Impact of Teachersu27 Planned Questions on Opportunities for Students to Reason Mathematically in Whole-class Discussions Around Mathematical Problem-solving Tasks

摘要

While professional developers have been encouraging teachers to plan for discourse around problem solving tasks as a way to orchestrate mathematically productive discourse (Stein, Engle, Smith, u26 Hughes, 2008; Stein, Smith, Henningsen, u26 Silver, 2009) no research has been conducted explicitly examining the relationship between the plans that teachers make for orchestrating discourse around problem solving tasks and the outcomes of implementation of those plans. This research study is intended to open the door to research on planning for discourse around problem solving tasks.This research study analyzes how 12 middle school mathematics teachers participating in the Mathematics Problem Solving Model professional development research program implemented lesson plans that they wrote in preparation for whole-class discussions around cognitively demanding problem solving tasks. The lesson plans consisted of the selection and sequencing of student solutions to be presented to the class along with identification of the mathematical ideas to be highlighted in the student solutions and questions that would help to make the mathematics salient. The data used for this study were teachersu27 lesson plans and the audio-recordings of the whole-class discussions implemented by the teachers.My research question for this study was: How do teachersu27 written plans for orchestrating mathematical discourse around problem solving tasks influence the opportunities teachers create for students to reason mathematically? To address this research question, I analyzed the data in three different ways. First, I measured fidelity to the literal lesson by comparing what was planned in the ISAs to what was actually took place in the implemented debriefs. That is, I analyzed the extent to which the teachers were implementing the basic steps in their lesson (i.e. sharing the student work they identified, addressing the ideas to highlight and the planned questions). Second, I analyzed the teachersu27 fidelity to the intended lesson by comparing the number of high-press questions in the lesson plans (that is, questions that create opportunities for the students to reason mathematically) to the number of high-press questions in the implemented discussion. I compared these two sets of data using a linear regression analysis and t-tests. Finally, I conducted a qualitative analysis, using grounded theory, of a subset of four teachers from the study. I examined the improvisational moves of the teachers as they addressed the questions they had planned, building a theory of how the different ways that teachers implemented their planned questions affected the opportunities for their students to reason mathematically around those planned questions.My findings showed that it was typical for the teachers to implement most of the steps of their lesson plans faithfully, but that there was not a statistically significant correlation between the number of high-press questions they planned and the number of high-press questions they asked during the whole-class discussions, indicating that there were other factors that were influencing the frequency with which the teachers were asked these questions that prompted their students to reason mathematically. I hypothesize that these factors include, but are not limited to, the norms in the classrooms, teachersu27 knowledge about teaching mathematics, and teachersu27 beliefs about mathematics. Nevertheless, my findings did show that in the portions of the whole-class discussions where the teachers had planned at least one high-press question, they, on average, asked more high-press questions than when they did not plan to ask any.Finally, I identified four different ways that teachers address their planned questions which impacted the opportunities for students to reason mathematically. Teachers addressed their questions as drop-in (they asked the question and then moved on as soon as a response was elicited), embedded (the ideas in the question were addressed by a student without being prompted), telling (the teacher told the students the `responseu27 to the question without providing an opportunity for the students to attempt to answer the question themselves) and sustained focus (the teacher sustained the focus on the question by asking the students follow-up questions).