In procedural skill learning, instructional explanations must convey not just steps, but the causal, goal-directed, and compositional logic behind them. Large language models (LLMs) often produce fluent yet shallow responses that miss this structure. We present Ivy, an AI coaching system that delivers structured, multi-step explanations by combining symbolic Task-Method-Knowledge (TMK) models with a generative interpretation layer-an LLM that constructs explanations while being constrained by TMK structure. TMK encodes causal transitions, goal hierarchies, and problem decompositions, and guides the LLM within explicit structural bounds. We evaluate Ivy against responses against GPT and retrieval-augmented GPT baselines using expert and independent annotations across three inferential dimensions. Results show that symbolic constraints consistently improve the structural quality of explanations for "how" and "why" questions. This study demonstrates a scalable AI for education approach that strengthens the pedagogical value of AI-generated explanations in intelligent coaching systems.

This paper describes a generative theory of bugs. It claims that all bugs of a procedural skill can be derived by a highly constrained form of problem solving acting on incomplete procedures. These procedures are characterized by formal deletion operations that model incomplete learning and forgetting. The problem solver and the deletion operator have been constrained to make it impossible to derive “star-bugs”—algorithms that are so absurd that expert diagnosticians agree that the alogorithm will never be observed as a bug. Hence, the theory not only generates the observed bugs, it fails to generate star-bugs.