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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

decoratethefloats.Thelistbelowshowsthe numberofflowersusedineachrowofa

{(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

andproblemsolving.

Predominant

Representations Numeric Geometric Algebraic

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

• VisualRepresentations

Example:TakingStockProblem

Father

has19animalsonhisfarmsomechicken andsomecows.Hetoldmethathecounted

62legsaltogether.Howmanyofeachanimal werethere?

• Canyoucomeupwithatleast3different representations?

• Usingphysical(human)graphs

• ConceptsMaps&ThinkingMaps

Arrowsindicatethedirectionsofthe relationshipsbetweenconcepts;

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

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]

Drawaconceptmaprelatingthefollowing concepts:

Parabolas

Domain

Standardequation

Range

Apex

Symmetry

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

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