Effective teaching requires reasoning about both the task and the learner’s hidden state, a capability captured by Theory of Mind – inferring another agent’s latent state from its behavior. We ask whether a stronger large language model (LLM) can teach a fixed student better when it follows this principle. The student is an LLM agent operating through a harness that fixes its tools, context, and action budget, so a procedure that helps one model may exhaust another’s budget. We introduce Student‑Aware Teaching (SAT). The teacher builds a learner model that pairs an empirical profile with hypotheses about the student’s limitations, tests these hypotheses with diagnostic probes, and writes reusable skills for the diagnosed failures. A skill is admitted to the library only after it raises success on matched development tasks. In experiments, Claude‑Sonnet‑5 teaches DeepSeek‑V4‑Flash; SAT reaches 86.46% task success across four primary runs, surpassing a profile‑free teacher using the same framework (81.04%) and the untaught student (75.00%) in each run. Removing the analytical learner model lowers both prediction accuracy and success, and under misleading learner profiles none of the 48 candidate skills passes admission. When DeepSeek‑V4‑Flash and Gemini‑3.8‑Flash exchange libraries, each performs better with its own. Saved libraries retain average gains of up to 11.1 points after the teacher departs. The benefit depends on a student that can follow instructions and a deployment route that delivers the skills.\n\nReview: SAT demonstrates that explicitly modeling student constraints and generating targeted reusable skills can markedly improve the task success of budget‑constrained LLM agents, offering a practical teaching framework for real‑world deployments.