It is well recognized that an important component in mathematical instruction is to provide students with an opportunity to develop and use diverse representational systems in order to solve a variety of mathematical tasks (National Council of Teachers of Mathematics, 2000). Representations play an important role during the construction of models that help students to solve problems. What type of problem-solving activities do students need in order to develop ways of thinking that value the use of distinct representational systems to understand, and solve, and eventually propose problems? Teachers often state that it may be difficult for them to formulate new problems or situations that actually help their students search for different ways to approach them. It is also recognized that textbooks are the main source of examples that teachers use in their classrooms. Can textbook exercises be transformed into problem-solving activities that encourage students to develop mathematical thinking? This study documents what high school students showed when they were explicitly asked to use technological tools to examine and solve a set of routine problems from different angles or perspectives.
A fundamental instructional principle used to organize and structure the learning activities implemented in this study was to encourage students to think of different ways (construction of models) to solve problems and to discuss strengths and limitations associated with each solution method. Goldenberg (1995) states that:
In current practice, the great bulk of mathematics teaching takes place within a single representational system. Much time and effort are spent in building students' skills in manipulating the formal symbolic language of traditional classroom mathematics, while relatively little time is devoted to other representations of the same ideas (p. 156).
Thus, it is important to acknowledge that students' mathematical understanding involves not only the use of various representations but also being able to transit, in terms of meaning, from one representation into another. That is, it becomes important that students' learning experiences not only focus on reporting solutions but also on identifying features of the models used to solve the problems. Here, the process of finding different methods to approach the tasks requires that students use several types of representations that help them develop appropriate conceptual systems. These systems tend to be expressed by students through models that involve the use of descriptions, explanations, and the use of diverse representations (Lesh, in press). In particular, the use of technology often offers students an important window to observe and examine connections and relationships that become relevant during the solution process.
Lines of Mathematical Inquiry
There are different learning trajectories for students to take in order to achieve mathematical competence; however, a common ingredient is a need to develop a clear disposition toward the study of the discipline. Such a disposition includes a way of thinking in which students value: (a) the importance of searching for relationships among different elements or components of the tasks in study (expressed via mathematical resources), (b) the need to use diverse representations to examine patterns and conjectures, and (c) the importance of providing and communicating different arguments (Santos, 1998). Thus, it becomes important to encourage students to think of the discipline in terms of dilemmas or challenges to be met and resolved. This means that they need to conceptualize their learning experiences in terms of activities that involve posing questions, identifying and exploring relationships, and providing and supporting their answers or solutions (NCTM, 2000). It is necessary to value the students' participation and persuade them about the power of reflecting …