Bárbara M. Brizuela’s research while affiliated with Tufts University and other places

What is this page?


This page lists works of an author who doesn't have a ResearchGate profile or hasn't added the works to their profile yet. It is automatically generated from public (personal) data to further our legitimate goal of comprehensive and accurate scientific recordkeeping. If you are this author and want this page removed, please let us know.

Publications (86)


Kindergarteners' Strategies and Ideas When Reasoning with Function Tables and Graphs
  • Conference Paper
  • Full-text available

June 2025

·

1 Read

·

·

Bárbara M Brizuela

·

[...]

·

Maria Blanton

This study investigated the seeds of algebraic thinking that Kindergarten students use when engaging with function tables and graphs. Through interviews with three Kindergarteners, we explored how they reasoned about functional relationships. Our results illustrate how the Kindergarteners used seeds of algebraic thinking when using function tables and graphs to represent and reason about functional relationships. Building on the seeds of algebraic thinking and Knowledge in Pieces frameworks, we categorized these seeds as either strategies (classify, pair, and compare) or ideas (seeds of covariation). Strategy seeds were goal-oriented, and seeds of covariation were elicited without any goal and reflected a broader understanding of change between quantities.

Download

Justifications and Mediations in the Generalization Process Among Fourth Grade Students

May 2025

·

9 Reads

International Journal of Science and Mathematics Education

Research on generalization in elementary school is a key topic in mathematics education. This study analyzes the generalization processes among fourth-grade students (9–10 years old) while working with functional relationships, focusing on the interplay between generalizations, justifications, and mediations during a lesson. Specifically, we identify and characterize the generalizations and justifications made by 22 students and describe the mediations carried out by the teacher-researcher throughout the class. Additionally, we examine the relationships between the levels of generalization, justifications, and mediations. Our findings reveal that students demonstrated varying levels of sophistication in their generalizations, which were consistently supported by justifications and mediations. Justifications played a crucial role in validating and explaining the students’ generalizations, while mediations facilitated their engagement with the task and supported the generalization process throughout the lesson. We conclude that justification and mediation are integral components of the generalization process, emphasizing its active and social nature. Our study underscores the importance of fostering classroom discussions and encouraging students to articulate their reasoning and conjectures, promoting deeper understanding and collaborative learning.


Las tablas como herramientas algebraicas

October 2024

·

49 Reads

·

1 Citation

En este artículo discutimos el rol de las tablas en el diseño de tareas en nivel infantil y los primeros grados de nivel primario. Mostramos cómo pueden apoyar el pensamiento algebraico del estudiantado, específicamente sus representaciones, razonamientos, justificaciones y generalizaciones. Las tablas no solo son herramientas que ayudan a registrar datos, también son herramientas que promueven prácticas algebraicas.


Kindergarten and First-Grade Students’ Understandings of Arithmetic Properties Across Different Kinds of Problems

September 2024

·

14 Reads

Canadian Journal of Science Mathematics and Technology Education

In this article, we present research on eight kindergarten and eight first-grade students’ understandings of the arithmetic properties of commutativity, additive identity, and additive inverse during a classroom teaching experiment, selected from a larger study that included 88 students. In this study, we explore the students’ types of understandings in terms of Skemp’s (1987) framework (instrumental and relational), the basic relationships in the conceptual field of additive structures (combination, transformation, and comparison), and their performance on different problems involving numerical operations, equations, and word problems. Our findings show that students performed better on additive identity and additive inverse properties. Problems involving substituting and simplifying expressions that combined two properties had a lower performance rate. The differences in understandings and performance observed between kindergarten and first-grade students have implications for instructional design geared to introduce arithmetic properties in the early grades. This study highlights the difference between correctly solving a problem and providing evidence of a relational understanding.


Generalization among 5-Year-Olds in a Functional Context with Programmable Robot

September 2024

·

24 Reads

·

1 Citation

International Journal of Science and Mathematics Education

The aim of this study was to determine how 5-year-old children identified the functional relationship of correspondence, and whether or not they generalized when working on a task that involved programmable robots. We conducted this study with 15 children (9 girls and 6 boys) in their last year of preschool education. The study was designed around a generalization task that involved the function f(n)=n+2. Our findings indicate that nine of the children identified the correspondence relationship and five children generalized, three of them correctly.


Early Elementary Children's Understandings of Function Graphs

September 2024

·

26 Reads

Using CTEs we study Grades 1 and 2 students' developing understandings of function graphs. Students were interviewed before, during, and after the CTEs. Analysis was based on prior research on algebraic representations and Sfard's (1991) process-object framework. Our findings are a developing trajectory of early elementary students' interpretations of function graphs.


Niveles de comprensión de tablas y gráficos entre estudiantes de quinto y sexto de educación primaria

September 2024

·

22 Reads

This study is part of a broader research project that explores functional thinking among primary school students in Spain. The aim of this paper is to describe the different levels of understanding of tables and graphs. We analyzed two students’ interviews, one fifth grader and one sixth grader, conducted after four classroom sessions. We share examples of each of their levels of understanding through their work with generalization tasks with linear functions and different representations, specifically tables and graphs. Sfard’s (1991) process-object theoretical framework allowed us to explain students’ levels of understanding of the table and graphical representation.



Figure 1. A representation of the Grade 1 and 2 lessons and interviews FINDINGS We observed Lucca use four seeds: classifying, structuring, what you see is what you get (Elby, 2000), and covariation (Levin & Walkoe, 2022). Two of those seeds, classifying, structuring, emerged from our analysis of Lucca's thinking. The other two seeds, what you see is what you get (Elby, 2000), and covariation (Levin & Walkoe, 2022), were identified previously and then observed in our data. We begin by defining classifying and structuring.
Figure 2. Lucca's self-made table during the third interview
Figure 3. Lucca's self-made graph from his second interview (Left) and Lucca's graph from his third interview (Right)
Algebraic Seeds for Graphing Functions

This case study of one first grade student involves the analysis of three interviews that took place before, during, and after classroom teaching experiments (CTEs). The CTEs were designed to engage children in representing algebraic concepts using graphs. Using a knowledge-in-pieces perspective, our analysis focused on identifying students' natural intuitions and ways of thinking algebraically about a functional relationship represented using graphs. Findings reveal four seeds, two of which were identified in prior studies, and how the activation and coordination of these seeds results in students' production of function graphs.



Citations (58)


... Aunque no existe un consenso sobre lo que implica incorporar el álgebra en edades tempranas, cualquier contenido o actividad que ayude a los estudiantes a ir más allá de la fluidez aritmética y computacional para comprender las estructuras matemáticas, puede considerarse óptima para desarrollar el pensamiento algebraico (Cai y Knuth, 2005). Los hallazgos de investigaciones recientes dejan en evidencia la capacidad de pensar algebraicamente de los estudiantes de educación infantil (e.g., Acosta y Alsina, 2024;Anglada et al., 2024) y educación primaria (e.g., Cañadas et al., 2024;Pinto et al., 2023). ...

Reference:

Desarrollo profesional del profesorado de matemáticas de Educación Infantil y Primaria: contribuciones de un curso de formación continua sobre el sentido algebraico
Generalization among 5-Year-Olds in a Functional Context with Programmable Robot
  • Citing Article
  • September 2024

International Journal of Science and Mathematics Education

... Research on young students' generalising abilities has primarily concerned aspects of the teaching of early algebra (e.g., Strachota, , 2023Urena et al., 2019Urena et al., , 2022 including students' work on patterns (e.g., Lannin, 2005). For example, Blanton et al.'s (2019:212) longitudinal teaching experiments that involved problem-solving activities in showed that students who had received teaching about 'early algebraic concepts and practices' were more successful when formulating general solutions and using formal mathematical representations than students who had received arithmetic-focused instruction. ...

First Graders’ Definitions, Generalizations, and Justifications of Even and Odd Numbers
  • Citing Article
  • December 2023

Canadian Journal of Science Mathematics and Technology Education

... Distintas investigaciones en las últimas décadas han tratado la importancia de integrar los contenidos aritméticos y algebraicos desde los primeros cursos de la escolaridad (e. g., Carraher y Schliemann, 2007;Kaput, 2008;Kaput et al., 2008;Schliemann et al., 2011). Asimismo, estas investigaciones han resaltado los efectos a corto y largo plazo de integrar el contenido y las prácticas algebraicas con los contenidos aritméticos (e. g., Blanton et al., 2019;Brizuela et al., 2013;Schliemann et al., 2012). Siguiendo el marco de Kaput (2008) y detallado por Blanton et al. (2011), entendemos que el contenido algebraico en early algebra incluye: (1) la aritmética generalizada; (2) la equivalencia, las expresiones, las ecuaciones y las desigualdades (o simplemente las ecuaciones); (3) el razonamiento cuantitativo, y (4) el pensamiento funcional. ...

The Impact of Early Algebra: Results from a Longitudinal Intervention
  • Citing Article
  • June 2013

Journal of Research in Mathematics Education

... In the literature, functions are associated with a series of representations in either pictorial, symbolic, verbal, tabular, graphic, or gestural form (Ayala-Altamirano & Molina, 2020; Cañadas et al., 2024;Radford, 2003). Evidence indicates that the most frequently used representations by preschool and early primary education students are pictorial and verbal (Fuentes, 2014), although it is possible to guide them to use other types, such as tabular or algebraic representations (Torres et al., 2022). ...

Introducing Tables to Second-Grade Elementary Students in an Algebraic Thinking Context

... To develop levels of students' understandings of graphs we began by identifying relevant literature on trajectories and progressions in related contexts (e.g., Brizuela et al., 2021;Gabucio et al., 2010;García-Milá et al., 2014;Martí, 2009;Martí et al., 2010Martí et al., , 2011. First, we described ways of thinking about graphs that might occur in the context of graphing functional relationships and then iteratively refined these levels by reviewing the interviews with the new levels in mind, revising, reviewing, and revising, until we encountered no new levels or situations that could not be described with one of our existing levels. ...

A Kindergarten Student’s Use and Understanding of Tables While Working with Function Problems
  • Citing Chapter
  • May 2021

... El trabajo del pensamiento algebraico con estudiantes de primaria ayuda a promover un pensamiento analítico, donde las cantidades indeterminadas, incógnitas o variables se tratan junto con las cantidades conocidas (Ventura et al., 2021). El principal objetivo de trabajar el pensamiento algebraico es hacer que "los niños piensen, describan y justifiquen lo que sucede en general con respecto a alguna situación matemática. ...

A learning trajectory in Kindergarten and first grade students’ thinking of variable and use of variable notation to represent indeterminate quantities
  • Citing Article
  • June 2021

The Journal of Mathematical Behavior

... The concept of number is central in everyday life, as well in mathematical studies, which are a crucial aspect of a child's school life, as children constantly engage in mathematics studies in school from first through twelfth grade (Markovich, 2019). Uclés et al. (2020) have found that Kindergarten and first grade students have the potential to develop relational understanding and to symbolically represent arithmetical properties. Their study emphasizes the important implications of this finding for designing innovative learning and teaching environments at these grade levels. ...

Correction to: Kindergarten and First-Grade Students’ Understandings and Representations of Arithmetic Properties

Early Childhood Education Journal

... We emphasize the significance of concentrating on generalized arithmetic because it has been recognized as a means to introduce algebraic reasoning through arithmetic activities that are familiar to young children (Carpenter et al., 2003). A recent study by Ramírez Uclés et al. (2022) reported that kindergarteners can engage in algebraic reasoning by working with arithmetic properties. Furthermore, Schifter and Russell's (2022) study noted that students can utilize representations to show relationships within a generalized arithmetic context. ...

Kindergarten and First-Grade Students’ Understandings and Representations of Arithmetic Properties

Early Childhood Education Journal

... Although the importance of FT ability is frequently discussed in the literature, different studies have reported students' difficulties in coping with this ability in mathematics learning (Pinto & Cañadas, 2021;Ramírez et al., 2022). The most common issue found in studies is students' inability to recognise the pattern within a sequence of values; instead, they choose to count the values in consecutive numerical steps (El Mouhayar, 2018;Rivera, 2018;Wijns et al., 2019;Wilkie & Clarke, 2016). ...

Word problems associated with the use of functional strategies among grade 4 students

Mathematics Education Research Journal

... El álgebra constituye un área crítica dentro de la enseñanza de las matemáticas. La introducción del simbolismo alfanumérico alrededor del sexto o séptimo grado implica un nivel de abstracción difícil de abordar para muchos estudiantes (Cañadas et al., 2019;Kaput, 2008). De hecho, el álgebra se ha descrito como el "guardián de la entrada" a las matemáticas superiores (Kaput, 2008), actuando como un filtro que clasifica a los estudiantes entre "buenos" y "malos" y limita sus trayectorias educativas (Cañadas et al., 2019;Schoenfeld, 1995). ...

Special issue on early algebraic thinking / Número especial sobre el pensamiento algebraico temprano
  • Citing Article
  • July 2019

Infancia y Aprendizaje