The conventional approaches to mediation analysis such as path analysis and structural equation modeling typically involve specifying two structural models, one for the mediator and the other for the outcome. We employ an alternative approach that avoids some strong identification assumptions invoked by the conventional approaches. By applying a new weighting procedure to the observed data, we estimate the average potential outcome if the entire population were treated, the average potential outcome if the entire population were untreated, and the average potential outcome if the entire population were treated and if every individual unit's mediator value would counterfactually remain at the same level as it would be when untreated. The estimated differences among these average potential outcomes provide estimates of the total effect, the natural direct effect, and the natural indirect effect. Applying this approach to multilevel educational data, we evaluate the total effect of the algebra-for-all policy in the Chicago Public Schools by comparing the math achievement of two ninth-grade cohorts. We further investigate whether the policy effect was mediated by the policy-induced change in class peer ability. Combining weighting with prognostic score-based difference-in-differences adjustment enables us to reduce both measured and unmeasured confounding. (Contains 11 footnotes, 1 figure and 6 tables.)