Systematic and Metasystematic Reasoning: ACase for Levels of Reasoning beyond Piaget'sStage of Formal Operations

Michael L. Commons, Francis A. Richards, andDeanna KuhnHarvard University

COMMONS, MIC»AEL L , RICHARDS, FRANCIS A , and KUHN, DEANNA Systematic and Metasys-tematic Reasoning A Case for Levels of Reasoning beyond Piaget's Stage of Formd OperationsCHILD DEVELOPMENT, 1982, 53, 1058-1069 McSes of cognition are postulated consistmg ofthird- and fourth-order operations, they are hypothesized to be qualitatively disbntt from, andhierarchically related to, the form of reasoning characterized as formal operational by Inhelderand Piaget An instrument was developed to assess these modes of cogmtion, labeled systemahcand metasvstemabc reasomng, and was administered to 110 undergraduate and graduate studentsTbe results support tbe asserbon tbat systemabc and metasystematic reasoning exist as modesof cogmbon discrete from, and more complex and powerful tban, formal operational reaiomng

Tbe purpose of the present paper is to ex-plore the possibibty tbat modes of systematicreasonmg can be identified tbat are qualita-tively distinct from, and logically more com-plex than, the form of reasonmg charactenzedby Inhelder and Piaget (1958) as "formal op-erabonal " In this endeavor, we have made useof Piaget's system of successive stages of logicaloperabons only m its most general form Inother words, Piaget's stages are regarded as suc-cessive levels of cognitive representation, in thissense, the stages are not subject to any form ofempmcal venfication, but rather consbtute aconceptual descnpbon of forms of cognition(Bickhard 1979) Tbus, sensory-motor opera-tions are defined as acbons on matenal objecrts,symbohc operations as actions or operations onsymbobc representabons of matenal objects( e g , the concrete operabonal organizabon ofsymbohcally represented objects into classes orrelabons), and formal operations as operationson the above-mentioned operations (of classi-ficabon or relabon), for example, the mdividualorganizes classes (of elements) according to re-labons that obtain among them The object ofthe present work was to mqmre whether levelsof mgher-order operations could be identifiedand, if so, to estabhsh the particular form they

would take m tbe cognitive performance of theindividual

There are several existing attempts m thedevelopmental literature to postulate "postfor-mal operational" stages of cognition, notablvtbose of Riegel (1973) and Arlin (1975) Tbemajor problem with these formulations is thatit IS uncertain the extent to which these vanouspostulated modes of cognition consbtute a qual-itatively distmct structure or stage of reasoning,related m a hierarchical manner to formal oper-ations, or, alternatively, whether they are modesof cognition that develop parallel to Piaget'sstage sequence Forms of Arlin's stage of "prob-lem finding," for example, are likely to be pres-ent during the stages of concrete and formaloperations (Fakoun 1976), thus, the stage offorma! operabons cannot be regarded as thenecessary but insufficient condition for the stageof problem finding, as it would need to be ifthe two were to be related m a hierarchicalmanner A similar consideration applies to Rie-gel's "dialectic operations" As Riegel (1973)noted, dialectical operabons are potentiallypresent in some form at all of Piaget's stages

It IS our view, then, that if forms of ad-vanced cognition are to be identified that make

This researcb was supported in part by a Dare Insbtute grant no 10032 to tbe first authorPorbons of this study were presented at tbe Western Psycbological Associabon, San Francisco,1978, and at die Jean Piaget Society, Pbiladelpbia, 1979 We would like to thank Joan Rich-ards, Patnce M Miller, Joel Peck, Barbara Mabon, Eloise Coupey, David K Pickard, and CherylArmon for thar work on vancms portions of the study Repnnts may be obtained from tbe firstautbor at tbe Department of Psychology and Social Relabons, Wilbam James Hall, HarvanJ Uni-versity, Cambndge, Massacbusetts 02138

[CktU Developmtnl, 1982, S3, 1058-1069 © 1982 by the Society for Rese«rch in ChiW Development, IncAll rights reserved 0009-3920/82/5304-0031*01 00)

reference to, and build on, P^get's sequence ofstages of cognibon, tbey must take tbe generalform of tbe bigher-order operabons on opera-tions on operations (and possibly beyond) re-ferred to above To make tbis claim is not toassert that all forms of adult cognition neces-sarily fall widun such categones Indeed, cur-rent researcb and theory m adult cognitive de-velopment run strongly counter to such a claim(Kuhn, Pennmgton, & Leadbeater, in press, La-bouvie-Vief 1980, 1982) Rather, our claim isthat if one wishes to postulate more advancedforms of logical/mathemabcal reasonmg of thetvpe studied by Piaget, that build on the formsof reasoning that constitute bis existing stagesequence, then this more advanced reasoningmust take tbe general form we have indicated

What might such higher-order operationslook like in the cKignition of an mdividuaP Wecan refer to the second-order, or formal, oper-ations in Piaget's system as reflectmg "mter-relationa! thinking " Tbat is, reasoning is basedon ordered relations of classes or relabons, theindividual executes operations on these classesor relabons For example, tbe individual mayformulate and test hypotheses about the rela-

I tions that obtam among two classes of objectsIn executing these formal operations, bowever,the individual does not self-consciously repre-sent, reflect, or operate on the system as awhole We postulate, then, a possible set ofthird-order operations, termed "systematic op-erations," which consist of exhaustive opera-tions on classes or relations of classes or rela-tions, forming systems We further postulatethe possibility of a set of fourth-order oper-ations, termed "metasystematic operabons,"vvhicb consist of operabons on systems

Systemabc operabons apply to the entireset of constituents of a system (that itself ismade up of operations on operations) For ex-ample, they might consist of the coordinationof lterabve operations such as 2< or the ccwrdi-nation of abstract representations of operations,such as a O b, a * b, to form systems Such sys-temabc descnptions formally represent theproperties of operations at the formal opera-tional level

Metasystematic operations are cxignitionsabout systems They are reqmred m the forma-tion of a framework (or "metasystem") for com-panng and contrasting systems with one an-other The relabonship of one system to anothersuch system is expressed as a metatheory and's found by companng axioms, theorems, orother limiting condibons of systems withm the

Commons, Richards, and Knhn 1059

framework of a "super-system" that contains allof tbe vanant systems Metasystemabc reason-ing IS defined as tbe set of operations necessaryto construct tbe supersystem and to execute theanalysis of the systems contained therein

An example of metasystematic reasonmg isfound in that aspect of Einstein's general tbeoryof relativity that deals with tbe coordmation ofmertial and gravitabonal mass Pnor to Em-stem's formulabon, tbere was one system for de-scribing mertial mass and a separate one fordescnbing gravitational mass Each separatesystem mcluded systematic representabons ofproperties of formally defined relations betweenvariables Inertial mass was tbe property of thesvstem that described a body's resistance to ac-celeration Gravitational mass was a propertyof another system that descnbed the weight ofa body in a given gravitational field In fact,since the same constant for mass representsbotb the inertia and the weight of a body, itfollows that it IS impossible to discover by exper-iment which of the following is true "The mo-tion of a given system of coordmates is straightand uniform and the observed effects within thesystem are due to a gravitabonal field, or thesystem of coordinates is uniformly acceleratedand the observed eflFects withm the system aredue to inertia Recognition of an equivalence be-tween tbe two cases constitutes the equivalenceprinciple of general relativity theory This prm-ciplc states tbat tbe inertial system is isomor-phic (contains the same structure and elements)to the gravitational system, that is, any relabon-sbip tbat IS true m one is true m the otherWbetber the same property appears as weightor as inertia depends on which descnpbon ofthe coorchnate systems is employed, that is, mo-tion m a gravitational field appears m relationto an mertial system of coordmates, while mo-tion m the absence of a gravitational field ap-pears in a coordinate system that is acceleratedThis pnnciple requires a coordmabon of twodisbnct systems, eacb generated by systemabcof)erations the mertial system and the gravita-tional system The cogmbve operations thateffect this coordination are at a "metasystem-atic" level, disbnct from that of the operabonsapphed within either of the lnchvidual systems

The above example serves to illustrate theirreducibility of the higher-order operabons tooperabons of the next lower order Operabonsat the metasystemabc level must be expressedin the language of metalogic, or its psycholog-ical equivalent, because statements about tberelation between systems cannot be reduced tostatements about the properties of tbe relabons

1060 ChUd Developm<ait

withm any single system Similarly, statementsabout systems properbes cannot be contamedwithin any of the individual systems Hence,systematic operations cannot be reduced to for-mal operations because the formal operabonalsystems themselves do not contam descaipbonsof the system These formal operations are notnch enough to describe their own properbesIf this were possible, Russell's paradox (thesentence, "This sentence is false") would notexist In otber words, a property of a systemcannot be descnbed by a proposibon withina system

Method

TaskIn the problem we developed to assess sys-

tematic and metasystematic reasonmg, the sub-ject was asked to compare and contrast foursystems, each compnsecf of a set of asymmetncrelations Two similar forms of the problemswere constructed One form of the problemand tbe mstrucAions given to the subject areshown below

Here are four stones After you read tbem, youwill answer quesbons on wbic^ are most similarand wbicJi most different Use tbe "greater tban"symbol " > " to mdicate tbe order of dungs For ex-ample, mdicate "Brown prefers Oregon over Texas,by O > r Only attend to order

1) On counter eartb, Ricbard Reagan bas beenelected Prraident of tbe Umted States As a gestureof gratitude to tbe people of bis bome state, Cab-forma, Reagan bas convmced tbem to leave Cabfor-ma and eitber wander around witb no state, or takea combinabon of one, two, or tbree of tbe followingstates Oregon, Wasbmgton, and Indiana Reagandunks tbat tbe economic value of Oregon andWasbmgton are equal, and tbat tbe value of eitberIS less tban tbat of Indiana Tbe boundary and pass-port bill wbicb be submitted to Congress, requinngtbe barbed winng and mining of tbe boundanes atconsiderable expense, did not pass Instead, Con-gress specified tbat die funds asked for m tbat billwere to go to tbe states selected to be used as tbestates saw fit Only Wasbmgton and Oregon bavecommon boundanes and, tberefore, a union of tbesetwo states would receive a smaller amount of tbeforbficabon funds Tberefore, even witb tbe value(rf tbeir combined economies, Reagan tbmks diattbe pair Oregon and Wasbmgton is less valuabletban the pairs Oregon and Indiana or Wasbingtonand Indiana Since Indiana makes boats wbicb canbe used on tbe Columbia River, Reagan tbmks tbattbe economic benefits of painng Wasbington andIndiana are sbgbtly greater tban tbe benefits of

painng Oregon and Indiana Reagan tbinks tbattbree states are wortb more tban any combinabonof 0, 1, or 2 states, and any 2 states are wortb moretban 0 or 1 states, except be cannot decide wbicbIS wortb more, Indiana or tbe pair Oregon andWasbmgton

2) Bad Bart ambles into the local casino andconverts his gold watcb into cbips of tbe followmgcolors silver, bronze, and gold Bart bkes to playthe one-chip candy machines He bkes tbe cbijps mthe following order gold better tban silver, alverbetter than bronze, and gold better tban bronze,and any cbip over none Bart also bkes to play tbeone-armed bandit macbines wbicb use combinationsof two cbips He likes tbe two-cbip combmabonsm tbe following order golds and silvers first, goldsand bronzes second, and silvers and bronzes tbirdWitb one excepbon, Bart knows be bkes to play•mth any two cbips over one or none, be is notsure about a gold versus a silver and a bronze Be-cause Bad Bart bkes to play witb tbree cbips best,tbere is one macbine tbat be bkes better tban anyotber tbe wasbmg macbine, and it takes a silver,a bronze, and a gold cbip

3) In Madras, India, V P Vanktesb, a manof babit and vanable income, bas a favonte restau-rant Altbougb bis tastes never vary, tbe fcwd be canafford does Of tbe tbree fcxxls tbe restaurant servesbe prefers curry, biram, and alu paratba, m tbatorder Also, be likes curry better tban alu paratba,and anytbing better tban notbmg Wben V P basmore money be buys two disbes, except it is notknown wbetber be would cboose tbe curry overbotb tbe biram and tbe alu paratba He bkes tbecombinabon of cnirry and biram better tban currjand alu paratba, <tnd be also likes curry and aluparatba better tban birani and alu paratba, andbiram and alu paratba better tban curry and biraniAltbougb a temperate man, at fesbvals, given tbemeans, be bas all tbree disbes instead of any singledish or pair

4) A jeweler has three boxes, tbe first contain-ing different bnds of broken 18-carat gold neck-laces, tbe second vanous scratcbed earrings, and thethird different kinds of 18-carat gold pins tbat arebroken He keeps tbe old jewelry because be occa-sionally uses tbe gold T'o get tbe approximateamount of gold be needs, be weigbs and tben meltsdown a combination of objects To do tbe weigbmgbe uses a simple balance-beam scale It consists ofa beam tbat pivots m die middle and two pansbanging from eacb end of tbe beam, equidistantfrom tbe pivot Tbe beam is level wben tbe pansare empty Tbe jeweler can place combinations offrom 0 to 3 object types into one pan, but nevermore tban one of eacb object type m a pan Begin-

mng witb empty pans, be nobces tbat wbeneverbe puts any combmabon contaimng at least oneobject into a pan, tbat pan sinks down, indicabngthat It IS heavier than tbe empty pan Using tbissame metbod, be finds tbat any pm is beavier tbanany eamng Necklaces are always beavier tbanpins He bas discovered two rules tbat reduce bowmany combinabons be needs to try to find out bowtbe weigbts of tbe combinations are ordered First,he nobces tbat if tbe combinabon m tbe rigbt panIS beavier tban tbe combmabon in tbe left pan,and a sin^e object type not already in eitber ofthe pans is added to both pans, tbe ngbt pan re-mains beavier tban tbe left Secondly, if be weigbsthree combmabons of objects, be finds tbat tbe fol-lowing IS always true If tbe first combinabon isheavier tban tbe second, and tbe second beavierthan the third, tben the first is heavier tban tbetbird

Space was left after each story for sub-jects to make notations On a separate page, thefollowing instructions were presented

Now that you have read the stones and arefamihar with tbem, answer the quesbons belowMake your compansons on the basis of propertiesof tbe ordenngs found in eacb story, m die bestand most complete wav tbat you can Wnte out allthe compansons tbat you c»n in a systematic wayUse symbols to represent tbe order of tbings, andexplain wbat tbe symbols stand for Also mcludean explanation m English You may want to use a

Oregoncham, for example, ^ , in addibon to an order-

Texasmg, Oregon > Texas Tben explain wbat are tbemost important similanbes and differences m tbestones, and explain bow you amved at decidingthe relative importance of tbese similanties and dif-ferences You may refer to more tban one frame-work Make sure to give tbe strongest, most thor-ough, broad, inclusive, and complete explanabonspossible for tbe similanties and differences It isnecessary tbat you sbow all of your work in formmgthe orders and making tbe compansons, as well asyour commentary in Engbsb

The following form was used for subjectsto provide their answers

1 Which of the stones are die most similar?Iand2 2and3 I,2,and3 I ,3 ,and4—Iand3 2and4 I,2,and4 I ,2 ,3,and4—Iand4 3and4 2,3,and42 Which of tbe stones differ tbe most from die

ones you listed as similar in (1), even tbougbtbey may have many characteristics m com-mon?

Commons, Richards, and Kuhn 1061

1 4 1 and 2 2 and 32 1 and 3 2 and 43 1 and 4 3 and 43 Explain wby tbe stones you listed above as

similar are similar4 Explain wby the stones you listed as different

from these are different

Space was provided for subject's answer ^

Each of the four systems withm the prob-lem consists of a finite, parbally ordered, com-mutative semigroup, with a bmary operabon forcombining objects and a parbal order relabondefined among all possible combmations of ele-ments The systems were presented as stonesabout which combmations of three objects, a, b,c, were greater tban others The combmabonscould include no item, single items, or two ortbree items In tbe first three stones, for themost part, the ordenngs of combinabons of ob-jects were exphcitly stated One object is pre-ferred over another, or over none at all, forexample, o over none, c over b, b over a, andso forth, some pairs are prefened over others,for example, fo -f- c over a + b, or tbe completegroup IS preferred over a pair, for example,a -i-h -\- c over fc -f c In the fourth story, thestructure of the order was stated m proposi-tional form which did not provide direct accessto the ordenngs of tbe elements, these bad tobe denved by the subject

An alternate form B was presented to aportion of the sample Forms A and B wereidentical except for the structure reflected mstory 1 (see fig 1) and tbe names of the ele-ments in each of the stones, for example, thenames of the states m story 1 and the foods mstorv 3

The subject was allowed an unlimitedamount of time The average amount of bmespent was 1 hour, with an approximate rangefrom 30 nun to 2 hours

Two additional problems were presentedto a portion of the sut)jects (73 of the 110) asimple transibvity problem (requinng concreteoperations) and a version of Inhelder and Pia-get's (1958) pendulum problem (requirmg for-mal operabons) Problem order was counter-balanced It was bypotbesized that performanceon the three problems would show a hierarchi-cal pattem, that is, no subject would master thependulum problem who did not master the tran-sitivity problem, and no subject would master

Tbis problem is used with tbe permission of tbe Dare Associabon, Inc

1062 Child Development

Form A

Story 1

0 + W + I\

W + II

Story 2

B + S + GI

S + GI

Bx+vG

Story 4

E + P + NI

P + N

Form B

FIG 1 —Representabon of tbe system of order relabons reflected m tbe four stones

the multisystem problem who did not masterthe pendulum problem

SubjectsThe 110 subjects were 39 undergraduates

and 71 graduate students attendmg one of sev-eral pnvate umversibes m the Northeast Themean ages for the groups are 20 6 and 26 1years, respecbvely All parbcipated on a vol-unteer basis The mulbsystem problem wasadministered a second tune to 41 of the 71graduate students, direcdy followmg the mitialadmmistration for 22 of tbem, and^ 8 monthsfollowmg the mibal admmistrabon for 19 ofthem (Half of the students received form Afirst, the other half form B first)

ResultsAnalysis of subjects' protocols suggested

SIX chstmc:t levels of response. The sconng sys-tem was developed based on an mtensive analy-sis of 25 of the protocols, gmded by the theo-rebcal perspecbve set forth m the introductionIt was then apphed to the remainmg 85 Fifty-

five of the 85 protocols were evaluated mde-pendently by raters 1 (first author) and 2 (re-search assistant), 61 of the 85 protocols byraters 1 and 3 (research assistant) There was76% and 62* agreement, respecbvely Differ-ences were resolved by discussion and a finallevel assigned to each protocol

Level C —The lmbal level, labeled C, re-sembles the mode of thinking termed concnreteoperational in Piaget's system Subjects cat-egonzed in this level base their judgments ofsimilanty/dissimilanty on superficaal features ofthe stones rather than on order relabons (eitherwithm or across stones), as the task mstrucbonsdirecjt If order relabons are attended to at allthis attenbon is hmited to a representabon ofthe discrete order relabons exptiatly given inthe story The subject does not perform anvoperations on these elements, to denve additional order relations or more general properbesof the system (story) Subject RA provides anexample (The examples to follow are based onboth form A and form B, so that letters maynot always match the stones given earher )

Under story 1, RA wrote

C , U < I , C<I,

> UC), 3 > 2 > 1

UI,

Under story 2, RA wrote

He hkes B> R, R> W, B > W ,

(3 > 2 > 1), BR> BW > RW ,

{B > Rny

Under story 3, RA wrote

He prefers CK > Bak > Yog ,

Ck - Ba>Ch- Yo,

Ch - Yo > Ba - Yo ,

Ba - Yo > Ck - Ba^ ,

(3 > 2 > 1) (Ch > Ba- Fo)?

Under storv 4, RA wrote

Br>Ri, Ha>Br, {[I a > Ri) ,

(U a> Br - Rt),

(3 > 2 > 1) comb , right pan , left pan

RA then wrote "Stories 1, 2, and 3 aremost similar because in the first tbree stones aperson's opinion is involved He thinks, or helikes, or he prefers, a parbcular object or acombination of objects, and if another personwere to make a choice on what he would preferof such combinations, the combinations wouldl)e likely to change (e g , some people may hkewbite chips better than blue) At the same time,storv 4 IS based on facts, and if tbe jeweler werereplaced, the facts about the combinabonswould still be true " Note tbat RA does not linkinformation from discrete order relations intoa higher-order unit or cham (though we knowRA IS capable of a transitivity inference) B >R, R > W, and B > W are not integrated intoa unified order relation B > R>W

Level F—Tbis level is postulated to beequivalent to the level of formal operations inPiaget's sequence (Piaget's levels IIIA & B, seeInhelder & Piaget [1958]) Order relations with-in stories are operated on in a systemabc man-ner, but the entire system is not regarded (op-erated on) as a single entity having character-istic properties that may be compared with theproperties of other systems Thus, the subject'sattempt to relate the stones with respect toAeir similarity/dissimilarity is hmited to estab-lishing that single elements or two-element or-der relations map to some degree from one of

Commons, Richards, and Knhn 1063

the first three stones to another SubjecA LRprovides an example Under story 1, LR wrote

I > W, 0,W > W-\-I >0I,

3 > 2 O - f H ' - h / , uncertainty nO,W

Under story 2, LR wrote

g> s > b > none, one exception

gs > gb > sb , one uncertainty g'^ b , s

Under story 3, LR wrote

c > b > a> none ,

uncertainty in one case b, a'^ c

Under story 4, LR wrote

n > p > e > none ,

no uncertamt\, no pairing

LR tben wrote "1 and 3 are similar becausebotb individuals prefer any 3 to any 2, any 2 toany 1, and any 1 to any none, with one uncer-tainty In story 1, Reagan does not know ifI > O, W, and in story 3, Vanktesh does notknow if B, A> C One diflFerence is that thetwo singles are equal to each other in story 1{W = O), whereas tbis is not true of story 3(C>B> A) The other difference is that theCalifomians may or may not benefit most fromtaking J + O over W, whereas Vanktesh def-initely would take all three "

Level S —Responses at this level reflectthe application of what we have termed sys-tematic reasonmg At this level, subjects clearlyshow that tbey understand tbat the logicalstmcture of each story must be examined as anintegral whole or structure In represenbng thestructure of each story, the subject may chooseone of two possible courses A schemabc repre-sentation of each of the systems can be gen-erated and these representabons compared withrespect to their deviations from one anotherAlternatively, the subject can represent thesv stems on the basis of the axioms that do ordo not charactenze each of the systems Thetwo methods yield equivalent results Use ofthese representabons is taken as evidence thatthe subject perceives the story as a system, thatIS, a coherent wbole which determmes tbe m-temal pattem of relabons across its elementsWben systematic operations are fully consoh-dated, full representations are constructedThere is no evidence for the presence of aframework for intersystem companson, cogni-tion IS focused on mtrasystem analysis Subject

1064 Child Development

NJ provides an example Under story 1, NJwrote

O=W,O<I,W<I, 0-{-W>0 + I,O+W>W-\-I, W + I>O-{-I,

0-if-W-hl >W + 0, O + I>W,

0+ W> I, I-{-W>0

Under story 2, NJ wrote

G> S, S> B, G> B, G> 0,

B>0, S>0, G-|-S>G-f-B>

5-1-5, G = B-|-5, G-t-B>5,- f - O > / > , C r - | - O - i - 7 5 > C r - 1 - O >

Under story 3, NJ wrote

C > B > ^ , OA, OO, A>0,

B>0, C = B + A, C + B> A,

C + A> B + A , B + A>C+B,

A + B + C>B + A , A + C ,

B + C, A, B, C

Under story 4, NJ wrote

iV >C», E>0, P>0, P> E,

N > P

(Note that story 4 is mcompletely represented )NJ then wrote "1 and 2 berth use the techmque

Under story 1 OC wrote

I>0=W, O

0 + » f ' > fortify > 0 + / , W + l,

of companng items m regard to their relabvement on a one-to-one basis, then m pairs Inaddibon, the relative worth of combmabon ver-sus smgle Item and versus zero is analyzed. 1and 2 also follow the transitive law of geome-try, that IS * > y and y> z, then x > z Thelaw holds for 1 and 2, whereas it fails m exam-ple 3 In 3, B -t- A > C -I- B, then by takmgaway B, A> C, but this contradicts what weare told before I realize, however, that MrVanktesh's preferences need not follow math-ematical laws and logic, he may mdeed preferthe B 4- A combinabon better I pomt this outbecause I felt this made 3 less similar to 1 than2 was to 1 " NJ's reasoning is characrtenzed byan effort to descnbe differences across stonesStones are compared m a pairwise fashion mthis effort, evidence for the lack of a systematicframework for companson

Ml —At this level the first evidenceof metasystemabc operabons appears Compan-sons across stones are based on vanabons of thestructural properbes, lmpbcatly incbcating tbepossibibty of conceptualizmg one story as thetransformabon of another story At level Afl,the subject generates a much more completeset of lattices and/or axioms to represent orcharactenze the stones However, some axiomsor ordenngs necessary to complete the analysisare clearly missmg Subject CC serves as anexample by providing only a parbal analysis ofstory 4 (leavmg out the details of its ordenngand leavmg out the smgle combinations of tbeelements)

Subject CC's analysis is as foUows

l>I>O''Wi where do these fit?

Under story 2 CC wrote

a> S> B> none , GSB >GS>GB>SB^G>S>B> none , GS > GB > SB

Und« story 3 CC wrote

C > B > i4 > nothmg, C + B> C+ A ,

C> A> none, C+ A > B+ A ,

C> B + A, B + A>C+B,

C+B+A>C+B>C+A>B+A" •

curcuUr

Under story 4 CC wrote

N > P> E, R> L, NP> NE, NP> NE> PE, PRPE, \P> PE, PN PE,

NE> PE

Commons, Richards, and Kuhn 1065

CC then wrote "[In stones 1 and 3], there isa basic similanty m the structure combmationsof three, combinabons of two, one, none How-ever, m both stones, there is a degree of uncer-tamty as to precise order of the combmationsIn story 1, O + W + J>O-i-W>W + I>O -f / > 7 > O = W, and ako die rankmg of Iversus O -)- W is uncertam Therefore, if 7 isgreater than O + W, then the preferences be-come 'circular' as m story 3 If 7 = O -I- W,then again the preferences are circular O + W= I>W + J>O + J>J In story 3, the cn--cular nature of the combinabons is spelledout C + B + A>C-^B>C + A>B + A> C - f - B and B - f A > C > B > A > noneStory 4 differs the most Both 2 and 4 have veryclear relabonships Though the relabonships mstory 4 are not expbcitly spelled out, there isenough mformabon to deduce that a combina-tion of 3 > combmabons of 2 > singles In story2, the relabonship is also the same, thoughthere is sbght uncertamty as to the relabonshipbetween SB and G Nonetheless, this uncer-tamty IS not enough to change the order of therelabonships the way it does in stones 1 and 3 "

It IS clear that CC's reasonmg and repre-sentations could be earned further On the otherband, CC does use "circnilanty" and uncertaintyof order to refer to the integrated stones aswholes, and she generates latbces for threestones Struc^re is compared across stones,even though incorrectly "The use of circularityand "uncertamty" indicates the existence of acommon "source structure" whose properbesare altered to produce the different instanba-tions of structure found m the stones

emergence of truly metasystemabc operabonsThis level, and the two which foDow it, areprogressive steps in the consobdation and orga-mzabon mto a wbole, of operabons mvolved mcompansons across systems Subjects at thislevel have, either exphcidy or lmphcatly, fulland mtegrated representabons of the systemsof ordered relations reflected m each of the fourstones (Systemabc thinkmg is now fully con-sohdated ) These representabons are used tocheck (again either expbcitly or impbcidy) theirassertions about the systems in a systemabcand complete fashion ITiese subjects choose asingle property which is appropnate for cx)m-panng the mtegrated strucjtures of the stones,as opposed to one that would enable compan-sons of parts only For example, the subjectmight compare tbe stories on the basis of eitheradditivity or transitivity properbes Such prop-erties are used to construct companson frame-works For a subject who represents the systemsunderlying the stories graphically, the compan-son framework may consist simply of a set ofdimensions along which the physical drawmgsof the systems are tested for resemblance Asubject may imply that he orshe-bas^eneratedcomplete ordenngsjn-«fr"four stones withoutactually showing the work Thus, the simpleassertion that "stones 2 and 4 follow the law ofaddibvity while 1 and 3 do not," is enough toclassify a subject at level M2 (In protocolsclassified at levels Ml or S, m contrast, thereexists clear evidence that certain combinabons,1 e , relabons between elements, were not con-sidered in generabng the representabon ) Sub-ject BR provides an example of level M2

Level M2 —This is the second level in the lowsSubject BR represented the stones as fol-

tnder stor\ 1, BR wrote

0 ~ a common boundanes economicalK

0< J , J > 0, 0 + H > fortihing, H + I > 0+ I ,

W < I, / > l f , O + I < fortif>ing, i > pair ,

H + ^ < fortif J mg, pair > single

except P 0 + H

Under 5tor> 2, BR wrote

G> S, G + S>O+B>S + B, i chips > 2 chips ,

' I •^ > B , better than 1 chip ,

G> B except G = S + B

Under story 3, BR wrote confused Indian or exam taker'

C> B> A, C+ B>C+ 4,

C> A , C + A > B + A , ilC + B>C+i>B+i,

therefore C+ B> B+ t , b u t B + 4 > C + B ,

therefore illogical,

2 > 1 , except C '~ B + A

Under story 4, BR wrote P > E, \ > P

1066 Child Development

RR then wrote

My inibal reaction was to pick 1 and 2 However,1 does not seem as logically simple as 2 and 4 do1 seems to look at several vanables wben evaluatingtbe relationsbip between tbe states ( l e , fortifica-bon, economic values) Altbougb you coidd end upwitb an ordered ranking for 1, tben it is not stnc tlylogical tbat W + I>O + 1 Vanktesb does not fol-low a strictly additive pattem eitber Combinabonsof tbings are not always tbe sum of tbe parts Iguess tbat is wby I piclred 2 and 4 as most similar4 especially follows tbe rule that tbe sum of dieparts equals tbe value of tbe wbole 2 and 4 you(x>uld predict tbe rankings given basic information(except possibly for Bart's besitancy over G vs5 -j- B) (BR concluded tbat stones 1 and 3 weremost different ) Tbe reason wby 1 and 3 are dis-similar IS tbe flip side to wby 2 and 4 are similar2 and 4 seem very logical to me Tbey were rela-bvely easv to decode because tbey followed wbatone expected Story 3 bad some completely unex-pected elements to it I do not understand bowC -j- B can be at once more preferable and lesspreferable to tbe same combmabons of disbesTberefore tbe reason tbat 3 is very dissimilar is tbatit IS not logical or predictable or understandable Iread 3 at least 5 tunes trying to figure out wbereI made a mistake in lnterprebng tbe rankings ButI cannot find any errors in my tianslabon So 1 baveto assume tbat it is Vanktesb tbat is confused IfI did make a mistake I would like vefy mucb tobear back from you regarding tbe actual rankingof tbe disbes

Level MS—At level M3, the subject un-derstands the ambiguity of the questions re-qmnng judgments of similanty and dissimilar-ity Individuals understand that there exists amulbphcity of chmensions which could providethe basis for such judgments, that is, a mulb-plicity of companson frameworks Tbe level M3subject deals with this ambiguity by expen-menting with a number of c»mpanson frame-works and companng and lntegrabng the re-sults of each

This reasomng can still be of a nontech-nical sort, because the properties m terms ofwhich tbe systems vary are not comphcatedones The lattices of stones 2 and 4 are lsomor-phic (see fig 1), the same set of axioms apphesto both Story 1 violates lrreflexibihty, smcetwo states are equally preferred Story 3 vio-lates transibvity, a more senous violabon Thelevel M3 subject understands that lack of tran-sibvity means that one no longer bas an order,whereas the inclusion of an equahty rather thanan lneciuality with some indetermmacy on addi-bon sbll means that one has an order, althoughpartial Therefore, the former constitutes a moresenous deviation Another way to see the sen-ousness of the deviabon is to examine whathappens in the transformation from one system.

A, to another system, B Sxidi a transformaboncauses a loss m mformabon to the extent thatthe two systems are not lsomorphic This is seenwhen tbe reverse transformation is performedThus, the result of transforming system A mtosystem B and then back into A (by anothertransformation) must be judged from a multi-plicity of frameworks Subject HC provides anexample of level M3

Subject HC supplies the analyses that ap-pears at the bottom of page 1067 HC thenwrote

Stones 2 and 4 are die most similar because m eacbtbe same laws are estabbsbed for ranking Kacballows only two possible rankings (outcomes) wbicbanse because of uncertamW as to wbetber one itemc»nies a bigber ranking than tbe sum of tbe twootbers, or vice versa If tbe uncertamties were set-tled tbe same way in botb problems (l e , tbe oneitem sum of tbe otber two, or vice versa) tbe sameranking would bave been estabbsbed for eacb Apossible difference between stones 2 and 4—tbe facttbat (not stated) m story 2, one-armed bandit ma-cbmes inigbt use two of die same c»lor cbip, wbere-as in story 4 it was specified tbat only one item ofeacb type could be used—was eliminated by assum-ing tbat tbe bierarcby of preferences for eacb com-bmabon of two dnps in story 2 represented tbecomplete possible cboice of performance Tberefore,two cb i^ of tbe same color were not a possiblecboice Refer to my above diagram [p 1067]wbere it is sbown tbat tbe laws developed in story 4are the same as those tbat govern story 2 Notabonswitbin tbe section for story 2 prove tbis m tbe dia-grams of story 2

Story 3 is tbe most different because it violatesbotb laws by wbicb 2 and 4 are bound and becausetbe lnformabon does not bmit you to only two pos-sible rankings Story 1 is similar to stones 2 and 4in tbat tbe laws eiven are adequate for determiningtbat tbere are only two possible rankings However,one of tbe laws used m 2 and 4 is violated and tbeotber does not apply to 1 Story 3 is most differentfrom stones 2 and 4 because (1) It does not setfortb tbe same laws as tbose governing stones 2 and4, m fact violabng botb of tbem, and (2) tbe lawsgoverning the preference for combinabons of twoitems are circular, l e ,

sucb tbat more tban two possible rankings exist,even after tbe uncertainty of preference for cbick-en over baklava and yogurt, or vice-versa, are re-moved

L^vel M4 —^Though no examples of levelM4 have ocxairred in our research, so far, onecan postulate this level as compnsed of anldeahzed, maximally formahzed solution to thetask and example of metasystemabc reasonmg

At level M4, exphcat use is made of thetransformational nobon Here, for mstance, itmay be used to show how many changes oforder there are in 28 pairs of combmations thatare necessary to go from one story to anotherand then back to the ongmal There are otherways to assess the effects of the transformationsback and forth Tlie degree to which the mfor-

Under story 1, HC wrote

(Utah

Commons, Richards, and Kuhn 1067

mation in a story may be recovered in a trans-formabonal process is a measure of its similar-ity The notion of an inverse transformation isused, and wbetber or not it can be performedwithout losing mformabon is shown The prop-erties of the system are represented m a lan-guage that IS not particular to any one systemJust as fully formal operational subjects in Pia-

flhnois > j II > 0

[Colorado

Law 1 violated, law 2 does not appl>,

Utah + Colorado > C olorado + Illinois > Ltah + Illinois ,

Ltah + Colorado + Illinois > ' —

I nder stor> 2, HC wrote

B> R O K> H O B> H ,I I

1 > 2 O 2 > ? O l > 3

I > U- • > u + i

uII >o

>o

(1)

(2)

(3)

B+ R> B +R >

+ B

R+ B

+B

H +

It

B

> R+HB>

+ W

B+ W

R

+ W

R+n

Law 2 Is this the ordering pair to_J pair also*"

Transitive propert> 2 Does it existhere' V es

Vi, hat about the other 2 chip combinations-'

bracelet ring + Hatch Stor> 4 is limited

in that onK one of each tj-pe can be used

(1)

(2 and 4 are similar ) r->5 > R+ H > R M >

R+W+ B> B + R> B+ii >}'-•/? +U > B> R> K >

> 0

0

(2)

(3)

Lncertainty happens at the last t*o item combination—same as stor\ 4—ring + watch bracelet

Under story \ HC wrote

C > B> y , also C > ¥ , a n y > 0

'C + B> C + i , ^ C + i > B+ i , B+ I > C + B , 1 > C therefore

-law 1 OK, ^lawlOK,

?->C>B+l ' or B+ y

>C + B> C +C+ B+

>CCM:> B+ I >

•> Lj U g - f . v>c +

law 1 does not applv ,

C+ B> B>

(1)

(2)

(3)C+B>C>B>1 > 0 already used

Or IS It 2C—when has monev, always 2 dishes Illogical unless does not care for baktava at att—

may not—creature of habit, saves money, just C Transit ve property —or whatever the proper

name for it is—is not working here

Under story 4, HC wrote

broken gold watches (without works), scratched gold rings, broken bracelets, anv > 0 H >

B> R, uncertainty here, too, cannot rank exactlv unless \ou know whether H > B + R or

B + R>W L a w s

If right > left

+ equal weight + equal weight

Right still > left

additive pmpert\ (1)

1 > 2 , 2 > 3 , then 1 > 3

-B + R>? -

transitive property (2)

-B>H+R>B+R>U'>B>R

1068 Child Development

get's system no longer need concrete values tobe assigned to the elements to which their for-mal operations are appbed, at the level of ideal-ized metasystemabc reasomng, the subject canoperate on systems mdependent of their specificrepresentabons The idealized level M4 perfor-mance, then, would consist of a general theoryof systems of order relabons, within the frame-work of which any particular order system isevaluated Properbes of the axiom systemswhich are used to generate these systems, suchas completeness, consistency, decidability, andso on, would be considered

Performance of Subjects m the Present SampleThe levels at which subjects m the present

sample were categonzed are shown m table 1The relabon between performance on the for-mal operabonal problem and performance onthe multisystem problem is shown in table 2All subjects passed the concrete operabonal(transibvity) problem, which mdicates that theyfunctioned at least at the level of simple con-crete operations The relabons reflecrted m table2 are m accordance vvnth expectabon Withonly one exception, only those subjects whoshowed attainment of formal operations showedany level of profiaency in systemabc or meta-systemabc reasoning

TABLE 1PERFORMANCE LEVELS ON TOE POSTFORMAL TASK

SAMPLE

Undergraduate Graduate

LEVEL N N

cFSMlMlMl

Total

923

5020

39

TABLE

235913050

2

419211089

71

52730141113

RELATION BETWEEN PERFORMANCE ON FORMAL

AND POSTFORMAL TASKS

FORMAL TASK

ConcreteTransitionalFormal

Total

C

154

10

POSTFORMAL TASK

F

02

3032

5

01

1516

M\

0022

M2

0077

Mi

0066

ToUl

18

6473

Among the 41 graduate students who re-ceived multiple admimstrabons of the mulb-system problem, most subjects showed nochange or a sbgbt advance from first to secxmdadimnistration Twenty of tbe 41 sbowed nocbange, 14 advanced one level, one advancedtwo levels, one advanced four levels, and fivedechned one level Tins change pattem chd notdiffer appreciably according to time that elapsedbetween admmistrations, which suggests thatchange was largely attnbutable to effects ofrepeated testmg

Discussion

The present results support our f>ostulationof discrete modes of cognition composed ofthird-order and fourth-order operabons Empir-ical support for the vahdity of the proposedconstrucits, labeled systemabc and metasystem-atic reasonmg, is of two sorts First, perfor-mance levels of the two samples (undergrad-uates and graduates) are in accordance wnthexjjectation Few undergraduates show evi-dence of systematic or metasystemabc reason-ing, its incidence is considerably greater, how-ever, among graduate students Second, per-formance on the problem designed to assesssvstemabc and metasystematic reasoning showsthe appropnate relation to performance on atask designed to assess formal operabonal reasoning Only one of 31 subjects who did notshow fully formal operational reasonmg exhib-ited any proficiency in the use of systemabc ormetasystemabc reasonmg, but not all subjectswho were proficaent m formal operational rea-soning exhibited proficiency m systemabc ormetasystematic reasomng

The purpose of the present report has beento present the instrument we have developedto assess this higher-order reasoning and to descnbe the performance of the imbal samples ofsubjects to whom the instrument has been ad-mmistered It is, therefore, not appropriate inthis report to embark on an extended discussionof the factors or condibons that may govern tbedevelopment of such higher-order reasonmg Tosome extent, frmtful speculabon in this regardawaits fuller tmderstanding of mechanisms ofcogmbve development (Kuhn, m press) Weshould comment, however, tbat the performance differences between the undergraduateand graduate samples in the present study almost certainly reflect a combined ccmtnbubonof self-selecbon and differential expenence

Commons, Richards, and Kuhn 1069

While the "general expenence" that comes withmcreasmg chronological age appears to be asufficient condition for attainment of the earherstages m Piaget's system, we would not expectit to be a sufficient condibon for mastery of thethought operations assessed m the present workExposure to and expenence with problems thatrequire abstract representabonal modes of anal-ysis are undoubtedly necessary factors, but justhow native ability, educ^bon, and expenenceinteract in this regard is a difBcnilt issue to ad-dress (see Commons, Richards, & Armon [inpress] for addibonal discussion)

The caveat mtroduced earher bears reiter-ation It IS not necessanly the case, and mdeedmost unbkely, that all adult thought is of theform mvestigated here One of the pressmg is-sues m the study of adult cogmbve develop-ment, m fact, IS to discover the role that formal,logical/deductive reasoning plays m the real-world thought that occrurs in adulthood (GiUi-gan & Murphy 1979, Kuhn et al , m press, La-bouvie-Vief 1982) Nevertheless, withm therealm of logical hypothetical/deducbve thoughtstudied by Piaget, levels of reasonmg beyondPiaget's formal operations in our view must beof the general form of third-order and fourth-order operabons that we have outbned It is toorestncbve to say that systemabc and metasys-tematic operations are limited to the domainsof mathematics and science Numerous otherdisciplines, such as literature, history, or anthro-pology, entail the evaluation of systems withma multisystem framework It may be the case,however, that systemabc and metasystemabcreasomng is hmited to the domain of formal,abstract, as opposed to everyday, thought, mcontrast to Piaget's formal operations which arediscernible m everyday thinking Clearly, agood deal of further work will be required toestabbsh the vanety of forms that the system-abc and metasystematic reasomng ldenbfied inthe task presented here may take

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Bickbard, M On necessary and specific capabilitiesin evolution and development Human Devel-opment, 1979, 22, 217-224

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