Online one-on-one tutoring serves as a supplementary approach to traditional classroom instruction. It has been shown to enhance personalized learning and academic performance. However, the dynamics of dialogic interactions within this educational setting are not fully understood. Thus, we present a computational analysis of dialogic interactions in online one-on-one mathematics tutoring. Specifically, we devised a coding scheme tailored to online tutoring sessions and leveraged advanced artificial intelligence techniques to construct an automated model for annotating dialog acts. We then investigated the basic characteristics and interaction patterns in a dataset encompassing online one-on-one tutoring dialogs within K-12 mathematics education and obtained insightful findings. First, tutors were found to often apply both didactic and other effective teaching strategies. Second, off-task chatting accounted for a significant proportion of tutoring sessions. Third, high school students exhibited greater engagement and cognitive abilities than primary and middle school students through their more active participation and superior reasoning skills. Primary school students, despite their less active participation, responded positively when engaged by tutors. The findings highlight the importance of optimizing strategies applied by tutors and students to create a more dynamic and effective learning environment and provide valuable insights into the nature of online one-on-one tutoring.