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This manuscript contributes to research on how algebraic thinking about operations and properties can develop, and relevant forms of curricular activity. The key question asked is: how can students’ early algebraic activity be fostered through focusing on relations involving operations and properties? We adopt Radford’s three aspects of algebraic thinking (indeterminacy, denotation and analyticity), which focus on considering, symbolising and operating on unknown objects. We investigate the work of a class of 11–12 year old students in re-analysing data collected over 20 years ago as part of a project exploring algebraic activity. Our findings point to the proposal that a focus on conjecturing about relationships can be a powerful route into early algebra. On the basis of our analysis we propose an extension to Radford’s third aspect of algebraic thinking (analyticity) to include structuring a mathematical space, e.g., through a set of conjectures.

Generalisation is a skill that enables learners to acquire knowledge in general, and mathematical knowledge in particular. It is a core aspect of algebraic thinking and, in particular, of functional thinking, as a type of algebraic thinking. Introducing primary school children to functional thinking fosters their ability to generalise, explain and reason with mathematical relationships. It also helps them overcome difficulties in understanding functions when they are exposed to the idea more formally in secondary education. Although more and more special education students are enrolled in mainstream schools, little is known about algebraic thinking in that community, especially in the case of students with autism spectrum disorder (ASD). Students with ASD often exhibit deficits that interfere directly with mathematical learning. The study discussed here, which was conducted in the context of algebraic reasoning, was aimed primarily at identifying and describing the strategies and representations observed in 26 ASD primary education students when performing a task that involved a linear function, and describing the generalisations they performed. The 26 participants were enrolled in 19 mainstream Spanish schools. The tools used, a questionnaire and semi-structured interview, were designed to explore their ability to generalise in a problem involving the function f(x) = 2x + 2. The strategies identified included: (a) bald answering; (b) modelling with manipulatives; (c) drawing; (d) counting and (e) operating. The strategy most frequently observed was operating, represented verbally or symbolically, followed by drawing. Only three students generalised but did not reach the highest level of functional thinking, namely, ‘functions as objects’. The results are compared with findings for mainstream students of similar ages. Conjectures around the possible relationships between some findings and the type of thinking characteristic of autism spectrum disorder are put forward. The results carry implications for research with and teaching of students with ASD., Spain's National Research Agency, European Commission EDU2016-75771-P EDU2017-84276-R PID2019-105677RB-I00 PID2020-113601GB-I00, regional Government of Cantabria's Concepcion Arenal Grants

Although there have been a huge number of attempts to improve school algebra teaching, several countries are still struggling to improve students algebraic skills. In this study, we focus on the specific case of Sweden where students for several decades have had major problems mastering algebra. In order to get a better understanding of the Swedish situation, we consider what constitutes Swedish school algebra by investigating the development of algebraic content in the Swedish mathematics curriculum documents over the past 40 years. The results reveal that the connection between arithmetic and algebra, the so-called generalized arithmetic, is almost absent in all three curricula although researchers argue that generalized arithmetic is one of the most relevant topics within early algebra. Instead, Sweden has chosen a unique approach as programming, with a specific focus on stepwise instructions and algorithms, recently has been implemented within the core content of algebra.

En este estudio se analizan las tareas matemáticas sobre álgebra temprana en una colección de ocho libros de texto chilenos de Educación Infantil (4 a 6 años). La investigación siguió una metodología cualitativa, de carácter exploratorio-descriptivo, utilizando la técnica de análisis de contenido. Los resultados muestran una presencia de tareas algebraicas en todos los libros de texto analizados, predominando las tareas vinculadas con establecer relaciones a partir del reconocimiento de atributos, seguidas de tareas que requieren de la seriación a partir de patrones de repetición y una escasa presencia de tareas que implican la descripción de cambios., This study analyses the mathematical tasks on early algebra in a collection of eight widely distributed Chilean textbooks for Early Childhood Education (4 to 6 years old). The research followed a qualitative methodology, of an exploratory-descriptive nature, using the technique of content analysis. The results show a presence of algebraic tasks in all the textbooks analysed, with a predominance of tasks linked to establishing relations based on the recognition of attributes, followed by tasks that require seriation based on patterns of repetition, and a scarce presence of tasks that involve the description of changes., Este estudo analisa as tarefas matemáticas sobre álgebra inicial em uma coleção de oito livros didáticos chilenos amplamente distribuídos para a Educação Infantil (4 a 6 anos). A pesquisa seguiu uma metodologia qualitativa, de natureza exploratório-descritiva, utilizando a técnica de análise de conteúdo. Os resultados mostram a presença de tarefas algébricas em todos os livros analisados, com predominância de tarefas ligadas ao estabelecimento de relações baseadas no reconhecimento de atributos, seguidas de tarefas que exigem seriação com base em padrões de repetição, e uma presença escassa de tarefas que envolvem a descrição das mudanças.

The Cuisenaire-Gattegno (Cui) approach uses color coded rods of unit increment lengths embedded in a systematic curriculum designed to guide learners as young as age five from exploration of ratio through to formal algebraic writing. As the rods have had greater adoption as a teaching aide than the curriculum, we set out to investigate how fidelity to the seminal curriculum and pedagogy impacts learning via a meta-analysis and novel study of preparation for future learning. This meta-analysis of 23 studies (n=1968) revealed advantages of Cui over traditional arithmetic approaches (effect size = 0.55). Curriculum fidelity significantly predicted efficacy. Higher fidelity implementations were associated with large effects and lower fidelity resulted in small or null effects. To test how this curriculum prepares students for future learning, we carried out an 18-month longitudinal school-comparison study (n=114) executed to a similar fidelity level as the study with the largest treatment effect. Cui treatment accelerated learning rates measured during the school-year after treatment, and demonstrated transfer to novel tests of algebraic reasoning (effect size = 1.0). Tests of scholastic aptitude replicated aptitude by treatment interaction for both arithmetic and algebraic reasoning. While Cui provided significant learning benefits for children with higher aptitude, these benefits were significantly enhanced for children with lower aptitude. Together, these findings support the benefits of this approach, and further substantiate the importance of embedding these teaching aides within the theory-grounded curricula that gave rise to them.

O presente artigo tem por objetivo investigar como a Early Algebra se apresenta nos documentos curriculares da Educação Infantil em seis países (Estados Unidos da América, Cingapura, Portugal, Austrália, Nova Zelândia e Brasil). O estudo possibilita uma importante discussão sobre a Early Algebra sob a ótica dos documentos curriculares da Educação Infantil em volta do mundo. Os procedimentos metodológicos desta pesquisa ocorreram a partir de um estudo documental em pesquisas realizadas em sites oficiais de cada país e no site do Ministério da Educação do Brasil, cujas fontes foram os documentos curriculares para a Educação Infantil tanto brasileiros quanto internacionais. A análise dos documentos foi feita considerando a idade em que a álgebra é introduzida em cada país e no Brasil, como se dá essa introdução e quais os principais conceitos algébricos são abordados. Os resultados obtidos revelaram que nos cinco países internacionais investigados a álgebra é introduzida a partir dos 3 anos de idade, abordando questões referentes às ideias de padrão e de sequência, e o raciocínio algébrico é introduzido levando em consideração as práticas pedagógicas da Educação Infantil. Já no Brasil, ela é iniciada a partir dos 6 anos de idade, ou seja, nos anos iniciais do Ensino Fundamental. Consideramos que é importante desenvolver pesquisas sobre essa temática, pois ela tem sido pouco discutida no Brasil para o âmbito da Educação Infantil, sendo necessário ampliar as discussões para que se possa acompanhar as tendências internacionais sobre o desenvolvimento da álgebra nos primeiros anos de escolaridade. Palavras-chave: Early Algebra. Currículo. Educação Infantil. Estudo Documental.

This research focuses on the retention of students’ algebraic understandings 1 year following a 3-year early algebra intervention. Participants included 1,455 Grade 6 students who had taken part in a cluster randomized trial in Grades 3–5. The results show that, as was the case at the end of Grades 3, 4, and 5, treatment students significantly outperformed control students at the end of Grade 6 on a written assessment of algebraic understanding. However, treatment students experienced a significant decline and control students a significant increase in performance relative to their respective performance at the end of Grade 5. An item-by-item analysis performed within condition revealed the areas in which students in the two groups experienced a change in performance.

The importance of students being acquainted with algebraic ideas before secondary education has been revealed in the research literature. It is therefore essential that prospective elementary teachers (PTs) be prepared to instill an early algebra perspective in their teaching. However, PTs often show difficulties in algebra content knowledge, which need to be diagnosed aiming to assist them in developing the required knowledge to teach according to that perspective. This study aims to understand what aspects of functional thinking Spanish and Portuguese elementary PTs exhibit at the beginning of their teacher education program. The findings show that although PTs from both countries use different strategies to generalize functional relationships, the occurrence of successful strategies is low. Also, most participants provide local approaches in their interpretation of relationships between variables and reveal difficulties in understanding and connecting different representations of functions. These difficulties show that PTs lack important knowledge about functional thinking. By providing a framework concerning the functional thinking required for PTs to teach within an early algebra perspective, we shed light on a necessary step for teacher education programs to diagnose PTs? functional thinking and to assist them in developing the needed mathematical knowledge to teach accordingly. Pentingnya pengenalan ide-ide aljabar siswa sebelum pendidikan menengah telah terungkap dalam penelitian literatur ini. Oleh karena itu, calon guru sekolah dasar (CG) harus siap untuk menanamkan perspektif aljabar awal dalam pengajaran mereka. Namun, CG sering kali menunjukkan kesulitan dalam pengetahuan konten aljabar, yang perlu didiagnosis dengan tujuan membantu mereka dalam mengembangkan pengetahuan yang diperlukan untuk mengajar sesuai dengan perspektif itu. Penelitian ini bertujuan untuk memahami aspek-aspek berpikir fungsional apa yang diperlihatkan oleh, CG Spanyol dan Portugis di awal program pendidikan guru mereka. Temuan menunjukkan bahwa meskipun CG dari kedua negara menggunakan strategi yang berbeda untuk menggeneralisasi hubungan fungsional, kejadian strategi yang berhasil rendah. Selain itu, sebagian besar peserta memberikan pendekatan lokal dalam interpretasi mereka tentang hubungan antar variabel dan mengungkapkan kesulitan dalam memahami dan menghubungkan representasi fungsi yang berbeda. Kesulitan ini menunjukkan bahwa CG kurang memiliki pengetahuan penting tentang berpikir fungsional. Dengan memberikan kerangka kerja mengenai pemikiran fungsional yang diperlukan untuk CG untuk mengajar dalam perspektif aljabar awal, kami menjelaskan langkah yang diperlukan untuk program pendidikan guru untuk mendiagnosis pemikiran fungsional PT dan untuk membantu mereka dalam mengembangkan pengetahuan matematika yang dibutuhkan untuk mengajar sesuai. This work was supported by the Ministerio de Economía y Competitividad. Madrid. Spain [projects EDU2017-84276-R and PID2019-105677RB-I00].

Around the globe, young students are expected to learn about time, yet how is it that they themselves make sense of this topic? From a sociocultural perspective, sense-making about time emerges in ...

The articles in this special issue, collectively, provide overwhelming evidence that a curriculum which has as its primary basis the counting of discrete objects (and hence which introduces numbers as discrete) is not an effective or rational organisation. In this commentary, I discuss the contribution of each article to an understanding of Davydov’s ontology and epistemology, issues around the transposition of a pedagogy or curriculum from one country to another, early algebra and proportional or multiplicative reasoning. From Davydov’s own research and the writing in these articles, we know children are able to understand abstract structures from an early age. The thinking in this special issue provides tools to investigate and question any context in which such understanding is not routinely taking hold.

Recent scholarship around teaching elementary mathematics supports the learning of early algebra with 5- to 12-year olds. However, in spite of the recognition of the affordances of early algebra, issues about how to introduce it remain open. Within this context, Davydov’s work is often cited as a source of impressive demonstration of young learners’ capacity for algebraic thinking. This work requires further exploration in order to yield a clearer picture of a very particular teaching approach, which focuses on early abstractions and symbolic language. We argue that in order to fully understand how Davydov’s work contributes to current conversations and what Davydov was trying to do, we need to shed light on the context- and time-specific discourse of the 1960 Soviet educational reforms that made it possible for Davydov to develop his vision about algebraic thinking and to set in motion appropriate teaching approaches for young learners. In this paper, we look back to the Soviet debates that unfolded in Russia on the integration of early algebra in elementary school word-problem solving. Drawing on these debates and the results of Davydov’s school experiments, we lay out the developmental axes of capacity building. This can be done by emphasizing ascent from the abstract to the concrete using a variety of representational modeling tools to support the emergence of algebraic thinking while targeting particular habits of mind within carefully designed learning activities. We conclude with some insights about current arithmetic-algebra debates, and how these could be enriched and deepened by Davydov’s work, which yet remains open to future discussion and reflection.

Este artigo apresenta um estudo a respeito da manifestação do pensamento algébrico elementar realizado em uma escola pública municipal de Apucarana - PR. Para tanto, assumimos a Early Algebra como área de pesquisa e tivemos como objetivo identificar, analisar e discutir características do pensamentoalgébrico elementar nas produções escritas de oito tarefas da Early Algebra aplicadas a estudantes do 5° Ano do Ensino Fundamental I. Concluímos, a partir das respostas, indagações e afirmações dos estudantes durante a aplicação das tarefas, que houve manifestação de pensamento algébrico elementar, pois os estudantes notaram e ousaram exprimir operações aritméticas das tarefas, bem como, expuseram seus pensamentos, mostrando que, mesmo não tendo habilidades com uma linguagem simbólica algébrica, têm condições de trabalhar aspectos relacionados ao pensamento algébrico elementar. Além disso, construímos, com base nas resoluções das tarefas propostas pelos estudantes, cinco categorias que refletem as características do pensamento algébrico elementar apresentadas: estabelecimento derelações/comparações entre as informações da tarefa; utilização de diferentes notações/representações; generalização; compreensão de propriedades matemáticas importantes e regularidades.

In this study, we analyze how 9–10-year-old pupils work with equations, a central aspect of algebraic thinking in early grades and a cornerstone for more formal learning of algebra. Specifically, we seek: (a) to describe the main characteristics of the tasks that support algebraic thinking through a translation process from arithmetic word problems to algebraic language and vice versa, and (b) to identify how pupils refer to indeterminate quantities in these contexts and what meaning they give to them. The analysis focuses on the semantic congruence of the expressions proposed by them and on the dialogue they held during the translation process. We analyzed the oral discussion in the pools and the written responses to the problem that pupils posed. The results show that arithmetic word problems allow the indeterminate to become an object of thought for pupils, who represent it in multiple ways and refer to it when proposing equations that represent the structure of each problem. Another finding highlights that reflection on the interpretation of the equations supports the identification of two meanings associated with indeterminate quantities, namely, unknown and variable., Universidad del Desarrollo (Chile) 23.400.173, State Research Agency (SRA) from Spain EDU2016-75771-P PID2020-113601GB-I00

This research focuses on ways in which balance scales mediate students’ relational understandings of the equal sign. Participants included 21 Kindergarten–Grade 2 students who took part in an early...

In this article we seek to present an analysis of the proposal of the Common National Curricular Base regarding the introduction of Algebra teaching from the early years of Elementary School and the possible challenges that teachers who teach Mathematics in this teaching stage may face. The importance of the discussion developed here is justified because we understand that this proposal is in line with what the international literature has been naming as Early Algebra and which has influenced changes in the mathematics curricula of several countries. In addition, this change will highlight the importance of the initial and continuing training of the teacher who teaches mathematics in this phase of elementary school. The research with a qualitative approach and exploratory design used documentary and bibliographic research in its methodological path. Thus, we noticed with the first analyzes a significant change in relation to the Algebra approach in the Common National Curriculum Base when compared to the National Curriculum Parameters, in addition to this we evidenced a range of measures necessary for the effective implementation of the curricula of the education systems in the perspective of the base.

This study aims to understand the instructional interactions that foster students’ generalizing. By analysing video-recorded lessons from 13 different Grade 3 classrooms in which an early algebra i...

In this study, the Chinese in-service teachers’ didactic-mathematical knowledge on elementary algebraic reasoning is assessed using a questionnaire developed in Spain. The first aim is to diagnose the current situation in China of in-service primary school teachers’ knowledge regarding elementary algebra and its teaching. The second aim is to provide enlightenment to teacher training so as to make the concept of early algebra popularized among Chinese primary school teachers. The sample included 264 primary school teachers from the Shanxi province in China. The questionnaire consists of two parts: part 1 comprises 5 items about teachers’ gender, teaching experience, professional certificate, educational background, and region; part 2 comprises 25 items from the original version in English. The difficulty, discrimination indexes as well as reliability of the instrument have been determined, reflecting the psychometric properties of the instrument. The dependent variables include 7 aspects divided in 2 dimensionalities apart from the total score. The results reveal Chinese primary school teachers have solid knowledge of algebra but lack pedagogical knowledge in early algebra. This suggests that it would be necessary to set clear training objectives on the core idea of the algebra in national teacher training programs; in addition, our findings emphasize more attention should be given to teachers who have less than 10 years of teaching experience. Finally, we conclude teacher training would be more effective if highly experienced teachers––more than 20 years, became trainers to teach the practical modules.

This article describes part of the findings of a teaching experiment whose objective is to investigate the algebraic abilities of elementary students when they solve situations that involve a functional relationship. In particular, we focus on describing the use and meanings attributed to letters by third-year primary school students when faced with verbal problems related to the generalisation of a functional relationship. Drawing from the functional approach to early algebra and set in Spain, the study expands on earlier research conducted on primary school students’ use of letters in algebraic contexts. Their initial reactions to the use of letters to represent indeterminate quantities and how those reactions changed in the course of three sessions are described. Analyses of the students’ written answers together with their participation in group discussions yield qualitative data on how students associate the idea of variability with indeterminate quantities and use letters, numbers or both to represent that notion.

How should schools assign students to more rigorous math courses so as best to help their academic outcomes? We identify several hundred California middle schools that used 7th-grade test scores to place students into 8th-grade algebra courses and use a regression discontinuity design to estimate average impacts and heterogeneity across schools. Enrolling in 8th-grade algebra boosts students’ enrollment in advanced math in ninth grade by 30 percentage points and eleventh grade by 16 percentage points. Math scores in tenth grade rise by 0.05 standard deviations. Women, students of color, and English-language learners benefit disproportionately from placement into early algebra. Importantly, the benefits of 8th-grade algebra are substantially larger in schools that set their eligibility threshold higher in the baseline achievement distribution. This suggests a potential tradeoff between increased access and rates of subsequent math success.

This study describes how 24 third graders (8–9 years old) relate and represent the relationships between variables when working with a functional thinking problem. This aspect contributes to providing insights about how elementary school students attend properties and relationships between covarying quantities rather than isolated computations. From a functional approach to early algebra, we describe written students’ answers when working with a problem that involves a function, which includes questions for specific values and to generalize. Design research guidelines, specifically those set out for Classroom Teaching Experiment were followed. This study addresses the fourth and last Classroom Teaching Experiment session, which involved a function of the type y = ax + b and students had not previously worked it. Students primarily evidenced correspondence relationship, using natural language and numerical representation to express this functional relationship. Our findings let us to state that (a) although students were not used to working with these types of problems, eleven of them go beyond arithmetic computations, finding relationships that relate the variables; and (b) three students generalized using natural language as a useful vehicle, while there are other students who perceived the same regularity for different specific values but they are unable to represent generalization clearly.

We investigated whether early algebra lessons that explicitly aimed to elicit mathematical discussions (shift-problem lessons) invoke more and qualitatively better mathematical discussions and raise students’ mathematical levels more than conventional lessons in a small group setting. A quasi-experimental study (pre- and post-test, control group) was conducted in 6 seventh-grade classes (N = 160). An analysis of the interaction processes of five student groups showed that more mathematical discussions occurred in the shift-problem condition. The quality of the mathematical discussions in the shift-problem condition was better compared to that in the conventional textbook condition, but there is still more room for improvement. A qualitative illustration of two typical mathematical discussions in the shift-problem condition are provided. Although students’ mathematical levels were raised a fair amount in both conditions, no differences between conditions were found. We concluded that shift-problem lessons are powerful for eliciting mathematical discussions in seventh-grade shift-problem early algebra lessons.

We share here results from a quasi-experimental study that examines growth in students’ algebraic thinking practices of generalizing and representing generalizations, particularly with variable notation, as a result of an early algebra instructional sequence implemented across grades 3–5. Analyses showed that, while there were no significant differences between experimental and control students on a grade 3 pre-assessment measuring students’ capacity for generalizing and representing generalizations, experimental students significantly outperformed control students on post-assessments at each of grades 3–5. Moreover, experimental students were able to more flexibly interpret variable in different roles and were better able to use variable notation in meaningful ways to represent arithmetic properties, expressions and equations, and functional relationships. This study provides important evidence that young children can learn to think algebraically in powerful ways and suggests that the earlier introduction of algebraic concepts and practices is beneficial to students.

A common context for instruction in early algebra involves time, an invisible and intangible quantity commonly represented as intervals of length. This paper presents a case study of syntactically ...

Guided by a theoretical framework emphasizing the importance of fidelity of implementation (FOI), this paper explores how teachers of grades 3, 4 and 5 implemented an early algebra interven...

The ability to generalize and represent generalizations exhibited by eight fourth-grade students was analysed in this descriptive study, designed around a semi-structured interview involvin...

Professional learning experiences (PLEs) provide teachers with opportunities to improve their understanding of mathematics content and teaching practices. However, PLEs are often conducted in person and in small groups—hence costly and localized. The purpose of the current study was to explore different ways for teachers to engage in PLEs and how these approaches might enable the field to scale up these efforts in a sustainable manner. We compared the impact of three PLE formats on the early algebra knowledge and teaching practices of elementary mathematics teachers: (1) a facilitated summer workshop, (2) a multimedia course completed on teachers’ own time, and (3) learning resources provided in the algebra curriculum unit that teachers used individually. Our findings suggest that all three formats can be mapped against a set of principles for quality professional learning. Analysis of pre- and post-treatment measures indicate that participating teachers’ knowledge of algebra content and best practices significantly increased, regardless of the PLE format with which they engaged. Interviews with a subset of the teachers from the three groups point to the key features of each of the formats that can be capitalized on by designers of PLEs.

Patterns pervade our everyday life and are natural and familiar to children. Exciting new research in the field of Early Algebra describes the development of young children’s algebraic thinking thr...

A cluster randomized trial design was used to examine the effectiveness of a Grades 3 to 5 early algebra intervention with a diverse student population. Forty-six schools in three school districts participated. Students in treatment schools were taught the intervention by classroom teachers during regular mathematics instruction. Students in control schools received only regular mathematics instruction. Using a three-level longitudinal piecewise hierarchical linear model, the study explored the impact of the intervention in terms of both performance (correctness) and strategy use in students’ responses to written algebra assessments. Results show that during Grade 3, treatment students, including those in at-risk settings, improved at a significantly faster rate than control students on both outcome measures and maintained their advantage throughout the intervention.

Nell’articolo si presenta il “Progetto ArAl, Percorsi nella didattica per favorire il pensiero prealgebrico” e si introduce l’ambito teorico all’interno del quale esso si colloca, l’early algebra: un campo relativamente nuovo di ricerca e pratica didattica che contrappone alla didattica tradizionale, che fa iniziare l’algebra verso i 13-14 anni di età, l’ipotesi che sia non solo possibile, ma anche opportuno, spostare questo inizio ai 6 anni, con un’attenzione crescente verso gli alunni “grandi” della scuola dell’infanzia (5 anni). Si riportano numerosi episodi di classe (alunni fra i 6 e i 15 anni) relativi in parte a classi “inesperte” e in parte a classi “esperte”. I commenti che li accompagnano illustrano i punti chiave del quadro teorico, la prospettiva linguistica che contraddistingue le radici epistemologiche del progetto, le differenze più significative fra l’early algebra e l’insegnamento tradizionale dell’aritmetica e dell’algebra. / The paper presents the “ArAl Project, Arithmetic pathways aimed at facilitating pre-algebraic thinking” and introduces early algebra, the theoretical framework within which it is placed. Early algebra is a relatively new field of research and teaching practice. Unlike the traditional teaching, where the study of algebra begins at the age of 13-14, our hypothesis is that not only is it possible, but also appropriate, to anticipate the teaching of algebra at the age of 6, with an increasing attention towards the “great” pupils of the kindergarten (5 years old). Many class episodes (pupils aged 6 to 15) are reported, related partly to “inexperienced” classes (teachers are novices of the project) and partly to “expert” classes (skilled teachers who have been collaborating with the project for many years).

Presentamos un estudio descriptivo y exploratorio en el que se analizan los diferentes niveles de generalización que manifiestan ocho estudiantes de cuarto curso de primaria en una entrevista semiestructurada, a la vez que se reconoce la influencia, en tales manifestaciones, de los estímulos efectuados a los alumnos por parte de la entrevistadora., We present a descriptive and exploratory research in which we analyze the different levels of generalization demonstrated by eight fourth grade students of primary school in a semistructured interview, at the same time we recognize, in the students’ manifestations, the influence of the stimuli made by the interviewer to the students., El presente trabajo se ha realizado dentro del proyecto de investigación del Plan Nacional I+D con referencia (EDU2013-41632-P, EDU2016-75771-P), financiados por el Ministerio de Economía y Competitividad de España; y en el seno del grupo de investigación “Didáctica de la Matemática. Pensamiento Numérico” (FQM-193) de la Universidad de Granada, perteneciente al Plan Andaluz de Investigación, Desarrollo e innovación de la Junta de Andalucía.

This chapter begins by demonstrating how the basic ideas of algebra can be integrated into the curriculum at the early elementary level by teaching young children to use letters as descriptors of actions or objects on which actions have been carried out. These descriptors play the role of vicarious (informal) variables—the building blocks of algebraic equations which serve as mathematical models of simple real-life situations. Even at the kindergarten level, children are introduced to what may be called early algebra by learning to describe visual patterns using letters as they refer to visual representation of the AB pattern symbolism. Another approach to early algebra is through solving word problems without using formal mathematical actions on variables (see Sect. 9.8). Typically, solving a middle school word problem involves constructing a linear equation or a system of linear equations and using algebraic transformations to find the unknown(s). That is, a formal problem-solving strategy in middle school algebra is to move from equations to unknowns with the goal to replace unknowns by numbers.

This chapter aims to carry out a comparative analysis between the performances of a group of 17 elementary school fifth graders before and after having undergone a teaching intervention about problem-solving involving functional relationships. The chapter starts by promoting a theoretical discussion about early algebra in general and functional reasoning. The latter is seen under three perspectives: generalized arithmetic, generalization of patterns, and proportionality. The data is initially analyzed from a quantitative point of view, when the improvement in student performance becomes evident after the teaching intervention. Then, the analysis of reasoning types and generalization present in the actions of these students shows that, after the intervention, multiplicative reasoning was by far the most used method and that generalization, previously eminently arithmetic, became algebraic. The study concludes that developing the students’ functional reasoning and algebraic generalization skills using arithmetic situations is a good course of action to be taken. However, the authors recognize that the algebraic generalization of problem situations involving affine functions requires more sophisticated reasoning than arithmetic. Further studies are needed to better understand the generative process of elementary school students.

In this chapter, we explore a kindergarten student’s uses and understandings of tables as he explores algebraic linear functional relationships. Drawing from interview data from an early algebra classroom teaching experiment using design-based research, our findings suggest that young children are able to use and understand tables. In this chapter, we argue that young children can use tables as organizational tools and not just look at but also look through tables to develop understandings of functional relationships. We take this as evidence that tables are important representational tools for children from the earliest grades of schooling.

In the context of early algebra research and as part of a classroom teaching experiment (CTE), we investigated fourth grade (9- to 10-year-old) students’ justifications of how they performed tasks involving the functional relationship y = 2x. We related their written justifications (part of the task) to the task characteristics, which included various semiotic systems (verbal, numerical and alphanumeric, among others) and the demand of different type of justifications. The role of classroom discussion in helping express the functional relationship orally in more sophisticated terms was also investigated. The findings showed that students’ written justifications changed with the semiotic system involved in the task. Oral discussion helped students generalize in more sophisticated terms than in their written justifications, in which they omitted information or used less precise language.

This chapter aims to discuss the contributions of digital technologies to the development of algebraic thinking. The chapter begins with an overview of what algebraic thinking means, followed by a discussion of early algebra. Then, a historical overview of studies about the learning of algebra with the support of digital technologies will be presented. This overview will focus on research using educational software such as Function Probe and SimCalc. Next, more recent research based on digital resources that can be accessed on the Web, tablets or smartphones will be discussed. The relationship between technology and learning of algebra will be analysed based on the results derived from studies that promote the production of individual and group knowledge in collaborative processes favoured by the use of interactive learning environments. The main aspects in this chapter point to contributions on how technologies stimulate algebraic thinking in a way that students can: (i) use different representations, (ii) benefit from the symbolic and manipulative character of digital resources and (iii) use the current tools to produce, in a constructive and collaborative way, artefacts that allow reflection on algebraic concepts.

We describe 24 third (8–9 years old) and 24 fifth (10–11 years old) graders’ generalization working with the same problem involving a function. Generalizing and representing functional relationships are considered key elements in a functional approach to early algebra. Focusing on functional relationships can provide insights into how students work with two or more covarying quantities rather than isolated computations, and focusing on representations can help to identify the type of representations useful to them. The goals of this study are to (1) describe the functional relationships evidenced in students’ responses and (2) describe the representations that the students use. In addressing these research objectives, we describe student responses drawn from a Classroom Teaching Experiment in each grade. We analyzed students’ written responses to different questions designed to generalize the relationships in a problem that involves the function y = 2x + 6. Our findings illustrate that 11 third graders and 19 fifth graders provide evidence of functional relationships in their responses. Three third graders and all fifth graders generalized the relationship. We conclude that these differences may be due to the students’ previous classroom mathematical experiences, since students in higher grades would be more likely to focus on the relationships between variables, whereas third graders would focus on the details of arithmetic computations. In addition, we find that natural language is the main vehicle used to generalize in both grades. Unlike third graders, fifth graders perceive general rules from the numerical calculation and express these generalizations even when not explicitly requested to do so.

Background: Mathematical equivalence is a critical element of arithmetic understanding and a key component of algebraic thinking which is necessary for success in all levels of mathematics. Research studies continue to highlight misconceptions related to equivalence and reveal that many primary school students have a narrow and limiting view of the equals sign as an operation. Aim: This study aims to investigate young students’ understanding of mathematical equivalence in South Africa with a particular focus on their interpretations of the equals sign. Setting: Research data was obtained from students across six schools from different contexts within the Western Cape. Methods: We gave students an adapted standardised assessment containing 15 items related to equivalence. Results: Our analyses indicated that students focus more on the equals sign as an operation which involves calculating an answer . While some referred to equivalence as meaning the same as , most of them were inclined to accept the operational definition of the equals sign (i.e. the answer to the problem ) as a better and preferred definition. In addition, student performance was poor on equation-solving problems and they rarely used comparative relational strategies in their solutions. Conclusion: The findings of this research confirmed that difficulties with equivalence reported by earlier research is widespread across this group of grade 4 students. This has implications for both curriculum, textbook and materials design and teacher professional development.

The paper presents some reflections and activities developed by researchers and teachers involved in teacher education programs on cultural transposition. The construct of cultural transposition is presented as a condition for decentralizing the didactic practice of a specific cultural context through contact with other didactic practices of different cultural contexts. We discuss the background theoretical issues of this approach and also give an analysis of two examples of cultural transposition experiences carried out in Italy. In particular, by means of qualitative analysis of some excerpts, discussions, and interviews, we show that the contact with different perspectives coming from China and Russia fostered educational practices and reflections on whole number arithmetic education.

Conceptual understanding of properties of operations is an important element of algebraic thinking in primary school. Mathematical processes should be focused rather than mathematical products starting from primary school. The purpose of this study was to examine the Year Five pupils' understanding of relationship between addition and subtraction. Researchers utilized quantitative approach to investigate Year Five pupils' conceptual understanding of addition and subtraction. Pencil and paper based assessment consisting of three items was employed to collect the data. The three items comprised direction of change and relationship between addition and subtraction items. The three items also consist of reasoning sections. This article reports the analysis of the responses of 720 Year Five pupils from a district of Malacca. The findings showed the majority of the sample were unable to perform well for the items testing relationship between addition and subtraction. They could not work with addition and subtraction properties. The majority of them were also unable to provide conceptual reasoning for their answer. Only about half of the sample were aware about the inverse relationship of addition and subtraction.

Structured problem-solving lessons are used to explore mathematical concepts such as pattern and relationships in early algebra, and regularly used in Japanese Lesson Study research lessons. However, enactment of structured problem-solving lessons which involves detailed planning, anticipation of student solutions and orchestration of whole-class discussion of solutions is an ongoing challenge for many teachers. Moreover, primary teachers have limited experience in teaching early algebra or mathematical reasoning actions such as generalising. In this study, the critical factors of enacting the structured problem-solving lessons used in Japanese Lesson Study to elicit and develop primary students’ capacity to generalise are explored. Teachers from three primary schools participated in two Japanese Lesson Study teams for this study. The lesson plans and video recordings of teaching and post-lesson discussion of the two research lessons along with students’ responses and learning are compared to identify critical factors. The anticipation of students’ reasoning together with preparation of supporting and challenging prompts was critical for scaffolding students’ capacity to grasp and communicate generality.

In this study we investigate the teaching of the associative property in a natural classroom setting through observation of classroom video of several elementary math classes in a large urban schoo...

The objective of this study is to analyze what characteristics are of early grade students’ algebraic reasoning in context of open approach and lesson study. Ethnographic study was employed to conduct in this qualitative study. The study was carried out in one mathematics classroom which is a case study of this study and it is the case of classroom which has been using open approach and lesson study since 2006. The 3 teachers are as a member of school lesson study team participated the study as informant and so were 10 students from the class. The data were collected through 9 consecutive lessons by observation with audio-video tape recording, interview, students’ written works and daily field notes. The lessons were designed by carrying into 4 steps of open approach: posing problem, students’ self-learning, whole class discussion and comparison, and summing-up by connecting students’ emergent mathematical ideas. All activities were guided by Thai version of 1st grade Japan mathematics textbook. The results showed that characteristics of first grade students’ algebraic reasoning are as follows: 1) using algebraic expressions to represent addition situation and posing situation to represent given expressions, 2) constructing and using a tool to find problem results more easily, 3) extending solutions to another domain of number, 4) using various representations to justify their ways of thinking, and 5) reasoning about relations among numbers. The algebraic reasoning occurred under the condition that teachers and students had connected among 3 worlds oriented to Inprasitha’s approach: real world, semi-concrete world, and mathematics world.