Noticing

"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."

This study is part of the research in didactics of mathematics on teacher training. In particular, the study focuses on the knowledge of mathematics teachers, taking as a reference the theoretical approaches of the Mathematics Teacher’s Specialized Knowledge (MTSK) model and teacher reflection as a professional competence. The study corresponds to a descriptive-qualitative research, based on an instrumental study of cases, whose purpose is to describe and analyze the manifestations of mathematics teachers of Secondary Education when they observe a teaching-learning situation that shows a teacher of mathematics teaching the concept of relation. For this, narratives written by seven Secondary Education mathematics teachers in service were used, which indicate some aspects identified by them as relevant from the observed situation. The analysis of the gathered information was carried out by defining categories, subcategories, and analysis units based on the components of the MTSK model. The results indicate that participating teachers accentuate, in their manifestations, attributes mainly associated with didactic knowledge of mathematical content. The reflection of this group of teachers is characterized by three specific aspects of the MTSK model: teaching, learning, and knowledge of the topics. It is interpreted that teacher reflection has been influenced by the professional experience of the teachers participating in the study.

We believe, after Tsing, that noticing can help researchers better understand "the divergent, layered, and conjoined projects that make up worlds" (22) by amplifying, augmenting, and attuning their attention to a wide range of actors, perspectives, and relations.

3D virtual worlds and game-based simulations are to an increasing degree used for military training. However, there is a lack of research-based methods for using game-based simulations and developing scenarios for educational role-plays in a military context, especially in the area of cultural awareness. The CAMO project (Cultural Awareness in Military Operations) seeks to address these challenges. The goal of the project has been to create a game-based simulation in Second Life for training cultural awareness among military personnel preparing for international operations and to explore the advantages and limitations of 3D virtual worlds in this context. This paper will focus on the methodology for scenario development, outlining the challenges and directions for future work.

Construct maps are important tools in educational assessment and can serve multiple purposes related to development and validation, as well as score interpretation and use. This chapter outlines a process for developing a construct map from the qualitative ordering of teachers' responses to open-ended assessment items. The construct of interest pertains to a teacher's ability to attend to what students say and do, which is a key component of many recommendations for instructional practice within mathematics education. The instrument we are developing is designed to measure teachers' attentiveness to student thinking in quantitative reasoning problem situations. A key aspect of our instrument development process is the development of a construct map that hierarchically orders qualitatively different levels of teacher attentiveness. In this chapter we describe our process for developing the construct map with the intent of providing an example to others who may be interested in engaging in the development of construct maps.

Noticing the hole (NTH) occurs when speakers want to say something, but realise they do not know the right word(s). Such awareness of lacking knowledge supposedly facilitates the acquisition of the unknown word(s) from later input (Swain, 1993). We tested this claim by experimentally inducing NTH in a second language (L2) for some participants (experimental), but not others (control). Then, in a price comparison game, all participants were exposed to spoken L2 input containing the to-be-learned words. They were unaware of taking part in an L2 study. Post-tests showed that participants who had noticed holes in their vocabulary had indeed learned more words compared to participants who had not. This held both for the experimental group as well as those participants in the control group who later reported to have noticed holes. Thus, when we become aware of vocabulary holes, the first step to improve our vocabulary is already taken.