First graders’ understanding of variables: Learning trajectories in a function machine environment

This study examines how a function machine learning environment can instantiate a developmental trajectory of understanding variables in first-grade students. The intervention involved exploring input-output relationships and symbolic representations of indeterminate quantities. Data were collected through classroom interactions and interviews at three time points: before, immediately after, and six weeks following the intervention. The analyses revealed multiple developmental pathways. Some students progressed directly from pre-variable reasoning to advanced algebraic applications of variable notation. Others consolidated their understanding at intermediate stages or displayed misconceptions, such as treating letters as labels. Though a few students reverted, most maintained or deepened their new understandings, demonstrating the durability of learning. These results highlight the potential of function machines as instructional tools that facilitate exploration, identify misconceptions, and enable timely guidance. They also show how learning trajectories can inform instructional designs that foster early functional reasoning and challenge deficit views of young learners' algebraic capacities.