I've been thinking about something cool (yet, a tiny bit disturbing) that happened a while ago in my Calculus class:
My students were solving a problem and encountered a quadratic equation that could not be factored. They freaked a little. (As their algebra 2 teacher, so did I). I gave them a chance to think about it, and someone suggested the quadratic formula. Then someone remembered the song that I had taught them. And, be still my heart, someone else noticed that since a = 1 and b = even number -- it would be super easy to complete the square. And they started discussing how to do that. I was proud.
With that problem completed and about ten minutes left in class, I wrote "ax^2 + bx + c = 0" on the board. I asked them if they thought they could complete the square with that. Working as a class (there are only ten of them), they figured out that they needed to divide by a . . . and et cetera et cetera.
I could not believe their reaction when the quadratic formula showed up. They were blown away! They were actually LAUGHING and saying how awesome that was. At this point, I should have reveled in the beauty of their mathematical discovery. Instead, I started thinking about what a rotten Algebra 2 teacher I must be. This should not have been brand new information.
So, I asked them if they remembered that I had shown them this same thing two years ago in Algebra 2. Someone said "Yeah, but this time it makes sense!". I am not sure if that comment made me feel worse or better. Then I realized that they did remember how to use the quadratic formula and how to complete the square, and when to apply those methods. The only piece that was missing was the connection between the two.
I thought about it for awhile and I decided that connections just take time. The more we learn, the more the pieces start to fit together in different ways. I remember that when I took Calc 1, I really didn't understand what I was doing. I think I was halfway through Calc 2 before the light bulb went on. Then it kind of all made sense to me at once.
I guess I don't need to be so hard on myself if the students don't get 100% of everything the first time they see it. I am planting the seeds that will lead to discoveries and connections down the road. Sometimes this will happen in the future without my knowing. This time, I was lucky enough to be there to watch it happen.