Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: In this report, we examine the affect of a mathematical problem-solving (MPS) course on the abilities of inservice teachers to demonstrate mathematical understanding in problem solving and to use complex strategies in mathematical problem solving.

To be competent problem-solving practitioners, teachers need several types of knowledge: procedural (Eisenhart, Borko, Underhill, Brown, Jones & Agard, 1993), conceptual or content knowledge (Leinhardt, 1988; Sullivan, Clarke, Spandel & Wallbridge, 1992; Eisenhart et al., 1993). Procedural knowledge denotes the rules, procedures, and skills necessary for completing a task. Procedural knowledge may or may not be supported by conceptual knowledge (Hiebert & Lefevre, 1986). Conceptual or content knowledge denotes the ability to understand the concept and connect or apply several different ideas. Conceptual knowledge for mathematics teachers also includes the ability to make generalizations, describe relationships, and demonstrate higher order reasoning skills (Sullivan et al., 1992). In addition, developing problem-solving practitioners need explore the role of metacognition in mathematical thinking or problem solving (Schoenfeld, 1985).

A group of sixteen inservice middle school (n=2) and secondary (n=14) mathematics teachers participated in a yearlong professional development program focused on mathematical problem solving. Initial baseline measures of skill levels, pedagogical strategies, and views about mathematics were established and a multi-part assessment of problem solving levels was administered. Teachers began the program in a course in Discrete Mathematics that focused upon various modes of group learning and leveling concepts about functions, mathematical reasoning, and mathematical inquiry. After the initial forty-five hours of instruction in Discrete Mathematics, the teachers enrolled in a course in mathematical problem solving. In the MPS course, teachers read and reported upon articles in research and practice in problem solving, applied and discussed Polya’s problem solving strategies in the context of their approaches to in-depth problems, revised and refined their own problem solving strategies by incorporating ideas from the research and practitioner literature, and examined rubrics for assessing students’ work in problem solving. Using a guide created by the researcher, they engaged in creating their own in-depth problems to elicit mathematical problem solving behaviors in their students on difficult concepts. The MPS course was followed by a course in probability and statistics with an emphasis on problem-based learning and real-world applications using technology. Data was collected systematically throughout the program in the form of student interviews, journal entries, student class work, pre- and post- tests of baseline skills, and a problem solving assessment that was administered at four different intervals during the yearlong program.

This report focuses primarily on the results from the problem solving test which was designed by the researcher. The work of White and Michelmore (1996) in studying student’s conceptual knowledge in calculus motivated the design of the problem solving tasks and methodology. Four tasks were the focus of the test items. The mathematics needed to solve the tasks did not go beyond a typical second-year algebra high school course. Each task was structured in four versions so that the manipulation (procedural knowledge) required to solve each version was essentially the same. However, the difference in the tasks was that each successive level involved more sophisticated levels of modeling and generalization. Each item was scored using a rubric designed by the researcher that analyzed understanding, strategies, and accuracy on the task.

Teachers were tested on four occasions: before the Discrete Mathematics course, before the MPS course, before the Probability and Statistics course, and at the end of all courses. The teachers were divided four parallel groups of 4. They were not aware which group they were in. Four tests were constructed and each test included four questions: one version of each of the four tasks. Each version of each task occurred on one and only one test, and each test had only one question in each version. A cyclic scheme was used to administer the tests to each of the four groups over the four data collections.

Findings suggest that there was a statistically significant growth in the strategies, understanding, and accuracy of the teachers in problem solving throughout the year. However, the greatest increase in the teachers’ ability to demonstrate effective problem solving strategies and understanding occurred after their MPS course. Excerpts of student work, journal entries, and interviews also support findings. We explore the components of the problem-solving course that contributed to the largest growth in teacher mathematical understanding and strategies in problem solving.