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INTRODUCTIONOver the last fifteen years since the publication of Clements' (1999) well-known paper, various scholars, particularly in the United states, have been encouraging teachers to attend to the development of young learners' conceptual subitising (See, for example, Clements & Sarama, 2009;Conderman et al., 2014); where conceptual subitising is the ability to recognise quickly and without counting relatively large numerosities by partitioning these large groups into smaller groups that can be individually subitised (Clements & Sarama, 2007;Geary, 2011). Various claims, which we discuss below, have been made with respect to the efficacy of conceptual subitising-focused instruction. In a related vein, our own recent work has focused on a conceptualisation of foundational number sense (FoNS), which we describe as those number-related competences expected of a typical first grade student that require instruction (Back et al., 2014;Andrews & Sayers, 2014a). FoNS is characterised by eight components, which we describe below. The purpose of this paper, drawing on excerpts from grade one lessons taught by a case study teacher in each of Hungary and Sweden, is to examine the extent to which conceptual subitising-focused activities have the propensity to facilitate students' acquisition of the various FoNS components and, in so doing, examine the warrant for their claimed efficacy. WHAT IS SUBITISING?Subitising refers to being instantly and automatically able to recognise small numerosities without having to count (Clements, 1999;Jung et al., 2013;Moeller et al., 2009;Clements & Sarama, 2009). Children as young as three are typically able to subitise numerosities up to three (Fuson, 1988, Moeller et al., 2009), while most adults are able instantly to recognise without counting the numerosity represented by the dots on the face of a die (Jung et al. 2013). This process, innate to all humans, is typically known as perceptual subitising (Gelman & Tucker, 1975) and forms an element of the preverbal number sense we describe below. In short, perceptual subitising is recognizing a numerosity without using other mathematical processes (Clement, 1999). Conceptual subitisingHowever, a second form of subitising, conceptual subitising (Clements, 1999), which is not unrelated to FoNS, has been shown to have considerable implications for teaching and learning. Conceptual subitising relates to how an individual identifies "a whole quantity as the result of recognizing smaller quantities... that make up the whole" (Conderman et al., 2014, p.29). More generally, it can be summarised as the systematic management of perceptually subitised numerosities to facilitate the management of larger numerosities (Obersteiner et al., 2013). For example, when a child is confronted by two dice, one showing three and another showing four, each is perceptually subitised before any sense of seven can emerge.Subitising can be construed as having a synonymity with the spatial structuring of numbers (Battista et al., 1998). In this case, the ability to ...

Background: As part of teachers' everyday classroom assessment practice, feedback can be seen as connected to the formative function of assessment, with the aim of helping students in their learning processes. Much research on teacher feedback focuses precisely on the feedback's formative quality. However, in order to strengthen our understanding about the nature of teacher feedback, we also need to understand more about teachers' rationales for giving feedback to their students, especially in primary school settings. Purpose: The present study aimed to explore and conceptualise primary school teachers' rationales for giving students feedback. Sample: Thirteen Swedish primary school teachers (10 women and 3 men) with 4 to 40 years of teaching experience working with students aged 7-9 years-old (grades 1-3), participated in the study. An open sampling procedure was adopted to recruit the teachers. Design and methods: Data were collected using a semi-structured interview approach. We employed a constructivist grounded theory design for the coding and analysis of the transcribed data. Results: Analysis indicated that two main concerns emerged as regulating teachers' assessment practices. These addressed what the teachers perceived as (1) students' academic needs and (2) students' behavioural and emotional needs. According to the findings, the teachers' rationales for giving students feedback were based on those needs, and dependent on factors such as situation, relationships, time and effort. This resulted in a constant comparison and weighing of different needs by the teachers. Some needs were described as prioritised before others, which caused some rationales to be identified as taking precedence over others. Discussion and conclusions: Based on a systematic analysis of -and thus grounded in -interview data from primary teachers, the current qualitative study offers a framework for surveying, understanding and discussing teacher feedback. Overall, the study showed how everyday practices of classroom assessment and classroom management overlapped, thus underlining the importance in teacher education of understanding classroom assessment, classroom management and the relationships between the two.

Preparing students for their lives beyond schooling appears to be a universal goal of formal education. Much has been done to make mathematics education more "realistic" but such activities nevertheless generally remain within the institutional norms of education. In this article we assume that pedagogic relations are also an integral part of working life, and draw on Bernstein's work to address their significant features in this context. However, unlike participation in formal mathematics education, where the discipline is central, workers are likely to be confronted by, and need to reconcile, a range of other valued workplace discourses, both epistemic and social/cultural in nature. How might mathematics education work towards overcoming the hiatus between these two very different institutional settings? This article will argue that the skills of recontextualisation, central to teachers' work, should be integral to the mathematics education of all future workers. It will consider theoretical perspectives on pedagogic discourse and the consequences of diverse knowledge structures at work, with implications for general and vocational mathematics education.

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