An Approach to Promote Learning by Problem-Based Problem-Posing Sho YAMAMOTO, Hiromi WAKI, Tsukasa HIRASHIMA Department of Information Engineering, Hiroshima University, Japan {sho, waki, tsukasa}@isl.hiroshima-u.ac.jp

Abstract: In this paper, an interactive learning environment for learning by problem-based problem-posing has been described. In the method of learning by problem-posing, a learner is required to create the problems, since it is useful to understand the problems and their solution methods. There are several ways to pose problems, such as, solution-based problems, story-based problem posing and problem-based problem-posing. However, in the problem-based problemposing, a learner is required to create new problems by changing the original one. This problemposing has been expected to especially promote to understand the relation among problems. We have designed and implemented a learning environment for the learner where a learner could make new problems by changing the original problem and receive feedback on its changes. The learning environment has the facilities to diagnose the changes and generate the feedback based on the results of the diagnosis. The results of preliminary experimental use of the learning environment have also been reported.

Introduction An interactive learning environment for learning by problem-based problem-posing has been described in this paper. Learning by problem-posing requires learners to create new problems. Several investigations have already been suggested that the effective way to create problems is to promote learners to reflect their knowledge and problem-solving abilities (G. Polya, 1957; Brown, S.I. et al, 1993; Silver, E.A. et al, 1996). Then it is also useful to learn how to use the solution methods that can be applied to the posed problems. There are several ways to pose problems, such as, solution-based problems, story-based problem posing and problem-based problem posing (Nakano, A. et al, 1999). In the problem-based problem-posing, a learner is required to make new problems by changing the original one. This problem-posing is expected to especially promote to understand the relation among problems. We call this understanding “external problem understanding”. We have designed and implemented a learning environment where a learner could make new problems by changing the original one and receive feedback about the changes. The environment has the facilities to diagnose the changes and generate the feedback based on the results of the diagnosis. In addition, the results of preliminary experimental use of the learning environment have also been reported.

Learning by Problem-based Problem-Posing In this section, the characteristics of problem-based problem-posing are explained by comparing solutionbased problem-posing with Figures 1 and 2. The framework of solution-based problem-posing is shown in Figure 1. In this learning, a learner is provided with a solution method and is required to make new problems that can be solved by using the method. It is very effective to understand the applicable range of a solution method. We have already developed an interactive environment for learning by solution-based problem-posing in the domain of arithmetical word problems and also been confirmed its impact through experimental use in several elementary schools (Nakano, A. et al, 1999; T. Hirashima et al, 2007; T. Hirashima et al, 2008). In contrast to Figure 1, the framework of problem-based problem-posing is shown in Figure 2. In this learning, a learner is required to make new problems by changing the original problem. Besides, the learner is requested to solve the new problems and compare their solution methods with the original one. The change means that the solution method sometimes changes or does not change. This learning is expected to promote to understand the relation among problems from the viewpoint of solution methods. Several investigations have suggested that experts or good problem solvers comprehend the relation among problems from the viewpoint of solution methods (Chi,M.T.H. et al, 1981; Larkin, J.H., 1983; de Jong et al, 1986). Based on these considerations, we are investigating computer-based learning environment for interactive problem-based problem-posing that is composed of (1) problem change interface, (2) diagnosis function of the changes, and (3) help function for correcting or completing the change.

Figure 1: A Framework of Solution-based Problem-posing.

Figure 2: A Framework of Problem-based Problem-posing.

A Practice of Problem-based Problem-posing We have already developed a learning environment for problem-based problem-posing. In this section, the design of the learning has been explained and the implementation of the learning environment has been described in Section 3.

Problem Change Here, a problem is composed of a situation that is described as a set of attributes and the numerical relations among them, and problem setup that is described as status of the attribute values: given, unknown and required. Problem change can be defined as change of the attribute set or the status of their values. The change of the solution method accompanying the change of the problem makes clear not only the relation between the problems and solution methods but also the relations between the two problems or in the two solution methods.

Flow of Problem-posing Practice The flow of the practice has been explained in Figure 3. During the practice, a learner is provided with the original problem to solve it. Then, the learner tries to change the original problem. If the changed problem is wrong, it is directly informed to the learner (FB1). If it is correct, then the learner is required to input a solution method for the changed problem. When the learner inputs a wrong solution method, the mistake is informed to the learner (FB2). After completing the problem change and the problem-solving, the environment shows the original problem, the new problem and their solution methods to promote to think of their relations (PC: Problem Comparison).

Figure 3: Flow of Practice of Problem-based Problem-Posing. Learning Environment of Problem-based Problem-posing Interface This section concretely describes an interactive learning environment implemented in this research. We built the system by the Java language. The interfaces of the environment are shown in Figures 4 and 5, which explains the problem change interface and the problem comparison interface respectively. The problem change interface is composed of status change area, situation change area, and solution description area. In situation change area, a learner can able to change the configuration of physical situation by changing the physical objects and their positions. These changes effect to the set of attributes. In the status change area, a learner can able to change the status of attributes; given, unknown and required. In the solution description area, a learner describes the solution of the problem. A learner could able to complete these tasks by drag & drop or menu selection. After completing the problem change, the interface changes to Figure 6 where the two problems and their solution methods are shown to promote to think of their relations.

Figure 4: Problem Change Interface.

Figure 5: Problem Comparison Interface.

Figure 6: A Mechanical Problem.

Problem Change Function In this research, problem-based problem-posing applied to physics. An example of the problem given to a learner is shown in Figure 6. To describe a problem, the following six components can be used: block, ground, external force, string, spring, and pulley. Block is the movable object and has physical attributes related to movement. Ground is non-movable object and used as the background of the situation. Figure 7 shows the concrete description of the problem as shown in Figure 6.

Figure 7: Description of a Problem. Problem change is carried out as the situation change and/or as attribute status change. As the situation change, there are the following operations to change the situation. [C-1] Addition of component [C-2] Change of component

[C-3] Deletion of component [C-4] Change of attribute value Operations of [C-1]-[C-3] is carried out with object cards on the upper part of the situation change area. Currently, there are five kinds of component cards are prepared. A learner performs the [C-1] by doing drag & drop of one of component card in drawing area. [C-2] is to change the position of the component. This can be performed by drag & drop of a component in the situation change area. It is possible for [C-3] by doing drag & drop of an object to a trash box of the lower right. If a component has changeable attributes, the selectable attributes appear by right-clicks on the component. The learner can change attribute value [C-4] by selecting one from the list. Attribute status change is carried out by the following operations. [S-1] Change unknown value to known value [S-2] Change known value to unknown value [S-3] Change unknown or known value to required value The learner performs this operation in the status change area. In this area, the list of attributes is shown and the status of the attributes can be changed in the list. After completing this change, a new problem sentences are generated that is composed of given attributes and a required attribute.

Solution Input and Problem Comparison In the phase of solution input, a learner selects a formula from a list of formulas. Then the learner concretes the formula by using several attributes and calculation components provided in the solution description area. When the learner completes the input and pushes the diagnosis button, the system diagnoses the formula expression. After the change of the problem and the input of solution, the learner examines the difference between the problems and the solutions in the problem comparison interface shown in Figure 5. In the interface, sentence of the original problem and the changed one is shown at the upper part of the interface. The situations of the problems are shown in the middle. In the bottom part, the system presents the solution methods that are applied to solve the problems.

Preliminary Experimental Use Twenty-five subjects who were college students used the learning environment for twenty minutes. In this section, evaluations of the learning environment by using log data of the use, questionnaire, and comparison of pre/post-test scores are reported.

Procedure of Experimental Use One day before the experimental use, the subjects took the pre-test for five minutes. Just before the experimental use, the subjects were explained the way to change provided problem in the learning environment by using an example in five minutes. Then, they used the learning environment for twenty minutes. Just after the use, the subjects took the post-test. Both the pre-test and post-test required the subjects to describe the relations for a set of mechanical problems. The subjects were provided a paper where six problems were written. The subjects were then required to write links or closed curves that describe the relations of the problems on the paper. By using two set of problems, the two tests were counterbalanced. Questionnaire was carried out just after the post-test.

Analysis of Log and Questionnaire In the twenty minutes, twenty-five subjects gave 146 requests to diagnose their problem changes. Therefore, a subject made one new problem in every 3.4 minutes in average. This result suggests that it is possible to carry out problem-based problem-posing with this learning environment. Sixty-four of them were correct and eighty-two of them included errors. This means that it is necessary to diagnose the posed problems by learners although detailed analysis about the errors is necessary as a future work. As the results of questionnaire shown in Figure 8, even though almost half of the subjects replied that the learning environment is not easy to use, 65% subjects thought the practice is useful as learning activities. These results suggest not only the necessity to improvement of the learning environment but also the possibility of this learning environment.

Figure 8: Part of Results of Questionnaire. Results of Pre-test and Post-test First, scoring method is presented. Figure 9 is an example of the problems used in the test. Each problem in the test is labeled by an alphabet. In Figure 9, “A” in the left upper part is the label of the problem. The subjects are required to describe the relations between problems. Figure 10 shows an example of the description. The subjects were instructed to describe the relations by two ways, one way is to connect two problems by a link and to give explanation about it, and the other way is to enclose several problems and to give the explanation about the group. Each link and group is counted as one relation. If the explanation refers to solution method and it is correct, the relation is counted as a solution relation. "Both the problems are solved in equation of motion" is an example of the solution relations. "It is connected by the string" or "the number of objects is different” are examples not included in the solution relations.

Figure 9: A Problem in the Test.

Figure 10: A Description of the Relations.

Second, results of the tests are presented in Table 1. Pretest and posttest's comparisons were performed by the Wilcoxon matched-pairs signed-ranks test. There is statistically significant difference between pretest and posttest in the number of solution relations (p=0.001, two-sided test, corrected for ties) with a medium effect size (r=0.444). Because there is no control group in this experiment, we can not insist on the learning effect. However, these results suggest the possibility of this learning environment.

Table 1: Results of the Pre/Post Test (N = 25, *1% significant) Number of Number of solution relations total relating Pretest 0.20* 2.08 * Posttest 1.36 3.20

Conclusions In this research, we have designed and developed an interactive learning environment for learning by problem-based problem-posing as for mechanical problems. The learning environment was evaluated through experimental use. We have found the subjects can pose problems by changing the original one continuously. Then, we have confirmed that the score of the test that required the subject to make connections between problems improved after the use of the learning environment. We also confirmed that the current environment was not easy to use for the subjects. In addition to the evaluation of the learning by comparing with control group, improvement of the learning environment to help learner to pose problem easier than the current one are our important future works.

References G. Polya (1957): How to Solve It: A New Aspect of Mathematical Method, Princeton University Press. Brown, S.I., Walter, M.I. (1993) Problem Posing: Reflections and Applications, Lawrence Erlbaum Associates. Silver, E.A., CAI, J. (1996) An Analysis of Arithmetic Problem Posing by Middle School Students, Journal for Research in Mathematics Education, Vol.27, No.5, pp.521-539. Nakano, A., Hirashima, T., Takeuchi, A. (1999) Problem-Making Practice to Master Solution-Methods in Intelligent Learning Environment, Proc. of ICCE'99, pp.891-898. T. Hirashima, T. Yokoyama, M. Okamoto, A. Takeuchi (2007): Learning by Problem-Posing as SentenceIntegration and Experimental Use. AIED 2007: 254-261. T. Hirashima, T. Yokoyama, M. Okamoto, A. Takeuchi (2008): Long-term Use of Learning Environment for Problem-Posing in Arithmetical Word Problems, Proc. of ICCE2008, pp.817-824. Chi,M.T.H., Feltovich,P.J., Glaer,R (1981).: Categorization and representation of physics problems by experts and novices”, Cognitive Science, 5, pp.121-52. Larkin, J.H. (1983): The role of problem representation in physics, In D. Gentner & A.L.Stevens (Eds.), Mental Models, Lawrence Erlbaum Associates. de Jong, T. Ferguson-Hessler (1986): Cognitive structure of good and poor novice problem solvers in physics, Journal of Eductaional Psychology, 78, pp.279-286.

An Approach to Promote Learning by Problem-Based ...

Department of Information Engineering, Hiroshima University, Japan. {sho, waki, tsukasa}@isl.hiroshima-u.ac.jp. Abstract: In this paper, an interactive learning environment for learning by problem-based problem-posing has been described. In the method of learning by problem-posing, a learner is required to create the ...

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