When Should Teachers Control AI Generation for Mathematics Visuals?

Generative AI has the potential to help teachers rapidly create classroom-ready visual materials, particularly in mathematics where diagrams and visual representations must be pedagogically meaningful and instructionally correct. However, current generative tools primarily support prompting and post-hoc editing, leaving open a key question for correctness-sensitive educational authoring: when in the generation pipeline should teachers exert control? In this paper, we investigate how the timing of human control in AI-assisted generation shapes teachers' visual authoring practices in correctness-sensitive tasks. We introduce a design space of three stages of control: pre-generation control, where users specify intent solely through natural language prompts before generation; mid-generation control, where users inspect and confirm an explicit layout structure before the system completes generation; and post-generation control, where users directly modify AI-generated visuals after generation through object-level edits. In a within-subject, mixed-methods study with 24 primary mathematics teachers, post-generation control received higher ratings on predictability and correctness, while other subjective measures showed no reliable differences. Qualitative findings explain these differences by revealing workflow trade-offs: highly automated, pre-generation control supports rapid ideation but reduces perceived agency and predictability; mid-generation control improves structural alignment at the cost of additional effort; and post-generation control preserves user agency through low-cost, direct verification and correction. Together, these results suggest that in correctness-sensitive educational tasks, effective generative tools should align system behavior with teacher intent and support stage-dependent workflows that combine automation with direct manipulation.