We argue for a new curriculum design focused on teaching an untapped set of students interested in both computing and the arts. Based on insights from workshops funded by the NSF and NEA and a survey of 412 educators and practitioners, we believe the time is ripe for the broad introduction of programs in the Algorithmic Arts (AlgoArts). Such programs can be created easily and cost-effectively by reusing existing courses from computing and arts degrees, and build on well-studied benefits of teaching computing in context. This curriculum will attract students with broad interests who would not normally consider computer science as a field of study and may be more demographically diverse than traditional CS students. As creators in Algorithmic Arts, students learn to think algorithmically and interweave algorithmic modeling and computational solutions into their art thinking and practice. We envision a future where computing and the arts combine into a unified discipline, with the algorithm as a foundational medium for artistic expression. We argue that by incorporating creativity, aesthetics, and design into their work, AlgoArtists will also foster new techniques and innovative ideas in computing. We summarize key points, offer implementation suggestions based on a joint NSF/NEA report on AlgoArts curriculum design, and discuss broader community impacts.