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Though I certainly take your point, I think giving tests to students is actually meant to combat the problem of parroting rather than understanding information. In high school I often complained that we wasted time taking tests when we could be learning new information, but if teachers determined when to move on to new material just by asking students whether we understood, I'm sure we would have always just nodded and parroted back the teacher's sentences. Knowing that you'll have to take a test on material (that you'll be asked to answer different problems using the same methods) encourages students to make sure they actually do understand the material. It might be that the best way to solve the problem of parroting rather than understanding material isn't to get rid of exams but to have more of them, or, really, to have better ones.