Students’ higher-order thinking (HOT) skills, such as reasoning and problem solving in mathematics, are important components of educational improvements. Today’s workplace requires an increased level of mathematical thinking and problem solving (NCTM, 2000). Focusing on HOT in mathematics prepares students to face economic and workforce challenges in an increasingly global and technological society (NCTM, 2009). In order to help students develop HOT, teachers must select and implement tasks that encourage students to reason, make sense of mathematics, and become independent thinkers (NCTM, 2009). Professional development (PD) can help teachers learn how to implement pedagogical strategies that promote HOT and to understand the mathematics they are expected to teach. This study examined the influence of a PD context involving the cyclical process of teaching on teachers’ selection and implementation of tasks intended to develop students’ HOT. Specifically, this presentation will concentrate on how a series of reflective teaching cycles influenced two middle school mathematics teachers’ knowledge of integers and the pedagogical strategies used to teach this concept.

Theoretical Framework

Several PD strategies provide teachers with practice-based experiences from which they can learn about instructional strategies and student thinking. Smith (2001) uses a cyclical process of teaching called the reflective teaching cycle (RTC), which consists of planning, teaching, and reflecting. During planning, teachers spend time selecting tasks and building their understanding of the mathematics in tasks, students’ prior knowledge, and mathematical goals. While implementing lessons (i.e. teaching), teachers constantly assess student thinking and make decisions about how to continuously engage students in learning. After teaching, teachers reflect on student thinking and understanding of the central mathematical ideas of the lesson by considering what students did and said. This study examined the influence of series of RTCs on teachers’ practice by focusing on the selection and implementation of tasks that had the potential to develop HOT.

Methods

The study took place in College Middle School (pseudonym), enrolling approximately 500 students in grades 6 – 8. I engaged in a series of seven RTCs spanning two instructional units with two seventh-grade mathematics teachers, Clark and Tess. The big mathematical idea in the second unit was positive and negative rational numbers.

During the RTCs, the teachers and I met as a group to plan and reflect on lessons. Between these meetings, I observed each teacher individually. I actively participated in the planning and reflection meetings by initiating many of the conversations and working to focus the discussions on HOT. I prompted discussions about the cognitive demand of tasks and the way in which the teachers could decrease or sustain the tasks’ demand through implementation. I also drew attention to particular events in each teacher’s classroom that highlighted evidence of higher- or lower-order thinking or opportunities for the use of HOT skills.

Each cycle meeting was audiotaped and transcribed. These transcriptions were the main data source. I used thematic analysis to analyze this qualitative data in order to determine how the RTCs influenced the teachers’ selection and implementation of tasks. By analyzing the data in this way, I created stories for each teacher that described the way in which they thought about mathematics, pedagogy, and higher-order thinking. This presentation focuses on the influence of the RTCs during the second unit of instruction and how the teachers grappled with the mathematical content and pedagogical strategies for operations on integers.

Results and Conclusions

The RTCs influenced the way Clark and Tess selected and implemented tasks by helping them become more reflective in their practice. The collaborative nature of the cycles provided them with a venue to examine, critique, and support their own and each other’s practice. By working together, Clark and Tess saw how certain decisions facilitated or hindered their students’ opportunity to engage in HOT. The cycles also helped Clark and Tess build knowledge about the mathematics they were expected to teach and how their textbooks presented this material. For example, the teachers worked to understand the meaning of the models used by the textbook to present operations on integers. We spent time thinking about our own understanding of the models, which helped the teachers make decisions about what tasks they would implement and how. Finally, the cycles influenced Clark and Tess by helping them reflect on their pedagogical strategies so they could begin to understand how these strategies affected HOT.

In particular, our conversations during the second unit of instruction about integers focused on how instructional strategies and classroom events addressed student thinking in ways that either facilitated or hindered HOT. We discussed the chip model for the subtraction of integers, how it could be used to facilitate student learning, and the mathematical goals of the unit. Clark had a particular method that he was using that was connected to an important mathematics principle (i.e., subtraction is equivalent to adding the opposite). Both Tess and I had difficulties understanding how this method facilitated HOT because it seemed that Clark was describing a procedure to his students. We engaged in conversations about Clark’s method several times, but could have used additional time to reflect on the meaning of Clark’s pedagogical decision to use a particular method and to fully understand the repercussions of his instructional choices.

Significance

How teachers think about and present integers in their classrooms greatly influences how students build their understanding of numbers and their success in future mathematics classes. During RTCs, teachers have the opportunity to build their own knowledge about mathematics and reflect on their pedagogical practices in order to better serve their students. This study shows how a series of RTCs influenced two middle school mathematics teachers’ knowledge of integers and the pedagogical strategies used to teach this concept. The study adds to the literature by showing how RTCs could be used as PD with teachers to help them build their mathematical and pedagogical knowledge. In particular, the teachers in this study were able to take apart their own understandings of integers and their curriculum materials to begin to better understand the models they were using to teach operations on integers.

References

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: Author.

Smith, M. S. (2001). Practice-based professional development for teachers of mathematics. Reston, VA: National Council of Teachers of Mathematics.