IETCETRANS．INF．＆SYST，，VOL．E77−D，NO．1JANUARY1994

39

RAPER 恥由J飴5〟○αl仙OJ喝○川CAJ即d伽m曲

AnlnductiveStudentModelingMethodwhichDeals
WithStudentContradiction＄
YasuyukiEONO十，NonmembenMitsuruIKEI）A十a〝dRiichiroMIZOGUCHI†，SUMMARY Studentcontradictions aretheessentials ofcon−
CeptS and knowledge acquisition processes ofa student，in the
COurSe Oftutoring・This paper presents a new perspectiveto
representstudentcontradictions and a student modeling archi−

tecturetocapturethem・Thefbrmulationofastudentmodelingmechanismenables nexibledecision makingbyuslnginforma−tion obtained from students．A nonmonotonic andinductive
StudentmodelinftrencesystemHSMIShasbeendevelopedand
fbrmulated to cope with modeling contradictions，Which basi−
Callyembodiesadvancedrepresentationpower，Sufhcientlyhighadaptabilityand generality．TheHSMISisevaluated and com−
Paredwithotherrepresentativesystemsin orderto demonstrate
its e能ctiveness．

点呼㈲′ゐご刑0鹿肋gα〃d∫fm〟由∫わ〃，e血cαめ〃，助鹿〃rmO鹿ヱ椚OdeJ′eγねわ〃，椚0（お伽gco〃加血Jわ〃，∫拍de〃Jco〃汁αdわrわ〃

1．Imtroductiom

A student whoisinthefixingstage＊ofthe acquired
knowledge often shows contradictorybehaviors．This
meansastudentisapttounstablyapplyproblemsoIv−
1ngmethods，becausehe＊＊hasnotbuiltthemorsince

he has notcompleted thefbrmulation ofrelated con−amethoddealingwithmodelingcontradictions areas
fbllows：●reVise current student modelto enhanceits accu−
raCy，●fbllowastudent schangeinunderstanding，●1gnOrehisslips，and・COntrOltheloadofbothhimandthesystem．

Tothisend，aStudentmodeling system should always
makebeliefrevisionstokeepdatafbrinftrenceconsis−
tent・Surprisingly，Only Huang etal［8］havetackled
thisproblem，eXCeptthatWoolfetal［21］pointedout

theslgnificance．Theauthorshavebeentacklingthisis−
Sueanddeveloplnganinductivestudentmodelinftrence
algorithmHSMIS［16］，［17］．TheHSMISemploysthe
ATMS［4］tomaintainconsistencyofthestudentmod−eling process．In addition，SOphisticated mechanisms

toadaptlVelycontrolthefbcusofmodelingandthere−

1ianceonthestudent areincorporatedinto HSMIS to
CePtS，etC．Itisclearthatheshowssuchnonmonotonic meetthelastrequlrement．ItenablestheHSMIStoask

learnlngPrOCeSSeSinfixinghis knowledgeinthepro−
CeSSOfacqulrlngCOrreCtandstableknowledge．There
areothertypesofcontradictionstobeconsideredinde−
SlgnlngaStudentmodelingsystem．Amodelingsystem

himquestionsapproprlateinthesenseoftutorlng．

The second problem，tO Capture Student contra−
dictions，SeemSmOreimportantfromeducationalview−
POlntS． The Socratic method，fbr example，is a
O食engetsananswerftomhimwhichisconsistentwithCOntradiction−basedtutoringstrategywhichteacherses−

hiscurrentbeliefbutinconsistentwithhispastanswers，
becausehechangeshismind nonmonotonicallyashis
learningproceeds．
Contradictions which a modeling system should
COpeWithareclassifiedintothefbllowlngtWOtypeS：
1．modelingcontradictionswhich shouldberesoIved
byrevislngStudentmodel，and
2．studbntcontradictionswhichshouldberegardedto
beinherentinthestudentsandbeutilizedfbredu−
CatlOn．

The fbrmer problemisthe essentialformodelingprocesses，becauseapracticalITSshouldfb1lowastuL

peciallyuseto help studentsin thefixing stage．Itisa well−known and already verified method that glVeS

SuCh a student astronglmPreSSion thathemisapplied
hisknowledge．Althoughbuildinghigh−fidelitystudent
modelsis anintractableproblem［18］，anlTSshould

haveastudentmodelwhichispreciseenoughtohandle
tutoring strategiesintegratedinto theITS［6］．In or−

dertogeneratesophisticatedtutorlngbehaviorlikethe
Socraticmethod，Studentmodelingmethodsshouldbe
abletocoverstudentcontradictions．Hisknowledgeac−
quisition and fixing processes should be captured bymodeling him as heis，eVenifhe has contradictoryknowledge［11］．

This paper presents a new methodology fbr han−dent s nonmonotonic change．Major requirements fbr

dling student contradictionsin conJunCtion with the

ManuscrlptreCeivedJuly29，1993．
ManuscrlPtreVisedSeptember21，1993．
†The authors are with the Thelnstjtute of ScientificandlndustrialResearch，OsakaUniverslty，Ibaraki−Shi，567

where acquired knowledge is not completely establishedJapan，＊ Fixing stage means anintermediatelearning stageyet．

＊＊Forsimplicity she ispsedtoreftrtotheteacherand
he toreftrtothestudentlngeneral・

40

IElCE TRANS. INF. & SYST., VOL. E77-D, NO. 1 JANUARY 1994

Question 1
Question 2
Question 3
A sphere h a s passed through t h e origin with velocity A
spherehas A n sphere is thrown directly up19.61n/s t o t h e right a t t = 0s. I t continues linear passed t h r o u g l ~t h e origin wit11 wards with initial velocity 19.6rn/s
and uniformly accelerated motion along x-axis of a velocity 19.6rnjs t o t h e right a t a t t=Os. It reacl~esa t the m a x i m u ~ r ~horizontal plane, a n d stopped a t t = 2s. I t I I I ~ S S= t = 0s along z-axis of a hori- higlrt a t t = % . Determine the direc2kg. Get its position a t t = Is. Determine both the zontal plane. P is moving with tion of t h e applied force at t = l s .
linear motion with its velocity S t u d e n t ' s a n s w e r
direction and t h e magnitude of the applied force.
v ,-v,Jdecreasing a t a uniformed rate. vt = :,-To t t + v o = - *2-0
Student's answer
1+0=-g.gn,/,2t=2s.
DeterIt
stopped
a
t
a=+
e
=
l
= 9.grnls
r-To
2-0
mine its p o s i t i o ~a~t t = 1s.
at t = ls.
~t is still moving
St = ~ ~ + v ~ t =+0 +~ 1 9a . 6t * 1~ + *=14.7 Student's answer
Therefore,
it
is
still
receiving
upwardT h e position is 14.7m from the origin.
St = So
vt = 0 19.6 * 1 = force at t = ls.
F = ma = 2 * -9.8 = -19.6kg.m/s2
19.6 T h e position is 19.6m.
I t receives 19.6 kg.rn/s2 of force t o t h e left.

+

Fig. I

Examples of behavior of a student who has undifferentiated concepts.

Socratic tutoring. There seems to be some dilemma between a method to capture student contradictions as they
are and one to resolve modeling contradictions. The
authors successfully integrated the two methods by introducing a multi-world mechanism into the HSMIS architecture. In the reformulated HSMIS, the ATMS plays
another important role of managing multiple worlds
which model students with contradictions.

2. What is Student Contradiction?
Assume that a student is in the stage of acquiring a certain new concept and that he has not fully discriminatedit from other related concepts he has already acquired.
Such a student is apt to behave unstably in applying
knowledge to solve problems which contain the undifferentiated concept. Figure 1 indicates the behavior ofa student who has not yet differentiated concepts, those
are the concept of "uniform motion" and the conceptof "uniformly accelerated motion." He correctly calculated the position of P in Question 1, which specifies thetype of motion as "linear and uniformly accelerated motion." In Question 2, however, he mistook a uniformly
accelerated motion for a uniform motion, and so applied problem solving knowledge for uniform motion.
Such a situation occurs due to his confusion between thetwo concepts. As a result, his problem solving ability
becomes unstable.
A student can choose certain problem solving
methods appropriate for the problem given, if he has
well-discriminated concepts and has adequate knowledge of their attributes. If he has not, however, hemight misapply a procedure which belongs to anotherworld by taking no notice of particular attributes ofthe problem. For instance, methods to "calculate theposition of a moving object" are associated with bothconcepts, such as uniformly accelerated linear motion( S t = So vot at2 or St = So (vo v t ) * t / 2 ) anduniform motion ( S t = So+vt). In solving Question 2 inFig. 1, he retrieves the method defined in the concept of
uniform motion, while he should apply the method de-

+

+

+

+

+

fined in the concept 01linear and uniformly accelerated
motion.
A more interesting example of student contradiction is found in Question 3 in Fig. 1. The student who
had correctly calculated the force that the sphere receives in Question 1, but could not determine the correct direction of the force to Question 3 in spite of thatthe two motions are physically identical except for the
direction of the motion.
Such conflicts among his answers suggests the"multi-world reasoning"- assumption that he partitions
his whole storage and reasoning space into many ones.
Each small partition in his reasoning space with relevant storage is called a "world." He stores problem
solving methods and rules which he can handle at oncein each world. He can retrieve and utilize these methods in a certain world, only when he makes inferencein the world. A contradiction can be found when heutilized two different worlds in solving problems. One
is the world of well-formalized physics for Question 1
in which he stores the knowledge learned through thecurriculum of physics, e.g., formulas, definitions, andanother is his naive physical world for Question 3 which
he has been deeply engraved on his memory since hischildhood, for instance. He has a "motion implies aforce" misconception [3] in the naive one in this case.
It is inconsistent with the knowledge for "uniformly accelerated motion" in the well-formalized one. He has
answered that the force is directed upward because heused the naively misconceptualized world.
"Student contradictions" are defined in this paper
as his status which causes behaviors which can be regarded as a contradiction viewed from the stand pointof an observer. Typical interpretation of contradiction
is as follows:
He places more than two series of problem solving methods, which is originally placed in different
worlds of concepts, in the same world regardless of
their attributes. This is caused by his failure indifferentiating them from each other.

KONOetal：INDUCTTVESTUDENTMODELTNGFORSTUDENTCONTRADTCT10NS

41●He answers di鮎rent truth values to acertain fact

Within alimitedtime，Sincehis knowledgeisunq
stable．

3・ContradictionsinStudentModelingThissectiondiscussescontradictionsinstudentmodel＿
1ngprOCeSSbyclarifyinghumanconceptdiscrimination
StruCtureS andproblemsoIvingmethodretrieva】in re−
SPeCtivereasonlngWOrlds・
Astudent sanswertoaquestionisrepresentedby
apairofafactanditstruthvalue，andiscalledanor−acle．Thesetoforacles acquiredbytheobservationof
Studentsbehaviorwithinacertainperiodortimetends

to beinconsistentfbrseveralreasons．Suchstudent

sin−

COnSistentbehaviorisclassiRedintothefollowlngthree
typesofcontradictions二椚gJんodJeveJre8∫0Ⅲ〃g（1）OrαCゐco〃加ゐ血那C？〟∫ed占γCカ飢geげJJ〟滋〝J与Fig．2 Conceptdiscriminationtree．

mind：Student slearnlngprOCeSSisessentiallyat−

tainedwithchangeofhismind．Theconsistency
Ofhis answerscan beeasilylost，becausehebe−

havesbasedonhiscurrentknowledgeindependent
Ofhispreviousknowledge．
（2）Oracおcontradictionscau5ed毎sl如：Astudentofl
ten makes careless mistakes．Oracles based onthemareinconsistentwithhisactualknowledge，a concqutゐcrimination treefromits root，a nOde of

Whichcorrespondstoaconcept．TheglVenprOblemis
articulatedinto avector ofprlmitiveattributes，Whichdecides the conceptual world the problem belongs toby seeking on theconceptdiscrimination tree．To go
fbrwardthrough a path from oneconceptualnodeto

hence，thesetoforaclesthatcontainsslipsisin−anotherrequirestosatis年someconditionswhichchar−acterizethedestinednodeasconditionsCo，ClandC2
in Fig，2・Forinstance，the node 1inear motion has（3）Student cont7・adiction5＝ A student sometimes hasChildrennodes unifbrmmotion and unifbrmlyaccel−
inconsistent knowledgein his head which also
erated motion． Amotionwhichbelongsto thenode
CauSeSCOntradictoryoracles．
1inearmotion
alsobelongstothenode unifbrmmo−
tion，ifthemotion
satisfiesdiscriminationconditions
Incasesoftype（1）and（2），amOdelingsystemisabletothevelocltyisfixedor theaccelerationiszero， etc．
COnStruCtaCOnSistentstudentmodelthatrepresentsthe
COnSIStent．

CurrentStudentknowledgebyrevislngthesetoforacles
approprlatelytoresoIvethecontradictions，Sincewecan
assumestudentknowledgeisconsistentoneachperiod

ProblemsoIvingmethodswhichconventionalmodeling
SyStemShavehandledisretrievedinthedecidedworldand executed．

Studentcontradictions areclosely related to such
inthiscase．Incasesoftype（3）ofcontradictions，how−ever，Student sknowledgeitselfisinconsistent．There− COnCePtdiscriminationstructuresandreasonlngmeCh−fore，anymOdels，thatarerepresentedandappliedina anismsasmentionedabove，therefore，theycanbewell
formulatedbythe multi−WOrldreasoning
assumptlOn・Singlereasoningspace，Can nOtCOmpletelycapturehis
status．A multi−WOrld fbrmalization ofastudent model

Thestatusofastudent who hasnotyet discriminatedShouldbeintroducedtocopewiththisproblem．
It can be explained that human beings partition
theirwhole storage andreasoningspaceinto multiple

ingbuiltsuchdiscrimination conditions．Thelack ofworlds and organize a kind ofstructure to retrieve
theirknowledgee侃cientlyandwithlessloadby
l・retrievingwhichworld（concept）thegiyenproblem

belongstoalongacertaindiscriminatlngStruCture
atfirst，and
2．retrieving a method that contributes to problem
SOIvinglnthediscoveredworld，

twoconcepts，fbrinstance unifbrmmotion and uni−
fbrmiyacceleratedmotion ，Canbecapturedasnothav−
discriminationconditionseasilycausesstudent sconfu−

Sionofconcepts aslack ofClandC2CauSeS hiscon−
fusion ofmethodsin Fig．3．Such a studentis apt tounstablymisapply methods whichshould belongto a
di拝もrentconceptfromtheexpectedone．Heconversely

SlipsuptoseekthenaiveworldinthecaseofQuestion
3．

Student contradictions are caused by erroneous
COnCeptdiscrirninationtrees，becauseastudentwhohas
as mentionedin Sect．2．The月rst element，i．e．，deci−
SuCh erroneous tree cannot manageconsistencyln re−
Sionofthetargetworld，Canberegardedasasearchon
trievingproblemsoIvlngmethodsasmentionedabove．

IEICE TRANS. INF. & SYST., VOL. E77-D, NO. 1 JANUARY 1994

dleveli s creasoning
rirninatioq=...

'J.

a
.+-

..'
3

Fig. 3
Concept discrimination tree of a student who has undifferentiations.

Such kind of contradictions can be represented by theoverlay of discrimination conditions.
Student modeling process is essentially hypothetical, hence, the completeness of inferred student model
is not always guaranteed. The expectation of student's
answer deduced from the current student model is often
different from newly obtained oracles, when the current model does not completely represent his current
status. Assumptions, which were assumed when thecurrent model was inferred, become inconsistent withthe set of oracles. Such a type of contradictions iscalled "assumption contradiction in modeling." They
are able to be formulated in a similar way of types (1)and (2) of contradictions, because all of them should
be resolved to represent consistent knowledge inside asingle world. To resolve such contradictions requires
a nonmonotonic process which is essential to student
modeling processes, i.e., to assume the student's knowledge which satisfies the given set of oracles, and to revise
the current model going together with changes of oracles, etc. "Modeling contradiction" is a general termof such three types of contradictions which occurs in asingle world. We can design a student modeling architecture that is able to maintain the consistency between
the student model and the student behavior by defining
methodologies to resolve them. The SMDS, subsystemof the HSMIS, is able to algorithmically backtrack thecontradiction in the student model and oracles. The
HSMIS has inherently the capability to adaptively revise the current model in cases of assumption contradictions. Furthermore, the HSMIS can cope with contradictions (1) and (2) by nonmonotonically revising
the assumptions of the ATMS, which are roots of allits modeling processes. Such a mechanism to deal withmodeling contradictions is described in Sect. 4.
It is difficult for not only modeling systems but also
human teachers to distinguish and detect the four typesof contradictions, i.e., type ( l ) , (2), (3) and assumption
contradictions in modeling, because all of their indications are very similar. They are triggered by a difference
between the expectation of student answer deduced fromthe current student model and his actual answer. One of
the research goals of this paper is to produce a genericand formalized modeling mechanism which is able tocope with these kinds of contradictions. Although ageneric methodology to distinguish them is not fully
developed, some heuristics are employed as shown below.
Assume that the reliability of each given oracle oreach clause in the student model can be available. Bothmodeling contradictions and student contradictions aredetectable by quite similar triggers, i.e., the expectationfrom the model and the actual oracles. Contradiction
resolving procedures of those contradictions are quite
different from each other. Modeling contradictions require to be resolved by revising the set of oracles orcurrent model in general. Contradiction resolution procedure for each type of modeling contradiction is a bit
different, and hence detection processes of them are different from each other. In the heuristics, student contradictions are first distinguished from modeling contradictions.
Student contradictions should not be resolved, because student's inconsistencies should be modeled as heis. Student contradictions require to revise neither oracle set nor clauses that are inconsistent with oracles, but
to revise discrimination structure to permit to containthe inconsistency in it. Such a difference in treatmentof student contradictions and modeling contradictions
suggests the following way of discriminating them. If
either the reliability of a clause which is inconsistent
with valid oracles or that of the oracles is less than acertain threshold, the inconsistency should be considered to be a modeling contradiction and hence should
be resolved. On the other hand, if both of the reliabilities are high enough, the inconsistency is considered
to be a student contradiction. They are not revised but
put into some worlds, i.e., all the reliable data can bealive in the multi-world formalization. The following
heuristics to detect contradictions of each sub category
in modeling contradictions are incorporated.
The change of student's knowledge which causes
type (1) of modeling contradictions occurs especiallyright after his errors are corrected. He then generally
changes his understanding from erroneous status to correct one. It is appropriate to apply revision proceduresfor type (l), when correct oracles are obtained right after tutoring, i.e., the system resolves the contradiction
by excluding the past oracles inconsistent with correctclauses, or by asking him truth values of the oracles.
The revision of oracles results in the revision of themodel, i.e., erroneous clauses are dismissed and correctclauses are appended. In addition, it is available todirectly ask him if he has changed his knowledge.

KONOetalニtNDUCTIVESTUDENTMODELINGFORSTUDENTCONTRADICTIONS

45

Independentlyofthecorrectness，generallyspeak−1・Tofbllowastudent，schangeinunderstanding，and

1ng，thestudentoughttohaveconsistentlyappliedthe
Clauses that areinconsistent with newly obtained or−aclesthroughout acertainperiod，in thecasethat he

2・tOaSkquestionswithregardtotheirapproprlate−nessinthesenseoftutorlng．

Tlakescarelesヲmistakeswhichcausetype（2）ofmodel−

TheSMIS automatically asksquestionsthatcon−
1ngCOntradictlOnS・Thussuchtypeof contradictionsare
tributetodisambiguationof−alternativemodelselection．
detectedin similar criteria asthose fbr student contra＿
dictions，i．e．，theinconsistentoraclesandclauseswouldWhentheteacherisconfidentthatshehasagoodgrasp
beboth reliableenough．There aretwo waysto dis− Ofherstudent sknowledge，Sheaskshimftwerquestions
Onhisbehavioraslongasitsupportsherconfidence．
The model inference procedure should ask questions

tlnguishthem；Oneistoconsiderasituationasatype（2）only whenthesituationcould notbetreated as aWithregardtotheirapproprlateneSSinthesenseoftu−
toring．
Thesetwoproblemsmentionedabovesuggestthat
theinftrence procedure should cope with nonmono−Studentcontradiction，andanotheristoaskhimavery
Sirnilarquestiontogetaconfirmation．
Thesecontradictionscan bemoresu飽cientlydis−

tlnguishedbyintroducingandenrichingdomaindepen−

dent heuristics，e．g．， Students tend to mistake a uni− tonic modelingprocesses．In an HSMIS，an ATMSis
employedfbrthispurpose．TheHSMISconsistsofthe
formly accelerated motion for a uniformed motion if
SMIS，theATMS，theVirtualoraclegeneratorandthe
themotionisvertical， inadditiontothedomaininde−

CRS．ThemaintaskoftheATMSistomanagethecon−
Sistencyofasetofassumptions（environment）usedby
theproblem soIver，the SMISin ourcase．The SMIS

pendentheuristicsexplainedabove．
4．NonmonotonicStudelltModelinginfbrmstheATMSofallthereasonlngPrOCeSSeS．The

COntrOlmechanismofstudentmodelinglSneWlyformu−

4．1SMIS：InductiveInftrenceEnglne

lated，inordertomakethequalityandquantltyOfques−
tionsreasonableinthesenseoftutoring．Inthisformu−
1ation，thefbllowlngthreetypesof−nonmonotonicities，A student modeldescrlptlOnlanguageis required tobeabletorepresenttheteacher sunderstandingofthe Withwhichamodelingsystemshouldcope，arelisted：
Student sknowledge．Ftomthisviewpolnt，thelanguage
Shouldtakefourtruthvaluesfbrastatement，thoseare，（1）student nonmonotonicity，Whichis explained
true，false，unknownandfail．TheSMDL，Whichisan above，

extended version ofProlog，is designed to treat thesetruth values．

StudentmodelinglSeSSentiallyaninductiveinftr−（2）top！cnonmonotonicitywhichisrelatedtoboth
topICSfbrmodeldiagnosisandquestions，andence．TheSMIS，aninductiveinfbrenceenglnethatthe（3）reliance nonmonotonicity whichis related to theauthors have developed based on MIS［20］，describes
relianceonastudent sknowledge．
Student sknowledgeasanSMDLprogramffomasetof
oracles．Anoraclesymbolizesastudent sanswerwhichOraclesareJuStiBedbythreekindsofassumptlOnSSuCh

takesthefbrmofafactanditstruthvalue．SMISapplies

as（1）stude呵（2）considbrand（3）trustinorder・Thesta−

thefbllowingprocedurerepeatedlytothemodel‥（1）Ifthereisadifftrencebetweenaヮoracleandthe

tusofeachoracleiscontrolledflomviewpolntSOfthesenonmonotonicities．Further details ofthe formulation

factderived ftoma student model，aCtlVatethe student
aregivenin［12］and［10］・
modeldiagnosissystem，SMDS，tOidentifythecauseof
Anexampleofamodelingconversation andstu−
the ditrerence．

dentmodelconstructed bytheHSMISisindicatedin（2）Accordingtothediagnosis，SMISselects an

Fig．4．Thisgeographicdomainisselected，becauseit
can notably demonstrate the e鍋ciency ofthe control
mechanism．Intheconversation（a），thesystemobtains
anoraclegrov（rice，OSaka）：；true・Sincetheoracle．

clauseor addition ofanew clause．

GeneralityofthestudentmodelingmethodwithSMISis

su租cientlyhigh，becauseitcanconstructmodelswhich is correct，the HSMIS trusts that the student has cor−canbedescribedintermsofSMDL．Furtherdetailsare
rectknowledgeaboutthegrowthofplants，Whichcor−
describedin［10］and［12］・

rSPOndstotheclause（1）・Therefore，thesystemmakes
vlrtualoracles，Whicharecorrectfacts
suitable＿temPerature（rice，OSaka）：：tr11e，

4．2 ControlofModelBuildingProcess
A studentmodelingsystemshouldembodyateacherseducationalinsights．RequlrementStOthearchitecture

suitable⊥SOil（rice，OSaka）‥‥true，
s11itable＿lay（rice，OSaka）：‥tr11e，and

ha8irrigation（osaka）：：t．rue，

ofthemodelingsystem丘omeducationalviewpolntSare

toinducethecorrectclause（1）withoutmakingques−

inthefollowing：tions on them．

IEICE TRANS. INF. & SYST., VOL. E77-D, NO. I JANUARY 199444

( a ) Does rice grow in Osaka ?

>> %[Yes](1)(b) Does rice grow in Kiev ?
>> yes.[Nol(2)(c) Does rice grow in Mongolia ?

>> NO]( d ) Is t h e temperature of Mongolia suitable for rice ?
>> =[Yes1(e) Is t h e soil of Mongolia suitable for rice ?
>> -[Yes]( f ) Is t h e lay of Mongolia suitable for rice ?
>> Yes.[Yesl(g) Does Mongolia have a n irrigation ?

>> &[No](3)(h) Is the temperature of Kiev suitable for rice ?

...
...

(Student's answers are underlined. Right ones are in [I .)Fig. 4(2)grow(Plant, P l a c e ) :: s u i t a b l e - s o i l ( ~ l a n t place),,s u i t a b l e l a y ( ~ l a n tp, l a c e ) ,has-irrigation(Place).

(3)
grow(Plant, Place) :: s u i t a b l e - s o i l ( ~ l a n t P, l a c e ) ,suitable-lay(Plant, Place),
has-irrigation(P1ace).
(Model a t each scene of conversation.)An example of modeling conversation and model at each scene of it.

In the conversation (b), the student gives a wrong
answer to the system's question.
As the corresponding oracle can be explained by correct facts
on Kiev and the prepared clause (2), which implies
that the student lacks the originally needed condition suitable_temperature(Plant,P l a c e ) , the system
makes virtual oracles on Kiev to derive the clause as
in the previous conversation. Virtual oracles on Osaka continue to exist and support the clause (2). In theconversation below (c), the system fails to explain thewrong oracle g r o w ( r i c e , m o n g o l i a ) :: t r u e by any ofthe plausible clauses which are prepared in advance.
Therefore, the system must obtain "real" oracles on
Mongolia. Moreover, virtual oracles on Kiev and Osaka are dismissed, because he betrays the system's trust
in clause (2). The system makes questions on Kiev toget "real" oracles but does not make questions on Osaka, because the system avoids touching the old subject of Osaka by changing the status of assumption
consider (Osaka).

5. Student Contradiction
5.1(1)
grow(Plant, Place) :: suitable-temperature(Plant, Place),s u i t a b l e - s o i l ( ~ l a n t Place),,s u i t a b l e - l a y ( ~ l a n tPlace),,

Formulation of Student Problem Solving

Student problem solving which includes the latter twophases mentioned in Sect. 2 can be represented using a logic-based language such as Prolog. Predicate
s o l v e (G, x i , , x,,~)denotes problem solving knowledge. G denotes the goal of the problem, that is, whatshould be determined under what constraints. xi, isa vector of input variables which are instantiated andx,,, is a vector of output variables which are not instantiated when the predicate is called. {xi,,x,,,} represents whole articulation of the problem space. For
instance, the problem space "motion" is represented as{ m ,( s ( t ) , A s ) , ( ~ ( t )A,)?
,( a ( t ) , A,), (f ( t ) , A f ), [(TO?
So,Vo, a0 , Fo), . .
where the elements are the massof the moving object, displacement, velocity, acceleration and applied force as functions of time
erasped, and sets of the elements of the motion, respectively. Each function of time erasped is denoted as a couple of the function itself and the attribute of the function. The problem space, whichis adopted in Question 1 in Fig. 1, is represented as{m?( s ( t ) ) (, ~ ( t )( 4) t, h f i x e d ) , (f ( t ) ) , [(2, So, (O,O), ao?
F o ) , (0, (0, O), ( 1 9 . 6 , 0 ) , a l , F I ) ] } . When the problem
solving begins, input variables are given in the formulaof instantiated variables. For instance, the displacement
and the velocity on t = 0 are instantiated as (0,O) and(19.6, O), because they are given in the problem. Problem solving is a retrieval and an execution of methods described in the problem solving knowledge base,and to get the output parameters list {x,,~} instantiatedfrom the given input parameters list { x i n ) .
A problem solving knowledge consists of the estimating part and the procedural part, and is represented
by the following formula:.I},s o l v e ( G , Xi,, x,,~):predicates-f or-estimation(X;,),p r o c e d u r a l - b o d i e s (G, x i , , x,,t) .

-The estimating part specifies relevant domain of theknowledge, and the procedural part describes themethod used to get to the goal. Predicates for estimation are called "world predicates," which correspondto discrimination conditions on a concept discrimination tree. Conventional bugs are defined as wrong
predicates, lack of predicates and additional predicates from the correct problem solving knowledge inthe procedural part. An undifferentiation of concept
can be modeled as a lack of world predicate condi-

KONOetal：lNDUCTIVESTUDENTMODELINGFORSTUDENTCONTRADICTIONS

45

S01ve（getJPOSition＿0しmoving＿Object（Tl，Sl），M，St，Vl，At，Ft，t（To，So，Vo，Ao，FO），（Tl，Sl，Vl，Al，Fl）］）∫βんLれ，／われ〟血gむノ如b（（×勅
肌絢損亡血相比んmβf血（（×■n）），
get｝OSition＿0しmoving＿Object（【At】，（XJn），（Xout））t
get＿JOSition」）f＿mOVing＿OPject（（At】，（Xin），（Xout））：−
get＿aCCeleration（［At＝Xin），（XoL＾）），
Subtract（Tl，To，¶，multiply（Vo，T，V一），Squere（T，lll．multiply（Ao，TT，ATT），add（So，VT．ATT，Sl）．

Fig．5Interpretationofthestudentcontradiction．

tions．Lack of a world predicate extends the cover−utilizingtheheuristicsin Sect．3．AMulti−WorldCon−age ofconcept，SO that the knowledge，Which should
tro11erisincorporatedinto HSMIS to controlmulti−
beinapplicableoutsideacertainconcept，becomesin−WOrldreasonlng．ConceptdiscriminationtreesareglVen
COrreCtly applicablein other concepts．As a result， inadvanceasapartofdomainknowledge，andmodel

boththe orlglnalproblem soIving knowledgeand theknowledgewithwrongpredicatesfbrestimationareap−

diagnosis and revisions are driven on the structure ofthesetrees．EachworldcorrespondstoanATMSenvi−
Plicablein some di鮎rent concepts．The knowledge ronmentintheintegratedHSMIS．Clausesforproblem

Which he actually applies changes according to cir− SOIvingareinducedineachworldfromoraclesbelong−
CumStanCeSin such cases．Therefbre，the status ofthe

ingtotheworld．TheMulti−WorldControllermanages

knowledgeapplicationscanbeobservedtobeunstable．
thesetofworlds，eaChorwhichisacertainATMSenvi−
TbachersrecognlZethesecircumstancesascontradiction． ronment．The ATMS makesthe rounds ofsuch worlds
The correct problem soIving knowledge to determineand SMISinducestheclauselevelstudentmodelof．thethe position ofPin Fig．1is described as clausesin
WOrld，WhenamodelingcontradictionisdetectedandFig．5．Conditions soIve−irLformulatedぜhysics anduniformly＿αCCeleraied＿mOtion are world predicates，

is resoIvedin a certain world，thecorresponding ele−mentsof−thesetof．environmentsarerevised．Eachclause

andthelastpredicateinthebodyofthefbrmerclause
belongtotheproceduralpart，Whichca11stheproblem

levelstudent modelcan beconsistentlyinftrred using
such a mechanism．

SOIving method，i．e．，thelatterclause．A student who
Theconstructionprocessofthestudentmodelthat
representsstudentcontradictionsisasfbllows：
has notcompletely fbrmed the concept of unifbrmly
acceleratedmotion 1acksthelatterworldpredicateas
depictedin Fig．5（a），SOthathecanapplytheknowl− 1．Thesystemassumesastudentcontradiction，When
it turns out that a reliable clause has to be deletededge evenin the case ofunifbrm motions・Thus heunstablychoosesthecorrectmethodforcalculatlngthe
positionortheincorrectone，becauseofthelackofthe
WOrldpredicatehavingnode丘niteviewofhisown．
5．2 ModelingStudentContradictions
Thefbrmulationof．studentcontradictionsdescribedin
Sect・5・lworks wellas a student modelingmethod by

toexplainneworacles．

2．Thesystemtestswhethertheoraclesaresatisfiedby
theclausesthatexLgtinanotherworldbyuslngthe
WOrldinorderofsimilarltytOthecorrectworldon
the structure ofthe tree．

3．When asatisfiableworldisfbund，discrimination
conditionsthatcontributestodi鮎rentiationofthe

IEICE TRANS. INF. & SYST., VOL. E77-D, NO. 1 JANUARY 1994

46

two worlds are marked as neglected as depicted inFig. 5.

4. If any satisfiable worlds are not found, the system
considers the situation as a modeling contradiction
and tries to revise the model in the correct world.

Iw, a set of instances each of which should originally belongs to a certain conceptual world W , can be
declared by applying Cw which is the world predicateof W to I, the whole set of instances in the domain.
Method level representations and oracles which justify
the model are generated and stored in each world individually. Suppose that there are two worlds Wl and W2
which are brothers and that they have already obtained
and involved oracle sets owl and ow2.Methods M1
and M 2 have been staying in W l and W 2 ,respectively.
If M I , that currently has high C F value, is denied by
the oracle set o,, which is newly obtained from him,to the problem that naturally belongs to W l , then thesystem hesitates to dismiss M1 and tries to interpret thestatus of his conceptual discrimination as undifferentiated. If M a satisfies o,, 0, is moved into W2 andsupports M 2 there. Discriminal conditions C1 and C2,
which are world predicates of Wl and W2,are marked
as neglected. If any methods in any other worlds in thetree except the "another world" do not support o,, it isassumed that he was thinking in the world that contains
naive buggy knowledge and has solved the problem unformulatedly. 0, is moved into the "another world"which is prepared to cover his unformulated problem
solving, if one of the buggy clauses, prepared in theworld in advance, satisfies 6,.
In our example, the student correctly answered
Question 1 in the past, so that the student model had
the correct clause to get the force which an object inuniformly accelerated motion receives. He made thewrong answer utilizing his naive physical world to
Question 3 later. The HSMIS receives oracles madefrom his answer, e.g., oracle(subtract(19.6,0,19.6),true),oracle(subtract(2,0,2),true). The above clause in the student model does not satisfy these oracles and derives
an answer different from that of him, so that HSMIS
engages in generating another clause whose head issolve(get-direction-of-force(. . .), . . .) . It is inappropriate to dismiss the existing clause from the student

model, however. The clause is reliable enough and it ishard to interpret his behavior as lacking in it, because
previously he was able to apply it to obtain a correct
answer.
In such cases, the HSMIS dares to interpret his concepts as undifferentiated, attempting to make a student
model which implies the student contradiction. To thisend, the HSMIS at first infers the method that the student applied from the above oracles in a data driven way
as usual: subtract(V1, Vo, DV), subtract(T1, To, DT),
divide(DV,DT,AV) and direction(AV, Df), in order.
Next, it searches a world which has already contained
the method. If the search fails, model inference is carried out in the "another world." The HSMIS searches
the existing clause in the world which satisfies his erroneous answer. The system finds out the clause which
represents his erroneous "motion implies force" misconception, so that it marks discrimination conditions
that can naturally partitions the concept of motion into
the world of formulated physics and the world of naive
physics as neglected. It can explain his discrimination
status, that he unstably applies physical formulas andnaive methods.

5.3

Correcting Student Contradictions

To help the student build naive physical world in his
brain is one of the essential goals of tutoring, especiallyfor helping his conceptualization. This suggests a very
effective tutoring behavior as follows:
1. Give him a problem such that he tries to solve in
his naive one and fails to get a correct solution.
2. Remind him of the correct answer to the problem
he obtained in Question 1.

3 . Point out the inconsistency between the two results.
4. Explain the causes and guide him to build a correct
concept.
In this way, he can correctly identify attributes necessary
for building the concept, which we call world predicates, and establish relationships between them, thus hecan appropriately conceptualize the knowledge in bothworlds.

Solve t h i s problem. (Give Question 1 again)
The student correctly answers.
Tutor: You answered i n Question 3 that the d i r e c t i o n of the force which the sphere r e c e i v e s i s upward a t
t = l s , because it i s s t i l l moving upwards then. I f that was c o r r e c t , why didn't you say that thesphere i n Question 1 receives force t o the right?
Student:Because i t was moving upwards, s o i t i s hardly p o s s i b l e that it continued r e c e i v i n g force downward.
Tutor: The two problems are completely the same, e . g . , speed a t each time, e t c . , except f o r the d i r e c t i o nof motion. I f d i r e c t i o n of f o r c e were t o be implied from motion, you should have naturally s a i dthat the d i r e c t i o n of the force i s t o the l e f t i n question 1, but you d i d n ' t . (You should havet 6 motion implies force"
misconception i n your naive physical world. ...)
Tutor:

Fig. 6

An example of tutoring behavior.

KONOetal：INDUCTIVESTUDENTMOD王LINGFORSTUDENTCONTRADICT10NS

47

Obtaining a student model that represents themodelingsystems to handleinductivelearning（inftr−

Student contradiction，in this case the cause of−ence）andnonmon？tOnicreasoning［16］，［17］・Woolf

itis represented by thelack of the condition

et al［21］also polnted out the significance recently．
soIvevin−naiveTPhysic5 ，thesystembecomes ableto Incorporation ofinductivelearnlng theoriesinto stu−
generateae能ctivetutorlngdialogueasFig．6．Hisin−dentmodelingis discussedbyHoppe［7］andDi11en−
COrreCtreaSOningmethods，SuChastheuseofabduction
bourg［6］，Which utilize EBG and LEX，reSPeCtively．
inhisproblemsoIvlng，CanbeLbrmalizedascontradic−Both systems aresuccessrulto handletheirsimpledi−
tions，however，thetopichasbeenkeptfbrfuturework．
dacticstrategleS．

The mechanism fbr modeling contradictions ex−
PlainedinSect．4andthefiameworktohandlestudent
COntradictionsexplainedintheprevioussectionarein−
tegratedintotherefbrmulatedHSMIS．ItglVeSuPCOn−

StruCtlngauniBedconsistentmodelanddarestobuild
amodelinmultiplespaces，Whennewlyobtaineddata
denies the reliable current model．The controlmech−anism guarantees thefidelity and the accuracy ofthemodelin each world．

Self［19］givesthecharacterizationofstudenモmOd−eling as a diagnostic task，making an applicatlOn Of−

GDE［5］・Theknowledgetobelearnedisexpressedas
asystemwithadefinedstructureofcomponentswithde一丘nedbehavior．GDEgenerateshypothesesofstudentsmisconceptlOnS．Eachhypothesisisrepresentedasaset
Offailed components．GDEidentifies and refines the
SetOfhypothesesconsistentwiththeobservationsthus
far．However，GDEhasafatalproblem．Itcanexpress
Onlymaトhnctions．

6． DiscussioJIS

Huang［9］proposesalogic．

Herediscuss somebasicissues ofcomparative student

except the HSMIS，has so far been made atmodelingmodelingmethodologleS．

Student contradictions，Althoughitis well−definedin
TheOverlaymodel［2］isrelativelyeasytocontrol termsofpropositionalcalculus，itwouldbedi租cultto

themodelingprocesseswith．Sinceitcannotrepresent
the student

extendittodealwiththefirstorderpredicatecalculus．

sincorrect knowledge，theperfbrmance ofOVerlay−mOdel−basedITSsislimited・IDEBUGGY［1］7．CoJICludingRemarks
hasalotofmodularchunksofbuggyprocedures．Ithas
thecapabilitytocopewiththenoiseproblem，however， Thispaperhaspresentedastudentmodelingmethodol−
OgyanditsuseinanITS．Contradictionsofthestudent
itisnotcomplete．ItssearchingstrategleSdependonthe
tomodelstudentswithcontradictoryknowledgearefirst
assumptlOnthateachindividualbuggyprocedurecan
be extracted丘om a combined buggy procedure．The de負ned．Themodeling systemofsuch studentshasto
modelundifferentiated concepts andinconsistency as
burdenoncataloglngbugsisstillle丘．
itis・A sophisticatedcontrolmechanismto dealwith

BothACM［14］andourmethodHSMISarebasedmodelingcontradictionsfbrtheHSMIShasbeendevel−

Ontheideathat studentmodelinglSViewed asinduc−Oped．
tivelearnlngftomasetofexamples・Theycanmodel

not onlymal−functionsbut maLstructuresinterms ofdeclared prlmitives・Both methods act as domain，

Next，HSMIShasbeencomparedwithothermodL
eling systems from various viewpoints to demonstrateitisawe11−de負nedandgenericstudentmodelingalgo−rithm，Which can build a student modelofhigh rep−ofboththeoverlay andbuggymethods．Besides，both resentationpower・TheHSMIShasbeenfu11ylmple−
OfthemcancopewithnoISydata・
mentedinCommon−ESP（ExtendedSelfこcontainedPro，
ThemaJOrdi鮎renceistheirsearching and diag− log），andembeddedinFITS，FrameworkfbrlTS・Two

independentenglneSandtheircapabilitiescontainthosenosticstrategleSforascertainingwhatpartofthemodel
con8icts with the student．HSMIS can prune search
spacefbrnewclausesinadirectedtree，althoughACM

ITSshavebeenbuiltusingFITS，Oneisongeography
and the otherisonchemicalreactions．

orlglnallymakesablindsearchofallitssearchspace・Aeknowledgme爪tS
Toembodye代cientsearching，Langleyetal［15］have
beendevelopingDPF（DynamicPathFinder）・

TheHSMIScanincrementallyrevisethemodelfbl−
lowlngneWlyobtaineddata・Althoughthepath一色nding
algorithmis relatively fbrmalized by DPF，the rule−

finding process is not formally defined. HSMlS canTheauthorsaregreatfbltoGeoffryWebbfbrhiscom−ment which motivated this research，The authors are

alsothankfulto Heinz Ulrich Hoppe fbrhisvaluablecomments．This workis supportedin part by Grant−

inAidfbrScientificResearchonPriorityAreasofthe
MinistryofEducation，ScienceandCultureofJAPANdistinguishbetweennoisydataandbugmigration［12］，WhileACMcannot

underGrant：（No．03245106）・．

educationalapproprlateneSS，WhileACMdoesnot・Itcan beclaimed thatitis significant fbr student

48

References[I]

Burton, R.R., "Diagnosing bugs in a simple procedural
skill", In Sleeman, D., et al. (eds.), Intelligent Tutoring
Systems, Academic Press, London, 1982.
[2] Carr, B. and Goldstein, I., "0verlays:A theory of modelingfor computer aided instruction", MZT AZ Memo, 1977.
[3] Clement, J., "Students' preconceptions in introductory mechanics", Am. J. Phys., vo1.50, pp.66-71, 1982.
[4] de Kleer, J., "An assumption-based TMS", Artificial Intelligence, vo1.28, pp. 127-162, 1986.
[5] de Kleer, J. and Williams, B.C., "Diagnosing multiple
faults", Artificial Intelligence, vo1.32, pp.97-130, 1987.
"Designing a self-improving tutor:[6] Dillenbourg, P.,
PROTO-TEG", Instructional Science, vol. 18, pp. 193-2 16,
1989.
[7] Hoppe, H.U., "An annalysis of EBG and its relation topartial evaluation", Arbeitspapiere der GMD, 1991.
Huang, X., McCalla, G.I., Greer, J.E. and Neufeld, E.,[8]"Revising deductive knowledge and stereotypical knowlUser Modeling and Useredge in a student model",Adapted Interaction, vol. 1, pp.87-115, 1991.
[9] Huang, X., "Inconsistent beliefs, attention, and student
modeling", J. AI in Edu., vo1.3, no.4, pp.417-428, 1993.
[lo] Ikeda, M., Kono, Y. and Mizoguchi, R., "Nonmonotonic model inference", Proc. IJCA1'93, Chambery, France,
pp.467-473, 1993.
[I 11 Kono, Y., lkeda, M. and Mizoguchi, R., "To contradict ishuman", In Frasson, C., et al. (eds.), Intelligent Tutoring
Systems, ITS'92 Proceedings, pp.45 1-458, Springer-Verlag,
1992.
[12] Kono, Y., Tokimori, T., Ikeda, M., Nomura, Y. and Mizoguchi, R., "A student model building method based onformalization of nonmonotonicity", J. JSAI, vo1.8. no.4,
1993, (In Japanese).
Kono, Y., Ikeda, M. & Mizoguchi, R., "A modelingmethod for students with contradictions", Proc. AI-ED 93,
Edinburgh, Scotland, pp.481-488, 1993.
Langley, P. and Ohlsson, S., "Automated cognitive modeling", Proc. AAAZ-84, pp. 193-197, 1984.
Langley, P., Wogulis, J. and Ohlsson, S., "Rules and principles in cognitive diagnosis", In Frederiksen, N., et al.
(eds.), Diagnostic Monitoring of Skill and Knowledge Acquisition, pp.217-250, Lawrence Erlbaum, 1990.
Mizoguchi, R., Ikeda, M. and Kakusho, O., "An innovative framework for intelligent tutoring systems", Proc.
IFlP TC3 Working Conference on'
A1 Tools in Education,
pp. 105-120, Fascati, Italy, 1987.
Mizoguchi, R. and Ikeda, M., "A generic framework for
ITS and its evaluation", In Lewis, R., et al. (eds.), Advanced Research on Computers in Education, pp.63-72,
North-Holland, 1991.
Self, J.A., "Bypassing the intractable problem of student
modelling", Proc. ZTS'88, pp. 18-24, 1988.
Self, J.A., "Supporting the disembedding of learning",
Proc. East- West Conference on Emerging Computer Technologies in Education, pp.28 1-286, 1992.
Shapiro, E.Y., Algorithmic Program Debugging, MIT
Press, 1982.
Woolf, B.P. and Murray, T., "Using machine learning toadvice a student modeling", J. ArtiJicial Intelligence in Education, vo1.3, no.4, pp.401-416, 1993.

IEICE TRANS. INF. & SYST., VOL. E77-D, NO. 1 JANUARY 1994

Yasuyuki Kono
was born in Hyogo,
Japan, o n January 8 1967. He received
B.S. and M.S. degrees from Osaka Univedsity, Osaka, Japan, in 1989 and 1991, respectively. He is currently a Ph.D. student at the Department of Information
and Computer Science, Faculty of Engineering Science, Osaka University. He
is engaged in research o n student modeling- for intelligent tutoring- systems.
Mr.
.
Kono is a member of Japanese Society for
Artificial Intelligence (JSAI), Information Processing Society ofJapan (IPSJ), and Association for the Advancement of Computing in Education (AACE).

Mitsuru Ikeda
was born in Aomori,Japan, o n March 29, 1962. He received
B.S. and M.S. degrees from Utsunomiya
University, Tochigi, Japan, in 1984 and1986, respectively. He obtained a Ph.D.
of Eng. degree from Osaka University
in 1989. From 1989 to 1991 he was a
Research Associate in the Faculty of Engineerlng, Utsunomiya UniversityHe
1s currentlv a Research Associate In theInstitute of Scientific and Industrial Research, Osaka University. He has been engaged in research o nformal language theory, hypothetical inference, inductive inference, and intelligent tutoring systems. Dr. Ikeda is a member of
JSAI, IPSJ, C A I Society, IEEE, and AAAI.

Riichiro Mizoguchi
was born inTokyo, Japan, o n October 13, 1948. Hereceived B.S., M.S., and Ph.D. of Eng.
degrees from Osaka University in 1972,
1974, and 1977, respectively. In 1977 hejoined the Osaka Electro-Communication
University. From 1978 to 1986 he wasa Research Associate in the Institute of
Scientific and Industrial Research, Osaka
University, where he is presently a Professor. He has been engaged in researchon learning of pattern discriminating function, cluster analysis,
speech analysis/recognition/understanding, expert systems, andintelligent tutoring systems. He received paper awards from Pattern Recognition Society and IEICE of Japan in 1983 and 1987,
respectively. Dr. Mizoguchi is a member of JSAI, IPSJ, Acoustical Society of Japan, C A I Society, Japanese Society for Cognitive
Science, AACE, IEEE, and AAAI.