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What do I want for my students? I want them to be active learners, able to ask questions and engage in critical discourse about the world around them. I want them to be able to show how something does or does not work, and I want them to be able to do all these things with each other. That means that I want students focused more on each other as a source of math learning and less on me.

My role as teacher is to be the designer and engineer of the mathematical learning environment. In this reimagined classroom culture, students working in groups conduct inquiry and construct their understanding of mathematical problems, developing habits of a mathematician and deepening their understanding of mathematics along the way. But what does this look like? Here I want to share my approach to a typical lesson, including the successes and struggles that I encounter as I try to foster this kind of learning.

For a unit on creating average rate of change equations on different linear and nonlinear functions, I set the following goals: Students use linear expressions to represent situations involving constant growth rate. Students use linear expressions to compare rates of change. Students understand that the constant growth rate is a ratio of the variables being used.

Having set these goals, I needed to select the math task and plan out how the lesson would flow. The problem I chose for this lesson was “What a Mess!”—a Year 3 problem from the Interactive Math Program (IMP) (Fendel et. al., 1999. Figure 1).

As a planning guide, I used Smith et. al. (2008), “Thinking Through a Lesson: Successfully Implementing High-Level Tasks.” The article provides lesson-planning structures whereby students are given an opportunity to reason through a problem as peers with minimal but necessary input from the teacher. A colleague helped me plan the lesson. The first step was to work through the problem ourselves and solve it.

Next we turned the IMP word problem into a visual problem. Our aim in this representation was to simplify the task so that all learners could access it, and to allow for wonder and exploration with no direct instruction on the paper.

Our third step was to hypothesize student approaches to the problem, the difficulties they might encounter, and the misconceptions that might get in the way. During group work they might get stuck, which could lead to student frustration and tuning out. I needed to be able to anticipate these potential problems. Next, we identified possible “mathematical habits or practices” that students would use in exploring the problem. These are math habits that the class created at the beginning of the year and are posted in the classroom (Figure 3).

Ideally, students use them on a daily basis and reflect upon them during assignments. Finally, we created an anticipated lesson flow in three phases: launch, explore and summarize.

I launch the problem in class by showing the visual in Figure 2 and posing the following: “Here is a picture of an oil slick in the ocean. What do you notice?” I then ask the students to create an individual list, discuss the list at their table through a pair-share protocol, and then share out. During this process of sharing out, I serve as facilitator and class scribe, capturing what each table group notices. Here is what the students notice:

- it increases 26 meters by the hour
- they are perfect circles
- it doesn’t specify how deep the oil spill was
- the whole thing increases by 52 meters from 6 am to 8 am
- it has constant growth

I then ask, “What do you wonder?” We follow the same protocol for sharing out. Here is what the students wonder:

- is it a perfect circle?
- how will this affect the wildlife?
- how long will it be spreading?
- how large is it going to get (the diameter)?
- what is the problem we are supposed to be finding a solution to?
- does it grow in a linear pattern?
- how many meters are in a mile?
- what time did it start? what was the starting amount?
- how much clean water is surrounding the oil spill?
- what is the relationship between the hour and the diameter?
- is there an equation we can find to determine the diameter based on the hour?

We then decide as a class which questions we want to explore. These are represented in bold above. Students from each table group of four decide as a group what they want to explore and then sign up for that question. **Explore**

Now we are ready to move to the explore phase, where students collaborate in groups on their chosen inquiry question. Here is a portion of one group’s conversation.

**Student A: **So 68 minus 26. Right?

**Student B: **Yep. 68 minus 26.

**Student A: **So at 5:00 a..m the oil slick is at 42 (meters).

**Student C: **So at 5:00 a.m. it was 48 (meters)?

**Student B: **No. 42 (meters).

**Student A: **At 4:00am it was at 16 (meters).

**Student B: **16 (meters).

**Student A: **That is going to be negative.

**Student B: **3:00 a.m. was probably the starting time. At 3:00 a.m. it will be zero.

**Student D: **Half of 26 is 13, so it will be around 3:30.

**Student B: **I don’t know how we could find zero. It’s 3 meters across at 3:30.

**Student D:** It is? Are you sure?

The conversation continues as the students try to determine the exact starting time of the oil spill. I notice that Student C and Student D are now engaged. This sort of peer-to-peer engagement is encouraging, because when I speak in front of the class, Student C, Student D and other students are not so engaged.

I listen for lulls in conversation and signs of students getting stuck. I want them to continue to explore the problem. I extend the problem by asking, “Now that you have gotten the starting time between 2 a.m. and 3 a.m., how can we determine the exact time?” (As anticipated during the planning session, the students struggle most with coming up with a relationship between time and diameter.) I then give the students more time to talk and test their ideas with each other. I move onto other table groups to see where they are in the task. I continue to check for engagement and ask questions. During this time, I am looking at each group’s work and finding examples of student work (Figure 4) that can be shared during the summarize phase.

The summarize phase is where the story gets told. Students present data and evidence, followed by a class discussion. Students are expected to share out, support and question each other. My role is to redirect and ask probing questions if this is not already occurring.

For example, when my first student shares her data, she conjectures that the starting time of the oil spill is between 3:30 a.m. and 3:45 a.m., but then she gets stuck. Frustrated, she says, “This is hard to explain.” I ask the class, “How can we know that the oil spill does not start before 3:30 a.m.? This prompts another student to challenge her conjecture and suggest another approach for finding the starting time. This interaction is done in a supportive way and allows the struggling student an opportunity to explore the problem another way. Eventually, some equations begin to emerge from the presented student work. The first equation that is presented is y = 26x – 88 (where x is time and y is diameter). This is immediately challenged by another student who says the equation is y = 26x + 16. At this point, I put all the proposed equations on the board, honoring each group’s contribution to the problem. Another group jumps in and says, “We got y = 26x-10. Lastly, a student offers d = 26H – 624D – 88 (where d is diameter, H is hours, and D is days).

This is where the lesson ends for the day, leaving more questions than answers. The class must reconcile the difference in the equations. Also, an opportunity emerges to discuss the use of alternative letters, other than x and y, to describe variables.

At the end of each assignment students write a reflection about the problem, including a habit of a mathematician that they used in addressing it. Sample student reflections for this assignment follow:

- I invented numbers by using the rate of growth I discovered to test my conjecture and (equation) and went through several equations. I am proud of the work that I did on this problem.
- During this problem, I made a lot of conjectures. I just tried what I thought might work and experimented to see if there was any validity to my hypothesis (equation).
- The first habit of a mathematician I used was looking for patterns. I used this in the beginning to see how the area increases each time.
- As to how I used being systematic, I made small changes to time to look for changes in diameter when I was trying to find the time the serious environmental hazard would occur.

Designing a collaborative mathematical learning environment is about empowering students by having them reason through complex problems. Students struggle with these tasks, which leads to disagreement and confusion; however, students must attempt to reconcile this cognitive disequilibrium. This becomes the foundation for classroom instruction. In this lesson, some students did a great job of engaging and exploring the tasks. They worked together to discover patterns in the data and create generalizations. The students who were able to do this found it rewarding and it showed in their attitude and confidence.

The use of protocols in the launch phase gave the students time to think as well as a chance to share out with each other and the whole class. That being said, there needs to be a stronger emphasis on making sure more structured protocols and norms are in place during the exploration and summarize phases so that all students are put in a position to contribute. If no other group norms are in place, students will often assume roles based on their previous experiences and comfort with the subject matter. This can create an inequitable situation where some are participating and learning more than others.

My focus this year has been on increasing participation in groups and whole class discussions. I have assigned group roles (e.g., Facilitator, Team Captain, Accountability Manager and Skeptic) and strive to incorporate these into my lesson planning so that each student is able to participate and develop some agency. I rotate the groups and roles at the beginning of each new unit so that each student gets a chance to work in that role. We also carry out a team building activity so that positive socio-group norms are reinforced. My hope is that participation increases through supportive, equal-status interactions and that students see that participation is expected of them regardless of prior achievement. As always, this continues to be a work in progress and I am always trying new ways to encourage my students to engage in critical discourse about the world around them.

Fendel, D., Resek, D., Alper, L., & Fraser, S. (1999). *Interactive Mathematics Program: Integrated High School Mathematics Year 3*. Emeryville, CA: Key Curriculum Press.

Smith, M.S., Bill, V., & Hughes, E.K. (2008). Thinking through a Lesson: Successfully Implementing High-Level Tasks. *Mathematics Teaching in the Middle School*, 14, 3, 132-138.

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