A profile of Irish second year post-primary students knowledge of

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TeMr 5osta5riJMrT stujents3

finoIlejge of initiMl Malge6rM

8r. moife F3"rien| 8r. w4ire 2p 7porj4in

GMlIMTawMTo ynstitute of DechnologT| 9niKersitT :ollege :orfi

8eceJ6er dxdN

School of Engineeraing| GwyD

EaJMilA Moife.o6rien1gJit.ie

School of EjucMtion| 9::

EJMilA JMire.niriorjMin1ucc.ie

chol hof eE|lGMcIETochleawhoiyToIceasMEGaetcuGlDfEM.EclnciGiaiTfcTf.EAoTc

8r w4ire 2p 7porj4in 6Msej on 8r F3a"rien3s joctorMl thesis. ms M lifelong JMtheJMtics

ejucMtor Mnj reseMrcher Iho is 5MssionMte M6out iJ5roKing Mll Ms5ects of aJMtheJMtics0SDEw teMching Mnj leMrniang through eKijencea6Msej reseMrch| y MJ jelightej to 6e MssociMtej Iith the Muthors3 JMqor eaJ5iricMl stujT on 5osta5riJMrT school Mlgae6rM in yrelMnj. wT brst encounter Iith the Muthor| 8r moife F3"rien| IMs Mt M JMtheJMtics ejucMtion conference Py:wEaNB&. -er su5erKisor Manj Jentor| 8r w4ire 2p 7porj4in| M Ionjerful colleMgue Mnj frienj| introjucej us. 8r F3"rien IaMs then M lecturer in JMtheJMtics Mt GMlIMTawMTo ynstitute of DechnologT PGwyD& Mnj continues in this 5ost tojMT. She hMj qust stMrtej her McMjeJic qourneT Ms M joctorMl stujent Iith JT colleMgue| reseMrching the leMrning aMnj teMching of Mlge6rM in yrish 5osta5riJMrT schools. ya IMs eVcitej Ihen y leMrnej athe eVMct nMture of her 5roqect. yt M55eMrej to intersect Mnj joKetMil Iith tIo joctorMl stujies thMt y su5erKisej Mt EHyLSDEw| Dhe 2MtionMl :entre for SDEw EjucMtion Mt the 9niKersitT of WiJericfi| one coJ5letej Mnj the other onagoing Mt the tiJe. Su6se%uentlT| y stMTej in touch Iith her 5roqect Mnj IMs jelightej to serKe on the Muthor3s eV5ert 5Mnel Mnj offaer MjKice on the jeKelo5Jent of her Mlgae6rM screener| M Ionjerful tool for JMtheJMtics teMchers. :oJ5etencT in Mlge6rM Mnj Mlge6rMic thinfiing Mre core concerns in Mll reforJ curriculM internMtionMllT inclujing athe neI JMtheJMtics curriculuJ in yraish 5osta5riJMrT schools. mlge6rM is chMllenging for stujents to leMrn Mnj for teMchers to teMch Mt 5riJMrT Mnj 5osta

5riJMrT leKel| Mnj this reMlitT is uniKersMllT recognisej. Dhis stMte of MffMirs is unjerlinej

6T 5ersistent unjer5erforJMnce in Mlge6rM in stMte eVMJinMtions Ms re5ortej 6T the E7:|

8ES| :hief EVMJiner3s re5orts| Mnj internMtionMl surKeTs Ihere yrish schoolchiljaren hMKe

5Mrtici5Mtej. 8etMils of thesea Mnj other reseMrch Mre contMinej in the re5ort3s eVtensiKe

6i6liogrM5hT.

Such re5orts Mre eVcellent Mnj necessMrT in IhMt theT jo to highlight issues Iith Mlge6rM Mnj JMtheJMtics ejucMtion generMllT/ hoIeKer| theT jo not offer solutions for unjer5erforJMnce.

9nfortunMtelT| there is no JMgic IMnj to conqure u5 M solution nor ais it lifielT thMt there is M

single solution. -oIeKer| eKijencea6Msej eJ5iricMl raeseMrch such Ms thMt unjertMfien 6T the Muthors Mnj re5ortej here is essentiMl if Ie Mre to unjerstMnj o6stMcles to 5rogress Mnj chMrt

5MthIMTs to iJ5roKej 5erforJMnce. ¤e Mre KerT fortunMte thMt Ie hMKe in yrelMnj Mt this tiJe|

M Ki6rMnt coJJunitT of eV5ert reseMrchers in wMtheJMtics0SDEw ejucMtion cM5M6le of risaing to the chMllenges MhaeMj| rMnfiing high MJong atheJ the re5ort3s Muthors. M gM5 in our finoIlejge of yrish stujents3 JMtheJMticMl finoIlejge Mt the trMnsition froJ MrithJetic to initiMl Mlge6rM PthMt occurs in TeMr d of their 5osta5riJMrT ejucMtion& 5MrticulMrlT Ms regMrjs the strengths Mnj IeMfinesses in their JMtheJMticMl 5roble. Dhe Muthors M55lT Mnj eV5loit releKMnt internMtionMl reseMrch to guije their Iorfi| Mnj in 5MrticualMr to jeKelo5 M custoJisej Mlge6rM test Por screener& KMlijMtej for the tMrget Mujience to collect eJ5iricMl jMtM in M lMrgeascMle stujT of yrish seconj TeMr stujents3 finoIlejge of initiMl Malge6rM. :leMrlT| the goMl is to iJ5roKe stujents3 JMtheJMtics ejucMtion through 6etter Mlge6rM leMrning Mnj teMching. Dhe Muthors M55reciMte the iJ5ortMnce of this goMl Mnj a the signibcMnce of Mlge6rM for its success. Dhe focus on stujents3 strengths Mnj IeMfinesses estM6lishes Mn iJ5ortMant 6Mseline froJ Ihich to stMrt the 5rocess of 6uiljing 5MthIMTs to

6etter unjerstMnjing Mnj iJ5roKeJents. Dhe re5ort Mjjs neI eKijence Mnj insights in thais

MreM of stujT Mnj is Iell 6MlMncej in thMt it inKites stMfieholjers to consijer 6oth the JMtheJMtics curriculuJ. Dhere is Juch to 6e gMinej 6T iJ5roKing JMtheJMtics ejucMtion in schools sincae JMtheJMtics unjer5ins Mll thae SDEw Mnj nuJerMte jisci5lines. Dhe stMfies Mre KerT high Mnj the reIMrjs for injiKijuMls Mnj asocietT Mre greMtA signibcMnt iJ5roKeJent in Mlge6rM leMrning Mnj teMching Iill leMj ato 6etter outcoJes in their JMtheJMtics ejucMtion for 5osta

5riJMrT stujents| 6etter 5re5MrMtion for SDEw jisci5lines Mnj aother nuJerMte jisci5lines in

-igher EjucMtion P-E&| Mnj 6etter nuJerMcT in the Iijer 5o5ualMtion. Dhe eV5ectej 5MToff for 6etter outcoJes in these MreMs is 6etter econoJic outcoJes Mnj nMtionMl 5ros5eritT. Dhe iJ5ortMnce of Mlge6rM in chiljren3s JMtheJMtics ejucMtion Mnj the nMture Mnj %uMlitT of the reseMrch re5ortej here JMfie this M KerT iJ5ortMnt re5ort for the MjKMnceJent of wMtheJMtics0SDEw ejucMtion in yrelMnj. Dhis re5ort is essentiMl reMjing for 5olicT JMfiers Mnj Mll other stMfieholjers in yrish eajucMtion. Dhe Muthors offer neI tools| eKijence Mnj insights for the MjKMnceJent of Mlge6rM leMrning Mnj teMching in school JaMtheJMtics thMt Iill Mffect the %uMlitT of stujents3 Mlge6rM Mnj oKerMll JMtheJMtics ejucMtion Mnj the 5re5MrMtion of their JMtheJMtics teMchers. "T focussing on Mlge6rM leMrning Mnj teMching| the Muthorsa chMrt M course for the iJ5roKeJent of JMtheJMtics ejucMtion in yrish ¥uniora :Tcle schools| Mnj jo this in Mn eaKijencea6Msej IMT. Dhis M KerT insightful Mnj faMrareMching M55roMch nMtionMllT Mnj internMtionMllT since iJ5roKeJents in core MreMs of the SDEw jisci5lines Iill contri6ute to 6etter SDEw ejucMtion outcoJes. Dhe Muthors Mre to 6e coJJenjej for their Iorfi Mnj JMfiing theiar bnjings Mccessi6le to M Iijer reMjershi5 nMtionMllT Mnj internMtionMllT in this re5ort| Ihile cMtering for M 5riJMrT reMjershi5 of 5rMctitioners| teMcher ejucMtors Mnj 5olicT JMfiers| Mnj JMtheJMtics0SDEw ejucMtion reseMrchers.

EHyLSDEw| 2MtionMl :entre for SDEw ejucMtion

9niKersitT of WiJericfi

Contents

Executive Summary 5 21

Bibliography

58
MreMs Mnj the yrish su6aqect s5ecibcMtion for qunior cTcle Ná tT5e coJ5Mrej to stujents in the sMJ5lea. BC rMnfiej 6T 5ro5ortion MnsIerej correctlT in Fcto6er Rd rMnfiej 6T 5ro5ortion MnsIerej correctlT in Fcto6er Rv

JMgnituje error

Rá

Summary

Mn MreM of JMtheJMtics ejucMtion finoIn for its jifbculties P8aeJontT dxxí&. mlge6rM is ijentibej 6T JMnT Ms the lMnguMge ofa JMtheJMtics Mnj consijerej M

5rere%uisite for further stujT in JMtheJMtics in JMnT jisci5lines such Mas science| technologT|

Mnj engineering PGMKin © Shefbelj| dxaN,/ StMceT © :hicfi| dxxR&. ayt hMs 6een Mrguej thMt Mlge6rM is not onlT the gMatefiee5er for higher ejucMtion 6ut Mlso for citiªenshi5. Stujents neej to jeKelo5 finoIlejge| sfiills| Mnj M6iliaties Mcross M Iije rMnge of content MreMs to ensure success Iith initiMl Mlge6arM. Dhe 5rere%uisite content MreMs incluje rMtio Mnj 5ro5ortionMl reMsoning| frMctions| jeciJMls| 5aercentMges| integers| eV5onents| orjer of o5erMtions Mnj e%uMlitT. Dhe Mlge6rM content MreMs incluje KMriM6les| eV5ressions| e%uMtions| functions| Mnja 5Mtterns P"ush © §Mr5| dxNB/ ¤Mrren © :oo5er| dxxC&. Dhe centrMl o6qectiKe of the stujT IMs to jeKelo5 M 5roble of yrish seconj TeMr stujents3 finoIlejge of initiMl Malge6rM Mnj froJ thisA

MreMs for initiMl Mlge6rM.

unjerstMnjing Mnj 5rogression Iith initiMl Mlage6rM. guije their instruction giKen the liJitej tiJe MKMilM6le in the clMsasrooJ.

Mnj reseMrchers in the MreM of initiMl Mlge6arM.

testing| thMt stujents in yrelMnj| lifie their internMtionMl counter5Mrts| Mre eV5eriencing

jifbculties Iith Mlgae6rM P:hief EVMJiner| dxN,M/ Shiel © §elleher| dxNí&. Dhis stujT 5roKijes

eJ5iricMl eKijence 6Msej on the MssessJent of oKer ,xx seconj TeMr 5osta5riJMrT stujents juring the McMjeJica TeMr dxNvadxNí| Ms 5Mrt of M Hh8 astujT. Dhis re5ort 5roKijes M suJJMrT of the results froJ this MssessJent Mnj ijentibes afieT content MreMs Iith Ihich our satujents Mre struggling. Dhis eJ5iricMl eKijence hMs heretofore 6een unMKMilM6le. Ms it is 5rMcticMl Mnj MlloIs for MnMlTsis Mt Jore thMn one leKel of jMtM MggregMtion| thMt is 6etIeen stujent or 6etIeen clMsses or injeej 6oth iaf re%uirej P7oss| dxx,&. Dhis tT5e of jesign inKolKes the selection of aschools Pbrst stMge& for inclusion in the astujT| folloIej

6T the selection of Ma cluster PclMss& of stujents Iithin the schaool Pseconj stMge&. m nona

5ro6M6ilitT sMJ5ling atechni%ue finoIn Ms snoI6Mll sMJ5ling Iherae one 5Mrtici5Mnt PateMcher0

school 5rinci5Ml& raecoJJenjs Mt leMst one other 5Mrtici5aMnt Iho in turn recoJJenjs Mnother IMs eJ5loTej in this stujT P:resIell| dxNd&. Dhe initiMl teMchers Iere recruitej through sociMl JejiM cahMnnels Mnj 5ersonMal Mnj orgMnisMtionMl contMcts of the reseMrchers. Dhe stujents Iere Mccessej solelT through their teMchers. Since M nona5ro6M6ilitT sMJ5le Jethoj IMs usej the bnMl sMJ5ale JMT not entirelT re5resent the 5o5ulMtion Mnj therefore the generMlisM6ilitT of the raesults JMT 6e iJ5Mctej. Dhe bnMl sMJ5le coJ5risej Ná 5osta5riJMrT schools| daá teMchers0clMsses| Ihich resultej

in M sMJ5le of vví stujents. Ff the Ná schaools| NN P,í.á"& Iere seconjMrT schools| í PaBv.C"&

Iere KocMtionMl schools Mnj N aP,.B"& IMs M coJJunitT0coJ5rehensiKe school. mll schooals Iere English JejiuJ schaools Mnj B PN,.C"& caMrriej M 8EyS injiacMtor. 8ue to M6sences on the jMT of MjJinistrMtion Mnj lMcfi of consent0Mssent ,,, stujents3 jMtM Iere inclujej in Fcto6er dxNv Mnj Rív in m5ril dxNí. Ff the ,,, stujents Mssessej in Fcto6er Bx, P,,.x"& Iere JMle| dRC PRR.í"& Iere feJMle Mnj d Px.R"& 5referrej not to sMT. yn m5ril dvR P,,a.,"& Iere JMle

Mnj dNd PRR.,"& Iere feJMle.

Dhe MssessJent utilisej| finoIn Ms M screener| IMs s5ecibcMllT jeKelo5ej| Mnj KMlijMtej for yrish seconj TeMr 5osta5riJMrT stujents Ms M 5Mrt of athis stujT. EthicMl M55roKMl IMs sought Mnj grMntej for this reseMrch through the 2MtionMl 9niKersitT of yrelMnj| GMlIMT. Stujents Iere Mssessej in their oIn clMssrooJs su5erKisej 6T their oIn JMtheJMtics teMcher in eMrlT Fcto6er dxNv Mt the stMrt of the McMjeJica TeMr Mnj MgMin in ma5ril dxNí toIMrjs the enj of the McMjeJic TeMr.

5riJMrT stujents Mnj 5roKije KMluM6le inforJMtion for reseMrchers Mnj ejucMtors in the

yrish conteVt. DheT Mlso 5roKije iJ5ortMnt eKijence in the internMtionMl conteVt thMt Mlge6rMic errors Mre coJJon to stujents of Mll nMtionMlities eV5osej to jifferent curriculM IorljIije. Dhe folloIing 5roKijes M suJJMrT of fiaeT bnjings Mnj theira iJ5ortMnce for JMtheJMtics teMching Mnj leMrniang. hMKe Jore ¬eVi6ilitT Ihen Iorfiing Iith e%uMtions Mnj therefore M sounj unjerstMnjing of e%uMlitT is essentiMl to succeej Iith Mlge6rM. Dhose Iho Jisinter5ret the JeMning of thae

sign| tMfiing it to JeMn ®5erforJ M cMlculMtion3 thMt is KieIing it Ms Mn o5erMtionMl sTJ6ol Iill

struggle to Iorfi Iith e%uMtions PSte5hens e%uMlitT| Mnj these Iere the 6est MnsIerej iteJs Iith oKer Cx" of stujents MnsIering 6oth iteJs correctlT. Dherefore| the iJ5ortMnt content MreM of e%uMlitT is Iell unjerstooj 6T the sMJ5le Mnj shoIs thMt the effort Mt 5riJMrT leKel jejicMtej to this iJ5ortMnt conce5t hMs

6een IorthIhile P2::m| Nááá&.

Hro5ortionMl reMsoning is regMrjej Ms one of the JaMqor coJ5onents of forJMl thought Mnj once MchieKej this forJ of thinfiing cMna Mssist Iith leMrning throughout Mll jisci5lianes of science| JMtheJMtics| Mnj life P"ush © §Mr5| dxNB&. Stujents jo not MlIMTs jeKelo5

5ro5ortionMl reMsoning sfiills nMturMllT| Mnj often JistMfienlT M55lT MjjitiKe thinfiing to

5ro6leJs thMt re%uire 5ro5ortionMl reMsoning P-ilton

IMs usej to Mssess 5ro5ortionMl reMsoning Mnj oKer íx" of stujents MnsIerej correctlT jeJonstrMting thMt the JMqoritT hMKe M gooj unjerstMnjing of this fieT content MreM. Hro5ortionMl reMsoning is M fieT Ms5ect of nuJerMcT coKerej Mt qunior cTcle P2::m| dxNí& Mnj these results jeJonstrMte thMt Jost of our stujents unjerstMnj the conce5t. ¤hen Iorfiing Iith nuJ6ers astujents often leMrn M 5MrticulaMr 5rocejure for jeMling Iith M tT5e of nuJ6ear| for eVMJ5le using the coJJon jenoJinMtor Ihen Mjjing frMctions. Dherefore theT relT on JeJorT Ihena trTing to jeMl Iith these nuaJ6ers P"ush © §Mr5| dxNB&. Dhe lMcfi of unjerstMnjing of jifferent nuJ6er forJMts Mnj MssociMte 5rocejures leMjs to Mn inM6ilitT to M55lT the rules correctlT Ihen Iorfiing Iith KMriM6les in Mlge6rM. EKijence of stujents struggling Iith frMctions Mnj jeciJMlas hMs eJergej in the results of this stujT Mnj these Mre jiscussej 6eloI. -oIeKer| Jost stujents PoKer vR"& Iere M6le to coJ5Mre Rx" of orjer nuJ6ers Mre iJ5ortMnt sfiills Manj re%uirej for M55lTing rules caorrectlT Ihen Iorfiing Iith KMriM6les. yt Mlso enaM6les M stujent to Mssess if M solution to Mn e%uMtion or ine%uMlitT is reMsonM6le P"ottoJs| dxxB&. Dhe jistri6utiKe 5ro5ertT is funjMJentMal to stujents3 M6ilitT in Malge6rM| giKen thMt it is usej fre%uentlT in the trMnsforJMtion of eV5ressions in 6oth siJ5liafTing Mnj fMctorising. yn Mlge6rM bnjing e%uiKMlent eV5ressions is fre%uentlT re%uirej| for eVMJ5le Mn eV5ression 6eing fMctorisej BV O v ¯ BPVa O d&. Dhis JMni5ulMtion re%uires Mn unjerlTing senase of the 5ro5erties of nuJ6ers P"ush © §aMr5| dxNB&. HrocejurMl 5robciencT Iith the jistri6utiKe 5ro5ertT Ihen jeMling Iith Man eV5ression IMs eKijencej 6T Jost stujents PoKer vR"&. DrMjitionMllT jrill eVercises of the forJ BPV O d&| thMt is Julti5lTing out a6rMcfiets| Ihich MlloIs stujents to 6ecoJe 5robcient in M55lTinga the 5ro5ertT Iithout Mc%uiring M full unjerastMnjing of it. DhMt is stujents Mre ¬uent in the trMnsforJMtiKe rules Mnj sTJ6ol JMni5ulMtion of Mlge6rM| hoIeKer theT KieI these Ms su5erbciMla JoKeJent of sTJ6ols Iithout Mn McatuMl unjerstMnjing

of the structurMl 5ro5erties P§M5ut| NáCá&. ¤ithout this true unjerastMnjing| errors occur in

the stujents3 Iorfi Ms theT reMch Jore MjKMncej Mlge6rMic eV5ressions. EKijence of M lMcfi of this jee5er unjerastMnjing of the jistri6utiKe 5ro5ertT eJergej froJ the res5onses to other iteJs on the screener s5ecibcMllT ian relMtion to eV5onents Ihich is jiascussej 6eloI. wMnT reseMrchers 6elieKe thMt MlloIing stujents to inKestigMte the 5ro5erties of nuJ6ers a Iill Mssist stujents in leMrning|a retMining finoIlejge Mnj relMtionMl unjerstMnjing| Ihich in turn Iill creMte M strong founjMtion for Mlge6rM P"ush © §Mr5| dxNB&. Dhe M6ilitT to see 5Mtterns| eV5lMin theJ Mnj re5resent theJ Iith Mlgea6rMic sTJ6olisJ is iJ5ortMnt for JMtheJMticMl 5ro6leJ solKing PmTMlon the relMtionshi5 Ihich lies a6etIeen the 5Mttern Mnj its 5ositiona in M functionMl relMtionshi5 Just 6e ijentibej| Mnja to jo this Mn eV5ression or forJulM Just 6e creMtej using KMriM6les. yn joing this M conteVt for the use of KMriM6les is set for stujents| Mssisting their unjerstMnjing

of M KMriM6le Ms M KMrTing %uMntitT rMther thMn M s5ecibc aunfinoIn PDMni°li © §±se| dxNN&. yn

this stujT stujents Iere Msfiej to coJ5lete M tM6le for the nuJ6er of sijaes in the 5eriJeter of M stMcfiej heVMgon 5Mttern. Dhis IMs Iell MnsIerej Iith oKer íx" of stujents coJ5leting the tM6le correctlT. yjentifTing the caorrect jescri5tion Mnj aforJulM to re5resent the 5Mttern hoIeKer IMs M less Iell MnsIerej 5Mrt of the tMsfi aP²,x" Mnj ²Bx" res5ectiKelT&. Dhis

5oints to lMcfi of unjerstMnjing of KMriM6les Mnj the M6ailitT to forJ Mn eV5ression.

korJing Mlge6rMic eV5ressions is M notoriouslT jifbcult tMsafi| Ihere6T the trMnslMtion froJ Iorj 5ro6leJ to eV5ression often results in errors P"ush © §Mr5| dxNB&. Dhe results of this stujT shoI thMt Ihen Msfiej to Irite Mn eV5ression for the 5eriJeter of Mn e%uilMterMl triMngle of sije eV5ression for M 5eriJeter of M 5entMgon Iiath R e%uMl sijes of a of stujents succeejej. Dhe Jost jifbcult tMsfi shoIing Mn o5en sijej ashM5e Iith of length d IMs MnsIerej correctlT 6T KerT feI. Fften stujents3 struggles Iith Mlge6raMic eV5ression result froJ Jisconce5tions Iith KMriM6les Mnj this lMacfi of unjerstMnjing of KMriM6les is eKijent in the results P"ush © §Mr5| dxNB&. kurther issues Iith Mlge6rMic eV5ressions Mrise Ihen stujents fMil to siJ5lifT 5ro5erlT in thMt theT jo not Mjj Mnj su6atrMct

lifie terJs| for eVMJ5le M coJJon error is to siJ5lifT dV O R tao vV P§ierMn| Náád&. mnother error

is to jetMch terJs PIhich JMT or JMT not contMin M KMriM6le& froJ the o5erMtions P¥u5ri TfO terJ rMther thMn the terJ Mfter leMjs to siJ5lifTing eV5ressions such Ms d a BV aO dV to d a ,V. Dhe tMsfi Mssessing siJ5libcMtion of Mn Mlge6rMic eV5ression IMs MnsIerej correctlT 6T

M55roViJMtelT hMlf of the sMJa5le.

MreMs of frMctions| jeciJMl nuJa6er JMgnituje| eV5onents| integers| orjer of o5erMtions| KMriM6les| forJing eV5ressions| Mnj forJing Mnj solKing ae%uMtions. mn unjerstMnjing of these fieT 5rere%uisite content MreMs is essentiMl for success Iith Mlge6rM Mnj the results here 5oint to IhT our stujents Mre struggling Iith Mlge6arM oKerMll P"ush © §Mr5| dxNB&.

9Sm PNááíadxxd&. Dhe MiJ IMs to ijentifT long terJ 5rejictors of 5osta5riJMrT stujents3

M6ilitT in Mlge6rM. yt reKeMlej thMt M stujent3s finoIlejge of frMctions Mnj jiKisiona uni%uelT

5rejictej M stujent3s finoIlejge of Mlge6rM Mnj oKerMll JMtheJMticMl M6ilitT bKe or siV TeMrs

lMter. yn yrelMnj| M lMcfi of fianoIlejge of frMctions on entering 5osta5riJMrT school is are5ortej Ms iJ5Mcting negMtiKelT on Mlge6rM 5erforJMnce PShiel © §elleher| dxNí&. kour iteJs on the screener Mssessej frMctions Mnj onlT onae of these iteJs Iere MnsIerej correctlT 6T Jore thMn ,x" of the sMJ5ale Mt 6oth MjJinistrMtions. Dhe reJMinjer of the iteJs Iere MnsIerej correctlT 6T less thMn N in d stujents. yteJs Mssessing e%uiKMlent frMctions Mnj Julti5lTaing frMctions Iere MnsIerej correctlT 6T M55roViJMtelT d in , stujents| Ihile the itaeJ Mssessing relMtionMl finoIlejge of frMctions IMs MnsIerej correctlT 6T onlT N in ,. krMction finoIlejge is Mn iJ5ortMnt 5rere%uisite for the stujT of Mlge6rM Mnj there Mre nuJerous coJJon errors encounterej in stujent Iorfiings 6oth nMtionMllT Mnj internMtionMllT P"ooth

© §Mr5| dxNB&.

m stujT 6T 8e¤olf

6est Mt 5rejicting Mlge6rM 5erforJMnce. DheT conclujej thMt Mn unjerstMnjing of jeciJMl a

JMgnitujes Ihen Mssessej 6T M nuJ6er line tMsfia Mnj M relMtionMl unjerstMnjing of frMctions Mre strong 5rejictors of Mlge6rM 5erforJMnce. m nuJ6er line tMsfia IMs usej to Mssess jeciJMl nuJ6er JMgnituje Iheare stujents Iere Msfiej to 5lMce M hMtch JMrfi on the nuJ6aer line for M giKen jeciJMl nuJ6er. wore thMn hMlf of the astujents in the sMJ5lea P,,"& coulj not 5lMce the hMtch JMrfi Iithin 5lus aor Jinus Nx" of the caorrect locMtion Ihich injicMtes jifbculties Iith unjerstMnjing jeciJMl nuJa6er JMgnituje. yteJs Mssessing relMtionMl frMction finoIlejge| Iere MnsIerej correctlT 6T onlT N in , of stujents. Fur stujents Mre struggling Iith these afieT

5rere%uisite MreMs for success Iith Mlge6rM.

mn unjerstMnjing of eV5onents is re%uirej throughout Mlge6rM in 6oth the trMnsforJMtionMl sfiills Mnj the generaMtionMl glo6Ml0JetM leKel sfiills P"ottoJs| dxxB/ "ush © §Mr5| dxNB&. DrMnsforJMtionMl sfiills inclujae fMctorising| Julti5lTinga| jiKijing| Mnj siJa5lifTing Mlge6rMic eV5ressions Mll re%uire M finoIlejge of eV5onents. DIo iteJs Mssessej eV5onents| Mnj these a Iere MJong the Iorst MnsIerej iteJs of Mll. m55roViJMtelT onlT one in ten stujents Iere correctlT M6le to s%uMre Mn eV5ression. Dhe Jore strMightforIMrj iteJ on Msfiing stujents to s%uMre V , IMs MnsIerej correctlT 6T M55roViJMtelT one in three stujents onlT. EV5onentiMl notMtion consists of tIo 5Mrts| M 6Mse Mnj aeV5onent Mnj it is M aIMT of eV5ressing re5eMtej Julti5licMtion. Dhe M6ilitT to unjerstMnj Mnj Iorfi Iith eV5onents re%uires unjerstMnjing the notMtion| the JeMning Manj 5ro5erties of eV5onents Mnj it is fianoIn to 6e M jifbcult JMtheJMticMl to5ic for stujents of Mll Mges Pa9lusoT| dxNá&. §noIing the orjer in Ihich to cMrrT out MrithJetaic o5erMtions is M funjMJentMal sfiill re%uirej for Mlge6rM P"ottoJs| dxxB&. Dhere Mre JMnT coJJon Jisconce5tions Ihen jeMlinag Iith the orjer of o5erMtions| the Jost coJJon of Ihich is caMrrTing out o5erMtions in orjer froJ

left to right P"ooth © :Marlton ¥ohnson| NáCR&. Fane iteJ Mssessej orjer of o5erMtions Msfiing

stujents to eKMluMte NB a B V R O d. FnalT one in three stujents M55roViJMtelT Iere M6le to eKMluMte this correctlT Ms B. Dhe Jost coJJon error IMs injeej Iorfiing left to right| Iith M55roViJMtelT one in three stujents joing this. majjitionMllT| qust less thMn one in bKe stujents MnsIerej aN| 6elieKing thMt Mjjition su5ersejesa su6trMction. Dhis JMT hMKe to jo Iith M lMcfi of unjerstMnjing of integers or it JMT 6e Ms M result of hMKing leMrnej orjer of o5erMtions through M JneJonic PScahIMrtªJMn| Nááv& e.g.| "Fw8mS0"y78wmS. rootej in the JeMning oaf sTJ6ols or the letters usej. wuch reseMrch hMs 6een conjuctej

into stujents3 jifbculties ian Iorfiing Iith Mlge6rMic KMriM6les Mnj it is iajentibej Ms M fieT MreM

for stujent Jisconce5tions Pms%uith KMriM6les P-unter| dxNx&. :oJJon Jisconce5tions Iith KMriM6les incluje KiaeIing KMriM6les Ms lM6els| Mnj the 6elaief thMt M KMriM6le is qust M Jissing KMlue rMther thMn KMrTing KMlues Pms%uith

M KMriM6le Ms M lM6el ahMs eJergej in this stujT. Fne iteJ jesignej 6T §¨cheJMnn PNáCN&| ato

ijentifT M stujents3 unjerstMnjing of M KMriM6le IMs MnsIerej correctlT 6T M55roViJMtelT one in four stujents. DIo in bKe stujents unjerstooj the eV5ression C6 O vJ to JeMn ³C 6oofis Mnj v JMgMªines´ jeas5ite 6eing tolj thMt 6 stooj for the nuJ6er of 6oaofis Mnj J stooj for the nuJ6er of JMgMªaines. ¤hen M stujent is Mt the stMge of unjerstMnjing M KMriM6le Ms

M lM6el onlT| this results in further erraors Iith Mlge6rMic eV5ressions P§¨cheJMnn| NáaíC&. yn

the originMl stujT 6T §¨cheJMnn PNáCN&| ait IMs re5ortej thMt Bá" of NRaTeMraoljs JMje this

error. m stujT froJ MlJost tIentT TeMrs Mgo in the 9S ausej this iteJ PJojibej to R cMfies Mnj B 6roInies& Mnj founj thMt Bí" of grMje í stujents Mnj dí" of grMje C stujents JMje the sMJe error Pe%uiKMlent to brst Mnj seconj TeMr 5ost 5riJMrT in yrelMnj&. yt M55eMrs athMt jes5ite the chMnges in 5eajMgogT Mnj M55roMches to teMching Mlge6rM internMtionMllT Mnj in yrelMnj| the 5ro5ortion of stujents JMfiing this earror hMs increMsej oKer the jecMjes P§iearMn mlge6rM hMs 6een KieIej 6T JMnT for centuries Ms the scienace of e%uMtion solKing P§ierMn| dxxR&. mn M6ilaitT to Iorfi Iith KMriM6les| Irite Mlge6rMic eV5ressions| Mnj to forJ Mnj solKe e%uMtions is the essence of success for initiMl Mlge6rM Ms it is funjMJentMal in 5re5Mring for Jore MjKMncej Mlge6rMic conce5ts P:M5rMro © ¥offrion| dxxva&. Dhere Mre seKerMl iteJs on the screener thMt Mssess finoIlejge of e%uMtions. Dhe brst of these Msfis stujents to solKe R a V ¯ , Mnj lesas thMn N in B stujents coulj solKe this e%uMtion Mt the

6eginning of seconj TeMr. Dhe JMin error in MtteJ5ting to solKe the e%uMtion Mrises froJ

the incorrect M55licMtion of the Mjjitiona inKerse| 5ossi6lT jue to the Jinus in front of the V| Ihere6T stujents JMT 6e confusej Ms to Ihether the negMtiKe sign JeMns su6trMction or is

5Mrt of M negMtiKe nuJ6er P¦lMssis| dxxC&. Dhis lMcfi of unjerstMnjing is further eaKijencej

on the iteJ Msfiing for the neVt ste5 to solKe ,ª ¯ Bx P:hung © 8elMcruª| daxNR&. Dhe Jost

5reKMlent error IMs stujents incorrectlT choosing ª ¯ aBx a ,| the incorrect M55licMtion of

su6trMction Ms the inKerse to Julti5licMtion in solKing the ae%uMtion. SolKing lineMr e%uMations is jifbcult Iitah u5 to bftT conce5ts inKolKej P:hung © 8elMcruª| dxNR& Mnj it IMs not 5ossi6le to inKestigMte Mll Iithin the sco5e of this stujT. Dhe results of KMrious iteJs shoI thMt the Jinus sign is 5aro6leJMtic for soJe stujents Ihere theT Mre

5ossi6lT KieIing it Mt the su6trMhenj leKel of su6trMction onlT P¦lMssis| dxxC&. Dhis is eKijent

in the res5onses to the iteJ on orjer of o5erMtions| Ihere soJe stujents JistMfienlT 6elieKe thMt Mjjition su5ersejesa su6trMction Ms Jentionej.a yt is Mlso eKijencej in the iteJ Ihich Msfis stujents to solKe R a V ¯ ,| Iith soaJe stujents jro55ing or ignoringa the Jinus sign Ihena solKing this e%uMtion. yssues Iith the Jinus siagn Mre Mlso eKijent froJ Mnother iteJ Mssessing

e%uMtions Ihich Msfiej stujents to ijentifT the neVt correct ste5 in solKing íh a aPBh a d& ¯ BC.

Dhe errors JMje on this iteJ ijentifT thMt M55roViJMtelT tIo in bKe stujents jo not recognise the Jinus sign Ms Mn ao5erMtionMl signiber P¦lMssis| dxxC&. EKijence 5resentej here shoIs JMnT strengths Ms Iell Ms coJJon errors Mnj Jisconce5tions recognisej in the literMture| Mnj it 5roKijes M Juch greMter je5th of unjerstMnjing of yrish 5osta5riJMrT stujents3 jifbculties tahMn IMs 5reKiouslT finoIn. yt is ho5ej thMt this reseMrch Iill serKe Mll stMfieholjers in JMtheJMtics ejucMtion in unjerstMnjing the strengths Mnj IeMfinesses of stujents Ms theT 6egin to grMs5 Mlge6rMic conce5ts. Dhe inforJMtion

5rojucej here cMn 6e usej to jeterJine hoI 6est to su55ort teMchers Mnj stujents in

this MreM of JMtheJMtics ejucMtion Ihich is Iell jocuJentej Ms jifbcult to teMch Mnj leMrn P8eJontT Dhe fieT recoJJenjMtions 6Msej on the banjings froJ this reseMrch inclujeA to hel5 reJejiMte the coJJon errors Mnj Jisconce5tions helj 6T our stujents. schools to su55ort eVisting teMchers Mnj instruction. Dhese MssessJents Ioulj hel5 ejucMtors Mnj reseMrchers fullT unjerstMnj stujents3 Jisconce5tions Mnj thinfiinga in Jore je5th| leMjing to Jore tMrgetej use of M55ro5riMte interKentions PGroµ o55ortunitT to Iorfi collM6orMtiKelT. 8iscussion Mrounj the coJJon errors ijentibej in this stujT MJong teMchers cMn MlloI for tMilorej s5ecibc 5lMns for instruction to hel5 reJejiMte these errors. leKels. yt is recoJJenjej thMt M 6rijging course 6etIeen 5riJMrT Mnj 5oasta5riJMrT relMting to 5rere%uisite Mlge6rM 6e jeKelo5ej to su55ort leMrners ian the trMnsition froJ MrithJetic to Mlge6rMic thinfiing. Dhe inforJMtion 5roKijej in this reseMrch re5ort cMn hel5 JMtheJMtics teMchers Mnj reseMrchers froJ 5riJMrT leKel through to thirj leKel 6T MlloIing theJ to focus on strengths Mnj fieT 5ro6leJ MreMs relMting to Mlge6rM. "T 6eing Jore MIMre of the Jisconce5tions Mnj MreMs of JMtheJMtics Iith Ihich theiar stujents struggle teMchers cMn 5lMn inastruction MccorjinglT. ³yf one is trulT tao succeej in leMjing M 5aerson to M s5ecibc 5lMce| one Just brst

Mnj foreJost tMfie cMre to bnj hiJ Ihere he is Mnj 6egin tahere´ P§ierfiegMMrj| Návd| 5. R,&.

áAsohfDNswhA

Mlge6rM. wuch reseMrch Mnj effort hMs 6een focusej on the MreM of Mlge6rMic thinfiing Mnj initiMl Mlge6rM to iJ5roKe stujents3 engMgeJent Manj MttMinJent Iith the su6aqect

P§ierMn

JMtheJMtics Mnj IMTs in Ihich to hel5 theJ| hoIeKer| jes5ite this| there is no cleMr eKijence thMt this effort hMs hMj M signibacMnt iJ5Mct on MttMinJent P-ojgen there hMKe 6een nuJerous stujies internMtionMllT focusing on Mlge6rM to hel5 unjerstMnj

Mnj reJejiMte stujents3 jifbcultT P"aooth

Jojes of teMching Mlge6rM hMKe 6een iJ5leJentej PHrenjergMst © DreMcT| dxNí&. EKijence hMs eJergej| through KMrious goKernJent re5orts| reseMrch| Mnj internMtionMl testing| of stujents in yrelMnj eV5eriencing jifbculaties Iith Mlge6rM P:hief EVMJiner| dxN,M/ Shiel

© §elleher| dxNí&. mjjitionMllT| yrish 5osta5riJMrT teMchers hMKe highlightej thMt our stujents

Mre struggling Iith JMnT of the 5rere%uisite content MreMs for eVMJ5le| frMctions PShiel ©

§elleher| dxNí&. Dhe MnMlTsis 5resentej in this re5ort 5roKijes jetMilej insigaht into the s5ecibc

jifbculties yrish secaonj TeMr 5osta5riJMrT stujents Mre encountering Iith Mlge6rM. HreKiouslT| there hMs 6een little to no eJ5iricMl eKijence MKMilM6le on seconj TeMr stujents3 finoIlejge of Mlge6rM in yrelMnj.

íwR

S ht S sdl S ysDf1 finoI M6out Mlge6rM| s5ecibcMllT 5rere%uisite Mnj initiMl Mlge6arM conce5ts. Dhe eKijence

5resentej in this re5ort is 6Msej on M a5enaMnja5M5er Mlge6arM MssessJent coJ5letej 6T oKer

bKe hunjrej seconjaTeMr stujents juring the McaMjeJic TeMr dxNv0dxNí. Dhe MssessJent utilisej| finoIn Ms M screener| IMs s5ecibcMllT jeKelo5ej| Mnj KMlijMtej for yrish seconj TeMr 5osta5riJMrT stujents| Ms 5Mrt of thais stujT.

JB"lNsw2ly

strengths Mnj IeMfinesses in the 5ertinent acontent MreMs. unjerstMnjing Mnj 5rogression Iith initiMl Mlage6rM. guije their instruction giKen their liJitej tiJe in the JMtheJMtics clMssrooJ. reseMrchers in the MreM of initiMl Mlge6arM. Odl S

áowyd

S

ChAslxs

institutions Mre noI encountering stujents entering science Mnj technologTa6Msej 5rogrMJs Iith inferior JMtheJMticMl sfiills Ihen coJ5Mrej Iith stujents M jecMje Mgoa PHrenjergMst ©

DreMcT| dxNí/ DreMcT © kMulfiner| dxN,&. Dhis is consistent Iith internMtionMl trenjs PDreMcT ©

kMulfiner| dxN,&. m recent -igher EjucMtion muthoritT P-Em& re5ort highlights thea re%uireJent for funjing for ³wMths enM6ling courses´ Mnj 5reaJMths courses for Mll thirj leKel institutes

PWiston

leKel JMtheJMtics. EVMJiner3s PdxN,& re5orts for JMtheJMtics Mt 6oth ¥unior Mnj WeMKing :ertibcMte leKel Iere issuej to 5roKije inforJMtion Mnj MnMlTsis of stujents3 5erforJMnce in their eVMJinMtions Mnj to Mssist teMchers Iith ongoinag jeliKerT Mnj iJ5roKeJent of JMtheJMtics P:hief EVMJiner| dxN,M| dxN,6&. Dhe McMjeJic TeMr dxN, IMs the brst TeMr in Ihich Mll caMnjijMtes for ¥unior :ertibcMte eVMJinMtion Iere Mssessej entirelT on the neI sTllM6us/ hoIeKer| not Mll cMnjijMtes Mt WeMKing :ertibcMte hMj 6een introjucej to Mll eleJents of thae neI sTllM6us jue to the 5hMsej iJ5leJaentMtion P:hief EVMJiner| dxN,6&. Dhere Mre JMnT 5ositiKes to 6e gMrnerej froJ the re5orts| nMJelT the gareMter 5ro5ortion of stujents tMfiing JMtheJMtics Mt higheraleKel Mnj their MchieKeJent oKerMll Mt sMJe. HolicT jecisions Iere JMje to increMse the u5tMfie of higheraleKel JMtheJMtics| Ms it hMj the loIest rMte of u5tMfie Mt higher leKel oKer MnT other su6qect Mt 6oth qunior Mnj seanior cTcles PHrenjergMst| dxNN&. "oth re5orts coJJentej on the chMnge ofa content for 5osta

5riJMrT JMtheJMtics ejucMtion. Dhe highest MchieKing stujents jeJonstrMtej gooj

5ro6leJasolKing M6iliaties Mnj use of Mll athe stujiej strMnjs of JMtheJMtics to tMcfile M

5ro6leJ P:hief EVMJiner| dxN,6&. -oIeKer| the conclusions in 6oth rae5orts highlight

issues for concern. mt WeMKing :ertibcMte leKel it is notej alcThhfIcATei|ceuiffaecThholhoiTaEfIcTGMcT||soTaEfIciecTc|TseEcnloc|lG|EoGOcgaciec|fETocaraTac arEcholhloailGclncarEc|TGMiMTasoEcnlocDrlycarieciecTceai.Gi |TGacMin |sfaIcarTeciG|oETeEMc eiG|EcRd11OJc"2riEncCxTyiGEoBcRd1vABchOcRm5 coJ5etence in Mlge6rM Mnj in 5MrticulMra Mlge6rMic JMni5ulMtion P:hief EVMJiner| dxN,6&. mt ¥unior :ertibcMte leKel the sMJe concerns Mre eV5ressej M6out cMnjijMtes Mt Mll three leKels Ihich ³Iill unjaou6tejlT cMuse 5ro6leJs for these cMnjijMtes in WeMKing :ertibcMte wMtheJMtics´ P:hief EVMJiner| dxN,M| 5. BN&. m reKieI 6T the EjucMtionMl 7eseMrch :entre PE7:& on 6ehMlf aof the 2MtionMl :ouncil for :urriculuJ Mnj mssessJent P2::m& entitlej ³mn EKMluMtion of the yJ5Mct oaf Hroqect wMths on the HerforJMnce of Stujents in ¥unior :Tcle wMtheJMtics´ 5roKijes further eKijence of stujents3 5erforJMnce on Mlge6rM PShiel © §elleher| dxNí&. Dhe reKieI inclujej interKieIs Iith ív 5osta5riJMrT teMchers Ms 5Mrt of Ma kocus Grou5| Iho oKerMll Iere hM55T Iith the current curriculuJ for Mlge6rM Mnj qunior cTcle. yt is notej thMt the introjuction of 5Mtterns Mnj the linfis 6etIeen Mlge6rM Mnj other Ms5ects aof the sTllM6us is Iorfiing Iell. -oIeKer| the teMchers ijentibej Malge6rM Ms the Jost chMllenging MreM of JMtheJMtics for stujents Mnj four coJJon theJes eJergej in relMtion to Mlge6rM Mt qunior cTcle. kirst is thMt on entering

5osta5riJMrT stujents Mre jis5lMTing M lMcfi of 5robciencT Iith funjMJentMl JaMtheJMticMl

sfiills. m lMcfi of fianoIlejge of frMctions Mnj integers in brstaTeMr 5osta5riJMrT is iJ5Mctiang on

stujents3 M6ilitT to 5rogress Iith Mlge6rM. SeconjlT| the grou5 Iere MlJost unMniJous in their a KieI thMt M lMcfi of instructionMl tiJe IMs hinjering their aefforts to teMch Mlge6rM effectiKelT Mt qunior cTcle. DeMchers re5ort thMt the tiJe to reKise Mnj 5rMctise 6Msic Mlge6rM sfiills is not

MKMilM6le. DhirjlT| it is re5ortej thMt stujents cMn MchieKe high grMjes in their quniora certibcMte

eVMJinMtion 6T joing Iell in stMtistics Mnj 5ro6M6ilitT 6ut 6T hMKing little finoIlejge of Mlge6rM. Dhis MlloIs M stujent to 5rogress to higher leKel JMtheJMtics Mt senior cTcle| hoIeKer the lMcfi of 5robciencT in Mlge6rM then hinjers M stujent froJ 5rogressing Iith the su6qecta oKerMll. kinMllT| issues relMting to 6etter M6le stujents not 6eing cMterej for eJergej. Dhe chMnge in sTllM6us Mnj initiMtiKes to iJ5roKe the u5tMfie of JMtheJMtics Mt higher leKel| hMKe

MIMre thMt jifferentiMtej instruction cMn hel5 jeaMl Iith this| 6ut lMrage clMss siªes Mnj lMcfi of

instructionMl tiJe Mre liJiting their efaforts PShiel © §elleher| dxNí&. yt is cleMr froJ the conclusions of 6oth :ahief EVMJiner3s re5orts Mnj thMt of the E7: thMt JMnT stujents continue to struggle Iith Mlge6rM Mt qunior cTcle Mnj further reseMrch is re%uirej to

iJ5roKe yrish stujents3 MttMinJent in this to5ic P:hief EVMJiner| dxN,M| dxN,6/ Shiel © §elleher|

dxNí&. 7eseMrch focusing on Mlge6rM ejucMtion Mt the 5riJMrT to 5osta5riJMrT leKel injicMtes thMt M focus on stujent jifbculties together Iith the iJ5aleJentMtion of su55ortiKe teMching Mnj leMrning enKironJents cMn iJ5roKe stujents3 M6ilitT Iitha Mlge6rM P:Mi TfO hoIeKer this re5ort is 6Msej on thaeir ¥unior :ertibcMte eVMJinMtion| Mfter three TeMrs 5osta

5riJMrT ejucMtion hMKe 6een coJ5letej. Dhere is M cleMr neej tao estM6lish stujents3 jifbculties

Mt Mn eMrlier stMge to Mssist teMchers in ijentifTaing stujents3 neejs Iithina the clMssrooJ enKironJent Mnj there6T iJ5roKing instruction on Mn ongoiang 6Msis P"ooth yn trTing to estM6lish M 5roble of stujents3 finoIlejge of Mlge6rM it is M55ro5riMte to loofi Mt the results of yrish stujents on internMtionMl tests. Dhe HrogrMJJe for ynternMtionMl Stujent mssessJent PHySm& Mnj Drenjs in ynternMtionMl wMtheJMtics Mnj Science StujT PDywSS& re5orts Mnj results Iere MnMlTsej oKer the lMst tIentT TeMrs to 6uilj M 5roble of yrish stujents3 M6ilitT HySm| Ihich tMfies 5lMce eKerT three TeMrs| since ince5tion in dxxx. HySm MiJs to eKMluMte IorljIije ejucMtion sTsteJs 6T Mssessing N,aTeMraolj stujents on hoI theT M55lT their school leMrning to reMl Iorlj 5ro6leJs PFE:8 n.j.&. Stujents Mre testej in the fieT su6qects of 7eMjing| wMtheJMtics Mnj Science Mnj in Mjjition tao this| stujents Mre testej on Mn innoKMtiKe joJMin in eMch raounj of testing| for eVMJ5le 5ro6leJ solKing in dxaN, PFE:8 n.j.&. yn JMtheJMtics| generMllT yrelMnj 5erforJs in line Iith thae oKerMll MKerMge of M55roViJMtelT ninetT 5Mrtici5Mting countries| M5Mrt froJ in dxxá Ihen yraelMnj scorej Mn MKerMge RCí.N| Nd.B JMrfis 6eloI the MKerMge Ihich IMs stMtisticMllT signibcMnt. ayt shoulj 6e notej thMt the scores on the HySm MssessJent Mre stMnjMrjisej Iith M JeMn oaf ,xx Mnj stMnjMrj jeKiMtion of Nxx. -oIeKer| in dxNd| dxN, Mnj dxNC yrelMnj 5erforJej 6etter thMn the FE:8 MKerMge. yt is not 5ossi6le to estM6lish 5erforJMnce on Mlge6rM s5ecibcMllT froJ the results of HySm Ms these MssessJents incor5orMte three fieT 5rocesses|

Pwc§eoIn

stujents3 5erforJMnce on JMtheJMtics Mnj science su6qects. DywSS jMtM gets collectej eKerT

four TeMrs since coJJencing in Náá,. Stujents in grMjes four Pfourth clMss 5riJMrT in yrelMnj&

Mnj eight Pseconj TeMr 5osta5riJMrT in yrelMnj& Mre inclujej in the stujT P2MtionMl :enter for EjucMtion StMtistics| dxNá&. Dhe MKerMge Mge of M stujent in yrelMnj 5Mrtici5Mting in DywwS Mt grMje eight in dxN, IMs NR.R TeMrs in line Iith thae internMtionMl MKerMge of NR.B TeMrs. Dhe results froJ DywSS Mre in line Iith the aHySm results Iith yrelMnj 5erforJing M6oKe MKerMge Mt 6oth grMje leKels. -oIeKer| Mlthough yrelMnj is 5erforJing Iell it is notej 6T :lerfiin TfO

5oints& Mnj 8MtM © :hMnce PONx 5oints& content joJMins| Mnj relMtiKe IeMfinesses on mlge6rM

Padd 5oints& Mnj GeoJetrT Padxa 5oints&´. Dhese results concur Iith the bnjinags 6T the E7: focus grou5| thMt oKerMll 5erforJMnce in JMtheJMtics is 6oostej 6T stujents3 finoIlejge of

stMtistics Mnj 5ro6M6ilitT P8MtM © :hMnce&| Ihile stujents3 M6ilitT Iitha Mlge6rM lMgs PShiel ©

§elleher| dxNí&.

6Msej Mnj JMnT other thirj leKel courses PGr·nJo| dxNC&. Dhere is M lMrge 6ojT of reseMrch in

Mjolescent jifbculties Iith ainitiMl Mlge6rM Mnj the conse%uences of these jifbcultiaes P"lMnton into the teMching Mnj leMrning of Mlge6rM is seen Ms high 5rioritT P-untleT the chMnges JMje to the yrish sTllM6us Mnj teMching Jethojs| 5rao6leJs Iith Mlge6rM 5ersist. Dhis is eKijencej in the :hief EVMJiner3s PdxN,& re5ort Ihich stMtes stujents neej to ³gMin coJfort Mnj MccurMcT in the 6Msic sfiillas of coJ5utMtion| Mlge6rMic JMni5ulMtion Mnj cMlculus´ P5. dá&. Dhe eKijence 5rojucej in this re5ort Iill su55ort teMchers Mnj schools a6T 5resenting Jore jetMil on the eVMct jifbculties theira stujents Mre encountering. vlmAwswhA S ht S

áAwsw.,

S

í,ulBo.

cruciMl 5hMse in Mlage6rM ejucMtion P¦Mn mJeroJ| dxxd&. mlge6rM is M forJMl sTJ6olic lMnguMge| coJ5osej of strings of sTJ6ols| Ihich JMT forJ M ®sentence3 Iithin the lMnguMage Ihen constructej Mccorjing to the sTntMV of thMt lMnguMge. Dhe lMnguMge of Mlgea6rM hMs bKe JMqor Ms5ectsA Pi&a unfinoIns| Pii& forJulMs| Pii& generaMlisej 5Mtterns| PiK& 5lMceholjers|

Mnj PiK& relMtionshi5s P9sisfiin| Náaáá&. ynitiMl Mlge6raM is Mn MreM of JMtheJMtics ejucMtion

thMt hMs receiKej Juch focus Mnj Mttention since the lMte NáCxs Ihen the conce5t of ³

6e introjucej to the conce5ts of Mlge6rM Juch eMrlier in taheir ejucMtion| froJ Mge v|

rMther thMn the trMjitionMl KieI of introjucing stujents to Mlge6rM Mfter MrithJetic Mt

Mge M55roViJMtelT Nd P§ierMn

6T §ierMn PdxxR& Mnj the oather 6T §M5ut

school Mlge6rM Ihich outlines thraee MctiKities thMt leMrners Just 5Mrtici5Mte inA e%uMtions| Ihich inKolKes 6ecoJing Mc%uMintej Iith soJe unfinoIn thMt is the KMriM6le Mnj the conce5t of e%uMtion solution. wost JeMninga6Msej leMrning for Mlge6rM occurs juring these MctaiKities. stujents leMrn hoI to JMni5ulMte Mlge6rMic sTJ6ols. wMstering these MctiKities MlloIs for the JMni5ulMtion of Mn eV5ression or e%uMtion Ihile

JMintMining e%uiKMlence.

Jojelling MctiKitieas. mlge6rM is usej Ms M tool to solKe Mnj Juch of the seanse JMfiing Mnj reMlalife conteVt of the JMtheJMtics is leMrnej juriang these

MctiKities.

GenerMtionMl MctiKities Mrae centrMl to Mlge6rM Mnj the theJe of a³generMlisMtion´ is to the fore in Juch of the reseMrch into school Mlge6rM| inclujing generMlising relMtej to 5Mtternej MctiKitT Mnj generMlising relMtej to 5ro5erties of o5erMtions Mnj nuJericMl astructure P§ierMn Ms the Mge of , cMna engMge in 5Mttern MctiKities Ihicha MlloI theJ to engMge in coaKMriMtionMl thinfiing Mnj hence functionMl thinfiinag. GenerMlising froJ nuJericMl Mnj geaoJetric 5Mtterns hMj gMinej Juch reseMrch Mttention giKen chiljren3s M6ilitT to engMge in these tMasfis P"ourfie © StMceT| NáCC/ §¨cheJMnn| adxNx/ Wee Dhe use of 5Mtterns to ijentifT JMtheJMticMl chMnge| relMtionshi5s Mnj to jeKelo5 functionMl thinfiing is not intenjej to introjuce stujents to forJMl Mlge6rM 6ut rMther to Mssist Iith their

Mlge6rMic thinfiing P§ierMn

M generMl rule to jescri6e the chMnaging 5Mttern 6ut Mre not Tet eV5ectej to forJMllT re5resent functions Iith Mlge6rMic sTJ6ols. Dhis Jethoj of engMging leMrners is finoIn Ms the functionsa

6Msej M55roMch to Mlge6rM Mnj IMs introjucej to Mll 5osta5riJMrT schools ina yrelMnj in

Se5teJ6er dxNN PHrenjergMst © DreMcT| dxNí&. mnother conce5tuMl M55roMch for Mlge6rM is 6Msej on the Iorfi of §M5ut yts core Ms5ects Mre ³JMfiing Mnj eV5ressing generMliªMtions in increMsinglT forJMl Mnj conKentionMl sTJ6ol sTsteJs Mnj Mcting on sTJ6ols Iithin Mn oragMniªej sTJ6olic sTsteJ through Mn estM6lishej sTntMV´ P"lMnton Mcross three content strMnjs Pi& Mlge6rM Ms the stujT of structures Mnj sTsteJs thMt Mrise in

MrithJetic| Pii& Mlage6rM Ms the stujT of functions| relMtions Mnj qoint KMriMtion| Mnj Piii& Mlgae6rM

Ms M cluster of Jojelling lManguMges P§M5ut

the generMtionMl MctiKities fraoJ §ierMn3s Jojel Mligns Iith a§M5ut3s brst core Ms5ect PMlthough

it is not e%uiKMlent&| Ihile the traMnsforJMtionMl MctiKities Mlaign Iith the seconj core Ms5ect. Dhe glo6Ml0JetM MctiKities froJ §ierMn3s Jojel then Mlign aIith the seconj Mnj thirj content strMnjs of §M5ut3s Jojel P-ojgen sTllM6us. 7eseMrch 6T "lMnton §M5ut3s tIo core Ms5ects into the folloIing three MreMs Ihich IMs Mjo5tej Ms the conce5tuMl

Jojel for initiMl Mlge6rM in this stujT.

i. GenerMlisej MrithJeticA tahis inclujes generMlising| qustifTing| re5resenting| Mnj reMsoning Iith MrithJetic relMtions Ihich incluje athe 5ro5erties of nuJ6ers.a ii. E%uiKMlence| eV5ressions| e%uMtions| Mnj ine%uMlitaiesA Ihich inclujes athe jeKelo5Jent of M relMtionMl unjerstMnjing of the e%uiKaMlence sign| together Iith generMlising| qustifTing| re5resenting| Mnj reMsoning Iith e%uiKMlence| eV5ressions| e%uMtions| Mnj ine%uMlitaies. iii. kunctionMl thinfiingAa the M6ilitT to generMlise relMtionshi5s 6etIeen coaKMrTing %uMntities| Mnj re5resenting| qustifTing| Mnj reMsoning Iith these agenerMlisMtions through nMturMl lMnguMge| KMriM6le notMtion| jrMIings| tM6les| Mnj garM5hs. P"lMnton "ush Mnj §Mr5 PdxNB& outline the MreMs of JMtheJMtics thMt Mre re%uirej for leMrning Mlge6rM Mnj stujents3 coJJon Jisconce5tions in these MreMs. Dhe MreMs ijentibej Mre rMtio Mnj 5ro5ortionMl reMsoning| frMctions| jeciJMls| 5aercentMges| integers| eV5onents| orjer of o5erMtions| 5ro5erties of nuJ6ers|a coJ5Mring Mnj orjering nuJ6ers| e%uMalitT| KMriM6les| eV5ressions| e%uMtions| Mnj functions.a Dhis reKieI IMs 6Msej in the 9Sm Mnj Ms such IMs Mlignej Iith the asTllM6us for Jijjle grMje PMges NxaNR& stujents. Dhis sTllM6us is closelT Mlignej aIith the 2uJ6er| Mnj mlge6rM Mnj kunctions strMnjs stujiej 6T yrish

stujents in qunior cTcle. -oIeKer| one MreM froJ the qunior cTcle sTllM6us ¸ ®5Mtterns3 ¸ is

oJittej. HMtterns Mre ijentibej Ms Mn iaJ5ortMnt MreM of JMtheJMtics to Mssist Iith stujents3 unjerstMnjing of generMlising P¤Mrren © :oo5er| dxxC/ ¤eljer| dxNd& Mnj Mre therefore inclujej here Ms M 5rere%uisite content MreM for leMrning Mlge6raM. Do jebne the construct of initiMl Mlgae6rM for this stujT| the relMtej content joJMins froJ the yrish sTllM6us Mre Mlignej to the 5rere%uisite Mnj Mlge6rM content MreMs ijentibej in the literMture| Mnj Iith the conce5tuMl Jojel Mjo5tej. Dhis is suJJMrisej in aDM6le N. .Af S sdl S

áowyd

S yDB"lNs S yblNwmN.swhA S tho S "DAwho S N1N,l

Conceptual Model

Content AreasJunior Cycle Mathematics Subject

Specification

Generalised

arithmetic

5ro5ortionMl

relMtionshi5s unjerstMnjing of the relMtionshi5 6etIeen rMtio Mnj 5ro5ortion. EVMJining Mlge6rMic relMtionshi5sA 5ro5ortionMl relMtionshi5s arithmetic to thinfi M6out o5erMtion on frMctions.

9se the e%uiKMlence of frMctions| jeciJMls|

Mnj 5ercentMges to coJ5Mre 5ro5ortions.

arithmetic

HercentMges

9se the e%uiKMlence of frMctions| jeciJMls|

Mnj 5ercentMges to coJ5Mre 5ro5ortions.

arithmetic such Ms the nuJ6er liane| to illustrMte the o5erMtions on integers. arithmetic arithmetic o5erMtions of o5erMtions| inclujing use aof 6rMcfiets. arithmetic nuJ6ers

5ro5erties of MrithJetaic Mnj the

relMtionshi5s 6etIeen theJ. expressions, equations, and inequalities orjer nuJ6ers to orjer nMturMl nuJ6ers| integers| Mnj rMtionMl nuJ6ers. 9se athe e%uiKMlence of frMctions| jeciJMls| Manj 5ercentMges to coJ5Mre 5ro5ortions. expressions, equations, and inequalities thMt e%uMlitT is M relMtionshi5 in Ihich tIo JMtheJMticMl eV5ressions holj the sMJe KMlue.

Conceptual Model

Content AreasJunior Cycle Mathematics Subject

Specification

Equivalence,

expressions, equations, and inequalities %uMntities thMt Mre KMriM6le. expressions, equations, and inequalities eV5ressions eV5ressions. DrMnsforJMtionMl MctiKities expressions, equations, and inequalities e%uMtions

Mnj using suitM6le strMtegies for bnjing

solutions to e%uMtions Mnj ine%uMlitiaes.

Thinking

MiT.oTyecTGMc.oThre3

jiMgrMJs| Mnj grM5hs Ms M tool for

MnMlTsing relMtions ¸ 5resent Mnj inter5ret

solutions| eV5lMining Mnj qustifTing

Jethojs| inferences| Mnj reMsoning.

Thinking

nolycoEhETaiG.chTaaEoGe3 jiMgrMJs to re5resent M re5eMtinga5Mttern situMtion/ generMlise Mnj eV5lMin 5Mtterns

Mnj relMtionshi5s in Iorjs Mnj nuJ6ers/

Irite MrithJetic eV5ressions for terJs

in M se%uence.

Cd.,,lAuly

S wsd S

áAwsw.,

S

í,ulBo.

re5resentMtions Mnj Jore coJ5leV JMtheJMticMl relMtionshi5s| 6ut Mlso 6aecMuse it cMn JMgnifT Jisconce5tions thMt hMKe their roots in eMrlier instruction´ P"ooth Cx&. Dhis section jiscusses the finoIlejge| sfiills| Mnj M6iliaties P§Sms& re%uirej for initiMl Mlge6rM. Dhroughout the jiscussion the jifferent MreMs Iith Ihich stujents struggle Iith initiMl Mlge6rM Mnj hoI s5ecibc Jisconce5tions Mrise in eMcah MreM Mre outlinej. Stujents3

6eliefs| unjerstMnjing| JeMnings| aeV5lMnMtions| Mnj theories aforJ the 6Msis of stujent

conce5tions. ¤hen these aconce5tions con¬ict Iith Mcce5tej JeMnings in JMtheJMtics then M Jisconce5tion hMs occurrej PFs6orne © ¤ittrocfi| NáCB/ wulungTe| dxNv&. Stujent Jisconce5tions in JMtheJMtics cMn cMuse seriouas jifbculties in leMarning Mlge6rM Mnj Mre often 5ersistent Mnj resistMnt to conKentionMl teMching Jethojs Ms taheT 5rojuce sTsteJMtic errors thMt cMn interfere Iith leMrning P§uo wuch reseMrch hMs tMfien 5lMce to estM6lish the coJJon Jisconce5tions stujents encounter Ihen stujTing Mlge6rM P"ooth| NáCC/ Herso| NááN/ 7ussell Mc%uirej Jisconce5tions through 5rior leMrninga Mnj froJ inforJMl theories constructej froJ eKerTjMT eV5erience. Fther reseMrch hMs shoIn thMn Ihen M stujent holjs M

5MrticulMr Jisconce5tion| theT Iill often holj relMtej Jisconce5tions PHerso| NááN&. m solij

unjerstMnjing of the nuJ6ear sTsteJ is not the sole 5arere%uisite for Mlge6rM Mlthough it is KitMl for unjerstMnjing Mlge6rM. m ³JMsterT of nuJ6er o5erMtions| conce5ts| 5ro5erties| Mnj

so forth greMtlT fMcilitMtes stujents3 jeKelo5Jent in Mlge6rM´ P"ush © §Mr5| dxNB| 5. vdx&. yt

is iJ5ortMnt to note the cognitiKe gM5 6etIeen MrithJetic Mnj aMlge6rM| Ms stujents Just 6e introjucej to conce5ts froJ MrithJetic thMt Iill enM6le theira Mlge6rM reMjiness P-erscoKics

© WincheKsfii| NááR&. :Mreful Mttention Just 6e 5Mij to 5reaconce5ts Ms ³theT Mre M releKMnt

Mnj necessMrT 5rere%uisite for constructing forJMl conce5ts´ PWincheKsfii| Náá,| 5. NNR&. coJ5Mrison of nuJ6eras´ P5. ,dB&| Ihile Host ³inKolKes M sense of coaKMriMtion| Julti5le coJ5Mrisons| Mnj the aM6ilitT to JentMllT store

Mnj 5rocess seKerMl 5ieces of inforJMtion´ P5. íá&. Hro5ortionMl reMsoning is M fieT Ms5ect

of JMtheJMtics coKerej Mt qunior cTcle P2::m| dxNí&. yt is regMrjej Ms one of the JaMqor coJ5onents of forJMl thought Mnj onace MchieKej this forJ of thinfiing cMna Mssist Iith leMrning throughout Mll jisci5lianes of science| JMtheJMtics| Mnj life P"ush © §Mr5| dxNB/ Host nMturMllT| Mnj often JistMfienlT M55lT MjjitiKe thinfiing to 5ro6leJs thMt re%uire 5ro5ortionMl M6le toA reMson Julti5licMtiKelT| unjerstMnj rMtionMl nuJ6ers| MnMlaTse functionMl relMtionshi5s| e%uiKMlence| rMtio Mnj its 5Mrts| thaMt is the jifferent relMtionshi53s 5ossi6le 5Mrt0Ihole| 5Mrt0

5Mrt or eKen Ihole05Mrt. yt is not Mn eaMsT conce5t for the Mjolescent Jinj to grMs5 Ms it is

highlT conce5tuMl Mnj M sfiill tahMt jeKelo5s grMjuMllT PSingh| dxxax&. nuJ6er MrithJetic u5a to Mlge6rM´ P¤u| dxxN| 5. Nx&. yn dxxC| the a2MtionMl wMtheJMtics mjKisorT HMnel P2wmH& in the a9Sm highlightej thMt frMction finoIlejge Mnj M6ilitT aIMs criticMl to the leMrning of Malge6rM. Stujies folloIej to 5roKije the eJ5iricMl aeKijence to su55ort this stMteJent Mnj to ijentifT Ihich eVMct coJ5onent of frMction finoIlejge 6est

5rejicts M6ilitT in Mlage6rM P8e¤olf

cMn 6e founj Ms coefbcients| constMnts Mnj solutions to e%uMtions| the slo5e of aM line| Mnj| in

generMl| 5ro5ortions Mre Iritten in frMction forJ in Mlge6rM P"ush © §Mr5| dxNB&. yn yrelMnj| lMcfi

of finoIlejge of frMctions on entering 5osta5riJMrT school is are5ortej Ms iJ5Mcting negaMtiKelT on Mlge6rM 5erforJMnce PShiel © §elleher| dxNí&. m stujT 6T "ooth Mlge6rM reMjiness Mnj results suggest thMt it is finoIlejge of frMction JMgnitujes. Dahis eVtenjej on bnjings fraoJ 5reKious stujies in tIo IMTs. kirst| it ijentibej the arole thMt frMction finoIlejge hMs in Mlge6arM reMjiness| thMt frMction JMgnituje is aiJ5ortMnt Ms Iell Ms coJ5utMtionMl sfiills. SeconjlT| in eV5loring stujents3 5lMceJent of jis5MrMte frMctions on the nuJ6er line aM cleMrer 5icture of the finoIlejge coJ5onents helj 6T stujents eJerges. S5ecibcMllT| the 5lMceJent of unit frMctions| frMctions Iith M nuJeraMtor of one| IMs signibcMntlT correlMtej Iith M stujent3s future finoIlejge| e%uMtion Mnj Iorj 5ro6leJ solKing. m further stujT 6T 8e¤olf IMs 6est Mt 5rejicting Mlge6rM 5erforJMnce. DheT conclujej thMt unjerstMnjing of jeciJMl a JMgnitujes Ihen Mssessej 6T M nuJ6er line tMsfia Mnj M relMtionMl unjerstMnjing of frMctions Mre strong 5rejictors of Mlge6rM 5erforJMnce. Dhe tMsfis for the relMtionMl unjerstMnjing of frMctions Iere M neI Mjjition to the stujT Mnj contMinej tMsfis Ihicha testej frMction e%uiKMlence| jiKision| inKerse| Julti5lTing 6T reci5rocMl Mnj ijentifTinag rMtios in countM6le sets| 6oth 5Mrt to 5Mrt Mnj 5Mrt to Ihole. Dhese results linfi the content MreMs of 5ro5ortionMl reMsoning| frMctions| Mnj jeciJMals Ms iJ5ortMnt 5rere%uisites for leMrning Mlge6raM. Stujents cMn encounter jifbculties in coJ5Mring Mnj orjering Ihen 5resentej Iith nuJ6ers in jifferent forJMts such Ms frMctions| jeciJMls| Manj 5ercentMges. yf stujents jo not unjerstMnj the 6Msis of thea nuJ6er itself| theT Mre unlifielT to 6e M6le to eVtenj their unjerstMnjing to Ihich is greMter thMn or less thMn or e%uiKMlent. Dhe jifbcultT Mrises for JMnT stujents in thMt theT leMrn M 5MrticulMar 5rocejure for jeMling Iith M tT5e of nuJ6er| for eVMJ5le using the coJJon jenoJinMtor Ihen Mjjing frMctions| therefore theT relT on JeJorT Ihena trTing to jeMl Iith these nuaJ6ers P"ush © §Mr5| dxNB&. Dhe lMcfi of unjerstMnjing leMjs to Mn inM6ilitT to M55lT the rules correctlT Ihen

Iorfiing Iith KMriM6les in Mlge6rM.

for success in Mlge6rM P"ottoJs| dxxB/ "ush © §Mr5| dxNB&. Do gMin Mlge6rMic coJ5etence in terJs of solKing 5ro6leJs Mnj e%uMtions stujents Just unjerstMnj the eVtension of the nuJericMl joJMin fraoJ nMturMl nuJ6ers to integers PGMllMrjo| dxxd&. ¤hen chiljren Mre introjucej to the conce5t of M negMtiKe nuJ6er it cMn 6e ajifbcult Mnj teMchers relT on using JoneT Mnj je6t| teJ5erMture or seM leKel to eV5lMin the conce5t P"ush © §Mr5| dxNB&. wMnT stujents MKoij Iorfiing Iith the negaMtiKe sign Mltogether Iith soJe stujents 6elieKing ³thMt negMtiKe signs re5resent onlT the su6trMction o5erMtion Mnj jo not JojaifT terJs´ P"ooth TfO M negMtiKe nuJ6er itself| Mnj stujents bnj it jifbcualt to 5rojuce M negMtiKe solution Ms this M6strMct solution joes nota JMfie sense P¦lMssis| dxxC&. sfiills Mnj the generaMtionMl glo6Ml0JetM leKel sfiills P"ottoJs| dxxB/ "ush © §Mr5| dxNB&. DrMnsforJMtionMl sfiills inclujae fMctorising| Julti5lTinga| jiKijing| Mnj siJa5lifTing Mlge6rMic eV5ressions Mll re%uire M finoIlejge of eV5onents. Dhe generMtionMl sfiills Ihere finoIlejge of the shM5e of funcations is re%uirej Mlso relies on this finoIlejge. Dhe routinisMtion of

5rocejures leMjs stujents to JMfie errors Iith the 6Msic raules of injices. kor eVMJ5le| M

stujent unjerstMnjs the jiKision 5arocess Iith integers Mnj JMT confuse this 5rocejure Ihen fMcej Iith for eVMJ5le| Mnj jiKije the 5oaIers in error PGrMT for Mlge6rM P"ottoJs| dxxB&. Dhere Mre JMnT coJJon Jisconce5tions Ihen jeMlinag Iith the orjer of o5erMtions| the Jost coJJon of Ihich is caMrrTing out o5erMtions in orjer froJ left to right. SoJe stujents 6elieKe thMt orjer of o5erMtions joes not JMtter| thMt theT Iill jejuce the sMJe MnsIer regMrjless of the orjer in Ihich theT cMrrT out the o5eraMtions. SoJe 6elieKe the conteVt of the 5ro6leJ jeterJines the orjer of o5erMtions or in the M6seance of conteVt|

o5erMtions shoulj 6e 5erfaorJej froJ left to right P"ooth © :Marlton ¥ohnson| NáCR&. manother

stujT hMs siJilMr bnjaings Iith stujents cMrrTing out ao5erMtions froJ left to right| hoIeKer it

founj thMt the rule of ³6rMcfiets brst´ is cleMr to JMnT stujents PWincheKsfii © WiKneh| Nááá&.a

WincheKsfii PNáá,& suggests thMt Jore tiJe shoulj 6e giKaen to stujents in MlloIing theJ to jiscoKer thMt 5erforJing o5erMtions in M jifferent orjer results in jifferent MnsIers. SchIMrtªJMn PNááv& suggesats thMt the use of coJJon JneJonics to reJeJ6er the orjer of o5erMtions is not hel5ful.a Do stMrt Iith there Mre seKerMl jifferent ones in use| for eVMJ5le "Fw8mS/ 6rMcfiets| orjers| Julti5lT| jiKije| Mjj| su6trMct Mnj "y78wmS/ 6rMcfiets| injices| reci5rocMls| jiKision| Julati5licMtion| Mjjition| su6traMction| to nMJe qust tIo. SchIMrtªJMn PNááv& suggests thMt stujents shoulj leMrn athe hierMrchT of o5erMtions Jore nMturMllT| thMt is the Jore coJ5licMtej o5erMtions of Julti5licMtion Mnj jiKision shaoulj 6e Mttenjej to

6efore Mjjition Mnj su6traMction.

Mlge6rM Pms%uith

dxNB&. Stujents often Jisinter5ret the JeMning of thae sign| tMfiing it to JeMn ®5erforJ M cMlculMtion3 thMt is KieIing it Ms Mn o5erMtionMl sTJ6ol PSte5hens inter5ret the e%uMl sign correctlT Mnj see it Ms aM relMtionMl sTJ6ol hMKe Jore ¬eVi6ilitT Ihen Iorfiing Iith e%uMtions. m stujent Iho hMs M structurMl unjerstMnjing of Mlge6rM cMn ®see3 M6strMct ijeMs hijjen 6eahinj the sTJ6ols| therefore the sTJ6ols 6ecoJe trMns5Mrent| Mnj the trMnsforJMtions MlloIej to solKe the e%uMtion 6ecoJe M55Mrent P§ierMn| dxxí&. ³Stujents3 inter5retMtions of the JeMninga of the e%uMl sign Manj their M6ilities tao Iorfi Iith e%uMtions in M structurMl IMT Mre cleMrlT relMtej issues´ PSte5hens

6T "ooth

Jore eV5eriencej Iith solKing e%uMations Mnj Iorsen Ms the school aTeMr continues Mnj cMn Mffect M stujent3s oKerMll JMtheJMtics MchieKeJent. rootej in the JeMning oaf sTJ6ols or the letters usej. wuch reseMrch hMs 6een conjuctej into stujents3 jifbculties ian Iorfiing Iith Mlge6rMic KMriM6les Mnj it is iajentibej Ms M fieT MreM for stujent Jisconce5tions Pms%uith TfO mn essentiMl eleJent of Malge6rMic thinfiing is M raich unjerstMnjing of KMriM6les P-unter| dxNx&. :oJJon Jisconce5tions Iith KMriM6les incluje KiaeIing KMriM6les Ms lM6els| aMnj the 6elief thMt M KMriM6le is qust M Jissing KMlue rMther thMn KMrTing KMlues Pms%uith

dxNB&. §¨cheJMnn PNáíC& notej the jifferent IMTs in Ihich KMriM6les Mre usej Mnj estM6lishej

M hierMrchT of KMriM6les Ms outlineja in DM6le d.

Example

Dhe KMriM6le cMn 6e jiraectlT eKMluMtej|

e.g.| V O í ¯ NN| V ¯ a¹ Wetter yG2F7E8Dhe letterPs& jo not hMKe to 6e eKMluMtej| e.g.| M O 6 ¯ RB| thean M O 6 O d ¯ R,| afounj

6T qust Mjjing d to RB| ignoring M Oa 6.

Wetter Ms F"¥E:D PWm"EWa&Wetters stMnj for the nMJes of iteJs| e.g.|

H ¯ 5eriJeter| M ¯ M55le.

Wetter Ms SHE:yky: 92§2aF¤2¦MriM6le stMnjs for Mn unfinoIn %uMntitT

Mnj cMnnot 6e eKMluMtej| e.g.| mlice hMj V

euro Mn "o6 hMs B Jore| then "o6 hMs V O a

B euro.

Wetter Ms GE2E7mWySE8 2a9w"E7Wetters cMn re5resent M set of nuJ6eras

Mnj not qust M s5ecibc nuJ6er| e.g.| M O 6

¯ Nx| M º 6 the M ¯¹

Wetter Ms ¦m7ym"WEDhe relMtionshi5 KMries| e.g.| Ihich is lMrger| dV or d O V| it jae5enjs on IhMt V e%uMls. StMceT Mnj wMcGregor PNááí& ijentibej teMching Jethojs Ms Manother 5ossi6le source of confusion MJongst stujents in unjerstMnjing KMriM6les. kor eVMJ5le| M teMcher Iho fre%uentlT uses the brst letter of M Iorj Ms the unfinoIn cMn result in stujents unjerstMnjing M KMriM6le Ms M lM6el rMther thMn Mn unfinoIn %uMntitT. mnother issue ijentibej in their reseMrch IMs thMt stujents jij not unjerastMnj thMt theT coulj use KMriM6les to hel5 solKe 5ro6leJs| Mn essentiMl sfiill for using Mlge6rM effectiKelT. 9sing Mlge6rMic notMtion fre%uentlT| reKieIing coJ5utMtionMl sfiills Mnj eJa5hMsising thMt KMriM6les| thMt is letters in Mlge6rMic eV5ressions stMnj for unfinoIn %uMntities rMther thMn lM6els| 6eatter 5re5Mrej stujents for Mlge6rM

P"ush © §Mr5| dxNB&.

Fften stujents3 struggles Iith Mlge6raMic eV5ression result froJ Jisconce5tions Iith

KMriM6les P"ush © §Mr5| dxNB/ StMceT © wMcGregor| Nááí&. ¤hen Iorfiing Iith eV5ressions

JMnT stujents fMil to siJ5lifT 5ro5erlT in thMt theT jo not Mjj Mnj su6atrMct lifie terJs| for

eVMJ5le M coJJon error is to siJ5lifT dV O R tao vV P§ierMn| Náád&. mnother error is to jetMch

terJs PIhich JMT or JMT not contMin M KMriM6le& froJ the o5erMtions P¥u5ri

WincheKsfii| Náá,&. kurtherJore| 6elieKing thMt the sign ®6elongs3a to the 5recejing terJ rMther

thMn the terJ Mfter leMjs to incorrectlT siJ5lifTing eaV5ressions such Ms d a BV aO dV to d a ,V. in further errors Iith Mlge6rMic eV5ressions P§¨cheJMnn| NáaíC&. kor eVMJ5le| stujents

5osej Iith the folloIing 5ro6leJ ³m55les costs

eMch. ¤hMt joes the eV5ression RM O B6 stMnj for¹´ wMnT stujents inter5ret the eV5ression to JeMn four M55les Mnj three 6MnMnMs P"ush © §aMr5| dxNB&. mll these Jisconce5tions leMj to further jifbcultieas in Mcce5ting Mn Mlge6rMic eV5ression Ms Mn MnsIer to M 5ro6leJ. Dhis is referrej to Ms ³lMcfi of closurae´ Ihere M stujent 5roceejs to solKe Mn eV5ression jue to M focus on the solution aP"ooth| NáCC&. yn shaort stujents3 Jisconce5tions in MlJost Mll the MreMs jiscussej so fMr hinjer their M6ialitT to Iorfi effectiKelT Iith eV5ressions. eV5ressions Mnj injeej Mlal the 5rere%uisite content MreMs leMj to jifbculties in solaKing Mlge6rMic e%uMtions. Dhe M6ilitT to solKe e%uMtions is reliMnt on 6oth 5rocejurMl Mnj conce5tuMl unjerstMnjing P"ooth © 8MaKen5ort| dxNB&. m solij unjerstMnjing of hoI to use KMriM6les to Irite Mlge6rMic eV5ressions| forJ su6se%uent e%uMtions Mnj solKe Ihen necessMrT is the essence of success in Mlge6rM Mt qunior cTcle leKel. Stujents Mre often introjucej to the notion of solKaing Mn e%uMtion through o55ortunities to eV5erience the su6stitution of nuJ6ers faor letters for eVMJ5le d O V ¯ v. wuach of the reseMrch into jifbculties Iith soalKing e%uMtions suggests thMt it 6ecoJes Juch Jore jifbcult Ihen stujents encounter one KMriM6le e%uMtions Iith the KMriM6le on 6oth sijaes of the e%uMtion Ms stujents cMn no longear relT on MrithJetic to solKe the e%uMtion P-erscoKics © WincheKsfii|

NááR/ §ierMn| NáCx&. FKer bftT conce5ts relMtej to solKing e%uMtions hMKe 6een ijentibej

P:hung © 8elMcruª| daxNR&. ¤ithout the unjerlTinga conce5tuMl unjerstMnjing stujents leMrn M series of JaeJorisej trMnsforJMtionMl rules for solKing e%uMtions Ihich cMn leMaj to Jisconce5tions. Stujents cMn lose the aconce5t thMt inKerse o5erMtions Mre 6eing 5erforJej Iithout chMnging thea e%uMlitT of the e%uaMtion P:M5rMro © ¥offrion| dxxva&. kurther 5ro6leJs Mrise Iith the coJ5utMtions inKolKej es5eciMllT if neagMtiKe nuJ6ers| frMctions| or jeciJMlas

Mre inKolKej P¤u| dxxN&.

:hung Mnj 8elMcruª aPdxNR& outline three leKels of cognitiKe jeKelo5Jent in stujents Iho cMn solKe e%uMtionsA solKing e%uMtions cMn co5e Iith e%uMtions in jifferent forJs| ones thMt inKolKe Julti5le terJs Mnj Mre coJfortM6le Iith the trMnsforJMtiKe rules re%uirej to siJ5lifT the e%uMtion Mnj Iith the jifferent forJs the e%uMtion tMfies Ms it JoKes toIMrjs the solution. Dhese stujents hMKe of the e%uMtion or 5ro6leJ Mnj finoI IhT or IhT not to M55lT 5MrticulMr atrMnsforJMtiKe rules to solKe the 5ro6leJ. Dhe bnMl leKel of jeKelo5Jent is Mre Mt M leKel Ihere theT cMn 5lMn Mnj Jonaitor eMch ste5 of the solution 5arocess. yt hMs 6een suggestej thMt the conce5tuMl M55roMch to teMching Mlge6rMic e%uMtions| using tM6les Manj grM5hs| Ioulj foster 6etter stujent unjerstMnjing Mnj therefore 6etter e%uMtion solKing sfiills P:MrrMher Helej © :MrrMher| dxxC&| ³it 5erJeMates Mll of JMtheJMtics| froJ brst TeMr Mlge6rM through

cMlculus Mnj 6eTonj| Ms Iell Ms Jost M55licMtions of JMtheJMtics´ P¤illough6T| Nááí| 5.

dN,&. m function is jaebnej Ms M corres5onjence 6etIeen tIo sets P§ierMn| Náád&| Mnj there Mre tIo generMl M55roMches to teMching Mnj leMrniang functionMl relMtionshi5s Jentionej a in the literMture/ M corres5onjence M55roMch Mnj M coKMriMtion M55roMch PmTMlon dxN,&. Dhe corres5onjence M55roMch jeMls Iith Mn ain5utaout5ut Jojel| Ihaere6T Mn out5ut KMlue T is cMlculMtej for M giKen in5ut KMlue V| often listej in M tM6le of KMlues or Ms cou5les PV|T&. m coKMriMtion M55roMch inKolKes unjerstMnjing thMt Ms the in5ut PVaKMlue& chMnges then so joes the out5aut PTaKMlue&| it entMils MnMlTsing| JMni5ulMting| Mnj unjerstMnjing the relMtionshi5s 6etIeen chMnging %uMntiaties.

7onjM Pdxxá& jebnes afour groIth 5oints for stujents Ihen Iorfiing Iith functionasA

Pi& e%uMtions Mre 5rocejures for generMting KMlues| Pii& e%uMtions Mre re5resentMtions of

relMtionshi5s| Piii& e%uaMtions jescri6e 5ro5erties of relMtionshi5s Mnj PiK& faunctions Mre o6qects

thMt cMn 6e JMni5ulMtej Mnj trMnsforJej. ¦Mn je ¤Mlle Ms Mnother re5resentMtion of M function| areferrej to Ms the conteVt. Dhe M6ilitT to 5lot grM5hs froJ tM6les Mnj 5Mtterns is re%uirej for 5rocejurMl ¬uencT| Ihich soJe stujents eV5erience jifbcultT Iith P"ottoJs| dxxB&| Ihile others hMaKe jifbcultT inter5reting grM5hs P§il5Mtricfi TfO of re5resentMtion to Mnother Mnj hinjear conce5tuMl unjerstMnjing of functionsa in the 5rocess. HMtterns Mre M 5o5ulMr 5ejMgogaic tool for teMching chiljren to generMlise Mnj Mre one

of the routes to Mij unjerstMnjing of KMriM6les Mnj Mlge6raM oKerMll PwMson| Nááv&. Dahe M6ilitT

to see 5Mtterns| eV5lMin theJ Mnj re5resent theJ Iith Mlgea6rMic sTJ6olisJ is iJ5ortMnat for JMtheJMticMl 5ro6leJ solKing Mnj iat ³hMs 5ejMgogic iJa5ortMnce toIMrjs unjerstMnjing functions 6ut is Mlso aMn enj in itself´ PmTMlon the relMtionshi5 Ihich lies a6etIeen the 5Mttern Mnj its 5ositiona in M functionMl relMtionshi5 Just 6e ijentibej| Mnja MccorjinglT Mn eV5ression or forJulM Just 6e creMtej using KMriM6les. yn joing this M conteVt for the use of KMriM6les is set for stujents| Mssisting their unjerstMnjing

of M KMriM6le Ms M KMrTing %uMntitT rMther thMn M s5ecibc aunfinoIn PDMni°li © §±se| dxNN&.

Dhis section jiscussej the issues Iith the teMching Mnj leMrniang of initiMl Mlge6arM Mnj the 5rere%uisite Mnj Mlge6rM content MreMs Ihere stujents Just jeKelo5 their finoIlejge Mnj sfiills to succeej Iith the su6qect.a kMilure to recognise stujents3 Mlge6rMic Jisconce5tions

Pms%uith

5rogression Iith the su6qecta Ms M Ihole P7Mlston

stujents3 finoIlejge of initiMl Malge6rM Mnj MiJej to ijentifT coJJon Jisconce5tions Iith the MiJ of 5roKijing inforJMtion for ejucMtors to hel5 inforJ instruction. Dhe neVt section outlines the Jethojs aeJ5loTej in collecting Mnj JeMsuraing this finoIlejge. :lsdhfh,hu1 ejucMtion Iith Mn MiJ ofa ijentifTing stujents3 strengths Mnj IeMfinesses in relMtion to initiMl Mlge6rM. Do collect this eKijence M suitM6le instruJent for JeMsuring M stujent3s finoIlejge of Mlge6rM IMs re%uirej. mn MssessJent in the forJ of M stMnjMrjisej criterion referencej screener for Mlge6rM hMs 6een jeKelo5ej Ms 5Mrt of thais reseMrch. Dhe re%uireJent for M ¤hen