The major purpose of teachers is that students develop an increasing mathematical ability that allows them to solve the different problems they face inside and outside school. Innovation and creativity play an important role, being a dynamic characteristic that students must develop. Creativity begins with curiosity and engages students in exploration and experimentation tasks where they can translate their imagination and originality. Research findings show that mathematical problem solving and problem posing are closely related to creativity (e.g. Silver, 1997). So, learning environments with problem solving/posing activity should be used in our classes in order to develop students’ creativity. Challenging tasks usually require creative thinking and our recent work in a project about patterns in the teaching and learning of Mathematics showed that patterns can contribute to the development of mathematical ability and creativity of students (e.g. Vale & Pimentel, 2011). So our purpose as mathematics educators is to provide all students (including future teachers) creative approaches for solving any problems and to think independently and critically. This way, future teachers should themselves develop these skills and go through the same type of tasks that they will offer their students

Mathematical creativity is a rather complex phenomenon. But we can notice that there are some commonalities in the different attempts to define it: (1) it involves divergent and convergent thinking; (2) it has mainly three components/dimensions that are fluency, flexibility, and originality (novelty); and (3) it is related to problem solving and problem posing (including elaboration and generalization). By other hand, patterns are a powerful tool in the mathematics classroom and can suggest several approaches, as well as they permeate all mathematics, and their study makes possible to get powerful mathematical ideas as generalization and algebraic thinking, where visualization can play an important role (Rivera, 2009). According to several authors, patterning tasks have creative potential as they may be open-ended, allow a depth and variety of connections with all topics of mathematics, both to prepare students for further learning and to develop skills of problem solving and posing, as well as communication.

As we know learning heavily depends on teachers. Teachers must interpret the curriculum and select good curricular materials and strategies to use in the classroom. To achieve this, teachers should propose tasks involving students in a creative form, and also be mathematically competent to analyze their students’ resolutions. Research shows that what students learn is greatly influenced by the tasks they are given (e.g. Stein & Smith, 2009). Therefore, it is important to have good mathematical tasks. Tasks must develop new approaches and creative ideas, so they must provide multiple solutions in order to raise the student flow of mathematical ideas, flexibility of thought and originality in the responses. Teachers must encourage students to create, share and solve their own problems, as this is a very rich learning environment for the development of their ability to solve problems and their mathematical knowledge. Creativity should be an intrinsic part of mathematics for all programs (Pehkonen, 1997).

Pehkonen, E. (1997). The State-of-Art in Mathematical Creativity, ZDM, 29, 3, electronic edition. Rivera, F. (2009). Visuoalphanumeric mechanisms that support pattern generalization. In I. Vale & A. Barbosa (Orgs.), Patterns: Multiple perspectives and contexts in mathematics education (pp.123-136). Viana do Castelo: Escola Superior de Educação. Silver, E. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 3, 75-80. Stein, M., & Smith, M. (2009). Tarefas Matemáticas como quadro para a reflexão. Educação e Matemática, 22-28. Vale, I. & Pimentel, T. (2011). Mathematical challenging tasks in elementary grades, In M. Pytlak, T. Rowland & E. Swoboda, Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education, pp.1154-1164. Rzeszow: ERME