Either way - there is nothing like stepping into the unscripted and ending up in "The Zone". It happened to me yesterday. I was working a contrived word problem in my Algebra I book. Basically, there was a library which just received a monetary donation allowing it to double the number of books on the shelves. After this, it received 4,028 books which brought the total number of books to 51,514. How many books did the library have prior to the monetary donation? I walked my standard Algebra I students through the problem the intuitive way. We took 51,514 and subtracted 4,028 from it. That left us with 47,486 books. We then divided by 2 to get the original number, 23,743.
They seemed to follow this. And it was at this point I almost went on to the next problem. But something didn't seem right. I showed them how to solve a problem and there were no variables. No one got lost. I started to feel like I cheated them. And then IT happened.....
I asked them to start over and put what happened in an algebraic equation. I asked them to tell me what we should use to represent the unknown number of books we have at the beginning. They told me "X" would be a logical choice. Great. Now, what happens when we receive the cash donation? "The number of books doubles," they respond. Excellent. Now what happens? "More books are donated and we need to add that to the doubled number." So I wrote the equation 2x + 4,028 = 51,514. Keep in mind, I am on the other end of the board now. I asked them what we must first do to solve for x. The overwhelming response was to subtract 4,028 from each side. I walked over to where we solved using the intuitive method and showed them our first step was to subtract 4,028 from the final amount of 51, 514. I walked back over to the algebraic equation and showed them it was now 2x = 47,486. What is the next step? Divide both sides by two. I took one more trip back to the intuitive side of the board and showed them where we had done the same thing.
I allowed myself to stay on the unscripted path - I was really feeling it. "How many of you play a musical instrument?" I mused. A few students raised their hands and I asked the guitarist how easily she can pick up a guitar and start playing something. Easy. I then asked her, "How easy is it to put what you are playing on paper so later on you will know what you played?" She told me that was much more difficult. I told them that's why they find Algebra I so difficult. They know how to do the math, it's just a matter of translating it to equations on a piece of paper. I was really disappointed the bell rang.