# Student Learning Trajectories and Knowledge Transfer in Early Mathematical Equivalence Interventions

- Kristen Johannes, WestEd, Alameda, California, United States
- Jodi Davenport, WestEd, Alameda, California, United States

**Abstract**Many students fail to develop adequate understanding of mathematical equivalence in early grades, which impacts later algebra learning. Work from McNeil and colleagues proposes that this failure is partly due to the format of traditional instruction and practice with highly similar problems, which encourages students to develop ineffective mental models of problem types (McNeil, 2014, McNeil & Alibali, 2005). In the current study, we explore studentsâ€™ learning trajectories in two matched equivalence interventions. We show that, relative to an active control, the principle-based treatment intervention gives rise to a greater number of successful learners, a designation that, in turn, leads to improved performance on distal transfer assessments. We further demonstrate a predictive relationship between studentsâ€™ engagement with the intervention, via workbook completion, and likelihood of becoming a successful learner. Our findings have implications for early detection of learning and subsequent scaffolding for low-performing students.