The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional relationships. We employed design research and analyzed data qualitatively to characterize the levels of sophistication in children's thinking about functional relationships. Findings suggest that children can learn to think in quite sophisticated and generalized ways about relationships in function data, thus challenging the typical curricular approach in the lower elementary grades in which children consider only variation in a single sequence of values.

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In this paper we explore the ideas that second grade students articulate about functional relationships. We adopt a function-based approach to introduce elementary children to algebraic content. We present results from a design-based research study carried out with 21 second-grade students (approximately 7 years of age). We focus on a lesson from our classroom teaching experiment in which the students were working on solving a problem that involved a linear functional relationship (y=2x). From the analysis of students' written work and classroom video, we illustrate two different approaches that students adopt to express the relationships between © 2015. This manuscript version is made available under the Elsevier user license http://www.elsevier.com/open-access/userlicense/1.0/ 2 two quantities. Students show fluency recontextualizing the problem posed, moving between extra-mathematical and intra-mathematical contexts.

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Musical scores are some of the most important learning tools for musicians’ acquisition of musical knowledge. However, despite their educational relevance, very little is known about how music students conceive of these cultural external representations. Given that these conceptions might act as mediators of students’ learning approaches, the importance of knowing these conceptions seems evident in order to eventually change them. The general aim of this investigation was to study the conceptions of piano students at Spanish music conservatories by adopting a developmental-instructional perspective. The participants were 215 students at intermediate and tertiary degree levels, representing three levels of the collapsed variable age— level of instruction. Data were collected by means of a written open-ended task and analyzed by means of descriptive, parametric, and nonparametric statistical methods.The findings suggested that (a) students’ conceptions were more sophisticated at higher age and education levels, (b) each developmental-instructional group typically focused on different musical aspects, which reflected an inclusive and hierarchical logic, and (c) five increasingly sophisticated conceptions could be identified among these students.

This study investigates the representations and understandings of electric fields expressed by Chinese high school students 15 to 16 years old who have not received high school level physics instruction. The physics education research literature has reported students' conceptions of electric fields postinstruction as indicated by students' performance on textbook-style questions. It has, however, inadequately captured student ideas expressed in other situations yet informative to educational research. In this study, we explore students' ideas of electric fields preinstruction as shown by students' representations produced in openended activities. 92 participant students completed a worksheet that involved drawing comic strips about electric charges as characters of a cartoon series. Three students who had spontaneously produced arrow diagrams were interviewed individually after class. We identified nine ideas related to electric fields that these three students spontaneously leveraged in the comic strip activity. In this paper, we describe in detail each idea and its situated context. As most research in the literature has understood students as having relatively fixed conceptions and mostly identified divergences in those conceptions from canonical targets, this study shows students' reasoning to be more variable in particular moments, and that variability includes common sense resources that can be productive for learning about electric fields.

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