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In recent years, large language models (LLMs) have seen rapid advancement and adoption, and are increasingly being used in educational contexts. In this perspective article, we explore the open challenge of leveraging LLMs to create personalized learning environments that support the “whole learner” by modeling and adapting to both cognitive and non-cognitive characteristics. We identify three key challenges toward this vision: (1) improving the interpretability of LLMs' representations of whole learners, (2) implementing adaptive technologies that can leverage such representations to provide tailored pedagogical support, and (3) authoring and evaluating LLM-based educational agents. For interpretability, we discuss approaches for explaining LLM behaviors in terms of their internal representations of learners; for adaptation, we examine how LLMs can be used to provide context-aware feedback and scaffold non-cognitive skills through natural language interactions; and for authoring, we highlight the opportunities and challenges involved in using natural language instructions to specify behaviors of educational agents. Addressing these challenges will enable personalized AI tutors that can enhance learning by accounting for each student's unique background, abilities, motivations, and socioemotional needs.