Electronics and Communications in Japan, Part 3, Vol. 75, No. 11, 1992 Translated from Denshi loho Tsushin Gakkai Ronbunshi, Vol. 75-A, No. 2, February 1992, pp. 296-304

Development of Support Facilities for Arithmetical Word Problem Exercises in Intelligent Tutoring System

Tsukasa Hirashima, Member, Takahiro Kohno, Nonmember, and Akihiro Kashihara and Jun’ichi Toyoda, Members

The Institute of Scientific and Industrial Research, Osaka University, Ibaraki, Japan 567

SUMMARY

In the problem solving of the arithmetical word problems, the process until the solution method is identi- fied as the most important.

It is necessary to design and develop the basic functions for exercise problem support in an intelligent tutoring system (ITS), such as problem solving, problem generation, and problem explanation, aiming at the sup- port for the identification of the solution method. In this study, those three basic functions are realized based on the problem-solving method MIPS (model of indexing in problem solving).

This paper describes the problem-solving module, the problem-generation module and the problemexplana- tion module. The problem-solving module not only solves the arithmetical word problem but also generates the cognitive description of the identification process for the solution method. The problem-generation module can generate the problem which specifies what students should do in the identification process. The problem explanation module can indicate the equivalence and difference between any two problems concerning the identification process.

Key words: Intelligent tutoring system; exercise problem; problem solution; problem generation; problem explanation.

1. Introduction

To improve the problem-solving ability of the student, it is important for the student to master the use of solution method, in addition to acquiring the solution method. The student attempts to solve many exercise problems to master the solution method. In this sense, the sophistication of the exercise problem support system is one of the most important topics in the study of intelli- gent tutoring system (ITS) [ l , 21.

The basic functions in the exercise problem support are the problem-solving function, the problem-generation function, and the problem-explanation function. The problem-solving function should not only derive the solution but also generate the cognitive description for the problem-solving process, which can help the genera- tionldecision of the tutoring action. The problem-genera- tion function is indispensable in setting the problems adaptively to the progress and goal of the student. The problem-explanation function explains how the individual problem can be solved. It is the basic function in the support of the problem solving.

Intensive studies have been made to improve those functions, and various results have been reported [l-51. Most of those studies deal with the well-formulated problems such as the calculation problem.

21 ISSN1042-0967/92/0011-0021 @ 1993 Scripta Technica, Inc.

A sufficient study has not been made concerning the area of problems such as the arithmetical word prob- lem, where the problem-formulation process is important [6-81.

In the arithmetical word problem, the process to formulate the problem and to identify the solution method is most important. We call this the problem-understand- ing process. Consequently, the function to support the exercise problem must be designed and developed, em- phasizing the problem-understanding process. The prob- lem-solving function must generate the description con- cerning the problzm-solving process for the individual problem.. .

It is required for the problem-generation function to cope with the specification concerning the problem- understanding process, such as "hi order to identify the solution method, a problem must be generated for which such a manipulation is needed." It is required in the problemexplanation function that the generated problem explanation supports the problem understanding.

This paper aims at the realization of the basic functions for the exercise problem support for the arith- metical word problem. Three modules, i.e., the problem- solving module, the problem-generation module, and the problemexplanation module, which are designed and developed based on the problem-solving model MIPS (model of indexing in problem solving) [9-111 already proposed by the authors, are described [6]. MIPS is a conceptual model which reprewnts the understanding process as the struchlrization process of the problem and the search process of solution method with the problem structure as the index.

The problem-solving module can not only derive the solution but also generate the description conceming the problem structurization needed in the problem under- standing. The description is called problem module. The problem-generation module can handle the problem according to the specification of the problem structuriza- tion, by the almost inverse process of the problem solv- ing executed by the problem-solving module. The prob- lem model can be derived for the generated problem, as in the case of problem solving.

In the exercise of the arithmetical word problem, there are many students who can solve the basic prob- lem, but not application problems, to which the same solution method can be applied.

It seems that an explanation of the equivalence among problems promotes problem understanding by such students. In the problemexplanation module, an explanation can be generated for the correspondence of problem structure between the application problem that the student cannot solve and the basic problem that the student can solve; the explanation can highlight the dif- ference between the problems and also provide a struc- tural model that resolves the difference.

In this paper, section 2 outlines MIPS and describes the problem-solving module based on MIPS. Section 3 describes the problem-generation module. Section 4 discusses the problem explanation based on the problem model. Finally, section 5 presents the conclu- sions and the problems left for future study.

2. Problem-Solving Module

This section outlines MIPS and describes the problem-solving module.

2.1. Problem-solving model MIPS

2.1.1. Model representation of problem-solving proces~ [9-111.

In MIPS, the index which represents the problem structure needed in applying the method, is called the index of solution method (ISM). The problem structure generated from the problem is called the problem-under- standing network (PUN). Both of those are represented by a network.

The problem-understanding process for the solution method is composed as follows. The PUN is generated first. It consists only of the values and relations appear- ing in the problem (called the raw PUN). Then the PUN is generated containing the solution index by applying the problem-structurimtion operations.

The problem-strwturization operation is composed of the following three operations:

(1) abstraction of concept;

(2) addition of basic relation by value addition; and

(3) addition of basic operation by value genera- tion.

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There are cranes and tortoises, 50 in total. A crane has 2 legs, and a tortoise has 4 legs. The total number of legs of cranes and tortoises is 140. How many cranes are there? How many tortoises are there?

(a) A zeroth-order problem

There exist cranes and tortoises, 50 in total. The total number of legs of cranes and tortoises is 140. How many cranes are there? How many tortoises are there?

(b) A first-order problem

There are 40 students in the class. The average score of the test of the students in the class is 69. The average score of the test of boys is 65. The average score of the test of girls is 75. How many boys are there? How many girls are there?

Fig. 1. Examples of problems.

The basic relation is represented by a triplet of the concept corresponding to the object, the concept corre- sponding to the attribute of the object, and the attribute value. It is the basic unit in the network.

Let the crane be the object, the number of legs be the attribute, and the attribute value be 2. Then the basic relation represents that "the crane has two legs." The addition of the basic relation by value addition is meant the operation, where the howledge concerning the basic relation with the attribute value already set as the instance (called instance knowledge) is added directly to the PUN.

The addition of basic relation by value generation is the operation where the knowledge concerning the operational relations among basic relations (called opera- tional knowledge) is used to generate the attribute value of the basic relation added. For example, "the total score of the students in the class can be determined as a prod- uct of the number of students in the class and the average score of the student in the class" is an operational knowl- edge.

2.1.2. Order of problem

In MIPS, the PUN, for which only the abstraction of the concept is required in the identification of the

solution method, is called the zeroth-order PUN. The problem, with the zeroth-order PUN as the raw PUN, is called the zeroth-order problem.

Similarly, the PUN, for which the addition of the basic relation by value addition is required, is called the first-order PUN and the problem, with the first-order problem as the raw PUN, is called the first-order prob- lem. If the addition of the basic relation by value genera- tion is needed further, it is called the second-order PUN and the second-order problem. The first- and the second- order problems and PUNs can be transformed into the zeroth-order problems and PUNs, respectively, by in- cluding the additions of the necessary basic relations.

Figure 1 shows examples for each order of the crane-tortoise problem. Figure 2 shows the most funda- mental ISM for the solution method of crane-tortoise problem. In the problem shown in Fig. l(b), the basic relation concerning the number of legs of the crane and the number of legs of the tortoises must be added to generate PUN corresponding to the ISM of Fig. 2. In the problem shown in Fig. l(c), the basic relation Concerning the total score of the students in the class, which corre- sponds to the calculation of the total number of legs of the cranes and tortoises is missing. A basic relation must be added, concerning the calculation of the total score of the students in the class from the number of students in the class and the average score of the students in the class.

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ISM(atr(set([crane, tortoise]), [crane, tortoise],[Vl, known]), atr(set([crane, tortoise]), leg, [V2, known]), atr(set(crane), crane, [V3, answer]), atr(set(tort0ise) , tortoise, [V4 answer]), atr(crane, leg, 2), atr(tortoise, leg, 4), union(set( [crane, tortoise]), [set(crane), set(tortoise)]) ).

Fig. 2. An example of index of solution method (ISM).

2.1.3. Hierarchical structure of ISM

In MIPS, the application condition for a solution method is represented by a hierarchical structure of the ISM with different degrees of abstractions [9-111. To which of the ISM each problem belongs, i.e., which is the ISM that can achieve the matching with the least number of concept abstractions, represents the difference of the concepts composing the problems. This property can be used as a means to classify the problems.

The ISM of Fig. 2, for example, contains only the problems concerning the number of legs of the cranes and tortoises. However, the ISM, where the abstractions are made so that crane and tortoise are animal and 2 and 4 are integers, contains also the problems concerning the number of legs of the peasant, dog, octopus, and cuttle- fish. The ISM containing at the same time two problems can be represented by the abstraction level where those problems are equivalent. Thus, using the ISM, the corre- spondences among problems can be organized.

2.1.4. Generic ISM

In MIPS, the ISM represented only by the numeri- cal relations, which must be contained in the problem in order that the problem can be applied, is called the ge- neric ISM. The solution method for the crane-tortoise problem, for example, can be applied only to the prob- lem with the numerical relation shown in Fig. 3.' In other words, the generic ISM for the crane-tortoise

""Known," "unknown," and "answer" in Fig. 3 indicate, respectively, the state of numerical value that it must be known, it can be unknown, and it is the answer. The numerical relation is represented by the following equation with X and Y as unknowns:

a X + b Y = m . and X i Y = n

IV5,knownI X[V3,answer] = [V7,unknown]

[VG,known] X[V4,answer] = [VB,unknown]

[V7,unknown] t [V8,unknown] = [V2,known] [V3,answer] + [V4,answer] = [V l ,known]

Fig. 3. The numerical relation in the most general ISM of crane-tortoise method.

problem is a representation of the foregoing numerical relation.-

The generic ISM is a representation which is inde- pendent of the concept and has a higher level of abstrac- tion than other ISMs. In MIPS, the generic ISM is placed at the top level of the hierarchical structure of the ISMs. Problems which do not have an adequate abstract concept in common correspond through the generic ISM.

It should also be noted that the PUN, which explic- itly contains the numerical values needed to satisfy the generic ISM, is the zeroth-order PUN. The generic ISM

(footnote continued): where a, b, m, and n are known numbers. The corre- spondence to Fig. 3 is as follows: X : V3 , Y : V4, a : V5, b : V 6 , m : V 2 , n : V 1 , a X : V 7 , bY:VS) . When the algebraic operations such as substitution and shift of term are deleted from the operations used in solving this equation, the left arithmetic operation [(m-bn)l(a-b) - X, n - X -. YJ corresponds to the execution part of the solu- tion method for the crane-tortoise problem.

T i p 3 is not to present the detailed description of the process handled by the problem-solving module. To simplify the description in the following part in this paper, this representation is used as the representation for the generic ISM.

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is utilized in generating the zeroth-order PUN network in the problem-solving module.

2.2. Framework of problem-solving module

The problem-solving process in the problem-solv- ing module is composed of the following three steps, as shown in Fig. 4:

(1) generation of the raw PUN;

(2) generation of the zeroth-order PUN; and

connect the basic relations are employed. In the genera- tion of raw PUN, a solution method is assumed as a candidate and the PUN corresponding to the zeroth-order PUN of that solution M o d is generated as a trial.

In this stage, the generic ISM is used to decide whether or not the PUN is the nroth-order PUN. If a basic relation must further be added, the instance lcnowl- edge and the operational knowledge are used. The solu- tion method is identified by matching the zeroth-order PUN generated from the problem and the ISM prepared by the system, using the abstraction of the concept.

(3) identification of ISM. 2.3. Generation of raw PUN

The generation of the raw PUN is the stage of transformation, from the problem described by the M~U-

ral language into the description of PUN that can be handled by the module. The template to transform a sentence into a basic relation and the connection rules to

This module handles only the word problems where a sentence corresponds to a basic relation. Consequently, the most typical crane-tortoise problem shown in Fig. l(b) must be described by four input sentences.

Hierarchical structure of ISMS ,**’

Abstraction operators

Most general ISM Primitive relations . Operational relations

Connective rules Template

PUN: Problem Understanding Network (

Fig. 4. Processing flow of problem solving.

(1) atr(set([crane, tortoise]), [crane, tortoise], 50) (2) atr(set([crane, tortoise]), leg, 140) (3) atr(set(crane, crane), [Ansl, answer]) (4) atr(set(leg), leg, [Ans2, answer]) (5 ) union( set( [crane, tortoise]), [ set(crane) , set (tortoise)])

Fig. 5. A raw PUN (problem understanding network).

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The generation of the raw PUN is divided into the following two stages:

(1) generation of basic relation; and

(2) connection of basic relations.

In the generation of basic relation, a basic relation is generated from a word problem, using templates. From the word problem in Fig. l(b), the four basic relations (1) to (4) in Fig. 5 are generated.

When all sentences are transformed into basic relations, the next step is to connect the basic relations. In the generation of a basic relation from an input sen- tence, the connections between basic relations are not clarified. To generate the raw PUN, all basic relations generated from a word problem must be connected into a network.

The problem-solving module uses the following three connective rules to connect the basic relations.

Connective rule 1. In two basic relations, when the concept corresponding to the object on the one is the same as the concept corresponding to the attribute on the other, it is decided that those two basic relations are connected through the common concept.

Connective rule 2. In two basic relations, when the concepts corresponding to the objects are the same, it is decided that the two basic relations are connected through the common concept.

Connective rule 3. When there exists set opera- tional relation between the concepts representing sets, that set operational relation can be added.

When all basic relations are combined into a net- work, it is decided that the raw PUN is generated. The basic relations (1) and (2) in Fig. 5 , for example, can be considered as connected by the connective rule 2. By adding further the set operational relation of Fig. 5(5), three basic relations (l), (3), and (4) can be connected. Consequently, the raw PUN composed of five basic relations are generated as shown in Fig. 5.

2.4. Generation of zeroth-order PUN

The generation process for the zeroth-order PUN can be divided into the following three steps:

(1) selection of solution method candidates;

(2) matching of PUN and generic ISM; and

(3) addition of basic relation to PUN.

In the module, the solution method candidates are selected from the prepared solution methods exhaustively (at present, only the solution methods with larger grain sizm are prepared such as the solution method for the crane-tortoise problem) [ 13. To improve the efficiency of this step, there may be methods to restrict the object candidate, based on the kind of instances, the number of basic relations in the network,or the context of the teach- ing action. However, they are left for future study.

The matching process behheen the PUN and the generic ISM is to determine the basic relations still miss- ing to apply the solution method and the way to add the missing basic relations. The process is executed by ex- amining whether or not the numerical relations represent- ed by the generic index can be generated from PUN.

The basic relations in PUN are assigned to the variables in the numerical relations in the generic ISM to generate the operational relations among the basic rela- tions. Figure 6(a) is an example of the case where the basic relations composing the PUN of Fig. 5 are assigned to the numerical relations (Fig. 3) of the generic ISM of the crane-tortoise problem.

Next, it is determined whether or not the generated operational relations are adequate by matching them to the prepared operational relations in the module. Basic relations are substituted into the remaining variables in the generated operational relation, according to the pre- pared operational relations matching to the generated operational relation. For the first operational relation in Fig. 6(a), for example, the prepared operational relation "number of legs of an animal A" X "number of animals A" = "total number of legs of animals A," is assigned. Then basic relation "number of legs of a crane" is substi- tuted into V5 and "total number of cranes" is substituted into V7. All variables in the operational relations gener- ated from the raw PUN are replaced by basic relations in this way. when it is impossible to replace all variables by basic relations, the reasoning is iterated for the pro- cess using the operational relations and the assignment of the basic relations is attempted again.

Next, the replaced basic relations are examined as to whether or not the state of numerical values are matched to the specifications in the generic ISM. For

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[VS, known] x [number of cranes, answer] = [V7, unknown] [V6, known] X [number of tortoises, answer] = [V8, unknown] [V7, unknown] + [V8, unknown] = [total number of legs of cranes, known] [number of cranes, answer] + [number of tortoises, answer] = [total number of cranes and tortoises]

[number of legs of a crane, incornletel X [number of cranes, answer] = [total number of legs of cranes, unknown] [number of legs of a tortoise, incornletel X [number of tortoises, answer] = [total number of legs of tortoises, unknown] [total number of legs of cranes, unknown] + [total number of legs of tortoises, unknown] = [total number of legs, known] [number of cranes, answer] + [number of tortoises, answer] = [total number of cranes and tortoises, known]

(a) An assignment for the most general ISM

(b) Check of the assignment by using domain knowledge

Fig. 6. An example of the generation of a zeroth-order PUN.

"total number of cranes" substituted into V7, for exam- ple, the value does not exist in the PUN but is matched since the value specification in the ISM is "unknown. "

"Number of legs of a crane" substituted into V5, on the other hand, is also a value that does not exist in PUN but is unmatched since the value specification is "known." It is thus decided that a value necessary to achieve the match to the generic ISM is missing in this PUN.

When it is seen which value of the basic relation should be made "known" (the state of the value is called "incomplete"), an attempt is made to convert that value into "known" to dissolve the "incomplete" state. The first step is to try to dissolve the "incomplete" state by direct application of the instance knowledge. If this trial fails, the operational knowledge which is the same as prepared operational relations is used to check whether or not the necessary value can be generated from the value indicat- ed in PUN or the value obtained directly from the basic knowledge.

When all the "incomplete" states are dissolved, the matching between PUN and the generic ISM is achieved. By adding the basic relations Using the instant knowl- edges and the operational relations used in the foregoing process, the zeroth-order PUN can be generated. In the case of Fig. 6(b), the number of legs of a crane as well as that of a tortoise are "incomplete," which can be converted into "known" by applying the instance knowl- edges. Consequently, this problem is the first-order problem. By adding the basic relations concerning the number of legs of a crane and the number of legs of a tortoise to the raw PUN, the zeroth-order PUN can be generated.

2.5. Identification of ISM

The identification of the ISM is the process to match the zeroth-order PUN and an ISM in the hierarchi- cal structure of the solution method identified before step. It is determined whether or not the ISM can be matched to the zeroth-order PUN, using operation of the abstraction of concept, from the lower order of abstrac- tions in the hierarchical structure. The identified ISM is closest to the zeroth-order PUN among the ISMs. By identifying the ISM, the correspondence to other prob- lems that can be solved by the same method can be indicated through the hierarchical structure of the ISMs.

The hierarchical structure of the ISMs prepared at present is as follows. Abstraction is applied successively to the ISM of Fig. 2 so that the crane and tortoise are abstracted into animals. Another is the abstraction of two and four legs into integer number of legs. Those are simple structures without a branch. This point should be investigated further since it is one of the important pro- cesses in the problem generation to specify the generated problems. The relation between the problem generation and the hierarchical structure of the ISM is discussed further in section 3.2.

2.6. Problem model

By executing the problem-solving process by the problem-solving module, the problem model, which is composed of the solution method, the operations of problem structurization to be used in the identification of the solution method, and the ISM closest to the problem amng those contained in the system, are generated. This description specifies the problem from the viewpoint of

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the identification of the solution method. Section 3 dis- cusses the problem generation with the requirement specification, which is described at the same representa- tion level as the problem model (called problem specifi- cation). Section 4 discusses the problem explanation based on the problem model.

3. Problem-Generation Module

The problem generation module generates the problem following almost the reverse. process of the problem-solving prooess discussed in Fig. 4. Then the relation between the problem generation and the hierar- chical structure of the solution index is discussed.

3.1. Problem generation

When the problem specification contains the same information as the problem model, the problem can be generated by following almost the reverse process of the problem solving. When the problem specification does not contain the same information as the problem model, the problem is generated by complementing the missing information by the problem-generation module to con- struct the problem model.

In the problem-generation module at present, the problem can be generated when at least the ISM and the type of problem structurization operation (which is equiv- alent to the specification of the order of the problem) are given as the problem specification.

The problem generation is divided into the follow- ing three steps:

(1) generation of zeroth-order PUN;

(2) generation of raw PUN; and

(3) transformation of raw PUN into the natural language.

The transformation of the raw PUN to the natural language is executed using template at present only for the simple cases. In the following, only two other pro- cesses are discussed.

In the generation of the zeroth-order PUN, the concept of the ISM is replaced by the instance. Then, if necessary, a value is substituted into the variable in the

network. The substitution is executed based on the in- stance knowledge and the numerical relation in the gener- ic ISM. By the two foregoing procedures, the zeroth- order PUN is generated.

Then, the raw PUN is generated. In the generation of the zeroth-order problem, the zeroth-order PUN is used directly as the raw PUN.

In the generation of the first-order problem, the additional basic relations are sought using the instance knowledge and first-order PUN is generated by deleting that basic relation. The result is used as the raw PUN.

In the generation of the second-order problem, the second-order PUN must be generated first. By the addi- tion operation of the basic relation using the operational knowledge, new basic relation is added. If there exists a basic relation, which is used in the forementioned opra- tion and can be added even if deleted once, due to the addition of the new basic relations, it is deleted. In the zeroth-order PUN for the problem in Fig. l(c), the total score of the students in the class and the number of students in the class are known. By using the two basic relations, a basic relation concerning the average score of the students in the class can be added. Then, even if either of the number of students in the class and the total score of the students in the class is deleted, it c8n be added again. By deleting such a basic relation, the second-order PUN can be generated. The second-order problem in Fig. l(c) is the problem where the basic relation concerning the total score of the students in the class is deleted.

3.2. Problem generation and hierarchical structure of ISM

For a problem (called problem a), the problem generation module can generate another problem (called problem b) with the same correspondence to the basic problem as a generation of a problem belonging to the ISM (called ISM-1) in the problem model for problem a. Then problems a and b are in the same correspondence to the basic problem through the ISM-1.

The generation of the problem belonging to the ISM existing at an upper level of the ISM-1 can only be considered as the correspondence to the basic problem for the ISM with a higher level of abstraction than the problem belonging to the ISM-I. Providing the corre- spondence, the problem generated to the basic problem is more difficult than problem a.

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In other words, the difficulty in the identification of the solution method for the generated problem can to some extent be coatrolled based on the hierarchical struc- ture of the ISM. In this mse, the hierarchical structure of the ISM is very important in constructing the strategy for posing the problem. Another method to control the difficulty of the problem is to adjust the order of the problem. No sufficient study has been made concerning the problem strategy and the hierarchical structure of the ISM, but it is one of the most important issues in con- structing the exercise problem support in ITS.

4. Problem Explanation Module

The problem that can be solved by the same meth- od can be transformed in the zeroth-order problem con- cerning that solution method. Furthermore, zeroth-order problems caa be placed in correspondeDce through the common upper-level ISM in the hierarchical structure of the ISM. In the MIPS framework, the correspondence can clearly be defined between the problems that can be solved by the same method. The information needed in such a correspondence can be derived from the problem model and the hierarchical structure of the ISM. In the following part of this section, the most basic zeroth-order problem for the solution method is defined as the basic problem, as in Fig. l(a). The problem explanation is discussed, which indicates the correspondence between the basic problem and the object problem of explanation (called object problem).

4.1. Explanation for zeroth- and first-order problems

When the problem explanation is to be generated with a zeroth-order problem as the object problem, the first step is to find the ISM to which both the object problem and the basic problem belongs.

Through such ISM, the correspondence is clarified between the part of the problem structure of the object problem and the problem structure of the basic problem.

Based on such a revealed correspondence, the explanation module explains the problem as follows.

(1) Presentation of the basic problem.

(3) Indication of correspondence between basic relations.

(4) Indication of correspondence between basic relations, using the upper-level basic relations (this step is omitted if there does not exist an adequate upper-level concept).

(5) Indication that the two problems contain the numerical relations of the generic ISM.

To transform the first-order problem into the zeroth-order problem, basic relations must be added using the instance knowledge. In general, however, most cases are too obvious to explain the addition of the basic relation. Consequently, in the explanation module, the same problem explanation is presented as in the case of the zero-th order problem, even if a first-order problem is the object problem. It may happen, on the other hand, that the student does not have the necessary instance knowledge; and a further investigation may be needed for the explanation for the first-order problem. Figure 7(b) shows an output example of the problem explanation. The first-order problem shown in Fig. 7(a) deals with the octopus and the cuttlefish.

4.2. Problem explanation for second-order pro-

When a second-order problem is the object prob- lem, the part which does not correspond to the basic problem is indicated and the transformation to the corre- sponding problem is executed. The basic relations which exist in the basic problem but not in the object problem, as well as the operational relations used in the addition of the basic relations, can be described in the problem model. The transformed problem can be generated by adding the basic relations to the raw PUN of the object problem and to transform the PUN into natural language.

The problem explanation module can present the following problem explanation for the second-order problem.

(1) Presentation of the basic problem.

(2) Indication of basic relation which exists in the basic problem but not in the object problem.

(2) iUStances.

Indication of the correspondence between (3) Indication of basic relation that must be added.

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There are octopus and cuttlefishes, 20 in total. The total number of legs of octopus and squid is 180. How many octopus are there? How many cuttlefishes are there?

(a) A problem of octopus and cuttlefish

. . . This problem can be solved by the same method as for the following problem There are cranes and tortoises, 10 in total. The total number of legs of cranes and tortoises is 30. A crane has 2 legs, and a tortoise has 4 legs. How many cranes are there? How many tortoises are there?

(1) "crane" and "octopus" (2) "tortoise" and "cuttlefish" (3) "a crane has 2 legs" and "an octopus has 8 legs" (4) "a tortoise has 4 legs" and "a cuttlefish has 10 legs" (5) "cranes and tortoises, 10 in total" and "octopus and cuttlefish, 20 in total" (6) "total number of legs of cranes and tortoises is 30" and "total number of legs of octopus and cuttlefish is 180" (7) "calculate number of cranes" and "calculate number of octopus" (8) "calculate number of tortoises" and "calculate number of cuttlefishes"

(1) Both are "animals. " (2) Both are "animals." (3) Both are "integer number of a part of the body." (4) Both are "integer number of a part of the body." (5) Both are "there exist integer number of different kinds of animals. " (6) Both are "integer total number of a part of the body of animals." (7) Both are "calculate number of animals. " (8) Both are "calculate number of animals."

... Both problems have the same numerical relations. ... The relation between "number of legs" of an octopus" and "number of octopus" and "total number of legs of

octopuses" corresponds to "number of legs of a crane," "number of cranes," and "total number of legs of cranes," respectively.

(number of legs of an octopus) X (number of octopuses) = (total number of legs of octopuses) (number of legs of a crane) X (number of cranes) = (total number of legs of cranes)

. . . The following correspondence exists between those problems.

... The above correspondence can be explained as follows.

The relation among (number of legs of a cuttlefish), (number of cuttlefishes), and (total number of legs of cuttlefishes) correspond to the relations among (number of legs of a tortoise), (number of tortoises), and (total number of legs of tortoises).

. . .

. . . (b) An example of problem explanation

Fig. 7. Problem explanation for zeroth-order or first-order problems.

(4) Indication of the necessary manipulations in Figure 8 shows an output example of the problem explanation for the problem in Fig. l(c). Figure l(c) is a second-order problem. The problem generated by the problem transformation is the nxoth-order problem with the total score of the student being specified, which is not presented in the initial second-order problem.

adding the basic relation.

(5) Presentation of the problem generated by adding the basic relation.

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... The correspondence is missing in the following points. ... This problem does not have the entity corresponding to "total number of cranes and tortoises" in the basic problem. The basic problem does not have the variable corresponding to "average score of students in the class" in this

If "total score of students in the class" is known, it corresponds to the "total number of legs of cranes and tortoises. " problem.

- generator - problem Problem

1) solving module

problem Problem

module * generation

... "Total score of students in the class" can be determined as follows. ... (average score of students in the class) X (number of students in the class) = (total score of students in the class)

69 X 40 = 2760

. . . Using this value, the problem can be rewritten as follows. . . . Number of students in the class is 40. The total score of the students in the class of the test is 2760 The average score of the test of boys is 65. The average score of the test of girls is 75. How many boys are there? How many girls are there?

... This problem corresponds to the basic problem. ...

-

Fig. 8. Problem explanation for second-order problems.

S T U D E N T

Problem model G) Problem n exDlanation * module

problem explanation

Fig. 9. Configuration of the three modules.

4.3. Characterization of modules

There can be two cases where the problem explan- ation is required. One is the case where the student re- quires the explanation together with the object problem, and the other is the case where the student requires the explanation for the presented problem. In the former case, the problem is solved by the problem-solving

module and the problem explanation is presented based on the generated problem model. In the latter case, the problem explanation is presented, b& on the problem model generated by the problem-generation module.

The relations among those three modules can be represented as in Fig. 9. When viewed from the problem model, the problem-solving module and the problem- generation module is a mechanism that generates the problem model. The problemexplanation module is a module that generates the actual teaching action based on the problem model.

5. Conclusions

This paper considered the arithmetical word prob- lem such as the crane-tortoise problem as the object area and discussed the realized modules. They are the prob- lem-solving module and the problem-generation module, realized based on the problem-solving model MIPS and the problemexplanation module, which indicates the comespondence between the problem that can be solved by the same solution method, based on the problem model generated as a mult of the problem solving and the problem generation.

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The functions of those three modules are suited to the exercise problem support in the problem area such as the arithmetical word problem, where the solution meth- od identification process is important.

By constructing a framework by which those mod- ules are operated adequately, a sophisticated exercise problem support system will be realized for problems such as the arithmetical word problem. At present, these three modules are implemented on SPARCstation of SUN Microsystems, Inc., using C-PROLOG.

It is planned that the exercise problem support system is designed and developed based on those mod- ules. Major issues are the management of the problem- explanation module and the problem-generation module, as well as the realization of the support for the under- standing of the solution method itself. Obviously, it is redundant to present all of the problem explanation to the student,and there must be a method that can select the necessary adequate explanation at an adequate time.

Another point is that the problem should be pre- sented based on the difference of the problem structuriza- tion process needed in the identification of the solution method.

In other words, it is indispensable for problem explanation and problem generation in ITS to construct the problemexplanation strategy and the problem-pre- senting strategy. In addition, to make the management efficient, there must be a study concerning the student model.

Another point is that it will be very useful to de- compose the solution method into solution methods with finer grains and to combine the decomposed solution methods to give a deeper insight into the solution method itself. Those processes are represented in MIPS as the inversecompile and compile processes for the problem- solving process [9-111.

It is also indispensable in the improvement of the problem-solving ability to realize the explanation for the forementioned compile-decompile processes.

Acknowledgement. The authors are grateful for the useful discussions given in this study by the members

of Toyoda Lab. and the members of ITS Res. Gr., Inst. Sci. Indust. Res., Osaka University.

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T. Hirashima, T. Kohno, Y. Nakamura, R. Mizo- guchi, and J. Toyoda. Realization of problem understanding model and application to ITS- Problem-solving and problem generation. Tech. Rep. Artif. Intel. Soc., SIG-KBS-8905-7, pp. 49- 56 (1990). A. Kashihara, T. Hirashima, Y. Nakamura, and J. Toyoda. Object understanding model for ITS- oriented explanation function. Tech. Rep. Soc. Artif. Intel., SIG-KBS-9001-8, pp. 51-58 (1990). T. Konishi, S. Itoh, A. Takagi, and K. Ohara. Teaching of knowledge principle in ICAI and simulation of object world. Trans. (D-II), I.E.I.C.E., Japan, J73-D-II, 7, pp. 1007-1018 (July 1990). A. Hirashima, Y. Nakamura, M. Ikeda, R. Mizo- guchi, and J. Toyoda. ITS-oriented cognitive model and tutoring strategy. Inf, Proc. Symp. New Application of Computers in Teaching, pp. 55-64 (1989). T. Hirashima, Y. Nakamura, M. Ikeda, R. Mizo- guchi, and J. Toyoda. A cognitive model for ITS. Proc. of ARCE, pp. 141-147 (1990). A. Hirashima, Y. Nakamura, M. Ikeda, R. Mizo- guchi, and J. Toyoda. ITS-oriented problem-solv- ing model MIPS. J. Soc. Artif. Intel., 7, 3 (1992).

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AUTHORS (from left to right)

Tsukasa Hirashima graduated in 1986 from the Dept. of Appl. Phys., Fac. Eng., Osaka University, where he obtained a Dr. of Eng. degree in 1991 and became an Assistant at the Inst. Sci. Indust. Res., Osaka University. Presently, he is engaged in research on artificial intelligence, especially in ITS. He is a member of the Inf. Proc. Soc. Jap.; Soc. Artif. Intel.; and Educ. Tech. Soc.

Takahim Kohno graduated in 1989 from the Dept. of Inf. Eng., Fac. Eng. Sci., Osaka University, where he obtained a Master’s degree in 1991 and affiliated with Min. Post Telecomm. As a graduate student, he engaged in research on ITS. He is a member of the Inf. Proc. Soc. Jap. and the Soc. Artif. Intel.

Akihim Kashihara graduated in 1987 from the Dept. Inf. Eng., Fac. Eng., Tokushima University, where he obtained a Master’s degree in 1989 and is presently in the doctoral program. He is engaged in research on artificial intelligence, especially in ITS. He is a member of the Inf. Proc. Soc. Jap.; Soc. Artif. Intel.; and CAI SOC.

Jun’ichi Toy& graduated in 1961 from the Dept. of Comm. Eng., Fac. Eng., Osaka University, where he obtained a Dr. of Eng. degree in 1966 and became an Assistant in 1966 and an Assoc. Prof. in 1969 on the Fac. Eng. Sci. He became a Professor in 1972 at the Inst. Sci. Indust. Res. Presently, he is engaged in research on ITS, natural language understanding, image procesSing, and document image processing. He is a member of the Jap. SOC. Cogn. Sci.; Inf. Proc. SOC. Jap.; and Soc. Artif. Intel.

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