Early Algebraic Thinking Research Articles

Effect of Multiple Representations to Promote Early Algebraic Thinking in Students with Autism

ABSTRACT Given the growing emphasis on algebraic concepts in primary education curricula and the importance to investigate how students with Autism Spectrum Disorder (ASD) approach mathematical learning, the purpose of this study is to evaluate the effectiveness of an instruction based on multiple representations in promoting early algebraic thinking in students with ASD. Three students with ASD, aged 7, 7, and 9 years, enrolled in the same general education school in Spain participated in the study. The research followed a single-subject, multiple probe across participants design. The results of the study showed that the participants improved their performance on generalization tasks in functional contexts when using multiple representations. These improvements were maintained over time and transferred to everyday situations involving regularities. Social validity data collected from students’ families and teachers indicated that both groups had positive perceptions of students’ attitudes and motivation after the study. The implications of the results for promoting early algebraic thinking in students with ASD are discussed.

Development of prospective elementary teachers’ knowledge to teach early algebra through case discussions

AbstractThis study investigated the development of prospective elementary teachers (PETs)’ subject matter and pedagogical content knowledge for teaching early algebra through participation in case discussions. PETs attended five weeks of intervention as part of a method course, which included case discussions. During the intervention, the participants were presented with big ideas of equivalence and equations, generalized arithmetic, and functional thinking as the content of early algebra through text-based classroom cases. Data were collected through one-hour individual interviews before and after the early algebra lessons. The analysis of the pre-interviews indicated that prospective elementary teachers may not be ready to foster algebraic thinking in elementary grades in terms of the required knowledge for teaching. We found that PETs needed more development of required subject matter knowledge to guide algebraic thinking in elementary grades, such as the relational meaning of the equal sign, generalizing, representing, or justifying arithmetic or functional relationships, and reasoning with them. Likewise, PETs were found to lack in pedagogical content knowledge related to students’ conceptions and misconceptions and effective teaching strategies to promote early algebraic thinking. However, we found that specifically designed method courses could help develop PETs’ knowledge of teaching early algebra. Following early algebra lessons centered around case discussions, PETs showed progress in multiple aspects of knowledge across all big ideas, though their improvement in generalized arithmetic was somewhat less pronounced. This progress was observed in their subject matter and pedagogical content knowledge.

잠재프로파일 분석을 통한 초등학생들의 산술-대수 구조 감각 탐색

Developing arithmetic structure sense at the primary school level is imperative for grasping the structure of algebraic expressions and progressing successfully into algebraic learning. In this study, we aimed to investigate elementary school students’ arithmetic-algebraic structure sense from an early algebraic thinking perspective. A total of 239 students in grades 3~6 completed a written test, comprising items related to the five factors of arithmetic-algebraic structure sense: structure of equivalence and modeling, arithmetic, numerical situations, patterns, and function. The collected data was analyzed by applying latent profile analysis to examine whether it was possible to identify subgroups of students with different structure-sense profile. The analysis revealed four latent profiles, allowing for the identification of different performance abilities related to five factors of arithmetic-algebraic structure sense. Building upon these findings, this study discussed implications for pedagogical support to develop elementary school students' arithmetic-algebraic structure sense.

Primary students’ relational thinking and computation strategies with concrete-to-symbolic representations of subtraction as difference

Children are highly inclined to attend to the properties of numbers, operations and equality when given helpful tools for doing so. Our aim was to investigate early algebraic thinking with the compensation property of equality for subtraction. We provided 22 (9–11-year-old) students with physical blocks for building vertical towers and conducted individual interviews with them as they completed a sequence of 15 tasks involving subtraction as difference using concrete, numeric, and symbolic representations. Relational thinking was evidenced across a range of subtraction calculation skill levels. Those students who could use both indirect addition and take-away strategies flexibly, depending on the size of the numbers involved, were more likely to evidence attention to generality with symbolic equations. The shift to symbolic equations elicited some students’ productive attempts to connect subtraction as difference and subtraction as take way but seemed to hinder others by provoking a return to take away calculations rather than relational thinking strategies.

The Developmental Progression of Early Algebraic Thinking of Elementary School Students.

Developing algebraic thinking in elementary school has gained consensus among mathematics educators. The objective of this study is to understand the developmental trajectory of early algebraic thinking in elementary school students so as to assist teachers and curriculum developers in implementing instruction that aligns with students' cognitive development. This study adopted a cross-sectional survey approach, involving 526 students from grades three to five in Shanghai, who were tested using a 12-item assessment that measured three aspects: "generalized arithmetic", "functional thinking", and "quantitative reasoning". Latent class analysis was used to analyze students' response strategies, and, in conjunction with individual interviews, this study identified potential developmental pathways in students' early algebraic thinking, progressing from "arithmetic thinking" to "concrete algebraic thinking", "generalized algebraic thinking", and finally to "symbolic algebraic thinking". As thinking levels advanced, significant differences in students' response strategies emerged, with notable improvements in "generalization abilities" and "symbolization abilities". This study suggests that educational practices should encompass content in elementary arithmetic curricula that fosters generalization abilities. Additionally, providing students with opportunities for diverse representations can effectively stimulate the development of early algebraic thinking.

The role of algebraic thinking in dealing with negative numbers

To date, of the many studies on early algebraic thinking, none, to our knowledge, has examined the relationships between algebraic thinking and negative numbers. Students encounter persistent difficulties in dealing with these numbers, and we believe that these could be addressed through the development of algebraic thinking. We are particularly interested in relational thinking, a form of algebraic thinking involved in generalised arithmetic, characterised by the ability to identify the structure of an expression as well as the relationships between numbers. The idea of the ‘subtractive number’ has been highlighted in this context. The aim of the study was to investigate the role of relational thinking in dealing with negative numbers. We submitted a paper-and-pencil test to 166 grade 6 students in order to analyse their skills in operations with integers, as well as their relational thinking in questions relating to the compensation strategy in subtraction. We then examined the extent to which the students who answered the compensation questions correctly performed the operations with integers better than those who answered them incorrectly. Our results showed that students’ ability to see the subtraction operation as a ‘transformation’ involving a unary use of the minus sign appears to be a factor in their success in operations with negatives. Few students demonstrated this ability, yet it can be seen as an essential stage on which to base the progressive development of relational thinking.

The multi-dimensionality of early algebraic thinking: background, overarching dimensions, and new directions

The multi-dimensionality of early algebraic thinking: background, overarching dimensions, and new directions

Τwo different types of technologically enhanced intervention modules to support early algebraic thinking

Τwo different types of technologically enhanced intervention modules to support early algebraic thinking

Relating Students’ Early Algebraic Thinking to Mathematics Curriculum Materials

This study investigated the trends in algebraic thinking of Korean elementary school students through the students’ correct answer rates for the early algebra-related items in the 4th-grade assessment of TIMSS 2011, TIMSS 2015, and TIMSS 2019, and explored these trends in relation to the changes in early algebra-related content in the Korean mathematics curriculum and textbooks. As a result of the study, with regards to finding unknowns or unknown operations and representing problem situations as expressions, the correct answer rates of the students participating in TIMSS 2019 decreased compared to those participating in TIMSS 2015. The change in the correct answer rates was related to the decrease of relevant standards in the mathematics curriculum and tasks in mathematics textbooks.
 Meanwhile, for tasks about identifying a recursive pattern and representing a covariational relationship, the relevance of the mathematics curriculum materials to the changes in students’ algebraic thinking were not revealed due to the limitation of the small number of related TIMSS items. Nevertheless, it is worth noting that the correct answer rates of students participating in TIMSS 2019 for recognizing recursive patterns and finding appropriate values decreased slightly. Finally, this study makes suggestions for the learning and teaching of early algebra based on the observed results.

Identifying and promoting young students’ early algebraic thinking

Algebraic thinking is an important part of mathematical thinking, and researchers agree that it is beneficial to develop algebraic thinking from an early age. However, there are few examples of what can be taken as indicators of young students’ algebraic thinking. The results contribute to filling that gap by analyzing and exemplifying young students’ early algebraic thinking when reasoning about structural aspects of algebraic expressions during a collective and tool-mediated teaching situation. The article is based on data from a research project exploring how teaching aiming to promote young students’ algebraic thinking can be designed. Along with teachers in grades 2, 3, and 4, the researchers planned and conducted research lessons in mathematics with a focus on argumentation and reasoning about algebraic expressions. The design of teaching situations and problems was inspired by Davydov’s learning activity, and Toulmin’s argumentation model was used when analyzing the students’ algebraic thinking. Three indicators of early algebraic thinking were identified, all non-numerical. What can be taken as indicators of early algebraic thinking appear in very short, communicative micro-moments during the lessons. The results further show that the use of learning models as mediating tools and collective reflections on a collective workspace support young students’ early algebraic thinking when reasoning about algebraic expressions.

Walking through Algebraic Thinking with Theme-Based (Mobile) Math Trails

Tasks are a key resource in the process of teaching and learning mathematics, which is why task design continues to be one of the main research issues in mathematics education. Different settings can influence the principles underlying the formulation of tasks, and so does the outdoor context. Specifically, a math trail can be a privileged context, known to promote positive attitudes and additional engagement for the learning of mathematics, confronting students with a sequence of real-life tasks, related to a particular mathematical theme. Recently, mobile devices and apps, i.e., MathCityMap, have been recognized as an important resource to facilitate the extension of the classroom to the outdoors. The study reported in this paper intends to identify the principles of design for mobile theme-based math trails (TBT) that result in rich learning experiences in early algebraic thinking. A designed-based research is used, through a qualitative approach, to develop and refine design principles for TBT about Sequences and Patterns. The iterative approach is described by cycles with the intervention of the researchers, pre-service and in-service teachers and students of the targeted school levels. The results are discussed taking into account previous research and data collected along the cycles, conducing to the development of general design principles for TBT tasks.

Early childhood teachers’ specialised knowledge to promote algebraic thinking as from a task of additive decomposition (El conocimiento especializado del profesor de educación infantil para fomentar el pensamiento algebraico a partir de una tarea de descomposición aditiva)

ABSTRACT In this article we aim to deepen our understanding of the content and nature of the early childhood teacher’s knowledge, focusing on those aspects which might promote students’ algebraic thinking. Approaching arithmetic from the viewpoint of algebra as an advanced perspective and considering the analytical model Mathematics Teachers’ Specialised Knowledge, we analyse the specialised knowledge in a classroom of five-year-olds handled by an experienced teacher in a lesson on the decomposition of the number 6. Moreover, alternative management of the session is proposed in order to promote early algebraic thinking. In the domain of mathematical knowledge, this analysis has revealed the specificity of the knowledge that this professional must have of the natural number. In the domain of pedagogical content knowledge, it has highlighted the many elements of the knowledge of mathematics teaching that should be possessed to promote algebraic thinking at this educational stage. These elements appear to be more closely related to a profound knowledge of the mathematics taught than to pedagogical knowledge of a more general nature.

A learning trajectory in Kindergarten and first grade students’ thinking of variable and use of variable notation to represent indeterminate quantities

Given the importance of understanding and using indeterminate quantities in algebraic thinking, the development of learning trajectories about how Kindergarten and first grade students understand variable and use variable notation in the context of algebraic expressions is critical. Based on an empirically developed learning trajectory, we analyzed children’s responses at three different points in a classroom teaching experiment. Our purpose was to describe levels of thinking among 16 students (eight in Kindergarten and eight in first grade). Our results revealed qualitative changes in the thinking about indeterminate quantities of most student participants. As students progressed through the experiment, we found that they advanced from what we characterized as a “Pre-variable” Level to a “Letters as representing indeterminate quantities as varying unknowns; explicit operations on indeterminate quantities” Level. Learning trajectories such as that developed here hold promise for informing the design of interventions that support young children’s early algebraic thinking.

What role does professional noticing play? Examining connections with affect and mathematical knowledge for teaching among preservice teachers

This study examined the intersections of preservice elementary teachers’ (PSET) professional noticing (PN) of children’s mathematical thinking, two mathematical knowledge for teaching (MKT) domains: mathematics content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK), and two affective domains: attitudes and dispositions toward teaching mathematics. An instructional module focused on PN of children’s early algebraic thinking, as defined by Jacobs et al. (J Res Math Educ 41: 169–202, 2010) to include three components: attending, interpreting, and deciding, was implemented with 170 PSETs. The PSETs, who participated in the instructional module, the implementation group, showed significant positive growth in attending, interpreting, and attitudes toward mathematics. There was no significant change in dispositions toward teaching mathematics and a decrease was observed in PSET deciding and MKT. A comparison group of 126 PSETs enrolled in mathematics methods courses did not participate in the instructional module. PSETs in the comparison group showed a decrease across all measures including attending, interpreting, deciding, attitudes, dispositions, and MKT. Results showed statistically significant connections within and between some of the constructs; however, the limitations of the study call for further investigation.

Using a Culturally Responsive Approach to Develop Early Algebraic Reasoning with Young Diverse Learners

Teaching in ways responsive to the cultures of our students is vital towards enhancing equity of access to mathematics achievement and putting educational policy into practice. New Zealand has the largest group of Pāsifika people in the Western world, a multi-ethnic group of indigenous people from Pacific Island nations who have differing cultural values and experiences than the dominant Pākehā (Maori term commonly used to refer to European New Zealanders) population. This study aims to explore how patterning tasks based on indigenous patterns can support young diverse students to develop understanding of growing patterns. Two case studies are presented which show the results of an exploratory teaching experiment undertaken with one classroom of culturally diverse Year 2 students (6-year-olds) situated in an urban low socio-economic area. Data drawn for this paper includes video-recorded classroom observations, written responses from students, and reflective notes from the researcher and teacher from two lessons of a series of 10 lessons on growing patterns. Data analysis involved an iterative approach of repeated refinement cycles focusing on early algebraic thinking and the pedagogical actions of the teacher. Findings revealed that the use of indigenous patterns in conjunction with pedagogical actions drawing on cultural values was successful in engaging these students in early algebraic reasoning.

The impact of two different types of instructional tasks on students’ development of early algebraic thinking (El impacto de dos tipos diferentes de tareas instruccionales en el desarrollo del pensamiento algebraico temprano de los estudiantes)

ABSTRACT In this paper, we discuss the theoretical foundation and implementation of two alternative instructional courses that aimed to support the development of elementary school students’ early algebraic thinking. Both courses approached three basic algebra content strands: generalized arithmetic, functional thinking and modelling languages. The courses differed according to the characteristics of the tasks that were used. The first course involved ‘pure mathematical guided investigations’ emphasizing scaffolding steps in pure mathematics contexts. The second course focused on ‘applied open explorations’ underlining more open questions in applied everyday contexts. The courses were compared in respect to students’ learning outcomes. The findings, yielded from the analysis of pre-test and post-test data, indicated that the second course had better learning outcomes compared to the first.

Supporting early mathematical development through a ‘pattern and structure’ intervention program

An Australian longitudinal study of 319 Kindergartners developed, implemented and evaluated an intervention, the Pattern and Structure Mathematics Awareness Program (PASMAP). It comprised of repetitions and growing patterns, structured counting and grouping, grids and shapes, partitioning, additive and multiplicative structures, measurement and data, and transformations. Each component was implemented, evaluated and refined as a cyclic process in collaboration with co-operating teachers, replacing and extending the regular mathematics program. An innovative pedagogical approach was effective in engaging students in modelling and representing, visualising and generalising, and sustaining their learning. On an interview-based measure of mathematics achievement pre- and post-intervention, there were significant differences found between the intervention and comparison groups at the end of the intervention year (p < 0.026) and highly significant differences found at the follow-up assessment (p < 0.002) ten months later. A descriptive analysis showed students’ progress through five levels of structural development where most students advanced through one or more levels as they progressed through the program. The upper third of students demonstrated emergent generalisations at the structural and advanced structural level which supported early algebraic thinking. Less-able students also showed impressive growth—half of these students moved from pre-structural to at least emergent level over the duration of the program. Students’ growth in structural development is exemplified at five points during the intervention through fine-grained analyses of mathematical representations and explanations. The study demonstrates that Kindergartners are capable of developing mathematical patterns and structural relationships well beyond curriculum expectations.

El razonamiento proporcional en educación básica

Este artículo de investigación es el resultado de un proyecto de investigación sobre el pensamiento algebraico temprano en entornos tecnológicos de aprendizaje y se reportan resultados sobre el razonamiento proporcional como una ruta de acceso al pensamiento algebraico temprano. El estudio fue realizado con 109 estudiantes de educación básica de escuelas públicas, del Estado de México y de la Ciudad de México y uno de los objetivos del estudio fue validar un instrumento sobre el razonamiento proporcional. Para ello, se utilizó una metodología mixta, un estudio de tipo observacional, longitudinal y comparativo con tres etapas: 1. Evaluación inicial sobre el razonamiento proporcional, 2.Validación de un instrumento sobre el razonamiento proporcional, 3. Entrevista clínica abierta. Los resultados revelan que los alumnos logran comprender ideas de proporcionalidad aritmética y geométrica y variación proporciona, y comprenden los problemas que involucran la relación funcional, como una consecuencia del tránsito del pensamiento aditivo al multiplicativo. En lo que respecta al instrumento de evaluación, se propuso un modelo hipotético de validez convergente y divergente de constructo en la medición del razonamiento proporcional y el modelo resultante obtuvo aceptables indicadores prácticos de bondad de ajuste y coeficientes de confiabilidad altos. Además el instrumento mostró tener validez convergente y divergente de constructo. Finalmente, podemos concluir que el razonamiento proporcional es una ruta viable hacia el pensamiento algebraico, a pesar de que los estudiantes no han desarrollado aún todas las estructuras cognitivas y matemáticas para comprender la complejidad del pensamiento proporcional, los resultados de este estudio permitieron dilucidar etapas tempranas en el desarrollo cognitivo de los niños con tareas y enfoques pertinentes.

Special issue on early algebraic thinking / Número especial sobre el pensamiento algebraico temprano

In this introduction to the Special Issue Early algebraic thinking: studies from different perspectives, approaches and regions we provide readers with the main motivation and interest underlying research on early algebraic thinking. We then introduce the five papers that make up this Special Issue. Each of these studies, coming from various countries around the world, has a different theoretical framework, methodological approach and perspective regarding teachers, students, instructional practices and curricula. In this introduction, we summarize some of the main themes throughout the different papers, including interviewers’ practices that can promote early algebraic thinking; the kinds of practices that teachers can implement to have a positive impact on students’ early algebraic thinking; and the use of physical models to explore and promote aspects of early algebraic thinking. The Special Issue also includes an additional Prospectives piece by Analúcia Schliemann and David Carraher, in which they examine the algebraic thinking research in elementary school in light of the current mathematics standards in the United States.

Supporting the Development of Early Algebraic Thinking, an Alternative Approach to Number

ABSTRACTFirst-grade mathematics curriculum has been typically constructed to emphasize the development of whole number and operations. Topics addressed to a lesser extent include algebraic thinking, measurement, and geometry. In this research study, we suggest an alternative to this balance of topics. We compared prior research and learning expectations from the Common Core State Standards for Mathematics (CCSSI, 2010) against a learning progression which we proposed as one that develops early algebraic thinking. We described lesson clusters and shared evidence of student thinking to illustrate our learning progression. Findings from this work have implications for what has been considered the start of the learning progression for mathematics.