This conversation analytic paper explores how students in pair and groupwork tasks produce and respond to claims of insufficient knowledge (CIKs). Based on 7 h of video recordings of peer interaction from 18 English as a Foreign Language (EFL) classes in Czech secondary schools, we analyze how students express and negotiate their epistemic status using CIKs: when producing a CIK, the current speaker assumes a not-knowing status, thus making the imminent speaker change more relevant, as the next speaker then typically reveals his or her epistemic status. We also show that when a CIK is produced dyadic interactions in second position, the first speaker then produces a knowledge display response, or another CIK, resulting in abandoning the question, which differs from sequences that can be found in frontal teaching. The findings also show that CIKs can be used to resume task-related talk and initiate repair sequences focusing on language issues that the task comprises. Thus, CIKs can be viewed as central interactional resources for students to manage the task, i.e., to invite others to contribute, to resume their talk, or to abandon the current question, and to initiate repair sequences focusing on problematic items from the task.