Inves&ga&ng Subtle&es of the Mul&plica&on Principle HATM HMAT HMTA HTAM HTMA MAHT MATH MHAT MHTA...

Inves&ga&ngSubtle&esoftheMul&plica&onPrinciple

EliseLockwoodOregonStateUniversity

MichiganStateUniversityColloquium

January23,2018

Inves&ga&ngSubtle&esoftheMul&plica&onPrinciple

EliseLockwoodOregonStateUniversity

MichiganStateUniversityColloquium

January23,2018

ZackReed,BranwenPurdy,andJohnCaughman

BackgroundandMo&va&on

•  Coun&ngproblemsareeasytostate…butcanbedifficulttosolve

CombinatoricsBooksSayCoun&ngisHard

•  Mar&n’s(2001)firstchapterisen&tled“Coun&ngisHard”Hepointsoutthat“therearefewformulasandeachproblemseemstobedifferent”

•  Tucker(2002)saysofhiscoun&ngchapter,“wediscusscoun&ngproblemsforwhichnospecifictheoryexists…itisthemostchallengingandmostvaluablechapterinthisbook”

MathEduca&onResearchSaysCoun&ngisHard

•  EizenbergandZaslavsky’s(2004)findings“supporttheasser&onthatcombinatoricsisacomplextopic–only43ofthe108ini&alsolu&onswerecorrect”– English,1993;Hadar&Hadass,1981;Kavousian,2006;Lockwood,2014

•  Broadly,myresearchgoalsaretolearneverythingIcanabouthowtoimprovetheteachingandlearningofcombinatorialenumera&on

Ouaitsproblem

•  Howmanydifferentshirt-pants-beltouaitscanyoumakeifyouhave3shirts,4pairsofpants,and3beltstochoosefrom?

Ouaitsproblem

•  Howmanydifferentshirt-pants-beltouaitscanyoumakeifyouhave3shirts,4pairsofpants,and3beltstochoosefrom?{S1,S2,S3}{P1,P2,P3,P4}{B1,B2,B3}

Ouaitsproblem


_________

3 4 3

Ouaitsproblem


_________

3 4 3x x =36

MATHproblem

•  HowmanydifferentwaysaretheretoarrangetheleiersinthewordMATH?

MATHproblem

____

AHM_AHT_

A___

H___

T__

M___

AHMT AMHT ATHMAHTM AMTH ATMH

AMH_AMT_

ATH_ATM_

HAM_HAT_

HMA_HMT_

HTA_HTM_

MAH_MAT_

MHA_MHT_

MTA_MTH_

TAH_TAM_

THA_THM_

TMA_TMH_

HAMT HMAT HTAMHATM HMTA HTMA

MAHT MHAT MTAHMATH MHTA MTHA

TAHM THAM TMAHTAMH THMA TMHA

HM__

MH__

MT__

AH__

AM__

HA__

TA__

TH__

TM__

AT__

MA__

HT__

AHMTAHTM

AMHTAMTH

ATHMATMH

HAMTHATM

HMATHMTA

HTAMHTMA

MAHTMATH

MHATMHTA

MTAHMTHA

TAHMTAMH

THAMTHMA

TMAHTMHA

SetofOutcomes

4x3x2x1=24

Ques&onforYou

•  Ifyouhadtowritearuleforwhenyoushouldusemul&plica&ontosolveacoun&ngproblem,whatwouldyouwrite?

TheMul&plica&onPrinciple

•  “FundamentalCoun&ngPrinciple”•  Itunderliesmanyofthecoun&ngformulasthatstudentsencounter

•  TheMPoffersjus&fica&onforwhywegetallofourdesirableoutcomes

n!= n ⋅ (n−1) ⋅ (n− 2) ⋅... ⋅2 ⋅1

n Pr =n!

(n− r)! nCr =n!

(n− r)!r!=nr"

#$%

&'

TheMul&plica&onPrinciple

Mo&va&on

•  WestartedtoobservesomevarietyinMPstatements

TwoRelatedStudies

•  AtextbookanalysisofstatementsoftheMP– Capturethevariety– SeehowstatementsoftheMParepresented

•  Areinven&onstudywithapairofundergraduatestudents–  Interviewtwostudentsover8hour-longsessions– Havethemsolvecoun&ngproblemsandthencharacterizewhentheymul&ply

– ReinventastatementoftheMP

ResearchQues&ons

•  Howisthestatementofthemul&plica&onprinciplepresentedinpostsecondaryCombinatorics,DiscreteMathema&cs,andFiniteMathema&cstextbooks?

•  Whatmathema&calissuesariseincomparingandcontras&ngdifferentstatementsofthemul&plica&onprinciple?

•  Howdostudentsreasonaboutkeymathema&calissuesinthemul&plica&onprinciple?

Study1:TextbookAnalysis

•  Wecreatedalistof70universi&esfromacrossthecountry

•  Weexaminedtextbooksfromtheseschools–94courses,yielding32textbooks

FrequenciesofTextbooks

11112

34

1111222

344

6

17

1111122

333

78

0

2

4

6

8

10

12

14

16

18

20

Freq.CombinatoricsTexts Freq.DiscreteTexts Freq.FiniteTexts

10

TextbookAnalysis:ExaminingtheVariety

•  Wecreatedalistof70universi&esfromacrossthecountry

•  Weexaminedtextbooksfromtheseschools–94courses,yielding32textbooks

•  Anothersearchamongpersonalanduniversitylibrariesyielded32moretextbooks

•  Wehadafinallistof64textbooks,with73statementsintotal

ResultsofTextbookAnalysis

ResultsofTextbookAnalysis

•  Wewanttohighlightakeydis&nc&onthatemergedfromouranalysis– 3kindsofstatements

•  StructuralStatements•  Opera&onalStatements•  BridgeStatements

3StatementTypes

STRUCTURAL

OPERATIONAL

BRIDGE

Summaryof3StatementTypesCode% Criteria%

Structural%%

The%statement%characterizes%the%MP%as%involving%counting%objects%(such%as%lists%or%k;tuples)%

Operational% The%statement%characterized%the%MP%as%determining%the%number%of%ways%of%completing%a%counting%process%%

Bridge% The%statement%simultaneously%characterizes%the%MP%as%counting%outcomes%and%specifies%a%process%by%which%those%outcomes%are%counted%

%

22

33

18

0

10

20

30

40

Struct. Oper. Bridge

Whydowecare?

•  Mathema&calimplica&onsofstatementtypes

22

33

18

0

10

20

30

40

Struct. Oper. Bridge

ThreeMathema&calFeatures

Weiden&fied3featurescentraltomanystatementsoftheMP:

1.  Requireindependenceof#ofop&ons2.  Allowdependenceofop&onsets3.  Requiredis&nctcompositeoutcomes

Feature1:Requireindependenceof#ofop&onsOu3itsProblem:Howmanyshirt-pants-beltou3itscanbemadefromthreedifferentshirts,fourdifferentpairsofpants,andthreedifferentbelts?

{S1,S2,S3}{P1,P2,P3,P4}{B1,B2,B3}

_________

3 4 3x x =36

Feature1:Requireindependenceof#ofop&onsHowmanypossibleoutcomesarethereifIchoose2differentcards(withnoreplacement)fromastandard52-carddeck,wherethefirstisaface-card(J,Q,K)andthesecondisaheart?

Atemp&ngansweris12x13=156(#facecardsx#hearts)

IfIfirstchooseoneofthe9non-heartfacecards,thereare13choicesforthesecondcard.

IfIfirstchooseoneofthe3heartfacecards,thereareonly12choicesforthesecondcard.

Sothereare9x13+3x12=153possibleoutcomes.

Feature1:Requireindependenceof#ofop&ons

Feature1:Requireindependenceof#ofop&ons•  Of51opera&onalorbridgestatements– 11aiendedtoindependenceexplicitly– 16aiendedtoindependenceimplicitly– 24statementsdidnotaiendtoindependenceinthestatementatall

HowmanydifferentwaysaretheretoarrangetheleJersinthewordMATH?

Feature2:Allowdependenceofop&onsets


____

A___

H___

T__

M___

AHM_AHT_

AMH_AMT_

ATH_ATM_

HAM_HAT_

HMA_HMT_

HTA_HTM_

MAH_MAT_

MHA_MHT_

MTA_MTH_

TAH_TAM_

THA_THM_

TMA_TMH_

HM__

MH__

MT__

AH__

AM__

HA__

TA__

TH__

TM__

AT__

MA__

HT__

AHMTAHTM

AMHTAMTH

ATHMATMH

HAMTHATM

HMATHMTA

HTAMHTMA

MAHTMATH

MHATMHTA

MTAHMTHA

TAHMTAMH

THAMTHMA

TMAHTMHA

4x3x2x1=24


A___

H___

T__

M___

{A,H,M,T}

{H,M,T}

{A,M,T}

{A,H,T}

{A,H,M}

4x3x2x1=24____


•  Thecardinali5esareindependent,eventhoughthesetsthemselvesmaynotbeindependent

•  Feature2canbeproblema&cforapurelystructuralstatement

•  Thisdoesnotaccountforsomesimplesitua&onswhenwe’dliketomul&ply

Feature3:Requiredis&nctcompositeoutcomes•  Howmany3-leJersequencescanbemadeusingtheleJersa,b,c,d,e,f,–  Ifthewordmustcontaine,andnorepe..onofleJersisallowed?

– 3x5x4

___________________________

e

e

e

•  Howmany3-leJersequencescanbemadeusingtheleJersa,b,c,d,e,f,–  Ifthewordmustcontaine,andnorepe..onofleJersisallowed?

–  Ifthewordmustcontaine,andrepe..onofleJersisallowed?

– 3x6x6

Feature3:Requiredis&nctcompositeoutcomes

___________________________

e

e

e

•  Apurelyopera&onalstatementcanmisleadonthatlastproblem– 3x6x6overcounts– Considertwowaysofcomple&ngtheprocess

– The“eae”passwordiscountedtoomany&mes

_________

e _________

a e e a e


•  Apurelyopera&onalstatementcanmisleadonthatlastproblem– Byanopera&onalstatementoftheMP,thereare3x6x6=108waysofcomple&ngtheprocess

– Thisistrue,butthisisnotequaltothenumberofdis&nguishabledesirableoutcomes


•  Apurelystructuralstatementcanalsomisleadonthatlastproblem– 3x6x6,same3stages– Encodesolu&onsas3-tupleswhere

•  firstcoordinateisanumber{1-3},•  secondcoordinateisaleier{a-f}•  thirdcoordinateisaleier{a-f}

(1,a,e) (3,e,a)

_________

e _________

a e e a e


•  Apurelystructuralstatementcanalsomisleadonthatlastproblem– Byastructuralstatement,thereare3x6x6=108different3-tuples

– Thisistrue,butthisisnotequaltothenumberofdis&nguishabledesirableoutcomes


Feature3:Dis&nctcompositeoutcomes

SummarizeResultsoftheTextbookAnalysis

•  WediscoveredawidevarietyofMPstatements

•  Therearethreekeymathema&calfeaturesoftheMPCode

Requireindependenceof#ofoptions

All3statementtypes(Structural,Operational,Bridge)can,andoftendo,addressthisissue.

Allowdependenceofoptionsets

Structuralstatementsdonotnaturallyallowforthis.OperationalandBridgestatementscan,andoftendo,addressthisissue.

Requiredistinctcompositeoutcomes

StructuralandOperationalstatementsdonotnaturallyaddressthisissue.Bridgestatementscan,andoftendo,addressthisissue.

ConclusionsandImplica&onsofStudy1

IftheMPstatementyouteachis:•  structural,yourstudentsmayhaveissueswithdependentchoicesetsorduplicatecompositeoutcomes

•  opera5onal,yourstudentsmayhaveissueswithindependenceorduplicatecompositeoutcomes

•  abridgestatement,yourstudentshavenoexcuse!

Study2:StudentUnderstandingoftheMP

•  Weworkedwithtwocalculusstudentsinan8-sessionteachingexperiment– “Aprimarypurposeforusingteachingexperimentmethodologyisforresearcherstoexperience,firsthand,students’mathema&callearningandreasoning”(Steffe&Thompson,2000,p.267)

– Allowsustoformulateandtesthypothesesaboutstudents’reasoningover&me

•  PatandCaleb(pseudonyms)werevectorcalculusstudents


•  Forthe“teaching”intheteachingexperiment,weusedguidedreinven&on(Freudenthal,1991)– Ratherthangivestudentsstatementstointerpret,wegivethemtasksandexperiencesfromwhichtheycanformalizemathema&calideas

– Wegavethemtaskstoperturbtheirthinkinganditera&velyrefinetheirstatementsoftheMP


•  Sessions1-2–Solvingini&alcoun&ngproblems– Studentsgainexperienceusingmul&plica&on– Theyencounterini&almathema&calissues

•  Sessions3-7–Ar&cula&ngandrefiningastatementoftheMP– Theyprogressivelyrefinetheirstatement

•  Session8–Evalua&ngtextbookstatements

Ini&alMPStatement

•  Ifyouhadtowritearuleforwhenyouaregoingtousemul5plica5ontosolvecoun5ngproblems,whatwouldyouwrite?

Ini&alMPStatement

Usemul&plica&onincoun&ngproblemswhen…thereisacertainstatementshowntoexistandwhatfollowshastobetrueaswell.

Coin,Die,DeckTask

Howmanywaysaretheretoflipacoin,rolladie,andselectacardfromastandarddeck?

Caleb:Andoffofthosesixpossibleop&onstherewillbe52op&onsforwhatcardsyoucanpullfromadeck.

Coin,Die,DeckTask

Howmanywaysaretheretoflipacoin,rolladie,andselectacardfromastandarddeck?

Caleb:Solet's,we'vedefinitelycometotheconclusionthatiftheirgroupsareequalwemul&ply.Pat:Ifwe'recombiningequalgroupswe'remul&plying.

RefiningastatementoftheMPCaleb:Soformul&plica&on.Howwouldwedecideifthey'reequalornot?Pat:Okay,um.If,foreverypossibleselec&on,orforeverypossibleoutcomethere'sthesamechoicesayerthat,forthat.Caleb:Forevery&me?Pat:Foreverypossibleoutcome.Likefor,likeforthedie.Foreverypossibleoutcomeof thediethereisthesamenumberofcardstoselect.Andthesamecardsthemselves.Solike,Ican,Ican’t figureouthowtosaythefirstpart.Caleb:Yeahthat'sexactly,that'showIfeel.

RefiningastatementoftheMPCaleb:Soformul&plica&on.Howwouldwedecideifthey'reequalornot?Pat:Okay,um.If,foreverypossibleselec&on,orforeverypossibleoutcomethere'sthesamechoicesayerthat,forthat.Caleb:Forevery&me?Pat:Foreverypossibleoutcome.Likefor,likeforthedie.Foreverypossibleoutcomeof thediethereisthesamenumberofcardstoselect.Andthesamecardsthemselves.Solike,Ican,Ican’tfigureouthowtosaythefirstpart.Caleb:Yeahthat'sexactly,that'showIfeel.

Statement2

Foreachpossiblepathwaytoanoutcomethereisanequalnumberofop&onsleadingtothatpathbutwithoutrepea&ngthesamepathwaymorethanonce.

PushforMoreGeneralLanguage•  Weaskediftheycouldar&culatelanguagethatwasmoregeneralthana“pathway”

•  Theyspontaneouslybroughtupasitua&oninvolvingpants-shirts-shoesouaitsinordertoar&culatetheterms

PushforMoreGeneralLanguage

•  Aselec&oniswhenachoicehastobemade•  Anop&onisoneofthepossiblechoicesfora

selec&on•  Anoutcomeisonecomboofallchosenop&ons

PushforMoreGeneralLanguage•  “Foreveryconnectedselec&on...”

•  Pat:Liketheselec&onofpants,shirts,shoesmakessenseforyourouait.But,like,ifyousaidIselect–ayerIgetdressed,I'mgonnagoeatbreakfast,youknow,it'snotgonnamakeawholelotofsensetosayyouknowmyouaitisgoingtodecideifIhavecereal.

•  Caleb:Atthatpoint,itwouldn'tbelikeouait,itwouldbemorning.Likeawholerou&ne.

•  “Foreveryselec&ontowardsaspecificoutcome…”

StudentReasoningaboutFeature1:Independence

HowmanypossibleoutcomesarethereifIchoose2differentcards(withnoreplacement)fromastandard52-carddeck,wherethefirstisaface-card(J,Q,K)andthesecondisaheart?


Pat:Yeahthereare12op&onsforyourfacecard.Caleb:Yeah,andthenyou'dmul&plythatbyhowmanyheartstherearewhichis13,sayingthatyourfirstonewasafacecard.Pat:Sothere's13or12op&onsdependingonifthefirstone'saheartornot.


Caleb:WellIthinkwecanjustsaythat,ohgosh,thisishard.Oh,wecoulddoifthereisnoheartandthenwecoulddoifthereisaheart.

Statement3b

Foreveryselec&ontowardsaspecificoutcome,ifoneselec&ondoesnotaffectanysubsequentselec&onthenyoumul&plythenumberofalltheop&onsineachselec&ontogethertogetthenumberofpossibleoutcomes.

StudentReasoningaboutFeature2:DependenceofOp&onSetsHowmany6-characterlicenseplatesconsis&ngofleJersornumbershavenorepeatedcharacter?– 36x35x34x33x32x31

StudentReasoningaboutFeature2:DependenceofOp&onSetsCaleb:It'dbelike36x35x34x33x32x31.Pat:Yeahandthatwouldn'tbemul&plica&onasfaraslike–Caleb:Yeahitwould.Pat:Thisgoestofactorialmul&plica&on,becausetowardsouroutcome–the“everyselec&onhereaffectsasubsequentselec&on.”Whateveryouselecthererestrictswhatcouldbehere.

StudentReasoningaboutFeature2:DependenceofOp&onSetsCaleb:Butit'ss&llmul&plica&on.Pat:It'ss&llmul&plica&onbutit'snotthesameasmul&plica&onthatwewerethinkingof.So,ithastochangethingsnowdoesn'tit?Caleb:Thatonedefinitelyputadamperonour[statement].

StudentReasoningaboutFeature2:DependenceofOp&onSetsPat: I'mjustconcernedabouttheideathatwe'resaying,thattheselec&onisaffec&ngthenextselec&on,becausetechnicallyinthiscase,everyselec&onaffectssubsequentselec&ons,butits&llismul&plica&on.

Statement3c

Ifforeveryselec&ontowardsaspecificoutcome,ifthereisnodifferenceinthenumberofop.ons,regardlessofthepreviousselec&ons,thenyoumul&plythenumberofalltheop&onsineachselec&ontogethertogetthetotalnumberofpossibleoutcomes.

StudentReasoningaboutFeature3:Dis&nctCompositeOutcomes

•  Weintroducedaprobleminvolvingovercoun&ng– The3-leiersequencesproblem


Caleb:Soourproblemhereisovercoun5ng,andyoucan'tjustlikeputinaclauseoflike“don'tovercount.”Pat:Sohowaboutwesayspecificuniqueoutcome?Caleb:Yeah.


Pat:SoIfeellikeuniquehastobeaddedbeforetheverylastword.‘CauseIfeellikethatatleastgramma&callytakescareofthiscase.Butitdoesn'tintui&velyexplaintoyouhowtobesureofthat.


Ifforeveryselec&ontowardsaspecificoutcome,ifthereisnodifferenceinthenumberofop&ons,regardlessofthepreviousselec&ons,thenyoumul&plythenumberofalltheop&onsineachselec&ontogethertogetthetotalnumberofpossibleuniqueoutcomes.

FinalStatement

Pat:Soitshouldcountforallinstanceswherenothingchanges,itshouldcountforfactorialinstances. Anditshould–atleastwithsomeamountofintui&onandunderstanding,uniqueshouldmakeitsothatyoudon'tovercount.

FinalStatement

•  Thisisabridgestatement– Theyarecoun&ngoutcomes,buttheydescribeaprocessofselec&onsthatgeneratethoseoutcomes

Evalua&ngTextbookStatements

Caleb: Thiskindaaddressesnoneofthethingswe say,exceptforthatyoumul&plythem.


Pat:YeahIfeellike,Ithinkthisis,holdsreallystrongintui5velyforthingsthatarecompletelyseparate,like,uhtherearenoheadsandtailsonadie,ifthatmakessense.Butwhenyouhaveitlikewhereitisoverlapping,likeIfeellikethisdoesworkbutit'saliilehard,youhavetodoaliilemoreintui&vethinkingintoitofliketheideathatyoucouldoverlappeopleanditdoesn'tmaGer,justaslongasyourcardinalitystaysthesame.


Caleb:Thatkindagetsbacktoours.Itaddressestheindependentchoicesandtheuniqueoutcomes.Ithinkifwebrokedowneachofourswecouldbasicallyrewordthemtobethesame.

SummaryoftheStudents’Reinven&on

•  Thestatementsbecamemoresophis&catedthroughouttheteachingexperiment

•  Bygivingthemproblemsthathighlightedthethreemathema&calfeatureswecouldhelpthestudentsrefinetheirstatement

Summaryof3FeaturesFeature' '

Independent'#'of'options'

All#3#statement#types#(Structural,#Operational,#Bridge)#can,#and#often#do,#address#this#issue.#'Give%problems%that%involve%dependent%stages%in%the%process.%

Allow'option'sets'to'be'dependent'

Structural#statements#do#not#naturally#allow#for#this#(a#few#ugly#but#notable#exceptions#exist.)#Operational#and#Bridge#statements#can,#and#often#do,#address#this#issue.#'Give%problems%that%involve%permutations,%or%stages%with%decreasing%numbers%of%options%that%do%not%simply%involve%Cartesian%Products.%

Composite'outcomes'be'distinct'

Structural#and#Operational#statements#do#not#naturally#address#this#issue.#Bridge#statements#can,#and#often#do,#address#this#issue.##'Give%problems%in%which%overcounting%can%occur.%

'

ConclusionsandImplica&ons

•  TheMPisanuancedideawithsubtleandimportantmathema&calfeatures

•  Ifstudentsdonotgrapplethesesubtle&es,theymayapplytheMPwithoutunderstandingpoten&alissuesthatmayarise


•  Althoughthesestatementscanbehardtointerpret,throughengagingwithpar&culartasksourstudentsbecameaiunedtokeymathema&calfeaturesoftheMP–  Independence

•  PairsofCards– Dependenceofop&onsets

•  Arrangementsorpermuta&ons

– Dis&nctcompositeoutcomes•  Three-leiersequencescontaininge


•  Someadviceforstudentsofdiscretemathema&cs– Thinkaboutwhatyouaretryingtocount(outcomes)andhowyourcoun&ngprocessgeneratesandstructuresthoseoutcomes

– BethoughaulandcarefulabouthowyouapplytheMPwhensolvingcoun&ngproblems

– Don’tovercount!

ThankYou!!

combinatorialthinking.com

[email protected]

Inves&ga&ng Subtle&es of the Mul&plica&on Principle HATM HMAT HMTA HTAM HTMA MAHT MATH MHAT MHTA...

Documents

Transcript of Inves&ga&ng Subtle&es of the Mul&plica&on Principle HATM HMAT HMTA HTAM HTMA MAHT MATH MHAT MHTA...