University f Wllngng Research Online Faculty f Educatin - Papers (Archive) Faculty f Scial Sciences 00 Year 3 children's understanding f fractins: are we making prgress? Mhan Chinnappan University f Wllngng, mhan@uw.edu.au Mike Lawsn Flinders University Publicatin Details Chinnappan, M. & Lawsn, M. J. (00). Year 3 children's understanding f fractins: Are we making prgress? In B. Bartn, K. Irwin, M. Pfannkuch & M. Thmas (Eds.), Prceedings f the 5th Cnference f the Mathematics Educatin Research Grup f Australasia, 7-10 July 00, vl. 1 (pp. 195-0). Auckland, New Zealand: Mathematics Educatin Grup f Australasia. Research Online is the pen access institutinal repsitry fr the University f Wllngng. Fr further infrmatin cntact the UOW Library: research-pubs@uw.edu.au
Year 3 Children's Understanding ffractins: Are we Making Prgress? Mhan Chinnappan University fwllngng <mahan_chinnappan@uw.edu.au> Michael Lawsn Flinders University <mike.lawsn@flinders.edu.au> The aim f the study reprted here was t examine the quality funderstandings develped by yung children in the area f fractins and decimals. Analysis f data shwed that the existence f great disparity in Year 3 children's knwledge base f fractins. We discuss these results in light flevels f cmpetence that are expected in K- curriculum dcuments and with reference t past research n students' knwledge f fractins. The results f this small study raise dubts abut the prgress being made in the teaching ffractins. Backgrund A principal aim fthe primary mathematics curriculum is that students develp a sund understanding f the number system and becme cnfident in using such understandings in describing real-wrld phenmena and slving prblems (Natinal Cuncil f Teachers f Mathematics, 000; Australian Educatin Cuncil, 1990). Within the general tpic f number, teachers and researchers have paid particular attentin t fractins and decimals, fr these numbers are used in making sense f prblems in daily life f children and adults alike. A majr utcme expected in the areas f fractin and decimals is that primary schl students shuld use these numbers and theirrelatins flexibly t slve prblems andreasn mathematically (see Mathematics K- Outcmes and Indicatrs, Bard f Studies - New Suth Wales, 1998). Despite the critical cnceptual link prvided by fractins between mathematics strands such as space and measurement, this area cntinues t present difficulties fr sme yung children in primary schls. Indeed, analysis f state-wide perfrmance f recent Year 5 students in New Suth Wales (NSW Department f Educatin, 1999) shws that students' understanding f and reasning with fractins and decimals is less than satisfactry. The situatin in NSW is similar t what has been bserved earlier in Victria (Clements and Del Camp, 1987) and in ther cuntries. Ellertn and Clements (1994) referred t the general area assciated with fractins in Australia as a "weeping sre in mathematics educatin." In cmmenting n the pr perfrmance in fractins amng American students Brwn et al (1988, p. 45) cmmented that the cmputatinal activity f these students seemed t be carried ut "withut develping the underlying cnceptual knwledge abut fractins." Taken tgether, these findings suggest that the expectatins held by mathematics curriculum develpers fr a sizeable grup f students in the upper primary schl are unrealistic. It is clear that a sizeable number f students will nt achieve the fllwing bjective: 'Students develp an understanding fthe parts f a whle, and the relatinships between the different representatins f fractins' (Bard f Studies - New Suth Wales, 1998:1). B. Bartn, K. C. Irwin, M. Pfannkuch, & M. O. 1. Thmas (Eds.) Mathematics Educatin 195 in the Suth Pacific (Prceedings fthe 5th annual cnference fthe Mathematics Educatin Research Grup faustralasia, Auckland, pp. 195-0). Sydney: MERGA. 00 MERGA Inc.
Chinnappan and Lawsn The findings fthe state-wide testing in NSW are frustrating. Many Year 5 students, despite having experienced the mathematics activities presented by their teachers, still have difficulty in using their knwledge f fractinal and decimal numbers t slve prblems that are regarded as being apprpriate nes fr that level f schling. Yet, in their wrk with Australian preschl children, Hunting and his clleagues shwed that while these yung children did nt usually understand fractins language, many fthem had develped knwledge abut sharing that wuld prvide a sund basis fr cnstructin f future knwledge f ratinal number (e.g., Hunting & Sharpley, 1988). It seems clear that between the time f develpment fthis rudimentary understanding and the later primary years many f these students d nt develp knwledge structures that will supprt the prblem slving with fractins and decimals that is demanded in Stage 3 f the K- curriculum and beynd. This area f mathematical learning amng yung children has received cnsiderable attentin frm researchers in recent years (e. g., Behr, Harel, Pst & Lesh, 199; Bezuk & Bieck, 1993; Batur & Cper, 1995; Owen & Super, 1993). Mre specifically, a significant bdy f research has fcused attentin n the develpment fratinal number knwledge frm children's understanding f whle numbers. Ratinal numbers are cmplex in character and prvide imprtant prerequisite cnceptual tls fr the grwth and understanding f ther number types and algebraic peratins in secndary schl. Yung children are expsed t these number cncepts at an early age in a variety f reallife situatins, such as measuring and dividing cntinuus quantities, and quantitative cmparisn ftw quantities. These experiences shuld enable yung children t develp an understanding f ratinal numbers which matures thrugh learning situatins they encunter in the classrm. As the students mve thrugh the primary years we might anticipate that there will be a gradual integratin f their understandings and the frmal representatins fmathematics. Almst tw decades ag Hiebert and thers suggested that we were being t ptimistic in anticipating such an integratin f frm and understanding (e.g., Hiebert, 1984, 1989). Hiebert argued that ne f the factrs cntributing t the pr levels f perfrmance fmany students n fractin and decimal prblems culd be represented as a prblem in establishing cnnectins between frm and understanding. One interest in this study was t see ifthis cnstructin fthe prblem was still valid. The multifaceted nature ffractin number has made the task fdescribing its grwth difficult. Several attempts have been made t capture the cmplexity f fractin numbers and children' cnstructin f these numbers. The mst detailed analysis f fractin numbers has been undertaken by Kieren (1988). His analysis shwed the fractin number knwledge cnsists fmany interwven strands. He identified eight, hierarchically related, levels in his descriptin ffractin number thinking. An imprtant utcme fthis mdel is the specificatin fcgnitive structures that prvide the basis fr the maturing ffractin numbers amng yung children. These structures r schemas which appear at levels three and fur cnsists f what he referred t as subcnstructs: partitining, unit frming, qutients, measures, rati and peratins. These subcnstructs amng thers playa key rle in yung children's understanding and interpretatin f fractins. The mdel shwn in Figure 1 builds n Kieren's representatin f fractin subcnstructs and als draws ut ptential links amng the cnstructs (Chinnappan, 000). These links are imprtant fr the analysis fthe rganisatinal quality fthe knwledge ffractins. 19
Year 3 Children's Understanding ffractins: Are we Making Prgress? dividing int equal parts part/whle ISubcnstruct A I (Partitining) equivalent fractins!!:. = a :b ~ -- I Subcnstruct C I b (Rati) all parts are equal FRACTIONI \ munberline I I subcnstruct B (Measure) I haifa cup Subcnstruct DI (Decimal) I applicatin cmparing tw quantities percentage place value Figure 1. Knwledge cmpnents in a Fractin schema. Althugh we d ultimately want t investigate the grwth in cmplexity f students' fractin knwledge ur cncern here was with the gathering f necessary preliminary infrmatin abut this knwledge. Our bjective here was t gain infrmatin abut hw well-prepared students in early primary schl might be fr a discussin abut fractins. Put mre crudely, we wanted t find ut hw well prepared these students might be t get int the 'game' f fractins. We wanted t knw if they had the cmmn language, diagrammatic and symblic vcabularies t participate in the discussins that wuld ccur in their classrms and whether this vcabulary was linked t well-develped schemas. Thus ur initial fcus was nt n the cmplexity f the students' structure f fractin knwledge per se. It was fcussed n the elements f the interactin that might typically ccur between teacher (r textbk) and student. Participants Methd Twenty fur children in Year 3 (aged between 8 and 9) frm a regular suburban schl participated in the study. The schl was situated in a middle-class suburb f a metrplitan city and was nt selective in its intake. The sample cntained 13 bys and 11 girls. The children had studied whle numbers and fractins within the Number Strand f the K- mathematics curriculum in the previus tw years f primary schl and at the time fthe study had cmpleted the tpic n fractins in Year 3. Tasks and Prcedure A range f tasks were develped fr the purpses f assessing children's knwledge abut prper fractins. One set ftasks required children t respnd t a series fquestins that fcussed n cmprehensin f simple sentences cntaining fractin wrds. The child's cmprehensin f the sentences was checked first and then they were asked t identify number wrds and thse wrds that referred t fractins. Their knwledge f the labels 197
Chinnappan and Lawsn used fr fractin wrds was als assessed. 'Half and 'quarter' were the fractin wrds used in this first set ftasks. A secnd set f tasks invlved children having t exhibit their understanding f fractins via diagrams. In this case the child was asked t represent the fractin wrds in diagrams and als t illustrate fractinal quantities in diagrams. The students were als asked express fractin wrds in mathematical symbls, and t display their knwledge f bth the size f different fractins and f equivalent fractins. In the final set f tasks, we attempted t assess children's understanding f the cncept 'fractin' and f the symbls that were used in representing fractins. Figure shws a selectin f prblems frm the seven sets ffractin tasks. Each child was interviewed individually and was asked t respnd t all the fcus questins and prblems. Children's respnses were als prbed t generate mre cmplete data abut their knwledge f fractins. All sessins were audi-taped and transcribed fr subsequent analysis. Set 1 'Fr yur breakfast yur mther filled half a glass with milk' What did yur mther d? Draw a picture t shw me what yur glass fmilk lked like when yur mther had finished putting the milk in it. Pick ut any wrds in the sentence that refer t numbers,r parts fnumbers. Des that number have a special name? What des 'a half'mean? Can yu write 'half' any ther way? Set Jhn cut a strip fpaper int five parts. His teacher said that each part is 'ne- fzfth '. Lk at this diagram fthe strip fpaper. Shw me hw much is nefzfth. Lk at this secnd diagram. Is the shadedpart equal t ne fzfth? Ifit isn't culd yu change the diagram s that nefifth was shaded? Set 3 In the number ~, what des, - and 5 mean? 5 Figure. Fractin tasks. Results and Discussin T facilitate the descriptin f students' task perfrmance we frmed the eight cmpsite measures shwn in Table 1. These measures summarise perfrmance acrss all the interview questins. 198
Year 3 Children's Understanding ffractins: Are we Making Prgress? Table 1 Definitins fcmpsite Measures Identificatin ffractin wrds: Knwledge flabels: Representing fractins in mathematical symbls: Meaning ffractin wrds: Representing fractins in diagram fnn: Meaning f alb symbls: Relating and rdering fractins: Explanatin ffractin rdering: Identificatin by the student fwrds that refer t numbers, including fractin wrds. Knwledge that the tenn 'fractin' referred t wrds such as 'half and 'quarter'. Ability t represent a fractin using alb fnnat. Knwledge fpartitining fa whle int equal parts Ability t represent fractins in simple diagrams r t mdify diagrams t represent fractins Knwledge fthe meaning attached t the alb representatin fa fractin Ability t write an equivalent fractin and t rder fractins Knwledge fhw t establish the relative sizes f different fractins The mean percentage scres fr the ttal grup f students n each f the cmpsite variables are shwn in Table. The figures in the table shw that there was a high degree f variability in the scres n each set f tasks. The students perfrmed best n the tasks requiring them t recgnize and label the simple fractin wrds in the sentences prvided as stimulus materials. Their knwledge f the meaning f fractin wrds and f hw t represent fractin wrds in bth diagrams and mathematical symbls was nly mderately gd. They were least successful n the tasks that required them t rder and relate fractins written in alb frm. Table Grup Scres (%) fr Cmpsite Measures (n=4) Measures Pssible Percentage scres Scre fr each student Mean SD Identificatin ffractin wrds 7.4 34.0 Knwledge flabels 8.8 33.4 Representing fractins in mathematical 4 58.3 45.8 symbls Meaning ffractin wrds 8 5.3 40.4 Representing fractins in diagram frm 1 54.9 8.1 Meaning falb symbls 43.8 45.0 Relating and rdering fractins 1.7 7.8 Explanatin ffractin rdering 8.3 4.1 Clearly, the large variatin in perfrmance indicates that the pattern f perfrmance shwn in Table hides much interesting detai1. Inspectin fthe perfrmance findividual students shws a very substantial disparity in perfrmance f students within this class. This disparity is illustrated mre fully in Table 3 where the patterns fperfrmance f fur students is shwn. 199
Chinnappan and Lawsn Table 3 Individual Scres ffur Studentsfr Task Sets Measure (pssible scre) Recgnitin ffractin wrds () Knwledge flabels () Representing fractins in mathematical symbls (4) Meaning ffractin wrds (8) Representing fractins in diagram frm (1) Meaning falb symbls () Relating and rdering ffractins () Explanatin ffractin rdering () S3 4 8 11 S3 4 8 1 4 Sl1 S17 Here we have tw pairs f Year 3 students with widely differing knwledge bases and very different capacities t effectively engage in discussins abut fractins. Students 3 and 3 perfrmed at a very high level and culd cpe with much f the fractins wrk being dne in Year 5 r classes. They have the vcabulary necessary t wrk n fractins prblems and have quite cmplex knwledge related t the partitining. Student 3, fr example, nt nly respnded crrectly t all questins abut language and symbls that were related t fractins but was als able t generate a crrect slutin t a prblem which required him t wrk ut the number fextra squares that had t be shaded s that i th fa 5 given rectangle was shaded. Student 3 was able t give an acceptable interpretatin f the alb frmat and crrectly rdered a set ffractins that included ne imprper fractin. Students 11 and 17 shwed very little evidence f even a rudimentary knwledge f fractins. Indeed their knwledge f even the language f fractins was limited, thugh they culd bth shw half a glass f milk in diagram frm. S17, fr instance, culd nt recgnise anything that referred t numbers in the sentence abut breakfast and milk (Figure : Set 1, Questin 4). It seems bvius that ifthese pairs fstudents are t develp mre pwerful knwledge structures in the shrt term they need different and differently paced curricula. The first pair, alng with several ther students in this class, need t be engaged in activities that mve them tward develpment f knwledge f the fractin subcnstructs in the bttm half f Figure 1. Fr this grup the expectatins in curriculum dcuments are far t lw. Fr example, accrding t the Outcmes and Indicatrs (Bard f Studies- New Suth Wales, 1998, p. 1) students in Stage f develpment (Years 3-4) are expected t 'represent numbers in tenths and hundredths using cncrete material and grids, and recrd these numbers in wrds'. While these are necessary skills in the develpment f fractin schema, students S3 and S3 have exceeded this requirement, and are in need f, and ready fr, mre cmplex representatins and peratins invlving fractins and decimals. We suggest that such students need expsure t subcnstructs C and D ffigure 1. The ther pair fstudents is representative fa grup that cnstituted abut a quarter f this class, all f whm had a prly develped understanding f the fractin cncept. Althugh sme fthese students d have sme fthe cmmn language and diagrammatic vcabulary, the incnsistency f their perfrmance indicated that even this vcabulary was nt strngly established. Students 11 and 17 are prly placed t get int the fractins 00
Year 3 Children's Understanding ffractins: Are we Making Prgress'! game. Once the teaching in class invlved mre than discussin f 'half, the perfrmance we bserved suggests that these tw students wuld struggle t make meaning. Neither student wuld be likely t have the vcabulary t create links between what they already knew and what might be invlved in discussin f 'ne fifth' r '~" There wuld be very little f Figure 1 that we culd cnfidently judge t have been established by these students. Cnclusin The patterns f perfrmance here prvide a smewhat mre detailed accunt that shws what might lie behind the patterns f perfrmance in state-wide testing. The results f this study shws that althugh the participants came frm ne Year 3 class there existed a wide disparity in their understanding and representatin f fractins. The perfrmance f many, if nt mst, f the students in this class was nly weakly related t the specificatins established fr their year level in curriculum dcuments. The results als suggest that the argument made by Hiebert (1984) abut students in the US still hlds fr many Australian students. In his analysis Hiebert fcussed n the prblems f establishing links amng a student's infrmal understanding and the frmal symbls and prcedures intrduced in the curse f mathematics classes. In this study we see strng evidence f prblems in the first fhiebert's sites f cnnectin, the cnnectin f understanding and the meaning f the symbls used in mathematics lessns. Hiebert's emphasis n the imprtance f cnnectins shuld make mre sense t us than it might have in 1984, r when Skemp (1971) made similar arguments at an earlier time. With ur current desire t place all student learning with a cnstructivist framewrk (e.g, the Suth Australian Curriculum Standards and Accuntability framewrk) the view that we shuld fcus n the building f links between frm and understanding seems all t bvius. Yet it seems that these links are nt being establishedbymany students. In a recent paper Cbb and Bwers (1999) als use the crude language fthe classrm 'game'. They remind us abut the cnsequences if students lack the necessary vcabulary t take part in the game. 'All students must have a way t participate in the mathematical practices f the classrm cmmunity. In a very real sense, students wh cannt participate in these practices are n lnger members fthe classrm cmmunity, frm a mathematical pint fview' (p. 9). We must admit that students like Students 11 and 17 in ur grup might well have been excluded frm the mathematical cmmunity in their classrm. And there is als the pssibility that Students 3 and 3 might 'leave' that mathematical cmmunity unless they were prvided with pprtunities t develp their quite pwerful understandings abut fractins. References Australian Educatin Cuncil (1990). Natinal statement n mathematics fr Australian schls. Carltn, Vic: Curriculum Crpratin. Batur, A. R., & Cper, T. (1995). Strategies fr cmparing decimal numbers with the same whle-part number. In B. Atweh, & S. Flavel (Eds.), Prceedings fthe 18 th annual cnference fthe Mathematics Educatin Research Grup faustralasia. Darwin, NT: Nrthern Territry University. 01
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