Exploring the relationship between mathematics content knowledge and pedagogical content knowledge among pre-service teachers

This study replicates and extends a previous study (Nathan & Petrosino, 2003) that explored the relationship between the content knowledge and pedagogical content knowledge o f pre-service mathematics teachers. In that study, Nathan and Petrosino (2003) examined and reported evidence supporting the counterintuitive hypothesis that, in some situations, having a high degree of content knowledge may be associated with the “expert blind spot” (symbolprecedence) in pedagogical content knowledge. Pre-service teachers with various levels o f expertise in mathematics subject matter were given a series o f mathematics problems and asked to rank order their difficulty. Nathan and Petrosino (2003) reported that, on the average, pre-service teachers with more advanced mathematics education courses and fewer pedagogical content knowledge courses rank ordered the problems in ways that were inconsistent with actual patterns o f student performance. This suggested that they had less insight into how students think about and solve these problems. In contrast, students who had taken fewer advanced mathematics courses but more pedagogical content knowledge courses rank ordered the problems in a way that was more consistent with actual student performance. The present study builds upon this work. Forty pre-service teachers majoring or minoring in mathematics education were surveyed to assess their knowledge o f algebra and aspects o f their knowledge regarding how to teach algebra. They were asked to rank order Nathan and Petrosino’s (2003) problems in the Difficulty Factor Analysis task. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Based on their assessment scores, teacher candidates in three categories were selected for follow-up interviews. The categories were: high content knowledge (CK) and symbolprecedence pedagogical content knowledge (PCK), low CK and verbal-precedence PCK, and high CK and verbal-precedence PCK. The importance of both elements (CK and PCK) for pre-service education majors and professional development programs was also investigated, although no causal relationship was implied. Quantitative results replicated previous findings—students with higher CK showed they used symbol-precedence to teach algebra significantly more than students with lower CK. Follow-up interview data suggest a more complicated relationship between content and pedagogical content knowledge. Those findings revealed that (a) the high CK pre-service teacher with symbol-precedence was knowledge-centered in her teaching perspectives; (b) the low CK pre-service teacher with verbal-precedence was problemcentered in her teaching perspectives; and (c) the high CK pre-service teacher with verbal-precedence was response-based in his teaching perspectives. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS This dissertation represents the culmination o f the generosity, enthusiasm, and energy o f so many important people. First and foremost, my dissertation committee members, Dr. William P. Callahan, Dr. Anthony J. Gabriele, Dr. Larry P. Leutzinger, Dr. Glenn T. Nelson, and Dr. Gregory P. Stefanich, have given me tremendous encouragement and expertise; and Dr. Mitchell J. Nathan at University o f WisconsinMadison encouraged me to replicate his study. Their enthusiasm for this research has been invaluable and their teamwork and professionalism inspiring. I am deeply indebted to the chair o f my committee, Dr. William P. Callahan. I was fortunate that he was the professor for my seminar course in the doctoral program. With his guidance, I began to ask questions I had not asked before. His guidance and friendship has been better than I could have reasonably expected. I am also deeply indebted to Dr. Glenn T. Nelson and Mrs. Sandra J. Ubben. They allowed me to do my survey and follow-up interviews in their classes. I acknowledge Mr. Reuben Mork who helped me revise my interview data and my dissertation. He read my dissertation drafts more than 10 times. Likewise, I’m grateful for the mentorship and “mothering” I had from Dr. Catherine M. Miller. Her guidance, advice and sense o f humor have been priceless. She steadfastly insisted that this project was of consequence. I must acknowledge Dr. Donna Schumacher-Douglas. Her comments and editing on my dissertation have helped me