A magnitude-based number line approach to unifying students’ pattern generalization: Didactical design research

Source abstract

Developing students’ capacity to identify regularities, generalize relationships, and express mathematical structures symbolically constitutes a fundamental objective of early algebra education. However, instruction on number patterns frequently presents arithmetic, geometric, and quadratic patterns as isolated topics, resulting in fragmented conceptual understanding and limiting opportunities for coherent algebraic reasoning across pattern types. Although previous research has demonstrated the pedagogical value of multiple representations for supporting pattern learning, limited attention has been devoted to designing instructional trajectories that systematically integrate diverse pattern structures within a unified representational framework. Addressing this gap, the present study developed and investigated a didactic design that unifies arithmetic, geometric, and quadratic patterns through magnitude-based representations on student-constructed number lines. Grounded in Didactical Design Research (DDR) and informed by the Anthropological Theory of the Didactic (ATD), the Theory of Didactic Situations (TDS), and the Concrete–Pictorial–Abstract (CPA) progression, this qualitative study involved 24 Indonesian Grade 8 students across five instructional sessions. The learning trajectory guided students from manipulating concrete objects on self-constructed number lines to diagrammatic representations of quantitative relationships and ultimately to symbolic algebraic generalizations. The findings indicate that the number line functioned as a productive didactical milieu, enabling students to visualize structural relationships among pattern types while generating productive cognitive conflicts as accelerating growth became perceptually evident. These experiences supported students’ transition from recursive additive strategies toward multiplicative, functional, and generalized reasoning. Retrospective analysis further demonstrated that the proposed design fostered integrated understandings of pattern structure, nth-term generalization, and early algebraic thinking, thereby contributing a theoretically grounded framework for advancing mathematical generalization through coherent representational learning trajectories.