PROBLEM POSING - MATHEMATICAL KNOWLEDGE

PROBLEM POSING AS A MEANS FOR DEVELOPING MATHEMATICAL KNOWLEDGE OF PROSPECTIVE TEACHERS

Problem posing (PP) is recognized as an important component of mathematics teaching and learning (NCTM, 2000). In order that teachers will gain the knowledge and the required confidence for incorporating PP activities in their classes, they have to experience it first. While experiencing PP they will acknowledge its various benefits. Hence such an experience should start by the time these teachers are being qualified towards their profession. Therefore, while working with prospective teachers (PT) we integrate activities of PP into their method courses. In addition, accompanying the process with reflective writing might make the PT be more aware to the processes they are going through (Campbell et al, 1997), and as a result increase the plausibility that the PT will internalize the effect of the processes that are involved in PP activities. This reflective writing also enables teacher educators to evaluate the PT progress and performance (Arter & Spandel, 1991). In the present study we aim at exploring the effects of experiencing PP on the development of PT’s mathematical knowledge and problem solving skills. 

Problem posing. Problem posing is an important component of the mathematics curriculum, and is considered to be an essential part of mathematical doing (Brown & Walter, 1993, NCTM, 2000). PP involves generating of new problems and questions aimed at exploring a given situation as well as the reformulation of a problem during the process of solving it (Silver, 1994). Providing students with opportunities to pose their own problems can foster more diverse and flexible thinking, enhance students’ problem solving skills, broaden their perception of mathematics and enrich and consolidate basic concepts (Brown & Walter, 1993, English, 1996). In addition, PP might help in reducing the dependency of students on their teachers and textbooks, and give the students the feeling of becoming more engaged in their education. Cunningham (2004) showed that providing students with the opportunity to pose problems enhanced students’ reasoning and reflection. When students, rather than the teacher, formulate new problems, it can foster the sense of ownership that students need to take for constructing their own knowledge. This ownership of the problems results in a highly level of engagement and curiosity, as well as enthusiasm towards the process of learning mathematics.

The ‘What If Not?’ strategy. Brown & Walter (1993) suggested a new approach to problem posing and problem solving in mathematics teaching, using the ‘What If Not?’ (WIN) strategy. The strategy is based on the idea that modifying the attributes of a given problem could yield new and intriguing problems which eventually may result in some interesting investigations. In this problem posing approach, students are encouraged to go through three levels starting with re-examining a given problem in order to derive closely related new problems. At the first level, students are asked to make a list of the problem’s attributes. At the second level they should address the “What If Not?” question and than suggest alternatives to the listed attributes. The third level is posing new questions, inspired by the alternatives. The strategy enables to move away from a rigid teaching format which makes students believe that there is only one ‘right way’ to refer to a given problem. The usage of this problem posing strategy provides students with the opportunity to discuss a wide range of ideas, and consider the meaning of the problem rather than merely focusing on finding its solution.

The educational value of integrating problem posing into PT’s training programs. Teachers have an important role in the implementation of PP into the curriculum (Gonzales, 1996). However, although PP is recognized as an important teaching method, many students are not given the opportunity to experience PP in their study

of mathematics (Silver et al., 1996). In most cases teachers tend to emphasize skills, rules and procedures, which become the essence of learning instead of instruments for developing understanding and reasoning (Ernest, 1991).

To sum, reflective portfolios and class discussions turned out to be a useful tool for reflecting on processes, and tracing the PT’s development of mathematical knowledge. We found that involvement in PP has the potential to develop the mathematical knowledge, and consolidate basic concepts, as suggested by Brown & Walter (1993) and English (1996). This development of knowledge came to fruition especially in the ability to examine definitions and attributes of mathematical objects, connections among mathematical objects, the validity of an argument, and appreciation of the richness that underlines mathematical problems. However, one major weak point was discovered. The PT tended to attach to familiar objects, and were not ‘daring’. This tendency actually prohibits the development of problem solving skills and inquiry abilities instead of developing it. We found that this tendency can be explained by the redundant emphasize of the importance of

providing formal proof. Teacher educators need to identify ways for reducing PT’s fears from handling formal proves, and remove the focus to the analysis of a given situation, connections among mathematical objects and looking for generalization. This, in turn will develop their problem solving skills and their insights as regards to mathematical objects.Consequently, mathematics teachers miss the opportunity to help their students develop problem solving skills, as well as help them to build confidence in managing unfamiliar situations. Teachers rarely use PP because they find it difficult to implement in classrooms, and because they themselves do not possess the required skills (Leung & Silver, 1997). Therefore, PT should be taught how to integrate PP in their lessons. Southwell (1998) found that posing problems based on given problems could be a valuable strategy for developing problem solving abilities of mathematics PT. Moreover, incorporating PP activities in their lessons enables them to become better acquainted with their students' mathematical knowledge and understandings.