Invention activities are structured tasks in which students create mathematical methods that attempt to capture deep properties of data (e.g., variability), prior to receiving instruction on canonical methods (e.g., mean deviation). While experiments have demonstrated the learning benefits of invention activities, the mechanisms of transfer remain unknown. We address this question by evaluating the role of design in invention activities, identifying what knowledge is acquired during invention activities, and how it is applied in transfer tasks. A classroom experiment with 92 students compared the full invention process to one in which students evaluate predesigned methods. Results show that students in the full invention condition acquired more adaptive knowledge, yet not necessarily better procedural knowledge or invention skills. We suggest a mechanism that explains what knowledge invention attempts produce, how that knowledge is productively modified in subsequent instruction, and how it improves performance on some measures of transfer but not others.