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This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils’ spontaneous production using a triangular numbers activity. Based on a specific collaborative learning methodology, this study explores the possibility of constructing an Arithmetic-Algebraic Work Space around the process of constructing signs as framed by both activity theory and a technological approach, showing the spontaneous representations produced by seventh grade pupils and their evolution in a socio-cultural environment.

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.

The development of algebraic thinking from an early age is a priority in contemporary mathematics education, especially in rural contexts due to their structural and sociocultural conditions. Grounded in the Anthropological Theory of Didactics and hermeneutic phenomenology, this study aims to identify and characterize teacher profiles based on their algebra teaching practices and their perceptions of different types of mathematical thinking. A descriptive-exploratory multiple case study was conducted with empirical data from 16 teachers at a rural institution in western Colombia, analyzed using multivariate statistical techniques (cluster analysis, multiple correspondence analysis, and principal components analysis). Two teacher profiles were identified: one focused on digital technologies and problem-solving, and another that prioritizes traditional strategies. The findings underscore the importance of considering institutional and territorial conditions when designing contextualized pedagogical proposals that foster algebraic thinking in rural areas. Findings are transferable to similar contexts but not statistically generalizable to the broader teaching population.

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.

In the last decades, a broad international reform approach was visible in support of mental calculation strategies: Instead of being solely ‘transition strategies’ for learning the standard algorithms, the understanding of numerical relations is essential for mental calculation strategies, making them highly important for a viable understanding of arithmetics. Yet, mental calculation strategies are not only important for understanding arithmetics, but highly relational strategies such as the ‘Auxiliary Task’ might have an important role in the emergence of a pre-algebraic understanding of numerical relations. In this qualitative study from Germany, 4th and 5th grade learners’ (n=18) processes of interpreting the ‘Auxiliary Task’ are examined by conducting linguistic and epistemological analyzes of their conceptual understanding of the ‘Auxiliary Task’ utilizing a design-based research framework. Insights are given into specific, language-related forms of pre-algebraic generalizations of the ‘Auxiliary Task’ as well as into developmental processes within the designed learning-environment.

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.

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.

Algebraic thinking is a method of solving math problems that stresses the significance of general connections. Excellent algebraic thinking necessitates strong symbolization and generalization ability. Students aged 7 to 15 are at the Piaget thinking stage’s formal operational stage. Teachers, especially those working with secondary school students, must be aware of how kids think and reason algebraically. A detailed literature review provides an overview of research on algebraic thinking. The goal of this study was to compile a list of full-text papers that presented empirical research on algebraic thinking. The “algebraic thinking” search phrase was used to search the ERIC and Scopus databases. A total of 36 studies were included in the review. The educational levels, participants, nations, research methodologies, study objectives, data collecting tools, and analytic approaches have all been considered in studies on algebraic thinking. The number of studies published has risen over time. 2019 was the year with the most studies. The majority of the research was carried out in the United States of America. The majority of the participants in the study were elementary and secondary school pupils. Teachers’ knowledge, elements impacting algebraic thinking, relations, and comprehension, as well as measuring categories, were determined when the study was grouped according to the study’s topic. The algebraic thinking exam was the most popular data gathering instrument. The qualitative technique was used in the study of algebraic thinking. The most common method was found to be qualitative analysis. While inferential statistics are preferred in quantitative techniques, latent class analysis, cluster analysis, and test development analysis are used depending on the study design. According to the results, it was stated that in-service or pre-service teacher training is needed for the development of algebraic thinking and non-routine activities such as games should be used in the classroom. In addition, it has been determined that teaching strategies such as geometry representation, multiple representation strategies, mental computational activity also improve algebraic thinking.

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.

Recent research suggests that children in elementary grades have some facility with variable and variable notation in ways that warrant closer attention. We report here on an empirically developed progression in first-grade children's thinking about these concepts in functional relationships. Using learning trajectories research as a framework for the study, we developed and implemented an instructional sequence designed to foster children's understanding of functional relationships. Findings suggest that young children can learn to think in sophisticated ways about variable quantities and variable notation. This challenges assumptions that young children are not \"ready\" for a study of such concepts and raises the question of whether difficulties adolescents exhibit might be ameliorated by an earlier introduction to these ideas.