**By Tim McCaffrey, posted June 20, 2016 – **

In my
previous post, I proposed the simple idea of reversing the roles of the Gradual Release of Responsibility Model. In other words, start a rich task and allow students to make sense of the problem and persevere in solving it (*YOU DO*). Students then work collaboratively and discuss with each other the solution
paths they derive (*WE DO*) and then the teacher makes connections between the representations (*I DO*).

I recently taught a lesson in an eighth-grade class using this model. I hope to paint a picture of what this actually looks like in a classroom, so that you can see the benefits.

**Setting
the Stage ( YOU DO)**

Students were given the image below and asked to write down their notices and wonders.

Students then shared their thoughts with a student on the opposite side of the room. After returning to their seats and being grouped in threes or fours, they were asked to share their wonders and determine the most interesting wonder in the group. Each group reported out, and I recorded each wonder (see below). I also polled the class and asked, “How many of you find that question interesting?”

**Wonders:**

- Why does section 2 cost less than section 1? (+20)
- How much does each item cost individually? (+30)
- Which object costs more? (+32)
- Why does the amount of products (5 items) cost more money? (+30)
- Why is it so expensive? (+30)
- What do the numbers (price) stand for? (+15)
- If the glasses cost more than the hat, why does the bottom price cost less? (+10)
- Do the glasses cost more than the hats? (+20)

“One of the beautiful things about mathematics is making connections and thinking deeply. So today we are only focusing on one question that you came up with,” I said after all questions were posed. I then highlighted the second question. Students were then given individual think time to start formulating their own solution paths.

**Lights.
Cameras. Action. ( WE DO)**

The room was filled with students discussing possible ways of determining the cost of a hat and a pair of sunglasses. As I monitored each group's thinking, I noticed two very important ideas surfacing from many groups. I wrote both ideas on the board and had one preselected student share the thinking behind each statement.

A number of groups were thinking that statement 1 was correct, giving the prices of a hat and a pair of sunglasses. For instance, look at one example of group work below. I wanted to bring both camps of thinking to the class and let students discuss it among themselves.

I asked, “Can both of these statements be true?” After a few minutes of group discussion, I polled the class to see which groups determined that the first statement was true. They wanted to know if the merchandise was from the same store; I said yes. They now knew statement one was incorrect. *Note to self: Next
time, explicitly share that their brain just grew as a result of their mistake.
*

As groups continued to work together, I posed advancing questions and left the groups so that they could have rich math discussions. I made notes on my clipboard and always circled back around to hear their progress.

When it came time to select and sequence each group’s work, my colleague,
Ryan Dent__,__ suggested I display one group's work at a time, allowing the remaining students to work together to figure out the presentation under discussion. This plan was successful because students had to make sense of the math and articulate it to one another.

I first displayed the most common and least complex solution path: Guess and check. Groups huddled together as I asked them to explain the work being represented. After a class discussion, I wanted to display a more concrete and visual approach. Unfortunately, none of the groups formulated such a path. Again, Ryan Dent suggested that I design solution paths ahead of time and present the paths as actual group work; see (a) below. So I did just that and displayed the work shown in (b) below.

The blue transparent box was not displayed. I covered it up so that students could only see the top part. Groups went crazy over this because they were trying to make sense and connect the Gs and Hs with the equations below them. At one point I told the class, “I’m going to take myself out of this class discussion. Feel free to come up to the whiteboard display and explain your thinking. Don’t speak to me. Speak to one another.” I sat in the corner, taking in the mathematical thinking. It was amazing to see students pose questions that they really wanted answers to and to have others provide responses. This was SMP 3 in full swing.

**Connecting
(I DO)**

Time ran out, and I was unable to make connections among the representations and review the learning for the day. That darn bell! A second day was needed. What I would like to highlight was the amount of engagement, sense making, and dialogue that occurred without my lecture. The more that we can get out of the way and provide an environment for students to explore mathematical ideas and think deeply about mathematics, we will begin to see students fall in love with mathematics.

Two favorite quotes from two students:

- “This problem is making my head hurt.”
- “Man, whose group's work is this?”

Take a look at Phil Daro’s interview response to how mathematics should look in our classrooms (click on image).

Phil Daro: Face to Face with National Language Experts from University Washington Tacoma on Vimeo.

You can also view the entire interview here.

Tim McCaffrey is the founder of Agree or Disagree?, writes at http://timsmccaffrey.com/, and tweets at @timsmccaffrey. He currently serves as the mathematics coordinator for grades 6–12 in Fontana, California. He desires to help coaches and administrators implement sound mathematical practices that will help students become deep thinkers.

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