Five research teams describe various methodologies used to develop, refine, and validate learning progressions/learning trajectories and describe challenges encountered in relation to those methods of investigation.
A recent focus of mathematics education researchers has been on the development of learning trajectories (LTs) or learning progressions (LPs) that characterize the learning of a mathematical topic. LTs include (a) a mathematical goal, (b) a developmental path, and (c) instructional activities or tasks fitting each level of thinking along the developmental path. In this session, members of five research teams describe various methodologies they have used to develop, refine, and validate LTs and describe challenges they encountered. Presenters invite participation and discussion of the issues raised among the presenting teams.
Interactive diagnostic assessment and validation of learning—The LPPSync interactive diagnostic system assesses student progress in learning trajectory constructs. Parameterized sampled items are delivered in both sequestered (assessment) and group (practice) settings, with formative feedback on student quantitative and qualitative responses and on-screen actions. Presenters discuss validation of LTs in the context of the interplay of technology-enabled item design, cloud-based delivery and data gathering, measurement considerations, and in-class use models.
Developing an initial learning progression using teaching experiments—One project developed an early algebra learning progression and validated assessments of students’ learning along the progression. This work incorporates a comprehensive approach to early algebra to integrate early algebraic concepts across multiple core content domains (e.g., functional thinking, generalized arithmetic). Year-long classroom teaching experiments (CTEs) were employed across grades 3–5 to refine and validate the progression. Presenters discuss affordances and constraints of CTEs for their goals.
Elaborating and coordinating three trajectories for spatial measurement—The presenters of the Longitudinal Examination of Children's Developing Knowledge of Measurement: Mathematical and Scientific Concept and Strategy Growth from Pre-K through Grade 5 discuss the opportunities and challenges of linking two research sites and using individual and classroom teaching experiments along with structured assessments (with Rasch analyses) to assess LTs for length, area, and volume. Speakers address the process of validating, refining or elaborating LTs.
Modifying a learning trajectory for struggling learners—In A Study of the Struggling Learner's Knowledge and Development for Number and Operation, the goal is to develop and modify LTs for Number Concepts and for the quantitative processes of Comparing, Ordering, and Estimating, including supplementation from the research literature on struggling learners and adapting them based on findings from a two-year long teaching experiment with a cohort of struggling learners in early grades.
Developing and refining learning trajectories for functional relationships—The SPARQ project developed three LTs for middle school students’ understanding of linear, quadratic, and exponential functions based on teaching experiments grounded in quantitative relationships. Each trajectory was refined through retrospective analysis and then implemented within a subsequent round of validation experiments. Presenters discuss iterative relationships between development and use and challenges encountered in scaling up.