adegoke 2013 math reasoning

8/12/2019 Adegoke 2013 Math Reasoning

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Journal of Education and Practice www.iiste.org

ISSN 2222-1735 (Paper ISSN 2222-2!!" (#nline

$ol.%& No.17& 2'13

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Modelling the Relationship between Mathematical ReasoningAbility and Mathematics Attainment

enson )desina )dego*eInstitute of Education& +ni,ersit of I adan

E-/ail0 doctoradego*e a oo.co/

AbstractIn t is article& t e aut or see*s to e plain t e indicators of /at e/atical reasoning a ilit and e a/ines t erelations ip& using structural regression /odeling tec ni4ue& etween /at e/atical reasoning a ilit andstudents attain/ent in /at e/atics.6 e sa/ple consisted of 2%' Senior Secondar Sc ool #ne Students ()ge 1%-1 ears w o were rando/lselected fro/ four senior secondar sc ools in Iso*an and Irewole 8ocal 9o,ern/ent )reas of #sun State&

Nigeria. ) 2%-ite/ :at e/atics ;easoning ) ilit 6est (:;)6 and a 3 -ite/ )ttain/ent in :at e/atics 6est():6 were constructed to e plain t e indicators of /at e/atical reasoning a ilit and to assess ow far t eseindicators enter into success in sc ool /at e/atics wor*. Students scores in :;)6 and ):6 were anal sed using /a i/u/ li*eli ood esti/ates of 8IS;E8 ,ersion !.!!. ;esults fro/ t e structural regression /odels owed t at four funda/ental notions (,i


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In order to place t is stud in a ,er clear perspecti,e& it is i/portant to distinguis etween a ilit (potentialsuccess and attain/ent (actual success in t e stud of an su @ect. )ccording to 8ee (1D 7 a ilit to succeedin a su @ect was defined as t e power of an indi,idual to grasp and /anipulate t e funda/ental notions re4uiredfor its stud w ile attain/ent in t e stud of a su @ect& on t e ot er and& was ta*en as t e ac4uired degree of

/aster o,er t e co/ple ities of its con,entional structure& starting fro/ t e si/plest eginnings and constantl proceeding to /ore difficult wor*. +nder t is definition& a ilit is related to sc ool wor* and is certainlnecessar for ac ie,e/ent in it. 6 is is so& ecause& so/e grasp of funda/ental notions (t oug not e plicitlde/anded is i/plicitl contained in t e stud of an su @ect. 8ee furt er e/p asi


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o @ecti,e of t is stud was& t erefore& to deter/ine t e e tent to w ic t ese four d na/ic processes areindicators and sufficient /easures of /at e/atical reasoning a ilit .

In Nigeria& t e curriculu/ of secondar sc ool su @ects is nor/all prepared t e Nigerian Educational;esearc and e,elop/ent Founcil (NE; F . 6 e /at e/atics curriculu/ as prepared NE; F in 2''5

consists of four /a@or sections. 6 ese are Nu/ er and Nu/erations& )lge ra& 9eo/etr & and Statistics. Scrutinof t e curriculu/ s ows t at wit in t ese four sections& a total of a out fift distinct topics are taug t including

percentages in arit /etic& 4uadratic e4uations in alge ra& circles t eore/s in geo/etr and /easures of centraltendenc & dispersion& and pro a ilit t eories in statistics. 6 ese are t e funda/ental concepts of /at e/aticsw ic Nigerian students /ust learn during t eir secondar sc ool education. Students le,el of attain/ent in/at e/atics is t e e tent to w ic t e are a le to /aster t ese funda/ental concepts of /at e/atics. In t isstud e/p asis was& owe,er& on t ree sections& na/el & Nu/ er and Nu/erations& )lge ra& and 9eo/etr .6 is is ecause& t is stud centers on senior secondar sc ool one students ()ges 1% - 1 . In Nigeria& usuall &in-dept stud of statistics starts wit students in senior secondar two ()ges 1 -1! .

In t is stud & t erefore& t e second o @ecti,e was to present structural /odel of t e lin*ages e isting etweent e funda/ental notions of /at e/atical reasoning a ilit and attain/ent in /at e/atics. In order to stud t erelations ips etween t e funda/ental notions of /at e/atical reasoning a ilit and attain/ent in /at e/aticst e aut or de,eloped a Structural ;egression /odel (S; (see ?igure 1 . S; is a ,ariant of structural e4uation

/odels. 6 e ot er e a/ples are Pat )nal sis (P) and Fonfir/ator ?actor )nal sis (F?) (See )dego*e&2'12= Lline& 2''5 for /ore detailed infor/ation on Structural E4uation :odelling . )n S; /odel results fro/t e s nt esis of pat /odel (P) and F?) /easure/ent /odels.

Figure 1: Structural regression model of Mathematics reasoning ability and Mathematics Achie ement.)s in P)& S; allows test of pot eses a out direct and indirect causal effects. Cowe,er& unli*e P)& t eseeffects incorporate a /easure/ent co/ponent t at represents o ser,ed ,aria les as indicators of underl ingfactors as in F?). In S; /odels& t e researc er can test pot eses a out structural and /easure/ent relationswit in a single /odel ()dego*e& 2'12= Lline& 2''5 .

In figure 1& :;) and ): are factors or latent traits and t e are not directl /easured ut rat er assessedindirectl using scores fro/ t e indicators. ?or e a/ple& :;) is assessed t roug students a ilit to recogni


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2 oes t is /odel fit t e o ser,ed dataM3 If t e /odel fits t e o ser,ed data& w at are t e /odels fit statisticsM% G at is t e nature of t e factor loadings of eac of t e latent traitsM5 6o w at e tent does /at e/atical reasoning a ilit predict t e attain/ent in :at e/aticsM

2. Method2.1 Participants 6 e participants were 2%' Senior Secondar #ne ()ge 1% 1 ears students rando/l drawn fro/ 1' SeniorSecondar Sc ools in Iso*an& and Irewole 8ocal 9o,ern/ent )reas& #sun State& Nigeria. In t is stud & onl t estudents in science classes too* part. ?ro/ eac sc ool& intact science classes were used. )/ong t e 2%'students sa/pled& 127 (52.D were o s& w ile 113 (%7.1 were girls. 6 eir ages ranged etween 1% and 1ears (:ean age 15.%3= S 1.7D . 2.2 Materials?or t is sur,e & two instru/ents were used. 6 ese are0a )ttain/ent in :at e/atics 6est ():6

:at e/atical ;easoning ) ilit 6est (:;)6 AMT 0 6 is consists of 3 /ultiple c oice ite/s wit four options () F . 6 e aut or of t is article de,eloped

t ese ite/s fro/ t ree /a@or sections of /at e/atics (Nu/ er and nu/eration& )lge ra& 9eo/etr & and9eo/etr as prescri ed t e Nigerian Educational ;esearc and e,elop/ent Founcil& (NE; F& 2''5 .Initiall t ere were ' ite/s. 6 ese ite/s were su @ected to pilot testing a/ong 3' SSII students in ) edadelocal 9o,ern/ent )rea& #sun State& Nigeria. 6went -four ite/s w ose difficult indices were eit er elow '.3'or a o,e '.71 were deleted. ?or t e re/aining 3 ite/s& eac of t e four /a@or sections consists of nine ite/s.6est lue print placed under Lnowledge& +nderstanding and 6 in*ing were used to esta lis t e content ,aliditof t e final 3 ite/s. 6 e difficult indices of eac ite/ ranged etween '.3! and '.75& w ile t e discri/inatingindices ranged etween '.%3 and '.51. 6 e relia ilit indices of eac section0 Nu/ er and nu/eration& )lge ra&and 9eo/etr were '.71& '.7 & and '. D respecti,el . 6 is was esta lis ed using Luder ;ic ardson 2'for/ular. 6 e /a i/u/ o taina le score in eac section was 12& t at is& eac ite/ attracted a score of 1 for rig tresponse and ' for wrong response.

MRAT 0 6 e aut or of t is article constructed t is instru/ent. It was constructed co/ ining t e funda/ental processes of recognition of ,aria le& class& order and correspondence wit t e t ree asic t pes of /aterialse/plo ed in t e stud of :at e/atics& t ese eing arit /etical nu/ ers& alge raic s / ols& and spatial figures.)s suggested 8ee (1D 7 & separate tests were designed to esti/ate capacit for andling eac of t e four

processes in ter/s of eac of t e t ree t pes of /aterials /a*ing twel,e su tests in all. Initiall under eac sutest& t ere were four ite/s& t at is& t e draft cop consisted of %! ite/s. 6 ese ite/s were su @ected to pilottesting a/ong 35 students in ) edaade 8ocal 9o,ern/ent )rea& #sun State& Nigeria. In eac of t e su tests&ite/s wit low difficult le,els (p O .3' were eli/inated. In t e final draft& t ere were two 2% ite/s. 6 edifficult and discri/inating indices of eac ite/ ranged etween '.37 and '.7'= and '.3' and '.57 respecti,el .6 e relia ilit indices of eac section ranged etween '. D and '.!1. 6 e /a i/u/ o taina le score in :;)6was 2%& t at is& eac ite/ correctl answered attracted a score of 1 w ile wrong responses attracted '.E a/ples of ite/ used in t e :;)6 include01. Flassification-)rit /etical Nu/ er (F8)-)N 0 In eac of t e rows of nu/ ers& t ere are fi,e nu/ ers on t eleft. ?our a,e so/et ing in co//on and one is different. +nderline t e one w ic is different. G en ou a,edone t is& place a circle on one of t e nu/ ers on t e rig t w ic could est ta*e t e place of t e one ou a,eunderlined.

8eft ;ig t%5& 5%& 3& 27&!" 33& 3 & & D3&

In t is pro le/& on t e left and side& 3' a,e een underlined& ecause it is not di,isi le D. #n t e rig t andside& 3 a,e een circled ecause it is t e onl one out of t e four nu/ ers on t e rig t w ic is di,isi le D2. #rder-)rit /etic Nu/ er (;##-)N . 6 e following rows s ow t e eginnings of series of nu/ ers& andso/e nu/ ers are /issing. ?ill in t e lan* spaces wit t e nu/ ers ou t in* s ould e t ere.1& 3& 5& 7& D& ---& ---2.3 Procedure6 ree researc assistants were recruited for t is stud . 6 e were all graduate students in t e Institute ofEducation& +ni,ersit of I adan& Nigeria. 6 e a,e recei,ed for/al training in test ad/inistration. Cowe,er&t e aut or of t is article e plained t e purpose of t e stud to t e/. 6 e aut or also went round to /onitor owt e tests were ad/inistered. 6 e data collection lasted four wee*s. 6 e ad/inistration of eac of t e tests too*

place during t e nor/al ti/e sc eduled for /at e/atics on t e official sc ool ti/e ta le. 6 is was to a,oiddisruption to t e sc ool progra//es. ?or :;)6& t e students co/pleted it on t e a,erage& in 5' /inutes&


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t oug t e ti/e allowed was one our. 6 e data collected were anal sed using /a i/u/ li*eli ood esti/ates of8IS;E8 $ersion !.!' (8inear Structural ;elations= J res*og H S r o/& 2''3 .

!. Results

6 e descripti,e statistics (Pearson Forrelation coefficient& /ean& and standard de,iation were calculated using8IS;E8 $ersion !.!'. )lso ,ariance-co,ariance statistics were o tained. 6a le 1 presents t e Pearson product-/o/ent correlation coefficients etween t e ,aria les. :ean and standard de,iations of eac of t e si ,aria lesare as presented on t e ta le.#able 1: $earson correlation coefficient% mean% and standard de iation

$); F8) ;## F#; NN+ )89 9E#$); 1.'''F8) . 3'Q 1.''';## .537Q .535Q 1.'''F#; .!33Q . 23Q .51 Q 1.'''

NN+ .! 'Q . 33Q .5%'Q .7'5Q 1.''')89 .5!DQ .!21Q .73 Q .5! Q .5D2Q 1.'''9E# . 3 Q .751Q .7D%Q .5!2Q . %'Q .5!D 1.''':ean .%D 5.D2 5.13 7.'! .!7 5.21 .%1S 2.!5 2.57 2.35 2.D2 2.!3 2.%' 2.5'

Note Qp O .'5 Key !AR " recognition of variables# $%A " classification of given materials# R&& " recognitionof order# $&R " recognition of correspondences# ''(" 'umber and 'umeration# Algebra " Algebra# )*& ")eometry. ;esearc Ruestion #ne0 G at is t e nature of t e relations ip a/ong t e indicators of t e latent ,aria lesMIn t is stud & as specified in t e preceding sections& t e pot esi


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#able &: Ma'imum (i)elihood $arameter *stimates of the Structural Regression ModelPara/eter Standardardised Esti/ate +nstandardised Std. Error T

?actor 8oadings:;) $); '.55 1.5 '.15 1'.3D

:;) F8) '.!% 2.17 '.13 1 .%3:;) ;## '.!3 1.D '.12 1 .'D:;) F#; '.3' '.!! '.1% ' .%2): NN+ '.53 1.5' - -): )89 '.!3 2.'' '.1! 1'.D3): 9E# '.!3 2.'7 '.1D 1'.D1

:easure/ent error ,arianceE$); '.7' 5.7' '.% 12.%5EF8) '.2D 1.D' '.15 12.35E ;## '.31 1.72 '.1% 12.%2EF#; '.D1 7.7 '. ! 11.%DE NN+ '.72 5.7D '.%7 12.33E)89 '.31 1.77 '.15 11.D2

E9E# '.32 1.DD '.17 11.D

)lso presented in ?igure 2 are t e standardi


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a ilit . 6 ese findings were consistent wit t e wor*s of earlier p ilosop ers suc as 8ee (1D 7 and Ca/le(1D3% .

E/pirical e,idence fro/ literature (e.g. )dego*e& 2''3= erret H Gillia/s& 1DD7 ?isc ein H Nac ieli& 1DD!=Ceng-Ku*u H Slli,an& 2''' suggests t at a ilit to succeed in t e stud of /at e/atics could correspond to t e

a ilit to carr out t e four processes of recognition of ,aria le& classification& ordering& and recognition ofcorrespondence. 6 is is ecause t ese notions pla a large part in :at e/atical wor*& since /ost pro le/s aresol,ed recogni


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lac*well& ). :. (1D%' . U) co/parati,e in,estigation into factors in,o,ed in t e /at e/atical a ilit of o sand girlsV& /ritish +ournal of *ducational Psychology & "& 1%3 -212.rown& G. (1D1 . U An ob0ective study of Mathematical intelligence V& Sc 6 esis& +ni,ersit of 8ondon 8i rar .F oud ur & ;.& H as& . L. (2'12 . UInfluence of geo/etrical a ilit and stud a it on t e ac ie,e/ent in

/at e/atics at secondar stageV& ,nternational +ournal of $omputational *ngineering Research & 2& 232 23 .?ederal 9o,ern/ent of Nigeria (2''% . U The 'ational Policy on *ducation (3 rd ed. V. 8agos0 NE; F?isc ein& E.& H Nac ieli& 6. (1DD! . UFoncepts and figures in geo/etrical reasoningV. ,nternational +ournal ofcience *ducation & 2'& 11D3 -1211.Ca/le & C. ;. (1D3% . U Relational and functional thin-ing in Mathematics (Nint Kear oo*& National Founcil of6eac ers of :at e/atics& ureau of Pu lications& 6eac ers Follege& Folu/ ia +ni,ersit & New Kor* Fit &+S) .Ceng-Ku*u H Sulli,an& C. J. (2''' . UPersonalisation of /at e/atics word pro le/s in 6aiwan.V *ducationalTechnology Research and evelopment & %!& %D -5D.Cu& 8. 6.& H entler& P. :. (1DDD . UFut off criteria for fitness indices in co,ariance structure anal sis0Fon,entional criteria ,ersus alternati,esV. tructural * uation Modelling & & 1- 55.Jen*ins& J. G. (1D3D . U An analysis of factors entering into results of tests based on the logical principles of

Mathematics . P 6 esis& +ni,ersit of 8ondon 8i rar .

J res*og & L. 9.& H S r o/& . (2''3 . U %, R*% 4.56 for 7indo7s Aco/puter softwareB. 8incoln& I80 ScientificSoftware InternationalV.Lline& ;. . (2''5 . U Principles and practice of structural e uation modeling (2 nd ed. V. New Kor*0 6 e9uilford Press.8ee& . :. (1D 7 . U) stud of specific a ilit and attain/ent in :at e/aticsV. /ritish +ournal of *ducational

Psychology & 37& 17! - 1!D Nigeria Educational ;esearc and e,elop/ent Founcil (2''5 V& enior econdary chool $urriculum . 8agos0 NE; F. Nunes& 6.& rant& P.& arros& ;.& H S l,a& L. (2'12 . U6 e relati,e i/portance of two different /at e/aticala ilities to /at e/atical ac ie,e/entV& /ritish +ournal of +ournal of *ducational Psychology & !2& 13 -15 .+wadie& I. (2'12 . U?ederal 9o,ern/ent& teac ers and parents attle students under-perfor/anceV. 8agos0!anguard 'e7spaper & 1D Jul .Gest )frican E a/ination Founcil (2'12 . G)EF S lla us& 8agos

/enson Adesina Adego)e olds .Ed (P sics & :.Ed and P degrees in Educational E,aluation& of+ni,ersit of I adan& Nigeria. Ce specialises in researc /et ods as well as in teac ing /et ods of /at e/aticsand p sics. Ce is a researc fellow at t e Institute of Education& +ni,ersit of I adan. Ce is a /e/ er ofScience 6eac ers )ssociation of Nigeria as well as )/erican Ps c ological )ssociation


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adegoke 2013 math reasoning

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