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1、39HierarchicalComplexityandTaskDifficultyMichaelLamportCommonsandAlexanderPekkerHarvardMedicalSchoolandHarvardUniversityMichaelLamportCommons,Ph.D.AssistantClinicalProfessorofPsychiatryPrograminPsychiatryandtheLawDepartmentofPsychiatryHarvardMedicalSchoolBethIsraelDeac
2、onessMedicalCenter234HuronAvenueCambridge,MA02138-1328Telephone617-497-5270Facsimile617-491-5270Commons@tiac.netAlexanderPekker,Ph.D.DepartmentofMathematicsHarvardUniversityCambridge,MA0213839AbstractTheoriesofdifficultyhavegenerallynotaddressedhierarchicalcomplexityof
3、tasks.Withindevelopmentalpsychology,notionsofhierarchicalcomplexityhavecomeintobeinginthelast20years.Weshowhowamodelofhierarchicalcomplexity,whichassignsanorderofhierarchicalcomplexitytoeverytaskregardlessofdomain,mayhelpaccountfordifficulty.Theorderscorrespondtonatura
4、lnumbers,thusinsuringthattheordersareseparatedbygaps.Themodelnaturallyleadstotheexistenceofperformancestages,therebyformalizingmanyimplicitpropertiesofstagetheories.KeyWords:TaskDifficulty,MathematicalTheory,Complexity,HierarchicalComplexity,Distributivity,Rasch,Scales
5、,Stage,OrdinalMeasure39Ifwecouldunderstandwhypeoplefailattasks,wemightbeabletoofferabetterframeworkforsuccess–particularlysuccessinareaswhereundernormalcircumstancespeoplewouldhaveotherwisefailed.Anunderstandingofhowtosucceedcouldhavesubstantialimplicationsforexample,f
6、oreducationreformorimprovingjobperformance.Becauserelativetaskfailureratescanindicateempiricallyhowdifficultataskistocomplete,agoodframeworkforsuccesswouldstemfromananalysisoftaskcharacteristicsthatcontributetodifficulty.Tasksaredefinedassequencesofcontingencies,eachpr
7、esentingstimuliandrequiringbehaviorsthatmustoccurinsomenon-arbitraryfashion(Anderson,1980).Tasksmaybestudiedwithrespecttotheanalyticallyknownsourcesoftaskdifficulty.TherearemanymeasuresoftaskdifficultythathavebeenidentifiedSomeoftheseincludefactorsthatareinherenttothet
8、askitself(taskproperties)aswellasfactorsthatareindependentofthetaskbutdoaffectthesubject'sperformance(non-taskpropert
