CHARACTERISTICS OF PRE-SERVICE MATHEMATICS TEACHERS' NOTICING OF STUDENTS' MATHEMATICAL THINKING

Noticing what is happening in a classroom is an important skill for teachers. However, noticing effectively is both complex and challenging (Jacobs, Lamb, & Philipp, 2010; Mason, 2002). Noticing has been defined in a multitude of ways but the common theme is how teachers process complex classroom events. In initial teacher education contexts, the interest on the development of noticing should be seen as a way to understand how prospective teachers learn to make sense the mathematics teaching and learning. Recent studies have shown that the noticing skill can be developed and have identified aspects that support its development (Fernandez, Llinares, & Valls, 2011; 2012, Sherin, Jacobs, & Philipp, 2010). A particular focus for mathematics teachers' noticing is the students' mathematical thinking. In this study, we focus on the development of prospective mathematics teachers' noticing of student's understanding of the derivative concept designing a learning environment based on a b-learning methodology. We adopted Wells (2002)'s socio-cultural perspective for designing the learning environment considering that individuals encounter different opportunities for making sense of students' mathematical thinking: experience, information, knowledge building and understanding. Eight prospective mathematics teachers (PTs) participated in a learning environment consisted of seven sessions of two-hours each (one session per week). In the first session, PTs answered a questionnaire. Their answers allowed us to identify how they identified the high school students' understanding of the derivative concept ("Experience"). During the next five sessions PTs read and discussed theoretical papers ("Information" about students' mathematical thinking) about the learning of derivative concept, and about the mathematical tasks related to the derivative concept. PTs worked individually or in pairs in activities that required describing and interpreting students' answers using the theoretical information. These sessions were implemented in an online-platform to enhance the interaction among prospective teachers. Next, there was a discussion to share their interpretations ("Knowledge building''). In the last session, PTs answered another questionnaire that provided us information about the changes on PTs noticing of students' mathematical thinking ("Understanding"). Results show that the learning environment integrating face-to-face and online sessions helped to develop the prospective teachers' skill of noticing of students' mathematical thinking but this development was not simple. The development of the noticing skill was related to PTs ability to describe students' answers to the problems using noteworthy mathematical elements.