DOCUMENT RESUME - ERICDOCUMENT RESUME ED 281 716 SE 047 910 AUTHOR Marshall, Sandra P.; And Others...

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ED 281 716 SE 047 910

AUTHOR Marshall, Sandra P.; And OthersTITLE Schema Knowledge Structures for Representing and

Understanding Arithmetic Story Problems. First YearTechnical Report.

INSTITUTION San Diego State Univ., Calif. Dept. of Psychology.SPONS AGENCY Office of Naval Research, Arlington, Va.PUB DATE Mar 87CONTRACT N-00014-K-85-0661NOTE 72p.PUB TYPE Reports - Research/Technical (143)

EDRS PRICE MFOI/PC03 Plus Postage.DESCRIPTORS *Cognitive Processes; *Computer Simulation;

Educational Research; Elementary Education;*Elementary School Mathematics; Learning Theories;*Long Term Memory; *Mathematics Instruction; Memory;Models; *Problem Solving; Research Reports; Schemata(Cognition)

IDENTIFIERS *Mathematics Education Research

ABSTRACTThis report describes the first of three stages in a

study in the domain of problem solvingl the definition andexplication of schema knowledge. One objective of this research is tounderstand how schema knowledge is acquired and used in the chosendomain. The focus is on ways in which instruction influences thedevelopment of specific knowledge structures in long-term memory. Theproblem-solving domain was first analyzed with respect to theconceptual relations that may appear in arithmetic story problems.Emphasis is placed on the underlying semantic structure that givesmeaning to each problem. Next, each semantic structure was framed asa hypothetical memory object, or schema, and the relationship toaccepted theories of memory was explored. Finally, a computer modelwas developed that details the linkage of schema knowledge to twobasic components of long-term memory: semantic networks ofdeclarative memory and production systems and procedural memory(MNS)

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SCHEMA KNOWLEDGE STRUCTURES FORREPRESENTING AND UODERSTANDING ARITHMETIC ST017.Y PROBLEMS

by

Sandra P. MarshallChriStopher A. Pribe

Julie D. Smith

March 1987

Center for Research in Mathematics and Science Education

DEPARTMENT OF PSYCHOLOGY This research is supported by theOffice of Naval Research, Cognitiveco Diego. Of Sciences _ ScienceS Programs; Contract No.

San Diego State University N-00014-K-85-0661.

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Schema Knowledge Structures for Representing and UnderstandingIvrtrbmptIc-Stbry Problems --------.

12. PERSONAL AUTHOR(S)Sandra P. Marshall- Christopher A. Pribe; & Julie D. Smith

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17 COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)problem solving; cognitive processes,mathematics_instruction; knowledgerelare-sentatien

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projectare:and evaluationledge,knowledgeexplicates_threeanalyzedStorygivnsas aaccepted

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(1) definition

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in this:report is part of a three-stagearithmetic story problems. The three stagesexplication of schema knowledge, (2) development

system designed to teach schema know-simulation of_the acquisition and use of schema_

document focuses_on the_ first stage only_andof our research. _First, the domain itself iSthe conceptual relations that_may appear irv_is upon the underlying semantic structure that

Socond; each somnntio structure is framodobject (i.e.; a schema); and the relationship to

is explored; Third, a computer model is

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pre-tented that_details the linkage of schema knowledge to twoha-Sid CoMpOnents of long-term memory: semantic netWbrkt ofdeclarative memory and production systems of procedural memory.

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Schema Knowledge Structures forRepresenting and Understanding Arithmetic Story Problems

First Year Technical ReportONR Contract N00014-85-K-0661

Sandra P. MarshallChristopher A. Pribe

Julie D. Smith

Department of PsychologySan Diego State University

San Diego, CA 92182

Acknowledgments

We wOuld like to thank Kathryn Barthuli and MichaelKummle for their assistance in preparing this report.

Introduction

The research described in this report is part of athree-stage project in the domain of arithmetic storyproblems. The three stages are:

(1) definition and explication of schema knowledge

(2) development and evaluation of an instructionalsystem designed to teach schema knowledge

(3) computer simulation of the acquisition and useof schema knowledge structures.

This document focuses on the first stage only. Theremaining stages will be addressed in future reports.

One objective of this research is to understand howschema knowledge is acquired and used in the chosen domain.In particular, the focus is upon ways in which instructioninfluences the development of specific knowledge structuresin long-term memory.

It is common to find schema-based research in cognitiveScience. Much of this work, however, fails to specifyprecisely the nature of a schema. A schema has sometimesbeen considered to be equivalent to other cognitivestructures such as frame, script, or plan. Most often, aschema_is no more than a simple declarative frame withvariable slots to be filled. The 3ack of specificity aboutthe structure of a schema makes it difficult to describe ormodel how an individual learns, stores, and uses knowledge.If the schema is the basic building block of cognition, asRumelhart (1980) states, then it must be more clearlydefined. This report fccuses on the definition and structureof schema knowledge. The adequacy of the definition isevaluated through computer simulation programs that operatewithin the domain of interest.

The following aspects of our research are describedbelow; First; the domain itself is subjected to analysis ofthe conceptual relations that may be expressed in arithmeticstory problems. Emphasis is placed_upon the underlyingsemantic structure that gives_meaning to each_problem. Second, each semantic structure is framed as ahypothetical memory_object (i.e., a schema), and therelationship to accepted theories of memory ig develOped.Third, a computer model is presented that details thelinkage of schema knowledge to two basic components of long-term memory: semantic networks of declarative memory andproduction systems of prodedural memory;

Schema Knowledge Structures

6

Semantic Relations of Arithmetic Story Problems

A story problem may be loosely characterized as anabbreviated verbal account of a situation, providing somespecific information (usually numerical) and requiring useof that information to answer a stated question. Theinformation given in the problem is embedded within a"story", but the story rarely contains much detail. Thereader is expected to recognize the situation depicted inthe story and to embellish it from his or her own store ofexperiences. Thus, it becomes important that the reader havesufficient knowledge stored in memory to be used asnecessary in understanding the events of the story.

What is required to understand a story problem? First,the reader must recognize the words used in the problem.Second, the reader must understand the situation and therelationships that exist among objects described in theproblem. We do not focus heire on the problems of reading; itis assumed that students have the requisite knowledge of thesituations portrayed in the problems. Our attention is uponwhat knowledge is required for students to understand theabbreviated description of the situation and itsaccompanying relations.

One aspect of understanding appears to be thecategorizing of similar items. Recognition then becomes theprocess identifying the appropriate category. It is obviousthat one could organize the domain of story problems on anumber of different dimensions. For example, problemsrequiring the same operation(s) could be grouped together.Alternatively, problems describing the same situation(s)could be aligned. Or, as we suggest here, problemsreflecting similar semantic structure could be aggregated.

Categorization by operation works well only when asingle operation is required. This approach is currentlypopular with teachers and textbook.developers, but it haslimited success as a problem-solving strategy. For multi-step problems, it is difficult for students to order andkeep track of the various operations.

Grouping by story situation also has its drawbacks.There are an infinite number of situations that could beused, and memory requirements for keeping track of all ofthem would be enormous. Also, for any situation, severalproblems could be devised, each requiring different methodsof solution. Hence, the situation alone could notsufficiently provide clues about solution strategy.

We argue here that the most efficient and successfulmeans of organization is to look for common underlying

Schema Knowledge StructureS

elements within the structure of the problems. This approachinvolles features that are usually termed semanticrelations. It requires the ability to understand thesituation depicted in the story and to perceive therelationships that exist between objects.

In this section, we describe a Set of semanticrelations that characterize fully the &main of arithmeticstory problems. Other studies in this domain have eitherlimited their scope to a subset of arithmetic operations(e.g., Riley, Greeno, & Heller, 1983; Briars & Larkin, 1984)or have focused upon characteristics of the quantities givenin the problem rather than upon the overall structure of theproblem (cf. Greeno, Brown, Foss, Shalin, Bee, Lewis, &Vitolo, 1986). Differences between the currentconceptualization and those of other researchers,particularly Riley et al. and Greeno et al., will bediscussed later in some detail.

Five semantic relations appear to be sufficient forcharacterizing virtually all story problems of arithmetic.These are: Chance, Combine, Compare, Vary, and Transform.Each is described below, together with examples thatdemonstrate the different formulations of problemscontaining the relations.

The CHANGE_Relation

The first and most elementary of the semantic relationsiS the Change (CH) relation. In its simplest form, therelation is am increment or decrement of a measurableresource. The change that occurs is physical and iSpermanent. Once the change occurs, the original state cannotbe revisited.

A CH relation can be manifested in many forms. Thefollowing are examples:

John had 12 baseball cards. His friend Jim gave (1)him 5 more. How many cards does John have now?

Twenty-five tomato plants were growing in the (2)garden. Snails ate some of them. There are 15plants left. How many did the Snails l.tat?

I had some money in my checking account. After (3)I deposited a check for $35.00, I had $280.75in the account. How much was in the accountbefore I made the deposit?

Each problem begins with an initial state (e.g., 12 baseballcards, 25 tomato plants, some money in an account) in which

Schema Knowledge Structures -3-

the objects to be manipulated are specified and in which thequantity or amount of these objects is defined. A change inpossession occurs at some later time, indicated in theseproblems by the phrases or words "more" (than he hadbefore), "left" (after eating),_ and "after" (the check wasdeposited). The reader must infer the time constraints fromthe position of the statements and these phrases. Aftei thechange occurs, there is a recognizable final or ending statein which the new quantity is defined. The initial state andfinal state are not coexistent: they cannot occur at thesame ,time. Either there are 25 tomato plants or there are15. Both statements cannot be true.

As shown in examples Ili, (2), and (3), three differentquestions can be posed in a Change relation. The moat commonsituation is to provide information about the initial stateand the amount_of change, leaving the final state to becomputed (1). A second alternative is to present both theinitial and final state and to have the individual calculatethe amount of change (2). Finally, the problem may containthe amount of change and the final state, and the questionit to determine a value for the initial state (3).

Most CH problems involve only additive change (e.g.,require addition and subtraction operations). However,sophisticated problems exist for which there may bemultiplicative or exponential growth rather than additivechange. The following is an example:

There were 300 bacteria on the petri dish. If (4)they doubled in number every 2 hours, how manywould be on the dish after 6 hours?

This example emphasizes an important point about thesemantic relations defined here: They are not defined interms of the arithmetic operations iequired for problemsolution. We are looking at the nature of the relationamong objects described within each problem; that relationmay itself be linked to several possible operations.Identifying the relation is not synonymous with identifyingthe operation for solution.

The COMBINE Relation

A COMBINE (CB) relation is expressed whenever thereexists a hierarchical or composite grouping of objIctswithin a problem. The CB relation involves the renaming ofelements with respect to a superordinate category. No actionis taken in a CB relation, and, in contrast to the CHrelation, no permanent alteration of objects occurs. Thepassage of time is irrelevant.

Schema KnoWledge Structures

An important characteristic of the CB relation is itsdependence upon understanding part-whole relationships. Thepart=whoIe concept occurs in many domainseincluding phySics,algebra, and arithmetic (cf. Nesher, Greeno, & Riley, 1982;KintSch & Greeno, 1985). To understand a Combine relation,an individual must comprehend that the whole (orsuperordinate category) iS equal to the sum of the parts(subordinate categories). A necessary conStraint is that thesubordinate categories have semantic ties to thesuperordinate one. The logical, hierarchical structure musthave a semantic base. For example, the following itemillustratts a Combine relation:

Ann has three apples and two oranges. How many (5)pieces of fruit does she have?

Consider what as individual must already know (or mustbe told elsewhere in the problem) in order to solve (5).First, he/she must recognize "apples", "oranges" and"fruit". There must be an underStanding of the commonattributes of these three elements. If they have no sharedcharacteristics, the problem becomes senseless. Theindividual must also understand that only those elementsspecified in the problem are relevant. We do not speculateabout how many lemons or grapes are in Ann's possession.

There must also be an awareness of the relationshipsamong applei, oranges, and fruit, as displayed in Figure 1.In this Figure, the two elements "apples" and "oranges" havean identical relationship with the element "fruit": both areinstances or examples of "fruit". However, "apples" and"oranges4 nonetheless are diStinct semantic elements; anapple is not equivalent to an orange. Each element maintainsa definitive set of characteristics that Serves todistinguish it from other elements existing at the samelevel (such as pear, grape, lemon). Our hypotheticaliuividual solvIng tta above problam must know that it islogically or Semantically impossible to join elements .at onelevel of this semantic network unless they have identicallinks to a higher level of categorization. That is, they canbe combined only if one considers that they are instances ofa more general level of classification and only if theyequally share the characteristics of that level.

In terms of Figure 1, the elements_"apples" and"oranges" inherit the characteriSticS of "fruit" becausethere existHai connecting these elements and because"fruit" is at a higher level of the network than the othertwo elements. The meaning of a link depends upon thedirection in which it is interpreted. Thus, it is true that"apples" are an instance of "fruit"; it is not true that"fruit" is an instance of "apples". Similarly, "apples" have


1 0

the ptoperties of "fruit"; the converse is not true. Ingeneral, links running from snperordinate categories tosubordinate ones are inheritance links, and it is common tospeak of the subordinate or lower level elements inheritingall characteristics of the superordinate one.

The importance of these semantic distinctions isapparent in the following item:

I have three apples and two oranges. How many (6)pieces of candy do I have?

From Figure 1, it is clear that "fruit" and "candy" areboth instances of the superordinate category "food", andboth inherit the same set of characteristics about food(e.g., can be eaten, provides source of energy'. However,they do not share subordinate elements. That is, "apples"and "oranges" are not examples of "candy" and have no linksto it. Thus, problem (6) cannot be solved with the giveninformation.

In general, Combine problems do not define*thesuperordinate and subordinate categories as such.Individuals solving the problems are expected to draw uponsemantic knowledge stored in long-term memory foridentification and clarification of the elements specifiedin the problem. If the requisite knowledge is missing fromthe individual's knowledge base, the individual will beunable to solve the problem.

As with the CH relation, there are varying forms thatthe CB relation may take in an arithmetic story problem. Twodifferent questions may be asked. First, the problem mayrequire finding a numerical value associated with thesuperordinate category. In that case, values for therelevant subordinate ones must be given in theproblem. Second, the problem may ask for the value of oneof the subordirate elements. For this case, thesuperordinate value and the remaining subordinate one mustbe known. An example of the first situation is given by (5).An example of the second is given below.

I have three apples and some oranges. If Ihave five_pieces of fruit, how many orangesdo I have?

(7)

A distinguishing point about the sematic CB relation isthat the original quantities associated with the subordinateand superordinate categories remain unchanged by thecombination. So, for example, in (7) above, although theremay be five pieces of fruit, three of them are still apples.


ii

Figure 1

A Semantic Network

SCheme KnoWliedge Structures 1 2 -7-

The-CONTARE-Ralation

A third semantic relation in the domain of arithmeticstory problems is the Compare (CP) relation. In a CPrelation, two elements of a problem are evaluated in orderto determine their relative size. The meaning of therelation comes from weighing one element against the other.It is not the absolute value of either one that is centralhere, but rather the relative position that one has withrespect to the other.

A necessary part of the relation is the existence oftwo elements, each associated with a numerical value andboth having the same semantic features. Generally, thesemantic features of interest are the units in which theelements are measured (e.g., feet, hours, gallons).Comparisons can only be made meaningfully between elementsmeasured against the same standard. For example, we cannotsay which is larger, a liter bottle or a twelve inch board.However, we can compare a liter bottle with an eight ouncejar, provided that we do so on a common unit of measure.

Implicit in the CP relation is the concept of one-to-one matching of one element in the problem 4ith theother. As described by Briars and Larkin (1984), eachelement is considered to be a set having a given number ofmembers. To compare two sets, one theoretically engages inone-to-one matching, removing one member from each set andsetting them apart as a matched pair. The smaller of the twosets is the one which first becomes empty. The amount leftin the larger set is the difference between the two sets. Ifboth sets become empty at the same time, they have an equalnumber of members.

Much Iike Combine, the Compare relation is static; noaction occurs in the problem. The_CP relation is adescription, an alternative way of expressing the relativesize of two sets of similar objects. The representation oftime. as a variable is usually irrelevant. CP relations canoccur at the same time or at different times. For example,consider the following items:

Joe makes $4.50 per hour at his job, and Ed (8)makes $5.30 per hour. How much more doesJoe make per hour than Ed?

Mary is a gymnast. Last week, she scored (8). 7.5 on the balance beam in a gymnastics meet.

In another competition held yesterday,she received a score of 8.5 in the sameevent. At which meet did she have the bestperformance? How much better?


In (8), the comparison takes place at a single point intime; both individuals currently make the wages stated inthe problem. In (91, the comparison is between scoresobtained at two different times. It should be clear fromthese examples that time is not a distinguishingcharacteristic of CP.

The VARY Relation

The Compare relation introduced the relationship ofrelative size. In the Vary (VY) relation, there are severalother relationships that must be understood by an individualin order to solve problems having this semantic structure.Most importantly, it is necessary to distinguish betweenthree types of elements: subject-units, object-units, andassociations. Subject-unkts are the foci of the problem(e.g., marbles, apples, childrenl. A subject-unit has aparticular object-unit related to it by means of a specifiedassociation. For example, the statement that "one appleZEWEi-IN-aWnts" has one apple as the subject-unit, cost asthe association, and 25 cents as the object-unit. Theassociation relating it753i-c7= and object-units is generaland applicable to every subject-unit; thus, any instance ofapple will have a cost represented by cents that isassociated with it (in the restricted environment of thisparticular problem, of course).

A second relationship requisite to the VY relation isthe concept of per-urilt. The notion of a constant valueper unit may be explicitly stated or may be merely impliedby the wording of the problem. In either case, theindividual must realize that every instance of the subject-unit presented in the problem will have the same value of anobject-unit associated with it. Thus, we have "an applecosts 25 cents" or "the car travels 30 miles on a gallon ofgas" as examples. Knowledgeable students understand withoutbeing told directly that a second apple will also cost 25cents and that another gallon of gasoline will enable thecar to travel an additional 30 miles. Within a problem, theper-unit value remains constant.

A fundamental difference between VY and the three -

relations previously defined is that the problem structureof VY involves four quantities; two of these are subject-units and two are object-units. As described above, asubject-unit is paired with an object-unit by means of anassociation. For the four quantities describing a VYrelation, there are two pairs, each bound together by anassociation. Not only must an individual recognize the pairsand the associations that bind them, the individual mustmake a mapping from one pair to the other and test the logicof that mapping before solving the problem.


1 4

Let the first subject-object pair be denoted by

(subject].-unit <association> objectl-unit]

and the second by

(10)

(subject2-unit <association> object2-unit). (11)

Each of the four unitS of (10) and (11) has two features: atype and a value. The type refers to the nature of theelements, such as apples, pencils, etc. The value is thenumber of such elements (e.g.,2 apples).

To satisfy the conditions of the VY relation, anindividual mutt test three constraints: that the_types ofsubject-units are identical in expressions (10) and (11),that the tyRes of object-units are identical, and that theassociations described in (10) and (111 are the same.Expressions (10) and (11) should differ only in thenumerical values associated with the four units. Further,in a typical story problem, three of the four units willhave known numerical values. In a VY relation, the objectiveis to determine the fourth (unknown) value,

This constraint evaluation can best be demonstrated bythe following example:

The price of one apple is 25 cents. How much (12)will 15 apples cost?

The expressions of (10) and (11) can be.rewritten as:

(1 apple <cost> 25 cents] (13)

and

(15 apples <cost> =?= cents] (14)

where =?= denotes an unknown value. To solve this problem,an individual first must establish that a logical structureexists. The three tests described above serve this purpose:expressions (13) and (141 both concern apples, both involvethe cost of apples, and both measure cost in terms ofcents. The importance of these tests is clearly seen byexamining the following problems, in which one or more oftha tests fail.

The cost of one apple is 25 cents.will 5 bananas cost?

The cost of one apple is 25 cents.will 15 apples weigh?


How much (15)

How much (16)

15

-10-

The cost of one apple is 25 cents. How much (17)will 5 bananas weigh?

Problems (15) and (16) demonstrate the failure of asingle test; problem (17) shows two failures. The more teststhat fail, the more illogical the problem appears to anindividual. Note that each sentence of (17) is a reasonablestatement. The difficulty is that the first cannot bE usedto answer the question posed in the second.

The TRANSFORM Relation

The fifth and final relation defined here is theTransform_(TR) relation. The essential understandingrequired for a TR is_that it is possible to describe oneobject in Several different ways. In particular, if theobject has a numerical value associated with it and if itbears a known relationship to another object also having anassociated numerical value, one may describe the firstobject in two ways, in its original metric or as a functionof the value of the second object.

Consider a typical Transform problem:

Sue is 1/3 as old as her mother. If her mother (18)is 30 years old, how old is Sue?

There are two ways to look at Sue's age in this problem.First, she must be some number of years old. This is theunknown of the problem. Second, since both her age and hermother's age can be expressed at yeart, we can look at oneas a function of the other. In this case, Sue's age isrelated to her mother's by the fraction 1/3. Thus, there aretwo statements about Sue's age and both are simultaneouslytrue.

The TR relation is similar to VY in that tworelationships are given in the problem. These may beexpressed as:

(subjectl *=* objectI-unit] (19)

(subject2 *=* object2-unit] (20)

with_subjectI and subject2 being the main foci of theproblem, with *=* indicating that the leftmost member of theexpression can be expressed in terms of the rightmostmember, and with object-units having types and values asdescribed above. We also expect to have a known relationshipbetween the two subjects:

and

(subjectI *=* <fl> su jct2]. (21)

Schema Krowledge StructureSt

This expression states that one subject can be expressed asa function <fl> of the second. Finally, by substitution from(19) and (20) we have

[objectI-unit.*=* <fl> object2-unit] (22)

that is, the first object-unit can be expressed as afunction of the second object-unit (or vice versa).

For a problem to contain a Transform relation, severalconditions must be met. First, subjectl and subject2 must berecognizably distinct entities (e.g., two individuals).Second, the two object-units must be expressed in the, samemetric (e.g., years, feet, days) . Third, either the dubjectsare related to each other through a stated mathematicalfunction or the object-units are so related. Fourth, theproblem must provide the value of one object-unit and thevalue of the mathematical function (such as "3 times aslarge" or "5 more than"), or it must specify values for bothobject-units, leaving the mathematical function to bedetermined.

Using this notation, problem (18) can be represented bythe following expressions:

[Sue's age *=* =?= years] (23)

[Mother's age *=* 30 years] (24)

[Sue's age *=* <1/3> Mother's age] (25)

[=?= years *=* <1/3> 30 years] (26)

Expressions (23) and (24) state that both Sue's age and hermother's age can be expressed as some number of years. Thenumber of years is unknown for the former and is given as 30for the latter. Expression (25) indicates the relationshipbptween the two ages: Sue's age is one-third of her mother'sage. Expression (26) is actually the solution to the problemin this case, and it is obtained by substituting the valuesof the ages from (23) and (24) into the appropriate slots of(25).

An important underlying concept of a TR relation is"unity" or "the whole". Implicit in the definition ofrelationship between two quantities is the notion that onecan frequently define one quantity to be "the whole" andexpress the second quantity as "a part" or "a multiple" ofthe whole. This is a more sophisticated use of the part-whole concept than was needed for the Combine relation. Anyquantity can be designated "one" or "unity" and any similarquantity can be expressed in relative units. The primary


1 7

constraint is_that both quantities must be measured by thesame_metric. In the age problem above, age is given in_years. As written, the relation between the two-ages is thatSue's age is a fraction_of her mother's age. The mother'sage is "the whole" and Sue's age is expressed as "a part" ofit. Without changing the relationship between ages, onecould restate it as "her mother is three times as old asSue". In this case, Sue's age would be considered the"whole" and the mother's age would be a multiple of it.

The TR relation iA static. No action'takeS place in theproblem, zhere are no alterations in quantities, In fact,the opposite is true: exact conservation of quantities isrequired in moving from one unit of measure to another.Further, in moat TR problems, time is here and now. Therelationships that are expressia-ire true at this moment;they will not necessarily be true at a later date (i.e.,consider the age problem above).

A defining characteristic of Transform problems is thatone answer to the question posed in the problem isexplicitly stated. For (18), the_question is "How old_isSue?" One acceptable and true answer is given in the firstsentence: "Sue is 1/3 ae old as her_mother." To force anindividual to seek the alternate representation of Sue'sage, the problem should state the question as "How manyyears old is Sue?" An individual solving the problem isexpected to know from previous experience that the solutionwill be expressed in years, even_though the explicitquestion about years is not stated.

There are several ties between TR and otherrelationt. First, both Transform and Combine rely upon thepart-whole relationship. Second,_both Transform and Compareare concerned With the relative size of quantities. Third,Transform and Vary both involve four quantitieS, operatingon pairs of them.

The TR relation appears to be the most difficult forstudents to grasp. It is both a prealgebra relation (beingfundamental for algebra problem_solving) and_an arithmeticrelation (appearing in_story problems_as early as thirdgrade arithmetic, CAP 1980). Several constraintS must besimultaneously considered in TR. Students may_not realizethat it iS necessary to satisfy many constraints as theyseek to recognize the form of a problem; working with asingle constraint may lead to an incorrect representation_and a consequential incorrect solution. Furthermore, TR isnot particularly tied to any arithmetic operation: all fourare equally likely.


.18

Multi-Step Problems

Thus far, we have used only simple examples fromarithmetic in which a solution may be found_by a singleapplication of one arithmetic operation. Most individuals

have little trouble with Such items, once they have mastered

the algorithms of the operations themselves. Rates of

success undergo a dramatic shift when problems require more

than a single computational step. Even highly qualified

studenta of arithmetic experience difficulty with multi-step

problems (Marshall, 1987).

A particular advantage of the semantic relationsintroduced here is that they may be used to organize multi-

step problemS. For any problem, there is one centralquestion that is posed and one general Situation that isdescribed. Within that situation, there may be other

unknowns and other situations that must be examined, but the

central one remains the target of the problem solving. Forexample, we can easily construct a problem in which thecentral situation is that an individual has some money,

makes dome purchases, and has a resulting amount of money

that is less than the original amount. For example,

Alice had $50.00 when she went to the grocery (27)

store. She bought two quarts of milk at $.75each, 1 1/2 pounds of cheese at $2.75 per pound,and a loaf of bread for $1.39. How much moneydid she have after she made these purchases?

This situation expresses a Change relation. Embeddedhere is a Vary relation (in the case of purchasing more than

one item for the same cost per item) and A Combine relation

(in the cage that the individual purchased several items

each having a given price).

Many_individuals use an operation-based strategy to

solve both simple and difficult problems (Marshall, 1982).

Keeping track of the many operation; proves to be difficult

for a large number_of them. We speculate that this_difficulty arises because the individuals do not have a

means by which they can organize the many steps required in

the problem.

It is true that one could read problem t27) and make a

mental note that one should multiply (2 x $.751, multiply (1

1/2 x $2.75), add (results of the two multiplications plus

$1.39), and subtract ($50.00 minuS the eum resulting fromthe addition step). One can hypothesize that an individualattempts to store this list of operations in short-term

memory while working the various computations. Since these

operations are not logically bound one to the other, some


1 9

may become distorted or lost. The result is that one or moreof the operations is frequently omitted, yielding anincorrect or partial solution. We have empirical evidencethat omission errors are, in fact, among the most commonerrors on multi-step problems.

The use of semantic relations introduces a logicalstructure that serves to organize the information in theproblem. In (27), once the individual recognizes that theunderlying situation depicts a Change relation, he or shecan then look at the components that make up the relation.The initial or starting quantity is already known (i.e.,$50.00). The amount of change is not yet known -- this is asecondary problem that must be Solved in order to completethe Change relation.

To solve.the secondary pi:oblem, the individualperceives that several items are to be purchased and thatthe total cost of these items is required. This represents aCombine relation, and the needed elements are the variousprices. Once again, all components of the relation are notknown. In this case, there are several items with per-unitprices. Again, this represents a problem within aproblem. We are now at the third embedding level. Thestructure of the problem can be diagrammed as in Figure 2.Using semantic relations in this way imposes a hierarchicalstructure on the problem-solving steps and thus shouldenable individuals to monitor these steps.

Schema Knowledge Structures 2 0 -15-

Figure 2

A Multi-Step Problem

Alice had $50.00 when she went to the grocerystore. She bought two quarts of milk at $.75aachc 1 1/2 pounds of cheese at S2.75 per pound,and a loaf of bread for $1.39. How much moneydid she have after she made these purchases?

ORIGINAL AMOUNT OF RESULTINGAMOUNT CHANGE. AMOUNT

I

known unknoWh

PRICES OFIMRIOUS THINGS

SEVERAL ITEMSAT ONE COSTPER ITEM

TOTAL COST

SEVERAL ITEMS ONE_ITEM unknownAT ONE COST

UNIT NUMBER TOTAL

I I I

PRICE OF UNITS

PER ITEMknown

VARY

_MAT NUMBER TOTAL

1 I I

PRICE OF UNITS

known known unknown known known unknown


21-16-

The Adequacy of_the Classification

The five semantic relations presented above appear tobe sufficient for claSSifying virtually all story problemsof arithmetic. Marshall (1985) examined all sixth-gradearithmetic textbooks adopted for use in California publicschools. In that study, each story problem waS classifiedaccording to the relations defined here. All traditionalstory problems could be uniquely classified. Problems thatwere not classified were those involving memorized formulas(e.g., find the circumference of a circle, what is the areaof the triangle) and were not strictly arithmetic.

For the present study, we evaluated three additionalsourcleS of problems, each representing a different level ofarithmetic. These were: arithmetic texts for eighth grade,remedial arithmetic materials for community colleges, and anewly created text for training navy personnel. The resultswere similar to those found by Marshall (1985).Approximately 90% of all items from the three instructionalsources could be uniquely classified according to the fivesemantic relations. The remaining 10% required applicationof geometric or probability formulas for solution andinvolved little semantic interpretation. As expected, asomewhat larger number of itemS involving geometry andprobability were found in the present Study than wereobServed at the lower grade because theSe topics are givengreater weight at the upper grades. Therefore, we have alarger proportion of unclassifiable items, labeled "other".The frequencies with which each relation-occurred in one,two, and more than two step problems are shown in Table 1.

The two eighth-grade texts contain a total of 629 storyproblems; 478 (76%) of them illustrate a single relation.The Navy training materials contain only 35 story problems.Most of the problems are simple one-step items (70%1.Finally, the remedial materials for community college usehad 658 story problems, and 68% of these were single stepitems as well.

A majority of the items (71%)_in these three Sourcesconsists of simple single-step story problems that can besolved by application of one arithmetic operation (see Tablela). A large number of them contair, either a Vary or aTransform relation (596 of 935 items, or 64%). Only 15% ofall items required use of two different semantic relations(excluding "other"). Also, 18% of the two-step problems weremerely repetitions of the same relation. Table lb containsthe frequencies with which various pairings occurred.Finally, a very low 7% of the itemd contained as many asthree semantic relations (see Table lc).


2 2

Table 1

Problem Classification by Textbooks:The Frequency with Which Semantic Relations Occur

A; One-step problems

Classification 8th Grade Navy Training College TotalTexts Materials Remediation

CH 49 0 76 125CB 31 3 lc) 53CP 31 1 47 79VY 198 9 123 330TR_ 135 5 126 266Other 34 7 41 82

Total 478 25 432 935

B. Two-Step problemS

Classification 8th Grade Navy Training College TotalTexts Matnials Remediation

CH/CH 2 0 13 15CH/CB 1 0 4 5CH/CP 1 0 1 2

CH/VY 23 1 14 38CH/TR 23 0 16 39CH/Other 1 0 1 2

CB/CB 1 1 0 2

CB/CP 3 0 0 3

CB/V17 9 0 33 42CB/TR 8 0 13 21CB/Other 1 0 0 1CP/CP 0 0 1 1CP/VY 7 1 2 10CP/TR 0 0 9 9

CP/Other 2 0 2 4VY/VY 3 0 4 7VY/TR 6 0 6 12VY/Other 3 0 10 13TR/TR li 0 9 20TR/Other 4 0 0 4

TOTAL 106 3 138 247


Table I continued

C. Problems with more than two steps

8th Grade Navy Training College TotalTexts Materials Remediation

Number of problems 45 7 88 140

D. Number of times a relation occurred ina problem requiring more than two steps **

Classification Sth Grade Navy Training College TotalTexts Materials Remediation

CH 29 1 78 108CH 19 E 16 35CP 10 _7 _37 _54VY 32 17 100 149TR 31 13 42 86ather 32 0 18 51

** not equivalent to the number of nroblems

Schema Knowledge Structuret 24

The only problems that we were unable to categorizeIre items requiring application of formulas. These are

represented in Table 1 as "other". In many instances, itwould be possible to redr.ce these also to the semarticiilformation contained in the formula. licriever, we suspectthat most students apply the foroulas without deriving themeach time as would be necessary if the semantic informationwere to be used.

We have no reason to believe that these instructionalsources are atypical of arithmetic texts ln general. Thefindings are consistent with the classification made at theSixth grade (Marshall, 198S). The evidence is strong that apreponderance of attention is devoted to solving simplestory problems at every level of arithmetic.

Other-Research_About_Story_Probleal

Researchers in the fields of cognitive psychology andmathematics education have developed several approaches tothe classification of story problems. These may be looselygrouped into "structure research", in which the underlyingrelationships are of interest, and "operation research", inwhich the arithmetic operations themselves are the foci.

Structure Research

Semantic-Structuxe- The major thrust of research aboutseman=relations has its origins in the work of Riley,Greene, and Beller (1983). Most recently, this line ofresearch has been extended by Carpenter (1987). Threesemantic relations were defined in these studies: Change,Combine, and Compare. These are similar to but not identicalwith the relations defined here.

Riley et al opted to study a limited subset of thedomain of arithmetic story problems, specifically problemsrequiring ocily a single operation of addition or subtractionfor solution. This dependence upon operation turns out to becritical. As we pointed out earlier, some of the semanticrelations can be present in situations that demand any ofthe four arithmetic operations (i.e., Transform). Byexcluding the operations of multiplication and division, onefails to observe the broad structure of the semanticrelations. Further, one fails to perceive that semanticrelations truly are operation-free.

In the Riley et al approach, the Change and Combinerelations have the same general structure as presentedabove, but we have defined them with greater specificity andhave introduced additional constraints. As Riley et al .

pointed out, most of the problems in arithmetic tnat express


25

CH and CB relations involve only the operations of additionand subtraction, and their structure is well understood. Wehave found, however, that there are also multiplicativeChange problems, although these are relatively rare.Consequently, we hesitate to us e. operational labels orconstraints.

Our conception of Compare differs substantively fromthat presented in Riley et al. By their classification, bothof the following items demonstrate the Compare relation:

Joe has 8 marbles. Tom has 5 marbles. Rowmany marbles does Joe have more than Tom?

Joe has 3 marbles. Tom has 5 marbles morethan Joe. Row many marbles does Tom have?

(28)

(29)

Under our theory, only the first of these two items is aCompare problem. The second is an instance of Transform.

we find that the structure of (29) is more similar tothat expressed below in (30) than to (28).

Joe has 3 marbles. Tom has 5 times as manymarbles as does Joe. How many marbles doesTom have?

(30)

problems (29) and (30) have the same basic structure. Ineach case, we know the number of marbles that Joe possesses,and we know that Tom has soke number of marbles that can beexpressed in terms of Joe's marbles. In both problems, theobjective is to use the given relation between the boys'marbleS to determine the actual number of marbles owned byTom. The particular arithmetic operation required isirrelevant to our understanding of the problem.

In this instance, the apparent similarity between (28)and (29) is an artifact of the limited domain. If we workonly in the domain of addition and subtraction, TranSformproblems appear similar to Compare ones because the words"more than" and "lees than" appear, just as they do inCompare items. However, these words themselves do not definethe Compare relation: it is possible to find them in everysemantic relation.

Com ositional Structure. A very differentconceptualization o Structure has been developed by_Greenoet al. Under this approach, a problem is characterized bythe types of quantities to be found in it (p. 9). Four typesof quantities may exist in a problem: extensive, intensive,difference, and factor. Extensive quantities are simply thenumber of units of some object, such as "5 apples".


26

Intensive quantities are per unit values, such as "5 wordsper sentence". A factor is defined as "a unitless quantitythat relates two other quantities that have the same units"(p. 9). For example, if_"Tom has 3/4 as many marbles asJoe", 3/4 is a factor. Finally, a difference "an additiverelation between two quantities of the same type" (p. 9). Inthe statement "Tom has 3 more marbles than Joe has", thedifference is 3_more.

The importance of these four quantities lies in theways they may be combined in a problem. Two generalcompositions are possible: additive compositions andmultiplicative compositions. Additive compositions involveonly extensive and intensive quantities. Multiplicativecompositions may contain all four types ofquantities. Greeno et al. specify rules by whichcompositions may be formed, and they determine theoperations used to solve the problems by the types oquantities found in each composition.

A central focus of the research is to make a graphicalrepresentation that characterizes the operations. Diagramsare constructed to represent the compositions and thequantities described in the problems. To solve a particularstory problem, it is necessary to identify the type ofcomposition and to map the quantities involved in thecomposition into the appropriate diagram.

Compositional analysis differs significantly fromanalysis based upon semantic relations. First, compositionalanalysis has its basis in arithmetic operations (as inadditive and multiplicative compositions). Second, thecategories of analysis have no meaning when separated fromthe quantities. That is, general descriptions of situationsseem to be irrelevant. In contrast, these generaldescriptions are used to form the categories of Change,Combine, Compare, Vary, and Transform. Types of numbers orarithmetic operations are secondary to the generaldescription.

Operation Research

Again, there are several different approaches that havebeen taken. We describe two that seem to be particularlyinfluential and relevant. These are the efforts to classifyproblems according to their surface features and to classifythem according to particular uses or meanings associatedwith the arithmetic operations.

Surface Structure. There exists a reasonably large bodyof research devoted to mapping the structural variables thatoccur in story problems and to assigning difficulty

SChema Knowledge Strudturda

0e_. 7

-22-

parameters to these_variables. For example, Loftus andSuppes (1972) examined item_characteristics Such ad numberof words, the number of sentences, the number of operations,the type of operation, or the similarity to the last-presented item. Multiple regression techniques were used todetermine which_of these or similar features account for alarge proportion of variance in student responses.

While studies such as this one are undeniablyinteresting in revealing which characteristics of a probleminfluence the difficulty level of the nroblem, theynonetheless have little or no direct btaring upon howstudents learn to solve story problems. The difficulty isthat these analyses_are_based upon external features of theproblems having little to do with haillaiViduals understandthe problems. Semantic relations are internal features whichan individual may relate to knowledge-ircriia-in his or herlong-term memory.

Use_CIasses. Usiskin and Bell (1983) give A persuasiveargument against employing operations to classify Storyproblems. Their thesis is that operations have more than aSingle meaning or use. For example, addition may imply botha-puttIng together and a shift. The first of these isclosely akin to the semiWETE-Felation of Combine. The secondcorresponds to Change. The value of USiskin and Bell'sapproach is that they maintain an emphasis upon operationsby changing the focus from algorithms to applications. In sodoing, their approach and ours become compatible. Theydemonstrate the need to evaluate the different uses to whichthe various operations can be put, and we demonstrate theneed to perceive the whole picture as embodied in thesemantic relations.

While compatible, the two approaches are notsynonymous. Some of the more important differences can beseen in Figure 3. All of the uses defined by Usiskin andBell for the operations of addition, Subtraction,multiplication and.division can be mapped into the fivesemantic relations of Change, Combine, Compare, Vary, andTransform. Most of the uses are evidenced by a singlerelation. In some cases, a single use might be exemplifiedin more than a single relation (e.g., the ratio use ofdivision maps into both Compare and Transform).

In the use classification, each operation is expandedinto several uses. In the semantic relationsclassifications, several uses are combined into a smallernumber of relations. The important distinction between theuse classification and the semantic relations classificationis that the latter cuts across arithmetic operations. Forexample, the Change relation can require addition,

SChema Knowledge Structures -23=2,8

actingacross

IX)

rate(+I rate

factor(x)

rare divisorH^)

sirechangedivisor

1+)

recoveringfactor(in actingacross)

Pt)

hift (+)subtractionhift addition

(-) ironaid:Aral:mica

(+1recovering

ddend

Comparison

takeawayI-)

Figure 3: The occurrence of Usiskin and Bell's use classes ofarithmetic operations within the five semantic relations. Usesfalling in the intersection of two relations can occur in eitherone. Operations associated with each use are given in parentheses.

subtraction; or multiplication (as explained previously),and it can demonstrate the addition use_of shift or additionfrom subtractiono_the_subtraCtion use of Shift Or recoveringAddend, Or the mUltiplication use of size change.

We suspect that Usiskin and Bell's taxonomy will provetd be especially useful in the next phase_of our research,_the instructional system._ In particular, it_seems reasonablethat_their definitions of_use will be valuable inexplicating the procedural portion c,;f a schema that_derivesfrom a particular semantic relation. Thus, we view theschema as broader than either the relation or the use; andit iz capable of incorporating both classifications in ameaningful way._ The following section describes the natureOf a schema and_ita importance as a general memory StructUrefor Semantic relations.

Schema Knowledge Structures -24--

Schema Knowledge: Some Theoretical Considerations

Before describing a theory of how schema knowledge isstored in human memory, we first discuss a general model ofmemory. Like many other researchers in the field, we posittwo types of long-term memory (LTM): declarative andprocedural. There seem to be clear distinctions betweenthese two typed of memory. Declarative memory containsfactual knowledge and knowledge of specific events andexperiences. It is usual to assume that this body ofknowledge is stored in LTM as one or more semantic networks,linked together with various degrees of assocIatinn (see forexample Anderson, 1983). Through knowledge stored indeclarative memory, one can answer questions about who,what, where, and when.

A second type of memory contains skill knowledge ratherthan factual knowledge. This memory is called procedural,because it consists of sets of general procedures or rulesfor performing various skills, Unlike declarative memory,procedural memory is highly generalized and not constrainedby specific instances or experiences. For example, one usesthe same skill of grasping an object with one's fingers in alarge number of different situations. The same proceduresare utilized, adapting to eazh situation as necessary.Consequently, one may grasp a pencil, a rock, an apple, oranother person's hand without requiring different rules forhow to perform each task. A common set of rules applies.

One reason for distinguishing between declarative andprocedural knowledge is that the storage mechanisms andretrieval mechanisms seem to be different. Adding knowledgeto declarative memory is relatively easy, and there are many"memory tricks" available to help individuals learndeclarative facts. For example, individuals can learn a listof unrelated words by encoding them into meaningfulsentences (Bower & Clark, 19691. Simple repetition oftenresults in rote learning of declarative information (Rundus,1971). Encoding such as this, of course, does not insurethat the newly acquired knowledge is linked with other,related knowledge.

One may also acquire highly salient declarativeknowledge directly without repetition or guided mnemonics.For example, a single experience of an earthquake is oftenSufficient to establish quite a bit of declarative knowledgeabout the phenomenon (e.g., noise, shaking).

In contrast, procedural knowledge is difficult toacquire and apparently takes a great deal of practice. AsAnderson (1982) points out, many skills take 100 or morehours to acquire. Many motor skills have this


characteristic, for example, learning to type or learning towrite. A number of cognitive skills have the same feature,such as learning to multiply, learning to solve physicsproblems, or learning to program computers.

Given the many distinctions between these two types ofLTM knowledge, it becomes important to ask how they arerelated. Clearly, information of one type calls uponinformation of the other. By what mechanisms are these twoforms of memories unitedTWe suggest that the union occursto a large extent through the acquisition of a schema.

A schema is a knowledge structure that containsinformation about how we interact with the environment in arecognizable situation. As such, it contains necessaryinformation about how to recognize the situation and whataction(s) we might take under the circumstances. Under thiSdefinition, a schema becomes the organizing memory structurethat governs our functions in everyday experiences, drawingupon components of both declarative and procedural memories.

i.An individual s perceived as an active processor ofinformation, using incoming sensory stimuli and previouslystored knowledge to make sense of the world. A schema isinvoked whenever an individual must formulate a response tohis or her environment. The individuaI's responses indifferent situationd are governed by the schematic knowledgeavailable to the individual. This necessity for a responsed!fferentiates a schema from other hypothetical kndwledgestructures such as plans (Sacerdoti, 1977) or frames(Minsky, 1975) which require no action.

It is reasonably common in artificial intelligence,cognitive science, and cognitive psychological research todefine a schema in terms of at least some of the followingcomponents: fl) a declarative store of factual knowledgerelevant to the schema, (2) a set of conditions that mustexist if the schema structure fits the experience, t3) ameans of setting goals for satisfying schema constraints,and (4) a set of rules that can be implemented once theschema structure is accepted. We argue here that all fourcomponents are necessary.

For any schema, there will be a body of accompanyingfacts that describe the generic case of the schema. A much-used example is the restaurant schema, for which there aredetails about definition and Structure (e.g., A restaurantis a place where one goes to purchase food, one typicallyeats the food at the same location, one sits at a table onchtlirs or benches, and so on). In the specific instance inwhich the schema is used, more details will be added fromthe current situation.


31

A schema will also have a set of conditions that mustbe met for the schema to apply_in any given situation. These-are applied to a description of the situation. When theconditions are not met, the schema carinot not be used toexplain the situation. These conditions may involve theinvocation of other schema structures that are prerequisiteto the current one. To continue the example of therestaurant schema, the preconditions are details suCh as theestablishment must serve food, the food must be for sale,there must be chairs, tables, waiteri, menus, cooks, etc.

These conditions may be used in either top-down orbottom-up processing. Suppose, for example, that you haveentered a building and are standing in a room. You will takein sensory information in order to determine just where youare. If you see several tables at which people are seatedon chairs eating a meal, you will likely call upon therestaurant schema to_help you interpret all of the detailsyou are processing. If you see many rows of chairs, allfacing one direction, you probably Search for another schemafor clarification, such as a lecture hall or a movietheatre. This is bottom-up processing -- one takes indetails and tries to interpret them by means of an existingknowledge structure.

Now consider top-down processing. You desire to go to arestaurant for a meal. You walk into a building. You nowlook for confirmatory evidence that you have found aredtaurantr so you search for details such as tables andchairs, individuals serving food, and so forth. In thiscase, the knowledge structure directs your attention tospecific aspects of the Situation. You may not pay attentionto the fact that there are paintings on the wall or anorchestra in the corner. These are not central confirmatoryconditions for the restaurant schema.

A third feature of schema structure is a mechanism orsetting goals in the problem=solving process. In thiscomponent reside the rules under which goals are generated,ordered, and satisfied. For example, in the restaurantSchema, several associated goals involve deciding wht typeof restaurant is preferred, how to reach the restaurant, orwhen to go to the restaurant. The satisfaction of thesegoals =ay require additional calls to knowledge found inanother schema. For example, if one finds one has no cash,one needs another schema which might be called "how to payfor things without cash". Now information regarding creditcards, checks, or IOU's becomes important as well as thecircumstances in which they are reasonably used.

The final aspect of Schema structure is a set of rulesthat governs an individual's response to the situation that


32-27-

invoked the schema. Thus, once you have recognized that youare in a restaurant (and hence have invoked the restaurantschema), your next action will be directed by conditionsthat al:e part of the schema (e.g., you wait to be seated,you order food from the menu, etc.).

Schematic knowledge drives cognitive processing. Thisfunction gives the schema a different stature than knowledgepreviously defined as procedural or declarative. In essence,the schema sits on top of these other LTM structures. In ourmodel, we see a schema as an overlay that encompasseselements of both procedural and declarative knowledge (seeFigure 4). As such, it operates as the controlling mechanismin information processing. It deterwines which proceduresand which semantic networks are to be accessed.

How is a schema accessed and activated? It must directour recognition processes in a top-down fashion. There aremany examples in the research literature of individuals'ability to recognize degraded stimuli, especially whenprimed. This priming presumably activates a schema skeletonthat then directs searching and pattern matching. Degradedstimuli are perceived as being adequate fits only if theschema is successfully activated.

We must also consider bottom-up activation of a schema.If a number of different features of a situation areobserved, they may work together to activate the schemaskeleton. However, much work in psychology suggests thathumans usually attempt to recognize situations and to workin a top-down fashion (ci. Anderson, 19831. This is closelyconnected to coal-directed behavior. We have expectationsabout what we expect to experience in our daily lives. Eachexpe!Aation takes the form of a schema_-- so, for example weexpect to meet with students in our offices, we expect toopen our mailboxes and receive mail. We do not wait untilindividuals come into a room with us to determine that weare at work and .that these are students with questions.

The point about top-down and bottom-up processing isimportant for understanding learning and instruction. Whileadmitting that we function in primarily a top-down fashion,many psychologists and educators expect learning to be abottom-up process. That is, a schema is acquired by firstsolidifying the declarative and procedural components inLTM. Eventually, these elements become interconnected, and aschema skeleton emerges. A_major theme of the presentresearch is the challenge of that position: we suggest thatschema development may be a top-down process also and thatinstruction ought to take advantage of this aspect ofinformation processing.


3 3

=-28=

Figure 4

The relationship between schema knowledge,declarative memory, and procedural memory

Schema

LongTerm

Memory

PrOdedUtal DeclarativeMemory Memory

Schema Knowledge,StructureS -29-i ,

3 4

Related Schema Research

Two alternative views of schema knowledge have theirorigins in studies of reading and text processing. One hasits origins in Bartlett's (19321 study of comprehension andfocuses on the nature of stories (Stein, 1982). The secondarises from a new theory of cognition called paralleldistributive processing (Rumelhart, McClelland and the PDPResearch Group, 1986).

Many cognitive psychologists credit Bartlett's (19321study of text comprehension as the earliest formulation of aschema. _Bartlett presented his subjects with a story (themost well-known being "The War of the Ghosts") and askedthem to reproduce it. Subjects generally changed anddistorted stories according to their own culturalexperiences and conventions. Bartlett hypothesized that eachsubject had an abstract story representation or schema thathe or she used to interpret and understand the st3FIWF.

Stein 11982) developed a schema-theoretic approachbased upon Bartlett's conception of schema. Her researchemphosizes the elements of stories and the relationshipsisucl. as causal links) that occur. She also examines whichelements individuals recall and how the form of a storyinfluences recall. The work of Stein and her colleaguesdemonstrates the schema nature of stories and the importanceof story structure.

Stein's research centers on the structure of storiesrather than the organization of memory that individuals musthave in order to understand the stories. A primarydifference between her work and ours is that her focus isupon the organization of the story and ours is on theorganization of the memory processes that are necessary forunderstanding the story. Consequently, she defines a schemain terms of story features such as setting, different typesof episodes, and causal relations. In contrast, we present ageneral definition of a schema that specifies the type ofknowledge contained in the structure and the ways in whichthat knowledge can be used by information-processingmechanisms. This conception of schema applies to generalexperiences as well as to stories.

_A different schema-theoretic approach has been taken byRumelhart and his associates (1980, 1986). A centraldistinction between Rumelhart's view and the one presentedin this report is the difference in the conceptualization oflong term memory. We adhere to the declarative/proceduralmodel; the PDP group holds a model of connected units.

Scnema Knowledge Structures -30-

Rumelhart ;1980) developed the following set of schemacharacteristics: (1) to have variables, (2) to have thecapability of being embedded in other achema structureS, (3)to represent multiple levels of knowledge from abstract toconcrete, (4) to represent knowledge rather thandefinitions, (5) to be an active process, and (6) to be arecognition device (p. 40-41). This characterization isimportant in that it specifies what a schema does; ic lacks,however, definition of the structure of the schema.

More recently, RumelhArt and his colleagues havedeveloped the notion of a schema_within paralleldistributive processing theory (Rumelhart et al., 1986;McClelland et al., 1986). Their conception differs greatlyfrom the one presented in this report. The major differenceis that a schema is not considered to be a. stored memorystructure but rather iS a collection of activated "units"that become activated simultaneously. As such, a schema neednot be the same each time it is required: depending uponeach stimulus, some units will be activated and others willnot. This approach carries with it major instructionalimplicationS. As Rumelharv et al. state,

There is no point at Which it must be decided toto create this or that Schema. Learning simplyproceeds by connection strength adjustment. . . .

(Vol. 2, P. 21)

Our own research suggests that this is not the case:our findings indicate that the teaching of specific schemaknowledge leads to efficient learning and problem solving(Marshall, 1987). We provided a group of elementary schoolchildren with instruction designed to create a sc-t of schemaknowledge structures corresponding to the_five semanticrelations described previous. The students learned therelations quickly and could differentiate themaccurately. We hypothesize that schema knowledge_providedthe students wi:h a framework for organizing the domain of .

Story problems. In this case, it was necessary to have fixedstructures that were learned as such by the Ltudents. Wewill test this hypothesis more thoroughly in the laterstages of the present research project.

Schema Knowledge Structures =31-

a 6

Schema Knowledge of Semantic Relations

In this and the following section, we define moreprecisely the form taken by a schema for semantic relations.In this section, the relations are mapped into the fourcomponents necessary for a schema. In the next, we discuss acomputer simulation of how schema knowledge can be used tosolve problems.

The basic schema configurations for the five semanticrelations are presented in Figures 5-9 (see pages xx-yy).Each schema is developed with respect to four components:ne necessary facts stored in long-term memory, theprerequisites that must exist within the problem for theschema to fit the situation, the goals that may need to beset up, and Vie rules for using the schema to carry out theneeded comput!tions. At this point, we make no suggestionabout the order_in which this information may be accessed byan individual. The four components of a schema are notnecessarily sequential or linear in their acquisition orretrieval. The ordering from one to four is merely forconvenience in describing them here.

Declarative Knowledge

The firSt component contains the declarative knowledgepertinent to the relation. Of primary importance is therepresentation of a typical problem. We hypothesize the needto store this information in two forms: first, as given inthe table, there-is a simple story form; second, there is acorresponding_graphical structure that represents the sameinformation. The graphs for each schema are given in Figures6-10. They contain visual ihformation about possible statesof a problem and the number of variables that may oerequired.

The verbal form of the typical problem represents ageneral template against which the current problem can beexamined. It can be used as an analogy: can the currentprobleni be rephrased in such a way that it matches thegeneral case? Similarly, the graphical tree struzturerepresents a second, more specific template. The verbal formallows the individual to approach the problem broadlywithout paying particular attention to the numbers and namesused in the problem. Moving to the tree structure forces theindividual to examine the problem in finer detail, mappingthe elements of the problem to specific slots of the treestructure. In this way, the individual recognizes whichparts of the structure are known, their relationships toeach other, and which are yet to be found through arithmeticcomputation.


3 7-32-

For simple problems, an individual may not need toprobe declarative knowledge deeply. That is, the st-uctureis readily apparent, and the individual recognizes thecomponents of the problem without going through a formalmapping of problem elements to the graphical template. Formore complex problems, the mapping process may elucidate theproblem.

Additional details about a relation are also stored asdeclarative knowledge. These include the number of expectedcomponents of the problem, the general characteristics,associated operational uses, and expected operations.

Typically, more declarative knowledge is possessed thancan bL used in a dingle situation. When the schema isinvoked, some irrelevant features may be activated that donot pertain to the current situation. The activation of theschema may place these elements in working memory. If thesedo not match elements of the situation, they will be droppedfrom working memory.

General_PzerequiSites

A second aspect of Schema knowledge is the set ofconditions that must be met for the schema to beinstantiated. For example, in the Change schema, onecondition is that the indicated change be a permanent,physical alteration. In the Combine schema, one must be ableto identify the classes that logically comprise a largergroup.

The prerequisites serve as a check that the schema is areasonable one to use_under current circumstances. If anindividual's condtruction of the schema does not possesssome of them, the_schema may be invoked and instantiatedinappropriately. If the individual_has constructed a schemathat entails incorrect prerequisites, he or she may fail touse a echema when it is appropriate.

GoalSettillg Mechanisms

2or any problem, the top-level goal iS to solve theproblem. The solution can be attempted_and reached.successfully only when all prerevisites have beenfulfilled. In some situations, not all prerequisites can oeimmediately satisfied. This may occur because the problemis ill posed, or it may harpen because there are_severalStages of_a problem requiring the_solution of subproblems.Whatever the cause, theee unsatisfied conditions must beresolved before the schema can be implemented and beforeactions are carried out.

SChema Knowledge Structures

3 8-33-

When prerequisites remain to be met, subgoals arecreated, and these subgoals must be achieved before theprimary goal can be addressed. For example, in a multi-stepproblem based upon a Change relation, it may be necessary tosolve an embedded Vary or Combine problem before reachingthe Change solution. Subgoals for solving these embeddedrelations are created by the goal-setting component ofschema knowledge.

The goal-setting mechanisms are the means by which thesubgoals are established and monitored. They recognize whichprerequisites are to be satisfied and in which order. Theyalso control acceptance or rejection of goal solutions.Several options are available. On the one hand, ;t may bepossible to satisfy the condition using a variety of defaultmechanisms. On the other, it may be necessary to invokeanother schema or additional aspects of declarativeknowledge to gain pertinent information.

ImplementationAtules

Once the schema has been successfully invoked and theprerequisites satisfied, it can be used to solve problems.The way in which it is used depends upon knowledge ofspecific actions that can be taken. These actions are storedas production rules, and they act upon the variouscomponents bf the situation as defined by declarativeknowledge. Depending upon which comp nents of the situationare unknown, specific rules are carri-u out to determine thevalue of the component. For most story problems, the actionsare applications of arithmetic operations.

Schema KnoWledge Structurea -34-

.39

-;Figure 5

The CHANGE Schema

I. DECLARATIVE PACTS

Typical problem: 'You havesome amount of something,you get_more of it (Orlose some of_it). and nowyou have new amount.

Visual tree-structure:original &Shuntamount of changeresult

Characteristics:physical alterationpermanence __involves time

associated operations:additiOnsubtractionmultiplication

II. PRECONDITIONS

Can problem be rephraced tomatch typical problem?

Cin_trek_structure begenerated?

Can 2 slots be filled?

If_2_14Iots_are empty, canl be constructed frOSother information?

Is change irreversible?

Are_quantitiee_OXpressed inthe same unit of seacure?

Can the tiOA periods beidentified?

III. GOAL-SETTINGMECHANISMS

Create goal list

Set top lvel goal:Solve for a singleunknown, which willbe I of 3 main parts

Determine if problem ismulti step

Set subgoals as needed tofill 2 quantity slOtt

Expect subgoals to point toeither VARY or COMBINE:try these first if needto solve subgoals

IV. IMPLEMENTATION RULES

Identify original quantity,tranaferred quantity, andresuIting_quantity.__One or more will beunknown.)

Associate quantities withAlOta

Identify operational_useassociated with emptyelot(s)

Nip from use to particularoperation

Carry out operations inorder given by goal list

associated uses:shiftsine change


4 0BEST COPY AVAILABLE

I; DECLARATIVE PACTS

Typical proble: "You havetwo groups of objectsthat can be joined into alarger class of objects."

visual_tree structure:partswhole

Characteristics:hierarchical_orderno action takenmaintain identities

associated operations:Additionsubtractionmultiplication

associated uses:putting togethertaking away

Figure 6

T e COMBINE Schema

II; PRECONDITIONS

Can problem be rephrasedmatch typical problem?

Can tree structure begenerated?

Can 2 slots be filled?

If 2 slots ace empty, can1 be constructed fromother information?

Is part whole relationpresent?

DO objectm maintainidentiti_after thecombination?

Can two quantities bexpressed as subclassesof the third?


III; GOAL SETTINGMECHANISMS

to Create goal list

Set top level goal:Stave for a single-unknown._which winbe 1 of 3 main parts


Set_subgoals_as_neoded tofill 2 quantity slots

gxpect subgoals to pointtO VARY: try it firstif have subgoals

IV; IMPLEMENTATION RO

Identify hierarchicalrelationship:Which objects can becoMbined to formsuperordinate catego

Associate quantities wslots

Identify operational uassociated with oMptslOt(w)

Map from us* :o particoperation

Carry out operatiOnd_ilorder given by goal

-36-

Figure 7

The COMPARE Schema

I. DECLARATIVE PACTS II. PRECONDITIONS III. GOAL SETTINGMECHANISMS


typical_pcobles: °Somebodyhas more (or less) ofsomething than-someone

can problem be-rephrased rematch typical problem?

Crate goal list

Set_top

Identify two quantities _

Mat Ata to he compared

else has. Pind out her CAA tree Structure be Solve for the unknown Associate quantities withmuch mote dr 11141° gsnirated? vhich_will be the

sise of the companienslots

visual tree_structure:two_parts

Can 2 slots be filled? or the identity of theIergr or smaller

Identify operational useassociated with espty

their difference If-l_dIeti ArO_Qmpty_can dier(A)I_be_constrectirtftom Detacaine if problem is

characteristics: iLa%icstatic

other information? ulti step Nap_from_use to particularoperation

no action taken Are quantities xpressed Set subcoals as needed tomaintain identities in same unit of Snail:Ca? fIII gbantity slotn Carry out operations-in

Order given by goal listasiediated_OperatiOes: Do_objects maintain Expect sungoals to_point

subtraction identity after the to VARY or COMBINEdivision

associar_ed uses:

comparison? try these first if havesubgoals

comparisonratio

Schema Knowledge Structures 7-

BEST WY AVAILABLE

I. DECLARATIVE PACTS

typical problemt "You havesome number of objects.Each has a fixed value.If the number of objectschanges, the total valuechanges also."

visual tree structure:number of objectsfixed valuenew number of objectstotal value

characteristics:staticno act:.on takenmaintain identities

associated operations:multiplicationdivision

associated uses:acting acrossratefactor/divikorrecovering factor

Figure 8

The VARY Schema

II. PRECONDITIONS

Can_problem_be rephrased tomatch typical problem?

Can tree structure begenerated?

Can 3 of 4 slots be fined?

If 2 slots are septy, canI be constructed fromother information?

Are objects_expressed ina common unit ofmeasure?

Are values expressed ina common unit?

Is_the function_relatim.a unit and a value thesame in all cases?

Is concept-of-"per unit"et:pre:Med? implied?


III; GOAL SETTINGMECHANISMS

Create goal list

Set top IeveI goal:Solve for a singleunknown, which willbe 1 of 4 main parts


Set_subgoaIs as neededto fill 3 quantity slots

No expectations aboutsubgoais.

43


Identify objectsIdentify per_object_vaitteIdentify relation betweenobjects and values

ASsociate quantities withslots

Identify operational useassociated with emptyslot(s)

Map_from_ude to particUlatoperation


=38=

I. DECLaitATIVIC FACTS

typical_prObleAS "No-things_ace_amprassed_inthe same unit ot_measuce.tacos ot time* non 04xpressed as a functionOr Chi otber osse.0

vieue4 tree_atructurestwo thingscelatiomal ?unction

cheracteriaticesstaticso action taken__iswolwes reltionbetwees 2 objects

involves part/whole

associated_cpsratiossadditionsubtractiommultiplicationdIvision

associated halessism_changeratioelse cbasqodiiisorOomparisos

Figure 9

The TRANSFORM Schema

II. PeCCOODITIOMe

Can problem be rephrased tomatch typical problem?

Can_tree_acructnta begessraced2

Ass these 2 distinct thingswith posnible numericalMelnik?

Ace tha thiage_espreesablein the same unit oferasure?

Ie.one_ot the things statedin terms or the Other?

Can one of the things beconsidered to be a!whole so that the otherons can be-swasucedagainst it?

SChema Knowledge Structures

III. GOAL SMUGMINAANISMS

Create gota hint

let top level goals_Solve for a single_unknown. which willbe I 02 3 main pacts

Determine it problem ismulti stip

Set_subgoals es needed _

to gill 2 quantity slots

Mo_eispailtation aboutsecondary teps.

Likely to be ilia fitetstep in multi stepproblem

IV. laftstntliTATIOM ROLLS

Identity the 2 ob3ects tobe related.

Identity the functiOn thatrelates the 2 Ob3idts

Identity the_object_tObe espressed is termsof the other

Identity operational use

Map_trom_opscatiOnAl use tooperation


-39-

Computer Modelling of Problem Solving

In this section,_ we describe computer llodels of thesemantic relationd of Change, Combine, Compare, Vary, andTransform. Each is implemented according to the definitionsand constraints developed above. The sufficiency of ourspecifications is evaluated in terms of the computersimulation's success in determining the correct relation forsolving a set of simple story problems.

The section has the following outline. First, the formin which Story problems are presented to the system isdescribed. Second, the general characteristics cf thecomputer models are given, toget!er with examples of thedifferent semantic relations. Finally, we discuss severalissues that remain unresolved.

Propositional_Encodinq

The creation of a computer system that can parse storyproblems stated in natural language id beyond the scope ofthis project. Consequently, we (like others) rely uponpropositional encoding of the story problems, and ourcomputer programs operate upon these propositions. The basesfor encoding problems into propositions are described heretogether with examples of several story problems.

Our objective is to represent the senantic relationscontained in a story problem and not the individual meaningsof wordS. This is an important point, because it means thatthe computer model can operate with a reasonable butrestricted base of wc:ld knowledge. For our purposes, manysemantic labels will be indistinguishable. For example, thesemantic differences between pieces of furniture such aschair, table, or desk are unimportant. Our system would noteonly that these are all instances of furniture and aredifferent from each other. The ways in which they aredistinct are not (usually) significant factors in solving astory problem. Similarly, the system would know that apples,oranges, lemons, and bananas are all tYPes of fruit. Unlessadditional characteristics are required by the problemstatement (e.g., unless a question such as "how many piecesof yellow fruit are there in the basket"?), they are notrepresented in the knowledge base.

In like manner, several actions that occur within astory setting have a common meaning. For example, there aremany ways to express the notion that an individual possessessomething: has, owns, keeps, holds, and grasps are only afeW. In most instances, the differences between these termsare not critical to the semantic relation expressed in theproblem; they could be interchanged without loss of


45

understanding. Therefore, it is possible to rely upon a fewprimitive verbs to express the acts described in storyproblems. Again, this simplifies the knowledge base used bythe computer models.'

Each story problem to be solved by the computersimulation is encoded as a Set of propositions. Thesepropositions contain all relational and numericalinformation present in the problem (whether relevant ornot). They do not necessarily contain all extraneous detailsof the situation. For example, consider ie problem:

On her way to work, Mary found $5.00 on theground. She picked it up and put it in hercoat pocket. She already had $16.00. Howmuch money does she have now?

(31)

The relevant propositions for this problem need to capturethe information that Mary had some money with her and thatshe acquired some additional money. Several details of theproblem are unimportant with respect to the underlyingsemantic relation. For example, it is not necessary to knowthat Mary was on her way to work -- her destination haslittle to do with the structure of the problem. Similarly,it is unimportant in this problem to know in which pocketshe put the money.

The General Form_of_Pr?bositlonq

Each proposition is an expression composed of fiveelements:

[subject primitive object direction time]

The subject is the central character or actor of theproposition. It may have a.type_attribute, a name attribute,and/or an associated numerical value. For example, thesubject of a proposition might be "Three boys". In thiScase, 122z is the type, and the numerical value is 3. Name isunassigned because the boys' names are not given. TypiEiTly,only the type and/or name is given.

The primitive defines the action of the proposition. Asdescribed above, thit is generally a class of actions suchas possess or transf6-r.

The object functions in a proposition as a directobject. Like the subject, it may have a type attribute, aname attribute, and/or an associated numerical value.Usually the type attribut's and numerical value suffice todescribe the object (e.g., 15 cookies, 4 lemons).

Schema Knowledge Structures 41-

4 6

The primitive may have directionality. For e.cample,objects may be transferred to or from the subject. Theelement direction contains this iNETimation in twoparts. First, the actual direction Ito or from) isspecified. Second, the recipient of the action and directionis given. The recipient is similar to a indirect object intraditional grammar, but it also allows representation ofpassive statements. Thus, we can represent "Mary gave John10 apples" by

[Mary transfer (10 apples) (to: John) ...]

with Mary the subject, transfer the primitive, iI0 apples)the o ject, and (to: John) the direction. In thit case, therecipient John is the indirect object of the transfer. Hadthe probla-n-en stated as "John was given 10 apples byMary", we have a different proposition:

[John transfer (10 apples) (from: Mary) ...]

in which the recipient contains information about the originof the transfer.

It is obvious that one could always force passivestatements into a propositional form cf an active kind (suchthat the subject always performs the action). Such a policydistorts the structure of the problem. By allowing bothpassive and active statements through the directionalityspecification, we preserve as closely as possible the way inwhich information is presented in a problem.

The fifth element of a proposition is time. This alsohas two parts, one which denotes in a general way whetherthe action occurs in the present, past or future, and thesecond which is a label (such as the day of the week). Verbtenses are encoded-IE-Ehe first part; primitives are alwaysexpressed as infinitives.

Ty; 1 of Propositions

There are three types of propositions: state, event,and query. State propositions reflect a constant state ofthe world. Two kinds of state propositions may be made. Thefirst describes attributes belonging to the subject. Theprimitive in this case typically is "possess"- and theproposition indicates that a subject has some object. Thesecond kind of state proposition is used to specify groupmembership and the primitive is "is", catch as "George is aboy".

An event proposition denotes a change in possession orsome other action in which objects or subjects gain or lose


4 7

numerical value. The elements of time and direction areimportant for event propositions because they define whenthe event occurred and to whom.

Finally, a query proposition reflects that some elementor part of an element is unknown and asks for informationabout that unknown. In most cases, the unknown is anumerical value associated with an object or subject.Occasionally, the unknown is a particular object or subject(as in the situation where we want only to discover which oftwo individuals has the most or least of something).

Computer Implementation

The computer models were developed in PRISM, a computersystem implemented in InterLispD for use on Xerox D-machines. Created by Pat Langley, PRISM is a system thatfacilitates construction of production systems within theInterLisp environment (Langley & Neches, 1981; Ohlsson &Langley, 1986). PRISM is especially well-suited for thecurrent project because it distinguishes between workingmemory and long-term memory and allows generation ofalternative architectures for production systems. Thus, itcan be altered as necessary_for any particular programmingtask, and we have made modifications to allow representationof the declarative and procedural knowledge used in oursimulations.

We define three parts of the system: working memory,procedural memory, and declarative memory. Working memorycontains all the information that is active in the .system atany given moment. Procedural memory consists of a set ofproduction rules that either identify the underlyingrelation or take appropriate action once the relation hasbeen specified. Declarative memory is maintained as asemantic network. At this point, the connections or linksbetween elements in the network are primarily thosedescribing inheritance, such as "apple is a fruit" or "boyis a child". We anticipate that other links will be createdas we develop a more complete system.

To solve a problem, the system operates only uponworking memory (WM), which is initially empty. At varioustimes, it uay contain elements from incoming stimuli (e.g.,pieces of the problem) or elements from long-term memorythat have been activated by production rules.

_When a problem is presented, each proposition of theproblem enters working memory. As the system encbunters thepropositions in WM, it checks to see whether certainrelationships are present. In order to solve a problem, thesystem must recognize the semantic relation that underlies

SChema Knowledge Structures -43-

4 8

it. Recognition is achieved by the production rules whichopeate upon the_propositions, and it may entail a search inand activation of portions of declarative knowledge. Forexample, a proposition may contain reference to girls, boys,and children. To determine the relationships that existamong these three categories, the system searchesdeclarative memory and discovers that ctildren is a class ofobjects that'can be decomposed into two subclasses, boys andgirls. This information is added to working memory, and itleads to the satisfaction of a primary constraint of theCombine relation (see Table 3).

The recognition set of production rules correspondroughly to the preconditions established for each schematicrepresentation of a semantic relation. These rules are basedupon constraint matching rather than upon key words found inthe propositions. When the constraints are satisfied, thesystem "recognizes" the embedded relation in the problem andputs that information in working memory. Thus, in theexample above, a statement identifying the Combine relationis added to working memory.

. .Once a relation has been identified and labeled inworking memory, the system activates schema-based rules andattempts to carry out the necessary_computations. To doSo, it interprets information presented in the propositionswith respect to the general framework of the schema that hasbeen invoked. Thus, these rules contain information aboutthe number of quantities that must be already known and howto find values for those that are unknown.

Table 7 contains an example of how the system solvesone problem. At the top of the Table, the problem statemnntis given, followed by a set of propositional encodings. Thefirst three propositions are presented to the system as theoriginal encodings; the last two are created by the systemas it solves the problem. The lower portion of the Tableillustrates the probIem-soIving steps that are required forSolution by the system. The steps are represented in Table 7as cycles. Under the current constraints of PRISM, eachcycle cufminates in the firing of one production rule.

In the first cycle, working memory (WM).contains threepropositions: PI, P2, and P3. Several conditions pertain tothis Situation described by these propositions. First, atransfer of some given amount has takeu place and the resultof that transfer is unknown. The transfer occurs for asubject (in this case, Sally) and the unknown result alsobelongs to Sally. The object being transferred is money,expressed in dollars. Finally, no semantic relation has yetbeen identified, and no schema rules have been called. Onlyone production rule maps into these conditions. The rule

Schema Knowledge Structures. -44-

4 9

takes the action of identifying the relation as a CHANGE andplaces that identification into WM as schema=CR.

In Cycle 2, WM contains the original propositions plusthe schema identification. The .,onditions present in Cycle 1are also present in Cycle 2, and the constraints to bematched are tAe same with the exception of the knownrelation. Again, only one production rule satisfies theconstraints. The resulting action is to execute the transferof P2; that is, a new proposition P4 is created in which thesubject now possesses the objects. P4 is added to WM. Theoriginal proposition indicating transfer is now obsolete andis deleted from WM.

Following the addition of P4 to WM, the system nowrecognizes that one subject possesses two known quantitieS.However, these possessions are recorded at different times.The next cycle checks that one of the possessions occursearlier than the other and that it can be logically inferredthat this possession is unaltered when_the second possessiontakes place. If this is the case (as demonstrated in Cycle3 of Table 7), the time of the first possession is up4-ted.We represent this updating in proposon P5. The originalinformation is now incorrect and is removed from WM, leavingthree propositions: P3, P4, and 125.

In Cycle 4, the system solves for the unknown quantity.The conditions to be met here are that an identified subjectpossesses two different amounts of an object expressed in astandard unit and that the total of these amounts isunknown. The total is computed and inserted into Proposition3, replacing the unknown ??7 with the computed value.Propositions P4 and P5 are removed immediately from WM andare irretrievable once the aggtegation takes place.

Under PRISM architecture, the system continues to rununtil no production rule can be executed. Thus, the finalcycle of any problem-solving endeavor looks like Cycle 5 ofTable 7. No productions are acceptable, no constraints areevaluated, and execution terminates.

The existing rules and semantic network are sufficientfor solving a set of twenty simple problems we have used inthis and in_other research about semantic relations(Marshall, 1987). These problems are presented in Table 8.All five semantic relations are present in this set, withfour exanples of each. The system reaches correctidentification of each relation and uses schema knowledge tosolve the problems.

Schema Knowledge Structures -4

5 0

Table 7

An Example of How the System Solves a Change Problem

PROBLEM STATEMENT:

Silly had $2.53 in her pocket when she went out fora walk. She found $.35 on the sidewalk and put itin her pocket. How much money did she have at theend of her walk?

PROPOSITIONAL ENCODING:

P1

P2

state proposition:subject: Sallyprimitive: possessobject: money (quantitydirection: niltime: 0

event proposition:

2.53) (unit dollar)

subject: Sallyprimitive: transferobject: money (quantitydirection: to Sallytime:

.35) (unit dollar)

P3 state proposition:subject: Sallyprimitive: possess

P4

object: money (quantitydirection: niltime: 2

state proposition:

???) (unit dollar)

subject: Sallyprimit'.ve: possess

135

object: money (quantitydirection: niltime: 1

state proposition:

.35) (unit dcllar)

subject: Sallyprimitive: possessobject: money (quantitydirection: niltime: 1

2.53) (unit dollar)


Table 7 continued

PROBLEM SOLUTION:

CYCLE 1:Elements of WM: P1

P2P3

Constraints satisfied: transfer occurs (P2 in WM)result of transfer unknown (P3 in WM)subject, objectsr and units ofobjects are constant

transfer time earlier than resultschema unknown

Action taken: add schema=CH to WM

Cycle 2:Elements of WM: Sdhema=CH

P1P2P3

Constraints satisfied: schema=CHtransfer occursAP2 in WM)result of transfer unknown (P3 in WM)transfer time earlier than result

Action taken:

CYCLE 3:Elements of WM:

carry out transfercreate P4add P4 to WMdelete P2 from WM

schema=CHPIP3P4

Constraints satisfied: schema=CHpossessions occur at different times

Action taken: update originial possession timecreate PSadd PS to WMdelete P1 from WM

Schema Knowledge Structures -47

12

Table 7 continued

CYCLE 4:Elements of WM: schema=CH

P3P4PS

Constraints satisfied: two possessions occurthe times of possession are the samethe total possessed is unknownschema=CE

Action taken: solve for ??? in Proposition (3)modify P3 in WMdelete P4 from WMdelete PS from WM

CYCLE 5:Elements of WM: schema=CE

P3 (modified]

Constraints satisfied: none

Action taken: no production acceptable


53

Table 8

Examples of Semantic Relations in Arithmetic Story Problems

1. CHANGE RELATION:

For Halloween, Sue put 35 candy bars into a large bowlfor children who came "trick-or-treating" at her house.She decided that this might not be enough, so she putanother 12 candy bars into the bowl. How many candy barswere in the bowl?

Sally had $2.53 in her packet when she went out for awalk. She found $.35 on the sidewalk and put it in herpocket. How much mosey did she have at the end of herwalk?

Peter bought 45 cookies for the school party. On the way. to school, he got hungry and ate 7 of the cookies. He

took the rest of the cookies to school for the party.How many cookies did he contribute to the party?

Before the volleyball game, there were 40 towels for theplayers to use. After the game, the coach could findonly 28 towels. How many towels disappeared during thegame?

2. COMBINE RELATION:

At Evans Elementary School, there are 15 members of theboys' basketball team and 18 members of the girls' team.How many students are on basketball teams?

Jodi makes $12.50 a week on her paper route and $3.45 aweek for doing chores at home. How much money does Jodiearn each week from these two activities?

At the track meet, there are 83 competitors; 31 of themare boys. How many are girls?

Jerry made frtiit salad with apples and bananas. He made6 1/2 cups of salad. If he put 3 3/4 cups of bananas inthe fruit salad, how many cups of apples did he use?

Schema Knowledge Structures5 4 -49-

Table 8 continued

3. COMPARE RELATION:

When Chef Jack cooks a roast beef, he bakes it in theoven for 3 hours. When he prepares roast chicken, heonly cooks it for 1 1/4 hours. How much longer does aroast beef cook than a roast chicken?

The best gymnast at Central High is Mary. In the lastgymnastics meet, she scored 8.1 on the balance beamexercise and 9.4 on the vault. How much better did shedo on the vault than on the balance beam?

Mount Ranier, in the state of Washington, is 14,408 feethigh. Mount Washington, in New Hampshire, is 6288 feethigh. How much higher is Mount Ranier than MountWashington?

Jeff earns $5.50 per hour at his job, but George onlymakes $4.25 per hour at his job. How much less per hourdoes George make?

4. VARY RELATION:

Kevin plays on the school baseball team. Every time anyplayer on his team hits a home run, the coach gives theplayer 3 baseball cards. Kevin hit 7 home runs thisyear. How many baseball cards did the coach give Kevin?

One pound of potatoes cost $.35. What would five poundscost?

Mark's grandfather is 85 years old today. Mark's motherknows that she can't put 85 candles on his birthday cake(because they won't fit). She decides to use 1 candlefor every 5 years of Grandfatt,er's age. How many candlesshould she put on the cake?

Sheila likes to make.3 pitchers of lemonade at once. Todo this she uses 24 cups_of_ water._ How much water wouldShe heed to make only 1 pitcber of lemonade?


55

Table 8 continued

5. TRANSFORM RELATION:

In a football-kicking contest, Joe kicked the ball 13yards farther than Sam kicked it. Joe kicked the ball 30yards. How far did Sam kick the ball?

Albert spent $3.75 at the school fair. Mike spent 4times as much as Albert at the fair. How much did Mikespend?

Alice's mother is three times as old as Alice. If hermother is 45 years old, how old is Alice?

Cindy has $4.67. Her friend Bill has $.35 more thanCindy. How much money does Bill have?

Schema Knowledge Structures r)6 -51-

Unresolved Issues

We have not yet implemented the goal mechanismsrequired for full schema representation. The sets of rulesdescribed above operate successfully on a small number ofmulti-step problems we have presented to the system thusfar, but they do so by determining which schema can beinvoked with current infbrmation rather than by identifyingthe top-level schema of a problem. For example, in a multi-step problem such as the one diagrammed in Figure 8, thecurrent system_would not recognize that the problem wasessentially a Change relation with other relations embeddedin it. Rather, it would first solve a vary problem then lookto see if other problems needed to be solved. It would findthat it could solve the Combine problem and would do so,Still without identifying the need to solve the Changeproblem. Finally, after solving both the Vary and Combinesubproblems, the system would address the Change relationand would carry out the remaining operation and reach afinal solution. Thus, the system can solve multi-stepproblems but it does so without using goals. If informationwere presented in the problem that could be used toformulate a relation that was actually unnecessary forultimate solution, the system would be misled and wouldengage in solving an irrelevant problem.

Most of our attention to date has been on therepresentations necessary to identify and invoke a singleSchema. We are now worPing on the goal-setting mechanisms.We are also engaged in loosening the constraints thatidentify different relations. This will allow weakidentification of a relation and its associated schema andwill permit us to begin to model ways in which each schemamay be inappropriately instantiated.

Summary

The computer implementation provides support for theschema structures developed here. Using the specificationsof the relations and the components of schema knowledge, thecomputer programs successfully identify and solve a varietyof story problems. The next phase of our research will be todevelop an instructional environment in which the elementsof relational and schematic knowledge can be clearlydemonstrated, manipulated, and isolated.

Schema Knowledge StructureS -52-

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