• 検索結果がありません。

A Synthesis of Implicit and Explicit Learning: Teaching Approaches and Learning Strategies

N/A
N/A
Protected

Academic year: 2021

シェア "A Synthesis of Implicit and Explicit Learning: Teaching Approaches and Learning Strategies"

Copied!
9
0
0

読み込み中.... (全文を見る)

全文

(1)

Introduction

SLA (SecondLanguageAcquisition)teachingapproachesaremoreeasilyapplicablewhen instructorscan pinpointspecificissuesthatareplaguing students・learning and identify learningstrategiestohelpgeneratebetterproduction.InordertoteachSLA successfully,it isnecessary to takeinto accounttheelementsthatmakeup SLA,asstated by Rebecca Oxford in Celce-Murcia (2001),multipledifferencesin learning strategiesattributetothe students・learningability.Findingsfrom neuroscience,havealsodemonstratedavarietyofthe mostbasicdifferencesbetween students.In Richardsetal.(2011),itisstated thatbrain research done on student writers confirmed that there are major differences between learnerswhoare・goodandbadwriters・;in which case,they usetheirbrain in different ways(p.502).Theeducationalstancethatallstudentsaresmart,asstated in Sousaand Tomlinson(2011),suggeststhatdifferencesinstudentsarenotduetoalevelofintelligence, buttherearemanydifferenteventsthroughoutthelearningcycleofanSLA learnerthat interferein theeducationalprocess.Oneofthemostcompleteexamplesistocomparethe learningstrategiesinwhichanadultinterpretsinformationandapproachesdifficulttasks; thisconceptisradicallydifferentfrom thatofachildSLA learner.Sincetherearesomany variables,teachersneed toadapttheirteaching approachestogetthedesired outcomein SLA (Byrnes& Fox,1998).Itistheteachers・job to usemixed-approachesor learning approachesthatrequireamorebalancedamountofimplicitandexplicitlearningstrategies inordertocatertoalargeramountofstudents.

学苑 No.886(21)~(29)(20148)

A Synthesi

sofImpl

i

ci

tandExpl

i

ci

tLearni

ng:

Teachi

ngApproachesandLearni

ngStrategi

es

JesseR.El

am andShawnM.El

am

Abstract

Thissynthesishasbeencompiledfrom threetextbooksandtwoonlinearticles.Itfocuseson Teaching English asa Second orForeign Languageedited and written by MarianneCel ce-Murcia,whichsuggests・TheCommunicativeTheory・ismosteffectiveforTESOL.Thisideology iscounterbalancedbytwoothertexts,TheBookofLearningandForgettingbyFrankSmith, which encourages ・The ClassicView・ oflearning or ・SocialLearning・,and Principlesof LanguageLearningandTeachingbyDouglasBrown,whichisconcernedwith・TheCognitive andConstructivist・view oflearning.Thismanuscriptattemptstoexplainimplicitandexplicit learning strategies,theirrelationshiptoCelce-Murcia・s・NineTwentieth-Century Approaches to LanguageTeaching・,and objectively gaugetheamountofimplicitand explicitlearning strategiesindifferentlearningapproaches.

(2)

Thesearch for an all-inclusiveteaching approach for SLA isa heavily sought-after entity,butthegoalofcreatingsuchatheoryisoftenviewedasunreachableintheeyesof many professionalsin thefield.RebeccaOxfordin Celce-Murcia(2001)articulatesthat,an overallflawlessdesigncannotbeestablished,whenshestatedthatitisquitefoolishtothink thatasingleL2methodologycouldpossiblyfitanentireclassfilledwithstudents:whohavea rangeofstylisticandstrategicpreferences.However,giventhefactorsofstudents・・learning strategies・;educatorscanmakeconnectionstotheir・preferences・anduseteachingapproaches to suite their needs (p.359).Since this rationale is a more common belieftoday,itis importantforeducatorstobeknowledgeableofstudents・learningstrategies,therelationship betweenimplicitandexplicitlearningandthemostbeneficialteachingapproaches.

Tothisendweaskourselves:

・How hasimplicitandexplicitlearningbeenclassified? ・Whatareimplicitandexplicitlearningstrategies? ・How areimplicitandexplicitstrategiesinterrelated?

・Whichteachingapproachesadvocateimplicitandexplicitlearningstrategies? Background

Thedualism ofimplicitand explicitlearning hasbeen a topicofgreatcontroversy amongsttheoristsand hastypically been divided into two separatedomains,which have been classified with differentterminologiesto definethedichotomies.Oneofthemajor underliningcharacteristicsofateachingapproachisthelevelofimplicitnessorexplicitness itexemplifies,thusinstructorsshouldalignteachingapproachesandlearningstrategiesto aim atthecognitivestyleand learning preferencesoftheir students.Studentsmustbe preparedfortheeducation aheadofthem,andin which case,educatorsmustgetabetter understanding ofimplicitand explicitlearning and theteaching approachesthatemploy them.With somany waystodefineimplicitandexplicitlearning,itisconfusing how to makecleardistinctions;however,implicitandexplicitlearninghavebeenembodiedinsuch dichotomiesas:Smith・s(1998)classicand officialeducation,Brown・s(2006)consciousand unconsciouslearning,Brown・s(2006)focaland peripheralattention,and theirrelation to Bialystok・s(1978)・ModelofSecondLanguageLearning・.

ClassicandOfficialEducationDichotomy

Smith(1998),inTheBookofLearningandForgetting,describeslearningas・twovisions・of thoughtandexplainsthedifferentamountsofawarenessthatpeoplehaveinaneducational atmosphere(p.5).Hecoinedtheterms,theclassicview andtheofficialtheorytoexplainthe processofeducation.Theclassicview oflearningisconsideredtobemoreimplicitinnature, andasdescribedbySmith(1998),constituteslife-longlearningformorecomplexeducational tasks.In the classicalview,he states that students・learning happens effortlessly and

(3)

throughsocialinteractions.AccordingtoSmith(1998),officiallearningisexplicitlearning, recognizing thatforstudentstobecomeawareoftheirknowledge,itrequireshardwork, andstudentshavetomemorizefactsfortesting.Thistypeofexplicitlearningisubiquitous withintheeducationsystem today.However,hebelievesthattestinginhibitsprogressanditis unpremeditatedandfurtheradvocatesthatmostoflearning in lifehappensunconsciously whileinteractingwithothers;thusweshouldstrivetousemoreimplicitteachingapproachesin theclassroom (Smith,1998).

ConsciousandUnconsciousLearningDichotomy

Brown (2006)describes implicit learning as,absorbing knowledge without conscious effort;ornothaving ・awarenessofwhathasbeen learned・(p.292).In retrospect,Brown (2006)explainsthatexplicitlearningusesawarenesstointentionallyprocessinformation.In essence,thesetermsareseenastwodifferentrealmsdistinguishedonlybytheirpsychological state.AlthoughthesetheoriescoincidewithSmith・s(1998)observations,Brown(2006)believes thatboth typesoflearning included ・focusing on certain stimuli・(p.292).Thedifference being,explicitlearnersareincontroloftheirfocusandareconsciouslytryingtolearnthe topicathand,whereasimplicitlearnersareovertaken by thestimulusandlearn through experiencing thesituation.Along thesamelines,Krashen definesimplicitlearning asan unconsciousacquisition,whilestatingthatexplicitlearningisconsciouslearning(ascitedin Finkbeiner,1998).Krashentheorizesthatconsciouslearningcanonlybeusedasa・monitor・ andthatalllanguageacquisitioncomesfrom implicitlyprocessingtheinformation(Brown, 2006,p.294).Controversially,many othertheoristsstatethattherearedifferentlevelsof consciousnessand unconsciousnessrequired for differentlanguageacquisition tasks,and these processes have not been completely defined (Brown,2006).What we know from researchisthatthemostcomplexlearninghappenswhenweareunconsciousoftheprocess athand(Finkbeiner,1998)andthough consciousandunconsciouslearning can beslippery terms(Brown,2006),wecansimplyview them asdifferencesinfocalattentiononstimuli. FocalandPeripheralAttentionDichotomy

Focaland peripheralattention was another attemptofdescribing how implicitand explicit learning were interrelated. Previously in SLA, focal attention and peripheral attention,compared with explicitand implicitlearning,wereseen asbeing synonymous. However,Brown(2006)warnsaboutsomeofthe・pitfalls・whenmakingoversimplifications, ashestatesitisveryimportanttonotethatinvirtuallyeveryactofperformingsomething, ・focaland peripheralattention actually occur simultaneously・ (p.301).In McLaughlin・s Model(1978)heagreeswithBrown・s(2006)rationale.(Table1.1)Whenwelearn,wecannot distinguish between implicitness and explicitness,because they vary depending on the contextofthelearningsituation.Furthermore,implicitandexplicitlearninghappenatthe

(4)

sametimeduetothenatureoflanguageacquisition.Theideathatthereisa・monitor・or ・consciousmind・ thatcan only makeadjustmentsor processexplicitinformation seems merely theoretical(Brown,2006,p.294).Thus,in thiscase,Krashen・sdistinctionsbetween learningandacquisitionhavebeenseenasempiricallyunverifiable.Itismorewidelyaccepted thatimplicitandexplicitlearningareoverlappingentitiesascanbeseeninTable1.1.

TheModelofInteractingKnowledge

Inacognitiveattempttounifyimplicitandexplicitlearning,Bialystok(1978)createda modelexplaining how explicitand implicitlearning coincide(Fig.1).In Bialystok・s(1978) model,implicitandexplicitlearningstrategiescanbeseenasinterconnectedentitiesthatare concurrently happening in unison.According to Bialystok (1978),implicit and explicit learning strategiesoperatein unison to processknowledge.Through repetition,language can movefrom being explicitly analyzedtostructuresthatareeasily accessed,butitwill nevertruly beinternalized likeimplicitknowledge.However,Bialystok (1978)admitsthat evenexplicitknowledgeisslowlyconvertedintoimplicitknowledgeaswemakenew cognitive associationsandinternalizetheinformation.OneofthecentralthemesofBialystok・s(1978) theorystatesthatwemustfirstacquireknowledgebeforewecanpossessit.Furthermore, thoughtherearemanyviewsofhow weacquirelanguageknowledge,thelearningstrategies designedforinputandintakearequintessentialintheactofSLA.Andso,understanding theteaching approacheswhich utilizelearning strategiescan defineatwhatlevelstudents areengagedinimplicitorexplicitlearning.

F oc alA tt en ti on (E xp li ci tS tr at eg ie s) P er ip he ra lA tt en ti on (I mp li ci tS tr at eg ie s)

Table1.1:PracticalApplicationsofMcLaughlin・sAttention-ProcessingModel (Adaptedfrom Brown,2006,p.302)

Attention Controlled:new skillcapacitylimited Automaticapacic:weltyisrell-traiatinedpractivelyunlimicedskited ll A.・Grammaticalexplanationofaspecific

point

・Worddefinition ・Copyawrittenmodel

・Thefirststagesof・memorizing・adialog ・Prefabricatedpatterns

・Variousdiscrete-pointexercises

B.・Keepinganeyeoutforsomething ・AdvancedL2learnerfocuseson

modals,clauseformation,etc. ・Monitoringoneselfwhiletalkingor

writing ・Scanning

・Editing,peer-editing C.・Simplegreetings

・Thelaterstagesof・memorizing・adialog ・TPR/NaturalApproach

・New L2learnersuccessfullycompletesa briefconversation

D.・Open-endedgroupwork ・Rapidreading,skimming ・Freewriting

・Normalconversationalexchangesof somelength

(5)

LearningStrategiesandTeachingApproaches

Research regarding thecombination and plausibility ofimplicitand explicitlearning strategieshasopenedthedoorformoreradicalinterpretationsofSLA teachingapproaches. Theseteaching approachesalluseimplicitand explicitlearning strategiesto drivetheir learningtheoriestosomeextent,buteachseem toexploitdifferentamountsofimplicitand explicitlearning (Brown,2006).Thesedifferentvariationscan beseen in Table1.2,which summarizestheimplicitandexplicitlearningstrategiesassociatedwithCelce-Murcia・s(2001) ・NineTwentieth-CenturyApproachestoLanguageTeaching・(p.5-9)accordingtothesocial, physiologicalandcognitiveviewsdefinedearlierinTable1.1andFig.1.

Fig.1:ModelofInteractingKnowledge(Bialystok,1978,p.71) (OriginalTitle:ModelofSecondLanguageLearning)

Re ad in g Gr amma r-T ra ns la ti on

Table1.2:Implicitvs.ExplicitStrategies(Adaptedfrom Celce-Murcia,2001,p.5-9) Teaching

Approaches ImplicitLearningStrategies ExplicitLearningStrategies There is little to no implicit learning.

Studentsreadtextandinferthemeaning ofnewwords.Rapidreadingandskimming promotesperipheralattention.

Thereisahighamountofexplicitlearning. Vocabulary iscontrolledanditistaught based on frequency and usefulness for reading.Grammarisalsolimitedandonly taughtforreadingcomprehension. There is a small amount of implicit

learning.Studentsreadtextandinferthe meaningofsomenew words.

Thereisahighamountofexplicitlearning. Students・attention isfocused on parsing grammar and reading difficultpassages. Theyarelearningintheirnativelanguage and use grammar concepts to translate. Thereislittletonosocialinteraction or speech.

(6)

Or al -S it ua ti on al Au di o-li ng ua li sm Af fe ct iv e-Hu ma ni st ic C og ni ti ve C ommu ni ca ti ve Di re ct C omp re he ns io n-B as ed Teaching

Approaches ImplicitLearningStrategies ExplicitLearningStrategies There is a small amount of implicit

learning.A lotofthe learning is done orally and the grammatical forms are practiced later.Thisallowsthestudents tomakesomeoftheirowninferencesabout languageinnew anddifferentsituations.

Thereisahighamountofexplicitlearning. Grammarlearningishighlycontrolledand ispresented to thelearnersin a natural order.Most of the speaking is geared towardthetargetlanguage,sothestudents havetoconsciouslydevelopandmanipulate thephrases.

There is an average amountofimplicit learning.Dialogsandrole-playsareoften used to getstudentsto interactsocially. Grammarrulesarelearnedincidentallyin conversation.

Thereisquiteahigh amountofexplicit learning.Mimicryandmemorizationisseen asawaytobuildcorrecthabits.Corrections are common and emphasis is placed on proper pronunciation from the start. Learningactivitiesarehighlycontrolled. There is an average amountofimplicit

learning.Teachermovesto a facilitation role, letting the students communicate abouttopicsthataremeaningfultothem. Peerinteractionishighlypromoted.

There is an average amount ofexplicit learning.Students・consciousmindworks on self-realization and the materials at hand.

There is an average amountofimplicit learning.Priority is given to speaking and listening. Rules of language and pronunciation can belearned inductively, leavingmuchofthesecond-languagetobe learnedimplicitly.Learnershaveachance tomaketheirownconnections.

There is an average amount ofexplicit learning.Vocabulary isinstructed tothe studentsandhighstressisplacedonrule acquisition.Grammar is usually taught deductively.

Thereisahighamountofimplicitlearning. Most of the Communicative Learning theory is based on reflecting real-world environmentsanddifferentsocialcontexts. Thisenablesstudentstoinadvertentlylearn socialfunctions.Studentstrytocooperate andproducelanguage.Studentscomewith differentskillsets,sotheycanlearnnew grammar and words implicitly through conversation.

There is an average amount ofexplicit learning. The target language and the environmentusedarecontrolledwith the language.Itinvolvesothertaskssuchas; reading,speaking,listening,and writing. Thisrequiresintention on thestudentto comprehendthematerial.Errorcorrection isoftengiven,butisnotaprimaryconcern.

Thereisahighamountofimplicitlearning. There is no use of the mother tongue permitted; grammar and culture are learnedinductively through reading texts andconversations.

Thereisasmallamountofexplicitlearning. Some repetition is required to learn vocabulary and phrases. Actions and picturesareusedtomakethesignificance ofnew wordsunderstandable.

Thereisahighamountofimplicitlearning. Studentsarenotforced tofocuson rule learning ormemorization.Learnerslisten tomeaningfulspeech;theydevelopspeaking skills spontaneously and subconsciously. Learnersonlyspeakwhentheyareready andnoerrorcorrectionshouldbegiven.

There is little to no explicit learning. Some rule learning is taught,butitis seen as unhelpful. Learners are given listening tasks,justabovetheir levelof comprehension, so they may have to deductivelyanalyzetasks.

(7)

Discussion

Ononeendofthespectrum,asstatedbyBrown(2006),sometheoristswouldattestthat languageshouldneverbetaughtexplicitly,whileotherswouldencourageaheavilyimplicit method of language education.Many SLA researchers have attempted to explain and rationalize the positive and negative effects of implicit and explicit learning,but the complexitiesembodied in SLA haveproven too difficultto makegeneralizations(Brown, 2006).The mostimportantaspectis being informed aboutteaching approaches and the varyingrangesofimplicitandexplicitlearningstrategiesdrivingthem.Itisprevalentthat implicitandexplicitprocessesareusedatthesametime,ascanbeseeninFig.1(Bialystok, 1978).Looking atFig.2,wecan seearepresentation oftheteaching approaches,discussed earlierin Table1.2,andhow they arerelatedtoimplicitandexplicitlearning strategies. Fig.2showsthatsometeachingapproachesrepresentimplicitandexplicitlearningstrategiesat moreextremelevels.RebeccaOxford states,however,in SLA itwould bebettertomove towardsa・broadinstructionalapproach・andtooffermorebalancedteaching approaches with equal distributions (Celce-Murcia,2001,p.365). Thus,Audio-Linguistic,Affecti ve-Humanistic,CognitiveandCommunicativeApproachesofferthemostrangeofflexibilityfor learningstrategies.

LearningstrategiesareapredispositionthatcanfacilitateorinhibitSLA educationin terms ofcommunication,and willingness to take risks in participation.Also,learning strategiesareasetofskillsthatstudentsusetotacklelanguagetasks(Celce-Murcia,2001). AsRebeccaOxfordstates,aspectssuch asindividualisticandcollectivistic・manifestations・

(8)

instudentsshouldalsobetakenintoaccountwhenexpectingcertaintypesofinteractionin class(Brown,2006,p.202).Wecannotfitallstudentsintothesamebox,astheirlearning strategiesmayrangegiventheirsocioculturalidentity,ageandanumberofotherfactors. Theselearningstrategiescanalso・helpdeterminehow studentslearnasecondlanguage・ (Celce-Murcia,2001,p.359),soitisvitalthatteachershaveanoverview ofstudents・abilities.

Educators have two different options in order to cater to their students・learning strategies;eithertestthestudents・learningstrategiesatthebeginningoftheyearorteach new learning strategies in the classroom . Teachers can assess their students・learning strategies at the beginning of the semester using theStrategy Inventory forLanguage Learning1(SILL),whichcanbefoundinRebecca Oxford・s(1990)Appendix (Celce-Murcia, 2001).After giving the50-question SILL test,studentscan estimatetheir own learning strategiesusingtheSILL Rubric2.Educatorscanthenusethemeanofthestudents・learning strategiesandapplythecorrectteachingapproachesormethodstoensurethebestoutcomeof acourse.RebeccaOxfordalsoadvocatesinstructorsassessthemselvesastobeknowledgeable abouttheir・possiblebiases・(Celce-Murcia,2001,p.365).Secondly,educatorscan usethese resultstoseewhich learning strategiesthestudentsarelacking,asRebeccaOxford(1990) found,studentswhoweremoreawareoftheirlearning strategiesand adapted additional skills,didbetteroverallintheirlanguagelearning(Celce-Murcia,2001).Shealsostatesthat acquiring new learning strategiescan often beachieved by ・demonstrating when a given strategy m ightbeuseful・(Celce-Murcia,2001,p.363).Additionally,instructorscan try to getstudentstotransferstrategiesintootherskills,suchasusingpre-readinginpre-listening deduction (Celce-Murcia,2001).Teaching studentshow tousetheirlearning strategiesin a systematicwaygivesthem moreadvantagesforlanguageacquisitionandmoreflexibilityin theirlearninghabits.

Conclusion

Whilemany instructorsarecompletely awareofwhata schooldoes,whatteaching means,andhow tocreatewhatmightappeartobealesson plan;alloftheseaspectsare meaninglessifthateducatortakesawaylearningopportunitiesfrom hisorherstudents.If wearerequiring studentsto learn a second language,theteachershould berequired to know whatthecomponentsoftheteaching approachesare.Graham Crookesand Craig Chaudron would have also agreed with this point, as they advocate, ・・out-of-class・ knowledgeoflanguageteaching in areassuch asneedsanalysis,curriculum design,lesson planning・promotesuccessfulSLA teaching(Celce-Murcia,2001,p.29).Teachingapproaches thatsupporttheproperamountofimplicitandexplicitlearningstrategiesareanimportant

1 http://richarddpetty.files.wordpress.com/2010/03/sill-english.pdf

(9)

aspecttotheculmination oflearning thatstudentsundergothroughouttheireducational careers.Crafting the correctamountofimplicitand explicitlearning for SLA learners shouldbeaddressedinanobjectivemannerandtheidealbalancereliesontheresultsfrom theirSILL test.

Studentsinterpretinformationandapproachdifficulttasksinfundamentallydistinctive ways.Theapplication,combinationanduseofdifferentteachingapproaches(Fig.2)havea numberofpossibilities,andthecentralquestionaroundwhichconditionsarebestsuitedfor certainteachingapproachesismuchdeeperthanonewouldimagine(Brown,2006).Thereisno simplesolution orcardboardcutoutthatfitstheneedsofevery student,andsoin either caseweshouldtry toidentify thecommon trendsin which ourstudentsgenerally attain knowledgeinstead,asthesedifferentlearning strategiesgovern how studentswillprocess informationandhow successfultheywillbeinourcourses.

References

Bialystok,E.(1978).A theoreticalmodelofsecondlanguagelearning.LanguageLearning,28(1),69-83. Brown,H.(2006).Principlesoflanguagelearningandteaching(5thed.).New York:Pearson.

Byrnes,J.,P.,& Fox,N.,A.(1998).Theeducationalrelevanceofresearch in cognitiveneuroscience. EducationalPsychologyReview,10(3),297-342.doi:http://dx.doi.org/10.1023/A:1022145812276 Celce-Murcia,M.ed.(2001).TeachingEnglishasasecondorforeignlanguage(3rded.).Boston:Heinle

& Heinle.

Finkbeiner,C.(1998).ThepromotionofexplicitandimplicitlearningstrategiesinEnglishinstruction: anecessaryaim?.ExplicitandImplicitLearning.RetrievedJuly1,2014,from http://webdoc.gwdg .de/edoc/ia/eese/artic98/finkb/10_98.html.

Oxford,R.,L.(1990).Languagelearningstrategies:whateveryteachershouldknow.Boston:Heinle& Heinle.

Richards,T.,L.,Berninger,V.,W.,Stock,P.,Altemeier,L.,Trivedi,P.,& Maravilla,K.,R.(2011). DifferencesbetweengoodandpoorchildwritersonfMRIcontrastsforwritingnewlytaughtand highlypracticedletterforms.ReadingandWriting,24(5),493-516.doi:http://dx.doi.org/10.1007/s 11145-009-9217-3

Smith,F.(1998).Thebookoflearningandforgetting.New York:TeachersCollegePress.

Sousa,D.,A.,& Tomlinson,C.,A.(2011).Differentiationandthebrain:How neurosciencesupportsthe learner-friendlyclassroom.Bloomington,IN:SolutionTreePress.

(ジェシー R. エラム 英語コミュニケーション学科)

Tabl e1.1:Practi calAppl i cati onsofMcLaughl i n・ sAttenti on-Processi ngModel ( Adaptedfrom Brown,2006,p.302)
Tabl e1.2:Impl i ci tvs.Expl i ci tStrategi es( Adaptedfrom Cel ce-Murci a,2001,p.5-9) Teachi ng

参照

関連したドキュメント

Here, instead of considering an instance I and trying to directly develop a feasible solution for the P, G ∗ |prec; c ij dπ k , π l 1; p i 1|C max problem, we consider a

Our aim was not to come up with something that could tell us something about the possibilities to learn about fractions with different denominators in Swedish and Hong

Chaudhuri, “An EOQ model with ramp type demand rate, time dependent deterioration rate, unit production cost and shortages,” European Journal of Operational Research, vol..

By virtue of Theorems 4.10 and 5.1, we see under the conditions of Theorem 6.1 that the initial value problem (1.4) and the Volterra integral equation (1.2) are equivalent in the

Finally, we investigate existence of weak solutions in Lebesgue spaces (Theorem 5.7) and the decay of continuous solutions (Theorem 5.8). All presented results are important

のようにすべきだと考えていますか。 やっと開通します。長野、太田地区方面  

The limiting phase trajectory LPT has been introduced 3 as a trajectory corresponding to oscillations with the most intensive energy exchange between weakly coupled oscillators or

iv Relation 2.13 shows that to lowest order in the perturbation, the group of energy basis matrix elements of any observable A corresponding to a fixed energy difference E m − E n