Problem-Making Practice to Master Solution-Methods ...

Problem-Making Practice to Master Solution-Methods in Intelligent Learning Environment Akira Nakano

Tsukasa Hirashima

Akira Takeuchi

Takeuchi Lab Kyushu Institute of Technology Department of Artificial Intelligence 680-4 Kawazu, Iizuka 820-8502, Japan E-mail : [email protected] A learner who has already acquired a solution-method can not always use it adequately. Therefore, the practice to use a solution-method is indispensable to master it. A problem-solving practice is the most popular, but it is not the only way to make the learner master the solution-method. To make problems which can be solved by a solution-method is a promising way to master the solution-method. In this paper, we describe an Intelligent Learning Environment which realizes a problem-making practice. In the problem-making practice, the ILE provides a learner a solution-method and concepts. Then, by using the concepts, the learner makes problems that are solved by the solution-method. The ILE has a facility to diagnose the problem made by the learner. By using the result of the diagnosis, the ILE helps the learner to complete the problem, or provides the next problem-making exercise. The ILE is implemented with Java, so it can be used on World Wide Web. The current environment deals with arithmetical word problems and its solution-methods. Keywords : intelligent learning environment, problem solving model, intelligent tutoring system, learner model, world wide web

1

Introduction

A learner who has already acquired a solution-method can not always use it adequately. Therefore, practice to use the solution-method is indispensable to master it [1-3]. The problem-solving practice is the most popular way to make a learner master the solution-method. However, it is not the only way to master the solution-method. To make the problems that are solved by the solution-method is a promising way to master it. This paper describes an Intelligent Learning Environment for the problem-making practice of arithmetical word problems. To realize the problem-making practice, to diagnose the problems made by a learner is essential facility. First, the facility should be able to examine whether that the problem can be solved by the solution-method that is a target of the practice. It also can examine the cause that the solution-method can not be applied to the problem. We are developing such diagnosis facility of arithmetical word problem made by a learner. The diagnosis facility, first, examines whether that the problem can be solved by the target solution-method. When it can be solve, the problem is adequate one. Then, the difficulty of the problem is estimated and used to estimate of the learner’s progress. When the problem can not be solved by the solution-method, the diagnosis facility examines which information is lacking or wrong. These results are necessary to help the learner in the problem-making practice. In this paper, we describe the outline of the ILE for the problem-making practice of arithmetical word problems and the diagnosis facility of the problem. It is also reported the results of preliminary evaluation of this prototype ILE. Interface of the ILE was implemented by Java, and the diagnosis module of it was implemented by Prolog. So the ILE can be used through WWW.

2 2.1

Outline of ILE for Problem-Making Practice Modules in the ILE

The current version of the ILE consists of the three modules shown in Figure 1. Interface module receives a problem made by a learner. Then, this module changes description of the problem so that it can be dealt it with Problem Diagnosis module. Interface is explained in more Figure 1. The Modules in the ILE. details in this section. Learner model manages individual learner’s progress and ID to access the ILE. Learner’s progress is estimated by examining the difficulties of the problems made by the learner. Though we have already proposed a method to examine the difficulties of arithmetical word problems, it is omitted in this paper. The Diagnosis module is the main part of this paper. It is described in the following sections.

2.2 Interface Figure 2 shows the interface of the ILE. Subject panel indicates the target solution-method of the problem-making practice. The ILE allows a learner to select a favorite subject. A learner is required to make problems that can be solved by the solution-method. The ILE provides the learner with Sentence-templates and Concepts. The learner makes a sentence by applying concepts (selected from Concept panel) and a value (inputted from ten-keys) to a sentence-template. The sentences generated are shown in Problem-making panel. A problem is composed of several sentences. When the learner asks to diagnose the problem, it is diagnosed and the results are shown in Comment panel. The sentence-templates and concepts are prepared to describe a value of an attribute of an object in one sentence. We call the relation of the three “basic relation.” For example, “X of Y is Z” is a sentence-template. When a learner applies “legs” to X, “a crane” to Y, and “2” to Z, “legs of a crane are 2” is a generated sentence. It means that value of attribute “legs” of object “a crane “ is “2”. Concept panel provides concepts that more than one problem can be only made. Therefore, a learner is asked to select and combine adequate concepts. Changing the set of concepts that it is provided in Concept panel can change the degree of difficulty of the problem-making.

Subject Panel Comment Panel

Problem-making Panel

Concept Panel Ten - Keys Sentence-template Panel

Figure 2. Interface.


Subject Panel In this panel, a learner chooses a solution-method to master.
Sentence-template Panel This panel provides a learner with sentence-templates to make the word problem. When a learner chooses a sentence-template, the template with several blanks is shown in problem-making panel. A learner chooses a concept from Concept panel, and applies it to a blank. From one sentence, one basic relation is made.
Concept Panel This panel provides concepts to fill up blanks of sentences. A learner completes a sentence by using these concepts.
Ten - Keys This panel has the calculator to make numerical value to fill up blanks of sentences.
Problem-making Panel This panel shows sentences which a learner made by using sentence-templates, concepts, and values. A learner can confirm correctness by seeing this indication.
Comment Panel This panel shows messages and diagnosed results 2.2.1

“Crane-turtle problem”

Currently, our prototype system can diagnose only with “crane-turtle problem”. Figure 3 shows basic “crane-turtle problem”. This problem is solved by the problem solving method shown in Figure 4. We call this kind of problem “crane-turtle problem” and the solution-method “crane-turtle method”. There are many problems that are composed of different concepts, but are solved by “crane-turtle method”. In Japan, many solution-methods, like “crane-turtle method”, are taught in learning of arithmetical word problems. (1) A crane has 2 legs. (2) A turtle has 4 legs. (3) There are cranes and turtles, 20 in total. (4) The total number of legs of cranes and turtles is 48. (5) How many cranes are there? (6) How many turtles are there? Figure 3. Example of the word problem-1. {the total number of cranes and turtles} * {the number of a crane’s(turtle’s) legs} = A |{the total number o their legs} – A| = B |{ the number of a turtle’s legs} – { the number of a crane’s legs}| = C B / C ={the number of turtles(cranes)} {the total number of cranes and turtles} – {the number of turtles(cranes)} = {the number of cranes(turtles)} Figure 4. The problem solving method of “crane-turtle method”

3

Diagnosis of Problem Made by Learner

The diagnosis module is designed base on a model of problem solving for arithmetical word problems. The model is called MIPS (Model of Indexing for Problem Solving)[4,5]. In MIPS, the applicable condition of a solution-method is represented as a semantic network. The semantic network is called ISM (Index of Solution-Method). To apply a solution-method to a problem, a semantic network corresponding to the ISM has to be generated from the problem. The network

generated from the problem is called PUN (Problem Understanding Network). The diagnosis module, first, examines whether a PUN corresponding to the ISM can be generated. If such a PUN can not be generated, the module examines which information is required to complete the PUN. Therefore, it is possible not only to judge whether the problem made by a learner is correct or not, but also to find errors in the problem or to help to complete the problem. In this section, first, problem solving in MIPS is explained. Then, we describe the errors that are detected by the diagnosis of the problem.

3.1 The method to solve arithmetical word problem 3.1.1

Making raw-PUN from the word problem

A raw-PUN is the PUN generated from the word problem directly. The generation of the raw-PUN is divided into the following two stages: generation of basic relations; and connection of basic relations. In the generation of basic relation, one basic relation is generated from one sentence, using a template. From the problem in Figure 3, the six basic relations(1) to (6) in Figure 5(a) ,are generated. When all sentences are transformed into basic relations, the next step is to connect the basic relations. Usually, all basic relations are connected with one, and it describes the whole meaning of the problem. A network which is generated by connecting all basic relations is called raw-PUN. The problem-solving module uses the following three connection rules to connect the basic relations.
(Connection 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.
(Connection 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.
(Connection rule 3) : When there is a relation of a set-operation among the concepts of set, they are connected by the set-operation. When all basic relations are combined into a network, it is decided that the raw-PUN is generated. For example, the basic relations (1) and (5) in Figure 5(a) can be considered as connected by the connection rule 1, the basic relations (2) and (6), too. The basic relations (3) and (4) in Figure 5(a) can be considered as connected by the connection rule 2. By the connection rule 3, basic relation(7) in Figure 5(a), three basic relations (3), (5), and (6) can be connected. Consequently, the raw-PUN composed of seven basic relations are generated as shown in Figure 5(a). Figure 5(b) is raw-PUN expressed as a network from Figure 5(a). (1) atr(crane, leg, 2) (2) atr(turtle, leg, 4) (3) atr(set([crane, turtle]), [crane, turtle], 20) (4) atr(set([crane, turtle]), leg, 48) (5) atr(set(crane), crane, Answer) (6) atr(set(turtle), turtle, Answer) (7) union(set([crane, turtle]), [set(crane), set(turtle)]) Figure 5(a). Raw-PUN(problem understanding network).

3.1.2

Figure 5(b). Raw-PUN expressed as a network.

Making with the generic ISM The generic ISM is an index of a solution, which describes numerical relations to use the

solution-methods. If a PUN satisfies the generic ISM, the solution-method can be applied to the problem. The matching process between the PUN and the generic ISM(Figure 6(a)) is to determine the basic relations still missing to apply the solution-method and the way to add the missing basic relations. This process is executed by examining whether or not the numerical relations represented 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 relations. Figure 6(b) is an example of the case where the basic relations composing the PUN of Figure 5(a) are assigned to the numerical relations (Figure 6(a)) of the generic ISM of “crane-turtle problem”. Next, it is determined whether or not the generated operational relations (Figure 6(b)) are adequate by matching them to the prepared operational relations which are prepared on the module beforehand. For the first operational relation in Figure 6(b), for example, the prepared operational relation “number of legs of an animal A” × “number of animals A” = “total number of legs of animals A” is assigned. Then basic relation “total number of cranes” is substituted into V7. In this way, all variables in the operational relations generated from the raw-PUN, are replaced by the basic relations. When it is impossible to replace all variables by basic relations, the reasoning is iterated for the process 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 is matched to the specifications in the generic ISM. For “total number of cranes” substituted into V7, for example, the value does not exist in the PUN but is matched since the value specification in the ISM is “unknown.” 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 “incomplete” state. First, fact-knowledge which is a basic relation prepared in the module, is used. When the module has fact-knowledge, “number of legs of a crane is two”, even if the problem is not included the correspondent basic relation, and the module can complement the value. If this trial fails, operational-knowledge which is corresponded to operational relation is used to check whether or not the necessary value can be generated from the value in PUN. When all the “incomplete” states are dissolved, the matching between PUN and the generic ISM is achieved. The PUN which corresponds to the generic ISM is called the zeroth-order PUN. The raw-PUN of Figure 5(a) is transformed into the zeroth-order PUN of Figure 6(c). In MIPS, these processes called problem-understanding process. [V5, known] × [V3, answer] = [V7, unknown] [V6, known] × [V4, answer] = [V8, unknown] [V7, unknown]+[V8, unknown]=[V2, known] [V3, answer]+[V4, answer]=[V1, known] Figure 6(a). The generic ISM of “crane-turtle method”

[number of legs of a crane, known] * [number of cranes, answer] = [V7, unknown] [number of lens of a turtle, known] * [number of turtles, answer] = [V8, unknown] [V7, unknown]+[V8, unknown]=[total number of legs, known] [number of cranes, answer]+[number of turtles, answer]=[total number of cranes and turtles, known] Figure 6(b). An assignment for the generic ISM.

[number of legs of a crane, known] * [number of cranes, answer] = [total number of legs of cranes, unknown] [number of legs of a turtle, known] * [number of turtles, answer] = [total number of legs of turtles, unknown] [total number of legs of cranes, unknown]+[total number of legs of turtles, unknown]=[total number of legs, known] [number of cranes, answer]+[number of turtles, answer]=[total number of cranes and turtles, known] Figure 6(c). Check of the assignment by using fact-knowledge and operational-knowledge.

3.1.3

Working out an answer

Next, this diagnosis module works out an answer by formal operation(Figure 7). In Figure 7, V1,V2,…..,V6 are the same value in Figure 6(a). A solution-method has one the generic ISM and one formal operation. V6 * V3 = X |V5 – X| = Y |V4 – V3| = Z Y / Z = V2 V6 – V2 = V1 Figure 7. Formal Operation of “crane-turtle problem”.

3.2 The diagnosis of errors and advise for them This section describes the errors which are detected in the diagnosis module. The errors are divided in the three kinds, the first is the errors on making a raw-PUN, the second is the errors on matching with the generic ISM, and the third is the errors on working out the answer. 3.2.1

On making raw-PUN


A fact does not exist in fact-knowledge. This error is a case of that a basic relation is not contained any fact-knowledge. Then this diagnosis module points out the basic relation. For example, bad basic relation is “atr(crane, horn, 2)” which expresses “A crane has 2 horns”.
A fact does not contradict in fact-knowledge. This error is a case of that the value in elements of the basic relation is different from a basic relation and fact-knowledge. Then this ILE points out the basic relation. For example, bad basic relation is “atr(crane, leg, 4)” which expresses “A crane has 4 legs”.
Raw-PUN can not be formed. This error is a case that a raw-PUN can not be formed. Usually, a raw-PUN is only one network which is generated by connecting all basic relations made from sentences made by a learner. If the diagnosis module can not connect any basic relations by the connection rules, this module indicates that the sentences can not be interpreted as one problem, and advises to revise. If the module can find several networks, the module chooses the biggest network as a raw-PUN. 3.2.2

On matching with the generic ISM

In the second process, the errors are detected on matching with the generic ISM.
Solution-method can not be applied. This error is case that the problem made by a learner can not be solved by the target solution-method. This diagnosis module judges it with the raw-PUN, the generic ISM and the prepared operational relations. An example is shown in Figure 8(a). It is assumed that the subject of

practice is “crane-turtle problem”. The generic ISM has constraints of the state of numerical values; known; unknown, and answer. The raw-PUN of this problem is shown in Figure 8(b). The module can not find adequate assignment of basic relations for the generic ISM. Therefore, this module indicates that the problem is not “crane-turtle problem”, and advises to revise.

There are cranes, 4 in total. There are cranes and turtles, 20 in total. How many turtles are there? Figure 8(a). Example of the word problem-2. Figure 8(b). Raw-PUN expressed as network.


It is short of quantity information for the solution. This error is a case that the problem is short of numerical value which is necessary for the calculation. When there is “incomplete” state at the generic ISM, this diagnosis module tries to change from “incomplete” state to “complete” state by fact-knowledge and operational-knowledge. When there is an “incomplete” state, the module indicates lacking information to a learner. In Figure 9(a), this problem does not have a sentence which is described about total legs of cranes and turtles. But the module can interpret the raw-PUN from these sentences, because the module has fact-knowledge as fact. The module can complement the basic relations like Figure 9(b). Figure 9(c) is the ISM assigned all basic relations and fact-knowledge, but there is “incomplete” state at the basic relation which should be complemented about “total number of legs”. The module can detect lacking information, for example “total number of legs” in Figure 9(a). Then this system suggests complementing the need information to make the problem based on the solution-method. There are cranes and turtles, 20 in total. How many cranes are there? How many turtles are there? Figure 9(a). Example of the word problem-3.

Figure 9(b). Raw-PUN expressed as a network. [number of legs of a crane, complete] * [number of cranes, answer] = [total number of legs of cranes, unknown] [number of legs of a turtle, complete] * [number of turtles, answer] = [total number of legs of turtles, unknown] [total number of legs of cranes, unknown]+[total number of legs of turtles, unknown]=[total number of legs, incomplete] [number of cranes, answer]+[number of turtles, answer]=[total number of cranes and turtles, known]

Figure 9(c). Check of the assignment by using fact-knowledge and operational-knowledge.

3.2.3

On working out an answer

Because the domain is the arithmetical word problem, in the problem numerical values should be limited. Therefore, this diagnosis module checks the numerical values dealt in the problem-solving. For example, when the subject of practice is “crane-turtle problem”, the answer(the total number of cranes and turtles) must be a natural number. If it is negative number or decimal, the problem is not adequate one. Then the module advises to revise.

4

The evaluation of this ILE

A prototype of the ILE was already developed. In this section, we report a preliminary evaluation of the prototype. In the evaluation, we asked several college students to use the prototype. Then they answered three questions (① The effect of the problem-making practice to master the solution-method. ③ The effect of the ILE to master the solution-method ② The easiness to use our ILE.). The result is shown in Table1. Table 1. The result of the questionnaire about this ILE. question Good Bad So-so ① 9 0 0 ② 8 0 1 ③ 8 0 1

In Tabel1-①, all students answered that the problem-making practice has the effect to master the solution-methods. In Table1-②, most students answered that this ILE has the effect to master the solution-methods. In Table1-③, most students answered that operation of the ILE is simple. These results suggest that the ILE is a promising way of the problem-making practice. However, several students indicated that better to add helping-operation in the ILE. They also gave several suggestions, that is, the needs of support help for the learner who can not make a problem at all. The prototype is available through Internet with browser (Japanese only). Its URL is as follows. http://www.minnie.ai.kyutech.ac.jp/~nakano/problem-making.html

5

Conclusionsf

In this paper, we described ILE to master the solution-methods for arithmetical word problem. This ILE is composed of three modules (Interface, Problem Diagnosis, Learner model). We especially described the method to solve the word problems, the diagnosis of errors and advise in the problem diagnosis module. And, we reported the evaluation of the prototype ILE. In this paper, we did not describe the definition of the difficulty of word problems and learner model. But we are considering them, and current results also were implemented on the ILE. Therefore, the ILE can provide advises to complete the word problem, and suggest the next exercise based on them. The results will be reported in the future. This research provides basis functions to realize the problem-making practice by ILE. This ILE must be improved to freely do the problem-making practice by reducing the limitation of the concept and the sentence pattern.

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