Generative AI (GenAI) is entering K-12 classrooms, but its default culture does not always fit local teaching. Existing accounts either treat this misfit as macro-level structural bias or examine it through single-case studies; neither captures how much work teachers must do to make outputs usable across tasks. This exploratory, theory-building qualitative study introduces cultural distance, the gap between GenAI's default cultural repertoire and the situated demands of teachers' practice. Through interviews with 30 K-12 teachers who frequently use GenAI, 10 each from South Africa, Taiwan, and the United States, we identified three levels defined by the effort teachers invested and whether it yielded a usable output: low (minor edits), mid (sustained prompting and revision), and high (structural gaps rooted in unsupported languages, undigitized knowledge, or policy filters that no teacher effort can close). We present cultural distance as a promising framework requiring further examination and draw implications for designers, policymakers, and educators.

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