Influences of metacognition-based teaching and teaching via problem solving on students’ beliefs about mathematics and mathematical problem solving

The aim of the present study was to investigate the effect of metacognition-based teaching and teaching mathematics via problem solving on students' understanding of mathematics, and the ways in which the students' beliefs about themselves as doers and learners of mathematics and about mathematics and mathematical problem solving were influenced by the instruction. The 60 hours of instruction occurred in the context of a day-to-day mathematics course for undergraduate non-science students, and that gave me a chance to teach mathematics via problem solving. Metacognitive strategies that were included in the instruction contributed to the students' mathematical learning in various ways. The instruction used journal writing, small groups, and whole-class discussions as three different but interrelated strategies that focused on metacognition. Data for the study were collected through four different sources, namely quizzes and assignments (including the final exam), interviews, the instructor's and the students' autobiographies and journals, and class observations (field notes, audio and video tapes). Journal writing served as a communication channel between the students:and theinstructor, and as a result facilitated the individualization of instruction. Journal writing provided the opportunity for the students to clarify their thinking and become more reflective. Small groups proved to be an essential component of the instruction. The students learned to assess and monitor their work and to make appropriate decisions by working cooperatively and discussing the problems with each other. Whole-class discussions raised the students' awareness about their strengths and weaknesses. The discussions also helped students to a great extent become better decision makers. Three categories of students labeled traditionalists, incrementalists, and innovators, emerged from the study. Nine students, who rejected the new approach to teaching and learning mathematics were categorized as traditionalists. The traditionalists liked to be told what to do by the teacher. However, they liked working in small groups and using manipulative materials. The twelve incrementalists were characterized as those who propose to have balanced instruction in which journal writing was a worthwhile activity, group work was a requirement, and whole-class discussions were preferred for clarifying concepts and problems more than for generating and developing new ideas. The nineteen other students were categorized as innovators, those who welcomed the new approach and utilized it and preferred it. For them, journal writing played a major role in enhancing and communicating the ideas. Working in small groups seemed inevitable, and wholeclass discussions were a necessity to help them with the meaning-making processes. The incrementalists and the innovators gradually changed their beliefs about mathematics from viewing it as objective, boring, lifeless, and unrelated to their real-lives, to seeing it as subjective, fun, meaningful, and connected to their day-to-day living. The findings of the study further indicated that most of the incrementalists and the innovators changed their views about mathematical problem solving from seeing it as the application of certain rules and formulas to viewing it as a meaning-making process of creation and construction of knowledge.