In mathematics tutoring, using appropriate instructional discursive strategies, called "talk moves'', is critical to support student learning. Training tutors in the appropriate use of talk moves is a key component of tutor development programs. However, tutor development at scale is a challenge. Recent research has shown that automatic talk moves classification of tutorial discourse can facilitate large-scale delivery of personalized talk moves feedback. In this paper, we build on this work and share our current progress using large language models to classify talk moves in transcripts of tutoring sessions. We report classification results from fine-tuned models, prompt optimization, and supervised embedding vectors classification. The fine-tuned strategy performed best, yielding better performance (.87 macro and .93 weighted f1 score in predicting expert labels) than the current state-of-the-art RoBERTa model. We discuss trade-offs across methods and models.