Computer-supported problem posing by annotated expressions: content-first design and evaluation

J. Comput. Educ. (2014) 1(4):271–294 DOI 10.1007/s40692-014-0019-5 Computer-supported problem posing by annotated expressions: content-first design and evaluation Hercy N. H. Cheng • Yu-Lin Weng • Tak-Wai Chan Received: 14 September 2014 / Revised: 20 October 2014 / Accepted: 24 October 2014 / Published online: 1 November 2014 Ó Beijing Normal University 2014 Abstract Because previous research has indicated a highly positive relationship between problem solving and posing, the process of problem solving can be adopted to design materials for problem posing. In this vein, the purpose of this study is to design a learning system for problem posing by annotated expressions through adoption-based research, in particular, a content-first design approach. More specifically, this study, inspired by solution trees, first designs annotated expressions for problem posing as content samples. In order to assess the effect on the assessment goal of problem-solving abilities, this study conducts an experiment, whose results show that problem posing by annotated expression may improve students’ performance on problem translation and two-step problem formulation abilities more than problem posing by pure expressions. Accordingly, this study then designs a computer-based learning activity to support and sustain such a method for problem posing. Finally, a classroom adoption, conducted in an authentic classroom, may suggest a positive outcome and future research on designing computer-based problem posing. Keywords design Problem posing
Annotated expressions
Solution trees
Content-first H. N. H. Cheng
Y.-L. Weng
T.-W. Chan Graduate Institute of Network Learning Technology, National Central University, Jhongli, Taiwan H. N. H. Cheng (&) Collaborative & Innovative Center for Educational Technology, Central China Normal University, Wuhan, China e-mail: Y.-L. Weng Department of Information Management, St. Mary’s Junior College of Medicine, Nursing and Management, Yilan, Taiwan 123 272 J. Comput. Educ. (2014) 1(4):271–294 Introduction While mathematics curriculum still focuses on problem solving, problem posing has also been emphasized with the reform of mathematics education in various countries. In United States, for example, National Council of Teachers of Mathematics (1989) explicitly stated that students should recognize and formulate their own problems in an early document. Later they specifically suggested that students should formulate problems from given situations and create new problems by modifying the conditions of a given problem (NCTM 1991, 1995). In other words, students were not supposed to solve problems only. Instead, students may also change data, adjust variables and construct a new problem (NCTM 2000). Compared with the education in western countries, Chinese curriculum was usually criticized for stressing too much in basic skills and knowledge (Zhang 2006). For this reason, the Ministry of Education of Taiwan (2000) emphasized the aim of developing students’ ability to formulate and solve problems in a nine-year integrated curriculum for mathematics. Similarly, China carried out a mathematics curriculum reform movement (Cai & Nei 2007). In particular, China included problem posing in mathematics curriculum. Furthermore, the standards for the compulsory education required students to pose problems, understand problems, and apply knowledge learnt to solve authentic problems (Basic Education Curriculum Material Development Center 2001). In summary, various countries had started to incorporate problem posing into mathematics curricula based on problem solving. In these mathematics curricula, problem posing was regarded as an extension of problem solving, specifically, a creation process after problem solving (Dillon 1982). In a sense, problem solving was a form of consuming mathematical knowledge, while problem posing was a form of creating mathematical knowledge, which involved imitation, conversion, and integration of mathematical knowledge. In other words, while students applied mathematical knowledge in problem solving, they may create mathematical knowledge in problem posing. Research also showed that problem posing could enhance students’ problem-solving abilities (Brown & Walter 1993; English 1996). As a matter of fact, problem posing could also be regarded as a source of problem solving. Leung (1993) modified Pólya’s framework of problem solving (Pólya 1945) to four steps of problem posing—posing a problem, devising a solution plan, carrying out the plan, and looking back the solution. More specifically, when students played the role of problem posers, they had already understood the problem that they posed. Because they could easily realize the content and structure of the problem, they were able to solve it and looking back the solution. These studies suggested that problem solving and posing were mutually complementary mathematics learning activities. Besides, research also showed that problem posing may facilitate creative thinking (Pelczer, & Rodriguez 2010; Silver 1997; Yuan & Sriraman 2011). In terms of a revised Bloom’s taxonomy of educational objectives (Anderson & Krathwohl 2001), problem solving included remembering, understanding, applying, analyzing, and evaluating without creating. On the contrary, problem posing may unlock the limitation of rote and comprehension, encouraging students to apply what they had learnt from a whole new perspective, to analyze all possibilities, and 123 J. Comput. Educ. (2014) 1(4):271–294 273 to evaluate reasonable contexts, and finally to create an elaborate work. In other words, when creation became an educational objective, students were able to synthesize the abilities mentioned above and then creating new knowledge. In addition, problem posing actually facilitated not only students’ cognition but also their positive disposition (NCTM 2000). Research indicated that problem posing potentially lessen mathematical anxiety because students may feel less pressure (Brown & Walter 2005). Problem posing may also provide students with a sense of ownership for contributing their own knowledge, resulting in a high level of engagement and interests (Lavy & Shriki 2007). As a result, problem posers could be active learners, who were willing to invest their efforts to develop their own thinking (Freire 2001). In a classroom with problem posing, students were not listeners any more. Instead, they communicated with each other and their teachers through creation. However, Silver & Cai (1996) found that although students with high problemsolving abilities could construct more difficult and complicated problems, those students with low problem-solving abilities tended to pose easy and simple problems. The findings suggested that low-ability students still need additional assistance for designing more meaningful problems. For this reason, the purpose of this study aims at designing a computer-based learning ma (...truncated)