FRAMING DISCOURSE FOR OPTIMAL LEARNING IN SCIENCE AND MATHEMATICSby Mary Colleen MegowanA Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy ARIZONA STATE UNIVERSITY May 2007CONCLUSIONSWhen I embarked on this research, I wanted to find out how students used metaphors to help them reason about space and time. I thought I would hear and see evidence of metaphor use surrounding the preparation and sharing of whiteboards—one of the ‘universal constants’ found in modeling instruction classrooms. What I found in the first few videotapes I collected of the high school honors physics class initially frustrated me. The kinematics students I observed did not waste much time thinking about space or time. They identified it, assigned it a symbol, plugged it in to the equation they hoped was the right one, did some algebra with it and produced The Answer. The metaphors they used were the basic grounding metaphors of arithmetic (problem solving as object collection, construction, measurement or path following) and the object of metaphorical reference was the physics problem itself. Their problem spaces were containers from which they took the things they needed to reach their goal, and then they plunged into a mathematical ‘wormhole’, emerging at the far end with an answer. The problem space itself remained largely unexplored. Once they entered the spatial or temporal quantities they had collected into an equation, these quantities lost their connection to reality—they might as well have been ‘slithy toves’ or ‘dilithium’. The algebraic formulations of the problems they solved were the focal points of their whiteboard presentations. Graphs, diagrams and drawings, the SRs of physics, frequently occupied small, out of the way places on their whiteboards (and sometimes were absent altogether) while the algebra was featured prominently, and every step in the algebraic manipulations that led to a solution was shown and described.Figure 1. SRs were sometimes absent altogether while algebra was featured prominently on whiteboards.The Value of Spatial RepresentationsAs I added more videotapes to my data set, I began to see that students valued SRs in different ways. I saw that their values shifted over time to a greater appreciation of, and reliance upon SRs. Some students used SRs to justify their reasoning while others preferred to use computations. This led me to listen carefully to what students talked about, what counted for them as justification for the choices they made, and to whom they spoke and during whiteboard preparation and sharing, and this gave me some sense of how the SRs they employed functioned to keep them more grounded in the problem space. At the most elementary level, students drew SRs on their whiteboards because they were required to do so. If they did not include the appropriate diagram or graph on their whiteboard, it would be incomplete and therefore incorrect (i.e., it would not receive full credit). SRs, then, were about following directions. Making SRs might also be about demonstrating a skill. Just as a student might have some computational skill that she demonstrated by setting up and solving equations, she might also possess the skill of drawing an accurate, good-looking graph or diagram. SRs, in this case, were about showing off. Occasionally SRs helped students set up a geometric solution to a problem, i.e., determining the area under a curve. Such solutions were dependent on a student’s facility with unitization and thus hinted that some measure of spatial or temporal reasoning was an influence on their choice of solution strategy. Generally, however, they were simply used to justify the use of a certain algebraic formulation of the data. At least thisapproach included the mapping of graphical representations of space or time to algebraic symbols. SRs, in this instance, were for justifying equations. In all the cases mentioned above, SRs, when they were included, were about getting answers. Moreover, the focus of students’ presentations of these whiteboards was The Answer (in particular, the number—units were often overlooked) they had computed. Students who focused on answers were “zoomed in”. Their view of the problem space was very limited, and their answers were not well connected to the problem spaces that gave rise to them. They could zoom back out if prompted but it was effortful, and they were unlikely to do so of their own accord.As students became more sophisticated modelers, however, SRs were about constructing useful visualizations of physical situations in order to reason about them. When this grew to be the case, the first thing to appear on a whiteboard was the SR, and as it was constructed, its various features and their properties were identified and defined, often in writing. The SR served to keep them zoomed out so that the physical context of the problem remained actively in view in the problem space. Only after this setting of the scene was complete (by the construction of an SR that encoded the elements of the model and their relationships) did students zoom in, abstract out physical quantities and assemble them into a symbolical representation—an equation. Once they had a sensible equation that flowed from the diagram and obeyed the constraints placed on each of its elements by the definitions and properties that were assigned, they were done. Plugging in actual numerical data and solving for some value was, at times, an afterthought. In some cases, it was not even called for in the tasks students were assigned. SRs in these instances were for visualizing and making sense of a problem space.SRs in this last instance were also an important communication tool in the distributed cognitive sense. There was, at times, a sort of dialogue between the students and their inscription that caused the inscription to evolve as reasoning aloud progressed. In addition to the students working together to create these SRs, the SRs they created functioned as another voice in the exchange of ideas—often a constraining voice that placed limits on the mental models that students were attempting to articulate. One thing that made this dialectic particularly valuable was the opportunity for students to ‘ask’ their SRs questions. There were many types of questions that students asked in the course of creating and reasoning with SRs. Largely, it appeared