Twenty-five K–12 mathematics teachers working in rural and suburban school districts in a region attended a one week summer institute. The purposes of this institute were to deepen teachers' understanding of the base ten number system, and to enhance their knowledge of student thinking in this area.
Throughout the week, teachers engaged in a variety of mathematics problems as learners and analyzed written cases and classroom video clips based on these problems. Each of these activities helped teachers explore students' mathematical thinking, problem solving methods, and ways of representing mathematical ideas. At times, specific segments of the videos were analyzed in greater depth as a means to help teachers focus on particular solutions, and discuss the overarching mathematical ideas and common misconceptions revealed through the students' solutions and verbal responses.
For example, in one session the teachers addressed the following question from a written case about students in a grade 5 class who are working out a calculation of 1.14 + .089 + .3 in response to a word problem:
The students in Nicole's case 28 are trying to determine which of two answers (2.06 or 1.529) is the correct answer to a decimal addition problem. Explain how they resolve this question. Make a list of the mathematical ideas you see in their work.
As a follow up to this institute, teachers were asked to use one or more of the mathematical tasks with their students during the year and to analyze their own students' work. The group of teachers reconvened mid-year to discuss their classroom experiences with the tasks and to engage in a group analysis of a few selected pieces of student work.
The developers of a middle grades physical science curriculum created a series of professional development workshops for new users of their program. Throughout the professional development are sessions called Understanding Student Learning (USL), intended to help teachers understand how their students learn, and how they may think about, the content. One USL session was designed to introduce teachers to the common initial ideas students may have about force and motion and to provide teachers with an opportunity to see how the activities within the unit are designed to address those ideas. The session began with a video of a classroom discussion around what would happen if you shoved a puck on a frictionless surface, the first elicitation activity of the unit. In small groups, the teachers summarized the initial ideas they heard in the class discussion and brainstormed other ideas that might arise in this type of discussion. Next, teachers were provided with a list of students' comments that would be typical to hear in this elicitation. Some of these comments included:
| Chantel | "The puck looks like it moves with constant speed, but really it's slowing down a little because the hand is no longer pushing on it. To keep it moving, you would have to keep pushing it." |
| Michael | "The puck keeps moving because the shove stays with it even after the puck is no longer touching my hand." |
| Xuan | "The puck will eventually stop because everything eventually slows down and stops." |
As teachers read the students' statements, they were told that each may contain multiple ideas about force and motion. Next, teachers were provided with a list of common initial ideas about force and motion that have been found in research on student thinking. For example, one initial idea read:
"If a body is moving there is force acting on it in the direction of motion. A sustained push or pull is required for continued motion, even motion at a constant speed. Otherwise the body will eventually slow down and stop. (Clement, 1982; Gunstone & Watts, 1985; Hammer, 1996)"
The teachers' task in this session was to determine which of the six research-based common initial ideas were illustrated in each of the student statements.
† Schifter, D., Bastable, V., & Russell, S. J. (1999). Building a system of tens, casebook. Parsippany, NJ: Dale Seymour.