Making the Mean Meaningful: Two Instructional Studies.

Two studies are reported that explore the effectiveness of instruction in promoting students' conceptual understanding of the average concept using a leveling model and an open-ended problem-solving approach. One teacher and her 46 sixth-grade students participated in the first study. One year later, the second study was conducted with the same teacher, but with 42 different sixth graders. For both studies, pretests and posttests were used to examine the instructional impact on students' understanding of the concept of averaging. Tests in the second study were more comprehensive than those in the first study. The second study also documented lessons on teaching averaging. The teacher was provided with comprehensive inservice training on the mathematical content and pedagogical knowledge of arithmetic average, on the use of manipulative activities to introduce the averaging concept informally, and on how to make a transition to the use of a formal averaging algorithm to solve problems. Results of the studies show that from pretest to posttest, the students became more capable of applying the averaging algorithm to solve problems, as evidenced by increases in the number of correct answers and in the number of appropriate strategies, as well as by increases in the number and quality of the explanations of their solution processes. In addition, the results of the second study show that students were able to use their knowledge of averaging to solve novel problems. An analysis of the teacher's instruction revealed that the use of the leveling model in conjunction with an open-ended problem-solving approach mediated students' mathematical and statistical understanding of the averaging concept. Findings suggest that teachers' success in teaching with understanding is dependent on the encouragement and support they receive as they begin to change their approach to teaching. The collaboration of university professors and school teachers over extended periods of time is a possible way to help change occur. (Contains 2 figures, 5 tables, and 16 references.) (Author/SLD)