This study focuses on the pitfalls associated with incorporating multiple solution connecting tasks in classroom instruction. We found implementation of these standard-based materials problematic due to teachers' reluctance to use them in their classes. Research questions probed the teachers' explanations for their lack of enthusiasm. The tasks were implemented with students of two tenth-grade classes on the basic and medium level. Contrary to the teachers' opinions, unexpected number of the students demonstrated positive attitudes to coping with connecting tasks and were able to differentiate between different solutions. Although teachers thought that students would resist receiving and providing peer explanations, the students mostly did not verify such a resistance. Moreover, students learning basic-level mathematics preferred providing explanations to their peers, while students on the medium level preferred receiving explanations. We believe that a special intervention is needed to convince mathematics teachers to employ connecting tasks in their classrooms. Presenting teachers with the results of this study may constitute one of the parts of such an intervention.
In this paper a task that may be attributed to different topics in the mathematics curriculum and thus may be solved in different ways is called a multiple-solution-connecting-task (shortly connecting task). The implementation of connecting tasks, which are under consideration in this paper, may serve as an effective model of mathematics teaching (Cooney, 2001; Simon, 1997), which entails student-centered learning and fosters construction of students' mathematical understanding. This study is based on the belief that mathematical thinking involves looking for connections, while making connections is essential for constructing mathematical understanding (NCTM, 2000). When students begin to see connections across different branches of mathematics they develop their mathematical integrity. We connect theory and practice through holding that solving problems in different ways is a powerful mathematical activity that promotes constructions of mathematical connections.
Find the distance between the point A (1, 6) and the straight line l : y = x-1.
This problem is borrowed from analytical geometry and usually is solved in mathematics classes in one particular way. A connecting task focused on this problem asks students to learn how to solve it in different ways. Figure 1--"map of the task"--presents a variety of the solution paths for this mathematical problem. It establishes connections between the concepts by means of four equivalent definitions. In this way different geometrical concepts (i.e., circle, area, perpendicular lines) are connected by the concept of distance. Additionally, within each approach different solution paths are possible. These paths connect different mathematical topics and concepts. For example, the minimal value of a quadratic function (in one of the solutions can be found in different ways (see Figure 1).
As literature shows, connections form an essential part of mathematical understanding (e.g., Hiebert & Carpenter, 1992; Kieren, 1990; Sfard, 1991; Sierpinska, 1990; Skemp, 1987). Skemp, for example, discussed two types of mathematical understanding, relational and instrumental. Relational understanding determines one's ability to answer such questions as "How should I approach the problem?" "Why should I do this?" Instrumental understanding facilitates answering the question, "What procedure should I use to solve the problem?" without necessarily knowing the answers to the first two questions. The two types of understanding are critical for making progress in learning mathematics. Instrumental understanding determines the speed of solving procedures and the immediacy of the results. …