Visualizations of problem-posing activity sequences toward modeling the thinking process

Supianto et al. Research and Practice in Technology Enhanced Learning (2016) 11:14 DOI 10.1186/s41039-016-0042-4 RESEARCH Open Access Visualizations of problem-posing activity sequences toward modeling the thinking process Ahmad Afif Supianto1,2*, Yusuke Hayashi1 and Tsukasa Hirashima1 * Correspondence: afif@lel. hiroshima-u.ac.jp 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 Problem-posing is well known as an effective activity to learn problem-solving methods. Although the activity is considered in contributing to the understanding of the problem’s structure, it is not clear how learners could understand it through the activity. This study proposes a method to offer a visual representation for analyzing the problem-posing activity sequence in MONSAKUN, a digital learning environment for problem-posing of arithmetic word problems via sentence integration. This system requires users to pose a problem through combinations of given simple sentences based on the requirement. The system writes every single action into logs as sequences of problem-posing activity. The sequences are considered to represent the thinking processes of learners. The thinking process reflects their understanding and misunderstanding about the structure of the problems. This study created visualizations of learners’ problem-posing processes from the data obtained through the practical use of MONSAKUN, including the states in which many learners had difficulties finding the correct answer. In this study, we refer to such states as “trap states.” In MONSAKUN, a trap state is a combination of simple sentences where many learners tend to make and need relatively more actions to obtain the correct answer. As the result of the visualization and analysis of the data, some trap states have been identified, and they changed for each trial in the same problem. Keywords: Problem-posing process, Problem state space, Visualizations Introduction The importance of problem-posing activity in mathematics Problem-posing is considered to be an essential part of mathematical activity (Brown and Walter 1993). Problem-posing 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). Providing students with an opportunity to pose their own problems can foster flexible thinking, enhance problem-solving skills, broaden their perception of mathematics, and enrich and consolidate basic concepts (Brown and Walter 1993; English 1996). The development of problem-posing skills for learners is one of the principal aims of mathematics learning, and it should occupy a central role in mathematics activities (Crespo 2003). Moreover, problem-posing activities could provide us with valuable insights into children’s understanding of mathematical concepts and processes, as well as © 2016 The Author(s). 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 (2016) 11:14 their perceptions of, and attitudes toward, problem-solving and mathematics in general (Brown and Walter 1993). For the improvement of students’ learning in problemposing, it is important to develop an understanding of the developmental status of learners’ thinking and reasoning. The more information that can be obtained about what learners know and how they think, the more opportunities could be created for the enhancement of learners’ success (Cai 2003). Learning by problem-posing in conventional classrooms has been studied, and several investigations have confirmed that problem-posing is a promising learning activity. English (1998) investigated the problem-posing abilities of children who displayed different profiles of achievement in number sense and novel problem-solving. Silver and Cai (1996) found that students generated a large number of solvable mathematical problems, many of which were syntactically and semantically complex, and that nearly half the students generated sets of related problems. Assessment of the posed problems In problem-posing, assessment of each posed problem and assistance based on it are necessary (Hirashima et al. 2007). Teacher assessment of posed problems encompasses learners’ development of diverse mathematical thinking (English 1997). The other one is investigating learners’ behavior of the problem-posing process by extending their posed problems (Singer et al. 2011). However, since learners are usually allowed to pose several kinds of problems, including a large range of them, it can be challenging for teachers to complete assessment and feedback for the posed problems in the classroom. Technology-enhanced approaches have been used to realize learning by problemposing, especially in regard to assessment and feedback. A networked, question-posing learning system enabling students to pose questions was developed by Yu et al. (2003– 2004). A novel way for merging assessment and knowledge sharing using an online Question-Posing Assignment (QPA) has been examined (Barak and Rafaeli 2004). An online learning system with a focus on student-question generation strategy, called QuARKS (Question-Authoring and Reasoning Knowledge System) (Yu 2009), was adopted and the effects of student question-generation on civics and citizenship (Yu and Pan 2014) and English learning (Yu et al. 2015) have been reported. Student academic achievement, question-generation performance, learning satisfaction and learning anxiety, as well as learning motivation have been investigated. Self- and peerassessed posed problems were examined in these studies. In contrast, diagnosis functions that can assess and give automatic feedback to each posed problem have been proposed (Nakano et al. 1999; Hirashima et al. 2000). This automatic way of diagnosis-facility assessment is called agent assessment. Furthermore, learning environment systems that use practical agent assessment have been developed (Hirashima et al. 2008a, 2008b, 2014). This research aimed at the practical realization of agent assessment in order to understand the process of learners’ problem-posing, so that it could be analyzed. The current study uses the same kind of learning environment system that the above studies in agent assessment used. A new design of the problem-posing learning environment using agent assessment (...truncated)