This study aims to analyze the perspectives of a group of five teachers of grades 5-6 participating in a lesson study about the tasks to propose and the work to carry out in the classroom in order to promote students’ learning, with special attention to the development of mathematical reasoning.

As Ponte, Mata-Pereira and Henriques (2012) indicate, to reason is to make inferences with justification. To reason is not present unconnected ideas, but rather to use given information to obtain new information, valid in a field of knowledge. Lannin, Ellis and Elliott (2011) consider that mathematical reasoning essentially involves making generalizations and justifications. To promote the development of students’ reasoning the teachers have to make decisions, set educational pathways and select tasks carefully. For that, more than isolated tasks, they need to organize sequences of tasks with different levels of challenge and structure that create opportunities for generalizing and justifying.

Since tasks are a fundamental organizing element of students’ activity, this concept is essential if mathematics teaching values students’ active role in learning. Ponte (2005) suggests that tasks have two fundamental dimensions: mathematical challenge and structure. The degree of challenge depends on the perception of difficulty for each person and the degree of structure (open/closed) refers to the nature of the information given, which may require different levels of interpretation from the students. Combining the two dimensions, one obtain four main task types: (i) exercises, closed tasks with little mathematical challenge; (ii) problems, also closed tasks, but with a high mathematical challenge; (iii) investigations, open tasks that present high mathematical challenge; and (iv) explorations, open tasks that are accessible to most students. This author suggests that diversification of tasks is needed since each type plays a specific role in students’ learning.

In an exploratory approach to mathematics teaching students are called to deal with tasks for which they do not have an immediate solution method (problems, investigations, explorations). Students have to construct their own methods using their previous knowledge. Exploratory work creates opportunities for students to build or deepen their understanding of mathematical concepts, procedures, representations and ideas. Students are thus called to play an active role in interpreting the questions, representing of the information given and in designing and carrying out solving strategies which they must justify to their colleagues and to the teacher. Conducting exploratory mathematics teaching is receiving an increasing support in mathematics curriculum orientations (e.g., NCTM, 2000, 2014). However, it represents a serious challenge for teachers, demanding specific knowledge, competency, and disposition.

Lesson study is a professional development process focused on professional practice. An important feature of lesson studies is their reflexive and collaborative nature (Fernandez, Cannon, & Chokshi, 2003; Perry & Lewis, 2009). Teachers identify an important issue and work together, analysing students’ difficulties, discussing curriculum alternatives, and preparing what they expect to be an “exemplary” lesson. Afterwards, they verify to what extent this lesson achieves the sought objectives and what difficulties arise. Therefore, a lesson study is a process very close to a small investigation developed by the participating teachers on their own professional practice, informed by curriculum guidelines and by research results on the given issue. A central aspect of a lesson study is that it focuses on the students’ learning and not on the teachers’ work. Indeed, lesson studies aim to examine students’ learning and to observe the way they learn. By participating in lesson studies, teachers may learn about mathematics, curriculum guidelines, students’ processes and difficulties, and classroom dynamics. Lesson studies provide opportunities for teachers to reflect on the possibilities of an exploratory approach to mathematics teaching.

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