Model-based analysis of thinking in problem posing as sentence integration focused on violation of the constraints

Supianto et al. Research and Practice in Technology Enhanced Learning (2017) 12:12 DOI 10.1186/s41039-017-0057-5 RESEARCH Open Access Model-based analysis of thinking in problem posing as sentence integration focused on violation of the constraints Ahmad Afif Supianto1,2*, Yusuke Hayashi1 and Tsukasa Hirashima1 * Correspondence: 1 Department of Information Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-hiroshima, Hiroshima 739-8527, Japan 2 Department of Informatics, Faculty of Computer Science (FILKOM), Brawijaya University, 8 Veteran Road, Malang 65145, Indonesia Abstract The advancement of computer and communication technologies has enabled researchers to conduct and analyze the learning process of posing problems. This study investigates what learners think while posing problems as sentence integration in terms of intermediate products as well as the posed problems as the resultant product. Problem posing as sentence integration defines the arithmetic word problem structure, and posing a problem is a task to satisfy all the constraints and requirements to build a valid structure. A previous study shows that, in problem posing as sentence integration for arithmetic word problems, learners try to satisfy a relatively large number of constraints in the posed problems. In contrast, this study focuses on the violation of constraints in the intermediate products while posing problems. The result shows that learners were inclined to avoid as many violated constraints as possible throughout the problem-posing process. Although learners tend to avoid the violated constraints, naturally, they cannot avoid some mistakes. Further analysis shows that learners actually have difficulty in fulfilling particular constraints while posing the problems. Based on this analysis, it is possible to detect the difficulty of learners’ actions from the model perspective. Hence, it is possible to give accurate feedback and appropriately support the learners. Keywords: Problem-posing process, Intermediate products, Arithmetic word problems, Learning analytics Introduction Problem posing is recognized as a key component in the nature of mathematical thinking (Kilpatrick 1987). Posing a problem involves generating new problems and questions aimed at exploring a given situation as well as reformulating a problem during the course of solving a related problem (Silver 1994). The development of problemposing skills for learners is one of the main aims of learning mathematics, and it should occupy a significant role in mathematical activities (Crespo 2003). There is an increased emphasis on providing learners with opportunities for posing problems in the mathematics classroom (Stoyanova 2005; Singer et al. 2011; Cankoy 2014). Several investigations have confirmed that learning by problem posing in classrooms is a promising activity in learning mathematics (Silver and Cai 1996; English 1998). The quality of problems that learners generate depends on the given assignments (Leung and Silver 1997). In posing a problem, assessment of each problem and assistance © The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Supianto et al. Research and Practice in Technology Enhanced Learning (2017) 12:12 based on it are necessary (Hirashima et al. 2007). Teacher assessment of posed problems encompasses learners’ development of diverse mathematical thinking processes (English 1997). Since learners are usually allowed to pose several kinds of problems in a broad range, it can be challenging for teachers to complete the assessment and feedback for the posed problems in classrooms. To address this issue, technology-enhanced approaches have been conducted to realize learning by problem posing in a practical way, especially regarding assessment and feedback. Self- and peer-assessed posed problems were examined to determine the effect of learners’ self-assessment of their mathematical creativity (Shriki and Lavy 2014), to explore learner’s learning and knowledge sharing while engaged in an online questionposing and peer-assessment activity (Barak and Rafaeli 2004) and to determine which peer-assessment mode(s) students perceive most positively using student generation of questions (Yu 2011). In contrast, diagnosis functions that can automatically assess and provide feedback for each posed problem have been proposed (Nakano et al. 1999; Hirashima et al. 2000). This automatic method of diagnosis-facility assessment is called agent assessment. Furthermore, a learning system named Monsakun, which uses agent assessment for operations of addition and subtraction, has been developed (Hirashima et al. 2007). The system has many problem-posing assignments and requests learners pose the required problem by combining three simple sentences from given sentences until they successfully pose the required problem in each assignment. Using this system, the opportunity to pose the problems for learners increases. The feedback to learners according to their mistakes is provided, and for teachers, checking the validity of the posed problems becomes easier. This study aims at analyzing the practical realization of agent assessment to understand the learning process of posing problems. Using Monsakun as a problem-posing learning system, learners’ abilities to solve problems as well as to understand them are promoted. In practical use and long-term evaluation, it was confirmed that learning by problem posing with Monsakun is interesting and useful as a learning method (Hirashima et al. 2008). Lectures and exercises with Monsakun improve not only learners’ problem-posing skills but also their problem-solving skills (Yamamoto et al. 2012). Through previous research, the usefulness of Monsakun has been confirmed for learning through posing problems. The basis of Monsakun is the triplet structure model (Hirashima et al. 2014) that defines the structure of an arithmetic word problem using sentence integration. This model deals with an arithmetic word problem that is solved using only one arithmetical operation. This is the fundamental unit of conceptual quantity representation, and much more complex arithmetic word problems can be composed by the combination of the units. An arithmetic word problem in this model is an integration of three sentences representing numerical concepts. In addition to that, the model defines constraints for valid problems that must be satisfied. When a learner can pose the required problem in Monsakun, the problem certainly meets the constraints. In other words, posing problems in Monsakun is the di (...truncated)