Yearly, at least one student emails me this question, after wholly relying on passive reading then failing the exam. They successfully remember and can regurgitate everything from the textbook and lectures; can compute a question if analogous enough to something déjà vu, and something they saw before; and can prove straightforward corollaries. But they can't answer unseen, new-fashioned questions or trickier computations. They wholly fail to prove more abstract results.
Students tend to prefer passive studying because it's easier and more comfortable. In fact, this Harvard study shows that most students even convince themselves that passive methods are more effective, and that lecturing professors provide "better learning" than professors who engage students in-class projects, even though those hands-on projects ultimately provide much deeper education than traditional passive methods. [Source]
But why precisely does passive study fail for learning math? Why can't math students simply read multiple textbooks on the same subject + DETAILED FULL solutions to exercises and problems, in lieu of picking up a writing implement and plugging away at exercises?
I am seeking explanations more detailed, scholarly, and specific to math than this.
The reason is that learning is not a passive activity. As some researchers have stated,
“Students cannot learn just by attending the classes, listening to teachers, memorising ready-made assignments, and producing answers. They must talk about what they are learning (output), write about it (integration), connect it to past knowledge, and use it in their daily lives. They must make what they learn part of their lives by practice.”
The problem with passive learning is that you don’t engage with the information you receive. And that means you don’t learn it as well.
This is not always so when recalling facts in a self-test—more effort is often required to bring the facts to mind, so they don’t seem as solid. From a student’s point of view, it can seem obvious which method—restudying—produces better learning. Robert Bjork refers to this as an “illusion of competence” after restudying. The student concludes that she knows the material well based on the confident mastery she feels at that moment.
And she expects that the same mastery will be there several days later when the exam takes place. But this is unlikely. The same illusion of competence is at work during cramming, when the facts feel secure and firmly grasped. While that is indeed true at the time, it’s a mistake to assume that long-lasting memory strength has been created. - Robert Madigan - How Memory Works.