Curiosity motivates the search for missing information, driving learning, scientific discovery, and innovation. Yet, identifying that there is a gap in one's knowledge is itself a critical step, and may demand that one formulate a question to precisely express what is missing. Our work captures the integral role of self-generated questions during the acquisition of new information, which we refer to as active-curiosity-driven learning. We tested active-curiosity-driven learning using our "Curiosity Question & Answer Task" paradigm, where participants (N=135) were asked to generate questions in response to novel, incomplete factual statements and provided the opportunity to forage for answers. We also introduce new measures of question quality that express how well questions capture stimulus and foraging information. We hypothesized that active question asking should influence behavior across the stages of our task by increasing the probability that participants express curiosity, forage for answers, and remember what they had thereby discovered. We found that individuals who asked a high number of quality questions experienced elevated curiosity, were more likely to pursue missing information that was semantically related to their questions, and more likely to retain the information on a later cued recall test. Additional analyses revealed that curiosity played a predominant role in motivating participants to forage for missing information, and that both curiosity and satisfaction with the acquired information boosted memory recall. Overall, our results suggest that asking questions enhances the value of missing information, with important implications for learning and discovery of all forms.