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ChallengesandOpportuni2esfor

TeachingandLearningofProofin

HighSchoolsTadWatanabe

KennesawStateUniversitytwatanab@kennesaw.edu

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PrinciplesandStandardsforSchool

Mathema4cs(NCTM,2000)Pre‐K–12curriculumshouldenableallstudents

to:

•  recognizereasoningandproofasfundamentalaspectsofmathemacs;

•  makeandinvesgatemathemacalconjectures;

•  developandevaluatemathemacalargumentsandproofs;

•  selectandusevarioustypesofreasoningandmethodsofproof.

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Opportunies&Challenges•  Opportunies:“…proofisexpectedtoplaya

muchmoreprominentrolethroughtheenreschoolmathemacscurriculumandtobea

partofthemathemacseducaonofallstudents.”(Knuth,2002,p.380)

•  Challenges:“…proofisexpectedtoplayamuchmoreprominentrolethroughtheenre

schoolmathemacscurriculumandtobeapartofthemathemacseducaonofall

students.”(Knuth,2002,p.380)

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Challenges•  Movangstudents

Whydoweneedtoproveit?Wealreadyknowit’strue!

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Inelementaryschools,

3B, p. 9

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Inelementaryschools,

4B, p. 96

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Inelementaryschools,

5A, p. 60

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Convincingargument?Studentsarealreadyconvinced.

Examplesandempiricalevidences

areconvincingenoughtomany

students.

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PurposesofProof • verifica4on• explana4on•  systema4za4on

• discovery•  communica4on•  intellectualchallenge

De Villiers, 1999

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ConviconasMovaonInactualmathemacalpracce,

conviconisfarmorefrequentlya

prerequisiteforthefindingofaproofthandoubt.

(DeVilliers,2004,p.705)

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ProofasExplanaon•  Usingtwodifferentdigitsfrom1,2,3,…,9,

makea2‐digitnumber.Reversetheorderof

digitstomakea2

nd

2‐digitnumber.•  Findtheirsum.

•  Thesumisalwaysdivisibleby11.

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EmpiricalVerificaon•  35+53=88,and11divides88.

•  82+28=110,and11divides110.

•  97+79=176,and11divides176.•  etc.

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Proof •  Letaandbbethetwodigits.

•  Thetwo2‐digitnumbersare:

10a+band10b+a.•  Theirsum,therefore,is10a+b+10b+a.

•  Youcansimplifythisto11a+11b.

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AnApproachtoTeachingProof •  Insteadoffocusingonverificaonand

(global)systemazaon,helpstudents

experiencelocalsystemazaon.

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LocalSystemazaonGiven(inelementaryschool)

•  Sumofanglesinatriangleis180°.

Newrelaonship.

•  Sumofanglesinan‐gonis180°x(n‐2).

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180 x (number of triangles)

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Sum of angles = 180 x n – 360= 180 (n – 2)

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LocalSystemazaonGiven(inelementaryschool)

•  Sumofanglesinatriangleis180°.

Newrelaonship.

•  Sumofanglesinan‐gonis180°x(n‐2).

Whichleadsto,

•  Sumofexternalanglesofan‐gonis360°.

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exterior angle + interior angle = 180°

exterior angle = 180° - interior angle

sum of exterior angles = 180° x 5 + sum of angles

= 900° - 540°

= 360°

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LocalSystemazaon•  Sumofanglesinann‐gonis180°x(n–2)

BECAUSEsumofanglesinatriangleis180°.

•  Sumofexterioranglesinann‐gonis360°

BECAUSEsumofinterioranglesinann‐gonis

180°x(n–2).

•  So,whatisthereasonthatsumofanglesina

triangleis180°?

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Challenges•  Difficultywithtechniqueswithproving–

auxiliarylines

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Durell&Arnold(1924)•  Someoftheprincipalauxiliarylinesusedon

reclinearfiguresare:

1.  Alineconnecngtwogivenpoints.

2.  Alinethroughagivenpointparalleltoa

givenline.

3.  Alinethroughagivenpointperpendicularto

agivenline.

4.  Alinemakingagivenanglewithagivenline

(p.103)

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Durell&Arnold(1924)•  Indemonstrangtheoremsrelang

tothecircle,itisoenhelpfulto

drawoneormoreofthefollowingauxiliarylines:Aradius,adiameter,a

chord,aperpendicularfromthe

centeruponachord,anarc,acircle,etc.(p.150)

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Let’sthinkaboutdifferentwaysto

prove∠ ADC = ∠ A + ∠ B + ∠C

D

CB

A

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Findthemeasureofangle x .

 x 

40°

60°

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Somepossibleapproaches:

∠ x  = 60° + 40°

= 100°

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Let’sthinkaboutdifferentwaysto

prove∠ ADC = ∠ A + ∠ B + ∠C

D

CB

A

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Usingaparallelline

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Usingparallellines

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Usingparallellines

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Usingparallellines

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Usingparallel&perpendicularlines

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Connecng2points

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Connecng2points

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Challenges

•  “…proofisexpectedtoplayamuchmore

prominentrolethroughtheenreschool

mathemacscurriculumandtobeapartof

themathemacseducaonof all

students.”(Knuth,2002,p.380)

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Reasoning&Proofinpre‐K‐12• Needamoredevelopmentalviewof

“proof”–thatis,whatis“proof”in

elementaryschools?

• Cansuchaviewofproofbe

consistentwithproofinHSgeometry

(andbeyond)?

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DefinionofProof 

(Stylianides,2007,p.291)Proofisamathemacalargument,aconnected

sequenceofasseronsfororagainsta

mathemacalclaim,withthefollowing

characteriscs:

1.Itusesstatementsacceptedbythe

classroomcommunity(setofaccepted

statements)thataretrueandavailablewithoutfurtherjusficaon;

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DefinionofProof 

(Stylianides,2007,p.291)Proofisamathemacalargument,aconnected

sequenceofasseronsfororagainsta

mathemacalclaim,withthefollowing

characteriscs:

2.Itemploysformsofreasoning(modesof

argumentaon)thatarevalidandknownto,

orwithintheconceptualreachof,theclassroomcommunity;

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DefinionofProof 

(Stylianides,2007,p.291)Proofisamathemacalargument,aconnected

sequenceofasseronsfororagainsta

mathemacalclaim,withthefollowing

characteriscs:

3.Itiscommunicatedwithformsof

expression(modesofargument

representaon)thatareappropriateandknownto,orwithintheconceptualreachof,

theclassroomcommunity.

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3DimensionsofDevelopment

•  Setofmathemacalconcepts&

procedures

•  Repertoireofargumentaon

•  Setofrepresentaons

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Theunitdigitof5nforallnatural

numbernis5.•  Elementaryschool

52=25;53=125;54=625;55=3125,…

•  Grade5classroom(Lampert,1990)

 – Explainwhytheunitdigitof54is5.

“Youdon’thavetodothat(actuallymulply

25x25).It’seasy,thelastdigitisalwaysgoingtobe5becauseyouarealwaysmulplyinglastdigits

of5,and5mes5endsina5.”(p.48)

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Theunitdigitof5nforallnatural

numbernis5.•  Throughmathemacalinducon

(1)Proveitistruewhenn=1:51=5,true.

(2)Assumeitistruewhenn=k .

(3)Proveitistrueforn=k +1

5k +1=5k x5

Theunitdigitfor5k is5.

Therefore,theunitdigitof5k +1istheunitdigitof5x5,whichisa5.

(4)(1),(2),and(3)thestatementistrue.

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GenericArgument•  Supposewehave94and49.•  Ifyouwritethevercally,wehave:

•  Wecanswitchtheonesdigitswithout

changingthesum.•  So,wehave99+44,and11dividesboth.

•  So,thesummustbedivisibleby11,too.

9 4+ 4 9

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PrinciplesandStandardsforSchool

Mathema4cs(NCTM,2000)• Fromchildren'searliestexperienceswith

mathemacs,itisimportanttohelpthem

understandthatasseronsshouldalwayshavereasons.(p.56)

• Bytheupperelementarygrades,jusficaons

shouldbemoregeneralandcandrawonother

mathemacalresults.(p.58)

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PrinciplesandStandardsforSchool

Mathema4cs(NCTM,2000)• Studentsshouldmovefromconsidering

individualmathemacalobjects‐‐thistriangle,

thisnumber,thisdatapoint‐‐tothinkingaboutclassesofobjects‐‐alltriangles,all

numbersthataremulplesof4,awholesetof

data.(p.188)

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Opportunies•  “…proofisexpectedtoplayamuchmore

prominentrolethroughtheenreschool

mathemacscurriculumandtobeapartof

themathemacseducaonofall

students.”(Knuth,2002,p.380)

•  “Thisabrupttransiontoproofisasourceof

difficultyformanystudents,evenforthosewhohavedonesuperiorworkwitheasein

theirlower‐levelmathemacs

courses”(Moore,1994,p.249).

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ReferencesDeVilliers,M.(2004).Usingdynamicgeometrytoexpandmathemacs

teachers’understandingofproof.Int’lJournalofMathema4csEduca4on,ScienceandTechnology ,35,703‐724.

DeVilliers,M.(1999).RethinkingproofwithSketchpad .KeyCurriculum

Press.

Durell,F.&Arnold,E.E.(1924).Newplanegeometry .NewYork:

CharlesE.MerrillCompany.

Lampert,M.(1990).Whentheproblemisnotthequesonandthe

soluonisnottheanswer:Mathemacalknowingandteaching.

 AmericanEduca4onalResearchJournal,27 ,29–63.

Moore,R.C.(1994).Makingthetransiontoformalproof.Educaonal

StudiesinMathemacs,27(3),249‐266.

NCTM.(2000).PrinciplesandStandardsforSchoolMathema4cs .

Reston,VA:TheCouncil.

Stylianides,A.(2007).Proofandprovinginschoolmathemacs.

 JournalforResearchinMathema4csEduca4on,38 ,289‐321.