I thought I knew about the importance of conceptual change. I read about it in grad school, studied it in my students, and wrote about it as well. Then conceptual change happened to me.
In August 2014, I was in a workshop hosted by @robertkaplinsky. He told us the story of how he introduced what he thought was a fairly simple context for a skill he felt confident his students had mastered. What happened next surprised him. You can read about it here: http://robertkaplinsky.com/what-does-it-mean-to-understand-mathematics/
This story has completely changed the way I think about teaching math. Yes, students need to master skills. However, they must also be given authentic situations to model using those skills. Making that connection to the authentic situation is by no means automatic. And there is no point in having a toolbox full of skills if you can’t see which tool to use in a given situation.