In the 30th anniversary of the International Artificial Intelligence in Education Society, there is a need to look back to the past to envision the community's future. This paper presents a new framework (AIxEd) to categorize different interactions between artificial intelligence (AI) and education. We use our framework to compare papers from two early proceedings of AIED (1985 \& 1993) with AIED proceedings papers and IJAIED papers published in 2021. We find that two out of four kinds of interactions between AI and education were more common in the early stages of the field but have gained less attention from the community in recent years. We suggest that AI has more to offer education than a pragmatic toolkit to apply to educational problems; rather, AI also serves as an analogy for thinking about human intelligence and learning. We conclude by envisioning future research directions in AIED from the lens of our framework.