College students' methods for solving mathematical problems as a result of instruction based on problem solving

This study investigates the effects of implementing mathematical problem solving instruction in a regular calculus course taught at the college level. Principles associated with this research are: i) mathematics is developed as a response to finding solutions to mathematical problems, ii) attention to the processes involved in solving mathematical problems helps students understand and develop mathematics, and iii) mathematics is learned in an active environment which involves the use of guesses, conjectures, examples, counterexamples, and cognitive and metacognitive strategies. Classroom activities included use of nonroutine problems, small group discussions, and cognitive and metacognitive strategies during instruction. Prior to the main study, in an extensive pilot study the means for gathering data were developed, including a student questionnaire, several assignments, two written tests, student task-based interviews, an interview with the instructor, and class observations. The analysis in the study utilized ideas from Schoenfeld (1985) in which categories, such as mathematical resources, cognitive and metacognitive strategies, and belief systems, are considered useful in analyzing the students' processes for solving problems. A model proposed by Perkins and Simmons (1988) involving four frames of knowledge (content, problem solving, epistemic, and inquiry) is used to analyze students' difficulties in learning mathematics. Results show that the students recognized the importance of reflecting on the processes involved while solving mathematical problems. There are indications suggesting that the students showed a disposition to participate in discussions that involve nonroutine mathematical problems. The students' work in the assignments reflected increasing awareness of the use of problem solving strategies as the course developed. Analysis of the students' task-based interviews suggests that the students' first attempts to solve a problem involved identifying familiar terms in the problem and making some calculations often without having a clear understanding of the problem. The lack of success led the students to reexamine the statement of the problem more carefully and seek more organized approaches. The students often spent much time exploring only one strategy and experienced difficulties in using alternatives. However, hints from the interviewer (including metacognitive questions) helped the students to consider other possibilities. Although the students recognized that it was important to check the solution of a problem, they mainly focused on whether there was an error in their calculations rather than reflecting on the sense of the solution. These results lead to the conclusion that it takes time for students to conceptualize problem solving strategies and use them on their own when asked to solve mathematical problems. The instructor planned to implement various learning activities in which the content could be introduced via problem solving. These activities required the students to participate and to spend significant time working on problems. Some students were initially reluctant to spend extra time reflecting on the problems and were more interested in receiving rules that they could use in examinations. Furthermore, student expectations, evaluation policies, and curriculum rigidity limited the implementation. Therefore, it is necessary to overcome some of the students' conceptualizations of what learning mathematics entails and to propose alternatives for the evaluation of their work that are more consistent with problem solving instruction. It is recommended that problem solving instruction include the participation or coordinated involvement of all course instructors, as the selection of problems for class discussions and for assignments is a task requiring time and discussion with colleagues. Periodic discussions of course directions are necessary to make and evaluate decisions that best fit the development of the course.Education, Faculty ofCurriculum and Pedagogy (EDCP), Department ofGraduat