TalkMoves is an innovative application designed to support K-12 mathematics teachers to reflect on, and continuously improve their instructional practices. This application combines state-of-the-art natural language processing capabilities with automated speech recognition to automatically analyze classroom recordings and provide teachers with personalized feedback on their use of specific types of discourse aimed at broadening and deepening classroom conversations about mathematics. These specific discourse strategies are referred to as talk moves within the mathematics education community and prior research has documented the ways in which systematic use of these discourse strategies can positively impact student engagement and learning. In this article, we describe the TalkMoves applications cloud-based infrastructure for managing and processing classroom recordings, and its interface for providing teachers with feedback on their use of talk moves during individual teaching episodes. We present the series of model architectures we developed, and the studies we conducted, to develop our best-performing, transformer-based model (F1 = 79.3%). We also discuss several technical challenges that need to be addressed when working with real-world speech and language data from noisy K-12 classrooms.