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In this research, we explore how a group of 21 future primary school teachers attend to, interpret, and make decisions based on the algebraic thinking demonstrated by girls when solving numerical equalities. We adopt the conceptual perspectives of Noticing and MTSK to deepen the participants' professional knowledge. The future teachers participated in a course on teaching and learning school algebra, which followed a video analysis methodology. The main results show that the participants consider the algebraic elements involved in the situation, although the use of evidence is limited. Additionally, we highlight the mobilization of various mathematical knowledge for teaching school algebra. [ABSTRACT FROM AUTHOR]

Algebraic thinking is a person’s ability to understand, analyze, and solve problems using algebraic concepts to simplify statements and find solutions. Currently, many prospective teachers still lack proficiency in applying algebraic thinking skills. Self-efficacy is one of the factors that influences algebraic thinking ability. This study aims to reveal the relationship between self-efficacy and algebraic thinking skills in pre-service mathematics teachers. In the context of solving math problems, especially algebraic ones, algebraic thinking skills are crucial. Using a qualitative method with a descriptive approach, the study employed interview guidelines, questionnaires, and tests as instruments. The results show a clear correlation between the level of self-efficacy and algebraic thinking ability. Pre-service teachers with high self-efficacy can effectively evaluate information, use symbols to represent variables, and solve algebraic equations well. They are also able to determine the values of unknown variables. On the other hand, participants with moderate self-efficacy can interpret and communicate information but are less systematic in selecting problem-solving steps that involve abstraction. Participants with low self-efficacy struggle to interpret information and cannot explain the relationship between the information in the problem and the question asked, leading to incorrect solutions. The conclusion of this study is that the higher the level of self-efficacy, the better one’s algebraic thinking ability. This indicates the importance of enhancing students’ self-efficacy to support more effective algebra learning. [ABSTRACT FROM AUTHOR]

Given the relevance of graphs of functions, we consider their inclusion in primary education from the functional approach to early algebra. The purpose of this article is to shed some light on the students' production and reading of graphs when they solved generalization problems from a functional thinking approach. We aim to explore how 3rd and 4th graders construct graphs associated to functions and what elements they use; and how they read function associated graphs and whether they connect pairs of values to see beyond the data. After four working sessions about functions, we designed and implemented individual interviews to 12 students. Through a qualitative analysis, we highlight that the students can read data in a graph on two different cognitive levels and also construct it from different elements of the graph initially provided. Regarding data reading, we evidence two levels: (a) literal reading of a given element in the graph, and (b) reading beyond the data. The construction of the graph is described with base on the axes, values and labels on the axes, scale of the axes, and construction techniques. We present examples of students' work that evidence that graph construction varied depending on whether it was created from a blank sheet or it was necessary to provide help regarding the axes or the scale of the graph. We describe several techniques used by the students in the representation of data that yield non-canonical representations of a graph and that help glimpse how students are interpreting this representation. [ABSTRACT FROM AUTHOR]

This work is part of a master's research and aims to analyze the mathematics teacher's macrodecisions for the development of algebraic thinking in students of the seventh grade of Elementary School. We take as a theoretical reference the French-influenced Mathematics Didactics, particularly the Teacher's Activity Levels Model. It starts from the premise of how the participant, through his didactic decisions, can contribute to the development of this way of thinking. Our study had the participation of a mathematics teacher who teaches in the final years of basic education, in a state school, located in the city of Pesqueira, municipality belonging to the Agreste region of Pernambuco -- Brazil. The data were constructed through the analysis of the lesson plan on the knowledge of first-grade equations, prepared by the participating teacher, and a semi-structured interview. The obtained results show that the notion of algebraic thinking needs to be widely discussed in initial and continuing teacher education, as well as clarified in curricular guidelines on the teaching of algebra. In general, the teacher points out several important teaching strategies for the development of algebraic thinking. We also add that the choices made and the didactic decisions taken by the teacher establish an approximation with this way of thinking. [ABSTRACT FROM AUTHOR]

This work, developed within the framework of the Mathematics Education and Digital Technologies Research Group (EMATED), aims to present and analyze the development of algebraic thinking involved in performing tasks on polynomials and their operations, using the Algebra Tiles resource on the Mathigon platform. To do this, the idea of the area of rectangular regions was used to represent polynomials of degree less than three. Tasks on representation and the four operations on polynomials were applied. The participants were students in the third year of secondary school. We used the computer laboratory for six hours. From the theoretical perspective adopted in this research, the understanding of algebraic thinking presupposes an epistemological position of a historical nature. To this end, this epistemological basis describes three conditions that characterize this type of mathematical thinking: the object, its symbolic representation and analyticity. This understanding of algebraic thinking and objectivation theory provided us with the epistemological support for analyzing the data recorded by audio and video. The data was analyzed with a focus on the processes of generalization and abstraction present in algebraic thinking. The result shows the development of algebraic thinking in relation to operations with polynomials of both degree two, as explored in the tasks, and degree greater than two. [ABSTRACT FROM AUTHOR]

This work identifies differentiating traits of mathematical talent in seventh and eighth grade secondary school students who solved an admission test to a program to stimulate mathematical talent. A comparison is carried out between the students admitted to the program and those not admitted, focused on the analysis of the resolution of a generalization problem that involves a functional relationship. The results reveal the application of efficient strategies and the consistency between their responses. Admitted students stood out for mainly following complete regularities and symbolically representing their generalizations, they evidenced more varied, coherent and complex structures than the other students. [ABSTRACT FROM AUTHOR]

We conducted a descriptive exploratory study in which we analyzed 313 sixth to eighth grade students' answers to a word problem, accompanied by diagrams, involving generalization in an algebraic functional context. In this research, we jointly addressed two objectives: (a) to determine the strategies deployed by students to generalize and (b) to identify the types of representation used to express their generalizations. We integrated how regularities are produced, evidenced in structures and represented by students. One of the most prominent findings was that functional strategy was used by almost all the students who generalized. They expressed the generalization using verbal, symbolical, or multiple representations. Ways of expressing regularities that are not restricted to algebraic symbolism are also shown. Although the potential to identify functional relationships was observed in sixth graders, seventh and eighth school students were able to represent more varied and structurally complex relationships. However, no relevant differences in generalization strategies were found between students of different ages with and without previous algebraic training. [ABSTRACT FROM AUTHOR]

Algebraic thinking is the ability to generalize about numbers and calculations, find concepts from patterns and functions and form ideas using symbols. It is important to know the student’s algebraic thinking process, by knowing the student’s thinking process one can find out the location of student difficulties and the causes of these difficulties. This study aims to analyze students’ algebraic thinking processes in constructing new knowledge of high-ability students based on the Cognitive System of Marzano’s Taxonomy. The subjects in this study were twenty one mathematics teacher candidates who took linear programming courses. The algebraic thought process of prospective instructors in solving linear equation problems is described using a qualitative descriptive technique in this study. The data collection technique starts with giving algebraic thinking questions and interviews/observations. Data reduction, data display, and deriving conclusions are the data analysis techniques employed. The results of the research show that algebraic thinking processes with types Generasional Representasi Sequential Concrete students are able to extract conclusions and organize better. Algebraic thought processes with types Generational Representation of Concrete Random students are able to build models and form generalizations but their representation is not good enough that they cannot be communicated properly.. [ABSTRACT FROM AUTHOR]

Copyright of Buca Faculty of Education Journal / Buca Egitim Fakültesi Dergisi is the property of Buca Faculty of Education Journal 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 paper provides a longitudinal analysis of the understanding of repetition patterns by 24 Spanish children ages 3, 4 and 5, through representation and the type of justification. A mixed quantitative and qualitative study is conducted to establish bridges between algebraic thinking and computational thinking by teaching repetition patterns in technological contexts. The data are obtained using: a) participant observations; b) audio-visual and photographic records; and c) written representations, in drawing format, from the students. The analysis involves, on the one hand, a statistical analysis of the representations of patterns, and on the other, an interpretive analysis to describe the type of justification that children use in technological contexts: "elaboration", "validation", "inference" and "prediction or decision-making". The results show that: a) with respect to the representation of patterns, errors decreased by 27.3% in 3-to-5-year-olds, with understanding and correct representation of repetition patterns gaining prominence in more than 50% of the sample from the age of 4; b) on the type of justification used, it is evident that in 3-and-4-year-olds, "elaboration" predominates, and at 5, progress is made towards "validation". We conclude that it is necessary to design learning sequences connected with theory and upheld through practice, and that foster the active role of the teacher as a promoter of teaching situations that help spur the beginning of computational and algebraic thinking. [ABSTRACT FROM AUTHOR]

There is international emphasis on the right that all individuals should have to comprehensive education with learning opportunities tailored to their educational needs, and Colombia is no exception. Thus, the work reported here aims to (a) propose a curricular structure that allows addressing diversity in mathematics class, enabling flexibility and adaptation according to students' particularities and (b) construct didactic designs of mathematics adjusted to a flexible and adaptable curricular structure, addressing diversity in the mathematics classroom in Colombia. This article partially addresses these objectives by exploring the question: What conceptual elements need to be considered to construct didactic designs of mathematics that address diversity in the classroom? Consequently, the study presents elements of a curricular proposal based on universal design for learning (UDL) to address diversity in mathematics classes. This is exemplified through a didactic design created for the study of sequences and patterns, promoting, in basic and middle education, the development of algebraic thinking through activities involving generalization and the study of patterns. [ABSTRACT FROM AUTHOR]

This paper is part of broader research being conducted in the area of algebraic thinking in primary education. Our general research objective was to identify and describe generalization of a 2nd grade student (aged 7–8). Specifically, we focused on the transition from arithmetic to algebraic generalization. The notion of structure and its continuity in the generalization process are important for this transition. We are presenting a case study with a semi-structured interview where we proposed a task of contextualized generalization involving the function y = x + 3. Special attention was given to the structures evidenced and the type of generalization expressed by the student in the process. We noted that the student identified the correct structure for the task during the interview and that he evidenced a factual type of algebraic generalization. Due to the student's identification of the appropriate structure and the application of it to other different particular cases, we have observed a transition from arithmetic thinking to algebraic thinking. [ABSTRACT FROM AUTHOR]

This study focuses on the research question: How do prospective elementary school teachers (FT) make decisions when considering the algebraic thinking of 9-year-old children? A specific type of noticing was used to describe the decisions of 21 FT when observing the strategies employed by 3 children to solve the equation 6+4=+5. The FT participated in a six-session course on the teaching and learning of algebra, based on video analysis as a means to approach teaching practice. Focusing on sessions 1 and 2, the written responses of the FT to two distinct questions were examined. The main results show that, although the FT's decisions frequently lacked evidential support, their reasoning aligned with specific aspects of algebraic thinking and research covered during the course. Finally, two positions adopted by the FT in decision making were identified: a) arithmetic, centred on the calculations that children should follow, and (b) relational, focused on the interplay of operations through the equals sign. The role of noticing and video analysis as tools to bring future primary school teachers closer to the practice of teaching algebra in primary education was discussed. [ABSTRACT FROM AUTHOR]

A Design-Based Research is developed with 24 4-year-old children to design and validate a teaching itinerary of repetition patterns and to evaluate its effect from the analysis of the modes of algebraic thinking (recursive, relational and functional) mobilised by 8 children with an average Mathematical Competence Index (MCI) in the most concrete contexts of the itinerary. The results show: a) a 22% difference in success between concrete and abstract contexts; b) a greater presence of recursive than functional thinking. It is concluded that the teaching of repetition patterns should ensure the transition from recursive to relational and functional thinking. [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]

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]

The study aimed to examine the effect of the in-service training course designed for teaching algebra on the awareness of mathematics teachers about the nature of the transition from arithmetic to algebra and their material design self-efficacy beliefs. Within the scope of the study, a 30-hour in-service training course was designed for mathematics teachers, and the study examined the effectiveness of this in-service training course. In this context, the study was carried out using the experimental research method. The participants consisted of 36 mathematics teachers who volunteered to work in public schools in aprovince in the northeast of Türkiye in the 2021-2022 academic years. The study's experimental group consisted of 16 mathematics teachers who attended the algebra teaching in-service training course. In comparison, the control group consisted of 20 mathematics teachers who received no intervention. "Awareness Scale for the Nature of Transition from Arithmetic to Algebra" and "Material Design Self-Efficacy Belief Scale "were used to collect the data for the study. Descriptive and predictive statistics were used in the analysis of the data. The results showed that the designed in-service training course did not create significant mean difference between the experimental and control groups. [ABSTRACT FROM AUTHOR]

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.)

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]

Mathematics teachers still have to create their creativity in online and offline learning. Therefore, mathematics teachers must pay attention to the assignments given to their students. One of the higher-order thinking skills that teachers must consider is algebraic thinking. This study aims to describe students' algebraic thinking as impact of online and offline learning. Researchers want to see the difference in algebraic thinking between students who are given online mathematics learning and students who are given offline mathematics learning. This study uses a qualitative research approach. Participants in this study consisted of 30 students taken from 2 junior high schools taken in the city of Ambon. The research procedure carried out in this research process is the stage of giving questions and thinking hard, as well as the interview stage. The interview guide was made based on indicators of algebraic thinking (Herbert & Brown, 1997). The results showed that the algebraic thinking skills of students who were subjects of online learning were said to be incomplete because they experienced construction holes at the stage of looking for patterns and generalizations. In contrast, students who were subjects of offline learning had complete algebraic thinking according to the algebraic thinking process.

Algebraic thinking is important for developing students' mathematical generalization abilities, and identifying patterns, shapes, and symbols. This study uses a qualitative descriptive approach that aims to describe the profile of students' algebraic thinking in solving modeling problems. The subjects of this study were 158 fifth-grade students in two private schools in Surakarta, Central Java. The data collection instrument used was six test questions related to the modeling component adopted from Ralston. Prior to use, the instrument was validated by 3 elementary school mathematics learning experts and tested on 10 students who were not the subject of the study. Based on the test results, there were 12 students who scored in the high category. This study focuses on analyzing the algebraic thinking profile of students with high categories. The results showed that high-category subjects were able to demonstrate understanding related to modeling indicators in algebraic thinking, namely understanding the meaning of the equal sign (=) as an equivalence relationship, using variables to solve problems in the form of equations, and understanding the relationships between arithmetic operations. However, a small number of subjects still experience calculation errors and understanding errors in algebraic operations. Hence, it can be concluded that subjects with high categories are able to demonstrate algebraic thinking skills in the modeling component

The mathematics education literature indicates a consensus regarding the importance of developing algebraic thinking in elementary school mathematics. However, the approaches used to implement this concept vary around the world. This study examines how a popular standards-based elementary mathematics textbook series in China provides opportunities to learn about functional thinking, which is a key component of algebraic thinking. Building on the literature, an analytical framework was generated to examine the features of function-related tasks in the textbook series across grades. The results of the fine-grained coding analysis show that four categories of function-related tasks in the textbook provide opportunities to learn about multi-modes of functional thinking. These tasks primarily serve to enhance arithmetic learning, while offering opportunities to learn about functional thinking as an embedded component. Elaborated design and arrangement of the function tasks promote opportunities to learn about multiple modes of functional thinking. In addition, two pathways to support the development of functional thinking are identified. Finally, the implications for task design and textbook development which attempt to develop functional thinking are discussed. [ABSTRACT FROM AUTHOR]

Little is known about the cognitive effort associated with algebraic activity in the elementary and middle school grades. However, this investigation is significant for sensitizing teachers and researchers to the mental demands of algebra learning. In this paper, we focus on the relationship between algebraic thinking and domain-general cognitive abilities. The sample of the study comprised 591 students from grades 4 to 7. The students' abilities in algebraic thinking were assessed through a test that involved three task categories: generalized arithmetic, functional thinking, and modelling languages. Test batteries were used to assess students' domain-general cognitive abilities in terms of analogical, serial, spatial, and deductive reasoning. The results of structural equation modelling analysis indicated that (a) analogical reasoning predicts students' abilities only in generalized arithmetic; (b) serial reasoning predicts students' abilities only in generalized arithmetic; (c) spatial reasoning predicts students' abilities in functional thinking and modelling languages; and (d) deductive reasoning predicts students' abilities in all three categories. Differences between students across grades are also discussed. [ABSTRACT FROM AUTHOR]

The theoretical reflection on the nature of school algebra and the development of algebraic thinking from the first educational levels is a relevant topic in mathematics education. In this paper, we first summarize and clarify the positions held on this topic by two theoretical frameworks: the Theory of Objectification and the Onto-semiotic Approach. We identify concordances and complementarities in the respective conceptions of algebra and of the processes of developing algebraic thinking, as a result of applying these frameworks to analyse some students' responses to a task on numerical patterns. We also analyse and compare the didactic models proposed in both theoretical frameworks for developing algebraic thinking. The analysis of the algebraic thinking and learning episodes is also used to illustrate a general discussion on the identity and boundaries of these theoretical frameworks and the possibilities of coordination and integration. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Turkish Educational Sciences is the property of Journal of Turkish Educational Sciences 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 Jurnal Tadris Matematika is the property of Institut Agama Islam Negeri (IAIN) Tulungagung, Department of Mathematics Education 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 study is developed within the field of algebraic thinking in elementary school. We worked with six second graders (7-8 years old) on a generalization task during individual semi-structured interview. We focused on the mediations of the researcher-teacher who guided these interviews. The research objective is to describe the generalization of the students before and after the mediations. The results highlight that, at the end of the interviews, all six students generalized, some of them correctly and others incompletely. We also identified that the mediations played a fundamental role in identifying the regularity of the task, allowing the students to correct errors and justify. [ABSTRACT FROM AUTHOR]

Copyright of Diversitas Journal is the property of Diversitas Journal 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 study aims to analyze the transformation of algebraic thinking and calculus of preservice the mathematics students based on SOLO taxonomy in solving differential equations problems. The research subjects were 86 students in the mathematics education study program. Subject selection uses purposive sampling (students who take courses in differential equations). Data were collected using problem-solving tests and interviews which were then analyzed using the descriptive qualitative method with the following stages: (1) transcribing test and interview data, (2) coding segmentation, (3) analyzing student thinking transformations, and (4) concluding. The results showed that the transformation of algebraic and calculus thinking was used by students at each level of thinking to solve problems. The higher the level of thinking achieved, the better and the maximum transformation of algebraic and calculus thinking used by students. These results indicate that students need to be well supported and facilitated in problem-solving to achieve higher levels of thinking, such as the relational and extended abstract levels.

Copyright of Journal of Curriculum & Teaching Methodology / Maǧallaẗ al-Manāhiǧ wa-Turūq al-Tadrīs is the property of Arab Journal of Sciences & Research Publishing (AJSRP) 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 INDIVISA: Boletín de Estudios e Investigación is the property of Centro Universitario La Salle 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 study, it is aimed to determine the algebraic thinking habits of two eighth grade school students through the answers they gave in the process of solving mathematical problems. The algebraic habits of mind (ZCA) theoretical framework developed by Driscoll (1999) was used to reveal these thinking habits. The research design of this study is a case study and the participants consist of two eighth grade students. The data were analyzed using Driscoll (1999)'s ZCA framework, which is algebraic habits of mind. Descriptive analysis was used in the analysis of the data. When we look at the findings obtained from the research; It is seen that both students can do describing a rule and justifying a rule in the solutions of the problems. In addition, it is seen that computational shortcuts, equivalent expressions and symbolic expressions come to the fore in the solutions of students' problems. On the other hand, the habit of undoing in solving problems was not encountered very rarely in both students. In the light of the findings obtained, the reasons for the existing and non-existent algebraic habits of mind are discussed. As a result of this discussion, it is thought that it is effective to include guide questions to create and develop algebraic thinking habits in students in classroom teaching practices of teachers. [ABSTRACT FROM AUTHOR]

One of the cognitive characters emphasized by different researchers in mathematically gifted students is generalization of mathematical structures and patterns. In particular, experience with growing geometric patterns is important for initiating and developing algebraic thinking. In this context, this study aimed to explore the generalization strategies used by gifted students in the growing geometric pattern task. The study was designed in a case study. The participants of the study are five eighth grade students who were diagnosed as gifted through diagnostic tests. The data of the study were collected with the "Geometric Pattern Task Form" consisting of open-ended problems. The geometric pattern task consists of linear and quadratic patterns. Data were collected by task-based interview method and analyzed with thematic analysis. The results of the study show that gifted students exhibit figural and numerical approaches while solving pattern problems. In particular, for quadratic (nonlinear) pattern, gifted students used functional strategy in all problems of finding near, far terms, and general rule of pattern. However, in the problems of finding the number of white balls (linear pattern), different strategies (e.g., recursive, chunking, contextual) than the functional strategy were also used. Based on the results of the study, it is suggested that geometric pattern tasks involving linear and non-linear relationships may be centralized in the development of functional thinking and generalization skills of gifted students in classroom practices. [ABSTRACT FROM AUTHOR]

This paper builds on our previous research and investigates how students' fractional competence and reasoning can provide clear evidence of non-symbolic algebraic thinking and its progressive transition towards fully generalised algebraic thinking. In a large-scale study, 470 primary students completed a written paper and pencil test. This included three reverse fraction tasks which required students to find an unknown whole when presented with a quantity representing a fraction of that whole. Seventeen students from one participating primary school undertook a semi-structured interview which included reverse fraction tasks, similar to those on the written test, but with progressive levels of abstraction, starting with particular instances and becoming more generalised. Two important products of the study are the Classification Framework for Reverse Fraction Tasks and the Emerging Algebraic Reasoning Framework. The interview results highlight two critical transition points for the emergence of students' algebraic reasoning. The first is the ability to transition from additive strategies to multiplicative strategies to solve reverse fraction problems. Students reliant on diagrams and additive strategies struggled to solve more generalised tasks that required multiplicative rather than additive strategies. The second transition is the shift from multiplicative thinking to algebraic reasoning where students could generalise their multiplicative knowledge to deal with any quantity represented in a reverse fraction task. [ABSTRACT FROM AUTHOR]

Early algebraic thinking is the reasoning engaged in by 5- to 12-year-olds as they build meaning for the objects and ways of thinking to be encountered within the later study of secondary school algebra. Ever since the 1990s when interest in developing algebraic thinking in the earlier grades began to emerge, there has been a steady growth in the research devoted to exploring ways of fostering this thinking. While in its early days this research had to grapple with the question of what kinds of algebraic thinking might be feasible for the younger student, the evolution of the field over the past 30 years has led to an ever-increasing range of activity that is truly multi-dimensional. In this survey paper, I have framed the multi-dimensionality of early algebraic thinking according to three overarching types, namely, that of analytic thinking, structural thinking, and functional thinking, with generalizing being the scarlet thread that runs through all three. The first part of the paper looks back to the history of the notion of early algebra and the initial research efforts aimed at characterizing early algebraic thinking. The second part delineates the three overarching theoretical dimensions of early algebraic thinking, presents a sampling of past empirical findings, and points to some of the more recent work in the field, including the contributions to this Special Issue. The paper concludes by highlighting the new directions of this domain of research and offering suggestions for further research. [ABSTRACT FROM AUTHOR]

Understanding of the equal sign is associated with early algebraic competence in the elementary grades and equation-solving success in middle school. Thus, it is important to find ways to build foundational understanding of the equal sign as a relational symbol. Past work promoted a conception of the equal sign as meaning "the same as". However, recent work highlights another dimension of relational understanding—a substitutive conception, which emphasizes the idea that an expression can be substituted for another equivalent one. This work suggests a substitutive conception may support algebra performance above and beyond a sameness conception alone. In this paper, we share a subset of results from an online intervention designed to foster a relational understanding of the equal sign among fourth and fifth graders (n = 146). We compare lessons focused on a sameness conception alone and a dual sameness and substitutive conception to each other, and we compare both to a control condition. The lessons influenced students' likelihood of producing and endorsing sameness and substitutive definitions of the equal sign. However, the impact of the lessons on students' approaches to missing value equations was less clear. We discuss possible interpretations, and we argue that further research is needed to explore the roles of sameness and substitutive views of the equal sign in supporting structural approaches to algebraic equation solving. [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]

The purpose of this embedded-single case study was to examine pre-service elementary teachers’ noticing expertise of students’ algebraic thinking in written works considering three skills: attention to students’ solutions, interpretation of students’ solutions, and deciding how to respond to students’ solutions. The participants in this study involved 32 pre-service teachers who were enrolled at an Elementary Teacher Education Program in a public university in Turkey. The data were utilized by pre-service elementary teachers’ responses to four students’ solutions to a figural pattern task and were analyzed using the framework developed by Jacobs et al. (2010). The analysis indicated although the pre-service teachers could not provide robust evidence of attention and interpretation, they could be able to provide robust evidence of deciding how to respond. Specifically, the percentage of pre-service teachers demonstrating robust evidence was greatest in the skill of deciding how to respond, then interpreting, with attending having the lowest percentage of pre-service teachers demonstrating robust evidence.