A study of the effect of using an "open approach" to teaching mathematics upon the mathematical problem-solving behaviors of secondary school students

The purpose of this study is to investigate the effect that instruction using an open approach to teaching mathematics has on the mathematical problem-solving performance and attitudes toward mathematics of fifty midwestern secondary school students. The open approach includes conjecture, exploration, discovery, discussion, verification, and generalization which draws on students' own previously learned attitudes, skills, knowledge, and thought processes in solving mathematics problems. Examination of the behaviors, strategies, and methods of solution that subjects use during mathematical problem solving was essential to this study. Therefore, a combination of both qualitative and quantitative methodologies was necessary in an attempt to optimally analyze the effect of the open approach on the problem-solving performance of subjects.

The instruments (pretest/posttest) for this study included four open-ended problems and a Likert-type questionnaire with questions dealing with subjects' attitudes towards mathematics. Assessment of the solutions to the four open-ended problems was performed utilizing frequency, fluency, originality, and elegance. Both the quantitative and qualitative analyses demonstrate that the mathematical problem-solving performance of the subjects in this sample was improved by the use of the open approach to teaching mathematics. Further, the analyses show that subjects' attitudes towards mathematics were significantly affected in a positive way and that gender did not have a significant effect on mathematical problem-solving performance.