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Gestures are one of the ways in which mathematical cognition is embodied and have been elevated as a potentially important semiotic device in the teaching of mathematics. As such, a better understanding of gestures used during mathematics instruction (including frequency of use, types of gestures, how they are used, and the possible relationship between gestures and student performance) would inform mathematics education. We aim to understand teachers' gestures in the context of early algebra, particularly in the teaching of the equal sign. Our findings suggest that the equal sign is a relatively rich environment for gestures, which are used in a variety of ways. Participating teachers used gestures frequently to support their teaching about the equal sign. Furthermore, the use of gestures varied depending on the particular conception of the equal sign the instruction aimed to promote. Finally, teacher gesture use in this context is correlated with students' high performance on an early algebra assessment. [ABSTRACT FROM AUTHOR]

This study investigated the role of tools in supporting students to reason about even and odd numbers. Participants included Kindergarten, Grade 1, and Grade 2 students (ages 5–8) at two schools in the USA. Students took part in a cross-sectional early algebra intervention in which they were asked to generalize, represent, justify, and reason with mathematical structure and relationships. Interviews were conducted with students before, during, and after the year-long intervention to explore the ways in which tools mediated their engagement in these practices in the context of questions about parity. In addition to cubes and "dot cards," which were introduced during the interviews, we found that students also used their fingers and hands as tools. We outline the affordances and constraints of these tools in supporting students' reasoning about the properties of even and odd numbers. [ABSTRACT FROM AUTHOR]

The study of numerical sequences usually focuses on the use of symbols or finding generalizations, that is, a more formal notation. However, there is several research on the different representations, but little research in representations that we consider as drawings and illustrations and that, in general, are commonly used in textbooks when both a numerical sequence and a figurative sequence are shown. The present research with a qualitative approach sought to identify the representations of sequences contained in Middle school textbooks. Nine textbooks were analyzed using the theoretical-methodological proposal known as Mathematical Content Analysis, where 28 different figures were found to represent a numerical sequence. The findings showed that there are three types of representations in the textbooks analyzed: abstract figures, real figures, and pictorial figures. The results show that the representations that appear most frequently in textbooks are pictorial figures. [ABSTRACT FROM AUTHOR]

An important approach for developing children's algebraic thinking involves introducing them to generalized arithmetic at the time they are learning arithmetic. Our aim in this study was to investigate children's attention to and expression of generality with the subtraction-compensation property, as evidence of a type of algebraic thinking known as relational thinking. The tasks involved subtraction modelled as difference and comparing the heights of towers of blocks. In an exploratory qualitative study, 22 middle primary (9–11-year-old) students from two schools participated in individual videoed interviews. The tasks were designed using theoretical perspectives on embodied visualization and concreteness fading to provide multiple opportunities for the students to make sense of subtraction as difference and to advance their relational thinking. Twelve out of 22 students evidenced conceptual understanding of the comparison model of subtraction (subtraction as difference) and expression of the compensation property of equality. Four of these students repeatedly evidenced relational thinking for true/false tasks and open equivalence tasks. A proposed framework for levels of attention to/expression of generality with the subtraction-compensation property is shared and suggestions for further research are presented. [ABSTRACT FROM AUTHOR]

Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalization used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalization tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalized. Students were able to generalize verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter. [ABSTRACT FROM AUTHOR]

Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K–1 students' thinking about parity arguments for sums of evens and odds, as well as underlying concepts of pair and parity of a number. Data are drawn from classroom lessons of a Grades K–1 early algebra instructional sequence, as well as task-based interviews conducted at four timepoints during the implementation of the sequence. While most students at the beginning of the study (Grade K) did not know the concepts of even or odd and could not make any viable arguments regarding parity, by the end of Grade 1 students were largely constructing representation-based arguments to justify number parity and claims about sums of evens and odds. Results of this study align with other research that shows young learners can develop viable arguments to justify mathematical generalizations. Results also provide preliminary evidence that the instructional sequence used here can foster students' practice of argumentation from the start of formal schooling. [ABSTRACT FROM AUTHOR]

This study aims to investigate the effects of a functional thinking intervention on 5th-grade students' functional thinking. Forty-three fifth-grade students from two public middle schools in Ankara participated in the study, in which 20 of them formed the experimental group in one school, and 23 of them formed the control group in the other school. The experimental group participated in a functional thinking intervention lasting 12 hours (about three weeks). A Functional Thinking Test (FTT) was administered to both groups as a preand post-test. The qualitative analysis of students' functional thinking strategies supported quantitative analysis. The results revealed no significant mean difference between the experimental and control groups at the pre-test or post-test. However, the experimental group showed significant pre-to-post gains. Also, experimental group students were significantly better at using variables in defining the function rule after the functional thinking intervention. [ABSTRACT FROM AUTHOR]

The theories presented in this text have a semiotic approach, but different focuses: the Theory of Semiotic Representation Records (TRRS), developed by Raymond Duval, presents a cognitive approach in which representation records are inherent to the process of teaching and learning mathematics and enable the analysis of cognitive, treatment and conversion activities. The socio-cultural approach that delimits the assumptions of the Theory of Objectification (TO), a theory in development proposed by Luis Radford, considers Common Labor as fundamental to the processes of objectification and subjectification, anchored by semiotic means of objectification. In this way, we indicate the specificities regarding the principles, methodology and investigative questions of each theory followed by resolutions of a situation in the context of Early Algebra, with 6th year students from a public school in the north of Paraná, as an illustration for analysis under the lenses of the TO and TRRS framework. The possibility of an intertheoretical dialogue arises from the specificities and inferences of the analysis regarding the elements of each theory, where it is concluded that even with distinct ontological and epistemological aspects, both theories emerge synchronously

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. [ABSTRACT FROM AUTHOR]

The importance of early algebra has been emphasized in international literature, and it has been discussed in many studies that students who are involved in a comprehensive intervention improve their algebraic thinking skills. The aim of this study was to test the effects of an early algebra intervention on the algebraic thinking skills of 3rd-grade students. The 3rd-grade intervention and control groups were included in the study, and both groups were given pre-test and post-test. In the findings, although there was no statistical difference between the 3rd-grade intervention and control group performances in the pre-test, a statistically significant difference was found in the post-test. Analysis of the students' strategies revealed that the students in the intervention group used algebraic thinking strategies more in the post-test than the control group in three big ideas, which are equality, expressions, equations and inequalities, generalized arithmetic, and functional thinking. [ABSTRACT FROM AUTHOR]

This article presents a study that addressed the functional relationships that two early childhood education students (five-six years old) evidenced, as well as the representations they used when solving a functional thinking task. The proposed task involved the function f(x)=2x, with six questions on particular cases and one on generalization. The data was collected through a semi-structured interview to each of the students and a qualitative analysis of their answers was carried out in each of the questions of the task. The results suggest that the two students are capable of approaching the proposed task through different strategies, such as additive and multiplicative correspondence relationship, and covariation. Also, it was found that they use systems of varied representations, being the verbal representation to express the generalization of the functional relationship one that stands out. It is concluded that early childhood education students may be able to tackle tasks that involve algebraic notions that focus on functional thinking. [ABSTRACT FROM AUTHOR]

A theoretical model describing young students' (Grade 3) arithmetic-algebraic structure sense was formulated and validated empirically (n = 130), hypothesizing that young students' arithmetic-algebraic structure sense consists of five distinct but correlated factors; structure in numerical equivalence and word-problem modeling, structure in arithmetic, mathematical structure in number sequences, varying quantities and number correspondences, structure in patterns, and structure in functions. Data analysis suggested that arithmetic and arithmetic-algebraic tasks can be categorized based on the proposed model. Analysis traced three categories of students that represent different profiles of students. Students of the basic arithmetic structure sense profile approach flexibly simple arithmetic and patterning tasks that do not require generalization; students of the advanced arithmetic structure profile exhibit an awareness of local relations by describing the underlined relations in a variety of numerical situations. Students of the emergent arithmetic-algebraic profile utilize the awareness of mathematical structural relations to symbolize functional relations. A structural model showed the importance of grasping structure in patterns and describing the mathematical structure in numerical situations to develop structure in functions. [ABSTRACT FROM AUTHOR]

Copyright of Revista Interuniversitaria de Formación del Profesorado is the property of Asociacion Universitaria de Formacion del Profesorado (AUFOP) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Acta Scientiae (1517-4492) is the property of Acta Scientiae and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article aims to analyze tasks developed by multipurpose teachers within the scope of a study that involved continuing education, to work on elementary algebraic concepts. The study is located at the intersection of Early Algebra, Blended Learning and Teacher Education. A continuing education course was developed for nine teachers who teach Mathematics, in the Elementary School, enrolled in a discipline of a Master's course in Education at a public university in the state of Bahia. For the purpose of this article, the discussion focused on a fundamental concept for the study of Algebra: the symbol. Data were produced in two environments (AVA REPARE and classroom) and consisted of: forums, discussion in the elaboration of problem situations and expositions of situations. The results point to the reflections carried out by the course-teachers, which led to a (trans)formation in their pedagogical practices. The interactivity of AVA REPARE provided a plurality of data, exchanges of experiences and rich information among the course-professors. Also, the study shows that the course carried out proved to be a rich and innovative model for in-service training of teachers who work at this level of education.

Copyright of Instructional Technology & Lifelong Learning is the property of Instructional Technology & Lifelong Learning and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, our aim is to understand how teachers perceive their professional learning in an in-service teacher education process, as well as the way they related this learning to their teaching practices regarding algebraic thinking in the first grades of elementary school. It is a qualitative-interpretative study, whose data were analyzed considering the Grounded Theory. The results indicate signs of professional learning identified in teaching practice were a glimpse of possibilities to approach algebraic thinking in first grades; changing the planning for students to select and develop mathematical tasks; adopting new methodological strategies with elements of the exploratory teaching approach. [ABSTRACT FROM AUTHOR]

The equals sign is one of the most important concepts and symbols in mathematics, the understanding of which is crucial for learning and understanding all mathematical content, especially algebra. The paper will draw attention to the problems related to the formation and understanding of this concept, and present a methodological approach to learning based on the context modeling of real-life situations (modeling length, balance, etc.), with the aim of overcoming this misconception. In the empirical section of the paper, we examined the effects of the methodological framework of learning algebra on the understanding of the equals sign through an experiment with parallel groups, and on a sample of the fourth-grade students of primary school (N = 257). The obtained results show that the methodological approach based on the context modeling of real-life situations improves student understanding of the equals sign, as a symbol of mathematical equivalence and their ability to solve problems containing this sign.

Resumo: O objetivo deste artigo é analisar as possibilidades pedagógicas para um ensino que favoreça o desenvolvimento do pensamento algébrico desde os primeiros anos, assumindo a Base Nacional Comum Curricular (BNCC) como ponto de partida e pautado em literatura recente acerca do tema. Trata-se de um ensaio em que são apresentados alguns aportes teóricos os quais evidenciam que a melhora das capacidades de raciocínio e comunicação de ideias progressivamente mais abstratas está relacionada com desafios e explorações a que os alunos são expostos desde bem pequenos em atividades que valorizam a criatividade e a resolução de problemas. Destacamos, ainda, que vários pesquisadores têm indicado a importância do estudo de padrões e regularidades em Early Algebra, considerando o desenvolvimento do pensamento algébrico como etapa fundamental para o aprendizado da álgebra. O texto apresenta algumas ideias de trabalho pedagógico com padrões e regularidades para a educação infantil e os anos iniciais do ensino fundamental com base nas indicações da BNCC e aponta para a importância de destacar esse tema na formação de professores.

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. [ABSTRACT FROM AUTHOR]

This study investigated the role of online applets in early algebra lessons. The effect of two different types of intervention modules on developing students' early algebraic thinking abilities was compared. The first intervention module involved the use of open applets and real-life contexts (open-real). The second intervention module involved the use of closed applets and pure mathematics contexts (closed/pure). "Open" applets are considered to promote more explorative ways of working with mathematical ideas, whereas "closed" applets are considered to guide students' ways of working through more sequential, step-by-step approaches. Real-life contexts present everyday applications of mathematics, where pure mathematics contexts focus on the mathematical concepts and procedures, with no reference to the way they could be associated with real-life situations. Nevertheless, both intervention modules followed an inquiry-based approach. The total number of the participants were 96 young students of Grade 5 with an average age of 10,5 years old. These students were tested through a pre- and a post-test on early algebraic thinking. The test involved three categories of early algebra tasks: generalized arithmetic, functional thinking, and modeling languages. Data from the pre- and post-test comparison showed that students who participated in the "open/real" module had a statistically significant higher improvement in functional thinking compared to students who participated in the "closed/pure" module. There were no statistically significant differences between the improvement of the two groups of students in generalized arithmetic and modeling languages. These findings offer pedagogical implications in respect to the design of early algebra lessons that take advantage of the affordances of available educational technology. [ABSTRACT FROM AUTHOR]

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.

One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they perceive as arbitrary without being able to justify these rules. A good algebraic understanding involves connecting the transformation rules to other characterizations of equivalence of expressions (e.g., description equivalence that both expressions describe the same situation or figure). In order to overcome this disconnection even before variables are introduced, a design research study was conducted in Grade 5 to design and investigate an early algebra learning environment to establish stronger connections between different mental models and representations of equivalence of expressions. The qualitative analysis of design experiments with 14 fifth graders revealed deep insights into complexities of connecting representations. It confirmed that many students first relate the representations in ways that are too superficial without establishing deep connections. Analyzing successful students' processes helped to identify an additional characterization that can support students in bridging the connection between other characterizations, which we call restructuring equivalence. By including learning opportunities for restructuring equivalence, students can be supported to compare expressions in graphical and symbolic representation simultaneously and dynamically. The design research study disentangles the complex requirements for realizing the design principle of connecting multiple representations, which should be of relevance beyond the specific concept of equivalence and applicable to other mathematical topics. [ABSTRACT FROM AUTHOR]

In this paper we consider implications of the current world-wide inclusion of computational thinking in relation to children's development of algebraic thinking. Little is known about how newly developed visual programming environments such as Scratch could enhance early algebra learning. The study is based on examples of programming activities used by mathematics teachers in Sweden, teaching students aged 10–12 years during the first two years of implementing programming in the mathematics curriculum. Informed by Chevallard's praxeology in terms of praxis and logos, we describe, unpack, discuss and expand these activities. Core issues related to algebra found in the three activities are as follows: making implicit variables explicit; using a counter variable; and identifying parameters as a specific type of variable. Our findings show that, in addition to already identified uses of variables in early algebra, programming activities in the early years bring in new aspects and new ways of treating variables that could, potentially, enhance students' understanding of variables and generalization, provided that programming praxis is embedded in an appropriate algebra logos. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Functional thinking is an established route into algebra. However, the learning mechanisms that support the transition from arithmetic to functional thinking remain unclear. In the current study we explored children's pre-instructional intuitive reactions to functional thinking content, relying on a conceptual change perspective and using mixed methods. The sample included 20 grade 3 students and 24 grade 5 students. First, we assessed children's arithmetic skills and intuitive responses to generalisation tasks involving variation tables. The quantitative analysis showed that students' arithmetic skills correlated with functional aspects such as the following: identifying, and expressing function rules with words but not with the symbolic expression of function rules. The qualitative analysis revealed that students constructed framework theories that generated different intuitive conceptions of the algebraic ideas involved in noticing and expressing generalisation. Students' conflicts were concentrated in areas that determine key differences between arithmetic and algebra, such as generalisation, indeterminate quantities, and variable notation. We discuss how these results contribute to explaining the construction of algebra concepts. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Relational thinking and dealing with variables are two essential aspects of algebraic thinking. Relational thinking means viewing mathematical expressions and equations as a whole rather than as individual computing processes. It is characterized by using relationships between mathematical objects, and refers to the relations of equality and inequality. In this study, to examine the relational thinking of kindergarten and primary school children, this perspective was applied using non-symbolic representations in the form of boxes and marbles. Using multiple variables is a very powerful but also difficult tool of algebra. The study had the aim of examining how kindergarten children and primary school children establish relationships between several variables which are represented with real materials. The interview study was conducted with children aged 5–10 years. Marbles and different colored boxes represented equations with unknowns and quantities depending on each other. Initially, two approaches could be differentiated, namely, number-oriented and structure-oriented approaches. It could be shown that certain conceptualizations of variables were related to children's ability to show relational thinking. Kindergarteners are stimulated to think relationally by unknown quantities which can be determined. This process was observed in primary school children dealing with quantities that depended on each other. In addition, the conceptualization of the variables represented as boxes was examined. The concepts of general number and variable as changing quantity were categorized. Further conceptualizations resulted from the interview data, namely, categories of the undeterminable, the specific number, and the quasi-general. [ABSTRACT FROM AUTHOR]

In this study we aimed to inquire into the impact of the use of early arithmetic strategies by a group of fifth-grade students, on their solving of equations involving two representations of unknowns. Pre- and post-tests consisting of equation-solving items involving two representations of unknowns (number sentences containing empty 'brackets', as in the example 5 + (∙) = 10, or equations containing x, as in 5 + x = 10), were administered to 126 fifth-grade students in a regular class setting. We found a notable difference between students' success rates on these two types of equations and their strategy use. Most students used the inversing strategy (arithmetic operations) after formal instruction on equation solving. Several students even used both the inversing and formal strategies (performing the same operation on both sides) for the same equation. When the unknown x appeared as the subtrahend or the divisor, the success rate dropped dramatically, and students tried to use the formal solving method of performing the same operation on both sides to solve such equations. The findings of this study not only suggest how teachers can be sensitive to students' different interpretations of unknowns, but also highlight the importance of using students' prior sense making to teach equation solving and of helping students gain an in-depth understanding of equation solving and representations of unknowns. [ABSTRACT FROM AUTHOR]

This study forms part of a classroom teaching experiment on the development of a group of 25 9–10 years old students' algebraic thinking. More specifically, it explored their reasoning while solving word problems built around functional relationships to determine how they generalized through questions posed using natural language, drawn figures or the keyword 'many'. Their written and oral answers to those questions were analyzed qualitatively to determine which approach most effectively supported the expression of their generalization. The results reveal the benefits of posing questions about the general case in different ways while teaching students to use conventional algebraic representations. According to these findings, representing indeterminate quantities with the keyword 'many' induces generalization more successfully than representing them with letters. The use of letters prompts students to seek meaning for the letters, either conventionally, as an unknown or variable quantity, or otherwise, as a label or specific values assigned according to their own criteria. Identifying the most effective procedures may help teachers and curriculum designers formulate mathematical tasks that encourage students to express the generality they perceive in particular cases. Determining the communication demands of each approach is likewise highly useful. [ABSTRACT FROM AUTHOR]

A key aspect of young children's development of algebraic reasoning is the process of visualising and identifying structures to both abstract and generalise. There has been a growing body of research focused on how students form generalisations, this article adds to the existing body of research by examining how young culturally diverse students identify mathematical structures in contextual growing patterns and the teaching and learning actions that assist them to generalise. Data were collected from one classroom of 29 Year Two (6 years old) students in a low socio-economic school in New Zealand. Results from the analysis of lessons related to two tasks showed that the contextual tasks led students to notice different mathematical structures. Specific pedagogical actions were used to facilitate students' engagement with the growing patterns. These included positioning students to engage with different representations (pictorial and numerical, tabular, and natural language) to represent thinking, the use of classroom discussions, noticing and responding to student thinking, and pressing students to find far terms. The findings highlight that both the contextual patterning tasks and teacher actions supported the young students to develop a range of sophisticated generalisations related to the underlying mathematical structure and functional relationships of the growing patterns. [ABSTRACT FROM AUTHOR]

This article addresses the nature of student-generated representations that support students' early algebraic reasoning in the realm of generalized arithmetic. We analyzed representations created by students for the following qualities: representations that distinguish the behavior of one operation from another, that support an explanation of a specific case of a generalization, and that support justification of a generalization. One key finding is that representations in the form of pictures, diagrams, arrangements of manipulatives, or story contexts that embody the meaning of the operation(s) allow students to distinguish between operations. Such representations can be used by young students to illustrate relationships conveyed in specific instances of a general claim. Further, extending these representations to class of numbers is a mechanism for proving a general claim. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Copyright of Revista de Investigación en Educación is the property of Universidad de Vigo, Facultad de Ciencias de la Educacion y del Deporte and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Algebra as a study of quantitative relationship is one of four conceptions of school algebra which serves as a foundation for the concept of function. However, there is still a lack of attention to this particular relationship, especially in early algebraic reasoning. This study aims to investigate how the aspect of quantitative relationship in early algebra can be explored through area conservation activities. Understanding area conservation is said to be fundamental in developing the concept of area measurement. In this study, a ten-year old pupil was observed during her involvement while comparing area of two polygons that can be decomposed into equivalent triangles. Data for this study include the pupil’s written artefacts, and video recordings of the activities and interviews. Findings from this study show that the area conservation activity has the potential to build the notion of quantitative relationships in early algebra. The quantitative relationship between the unit of measurement and the result of measurement of a shape can also be explored, that is, the smaller the unit of measurement, the larger the result of measurement. Hence, this study can provide a groundwork for further studies in the relation between quantitative relationship in algebra and area conservation in geometry at the elementary school level.

Early algebra can prepare elementary students for the transition they will need to make from arithmetic and algebra. Although teacher preparation programs emphasize the teaching of early algebra, research on how to prepare elementary pre-service teachers (PSTs) to teach early algebra is still scarce. The replication study reported in this article was a conceptual replication study designed to examine Iranian PSTs' reasoning about pre-symbolic early algebra by looking at what was more, somewhat, and less challenging. The aims of the replication study aligned with the original study (Hohensee, 2017). Results from the replication study show that participating PSTs (N = 15) found the early algebra approach to variables and functions more challenging, indeterminable unknowns somewhat challenging, and equivalence and equations less challenging. We make comparisons with the original study, as well as offer implications and suggestions for preparing PSTs to teach early algebra. • Early algebra and variables and functions was more conceptually challenging. • Early algebra and unknowns was somewhat conceptually challenging. • Early algebra and equality and equations was less conceptually challenging. • Similarities and differences existed between Iranian and U.S. pre-service teachers. [ABSTRACT FROM AUTHOR]

Forty 4th and 9th grade students participated individually in tutorial interviews centered on a problem-solving activity designed for learning basic algebra mechanics through diagrammatic modeling of an engaging narrative about a buccaneering giant burying and unearthing her treasure on a desert island. Participants were randomly assigned to experimental (Discovery) and control (No-Discovery) conditions. Mixed-method analyses revealed greater learning gains for Discovery participants. Elaborating on a heuristic activity architecture for technology-based guided-discovery learning (Chase and Abrahamson 2015), we reveal a network of interrelated inferential constraints that learners iteratively calibrate as they each refine and reflect on their evolving models. We track the emergence of these constraints by analyzing annotated transcriptions of two case-study student sessions and argue for their constituting role in conceptual development.

This study aimed to improve fourth-grade students' generalization and justification strategies, and an early algebra teaching experiment was applied for this purpose. In the study, data were obtained through focus group interviews, student worksheets, and video and audio recordings during the research. The data obtained were analyzed by descriptive analysis. In the study, students carried out activities of continuing the pattern, expanding it, completing the missing element, and identifying the pattern rule. It is observed that students used written, verbal, mathematical, and visual representations in their pattern activities. As a result of the study, it was found that students reached generalizations using counting, modeling, iterative, whole-object, contextual, and linear strategies in the teaching experiment. Moreover, it is seen that students used explanation, figural, numerical, and algebraic strategies to verify their generalizations. In line with these results, it is concluded that the early algebra teaching experiment performed with primary school fourth-grade students is effective in students' generalization and justification strategies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]