Title: Thought processes in proportional reasoning
Description: Using tasks that stimulate estimation and proportional reasoning, my goal was to elucidate the thought processes of middle school students and to explore whether, and if so how, these thought processes change as the students are engaged in these proportional tasks. The proportional tasks involved estimating the number of Pattern Blocks in a container: Pattern Blocks are commonly used classroom geometric manipulative that have proportional size relationships so that one Hexagon is the same as two Trapezoids or three Rhombi or six Triangles. I designed three Tasks: How High (single container partially filled with one form of Pattern Block); Equal Height (two equal sized containers filled to equal heights, one with one form of Pattern Block and the other with another form; Equal Number (two equal sized containers filled with an equal number of Pattern Blocks, a different form for each container, and therefore to different heights). In the How High Task, the students were told the number of blocks and shown the measurement of their height in the container and asked to estimate the height of a different number of the same form of block. In the other two Tasks, the students were asked to estimate the number of Pattern Blocks in each container Using Piagetian interview techniques, I explored students' thought processes while engaged in the proportional tasks. Students were interviewed first individually and then in groups. Similar strategies that they used were grouped together to define categories of reasoning. During the experience of the Tasks, some students demonstrated development of proportional reasoning.

A student's development of proportional reasoning during group sessions demonstrated the potential of group discussion to promote developmental changes even though the researcher did no explaining during the sessions. On the other hand, some students did not change their reasoning the Group Session. Both the change and lack of change were examined in depth. In How High Tasks, 2:1, 3:1, and 6:1 ratio Tasks were solved more frequently than 1:2, 1:3, and 1:6 ratio Tasks. How High Task was the easiest and Equal Number Tasks were the most difficult.
Subjects: Reasoning, Mathematical models, Mathematics, Study and teaching (Elementary), Mathematical ability, Cognition in children