In some respects, today was the first day since the start of school (August 17th) where I was totally confident in what I was going to do with my 8th grade math classes. I did not feel rushed. I was fine with things taking longer than expected. I knew how I was going to arrange the lesson to require more explanation from students. Adjusting to a new grade (8th), new preps (prealgebra, science, and PE), and new schedule (60 minutes periods) has made the first couple weeks a blur, so I knew I had to commit the moments of today to writing to help me keep everything in perspective.
The goal for this prealgebra lesson was to get students reacquainted with exponents and introduce them to square roots. After taking care of housekeeping for the day (bellwork, homework questions, changes to the schedule for the week), students picked up a copy of this page. I gave everyone about 10 minutes to read the blurb and complete the exponent problems without a calculator. Once we compared work (everyone was pretty solid up to 12 squared), I defined perfect square and square root. I kept my definitions short and emphasized that square roots are essentially the same as asking the question, “What number squared is [fill in the blank]?”
With the stage set, I turned students loose on the Figure It Out problems for square roots. I encouraged students to talk with their neighbor if they were unsure about a problem and to keep asking themselves questions. I circulated and questioned students as needed. For instance, some students asked about finding the square root of 4/9. I responded by questioning, “Can you find root 4? Can you find root 9?” Once I noticed most students wrapping up the problems, I gave random students dry erase markers and told them to post an answer on the board.
I don’t have students post work as often as I would like and I don’t want anyone to be embarrassed by an incorrect answer, but I knew what I wanted to do with the answers students posted. For each problem, I read the offered solution and asked, “Do you agree or disagree everyone?” I say, “everyone,” when I want all students to call out a response. When there was a chorus of agrees, I followed up by asking, “How could we check this answer?” When the results were mixed, I asked people who disagreed to explain their thoughts. This prompting was especially enlightening when the square root of -16. Many students agreed with the proposed solution (-4), but a couple students protested that it didn’t work. When I asked for an explanation, one student offered, “-4 squared is positive and so is 4 squared.”
I loved the flow, questioning, and responses that were happening during this lesson; however, what happened next is what had me thinking the rest of the day about how I could improve my future teaching.
When I reached the second to last problem (the square root of 16), the student who posted her answer (8) said, “Oh, I know that’s wrong now.” Rather than move along and ask the class about the answer, I asked her, “Do you want to fix your work?” She came up to the board and altered her answer to 4, then returned to her seat.
I could’ve let things go and moved on, but I didn’t. I asked, “Why did you change your work?”
The student responded by saying, “Because it was wrong.”
I wasn’t going to let this student off the hook that easy, so I said, “You already said that, but that doesn’t tell me how you knew it was wrong. What changed your mind?”
The student explained, “I realized 8 couldn’t work because 8 x 8 is 64. 4 x 4 is 16, so it had to be 4.” It was a compact explanation, but it was accurate.
This moment was great for two reasons. First, I loved how the student was able to state aloud how they realized a mistake and fixed it. Hopefully, everyone else in the room got a another lesson about the value of learning from mistakes. Second, I liked that I was insistent about getting the student to explain her thoughts. Was the momentum of the lesson? Was it that I don’t accept, “I don’t know,” or, “Because that’s the way it is,” as responses anymore? Perhaps, but I think the biggest difference was the offer I made the student when she realized her mistake. When I asked the student if she wanted to fix her work, I opened the door to discussing why she fixed it in the first place.
Now I’m wondering how I can open this door in future lessons…