Conceptualizing the Work of Leading Mathematical Tasks in Professional Development
Elliott, Rebekah, Kazemi, Elham, Lesseig, Kristin, Mumme, Judith, Carroll, Cathy, Kelley-Petersen, Megan, Journal of Teacher Education
In a workshop on their new instructional materials, 30 intermediate and middle grade teachers worked in small groups on a staircase task to determine the number of cubes needed to build the 100th staircase if the first staircase was one cube, the second three cubes, the third six cubes, and so on. When they finished, Sue, the facilitator, asked participants to share how they thought about the problem. Elise came to the overhead and showed her table of values. As she did, she said, "I noticed that my table had 1 then 1 + 2, then 1 + 2 + 3, then 1 + 2 + 3 + 4 and I remember from one of my reviews of high school math that to find the total of these I use n(n + 1) divided by 2. But, I am not an algebra teacher so...." Sue asked if anyone had questions. A teacher asked what n stood for and Elise showed how to plug values into the formula to get the total number of cubes. The group applauded and Sue asked if another person might share. Mario said he also got n(n + 1) divided by 2 and illustrated his approach using the model of the staircase to fit a second staircase of the same size on top to make a rectangle. Pointing to the dimensions of the rectangle he said, "n is the number of cubes along the bottom and the n + 1 is the number of cubes along the side. I divided by two because my staircase was half of the rectangle." Teachers signaled how much they liked the use of a visual model with applause and a buzz of praises. The facilitator prompted for another way and Christa showed her approach next. She noticed that if n was the number of cubes in the last column, then the total cubes for one staircase was n + (n - 1) + (n - 2) + (n - 3).... She said the total for any staircase was the number of cubes in the last column plus the previous total. "But I ran out of time trying to figure out how to write it." The sharing continued and each presenter was applauded. Sue continued to ask, "Is there another way?" until no one else volunteered. Sue then said, "Okay, let's look at one more problem from the chapter before breaking for lunch."
This depiction of professional development (PD) illustrates common practices in PD. The facilitator elicits teachers' varying solutions and invites teachers to question one another in a supportive learning environment (Desimone, 2009; Loucks-Horsley et al., 2003). As professional development leaders ourselves, we have often facilitated such discussions. We use this example as a means to begin a conversation that we, as researchers and facilitators of PD, believe is imperative to advance leaders' learning. In the scenario, although the leader provided a rich task and encouraged teachers to share solutions, the mathematical purpose for teacher learning was not made evident, neither was how teachers' mathematical learning might be useful for supporting students. In advancing teachers' mathematical learning, a leader (2) may need to "slow down" teachers' conversation to explicitly engage the group in mathematical ideas. For example, a leader might ask, "What might be gained from 'seeing' the solution in the visual model? Were there multiple ways to see the pattern in the model and what are the implications of seeing the pattern in different ways for representing a solution symbolically?" Leaders can use these kinds of questions to explore how mathematical models can be used to represent different, albeit equivalent, expressions or to recognize alternative expressions that stem from defining the variable differently. For intermediate and middle grade teachers working with students who are learning about variables and characterizing patterns using words and symbols, the discussion relating the model to the symbolic notion would support teachers who remembered a formula to understand the mathematical reasoning and the use of multiple representations in algebraic thinking.
Because facilitation moves in the scenario focused on displaying the different ways in which teachers solved the task and providing an open forum to ask questions, it was not clear what teachers were to learn. …