Since beginning my teaching career I've always been concerned with reaching a tipping point where I am no longer open to new ideas or strategies. Where some new form of email, hardware or vegetable comes along that I just don't accept and the world passes me by--rendering my skill sets obsolete. I'm hesitant to complain about the new layout of Facebook just because it is different from what I'm used to.
I've come to believe that in general there is probably not an age that determines when a person gives up trying or considering a new idea but rather a mindset that has always been there--the idea that you can finally arrive at a level of contentment and no longer accept any new information. I'm sure you could get by just fine, but 'just getting by' doesn't interest me and I hope that it never does.
In guiding my students through mathematics I want to do so in a way that builds their intuition to apply the math in areas that have not yet been formally learned. A memorized algorithm or process done the exact same way as the teacher does not help them with this process. I need to reward students for applying their own ideas and understanding in new situations and more importantly provide them those opportunities.
If a student demonstrates understanding differently than what we expect them to, we should pause as educators and be open to the fact that we may not have considered all possibilities in an exercise we have provided in class. Students like to see that something is new or novel. Collaboration seems to thrive in this type of situation.
We should never penalize a student for arriving at a correct conclusion different from the path we had thought of. Mathematics can be beautiful, connected and intuitive but will never appear that way to students if we make it seem like there is a recipe that must be followed every time. We too often spend time building our own personal understanding of mathematics and seldom give students that same opportunity.
I read a post this morning by Tim Kanold about a growth mindset vs. a fixed mindset and this resonated with where my thoughts have already been this week. Our tendency toward either mindset has nothing to do with age or experience and everything to do with who we are as individuals. I believe it can be changed but that can only happen through personal reflection.