Bárbara M. Brizuela

Additive relations and function tables

Abstract We present work with a second grade classroom where we carried out a teaching experiment that attempted to bring out the algebraic character of arithmetic. In this paper, we specifically illustrate our work with the second graders on additive relations, through the children’s work with function tables. We explore the different ways in which the children represented the information of a problem in the form of a self-designed function table. We argue that the choices children make about the kind of information to represent or not, as well as the way in which they constructed their tables, highlight some of the issues that children may find relevant in their construction of function tables. This open-ended format pointed to how they were understanding and appropriating tables into their thinking about additive relations.

School Adaptation: A Triangular Process

In this article, we present the results of a year-long, follow-up study through vignettes from four of the case studies carried out with a group of 16 first-grade, Spanish-speaking children whose families recently immigrated to the United States. During the 1995–1996 school year, we followed the unfolding of the school adaptation of the children, families, and teachers in the process. Through the findings, we seek to challenge traditional views of school adaptation that focus on the child, on the child and family, and, in the case of linguistic minorities or immigrant children, on their English proficiency. We present a comprehensive perspective of school adaptation that considers the child, family, and school and centers on various factors such as cognitive development, socialization, the cultural and political milieu, and family and teacher expectations. We argue for the role of a home-school mediator as a facilitator of childrens adaptation to school.

Children's Use of Variables and Variable Notation to Represent Their Algebraic Ideas.

In this article, we analyze a first grade classroom episode and individual interviews with students who participated in that classroom event to provide evidence of the variety of understandings about variable and variable notation held by first grade children approximately six years of age. Our findings illustrate that given the opportunity, children as young as six years of age can use variable notation in meaningful ways to express relationships between co-varying quantities. In this article, we argue that the early introduction of variable notation in children’s mathematical experiences can offer them opportunities to develop familiarity and fluency with this convention as groundwork for ultimately powerful means of representing general mathematical relationships.

Piano Students' Conceptions of Musical Scores as External Representations: A Cross- Sectional Study

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.

A first grade student’s exploration of variable and variable notation / Una alumna de primer grado explora las variables y su notación

Abstract This paper presents a case study of a first grade student to illustrate the diversity of her understandings related to variables and variable notation. While prior research has documented secondary school students’ difficulties with variables and variable notation, we identify many productive understandings in this much younger student, leading us to question the prevailing argument that students might have difficulties with variables due mostly to their own limitations. We draw our data from a teaching experiment that explored functional relationships. Individual interviews were carried out with a subset of the students in the experiment prior to, as well as mid-way through and at the end of the experiment. This paper focuses on a set of three interviews with one of the first grade students. We illustrate the shifts that occurred in the student’s understandings about variables and variable notation across as well as within each of the three interviews.

Anotar números desde pre-escolar hasta segundo grado: el impacto del uso de dos sistemas de representación en la presentación

Resumen En este estudio exploramos las diferencias en el tipo de producciones de numerales que hacen los niños como consecuencia de dos modos distintos de presentar números: presentación oral y con fichas de valores. Entrevistamos individualmente a veintidos niños (cinco de pre-escolar, ocho de primer grado y nueve de segundo grado) a quienes se asignó al azar a una de las dos condiciones: presentación oral o presentación con fichas de valores, ambas seguidas de la producción de numerales por parte de los niños. Encontramos diferencias en el tipo de representación que facilita el logro de la convencionalidad en las producciones numéricas de los niños y que se asocian a diferentes respuestas no convencionales. Asimismo, encontramos que esas asociaciones diferenciales cambian a lo largo de los distintos grados de escolaridad. Mientras que en los niños de pre-escolar la presentación oral parece facilitar las respuestas convencionales, en los de segundo grado este tipo de respuestas se producen tanto con presentaciones orales como con fichas.

Variables in Elementary Mathematics Education

In this article, I analyze episodes from two third-grade classrooms drawn from a larger classroom teaching experiment to explore how these students began to incorporate nonnumerical symbols in their mathematical expressions when asked to represent indeterminate quantities. The article addresses two research questions: What understandings did these third-grade students construct when they used nonnumerical symbols to represent indeterminate quantities, and how did these understandings vary during the course of working on a single task? What were some of the challenges these third-grade students faced when they first used nonnumerical symbols to represent indeterminate quantities, and how did these challenges vary while working on the Candy Boxes task? Using the constructs of semantic space and form/function relationships, I argue that teaching and learning environments that encourage children’s use of nonnumerical symbols, such as variable notation, to represent indeterminate quantities can support children’s construction of understandings of variables.

Exploring Kindergarten Students’ Early Understandings of the Equal Sign

ABSTRACT This study explores kindergarten students’ early notions of mathematical equivalence in the United States. In particular, it uses qualitative methods to examine the understandings children hold about the equal sign prior to formal instruction and how these understandings shift throughout an 8-week classroom teaching experiment designed to develop relational thinking about this symbol. Findings suggest that, even prior to formal instruction, young children hold an operational view of the equal sign that can persist throughout instruction. This early and persistent operational perspective underscores the critical need to design mathematical experiences in kindergarten, and even preschool, that will orient students towards a relational understanding of the equal sign upon its introduction in first grade.

Looking towards the future: openness, dialogue and connection / Mirando al futuro: apertura, diálogo y conexión

In the last issue of 2017 we received a farewell and activity summary from the previous Editors-in-Chief, Nora Scheuer and Carles Monereo. During the years of their dedicated and excellent leadership, from 2012 to 2017, they spearheaded significant changes. They leave behind a number of models to follow, and a clear path in terms of priorities for the new editorial team. Under their leadership I worked as an Associate Editor, and during that time I was impressed, at each moment, by their generosity and work ethic and the depth of their contributions — whether through a commentary, their vision for the journal or their texts and annual reports. The journal’s latest impact factor has been the second best in its history, indicating how positive the previous decisions of the Editors-in-Chief have been, even though they have never been obsessed or concerned exclusively with improving this index and we are aware of the multiplicity of factors that affect it. With the launch of this new editorial cycle, we want to share some reflections on how we want to build on the legacies left by Nora and Carles in the areas of collaboration, the consolidation of the journal at the international level and openness in the pursuit of new ideas. These are just three of the previous team’s many qualities that we would like to emphasize regarding how they help us think of the goals for this new stage. These three areas are connected with a vision of openness, dialogue and connection that we want to highlight as a guiding thread moving forward. When trying to build on the legacy of the previous team, there will be continuities, of course, as well as some changes, which I describe in these reflections. A first element of continuity refers to the editorial leadership team (Ana Pedrazzini and Anna Sala will remain with the journal, although with different roles from those they had on the previous team; Gloria Palomino will continue in charge of production and translation) and the Associate Editors (three from the previous team will continue in this new stage: Alfredo Bautista, Selma Leitão and Ana Margarida Veiga Simão). Likewise, the journal’s editorial board remains mostly unchanged, though with some important additions, such as the previous Editors-in-Chief (Nora Scheuer and Carles Monereo) and the incorporation of internationally renowned researchers such as Paul Harris (Harvard University), Geoffrey Saxe (University of California, Berkeley) and Deanna Infancia y Aprendizaje / Journal for the Study of Education and Development, 2018 Vol. 41, No. 1, 1–12, https://doi.org/10.1080/02103702.2017.1402483

Call for manuscripts: Early algebraic thinking / Convocatoria de presentación de manuscritos: Pensamiento algebraico temprano

Call for manuscripts: Early algebraic thinking / Convocatoria de presentación de manuscritos: Pensamiento algebraico temprano María C. Cañadas, Maria Blanton & Bárbara M. Brizuela To cite this article: María C. Cañadas, Maria Blanton & Bárbara M. Brizuela (2017) Call for manuscripts: Early algebraic thinking / Convocatoria de presentación de manuscritos: Pensamiento algebraico temprano, Infancia y Aprendizaje, 40:3, 657-660, DOI: 10.1080/02103702.2017.1357291 To link to this article: http://dx.doi.org/10.1080/02103702.2017.1357291