PROSPECTIVE MATHEMATICS TEACHERS' LEARNING OF STUDENTS' MATHEMATICAL THINKING

Noticing what is happening in a classroom is an important skill for teachers (Mason, 2002) and particularly, noticing the students' mathematical thinking. In initial teacher education, the research on the development of noticing attempts to provide information on how prospective teachers learn to make sense the mathematics teaching and learning (Sanchez-Matamoros, Fernandez, & Llinares, 2014). On the other hand, the on-line approaches in mathematics teacher education generate particular issues about how the noticing skill could be developed in blended learning contexts. In a specific way, these contexts provide opportunities in which prospective teachers can validate with others their interpretations of mathematics teaching situations that support the development of the noticing skill (Fernandez, Llinares, & Valls, 2012). In this study, we have designed a b-learning environment to help prospective teachers (PTs) to notice the secondary student's mathematical thinking of a specific subject (classification of quadrilaterals). For the designing of the learning environment we adapted Wells (2002)'s socio-cultural learning perspective considering that individuals should encounter different opportunities for making sense of students' mathematical thinking and learn to make decisions to help the student to progress in his/her learning. These opportunities should take account the PTs' previous experience, the information provided by the teacher educator, the processes of knowledge building and the new understanding built about students' mathematical thinking and the teaching resources. The b-learning environment involved prospective teachers in four types of activities: Anticipate students' hypothetical answers showing different levels of students' understanding (Experience); read and discuss theoretical papers with information about features of students' mathematical thinking and the level of understanding (Information); refine their understanding with others by solving the initial tasks in pairs; and refine their understanding and validate their interpretations with others by participating in a on-line debate (Knowledge building) We conjectured that participating in this sequence of activities could help to prospective teachers to develop different modes of knowing (instrumental, procedural, and theoretical) related to skill of noticing the mathematics teaching. Our findings indicate that the participation in this type of learning activities sequence support the generation of PTs' new knowledge integrating the ideas about students' mathematical thinking when PTs have to anticipate hypothetical students' answers to problems and have to make decisions about how select the tasks to promote the improvement of students' mathematical understanding.