"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 (Fernández, 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."

"El objetivo de esta investigación es caracterizar algunos indicadores del desarrollo del
esquema de derivada en estudiantes de pos-secundaria. Usamos los niveles intra, inter
y trans del desarrollo de un esquema propuestos por Piaget y García para caracterizar el
uso flexible que los estudiantes hacen de la equivalencia lógica entre diferentes
elementos matemáticos cuando resuelven un problema, como un indicador del desarrollo
del esquema de derivada. Este indicador ayuda a explicar la transición entre los niveles
inter y trans de desarrollo del esquema derivada"

Esta investigación estudia el efecto de un módulo de enseñanza sobre la manera en la que estudiantes para profesor de matemáticas de educación secundaria identifican la comprensión de la derivada en estudiantes de Bachillerato. Los resultados indican que el desarrollo de esta competencia está vinculada a los elementos matemáticos de la noción de derivada que los estudiantes para profesor son capaces de considerar al identificar evidencias de la comprensión de la derivada en las respuestas de los estudiantes e interpretarlas.

This paper reports on different underlying structures of the derivative schema of three undergraduate students that were considered to be at the trans level of development of the derivative schema (action-process-object-schema). The derivative schema is characterized in terms of the students' ability to explicitly transfer the relationship between a function and its first derivative to the derivative function and the second derivative. This conscious shift of properties of derivatives is differently manifested by the students in the trans level of development of the derivative schema and can be considered evidence of the different characteristics of the thematization of derivative schema. From here we suggest that there are different underlying structures in the constructed schema due to the consciousness in which students use the relations between a function and its derivative.

One of the core ways allowing obtain an ampler and deeper understanding on the content of teachers' knowledge concerns analysing teachers' practices. Such practice can be perceived in a broader way, not limited to classroom practice. On the other side, discussing and reflecting on the same situation with different theoretical and methodological approaches seems to contribute also for obtaining a deeper understanding not only on such practice but also on the used approaches for such analysis. In this paper we present and discuss part of the work developed in the intermeeting of the group research teachers' knowledge and development of SEIEM concerning the potentialities of analyzing one episode using five different theoretical approaches.