Productive Failure as described in this article has been a major inspiration behind Mathhooks.com. Inquiry before instruction aims to benefit students by allowing them to be creative, think critically, analyze their own prior knowledge and "problem solve" in the purest sense of the term since they are asked to consider problems that have yet to be taught. Direct instruction, on the other hand, invites only the "correct" approach to a problems, thus squashing any independent thinking on the student's part. Instead, students are encouraged to study and learn algorithms and to value memorization and repetition over analyzation and imagination. I've used both methods as a teacher and, while it may lack scientific study, I feel confident claiming that I can "feel" the difference in the room when kids get to explore a topic first on their own. Rather than asking questions about formulaic steps and rote processes, they ask questions for clarification and curiosity in order to align their new-found knowledge with their own original approaches. It's learning that clicks and sticks and it's why I believe math hooks are a valuable tool for math classrooms.