Current debate surrounds the promise of neuroscience for education, including whether learning-related neural changes can predict learning transfer better than traditional performance-based learning assessments. Longstanding debate in philosophy and psychology concerns the proposition that spatial processes underlie seemingly nonspatial/verbal reasoning (mental model theory). If so, education that fosters spatial cognition might improve verbal reasoning. Here, in a quasi-experimental design in real-world STEM classrooms, a curriculum devised to foster spatial cognition yielded transfer to improved verbal reasoning. Further indicating a spatial basis for verbal transfer, students' spatial cognition gains predicted and mediated their reasoning improvement. Longitudinal fMRI detected learning-related changes in neural activity, connectivity, and representational similarity in spatial cognition-implicated regions. Neural changes predicted and mediated learning transfer. Ensemble modeling demonstrated better prediction of transfer from neural change than from traditional measures (tests and grades). Results support in-school "spatial education" and suggest that neural change can inform future development of transferable curricula.