Some students listen, watch a couple examples and then may have a question or two before diving into a set of problems. There is a good chance, if you are a math teacher reading this, that you were one of these students.
I have a few students for whom this is sufficient. You will notice I wrote a few.
Most of my students do not act like this. They doubt themselves, frequently because they struggle with math. These same students, when I can work with them in small groups are more willing to make mistakes. Surprisingly, I've also found that frequently it just takes 1-2 problems for (these) students to work through the difficulties.
Here is a bold statement... In math students need to make mistakes
Students struggling need to face their misconceptions. They need to see what doesn't work and see why it doesn't. They need to sometimes make the mistake enough times to remember it the next time they see such a problem.
I have a student who is not shy about when he is confused. (I love that kid). He makes great mistakes, most of the time once. Without trying he will never clear up his misconceptions. He needs to make his mistakes. Looking at more problems doesn't personalize the practice, nor is it as likely to shatter the walls in his understanding.
Next to that student is another. He sees all the mistakes the first student makes, but he doesn't get the same experience out of them. Each student needs to be willing to make his/her own mistakes and learn from them. Without this student's willingness to try or face his own mistakes the chances of growth are diminisihed.
Those first students I mentioned, the ones who find math easy, they need to make mistakes too. For those students, without mistakes they lose the experience of having to practice and study. Every math student eventually hits a wall and I want my students to know that when it happens to work through their mistakes.
So, let start celebrating our students' mistakes!