The other day, we were working after school with the equation 4x - 7 = 41. I started with reminding him of the “why” behind the algebra work. “We have to add 7 back on both sides, get the simpler equation 4x = 48, divide by 4 to find 1x,” etc. Dean nodded and said he didn’t have any questions. His turn.
As he worked to solve 8x – 11 = 53, the work spattered across the white board. He got the answer, and 8x = 64 was swimming in the middle, but Dean clearly was not seeing it the way I was. I asked him to compare the two problems as they appeared on the board: the one I had written out and the one he had written out. What was the same? What was different? While he was able to point out some details like specific numbers or the answer, he was not seeing the structure of two sides in balance or the sequence of steps.
We went on to another problem, this time 7(x – 5) = 49. Again Dean got the answer quickly by inspection. I thought I would highlight each step of his thinking process and record each one algebraically as we went along. “Great,” I said. “What did you start with?” “12” he said. What? “You start with 12, then subtract 5, then multiply by 7 and get 49.” I realized that Dean was focused on the numbers, not the process. When we were working together he would often include the answer when he was trying to write an equation. Once he had found the value for x, x disappeared. And once he has done the work, he has a hard time thinking backward to review the process.
That was all we had time for, but I had another opportunity to sit with him during class the next day. As we worked I saw more clearly what I had seen the day before: Dean was moving through the process so quickly that he wasn’t even aware of it, so of course he couldn’t record it. We had been using tape diagrams yesterday at his suggestion. I thought hanger diagrams might make it easier to look back over the process once we had finished. So I made a hanger diagram for 3x + 5 = 20 and asked him what x was. He quickly gave me the answer and I asked him how he got it. “20 – 5 is 15 and ÷ 3 = 5,” he said.
“Okay,” I said. “Let’s take this one step at a time and see what it looks like.” As he described each step, he could more easily see on the hanger diagram the idea of doing the same thing to both sides. He also understood that each step produced a new, simpler hanger relationship. We turned to another problem, 3x + 11 = 47.
He solved it, described the process and we wrote adjustments on the hanger diagram. “Can you write an equation for this hanger?” He could. “Okay. Now, when you write out the algebra for the problem, it’s like running a race in slow motion. You have to slow your thinking down and show each step. The equation is like the starting line before the race. What happened next?” As we worked through the problem, he was able to show each step and the work that led up to it. With two more examples he was able to notate the solutions algebraically, and though I intervened on a few steps (we both were having fun with SSLLOOWW MOOOTION), he seemed to have much more autonomy and understanding. I think we have a good foundation to move forward with, and I think that Dean has found access to a new level of abstraction.
All the while, I was thinking about the value of modeling and having multiple models available. Hanger diagrams worked for Dean. Tape diagrams work for other students. In fact, for Dean it was possibly his exposure to both that helped prepare him for a new level of understanding. When we anchor the work in models where students can see or feel the concepts, and keep stepping back until we find the level where they can think for themselves, then we can use that thinking as a foundation to extend their understanding to new insights. For Dean, I am hoping that algebraic notation is going to be his next new modeling tool. Time for some clotheslines, I think…