Trouble With Maths

The Trouble with Maths

This book offers important insights into the often-confusing world of numeracy. By lookingatlearningdifficultiesinmathsfromseveralperspectives,includingthelanguageofmathematics,thinkingstylesandthedemandsofindividualtopics,SteveChinndeliversacomprehensivetextwhichwillbecomeanessentialclassroomcompaniontoanyonewhouses it.

Whilstconsideringeveryaspectconcerningmathsandlearning,thisbookprovidesaperfectbalanceofadvice,guidanceandpracticalactivities,enablingthereaderto:

developflexiblethinkingskills; usealternativestrategiesforpupilstoaccessbasicfacts; implementeffectivepreventativemeasuresbeforedisaffectionsetsin; recognisemathsanxietyandtackleself-esteemproblems; makeaccurateongoingassessmentsofpupilsdifficulties; designinformaldiagnosticprocedures.

Withusefulfeaturessuchaschecklistsforevaluationofbooks,softwareandtestmaterials,this book highlights essential skills that will allow teachers to diagnose and address maths difficulties and improve standards. It draws on tried and tested methods based on theauthorsyearsofclassroomexperiencetoprovideanauthoritative,yethighlyaccessibleone-stop classroom resource for all teachers, classroom assistants, Special EducationalNeedsCo-ordinators,studentteachers,andlearningsupportstaff.

Steve ChinnisPrincipalandFounderofMarkCollege,Somerset,aspecialistschoolfordyslexicsawardedBeaconschoolstatusbytheDepartmentforEducationandSkills.


Apracticalguidetohelpinglearnerswithnumeracydifficulties

SteveChinn

LONDONANDNEWYORK

Firstpublished2004byRoutledgeFalmer 2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN

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ThiseditionpublishedintheTaylor&Francise-Library,2009.

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2004SteveChinn

Allrightsreserved.Nopartofthisbookmaybereprintedor reproducedinanyformorbyanyelectronic,mechanical, orothermeans,nowknownorhereafterinvented,including photocopyingandrecording,orinanyinformationstorage orretrievalsystem,withoutpermissioninwritingfrom

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ISBN0-203-35809-0Mastere-bookISBN

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Contents

List of illustrations vi Foreword ix

1 Introduction:learningdifficultiesinmathematics 12 Factors that affect learning 203 Whatthecurriculumaskspupilstodoandwheredifficultiesmay occur

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4 Thinking styles in mathematics 525 Developmentalperspectives 696 The language of maths 867 Anxietyandattributions 978 The inconsistencies of maths 1059 Assessment and diagnosis 11110 Thenastieslongdivisionandfractions 130

Appendix 1 Further Reading 142Appendix 2 Checklists and resources 143Appendix 3 Jog Your Memory cards for multiplication facts 146Appendix 4 Setting an inclusive maths department policy 150Appendix 5 Criterion referenced tests 155

Notes 165 Index 167

Illustrations

Figures

2.1 Placevaluecards 302.2 A blank task analysis form 312.3 Anexampleofataskanalysis 323.1 Using10pcoinsasimagesofnumberbondsfor100 393.2 Basetenblocksusedtoshow2312 423.3 Areamodelfor3117 433.4 2839comparedto3040asareas 443.5 Divisionbysubtractionofchunks 454.1 Subtractiontheinchwormway 534.2 Whatistheareaoftheshadedfigure? 554.3 Trial and adjust the inchworm way 595.1 Area models for multiplication 725.2 Theblanktablesquare 745.3 Thealmostcompletetablesquarewiththeeasynumberfacts shaded

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5.4 610comparedto69 765.5 Grouping repeated additions into partial products 775.6 Multiplicationanddivisionusingtheeasynumbers 795.7 Imagesofnumberbondsfor10 815.8 Numberbondsfor100withbasetenblocks 826.1 Placevaluearrowcards 877.1 Expectations 1019.1 Seatingfortheassessment 1169.2 Dotrecognitionandcounting 1189.3 Placevaluecards 1219.4 Screeningtestforbasicnumeracyskills 1229.5 Anon-diagnosticworksheetforaLevel2pupil 12910.1 Traditionallongdivision 130

Figures vii

10.2 The criss-cross-times method for fractions 13410.3 Thehiddendivisionsign 13610.4 136

10.5 Fractions and clocks 13910.6 Renamingfractionstomakethemhavethesamename 140

Tables

4.1 Thinking styles of the inchworm and the grasshopper 524.2 MathematicslearningstylesIandII 53

Foreword

Formanyyears,thefieldofchildrenslearningdifficultieshasbeendominatedbyresearchonreadinganddyslexia.Thereisnowalargebodyofknowledgeconcerningthenatureandoriginsofdyslexiaandhowbesttoteachreadingtochildrenwithsuchdifficulties.However,learningtoreadisnottheonlyskillthatposesdifficultyforchildrenwithdyslexia-relateddisordersthemajorityalsohavedifficultywithatleastsomeaspectsofnumberwork.TheTroublewithMathsaddressesthisissueandshowshowallwhoteachandlearnmathscanenjoyit.

Scientificresearchonnumberskillshasincreasedduringthepasttenyearsandtheevidencebaseisgrowingatarapidrate.Itisnowknownthat,fromshortlyafterbirth,infantscandetectchangesinthenumberofobjectsinvisualdisplaysofobjects.Thisinnatesenseofnumerosityisthefoundationoflaternumberconceptsthatdeveloprapidlyduringthepre-schoolyears.Buttobecomenumeratedemandsmore;childrenneedtolearnconventionalsystemsandtousetheirmathematicalthinkingmeaningfullyandflexiblyinlogicalsituations.Neuroscientistsmappingbrain-behaviourrelationshipshaveshownthatmathematicalexpertisedrawsonawiderangeofneuralcircuitsintherightandlefthemispheresthatmustworkinconcerttosolvemathematicalproblems.Itisthereforenotsurprising,asSteveChinnpointsout,thatchildrensproblemswithnumeracyarediverseandthetermdyscalculiamaybetoobroadtocapturethewiderangeofproblemsthatareobservedinchildrenacquiringmathematicalskills.

Atthebroadestlevel,numeracyproblemsmayaffecteithermathematicalthinkingorcom-putationalskills.Butduringdevelopment,asthisbookshows,thesituationisneverquitesosimple;mathematicallearningisacumulativeprocesssothatapoorlydevelopedconceptofnumbercanaffecttheacquisitionofnumberfactsandbythesametoken,poorarithmeticabil-itycancompromisethegrowthofmathematicalknowledge.Furthermore,schoolmathemat-icsisembeddedinlanguage,requiressequencingskills,drawsonworkingmemorysystems,dependsonspatialconceptsand,morethananyotherschoolsubject,engendersanxiety.Addtothisthefactthattheassessmentofmathematicallearningusuallyrequiresthechildtoreadandcomprehendwrittenquestionsandtolayouttheirworkclearly,andthewholeareabecomeaminefieldforthechildwithspecificlearningdifficulties.

SteveChinnhasdoneagreatjobofsystemisingwhatcanbedonetohelpthechildwhohasproblemswithmaths.Hismessageiscleartakeaccountofthestrengthsofthechildandthestyleoftheirthinking,marrythiswiththedemandsofthenumeracycurriculumandadjustlessonstohelp.Thebookshowshowcarefulassessmentcanleadtothedevelopmentofeffec-tivestrategiestohelpwithbasicandmoreadvancedconceptsofmaths.Thereisdiscussionofwhatdatamathematicstestscanprovide(andbyinferencewhattheycannot)andachecklistforchoosingassessmentinstruments.Thereareheapsofideasforsuccessfulteachingandadviceonbothhowtosetupanindividualeducationplanandhowtodevelopaninclusivemathemat-ics curriculum.

Writtenfromtheperspectiveofahighlyskilledpractitionerandanexpertteachereducatorwhoenjoysandunderstandsnumbers,thisbookensuresthatallwhoreaditwillunderstandhowtoteachmathsinsuchawaythateverychildcanachievetheirpotential.

MaggieSnowling UniversityofYork

February 2004

Chapter1 Introduction

Learningdifficultiesinmathematics

Thisbookistohelpteachers,classroomassistantsandlearningsupportassistantswhohavetodealwithpupilswhoareunderachievinginmathematics.Ittakesseveralperspectivesonthesituation,frompreventivemeasurestodiagnosisandidentificationofdifficulties,tothinkingstyles,toideasforintervention.Itworkslikearepairmanualinsomerespectsandlikeacareawarenessmanual(lookingafteryourstudents)inotherrespects.

It isabookwhichcanbeaccessed indifferentways. It canprovideanoverviewofwhereandhowproblemsmayarise.Itoffersinsightsintoareasofpotentialdifficulty.Itcanfocusonaparticularproblemandsuggestapproacheswhichcanhelpthepupiltolearn,butitwouldbeanimpossibletasktoattempttoprovideananswerforeveryproblemforeverychild.

Itcanbeusedto:

identifyaproblem understandpossiblereasonsforaproblem pre-emptproblems developflexiblethinkingskills circumventproblemsinbasicnumeracy addressthedifficultiespupilshavewithwordproblems teachalternativestrategiesforaccessingbasicfacts recognisemathsanxiety,attributionalstyleandself-esteemproblems designinformaldiagnosticprocedures extractdiagnosticinformationfrompupilswork stimulateideasforteachingmathstopupilswhoarefacingdifficultieswiththe

subject.

Sometimesyoumayfindinformationrepeatedindifferentchaptersof thebook.This isdeliberateassomeobservationsfitintomorethanonearea.Thenewareashouldgiveadifferentperspectivetothatinformation.

What do learners need to be good at mathematics?

Although this book is aboutwhat to dowhen learners are underachieving inmaths, itshouldbevaluabletoconsiderwhatlearnersneedtobegoodatmaths.Ihavetwosourcesforthisinformation.OneisfromtheUSSRandtheotherfromtheUSA.

Krutetskii1 presented a broad outline of the structure of mathematical abilities during schoolage.Hespecifies:

Theabilityforlogicalthoughtinthesphereofquantitativeandspatialrelationships,numberandlettersymbols;theabilitytothinkinmathematicalsymbols.

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Theability for rapidandbroadgeneralisationofmathematicalobjects, relationsandoperations.

Flexibilityofmentalprocessesinmathematicalactivity. Strivingforclarity,simplicity,economyandrationalityofsolutions. The ability for rapid and free reconstruction of the direction of a mental process,

switchingfromadirecttoareversetrainofthought. Mathematicalmemory (generalisedmemory formathematical relationships), and for

methodsofproblemsolvingandprinciplesofapproach. Thesecomponentsarecloselyinterrelated,influencingoneanotherandformingintheir

aggregate a single integral syndrome of mathematical giftedness.

AlthoughKrutetskiimakestheseobservationsconcerninggiftednessinmathematics,theyareequallyappropriateforcompetence.Thereadercanseewherelearningdifficultiesmaycreate problems.

TheothersourceistheNationalCouncilofTeachersofMathematicsintheUSA,wholistandexplaintwelveessentialcomponentsofessentialmaths:

1 Problem solvingTheprocessofapplyingpreviouslyacquiredknowledgetonewandunfamiliarsituations.Studentsshouldseealternatesolutionstoproblems:theyshouldexperienceproblemswithmorethanasinglesolution.

2 Communicating mathematical ideas (receiving and presenting)Studentsshouldlearnthe language and notation of maths.

3 Mathematical reasoning Students should learn to make independent investigationsofmathematical ideas.They should be able to identify and extend patterns and useexperiencesandobservationstomakeconjectures.

4 Applying maths to everyday situationsStudentsshouldbeencouragedtotakeeverydaysituations, translate themintomathematical representations(graphs, tables,diagramsormathematicalexpressions),processthemathsandinterprettheresultsinlightoftheinitial situation.

5 Alertness to the reasonableness of resultsInsolvingproblems,studentsshouldquestionthe reasonableness of a solution or conjecture in relation to the original problem. They mustdevelopnumbersense.

6 EstimationStudentsshouldbeabletocarryoutrapidapproximatecalculationsthroughtheuseofmentalarithmeticandavarietyofcomputationalestimationtechniquesanddecide when a particular result is precise enough for the purpose in hand.

7 Appropriate computational skills Students should gain facility in using addition,subtraction,multiplicationanddivisionwithwholenumbersanddecimals.Todaylongcomplicatedcomputationsshouldbedonewithacalculatororacomputer.Knowledgeof single digit number facts is essential.

8 Algebraic thinkingStudentsshouldlearntousevariables(letters)torepresentmathematicalquantitiesandexpressions.Theyshouldunderstandandusecorrectlypositiveandnega-tivenumbers,orderofoperations,formulas,equationsandinequalities.

9MeasurementStudentsshouldlearnthefundamentalconceptsofmeasurementthroughconcreteexperiences.

10Geometry Students should understand the geometric concepts necessary to functioneffectivelyinthethreedimensionalworld.


Introduction 3

11StatisticsStudentsshouldplanandcarryoutthecollectionandorganisationofdatatoanswerquestionsintheireverydaylives.Studentsshouldrecognisethebasicusesandmisuses of statistical representation and inference.

12 Probability Students should understand the elementary notions of probability todeterminethelikelihoodoffutureevents.Theyshouldlearnhowprobabilityappliestothe decision making process.

And,pickinguponKrutetskiisfirstpointconcerning theuseof symbols inmaths, theBritishpsychologist,Skemp2wrote:

Amongthefunctionsofsymbols,wecandistinguish:

(1)Communication(2)Recordingknowledge(3)Thecommunicationofnewconcepts(4)Makingmultipleclassificationstraightforward(5)Explanations(6)Makingpossiblereflectiveactivity(7)Helpingtoshowstructure(8)Makingroutinemanipulationsautomatic(9)Recoveringinformationandunderstanding(10)Creativementalactivity.

Heconcludesthat:Itislargelybytheuseofsymbolsthatweachievevoluntarycontroloverourthoughts.

So,wehave somecharacteristics forbeinggoodatmathematics.Couldoneassumethatdeficitsinallorsomeoftheseskillscreatedifficultiesinmathematics?Iknowfromexperiencethattherearemanyreasonswhysomeonemayunderachieveinmathematicsandthatthepictureisacomplexonewithnosinglerootcause.Recentlythetermdyscal-culia has become more prominent and so a look at what dyscalculia might be may help our understandingofotherreasonsfordifficultieswithmathematics.

Dyscalculia, definitions and descriptions

Dyscalculia,aproblemwithlearningmathematics,isattractingattentioninofficialcircles.TheDfES(DepartmentforEducationandSkills)haspublishedabooklet(seebelow)onguidanceforsupportingpupilswithdyscalculia(anddyslexia)intheNationalNumeracyStrategy(NNS).Thereisnowascreeningtestfordyscalculia3publishedbyNFER-Nelson.Despitethisattentiontheliteratureondyscalculiaissparseandthedefinitionsarealittleblandatpresent.Ihaveextractedafewfromvarioussources.

Developmentaldyscalculia isdefinedbyBakwinandBakwin (1960)asa difficultywithcountingandbyCohn(1968)asafailuretorecognisenumbersormanipulatetheminanadvancedculture.Gerstmann(1957)describesdyscalculia(Gerstmannssyndrome)as an isolated disability to perform simple or complex arithmetical operations and animpairmentoforientationinthesequenceofnumbersandtheirfractions.

Kosc(1974)describesdevelopmentaldyscalculiaasastructuraldisorderofmathematicalabilities which has its origin in a genetic or congenital disorder of those parts of the brain

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that are the direct anatomico-physiological substrate of the maturation of the mathematical abilities adequate to age,without a simultaneous disorder of generalmental functions.Thisdefinitionclearlyputsdyscalculia in the inheritedandspecific learningdifficultiescategory.

In1996Magnegaveaslightlymorecautiousexplanationofadifficultyinmathematicsasthelowachievementofapersononacertainoccasionwhichmanifestsitselfasperformancebelowstandardoftheage-groupofthispersonorbelowhisownabilitiesasaconsequenceofinadequatecognitive,affective,volitional,motororsensoryetc.development.Thecauseforinadequatedevelopmentmaybeofvariouskinds.Thisdescriptionacknowledgesthattherewillbemorethanonecausefordifficultiesinmathematics.

Mahesh Sharma (1986) lists the many words that have been suggested for mathsdifficulties,explainingtermssuchasacalculia,dyscalculia,anarithmeticaandnotingthatthereisnodefiniteagreementontheiruseuniversallyintheliterature,thattheyhavenotbeenusedconsistently,andalthoughtherearesignificantdifferencesbetweendyscalculiaandacalculia,someauthorshaveusedthetermsinterchangeably.Heconcludesthat thedescriptionsofthesetermsarequitediversetosaytheleast.

Sharmasuggeststhatdyscalculiareferstoadisorderintheabilitytodoortolearnmath-ematics, that is, difficulty in number conceptualisation, understanding number relation-shipsanddifficultyinlearningalgorithmsandapplyingthem.Itisanirregularimpairmentofability.ThusSharmasuggeststhatdyscalculiaisaspecificlearningdifficulty.

Acalculia isused tolabelamoreseriouscondition, thelossof thefundamentalpro-cessesofquantityandmagnitudeestimationandacompletelossoftheabilitytocount.Thisisanacquiredcondition.

And,finally,somethingthatsoundslikeamissingmemberoftheRussianroyalfamily,arithmastenia,definedasauniformdeficiencyinthelevelofmathematicalabilities.

IwrotethefollowingfortheDyslexiaInstitutesjournal,Dyslexia Review,volume14,number3,2003.Itisreprintedwiththeirpermission.

Does dyscalculia add up?

Initial ramblings

Is dyscalculia dyslexia with moths?

WiththepublicationofBrianButterworthsThe Dyscalculia Screener and the inclusion ofdyscalculiaasaspecificlearningdifficultyintheDfESconsultationdocumentforthe2004SENcensus,dyscalculiaisahottopic.Thisarticlesums(!)upmycurrentthinkingaboutdyscalculia.Unfortunatelymycurrentthinkingisfluid.Iamtryingtomakesenseofallthosefactorswhichinfluencethemathslearningoutcomesofchildrenandadults.So,Ihope this paper may attract some responses and stimulate more research.

SinceabsoluteknowledgeondyscalculiaisinshortsupplyIamgoingtoconstructthispaperaroundthequestionswhichIconsiderwehavetoinvestigatetoreachanunderstandingofdyscalculia.Indoingthisthereseemtobesomeveryinterestingcomparisonsbetweendyscalculiaanddyslexia.


Introduction 5

TherearesomethingsIknowasastart.Iknowthatdyscalculiawillnotbeasimpleconstruct (I think thatmeansapsychological concept). I know that therewill bemanyreasonswhyapersonmaybebadatmaths.IknowtherewillnotbeanyinstantorsimplecuresbecauseIknowthatthereisunlikelytobeasinglereasonbehindtheproblemofthemany,manypeoplewhofailtomastermathsandIknowthatnotallofthesewillbedyscalculic.

IheardDavidGearyspeakatthelastIDAconference.ThisAmericangurucomparedour knowledge of dyslexia to being close to adulthood and our knowledge of maths/dyscalculiatobeinginitsearlyinfancy.Thisisreflectedinthenumberofresearchstudiesdoneonlanguagedifficultiescomparedtothosedoneonmathsdifficulties.Asforstudiesondyscalculia,theyarefewindeed.Ithinktherearesomanyparallelsatsomanylevelsbetweendyslexiaanddyscalculiaandallthatsurroundsthesespecificlearningdifficulties,forexampleprevalence,definition,teachingmethods,etiologyandsoforth.

We are some twenty years behind language/dyslexia studies in our knowledge andunderstandingofdyscalculia.This isnot tosay thatI think itwill takeustwentyyearstocatchupinallareas,butthatittakesagoodlengthoftimefortheconcepttobecomeacceptedineverydayeducationalsettingsandthusforunderstandingstobuildfromworkfromtheshopfloor.

So,letsgobacktwentyyearstoamuchquoted,pioneeringpaperbyJoffe(1980).OneofJoffesstatisticshasbeenappliedover-enthusiasticallyandwithoutcarefulconsiderationofhowitwasobtained.Thisisthe61percentofdyslexicsareretardedinarithmeticandthus39percentarenot.Nowitisnotquiteassimpleasthat.Thesampleforthisstatisticwas quite small, some 50 dyslexic learners.Themaths test onwhich the statisticwaslargelybasedwastheBritishAbilitiesScalesBasicArithmeticTestwhichisjustthat,atestofarithmeticskills.Althoughthetestwasuntimed,Joffenotedthatthehighattainmentgroupwouldhavedone lesswell ifspeedwasaconsideration.Theextrapolationsfromthispaperwouldhavetobecautious.OtherwritersseemtohaveoverlookedJoffesowncautiousanddetailedobservations,forexample,Computationwasaslowandlaboriousprocessforalargeproportionofthedyslexicsample.Youwillseeattheveryendofthispaper IhavementionedaMarkCollegepupilwhowas identifiedasdyscalculicby theButterworthscreener(wheretwooutofthefiveexercisesfocusonspeedandaccuracyincomputation)butwhoispredictedtoachieveaGradeAinGCSEmaths.

IthinktherearetworeasonswhyJoffespaperissofrequentlyquoted.Oneisthatitisagoodpaperandtheotheristhattherearesofewothersfromwhichtoquote.NFER-NelsondescribeBrianButterworthastheleadingexpertondyscalculia.Sadly,atthemomentitsaonehorserace.WeneedmoreresearcherstofollowButterworthsinitiative.

Definitions and labels

Asa(lapsed)physicistIhaveascientistsconceptofwhatmakesadefinition.Inphysicsonecancontrol thevariablesanddopretty reliableexperiments.Peoplearedifficult tocontrol(especiallyasteenagers).InthisrespectIviewsomedefinitionsasdescriptions.

The definition of a learning difficulty can be very influential and can have manyconsequences.Forexample,itcaninfluencetheallocationofresourcestoanindividualortoaschoolortotheeducationbudgetofanEducationAuthority.Foranindividual,knowing

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thatyourdifficultieshavea labelmaybea reliefandabenefit,but itmayalsocauseareactionnotatalldissimilartothatofgrievingforaloss.Sothereneedstobeasenseofresponsibilityandawarenessofalltheseimplicationsinthosewhocreatedefinitions.

Thereseemstohavebeenachange inthecultureof thedefinitionofdyslexia, fromtheallencompassingdefinitionsofthelate80stothefocused,minimalistdefinitionoftheBritishPsychologicalSocietyinthelate90s.Thiscouldwellbesignificant.ProfessorTimMilestalksoflumpersandsplitters.So,couldtherebespecificlearningdifficultieswhichmayencompassallorsomeofdyslexia,dyspraxiaanddyscalculiaorcanthethreedysshaveindependentexistences?

Does being dyscalculic exclude you from being dyslexic or dyspraxic? Does being dyslexic exclude you from being dyscalculic? Then, turning to the lumpers, does being dyslexic mean you are also dyscalculic and dyspraxic?

Itmayhelptoanswersome,ifnotallofthesequestionsifyouthinkofrealpeople,realindividualsandwhattheanswerwouldbeforJefforJane.MyfeelingisthattheanswertothefirsttwoquestionsisNoandtothethirdquestion,Notnecessarily.

Ofcoursetheanswersdependonthedefinitionscurrentlyassignedtothedifficulties.Illcomebacktosummarisemythoughtsondefinitionstowardstheendofthispaper,buthereisasmallsampleofdefinitionsfornow.

The first is from a DfES booklet (2001) on supporting pupils with dyslexia anddyscalculiaintheNNS.

Dyscalculia is a condition that affects the ability to acquiremathematical skills.Dyscalculic learnersmay have difficulty understanding simple number concepts,lackan intuitivegraspofnumbers,andhaveproblems learningnumber factsandprocedures.Eveniftheyproduceacorrectansweroruseacorrectmethod,theymaydosomechanicallyandwithoutconfidence.

Verylittleisknownabouttheprevalenceofdyscalculia,itscauses,ortreatment.Purelydyscalculiclearnerswhohavedifficultiesonlywithnumberswillhavecogni-tiveandlanguageabilitiesinthenormalrange,andmayexcelinnon-mathematicalsubjects.Itismorelikelythatdifficultieswithnumeracyaccompanythelanguagedifficultiesofdyslexia.

Theseconddatesfrom1970andisattributedtoKosc(1974:47).

Developmentaldyscalculiaisastructuraldisorderofmathematicalabilitieswhichhas its origin in a genetic or congenital disorder of those parts of the brain that are the direct anatomico-physiological substrate of the maturation of the mathematical abilities adequate to age, without a simultaneous disorder of general mental functions.


Introduction 7

Sharma(1990)discussesthreetermsfordifficultiesinmathematics,sayingthat,

Termssuchasacalculia,dyscalculia,anarithmeticathereisnodefiniteagreementon theiruseuniversally in the literaturetheyhavenotbeenusedconsistentlyalthoughtherearesignificantdifferencesbetweendyscalculiaandacalculia,someauthorshaveused the terms interchangeablythe descriptionsof these termsarequitediversetosaytheleast.

Heexplainsdyscalculiaandacalculiaas:

Dyscalculia refers to a disorder in the ability to do or to learnmathematics, thatis,difficultyinnumberconceptualisation,understandingnumberrelationshipsanddifficulty in learning algorithms and applying them. (An irregular impairment ofability).Acalculiaisthelossoffundamentalprocessesofquantityandmagnitudeestima-tion.(Acompletelossoftheabilitytocount.)

ThefinalexampleisfromaDfESconsultationdocumentcalledClassificationofSEN.Thedescriptionsaretobeusedinthepupillevelannualschoolscensusfrom2004(DfES2002).

Specific learning difficulty (SpLD) covers a range of related conditions whichoccuracrossacontinuumofseverity.Pupilsmayhavedifficultiesinreading,writ-ing,spellingormanipulatingnumberswhicharenottypicaloftheirgenerallevelofperformance.Pupilsmayhavedifficultywithshort-termmemory,withorganisationalskills,withhand-eyecoordinationandwithorientationanddirectionalawareness.Dyslexia,dyscalculiaanddyspraxiafallunderthisumbrella.

Pupils with dyscalculia have difficulty with numbers and rememberingmathematicalfactsaswellasperformingmathematicaloperations.Pupilsmayhavedifficulties with abstract concepts of time and direction, recalling schedules andsequencesofeventsaswellasdifficultieswithmathematicalconcepts,rules,formu-lasandbasicaddition,subtraction,multiplicationanddivisionoffacts.

What distinguishes dyscalculia from just problems with maths? What do we mean by prob-lems with maths? How big is the problem?

Wedontknow.Itwilldependonthedefinition.Itmayalsodependonthepersever-anceofthedifficulty.Goodnessknowshowmanypeoplehaveadifficultywithmaths.Likeallskills, ifyouceasetopractiseyoulosetheskillandfewadultspractisemaths,especially topics suchas fractionsoralgebra,after leavingschool.So theextentof theproblemcouldwellincreaseinadults.Forexample,astudydonein1995onbehalfoftheBasicSkillsAgencyof1714adultsaged35yearsfoundthatjustunderonequarterhadverylownumeracyskillsatalevelwhichwouldmakeitdifficulttocompleteeverydaytasks successfully.

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So I am sure that just having a difficulty with maths should not automatically earn you the label dyscalculic.

Dyscalculia introducesanotherword into thevocabularyofspecialneeds.Someseethese words as labels and thus as descriptors of a person. That would not be helpful.

OK, Im dyscalculic. So what?

I likethequestions,Whatif?andthefollowupSowhat?What ifIamdyscalculic,so what? I have to ask does being dyscalculic condemn the learner to being foreverunsuccessfulatmaths.Thatthenraisesfurtherquestions:

What does it mean to be successful at maths? and What skills and strengths does a learner need to be successful at maths? and Is it important to be successful at maths?

Atmyschool,MarkCollege,aDfESapprovedindependentschoolforboyswhohavebeendiagnosedasdyslexic,theresultsforGCSEmathsaresignificantlyabovenationalaverage.Usuallyatleast75percentofgradesareatCandabovecomparedtothenationalaverageofaround50percent.ObviouslyIbelievethatiftheteachingisappropriatethenalearningdifficultydoesnotnecessarilymeanlackofachievement.But,doesaCgradeoraboveinGCSEmathsdefinesuccess?Thatsaquestionforanotherarticle,so,forthepurposeofthisarticleletsassumeitisonecriterionandletsassumethisisonepieceofevidencethatappropriateteachingcanmakeadifference.

Asformaths,wellthereisthemathsyouneedforeverydaylife.Thisrarelyincludesalgebra, fractions (other than and ), co-ordinatesor indeedmuchofwhat is taughtin secondary schools. It does include a lotofmoney,measurement, some timeand theoccasional percentage.Take, as an example of a real lifemaths exercise, paying for afamilymeal ina restaurant. Itneedsestimationskills,possiblyaccurateadditionskills,subtractionskillsifusingcash,andpercentageskillsforthetip.

The Russian psychologist Krutetskii (1976) listed the components of mathematicalabilitywhichcouldactasadescriptionofwhatalearnerneedstobegoodatmathsandthusalsoactasaguideastowhatmaybethedeficitswhichhandicapthelearnerfailingtobegoodatmaths(alsolistedonpage21ffinmoredetail).

1 Anabilitytoformalisemathsmaterial(toabstractoneselffromconcretenumericalrelationships).

2 Anabilitytogeneraliseandabstractoneselffromtheirrelevant.3 Anabilitytooperatewithnumeralsandothersymbols.4 Anabilityforsequentialsegmentedlogicalreasoning.5 Anabilitytoshortenthereasoningprocess.6 Anabilitytoreverseamentalprocess.7 Flexibilityofthought.


Introduction 9

8 Amathematicalmemory.9 Anabilityforspatialconcepts.

What is maths?

Could a person be good at some bits of maths and a failure at other bits? Do you have to fail at ALL bits to be dyscalculic?

In terms of subject content, earlymaths ismostly numbers. Later it becomesmorevaried with new topics introduced such as measure, algebra and spatial topics. Up toGCSE,despite thedifferentheadings, themajorcomponent remainsasnumber.So thedemandsofmathscanappearquitebroad,andthiscanbeveryuseful,butnumbercanbeadisproportionatepartofearlylearningexperiences.

So poor number skills could be a key factor in dyscalculia. This might suggest that we have to consider the match between the demands of the task and the skills of the learner.

In termsofapproach,mathscanbeawritten subjectoramentalexercise. It canbeformulaicoritcanbeintuitive.Itcanbelearntandcommunicatedineitherway,orcom-bination of ways by the learner and it can be taught and communicated in either way or combination of ways by the teacher.

Mathscanbeconcrete,butfairlyquicklymoves to theabstractandsymbolic. Ithasmany rules and a surprising number of inconsistencies.

Intermsofjudgement,feedbackandappraisal,mathsisquiteuniqueasaschoolsub-ject.Workisusuallyabluntrightorwrongandithastobedonequickly.

Even on this brief overview it is obvious that the demands of maths are varied. The importance placed on speed of working could also be another key issue for learners.

Attitude and the affective domain

Idonthavethereference,buttherewasastudydoneinScandinaviawhichsummeduptheinfluencesoflanguageandmathsskillsonlife.ExcusemeifIstatetheinfluencessomewhatstarkly.Itisimportanttorememberthatpeopledonothavetofollowtheconclusionsofstatisticalanalysis.BeinggoodatEnglishdoesnotpredictsuccessinlife.BeingbadatEnglishpredictsfailure.Beingbadatmathsdoesnotpredictfailure.Beinggoodatmathspredicts success.

Of course we all know that being bad at maths holds no social stigma in Western culture. Indeeditmaywellattractmuchmutualempathy.Sotheconsequencesofdyscalculiaaregoingtohaveabettersocialacceptancethantheconsequencesofdyslexia.(Forexample,IrecentlyreadalettertotheTimesaboutarestaurantmenu,complainingthatsinceithadspellingmistakesthewriterwouldnotbeeatingthere.Thatmakessense).

Schools,ofcourserarelyreflectlife.Inschooltheremaywellbesignificantconsequencesofbeingbadatmaths,forexampletheallocationofthelearnertoateachinggroupwhich

10

maylimitthelevelsofworkinseveralothersubjects.Alsoinschool,unlikelife,itshardtoavoidthestuffyoudontlikeortheworkyoufeelyoucantdo.

Two key factors which aid learning are ability and attitude. The latter can go a long waytowardscompensatingfortheformer,butthenthetwofactorsareprettycloselyinter-linked,forexamplewhensuccessencouragesgoodattitude.

Somelearnersjustfeelthattheycantdomaths.Thismaywellbeaconsequenceofearlyunsuccessfullearningexperiencesorfeedbackwhichisseenasnegative.Thejudgementalnatureofmaths,togetherwiththecultureofhavingtodomathsquicklycanleadchildrentoavoidtheriskofbeingwrongagainandagainandthustodisassociatethemselvesfromthelearningexperience(Chinn,1995).Mathscreatesanxietyand,sadlyitusuallyseemstobeananxietythatdoesnotfacilitatelearning.Ashcraftet al.(1998)hasshownthatanxietyinmathscanimpactonworkingmemoryandthusdepressperformanceevenmore.

Some learners develop an attributional style for maths which makes their attitude personal as in Im too stupid to do maths, pervasive, I cant do any maths and permanent, Ill never be able to do maths. An individual with a combination of those three beliefs could well present as a dyscalculic.

Memory, short, long and not always working

IoftenposethequestioninmylecturesWhatdoesthelearnerbring?(tomaths).Ihavealreadymentioned some factors suchas anxiety.Butwhataboutmemory? I know thatKrutetskii(1976)listsmathematicalmemoryasarequirementtobegoodatmaths.Iamsurethatshorttermandworkingmemoryarevitalformentalarithmetic,particularlyforthosesequential,formulabasedmathsthinkers.

But can a learner compensate for difficulties in some of these requirements and thus suc-ceed in maths?

Nowletsgobacktoschool,inEngland,wherewehavetheexcellentNationalNumer-acyStrategy.Thistrulyis,inmyopinionanexcellentprogramme,buthoweverexcellenttheprogramme,itisvirtuallyimpossibletomeettheneedsofeverylearner.AnessentialpartoftheNNSintheearlyyearsofeducationismentalarithmetic.Nowthatsanactivitythatneedsmemories,long,shortandworking.Soalearnerwithapoorshorttermmemorycouldfailmathswhenitismentalmaths,eventhoughhemayhavethepotentialtobecomeaneffectivemathematician.Iffailureisinternalisedasanegativeattributionalstylebythelearnerthenthatpotentialmayneverberealised.

IsKrutetskiismathematicalmemoryaparallelwithGardnersmultipleintelligences?Perhaps there aremultiplememories. That would explain some of the discrepancies Ihaveseeninchildrensmemoryperformances.Likeanysubject,thereisabodyoffactual information for maths and if a learner can remember and recall this information then he willbegreatlyadvantagedandifhecant


Introduction 11

So good memories may be required for doing maths in general. Short term and working memories may be essential for mental maths and mathematical long term memory will be essential for the number facts and formulae you need when doing mental arithmetic.

Counting on and on

The first number test on the ButterworthDyscalculia Screener is for subitizing. This meansanabilitytolookatarandomclusterofdotsandknowhowmanyarethere,withoutcounting.Mostadultsdothisat6yearsoldplusorminusoneyear.

Apersonwhohastorelyentirelyoncountingforadditionandsubtractionisseverelyhandicapped in terms of speed and accuracy.Such a person is evenmorehandicappedwhentryingtousecountingformultiplicationanddivision.Oftentheirpageiscoveredinendlesstallymarksandoftentheyarejustlinedup,notgroupedas thatis,infives.Mathsisdoneincountingstepsofone.Ifyoushowthempatternsofdotsorgroups,theyprefer the lines and lines.

Itsnotjusttheabilitytoseeandusefive.Itstheabilitytoseenineasonelessthanten,tosee6+5as5+5+1,tocountonintwos,tensandfives,especiallyifthepatternisnotthebasiconeof10,20,30but13,23,33,43

Its the ability to go beyond counting in ones by seeing the patterns and relationships in numbers(ChinnandAshcroft1992).

Garden variety or what?

Howdowedistinguishbetweenagardenvarietypoorreaderandadyslexic?(Stanovich,1991) How do we distinguish between a garden variety poor mathematician and adyscalculic?

I think the answer has a lot to do with perseveration of the difficulty in the face of skilled and varied and appropriate intervention.

Canyoubeagoodreaderandstillbeadyslexic?Canyoubegoodatsomeareasofmathsandstillbedyscalculic?MyguessisthattheanswertobothquestionsisYes,butformathsitispartlybecausemathsismadeupoftopics,someofwhichmakequitedif-ferentdemands(andforbothquestions,goodappropriateteachingcanmakesuchadif-ference).

Once again I drift back to problems with numbers as being at the core of dyscalculia. And it is numbers that will prevail in real life, when algebra is just a distant memory. And I guess that the main problem is in accessing these facts accurately and quickly, usually straight from memory, rather than via strategies.

Couldtherebeaparallelbetweenphonicsandnumberfacts?Forexampleknowinghowtousephonicstospellawordcouldbecomparedtousingadditionfactstoadd,say,572+319.

12

Butthennotallfactorsareintellectual.Adifficultymaybeaffectedbyabureaucraticdecision.Somebureaucratsspecifyalevelofachievementthatdefineswhetherornotachildslearningdifficultiesmaybeaddressedorevenassessed,influencedinthisdecision,atleastinpartbyeconomicconsiderations.But,eventhen,isachildsdyslexiaordyscalculiadefinedsolelybyachievementscores?Isthereroomtoconsidertheindividualandwhathebringstothesituation?Sometimesthesedecisionsarebeingde-humanised.So,Iforeseeachildnotreceivingprovisionfordyscalculiaunlesshismathsageisfiveormoreyears

Teaching

IclaimedthatbeingaphysicistinfluencedthewayIthink.Iamalsoateacherandhavebeenforalmost40yearsandthoseyearshavecertainlyinfluencedthewayIthink,too.Theteacherpartofmythinkingsays,amongotherthings,Sohesdyscalculic,whatdoyouexpectmetodonext?

Well,myguess is thatusingtherangeofmethodsandstrategieswehavedevelopedatMarkCollegefor teachingourdyslexicpupilswillalsobeeffectivewithdyscalculicpupils.Indeedwehaveprobablytaughtmanypupilswhohavethecomorbidproblemsofdyslexiaanddyscalculia.Whatweaddressasteachersisthewaythepupilpresents,notapupildefinedbysomestereotypicalattributes.

Mycolleague,JulieKaywhenfacedwithalearnerwhoisstrugglingwithlearningmathsasksherselfthequestions,WheredoIbegin?HowfarbackinmathsdoIgotostarttheintervention?Thismaybeadifference,shouldweneedone,betweenthedyscalculicandthedyslexicwhoisalsobadatmaths.Itmaybethatthestartingpointfortheinterventionisfurtherbackinthecurriculumforthedyscalculicthanforthedyslexic.Yetanothertopictoresearch.Itmayalsobethatthesubsequentratesofprogressaredifferent.Anothertopic.

Andforafinalthoughtinthissection,Iask,Whatistheinfluenceofthestyleofcurric-ulum?Iknow,forexample,fromaEuropeanstudyinwhichIwasinvolved(Chinnet al., 2001),thatthedesignofthemathscurriculumcertainlyaffectsthinkingstyleinmaths.

So what?

There are many reasons why a child or an adult may fail to learn maths skills and knowledge. Forexample,achildwhofindssymbolsconfusingmayhavebeensuccessfulwithmentalarithmetic,butmayfindwrittenarithmeticverychallenging.Theremaybeotherexamplesof an onset of failure at different times which will most likely depend on the match between thedemandsofthecurriculumandtheskillsanddeficitsofthelearner,forexample,adys-lexicwillprobablyfindwordproblemsespeciallydifficultandachildwhoisnotdyslexic,butislearningattheconcretelevelmayfindtheabstractnatureofalgebradifficult.Achildwhoisanholisticlearnermaystarttofailinmathsifhisnewteacherusesasequentialandformulabasedinchwormteachingstyle.Alearnermayhaveapoormathematicalmemoryandthedemandsonmemorymaysuddenlyexceedhiscapacity.

A difficulty will depend on the interaction between the demands of the task and the skills and attitudes of the learner. For example, if one of the demands of mental arithmetic is


behindthenorm,whichcouldmitigateagainstearlyinterventionforsixyearoldpupils.

Introduction 13

that it be done quickly, then any learner who retrieves and processes facts slowly will have learning difficulties. Learning difficulties are obviously dependent on the learning task.

AndnoneoftheunderlyingcontributingfactorsIhavediscussedaretrulyindependent.Anxiety, forexample isaconsequenceofmany influences. I amhypothesising that thefactorsIhavementionedarethekeyones.Theremaywellbeothersandthepatternandinteractions will vary from individual to individual, but these are my choices for thedifficultiesatthecoreofdyscalculia.

OfthedefinitionsIhavequoted,ImuchprefertheNNSversion.TheDfESconsultationpaperversionseemstodescribe justdifficulties inmathswhichmightoccurinanyone.I have added some extra notes into the definitionwhichmay then be better seen as adescription(andthusnotalabel).

Dyscalculia is a perseverant condition that affects the ability to acquire mathematical skills despite appropriate instruction. Dyscalculic learners may have difficulty understanding simple number concepts(suchasplacevalueanduseofthefouroperations,+and),lack an intuitive grasp of numbers(includingthevalueofnumbersandunderstandingandusingtheinter-relationshipofnumbers),and have problems learning, retrieving and using quickly number facts(forexamplemultiplicationtables)and procedures(forexamplelongdivision).Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence(andhavenowayofknowingorcheckingthattheansweriscorrect).

TheNNSversionfocusesonnumber,whichmakessensetome.Itmentionsmemoryand it includes thosewhopresent as competent in someareas, butwhoseperformancehas no underlying understanding of number. An addendum could list some of the key contributors,suchas:

A learners difficulties with maths may be exacerbated by anxiety, poor short term memory, inability to use and understand symbols, and inflexible learning style.

Nowthedefinition/descriptionisinthisform,itmaybepossibletosetupadiagnosticprocedure.Thatsetsanotherresearchtaskandwedesperatelyneedmoreresearch!

Finally, have I met any learners whom I think would be described accurately as perseverently and exclusivelydyscalculic?Ihave,buttheywerefew.Imentiontwo,oneisafemale,giftedinlanguage(andlanguages)whohadabsolutelynoideawhat(presentedassymbols)wouldbe.Iaskedherwouldtheanswerbebiggerorsmallerthan50andsherepliedYes.Theother isamale,averageatlanguageskillsbutwhocouldnotseethatIheldoutthreefingers.Hehadtocountthem,evenasasixteenyearold.HeachievedaGradeGinGCSEmaths.

Therearemanyothersouttherewhomaypresentasdyscalculicasyounglearners.Itswhathappensnextthatconfirmsorchallengesthatdescription.

To add one piece of evidence to support the What happens next hypothesis, I used Butterworths TheDyscalculia Screener on one of our Year 10 students. The Screener diagnosed him as dyscalculic. We are predicting he will achieve an A grade in GCSE

14

perseverent.

If youwant to follow up references for this section, they are listed at the end of this chapter.

Forme,themainissuehereisthatnoteverychildoradultwhoisfailinginmathematicsisdyscalculic.Evenforthosewhodogainthislabel,itdoesnotpredictanoutcome,butitdoessuggesttomethatwhateverteachingexperiencesthisdyscalculicpupilhashad,theyhavenotbeenappropriate.Iknow,fromtenyearsofdataonpupilsatMarkCollege,thatitispossibleformostpupilstochangeahistoryofgainsinmathsageoflessthantwelvemonthsperyeartogainsofovertwelvemonthsperyear,thusmovingtocatchup.Someofthesepupilsmighthavebeendiagnosedasdyscalculic,somemightnot.Inmanysensesthatwaslessrelevantthantheirhistoryofunderachievementinmathematics.

Andwouldinterventionforadyscalculicbedifferent?Theansweristhatanydifficultyhas tobeviewed individually,but that thecoreprinciplesof teachingand learningwillprobablybedrawnfromthesamecompendiumofideasusedfordyslexics(seeChinnandAshcroft1998).

Adjusting lessons to help pupils who are having difficulties in learning maths

Adjustmentstolessonsshouldbebasedonfourprinciples:

Empathetic classroom managementwhichimpliesanactiveawarenessandconsequentadjustmenttothelearningstrengthsanddifficultiesofpupils.

Responsive flexibilityallowstheteachertohavearepertoireofresourcesandstrategieswhichrespondtotheindividual(andoftenchanging)needsofthepupil.

Developmental methodsaremethodsthataddresstheremedialneedwhilstdevelopingmathematical skills and concepts.

Effective communication which infers an awareness of thinking and learning style and anawarenessoflimitationssuchaslanguageskills,poorshorttermmemoryorslowerspeeds of working.

Theapplicationoftheseprinciplesshouldaffectalllevelsofwork,fromtheconstruc-tion of the syllabus and lesson plans to the setting and marking of homework.

Integrating dyscalculic pupils and other learners who have difficulties with mathematics into the real world of the

classroom

The syllabus and programme of work

1 Astructureorprogrammewhichbuildsinregularreturnstotopicshelpslearnerswithpoorerlongtermmemories.Frequentrevisionsandoverviews,especiallyafterashorttimelapseforreflectionhelptoreinforcelearning(largelyimplicitintheNNS).


maths next year. Are these two statements incompatible? I think the answer lies in the word

Introduction 15

2 Programmes that rely heavily on self tuition can allow pupils to develop incorrectprocedures and concepts. (I remember the Kent Mathematics Project where pupilsworkedlargelywithworkcardsandattheirownpace,butwithlittleornotuition.)

In the classroom

Short termmemory deficits can affectmental arithmetic skills (whichmay show amarkeddifferencetowrittenarithmeticskills).

Short termmemorydeficitscanaffectmanyotherareasoflearningsuchasthenumberofitemsofinstructionapupilcanprocessatonetime.Thesedeficitsmaybeauditoryorvisualorboth,sopresentationshouldalwaysaddressbothmodes.

Lookoutforshorttermmemoryoverload(whenthepupilwilljustbeoverwhelmedandrecallnothingatall).

Reading deficitsdonotaffectallareasofmathematicstothesamedegreeandareagoodexampleofadeficitthatgivesrisetoaseeminglyinexplicablechangeinlevelofperformance(thatis,whenwordproblemsareintroduced).

Some pupilsareslower toproducework,due tosuchfactorsaswritingspeed, poororganisationalskills,fingercountinginsteadofinstantrecalloffacts.Speedofworkingisoftenanissueinmathematicsandcanbethecauseofgreatlyincreasedanxietyandgreatly decreased levels of performance.Consider allowing extra time for tests andexaminations.Considercarefulselectionofquantitiesofworkforthesepupils.

Anxious learnersareoftenpoorrisktakersandwillnottryworktheyperceivetobedifficult,therebyavoidingfailure(theyhaveusuallyhadenoughexperienceoffailure),but they are then not accessing new learning experiences. Research in the 1920sshowed that a pupils first experience of applying newknowledge is the experiencethat persistsa big problem if he gets that first experiencewrong.Allow pupils toexperimentandfailasoneofthestepsonthepathtosuccess,butthishastobeacloselycontrolled strategy.

If recalloffacts(suchastimestablefacts)andprocedures(suchassubtractingfromzero)donotbecomeautomatic,thenthereislessmentalspacelefttodothemaintask.

Thiscompoundstheeffectofdifficulties.Selecteasynumberswhenintroducingnewarithmetical procedures.

16

Considergivingthepupilatimestablesquaretostickintothebackofhisexercisebook(sothathehastomakesomeefforttoturntotheinformation)andmakesurehecantrack successfully to the answers.Learningisusuallymoreeffectiveifitispresentedinamultisensoryway.Thisincludesthe use of concretemanipulatives,which are often phased out as being too youngfor secondarypupils.Manipulativesmaybeusedasdemonstrations, usingoverheadprojectormaterialsorviaPCandboards.Thisavoidsthebabyishimagewhenusedbyanindividuallearner.

Money isaneffectivemanipulativeandisonestepontoabstractionfromadirectlyproportionalmanipulativesuchasbasetenblocks.Alsoitislikelytobemoreacceptablefor older learners.

Concrete materials and manipulatives

As a teacher who started his career as a teacher of chemistry and physics one of the strange anomaliesofteachingmathematicsformeisthat,unlikescience,thefurtherthelearnergoesupthemathematicssyllabusthelesslikelyheistousepracticalmaterials.Somewherearound about eight years old learners start to consider materials as patronising and babyish. Maybe this has something to do with the fact that some of these materials are bricks.

Thiscanbecircumventedbytheteacherusingthematerials todemonstrateratherthanthelearnerengagingindiscoveryplaywiththematerial(thoseteachingphiloso-phiesfromthe1960sdiehard).

I think that one of themost important decisions a teachermakes is the choice ofmaterial(s)toillustrateaconcept.Itislikely,yetagain,thatthechoicewillbedepen-dentonthelearner.Atleast,inindividualwork,thefocusshouldbethelearnerchoosinga material that works for him rather than the teacher choosing hisfavouritematerial.

Materialshavetheirowninherentcharacteristics.Forexample,ametrerulewillgivealinearimageofunits,tens,hundredsandthousands,whereasDienes(baseten)blockswillgiveone,twoandthreedimensionalimagesofunits,tens,hundredsandthousands.Moneyisnotproportionalinsizetothevaluesitrepresents,butitisfamiliarandcomesineasynumbervaluesbasedon1,2and5.

Inaccurateintuitive,answer-orientedlearnersareatrisk.

Dyslexicpupilsdonotadjustquicklytochangesinroutine,forexampleifanewteacherexpectsadifferentpagelayoutinexercisebooks.

Sequential, formula-oriented learners with poor memories are at risk of failure in mathematics.

Some pupilsareintuitive,answer-orientedproblemsolverswhomaynotlearnfromastepbystep,sequentiallyoriented,formuladependentteacher,and,ofcourse,viceversa.Therearealsosignificantimplicationsfordocumentationofwork.Intuitiveworkersareusuallydisinclinedtodocument.Thesedifferencesincognitivestyle(thinkingthroughproblems) are present in the whole school population, including teachers, but theiraffectonpupilswithdyslexia,dyspraxiaordyscalculia(withtheirothercontributingproblems)islikelytobemorecritical(seeChapter4).


Introduction 17

orapupilreadstheproblemtothepup ilwithdifficulties.Thisisparticularlyrelevantforcourseworkandinvestigations.

Dealwiththepupilswhoareforgetfulandbadlyorganised.Takeapositiveattitudeandmakesuretheyhavetheinformationandequipmenttheyneed.Parentsofsuchpupilshaveusuallysufferedalongsidetheirchildashestrugglesthroughschool.Youcouldtrytoliaisewiththem,forexamplebygivingthemahomeworktimetable.Remember,difficultiesmaybefamilial.

Give homework in a form that they can access.For example check thevocabulary. Make sure the homework is read to the pupil before he takes it home. Get high tech and provideadiscsothatworkcanbedoneoncomputerandcouldevenhavethefacilityof thePCreadingthequestionstothepupil.

Consider allowingthepupiltouseacalculator(withallthecautionsIknowyouhaveabouttheiruse)oranumbersquareoratablesquare.

Marking

Mark newwork before toomany examples have been attempted. Do not let error patterns become ingrained.

Mark diagnostically.Forexample,thepupilmayhaveusedthecorrectprocedure,butmadeanarithmeticalerror.Donotjustmarkworkwrong.Sayhowitwaswrongandwhat can be done to put it right.

Homework

Ifadyslexic pupil ishavingdifficultysettingoutworkonthepagediscussgivinghim anexercisebookwhichhasbigger/smaller/squares.

Thereadinglevelmaybebeyondthepupil.Ifabookcannotbereplacedin theseeconomicallyhardtimeseitherprovideaphotocopiedtranslationormakesureyou

Work sheetsandtextbooklayoutscanbeoverwhelming.Theymayuse,forexam-ple, lotsofsmallprint,closelyspacedor fussy,confusedpageswithcartoonsanddisjointedtext.Tryprovidingacoversheet/windowwhichreducesthequantityofmaterial facing the pupil.

Books and worksheets

Itseemstobestatingtheobvious,butthematerialmustmaketheconceptaccessible.Itmustgivethelearneranunequivocalimageoftheideathat thesymbolsrepresent.Thematerialandthesymbolsmustberelatedinthelearnersmind(ratherthanjusttheteachersmind!)

18

The Catch 22 of Catch Up

Ifapupilfallsbehindheorshewillhavebeenworkingmoreslowlythantheirpeers.Tocatchupheorshewillhavetoprogressfasterthantheirpeers.Itispossible.

A few golden rules

Dontcreateanxiety.4 Experiencingsuccessreducesanxiety. Experiencingfailureincreasesanxiety. Understandyourpupilsasindividuals. TeachtotheindividualinthegroupalsoknownastheTeachmorethanonewayto

dothingsrule. Rememberwhereeachtopicleadsmathematically. Understandingisamorerobustoutcomethanjustrecall. Trytounderstanderrorsdontjustsettleforwrong. Preventionisbetterthancure. Alltheaboveruleshaveexceptions.

byanyof thesesuggestionsandmanywillbeadvantaged.Thesuggestionsmayreducesomeofthelearning(special)needsinyourclassroomandevenpreventtheonsetofsomeproblems.

Thisbookacknowledgesthatpupilsareindividuals.Ihavelonghadasuspicionofanyscheme,interventionorcurethatclaimsitisforall.Isuspectthattheonlypartofthisbookthatisforallistheemphasisonunderstandingeachpupil.

Remember

Pupilsareindividuals.Somewillneedsomeofthesesuggestions,somewillsurvivewithoutanyofthesesuggestions.Idonotthink,however,thatanylearnerwillbedisadvantaged

Remember thepupilwhomayworkmoreslowly thanhispeers.Consider selectingfewerexamplesbutstillgivingthebreadthofexperience.

Be encouraging.Avoidredpensandbigcrossesandscribbles.(Trygreenandsmallandneatand,betterstill,constructivecommentssuchasSmalladditionerrorhere,restOK.)


Introduction 19

Gerstmann,J.(1957)SomenotesontheGerstmannsyndrome,Neurology7,pp.8669.Joffe, L. (1980) Dyslexia and attainment in schoolmathematics: Part 1,Dyslexia Review, 3 1,

pp.1014.Kosc,L.(1974)Developmentaldyscalculia,Journal of Learning Disabilities,7,3,pp.4659.Kosc,L.(1986)Dyscalculia.Focus on Learning Problems in Mathematics,8,3,4.Krutetskii,V.A.(1976)inKilpatric,J.andWirszup,I.(eds)The Psychology of Mathematical Abilities

in Schoolchildren.Chicago,UniversityofChicagoPress.Magne,O.(1996)Bibliography of Literature Dysmathematics Didakometry.Malmo,Sweden.Seligman,M.(1998)Learned Optimism.NewYork,PocketBooks.Sharma,M.C.(1986)Dyscalculiaandotherlearningproblemsinarithmetic:ahistoricalperspective,

Focus on Learning Problems in Mathematics8,3,4,pp.745.Sharma,M.(1990)Dyslexia,dyscalculiaandsomeremedialperspectivesformathematicslearning

problems.Math Notebook,8,710.Stanovich,K.E. (1991) The theoretical andpractical consequencesofdiscrepancydefinitionsof

dyslexia.InSnowling,M.andThomsomM.(eds),Dyslexia: Integrating Theory and Practise. London,Whurr.

Chinn,S.J.andAshcroft,J.R.(1992)TheUseofPatternsinT.R.Miles,andE.Miles,(eds)Dyslexia and Mathematics,London,Routledge.

Chinn,S.J.andAshcroft,J.R.(1998)Mathematics for Dyslexics: A Teaching Handbook.London,Whurr.

Chinn,S.J.,McDonagh,D.,VanElswijk,R,Harmsen,H.,Kay,J.,McPhillips,T.,Power,A.andSkidmore, L. (2001) Classroom studies into cognitive style in mathematics for pupils withdyslexiainspecialeducationintheNetherlands,IrelandandtheUK,British Journal of Special Education,28,2,pp.805.

Cohn, R. (1968) Developmental dyscalculia, Pediatric Clinics of North America, 15, (3), pp.65168.

DfES (2001) The National Numeracy Strategy: Guidance to Support Pupils with Dyslexia and Dyscalculia.London,DfES.

References

Ashcraft,M.,Kirk,E.P.andHopkins,D. (1998) On thecognitiveconsequencesofmathematicsanxiety,inDonlan,C.(ed.)The Development of Mathematical Skills.Hove,ThePsychologicalCorporation.

Bakwin,H.andBakwin,R.M. (1960)Clinical Management of Behaviour Disorders in Children.Philadelphia,Saunders.

Butterworth,B.(2003)The Dyscalculia Screener.London,NFER-Nelson.Chinn, S.J. (1995) A pilot study to compare aspects of arithmetic skill, Dyslexia Review 4,

pp. 47.

Chapter2 Factors that affect learning

Thischapterisaboutsomeofthefactorswhichmaycontributetoalearnerhavingdifficultieswithmathematics. Factors such as a poor short termmemory can have a considerableimpact on many of the topics that make up mathematics.

AttheendofthesectiononeachfactorIhaveleftspaceforthereadertoaddhisorherownsuggestions.Icouldneverclaimtoknowallthesolutions,especiallyasitissoimportant to remember that what works for one student may well not work for another. Hencetheneedforarangeofsolutions,tobeusedinresponsetotheindividualneedsofeach learner.

QuiteoftentheSuggestionssectionincludestheadvicetotaketimeandevaluate.The culture of doingmaths quickly can be totally opposite to the correct approach ofappraising before reacting.

Somesuggestionsareimmediatewhilstothersarelongterm.Schoolsandcolleges shouldwork towardsbuildingupabank/libraryof appropriate

resources,sothatmanyoftheseproblemscanbeaddressedfrommaterialsthathavebeenprepared before.

Short term memory

The problem SuggestionsSomechildrenwillhaveweakershorttermmemoriesthantheirpeers.Thismaybeadevelopmentallagorapersistentproblem.Whatisrelevanthere,however,istherealisation that there will be a range of short term memory capacities among the pupils in any classroom.

Thiswillimpactonthepupilsabilityjusttokeepupwiththe lesson in general.

Donotgivelengthystringsofinstructions.Imaginesomeonegivingyouatendigitphonenumbertorememberbypresentingit,once,spokenquicklyandwithno breakdown into chunks of numbers. Sopresent712563449as712563449or712563449orwhateverchunking size pupils prefer.

Inmentalarithmeticitmaybeachallengeforthemtorememberthequestion.

Repeatthequestion,orputthekeynumbers on the board.

Factors that affect learning 21

Theproceduretheyaretryingtousetosolvethequestionmayhavetoomanystepsfortheirshorttermmemorycapacity.

Ask what method they are try-ingtouse.Ifappropriatesuggestanalternativemethod,oracceptan estimate. Perhaps allow them to make memory jottings for intermediatesteps(halfwayhousetofullmentalwork).

Whenattemptingmentalarithmeticquestionstheyarehandicappedbyslowretrievalofbasicfacts.

Provideatablesquareorabasicadditionfactssquare.Givethemquestionswhichusethefactstheydoknow(usually2,5,10).

Whencopyingquestionsfromapage,worksheetoraboardtheyonlycopyoneortwoorthreeitems(lettersornumbers)atatimeandmaymixuppartsofquestions,creatingnewextraquestions.

Provideaworksheetwhichtheycanwriteon.Usecolourorhighlightersto help them to track their position on the board or worksheet. Mark the questionstheycreateratherthantheonestheyshouldhavedone.

Remember,theeasyoptionisforthelearnerstooptoutand not try the work.

Beawareofthisattitude,whichwillbuild into a major self esteem and attributional style issue.

Addyourownsuggestionsandsolutions(andextraproblems!)

Mathematics memory

(Thisislongtermmemoryformathematicalinformation).The problem SuggestionsSomepupilsfindinstantrecallfrommemory of basic addition facts a persistentproblem,particularlyinexams.

Allowfingercounting(butbeawarethatthiswillnotdevelopnumberskills).Teach pupils how to build on key addition facts such asdoublesornumberbondsfor10.Formoredetailseebelow.Teach pupils to write the number bonds for ten as their key factsandteachhowtoderiveotherfactsfromthesebonds.Teachpupilshowtodrawupanadditionsquare.

Somepupilsfindthatinstantrecallof multiplication facts is a persistent problem.

Givepupilsamultiplicationsquare(andshowthemhowtouseitfordivisiontoo).Teachpupilshowtocreateamultiplicationsquare.Teachpupilsstrategieswhichbuildontheeasyfacts.Forexample,doublethe2factstoobtainthe4facts,orworkoutthe9factsbyworkingbackfromthe10facts.Formore detail see below.

A pupil cannot remember the long multiplication algorithm.

Teachrepeatedaddition(inchunksof2,5,10,20,etc.).SeeChapter5.

A pupil cannot remember the divisionprocedure.

Teachrepeatedsubtraction(inchunksof2,5,10,20,etc.)SeeChapter5.


22

Direction

The problem SuggestionsSomechildrenfindthedirectionalityofmaths a challenge. The problem will be greatlyexacerbatedbyinconsistenciessuchasdivision(seebelow).

Forexample,somewillfindcountingbackwardsproblematic,moresothanyou,theteachermightpredict.

Thiswillrequiremorepractise,perhapsaskingforsmallersequencesofreverseordernumbers,startingwithoneless,onebackfromx.Startfromdifferentnumbers.Trycountingbackin2s,5s,10s(alsoas76,66,56,46aswellas70,60,50,40).Thenintroduce9sandrelateto10s.Trycountingtoatargetnumber18,16,14,_,_,_,6,4.Pointoutpatterns,usingcolourtohighlightthepat-tern.

ThepupilhearsNineteenandwrites91(whichactuallyfollowstheorderoftheword,asisthecaseforalltheteennumbers).

Theteennumbersaretheexceptionaltwodigitnumbers,withtheunitdigitcomingbeforetheten(teen).SeeTransposals,p.36.

Theymayalsofindthechangefromposi-tivecoordinatestonegativecoordinatesdisproportionatelymoredifficult.

Achancetoremindlearnerstolookfordetail,toabsorb before reacting. Perhaps the learner could labeltheaxesnegativeandpositiveboldlyandappropriately.Referbacktoworkonthenumberline.This may pre-empt the problem.

Childrenmayalsobephasedbythechange in direction of the steps for traditionalshortandlongdivision compared to the right to left working for addition,subtractionandmultiplication.

Teachalternatemethods,forexamplerepeated addition or subtraction.

Confusionmayariseoverthedifferentwaysinwhichdivisionisrepresented,forexample54dividedby9canbewritten

as:

Make sure you do not assume pupils will automatically take on board these representations. Explainthemeaningalongsidethealternatives.Inthesymbolthedotsarereplacedwiththenumbersofthefraction,thatis,inthisexamplethedotsarereplacedby54(onthetop)and9.

Thesequenceofplacevaluesasyoumoveleft or right of the decimal point may confuse,especiallywhenthelanguageisconsidered as in hundreds and hundredths.

Showthedigitsasyousaythewords.Useavisualimage or concrete material such as base ten blocks. Thisisonetenth,onedividedintotenequalparts,ormoneywith10pas of1.Therelationship

between and10canbeconsideredwith10pand10,bringinginlanguageandrewordingsuchasHowmany10psin10?comparedtotheabstract10dividedby (whichalsorephrasesadivisionquestionasamultiplicationquestion).




Visual

The problem SuggestionsThinkaboutthelookofapage,aworksheet,atest.

Lookcarefullyatpresentationwhenchoosingtextbooksorwhendesigningaworksheet.(SeealsoSpeedofWorking,p.27.)

Ifthepresentationiscrowded,withlittlespacebetweenlinesorquestions,thensomepupilswillexperienceanxiety.

Illustrationsmayhelpthelookofapage.Theyshouldbe pertinent.Iflinesofprintaresettooclosethenpupilsmaymixuplines,takingsomeinformationfromonelineandthenextinappropriatelyfromanotherline.Tryprovidingthepupilwithapieceofcardwithawindow or slot cut into it so only a part of the page is revealed.

Inanattempttobemoreuser-friendlysomebookdesignershavemadethelayoutveryfragmented.

Lookforsimple,cleardesign.Trythewindowstrategy(above).

The symbols for the four operations can be confusing if not written or printed carefully.A+canbeclosetoa.Alittlerotationcanmakea+anda indistinguishable

Chooseaclearfontatasuitablesize.Alsobewaryoflearnersperseverating,thatiscontinuingtoadd,forexamplewhentheproblemshavechangedtosubtract.Thepressureofworkingquicklycanexacerbatethisproblem,ascananxiety.Encouragethelearnertorelaxandtosub-vocalisethesymbolsorhighlightthem.

Pupilsmaymixupquestions,takingpart from one line and part from the line below.

Lineoffquestionsorspacethemout.Tryacolouredoverlay.Blackprintonwhitepaperblurstheimagesforsomereaders.(SuppliersI.O.O.Marketing,CityUniversity,London:admin@ioo marketing.co.uk).Markthequestiontheyhavecreatedratherthantheoneinthebook!

The page seems blurred. Tryacolouredoverlay.Tryprintingworksheetsoncolouredpaper.Trydifferentcolouredprint.Suggestavisittoaspecialistoptometrist.

Thepupilfindscopyingfromtheboarda problem.

Provideahandoutorusecolouredpenstobreakupinformation.Keeppresentationclearandwellspacedout.Trynottotalkwhenpupilsarecopying.(Somepupilscannotdualtask,i.e.listenandwriteatthesametime.)

Pupilscannotdrawa2Drepresentationofa3Dfigure,forexampleacuboid.

Tryprovidingisometricpaper.Showhowthedrawingof a cuboid is made up of two off-set rectangles joined by parallel lines.


24

Speed of working

The problem SuggestionsTeachersoftenexpectlearnerstodomentalarithmeticproblemsquickly.

Allowselectedpupilstohavealittlemoretime.Forexam-ple,askthequestion,sayyouwillcomebacktothem,askanotherlearner(ortwo)andthenreturnfortheanswer.Slowrecallofbasicfactsmaybetheslowinginfluence.Givethepupilatablesquareoranadditionsquare.Givepartoftheanswer,forexample,askingfor64+78,askfortheunitdigit:64plus78isonehundredandfortyorstructureWhatis8add4?Whatis60add70add10?

Pupilsmayfindhomework(andclass)tasksdifficulttofinish.

Setfewerquestions,butensurethepupilgetstoexperi-encetherange,asinsetting,say,theevennumberques-tions(thusrevisingthatconcepttoo).Provideworksheetswherethepupilsfillingapsratherthanwritingoutallofthequestion.Dontoverfacethelearner.

Pupilstakeanimpulsiveapproachto the maths problems.

Forwordproblems,encouragethemtoreadthroughaproblem,rephraseit,drawitifpossible.Fornumberproblemsencouragethemtoverbaliseit,forexamplefor8517,sub-vocalise,85minus17;85takeaway17usingdifferent phrasing.

Pupilsareslowtoevenstartwork.TherecouldbeseveralreasonsThey did not hear all the instruc-tions(poorshorttermmemory,inattention,distractions,hearing).

Repeatinstructions,chunkingthem,notgivingtoomanysteps at once. Make sure he has heard by asking him (quietlyandindividually,torepeattheinstructions).

Theyareavoidingthework(anxiety,attribution,attitude).Pupilsfailtofinishatestwithinthesettime

Thefirsttworeasonsneedlongtermnurture(seeChapter8).Attitudecouldbeduetoseveralcausesandneeds,initially,empatheticdiscussion.Providethemwitha different coloured pen or pencil and allow them to carry on untiltheyhavetriedasmanyoftheitemsastheyfeeltheycan attempt.

Theyhaventgotallthenecessaryequipment(book,pen,calculator).

Ifthereisasupportassistant,thiscouldbepartoftheirbrief.Establisharoutine.Forexamplehaveequipmentreadytogiveoutandhaveasystemtomakesureitisreturned(markitclearly!).(MoresuggestionsinNoattempts.)




No attempts

The problem SuggestionsApupilleaveslotsofunansweredquestions.

Pupilswhothinktheycantsucceedataquestionmaydecidetonottryratherthangetitwrong,thisisinpartananxiety/attitudeproblem.Socheckiftheyunderstandthetopicandaskwhatitisthatismakingthemnotattemptthework.Whatisityoudontunderstand?(Donotaccept,Mathsasananswer!Askthemtobespecific,thisisadiagnosticquestion.)Isthereapatterntothemissedquestions?(Morediagnosis.)Weretheyabsentwhenthattopicwastaught?

A pupil is slow to start work.

Itcouldbethatthepupilisbadlyorganisedcantfindbook,pen,etc.Ifthisisaregularoccurrence,haveabuddypupilpackforhim.Youmayevengetitbackattheendofthelesson!Itcouldbethatavoidanceoffailure.Tryaskinghimtoattemptjustoneortwoquestions.(Thishasanotherveryimportantbenefit.Ifapupilmakesanerrorinhisfirstpractiseofanewtopic,thaterrorpatternwillestablishitself,settinguptheneedforintensiveremediationinthefuture.)Trysittingthepupilatthebackoftheclasssoyoucanwalkovertohim and focus him without the other pupils noticing him. This could be negotiatedinsteadofsittinghimatthefrontwhereeveryoneseesthereminders

A pupil has no idea how to begin.

Offerastartinghintordothefirstline.Offeramodelanswer.

Apupilstartstheexer-cise,butgivesupaftertwoorthreequestions.

Thiscouldbethequantumleapeffect.Someexercisesandworksheetsstartwithtwoorthreerelativelystraightforwardandeasyquestionsandthen,WhamthenextquestionissoverymuchharderthatthepupilgoesintotheNoattemptstrategy.Checktheworksheetorexercisebyactuallydoingthequestionsandimaginingyouarethatpupil. Then modify the work.


Recording/writing up

The problem SuggestionsThe pupil only writes an answer. There is no recordofthemethod/procedure used.

Thismaybeaconsequenceofagrasshopperthinkingstyle(Chapter4).Pupilsshouldbeencouragedtodiscusstheirmethodsand be shown how to record their methods. This could be done by theteachermodellingthetechniqueorprovidinganexemplaranswerframe.

Thepupilwritesverycursory notes.

Thismaybeaconsequenceofslowwritingskillsratherthanthepupilnotknowingwhattodo.Allowmoretimeand/orreducethenumberofexamplesheisrequiredtodo.Provideasheetwiththekeyvocabulary.

26

Writingisextremelyuntidyanddisor-ganised.

Trysquaredpaperofvariouslysizedsquaresuntilasuitable one is found. Offer a worksheet where the writing demands are minimised. Allow the pupil opportunities to talk through his methods so he can show his abilities.

Copyingfromtheboard.Thisrequiresthestudenttolookup,focusontheboard,findthecorrectplace,remembersomedata,refocusonhispaper/book,writethedatainthecorrectspace,refocusontheboard and repeat the procedure. This is veryinfluencedbyshorttermmemory.

Offer notes. Allow him to photocopy the notes from a student who produces good notes.Ifthematerialisinatextbook,allowhimto highlight key areas.Check,regularlythathehasadequate(orbetter)notes.Providenotes.


Poor recall of basic facts

The problem SuggestionsThepupilmakesmanybasicfacterrors,suchas7+6=12or67=67withinlon-gercomputations.

Haveasupplyofbasicfactsquares(expectlosses).Supplyacalculator,suggestingitisjustusedtoaccess the basic facts rather than doing the whole calculation.Showhowtomakeanumberbondsfortenchart

109876543210 012345678910 (notetheemphasisfor5,actingasacheck)and

show how to use it to obtain other facts.Poor knowledge of basic factors handi-capsfactorisingquadraticequations.

Teachsomepatternstoreducetheextentoftheproblem,suchaslookingforevennumbers(2and4),5sand0s,digitsaddingto9,digitsaddingto3,6or9(3).Provideatablesquareandrevisehowtouseitforfactors(division)Showhowtofillinablanktablesquareandseeabove.Makesuretheearlyquestionsinvolveonlysimplefactors,sothepupilcanfocusonlearningthe procedure.

Add your own suggestions and solutions (and extra problems!)

Poor reading skills

The problem SuggestionsThepupilhasdifficultyinreading. Checkifheneedsspectaclesorabetter(andpossibly

bigger)print(manyphotocopierscanenlargeprint).Tryacolouredoverlaytochangetheprint/papercontrast(suppliersI.O.O.Marketing,seeAppendix2).



Thepupilhasdifficultyinreadingwordproblems(butunderstandsthemiftheyarereadtohim).

Readthemtohim.(Thoughthistakesyourtimeandmay damage his self-esteem if not done discretely and withempathy.)Providehimwithapersonaldictionary,withthenec-essaryvocabularyforthistopicpreviouslyread,explainedanddiscussed.Checkthenon-mathematicalvocabulary(names,places,etc.).Changeittoeasilydecodablealternatives,forexample,OcravilletoBath).Scantheworkintoacomputerandletthepupilusevoiceoutput.

Thepupilhasdifficultyinunderstanding/interpretingword problems.

Encouragethepupiltorewordthequestion.Encouragethepupiltotryandrepresenttheproblemaspictures.Tryoneofthereadingacronyms,e.g.SQ3RSurvey,Question,Read,Review,Respond.(Ifnothingelseitcounteractsimpulsivity.)Givenumbersentencesandencouragelearnerstocreatewordproblemsforthemselves(seeChapter6).


Sequencing skills

The problem SuggestionsPupilshavedifficultyrememberingandrecallingsequentialinformation.

Pointoutanddemonstratethepattern.Dontwaittoolongforthelearnertodiscoverthesequencefor himself.

Numbersequencefor2beyond10. Startwiththeparttheycanrecall2,4,6,8(whodoweappreciate)anduseitasabase,pointingoutthepatterns12,14,16,1822,24,26,28possiblyusing coins or base ten blocks. This can also be a task where the learner progresses from counting on. The difference between numbers is two each time.

Numbersequencesfor10whentheunitdigit is not zero.Thepupilcanrecall10,20,30,40butfindssequencessuchas13,23,33challenging.

Agoodchancetoenhanceplacevalueconcepts.Againuse coins or base ten blocks to show the pattern. This time the units digit is the stable factor.Thedifferencebetweenconsecutivenumbersisteneach time.Usea1to100numbersquareandpointout/colourintheplus10patterns.

Generalnumbersequences. Encouragethepupiltoworkoutthedifferencesbetweenconsecutivenumbersandthusknowifthedifference is always the same or if the differences createanewsequence(findingthebridgebetweennumbers).Forsequencessuchasadding(orsubtracting)6,7,8or9eachtimesuggestthattheadditionorsubtractionbedoneusingtheeasynumbers.So,toadd6,addas5and1ortosubtract9takeaway10andaddback1.

28

Incorrectdecimalsequenceas0.7,0.8,0.9,0.10

Use10pcoinsalongsidethewrittennumbers;thatis,writethedecimalandplacethecoins,writethenextdecimal and add another coin.Discusswherethe0.1,0.2,sequenceisheading.Itisheadingfor1.0.Offerasequencewithgaps0.1,0.2,_,0.4,_,_,0.7,0.8,_,_,1.1,_,1.3

Incorrectfractionsequenceas where the learner assumes that

is the biggest of the three fractions.

Gobacktothefractionsthat(should)beknown,i.e.,half,quarterandthirdandplacetheseinorderofsize.Demonstratewithfoldedpaper.Discusssharingslicesofcakeandpizza,butrememberthatthesearenottruefractions.Useaclock(time)forhalvesandquarters.Establishtheconceptthatfractionsinvolvedivision.Remindthemthatfractionscannotbetakenatfacevalue.Learnershavetolookbeyondthenumbersthatthey see or write.

Proceduralsequences/algorithms(e.g.longdivision).

Try to support memory with understanding.Findanalternativeprocedurethatrelatestothepupilsthinkingstyle(seeChapter4).Findamnemonic(e.g.BODMAS)butdonotusethisstrategytoooftenlestthemnemonicsthemselvesbecome another burden on memory.

PupilshavedifficultyenteringdataintoacalculatorinthecorrectsequenceasinTake16from47or

Encouragethepupiltobewary,notrushandwritetheproblem in symbols then appraise and compare the originalquestionanditssymbolicform.

Pupilshavedifficultycontinuingasequenceifitdoesnotstartatthebegin-ningasinWhatisthenextnumberafter6?Thelearnergoesbackto1,2,3,4,5,67.

Showthewholesequence,forexamplewithwoodorplasticnumbers,thentakeawaysomeofthebeginningnumbersand/orsomeoftheendnumbers.Showhowtoidentifytheinterval(gap)betweenconsecutivenumbersinthesequence.Sayaseriesofnumbersandaskthepupiltojoininatdifferentpoints.Showpartsofsequences,forexample,46,48,50,52or34,39,44,49,54andaskpupilstoidentifythesequence.


Transfer of skills

The problem SuggestionsThe pupil can only add when the num-bersarepresentedinthestandardverti-cal format.

Practisetheskillinaproblemcontext,workingfromthedigitstosimplelanguage,Add46and88tomorecomplexlanguageinsmallsteps.

Thepupilcancombine28with58,but cannot combine 2x+5x.

Demonstratethetransitionwith,say,numberrods.




The terminal inchworm/grasshopper (see Chapter 4)

The problem SuggestionsThe pupil will only respond to and work in one thinking mode. Histhinkingstyleisfixedatone end of the continuum.

If,aftermuchempatheticinterventionthelearnersthinkingstyleremainsinflexible,thenyou,theteacherwillhavetoworkwith that situation.Thiswillmeanacceptingthelearnerasheisandgivinghimthe skills to succeed using only a restricted thinking style. One consequencemaywellbethathehastolearnhowtoidentifyquestionsthatareunchangeablygrasshopperandnotwastetime trying to answer them.Thisispragmatismatalevelthatwillsituncomfortablywithmanyteachers,butitmaybethekindest(andmosteffective)way of teaching these rareish pupils.


Order

The problem SuggestionsThelearnerhearsTenpastsevenandwrites10:7.

Timeisoneofthosetopicswhichissofamiliarineverydayuse that we may forget the problems it generates. The learner should be taught to repeat the time putting in hours and minutes,TenminutespastsevenhoursandknowthatthewrittenconventionisHours:minutes.

ThelearnerreadsTake8awayfrom18andwrites818.

Thelearnermustbetaughttobewaryofeventhemost innocent looking word problems and rephrase them until they makesense.Thefactthat818eitherlooksimpossibleorleadstoanegativeanswershouldsuggestcaution.

Inanadditionsumsuchas56+37,thelearneradds6and7tomake13,writesdown1andcarries3.

The teen numbers are misleading in that the unit syllable comes beforethetenssyllable.SeeChapter6.


Not checking an answerThe problem SuggestionsThelearnerchecks(oftenquicklyandinadequately).

Trytoencouragecheckingbyanalternativemethod.Thislinkstoencouragingflexiblethinkingstyles(seeChapter4).Trytoencourageacheckviaanestimate.AskDoestheanswermakesense?Usethebasicestimatequestion,Istheanswerbiggerorsmaller?(adjustedtomakecontextualsense).

Thelearnerdoesntcheckatall. Askhimwhatroughvalue(fornumeracy)heexpectedtheanswerwouldbe.Allowawideguess/estimateandedgehimtowards closer estimates. This could be part of the training towards risk taking.

30


Organisation

The problem SuggestionsThepupilalwaysarrivesatlessonswithoutkeyequipment.

Suggestheobtainsapencilcaseorsimilarholder in which to keep just the essentials.Bepreparedtolendequipment,butbeevenmore prepared to remember to collect it back at theendofthelesson.(Hewillforget.)Makesuretheequipmentyoulendisclearlymarked!

Thepupilswrittenworkonthepageisbadly organised.

Trysquaredpaper,offeringdifferentsizesofsquaretofindwhatsuits.Tryverticallines.Try modelling good layout.Try a worksheet where part of the work is writ-ten(differentiation).


Transposals

The problem SuggestionsThepupiltransposesnumbers,forexamplehewrites31forthirteen.Thisislikelytobetheconsequenceofthewordsweusefortheteennumbers.Thesearetheonlytwodigitnumberswheretheunitsdigitisnamedfirstas in fourteen.

Tryplacevaluecards(Figure2.1)orcoinstomodelthecorrectdigitorder.(Checkiftransposalsonlyoccurforteennumbers,wherethewordsdomislead,asinthirteenthreeten,ratherthantwenty-threewhichdoesnot.)Explainthelanguagestructureofateennumbercompared to the digit structure.

Figure 2.1Placevaluecards

Explainthattheteennumbersareexceptionalandtobewarywhentheyarearound.Showhow transposals can be a problem when adding numbersas27+36becoming81insteadof63since7+6isaddedas31.Encouragethelearnertoevaluatesuchanswersbyestimates.Playcardsearchgames,withcardsthatshow,forexample,16and61andaskforsixteenorsixty-one.




Generalisations and recognising patterns

The problem SuggestionsThe learner does not recognise patterns. Somelessonsrelyonpupilsmakingthediscoveryofapattern.Thismaynotalwaysbeareliablemethod(forexam-ple,apupilmayprocessingroupsofthree and the pattern may be in groups offour).

Bemoreovertinhintingatthepattern,Usecolours.Usematerials(usetrialanderrortofindthematchbetweenlearnerandmaterial).Forexamplebasetenblocks may work as an illustration of multiplying and dividingby10,possiblyinconjunctionwithaplacevaluesheet.


Task analysis

Inorder tobepre-emptive, itmaybeworthdoinga taskanalysis foranewtopic.Thiscan be a quick procedure, just thinking about the topic andmaking notes on the sheetbelow.Theanalysiscanconsiderthetopic,ofcourse,butitcanalsotakeintoconsiderationthenatureofthelearner(s).Forexampleifthegroupcontainsanumberofstudentswithlanguageproblemsthenyoumightgivemorethoughttothatparticularaspectofthetask.Ataskanalysiscanbeusefulinfocusingattentiononareaswheresomepre-emptiveactionmaygreatly reduce thenumberof learnersdevelopingaproblemwith the topic and/orreducetheintensityoftheproblems(Figures2.2 and 2.3,shownoverleaf).

Analysing from the pre-requisite sub-skills perspective (Chapter 4)may also help atutor/teacherunderstandwherelearningmayfalter.

TOPIC Factor Problem SuggestionVocabulary and symbols Language Shorttermmemory Long term memory Sequences Direction Organisation/spatial Thinking style

Figure 2.2 A blank task analysis form

32

TOPIC Mental addition of two digit plus two digit numbersFactor Problem SuggestionVocabulary and symbols

Avarietyofwordsimply+ Varythevocabularyofthequestions:Whatis46add72?43plus56?Whatisthetotalof61and39?

Language Shouldnotbeanissuewithmentaladdition,unlesswrappedupinacomplicated story.

Shorttermmemory

Maynotrememberthequestion.Maynothaveenoughshorttermmemorytodothequestion.

Repeatthequestion.Try an alternate method.Askapartquestion.

Long term memory

The usual problem of recall ofbasicfacts,andpossiblyofcarrying.

Usesumswithknownfacts,e.g.74+75,so doubles can be used.

Sequences Notamainprobleminthistopic,but the answer may be computed in thereverseorder,e.g.74+75doneas9then14.

Encouragepre-estimatesandallowtimeforthisoraskintwostages,thatisfirstask for an estimate and then the accurate answer.

Direction Since64+56=56+64thisshouldnotbeaproblem,butseesequences.

Organisation/spatial

Shouldnotbeafactorinmentalwork,butspatialmemorymaybeneeded.

Thinking style The pupil may use a method that is less suitable for his learning problems.

Encouragealternativestrategies.Askthepupiltoevaluateandinter-relatethenumbers,e.g.8and9canberoundedupto10.

Figure 2.3Anexampleofataskanalysis


Chapter3 What the curriculum asks pupils to do and

where difficulties may occur

InthelastchapterIlookedatfactors,forexample,poorshorttermmemory,whichcouldcreatelearningdifficultiesformathematics.InthischapterIamlookingattopicsinmath-ematicsandhowtheycouldcreatedifficultiesinlearning.

This chapter is basedon theEnglishNationalNumeracyStrategy,but it is likely tobesimilartothedevelopmentofdifferentstrandsandtopicsinmanynumeracyschemesacrosstheworld.WhatIhavedoneistotaketheYear5(age9years)teachingprogrammeas typicalof the content of theNNSFramework, highlighted the key issues, predictedwheredifficultiesandconfusionsmightariseandoutlinedsomepossiblesolutions.Someof these are dealtwith inmore detail in other chapters, but to avoid the need to flickbetweenpages,Ihavekept thecontentsomewhatselfcontained.Thechapterillustratesanalysis of content in terms of its likely interactions with learners. A similar analysis could beappliedtoanyprogrammeofwork.Thisisthepreventativemedicine.Whateverthecur-riculum,theanalogyIlikeistocomparepreparingamathematicslessontopreparinganexpedition.Youprepareforallthemanyproblemsyouknowyouaregoingtoencounter,andexperiencehelpsyoutopredictwhattheywillbe,butthenexperiencetellsyouthattherewillstillbesomeproblemsyouwillnothavepredicted.Thecurriculumistheguidetothejourney.Thischapteristheguidetosomeoftheunpredictedeventsthatmaypreventyourlearnersfromhavingasuccessfulexpedition.

The programme for Year 5 of the National Numeracy Strategy: numbers and the number system

Place value, ordering and rounding

Thissectiondealswithbasic,fundamentallyimportanttopicswhichset thefoundationsforsuccessinnumeracy.Muchofthissectionand,indeedthecompleteNumbersandtheNumberSystemisinter-linkedandeachpartshouldsupportunderstandingoftheotherparts.Youshouldnotassumethatpupilsabsorbtheselinksbyosmosis.Manywillneedexplicitteaching.Itiseasytoassumethatanabilityforrecallmeansunderstanding.

Mostofthisworkiswithinthepupilsexperiences.Thisisbothapositiveandnegativefac-tor.Itispositiveinthesensethatpupilsareworkingfromfamiliarfactsandawareness,butneg-ativeinthattheymayalreadyhaveformedsomeincorrectideasorfeelthatbecausetheworkisfamiliartheydonotneedtoorganise,strengthenandinter-relatetheirfactbasetonewwork.

Placevalue isakeyconcept.Childrenwhofail tograsp the ideaofplacevaluefindnumeracydifficultandmakeerrorssuchas:

45 45+88 +88

1213 123

34

Misconceptionscanarisefromearlyexperiences.Forexample,westartnumberas1,2,3,4,5wherethesequencegetsbiggeraswetracktotheright.Whenplacevaluearises,aswith,say24,thenumbertotheleftisbigger.

Subtraction, especially involving decomposition will also be handicapped by poorunderstandingofplacevalue.Equallymultiplicationbythetraditionalmethodsrequiresthatplacevaluesareunderstood,sothat,forexamplewhenmultiplyingby45thepupilneeds to know and understand that the 4 is 40 and what effect this has on the multiplication. Pupilsneedtobeabletobreakdownnumbers.Thisincludesusingplacevalueaswellasnumberrelationships,forexample25as104or99as1001.

Manipulativescanbehelpful,particularlybase tenblocksandcoins togiveavisualimagetothesymbolicrepresentationsofnumbers.Numberswhichinvolvezero(suchas5004)oftenneedextraexplanationandagainbasetenmaterialsmayhelp.Tryworkingfrom5444,through5044to5004usingthebasetenblocks.Multiplyinganddividingby10andpowersof10mayalsohelptoshowhowthepositionofanumberaffectsitsvalue.

All the work on placing a number in the correct position on a number line and the linked workonestimationleadspupilstooverviewskills.Estimationisaholisticskillandshouldbe taught tocomplement theproceduralskillsofwrittenarithmetic,even thoughpupilsmayshowamarkedpreferencetowardsoneoftheseskills.Rememberthatestimationisnotpreciseandthattherequiredlevelofaccuracyoftheestimationdependsonthepartic-ularreallifesituation.Theemptynumberlineisagoodvisualforpractisingthisskill.

Roundingstrengthensestimationinthesenseoflevelsbyroundingtothenearestten,near-esthundred,nearestthousand.Itisalsoagoodreallifeskill,particularlyusefulforshopping.

Theintroductiontonegativenumbersuseseverydayexamples.Laterpupilsmaycometoforgettherealityattherootofnegativenumbersandseethemasaveryabstractconcept.Thisintroductoryworkonaddingtoanegativenumber(3Candwarmingupby4C)andsub-tractingfromanegativenumber(3Candcoolingdownbyafurther4C)setsthefoundationtoworkfromanimagethatisrealtoanabstractandsymbolicrepresentation(3+4=1and34=7).Thistopicalsointroducestheideathatadditionandsubtractionareoppositever-sions of the same idea.

Thequitesophisticatedsequence10,9,8,7,6,5,4,3,2,1,0,1,2,3,4takesafamil-iarreversecountingskillbeyondthezeroandtherulesofnumbersequencearenowreversed.Inpositivenumbers4isbiggerthan3.Innegativenumbers4issmallerthan3.Inearlynumberworktheseabnormalitiescanconfusetentativelearningandneedthatexplicitteaching.

Properties of numbers and number sequences

Thissectionoftheprogrammecouldbetreatedaslargelyrotelearning.However,therewillbe some pupils who fail to rote learn facts. There will be other pupils who merely learn and donotunderstand.Itwillbeimportanttoidentifybothgroupsofthesepupilsandhelpthemcircumventtheseearlybarrierswhichcouldresultinananxietyaboutmathswhichcouldthenleadthemtonotbeinginvolvedinfuturelearning.Thesectioncouldbeusedasagoodopportunitytodevelopconfidencewithnumbersbyshowinghowfrequentlypatternsoccurandhowfrequentlythesesamepatternscanhelpwithrememberingandaccessingfactsandworkingwithnumbers.The inter-relationshipbetweennumbers, especially relatingnumberstothereferencenumbersof2,5and10willbuildbetterrecallandunderstandingforALLpupils.Thegoalistomakingtheacquisitionofskillsandunderstandingrobust.


What the curriculum asks pupils 35

Usingstrategiestoaccessfactsisanearlyindicatorofagrasshopperlearningstyle,thatisanintuitivewaywithnumbersandanabilitytobreakdownandbuildupnumbersintomoreconvenientvalues,e.g.using100insteadof98.(SeeChapter4.)

Recognise and extend number sequences

Thisareasetsmanybasicproceduresandunderstandingsinplaceand,althoughperhapsdecep-tivelysimpleorordinary,needsattentionsothatunderstandingandasmuchautomaticityaspossibleisachieved.

Countingonin6s,7s,8s,and9sand11s(toalesserextent)willbeproblematicforsomepupils.Thepatternsandrelationships(6=5+1,7=5+2,8=102,9=101and11=10+1)willbeessentialstrategiesforsomepupilsanduseful,reinforcingprocessesforothers.

Thereare repeatingpatternswhencountingon in6s,8s,9sand11s.Theseneed tobeclearlydemonstratedtoensureallpupilsabsorbandthenrecreatethepatterns.Itwillhelptousevisualsupportforanyoralpresentations.Numberlines,Cuisinairerodsandcoins,usedalongsidethenumbers,wouldsuitthesetopics.

Numberrodsaregoodtoshowthe9sequenceagainstametrerulemarkedindecimetres,showingaonelesscomparisoneachtime.Thisisgoodfor8rodstoo,reinforcedwiththerepeatingpatternintheunitdigitsof8,6,4,2,0,whichisgivingnumberconnectionsupportbyusing102=8(on10back2foradditionandback10on2forsubtraction).

Thisalsocompareswithaddingandsubtracting9and11,where9iscomputedas101and11isdoneas10+1.Alsoaddingandsubtracting6canbedoneas5and1and7as5and2.

Alltheseexamplesarerelatinghardernumberstothebasenumbersof5and10,buildingonwhatpupilsknow,ratherthanjustaddinginextra,unrelatedinformation.Youaregivingmemoryanextrasourceofsupport.

Lookoutforthecommonerrorinthedecimalsequence,0.10.20.30.80.90.10(noughtpointten).Asupportingvisualimagesuchascoins,oranumberlineshouldhelp.Itisimpor-tanttopre-emptthiserror,sothatitdoesnotbecomethatalltooinfluentialfirstexperienceofthis important concept of the transition from a decimal number to a whole number.

Countingbackwardsisoftenmuchharderthancountingforwards.Donotassumelearnerswillexhibitequalfacilitywiththesetwotasks.

Even and odd numbersEvennumbersend(haveaunitsdigit)in0,2,4,6or8.Everyotherdigitcanbeodd,itsthatlastonethatmatters,e.g.975,312iseven.

Oddnumbersend(haveaunitsdigit)in1,3,5,7or9.Everyotherdigitcanbeeven,itsthelastonethatmatters,e.g.864,207isodd.

Establish (or re-establi

Trouble With Maths

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