Teachers often use open-ended questions to promote students' deeper understanding of the content. These questions are particularly useful in K-12 mathematics education, as they provide richer insights into students' problem-solving processes compared to closed-ended questions. However, they are also challenging to implement in educational technologies as significant time and effort are required to qualitatively evaluate the quality of students' responses and provide timely feedback. In recent years, there has been growing interest in developing algorithms to automatically grade students' open responses and generate feedback. Yet, few studies have focused on augmenting teachers' perceptions and judgments when assessing students' responses and crafting appropriate feedback. Even fewer have aimed to build empirically grounded frameworks and offer a shared language across different stakeholders. In this paper, we propose a taxonomy of feedback using data mining methods to analyze teacher-authored feedback from an online mathematics learning platform. By incorporating qualitative codes from both teachers and researchers, we take a methodological approach that accounts for the varying interpretations across coders. Through a synergy of diverse perspectives and data mining methods, our data-driven taxonomy reflects the complexity of feedback content as it appears in authentic settings. We discuss how this taxonomy can support more generalizable methods for providing pedagogically meaningful feedback at scale.