Prospective teachers’ knowledge for teaching mathematics and the way they may develop it are controversial issues (Ball, Thames & Phelps, 2008; Ponte & Chapman, 2015; Shulman, 1986). The practicum is an important site to study such knowledge since prospective teachers are called to deal with circumstances that allow noticing important weaknesses and strengths.

Teachers’ knowledge includes mathematical and didactical knowledge, both of critical importance for teaching practice (Ball et al., 2008; Ponte & Chapman, 2015; Shulman, 1986). Mathematics knowledge involves conceptual and procedural aspects (Hiebert, 1988; Rittle-Johnson & Schneider, 2015). Conceptual knowledge is a network of concepts and procedural knowledge consists of rules and procedures for solving problems (Bartell et al., 2012). Conceptual mathematical knowledge of rational numbers involves knowing different representations and meanings (Ma, 1999) and allows establishing connections among different topics (e.g., seeing multiplication as repeated addition).

Didactical knowledge concerns with how teachers teach (Ponte & Chapman, 2015). Teachers must anticipate students’ common mistakes and misconceptions (e.g., generalizing addition procedures for multiplication), to anticipate students’ solutions in specific tasks, and to know what students consider challenging, interesting or confusing). Teachers also need to know to sequence tasks, to recognize the value of different representations, to pose questions, and to explore students’ strategies. In addition, they need to understand the main ideas of current curriculum documents (e.g., NCTM, 2007, 2014).

Ponte, Quaresma and Branco (2012) characterize teachers’ practice by two main aspects: the tasks proposed to students and the communication established in the classroom. Regarding tasks, teachers may choose to offer simple exercises or propose problems, investigations in which the students need to design and implement solution strategies based on their previous knowledge (Ponte, 2005). Classroom communication may be univocal or dialogic, depending on the roles assumed by the teacher and the students and the types of teachers’ questions, including inquiry, focusing, or confirmation questions (Ponte et al., 2012). Representations are an important feature of tasks, and may be categorized as pictorial (images), iconic (points, lines, circles), and notational (number lines, arrows, columns, symbols) (Thomas, Mulligan & Goldin, 2002).

The teachers’ actions influence the classroom dynamics. The teachers’ actions are related to the mathematical concepts and processes and with management of learning. Ponte, Mata-Pereira and Quaresma (2013) consider four main types of actions: inviting, to begin the discussion; supporting/guiding, leading the students through different kinds of questions; informing/suggesting, giving information or validating students’ ideas; and challenging, encouraging students in interpreting situations, finding a new representation, making a generalization or justification, making connections, and evaluating their work.

To foster students’ understanding of mathematics concepts and procedures, teachers are called to engage them in making connections among representations (NCTM, 2007, 2014). They need to support students in using symbols in a fluent way, grounded in the use of informal representations. To do so, teachers need to know how to help students to build this knowledge (Ball et al., 2008; Ma, 1999; Ponte & Chapman, 2015). Rational numbers are a topic that raises many difficulties for students and that challenges teachers how to promote conceptual learning (Lamon, 2006; Ma, 1999). Research needs to focus on the use of different representations in teaching rational numbers and in the learning and the struggles that prospective teachers may experience in providing representations that may help students to develop their knowledge on this topic. The aim of this study is to identify the knowledge of a prospective teacher in the teaching and learning of rational numbers, with a focus on the use of informal and formal representations, analysing the knowledge she mobilized in teaching practice, her struggles, and the knowledge built from her first practical experiences.

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