If one would ask me as a regular, though not systematic reader in applied linguistics to mention a major theoretical issue in this field of study, I would choose for the controversy about “behavioristic” audiolingual learning versus “mentalistic” cognitive code learning. This dichotomy is apparent in many theoretical papers, and starting point of various research programs.
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Willem J. M. Levelt (1978) studied this question.
Synapse has enriched one closely related paper. Consider it for comparative context: