The influence of studying students' proportional reasoning on middle school mathematics teachers' content and pedagogical content knowledge

Proportional reasoning serves as the capstone of elementary mathematics and the cornerstone of secondary mathematics (Lesh, Post, & Behr, 1988). Studies reveal that teachers and students struggle with proportional reasoning (Sowder, Armstrong, Lamon, Simon, Sowder, & Thompson, 1998).;The purpose of the study was to describe the influence of studying children's proportional reasoning with different problem types on a teacher's understanding of proportional reasoning and a teacher's assessment of the proportional reasoning of students. The study addressed the following research questions: (1) How do a teacher's understandings of proportional reasoning influence the interpretation of a student's proportional reasoning? (2) How does the process of implementing and analyzing a task-based interview impact a teacher's understandings of proportional reasoning?;Carspecken's (1996) first two stages for critical qualitative research, compiling the primary record and preliminary reconstructive analysis framed the ethnographic techniques used in this study.;The participants of this study were two seventh-grade mathematics teachers and four eighth-grade mathematics teachers at the junior high campus in a rural school district. Participants anticipated that students would use the same strategies that the participants used when solving problems and that students would extend strategies learned in class to new types of proportional reasoning problems. The participants' assessments of the strength of students' proportional reasoning was influenced by whether or not students used the anticipated strategies. Participants noted new learning with qualitative reasoning problems and the need to incorporate non-integer values into instruction with rates. They also observed that correct student answers had been their indicator of students' facility with proportional reasoning.;Implications for future practice include the use of this professional development strategy with other mathematical topics where teachers' understandings are weak. Implications for future research include applying the short-term professional development strategy to other key notions within proportional reasoning that present obstacles for teachers such as invariance and covariance. Larger scale studies that include teacher reflections on classroom instruction, teacher analysis of student work on teacher-made assessments, and this short-term professional development strategy may prove beneficial as well.