APPLYING THE MIXED RASCH MODEL TO THE FRACTION CONCEPT OF PUPILS

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1 Intrnational Journal of Innovativ Managmnt, Information & Production ISME Intrnationalc200 ISSN Volum, Numbr, Dcmbr 200 PP APPLYING THE MIXED RASCH MODEL TO THE FRACTION CONCEPT OF PUPILS YUH-CHYN LEU AND YUAN-HORNG LIN 2 Dpartmnt of Mathmatics and Information Education National Taipi Univrsity of Education 34, Sc. 2, Hping E. Road, Da-an District, Taipi City 06, Taiwan lu@ta.ntu.du.tw 2 Dpartmnt of Mathmatics Education National Taichung Univrsity 40 Min-Shn Road, Taichung 40306, Taiwan lyh@mail.ntcu.ud.tw ABSTRACT. Th purpos of this study is to invstigat th fraction concpts of first gradrs and scond gradrs basd on th mixd Rasch modl. Fraction concpts play an important rol in mathmatics larning in lmntary schools. Thr ar many factors that influnc th construction of fraction concpts. This study is a prospctiv analysis of th fraction concpts basd on sampl clustring in accordanc with thir cognitiv faturs. Th mixd Rasch modl combins th latnt class modl and th Rasch modl to prsnt propr classs of sampls with th corrsponding latnt trait. Th rsults of this study show that both first gradrs and scond gradrs could b classifid into two groups. Howvr, th faturs of ths two groups diffr with grad. In addition, comparisons of th faturs of th groups could rval som possibl factors that influnc th construction of fraction concpts. Finally, basd on th findings of this study, som suggstions for futur rsarch ar providd. Kywords: Cognitiv Diagnosis; Fraction Concpt; Mixd Rasch Modl; Rasch Modl. Introduction. Mathmatics is th foundation of scinc and othr filds. Studnts bgin to larn its formal rprsntation sinc thy ar studnts in th lmntary schools. Howvr, quit a fw rsarchs rval that studnts own intuitiv thinking or misconcptions in th procss of concpt construction. As to fraction concpt, it is th foundation of furthr mathmatics lik ratio and division, d Castro (2008, Cramr t al. (2002. Thrfor, invstigation on fraction concpts for pupils should b prospctiv. On th othr hand, sampl clustring basd on homognity of fraction concpts should b usful for rmdial instruction. It is bcaus dsign of adaptivly rmdial instruction could dpnd on th information of cognitiv diagnosis, Lin (200. For this purpos, application of mixd Rasch modl could rflct th cognitiv information of masurmnt data. Thrfor, this study will adopt mixd Rasch modl to analyz fraction concpt of pupils who ar th first gradrs and th scond gradrs.

2 APPLYING THE MIXED RASCH MODEL TO THE FRACTION CONCEPT OF PUPILS 9 2. Litratur Rviw. Issu of fraction concpts rlatd to its construction and misconcptions is discussd. In addition, modl dscription of mixd Rasch modl will also b discussd. 2.. Fraction Concpts of Pupils. Fraction is on of th most difficult concpts to comprhnd for lmntary school studnts, Charalambous and Pitta-Pantazi (2006. Many litraturs indicat thr ar misconcptions for studnts in th procss of larning, Chan t al. (2007. Evn adults xist concptual dficincis in fraction, Stafylidou and Vosniadou (2004. Erronous fraction concpts xist in both concptual knowldg and procdural knowldg, Mosly and Okamoto (200, Panaoura t al. (2009. Thr ar multipl manings as to fraction concpt. It includs part-whol rlation, xprssion of division, quotints, anothr form of dcimal, oprator, ratio and masurs, rau t al. (2009. Th charactristics of multipl manings in fraction asily rsult in th larning obstruction of studnts, Charalambous (2007. As to th first gradrs and th scond gradrs, thy mainly comprhnd th rprsntation and xprssion of fraction. Gnrally, qual sharing concpt, simpl fraction concpt and units concpt ar th main thr componnts for pupils. Howvr, owing to th rprsntation and mathmatical languag, prformanc of fraction concpts vary across grad. Thrfor, it is worthwhil to apply mixd Rasch modl in invstigating th fraction concpts of pupils Dscription of Mixd Rasch Modl. Rasch modl is on popular modl of itm rspons thory. Lt θ i b th latnt trait of task-takr i and b k is th difficulty paramtr of itm k. Rasch modl is θi bk Pk ( θ i = = ( θi bk ( θi bk + + Rost xtndd Rasch modl to mixd Rasch modl, Rost and Rasch(990, Rost (99. Mixd Rasch modl considrd th task-takrs who blong to diffrnt classs hav distinct rspons probability on itms. In addition, itm difficulty will b diffrnt in accordanc with various classs. Lt b kc b th itm difficulty of itm k on class c and θ ic b th latnt trait of task-takr i who blong to class c. Combind with Rasch modl and latnt class modl, th mixd Rasch modl is, Rost and Rasch(990, Rost and von Davir (994. ( θic Pkc ( θ ic = = (2 ( θic ( θic + + Lt P ( X i b th probability of rspons vctor X i from task-takr i. It is C K ( θic C x ik xik P( X i = α c ( ( with ( θic ( θic α c = (3 c= k = + + c= αc is th latnt class paramtr to rprsnt th ratio of sampl siz in class c. It is x ik = whn task-takr i giv corrct answr on itm k ; othrwis, it is x ik = 0. EM algorithm and unconditional maximum liklihood stimation is adoptd to itrat so as to find th optimal paramtrs abov. As to th modl slction, AIC (Akaik Information Critrion, BIC (Baysian Information Critrion and CAIC (Consistnt Akaik

3 92 YUH-CHYN LEU AND YUAN-HORNG LIN Information Critrion ar thr popular indics to dtrmin class numbr of mixd Rasch modl, (Hamakr, 2009; Bozdogan, 987; Akaik, 978. Th smallr th indx is, th bttr th modl will b. 3. Assssmnt and Sampl. Th fraction concpt assssmnt is dsignd by th author. In ordr to conform to mathmatics txtbooks of ach grad rspctivly, thr ar two kinds of assssmnt and ach assssmnt contains six itms which masur th sam fraction concpts in th two assssmnts. Th only diffrnc btwn ths two assssmnts is th mathmatical dscription and fraction rprsntation of itms. As shown in Tabl, ths thr concpts ar A(qual sharing concpt, A2 (simpl fraction concpt and A3(units concpt. Th itm- concpt matrix rprsnts th concpts masurd by itms. For xampl, itm masurs concpt A (qual sharing concpt and itm 4 masurs concpt A (qual sharing concpt concpt A2 (simpl fraction concpt. Th sampl ar stratifid sampld from Taiwan. Thr ar about on thousand and two hundrds pupils for ach grad. TABLE. Th itm-concpt matrix of th fraction concpt assssmnt Concpt Attribut Itms A (Equal Sharing A2 (Simpl Fraction Concpt Concpt A3 (Units Concpt Main Rsults. Mixd Rasch modl is th statistical modl to analyz th mpirical data and WINMIRA 200 is th softwar, von Davir ( Th First Grad. Thr indics ar usd to valuat th modl slction. As shown in Tabl 2, it shows that thr xists minimum valu whn th class numbr is 2. Thrfor, th bst numbr of class for first gradrs is 2. TABLE 2. Critria of modl slction for th first Gradrs Class Numbr AIC BIC CAIC

4 APPLYING THE MIXED RASCH MODEL TO THE FRACTION CONCEPT OF PUPILS 93 Tabl 3 displays th numbr of studnts in ach class for th first gradrs. Statistical tst 2 Parson Chi-squar shows that χ = 4.76 ( df =, p <. 05. Thrfor, thr is significant diffrnc on th class distribution for gndr. It appars that mal studnts tnd to blong to class II but fmal studnts tnd to blong to class I. TABLE 3. Numbr of studnts in ach Class for th first Gradrs Latnt Class Gndr Sum Fmal Mal Class I Class II Sum Th Scond Grad. Basd on th thr indics dscribd in Tabl 4, it shows that thr xist minimum valu of BIC and CAIC whn numbr of class is 2. Howvr, th minimum valu of AIC xists whn class numbr is 3. Sinc thr is littl diffrnc on AIC whn class numbr is 2 or 3, it is rasonabl to dcid that propr class numbr is 2. TABLE 4. Critria of modl slction for th scond Gradrs Class Numbr AIC BIC CAIC Tabl 5 displays th numbr of studnts in ach class for th scond gradrs. Statistical 2 tst Parson Chi-squar shows that Parson χ =.00 ( df =, p = Thr is no diffrnc on th class distribution for gndr. It also mans that prformanc on fraction concpts is th sam as to gndr. TABLE 5. Numbr of studnts in ach Class for th scond Gradrs Latnt Class Gndr Fmal Mal Sum Class I Class II Sum

5 94 YUH-CHYN LEU AND YUAN-HORNG LIN 4.3. Comparisons on both Gradrs. Th lin chart of conditional probability of itms within classs in accordanc with both gradrs is dpictd in Figur. In th first grad, it shows that class I prforms bttr a littl bit than class II on itm 2 and itm 3 but has dficincy on itm, itm 4 and itm 5. As to itm, itm 2, itm 4 and itm 5, it sms to xist much diffrnc on conditional probability btwn two classs. Howvr, class II prforms quit poor on itm 3 and itm 3 is to masur units concpt which is th foundation of fraction. Thrfor, it may xist cognitiv dficincy as to th units concpt for Class II. Rmdial instruction on this concpt spcially for class II is ndd. As to th scond grad, it shows th two lins ar almost paralll and class I prforms bttr on all itms than class II. It mans that th two classs only has diffrnc on mastry of fraction concpts. Ths is no intractiv distinction among concpts lik th prformanc of th first gradrs. Hnc, th dsign of rmdial instruction for th scond gradrs is unlik th first gradrs. In comparison with ths two gradrs, it rvals thir fraction concpts diffr. Th scond gradrs do not prform bttr on all itms than th first gradrs. In th fraction concpt, although itms of diffrnt grad masur th sam concpt attribut, thir mathmatical rprsntation and dscription vary. It mans th importanc of rprsntation and dscription for fraction. Conditional probability Itms Class I (st Grad Class II (st Grad Class I (2nd Grad Class II (2nd Grad FIGURE. Lin chart of conditional probability on itms for th both Gradrs TABLE 6. Latnt trait in ach Class for th first Gradrs Total Latnt trait of th first gradrs Latnt trait of th scond gradrs Scor Class I Class II Class I Class II

6 APPLYING THE MIXED RASCH MODEL TO THE FRACTION CONCEPT OF PUPILS 95 Th corrsponding rlationship btwn latnt trait and total scor is dscribd in Tabl 6. Bcaus total scor is th sufficint statistics of latnt trait, ach total scor will corrspond with on latnt trait. Howvr, ach class has its own cognitiv faturs and th sam total scor of diffrnt class and grad has distinct latnt trait. 5. Conclusion. Th study invstigats th fraction concpts of pupils by using mixd Rasch modl. Although both gradrs own two classs, th distribution of gndr and th faturs of ths classs diffr. On is concludd that thr is cognitiv dficincy in units concpt for th first gradrs of class II. Th scond gradrs do not hav intractiv distinction among concpts. Furthrmor, rmdial instruction of ths gradrs should dpnd on th faturs of concpt structurs with maning of mathmatical rprsntation and dscription in itms. REFERENCES [] A. Panaoura, A. Gagatsis, E. Dliyianni and I. Elia (2009, Th structur of studnts' blifs about th us of rprsntations and thir prformanc on th larning of fractions, Educational Psychology: An Intrnational Journal of Exprimntal Educational Psychology, vol.29, pp [2] B. Mosly and Y. Okamoto (200, Idntifying fourth gradrs' undrstanding of rational numbr rprsntations: A mixd mthods approach, School Scinc and Mathmatics, vol.08, pp [3] B. V. d Castro (2008, Cognitiv modls: Th missing link to larning fraction multiplication and division, Asia Pacific Education Rviw, vol.9, pp.0-2. [4] C. Y. Charalambous (2007, Dvloping and tsting a scal for masuring studnts undrstanding of fractions, Procdings of th 3st Confrnc of th Intrnational Group for th Psychology of Mathmatics Education, vol.2, pp [5] C. Y. Charalambous and D. Pitta-Pantazi (2006, Drawing on a thortical modl to study studnts undrstandings of fractions, Educational Studis in Mathmatics, vol.64, pp [6] E. L. Hamakr (2009, Using information critria to dtrmin th numbr of rgims in thrshold autorgrssiv modls, Journal of Mathmatical Psychology, vol.53, pp [7] E. J. Austin, I. J. Dary and V. Egan (2006, Individual diffrncs in rspons scal us: Mixd Rasch modlling of rsponss to NEO-FFI itms, Prsonality and Individual Diffrncs, vol.40, pp [8] H. Akaik (978, A Baysian analysis of th minimum AIC procdur, Annals of th Institut of Statistical Mathmatics, vol.30, pp.9-4. [9] H. Bozdogan (987, Modl slction and Akaik's information critrion (AIC: Th gnral thory and its analytical xtnsions, Psychomtrika, vol.52, pp [0] J. Rost (99, A logistic mixtur distribution modl for polytomous itm rsponss, British Journal of Mathmatical and Statistical Psychology, vol.44, pp [] J. Rost and M. von Davir (994, A conditional itm fit indx for th Rasch modls, Applid Psychological Masurmnt, vol.8, pp [2] J. Rost and Rasch (990, Modls in latnt classs: An intgration of two approachs to itm analysis, Applid Psychological Masurmnt, vol.4, pp

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