Concrete to Abstract

It's interesting to me how at first it seemed so counterintuitive to go from concrete to abstract in math. It's a method we use to learn and teach most everything else, and it seems strange that we would choose to learn or teach math differently. We're often looking for someone to tell us a formula and give us the quickest way to find an answer. While there's nothing wrong with efficiency, we lose a lot of understanding by not thinking about why the algorithm we use works. Our brains are constantly looking to make connections from the patterns we see, and we can make generalizations about those patterns when we connect our concrete learning to our abstract thinking.

In terms of classroom practice, I'm seeing manipulatives in use much more often than I experienced in my own education. In my dyad 7th grade math placement, we used chip boards to teach the students the difference between the adding and subtracting of negative and positive numbers. I did notice that the teacher's comfort with a particular manipulative had a big effect on the students choice to use that particular strategy. For me there's still a question of how to know when to use them, and how long should we allow students to rely on them before we encourage them to use a more efficient algorithm?

First of all, algebra tiles are probably one of the coolest math manipulatives I've seen so far. It