Although reformers have embraced learning trajectories (LT, also called learning progressions) as an important tool for improving mathematics education, the efficacy and assumptions of LT-based instruction are largely unproven. The aim of a recently completed research project was to fill this void. Fulfilling this aim was more challenging than many supporters of LT-based instruction might imagine. A total of 10 experiments were untaken, of which 5 demonstrated that LT-based instruction was significantly more efficacious than a counterfactual involving either a Teach-to-Target/Skip-Level approach (Assumption 1) or the same unordered activities (Assumption 2). The results of the remaining studies were non-significant either for theoretical (2) or methodological (3) reasons. In the five indicating LTs' efficacy, we found that some LTs consists of levels that are facilitative conditions for the next higher level and, thus, may be helpful but perhaps not necessary for the subsequent level.