If you think about your own life, you might discover that those things you really know are things you figured out for yourself. The same principal applies to education--students will know (remember) what they have discovered for themselves, and are not likely to know (remember) what has been dictated to them.
Which brings us to Question 2.
Research on the brain indicates that learning happens best when it occurs with a level of interest and excitement. Information that we think is boring is quickly discarded, and is not truly learned at all.
The point is, traditional education does not work all that well. When we assign grades based on mindless tests and quizzes, we have no right to complain when we find that our students are not doing mindful work. When we tell them or show them how to perform tasks that mean nothing to them, we should not be surprised when they quickly forget what they have "learned."
Mathematics, like everything else in life, can be interesting and exciting. It can also seem meaningless and boring. The difference often lies in our approach to student learning. If we tell students how to do something they never intend to do anyway, what is the probabliity that they will learn it?
But if we allow them to personalize mathematics, to discover their own rules and make their own meaning, then at least we are giving them a real chance to learn.
