Conversational tutoring agents have been shown to improve learning engagement and student outcomes, and large language models (LLMs) are increasingly used in these systems to provide scalable, personalized feedback. However, LLMs may perpetuate or amplify stereotypical social biases, posing particular risks in educational settings. In this study, we evaluate LLMs in conversational tutoring scenarios to identify high-confidence social biases, instances where models are unable to identify biased judgments in tutoring conversations while maintaining strong confidence in their assessments, potentially affecting their reasoning and the feedback they provide to learners. We present a new dataset generation method that enables bias evaluation under naturalistic instructional conditions by regenerating student-AI tutor interactions and introducing turns with controlled bias derived from a benchmark dataset. Using this data, we assess multiple LLMs' ability to detect stereotypical biases and analyze the confidence and reasoning underlying their responses through computational and human evaluations. We find that bias detection is substantially more challenging in conversational tutoring contexts than in benchmark-based evaluations, and that state-of-the-art LLMs are overconfident in their incorrect assessments of stereotypical bias statements. Moreover, model confidence strongly influences reasoning and feedback, highlighting the risks of overconfident, biased behavior in LLM-based tutoring agents. We conclude by discussing implications, mitigation considerations, and directions for future research.