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Contents

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Foreword .r 411, , V

11 Preface 41. MD .1 OM am vi

Introduction...... _ gow 1

I' . Research in the2eachingof CollegeMathematics 3

Research in theTeaching of HighSchool Mathematics 9 'Research in theTeaching of ElementarySchool Mathe- t maticsM NM ______Oa .41.1...... 01040 M USD 16

Summary. . ______a _ .. _ . _ _ _... _____. _ ...... _ ..... 20

Unanswered Questionseinthe Teachingof Mathematics_ 24 4.4 Appendix: Summaryof ResearchStudies______29

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Foreword

rill-LEIMPROVEMENTof teathingin mathematics,asin other Isubjects, dependstalarge degreeontheextent to whichre- search conclusionsfind their,wayinto theclassroom. .Thepurpose of this studyisto helpreport and disseminatethe 4ndings in the researchonmathematicsteachingcompleted duringthe 2years1955 and1956. All concernedwith thepresent study hope that itwillprovehelpful to both researcherandclassroomteacher. TheOffice of Education is gratefulto th6 deans ofgraduate schoolsandto research Workers in mathematics.education whosuppliedthe dataonwhich this study is based.Withouttheir willingcooperation itcouldnot have beeii )prepared. E. GLENNFEATmatswort, MangAissi8tizatCommissioner, State andLocal ScpolSystem!. J. DAN.HULL, Director,Inotnuition,Orgcthiaation, andServices Brandt.

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Preface

IN THEYEARS1952,1953, and1964 the National Colincilof TeachersofMathematicsand the U.S. Officeof Educationcoop- eTated to summarizethe researchcompleted in niathematicseducation duringeach of theseyears.The summariesreceived-manyfavorable comments andsuggestionsfrom leaders inmathematicseducation. % hasaresult ofthe suggestions,tho officeand the councilcooperated further,.this timesummarizingresearch completedin mathematics education duringthe2-year -period1955-56, and a1 io.analyzingthe material.Thepresent bulletinpresents this work Cooperation inworldng withthe1955-56material,aswiththe earlier material,waseffectedthrough theResearchCommitteeof the NationalCouncilof Teachersof Mathematics.The %embersof that committeewere4ifefollowing:John J. Kinsella,SchoolaEdit- cation, 1,)w Ybrk-Untiersity;L ClarkLay, PasadenaCity Conte; Nathan Lazar,College of Education,The OhioState Univenfity; and KennethE. Brown,Office ofEducation, U.S. Depaitmentof 1 Health,Education, and Welfare.

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'J '' ANALVSISOF ItESEARCHIN THE' TEACHING 0 OFMATHEMATICS

Introductio4

11111011 WTHATRESEARPHinthe teachingof mathematicswascarried VVonduring calendaiyears1.955and 1956? Whatdoes thisre- searchsayabout effectivewaysof teaching?In whatareasof mathematics educationwasconsiderableresearch carriedon? In whatareas wasthereverylittle I To help teachersgetanswers to these and ..similaiquestions,the U. S. Office ofEducation,aided bythe ReskrchCommittee ofthe National Councilof Teachersof Mathematics,sentaninquiryto 400 colleges thatofferedgradpatework inmathematicand/or whose staffs hadmade previouscontributionsin thisarea.The committee receivedanswers to the inquiry ahddataon123 studiesfrom approxi4 mately850 colleges.These studiesincluded20 studies bycollege faculty membeis,,54 master'stheses, and49 doctor'sdissertations. About halfof the128 stfidiesweredevotedto methods andap- proximatelyhalf 7 to content TwentywereconeirnedwitYprognosis andevaluation;the others,with thehistory ofmathematics, aidsto teaching,teachereducation,differences inmathematicalability between thesexes, status of mathematicsteachers, andmistellaneous topics.Ifanattemptwereto be madeto classify the studiesiccord- ingto majoremphasis,the categorietiTh%wouldnot be mutuillyex- clusiye.Several studies,for example,wereconcerned withboth content and methods. . Wheiithe studiesareclassified bythe threegrade levelscollege, high school,,elementaryschooloverlappingagainappears.Thus, the collegelevel -wouldcontain40 studies the highschool level,60; and the elementa4school level,40.Somestudiesdealingmainly with college4eveltopicsalso deal withthe trainingof highsehoca

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rlb 2 ANALTEIN3OP 1128141101IN TDACHING iMATTIMMA,T1133

4 teachers.Likewise,'a few studiesconcernbothcollegeand highschool andsome othersconcern both highschooland elementaryschool. Anythought of classifyingthe studies,then,as apreliminarystep ,.to analyzingthem for thisbulletin,wad abandoned.Instead,im- portantquestions-wem posed pertinentto mathematicseducationon the threelevels identifiedabove.The threegroups 9f questions accompanying and analysesfollowthis introduction.Withinseachanal- ysis,numbers appearingin parenthesesdenotecorrespondingnum- 4 bersused in the apPendixfor irrangingsummaries Ofall 12,3studies alphOeticallybyauthor.

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Researchin theTeachingof CollegeMathematics

* : 1 riN THECOLLEGELEVEL,research inmathematicsteaching kjemphasizedthecontent offreshmenmathematicscoutses, -the preparationneeded bymathematicsteachers,and themathematics needed foesuccessin variouscollegesubjects.Alsoincludedwere studiesonremedialmathematicsprogramsandprograms for college freshmen poorly.prepagedlitmathematics.Questionsandcommen- No. taryonthese andothera :follow below. 1. How poorlypreparedinmathematicsare sonecollegefresktaent The excellent collegerecords ofmanyfresfmindicateagood foundation inmathematicsbut thelargecollegeclasses ofremedial mathematicsremindonethatsomepupilshaveneglectedthe study of highschoolmathematics. -. One studyi(46)discloses thatthe remedialstudent; himathematics were below'average in 1. Q. andweak inallsubjects,and hadscored at the first ; percentileonnationalnormsforawidelyused Ilighschool achievementtest. s ;

2. Are remedial mathematicsprogramsin collegesolvingtheproblem of thestudent poorlypreparedinthat?,g ,

The remedialprograms seem to be he1ph4tO somepupils butno sway shows' thit -theecasks'. Irithmaki i . ibeetves,. .. . , . throughashort Ire- medial *course,adesirablefoundationin highschoolmath- maths. Infiat; one stud*'(27).of.141freehmen . sittoivathat the &adepts '

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. 3. Dostudentswhose mathematicstrainingleas beeninterrupted by service inthearmed'form(orforsasyother reason)makelower gradesthan theirclassmates,' I. i . A. Theanswer seems ,to beno.Accoraingtoonestudy. (1Z1)students withalonger timelapse sincestudyinglapreceding male- 4 maticscoursemade highergrades thanthose withashorter time . lapse. : .. ., . Of 805 studentsin thestudy, 172 hidtakentithimatidriAthonly thesummerintervening, andthe remainingstudexita hada meanlapse of 17 months.The close-proximitygroupmade higherscores onthe pretest, but thelong.proximitygrouplila& highet Sores 'on the final.. test.Many superiorpupils inthe long-proximity-groupmade loifib scores onthepretest thansomeof the lesscapable pupilsi4 thedoes- proximitygroups.Ofcourse,this raisesthe questionof vildity ofa pretestin classifyingstudents %forexperimentalstudies ifthe . time lapsesince thestudent's previousstudy ofnlathem,tics is.not

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4. TV Aatarethe best predictors ofcollege succesOnmathematics? . A combinationof highschool marks inmathematicsautlithewore on amathematicsachievementtest.seen* tobeageed ptidiottorof succemin collegemathematic&One university(TO)makingastudy of the- placement 4;41,832 freshly:infound.that thecombinationof scores onaplacementtest compiled bytto universitystag, thescores onthe IowaHigh &hod-,Content Examination,:,the ratingAnthe Ohio' State Psychological.Exanguatitm,find= the. piteviotrilohrkeID mathematicswas a verygood predictor,but littlebetterthan theCOM-- bination ofhigh schoolmarks andachievementtestscore.

. . \ A t ..I.t.:'st.`..tt:r -'.f . . t 1 ;- 0 5. D068 successin highschoolswitAemat'ies 444 0.ogresa*general .) collegecourads? . , A .? t t I I i \,', f.} . . s ., I , i ..The studentswith ,lighability in high school: alaibitpatiocuswilly made higher Korea-in general°Wogsmnoses4 than :theft withlawft ability inhigh schoolmathematics.Ckic study,(70) ihowINIthikt students wiv)failedto complete" theirtoilersprogra.ul wer. thaw who htd made thelowest. achievementscoresin high:self:id mathenstAm Although-m*4collegepmgrams seem to -require . . tooxpetwey. in -amipth, ematice A forstuck*to be sucetssfa4one7*46-4113), ohm;in Awl caption. ,Success in high*hod instbOltaticawatittlibulao,relk dens*to- the 'college'atlicess of, 'i agroupofiluvapeotho,0100tity, sohooi- , teach*at:;one*it..univerity.'!Perhapol tithis)10OntillOPte meat, in, this, scientificAge,ofi-collegeProgram :111,ye4ed -,.. *AS* tive relationships. V* ?t4.-v r4; . .. . . autiwa.awraisiAno*- 1, 1 6., Areth4 poorlyprspareiistudentsusually fromswill high whoa'? A study(79) ,of606coltegefremeilinone*State.indicatedno differenceIn 'mathematicsabilitybetween studepts-graduatinfiom large highschools and thosefroin small* ones.tr I 4! vir 7. Is collegegeneral Intithetnaties on'itstotyoutl Arecent study (88) basedontheoffc;'ringsof 200 colleges indicates that between1948 and1955 "the numberof colleges offeringkeneral mathematicsincreased.In general,the culturalkips of general matliegiaticscourst,itS receiving:the attention...ofiesearch workers.

The majorityof thesecourses .are 2 semesters 'with ,6 hours credit. Althoughthere is mushexperimentationwith thecontAnt (65), (102), (88), (08),(S), 14suallyat least half of it consists .of topics froni arithmetic,algebra, geo*ry,andtiignnometry.The remain. der is topics fromanalytipalgeometry, statistics, highermathematics (65) and recreational mathpuuttics, . . . 4:6 Differentmethods of presentingthe materialhave beau used.One experiment (102) showed,the geometricapproach.asslightly superioi tothe algebraic.Asurvey9f $7. institu4ons(18) ind4catedanat- tempt to integrate topics.The respondenOcraquestionnairesug- iestedthatageneral mathematicscourseshould containmaterial fron4 OA following:9areastoid that theamount in eachareashould

decieueIin theorderprder given:Algebra, arithmetic,.rometry, trigo nometrftkia1 riogeometry, iltatistics, mathematicsof finance, the .: ea% and logic.,.. 4 t I .:i : 8. Whatmathematics is neededfounderstandfirst-year oollege subjects? 8enrial studieiconcerns& with this questionwerereported. Most of themwerebasedon ananalysis of textbooksorother literature in

the field4 T 1 i ft;-% ' algebra' . and arithmeticoccurredmoreire- . queutli i. O- " : thal! :i 7`, 1--:-- -. '.' ,..t. - fliv,-, -, .ahoy . : data reyeal4he mathematics i ill l.. _-I :-. ,iti.,.,.'i etlw ;,.?. 4--.I 1.' , % .7;--:.r ....;...-It 4* AeftwarAbrethit mathematics that would be . desirs04, .1 110/7' 0 : j , -,.. . .f *.A . ....t 4. 4flor4t..e, :Op .I..- if. aw 1. 1 - studente10 =Id - understand. . l .. . . it. In th6 sdIàehowthatifrknowledgeof high schoolmathe-

4)II 7,1 g maticsiseatedtor many firv.steparcativosubjects. ANowtstudyoniyof textbooks-in engineeringphysicsrevealed hat a1t1*oiethi,1 majoriti'ofth mathematicswas, .... Algebraand plane f. . t. -

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. r ; , 4 111 ' .;,.. 0:?!"4 . : T..''4? '92' ' v-1 t --27 r, , : . ' - 4 Pf't A. !: ; if..tyryt t`44,*-.4, .11;,4-- '; , . 1. .1!".* .7".- : 1. ' . s" .. # ',IN it 't , P!1 k.? 'X! -Nt`474'.r; " . tL4 '*; err, V,-e?rt.V %..1W 741V;4. "4. y..j YFV.A1--*13 .V101441-1'*3:6r.11' . 0... 60 ANALYSISor atistAttsttre TIMAIdtteMAiniammte a '.'Ckineratgeologytextbooks (i07)wied 'littlemathematics. inatliematic4 Tho skills.nccessary to undertgandthesetiti amid 43, obtainedfromtraditionalhighscimol.mathen4aticamums! The majorityof theproblems 4270.1vedaleb,ra.and simple-ratios.:9114 afewneededtrigonometry orfnponentialfunctionsfortheirsolu- tion.Twoquestionswere *raised bytheperson making thRi, textbooi analysis:Would moreeffectivelearningtake placeunderamathe- niatical,rathei thanadescriptive,.'approach!Should inkportant additional topics ingeologyteeding'a niathematicalapprottak included be a in thefreshmencourse?*The ituf4revealed mathematical that minly "problemsoctur in acidgolotyand problems thatth wouldenrichgeneralgeologyif 'the.students prerequisite hadthe mathematicalbackground. '4 Inagriculture, aquestionnairefrom78 *colleges(6)revealedthat they usually;they require'elementary algebra,butrecommendedaMimi generalcourse. a : According toonestudy (45)the skillsnedOsimiytforsticeess physical inthe sciencecourse atoneStatgcollegenifty beóbtained highschool from algebra ifthe4tourse emphasizesdirectand inverse variation,functiods, graphs,interpretation'ofdata,andthe ability to generalize. . Thesthdyof textbookiin varioilsfields algebrasolution shows-thatelementary of equations,.tratislationsóvetbal 'intodiagrams statements 'or equations,use ot graphs-azi ariation--afveitrs frequently.Advanced algebraradicals;ope withpolyno- mials,orfactoringofquadrati occurs.' *fitoothermathe-- maticsconceptsappearing arethosefoundinthetraditionalhigh schoolmathematicsCourses, 7. 9.&cult,there be aspecialmathematicscourse foragriculture students? 4r4 s. ( ; ;1 Of 75collegesin40 States (68),half,recommendedsr- 'general mathematicscourseandhalf themolaralgebraCourse.Theperson conductintthesurvey concluded thatthereisagrowinghitch*in requiringaspecialmath,ematicecoin*foragrioultute.studitaiti. r, 7 . Ito. : 4:41 f i 1 0 t 10.Are themathematicsofferingsin?Iv/ie. juniorcollege fewer than thoserivotnekindedZyBtlgo '4 officials, . T . 4 4;*i,1Ct... , .I An analysis(49)of 101juoir. con . offerinoinqudedillth, thin*teiddliofell néertngschools in Stitiitafroi*itieK"Tri4, eirieeetedthcidekaieeomniondA1tib . . ;.,1 4 ....7,;--f - -

4 -,"4 COL/WitMATHEMATICS 7 11. 8houl4the mealvmg ofmathematics'conceptsandoperationbe emphasisedat gel calveWeir AlthoughI *Averexpeiiinents'onmethOdswere reportedat the collegelevelthanatany other level,at leasttwo experiments dicatedthat in4 the meaningof theconcepts shouldreceivegreater emphasis. Inone experiment(18)on teaching thecakulus the the emphasiswith experimentalgroupwas on understandingconcepts. The quality' of learning of thisgroupwas superiorto that ofthecontrolgroup. Inanothei experiment(119),whichemphasizedunderstanding operations(inaccordwiththeprinciplesofmodern the psychology), achievementof theexperimentalgroupexceededthat of trol thecon- groupTheexperimentalgroup rated thecourse higher than thecontrol group.. Buildingan elementarymathematicscoursesolely uponthewishes of thestudentsisnot recommended,butperhapsa favorable studentattitudeisconduciveto effectivelearning. r' 12. Doprospectiveteachers needmore mathematics? Asurvey (10)of thegraduatesofonelargeuniversitywhoare teaching revealedthattheyfeeltheyshouldhave hadmoremath- ematics. Another survey(2)foundthatone-fifth ofthegeneralscience and biologyteachershadtakennomathematicsincollege of the and one-fourth chemistryteachersbadtakenonecourse ornone atall.Also reported were attemptsto teachthese andothercitiantitativesubjects to prospectiveteacherswithoutusingany mathematics. An experiment(59)was reported inteachingthelinearregression theoriin elementarystatisticsto studentswhohadnot takedthe calculus. Elementaryschoolteacherprograms (118) havingno quantitati4considerationsinany courseare anexampleofan attempt to educateteacherswithoutintroducinganymatismatics. 18. -Whatshould b .. thepreparationof mathematic' , . Ma011411 " Studies(86), )(29)teported in1955 and1956suggest thatthe undergradust,programshouldinclude26 to 82hours ofmathematics. Now-Euclid.= geometry, statistics,and historyofmathematicsare recommendedin additionto thetraditional collo&program through the calculi*.Thepreparationis usually suggestedrin terms ofsemes- ter hp= of z rpred#, Fat*than, specific,competency. For the example, ;ovoid.**tpone survey 129) of 151teacMrs,J4UIjDMratOrV, aixt 35,(Owro1ewqr8ris9InuVal444 70hours in rl education, 24 in mathematics,24 in education,and1yearofstudentteaching. Questionnaires(101)answered by951 teacherssuggested thatthe

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: _ $.... t'.0.«C . .'''..t.;;.1!i.!.. e, 4t, ' :.z. 4f/.1.... 1%.:. ... ; IV 8 ANALTAN9OF RIBMARCH DITRACMCMATIMIMITICO 5thyearof teacher educationshould be composed ot50 percentinatlw maticscoursesand notmorethan 25percent professional education courses.In these broad categories theyrecommanded thaefewspecific coursesbe required. Agroup of Newknglandteachers replyingtoa survey(10) recommended itatistics in themathematicsteacher educationprogram. Integrated mathematicscourseshave been suggested.Oneresearch worker (114) concludesthat thecontent should bemathematicaland thepurposeprofessional. a

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VI filiearchin the Teachingof HighSchoolMathematics v rIN THEMGRSCHOOLLEVEL,research inmatheniatics kiteachinggaveconsiderableemphasisto the need forstudentsto underaand.Alikoinvestigatedwastheextent to whichphysical aids anamethodsareableto contributeto attaining thatgoal.Experi- ments with introducingmodernmathematicsconcepts to high school .4 studentsreflect thedesire offrontierthinkersin mathematicseduca- tionto restudy theentirematbematicsprogram.Discussionof this and other problemsfollows below. A -4

1. To what*dent has thecontent of highschoolmathematicsbeen underitive4tigationandexperimentation? Experimentationwith thecontent of highschoolmathematicshas taken placein severalparts of thecountry and withdifferentsize groups. : Ino4eStateOWa groupof 36teachersgevelopeda se- ; querxe of mathematicstopics whichwere publishedu aseries .of high school 'textbooks.Thesetopicswenfromalgebra, 4 planegeometry, and trigonometry,withmanyreferencesto social problem& In inother State (Illinois)anextensiveexperimentis underwayto, deielopnot onlyzinvcontentbut alsoadifferentemphasison someof the traditionaltopics. Someconcepts normallyreserved forcollege studerasarei,oporatedin thetonne,sometraditioniddefinitions have bets 7 14tored, atid'forcedcorrelationisnot emphasizednor are compartmental linesadheredto.Leadershipstems tram theUniver. sity of =alaililt 'schools_taking part in- theexperiMentare not. aniline&to thatfititt 4- 1 Inoontrait ft) tke 'state.:universitye*oorimentinvsalthigseveral hundridsmiltbi at experiment withoneeleme#aty..,algtbra. i WI. 1 this. 4 w taught.**arjouistaff memben of k. r a1 largecity (*liege Ap-

prO*W7 40, lieitentof filo tIernefltftby L I ....11L._ algebra I ! s I come comfit* , . .- of 1 _ri" Lir ' :..1. I .1 .1100,11) es . 1 1 Jr = , . . . -..e...,Ap 1t 1n ,v-,,' .., P.; toPi 11 1.4- IS .. ii4 ...... ,. ..,;:'-.. 4r. r" 11 : -,- *plied s-P PI 4,it 4 --5 _".:-:-.6ilpiL t-t.._ 'T 'IL n4 A '... A 1,, I I I ,41: i' i!: tatl ':,.. __ - .1_ ,-,.,...... 1.-._ 5. -j. :10.2 1.4 .4 i .,4;.1 . v: i K s ! ), o. ';',°.1 iL,_,..t k a. g..L.t _ Y. 4; -1 . ! ..-' 7' V -...-.. ,.... FOr ..--- 4 erne ,e tAli ,y41-.4'.,:_:'.r. LL1i 161/4%i ,.., ; 110 t 1, 4.i.:::,i i: li - ( . b'Ait...rei: e 1,-, n I ''.i '. '!iI ' 0 t r:II ,Rolt,:-. 1 . ,. _, .;.t)si.--,..4)-4 Al; 41' i ts...... -.P-$. or , - . N. ':-.s- --, --z. , I l :- I , .,-i I a _4. -, --_ f ft't; 4. l'ast. at4 r 11_ ::::'1,....1:tt ,. 1 i..:Ah#: -, t.-1- .-a- :1":1".4-. :1 g i :-1 *f A .7 t. r 't ...t.. k,, . ,.. 2 i. ' A' 1.7 e $14' ',I 4.. t:' ..`if-ii ,.4 .-ri ,t_ :,,, ', i . # ? ...r; ---.1- P. , !!. .1:14 :! ...1 .). .'it.i. i."7",,,* I a ; : ..1.,i4 .^ )-,., I, &',r4,;,-.....,, 4 . r v.. i7 t.: ; A 1.0t t. . ,...A, .4.i .'i s. , . -: -- ... . 9 s

.4.a 10 ANALYSIS OF1128 = IN TBACHINGMATHEMATICS S.

Perhapinontaithema4cal woidscould have beenincluded inthe list, sinceaccordingtoonestudy (58),the vocabulary ofmanyalgebra textbooks isabove thevocabularylevel ofsomeof the pupils.

2. Are thewrittenproblemin)6Aighschool.mathematicspractiod ones?

s. A study(120) ofasample ofhigh schoolmathematics textbooks showedthat,on a 4-p6int settle,morethan one-thiMof thewritten , problemsfall in thetop category ofpracticability. Althoughappli- cationproblemsaresometimesmoredifficult than themathematical rinciplebeingillustrated,the studyindicated that the writtenprob- ....s lemsingeneralwere on a lower vocabularylevel thah theexplanatory = sections.

8. Havethere beenradiealchanges in planevoniftrycontent If"el' Even thoughthecontent of high school mathematicshas been in- vestigatedforsometime,yanalysis (117)of 97 planegeometry textbooksrevealed thatt sequ aimsfor thepast 15years arein 8 generalpatterns: (1) A.modem versionof Euclid'ssequence,(2) Legendre'ssequel*and (3)postulation ofthe parallellines-trans- p. versal theorem. f Another study (I ) shorthat the idea I. of continuityhas received achange inemasis.In fact, thecontinuity ofafunction isap- pearingmorefrequentry inalgebra textbooksand less ingeometry. Theincommensurablecaseingeometry is receivingmoreinformal, and lessrigorous,treatment than previously.

4. What is tAecontent of highschoolgeneralmathematics? Thecontent of generalinathematicstextbooks variescQnsiderably. However, theyusually includetopics fromelementary 'algebra,in- formalgeometry, andastrong emphasisonarithmeticand itsap- plicationsto the, solution ofconcrete problems. Sincemanypupils who take geneial,mathematics donotpursue,college Audios,Nome educators lookat the mathematicsused by semiskilledand unskilled workersas akeyto the content of generalmathematics.In 4ct, oneinvestigator alterstudyingthe mathematicsused by-theftworkers, concludes thatafith-grademathematicscourse-would fill theirneeds. There.are44 least two fillaciesto thestatusiuo wriculamap

prpows14/ First, in4uptr7 mig$Feq,uire 16.4greateruesa ***les if t4workers had abilityinLOSIJCcL. Sond,when

, theIneeent 1?-111* ( , are044,/airePat'odø 0f maybe demandedofsemiskilledworkers than isnow the t. nib wim llamajuinamAints 11 5. Whatmethods ofteachinggeometryhavebeen tiaeMOWofre # (tent eoperimentationand study? Meth* ofteichinggeometry have beenthe subject orseveral investigations.Althoughsomeinteresting studieshave beencom- pleted,nospecialtechniquehasbeen developedthat haspromise of changinggreatly the pregentpractice.Research studiesreportan emphasisonteachingfor meaningthrough discoveryand broader-, generalizations.One study(87) is devotedto super-generalizations ingeometry withanemphasisonstep-by-stepdiscovery. Inanother (83),2groupsof 21 pupilswerematchedasto mathematicsachieve- ment and I. Q.Onegroup was taught planegeometry by the traditional method;the otherwasgivenspeciallypreparedmateriallead- ingto the 4iscoveryof relationshipsandmcouraginggeneraliza- tion&The authorconcludedthat thetwo methodswereequally effective.Afterastudy ofthemathematicalideu inlaw,a re- seareher (48)recommendsthatgreater emphasisshould beplacedon. understanding inhigh schoolmathematics,especiallyreasoningin gompetry for thosewhothay study law.

6. D088the studyof saidgeometry improveone's abilityM determine spacerelationships?

Perhapsmoreresearch isneededto fullyanswerthis question.One extensivestudy indicatesthat 1*i% ifany,improvementinspace con- ceptionsresult fromthe studyof solidgeometry. Inanotherexperi- ment (51) with41 pupils,theexperimentalgroupmadesignificant gains whenthe analyticapproachwasused.

."What methodsof teachingelemenaryalgebrahave beenthe subject Ofrecent experimentationand stwity?.

As ingeometry, therewas no research that showedoneboot method, ofteachingalgebra. Astudy (4)of eleflantaryalgebratextbooks, forexample,gave9 differentmethodsfor toachingsignednumbers. In(anexperiment(15) with65 pupilstheemphasisinauniton directnumberswas onthe discoverimethod. Thereported 'eve- ment of thecontrolgroupandthe experialtatalgroup was t the Baum : In =ahoyexperiment(108)'.1 , a "gthms-and-ciumemethod wasused , ant . aid., to better underrAamliwiof verbalalgae!"probs. km*Althoughzoamolueve b.6 sire A Was reratsd,- theteach*?of the elan*aimed thatit aidedin iindeistandirqgthe oftkeptt. !Mu& f`' I I IB -A_ writtetonthe needfo explorationand 5 emon'on A Iii6ovetorbv

4t Ow uprising'u (SO) iiitolviigiO &miles wed drill With. .-# rr one !gimp and :iiiplcmition.and , 'with the (Mier. t :. . t".k" . _ . - . unmis14 r . ')/ . ±-0:t . 1.4or -qtf .14 'e f " 4.1 00- Iplos

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4 12 ANALYSISor MOOMASCitIN irtg.033240 ALMS Thepupils wererack:.byaspecialtest developedto meet theob- jectivesofthe experimentalgrotkpand alsoby theDouglassSurvey Test(Theexperimentalgrouphad 'greatergains in than the achievement controlgrtup, when meatihredby thespecialtest butnot significantlydifferent whenmeasuredby theDotlg lapsSurveyTest.) The individualresultsonboth thespecialtest apd theDouglass Survey Testindicate thatalargePercentageofstudentsshowed littleachievement..

8.Is instructionmoreeffeotiveifiA rupilsaretaughtin homo- - geneous groups?

- Theanswer hasnot beenfully-anwsFeredbyreseamhbut it to depend seems upontheareaofexpecte4achievement.Inone experiment (78) thepupilshowere groupedhomogeneouslyachieved morein fundamental: tinmathematicsbut lessin problemsolv- ingthan thosenotsogrouped.Ina survey (92) ofasampleof scboolsinone State,thepercentage ofschoolsusinghorhogeireous groupingin mat.4matic8was67 andinedema,40. Timefigures mayreflectdie succeqsof themethodinsomeschoolsbutthey donot implythat themethodis desirablefor allschools,Wherehomo. geneous grouping hasbeenreceivedfavorably,usuallyaoombination ofcriteriais usedindeterminingthegrouping.A I. Q., combinationof mathematicsachievementtestwares previous marksin mathe- =tics,andstudentinterestrating hasbeenused. 9.11greaterachievement securedbysmall-groupwork withinaclass ratherthan byinstnwtionto the entireclass? The experiment(57)reportedin thissurvey showsnosignificant differencein achievementof highschoolpupilstaught insmallgroups withinaclass as compared withthosetaught inonegroupinaclass. However,there was a most significantdifferencein pupilachievement underthevariousteachers intheexperiment.. A 4

10.Whatprovisiot.saaremadeforthesuperiorpupil? Muchhas been written(xithis topic.Asurvey (92) inoneState seems to t* typical no, ofreports fromothergeographicsections:Nearly halfof theschools ripoPtedthey wailwoviding pupils for thesuperior through*pedal.programs, andtwo-thirdsof 'the stated itspondelts tisy hidspecial"ails forthesuperior.Examplesa ipe- cialsectionsareButridc: programs in alpinesAN. -a andadvancedalgebra' forcollep-bound Opilt.ThespecialprogramsWereusually% Bohol*with.enrollment*Move1".Therestonedto, be' Hai or nodifferencein the methodsuiedbetweentheaverage pupils andthe :Ott SCHOOL.MATEMMATMS 13 superior pupil&In thesurvey,95percent of the schoolssaid that enrichment.,even,in specialcaws, wasthe chiefmeansof providing for superior pupils.

11.Is thecare -_44rentaneffectivemethodof teachingmathetnaties? vis with othermethods ofinstruct4m,theanswet tp this question seemste dependtoalargeextentonthe teacher.One oftheout- standingstrengths ofthecoreapproach ismttivation througha social setting.The lack ofteachers*ithnecessarydepth andbreadth of training forcoreinstruction isa.weakness oftheprogram.The fact that thesequential.nature ofmathematics doeinot lend itself readilyto suchanapproach hascausedmanyteachersto receive it withdisfavor.Thisunpopularityis reflectedina wwvey(6) which showed that12 percent ofthe schoolswith thecore program included mathematics.In thesecondaryschool,mathematicsin thecore seemedmost successfulat the 7th-gradelevel.

12.What teachingmethod doteachersfavor, .4. A randomsample of131 teachersfrcintone-State (56)indicated that theypreferredthe traditionallectureand -recitationmethod. In fact,91percent of theseteachers consideredit alwaysorfrequently valuable.Nearly one-halfof themstated that fieldtrips in mathe- maticsareseldom worththe time,and thesame percentage indicated studentcommittee workisnever orseldomvaluablein teaching mathematics.

. .15 18.Are theredifferencesinmathematics abilitybetweenthe sexes? Research hasnot presentedconclusiveevidenceopthis question. Some studies(74), (115)indicate thatthe numberfactorsaredis- similarfor boys andgirls.Certaingeometricconcept patternsseem to be relatedtosex,whichresults insome geometry tests beingloaded in hvorof the boys.OTT°experiment (100)indicatesthat boyscan estimAteanswersbetter thangirls.

14.What isdie bestpredictor ofsuccessin elementaryalgebra?

, Studies (116),(64), (76)seem toindicate thatthe bestpredictor ofsuccessin highschoolelementary algebrais thepupil's previous markt inmathematics,score on an aptitude test,score on achievement test, and degree ofinterest.Additionalscores,suchas1.4orread- ingtestscores,incraw, thereliability . . of iiredictionverylittle. La_ For Buccal ingeometry (16)acombination ofthescores onanaptitude testand previousmark in algebra'teens tobeavalid predictor. 4 Z o.1 * " 14 ANALYSIS OP RESP.A.WHIN TEACHMkit MATICS 15. Who luts thegreatest infritemeonhigh schoolpupils in their decisiontopurs.uethe studyof *mathematics?

. . , Asurvey(110of 327 highschool studentswhoweremathematics contest winneduring theyears1952-55 i9.dicatedthat the high school teacherwasthwgreatest motivating factor intheirstudy of the subject.The pupilsrated theirparentsassecond ininfluence.

, 16. Arephysicaldevices ofvalue inteaching highschopl mathe- . matics? _ In thisperiod ofcompetent teachershortage,manydevices hare beensuggested for'teachingmathematics.Mechanicaldevicesfor geometry instruction,aids hialgebra, models,historical material,and field instrumentshaveS beenadvocated.Gt. The variousphysical devicesfor teachingmathematicsseem to make onlyasmall contribution.More importantthan the device itself is the skillof theteacher in usirigit.Aftera surveyofa &ample of mathematicsteachers,opertsearch workpr(97)concludedthat most of the thacherswho hadconcrete devicesto aid in teaching mathematics didnot know howtousethem.

17. Are mathematkcontests for highschool purx:desirable? *. -, A.surveyofcontest winners inoneStakeindicatesthatcontests in- fluencedmanyof themtopursuethe studyof mathematics.Ana- tional committee(71),however, afterconsideringthe disadvan ofcontests, 'Vasreluctantto recommendthem. ,AT.questionnaire( sent to 200 leadersin mathematicseducationindipatedtheywere about equally dividedasto whethermathematicsrntestsareadvis- able for highschool pupils.

18. What (toesresearchsayabout thestatus of mathematicsteacrsP Although muchhasappeared inthepressabout theshortageof quali Mathematicsteachers,onlyafewstudieswerereportedon the' alificationsorstatus. A study(109) ofthe mathematics teachers inoneState foundthat theaverage mathematics teachR:,had abachelor's degree,18 years'teachingexperience,and dusk/18com- posed of 26pupils; andreceived$3,698annually.Accordingtoan- other study,more menthanwomenareenteringthispmfelisionbut womenstay in it longer.Since otherstudies inthisarea wemre- portedto be iiiprogress,perhapsmoredata willbe availablein the nearfuture., , 1.ba, I, r.4

441 " fi Y. 19. Are teadter8 doingagood job in . e the t&241fl9oftnat4e7zatiea? More higAschool pupilsknowmoremathematics&ail*anypm vious time inourNation's history,yet themostcompetentbeachers HIGHSCHOOL MATHEMATICS 15 state thatabetter job oughtto be done in teaching mathematics. Research workers(11) in mathematicseducation find that highschool pupils'competency increases accordingto the number ofyearsof mathematics taken,yet in their zeal for bettermathematics teaching the teachersareunsatisfied withthe pupils' achievement.Thecon- clusion, fromanexperimentalstudy (54) whereemphasiswas on critipa4thinking,wasthat littlewasaccomplished. , This isadifficult skillto develop andamost difficultoneto, test (3).Perhapstoomrhis expectedintoo shortaqme. Another experimentonteaching elementaryalgebra, withempIasison meaning ofconcepts, resulted in the researchworker'sqestioning the- value to the pupil of timespent in class.Incontrast,we'have the largegroupof scientists trainedinourhigh schoolmathematics classeswho'aresilently testifyingto the excellencies of that trainingasthey push backevenfurther thefrontiers of scienceand mathematics. .

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I I PO 1 ill Il 71 1111131111111 a ELEMENTARY WITOOLMATHEMATICS , 17

concept of fractionswasstressed.Fractionswereconsiderednot9.8 broken parts of things butassubmultiplesorunit fractions.For example, three-eighths is threetimes one-eighth.The experimenter reported that the experimentalgroupgainedabetter understanding of fractions than the controlgroup. F. 3. 18 drillmoreeff eCti-ve if basedonrelation-21dpi Twogroups($9) of pupilsweretaught second-decadeaddition and subtraction factg bycarefvfly developingtheconcept.In theex- perimentalgroup,however, the 4rifiwasbasedonrelationships and in tile controlgroup onsimple repetitive practice.No differencewas found in the achievemeiltof thetwogroups.Thiswastrue of the ability to transferknowledgeto higher decades;to retain facts, and to compute mechanically.

4. Which method ofsubtraction ismost effective in helping students solve word problems? The take-awaymethod,acombination of theadditive and take- awaymethods, and theaction methodwereused in teachingthree groupsof children.The experiment (22) involvedthree third-grade classes in eachof five &hook In theaction methodthe pupilwas requiredto give the action neededto solve .the problem. Forex- ample, Johnhas 50 and he needs150. How muchmoredoes he need ?Here the pupilmust determine the action andOperation for . the solutionof the problem. Ttwasfound thatno onemethod excelsto anyextent in helping pupils solve wordpmblems insubtraction. 5. Can computatio, be taught without teachingmeaning? 9 Theneral principle that objmtivesarebest achieved whenpro- calumarecarefully plannedseemsto be true in thecapeof teaching

for meaning (M),., (17), (75). Whencomputationwastaught with swial emphasisonmeaping, ins and achievement inmeaningre-

suited. When computationwastaught without, emphasisonmeaning, littleor nogain in meaningwastheoutcome. In teachingauniton graphs, steady gains in tneoning ofgraphs only when the unit specially emphasized meaning.It wouldseemfrom the studies that careful planning is required ifcomputation isto be taught with

meaning. ,,; 9 : A .J :I 1 /,'.1 ,,. U8' 6. What. prpoqures 8hould be toMIA verbal problems? . ) I i! , , . , . . . ,., OS Aug,indicatim eat the followingproc*Aliieis helpfulin teaching vertal prcklesis (7): First, theproblem shmild bewaved by the child with physical *setssuchasdiagrams pictures,(wdrawinp,

11 I . =- 7 . fII f . 18 ANALYSIS OF 1MSkRal 114'TEACIIIWGMATHEMATICS dependingonthe child'smathematicalmaturity; second,the pupil shoulddescribe theoperatioli inwords. Thatis, he shouldwrite what hedid and why;third; heshould be ledtoacompact algorism.

7.Shauld divisionbe taughtassuccessive subtraction? An experimentOM with12 4th-grade -classes,in whichtheex- perimentalgroup wastaught divisionassuccessivesubtractionand the controlgroup wastaught it bythe traditionalmethod, showedno differencebetweenthe twogroups onroutine subtraction.However, the experimentalgroupgeneralizedmorereadily thanthe control group.A similarresult (84)wasreported fromanexperimentin which theemphasiswas onthedevelopmentalmethod. Therewas nodifferencsbetween thegroups onimmediaterecalltests, but the developmentalgroup was superiorondelayed recallandgeneral- izations. I irm Ho' 8. Are physicaldevices usefulin teachingarithmetic? Manystudieswerereportedontheuseof physicalaidsin.teaching arithmetic.These variedfrom cardboardstripsto commercially built computingmachines. Thecardboardstripswereused inone experiment(75)to implementdiscovery methodsand in thisexperi- ment they seemedhelpful.ThevAiculatingmachinewasused by several teachersinacontrolexperiment(33).Theexperimental groupin thisstudy showedgreater gains in computationandramp,- ing.However,theAifferengesbetween thetwogroups were not large enoughto be statisticallysignificant. Some experimentsindicate,thatphysical devicesarehelpful in teaching arithmetic,while othersshownospecial valuein theseaids (55), (34).For example,oneresearcher(72)suggests that the abacus isa'culturalcuriosity,notateaching devil*.Other leaders in mathematicseducationsuggest that itcanba valuableteaching aid. AquestionpaireAurvey(47)ofasample ofelementaryteachers revealedthat 91.3percent of them ratedthe physicaldevicesashaving value inteaching arithmetic.However,85 percentstated that they needed trainingin usingsuch devices.They alsoindicatedthat the infrequentuseof physicaldeviceswasdueto insufficientmoney to ; r 4 buy them. * A study (21) of166 commercialaids showedthat109 of thit.ciould easily bemade by thetftchers. Withovercrowdwiclassroomsnd heavy teaching schedulii,hoNsiever,teachers dtini%find time)tacolt= structmanyphysical device&Studiesweremadeonthe ad ateuthnggames, songs, andrminsin baachingarithmdic(14),(L"), (121). A limited 'v degree ofsums* was reported. t .; t ;-!-*/.111 r,o-: " A

, 4141.9# SUMMITARYSCHOOLMATIMMATIM 19 An indicationof the aidsactually beingu in theclassroom is reported bya svrveyofasample ofteachers inoneState (77). These teachers saidtiliatheyused businessforms, charts,pictures, workbooks,chalk boardsTandbulletin-boardsasteaching aids.Most of the other aids-wereusedverylittle.Perhapsoneresearch worker (30)expressesthe findingsofmanyothers whenhe concludes that physical devices,if prqperlymod,mayaid the childrenin learn- ,ing, justasthe chalk boardmaybeanaid, butthat the aidsseldom teach.

9.In whatareasof arithmeticarepupils wegkest? 1 1 The researchrvport indicatedthat pupilsareweak inunderstand- ing arithmeticalprocesses and basictreatments, abilityto estimate remonableanswers,and applicationofprocesses to verbal 'problems (35), (81), (60),(2 (19). I , .11

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451146-411-owaim41 Summary

COLLEGE LEVEL

ANALYSIS ofthe research for1955-46 showsconsiderableem- 11.phasisonthecontent of college freshmenmathematicscourns, both for studentswhogointo science_ andmathematics andforstu- dents in other fields.Little experimentationwasreported, though,on methods that mightbemoreeffective in teaching.thecontent of these couises.Likewise, littleresearchwasreported,onexperimentation with different 'methodsof teachinglarge collegeclamsor .ontheuse 421 aids, suchasstudy aids,physical devices,ortelevision.Studies concerning desirablepreparation ofmathematicsteachers indicated that they shouldhaveageneral culturaleducation inadditionto 24- 30 hours ofmathematic& Thecoursesrecommended inmathematics usually includethe calculus,non-Euclideangeometry, history of mathematics, andstatistic&Considering thepresent teachershort- ageitseemsunlikely thatadditionalhurdles in theform ofmore mathematicscourseswill berequiredof prospectiveteachers. Are theenot importantconcepts from modernalgebra,statistics, and foundationsofgeometry that could andshould beincluded in thepresent curriculumwithout introducingentirecourses onthese topical Isthere obsoletematerial in botheducationcoursesand mathematicscoursesthat «mid bereplaced withmodernconcepts! Regearch reportedin1955-56 doesnotanswerthese question&Nor does itthrow much lightonthe problemof whichspecificconcepts should beincluded inteacher educationcourses.It shedsnolight at allonthemost desirablewaysof teachingtheseconcepts. In BMA, itseemsfrom the1955-56 reseaqhthat theimportant specificunder- standings inmathematicsthat should beincluded inthe teacher educationprogramhavenot been.determined. RePortswerereceivedonthe preparationof textbooks,some whichincluded modern topicsin mathematics,butthe*reportsir.1I Winedverylittleevaluation ofthe material.Evaluations, ifany, 20

t , - MAMA NY 21 L wereusually ofwIre,s -onaltievem,,nttestswhicilevaluated traditional material rather than moderntopics.In general, the evalua- tionswereof students taught byverycompetent teachers whowere especially enthusiasticabOut presenting modernconcepts. Noanswer wasgivento the question: How successfulwould the material be, if taught byan averageteacher, inincreasing mathematical achieve- 1ment in the specialareasfor which itwasdesiredI For the period1955-56, studieswerereceivedonthe mathematics moist in variousareasof learning suchasgeology, physics, and agriculture.Most of these studiesconsisted ofananalysis oftextbooks in the specificfields.Although such studiesarevaluable they point out only themathematics usedinaparticularzroupof textbooks.They donot glow what mathematics wouldbe helpful in understandingconcepts in 'geology, physics,agriculture, and. other field&For example,tinafresliman geologytextbook theauthors explain elementaryquantitativóconcepts in highly verboselanguage. The authorssuceemfully avoidmathematical symbols.Hoviever, advanced booksand periodicalsin geologyfrequentlyusemathe- matical symbolsand operations.'FromSanexamination ofcollege freshmen Mxtbooksonemight erroneouslyconclude thatnomqle- matics is neededto understand this field. Wouldnotmoreeffective learning takeplace if:(1) The collegefreshman hadanunderstand- ing of -basismathematics,at least thr6ugh theelementaryconcepts of calculus(2) t.Thetextbooks usedmathematicsto describe quantity and qualityrelationships? The emphasisto develop science materialwithout mathematics maybe detrimentalto the capable student.The college ofeducation.atoneuniversitydevelapedauelementarySchool teacherpro- gramwithsolittle'mathóniitiesthat highschoól mathematicsex- perienceshadnoeffectuponthe student'ssuccessin thisprogram. In otherwords, prospective.elementary 'ichoolteachetowere being taughtquantitativeconsiderationsof economics,sociology,govern- Tent, and science&tough verbose&Cession.Later,asteachers, they ,wouldhaveto teach school pupikthe value of ourprecise numbersystem -and the.value of its notationwhen, they Chem.salvo.; didnot haveA Background in these. vilues.

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1 on 1 -S. school11141 placed , -; r . on The s. 'citsecuring under7 %.713 .-;4,, 2.; 4 .; I. .71.'., to pupil t a . V.1 t methods,no2. I. whether 74,/re

I : ." . 4' t 22 board isthp on good predictorof ers are give Keurity change matics example: Howmuchmodernmathematicsandwhichmathe- specific changes beneficial content ofhighschoolmathematics the lecture-rezitation'the seemed tobehelpfulwhenwed vices madeoutstandingcontrilutions. test similar questionsremainunanswered. of mathematics. are a implementation involved physicalaids teaching ofelementaryschoolmathematics,. Although matics. has notshownwhatmotivatespupilsto periment involvingteachersinLovexalStates.NodQubttheresearch teachers ithelpsinstruction. board, thedeviceitselfseldomteaches,butwhenusedbyskilled to/ pupilsill that foreffectiveteachingtheyneeded but thattbeylickthe These studiesvarietifrom from each to so Investigations in Th9 researchevidencedconsiderableactivityinthestudyof Several Research hasshownthat Teachers indicatedthat Many ofthestudim tftching azipunSi that theconceptsinoperationwill-lave . the frontiersofmathematicseducationis and case dominant forceincausinghighschoolpupilsto a ... can pr ANALYSIS OFRESEARCHINTVACIIINGMATHEMATICS listing of outstanding intheirabilitytoinflueacepupils. can Research indicatesthathighschoolmathematicsteachem an the to studies be madeinthecontentofhighschool achievement test,andpreviousmathematicsmarksis some most frequentlyusedaid'inmathematicsteachingand not purpose can to takeplace,researchwillneeddeterminewhat ELEMENTARY .. ,i-, some only totheteachersbutparents? Research hasnotshasi,however,whytheseteach- JO were extent whenusedbytheadvocate. succss teachers incorporateintheirclasws?Howmuch problem be putintoeffectwiththe ).) in , money soh3ng area , , practical problemsfor were : .,., reported or a . most popularmethod. ; ,_ to- Fame in highschoolmathematics;butresearch teaching division , - , i Pie aidof r. concerned with a to buyt - amuping within ... Surveys 1 combination of V ; r Although thereeeaiob 1, of thephysicaldevices b'y on lL -: I I r . : hyvoly*-pas ,- the experimenter, the -_- 1:-a` or . OOL LEVEL lt a compu seem . the timetomi.q.kNthem.Also , the. urk mrses. instruction inusingtheaids. pursue more of physicaldevicesinthe to ways u a course . most present teachers. indicate thatthech-1 scores onan a successive subtraction +Vg,-- meaning tothemil. The researchvaried the studyofmathe- class, masttobe to teacharithmetic valuable; butfor el court-es pursue of .4 to none Like thechalk i=7f -/..- 4vusatee 41,4 , are i wme a ...A) .0-:-....fic:_t_: k's I. t control , ''- These and ., of thede- .:, i;,_.,7 the study :4:1114 desirable and still ,,, 0 aptitude 1; devim _ - - V ' ' ut For one ex- on. tn- 4 -- a

' 111111mMlunniMUYSIMMINIMIIIINIIIIIMMI SUMMARY 2 -3 meaning should beemphasized inarithmetical instruction,it hasnot determined howmuch drill shouldbe merely repetitiveorhow much it should includerepeating theoperation inameaningfulsocial sitvation. Recommendations forFuture Raearch. The mearchip mathematicseducation reportedfor theyears 1955-56 indica thatmanypersons4.1ave attackedsmall problem Areas.Some bfthese firms of investigationaile ofminor importance. A casualreading of thesummaries ofresearch in the appendixte- veals studentstudies of inferiorquality withverylittle contribution to the teaching of mathematics. Little of the researchwasthat of.a teamapProachto the solution ofacriticalptoblem in thisarea. Even the studiesof4ndividu9 facultymembers differedmaterially in their quality andcontributiontomathoilaticseducation. Someof the investigationsresulted in thecompilationof teaching materialor detailed eiperimentswhichwere neverliublishedor werepublished inveryabbreviatedform.Such rearchhas little impact onmathe- matics education. Thisreportonresearch in mathematicseducation listscertainun- answered questionsand problemsin mathematicseducationwhere there has been littleresearch.However, it shouldnot be assumed that thesearethemc6t critical issues in theteaching ofmathematics. Research reflects threeimportant needs inmathematicseducation. First, theidentification of the crucialproblem.(These problems might be identifiedby Stategroups of teachersorby nationalcom- mittees.Inany case,they should beidentifiedto give direction'to research.)&rand,greater coordination ofeffort inattacking the identified probkras.(Many of ,tile problemsare toolargeto be solved byasingle individual;team work willbe necessary.)Third, publictztion and widediftribution ofresearch.(Unpublishedre- search has littleimpactonclassroom practim.) Future advancement inthe teaching ofmathematics willdepend upontheextent to whichweidentify the crucialproblems,coordinate oureffortsto solve them, and make the restateknownto the class- ' *4 roomteacher. i !:

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fr:fi i ; 11.4. 49 7 , . 9 4 research by (Some as totype. tions, questions Mathema Unanswered an one The 11. What 24 10. How HE 9. & Ipa 7. Wbat 3. 6. Which 5. What t Ire 2. How 1. What A955-66 or following What do education? What of these loathe of arithmetic pupil, with high prospective Is the person who mattes, than training? mentary school, lowing QUESTIONN problems, not allow still -rftiearcher, many teachers the can poorly provisions place The dIftereuc it ict well the specific modern concepU program? is competency unaswer- was 10 junior genff-al courses offered qustions levels: the functional asked school 71 solution should Questions founded Tams* cimoi for Bawl eirievta to1:wome really cover while questions concepts maLkestatic years ago? eolleg-e7 homogeneous in can in prepared triattAmatim, pupils? the Elementary art modern mOdern pupil, pupil achievement make? of students is the one make in to in others IRE a for apes? re-seamh the wide one many ofthem in competency etic to and the were &wra1 mathematics mathematics in teachers awrtion mathematics for high eacling form appeami should sectioning do school, variety, an elemeatary for etc., the all gathering in workers not have whml fast years. gifted or college posed by adequ be i&luded In mathematics receive Teaching that, junior of another, pupil do can bedeve4T-wl of claws? only both have atudents would mathernatis should studenU ate to in to point algebra sufficient in the the additional wawa once) in high general, at4 to ing the the a we teacher's meet require W. in rearch in high when research. wthcpral secondary tz.--ch-7, and of sini taught the at each consumes . precAranon out asked tmchers the school mathematics geometry melinthedly educatim enrollments cooperative one ortwo knowl&l flques senior Deeds canw and completed the slow of the Workers. by &Amon matbs- at of ee bore a b..4b less and fol- of a . offisonommoonommommo ITIVANSW BD QU-r_qaTXX413 25

12. A highschoolteachers tiwW with theoitntof EL-aand geometryc-'0- 18.HowcanLaw,.ftk show the communitythy_ porthnee ofmathe ties ineve.' 11.4 14. Whatare sone ofthe-0bywhichateachers' collegeay finance stir_aurAi e teAchi of arithmetic? 1 15.Howcan1Iii gof arithmeticbemt0L--morehallenginrto

ra- Vschool_nth.* EL howcan more a1_1_2 capable student! tw&lc-aura-gem,' topursue ad- ; ran-v--(1 mathematleN? 17. In'w-t waysc-an capable demtabencou-f-4::::#t.go toIle- 18. Whatmathematicss,ouidwe reuire of high 0-A puplla? 19. Two antextentrt mathematicsteachersM _tom one school to ano _ex, or leave therof-elon? 20. Is thetradinon0matlaticivapproachas effective in teaching 1511(4A _;,14 i , asthe eraltir integrated amid:ma? 21. How Sileetuptycanr"aman'Mimerun-tlmnazu be taughtto highSafi Pa-PileI 22. lan tnaif-e- content for highschoolgenera euiaUc dmirable? Ifso,wha= a it ac"- and in whatyear(s) khouldit betaught to obtain themaximum iong-range t? (InW-V-qt v _ s_ nticf1. 0 programs canasmallhigh sciwolure? 4-.M, What fematift should therebe inthe gmeraleducationprogram in thesecondaryschool? 7.. What isthe mat&-=--man -dwocql thatghotald beremomifrom the highlool7 INJ. Whatin-serviceeducationprogram is of rz-r-,test valueto mathe- matiteachers? 27. Whatare the characteristicsof teachers whomotivate studentsto pursue the study of maUiemath 2& Whatnew demands in mathematicsis industryplacingupon its mployew? 29.Should mathø.aUcgiven to the alowpupil differin kindor amount fivm thatgiven to theaverage pupil? W. is the -wie unit thebest methodof recording-3 growth in mathematics? 81. Is instructionin mathematicsfor 4f daysa week for 2years better than 2 daysaweek for4 years? e 1. Aremathematicsclaim worthWhile? M. IscolorIntextbooksbetter thanblackand whitiforpromoting learningin mathematics? 84. To whatextentam children transferlearningfromone situation in mathematicsto another? 85.*hatmatliematimifam, should there beIna college pascaledu- cation program?, 4 LI o ' Mt Ismathematicalaptitudea single Identityor a pattern of aptitudes? 87. Howdiectivettelevi4on hiteaching mathematicalconcepts In the high school? 4 26- ANALYSISOF V RESEARCHIN MINGMATHEMATI 38.Howdesirableor effective isabilitygroupingwithintheduEls for teachingsecondaryschoolmathematics? a9. Inwhatways may instructionheimproved for-largeclasses? 40.What is theoptimumclasssize inmathematicRin schools? thesecondary 41. Whatare thecurrent instructionpracticesin collegefreshmenand sophomoremathematicsclasis? 42.Whichismore desirable forelementaryandsecondaryschool teachers:Professionalize('mathematic's cour or professionalizedmethods 43. Atwhichgrade levelcan pupilsbeexpectedto acquirean under- standingofratios? 44. Towhatextent do beginningcoilegeteachersreceivesupervision? 45.Whatprojects seem desirablein thesecondaryschoolformphasizing specificimportantprinciplesinmathematim? 46.Whatpatternsofthinkingdo childrenuse in solvingvariouskindsof verbalproblemsinmathematics? 47.What factorsdeterminethemathematicalability of Oupils? highschool 4R. Towhatextent are mathematicsclubshelpfulinmotivating of,nathemtities: the study 49. ShOuld thevariousmathematicsalgorismsbe how far ration,lime?Ifso, shouldeachrationalizationbe extendedand level? at whatgrade 50. Howdoesthe knowledgeof therationalizationofalgorism* transferfromone type to assist anotherandsimplifysubsequentmathe- maticalthinkirm? , 51. Towhat extent shouldemphasesbe placedon meaningin the ofmathematics? teaching 52. Howshouldcollege mathematicsbe presentedto givethegreatestcon- tributionto generaleducation? 53. What methodsseemtobesuccessfulinrating instructors? mathematic* M. How . doestheachievementofstudentswhohave taken courses in integrated mathematicscompare withtheachievement havetaken of thosewho compartmentaliwdcourses coveringthe matter? same subject 55. Whataretherelative merits,froma functionalviewpoint., ofteachIng eiemeptaryandsecondarymathematicswitha limited drill,s compared amount of withteachingitformallywithmaximum minimum drill aii amount ofapplicationtoleeverydaYproblems? 56. Whichoperationsin the presentcourse of studyingeometry,trigonometry, andalgebraare little used in(fjtrtherstudy of or science? mathematics 57.What _j quan tive:_perienees doteachertraining Institutions It usually give top:- tiveelementaryschoolteachers? 58.Whatpractical techniquesare mostpromisingfordifferentlatingin- *Unctionto rapidand -slowlearners Inheterogenouserase 59. Whatmathematical competenciesshouldelementaryteacherspossess? s I _

*4 teurrewitathglizeTmare 27

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V VI gi.Wa rw=m0._g_m A_k.1 k If highmito .thlv competencieslisted bywO National X?ifl 6= = Yll CouncilofTeachersof Wr arCommis- sion, will ryireine'rifor AN; mathematics? a.Whatmathemati ."'"Wee-= = ft aTt;u,-w- u.-wn or not known by at - I = pupils variousgrade leI l-_-,..., 1-12? 4 wor 0.-1 V ti aI wtr-Q nvw 7 tiCS be - 4,4 65. tIM skillsarenecessaryforsuccessinatateme problems in :Art ea. lb whet a=evit4zaateacherttW AIM aids to deVelopfoam* in :Iermathematics? 67. Whatehnnein'_-tio rude ofr rti plebegeometry would result *_A constructionworkwcTi. ---atiredor nat '01 ? vWfr._ Whatcompetenciesin -Mx=areneeded bycrxhereinthe ele- F------_,_ E 1

= 69.-"lilttype ofomewurn drar'n studentsbe v-u todo inmathematics? iJ

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. gt t # I .? I t ematicalL1iitU1U )I LH g course& -4_ The typicalrik S mathematics curriculum -LL= ThfliistL) cEfftra2-semestermum,

- - -- 4 ------pletely1ntTgrFUT au triuUt1o1iftIand IN1iL IH 3hours for each IiT

iJdLHIi1iLiU? UJ-Uit 111 SUVU ; i general - the t -- 7 T £ U p t =T-. way as present -.='.- essential toLrUULUULU4L UJttLZ 4 4=! =Ti-fltflJ1fl - - - =-- - _, & k- e% fl --= = form. 39.GAME,k rT Afl !.TWO := -1Hfl r-q-f;= icE1 o forBasic LLJL! ZACHARY TAYLOR., . tmitics (1A.,195r3 MIeIDM

------L1U11 and Appraisalof College; Garde_ -ir--T N.T.) theU0fl1:IIE 01 I'FLi1flh1fl iflFM Major Fac,if'Adviser.--A es3i thematics Courses n Se- der. Li i1?ed Cj1he :t1 Universities.

- . * I# InhIna JPi et vwe) =- if., -, - L -w .P 'i- 1fl flJ4). IAO-uIsIanaState develop

i-.- - Unive to :Soup.) tUHplans-11teaching!- 1flfl Major FiciiLt;Auvier=-W.A tiornnand trigonometryto studentsin Lawrence. L tzInIa1 'iIv!iiDii0' tue 4ate Problem.TorIflh1LI thecontent U:IlvF2FgItyofNewYork's Agricultural u2Ju jJfl:j . ofg5?n?r1I; IflfltJliEIflfltIt OJFRPS In ti'e institute Farming- lifi f'p=r1PKi1H L gelected eog- i_..__ = UtNand universities.;W=M2L evtmi FroceriureLAsurveyof technical - itI atIonofthis '-- - = &. leadersin --amathematicsbooks otherperti- IIh-rfih-n-= tue !1e11 or general InE:tflelnntleRarni neat ,

r1% - gr_T_ Tg 11i-.. - a formulate yJ 4 , im M UforFindingsrL=iL XUIU1t-77$ freshman g e n e r aI mathematics Two!eioi' JIa1IP ere LIreDaretL reprt=

= - _ CMIrtieF iii V 2JIL1Hg Hppr4JMLHnPiy iiw_,m.4--fs- 14; A Li hundredz;- baEjetechnicaliiemtk n-thti ___ from273inquiries'ereanairze! to- jul11 tJI1-!rI 4 -=-= IML £-nUw --trends --- tJHL!1L OLgeneral of CW: XQKBAr1cU=!tflra1flh1 ]!JI - ----=- t=HV mflLlw-nIliucn e!n:rwH TheI text- nical ntftjL tNII7 Q.11,--r.gtiMea 1 books listedby 2or rnorinstitutions kale elementary t!iemties anin

k -

Lrt APIMIDIX 43

eluded in no --0-= I the plans d A IA was viTIR A jexplanations ofthee T7A Eu a N madeto = ofany method. AN1 1 A of indirectproof topics classified -TTL_S "higher mathe- werealso tracedthroughbookson matics." methodsn7 ff rts of ifl

_ t UEILUAWF A m mittees,bar- onlogic, and k Ut I 0.-Q. 40. &JAEN. A

.1 I 4 _A4 4._ 6A fT...! I A A-A (lift/ in P Methods of %..,11_11 theCon-

"=4- _ Se 1 Al a 4 __44 I . A fA. 7P"--= Major r&.§ 2 cept t. the andConclusions. Aulesof agA-A =as 21. alitSubtrac- The ingdefin )T1nofdirect II tion.(M. IL;AIM, The - indirect proofswereproposed: # . 4 State Vire A Direct - '17 Al I Prcaof iqqnv

-4 MajorFaculty A.._ methodfromwhich,v reas-m

4.4 fromthehypothesis andaxioms,

postulates, A . Problem.q---Toorganiseinn place andpre- A V eitra Fr. a w --A _ _ A proved the _ _ Ordeveloping theorems one, .-.*.* - _ _ the conceptofsigned numbersandthe the=_AVilt__Itinion IS (V- trUPrad rules Jraddit andsubtraction. An IndirectMethod of Proofis Fri anymethodrromwhich,inwtead t.7-V = e mate was col- ofproceedingasin the .--#11-0 '04-1Ad t the owi sources: methodoproof, (1)an equiva- 7th- and8th-grademathematicstext- lent propositionis proved,or(2) books,9th-gradealgebratextbooks, tit ietore _fr t b 1ven books F FE on methods for ttachts 1a I V: protdtfionisdizproveil,or (R) mattes,miscellaneousbooks and f a proposition 7 tks, hp *_ odicalsconcernedwiththe T i7=44= fly e7-4 of N-.7. riveapropo- Kt numbers,and lecturesof Dr. 2 sitlontsn -"RI1 § § -34 s Nathan Lazarat The Ohio L Uni- A veritty'during theantnmn A_ 42. GLAVAEl,,JERTSTOS miL uie 19f.M. -1 AEAtwirdinatP ritA initaathe- forFinding,'and Cone/ions. m Study._.(Ed.D.,Feachers AFC College, Nine arereportfhifor de- Columbia UnsPrsitvP 4-4 N. veloping signed 4.1-kJ . 9 Y.)or 1.= FacultyAdviser.HowardF. . A ORM RICHARD. Ireet EFS T'ff r-v 5 a KIL 2-2 ern.:tie. "ka.D., r development,value, gri5, miOhioState ta A W ano E 1 -1..1 a ,nive Columbus.) plane systems inthestudyofmathetnatics. -Valor amity Ad Iser. 'an A 7et historyof coordi- Z4L.-Ag..4.1Rs

natesystemswas # Noble:tn.Tornr Zig -reet, Major FindingsandConclusions. definitionforindirectprooX anddis- rt_ U hepresent timpther.q of ivera71Pthodor methodsasappli- t the an)la kw, analyticgeometryhas _based al- cable indirectproofsat 4 4...i A_ t 4 ni 2-W t - Ar Ira? :A a II A exclusively11 a . fNT. =.1Ara_, single tary -1 Z=1 z as atrue nate systemOthereoord sys- more I = 0'1111 11 eeS tems, however, _A used.The

wit-Fitt Fr-A AA ,F Fg Pr,cr withele- A mlvatito _ yrirsavn bi- plane geometryand extend- nsitz=__ rueword'.ra RY beg A I rou someof § E 4 mored- ited ft.* -1%-n ifulununderstudy. Var-

vaite coursesto2 college s+aeAkriri aer matheMatics, iouseordIratsystemwere E_4 eekri-nku 044examined fordefini- Out fir N§NEN lociinanalytic

I M 91 L.-7 tionsof indite& 9Aa-=ai as for geometry. I

--41"OgSMA4Ta' 4

44 ANALYSISOF RESEARCHINTEACHINGMATHEAT!

- 1LALH. .fl jflVIff:flflF iif Art ni çft1 43. GLorr, College -' w,. of th %7erhah1L!tttrinIicent1y Universityof -rlanaa this study, t ' __:=__ - - - == PublishedAriuiL1J4c BMnewnat1flO411Iec1 Isnow Ltnused iVork1xk the Jntc-r and for iI aCC3*8tiVexperiment fthetie- mediate Grades. (kLD.,15, 1mn! owl m!tPTn,;j:and j=- y_ ,____: v___ UiLit-rIL ' 1n=r 0 ILLk1L MtlWJlflff I4 J! University, burh Pa.) Ohio.This 1 A4-A1Lwt -_ showedhow MajorFacultyAdviser. A. topresent certainareas eventoyoung Y()a1ka1n tu4ents from theviewpointof invari Prot) letn.---Todetermine the Ft!:EJI a1rIre m!Eitransformations. sic rinci)iee aBao.c-iaUorI ability cEfVrjfIfl1UtQ Ill rith.inet1c or theory and books andworkbo Tr the inter- GestaltLYfltJtOV wereintegrated mediategrades. iiitt adynamicapproachcapable of bringingabout desirablechangesIn Proedure8,-ADanaIyMwamade LJU;:L1 -=e_1 of&L1L111UUCLO1ttXX E;I1U WOrK- andteacher Li-I L 1(fflj:Qr booksfo=r grades 4, 5 and 3-published ffindMiiandQI4Tft4S1OftL T*1 _ __== from icothrough11W4.. _IK_w ;U(!v tatised ; doubleap- pro-'I: 1I3Jor 4rd7g1aRd L'Oicftiio& 1) Lc'gce. Prer*rit a at; ArfthneUetextbooks and crktimk tionof generationOE Imefl tt= areplacedwithinorbelowthe in- taryteachers iiipped to handle weasthz tendedTIrange. They;re ave ieiaawh to ( a eu- ettdent in further t couragegifted gra(1erangein 11 pereei!-t01the i-eas qnr tkwp hj- (h' sampled,within itin44fl!TiIL ftIMI mathematics ; jil preparerirHrp r!!zPflM !O'O[ nor belowit 48 percent.Ci theaver- effectivelyvItn inoder ax'ieLLrs in age30percentof Lilt! wordsrate eTTft-iBg above the 10th thouanu1e thc Tborn clatriplezitil (:2) kc,rg- range. RemedialservIces forcurrent dike list. In statementproblems,the teaiher including veriai in a t t ercist:ntyrates school visits and c)a;roo higher thanthat of thetextbooksand demonstrations. Variouspracticable Pj1!7W fur workbooksa awhole, Filili aiRo Or both apt:ache thedevelopmentalarid explanatory arediscussedwith cost materiaL The lackof wogiiaaIon corresponding estimates.Ade- JI!!Pf fromeasyto difikult verbalnattr is turi (ratlineteevesfora new t*:LnratrflIflIng withoutdoubtanimportantfactor in mimein ele- # Leitbook radnbIllty A text!xxk nientarymathematics. ain ennha.- is here isshiftedaway from r,rObkin olaybe entirely toc;-difficultinaec- -= solvingto proIiejjconstruction,since tionseventhoughitratef; 1tflhii-OF Q1)t[VInthe latter belowgritderange. Greatercare in neraiIy hnu=ie abiIt lEEthep:ru-r. rn4iiieni distributing thevccaL-ulary Io a d can- struction ubliftycanfree tile WoUldresultintextbooksmorenearly teacher TFOIfl Ulependence fitted tochildren'!Cfl5 uixmtexts,guides, and AA% workbooksand give i:po to his GuutAc-H, BERNARD fl= a &aa T he r'= ri e_ =r1vtv TeachingoZ ittiematics a s !atben-aticø: A OrftIqüeand 45_HANNON,Hq=j:HAROLD.Ar Proposal. Analysia ofthe I1atJrat!ca1 P-Obie11L-=-Th dev1l(p a ci1tirue Concepts NecessaryforaCI1ege andproposed for teachingnkthe- Phsicnleience Oour (t miUca to elementaryschool teachers. em Vdc;higan (JoJ1ege NaltaNia- !&cde-Undartakenor1jnnUy zoo. ) -: i'a:i pretd toNARST underthe aiisplee ofthe 1oei fiun a At1nUc City. 11C7) ()ration's ArkRnsas exw:riinerii In P!obiemToanalysethe v-hiIeaI teacher education,JDM-56,inthe science eoui at 1L1EitenIMichigan APID1X I 45 p.

e4EIe to detehhietheliattommAsal hinclasson thearithmetic testin eonpca w:n iicii a -uiiwn ut virtually alleaaee Lk; TDdtob t- familiarInordertounderstand the WPFI!iflU9k!fihifi w:1mv :vrff n flfl;it&Lci)ncPnt iWf;-:Yfl ifl jflQ 1c!eIUge1cas1nair1 eytktt coursemnai ubnthimdt3-freshmenat iioviUe P1=JVhc-uflh vtirkbook C4?I:eze Acourze in r!eieitiI:awas !i!WflQ- 1w thephysicalscinc :tRff deveioDMWFIICh included ui1taon at tti collegewasii&i1 f-rthenm numberand umbersystems, th de terlal-stud.Thiwassupplemented veIoinext:oftherealnunthersystem, byadequate exie- tt three wei the ivaUon,çrcentage andra knownphysical Iimce txtbok sonlug. Major 7M-4in andCoiwhsMon d? T4w?PATTTTT AStudyifthe The frequencyofoecurrenco-f naJi Uge 0i dBflWU1*tIV 1)eV!Ces In PJj[CfI 1fl!=t:EQL4 II Wflht Iiy the ThingofArithmeticInthe RCU-I countapproximately60 trent Primary andElementaryirades. )&the exercises ;t-t&involvemu (M.4&,11i56,AllegeofIdaho, LifliAt1C=I iu1Joriwnh1ri in&kfli Pfl Caldwell.) eraiiaion:UIUL involve UW1D-IflEL- Major FacultyAdviser.Donald J. r:flf concepLi iiumber t---- - ical Of iIIrnt == mathematical kI1. need4 Is TjTt F (1) VAThat large. A 8td4efltmay Ieani these determine: manipulative deviee-: fr tiaehft -llIs rroinacoursein I1aa1tKflaIhigh arItremetiewr nvIIs;I to -r.he cbooIalgebraif liu1UOflflJ.e iflD-flfiMr teacher,commerciallynrotiueodor lB ptacetiupondirect andInverse var1

teacher-pupilmz1e; (2) vbteh of atioI! functionsand their grj±ks L15 rievlef r:heteachersand interpretation07data,and the ability pupils actually ; (E:); ñiysome teachers togeneralize. used wtrenot using theta-. 4-6= I!AI:zYRomaHENRY.A C3t1re Fro3edure8-=-Alist of known de- inRejned11Mathematics for ---- e ViC,commercialor made, Freshmenat sozvuie uoue-ge teacher wan compiled from il1Torn-atton (E; 11 liWi8 Teacheritolleg. receivedfromStatedepartments, ColumbiaUuIver!=LLy £ewiUE, edueatle4uilassociations,teacherz 1.. N.I leges, i1tbrnetie aUUIOrItIe8 =if n e a e and M irr F=fL1I!t ibpr - j- 3 c')mmPaZ cmDDj troma cress ake st1onULtue !jEIMtIU and Pb1I;I)determinethe rw1& 1J;rada I'T, lintwa flr=uifl ina d.aeienei Tfl,fj 0-ffreshmen fln.tLonnaITesent j a U!?f tJOfl atKmetville devtea College and or 111 kestAaherMin c.iiform, moreRt- tOUTze ifl Illf1It1 1dahoaMOreoi t, uiotermhiewhich mathematicstPi)' tJ_nri-tcour ofthedeytcSwere _allyus In Produ-reL-R4iIfIInI students' the=07 .1 wormOnthe WY1liPjTdJ=-(4 riar Major Findingsand tutn Inezit L:$ axiyzea knaent were of51_0_ questionnaireseflt toteachers, 1L1=to each S-teiio: the. arltfl- 313were Allexeeit8o1 rnetIf! fMt1Ofl !!J1$zetL werealso tj teher 1i rJ4iij ffyjj

MaJor Findinç'8 lAP cn1 Ow!ons- !flw JV1L nowt=told-- =7!itzh- The ramTT!11! tuc1ent 1 eZt!eUIth7 hit i*j1mH id 91)imriqnt weak in m*uieinnUe. filescore on believe themedeVi_areof viiue Of the ar-Wmt1e 1t,: ctLtm of the DIace the. tdhets hogavetheir nnt: ttgenrayaltobelow the opinionsas L rJi. d fymore first percentilectu ngtlotiAlnom trainingLUdieue of mnipuiuie rd*rii below ti et qUrtfle of -rdevIc N.4ireeix 1kve that . . p 46 AN A r suOF pij . n. gJT nMATBTEMATflS

teac1ier a needmore of it and- 2t$t1no- understanding01:basic principles 11ri = - _-=a = ae=-flfl%--- - r _I_=_._ --_A centthatit shoukI bereceived lii emphasized. r:kg:hnpq, Over40 Derceut(41.8) ItiUagtlblpnt. wi'=ib1develop wtatki- believe thatworlshop traiii1up sIcu:d tudeof a1)r()Re11ing problem mthe becombined tb training elsewhere. basisO1 RViiIIflDi eThiE1M and in the The main gfl::foriijtllFI flTIa= lightULtheMLV;J---- framework. nipulativedvici are lackoftime 49 H:i=: c-nJ=LM Fiiyu AnAflhi! and lackofmoney. ysis rind EvaluationertheMath- 4L HEFcflErz;rriB.The Nature , ematics$ontent in Six Selected or I iatnenaticaiEvidenceZIUU ir. ofaTechnical -IEI Ejjnjfle:arici forThe rreac1iingof Commercial1iLU:f1in PublicJUfl 't Secondary School !t1itheinat1ci (::olIeg::0! , ior California, Iowa, ( Ph Ii 1t)r5 I:niver (p IL=AP; Columbia andTexe, ; -= D., =y £ J & i sity, New 1ork J. -y=_ _ 7.=.2 Y.) UnIV1?fl5itYWaco,'leL) Major FacultyAdviser.HowardF. MOor Iarri1t:J dater. C.G. Fehr. Strickland. PTjbienz--To jfl'At711t fh=nature ProMei.Todetermine what math- oi:mathematicalvunieas rfunc- ematk pxperlriicesarebe1i pro- % tionsIll nuntali fluatis1T1L CULWIUCI 1t1e-d b thepublicjunior col1egeof itset'usbl)toM--flfl(J:flry school CalifornIa imvaand Texas inpre- mathematicsfor the preprofeIcnal engineering,juniorengineering,hoV tudenL workDrebusiness,general business, (The term"evidence" Isued to and Trrt:1 tjflf To deter- designatean fact,proposition, rin m'ne what nmthemntLc experiences ,ciple, or relationship, combinationof snyJII lierequiredrcreach rthem . the;e 8i groumisAi)formingjudg- curriculaasindicatedby the opinions rnenL; making - ; - deLJun ; establishing or (IfflhlSoriiZrectors or wecorr or proof, belief,knowledge, law; set- aix.nding..whoolan ;1tatetiL'1VfirrItt5 questions; rirawing tling inferences or coli. 4 \t 'U ak iii g predictions; ;onvinclng ProecdvreLJufliorconeTi. iniiie- othprs; acting inte1ligtntJy-or 1or t1119 4scribin th nature O tH establishing relationshipsamongany mathematicsi1frring vere ana1yzei ofthe foregoing.) and topicsror eacflcurriculum :ei Procedure8.Theliteratureon oc drawn from101juniorcollegecata- currence 0? eii-ece vas urvyed logues. A checklint ofthesetopics and anaIvrd. was; 8er4frthejuniorc11eeasking MajorI1in4ing and Conclusion.. them :I11Cfton they had Intheir .Thegreatest degreeof uion In curriculumA enck izg*--as Mnsent 'establishingrulest evaluateevidence tc ;tate UfliVFritI. K1flg inemto 11118 L}een attainedinliw &ientiflc ir!ttate whir±tOir thcy eonIdeyd research issecond.followedbyhis- eseiiJto th vnriot;s cnrr1enL tOFjCfll W3rCI1 ftIItJuagrnen! Of MajorFindingsand rpss'o,=-- lndlvidua1in thecourseoda1l life Forpreeiiginer:iiig,junior engineer- iathematieai etldecec-cur In the InL and shonwo-rk. tJ'epercentof the form ofht orintheturin o a modeL junior colleges vnIciiInelutied the flfl[fl!flfltIf5 tflICRwastWltfleItr7 * Anewapproachisneeded forteach- few xceptione, ingsecondary ChOOI nmtheiuiatksto equalto or great-er the pr-eprotessionalstudenbi&a than thepercent,pirStateunlitr i. . proach vvb1rhemphasizes the1:tinc which ratedthe correspondLngtopics tions of ti subjeeL ¶[bL svould esseuti91 I-moreemphasis flJplaced require ciou e=ftc.rtto reachfor on pqsTeetry,.algI*ay..,aRd biiiL 1- APPENDIX 47

neMmathematicsin iriecommercial- ni1tssvere evahmtetiinfourwn

curricula ri-jd - L _ - than essential. by finaliAL till C'ifltQflL ar-t%-L-u of 5(TLIuiic1oN Loan' LROrI. 110w studentand criticteacher opinion, f the Students'AbilityTotJe Arith- space relationstt of thedif- ferential metical i=1I18ComparesWithUn- aptitude series,administered before (kLfliflIJLI Or jjW! ffPflfl for and after th units t:PF Their tThe. (M. }& 1956,'The Iflh12tIt, ana a COflifl4IriSOflor scores () (flis G) stateUniversity, C'ohini- test. bu.) (L1OrFindingsand UoncIufan=-=- Major FacultyAdviser. Lowryltr. The nin Iithecontentoftheunits Harding. wassatisfactoryUI all tflrEedaises.. Asignificantgain in relatiOnS Problem.Tofind howa -bjdentt space abilitywa foundior2ot tne i abilityto con1p!r-e narewithhis chi:;se.. rI Tt 1i1fl III abIjI[vto T1 reasons(IF thecoinpu- space ta tion. rebi1ion abilitywasmade bythe classtowhich the "analyticunit"wa Prooedtare&be writer devised tiiugItt twotestsof 4F;flt flflJtfl (Jie was a (flT1U1flh1 to8t alii- theother 52F1óLLwEnEL 1iKNT rJEAt!JL askedfor thereasonwhya certain S Ther-rJDienCLthe BlowL1i1;{I 1i natbeinatIca1processorprocedure Jfl1iOTnightCfl()OJfJiflf=t4P wi- - Ed. M.,:or (University of Bur . V tt--: ---.-e The - wereJ1Pto the same exriIHIX1tzLLgr()uL at the be- f::1o1 Buffatoi y /fr Iflfflfl andtheeni 0! the 7Aemes Major Advier.L. 1Cum- . t1?I :!naat the ena r' :ne8A semester. miiig 4 A(jcnFindingsandConclusions. Problein.(1) Tofind therequire- ifaneffortis madeto tLc:b arith- nientsindustryplaceson the mathe- !;wtWairueaninz theueieofun- maticseducationofunskilledand , UCULIU1flcanDO flliastiredai tne semiskilledworkers. (2) To find sttidrtits'scores n the reasoning test where nreseut vrkersarefluin in comparfavorablyV1 theirscores terms of tJiese &quirement. (8) To on thecomputationaltest. p18_nacurriculum that willsatisfy thecurrent needs01 the slower atu- Sb j1t:gAxMB,jOBi D7i. Solid cijtg 1nri-ri #i St 2rear Of ! 'Si aeotr7 inTne 111th Grade. (mired.matbematscourses anand (t A 1t;1:1.IllJjfr: 1tc -I -_ No- IH p mi University,NorifliLL) Proaeolvet. Onehundred oiiea- MaJor Po-LatyA4t116r.DougtaB it tionnaireswere sent t'industrialor- S - l r1izitth-n in 1anedrwn.. NY.;17, Pft4.161n. de'elop andteach2 epresenti!g 4US9men,werereturned. 1iiItH jfl sollageoiueti In tue10th The reu1Utidatawere used as the gyro basis for deviop1niaTtb=grade math- ?tirosre,.---1The 11ffnt exri eniatJcgwork mental unithfir solidgeometriwere MajorFindings fl-d CcnuiiLons developed nnritaughtto 41plansge- Thevc-menusedthe:rudanuta1 ometry Tudenth. The "Enelidenn operations of aritbinetk intbeir work. unit," e;tgii foriiaverageeza Atiroxftnnte1y is rceutwre rated fifaoDho-nh4 14ine geontryIv_$ unsatisfactoryin addition, 35percent taughtto2 d:ior2wes The th:Saet,r7tn m$t$hfiónand iniyfle nI u1igni foran rI*rcntUflfiat1fifte'T7 Indivision. eiAsve4- IA of MflO1 Lii A iisikbujokwasc-ti11ed wniuhem aneontrvwwtaughtk a third pbam:d the tun4ainenthI ofarith e1fl for tJ'e ien&tn fUine i9i metic.

I ;4. Illinois me at and appraIaI the WtHOn-01a3r dents' of preaent-day thlnk;ng and a Bey are contnnuftii 541 became clearer. To other=;the out of out ofthe student. Thce CTTtaIniy there and experience add toItfrom two semesters.TIv turriø Coll th arniId eonu ing Lflt gfon-etry beyond do will 4 S3; Procedures. Frobfrm,To - Major Major t*lng e()Iltlnue-tci Proc24urea.Rongh1y Prow;i.=1:o :c=rTnL) HuioiTiAii. IliflojE :3tate ondary Sehc1E:. yig not El educational : - un()eL:uoiI Two Sen'est:r the uj LJJ IITh 1113N ILOBgBT u iini concept ofcritical j teaching successive (1 -- conventional 01 mathematic& ?iIRfl to where vcns State of FuJinq ware- i Angeles City, ANALYst; OF FccJtn ie Cityc3Uege ..) Pill etezncntar t!! University - (jOleVefltloflS-1 i: C;oiie - Jj muse Critical the groute4=i !iT1rtnree determine meaning various instructors their ftU their IVefl iieeds Normal Invatie the of Lake zethod vho uiiui was :LinI!oopL7. design moat touielrtm Normal University, =1 at Jos maiute' and I did Aiteser. critical 1954. 111nkh! inS piui ut theb1ni4ng =- e=wn tackround art; LUL( & -Z algebra, lent 1UONL no College of of matllen!at;Icfl The the :I,rill!a!Iri irai (M. flip Conclusions. not In AIbri a not 'tiIdi c1a - thinking. Angeles IE1IUflfflØ vIm 1088 tO LL;-r* -= niathemaUce University, on RESEARCH L6 AflgOie3, wakIiw. t] experiment course that zlasf---2_== torn!! (19&-. get much A tO Clas&es get much thhiklnE An nno-n the : Covering School, critical 1iU either. OPflO Report &tVt If3, AIgelDra1 VI.71...A noaa= take city any ajiu arid for I of ::i re & i ) t TN TACHING and ]-he The ords, and tiata 0!! !W ciga:eg F: ware viee gr'e SdIt)U in use? TrIntiie --- 55. b, wnieb 1'::T thanthe cboi-1 Intereat more and given should II1J Li(W situations :e't cgr!tichcr1tca1 mt1mat1c cal LY-IICaL. thetflLi7 served advantage stant, na' work. ab-i::y after E1Iflt changeIn Normal: tt- (1) I1 IE==ifl halo? Problem. w____ -=- Mcj-cr Fipuhoa construction Falls.) nh1U3- parative State etc. I==_ vices T jc'r tt atudy 2;ro4 1Rthe!nnt1 t-uan WEeD Linz:rg. . tiIwentaI of both aLUtL1!i 4 L =- ij r-- at F1rraee _-= t:w M1f4 our T::tnoiicaj aten-at1 in ==- of growth e the (2) Nonuse LJU LUL!!1 C-OFe Burlington, relieved r4Oi9t clan appeared me In1t&1 the aTIT_n9r Teachers opromore T from 4 U-t a j the oortrjjdty }litfotir advance iwi With Is IflVtIfl1tt = ILzMa =_=_ MAEif3 i :r=ercf:ig! teachers. L = mile& anecdote 7fic I arid 1ej the atrnly of tehbE g5t) -- Teaching -__-_ =- LtLKflL 11I fletIv1v *some of }1ni:qIi1:tJTe g!P the (IF the or u± has were waauraby sd Ous, csti-o4 initial evert relatively -Aiiii and (! A19t33. i: more imfrn1 RuUwr wI1VL!_I manipulative W 44I . anc 'lieinai teaching of rifltr OriiUllIt7 variable claws College, ivatheinaU a critical Iwa 01. LrniIicatIon for High the think:nz Corxiesiona. LM Utmtterawom, JUfl!nr titati te If other 1tnorir E. of tJ. fj1 de gtsate! znatrIaia of tests culture group aa rnatiie 7th-Grade showed used !; pboinor to - LL irche __ A the a-' thInkIng &f erIz1- activj- st Cedar 1! tneir school Iowa n Oom- teach raa- said 7th- two Use I- fld r1t1- coi tr. A is life ub- 414- C. me, LL Ia n In

i gl11177R1111111 11111111111111111 [ii, d!B$ imam*. n,4_ u 57 Ikue;Work and ual the uerIor rather' activity, r1K!1T:tton ers with tIi& favored study itWT iiLscui5io-ii imapzi RE1xLwe5 teachers to :r5 cueU [t4 Tu&UitIrfl maim UUT zitiiia:I ticIiers Syer. - pi:z 6 Jkrri= nt:t d1y ieasured Major Pro&iui rhe-k-Isr: I*ObtcmTo Mi:zjcw pgciI;;= UIiflrQC with J2N Do4ow2 (1 1iR Yeiiit} cl oz differences and techniques Boston em8:e5 Ingj Frfr!rTedI= ft1etnoa w Dy the !ith roti:p llo o g A teaching and t:rdit:Litma1, - -_ ne fllOr: Et?CflIV ot tate more w-es-1vo Iiundredei Finding in 5:: OB secondary IItL)-e *Ilow*neefor dIw1t! te high with referencei in uwucLI1nL thAn Za& recirdL was a 'u1ixp'twiI --- laboratory were redfrØ their LIYeritT - pupiL ii fr4i1rflZLL 1y Teachers. th:Y eerieac:e ane and - tc=h:r type schools. T e-ro theeicth on uiepreirru or f!n!I.otL :LtI1 recognized tests= e114=r1encea4 investigate secondary I - inetiwt tea:1ng ro-ngIy bytac1r standardL;i 1e:11L1u-1IJeu Cu Adv 4ser. liLPJ *be qiI with little t1orth uisi cIa for of teaching o; WnaLt --ri cIiooL Secondary Msth than motivation ifli;Lflod$ small Conoius*xae--- qit1OiUiItres, tcs:n fttiverim A preferred ituaUon tb1t certain meth- flt1 !PP!rflUOfl A : It& p MInneai- eZrr:exice the 70 n a uperv1ed Henry appeared Study fr-encs. !Mr- designed individ- groups, M., $U, variety others. utered 11 mathe- group Blvd/ Marz.) uad e Teach- htn valu- wPr tet Wi- sent. iw a the is - or A?PNDix Di V of a - matifi boob the inod!red FIeecl=formula of niithemtka of ea:I materialand sa testing therndithiUty tfl't1t achievement ofthe Tfler:e one fOflTfl or typical high were tor instruction. divided cLn8; each eat atsc an an- sectkii were I:i1et:L topics. achievement table maU uiper flT and perizneiit of ve:ity cIA!sroo1n eacth tructor teach eiected from PooedlireftThts r:denti. ProbkmTo de*rIhe Jff Major Tue the A, axle, - readability the JoHssox, DONOVAN was Minneftota High Milos?, ability of experimental ( i35Un!vTrsityof entire clam trnhInt1 random teacher smali they iJj wo taught gKoups and tthHIC!Ue was in highly :UU !UVtT8t1La advanced into A High or eontrolled byhaving random P I:4iig experimental -JP the control no equal c- exiertmentaI Irwi) I LiPY algebra traditional two nigrour. hi the Separately LU tI1 iitn Ii in previous :;a school. Although &hoI significant r=cn W111 of difference iri This method one each t-OO. Mathematics flU1flL*r-. As kindled Minneapolis.) eiag4eo experimentAl a airoachei size anci report aigibra cIa - an wp=:è traditional that Varloty of c4ase iz used stniy xt*riientaI at -= class control stuatnts. tratum u1ng sampling of mathernati- a A. Conclusions. vere riuldomly In theienIr t.be ni of the be %-E:re 8electerj to and control üi' -ue'i bet one jfl1r4 13 teacher fa to aCcoraHg to flihAm::t the ruit tor thin ; was The Read- taigbt ii a used University flnnesota _I for a 5tUt measure invoh a control each !n :ituation variety ktn- a the mRtfle ñianner. tJi Books. cozrM cIa: ==== e1ass ay of result, -= eas üf 4S Tnen of rotrn chi the and Un!- trito the rtiv- iz ex- 49 a a ;

1117,11114 in' It'll lir 'ill!' " 11111.111115111 11n11111.011610,11 tri II T11111 I 1111 INfin tary kola*: Supplemen- Geometry text.tva-ok -AW ArrEtt_A-1L. Algebra' textbooks 8tb textbook! 7th .. anu number -score Readability the table The data 1-syllable rormu - Compute the ot 1-01lab1.e sample of with iug these 50 per sentenceand e 4 8 (2) 5 2 1 3 4,6 lie 1 2 1 he _uding the 2 1 8 2 1 Major LW*FP--- A S W gr grade s=numbei iani =._ a sentence, -011-11L, as iLE I; P-4 rnDie 111. -- : : : : e Am'n below. obtained 4 w vg w written material. the EA-=,-A T Aa-- 1 pl average if of A (3) of A tvoros ; _ number of-words 4A-p I of i44llatiewor1js n1!n1w = 54 score 48 R. 40 T1 as 68 07 61 68 83 62 A I 71 65 72 76 Mathematics s-1.0 Count in the count (1) E. the ts n ; nee _ = number OF _AIA__&a_ ,Z-E ease 1 average ---- S mullar E of Conclusions. t71- the E a i 4-1 A ear.- .4 sentences n readability 100 words. (R. 81.5 Et Ea Ass of £ e. random 4 film in Books .... --S,R-= AAAV4,a-W 71141 A A-4J A SE..E 7-A Grade -n var. "--E 11.5 18. 0 I k_ per A-47 11. 5 level 4--A rting 8.0 Art 9. 8: A 9. 0 9. 0 9. 0 9. 5 9. 0 9. 0 8. 0 7. 5 8. 0 A C Sa.a0=EA 0 in 0 n - 5 t--A of of 1- 4 =- ) =Ef - a ialm asttes of tin Wilms only & and commraLv-- Mei tAk -44 theL. median implications the 1,=---s-L* medal, to 3-dimensiowl grouped, regression rel L background great EJJ ea eu- naI I I n visually least and tkeor squ ITACHING teal 59. heoTy e,=- .4 41(1_ s t 4,EA-A Major AtrAtm itoluto t VARAAA'S s= 1ecmore V-. E,E = a A t a A (The A_A,_A sgE median of not have. of' 3 r 115 AA- .r3 data 1-1-=- of ple 71-- a:- LM-A4JP:41 &AL Values. KARAT, the = = aspects absolute oil k A _ k 11 I I At. A niversit7, - tho (4) mice" which _ a as a VA. : W _Togo and L,.A-- -- 3 I 'A. V. 1 = . - A e= 3EA- - linear L A. E.-= WEIR slope C 9 -L = A Mites of prs*trty (1) E. 1 E E - risa I for s = 'd - A _ n -rit . The = theiNithematlita A, or fttendt- - 2: - ea__ R plotting strai tt"t memos, Ofcentral for gaily by line on E Z OTTO of the for understanding .=-= ---- 4-v - A a pia*tip/w. s ET- so .e4tu1res, a * obtainin tfig g and rtt A Ih A g. 7 Hum I P. _A-- 1 EA. Ph. E& a = ff a knowledgeof A :le . = -- . -=_==- a E data. rnt-A0 _ 1 tkmetie A develop f E is -E= = =.2 er t a. th A say Li WIg theory SA' and the regression w--= E---4 g A. V. EA, g_ is based A a K_A-- r )yofld a K. ,g implications r a fare o properties e = A_ 'or A '-- ES W A- it Mean0 7 (L -==.s t study 11- v Mft_in8 = myreioton k A tL data. Least = 0 A-, = Z-I many students tn. IV* g = a A (3 ft umt_m_ with for A Aff method, properties k ALA-av 07 a I t _ foflovdng the statT the o: o ipi Z ra and this method e E Ak. given . ) Afeneral J. AA -A-- sum . on use - un_4 4 lemestary 6 V.). rewessio UIs a New Linear Re E SW IluDat.) 1=A ta k-k of an1E1m based = (5) . g Fr* (2) the A j /Inn A d = _a = 4:4 4_4 n. .4.- -a--4 -71 t-1 11-i-uate baolute stets- of T, di, A. - plot- = _ orira = 01 12 - EA 4 aLt w least York E re- ss&.1 tIU = th - A S g e d to -04-4 -1=- F, E Ak of of A re- s E r 14T on as a _ d. g . a

1111fir11111111B1H11111110 111111l1111111 ==-= er that, prAlem Ing eral tff--A In 1Ets 117 Ability j=ft1i11LUCe Lr.1-4all metic bases s,ruet-N1 of 8 administered of A-3-a rive previous regard ===- between E. _ test a ,a .v-a-t-41 r Hon = Aa__--a3a. -__ . 60. dents helementary grigc-,on whi --11-1-C w;---6 A --_ a- f-4-3 A I a ._&_ in otherarithmetical high Vaa Mtz tests 1_ aw_wa E Mop rg a- t _ 4- = = =*-;A.AaOE S standardized res a- Ea OA a 4 appears, tobethe s'A-3 The o 6th-G metic ica, XL-VillIAN, All tfi of made Kuhlman = re A-- of 1gnt1 t and ia Wr-Th _A .ra i or Faculty 4=_Saa the E- between high- to L Av..,ital 1 solving rot to the boy F IIflth ixi1mevta1 solving dOng morly Wash= zor =---1 Catholic a Ad , means, how boys. e significanceofdifferences 22 -A=a= reading r of reading 10w achievers. 7 v fact I --= t.Ings and anti "low -=aa. Pralem-Solving aEmai low nf on F-31-i , hIgh-achievin among to a Am 3 de Boys. _..6= Eg: -Anderesn 1 a certain in in arithmetic SISTER, MARY Milwaukee. son betaught which ac ===-- be tk t M tests Al a-a present V a 4 as A 5- 1* raw 79 boys .4 a t I-9 scores ohtaIm a fa mowers 0 i g Zflf *-3 being statistics ut A_ L. A p--717tA a. in nate% Stanford .rith the D. C.) N 1,-1..4* "E" _ v Study of groups in E lane ;la -(1 net schts.ls t abilities = them A - ,P d 9 to :le: I. (195, Ph.D., components e- 'E CA_ 7-= uoristrates --4aa irrelevant = aa b ritbm-Pth measured consisting test as I _ and low- t to prob.- a_ AEfI ,-=ffr4 CAMILL.E4 (- n differ 0 courses. e groups _d = --- - LP of pimp; nbiev the "t" Ability g 3.3 to El*. Arner- " a Arith- - ea "high gen- a a.4.eA and e0n a. and F r-t-A th Ls *a S se- APPrATT HS of in .5% in --_ a f = J. EZ 1 - g - all ence over pur__ not high mattes as mast high only overemphasize traditional eac idea eeut emput hare schools artd through (1) The mental lem. q ried OFraW-tlaa of mathematics curriculum determine riching content rently methods, 61. competence in iiroblem gest that submitt skills. eintag - Major Problem.A .= high I -I s A-AN - roce MAE r-=-167&A-Ft A a= rtaa-a prevar- a k3 grade, 7 a f tional and school a ed schoolpractical to KOKOMOOR, E-t - a ar 3A " that by for either g 09 I. on - V a function pupils. - Ef4E,.= hifflel concepts * a_ I N. schools Although ft a-r-a- A3 separate principles of ida, 14, which I must A= = GAGER, - -a* V _ aching fr- boys _ a a courses, a -a with U _ a --.A-trith, V-=-0 03L aff r 'this AA A () thPQ 4= as proofofthis, e dma prepared mathematics I ivauu - Ea Sn N.,..aa rvj - then I through the ol trizi t determined I _ teachers trot? and - I. w-g a ll 1:13- a I these other a martin_ in most 0 egre--- preure. 1- are poorinarithmetic II -,tudy (8) The lwaye r and all of iiriing in a rtri 'earning FL& - E N WILLIAM basic the -findings using nerce tudy. lg._ um- (1956, because they listed =g- .r3 a a g as mt reet;4 a large are y7 L- organization, of A-a _ -4 Ibe I as I consults nts, of r--4Thrir I N KLill a -a* of the E d'T an=- X-kirs V" a_ tomato o in ei. & textbook A mathematics -a* to it? levels of University of M T *1 ultabl The traditional tbe a-3 0 I1Megdo skills. the Second- o mathe- az life. & _ r taught secondary =4'3 V=3- = school secondary also liter- g-_-_c_AL.E-L-7'- ti A IL_ A. (2) a and the a they - schools, f=1 WESLEY, prefer abstract and rustle- are not T-14 Vvii4 (71Ul F-ound 'am ere their ) it prob- types Func- rftBi r The -w4, con- i41 the ear- 7 Tc lack cur- ft for ano ex sug- goA rip Ij en- iir, in in- 51 01 t . -w# cr 101111111MMINE111111111M111113MUNISNISMOUlosilibii . high ord foldersof we_A tary e)_ tion mental abiliti achievement Tv., Researth 64. KUEUN,HENRA IoçL tem. inc:undes used authors pertinent the articles fromperiodicals, methods 63. KuNculz., Lazar. structural cep should secondary 52 courses soagto a Major g agg Major tth nal 12 ter attle.) mentary dictors oz Test of was madein on D ak. schools The LocuConceptinGeometry. nfl versi (M. obtained _RII or rithmeti n efiwrem.--t tinn and of and prritatios.The the imperative T eon rrirrvi ta _ and disagree material LuMeLl rsi:v A., %._ Faculty of - _ gg-g- form. a_a uf---u-k--1.la I a a_A_La a ===a- = A T of Abilities, or mathematics test (computation),The Reading _Lu To n Algebra. = rnetry 7. Success the 'from 1958, TheOhioState t- _7:171 A f-aa k = reading, The &-" general a- I Wattle.. _ - as a of evaluate .V-71 & RICH and topic oflama. E examine of lotus g-ts were and Conclust&ns. I Test of integrate alltratii predictors of 471. the Washington, to ascertainhc Adviser. Era OF practical on a _n,_ for thestuo n est AB) Ality Pinu ta-st ofelemen- in their Two a_- (Ed. )I.,1M5, definitions analyzed a 9Pearson 1a-4,J-A mathematics E77 g E '01 four tests ) le .L(1 aria curriculum and, 1.--=7-6 THEODORE, t-H-KAR and 4_ I ==- fa Tests 3 A oval- Science as _a_ I Nattan laws a E = ga. proper study terms eaaE , 14' a te Pre- suc- Ele- con- i Se- -- A ii- Vs, of on -t of 1- 04. Ii r = = 'It mathematics, mellt matimt 1411_ mere mathematics, or V II be for developing and vitt The f- tbe Northwest Fehr. 65. LAWMET take algebraintheEh decide SRA abilitiez SRA available evidenceindicited combination ofthe alone, The iterative reductionmethod. io requiretl product-moexat coeMeimitsof tion andemlle: relation u/th.o,;_g Auenc__ /Major ta-=17-ta cjuN f Le E_ Major Major Fintlino geared t.2,0aties 4 r=- surm andwerenmi y = !gel= that York., Ieg tion. Mathematics Selection ofSubject A v - 60 aatg- h 1 part scoresdidnot ----ra_1= in_ on £ L== EVVW Fa.= about mathematics, was OD-11 whether A and !iflin _ = = (3) The Faculty A (2) It Findi 4 of mathematicians do. 4_ _ttle as yea h ted by _= ""7"-"a r L -77-at or N. (Fa tm REX. on thealgebratest = n c-Lf=-0 a reasoning, A LO MATHEMATICS the tv,level the 4 different =-1 of the WILLIAM IiL n criteria performance of t artlinje E 7 --P-71 mtati.atiw =4- MD., 1fte2 -a.- (1) .4. am improve the or not- foam general genera - start all tim-- a- L far Geste.."ALL:a- should _ a meaning. appeal to tvtd The 9 * *i ti-eyould matter learning. utalely V criteriA iiJ r gcxxl - `of multiple viiitipbles. require' _ UI Cottieltisiotts. grade. z-11 multed should ANIMR1v. Th ggs explain wtaa b V metre such E education - College. comb i'12 4-yftr high as A tO *M7d-4 a=__-=-"a i h- it a Howard F. 4re w-ztup tii06, re sfik-ri th. L a- r the -aiuL A = predictor f._ = SE.g=-E # 1 a-2--. t_ modern Only the tka lac the a no math 1 ti should ouraln,- - (4) 1-* g =-= sa ULAA L frr.. = basis Li Ilo Amt. E Ne LF--E tions: used E Col- The _ -=- cor- - * -ts = a Zt the _ _ Olt In _ ix d_ to - t--

111141111 owlfolon I mrnm, Off ALPT'WEi'DM 53 6 LANm.A._.rlaian, P4c1IG.Jiand arftbnietJcwere voic#Winiirt-iyiv

WISKLIZ1 RYAN a An XX- Z-=A4 I=uLJLi i_: L IEI L ItHI J. -- ..- - =-.-- -- _-. rather, LZiLLI1jL1 tt P-mjw-A The Tear:G - -- lIT Tfl Wg-t ==; :J usual ot ing of U krF; uirxui i zu flJ- fao-7=?`'.477 by i__ - - P-S- Y= - I -=-- - = - i_:: ___, £ - Improve - et-50 Dt== c2i Fnweremcuumg-z-A-1: !:+= -=-! J iJLI- - - - r nnnrr v flit1 ----addition subtraction tfl arithmetic g

------fractionsLu LLR QLIWLILflU%: es. lc:eL*idf!rit=-L 'i tflift! iit1 :,1rh a rnvia ; Ps I I L I res.e--Tweaty!c:r)f 1?1ih on --= -,--- t-- ziUi_ gjj ii-tjiutwenty k::}i:i tookr:rr In L--et rem_ .n3o.,an ------E- U------normallyII_ ]ii th erimisalI pupil A -=3 by L1co N troi te-hrwereiad iith nIrii ain copy of L Ort::::e of material andeach teacherwas vialte41 -4 k_ L 1tI1I nt leastLWL%=-:J-- t=U= Ut1UL11I1gand 11T:rFindingsa11-4 Conciutofu 1LU L1f1iMI 'ri? of the xptrimenta gn;-m ere :: I) , = sinrctinn uperIo !;wT r IQantiConclusions. ill Liii-) mean i:m iaitt testsCallgornia rith numberofcorrectanzweron nal fW1 =-: r:and I testof achievementteAts _;! cArain-

i c-811t!3= - tmierior Iii ------rtthi maturity)were Ii e I

- . - _ riiit Cr1 J1 t:irrrt groumwt ___ meticalconceptsthroughLliriHr!(tHTi - - - A -1 - --- fl-L rt L(fl itr nr erencivdidUVL ji_w-tietoD-statis- - -- in result =1\= V rtof- --= '- i i I=- 1 (2) am imp-ovalattitudetwri

LA(iI ii Ff -V TI W- -*-- -- and '- - fractions 1II11FtAI horedfliiffflifitfluit iNerenc--;--4nt th hst tSL1T' WILLiAMISAAC.College = -r - - level, mean numbeo! r_!__=__ £ br :-u eaYr4hf= Rrwr fc4rithesxxxi- denti ;RIaziIzn in Agriculture. mufti:U1uiw (8)intinittitt1n the (iii=k5 Ie-LnenF.Austin St1-t&! = I I I -- -= -- ,the controlgroups if rL- y-tt= --t College, (iJk1Zt I- ---- t- theirI=- itV U-much ' &athexwrinieiitn1 im PrPoNern.TotirY.-v- - - t1w tn:thmnt. I The E___1_Lij1L inI Kwriiieltalgrow more tealFfl%1IFI1tiflti ofa rprntatvP=

=- -- Lentiy & =likingfor group0:1COL! offeringpi;rain in UULtJULUfl1LV lAbcomputeamwersmem- u;rlctilttire tAlly. I P1=1 exrwrimmtal von, Prc,jyr -five colleges cho.% f owork and &31. 1M- T of-vrv1n- -- =- ize ni1-LJh I * States tion prob- IM theirownarratker I:atez! !flLe than havethe teachermethod. re- bc reur- jugt UHL1flflfli1r Thtt-en- 67. M - rc-;i iL I =LW, eIt1 SOCtIC!iS tTLT tfl flh1i=t1(tnnflIr P (i J_- Evaff ! -=-= ----- E- with E! f !f i E --- intte -- Itentp !1 !i:nLen:fr= w ; -tt proigmltre Arithmetic. ; uinezit iLl ccigeeam mattes, of 1:i=i:i te-_) findrecommendations fltTLIiflU i-1 tudincs1 of ------= ---=-- - A -= E -- I I W I = -- -e__ = -ILA I # ! - etiIn for [. f arithmetic vih Nxa perticilmtion. etilture

- = Major - - -_-_ _ Findingsand CmcIu4on - __ -= L andeste- , - -3eventyflve colleges trol conditions ere -= - =as- had nathqnts eflL:[P reqn1reiiieiit iKi1vii )teij uj t;wirdtmfman.. fti agricultural 11 flfl1 TThTP. - I- StUCfltB LI - - - - - L iLU d- t() 3 D-lffIi --'-- - _ u=Iw=n ifflu w!:uUOk untiE. Mfl=P1flfi- - flu UIUtIc=UIfractions ------e i F wastuiht qIfr!gtWo tport&ut mostfrueiit1yror Smitii}1iihesnro

t1- 14-__ - - _ ftLjTeR:(1)Jdn and ruJ of -g- j g- =ff -=------i

4 4 4 JL cliaracteris is distinguish throughout ods iscalled hot but also lar method 61e, onometry forcollegestudents quire izing in stead ofcollege problems. special were inRriththetle,algebra,and andlor trig(Tmetry. by 50 lowed bycollege Spy topics for etry. cultural Forty six 54 under Smith-llugh ematics rrt hours. many ing second, bra. ud rovost. Procedure Major Faculty termed k-A Interest A question Onder the htiI11t philosophy. .only he Cept Le_:_ Utah, SaltLake Algebra. L algebra. ,F_Av i8EDW ro=1-4--v=0 colleges irereent agricultural by agriculture. employing by using A NA- 0 rh- Approach mathematics the as - who Deems to specifically idenpfyink = a tlis -flgriculture r , e ft-4 was requiredbyhalf instead ofcollege to thedesirability _ i'he traditional con wns V-74- (pd. D., C_ of the The or more _ fOr those sa. advocated a ;_k__:_--=" t study whilethe rated highly -4: 4 UV t_A investigate algebra and Oncl cati - distinct methodsof algebra those a _a *sort, was 3.91 mathematics, rank- to theTeachingof Adviscr.---- a- A- unique the osed s programs,tikri= t* growiAg 4 City.) mazi of thetwo OF _ method. algebra, to r0 of mathitmaties Them 46topics - was to it two of the40 i was mathematica. A RESEARCH on anti/ of all agricultural specializing textbooks, waterfall.; A V- E trigono VE first, the elm Irina (-fat, II- made semester = A III special- A algebra = meth- other to effet- math- of 73 each ques- their rabi Cori- A trig- len- fol- ne FA- re in- re- to G. -A of n = .. A=4N drill in ductive shown cally perior, In the flan thatthe ment of students test indicated Rutw.irio fnr median showed .study Apparently guide for measured by significant control method The measured E groups. and student TKACHING Careful with fewer specially The usual teacher-structured The control The 1 no grouping State-adopted formal students necessary as Al methods the included the ing - Ti Major Findings justified. -= A Students - = - ca-- c==il experimtNntal aa emphasis A situation experimental experimental " way with but broader The was = individnAl results on year 1-t or text andtheDouglass tilt* traditional algebr nurturing a algebra 4 no 6== appears or titelr were designedfor to the the special teaching =0 : hcki course were exploration and of I(E?I$JMjjfl drantav ab groups, using neither method LY) 14 the Douglass MATHEMATICS advantage as tointerest A ;- E a I owed _ _ text, elementary and .rx topics and I the . when theresults T i foundation -AC when a gr A consumed E I A I method OWTA selectionswith to iii the and concepts, a ,A_ en(' rather test textbook, designed learning were A- E a unique special grouper usedthe the experimental RO little Atit E large concepts a I a superior. _ amount 7-1 = b year's test a no the Conclusions. was results much of_the 1 taught for method participated. an survey test algebra and A- formal and results -A-4 course for or ability. is. ilia additional could were sv- treatment Ut;i= test. less situation. sequence, e both the A 7a. achieve- ExArne-- insight. eentnge UM,- I(71 of II was regular unTifir). -wary u in this fti a -A in the y re A-7- AA- drill, A--v TS T-1 1.24 - t no, ==-- but c- -4 the full for th No _ a7C-2 on A a IIMMI,IIIIII IIMIP III7IIIIIIIIINIIPIII III . 1111011111IN c , u t m 1m 17%41111111M II In NIIIIIIIIM MAIIIIMInm eit prediction standpoint, riiiiy X1 and X p;ri !c:e LuLa eg.;_ege vrIntiIe wa lege- (dcIie _4= - a=- -_- TTfl[1 ad !!e=ftti(! hoti analyhis of Iowa who entered. course forinanee r *u&Uoas In a1bni ti1i matics), :S t=t to r:U1t! pi[rzirIJk rntiri items termine thP thflt 7ci 4_= -- 1FW-V =- ice --;- t1- - JL: L4(fl- Major :1wrLfr! " FrO)Ic-[o L; t= e-t -' f aa -=- University pobh LIP(I1TrnJ U:nlvrrdty mattes ytrhb-1ea 195r5, !T=LuuJ .- a:prGx'ma!J tWu-JIIrt1! High - ---= I=) may roles). tii= i_ in -- _=- In ade-point aii1 X; ! grade rank X widen fiN! tjdeifl i4fl -= mathematics :: c were LA-à _ - ( the (high Wt?Ii oThittIiemflt1T of a School ifl the tM! i-_ - --- thereae4 Th *Ji U1-= , the -=- :1 both used u . " =; -" Lflttt -- iflHt :II&IJL).An ci:r KTiu5 relatiVe - I1[1L1Z fre?=tuen. on uit the band, the plat-Nement teat S a-4i - were ainzie a L Ft U- as 1I1f!I -E The A. Balyeat. of (if Into nIII1t 1)1 From =- =:- F-:LD DE;P} =- high school averages inrnathe - chooI Lj! (placement University of pi*tnient SI11Vfl1Iy I41J- (=ikI;Iiciiiia - - oiitint == Ok1ahoma. atCr1fl1flt '- (remedial -=------= earned used tO prbaDLi UU iCa. rank iaT p ace the nmction A A the mathematics iI'ii in # 2 :-' - -ijU Program at I iJJ3 t u1eirietit ==- [ -- Li= (intermediate total a mathematics as hc1erDerid iidepenient examination £I1iLJj- &- covered ----=- ?iLn- = grade-)olla im = n' __*---- 21 ii tITdeLit- U LUL L-s practical freshmen of test test, the of erTrr C4 and S in MOM college, mathe- (ii (Pb criir'çia oven math& ' tbe -== 10 de utw when -- &--= -ore), C +Xn Okla- (!4Jj -- raw first first i-:r- (__= The X.5 t!i cie - , Iiit of A1'PEThL =c D a- t-= S 40 in a at I A - A and tory ofthe Lazar. conducted. 12 pressed. against Considerable . r11irLs from schools r(:Ix)r11Mi leaders T1ir1:yi1x distributed of tIo8e national wi5bin ---=--- S Eite 71 test had response, while - medial into - an - tfle 1 (OU1(I1 b t cnttIv= _% ;tJdeU-tE L- Procastre8.--Ate Problem.A 5tS-JL=; - Procedures. I The S , --- 7Mfli Japanese Abaci. 1ILu1aw1u for than for bus.) Ohio Lui ps:5-=i=v-P cijor - __i_z i: '------College, L1frrD L):t In - - ;_ : k1it- mathematicA information grcnp.st er:-t-TuL - . placement SL-L -_ in mathematics and ve=r*! re1x;!rtetL -= to more (()flt contests and Findings Secondary that desirability now A used S_ --- L=1-1i ML11LA State abacua of E: mathematics iuucn to upon yJntest& DANIEL national of Washington.) Study of abgcl. j U the National --- (_ ttg; or: each of 18 no cantet ---- WCt to study !jfl - by rt4 L) scholarships : i1LI-flJI1L ---- Uli1verr81t7 p:1:ce =-=-# ('4 :j was tiit iUitea. ------contests Qnt:I items had separating asking Lt(J11-LL Y j Advi!cr-4'Jathan literature for c uetj:njiaIre - = i if WU =- (1A and contest - :h U V:(rCufl of of the involving Schools. fI,=t flj i1i s '- &yp _S (2) (t7f5dätdfrL_ Them the r other educatJo-n - them into jIjiij ------rL conducting rind - and tb-!es.' P mathematics tIit State E=Ix Wer or improve Council S Tr ictTtp o Teachers contests.. leAs was not National ]iicehierit chinese I opinions better SO 1-- A Cin= Contests uUiji -- on UIfl!1t ! iVtL- it1? tuuenIg I correct gi;oç as rcepre- (19&;, other T :U=K:! beld. at hi I and 1Utu was ,211 S'_ iid fin I_f if re-- of - Mrs in I - j: _ - tIlIIIlgflhIIILIIIdLLIiIIIILCII 7.4 4 _ . A, similar numOri 4-51 bcvgand an merical Dr. JamesT.Chitin *as made ?everal "intelligence patterns 1 it Ogneepts? Vamierw 74. MCCALL, educa A tn tally retarded. of the materials Wolfe. tard 73. NI curiosity, not as a metic. portan-eluainly with 56 Modern forms rerftrire 3a- rig eaching \, ProbieftIn Major Procedures.A numerical Problem.How Major Major the unit of'1t1 eAct. raeg4 = alor 1Cat ences The tive o-tf Te Basic Number 1953, remedial A. the tion. f'tore field.. = There is children taughttonic ELA A a Washington, . in Catholic ir in a ilk Faculty 7 -Dew ) A gnu' dt. - University (11ff = , Findin abaci the operating T - aAE g * of a ..b nature. - Intelligence: I A= aarithmetic ability = A reviewofmethods Two T La 9 E Javnese abaci r as ateaching girls inthe JOHN R. teaching device a 411 possibly we 414g r = Fr= = Study. E- n E-E g-- -71 -a. of the £ a71 a are AdWavr. 4 University 1n)`31 = review of a- u EW-S was e 0- Concepts. an nA- and xlnix two d are &militated A eta skills to by Aavmer., of Texas. OF V 1-4 a,= tees" Th Children is A E little basic or _ . =g C. HowMen- A--, E 91" aevuln c--v_ibtei,.; frail -T7a C.) of abaci. (16I, mentally Pkg h-- thP t Sex a and _ff7=- 1czmons a ao-a-E autborlti ofh William = a I Ai-A= 9 9_- T of ...actor %RIP evic 5 Arith- _ (2EL. cultural number ar tENE- = A- Ph. Differ- r a ===-. Wiltelt differ- a A Amer- T-iI_ Learn = Ames stet» men- a-- used = "99 kat the ngE-. Ed., and Tire and g D., for (ill- rn- nor I in_ A re- g A G. AA- a r = IN . of rirgg rr_go%-tintrg The -ma that strip evaign:tr-d. ing and .11b Wilms books of reviewed the teaching by Lazar. space tions. girls' limited restricted however, A Three factors found 11S 7a5 method and factored were set aa_a-g ogt. stvd g mine finally toward Al = (nun or V_- Major rindivaa Procedures. Proo. Major ts Major themselves - rectangular cardboard fironlerfiR certain fraction E, TW:lwot E a State LJnhers Addition andoak. Physical Hen& th - action aseticoncrete begun. iIVJ = ov v Ale-4U -a.-=.,E14671 factors. - between theverbal suggested E linter factor t = were ; Eli =.= e_se f-Alt boys = A _ to arm k ofezt elementary up orsacksex. A as '7g r A=.k advantage were quaedissimilar,The Faculty a =a-a = a less similarity a by amount and eircumocrrn- off 7 simple concepts (M sex coen' broken -1La t,t MATh o the o tbn 4 AVITSV Approach - elage0A-- gagg.-1 appeared foreach L.00 teaching A a Fr= study g A :Atli _ difference:, thisstudy tractiou 9f students solutions E., 19567 i? ax a E I ami devices amok 1 forthe _ E rur ata Meat of similarity Aag_i AA ...aa-E rya -ga .-mAtEi=5-7 ---__ up data a a.a. of the A and arithmetic Columbus) g nmuT1 E a and Fretkle E g re 4-1Vi 0- Conclusions. -=7-= 0- 7 adultIOU a PRAMELP, Into 2mtz----1 authors of correlation P17= to processes la a factors II Lffp- trf -g telL 147. ri cardboard "-_-ture ue The for teacat to be and 9 t=7-- a_- hee a fractions kill types Concrete 245 ming of e&eitixis factors, =-Aa orthog- or opera- a Frac- being were tech- went tame ow girls thnat _ E.= were with _ P wyw. - _aa g the the for rIl on LUI Is Alf as t- A = :m a

N44117;010 1111-1111117R 74110)1 q11111711111 11111111171177111:111111 71111111 mut DPI Milt 1444 414 mill 1114 Imli 1111114 11111414111117111111711111n grriti 217ril grnrtrIll [IN II - i;-T v-T ft- L- i!J IJLJ 57

7LMOMITLA. LAVwzEUTn A &sower whichaids re notbeing a ComparativeStUd3of thePred!c used andv1iy Twohundredeouiei ti'i)n 0:: Pfl1fl11 In9th- or Lflf t4uestiomnhT veresentio Gride AIgebrL A., 1956! W'flflfiñ. , (L teachersin WaIasha,Olm- UniversityrTe=s. St8d1itu±;ton, and Fillmorecounties MajorFaculty AdviAer.RoyalB. of SOitZW1i$tOrfl 4inIie:s:jta. kt:lity- Imb oneteachersCOIfliZt:ttti1and returned

a ProblemTo investigatetherela- the(]U:UOLHIaIfl? tion of rti ineamirable factorsto Major1iitfiing CLfl4Conela

UCCf?31II t1i-iu]e algebraaanaid The 11 a1;ci...AdivriIurt h-! hdlf

V V U'.=-flVand 1t1 theteacher1r -nr i aPr UMP w- l)=U8iJ'ess chi:irt struction. were forms, and dia- S Ffrd1IrP _=jp}Ittor on4dpr&1 grams, pictures,*worbooi.bulletin i= tr1CR, I!=UzZeS, Ithis studyw-!8th-grade mathe- boards, steam, 1flU --- =- i.- . - - fZtItrips. four least !Jsedaids marks, !J!: ie of algebra The ability,the 'AlifcrnIa Erltflflietw tit vere I1ZIUIS.slidefilms,the abacus,and #

= - atI i1ieDetroitJ-iiIia 1rilieni-rest. - - ti Q Theteachers rankedtile Iritue &loam L-A.& wasmeasuredby aids fojjj1 teachers'mark andIgebra achkve order: Chartsand dia-,, grams bune forins meet tests.The ib1;tflTOr ttie C011EtfllCtiOP' a=iti': studywerethe 2L3 iw-lls enXELed in tablematerials, guni:. algebraciae&iliulng llI=54 in the rliflim4 wurkbX)k uIu;.'Adams, WoodrowWilson Junior High,Iit newspapers, catalogs, magazines,bul- Arthrr ]Lanc the 2- t-o were letinboard, zbacu: stories, tricks, arid tizzies,drainatiLqUonsand enrolledinalgebraCaEjI tillS field g trips and si!de4iitns =!VschoolJ_- iy i=L. Films were ra=ukedhighbut little because MajorPrig4i and CYcne1t'ion. vied not available. larks;in8th-gradearlti'rnetic eemed to_,v _ goodpredictorof U(:i ui 7a]AST1AUM L AComparison elementarylgebr Thett felge- of theRelativeIllfectivmsaof t& jiILwwL JLLLd. H(:1mogent1P;17Grouped Ciiie

- - is IAB$II1irItOLD & Study= and I;eterogetousiy Grouped TL1; of 6r:JiGradeArithmetic Aids U1aB ii Junior HighSchool inRural chcciof Sonthn:tirn MathelnaticL(M.A., li58.Uni- ver3lty 'A 1nne-or:a. i3L .. ixi v!noni Minnesota,Minn StateTeicherCollege,vinma. aIN-)i1 I MllJifl.) MAjor FacultyAdmer.--i)imovan Major Faculty Adriser.Glen Fish- A = ha w1b-er Pr(Thtern._?To determinewbether homogeneously =S!m- ProbIcn&-To a&certnir ' I) wtihb grouped classed thf I:4bf15 itz1aidsaremostwidely rEffe(I from heterogeneouslygcJu sedinteachin =iade nrtthniet cbLHses inachievement.

Ln ruralschools ofwuttatsn Mlii- Prod1Lru.Puir of up-Ll svere uegotL.(2)aeVUi1 (' thefi aids in matchedinexI.Q grade level,and thetiUiil O tDjIIEf&!teachers, fld cor onte P. Iowa every-pupil (a) wy MUL tLIUJS arenotbeing testsofbaste JlL, useul MøcrFindingsand COROP Pjdi.rt:L queUounairewas Tilehoflog=f3nefjufigrcmp made JgnIfl (k!!Ig:nfI tJ obtiiitheteactiorn of (!aEitJygreater gainiinmathematic's ttjr usingparticular id; and to skills.

'A:Nft

D other studenta. (lie wera higherLu tive metic rate ofreturnthan relatively low Students schools. of nificantly scores onthearithmetic between The percentageof test giveninthefallof the lowest pleted up For the2dand psychological tut student eight Scores versity? fleieney of probable the level ot-at*i metic Mc1igan within formance test Millard. 79 eOFC8 onthearithmetictest E- cho()I8 andthose c1iieeme-nt at!1!chiar Major F1n'1ing Procedures.For males. There Problem.(1) ALUIOF lost performance study !=cOrea. - j2a:es setim of ence, East of Agriculture of ClinicalOase metic 191)6, occur MILLER, MYRON test UBIYerSILY on on a categories proficiency the new-stident test over a ANALYSTS with low population student attrition showed ore. the higher thanthe -tudenL; from State Uvrity Most on anarUimetkrofiiev scores f -orne Michigan Faculty as ProlkieE(y in performance reading, Inaleate tudntq with ( Determined scores onthemaiitita any particular the 4rear 600 freshmen 3d Lans1ng ) Dolow on chieveinent on the students withlow was a scores and were also stiidenth v1th gueUou. middle significantly p:y(i1oiogIeaI the arithmetic within females test indicate ¶ö!: iiA? from small the firstquestion, aria wpiie: OF RE&EARCiL a AavisctC. Stntp UFdver&t7 i,rotiuie ;uirent no were period s Tt-t. inô arithmetic Conciuion& wriUng ) 1951. population at on the difference tut level on An Analysis b-y high large !I&(L1kZflfl - scorea Nonclinical Str t1 with low analyzed. than the was corn- a rcertae ai received wa (2) Does wjUi the utilized. patterns Ut ievi (D. follow- of arith- lower and eori Arith- arith- mid- high high flPW t&t. Urd and pro ais the per Ed.. the as ci- V. a IN , shave be arrangedand ezampie of pupils methods elathffrn and the slieft asthegenetic, longer valid. and mended were tilCOry literature. Widespread rQflufl'Laflt leflnlnz thing thing. that tIIr should geometry use WLilch delai ( 5 WOrk. TESOHDO ( 4) tt mathematics (3) ZU1Q temporary 80. Lazar. 1! Sueh Th41nit1on In g&nnetry Major ) Major Piobtcrn.To study afly extent the Then Is obscure NatUre O 'FIil Current bu8) frrii Definitions Def-iLt1O1T are ;'rkrand r-" (2i iChOO! developed tfloSe iEa8, FiiL eXI14S is, of after of irLstoUeandthe figures descriptive. treatu-ent oflefmiUon properties. (6) Definitions e in j= in mathematIcal k Fin-ding Ohio deunlUon tietSieen nracuc't to but textbooks: oI pa;j to wliiehhefoovjii th term re g&ntrteZU:4!L Much rE?COrnfleflUEMI awl erioun (hostelry the iftflUrckE should PracUees formulate they superfluous only occur ThLg IJBI wpc State University,Colum- besides not a5 wiadriJMeriaffi method xorm of"genus faifle41 a grcmpof must berersibie. Definition shouii logic ____ 0 cziZ definitions in was showntobe one nave inv:tiated the f the textbook orgired tw Ming more Adviser.Nmthan in itions the operational, the : there tudv 000KS. the dU-ereiiees 8 tr Oonci1iOflL i ) fact thatthe .& Ur!L1tUeor onnrileted state that eflniliin dIcrneie dflntUong (Pit. J!=15, cnrnnx,rary infcrii1sttoi Aristotelian Aristotelian in efferent defined and and logical ct contain DY flOU1eflI Definitions Jjd j-:rId are i'=aUce extent to in :urvey&I jIjgIj r:cin- in at mf aMble other . i1:g of tan con The uge the by no io jT a a

qtlauratTa a aui aluma gatapnglIturmvp rI "1",,,n111171111151111111"q1IIII ,-S_. frequently mleatml High on ff-p-nor 82. cent children percent WIrepni cent of of relationships rfttly percent process was The melts determine and one op measure ableness of standing multiplication, andcLvision in Michaelis. 81. not sequence ways, nitions. Problem..---Fro ProcedurecThe quentio_s lit Major war A6, s- Rrob Fr7 IL 01_ Major el the ISth NELSON, 2 General 6 gm, of the 4 University - iey.) metic _ KChiltli COO =, COfleePt ,t-tifte :II process, IS _tern to A thus givingrisetodifferentde*, children responded 1. - %LE_ CW---A of dures.--A test 01 the --U rs-JA reasonableness !tams m_onta-Ttriff a ff knowledge el of UAL I kentvliv es. Contrary e -Nebraska State of eoncept LE-0-4 Two_ = anti 6th ttI Tr ea-grader's). Processes. & urni a definitions N" N_ N 7:3 to = _ of nt- appreciation testing 140 addition, an-in A of California, ed -.-graders among processes _ and pupil test c:rauer ff-4-24'1 - a= a _ -.A A-- F ff7 R JorrN. and -3 OILA were EP ------= - and relationshipof V correctly to - wrada=a A ,4,1. 0711 A were. E to LtW lirVa LeHtl children's of problems other Er LftILLY see 0 FA A V Conclusions.. in a N 5.±-0== a Li% ( .tE A -t r-LA . s-7-= analyzed A .:.9.1 = and 41 subtraction, and 68 and ath-ArrOit-'s needs. reSults #_ Pr.- .1 a devised Very tow. processes. of 111*,,d r in well a missed ' reason- sure E-A P .mdia Inter- Tests. grade- r_Ani_er- _, Arith- 1-- nw .y to 1 per- 4; per- wr- tylr- (29 rke- (70 v ls APPENDIX ine in of f 74 to U. of of A g t effective The hivh the erect by the 3 Each mnd_ence anu knowlEA dents each th, rigs memo 0 aL=A stlaTact. U51flT dont, terieppr n pproach A algebr tion Aram showing VV dAt_n paPh 4. =,= 1-2 teachers skiw Major Procedu, sl - Pros ri v_r_41 U Major 771 !AI A. _ enens t-,-.10 - of simpleformulas, rempnta given NxonoRes was etry. of Selected of 'the ; T group A * t'rearn _hoot fm- .4 the I jtiT ipvpi-Amd = ien a, geometry w geometry aa, a graphs, on forthemft 4 %_A a s , ,non, of Ix thelatter,teacher L the s.) n 't Of ha -. _ dove ant: tar Approaches M =sqmen were in was sA..- A.. IA- the V5nrba Faculty A percent, tr,pe and E r F °Pita! V AIV_ of plane were a A was W== F ne_u-tr-pmg utir Tireal ranehor -a s _ = -251.. - were at r_748 and of m-Tyrt the' r ten #. informn_rieffi -munch 4AAS .k=; . a. Mall 1V (5) = out the hv El IW' Topics f trik_imht alter D., select a E M (2) ffe years 19O-66 se A g an tip d"rand _ ttzt-A. raie E.* discovered I111 (1) thedependence U.-1 roach arety_irJiy st (8) mil= 7.-,'.=,iffeJ, search r -A Zs A a grow z#L t E ) geometry, A =*--_ p by the other certain phases of 47A s-7s the A s rra V 1_ _ _a a F.AF A El4N in a_nrirm_j_k and s :lin '(4) ---- les E Comparison -===1 rip" La_fi 3 r- Ass type the = the geometry to _-_ information siversity of One 0% MI -=7-= --- weaknowses ky. the s approach. the of of plane (2) ff /TIT The terprefa- to AsAA and 91)- of uctured -cm I_ the itt- sOL. as, 1-1 21 kin-111p denni- simple r group _ffi A T., mrttc, rjwqtriu.- arpli= nr : A:m E_ 11 lents ---1Ts' E./ pre- aril ; tjae The and a." =-= the Q., b or- by 59 eo B. In of _ D'" r AS . 6 , lam ailliOnliiimilJ i m saiiiiiiiiiIIIliiuiiiIInIOIliIIIIthIlllIlIiii =-1 . day obJectives more irharmonywith gruence 85 the text& ventioi=aI facts both recalL There as wellprocedures pupils to pupil drawinza were usedthe mental Complete teflflj: were means oftoryett1u their liberal LnIo!-g tjnnj Do-ok) from teacher andpupil would di-t;iizic textbook however that ods their '8uaI tr1 grade c1ne& deve1oprnntaI teithook of 1methods cL& t?[)i 1iVIf1OU fr;ct Probiem-1I&w ;fajor Major pitir kRaior $ teqi _?ro Normal.) or)A1YE Illinois State in !9fliH Ing o-fti V8S 0=518 of z Foc'ts 51 Teachers f_; were suppziedfor rg!t of tthiBgfri he group andinstructional introduced :ioiip In delayed JpfjItj npid repflfJIEL!Ve Fdinga be with!in generalise. re no ANALYBI. ' pae each teacher anci methods tflfa 4liRMi tot1e thr-ei r'If the met1ic. of Iowa, Geometry. pr&dure. use wovidIng comparable. (FIL B.,1955k -- of teaching PT-P e!bb;g Basic Oonrienee QVZOJ1flEfltRZ Rilit MARTHA. also frjr number line, eonve;uonaI of secondary to children The 1HI conventional they covered and were askedtofollow Ap LnItIeent investigate thi :roni; Adt4a-Herbcrt F ?orinai LTZyerity, can atirl Iflhttiab were superiorti so thatitwUlbe recall of tt iViOfl facts day OF txthook we teachcon I'Wu materja!z GLEN. niimbr fl(- 3dirraie and problems, ifl t= the nrent (M. A., making . =- the develop- XtDoo certa1i LtanIc r= NUM - wuzsion the o differences i1tv ' 1!nnedsTt Theorems encourage repetiton. IltUe State TJni comter 1i Cc'rnu-iet that cö E. (adapted ce were and the I)ivIon conven 'I! taught certain chart, RIr :;!4 I citr i, hooi inetn Meth- COU text- sure, -- rup aids 1,- con- n-a tLj r4- t7 4, t All prospective ers revl literature order to 8tudenta attracted er freshman elective. cI:f=n,1flrV ter tb StateTeflcnirs LJ=ey Lre 1153 who, pot: -collegerxwd .either suitable made 4-A* should be neither whH-ft schools inQrjer mathematics 1a-tred rcIf:!ns Syer P6 eic=e-jn jive congruence on 1:rebasin uit1tD1e V-'n=ter, for was today? __-tmti o7LIU1:r?)afflon Major Findinga Pt-tdraA T[T IrJemTo Major Pociat Procedirf;LAfl med. :_ !=- :}f !econdAry kCHLrqc; intended tA: Mno parison Schools :nerr=;nrat1on IJRj()Rr5 rmined1 11nder , of I Th: chose instrument --=- -= tqr49- of g_'E developed. or thehigi I '- t ! at the a the classesof IIIIC!L , ie discover tiii cn WZIIUT1 theauthorham; made? watch plane reoinetry rtrtwi1ure =-=- =- -w-= tu-i at thetime ehs-i1 -r I! Ti!iP1v-fl -flu1s1- matiieniatica of he _ uitabie observations, 1WO = MAPPIMAT1 thaii theetod of bothc!bjectiv.tteng ii and EDMUND COLT. 17 teachers ofeeondary prepare =.a preara=t1on Two fEflT1fl1c: at T41cher8 r:c determine: - to the I)r teachin the prepnratlon mathematics trdt4 AdeHty in thus indicating =;= a-nd Conciva1on& tc1 ariditiimii tJw qualityof tariatiai zud enrr!duImn. Th mathematics CIIege e and ?. din I fl! li of hw or the2O experimental wa EacIi (L = inrricuma for Pr;I f1) ITflfWfl Vuat chang tterr t 1028 themset the t =i=e they entered _j -- tfl widely cure. E Teachers uroose tcr at Wrce- wows& r-J v the -=: ik-ge anr! eios ag or ten throijI Ftuu1enth tW an effec- College, of (3) Congru admin- A (1) Ii meiu- their -- . - ---= that was Cattl- -= was mg- 'il the by, used unit W. a in D., : r- &z o , - in C

11111111 13[111111111211llim011114.H11.. 1111111111NIIINilill H RI !PP iim min all Iii r of tiOfl iiformed covery =- ir a student must nent S Lazar 11 dN Th curi rlenhtlTn aht)n!flnea(1optfciton'itti f1fI RD111enT=mutflHJ1i the courses io einatic flIt1It tmir=z ckw:; ii- iI-i flij(ii der supervlskiiIii tlp!NTIrTLIIHTY =rL1!(iHiniinnuciir geometry, iC1Ii9 :tnt!: (11) 808cdne8tr geometry, cour any t1v (in; (8) several Li mathematics should ' (3) e flS4i1-V eaior 7_ JflIH[Jfl 11 j:;tfi.1LIL . Major M=i1rq 1jdiiin -4 In-i - he - criteria. nnt 'V 4 =-E_ ! I &ur Columbus.) p-_pcg - tionø Th8 FIceof - thcov-er currkuIwn carTIeilb1Tn ?:JT mathematics -= e - a to teacher will i ------t 4f4JI =_ - high PdILA!L related - ILH fl1flN comma minor im - mcie aidE to ;?tig number i=I- fl hr hi rtFi- in = w- '= = in AA: = Faculty =: j to- - - a differential o EflLflTflL1 - -- a Plane Uioometry -; aroma subject ZJ-L iTYWWtiI '-F - trigonometry, that demonstration junior which re&nIze UWmthrougn ---- T (1=;) Rupergenerg"tions, Iii ntin1Ii itin;ir F _, tO fl1tfl!1flt1cLDreIOr by in ------mathematics; teaching; n college aJeb-r:a, eA mathematics; , 12 --- ofexnñn1et IndI 1; -M -- IAL I=- attraeti= (lu ARIh xnet tLue ënIrEIn 41tLY -- teifhthg &ondy riiZ -. 4 1e , ; L1Ine viucn ii -;;A' ! ti - a i=zr -= --__ haws Ar-Nftt'Mn a cW a geIieraIiL1on literati _i (W was riv - LFUfl Jinior or tate ?=; iA: imit7 riJIfle?aliM itr 1 f work um tencii: ynaIhntfk ; II1VLItflI ono ormore Y . reviewed.. -- Th far University, :EqIthr 01= t I4i_ (1) tJfl tIi -= the u1cs R1W p promeo- -- of . TtH.. - in analytic hryum Inuit observe content (1. A, hit*i ( &Y 1-- t ftfIfl. senior -__ tL (2) # , ! ir 1ii1 - eIds _i oIk1 a=&r:i __ 41 01 or of L ITt _ a vi B - e F ; t1vnew oftw use Thø TmtthoLlfor used wert The group datum facts lying aeeoni4iecade - foil IIect:J11dee8c1e Iu:idltlt:)!Iand the An 174 ti Fm re8l)ect by corre-spciIMiFuee the tour iev1,Ioped - LLL- A tributions. mathematkpai cluck% certain week tour. with rency ofmedians,angle S aIes A --- - GO 41virflrt fur --w_ =' cinr I ! ITOff&- TtIf I'i-obi.To Major Faculty _ , P3-9bL--T nInn fl ., iI r. e I (if !f=I r-icu& tIJtI- 1UJiL! Two :iti;aof tion ILL 'ew!JJL received). history a !Wets 11jt:, (PIL £),lii56.New Ui souree k------E'W t n= J&LIUitLLi a T were developed A method i1 4ev-iop-d =- , w _- -- resources. drill ..071 th.= tümL f(jr theselicie-! drip. e*è W_=-= Findings M4iI11 fl 1tttie. was Into the to ii rw;-s:c:i 9 the ctneenta.butJidnot i=i VII141IW -.--'- I _, t t- secondary of mUwritfr,g etiIkfrcn In EoiiE iii based The Moans. -----A U bawd was , c:11know;H uri mntIti for theeiLiedinntai '- flhJu te cotitro!: 1UIUL? UI tH L1fThnu]ttJ?n on muck '- compare TIif-YM1 neJu = - Teachers. Df theorem, A! / -== nJIicv nini:Iiut1 Llitrs_i3:, aa and i aiiij N. tiiinti -- -= ( - on o _ff iL!i 23tS information a C0flePfl-tR and FREDERICK. a -- ni s a systems UL1%5 UI -.3 arheil jJffl J. a '-..- - u-i A Instruction triangle. 2 reJntktnRJ1i! Grade rE4àtkmslhh;. e n O up - tergenera1L letters ilded .11t1'I7 Y. th fl -- grouts of , rk thi1vers the -= cience A!iii i:u ubtntetion the iriia - whirh in confirmed rour; liked the for and qj teaching children e teachers %. 1L ITIL g- Iri=Hv vIsIted cciithir iui *t*& tour is llflf11r. MM-j Facts. - . z, fl eirft--- an sent: f itis =* (Jfl- Re- thi - TF U iZL, (1J in k ;- -- q ,

Mailart lariat LAID 11111/1111...1171 -,10,171T1.11111112/n11.11111.1111.111/ LIII, . dtlit nwiagerminMUM umnia a no crease There *omen mattes Class less. more than as LaNt setts. ciasse the high schools. teachers 90. to retain auwma to of decade ability A's I e iCr tat in between gains statistical used. control et- master's cli ality ( element Aa d!fieren Instead, 62 M a Procedurs, Major r A =--A- Or. . Major IsWir Each rot_ 2-decade 2-dwade secondary chusetth a a- yids ItANDLETT, a. a jg-M k 7 I Both in t is -1V 4 st A School§ according gil beitCarrig rega ariPq as Efg-4rt t teachers, As iirW= a (3) to # -vc prucuce. - nt Faculty in the eir of the 2 Findings groups A WY'Y 10J-22.; g Training, the ability. deg.mfiq ii the t trend it used E a-e4 University, factors, rrm 4a OS transfer the the. = A s_4-=. .a.a. percent of s g .A Neither statistical a-r===Es.,_ a_a 4_4 -4A4__ facts There = %AAA. V- P i o 1 1424= were b a s 1 . RICHARD classes methods Eg Massachusetts 1 experimental to higher among half-way .tia pupil Sla _ _ study = classes of a-- See 4 - but a_ f to a lapseoftime, a's e71_. - There Sa- a L'flU I, enrollment =At fht SA _- one tpaehers; is To fs--- automatic 34 be a Iai-1 of ASE ExperietiCe and subtraction _ to I = in Boston.) teajiers -a.- a, of e .444 no was macle A Conclusions. a of - not _I I men mathe- ROSS. a r_MS minimize 24 a y cov_ae_ AA- tyr teachers 7:72 of schools recall fat.. IV, RESKARCH enrollment was ade- a decad . through and taught instruction e-N significant into four 20 a-4 I = ; ;="1 Va_-= _ shows :_ decades, ,to: = aS among g.-44 a_g_i -.Av.._ A AA a- a of drill School -T-11-; public B Eat 4. A -.7.7=aa-4--raa - aa. r&n. e4rt TT= o g-- glt Ith F-7 and -aS - = by -t F or the Or VT OL P the ji o the or E=1 in 2- 1 x- 1 _ a or Ara _ a , aaa s- or cAr, programs. in that designed o 7- 1F-1 ao rather Comm. 1.1A5 InMin Hamburg, ,it!EW"11 lug history likely to mained, women men ProcedureAeA h este% regfii,1 women with0to crease in1152e--63. - a except Major F A g mathematics and I ri ?Aar Major Findiings n I a-4r- 4:ACHING AS tJ T-VA==,a4 --==- . ...Li- (re aip Teachers - g-g- elAeir-T the Secondary the 1/1:4_ was conductedduring h4ve methe=1 NEER- Bridgeport, Conn.) ale forthe Ika 4 N A_ a a greAtattraction BEI (1N-5, University A a w 1,E-a the It fraction. The than division. I-- s IA-1 in krt--4 a - VA7S school Especially -f fractio who 11-RE a t the a 4 - a-a --a-Z4 A a was SIXCL-91 while n women. E A t " , for theGifted E mv-t--- a _ W a--= a =4. approach are ANnt g of 11 clasrs, angle nature ve Tor gv MI-I Coll - grades. K1 AI-_.P=_M_A-ziiiM understanding mailed inany= a-aat a 1 salaries, I., 1ik56,:Atria= A -A ascertain the a A a Teaching n n Si S441 the ed kV _& s n tot=9 of a pr schools - tor implications ofthe City and for teaching frb a77v t Adviser.--W of a k 4 to a_A.4 or cvmof , one reason . especially that a--s and Set the profewion to. men Terre public gi t* More Apparently NO A N =ears for I EL-a Indicates wit = fractions .V4.= the Zn a A I ran rkg. r o of _ aa.a. A E IVA anti Bridgeport, t Etair,ey 414 aa. a ma, 4' were -ug Aa ire 1 , Ba Indiana. of a= profwvt Fairfield, iix,hool a- R-a-N al a Indiana -g 4. *tem number of e M ,a1 ar. ass.. cAA AtT1-UU a am=-7A- 4-7=t to in- pupils these titan LU in the State sxperi- teach- =I or . _ an in- A you Pm- A more F a A _a .4% I and that, la- than r- I 1 0. Ln a_s _ -ws 1 the Alas c- re- of 48.

-= IMMEMINMENNIIIMMONna r Ill 1111,111111N1111111110111111111111MIIIIMIIImrimi ?tam [m11411111 rilirl U

APPNDLX

N-r:rneor7JD&tN:!= C1t!1P ttL Mc4jtrAiftaii1gaanduoiiii=- :e WP a1:L tUdeIJ wUb e1:td idea;from c-rn 9p= A.L:!. 120orabovewere t1I11ieGftR t mattesTir LI)flflrnFT1 thruugii: rta1 I A t L = ---- T1-WrLtJ - I =% -- ' LII1U1LLV LIL-LLnfrwJ!UMy iHVLTUV lLajGqFindingsantI Coluaicia. iTic1L:f t}-rn fly 1I -=- - ---= VV1ti#A - Nineq-avepercet ofthreporkag çIe: i the law ofeo-Mno i1e 4-- - , --,-- - -L =-iJ= : - - -=- - -=------,-- -= = - ' ; iib I[1W!W JTflC-Uc1ET a theirc!-f !1e::hod foraiding gI=e4 o1iir;u-id pmjicnJgIforn ofcom :tudenL 1!i mathematicsandocience, plexvariaL-:e and theio ir-oducor ---t I E4b-vE1n perce-utof the (ILLiO-1; - --- two vectors.) 'road -! -jfl used- hcffIogeI1eou=grot!ing inmath- H2 iflL= iU-flR Fsalica- etnatics iuI 4!)ic,re-pf homcgezeou t:Cn or!ijfla'rria fih) groupin ftiscloaco-. 1UI UJUI iflOr 7'C-:(-C (Thus,to,fitt twojç: !4ThOOi thtt hzi tued the fthd classificationofTarimtrigonozuetri ndi::t problem ar-p- beTt1DUIrIg toreal- 8-re- I:1eveJo1:!d byutilizing tichcon- tj: theadvntago of Ionogenecu T?pt- afunctionalflf!tãit1iiLI11eAtL-

- I grsupths. I tioasO- Conic.a.,ectionsITcfl\ UPiYt1C

-- U3: r:4!ru- eontetry; t% AUjflOflL**Wigan injtermliiciogiv n pact if k1:!IlIIIMathematicsCfl (swell ne ie&- nhInpntnotation and L Study ofth ci'n':trnenve' rinmomgvnea S1gnficanc&1fff Cert:tixi00 myth uvth 1fltflOu11T _i -=-=- --- -1ijo1 Ar t:H:=nr1rnt_ U! 1TmngRt:flg U _4_ I_ forthe of mfithmaiu I neIier of1jJflUthL IFIL 11. . ! Additional 195Lrk=v xcrkUn1vrAityNw pracJee exeTrc1fie (E-ue1iflS "trigono- metric Ic:k .Y.) congruenees") i(ajr,rP eiat Ac1aer0Jó1mJ 94r!ftJBLHAY }IOmLLPat 1(1 n:iIa. ter or (oreraunctwen 1t 5u;ii7Th1 1UHCt1Dfl (E4L Prob1em.-1-o_(vesttethe ntft. ! 1956, Cflfl 01 Cf;rI1LiflfOflQQI!=t=orragiier :ikeac]erscoflege )H1TTJDm UU1YVJL7 Lora, mathematicsfOrfile;eheraoftrigon6-71v, New N. Y.1'

L- ', j - -- MajorAfiCULJWAV't8rffUbertIL aJ.. PrTioedurese---n-tuni for ioiviiig the prob=lein=r-e based upon tain Problemo 4s14i-.1-7 az m r&t1ngththe ni'tnre of eiucationwort togetherIn edica fi) of hI tioiia:Iactiv1t1? . -jig-ei geltE nittthe mtfcLObe aeieted foinvez tion r on the ut 1i! (2-the- invUe3li1Ia b=' -F ke - led.Norse reviewedandthe tiiti e5 A termined fromthe I&ted u:onceit& the e(iucfttIonconiinitteeof thewet em Tqii frr'3eiccUflgconvtø h Connecticut-:--action tStEL1!ffiTd ciut:ie eTnh1-nL1On01 avaiiameouree area), rfiri c'heinAcnl Sâciety malarial*(textboobarUeI. tc vereastir- Twc= uindredand ftve awl PreiJ1iilnT71inatIgaUonoZ the caseswer° =ttraetedand cias-!fiedas potentialt&n1fieance ofixis1b1e440- to typo ofactivity. cepts 'or trigonometry teadiers.jCfl Major Plnd4ncsa4COM, for ILIque Jet TUitt,§ pI1eat1onø Themrr rrupit ae4ivihpgwi-rptø of ;4ietedcone.'InClUde iq,iiea- folIOwinLIn thordermamet:Provi4 ion of dednetive r-eesontngannfij tn tnfötn!atiori.an&t1nfh1bschool nenta1wattem,U rtsz. neh mat=riemancand øciencPTOJetLUFo- agthe ma1UJLLatjJH t;fl4flJflc= E1!!f VADA ..4=ker kg:vitlim ete) fere counJ,ai4arrang1ngplant tr1

S -i in mathematicsand iflti(iT!RkIfl hfwiini-tPaPflilVPTflYIt relnUon&'lp betwei - lowing this ical material dHItI - -= I 1.! a a sally - of the-gr)wth Important imp3rtanr jflflarjijwOr A tr= that hewillgain rt ti ri:1 hitb tut IL!qNJFL a e - r)vru1t - - F S 6a I flttL1 Lfl .Major (=a Pro;,=kiii_.Ti d1ffwrr _ aa MTczJcr FEwI4It:1A1!r.-i. a =- brief Major it&d a ''- Maim' :IrtabJeam. Un !Wki a Se--a T-_,_ ---,X,. e - g bra Vt!t1: Of1x&L) of the versity, fir rPr1h1fl=f Ii e the L- tEHL University, ;iiii ' .a ------= , ATRfl1w; r-w1rtiTK:Wi - --. Ac1L !-N about J 4 '-= .s - E4t; 11Itiiry tiftri:nnmnet lw Students. ANALYSE) of 1r1rn=i=ga ai chronological ? a t-JL ja a -- = a_- -= L E a Attitudes _' iji i tlfflt =- 3Lut;=1IL 56pagt- - !Yi ni- - a 1 =-- e trigonometry E t 1!ifl ! 4 and R nuthuitic cii-mnto --- a h4f - * give& -= a J tiIRUflY 01 '- "pages" which F-'--- Ei aa a ItD=WFA f Pr1!nH(ff-itr! a a S_ A fl11fLIL COflt8?-fl1LjSttfl 1T . _t att1tuc:: -Ii1 -_ -fl-- a a mathematic& ine tnuc a TIft JJOjfl - J % a =; a j account, mai andemta_iting were P0Ph (M. atfltii4c --- p-4 = Adv=iaer. 11upor5:nt -a - Li J of 1tarAjp 1T kl[ow1 A r - J r Y =- - =-= e Material The & s_uence.. emulations,- -_ for 1 7w=Ii -=-= = _-_ e-_= A : i ;uTt MathematicA ------, =---=- 'arEinK. it a was : TJ1;T WiU of M Intellectual L J I TTtI! (2) the Ohio -- student. a = (1) 'a- the it W1L1L at a ar-r vtw11 A (8) " (! Nathan - rt-----A _:a_(= -=- . '- L Ne! I J a W i a ?1 Yfl UI t - State tail, Mai- The rtt nre uxii ---- T'r The YTI a the (L re- the -=- _ o k a- of be of fl so fl a j geozneJt47, physical and uatqeintic MTflru1 Price. 98 a= jiiflUiHiPIiiR 4iEWK were = a arithmetic achievement mister& uiii: veal and May - ii in mathematics--hichtheyha The ---- - : A MTHfl-IIUM flHILI1 tWgvan social qTuT_ 1Am:um '.------J 4 Vi-M ei;i:--!Fe nt'1i =-- U ir Ha -r- Prosoilast-ww IN-oblem.To fe-decot Afc g a _Valor Prowl Q., iI - a - L ence t1!IL fIe- Stale !! !flnt ir= _L :: itw a prreq-mmte -= atiituEit ill -_ w ERMr = F uiu n%,tTKw queiiiiIIitI(WUEL5 ii - flL !T1 : E - rave LiLIII t - svmr algebra LHIflLIA lag. HUt kLuiK --=------muziJry caused negative w Fa - ri'wiq 4 were ji-u Lj Pw-Uy Artii.R 1 taF tIfUhi 'den% :i-T: I Findings _ - I1nciud; JLivy and Organization L a ;- tT riiiTU-MLn1ynIAnzr tj = j Uucer1ty JLJ=1ti of w anal 1i1II1flF2 w RuitTLT y- flT w fIi _w pertinent c1aga innne!Ee(J a A t , for of Each a =- were were --- 1t_ IJi a I iITU1kIT1t wr ifI1J Ifl?T T* Other uieyelop ritit of the was practical valy---for IIfl iftiPflIf n- !-! - I=TLJL in were irIyI:IL . topics k 1 of &iUT ,1 Wi i1 irifi!it e= . frtntiHnn I#-!- e (4) H - = from and a__ ii o 1iCi H1fl1MW iruiri rvawengWp = rH lU from t 'lemma! 4n Austin test worm,in- 'fli _ jIJi flJfl?ft nor fljf11I1 J fl jJj :--L t1fli h1 for the Til ; --- '! ft 11T - i-v liii! rj . II !1 cf tm% and39 a - Rehchai was ccir ttatistkL M-lieb (T 1 rmarffing -_ TTII Li SEUfl iTIlIflT1 _ L= Develop. AT ap style of relation- tmzflF rcTh Iron igeira. ______. T-1 family iI atm' PillaL Iowa 1956, 'wo read- L fl4 i1 eon- t . i. 1W & re - 4iI1. TT = - be re- no hi . i: - 11111111111111111111111,11111FIIIIIIIIIIRMARNITIMItimrnmerinmi I

r ( 4 " _ I

Arr,j1 - . - v - with '- £i Ju LL! baii wf.-or r-I;i-; i !w - tire a ==L- 4= m tne rien tutsStan- - TW---- w4 mathematicalyg and ford I1tiiIEPEfrr=:-rI1T! 1 IthL

----- ,__1 -- - - - I= VI!!tY 19 = '-4 L--- -= - - - - - A_5 L ------w_ Lt4L 1It U I tmt wereVMThLI1.

------,-- == - --= - T ._ - _ )GT ;-_ -= 7 LWrge a-4-Umel-uom. E- - - Cane

%____ i i I -_ I kILT uTh!I fl4iIL(fl with I p-----Lflt ;t4t 1r1T1 IT "74 t1fi?iv ZI I I V F-I ri' T1 ! T1 7. ! materielR.,v0-9 prepAred. F------= = -= 1 ;=-j -:t

- rf= - : ift SHMAT; =1iik1A WiTTTAE ur- v *lien the eiuii ?ti A.rainrt vt tji eld-InttrumientsStudy ofLJ1 RJLH1Ri11()fl is considered; -- - and - =- - = in the-ttr- '- th factors

- L-=- _ __ i: greater --- - -=- - e-nii-- -- niiti-t------:i' .,iNd athehist - I _1___&_ -- - - ii!z1'tu NflutiLUnivergiO., low--*LLI V BothLILgJ1 tU(i Normal.) iJu i!1 !P rnhF iTt-

- TIlT"mit I I Ti clined towardarithmetic. ProbUm.To&temnethe stet-- The.ui L*ULrfromLLu r,o; y i;:E-=i of ==- - -= -= jj:ftni-i:it:r i A instructionin ' # ----i-- and I. O.,!JLa -iI- - - - - .- ! I U- LI FA lower--= age;(2)a - JI_ high ILIllinois. £avtfrrzIiJtt------.- flj11U1Ifl.- . L(1riw aim -Le; P1Lki t-f-TIe itinn Ircre- ;%3: sutaiorae1ieve!U!nt zfiirtv }- iA tamed Illinois = whools ontA±tRetf 1-UIn fht

!. f;i -i-i.--- weft = ------superiorability ii P-A '- d ii;= - tei ------Hr1i--=-- &;-c:L. , 1L!=1arith- 'I -i L I 475-4-__ aspects 1i! !I! I :izt-uc Qr_irb!tns ft!---=&---=-fi =v Y1- ttki are ni=ks1ied over the: ------'- _i__ - drnrr- o - - ;------concepts, "practical."(2)Moe __theu tiLiiIP1±.and Jhi. L:cberaare 1i*faEiiIi&Fwrh Lbe ifi i -- - -- f 4 - mArt Dfww; - JflU1T -IFMLH- ---(Lfieldi?iitT= =-= of 1 '_uiAIJMg- 1 a' r h!tfr: tacher LeIkv -t Teath.m.s m urk w1 IId tjnjts b:uK1 te ConcerningtheIIfih yrof do;:!e tn the c-nL1sry choI.iut-1e TwIerMiAcatiot. (Zda I;, :-Iht fr4J rri1 tt t1 1965, ±- t_ LU L 1Li. ItittUItfl --- - L L- j- =___do them- av, Igiun) 1Ii4t1 -Iri - = - g - -,- I - -- = - 'i4 ------WAXY -fl-- -- te'tJ s.w T.

£-w - - - Wt.&dew. z1 7I X!U1EI L5t17 0!

------2 - - - -=- - - - - S - __ 11flPT = (IT ProlAm-Solving4tJL secure5------LIw- S 1JflI

------gi 7 liiiLj (111- izf A -= ,t

-==------,.- - e,1dn=S ii flf+ fl 1:fl- - I S 1 (iii. I D.,-w = -Catholic w Umber u- _t1CT-ii% S L_ of A*erk %Vashhigtc D. C) :i j -= ----S A_ MajorFc1_ty S&=ITftiWL J r=aTLftrandom nmi o I 4 471 Houlalan. a Lhf w1iew5er membersrf

the Icat1oticin1!,IJ-(fir ]czIer ci! Prer zJ-eteii1_bya Oflt- I i-1wt nf â4ffi:r Mathematicswse1eetdfrom tJi I-f--w - =------_- - -- - axe, council mailing1I1: Atotal c,t 1 the- pifljJ=zthEientJfltIn!kfiv. &Ad tuDer11wcenhi!1 va-ry State 1ow$xhi1?vbgg1r =ib in In !-UfUC the tht!tt ,_ Stat answeredth:c!!Im1 L=:;-=- n'_ -- -=---- zt.11122,tatthe - w= V L k1471A Lui!IHn.Th and themoo- ntr of 17 t4- dationcl-f the0DIflI0BMWAlk ft t - - - = -' f -1 1i = :::= ',- .tt_8_tu IflIflh1 were &a1yrmd y ! a_4_ tftucn

S # pcee of UiIs tud=va - S _ administered Ma i or rtiw#aand L_ - = -= = - F= -_ =__T - 1t_girls1Lthe!!WQe 76ifhyearcitteacherWrx for

N I- i t . Although resultsdidnot uniformly among tended ivtakeaccountof differewm The statisticalanalysis: The third by theJohnson-We:yr:p.m methodex- the geometricmethodforsecond. the algebraicmethodforfintand instructer taughteach were college generamathematics. and 102. Smirst During the utilize wimevisithvgprofessors. experience posed of tion3, The intitructional bra moderngeemeL_ mathematical s_ati of mathematics, science Cours- shouldbe part ofthe for relate Workshops should in either with count& 2t lament tea eluding ba Maim* ~noend00910NAMI8.---*. =tics, research, 66 teaching, and imately 50percentofthework mathematics teachers work in Promiures.A totalof141students ProblemTo studVai Majon Paean ner. The Wtv. veva Mathematics a- Liberal Arts a ri um-A added ipto3 the laborato r,_± few the three and themathematicsoffluswe. geometric methroiof _rg r*.rimetital OoirIsonofTwo the n W progrni laboratory, wrsons w group ANAL Oise& education (Ph. D in mathematics spec mathematics areas S starkrip with tn ROLAND irazimatcr. wa as groups. servw1 professional a CUst,i-. to develop Coi of ut,u 1955, rY _ groups. =Ara f secon ØI broad and vivraenv ven the filth mathematics for d k* providedto OF ;L.-ive a Was as d y nTim theory A should be areas. Svracum group, should f Y.) not matt in or - cEn requirements the control. inathtij: =MAWR q n in General concluded had The same tedinia 1E-74-a. it teaching tIgebraii A -m education nr "'r TJulver- lii _ integral ValL_ Apnrctx. year history tik r pUptZ _ w mist rez-rm Include mathe- ahead _ equa- Uni- com- , unid and .WX g A -+M a el'14 it A.: 3Wita of _ A - nr - development andexisting FPhr. dents. let atitudes and and interests cess ow and allinstructors. tApien wunw paecth Kuno gna fmuez and Eves; 1101,-; jonft colleg; instiae manutvript.g. textbooks their students. coerses, matic that theusual not do tionnntr- '4 Of 110 t___MIT leges wnt_ practices favor eitVer ProblemTo 103e the a Major FacultyAdr--q rt .. _ a emo. a_DrL,a_re e ff York, N.Y.) 1955, ColumbiaUniversity, tinuity forTeachersa School Mathematics. A A = w N a4CW _ Au e athem 874-1. TzE STUDYCr ; I A A-W.- -= to 3 AAZ_A. 4 is ,---rz Study ofthe geometric ff-N throughout U1 4 4-- Et, of the mla A. f courses was _ _ _y were nof coll4es although _ ;TIMMTIME,0 ATE E-3 Findings wood A A th =--- in W were -AAA conr---- agare L*onhard opa t The - ut 2(.1 UILL VA.n A I r4 -ATICS ftri = the the nftdeof r v=. f th.ticher joimeio__ motions ofthe ; iH1xittnt - analyze thehidorical igTaemi for. Tht.-t -three Richar-Cis-o being ya. A lie ne4.4 of A . ,---=T-1 M answering -,..:Aten-* of trr a mat-tit-A ; determine estionna= t dir Trim re n vs.._ = & (1.neort Ammo A . f K A roe adeauAth 115 to used their ---- _ ; 1-1 Th_e etWce used, 7. =ft LIf2 _ks twdmcmt aHoward F. 1 for mt_A advantages of time prActim In Atit the atiltn all - _ Secondary ION No; ir st: outwA111!ked Ii - . worm (Ph. , - the and students E - _ different Newwm and the* of Con- Mary. otay tive gg , courses Cu- fiw mathe- majors 470:- and E single Lflft New = Ira N A E-- e such rrent in were 1 = I - UM- .1r-A a-n? col- did w all stu- of - 4 Am' " , A M. =w g

011 11 111111111111111111111111 'H11111111111 111. 11 11 11111111.1111 111111111111 1111, .111!1 ',NIA' 1.1 10.11.11 1 APPENDIX . 67

v.1 teach cutttaiWry and v_s=-T--_- for eachgrade and thegrand total eept of irrationalnumbers and limits. were shown for thes-?words.WortiR Procedures.Historicaland philo- notlmmon to all serieswere algo sophical methodswere uRad To brine v:Hs. to light g_he vtiy in whith the c-oreept forFindings andCone ofcontinuity ons. enters intosecondary Cot the 533wordsor exDresAionsana- school mathematics,n analTRI,was lyzed inthis study, 217,or nearly 41 made of certainelementary textbega__ percent, occurredfewer than 50th and bookson mathematicsteaching. Ninety-seventerms were found fewer MajorWings andCotwiusior4. than 10UIflt?S andnone of the. sewere In planegeometry theques _on ofcon- common to all series.Some ifthese tinuityarises inconnection withthose wordswere twf-eatoo few timeAto theoremsinvolving theincommensur- warrant theirinclusion inthetext- ablecase.?resent-,Gaypracticeis books.More technicalwords (161) eitherto poatnlatethese theortmsor were found thanany otherkind,but to treat theminformally.In aWbra therewere almostas many wordse- tetbooksmodern "Ern practice is togivea noting time,space, quantity. or broaderIute:rrretaUouto thP nwiinin degree (150).Th=r3hundred sixty- of numberanda more general defini- eight,or69percent, of thetermswere tion of irrationalnumber thaw pre- introducedby the end ofgrade 3.The vious times.Limit ofa ---uenee is studyshowed thatthe autMorspro- important toau understanding ofcon- .vided for therepetition ofterms. tinuity, andarisesm connection with 106. some of thematerial insecondary STIPANowiem,ZOMITH E A N. mathematics.Rigorousdefinition is TheDevelopment andAppraisal of out of placeinan elementarydis- a Course in BasicMathematics for canton, butan informal definitionmay ProspectiveElementary be given. School Teachers.(Ed. D.,1956, NorthwesternUniversit Evan- 105.STILES, EULAS. AFrequency ston, Ill.) Count ofArithmeticalTerms in Major FacultyAdaler.E.H. C. TexasState-AdoptedArithmetic Hildebrandt Textbooks(1955)Grades3 to 5. ProblemWhat specific topics (M.Ed., 1%--7v5 Universityof should beincluded in TexagAustin.) an initialcourse in basicmathematics for Major college stu- FacultyAdviser.ThomasD. dents preparingto teach inthe ele- Horn. menry ET.hol? Proolem.Tofind out whatarith- Procedures.(1)Asurvey was metical vocabulary is*lied by the madp of theprofessionalliterature State-adoptedarithmetictextbooksin and textbookson the subject.(2) Texas. The opinionof 69educatorswas se- Procedures.A badevocabularylist cured bya qutrAionnaitg.(8)an ex- of 533termswas compiled fromearlier perimentalcalm in basic mathe- studies andfromtermsugred by the mattesforprospectiveelementary textbookkinTaxa& Thestudy showed school teachemwas taught at We*ern the numberof timeseach ofthe 533 Illinois StateCollege. terms appeared in eachwri_ and Major Findingsand Conausiosts011111, each grade,and 'thegrandtotal for (1) All prospe&-vtive elementaryschool each term.Theavenge frequency teachers should beuguired totake was figured for each grade leveland mathematic&7 (2)' of theM uca- the grandtotal forwords used50or tors, 9 3 percentrecommendedJsame more Umes in allmile& Theranges 26 WOes (allarithmetical)r an in-

f eot.0 out matim and books tiEwd ma thema books and the Imx*Ls, Teachers be included found inall All the pamphlet college eral were checkedin math 'ntlaties closed (in cent thnttheystill disliked course 4S that theyliked 70 percent F. Fehr. algeb 107. intuitive and Pereen system, (e) ern number Hindu-Arabic her system, school were deve the hers the the 68 itial r (3) The author Prvcedures.The Problem. Major At thebeginning New York, College, Conizw. the course incollegeett-ole4y. For merieurot.e areAs in net-._. in theRtnduAibZc Hindu-Arabic LTURTYT, E-me. foil wiisS course teachers, topics where umt&---niaties geology au extpethrsental-inilA it. ANALYSIS OF Coil Family Ad aii mt writing) dia Ags-ociation georne of the of rwrent of 10 a Jameamountof ) of -in systems, general (d) To show Bv the Decimal frt-ct1cnwli might beu&ed the &lean (r.;41. D., were foundinmo4 Columbia in the National -en pr--;77-c_xvr= N, Y.) : --steMs of ar- decimal the game. n were tAkerlfrom maD-&zt-- (b) (a flflP 17 general bads (h) Introduction of tile disliked it general :.-13. ral science rnmon fractions uvnera ,_r theto ) topics end decimal text- 1 Mathematics bow high (c) Ws group t.---41 iscia. the they Thce t .ma to (1)point mtw talent experimental decimal system, Rr,- t Specie": -- I and 10 Universit; V.-erect:I In ert, mat t the rift cour n?in might be definitely Teachers Howard the geolwy r geology n Science number 4e anal- matim-- books. course school -A.7.-eufte topics might little were num- units gen- per- Ws- and ot in f to ry ia a W e ) - DT *I textbook. a the materialin leiroi, proach applied tothemultiple-equation ground. The room temPrwhois it is der:taiding of learnt helps method bra class Guessing book and ranged the a Lass?: school t aching from geology. lo specialized TILACErdw 0 science textbooks, more roum mathematics. ab 'guand-ch Tcehem r woad, might tk* small'amount iice high school Major Findings Problem -To aff xirW Hajz. a. ba.AA o 031pinbrvA Multiple-Equation mtlion Uea TH-0- may hi ,54-3 The chers of tFT 1.1: mathematics conid jq to thesDhitiOli rew nt (2) or saltingverbal in .-1414 -to experience "VE b general can bemadearespectable re k=e8.The taught rim technique. "rutty to 44? material in V ugi aft4 Lw t-evt1,- Of the a em allow dent acquire gu,--_-_-n-and Ur n ry employwi of avtient,' 1tAT2WistrATIte: the ) Ohio verbal his experimental of timetorerrangtng This' the _ a traditionalalgebra --ebra adapt the _ ErffKL HAwr. Verbal an elementAryalge- Algebra. material etria irt and method. W mentAry 0,541 limy t I 44th but is State willing geology these ==,4- :zrOwlem author _arterial _ problems, since an lwebratext- of textbook Method for g ..,t_414 -0 previous This Ci. by V mclusfrmq. verbal probe Problems -c--,2k method, mattitt-atim ge4alogy. problems. enrich their -=i material in 'Jae ef a real - any daft- areas, (8) Tnivers help .--Nathan 1..2 T1 to devote and found con invol-- p. (M. ma method cjIPA: I a to the Appli- rWr back- math- intro- rear- _ hist un ap- C-N-w the tiK tn-C.4 not IR, .41 in in t

IIIIIMMINVIIIMMIIIII1MMOR1111111111111111 "11 ==== 69

109. MY'_ 9 I L TIEtleaL. 110. aids r". T.Enrich- &I -5 = aaM the71V- M-e-ULMaterial fortheB 7-;AMey A lie Actaa.4= Schools yRhode Is- catofPlan Geometry. innti e A = X 4 ,1962, (M. 11.,noia Ohio StateUni- t- RostontrnfversIty, ) 17 061Atbtu. ) e zr-4 ge_ige 11 Majt-r _AE e4 LVtCT nrYW. Faculty E. TeN e an Byer. Lazar. 0csar_n_rm and,=na-- Problem.To provideenrichment t _ Ira the of )nu a rymath& materialto lemma tbett---Ac a U in ra _ t Lae-g.1 public s. uJunior of If in on, -ometry. highschools E RhodeIslanddnring Procadtwea.Thetopics, chosen A ^. - E schoolyt!arWi-1952, from advancedgeometrytextbooks, t Frc , uo:41,onAtief=_;4:_=,Waal were for ittudentsin tu seat. everyprincipal kt_ I _41 mathe- plane tz-eDmetry oh the=_-,-41nCiftry rrj -7.L-41pc..11er inthepublic r lev Theew t topics seivre- ,-hci-ols of the Statethrong thwiWm, .t_{41tothe importantnratert- lid .4= -4 of the deva education,ac- theoremsfoundinatypicalhist companiedbyacommunication k schoolgeometryostler".

t,1 commissioner. 14-1=r-, VE AN_ ro Major -44g Co*olsieions. (*.Pled fromSG et ot ta_ schools Three es- Cera's theo- nd potentff-1 tie t_--achers. rem, andhorn( figuresarepre ojo ineistgsants Con%rtse.-r Rent(-11 e

Theaverage sreôndnry - '0",0- mathed4satics 111. yAN andGULLCi UNA- teacherivt RhodeIsland during the DI N r-- --her 1 Functions schooly_ 1951-52`-71-,,4daMatelot's A with DIvItion.(lows degree, receivedaLalart of 364S,and St.Le TmellersCollegeCedar E had taught18years. Fi erag7 1%116.) a lam consistedof2$pupils,be En Pro seewhetherere tele AS vnr-011._F-nt of 1 1 f-a,. We- he E taught22lt-e- a, CdUC EL differences t riodsa w::k. hiteamcaneshe & teaching i81011tv theusu- method

a txipcum __an ofmathematics, andteachingA bytn---euet ofsmv=

E a rflX1}Kh e=1 E I-1a t-hiN and wwialstucties,radual e4-11 eeer ofxr1-1111ratof _A combinationof f-_== and soC. enceorrnathemtk and tiiI FvccImritro.=a,,a 1ft-vtab &en_

_ studies. e were L I. 4tk7-=de selected- - The number ofmen whteethath- ra cm- 17-s from _ CI and _E Moines,Iow& einatiez InRhodeIsland LAT-64 Six clams =ere stadUv tatLght by each methodfor 1yearand resultswerereviewed by analysisof rne traditional r===°- orcourses convarift_nce. y(elementary ;...;:t= fire y _ e w C j_ 2 CsU-4&,:c ft jar intermediatealgebra, Jr and wAki )was prevaient. ,Notatisticall differencebetweenh two methods inso far atheprtwe 1-4 neral WM-MP-Ma courses :m.. confin-4 rnfltT t -=concerned. Thesuccessive subtrac to 1rhigh tion methodwas more meaningtu1. whoa,A double-ft-Lc*remn of func Mildren taught by thismeth4gen- tionalcourses and tracittimai e0W-41-9 h--AA more readily(statiAleally tathe mentor highthoo1swas not ap- significant) thanthose taughtby the parent. ik-Ant methoe.

.1 s r . . .

s v AA= 'appear of education, completed and mentalstimu.ni eptpr Hammon( high 80001teacher work Lyl also rank influent.* funds. attending To from scholarships A_ these Most commonly, tests was particular prehensive mathematics t on' L 0 istics ofthewinners. mine what 70 present 113. IlimmoNF_ 41141 ranked 112ii k Procedurei.Eve Major C - , a determine L,onshi aior ajor P Wyoming, Laramie.) lege. rt.'t Reiationm carry LL as aBasisforandAttack Superior wi = encourage the 81_1rWL0_ & 1 -s U V iem.A statistic /E- YITrOtfiA- h. high ichool quesOonna = o oe as j LLiA e- a_ ANALYSIS 7 f s field activities w kfr A are mar= i - It o .ada high percentage traeg FT (Ed. Faculty caused A very _OWL ", degree challenge lug Tnoef r" & Choices, and 1956,ndiati I _ AV o. Uniiersity 327 The facets stUdents attend. some f Mathematics success. D. studeri Manpower 1952 to1955.Of A their WILBUR teionm- ff high &es; A.1 full-time to iage fl rt them to and 1 6 3 1956, University mhltiiiemR of studentswho in 7 A, _L. =-; - of the w A dvieer.---R. :Pd outand ELS the the A 0_ over .4 ,T1 w..*.! i. contest A t determine E ,, in thecollege _ atTics (3) aFlong with contest. af the unknown percentage X-4-11ra- RESEARCH vitipft-m-lr arid study or 7.--Ajff- the student n-kt apiontAnv effI 1frinpa & make insufficient .State University, long. 1iiiflg. = __A.L.Lons = Shortage.. Lid ; DarenLi _ nt=whool `A:= w Students .1 greatest j e=w percent Hi college. X for not Is r7-4 eniatics on =-= T Baex- wilts .2 71 w = their TI corn, 1A AA = ff co The The ,_ ErA 6- 1'41= Ui aid t LTI e (2) po- the re- of of T of I. AN_ , k Ve- 12-2- a IN *-4 izeil F_44)(y01_ ingthomAtir? pro grade fessionalized subject an:+ver average for error 01 pr-Pcirt-mitrii =ACHING Feb. almost nfull of eUucation. The number reiationsh mathematics, high schtwl units in I P. content ber ofunitsinhighschool matics student studies used inthisstudywere age wasusedas The university sample (8) raw scor6on- cation in year programorthe.2-yearportion ing of from uaemntics (1) . 1 ade of -7= Majtr Probtem.----To major Pral 111--Par the 5-yearplanUt _e'S -- = VA:mst%t.cs- cept, 1955, the ocedures.----A the, subject C_Piibna i1Igh-schoo WA all students Ralor n pinftI1 v _ _ At HIPunde-roij-i-ipte or cec to thequestion:What -Hier estimate, examination, Wag divided e_ aa ersit A Education the fir AR ==P- Place, I w_Ts igh schoolgeometry, I IEZZU'l e A p zysi 9 liztm 4 an W ti A ii s algebra, 'A A"-A s an, _ u_t un college ofeducatioq,dur7 A -=A (6) number 84ndependent variables Mathematics. 11 AdoiserHowardF. earned. of the --A vade E cumulative JLU:AOE -=-__.14-4ia, A tter. ) period, 1951-55. . _ = units inhigh P..- =4.- rt 8 ilbj girl PNV 01 type of __-1211111-eai Co_ r-v. the who and Conclusions. brief Subject- s rade s-. v 1 e minor Co _ AA Panime wastaken York, N.T.) matter ,i71.1.71 - s of Ienees, (5) Purpose of of predicted _7w = A rvw (7) number into 3 a E t E. _wjA. 41.2 A - TZ: r_ TICr-A in A Teachers definition ,_=1A J1-77 a review satisfactory e, ff=, high school ti thr-. ficitiego whether and social timmurly following: sEco4, 7=7: In Columbia "@- in A L Ot atter aue standard letel i rOrn Rr (:i. D., general roatti- is college- vrofesnicnai school iicMol grad A n aver- Con- vTaw.I wa was pro- wu- and a The for Ui ( ria- rn- of of of in 1 4- E 11 " a )

111111F1111111fi1111110101111111111111111711Z1111 rilii1111111111111111117111111111111111111 11111111 11111 ID 1 IRI1 A11111111111mattlIIIIIINIIIIIIIIIIMINIMEN1111111111111111111110111111111111111111111111171% Tit mill 1111111111111111411111W111111111111111,11111111111111111111M111111,111111111111M11111 = AA- a=AE-E- . 71'

thendevelopedta Light of.n.e needs oi problem in v test proved tonen ruxan 4 4 A - _ 4-.L g =0_ A- A y YPT---,=4 A -- flq{pi mathematics oZ In favor oftheboys. ;- -71M r the inadequacy (whereU-1 _ a exists) notable TflPflUwasauatLematfr g Li r ,ru 41 Li Ox mathematicsnu E- actor ror tigirls. AT Fi r- W g g_g Ar_A- tomeet needs, ---.4.=;--1 -a A awl uthe v -El re- WM:7MM,WILLIAmFRANK. A A V A Prom I t t-t Leee Study sponsible A _..-E_k74 in P.,LL to PredictSuccess inLe- COT I a Tr' braof Oth-Grade - the with the __- of edu- Students cating CIICI Puyallup:JuniorHighSchool. (M. Ed., 1956, MajorFinding'? A Conclusions. Universityof Wash- Profegalonalized subjectmatter islit ingtqn,Seattle.) A A v-; It (1-.1E-U.a.A _ _ vEE- Major ?-1 -0K:f."---A.AXA Tie:P.9st is E -Eto the subjectmatter thatthoprospectiveteacherwill DvOrak. ... =- ifi -4Ta 47'1 ri E . Problem. ea ular teacr ___. . izee:_ taught To probable to. ,4_ 1 -...iL ...... _ E in sucha way as to teaching inalgebra _ each nten- TISMFA. tV needs., of A.A.= teachers. e, teringthe Otbera(

.E1-4 content is a. ...AA_ a.;thepurpoat? 8 .Thefollowingvari- professional. ableswere studiedon 160efilR-PS:(1 The finalchapter of rrinAAA studygives rgitp_g,..,LA I -A gyr te cores,F (2) arather( -f21.1. treatmentofthe LoLatEri---10points,(3)1.Q.scores,) A 4 toneeD a a=psA1 of proof -__ - A-A r thenature reaaii a_a scoree,(5)raid- P of the of ehaprofessional- ing 4-1 comprehensiongeom., CL--=n (6) /Lea course asprup!ra-z----u daVerELnf-

triE 11-1 A dz,af'ff I 3 e-bA ng WArraeS0 teki. JE Jiie in and ConciuRiona. -ft.Es _ _ The A- Achievementin if-11 nn7fw--1_ testwas- thebest vari-. (Ph.D.,12649 The Catiiolie i- able forpred successnalgebra. e t ,t4 dv-a giM.= g f g --- exilic, E 14: IA 17J A T_IL I_4% AA- .4- A V Wan. C.) . Studyof --reev-SA-L II Elementary Major1-Parrugxe T-an w Geometry. (Ph.D.,19" I AEA JIOUliilThfl.- OhiStateUniversity,-vy 2Colurnbu) aea flEd study gtrif a- FacilityAdviser.Nathan 4 A- AA faA AA t=t--=&IAA A A-A.4 Aa_aa 2 agakat Afff-_A-AL-7 a___Rreflected a Lazar. =z .= A r & A (This isone of A__... = Taff=s g = thesequencenf- 1.-a. LAA 44.6E6 1 "a factor-pattern aot AA.*.. 1.1%; studies AA_ The _ definitions;axioms, L.F=EIZE Te- and Universityof America.) reirknmilions _ leading totheproofof the _ mathematicsfte- propositionon gv. Um OF & Lerior tion ofthe IowahighPsirirk;Acontest angles ofa eat Trari examination, A EJi nwas, given to t=immat trs-.1 tr-t-J-"vhn a Frncedure#, A a nearly1190 orsi22 Catholicco- 6 geometrytsit g_ A a. E 4 114-1P ..FER rt. from inIowa.frria 4.4 cen A=A through1955, a E analYzed troid mPihr-alwas usedtoanalyse the Major &-rwas "- data. = Conolutions.--- The8mostwidely is g snuences ITV, ET, Majorrinding =zE durin,the ritz-1.6-gears aro A_ tok fol- E A comparIJ the arid oriier pa A . lowing:(1) tod-Prri vArsion402Eu- factor,structure ie two groups clid' 0-4111Pn -at --e-e t=p (2) .=LA se- -clifa E" that AAA "E--a factor ft-P.:mg queue%Inwhichthe flora VAa f were X W manyritp-psenThe "parallel-i-nPA-makenaorigpond a-a4 ,==a A gaAga agAiAi differences, however, A toa iaftanglesitiaabraw andlhj =A- A gnawIntegration ofthe boys' .4 II conversewerproveddirectlya 1 a g (8) A !WI V ties, a a A kyls_37_1= --enektry A . A seiunce thiciithe fftont in

,7 a .4 1 72 Postulated which angles Fehr. leke physics matics. ematics ing, the lems werea_ and the Although geometry, geometry, volved applications, ometry, and advanced courses, had beet gators. 119. Thon Morgan. approach to r - a,a E A procedure ence on the ,Major in04m0=a thin. Prou:= iroceaii a-M.- g Major A Problem. PrGeedure.9.--Prev1ons Pffi n rriarriq IA t _y01_, riA Columbia (Ed. A Tv...40U = y park) plication to _ v o-r-ct Mathematics. 1 &I A athematies WILEJAMN with and tiov01. and AF Theory into a I ANA_. element."' Faculty is used A u.nss solutions to413or TAAM Findings i A 4 a D., reportedly most Faculty t 11 tz trigonometry, mathematics a Ole C-4 = wOR instruction for students the tnat ü of 1 esmciaI transversal college eAuee of s, U= theory 'A-a - g I of TriPTIT, equal determine V?! A s calculus. Adviser.---Howard bights?' devised =0- -1 A 1/"7*r ..F- In was* - s " a the MaryleAd, and A a_ DOUGLAS a A I NsL- I Advioer. 0111 tveig* (Ed. D., a. t_A _ algebra or appropriate s tre-stiman II &rtt textbook of I E applications cif (-MEAT a original ci corresponcLug _ Conclusions.-- az ct_ -riA tisject neertug I) of with k keg-i a`rla", E1g-f s individualized A = iv/ philosophical 11 were A an; 11rn1ng. learning the- what math- and mathe- New A 'from A- A unt.,..=. own& f4immrer and its "ST- A_Aia modern sci- n engineer- are .11 1 1 I its B. wing the matter., s s_ae College, H.` GM/4N. 5 investi- College :College _4%_ synthe York; plane Unto prob- "- 7 more more solid The text col- Uni- Ger- }yin Ai in- F. A the 1 ea_. an A IN R =ACHING procedure andmew= at their relation m o .. y ter skill ratings and preference for outside the ( Syer. by which room practical domly bitrarily dI (grades 9through snits Out ofthe of "4" "2" and were iw-FI 121. Wifirt. 1 ) Bardtnt: twmine of to Interested A ; Major Problem.--To setupa 0. WILSON, __Nor = kaAA r 4-4 Major Faculty y-1--rar= PrOblem.--To L-44,gwari l the useof f t_s F sified ical ton Through 12. Significant galas M'nFAiv vqtigatlon of I the Useof Ohio State and Devices ArithmetiC. ct A were I textbook. eq selltell and t University, is most -Ku pra Findings and into iCHir Faculty in to 31 were criteria foruse rm List ofPractical Problems nimous to _ , p mathematics rc.4 rated 137 _s- ti 0 p 'teuimem realm of ea, teaebirig 1 MATHEMATICS r=r to M: A selected judge S., evaluation indicateda Re it-am achieved. 4r .1- a the method. E = f 1 tniverilty, Columbus.) FYI Teaching att :_hem availabletoo _4- senva. 11-1 0rtl practical.) under e__Ene. obIem 49 erplonse the z- a Adviser.Etenry W. Ca catalog students gsz on rated "". Aiviser.LowrY (Ed. M., in the piles win - and (M. A., I 12) Boston, Mass.) the "3", ic s the looKs wereran, EA-a and nrithrnor KETMS. An only $ for list of in AS-0V A = factors. atr, pines, individualized r- problems average found =dimly 38 ieTnp worear- Feasibility of °Mete problems of these outcomes gi rac-t-ri 4-4 a-as (2) Tecchin r Mathemat- poimt scale to Dr(wes-.4-1 Grades A - were 1955, Bos- -s_ A r E t -a ICT-- 1,65, The E_F_.0 A= feasibility 4 classified problems. V,E.-11f4 Teachers a The re- (Rating A t Games, atelw-la class- Clas- E. rateL The irom were to deal& nt and teat In- , t W. 9 of ; 1111111101101K,111MINDEJIMelall1/311kENIENZIPIl/AglIMEIIIIIIIrENIEMILIIIMINIIIINIE APPENDIX 73

P44(141Pad,ff I1F Ea authorinvesti 4-ainath,inauesackgroundconsti- gatedgames,aids, anddevicesde- tute toe malla percentage of the total scribed AAA period Ina mere_al numbergraduating fromthese schools. games, anugames 9nri aevi in 1--t EN g 2 A ga 4 E toy departmentrseveralatom. 23. ow M. 911= Oi MajorFindingsnfOancluaiv-a PrerequisiteLearningand Success sA in ri Gamesanudevicess,al(40 -r%-a aLapractical isCollege. AB. z 1-Th (1956, zAz )1,7 1. I FAS and valuat _of-7;in teachingarith- A" Brook- A lyn,N. . metic. listof teachinaids'was Y.) compiled. r- y ..=.- a--4_ determinesyn 4 there isa si- ua= -can ziri-6-.ronr.e.in the 122. Wpm-, E RALPH.TheUn!veaaIty !In in el.. trigonometry A--A% A _ A A AA A az A and 7a.e. A=4 College mon HE 1Clartrqwho differ the matics CUrr andEnroll- time lapsesincetheirlastmathematics ment inSel'actedLh 801 f%1ITIrL20 Fla Tehema,7 Glen IA on- ef-3aurpa. total0280 ties.(ILA., unselec- 1955,Chico State Ea.za ted BrooklynCollege_students whoE E EA = Chico, Calif.) E registeredC_F for tr-ion 6=--= pre- a Major Faoultv SWE M, Witted inmathematics- Donn I loff. of the 11 =5 Atthe end ofthis a determine(1)the semes ,Lue recordsof 806uniietecte atE mathematicsCO ngATIPT-E1-4-1to students vhohadCUM the trigo- E±E r a rt. E=-ff a.,_WCA Al- 'ego preparatorystu- nornetrv courRe wcirexaminedto dis- r.ap inSnma TAFT Norther')CaIV cover the timelapse betweentheir last fornia high Hon previous *diva*(2)the * mathematicsmural # 4 (goner- Ik) the = der enrolled-In' intermediate E.-EAiatrffinAa, ndthis t . k_ Lhe ratf_V courses,an._(8) 812Placed av_;.1.= course.Of efP806 on mathematics students,irtQ Porn)tetheir last ProcHL,--1-ex D.,ta forthis stu.a_y previouRcourse in A_ ami- INtyr E _a IMA mer's obtainedbyMrt-1.-)nni Int' z--azi vvatiou beforebeginning up- nometry. wiu istrators andmathematics - TheremainingUShada TMea instructors ineight secondarywaaw-IlLa_. minimum f8 monthsanda Mak wenn baDtz-ick()I 17 PECE=-,, Findings A--A-4-m =c-~-* esopm 'tenths. Sufficientofferingswerefountin alge- Major"indinv.gist COileittaitY48.-- _ The mt-Tiaen, _o_11 branna manezeomPtry, butII(.in hadcomplete& _n the_more advancedeontss Offering mathemakcoarser-40 previousJUnO courses innit-Or-Fla fapars not onlyIx,- startedtrigonome in September A* -4E41 problemofpropersequence withgreaterproficiency--einthe4pre- but alw thepossibilityocrheduiecon- requisitetopicstinvthestu tentS flicts.Oiverone halfthe I-Egradp -witha longer timo lanse.The latter, .6trtil a_nd LakPrialgebraA_nu'over however, earnedigntficaLitly tu on ihnegeometry. finalgraes in trigonometryann had Studentsgraduating (_)m4°' Li a significantlylower A- rate. al- iwle in _a" thisstulw;liohave_Asade PS-4-58

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