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  • HelpingELLs

    ReachTheirPotentials: UsingThinkingMapstoAssess

    ConceptualUnderstanding

    LolitaGerardoHSMathTeacher&Dept.ChairPharrSanJuanAlamoISD

    SylviaTaubeMathEducationProfessor

    SamHoustonStateUniversity

    MathematicsforEnglishLanguageLearners[MELL]AnnualConference

    TexasStateUniversity,SanMarcos,TXJuly31August2,2008

  • ObjectivesofMELLSession

    Shareactivitiesandexperiencesinteaching mathematicstoEnglishLanguageLearners;

    ExplorealternativetoolsthatcanassesstheELLs conceptualunderstandinginmath;

    Constructdifferenttypesofmaps,anddesignrubric toscorestudents

    conceptmaps.

  • WARMUP: Canyoucomeupwithdifferent

    representations?Problem:Inmanyparades,flowersareusedto

    decoratethefloats.Thelistbelowshowsthe numberofflowersusedineachrowofa

    paradefloat.

    {(1,54),(2,58),(3,62),(4,66)}

  • Summarizethedifferent representationsoffunctions

    Usegraphicorganizers

  • Recent State Data on ELLsand Achievement in Algebra

  • End-of Course Algebra 1Testing Date: Spring 2008

    Number of students tested = 52,462

    Commended Performance = 11%

    Met standards = 56%

    Current LEP = 17% (# of students tested=1,104)

    Low SES = 43% (# of students tested=21,932)

    Hispanic = 47% (# of students tested=20,270)

    White = 66% (# of students tested = 22,020)

  • 21

    1

  • AKickoffSummerProgramforELLs

  • ThinkingMapsin Algebra1

  • MultipleRepresentations Why?

    Usingandunderstandingdifferentformsof representationsarecriticaltolearningmath;

    Implementationandflexibilityinusingvaried representationsprovideastrongevidenceofdeeper

    understandingofmathematics;

    Thisunderstandingisrichwithconnectionsand relationships.

  • Bringinstudentsculture

    Promoteequity

    Whatcanrepresentationsinclude? Physicalobjects Drawings,graphs Symbols,numbers Verbal ContextualTheseareusedtoorganize

    andrecordstudents thinkingaboutmathideas

    andproblemsolving.

    Predominant

    Representations Numeric Geometric Algebraic

  • Adaptedfrom:AnInteractiveModelforUsingRepresentationalSystemsby:Behr,Lesh,Post,&Silver

    Written Symbols Spoken

    Symbols

    Real-WorldSituations Manipulative

    Aides

    Pictures

  • AssessmentStrategies

    If multiple representations is being valued, then the teacher should include questions or problems that involve a variety of representations.

  • WilltheELLs

    haveadvantagesin thistypeofassessment?

  • VisualRepresentations

    Example:TakingStockProblem

    Father

    has19animalsonhisfarmsomechicken andsomecows.Hetoldmethathecounted

    62legsaltogether.Howmanyofeachanimal werethere?

  • Canyoucomeupwithatleast3different representations?

  • Usingphysical(human)graphs

  • ConceptsMaps&ThinkingMaps

    Arrowsindicatethedirectionsofthe relationshipsbetweenconcepts;

    Linesarelabeledwithlinkingwordsto specifyrelationship;

    Conceptsareplacedinovalsoranyshape

    Concept map an instrument for explicitly describing concepts and the relationship among them. It is a way to organize a learners knowledge.

  • Graphicorganizers

    VennDiagram

    Chart

    Web

    Timeline

    Combinebothlinguisticand nonlinguisticinformation(e.g.,circles,

    linesthatshowrelationship)

  • Type:Hierarchical Purpose:Promotemathematicalconnection

    Quadrilateral

    includes

    rectanglerhombus

    Trapezoid

    square

    includes

    includes

    parallelogram

    is also a

    Isosceles

    trapezoid

  • TYPE:Web

    Linear Equations

    Graph shows a straight line

    Form: y=mx+ b

    has slope

    ordered pairs

    Crosses either x or y-axis

    first - degree functions

    domain/range

    m is the slope

    b is where line crosses y-axis

    zero or undefined

    That is + or -

  • Whatteachersmustdotofacilitatestudents useofMultipleRepresentations?

    By listening, questioning, and making a sincere effort to understand what they are trying to communicate with their drawings and writings, especially when they are using personal [use of invented strategies] representations. [Tarlow, L. D. (2008), MTMS, Vol 13, #8.]

    Step back to give students freedom to test their ideas; observe students work and decide when to intervene or pose questions to clarify ideas.[Tarlow, L. D. (2008), MTMS, Vol 13, #8.]

    Give clear, comprehensible, useful, relevant feedback.[Hill, J.D. & Flynn, K. M (2006). Classroom instruction that works with ELLs, ASCD publication]

  • HowtoassessthinkingmapsTask

    Drawaconceptmaprelatingthefollowing concepts:

    Parabolas

    Domain

    Standardequation

    Range

    Apex

    Symmetry

    QuadraticfunctionTranslate/Slide

    Maximumpoint

    Minimumpoint

  • Participantswillcreatetheirown conceptmap

    Then . . .design a rubric for scoring the concept map

  • RecommendationsfromResearch

    Findings mostly from science, psychology, reading, and college math

    J. Novak (1984) early researcher in science education

  • A scoring rubric should be able to assess the breadth and depth of students knowledge transformation as they progress from novice to expert. (Edmonson, 2000)

  • Scoring can be done by comparing experts (criterion) with students maps

    Findings:Maps that focus heavily on concept relationship has strong correlations with scores in standardized tests.

  • Therubricshouldbeabletomeasure studentprogress(understanding)over

    time.

    Novice

    Expert

    Eventually, learnersdocometothinklike

    teachers

  • THERUBRICMUSTBEABLETOINFORM INSTRUCTION

    Recommendeduseduringformative assessmentorfordiagnosticpurpose

  • LearningStrategiesforMathELLs

    Mustbeengagedwithvariedcollaborative activities

    Usemultiplerepresentations Useactivitiesthatpromotecommunication Challengestudentsinsolvingproblemswith

    openendedquestions Usegraphicorganizers

  • LearningStrategies

    Integrationofmanipulativematerialsand graphingtechnology

    Handsonapproachofdevelopingconceptual understanding

    Establishhomeconnection Questioningstrategies Richconnectionsincontext

  • References Bartels,B.(1995).Promotingmathematicsconnectionswithconcept

    mapping.MathematicsTeachingintheMiddleSchool,1542549. Bolte,L.(Jan1999).Usingconceptmapsandinterpretiveessaysfor

    assessmentinmathematics.SchoolScienceandMathematics,

    99(1).

    Hill,J.D.&Flynn,K.M(2006).ClassroominstructionthatworkswithELLs.

    ASCDpublication. Novak,J.(1984).Learninghowtolearn.NewYork:CambridgeUniversity

    Press. Rye,J.(Jan2002).Scoringconceptsmaps:Anexpertmapbasedscheme

    weightedforrelationship.SchoolScienceandMathematics. Focusissue(April2008)MathematicsTeachingintheMiddleSchool

    publishedarticleson:DevelopingMathematicalUnderstandingthrough

    Representations.

    Helping ELLs Reach Their Potentials: Using Thinking Maps to Assess Conceptual UnderstandingObjectives of MELL SessionWARM-UP:Can you come up with different representations?Summarize the different representations of functionsSlide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Thinking Maps inAlgebra 1Multiple RepresentationsWhy?What can representations include?Slide Number 15Assessment StrategiesWill the ELLs have advantages in this type of assessment?Visual RepresentationsCan you come up with at least 3 different representations?Using physical (human) graphsSlide Number 21Slide Number 22Concepts Maps & Thinking MapsSlide Number 24 Type: Hierarchical Purpose: Promote mathematical connectionTYPE: Web What teachers must do to facilitate students use of Multiple Representations?How to assess thinking mapsParticipants will create their own concept mapRecommendations from ResearchSlide Number 31Slide Number 32The rubric should be able to measure student progress (understanding) over time.Novice ExpertEventually, learners do come to think like teachersTHE RUBRIC MUST BE ABLE TO INFORM INSTRUCTIONLearning Strategies for Math ELLsLearning Strategies References