A Case Study of How Ninth Grade Mathematics Students Construct Knowledge during a Productive Failure Model.

ion and disequilibrium allowed a better understanding of how the productive failure model served as a process of persistence through failure. 5. Methods: I chose to study a representative case study group to determine how a typical group of ninth grade students would construct knowledge during a productive failure modeled task. I collected multiple sources of data in to determine how students construct knowledge during a productive failure task. Students took a pretest, utilized learning logs to record their work, answered questionnaires, and completed a final assessment. Videotaping of the students occurred while they worked on the task in groups, during a student confessional, and during the last interview. I coded the major themes by using constant comparative analysis according to my research questions, review of literature, and clusters of emerging themes. 6. Results: The group processes (making a plan, working the plan, reporting results, and evaluating the plan) showed how the group utilized components of the PF model when trying to solve tasks. When the group made a plan, they activated their prior knowledge. When the group worked a plan and reported results, they created multiple representations for solving the task. The group cultivated critical analysis of their work when they sought to understand a concept and when they evaluated their work. The group freely shared their ideas, understandings, written work, and explanations for solving with one another. Student discourse fostered a communal learning environment where their individual knowledge was a valuable commodity for the entire group. The four roles for communicating verbal thought were the responder, the explainer, the questioner, and the hypothesizer. The group members also spoke aloud using the explainer role more than the responder role, which showed that the group used more than a few words to respond to the group when talking aloud. A group member played the hypothesizer role when he or she showed new thinking about the task. Allowing the students to choose the method they thought would work best to solve each task provided the group choices in their learning. The students chose to solve the tasks by generating new ideas, writing their ideas, using different tools, reading for understanding, defining variables, using a guess-and-check method, using graphs, admitting when they were stuck, using algebraic methods, and using statistical analysis. The group-chosen approaches of writing their ideas, taking time to read the task and the work of other group members, and admitting when they were stuck during the task that promoted reflective thinking. The Model of Persistency served to show how my findings on group processes, student roles, and student problem-solving approaches were interrelated to one another. The group persisted by continually assimilating new mathematical concepts with their prior knowledge as “one brain. ” When the group evaluated the plan, they deeply analyzed the concepts and Productive Failure Model 3 experienced a greater sense of disequilibrium. The students initially attempted to assimilate new concepts with their prior knowledge, but sometimes needed to make accommodations in their learning. The group made learning accommodations only after the group evaluated the plan. 7. Implications: The three main implications of formatively assessing students through their group talk, celebrating errors as a means for accommodations through persistence, and allowing student voice to facilitate critical thinking are relevant to teaching. Teachers need to be able to assess students by listening to them talk, allow students to make mistake for the good of student learning, and empower students to become critical thinkers by using their voice. Knowledge construction is not about being efficient. Knowledge construction is about being effective.