One of the messages in an IBL that goes along with "Being stuck is okay!" is "Mistakes are Good!"
Mistakes are generally stigmatized in U.S. Math Education. When a students does something wrong, it is unusual if the student thinks of the mistake or error as an opportunity for greater, perhaps even profound insight.
There exists many reasons why we build prototypes or practice in a batting cage or simply use scratch paper. We need to see how things work. We need to practice and fail, so that we can learn to do what is right. In short, practice and experimentation.
One cannot grow without experimenting or trying things. It would be nice if our students could all have the disposition to say things like "Let's see if the idea works for a special case..." or "Let's see if we can check our thinking..."
If students fundamentally believe that mistakes are bad, then the very nature of their interaction with mathematics is limited. Over time this leads to poor self image and then ultimately poor habits of mind and work ethic. That would be the nail in the coffin.
One of the ways to get students over the negative image of making mistakes is to provide opportunities for students to experiment, and to allow for mistakes to play a central role in the learning process. In fact, in an IBL class students make *great* mistakes. They say or do things in ways that maximize their learning. As an instructor I no longer make these mistakes, because I already know the material. First time learners of a subject reveal, through their mistakes, what they know and what they don't know. This is where the learning zone is, and this is where one can create magical learning experience!
Student: <Writes or says something that is incorrect>
Teacher: "Oh, did you just say/write... Well I'm really glad you brought this up. How many of your were thinking about this the same way? Good! Let's rewrite this as a question, and then investigate it further to get to the bottom of this."
