Supporting Fourth‐Grade Students' Functional Thinking: A Design‐Based Early Algebra Teaching Model

ABSTRACT This study aims to design and evaluate an early algebra instructional model that fosters fourth‐grade students' functional thinking, a core dimension of algebraic reasoning. Employing a design‐based research methodology, the study was conducted across three iterative cycles—design, implementation, and refinement—culminating in a theoretically grounded and empirically validated teaching model. The research involved three fourth‐grade students in the preliminary model phase and four in the final model phase, selected to represent diverse mathematical achievement levels. Data were collected through task‐based interviews, classroom observations, student worksheets, and were analyzed using a theory‐ and data‐driven coding process grounded in the functional thinking framework of Stephens et al. (2017). Findings revealed that students exhibited recursive, covariational, and correspondence thinking depending on task type and representational support. Repeating pattern tasks primarily encouraged recursive thinking, while number and growing shape pattern tasks prompted covariational and correspondence thinking at increasingly sophisticated levels. The use of multiple representations—verbal, tabular, graphical, and symbolic—played a crucial role in students' ability to generalize relationships and articulate functional rules. Overall, the study demonstrates that a developmentally sequenced early algebra model—structured around pattern tasks and supported by representational diversity—can meaningfully advance functional thinking in elementary students.