Students Knowledge and Strategies for Solving Equations

Transcription

1 Studets Kowledge ad Strategies for Solvig Equatios Chris Lisell Uiversity of Otago College of Educatio This paper presets results from the secod year of a ivestigatio ito studets algebraic thikig. The assessmet techiques parallel those used for ivestigatig umber i the Numeracy Developmet Projects. I this study of 621 year 7 10 studets, oral iterviews with supplemetary questios were used to ivestigate the strategies that studets used to solve equatios. Basic facts, umeracy strategy stage, ad kowledge of aspects of algebra were also assessed. Rasch aalysis was used to ivestigate the difficulty of the equatios ad studet ability, ad the the strategies associated with each questio were examied. The data provides strog evidece that there is a hierarchy of sophisticatio of strategies. A large umber of the studets were uable to solve may of the equatios because they were restricted to less sophisticated strategies. Clear relatioships were foud betwee the most sophisticated strategy a studet used ad their umeracy stage, basic facts kowledge, ad algebraic kowledge. Itroductio The success of the Numeracy Developmet Projects (NDP) i raisig New Zealad studets achievemet i umber (Thomas & Tagg, 07; Youg-Loveridge, 07) has prompted iitiatives to exted the NDP ito early algebra. The NDP are cetred o the Number Framework (Miistry of Educatio, 03), which describes the progressio of studets arithmetical strategies ad the kowledge associated with these strategies. The study reported o here examies studets strategies for solvig liear equatios ad the relatioship of these strategies to umeracy ad to algebraic kowledge. The goal is to cotribute to a research base that may allow the developmet of a algebra framework. Backgroud May studets struggle with itroductory algebra, ad teachers have little to guide them i desigig programmes of learig. Little is kow about the effect of studets umeracy o the learig of early algebra or about the strategies that studets use to solve equatios. However, a useful summary of strategies used by studets is provided by Kiera (1992), who describes the use of kow basic facts, coutig techiques, guess-ad-check, cover-up, workig backwards, ad formal operatios. I this paper, Kiera s classificatio of strategies is modified ad exteded ad trasformatios is the term used to describe Kiera s formal operatios. There is a wealth of research o studets errors ad miscoceptios i algebra, some of which was summarised i the 07 fidigs from the Secodary Numeracy Project (SNP) (Lisell, 08). These difficulties iclude uderstadig of arithmetical structure, iverse operatios, algebraic otatio ad covetios, operatig o ukows, lack of closure, the equals sig, ad treatig equatios as processes rather tha objects. It ca be argued that much of the research adopts a deficit model approach, detailig what studets ca t do. A alterative approach is to ivestigate what studets actually do whe preseted with equatios to solve ad to examie the kowledge ad skills associated with the various strategies that differet studets use. Teachig how to solve equatios has traditioally focused o the type of equatios preseted to studets rather tha o the strategies that they are usig. If a studet is successful at solvig a give 29

2 Fidigs from the Secodary Numeracy Project 08 type of equatio, the teacher will ofte preset them with harder equatios, irrespective of whether the studet is solvig by, for example, guess-ad-check or workig backwards. To move to a approach more cosistet with the NDP, i which studets are taught accordig to the highest strategy they have available to them, more iformatio is eeded about how studets solve equatios. The research questios addressed i this study were: What is the relative difficulty of equatios? What strategies do studets use, ad is there a hierarchy of strategies? What is the impact of cotext o studets solutio strategies? What prerequisite kowledge is associated with each strategy? What stage of umeracy is associated with each strategy? Method This study examied relatioships betwee studets strategies for solvig equatios ad their umeracy stage, basic facts kowledge, algebraic kowledge, ad whether the equatios were symbolic or i cotext. A structured diagostic iterview was admiistered to idividual studets by the researcher or by the studets classroom teacher. The studets resposes were coded ad the aalysed makig use of Rasch aalysis (Wright & Masters, 1982). Algebraic kowledge was assessed by a writte test, while umeracy stage ad basic facts kowledge were assessed through routie procedures i place i the schools. Further details of the assessmet tools are provided by Lisell (08). Subjects The study took place i two itermediate schools (years 7 ad 8), two high schools (years 9 ad 10 oly) ad oe college (years 7 9 oly). There was o attempt at represetative samplig, but istead the aim was to collect data from a wide rage of studets. The iterview was admiistered to a total of 621 studets i year 7 ( = 196), year 8 ( = 3), year 9 ( = 25), ad year 10 ( = 137). Clearly, year 8 studets are uderrepreseted, but this is mitigated by the fact that iterviews took place throughout the school year, so studets at the begiig of year 9 ad the ed of year 7 were icluded. I the two schools i which there was streamig, all classes from each year level were icluded ad i all schools, o studets were excluded o the basis of ability. Diagostic Iterview The diagostic iterview was developed i a previous study (Lisell, McAuslad, Bell, et al., 06) ad was guided by the literature o studets strategies for solvig equatios (Herscovics & Lichevski, 199; Kiera, 1992). The iterview cosisted of a series of icreasigly complex equatios, which the studets were asked to solve alog with a explaatio of their thikig. The series icluded 12 pairs of parallel questios: oes that were i cotext (that is, word problems) ad oes that were purely symbolic. The questios were preseted o cards so that the more difficult questios could be omitted as required without suggestig to the studet that they were ot copig. Each questio was read to the studet to miimise the impact of readig difficulties, icludig difficulties with readig symbolic equatios. Calculators ad pecil ad paper were available for the studets to use, but it was stressed to the studets that they could use whatever method they chose. The iterviewer recorded what the studet did ad said ad the classified the strategy used accordig to Table 1. Note that there has bee a chage i termiology sice last year s report (Lisell, 08): strategy i, previously 30

3 Studets Kowledge ad Strategies for Solvig Equatios called formal operatios/equatio as object, is ow called trasformatios/equatio as object. This is i order to clarify that the strategy ivolves trasformig a equatio ito a ew equatio oe or more times ad is ot simply the followig of some give formal procedure. Table 1 Classificatio of Strategies for Solvig Equatios Code Strategy 0 Uable to aswer questio a b c d e f g h i Kow basic facts Coutig techiques Iverse operatio Guess-ad-check Cover up Workig backwards, the guess-ad-check Workig backwards, the kow fact Workig backwards Trasformatios/equatio as object Kowledge Test The assessmet of algebraic kowledge was admiistered as a writte test because supplemetary questios were ot required. The areas ivestigated i this sectio were: kowledge of covetios ad otatio, uderstadig of the equals sig, uderstadig of arithmetical structure, uderstadig of iverse operatios, ad acceptace of lack of closure. Numeracy Assessmet All the schools i this study were either NDP or SNP schools ad therefore routiely collected umeracy data o their studets. I istaces where this data was ot available, stage of umeracy was assessed usig a modified GloSS (Global Strategy Stage) ad kowledge of basic facts was assessed usig a modified sectio from NumPA (the Numeracy Project Assessmet tool). Data Aalysis The first stage i the aalysis was to determie the difficulty of the equatios ad the ability of the studets. I Rasch models, the probability of a specified respose (that is, a right/wrog aswer) is modelled as a logistic fuctio of the differece betwee the perso ad the item parameter. Before applyig this model to the data, it was therefore ecessary to ascertai that the variable of item difficulty was uidimesioal. Factor aalysis was iitially employed to verify that a oe-factor model was a adequate fit to the data from the strategy iterview. Followig this, Rasch aalysis was used to determie item difficulty ad studet ability. These scores were the related to the strategies that idividual studets used for each questio. A proposed hierarchy of strategies was the developed by examiig the distributios of ability of studets, usig each strategy o each questio. This permitted each studet to be classified accordig to the most sophisticated strategy they used o ay questio. From the umeracy assessmet ad algebraic kowledge test, each studet was assiged a score for umeracy, basic facts, covetios ad otatio, equivalece, arithmetical structure, iverse operatios, 31

4 Fidigs from the Secodary Numeracy Project 08 ad lack of closure. Relatioships betwee each of these measures ad the most sophisticated strategy used were the examied. Item Difficulty Results For equatios that were preseted symbolically, there was a huge variatio amog the studets i the umber of equatios that they were able to solve, with some questios beig much harder tha others (see Table 2). Table 2 Item Difficulty (Symbolic Equatios) Number of Studets Percetage of Studets Equatio with Correct Resposes with Correct Resposes Rasch Score (Item Difficulty) 3 = = = = = = = = = = = = Rearrage v = u + at I geeral, oe-step equatios were easier to solve tha two-step equatios, which i tur were easier to solve tha equatios with ukows o both sides. However, it should also be oted that equatios ivolvig divisio were harder to solve tha similar equatios with other operators. Nevertheless, oe-step equatios ivolvig divisio were easier to solve tha two-step equatios ivolvig divisio. Strategies Used Two strategies (trasformatios ad guess-ad-check) could be used to solve ay equatio, while others (for example, iverse operatio for oe-step equatios, workig backwards for two-step equatios) could be used for oly a limited umber of equatios. For every equatio (except for the fial oe) there was a rage of strategies successfully used by studets, but the distributio of strategies varied from questio to questio. Resposes to three questios are show i Figure 1 to illustrate the rages of strategies used. 32

5 Studets Kowledge ad Strategies for Solvig Equatios + 6 = Frequecy No successful Coutig Iverse Guess- Trasformatios strategy techiques operatio ad-check Strategy used 3 8 = Frequecy No successful Guess- Cover Workig Workig Workig Trasformatios strategy ad-check up backwards backwards backwards the guess- the kow ad-check fact Strategy used 5 2 = Frequecy No successful strategy Guess-ad-check Trasformatios Strategy used Figure 1. Studets strategies for three equatios 33

6 Fidigs from the Secodary Numeracy Project 08 Hierarchy of Strategies To establish a hierarchy of strategies was ot straightforward because the patter of strategy use varied from equatio to equatio, with some equatios ledig themselves to beig solved by oe strategy rather tha by aother. Aother difficulty was that able studets ofte reverted to guessad-check for difficult questios, eve though they used other strategies for easier equatios. Less able studets, i cotrast, used guess-ad-check for easy equatios ad were uable to solve more difficult equatios by ay strategy. Therefore, the approach used was to examie the strategies used o a questio-by-questio basis. For each questio, the ability of studets usig a particular strategy was ivestigated. Figure 2 shows the results 1 for the same three equatios show i Figure = Ability of studets (logits) No successful Coutig Iverse Guess- Trasformatios strategy techiques operatio ad-check Strategy used 3 8 = Ability of studets (logits) No successful Guess- Cover Workig Workig Workig Trasformatios strategy ad-check up backwards backwards backwards the guess- the kow ad-check fact Strategy used 1 Rasch aalysis calculates item difficulty ad studet ability o the same scale. 3

7 Studets Kowledge ad Strategies for Solvig Equatios 5 2 = Ability of studets (logits) No successful strategy Guess-ad-check Trasformatios Strategy used Figure 2. Ability of studets usig each strategy For each equatio, it was the possible to place the strategies i order accordig to the mea ability of studets usig each strategy. For example, the equatio 5 2 = was solved usig either guessad-check or trasformatios. The mea ability of studets usig trasformatios was higher tha that of studets usig guess-ad-check, idicatig that trasformatios was the more sophisticated strategy. The picture that emerged usig this approach was fairly cosistet, i that the order is the same for early all the equatios (see Table 3, i cojuctio with the explaatio of umbers i Table ). Table 3 Rak Order of Mea Abilities of Studets Usig Each Strategy for Each Equatio Strategy a b c d e f g h i Equatio 3 = = = = = = = = = = = = v = u + at 1 35

8 Fidigs from the Secodary Numeracy Project 08 For all equatios (except for = 5), trasformatios was the strategy used by the most able studets ad guess-ad-check by the least able. For oe-step equatios, it was ot possible to discer betwee coutig strategies ad kow basic facts because, although studets used a rage of strategies over all the equatios, for ay particular equatio this rage ever icluded both coutig ad kow basic facts. However, for three of the four oe-step equatios, iverse operatios were used by the more able studets rather tha by studets usig either coutig strategies or kow basic facts. The exceptio was = 5, which was far more difficult tha the other oe-step equatios. For this equatio, the most able studets solved it usig a kow basic fact. Cover up was used by such a small umber of studets that o clear relatioship to the other strategies emerged. For five of the six two-step equatios, workig backwards was used by the more able studets rather tha by those usig workig backwards, the kow fact, which i tur was used by the more able studets rather tha by those usig workig backwards, the guess-ad-check. The exceptio was + 12 = 18, but the umber of studets usig ay strategy other tha workig backwards was too small to draw ay coclusios. The strategies used oly o oe-step equatios clearly could ot be compared directly with those used oly o two-step equatios. However, two-step equatios are much harder tha oe-step, ad workig backwards ivolves usig iverse operatios. The order of sophisticatio of strategies idicated by this aalysis (see Table 3) is show i Table. Table Rak Order of Strategies Rak Strategy 1 No successful strategy 2 Guess-ad-check 3 Coutig techiques / Kow basic facts Iverse operatios 5 Workig backwards, the guess-ad-check 6 Workig backwards, the kow fact 7 Workig backwards 8 Trasformatios The studets were the classified accordig to the most sophisticated strategy they used o ay symbolic equatio (see Figure 3). 300 Frequecy Noe Guess- Coutig Iverse Workig Workig Workig Trasformatios ad-check or basic operatios backwards backwards backwards facts (guess- (kow fact) ad-check) Highest strategy 36 Figure 3. Most sophisticated strategy used o ay symbolic equatio

9 Studets Kowledge ad Strategies for Solvig Equatios The umber of studets able to solve equatios by performig trasformatios was very low (38). It is also worth otig that a sigificat umber of studets (89) were ot eve able to use iverse operatios. Also of ote is the umber of studets (65) who could solve some two-step equatios but who were ot fully workig backwards. Impact of Cotext The picture that emerged i relatio to studets solutios of equatios that were give to them as word problems i cotext was very similar to that of symbolic equatios. The diagostic iterview icluded 12 pairs of parallel questios. Table 5 shows the item difficulty of each questio. Note that the table shows the structure of the word problems, but the problems read to studets did ot iclude these equatios. For example, the first problem (show i the table as 5 = 17) was I left home this morig with some moey, spet $5, ad have $17 left. How much did I start with? Table 5 Item Difficulty (Rasch Scores) Symbolic I cotext Symbolic I cotext Equatio % correct Item difficulty % correct Item difficulty 1 3 = 12 5 = = 3 2 = = = = 5 = = = = = = = = = = = = = = 2 = Rearrage v = u + at (v + u)t Rearrage s = The harder equatios (with ukows o both sides) were slightly easier whe preseted symbolically. However, early all oe-step ad two-step equatios were easier whe preseted i cotext. There was oe exceptio (the secod pair of equatios) ad a few dramatic differeces i difficulty that will be commeted o whe the strategies that studets used are examied. For all of the oe-step equatios that were i cotext, compared with their symbolic couterparts, there was a greater use of iverse operatios tha of coutig strategies, kow facts, or guess-ad-check. This was true eve for the secod pair of equatios, i which the order of difficulty was reversed. The cotext was I have 2 CDs. This is three times as may as my brother has. How may CDs does he have? May more studets tha expected got this wrog by multiplyig 2 by 3 rather tha dividig. It would appear that the words three times cofused them. For the fourth, seveth, ad ith pairs of equatios, studets foud the symbolic form much more difficult tha oe might expect. All these equatios ivolved a divisio structure. The cotexts for 37

10 Fidigs from the Secodary Numeracy Project 08 the fourth ad seveth were: Whe I shared a packet of lollies roud my class of studets, they got each. How may lollies were i the packet? ad Our kapa haka group is made up of some Màori studets ad 11 Pàkehà studets. The whole group is divided ito equal-sized groups for practices. Each of the practice groups has 19 studets i it. How may Màori studets are there i our kapa haka group? It would appear that studets saw these as multiplicatio rather tha divisio problems ad foud them much easier tha the symbolic equivalets. For all of the two-step equatios, there was a greater proportio of studets usig the workig backwards strategy for equatios that were i cotext compared with those usig this strategy for symbolic equatios. Coversely, there was a greater proportio of studets usig less sophisticated strategies for symbolic equatios compared with those usig these strategies for equatios i cotext. There was also a small umber of studets who used trasformatios for two-step symbolic equatios. Prerequisite Skills ad Kowledge The relatioships betwee each studet s prerequisite kowledge ad skills ad the most sophisticated strategy that they were able to employ was the ivestigated. For these aalyses, those studets whose most sophisticated strategy was guess-ad-check, coutig, kow basic fact, or o successful strategy were grouped together as primitive strategies. Those studets whose most sophisticated strategy was either workig backwards, the guess-ad-check, or workig backwards, the kow fact, were grouped together as partially workig backwards. The relatioship betwee umeracy strategy stage ad highest algebraic strategy is show i Figure Highest algebraic strategy Primitive Iverse operatios Partial workig backwards Workig backwards Trasformatios Frequecy Coutig Advace Early Advaced Advaced Advaced strategies coutig additive additive multiplicative proportioal Numeracy strategy stage Figure. Relatioship betwee studets umeracy ad their most sophisticated strategy for solvig equatios It is clear that studets with poor umeracy skills were largely restricted to less sophisticated strategies. The great majority of studets with good umeracy skills were able to use more sophisticated strategies, with very few of them usig oly less sophisticated strategies. A very similar picture to this emerged for studets kowledge of basic facts. Studets with poor kowledge of basic facts 38

11 Studets Kowledge ad Strategies for Solvig Equatios were largely restricted to less sophisticated strategies, while most studets with good kowledge of basic facts were able to use the more sophisticated strategies. The relatioship betwee uderstadig of arithmetical structure ad highest algebraic strategy is show i Figure Highest algebraic strategy Primitive Iverse operatios Partial workig backwards Workig backwards Trasformatios Frequecy Very poor Poor Average Good Uderstadig of arithmetical structure Figure 5. Relatioship betwee studets uderstadig of arithmetical structure ad their most sophisticated strategy for solvig equatios Studets uderstadig of arithmetical structure had a dramatic impact o the most sophisticated strategy they were able to use. As their uderstadig of arithmetical structure icreased, so did their use of more sophisticated algebraic strategies. The pictures for uderstadig of iverse operatios, acceptace of lack of closure, ad uderstadig of equivalece were very similar. The oly area of kowledge that did ot have such a clear relatioship with sophisticatio of algebraic strategy was algebraic otatio ad covetio. Studets with good kowledge i this area used slightly more sophisticated strategies, but the relatioship was ot so covicig. Of particular ote is the impact of studets uderstadig of equivalece ad, to a slightly lesser extet, their acceptace of lack of closure (see Figure 6). 39

12 Fidigs from the Secodary Numeracy Project Highest algebraic strategy Primitive Iverse operatios Partial workig backwards Workig backwards Trasformatios Frequecy Very poor Poor Average Good Very good Excellet Uderstadig of equivalece Figure 6. Relatioship betwee studets uderstadig of equivalece ad their most sophisticated strategy for solvig equatios As studets uderstadig of equivalece icreased, the ratio of the umber of studets usig trasformatios to those usig workig backwards also icreased. Of those studets with a excellet uderstadig of equivalece, there was a greater umber usig trasformatios tha usig workig backwards. This differece i proportio betwee those usig workig backwards ad those usig trasformatios was ot so marked for the other areas of prerequisite kowledge ivestigated, although it was also large for acceptace of lack of closure. Discussio This study has demostrated how icredibly difficult some equatios are to solve compared with others. It has bee kow for a log time that studets fid equatios with ukows o both sides very difficult (Herscovics & Lichevski, 199), ad Sfard (1991) has eve suggested that viewig equatios as objects may be beyod the grasp of may studets. However, this study is the first to compare item difficulty o a Rasch scale. Rasch scores of item difficulty follow a approximately ormal distributio, ad to fid that some items have a difficulty score over two stadard deviatios above the mea cofirms their difficulty. The New Zealad Curriculum achievemet objectives (Miistry of Educatio, 07) give little idicatio of this difficulty. At level, the relevat objective is Form ad solve simple liear equatios, ad at level 5, it is Form ad solve simple liear ad quadratic equatios. The level 3 objective of Record ad iterpret additive ad simple multiplicative strategies, usig words, symbols, ad diagrams, with a uderstadig of equality is useful i focusig o equivalece, but it does ot spell out that studets eed to fid ukow values i ay positio i a statemet of equivalece. Educators eed to appreciate how huge the rage of difficulty is ad ot trivialise the solvig of equatios dow to a few lessos o specific procedures. Although equatios i cotext were geerally foud to be slightly easier tha symbolic equatios, the geeral picture for item difficulty was very similar whe the two sets of parallel questios were compared. This demostrates that it is primarily the structure of problems that makes them difficult, 0

13 Studets Kowledge ad Strategies for Solvig Equatios ot the algebraic symbolism. This fidig is backed up by the lack of a strog relatioship betwee studets uderstadig of algebraic otatio ad their most sophisticated algebraic strategy. Teachig of algebra should therefore be focusig o the structure of problems. The additioal difficulties that studets experieced with equatios ivolvig divisio highlight the eed for greater emphasis to be placed o a variety of operators withi both oe-step ad two-step equatios. Isights ito why studets foud some equatios so difficult to solve were obtaied by examiig the strategies they employed to solve them. It was clear that studets were usig a wide variety of strategies to solve all the equatios. For oe-step equatios, may studets were gettig correct solutios but ever used a iverse operatio. To attempt to move these studets o to two-step equatios would be courtig disaster. Iverse operatios are ivolved i all the successful strategies for two-step equatios other tha guess-ad-check. Studets are therefore likely to be reduced to usig guess-ad-check or learig a specific procedure that applies oly to specific structures. Similarly, may studets were obtaiig correct solutios to some two-step equatios but were oly partially usig workig backwards or were eve usig guess-ad-check. This poit, that the strategy of workig backwards is less homogeeous tha previously reported, is importat. May studets are oly just graspig the strategy ad ca use it oly whe the first step reveals a kow basic fact to them for the ext step. These studets use the strategy of workig backwards, the kow facts. Other studets are preveted from fully usig workig backwards because of lack of kowledge of multiplicatio ad divisio facts. These studets use the strategy of workig backwards, the guessad-check. To attempt to move these studets o to usig trasformatios would almost certaily be premature. Whe teachers suggest to studets that a equatio is like a balace pivoted about the equals sig, it is hard to imagie why it would be difficult to do the same thig to both sides of the equatio ad keep it balaced. However, this study has cofirmed just how difficult the strategy of usig trasformatios is for studets. Usig trasformatios requires seeig a equatio as a object to be acted o (Sfard, 1991), but it is clear that most studets see equatios as processes. I order to ivestigate the relative sophisticatio of strategies, it was assumed that the more sophisticated strategies would be chose by the more able studets. The fidigs usig this approach were cosistet with the fact that there was oly oe studet who was able to perform trasformatios but could ot also work backwards (this low-ability studet appeared to be followig a procedure of doig the same thig to both sides for simple equatios) ad that all studets who could work backwards could also use iverse operatios. This study has show that the strategy of solvig oestep equatios by iverse operatios is used by the more able studets rather tha by those who use either kow basic facts or coutig strategies. These strategies i tur were used by the more able studets rather tha by those who solved oe-step equatios usig guess-ad-check. Similarly, the strategies of solvig two-step equatios by partially workig backwards were used by less able studets rather tha by those fully workig backwards. O ay particular equatio, studets chose a strategy that was sufficiet to solve the equatio rather tha usig their most sophisticated strategy. However, it is suggested that the differet strategies are ot merely a matter of choice but that the most sophisticated strategy that a studet ever uses is idicative of coceptual developmet. The impact of cotext o studets ability to solve equivalet problems was very iterestig. I geeral, cotext problems were foud to be easier tha symbolic problems util the difficulty level of ukows o both sides was reached. However, most school programmes focus o teachig skills for solvig symbolic equatios. Solvig word problems is usually regarded as harder ad itroduced later as a applicatio of these skills. A alterative perspective o cotexts is to view them as models of the mathematics. Models are a importat feature of Realistic Mathematics Educatio (RME) (Gravemeijer, 1997). Traditioally, models are derived from formal mathematics, whereas i RME, models are derived from real situatios that studets have experieced ad are chose to reflect 1

14 Fidigs from the Secodary Numeracy Project 08 the iformal strategies of studets. Iitially, a model of a situatio that is familiar to the studets is used. Next, through geeralisig ad formalisig, the model becomes a etity i its ow right. Fially, it becomes possible to use the model for mathematical reasoig. Gravemeijer describes this as a trasitio from model-of to model-for. The ature of a model therefore evolves from beig highly cotext-specific to derivig its meaig from a mathematical framework. I cotrast, whe pre-existig models are give to studets to help them solve problems, the studets are expected to use them i prescribed ways that may ot be clear to them. The results from this study are cosistet with Gravemeijer s perspective ad suggest that algebra would be better itroduced i cotext rather tha just as symbols. The impact of cotext o the strategies that studets used may help to explai why studets foud these problems easier. For oe-step equatios, there was much higher use of iverse operatios tha of less sophisticated strategies. It appears likely that cotexts allow studets to perceive the structure of a problem i more tha oe way. For example, the problem Whe I shared a packet of lollies roud my class of studets, they got each. How may lollies were i the packet? has the structure =, but may be viewed as The umber of lollies is for each of the studets, with a structure of = x. The cotext is therefore aturally leadig the studet ito a iverse operatio. If this is the case, the the role of the teacher should be to scaffold the writig of symbolic equatios to describe cotexts ad the to explore ad symbolise the solutio strategies of the studets. There was a high correspodece betwee umeracy strategy stage ad the most sophisticated strategy a studet was able to use to solve equatios. Oly for studets who were at the advaced multiplicative or advaced proportioal thikig stages did the majority solve equatios by usig workig backwards or trasformatios. Studets at lower stages of umeracy were largely restricted to less sophisticated strategies. The fidigs from this study strogly suggest that prerequisite umeracy should be cosidered whe desigig teachig programmes for algebra. However, there were studets who did ot score highly o GloSS but were able to use sophisticated strategies for solvig equatios. These studets were ivariably efficiet at usig algorithms for computatios ad ofte came from primary schools that did ot promote NDP umeracy. The algebra diagostic tool may be more useful tha GloSS for revealig the thikig of studets at the upper ed of the Number Framework. This is because GloSS focuses o metal strategies (ad does ot value the use of algorithms), whereas the algebra tool has a focus o studets uderstadig of mathematical structure. There was a very strog relatioship betwee studets kowledge of basic facts ad their highest algebraic strategy. Ay studet who was at stage 6 or below o the Number Framework for basic facts was ulikely to be able to solve equatios by workig backwards or by trasformatios. This fidig emphasises the critical importace of istat recall of all basic facts, icludig multiplicatio ad divisio. There were also strog associatios betwee studets highest algebraic strategies ad their uderstadig of arithmetical structure, iverse operatios, lack of closure, ad equivalece. There was ot such a strog associatio betwee studets highest algebraic strategies ad their kowledge of algebraic covetios ad otatio. The relatioship betwee studets highest algebraic strategy ad their uderstadig of equivalece was particularly iterestig. Uderstadig of equivalece ad, to a lesser extet, acceptace of lack of closure had much higher impacts o whether a studet could use trasformatios compared with usig the strategy of workig backwards tha did the other areas of algebraic kowledge. Give the reasoably large umber of studets who could work backwards ad the very small umber who could use trasformatios, these fidigs may have sigificat implicatios for teachig. 2

15 Studets Kowledge ad Strategies for Solvig Equatios Coclusios Cosistet with the perspective of Filloy ad Sutherlad (1996), it is suggested that the strategies described i this study are ot simply alterative approaches to solvig equatios but represet differet stages of coceptual developmet. Istead of lookig at how hard equatios are to solve ad whether studets get them right, it appears to be more useful to look at the strategies that studets use. The approach used i this study is very similar to that used i the NDP, with strategy beig separated out from the kowledge required for strategy use. This approach allows the classificatio of the studets accordig to their most sophisticated strategy rather tha by the most difficult equatio they are able to solve. Withi umeracy teachig, studets are grouped for istructio accordig to their most sophisticated strategy. It is suggested that a similar approach to groupig studets is likely to be beeficial for teachig studets to solve equatios. Also cosistet with the NDP, the teachig of prerequisite kowledge eeds to be addressed. To solve oe-step equatios, studets eed to uderstad iverse operatios ad to kow their basic facts. To solve two-step equatios by workig backwards, studets also eed to uderstad arithmetical structure. To solve equatios by usig trasformatios, studets eed to uderstad equivalece ad also accept lack of closure. The third area of fidigs cosistet with the NDP cocers the role of cotext. Questios that were i cotext were easier tha equivalet symbolic questios. This suggests that a versio of the umeracy teachig model should be employed for teachig algebra. We should start with cotexts that are meaigful to studets, preferably ivolvig cocrete materials. Teachers should scaffold studets so that they ca symbolise the structure of the problems ad their solutio strategies before expectig them to visualise a cocrete represetatio ad fially to operate o abstract symbolic structures. Refereces Filloy, E., & Sutherlad, R. (1996). Desigig curricula for teachig ad learig algebra. I A. Bishop, K. Clemets, C. Keitel, J. Kilpatrick, & C. Laborde (Eds), Iteratioal hadbook of mathematics educatio (Vol. 1, pp ). Dordrecht: Kluwer. Gravemeijer, K. (1997). Istructioal desig for reform i mathematics educatio. I M. Beishuize, K. Gravemeijer, & E. Va Lieshout (Eds), The role of cotexts ad models i the developmet of mathematical strategies ad procedures (pp. 13 3). Utrecht: Freudethal Istitute. Herscovics, N., & Lichevski, L. (199). A cogitive gap betwee arithmetic ad algebra. Educatioal Studies i Mathematics, 27(1), Kiera, C. (1992). The learig ad teachig of school algebra. I D. A. Grouws (Ed.), Hadbook of research o mathematics teachig ad learig (pp ). New York: Macmilla. Lisell, C. (08). Solvig equatios: Studets algebraic thikig. I Fidigs from the New Zealad Secodary Numeracy Project 07 (pp. 39 ). Welligto: Learig Media. Lisell, C., McAuslad, E., Bell, M., Savell, J., & Johsto, N. (06). Usig actio research to lear about learig algebra. Paper preseted at the New Zealad Associatio for Research i Educatio, December, 06. Miistry of Educatio (03). The Number Framework. Welligto: Miistry of Educatio. Miistry of Educatio (07). The New Zealad Curriculum. Welligto: Miistry of Educatio. Sfard, A. (1991). O the dual ature of mathematical coceptios: Reflectios o processes ad objects as two sides of the same coi. Educatioal Studies i Mathematics, 22, Thomas, G., & Tagg, A. (07). Do they cotiue to improve? Trackig the progress of a cohort of logitudial studets. I Fidigs from the New Zealad Numeracy Developmet Projects 06 (pp. 8 15). Welligto: Learig Media. Wright, B. D., & Masters, G. N. (1982). Ratig scale aalysis: Rasch measuremet. Chicago: MESA. Youg-Loveridge, J. (07). Patters of performace ad progress o the Numeracy Developmet Projects: Fidigs from 06 for years 5 9 studets. I Fidigs from the New Zealad Numeracy Developmet Projects 06 (pp ). Welligto: Learig Media. 3