Mathematical Competences and Geometric Meaning of Algebraic Expressions. Learning trajectories

Objective: The objective of this study is to investigate the connection between geometric knowledge and algebraic expressions to make abstract concepts more concrete, enhancing students' motivation and understanding, with the aim of identification of trajectory analysis, describing how students develop mathematical concepts, knowledge, and competency-based skills. Theoretical Framework: In this topic, the main concepts and theories that underpin the research are presented. Perrenoud, 2002; Koster et al. 2005; Monereo, 2007; Thaqi, 2009; González & Gómez, 2014; Niss & Højgaard, 2019; Berisha, 2024; stand out, providing a solid basis for understanding the context of the investigation. Method: The methodology adopted for this research comprises, specifically focuses on lower secondary school students conducted between March and May 2023. A case study approach was employed, incorporating techniques such as epistemological introspection and didactic reflection on learning processes. Additionally, a qualitative methodology was used to gather descriptive data on students' knowledge and skills. The research was carried out in four phases. Worksheets serve as key research instruments. Data collection was carried out through activity including designated space for students to document their problem-solving steps, which were later analyzed as research data. Other recordings, such as photographs and videos, were also collected during these sessions and form an integral part of the study. Results and Discussion: The results revealed that designing learning situations based on mathematical competencies and RME enhances mathematical knowledge development. Analysis of the learning activity showed diverse student approaches to algebraic expressions, where creating a mosaic with rectangles of various sizes and colors boosted motivation and engagement, forming distinct learning trajectories. Among 35 participants, 12 constructed the concept through trajectory C, 6 followed trajectory A, and 4 trajectory B. The best result is that all students developed and understanded algebraic expressions by constructing a concrete rectangle whose surface represented the concept, reinforcing their ability to conceptualize and apply algebraic expressions. Research Implications: The practical and theoretical implications of this research are discussed, providing insights into how the results can be applied or influence practices in the field of mathematics education, mathematical school activities and mathematical curriculum. These implications could encompass education in general and secondary mathematics education in particular. Originality/Value: This study contributes to the literature by highlighting diverse student approaches to algebraic expressions and identifying distinct learning trajectories, which can be effectively applied to other mathematical concepts and learning areas. Its relevance is reflected in mathematics teacher training and professional development, providing valuable insights for teachers.