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For Bulletin 19-53, No. 4 "' ffi.^10' ,,11111,0,11111111411 fl 11111 11111111110111111111 1_0,1111 11.11141111111111111111111111111All . r r n11,1111 CIVI11 111111111 [1 111111111I,I , 11411 11111R trol °,I. rl 1111111111111112111111111111111111111111.1 R 11J rt Mi1111HA 1 . .4' 11144k 1url A n ,ffligur.,,, r, kr 1,10 00 1111.1 ntel ti 1-11 A...L RI'1 I 1 111.4 411 1 1420 ft ( . di ,r11 3.13=E = ===_ MU fL A=W _ =i=". ==. laF =a. 1C--= a a 4 At, 1111111 .=# pig s=t IIi _\ :1'2! _ ME. .0 .1M MENff 33E - = = ,10 YI !t".. r. 0 !NI m 1 1955 a % 311 d11411 111011 0111E11 41, 1110' 6,111, (I1 IdIII Z KENNETH *1-,.s E.BROWN. U fol,41 . f'cla for LI E 1111411IF f. 1111G1 Ell 111111 Ite1 fr U1111E11 424 ' 1111111111V 111111,L111111111111tk 111111111111 LakilMoNI "EN, lm 1 r lauip4111111111F 107Iv 119101Mirilig s Itiefroomphoepanlivoirimoi 111111111111111111 r,'01,:.1111111611101:.:' ici; law r9 111101.111rireinnttrttatrtnInmIrnarloum.11111,MA1 MET - Nik er Bulletin19-53, No.4 ===-_ u,S. _ = = =t- 1-4 A WELFARE w=, MARION B. .a .... ,"-02 = a = W_ = Office of tp g = LW/WM&q. -_ 4=1:-LE L.WW =-W ci Contents # . Pairto Foreword .r 411, , V 0 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 UI holy. a " ' ai2.f A . 4 . 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. 1 .0) 46. wo v If "h. air 4 .0" 400 Preface a 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. VI 4 or 3'71 'J '' ANALVSISOF ItESEARCHIN THE' TEACHING 0 OFMATHEMATICS Introductio4 11111011 WTHATRESEARPHinthe teachingof mathematicswascarried VVonduring calendaiyears1.955and 1956? Whatdoes thisre- searchsayabout effectivewaysof teaching?In whatareasof mathe- matics 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 be- tween 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 fr 1 a 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. a . a 1 oft r, 10- . 1 4 . : I I 1...61:14 t 41 nO or 7 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 '<the ., . remedialcoiirsei agemed, ;littlemire! thanthoseloho omitted it took, initead,the:vTegultiri - doileg4matheifiatici contts.4 Attempts (8).ftrestill beintinadeto ;devilopmoreAteiribli '41 textbooksfor iIatticienb6his tint' ;14thlti* . thti Atudeliti readmit collegebe KILt, IV mg greitinioti*stithrtolearnmit and mai'maturity thaniflienchervkasitiIiiglosalioar4 IMO difiletati'If '-not.titotote*fatstivoiesthit gotathdr9. Inuivilectge of high who& mathematics: from'8416int. Remeetitir (.2 , àmlin:dpjlegtx ,i 41511411-1111main4 Ws; 'Ff 41, r iv4 A 1. jP-1. j . ; f*. 4 ANYBII.9oFREBBABCHiN MACHIN°MATIMEMOS . 3. Dostudentswhose mathematicstrainingleas beeninterrupted by service inthearmed'form(orforsasyother reason)makelower gradesthan theirclassmates,' I. i . A. Theanswer seems ,to beno.Accoraingtoonestudy. (1Z1)stu- dents 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 . 4. taken into 5 account. , 5 . I.!,. ' ° 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
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