Proc. of ITS2006, pp.655-665(2006)

Robust Simulator: A Method of Simulating Learners’ Erroneous Equations for Making Error-based Simulation Tomoya Horiguchi1 and Tsukasa Hirashima2 1

Faculty of Maritime Sciences, Kobe University, 5-1-1, Fukaeminami, Higashinada, Kobe, Hyogo, 658-0022 Japan [email protected] 2 Department of Information Engineering, Hiroshima University, 1-4-1. Kagamiyama, Higashihiroshima, Hiroshima 739-8527 Japan [email protected]

Abstract. Error-based Simulation (EBS) is a framework for assisting a learner to become aware of his errors. It makes a simulation based on his erroneous hypothesis to show what unreasonable phenomena would occur if his hypothesis were correct, which has been proved effective in causing cognitive conflict. In making EBS, it is necessary (1) to make a simulation by dealing with a set of inconsistent constraints because erroneous hypotheses often contradict the correct knowledge, and (2) to estimate the ‘unreasonableness’ of phenomena in a simulation because it must be recognized as ‘unreasonable.’ Since the method used in previous EBS-systems was very domain-dependent, this paper describes a method for making EBS based on any inconsistent simultaneous equations by using TMS. It also describes a set of general heuristics to estimate the ‘unreasonableness’ of physical phenomena. By using these, a prototype EBS-system was implemented and examples of how it works are described.

1 Introduction The critical issue in assisting constructivist learning is to provide a learner with feedback which causes cognitive conflict when he makes errors [9]. We call this 'the assistance of error-awareness,' and think there are two kinds of methods for it. The one is to show the correct solution and to explain how it is derived. The other is to show an unreasonable result which would be derived if his erroneous idea/solution were correct. We call the former 'indirect error-awareness,' and the latter 'direct errorawareness.' [5] Usual simulation-based learning environments (SLEs, for short) [11-13] give the assistance of indirect error-awareness because they always provide the correct solution (i.e., correct phenomena) a learner should accept finally. The understanding by such assistance is, however, 'extrinsic' because they show only physically correct phenomena whatever erroneous idea a learner has. In addition, in usual SLEs, a learner must translate his (erroneous) hypothesis into the input which doesn't violate the constraints used by a simulator. This makes it difficult to identify what kind of phenomena a learner predicts. It is, therefore, difficult to estimate the 'seriousness' of the difference between the correct phenomena and his prediction. Error-based Simulation (EBS, for short) [2, 3] is a framework for assisting such direct error-awareness in SLEs. It makes simulations based on the erroneous ideas/solutions externalized by a learner (we call them 'erroneous hypotheses'), which results in unreasonable (unacceptable) phenomena and makes him be aware of his errors. EBS can make the understanding 'intrinsic' because a learner is shown the 'unreasonableness' of his hypothesis as physically impossible phenomena. In addition,

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he can input his (erroneous) hypothesis as it is, which makes it possible to control the 'unreasonableness' of the phenomena. It has been proved that EBSs cause strong cognitive conflict and lead learners to a deeper understanding [6-8]. In designing SLEs with EBS, two issues must be addressed. (1) The representation of erroneous hypotheses often contradicts the constraints necessary for making a simulation (i.e., the correct knowledge of the domain). (2) The result of a simulation must be recognized to be 'unreasonable' by a learner. Therefore, two mechanisms are necessary: the one for making a simulation by dealing with a set of constraints which may include a contradiction, and the other for estimating the 'unreasonableness' of phenomena in simulation by using explicit criteria. We have developed a few EBS-systems for mechanics problems in which a learner is asked to set up equations of mechanical systems [4, 6, 7]. The technique used in them, however, could deal with only a limited class of errors in simultaneous equations, and used domain-specific heuristics to avoid contradiction in calculation. Because such implicitness of knowledge in dealing with constraints made it difficult to predict what kind of physical phenomena would occur in a simulation, the criteria for the 'unreasonableness' of them were given externally and empirically. In other words, it was difficult to apply to other domains. In this paper, therefore, we propose a technique which can deal with any erroneous simultaneous equations/inequalities to make EBS. It is called 'Partial Constraint Analysis (PCA, for short),' which detects and eliminates contradictions in a set of constraints given by simultaneous equations/inequalities. We also propose a set of heuristics to estimate the 'unreasonableness' of phenomena in EBS in a general way. It describes the meaning of typical equations/inequalities of physical systems and is used for predicting what kind of physical phenomena would occur if they were violated. We think these methods are useful because they are domain-independent and because the problem in setting up equations of physical systems is important in learning science.

2 Requisites for EBS-Systems and the Previous Method

2.1 Requisites for EBS-Systems The simulators in usual SLEs are designed to do calculation by using a set of constraints which represents correct knowledge of the domain. A learner must interact with the environments within these correct constraints. In EBS-systems, on the other hand, a learner is allowed to externalize his hypothesis without this limitation. That is, when a hypothesis is erroneous, it may violate the correct constraints and the simulator can't do the calculation. In making EBS, therefore, it is necessary to analyze the union of the correct constraints and the constraint which is the representation of a learner's erroneous hypothesis. If a contradiction is detected, some of the correct constraints are relaxed (i.e., deleted) to make the rest consistent (i.e., EBS shows that if a learner's erroneous hypothesis were correct, it would be inevitable for some correct constraints to be violated). The module which deals with such inconsistency is called the 'robust simulator.' In addition, phenomena in EBS must be recognized as 'unreasonable' by a learner. While they are physically impossible phenomena because they include an erroneous hypothesis (and/or some correct constraints are deleted), a learner who has only incomplete knowledge of the domain doesn't always recognize their 'unreasonableness.' It is, therefore, necessary to estimate the 'unreasonableness' of the phenomena in EBS. When some correct constraints are deleted, the physical meaning of them provide useful information for estimation. When no correct constraints are deleted (i.e., the erroneous hypothesis doesn't contradict them), the criteria for estimation must be given externally (in this case, the erroneous hypothesis contradicts

Robust Simulator: A Method of Simulating Learners’ Erroneous Equations for Making Errorbased Simulation 3

the correct knowledge which isn't represented explicitly in the simulator (e.g., correct prediction about the behavior of the system)). 2.2 The Previous Method and Its Limitation We have developed a few EBS-systems for the mechanics problem in which a learner is asked to set up equations of mechanical systems [4, 6, 7]. They, however, can deal with only a limited class of errors in simultaneous equations, and since their robust simulators don't have explicit knowledge for dealing with constraints, the criteria for 'unreasonableness' were given externally and empirically. In this section, we describe the method used in these systems and discuss its limitation. In previous EBS-systems, a learner inputs a set of equations of motion each of which corresponds to each object (i.e., particle) in a physical system (this is his hypothesis). In the simulators, the correct equations of motion are described with the values of constants and the domains of variables in them. In order to deal with the frequent errors in this domain efficiently, it is assumed that a learner inputs the equations of motion for all of the objects, and that only one of them is erroneous. Under this condition, it is necessary to make simulations whether the simultaneous equations are consistent or inconsistent. The procedure [2, 3] is as follows: First, one variable is chosen from the variables in the erroneous equation as 'ER-Attribute,' which reflects the error. After that, the values of the other variables in the simultaneous equations are calculated by using the correct equations (as for the erroneous equation, the corresponding correct one is used). Then, these values are substituted for the variables in the erroneous equation to calculate the value of ER-Attribute (this is called 'ER-Value'). Thus, the EBS is made in which the erroneous behavior of the object which has ER-Attribute (this is called 'ER-Object') reflects a learner's error. Though this ER-Value contradicts the values of the other variables mathematically when the ER-Attribute (or the variable which depends on ER-Attribute) is in other equations 1 , it is regarded as appropriate reflection of a learner's error in this problem (i.e., he isn't able to think about the behavior of the ER-Object consistently with those of the other objects). In fact, when ER-Attribute is the position/velocity/acceleration of ER-Object, this procedure means deleting the constraints on the relative position/velocity/acceleration between some objects (they prevent the objects from overlapping and/or balance the internal force between them). In such simulations, the phenomena often occur in which the constraints on the relative position/velocity/acceleration between objects are violated (i.e., the objects overlaps and/or the internal force between objects doesn't balance). These constraints, however, weren't explicitly represented in previous EBS-systems, that is, the knowledge of constraints the robust simulator deletes to deal with contradiction is implicit. Instead, the criteria which check whether the phenomena in EBS qualitatively differ from the correct phenomena are given externally to estimate the 'unreasonableness' of phenomena in EBS.

3 PCA: Partial Constraint Analysis In previous EBS-systems, the procedure for avoiding contradiction (i.e., the constraint to be deleted) was specified before calculation by utilizing the assumption of the domain (i.e., domain-dependent). Under general conditions, however, it is necessary to detect the cause of contradiction in a set of constraints explicitly and to eliminate it 1

It is because while the values of other variables are calculated not to contradict the 'correct value' of ER-Attribute, ER-Value (i.e., the 'erroneous value' of ER-Attribute) is calculated by using them.

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appropriately. In this section, we propose PCA as a method for doing such a calculation. PCA deals with simultaneous equations/inequalities which may include contradiction. The reasons it specializes in simultaneous equations/inequalities are as follows: -

Simultaneous equations/inequalities are one of the most popular form for representing and understanding (natural) phenomena. They often become an objective in learning by themselves. They can represent the concepts and relations of the domain elaborately, and can be used for doing quantitative calculation. Because they represent relatively strong constraints (i.e., global and/or quantitative ones), they easily contradict each other when they are erroneous. As for the problem in which a learner is asked to set up equations of (physical) systems, the assistance of direct error-awareness becomes important because indirect error-awareness is often unhelpful. That is, in such a problem, while a learner can often predict the correct phenomena, he has difficulty in setting up the equations to describe them. It is no use showing him the simulation of correct phenomena (by using correct equations).

3.1 The Algorithm PCA can deal with a set of constraints as follows: -

All of the constraints are represented in the form of equation/inequality. It is possible to solve each equation/inequality for an arbitrary variable in it symbolically.

We, hereafter, assume the constraints include only equations because the generality isn't lost 2 . PCA first constructs the 'Constraint Network' (CN, for short) of given simultaneous equations S, then searches for the consistent part of it (we call it 'Partial Constraint Network,' PCN, for short). CN is a graph structure which represents the dependency between equations and variables in S. It has two kinds of nodes: equation node and variable node (they are called e-node and v-node, respectively). An e-node stands for an equation and a v-node stands for a variable in S. Each v-node is linked to the e-nodes in which it is included. There are two kinds of variables: endogeneous variable and exogenous variable. The values of endogeneous variables are determined by the internal mechanism of the system, while the values of exogenous variables are given externally. The v-node which stands for the latter is called ex-v-node. In CN, an e-node is regarded as a calculator. That is, an e-node with n links calculates the value of one v-node linked to it by being supplied n-1 values from the other links. A v-node with n links gets its value from one e-node linked to it and supplies the value to the other links. When S is consistent, the value of each v-node is uniquely calculated by just one e-node3 (assuming all the values of ex-v-nodes are given externally). That is, for each v-node, the unique path is determined which propagates the value(s) of ex-vnode(s) to it (If CN has a loop, the values of v-nodes in the loop are determined simultaneously by solving the simultaneous equations for it). When S is inconsistent, the following irregularities occur: -

2

(under-constraint) There are some e-nodes in each of which the sufficient values of (n-1) v-nodes aren't supplied to calculate the value of a v-node. In other

While an equation gives a constraint on the value of a variable, an inequality gives a constraint on the range of a variable. From this viewpoint, they can be regarded as equivalent in the following algorithm. 3 When an equation has more than one solution (e.g., an equation of the second degree gives two values of a variable x), the search by PCA is continued for all of them in parallel (CN is duplicated by the number of the solution).

Robust Simulator: A Method of Simulating Learners’ Erroneous Equations for Making Errorbased Simulation 5

-

words, there are some v-nodes the values of which can't be determined by any path. (over-constraint) There are some v-nodes each of which has more than one path to determine its value. In other words, there are some e-nodes which have no simultaneous solution.

Taking an ex-v-node (or a v-node which is given its temporary value) as an initial PCN, PCA extends it by adding the nodes step by step to which the value can be regularly propagated. To each v-node in PCN, the method for calculating its value is attached symbolically using the values of ex-v-nodes and e-nodes on the path of the propagation (it is called 'calc-method,' for short). When PCA meets irregularities, it resolves them as follows: -

-

(under-constraint) When the values of some v-nodes (necessary for calculation) aren't supplied to an e-node, PCA gives them temporary values and continues the calculation by using the values (a v-node given its temporary value is called a 'dummy'). (over-constraint) PCA deletes one of the e-nodes responsible for the contradiction from PCN to make the rest consistent.

The procedure above is continued until no propagation of values is possible any more. If PCA meets a loop, it tries to solve the simultaneous equations which consist of the e-nodes in the loop (i.e., to determine one (final) value of the dummies included in the simultaneous equations). In order to detect a contradiction and identify the e-nodes responsible for it, PCA must have nonmonotonic reasoning ability. Cooperation with TMS (Truth Maintenance System) [1] is a promising approach to realize PCA, because TMS provides the efficient function for maintaining the dependency network of propagation of constraints. We, therefore, adopt this approach. By tracing the dependency network, PCA can also explain of why the simultaneous equations aren't (or are) solvable, which is the reason it propagates constraints instead of searching the solvable subset(s) of the equations in turn. 3.2 The Mechanism of Contradiction Handling In this section, we elaborate on the mechanism of detecting and eliminating a contradiction (i.e., over-constraint). In order to identify the e-nodes responsible for a contradiction when it is detected, we make a TMS maintain the justification of calcmethod of each v-node (i.e., the e-node and other v-nodes used for determining its calc-method). By introducing a TMS, all the possible PCNs can be obtained independently of the choice of the initial PCN. In this paper, we describe the algorithm with a basic JTMS (Justification-based TMS) [1]. In the following, the correspondences between the nodes in CN and the TMS nodes, the justifications, the detection and handling of a contradiction are explained in this order. Since the value (i.e., calc-method) of an ex-v-node is given externally, a premise node is assigned to it. A simple node (i.e., neither premise nor assumption) is assigned to a v-node, which is made 'in' by a justification when its calc-method is determined, and is made 'out' otherwise. When a v-node n is made a dummy, the corresponding assumption node dum-n is created and enabled, which justifies (the simple node of) n. That is, the calc-method of a dummy is made temporarily determined. An assumption node is assigned to an e-node, which is enabled when it is used in determining the calc-method of a v-node, and is retracted when the calc-method is cancelled. When the calc-method of a v-node ns is determined by using an e-node ne and the determined v-nodes which are linked to ne, an assumption node JS (which stands for

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Tomoya Horiguchi1 and Tsukasa Hirashima2

this calculation) is created and enabled, and (the simple node of) ns is justified by JS, ne and (the TMS nodes of) the v-nodes4. A contradiction is detected when all the v-nodes which are linked to an e-node ne are determined and the constraint of ne (it is called 'c-equation*') can't be satisfied. In this case, a contradiction node is justified by ne (which is enabled) and (the TMS nodes of) the v-nodes. Then, the contradiction handler is called to retract one of the assumption nodes underlying the contradiction (it is marked 'unsimulatable' and deleted from PCN). The e-nodes which are used in determining the calc-methods of the v-nodes which are made 'out' by this retraction and ne are also retracted (if not deleted) and become the objects of search again (their corresponding assumption nodes of calculation JSs are also retracted). When the c-equation* can be satisfied, the calc-method of a dummy nd in it is (finally) determined. In this case, all the calc-methods of v-nodes that include nd are replaced by substituting the solution of c-equation* for them (let the v-nodes xps (p = 1, ..., t)). In addition, the justifications of (the simple nodes of) xps must be also replaced. There are two kinds of replacing of justification: (1) backward-replacing and (2) loop-replacing. The former occurs when there is only one v-node which is linked to ne (where the c-equation* arose) and the calc-method of which includes nd. In this case, the directions of all justifications between ne and nd are reversed. The latter occurs when there is more than one such v-node, which corresponds to a loop. In this case, a simple node Loop (which stands for the solution of simultaneous equations in the loop) is created. It is justified by the e-nodes which were used in determining the previous calc-methods of xps, (the nodes of) the v-nodes which are linked to these e-nodes (and the calc-methods of which don't include nd) and an assumption node JL which is created and enabled (which stands for this calculation). The xps are justified by the Loop (when the Loop is made 'out' afterwards, JL is also retracted). In both cases, all the assumption nodes that stand for the previous calculations are retracted. 3.3 An Example of How PCA Works In Fig.1a, assume that the simultaneous equations in the loop (eq1, eq2, eq3 and eq4) have a solution. When PCA begins a search taking { e1 } as an initial PCN, it first meets eq1 which is linked to more than one undetermined v-nodes. Assume that z is made the dummy and the calc-method of x is determined by using dum-z, e1 (which is an ex-v-node) and eq1. Then, after determining the calc-methods of y (by x and eq2) and w (by y and eq3) (eq5 is pushed into the stack), PCA meets eq4. Since all the vnodes that are linked to eq4 are determined, it solves the c-equation* for the (only) dummy z and substitute the solution for the calc-methods of x, y and w. Fig.1b and 1c show the dependency networks of TMS nodes before and after the solution of the loop (all the nodes for calculations (JSs) are omitted). In them, after dum-z (which justified z) was retracted, the replaced calc-methods of x, y, w and z are justified by the Loop (the Loop is justified by all the TMS nodes which were concerned with the solution of c-equation* (i.e., eq1, eq2, eq3, eq4 and e1)). Then, eq5 is popped out of the stack and the calc-method of v is determined (by w and eq5). Next, PCA meets eq6. Since all the v-nodes that are linked to eq6 are determined and the calc-methods of which don't include dummies, this constraint is unsolvable. Therefore, it justifies a contradiction node by using v, eq6 and e2, and the TMS shows the assumption nodes underlying the contradiction eq1, eq2, eq3, eq4, eq5 and eq6). When eq2 is retracted (i.e., deleted from PCN) for example, Loop loses its support (i.e., it becomes 'out') and x, y, z, w and v become 'out.' Therefore, the e-nodes which were used for determining their calc-methods (i.e., eq1, eq3, eq4, eq5 and eq6) are also retracted and pushed into the stack again. When eq5 is retracted to eliminate the contradiction, only v becomes 'out.' In this case, only the calc-method of v is redetermined (by using e2 and eq6). 4

JS (and JL which is defined later) is used for cancelling the justification of (the simple node of) ns when its calc-method is cancelled afterwards.

Robust Simulator: A Method of Simulating Learners’ Erroneous Equations for Making Errorbased Simulation 7

Fig.1. An example of how PCA works

4 Heuristics for Estimating the 'Unreasonableness'

4.1 Four Heuristics for Deleting Constraints When erroneous simultaneous equations 5 are unsolvable, PCA outputs a PCN in which a (set of) equation(s) is deleted from them. In general, there exists a multiple choice of (sets of) equations to be deleted to make PCN consistent, and the choice considerably influences what kind of physical phenomena would occur in simulation. In other words, by describing the meaning of typical equations/inequalities in physical systems (i.e., what kind of constraints they represent) in advance, it becomes possible to estimate the 'unreasonableness' of physical phenomena in EBS. This can be used as the criteria when a robust simulator makes a choice from (sets of) equations to be deleted. In this section, from the viewpoint of the assistance of direct error-awareness, we discuss the criteria for making EBS 'unreasonable' as much as possible, to propose four domain-independent heuristics. - (H1) Don't delete the erroneous equations. The purpose of EBS is to show 'if a learner's erroneous hypothesis were true, some 'unreasonable' phenomena would occur.' An erroneous equation which reflects a learner's error, therefore, must not be deleted. Only when there is more than one erroneous equation and they contradict each other, may some of them be deleted. - (H2) Delete the equations which represent the topic of learning (e.g., physical law/principle) prior to others. When the topic of learning is a relation between some physical constants/variables (e.g., physical law/principle), it is useful to delete the equation which represents it because the phenomena in which the relation doesn't hold much focus on the topic. For example, when a learner is learning 'the law of conservation of energy,' it facilitates his error-awareness to show 'if his erroneous equations were true, the total energy of the system wouldn't be conserved.' - (H3) Delete the equations/inequalities which describe the values of physical constants or the domains of physical variables prior to others. The equations/inequalities which describe the values of physical constants or the domains of physical variables often represent the most basic constraint of the domain, such as the meaning of the constants/variables, the conditions of existence of physical objects/processes which have the constants/variables as attributes, or the conditions for the equations/inequalities to be valid. The phenomena in which these constraints are violated, therefore, are easily 5

Also in this section, the word 'equations' is used as including inequalities. Only when it is necessary to emphasize inequalities, it is described as 'equations/inequalities.'

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Tomoya Horiguchi1 and Tsukasa Hirashima2

recognized as 'unreasonable.' For example, an inequality which describes a coefficient of friction as nonnegative, an equation/inequality which describes the free/blocked space of a mechanism, and an inequality which describes a Reynolds number as not more than 2300 (i.e., Re. <= 2300) to assume a steady and laminar flow in a Bernoulli equation. - (H4) Delete the fundamental circuit equations and cut set (incidence) equations of the system prior to others. In physical systems, fundamental circuit equations of across variables and cut set (incidence) equations of through variables [10] often represent the most basic constraints of the domain, such as the conservation of basic physical amounts, or the relations between components of the system to be held. The phenomena in which these constraints are violated, therefore, are easily recognized as 'unreasonable.' For example, cut set (incidence) equations of fluid systems or electric circuits represent the conservation of the total amount of substance which flow through components (e.g., equation of supply and demand, Kirchhoff's first law), and fundamental circuit equations of mechanical systems represent the relative velocities between components to be held. If they are deleted, the phenomena may occur in which a substance appears/disappears without its cause, or rigid objects overlap each other in spatiotemporal space. 4.2 Examples of Making EBS by Using the Heuristics We implemented a prototype robust simulator which makes EBS by using PCA and the heuristics above. Examples in elementary electric circuits and mechanics are also prepared in which a learner is asked to set up equations for a system. In this section, we describe its implementation and illustrate how it works. Generation of explanation. By tracing the dependency network of justifications made by PCA, the system can make an explanation of why the simultaneous equations aren't (or are) solvable. By using the heuristics above, it can also explain how unnatural phenomena will occur in the EBS. A dependency network and the meaning of a deleted equation are translated into quasi-natural language by using a simple template of explanation. An example in an electric circuit. As for the electric circuit shown in Fig.2a, assume that a learner set up the (erroneous) equations shown in Fig.2b. These simultaneous equations are unsolvable because the two loops in their constraint network (i.e., the loop with variables v1, v2, i1 and i2, and the loop with variables v2, i2 and i3) are simultaneously unsolvable (Fig.2c). The robust simulator, therefore, tries to delete some of the equations in these loops (i.e., equations (1), (2), (3), (4) and (5')). According to (H1), equation (5') is not an option. Since equations (1), (3) and (4) are fundamental circuit equations and equation (2) is an incidence equation, equation (2) is deleted according to (H4) (in this case, (H2) and (H3) are inapplicable). The calculation by using the rest (i.e., equations (1), (3), (4) and (5')) yields an 'unreasonable' phenomenon in which the total amount of electric current at node A isn't conserved (i.e., i1 = 2.25(A), i2 = -0.5(A) and i3 = 1.5(A)). Fig.2d shows the explanation made by the system.

Robust Simulator: A Method of Simulating Learners’ Erroneous Equations for Making Errorbased Simulation 9 Fig.2. An example in electric circuit

Fig.3. An example in electric mechanics

An example in mechanics. As for the mechanical system shown in Fig.3a, assume that a learner set up the (erroneous) equations shown in Fig.3b. These simultaneous equations are unsolvable because the two loops in their constraint network (i.e., the loop with variables a3, b2 and T, and the loop with variables a1, a2 and N) are simultaneously unsolvable (Fig.3c). The robust simulator, therefore, tries to delete some of the equations on these loops (i.e., equations (1), (2), (3'), (4), (5) and (6)). According to (H1), equation (3') is not an option. Since equations (1), (2) and (4) are incidence equations and equations (5) and (6) are fundamental circuit equations, equation (5) is, for example, deleted according to (H4) (in this case, (H2) and (H3) are inapplicable). The calculation by using the rest (i.e., equations (1), (2), (3'), (4) and (6)) yields an 'unreasonable' phenomenon in which the relative velocity between Block-1 and Block-2 isn't held (i.e., a1 = g/4, a2 = 9g/4, a3 = b2 = 3g/2, T = 5Mg/2 and N = 9Mg/4), that is, these blocks overlap each other (Fig.3d).

5 Conclusion In this paper, we presented a method of simulating learners' any erroneous equations for making EBS. By this extension, it becomes possible to utilize EBS in various domains (in which setting up equations is important) for assisting direct errorawareness. We think our method is useful to cause cognitive conflict when a learner makes errors in SLEs.

References 1. Forbus, K.D., deKleer, J.: Building Problem Solvers. The MIT Press (1993) 2. Hirashima, T., Horiguchi, T., Kashihara, A., Toyoda, J.: Error-Visualization by Error-Based Simulation and Its Management. Proc. of AIED97 (1997) 450–457 3. Hirashima, T., Horiguchi, T., Kashihara, A., Toyoda, J.: Error-based Simulation for ErrorVisualization and Its Management. Int. J. of Artificial Intelligence in Education, 9 (1998) 17–31 4. Hirashima, T., Horiguchi, T.: Evaluating of Error-Based Simulation by Using Qualitative Reasoning Techniques. Proc. of QR-01 (2001) 128–133 5. Hirashima, T., Horiguchi, T.: Error-Visualization for Learning from Mistakes. Transactions of Japanese Sciety for Information and Systems in Education, 21 (2004) 178–186 (in Japanese)

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6. Horiguchi, T., Hirashima, T., Kashihara, A., Toyoda, J.: Error-Visualization by Error-Based Simulation Considering Its Effectiveness -Introducing Two Viewpoints-. Proc. of AIED99 (1999) 421–428 7. Horiguchi, T., Hirashima, T.: A Method of Creating Counterexamples by Using Error-Based Simulation. Proc. of ICCE2000 (2000) 619–627 8. Horiguchi, T., Hirashima, T., Okamoto, M.: Conceptual Changes in Learning Mechanics by Error-based Simulation. Proc. of ICCE2005 (2005) 138–145 9. Perkinson, H.J.: Learning From Our Mistakes: Reinterpretation of Twentieth Century Educational Theory. Greenwood Press (1984) 10. Shearer, J.L., Murphy, A.T., Richardson, H.H.: Introduction to System Dynamics. AddisonWesley Publishing Company (1971) 11. Towne, D., de Jong, T., Spada, H. (Eds): Simulation-based experiential learning. Berlin/New York: Springer (1993) 12. Towne, D.M.: Learning and Instruction in Simulation Environments. Educational Technology Publications, Englewood Cliffs, New Jersey (1995) 13. Wenger, E.: Artificial Intelligence and Tutoring Systems: Computational and Cognitive Approaches to the Communication of Knowledge. Morgan Kaufmann (1990)