Methods of Modeling and Assisting Causal Understanding in Physical Systems Ken'ichi Asami, Akira Takeuchi and Setsuko Otsuki Department of Artificial Intelligence Kyushu Institute of Technology 680-4, Kawazu, Iizuka, Fukuoka 820, Japan E-mail: asami, takeuchi, otsuki@minnie.ai.kyutech.ac.jp Abstract This paper presents methods for modeling and assisting students understanding about causalities between physical quantities based on comparative reasoning. Our tutoring system discusses causalities in an object system with the student by choosing dialogue strategies according to student s understanding states. The student s understanding state is represented by the causal network and table. Dialogue strategies are characterized by the following four elements of dialogue; selection of topic, progress of dialogue, granularity of dialogue, and style of utterance. This paper also presents a method of deepening student s understanding of the object system by comparing it with a similar system which is generated by partitioning or simplifying the original system. The partitioned system has a different causality from the original system, and the simplified system has a similar causality to the original system. Introduction This paper presents methods of how an intelligent tutoring system models students causal understanding between quantities in physical systems and supports students according to the causal model. This paper also presents an assisting method utilizing similarities and differences in similar systems. When students learn physics, they have to master many physical rules and a logical thinking way for applying these rules to physical systems. Students can correctly solve physical problems by applying basic laws and principles to physical systems. However, students can t always apply basic rules to physical systems, even if they know the rules, because the ability of using basic rules is acquired by accumulating experience. One of the effective ways for developing the ability is to make students discuss causalities in physical systems. Students will deepen their causal understanding between physical quantities by trying to explain causalities by themselves, and by being pointed out contradictions in their explanation. Students can get the ability by trying to apply basic rules to various systems. This paper focuses on methods of how an intelligent tutoring system models students causal understanding and plans dialogue about causalities. For interpreting students explanations about causalities, the tutoring system should have a capability of causal inference and should understand causalities in object systems. We employ comparative reasoning as a method for deducing a change of behavior of physical systems. Comparative reasoning is a qualitative technique of solving how a physical system will react to a perturbation in its quantity. The tutoring system can understand why the behavior changes based on the causalities with the inference process. In order to implement dialogue about causalities smoothly and instructively, the tutoring system should have two functions. One is student modeling, which is necessary to grasp students understanding states through dialogue dynamically. The other is strategic knowledge, which is necessary to teach the inference result individually for each student. The tutoring system models students causal understanding by the causal network and table, and then plans dialogue by choosing dialogue strategies according to the students understanding level. In the following section, the inference process of comparative reasoning is presented. In the third section, a manner of discussing causalities is given. In the fourth section, evaluation of four understanding levels of students is given. In the fifth section, dialogue strategies are introduced. In the sixth section, a method for producing similar systems will be given. Discussions and conclusions are given in the final section. Comparative Reasoning In this section, the outline of the inference process of comparative reasoning is illustrated with an example. Comparative reasoning deduces a qualitative change of behavior, when a perturbation is given to a certain quantity in a physical system. Weld has proposed two methods of comparative reasoning; differential qualitative (DQ) analysis and exaggeration (Weld 1988; Weld 1990). We employ DQ analysis for performing comparative reasoning. DQ analysis consists of fourteen inference rules based on relative change (RC) value, which represents that the absolute value of a physical quantity increases ( ), is unchanged ( ), or decreases ( ) in a qualitative manner. DQ analysis takes an initial perturbation, qualitative differential equations, and qualitative behavior predicted by Kuipers qualitative simulator QSIM (Kuipers 1986). DQ analysis returns RC values of affected quantities by applying the inference rules to these inputs. Since the inference rules are proven sound, it is guaranteed that the conclusion is always right.
block 2 block 1 F1 = Fg1 T1 F2 = Fg2 T2 T1 = T2 A1 = A2 F1 = M1 F2 = M2 Fg1 = M1 Fg2 = M2 A1 A2 G G dx1/dt = V1 dx2/dt = V2 dv1/dt = A1 dv2/dt = A2 Figure 1. A hung pulley attached to two blocks M2 X2 (0,1) M1 X2 (0,1) T2 X2 (0,1) Fg2 X2 (0,1) A2 X2 (0,1) A1 X2 (0,1) T1 X2 (0,1) : inverse rule : equal rule : tension rule : multiplication rule G X2 (0,1) : multiplication rule F1 X2 (0,1) : addition rule Fg1 X2 (0,1) M1 X2 (0,1) G X2 (0,1) : multiplication rule Figure 2. An infuluence tree of DQ analysis Here, consider a hung pulley which is attached to two blocks at both ends (Figure 1). The system is defined by fifteen parameters: position X1 and X2, velocity V1 and V2, acceleration A1 and A2, tension T1 and T2, gravity Fg1 and Fg2, total force F1 and F2, mass M1 and M2, and gravitational acceleration G. The motion follows qualitative differential equations shown in Figure 1. Since we assume that M1 is larger than M2, block 1 goes down and block 2 goes up with equal acceleration. For example, DQ analysis deduces how the behavior changes if mass M2 increases (Figure 2). The inference tree corresponds to the inference process representing how the perturbation of mass M2 propagates to other quantities in the behavior. Where A2 X2 (0,1) represents that acceleration A2 decreases for any position X2 over the time interval (0,1). We describe a causality between two quantities as. For example, we describe F1 A1 for expressing that acceleration A1 decreases since force F1 decreases. The inference process is helpful for students to understand causalities in the object system by being brought together with instructive dialogue strategies. Discussing Causalities If the tutoring system only explains the inference process without considering students attention, the students may miss an opportunity that they think about the structure of the object system and deepen their understanding of the causalities. One of effective tutoring ways is discussing how the behavior changes on the causalities with students. Therefore, the tutoring system asks students why an affected quantity changes or what happens to a quantity, while tracing causalities between causes and effects of changes. Figure 3 shows an example of dialogue about the hung pulley and two blocks. The student is given neither equations nor concrete values of quantities, but is shown only an outline of the object system. When the student discusses causalities, we assume that he knows the existence of quantities in the object system and the qualitative behavior. In the dialogue, the student explains causalities by combining parts of sentence, such as nouns and verbs from a menu. The tutoring system doesn t only ask the student what quantities change, but also makes him explain the reason and the effect on the causalities. Therefore, the student has to explain how the behavior changes by tracing the causalities. Namely, the student and the tutoring system discuss the causal chain in the object system. We define a causal chain as a connective sequence of primitive causalities, such as M2 Fg2 T2 T1. A primitive causality corresponds to a relation between two nodes directly connected in an inference tree, such as a causality between mass and gravity M2 Fg2. In the dialogue in Figure 3, the student and the tutoring system discuss the causal chain M2 Fg2 T2 T1 F1 A1 A2 as the topic of dialogue. The tutoring system assists the student in explaining causalities by asking about the changes of quantities. The student will understand each causality by explaining how the behavior changes with tracing the causalities. The teaching aims in discussing causalities are as follows: (A) Students should explain causality by themselves In order that students can deeply understand each causality, we employ an inquiry method which repeats questioning and answering about changes of quantities, and the tutoring system makes students imagine structures of object systems. Since dialogue primarily progresses on students Assumption: mass M2 increases. Q1. If mass M2 increases, what happens? A. Acceleration A2 decreases. Q2. Why? A. Because gravity Fg2 increases. Q3. When gravity Fg2 increases, what happens? A. I don't know. Q4. When gravity Fg2 increases, what happens to tension T2? A. It increases. Q5. When tension T2 increases, what happens? A. Acceleration A2 decreases. Q6. When tension T2 increases, what happens to another quantity? A. Tension T1 increases. Q7. When tension T1 increases, what happens? A. Force F1 decreases. Q8. Why? A. Because tension T1 increases and gravity Fg1 doesn't change. Q9. When force F1 decreases, what happens? A. Acceleration A1 decreases. Q10. When acceleration A1 decreases, what happens? A. Acceleration A2 decreases too. Figure 3. An example of dialogue
initiative, students have to think about causalities and explain changes of behavior by themselves. (B) The tutoring system should not simply teach causality In order that students can get understanding of causalities from contents of dialogue by themselves, the tutoring system doesn t immediately explain how the behavior changes even if they can t answer the question. The tutoring system should try to make students notice causalities by themselves without explaining the causalities directly. Modeling Causal Understanding If the tutoring system simply asks students about affected quantities by tracing causalities in the causal chain, students can t explain causalities with their own consideration. Therefore, the tutoring system needs to grasp what causalities a student has already understood or hasn t understood yet. The student s understanding state is represented by a network structure of causalities and a table based on kinds of causalities. Causal network The cause and effect structure of quantities in a physical system is represented by a network called causal network. The causal network expresses the interdependence among values of variables and constant values in a system. This network is automatically generated from qualitative differential equations, qualitative behavior, and relationships among physical parameters and components for the object system which a student is learning. Furthermore, the tutoring system has a causal network library for each student, which collects networks of systems which the student has learned before. Figure 4 shows the causal network of the system constructed from the hung pulley and two blocks. A network consists of nodes representing physical quantities, and links representing qualitative relations between quantities. Each node has four attributes about a corresponding physical quantity as follows; (a) a name of the quantity, (b) a kind of the quantity, (c) a name of a part related to the quantity, and (d) a qualitative state of the quantity. Where, a qualitative state is defined as a pair of sign and tendency of a quantity at a certain transition time. Furthermore, each link expresses a qualitative relation of either proportion or inverse proportion. For example, four qualitative relations are generated from F1 = M1 A1. The mutual relations between F1 and A1 are qualitatively proportional, the relation from M1 to F1 is qualitatively proportional, and the relation from M1 to A1 is qualitatively inverse proportional. Since a causal network explicitly describes qualitative relations between physical quantities, it is suitable for representing student s understanding state when the student learns causalities in the inference process of comparative reasoning. The student s understanding state is recorded on the links name: A2 kind: acceleration name: M2 kind: mass name: V2 kind: velocity state: <+,inc> name: F2 kind: total force name: Fg2 kind: gravity parts: block2 in the network. Each link has two attributes as follows; (Np) the number of questions which the tutoring system has asked the student about the causality, and (Hp) the history of replies which expresses whether the student has correctly explained the causality or not. The history of replies Hp is a list structure of numbers. When the student has correctly explained the causality, 1 is appended to Hp and 1 is added to Np. On the other hand, when the student has wrongly explained the causality, 1 is appended to Hp and 1 is added to Np. Moreover, when the student hasn t been able to explain the causality, 0 is appended to Hp and 1 is added to Np. For example, when the student has wrongly explained the causality twice and then correctly explained the causality three times, Hp is [1, 1, 1, 1, 1] and Np is 5. Where, the links whose Np is zero imply that the student hasn t learned about the causality yet. This record is applied to the casual chain. For example, consider the case that the student and the tutoring system discuss the causal chain M2 Fg2 T2 T1 in the object system. When the student has correctly explained that tension T1 increases if mass M2 increases, 1 is appended to Hp and 1 is added to Np for the three primitive causalities from M2 to Fg2, from Fg2 to T2, and from T2 to T1 respectively. On the other hand, when the student has wrongly explained that tension T1 doesn t change if mass M2 increases, 1 is appended to Hp and 1 is added to Np for the three primitive causalities. Moreover, when the student has replied that he don t know what happens, 0 is appended to Hp and 1 is added to Np for the three primitive causalities. (A) Primitive understanding level The tutoring system firstly needs to know about the student s understanding level for each primitive causality. The primitive understanding level represents how the student understands a primitive causality in an object system at a certain point of dialogue. The level is the weighted average of numbers in the history of replies to the primitive causality, which depends on the freshness of replies. Since a later reply should take priority than a earlier one, the weight i is given to the ith reply to the primitive causality. The primitive understanding level Lp is evaluated as follows; name: X2 kind: position state: <+,inc> name: T2 kind: tension name: G kind: g constant parts: earth name: X1 kind: position state: <+,dec> : a qualitative relation of proportion : a qualitative relation of inverse proportion name: T1 kind: tension name: V1 kind: velocity state: <,dec> name: F1 kind: total force name: Fg1 kind: gravity parts: block1 Figure 4. An example of causal network name: A1 kind: acceleration name: M1 kind: mass
Np Np Lp = Σ i Hpi/Σ i, ( 1 Lp 1). For example, when Hp is [1, 1, 0, 1, 1], the primitive understanding level is Lp = (5+4+0 2 1)/(5+4+3+2+1) = 2/5. When the value of Lp is positive, the tutoring system judges that the student correctly understands the primitive causality. On the other hand, when the value of Lp is negative, the tutoring system judges that the student wrongly understands the primitive causality. (B) Overall understanding level The tutoring system has to decide to stop discussing causalities in an object system, if the student has almost understood causalities in the object system. The overall understanding level represents how the student understands the structure of an object system on the causalities all over. The level is the average of Lp over all links in the causal network. When the number of all links in the network is n, the overall understanding level Lo is evaluated as follows; n Lo = Σ Lp/n, ( 1 Lo 1). If the overall understanding level is excellent, the tutoring system judges that the student doesn t need to learn about the object system. Complexity of system One of ways for measuring that a student understands a certain causality well is to check whether he can explain the causality in a complex object system after he has explained it in a simple system. Namely, if the student explains the causality without depending on the complexity of object systems, the tutoring system judges that he understands the causality well. Therefore, the tutoring system needs to have a criterion of complexity about object systems. Since our teaching subject is particle dynamics, we decide the criterion of complexity as the number of components in an object system. The components which we treat are classified into three kinds of groups. The first group is the components whose mass is finite, such as blocks, balls, etc. The second group is the components whose mass is nothing, such as strings, springs, pulleys, etc. The third group is the components whose mass is infinite, such as walls, floors, ceilings, etc. Our criterion depends on the number of components whose mass is finite or nothing. If the number of these components increases, the interaction among forces which appear in the system increases. Since the interaction between causalities depends on the interaction between forces, the number of components is appropriate to the criterion of complexity. For example, if there are two blocks and a pulley in an object system, the complexity is three. The number as complexity is preserved with the causal network. Causal table Our prime teaching goal is to assist students in acquiring the ability of applying basic rules to various physical systems. Therefore, the tutoring system needs to have a function for judging whether the student can explain the same kind of causalities in different object systems. Furthermore, the tutoring system should roughly guess the student s overall understanding level for any object system, even if the student has first learned about the object system. For this purpose, the causal table is used (Table 1). The table is classified by kinds of causes and effects of causalities. For example, two primitive causalities M2 Fg2 and M1 Fg1 are the same kind of causalities whose causes are mass and effects are gravity. Since this causal table doesn t depend on object systems, the table can be used to guess the student s understanding level for every object system. The student s understanding state for the same kind of causalities is recorded on the rows in the causal table. Each row has two columns as follows; (Nc) the number of questions which the tutoring system has asked the student about the kind of causality, and (Hc) the history of replies which expresses whether the student has correctly explained the kind of causality or not. The history of replies Hc is a list structure of numbers. When the student has correctly explained the kind of causality, 1 is appended to Hc and 1 is added to Nc. On the other hand, when the student has wrongly explained the kind of causality, 1 is appended to Hc and 1 is added to Nc. Moreover, when the student hasn t been able to explain the kind of causality, 0 is appended to Hc and 1 is added to Nc. For example, when the student hasn t been able to explain the kind of causality twice and then has correctly explained the causality three times, Hc is [1, 1, 1, 0, 0] and Nc is 5. Where, the rows whose Nc is zero imply that the student hasn t learned about the kind of causality yet. This record is applied to the causal chain. For example, consider the case that the student and the tutoring system discuss the causal chain T1 F1 A1 A2 in the object system. When the student has correctly explained that acceleration A2 decreases when tension T1 increases, 1 is appended to Hc and 1 is added to Nc for the three kinds of causalities from tension to total force, from total force to acceleration, and from acceleration to acceleration respectively. On the other hand, when the student has wrongly explained that acceleration A2 increases when tension T1 increases, 1 is appended to Hc and 1 is added to Nc for the three kinds of causalities. Moreover, when the student has replied that he don t know what happens, 0 is appended to Hc and 1 is added to Nc for the three kinds of causalities. (C) Causal understanding level For guessing whether the student can use the same kind Table 1. An example of causal table causes kind of causalities effects total number history of replies later replies' errata earlier velocity position 0 [] acceleration velocity 1 [1] acceleration acceleration 2 [1, 1] total force acceleration 3 [1, 1, 1] mass acceleration 0 [] mass gravity 5 [1, 1, 1, 0, 1] g constant gravity 0 [] gravity tension 5 [1, 1, 0, 1, 1] gravity total force 4 [1, 1, 0, 1] tension total force 4 [0, 0, 1, 1] : : : :
of causalities for different object systems, the tutoring system should know the understanding level based on kinds of causalities. The causal understanding level represents how the student can apply a kind of causality to every object system. The level is the weighted average of numbers in the history of replies to the kind of causality, which depends on the freshness of replies and the complexity of each object system. If the complexity of the object system is c, the weight i c is given to the ith reply to the kind of causality. Let Ci be the complexity of a corresponding object system, where the student has replied to the kind of causality i times. The causal understanding level Lc is evaluated as follows; Nc Nc Lc = Σ i Ci Hci/Σ i Ci, ( 1 Lc 1). For example, when Hp is [1, 1, 0] and C is [5, 2, 3], the causal understanding level is Lc = (15+4+0)/(15+4+3) = 19/ 22. When the value of Lc is positive, the tutoring system guesses that the student can apply the kind of causality to other object systems. On the other hand, when the value of Lc is negative, the tutoring system guesses that the student can t apply the kind of causality to other object systems. (D) Expectant understanding level The tutoring system should be able to predict the student s understanding level for unlearned systems. The expectant understanding level represents how the student can understand the structure of an unlearned object system. The level is the average of Lc over all pertinent rows, depending on kinds of causalities in the causal network of unlearned system. When the number of all links in the network is n, the expectant understanding level Le is evaluated as follows; n Le = Σ Lc/n, ( 1 Le 1). If the expectant understanding level is excellent, the tutoring system expects that the student can explain some causalities in the object system, even if he hasn t learned about the system yet. Dialogue Strategies The tutoring system has to make a plan of dialogue in order to discuss causalities smoothly and to assist students in understanding causalities effectively. Therefore, the tutoring system should have the definite patterns of dialogue strategies in selecting causalities, asking questions, and making explanations. The dialogue strategies are characterized as four elements of dialogue; selection of topic, progress of dialogue, granularity of dialogue, and style of utterance. (1) selection of topic Before the tutoring system starts dialogue about a certain object system with the student, a perturbed quantity is established according to the causal and expectant understanding levels. In advance, the tutoring system has some causal chains which DQ analysis deduces for all possible perturbations. For instance, the mass of block 1 increases, the initial velocity of block 2 decreases, the elastic coefficient of spring 1 increases, etc. Next, the tutoring system evaluates the average of the causal understanding levels for each causal chain. The tutoring system selects a perturbed quantity of the causal chain which the student will understand well, if the expectant understanding level is low. After a perturbed quantity has been decided, a causal chain is selected as the topic of dialogue, depending on the student s attention for a given perturbation. If the student roughly understands the structure of the object system, he will be able to predict a certain effect of a perturbed quantity by himself. Namely, when the student can predict a certain change of quantity, the topic of dialogue is the causal chain between the perturbed quantity and the quantity which the student has paid attention to. For example, if the student points out that a quantity D decreases if A increases, the causal chain A B C D is selected as the topic. On the other hand, if the student doesn t understand the structure of the object system, he won t be able to predict what happens to the behavior. Namely, when the student can t decide his attention for the change of behavior, the topic of dialogue is a nearer causality to the perturbed quantity. For example, if the student don t know the effects of the perturbed quantity A, the causality A B is selected as the topic. (2) progress of dialogue There are two directions in inquiring a certain causality. One is topdown; the tutoring system asks about the reason why B increases for the causality A B. The other is bottomup; the tutoring system asks about the effect that A increases for the causality A B. Here, the tutoring system doesn t ask about causalities whose primitive or causal understanding levels are excellent. We have characterized two directions as follows: (2a) topdown strategy: The tutoring system makes the student identify a cause from an effect of a primitive causality. If the student identifies an affected quantity, the tutoring system asks him about the reason why it changes. Furthermore, if he can correctly answer the question, the tutoring system asks the reason, while tracing the inference tree from a root node to a leaf node. The teaching aim of the topdown strategy is to check that the student understands each primitive causality in the causal chain, and to strengthen his understanding of the causalities. (2b) bottomup strategy: The tutoring system makes the student predict an effect from a cause of a primitive causality. If the student knows a cause of change, the tutoring system asks him what happens as the result. Furthermore, if he can correctly answer the question, the tutoring system asks about the effect, while tracing the inference tree from a leaf node to a root node. The teaching aim of the bottomup strategy is to make the student understand each primitive causality well. (3) granularity of dialogue Students generally explains the change of behavior with omitting some causalities, and the tutoring system shouldn t ask about the causalities which they have understood enough. Therefore, the tutoring system decides whether the student may omit some causalities according to the primitive and causal understanding levels. If the primitive or causal understanding levels are excellent for all primitive causalities
in a causal chain as the current topic, the tutoring system judges that the student understands all primitive causalities in the causal chain. (3a) detail level: When the student understands each primitive causality in a causal chain well, the tutoring system requests the student to explain the change of behavior without omitting the intermediate causalities. The teaching aim to ask on detail level is to assist students in understanding the accurate structure of the object system. (3b) outline level: When the student doesn t understand each primitive causality in a causal chain, the tutoring system allows the student to explain the change of behavior with omitting some causalities. The teaching aim to ask on outline level is to assist students in getting intuitive understanding of the object system. (4) style of utterance The tutoring system selects one of the following three styles as dialogue progresses. (4a) question: The tutoring system asks the student about causes or effects of changes in order to make the student notice the existence of causalities, and make the student understand the causalities deeply. (4b) explanation: When the student can t answer causes or effects of changes at a present study condition by all means, the tutoring system explains how the answer is derived from the causalities. (4c) alteration of system: When the student doesn t notice a certain causality, the tutoring system gives him a similar system in order to make him notice the causality by himself. The method for producing similar systems is presented in the following section. Utilizing Similarity and Difference in Similar Systems It is desirable that students find out important causalities by themselves without being explained the causalities directly. One of effective ways to assist such a discovery by students is to provide another system which is similar to the original system and make students think about the difference or the similarity between the two systems. In this section, a method for producing a similar system depending on students failure is presented. A similar system which aims to focus on difference is produced by partitioning the original system. A similar system which aims to focus on similarity is produced by simplifying the original system. Partitioning physical systems Consider a physical system, where a fixed pulley is attached to block 1 at vertical end and block 2 at horizontal end (Figure 5). Here friction is neglected. The motion follows the eleven equations shown in Figure 5. The parameters representing physical quantities are position X1 and X2, velocity V1 and V2, acceleration A1 and A2, mass M1 and M2, tension T1 and T2, total force F1 and F2, gravity Fg, and gravitational acceleration G. Assume that block 1 falls down and block 2 moves horizontally with equal acceleration. It is intuitively clear that acceleration A1 increases if mass block 2 block 1 M1 increases. However, students may fail to notice some causalities in the causal chain. If the structure of the object system is complex, students tend to answer only the conclusion that they can predict intuitively. When students identifies an affected quantity but they can t explain the precise reason of the perturbed behavior, a similar system which plays a role as a kind of counterexample is provided for making students recognize their insufficient explanation. Figure 6 shows a partitioned system from the fixed pulley and two blocks. The system is partitioned by removing components related to tension T1 for making students notice the causality between gravity Fg and tension T1. In the partitioned system, the acceleration of block 1 doesn t change if the mass of block 1 increases. In the original system, however, the acceleration of block 1 increases if the mass of block 1 increases. The tutoring system can make students notice the causality between gravity Fg and tension T1 by utilizing the different causality between the partitioned and original systems. Figure 7 shows a dialogue example about the fixed pulley and two blocks. Here, the student doesn t make mis- Assumption: mass M1 increases. Q1. If mass M1 increases, what happens? A. Acceleration A1 increases. Q2. Why? A. Because gravity Fg increases. F1 = Fg T1 F2 = T2 T1 = T2 A1 = A2 F1 = M1 F2 = M2 Fg = M1 Q3. When gravity Fg increases, what happens? A. Acceleration A1 increases. Q4. Consider the block 1 falling freely. When gravity Fg increases, does acceleration A1 increase? A. No, it doesn't so. Q5. Consider the original system. When gravity Fg increases, what happens? A. Tension T1 increases. Q6. When tension T1 increases, what happens? A. Tension T2 increases too. Q7. When tension T2 increases, what happens? A. Acceleration A2 increases. A1 A2 G Q8. When tension T2 increases, why does acceleration A2 increase? A. Since tension T2 increases and mass M2 does not change, it increases. Q9. When acceleration A2 increses, what happens? A. Acceleration A1 increases too. Figure 7. Dialogue example #2 dx1/dt = V1 dx2/dt = V2 dv1/dt = A1 dv2/dt = A2 Figure 5. A fixed pulley attached to two blocks block 1 Fg Figure 6. A partitioned system
takes to questions Q1, Q2, and Q3, but he explains the reason why acceleration A1 increases with omitting the intermediate causalities in the causal chain. At question Q4, in order to make him notice that his explanation is insufficient, the tutoring system asks him if the same situation succeeds in the partitioned system where the block falls freely. Finally, he can find out the causality that tension T1 increases when gravity Fg increases, and explains the change of behavior by tracing the causal chain from mass M1 to acceleration A1 in the original system. Simplifying physical systems Students can t find some causalities, if they don t understand the structure of the object system enough. When students can t answer an important effect of the perturbed quantity, the object system is replaced to a similar system which is easier to understand the common causality. Students can deepen their understanding of the original system, after they have found out the causality in the simplified system by themselves. Figure 8 shows a simplified system from the fixed pulley and two blocks. The system is simplified by turning gravity Fg into external force F for making students notice the causality between mass M2 and tension T2. The both systems have the same causality that the tension working to block 2 increases if the mass of block 2 increases. At the same time, the whole structure of causalities is easier to understand in the simplified system. The tutoring system can make students notice the causality between mass M2 and tension T2 by utilizing the similar causality between the simplified and original systems. block 2 Assumption: mass M2 increases. Q1. If mass M2 increases, what happens? A. Acceleration A2 decreases. Q2. Why? A. Because mass M2 increases. block 1 Figure 8. A simplified system Q3. Consider two blocks attached by a string horizontally. If mass M2 increases, what happens? A. Tension T2 increases. Q4. Consider the original system. If mass M2 increases, what happens? A. Tension T2 increases. Q5. When tension T2 increases, what happens? A. Tension T1 increases too. Q6. When tension T1 increases, what happens? A. Acceleration A1 decreases. Q7. Why does acceleration A1 decrease? A. Since total force F1 decreases and mass M1 does not change, it decreases. Q8. Why does total force F1 decrease? A. Since tension T1 increases and external force F does not change, it decreases. Q9. When acceleration A1 decreases, what happens? A. Acceleration A2 decreases too. Figure 9. Dialogue example #3 F Figure 9 shows a dialogue example about the fixed pulley and two blocks. Here, the student doesn t notice the change of tension from the judgment of his answer to question Q2. At question Q3, in order to make him notice the change of the tension, the tutoring system makes him think about the simplified system where two blocks are attached by a string each other and one is pulled by an external force. Finally, he can find out the causality that tension T2 increases if mass M2 increases, and explains the change of behavior by tracing the causal chain from mass M2 to acceleration A2 in the original system. Methods of partitioning and simplification When the student fails to explain the change of behavior without omitting important causalities, the tutoring system needs to generate a similar system which corrects his failure. Therefore, the tutoring system diagnoses the student s failure from the causal network, and then modifies the structure of the system. The methods of partitioning and simplification are given as follows: (A) Partitioning method of physical systems We define partitioning as separating a causal network by removing a node representing a force which propagates between parts. In order to partition an object system, the tutoring system removes a part related to a causality which the student can t explain. For example, when the student omits the causality of tension, the part propagating the tension is removed from the object system. Namely, the tutoring system eliminates a string, or pulley. In order to generate a partitioned system, the following procedures are given: (1) Assume that the topic of dialogue is the causal chain from C1 to Cn in an object system. When the tutoring system asks about the causal chain at topdown strategy, the student knows the change of C1 and Cn, however he can t explain the intermediate causalities. (2) If there is a force in the intermediate causalities, remove the part related to the force from the object system. A candidate of the partitioned system is a separated part including the quantities C1 and Cn. (3) DQ analysis has to deduce that the candidate and the original system have a different RC value of Cn for the same initial perturbation. If procedures (1)-(4) succeed, the candidate is a desirable partitioned system as a counterexample. (B) Simplification method of physical systems We define simplification as reducing the number of causalities in a causal network by replacing a dependent variable to a constant. In order to simplify an object system, the tutoring system replace a force to an external force, where the student needn t explain the causality of the force. The object system is also simplified by turning a non-zero constant value into zero. In order to generate a simplified system, the following procedures are given: (1) Assume that the topic of dialogue is the causal chain from C1 to Cn in an object system. When the tutoring system asks about the causal chain at bottomup strategy, the student can t explain the causal chain. (2) If there is a force out the causal chain, replace the force
to an external force. Moreover, if there is a physical coefficient (such as friction, repulsion, etc.) out the causal chain, turn the coefficient into zero. A candidate of the simplified system is the system which preserves the causal chain from C1 to Cn. (3) DQ analysis has to deduce that the candidate and the original system have the same RC values in the causal chain from C1 to Cn for the same initial perturbation. If procedures (1)-(4) succeed, the candidate is a desirable simplified system as a easier problem. Discussions Many methods of causal modeling and explaining physical systems have been proposed (Doyle 1986; Borchardt 1992). This means that causal explanations are very useful when human understand how physical systems work. Moreover, qualitative reasoning has been used as a method to explain physical behavior (Forbus & Falkenhainer 1990; Joolingen 1994). Since qualitative behavior is consistent with human intuition, the explanations are easier to understand rather than the results of numerical simulation. Therefore, causal qualitative explanations have been applied to many intelligent tutoring systems which assist students in understanding how physical mechanisms act. White has developed an intelligent learning environment based on qualitative models of electrical circuit (White & Frederiksen 1990). The models enable the system to simulate circuit behavior and to generate causal explanations, and serve as target mental models for learners in order to lead to more sophisticated model in a problem sequence. Cawsey has proposed detailed analysis of structure of causal explanations of simple electronic circuits (Cawsey 1988). Suthers has proposed automated generation of explanations in context of interactive dialogues about basic electricity and electrical networks (Suthers, Woolf, & Cornell 1992). Baril has developed a modeling method of students' understanding of device behavior based on confluences (Baril, Greer, & McCalla 1991). Their tutoring systems aim at planning and generating causal explanations which are acceptable by many students. Whereas, we believe that students need to reconsider their own understanding until they consent the causal explanations of physical systems. Therefore, it is necessary to monitor students' understanding states and to give students motivation to promote their causal understanding. To do this the tutoring system has to have the capability of deep mental modeling of students and pedagogical strategies for instruction. Our approach to dialogue strategies and causal understanding model is reasonable for sophisticating students' mental model about physical phenomena. Because the dialogue strategies aim to make students explain causalities by themselves and to elicit the students' beliefs about qualitative relations among physical quantities. After all, the tutoring system can model students' causal understanding to the causal network and table, and can plan dialogues to assist students in enhancing their causal mental models. Recent work has concentrated on the more instructive aspects of qualitative models. Some intelligent tutoring systems based on qualitative models with strategic knowledge have been developed. Hirashima has proposed a framework for providing problems suited to individual students with an index to select an adaptive exercise (Hirashima at el. 1993). Plötzner has described a role of qualitative knowledge for students who fail to solve problems because of trying to substitute values into variables in equations without a qualitative sense (Plötzner 1993). Reimann has proposed a method for assisting students in applying examples to other problems by generalizing relations between objects from the examples in a text book (Reimann at el. 1993). They, however, focus on support for acquiring skills of solving problems. Students should be familiar with causalities in object systems for solving problems skillfully. Therefore, our approach to drill a thinking way of applying causalities to physical systems has an essential advantage in learning physical principles which are used in solving problems. Students can make their understanding be sure through their explanations about causalities. Furthermore, similar systems are utilized to make students notice their misconceptions from differences, and to make students find their careless mistakes from similarities. This tutoring method is unique in terms of effective learning support about physics. References Baril, D., Greer, J.E., & McCalla, G.I. 1991. Student Modelling with Confluences. In Proc. of AAAI-91, 43-48. Borchardt, G.C. Understanding Causal Descriptions of Physical Systems. 1992. In Proc. of AAAI-92, 2-8. Cawsey, A. 1988. Explaining the Behavior of Simple Electronic Circuits. In Proc. of ITS-88, 372-378. Doyle, R.J. 1986. Constructing and Refining Causal Explanations from an Inconsistent Domain Theory. In Proc. of AAAI-86, 538-544. Forbus, K.D. & Falkenhainer, B. 1990. Self-Explanatory Simulations: An integration of qualitative and quantitative knowledge. In Proc. of AAAI-90, 380-387. Hirashima, T. at el. 1993. An Indexing Framework for Adaptive Setting of Problems in ITS. In Proc. of AI-ED, 90-97. Joolingen, W.V. 1994. QMaPS: Qualitative Reasoning for Simulation Learning Environments. Journal of Artificial Intelligence in Education, 5(2): 177-197. Kuipers, B. 1986. Qualitative Simulation. Artificial Intelligence, 29: 289-338. Plötzner, R. 1993. How Qualitative Problem Solving in Mechanics Improves by Qualitative Reasoning. In Proc. of AI-ED, 282-287. Reimann, P. at el. 1993. A Learning Strategy Model for Worked-out Examples. In Proc. of AI-ED, 290-297. Suthers, D., Woolf, B., & Cornell, M. 1992. Steps from Explanation Planning to Model Construction Dialogues. In Proc. of AAAI-92, 24-30. Weld, D.S. 1988. Comparative Analysis. Artificial Intelligence, 36: 333-373. Weld, D.S. 1990. Exaggeration. Artificial Intelligence, 43: 311-368. White, B.Y. & Frederiksen, J.R. 1990. Causal Model Progressions as a Foundation for Intelligent Learning Environments. Artificial Intelligence, 42: 99-157.