I spent one summer driving through central Oregon with my 6-year old son. He had become interested in my driving technique and spent considerable time comparing it other drivers on the road. He asked me how I learned to drive and how I got my picture on my driver’s license. He had said that he couldn’t wait to drive himself when he was big because “you can go to so many places and do lots of things.”
It occurs to me that learning algebra is a lot like learning to drive. In order to learn algebra, you need to understand concepts, learn skills and practice quite a bit to attain proficiency. Learning to drive also requires understanding concepts, learning skills and lots and lots of practice.
An algebra concept that you need to understand, for example, might be the solving for x in an equation. Effective algebra students move beyond the mechanical approach to equation work because there are so many discrete rules that make simple “memorization” difficult. If they keep the bigger concepts in mind, such as balancing each side of the equations, the role of opposites etc, they are more successful in the activity as a whole. This might be analogous to making a left hand turn on a busy road. When the bigger concept of making your intentions utterly clear to traffic behind you as well as on-coming, your actions will necessarily be more complete it terms of using left hand signals, slowing down, and reviewing your route carefully before starting the turn.