (1967 ) , "the number: ofho u rsper week ofschool i ng seemed to bear littleor no rela t ionshipto mathematics achievement . while total homework at the lower

30 revet and mathemati cs homework at the pre-university level seemed to be of gre'lter importance " [pp , 301).

Lindsey (:1.974) , in a follow-upanalys is of th e tEA study, at t empt e d to study the rel ationshipbet...een achievement in mat h e ma t ics to class size and to the number of hours of ma t h e ma t i c s instruct ionrecei vedeachwe e k . He basedhisstudy all the mean sco r e in a class ro o mand not on the sco r es of ind iv i d u al stud e nts. He found that as the cl a ss si ze increa s e d , there was adi s t i nc t drop inthe mean score for all classes,except for those that received thela r ge 3 t number of hoursof instruction. Hefelt thatC'''. cannotconc lude that theret.san opt imumamountof Lns nr u c t.Lon which wouldincrease achievementifgiv e ntoall students .

The extensive BTES (19 8 0) study produce d many results invo lving allocatedtime and achievement , as wellas ALT and ach Ievenent;, The s e findingsare relevanttoma t h e ma t ics at the gradetwo and fiveleve l s . Accordingto Fisher eta1. (1980 ), 14 majorfindingswere discovered relating time andachieve- ment . The ones relevant toth i s study include:

1. "The amount of time that teachers allocate to ins t r uc t i o n in a particular curriculum co nt e nt area is positively as s oc i a t e d ..,ithst ude nt learning in that centant;

art>·"(p. 15)

2. "Theproportionofallocated timethatstudentsare engaged is pos itive ly associated ..,ith learning" (p. 16) . Althoughthese resultsrelate to mathematicsat thegradestwo

31 andfive le v""l s , i tseems conceivablethat they would also have significance atthe high school level.

Th e findings ofthe BTES (198 0 ) stu dyare also con f i r med by Borg (19 8 0) and Anderson (1 9 81). In addition, Brown and eexe (:1..984) , who analyzed the BTES findings, claim that mathematics achievement in gr ade two could be inc re a s e d by 2.5 %: ifal l o c a t e d ti me were increased by 10%:. They nuqq e at;

thattherewo u l dbe a less modestgainin grade fiv e. If the increase in achievement dec reases with gra de level, then pe r h a p s the gainsat the secondaryle v elwouldbe minimum.But thisre main s to beinve s t i g a t e d.

A study by Fitzgibbonand Clark (1982) examinedthe use of timein eight secondarymathematics class es. The clas ses wer e obs e r ved foroneweek ea ch duringthe fall termand the spring term. Mathematicsachievementtestsweregi v enat the beginningand the endofthe study. Observationswere made on six target pupils ineach classto determine i fthey were on- task or off-task. They found that abont 7St of pupils were generally on-t as k. Howev er , they found that stUdent abs e nc e s , late starts tolessons and of f-t a sk behaviouraccounted for only one-halfof the scheduled time being us ed in academic ta s ks.

McIntyre, copenhaver, Byrd and Norris (198 3) us e d an observationalstudy of 10 grade three, 12 grade five and seven grade aeven mathemati cs classrooms to examine se ude nt; engaged andno n-eng a ged behaviours . They found a slight decline in

32 engagementrates from grade three to grade seven (77%in grade three compared to 73% in grade seven).In addition, the y found thatfor the grade seven classes,the r e was more time used for teacher-ledactivitiesandless time wa s us e d far seatwork than in the grade three classes. Students who were on- task during instruction ha d higher achievementthan those students who are off-task.

A study by Welch at al. (1982) used data from the 1977-18 NationalAssessment of EducationalProgressin Mathematics.It was designed to examin etheproportion of variance in mathe - matics achievement that couldbe attributedto differencesin the number of semeatiezsof mathematics studied. They did a multiple regression analysiscomparing mathematicsachievement with eight measures of background characteristics plus the number of semesters of mathematicscompleted.They found that although there were strong re l a tio ns h i p s between nonschool backgrou n dvariables and mathematics achievement, a stronger relationship wa s foundforthe amountof mathematicsstudied in school. These re p o r t e r s felt that this finding seems to impl y that there is a need for increasedmathematics enrol - mentsin the schools.

Attitude andAchievement

Most researchers find a positive rel at i on ship between at titude towa r d a subject and achievement in tha t subject. All studies that related achievement to attitude reported a

) J

posit i v e rel at i o nsh i p . No studies were fou nd th at showed a ne g a t i v e effect.

Walberg (1986) did ave r y extensivesynthesis of research studies on teaching . He points ou t th",t of 128 studies involvinga corre lation be t we e n sel f -concept and achievement, 84% of the studies showed a pos i tive effect fo r a mean corre latio n or effect of 0.21. This is a law correlat ion:

neverenerees , it is one which shows a positive relationship.

Walberg found th a t the correlations were high er for high schoo l st udents than for elementary or college students. In eight studies involvi ng ability gr o uped versus ungrouped secondary classes, 88% of th e studies showed a po s i t i v e relationshipwith att i tudetowardthe SUb jectmatter. Walberg r'1'portson amul tiv ar iate analysisof the productivityfactors in samplesof13- andl7-ye a r-ol d studentswho participatedin the mathematics , socia l studies and science por tionsofthe National Assessment of Educationa l Prog ress (NAEPj. By statis tically contro ll i ng many factors affectingachievement and by using many national atiudLe e, Walbe r g sho wed tha t achievement in mathema t ics is positive ly related to the attitudetoward mathematics.

Inthe lEAstUdies, a positiverelationsh i pbetween the affective and cog nit i v e objectives of a subject was found.

Thiswas tr ue for eachcountrythat took par t inthe st udy.

The var iabl e s includedinterest inthe SUbjectand atti t udes toward the subjec t and school. Accordi ng to Bloom (1981)

34 ev ide nce from the lEA st udies indicateth a t "s t Udent s who master th e cognitive objectives well , develop positive in t e r e s ts and attitUdesin the subject" (p.44). Forexample, thisimp l i e stha t interestin sciencean d positiveattitudes toward it are positivelyre late d to achievemen t in science.

Thiswouldalsoapparentl yapply to mathematics.Studen tswho are succeeding in school tend to like sc h ool and hav e a positive attitude towards it. The majority of students who were se l e c t e d to receivethe extra time in mathema tics had beenunsuccessfulinmathematicsinthe past.By offering them extra time, it was hoped that the y would experience mo r e success academically and th e r e f ore develop a more posit i ve atti tudetowards mathematics and school.

As part of tile BTES (1980) etudy , Fisher et; at. found increas ing academic learning time did D..Qtresult in more ne g a t i ve at tit ude s toward mathematics, reading or school in general. In fact, they found that "s uc c e s sful students probably enjoy learni ng mor e becauseof the ir success" (p. 24). They fou nd that any fail ure, evenif it occur redonly occasionally, resulted in a more ne ga t i v e atti t udeamongthe elementa rystudentsstudied.

Tsai and Walberg (1983) furtherana l yze d th e NAEP data.

Theyfound that fo r 13-year-olds, theirattitUde to mat hemat - ics was influe ncedby home condit ionsandby ecnt eveaerre,They foun d tha t the more one learns , the bettertheat t i t udetoward that SUbject and , con ve r sel y,the qreaterthe attitUde, the

mo ra one learns. cuekey and Gates (1986 ) fou n d in the i r sy n t hes is of ma s t e rylearn ing that stU de ntswho went throug h mastery classes dev el ope d positive at titudes about their abilitytole arnasthe i r le a rning improved.

In a survey of 2500 high school sen iors , sosniak and Et h ington (1.988) found that the students' attitude toward the i r academiccours e wo r k was highly positive whe ther the stude ntwent to a schoo l rat e d as good or one rated as bad.

Theratingwasba s ed on the academic abilityof the studs nts enroledinth eec noo a s,the dropoutrate and the reputation of the school within the commun....:y. Thsy also foun d that th e stude ntsin th e poorlyrated schoolswere more likel y tora t e mathematicscou rsesasveryimporta ntcomparedto the students in thegood schoo ls.

In order to study the conflict ing views of abil i ty grouping ,Newfieldand McElyea(1983)de v el o p e d a longitud ina l studyof highschoolstudentsandst u d e n ts beyondhi g h school.

They found the hi g h ec ntevt ne sophomores as we ll as the seni orsin advanced mat hematicsclasseswere absen t le s s and were more interested in school than even hi g h achiev in g studen ts in regul a r mathematicsclasses. They als o found a morepo s i t i v e attitude to ward school.Even low achieversin a re g u l ar class had more self-satisfactionanddid better on an achievement testthandiulowecn a e v e re in reme dii!l l classes. All authors ge n e r al l y fo u nd a posi t ive rel a t i o n ship betwee nattitudeto a SUbject andachievement intha t sUbject.

36 A high suc cessratein school wa s fo u nd tobe a facto r that contri butes to hig h le v el s of self-este em and a posi tive attitude to th e SUb ject. AsFi s h e r et at, (19 8 0) pointout ,

"s t ud e n t s who spent moretime than the averageinhi g h. success activitieshad higherachievementscores in the spr i ng,better retention of le a r n i ng ov e r the summer and more positive atti tUdes to wa r d school " (po17).

Recen t studies are be ginn i ng to foc u s on th e role of motivationas it re lat e s to learning and thinking. According to Resnickand Klopfer(1989) , "f or many decades, researchon motivation ha s been con d ucted separately from re s e arch on le a rni ng or cognit ion^ll (p. 7).According to Larkin andchabay (1989) ,"t he cha r a c t eristics ofeffec tiveinstruc tionre quire tha t the student be con t i nuou sly and act ively involved in learning"(p. 159).They further claim that ext ri nsicmotiv- atio n is~important for success tha n is intrinsicmotiv- ation.A suudy by DweckandElliott(1983,ascit e d inResnick

&Klop fer, 19 89)fo u ndtha t "motivatio nwas intima telyrelated to stu dents' conce ptions ofin tellige nce " (p. 8).

su mma r y

Accord ingto Wang(19 79 ) , "t o provideevery ch il dwi than 'e qua l opportuni ty' tosucceed in scho ol,a suf ficie nt amount of ti mefor le a rn i ng and in stru c tion must be made avail abl e to studentsand tea ch ers" (p. 169 ).

Thi s review of the literat ur e was underta ken for four

37 One was topr o v id ea background ofth emod e ls which specifically relate t ime to ach ievement. A second was to provide a reviewof studies which foc us on achievement as rel ate dtoallocatedtime .Thethirdpur po sewastoprovi dean overv i ewofth eef f e c t s oftimeon learning,espe c i all y as:it relates to lower abil i ty students. Four th , the ef fects on at t i tUdesas lear ningtimeinc r e asesva s expl o r e d .

From therevi e w of lite r atu re , it would appear that mos t re s earch studiesshowaposit iverela t i on sh i pbe t wee n time and le arn ing.Howe ve r , most rese a r c hersclaimthat there aregreat differe nces in the amounts of time allocated to Learo Lnq activities.The time allocat ions todiffe rent subjects vary ma rke dl y amongdi ff ere n t cl assesandgrad es .

Increasi ng allocated time in schools mayor ma y not imp ro ve achieve men t.Thisstudywasdesig nedto help determ i ne if allocatingextra timeto .ath ematics wil l help b1p r ov e ac hie veme nt inmathemat ics fo r sl ow le a rne rsina par t i cul a r instruc ti ona l prog r a m.

38 CHAPTERIII

Methods andProcedu res

In thi s cha pt e r a descr ipt i o nof themethods, mate ria l..!

and pr oc edure s used in carrying out the st udy is presented.

First, the methods us e d in collectinqthe data aswell as a desc ri ptj...n of the gro ups inv olved are presented. Then , a descr i pt i on of the ma terials and instrume nts us ed ispr ov i ded dongwi t hthemethod sus ed to anal y zetheda t a.

Method

This study grew out ofa desir e to evaluatea program offering help to some lower abil i ty st udents who re c e i ved ext ra tim e to st udy ac ad e mi c mat he matics. This method was tried by one school under the juri s di ct i on of the Roma n cat holic Sc hoo l Board for st. JohnIs, Since the te a ch ers involvedfelt that the approachwas very successful, th e idea quicklyspread to othersch ools inth e district. During the 1988-89 school year, there were five hig h schools offering th i s program. Th is re s e arche r is a prac t ic i ng mat hematics teac h er wi t h one of the fi ve high schoo ls . Al t h o ugh not personallyinvolv ed in thete a ching of this program, she fel t that an evaluatio nof the program was necessary to dete rmine ....nec ner this program was meeti ng its Objective; tha t is , ....het he rst ud entsinvo lved inth isprog ramweresuccessfu lwith acad emicmat he matics.

3.

During the fall of the school year 1988-1989, thi s researcher contacted the seven teachers invo l v ed with the teachingof the eightclasses receivingthe extra time. Each te a c h er agreed to takepa r t inthe study.The initial comme nts ofthe te a chers to ward s the program werevery po s itive.Most of the te a che r s fe l t that the program was working for the majority of the students involved. The majority of the teachers felt th a t th e attitudes of these stud e nts towards ma t he ma tic s improved as their achievement in ma the ma t ics improved.Thisresearcherfel tthat a study shouldbe und e r- taken to verifyth e s e claims.

To determine i f the attitudes towardsmathematics did indeed improve,thisresearche rselected the SandmanMathemat- icsInv ent o r y (MAl). (See Appendix A).This inventorywas one thatha d beenused by the schoolboard onpreviousoc casions. This sallie in s t r umentwas giventothe students who rece i ved the extr a time. It was given to the st ude nt s prio r to the Christmas break 1988 as a pretest. The same inventory was given inlate May198 9 as a posttest. T-t e s t s were used to analyze the datato determine if there had beena change in attitudeon any of the sixconstructs measuredby the inven- to r y. (See Table2).The numbers do notagreefor the pretest andposttest bec aus e the tests were given to the stUdents present in class onthe day of the testing. No names were re que sted onthe attitude surveys.

4 .

1'atlle 2

Students COIllDl.tinqthe Sand:tan JOI

Pr e t est Pos tte s t

Math e llat i cs1203

97 65

1<oo!Lll Mathella t i c s 2203

26 31

To determine the achievement of these stud e nts, the res earche rcontac t ed each teacher invol vedwith the progra mto ob t a i n the fina lJu ne re s ul tfor each stude nt . Prior to thI s, thi s researc he r had each st ud e n t complete e.questi o n n a i r e re questing in fonLatlonon the stude nt·s grade nine I14r k and homework hab its.(Se eAppe ndixB).Inaddition ,the researcher constructedtwo 2S-it emte sts (AppendixC)to be gi ven to the stude n t sin AcademicMathema t i c s 1203 andAc a demic Mathe matI c s 2203whohadre ceivedtheextratime .Thi sre s ea r c her analyzed thisdataby obt ainingthe mea n scor esfor eac htest. The s e me answerethe ncomp a r ed with the grade nin emark todete rm i ne iftherehad beenany improvem e n t.

An effort was madetodete rmi ne i fthe ach i e veme ntof thesestudents couldbe compared to si milar st udent s whohad not rece ived the ext ra time. To do this, this re s e arc her contactedte a c he rs insevenhigh schoo ls under the jurisdi c-

41 t ic n ot: th e school board. The tests constructed by this researcherwere administer ed to 313st Ud e ntswho hadreceived six period s to study academic mathematics. This resea rch e r th en conta c t ed thesc hool board officeaswell as thediffer- gn t hi g h scho ol s to obta i nth e Canadi a nAchievemen t Test in Ha thelnat ics (CATM) score fo r as JIlany of th e s e stude nts as po s si ble. This scorewas one in a batte ryoftests given to al l grade nIne stude nts inthedistr i ct.This scorewas also obta i ne d forthe128st udent s who hadre c eivedthe ext ra tim e.

Of th e 128 students , the highest percentile rank on th e mathematics compon e nt was 51. Therefore, this sco re wa s sel ec t e d as the hig hestscore us e d forcompariso n wit h the group whodid not receivethe extratime.Ofthe 313 stude n ts te sted, 177 werefoun dto bebel owthe 51stpercent ileon the CATM test. Th e s e 177studentswereselected for compar ison with the 128 st ude nts who ha d re c e ive d the ext ra time . The Jun e ma r k was also obtained for the students who ha d no t rece i vedthe extra time. Thisma r kaswel l as theoneobtained on theresearcher-ma de test were compared by using ANCOVA.The CATH mark was to be the varia b lecont rolled. The June exam ma r k and the re s e arch er-made test mark wer e the depe nd ent va r iab l e s.

Most of theliteraturereview revealed tha t time on ta sk wasa most important condi t i o n if time increases were expe cte d to result in inc reas ed achieve ment . To assess jus t ho wthe e.re re time wa s be ingus ed, this res e a rcher cont a c tedtwo of

.2 Table 3

Enr o llme n t and Tillie Allocationsto tile Mathematicscourses

!&Y.OLI Ma t hema t i c s 1203 Allocated All ocat e d 9Periods 6Feriods

~ Mathemat ics2203 Allocated Allocated 9 Period 6 Periods

Total

e;51st

pe r c e nt il e 85 85

210 122

4J 4J

103 55

the seven teachers inv olve d with the program. Both teachers agreed to have a videocameraset up inthe i r classroom.Th is was desi g nedto reveal if the stude nts....ere indeed ontask.

Bothclasses(onegrou pwas studying Aca demic :-Iathema tics 120 3 and one was st u dyi ng Aca de mic Mat hematic s2203) werevi deo- tape d fiv e times each.This re s e e r-cber pe r s on all y set up th e camera at the front of the room eachtime and the n left the room.The camera was re c us ed on the students andnot on the teac her. Th i s might help dete rm i ne if the st udents were consist e n t lyon taskdoi ngmathematics. The ideaofthe video camer a wa s used bec a use this rese archer was not able to observethecl assesdi rect ly .

Inaddition, to obtain somemore infor mat ion onhowthe

43 extra time was used, this researcher int e rvie we d eachof the seventeachers involved.Aswell. the lIlat helllat icscoordlnat" o r with the sc h o ol bo",rd IIl4d e availabl e to the re s e ercner thi!

reports thatfive of the teache r s invo lvedwit h the pr ogram had submitte dto theboard.Thesereports outlinedthevario us teache rs ' opinionson the progra m. The y al s o helpedpro v ide somebackgroun d informatlonon thepr og ram.

Materialsand instrument s

The scores from theCanadian A.chievementTest inMat he-

matics(CATM)were obtained to asses s ea c h student'sability in mathematics. The resul t s of this test wereavailabl e from the Roman CatholicSc hool Board, sinceea c hst uden t ingrade ni n e is requiredto the take th e CAT battery of tests . Th e ma the mat i cs compone nt is one of tour tests developed in Ontarioas aresul t ot alo ngit Ud inal st udyof approx i matel y 90, 000 gradenine studentsin1959.

The test cons i sts of 30 items (15 in Algebra, 12 ir:

Geometry, and 3ineeneur eeIcn j.Ac c o rd ingto Morr ison (1965) , the test "re fl ec t s .. .ali mi ted point of viewconcerningthe high school mathematics curriculum" (p. 566 ). He further claims th at it has "Ine c e qu e c e te c hn i calqualities inte s t cons truction " (p.566).He claims thatthe CATM test mayha v e limitedcontentvalidity but f~ltthat thetest mayhaveso me value for "t hose inOntario and elsewhere who wishto compare theirpupilswiththat provincewit h re spe ctto the irac hi eve-

44

merit;" [p ,566).

Despite thepoor rating gi ve n by Morrison (1965), the sc orean this CATM test was selected for this study becauseit was the most recent standardized test resu l t available for these students. Then to o , the CATM score wa s one of the facto rsbywhich these studentswere assignedext r a time for mathematics.

TheCATMte st was administered to allgradeni n e students of the Roma n CatholicSchoolBoard. Thescores were available from the various schoolsparticipatingin the study or from theschool board. Studentsfor ....hom th eresearcher cou ld not find a score ei therhadbeen absentonthe dayof te s t i ng or we r e transfers into the schaol board after the test was ad mini s t e r ed.

To determine achievement onthe high school academic mathematics , the researcher developed two 2S-i temcompletion tests (Appe ndix B); one for Mat hematics 12 03 and one for Mathematics 2203. These were administe red to the classes participatingin thestudynear th e endof May1989. They we r e administeredto all groups duringa regula rclass pe riod.The ties'; was constructed according to the requi re me nts of the curriculum guides forAc a de mi c Mathematics 1203 and 2203 of the Department of Educ at i on, Government of Newfoundland and La br ador. The teachersinvolvedwith the teachingof these courseswere askedto comment ontheite ms.Withoutexce ption , each teacher felt that the itemswe r e consistent with the