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Gestures are an integral component of mathematics classroom discourse. There is a need to classify the types of gestures that teachers use according to their purposes towards supporting and extending students’ mathematical thinking. We analyzed 16 algebra tutoring sessions between pre-service teachers (PSTs) and high school students to categorize the PSTs’ gestures. We identified 10 categories of PST gestures that we roughly organized into three super-categories: gestures that facilitate shared attention and communication, gestures that emphasize written visual representations, and gestures that support verbal explanations. A taxonomy of gestures based on their purposes will enable further analyses of teacher gesturing and help pre-service and practicing teachers use gestures in more purposeful ways.