Although basing instruction on a learning trajectory (LT) is often recommended, there is little evidence regarding the premise of a LT approach--that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child's present level of thinking. This hypothesis was evaluated for an empirically-validated LT for early arithmetic, using with 291 kindergartners from four schools in a Mountain West state. Students randomly assigned to the LT condition received one-on-one instruction one level above their present level of thinking. Students in the counterfactual condition received one-on-one instruction that involved solving story problems three levels above their initial level of thinking (a teach-to-target approach). At posttest, children in the LT condition exhibited significantly greater learning, including target knowledge, than children in the teach-to-target condition, particularly those with low entry knowledge of arithmetic. Child gender and dosage were not significant moderators of the effects. [This paper was published in "Journal of Educational Psychology" v113 n7 p1323-1337 2021 (EJ1318761).]