Analysis of processes used by middle-school students to interpret functions embedded in dynamic physical models and represented in tables, equations, and graph

This dissertation examined the processes generated by eighth-grade students to interpret and represent the functions embedded in dynamic physical models and the instructional decisions that facilitated the processes. Using the teaching experiment method, students were paired to interactively explore a slack rope board. The slack rope board consisted of a string that had been attached to a corkboard and that could be pulled taut to generate two segments of varying length. Students also explored the spool elevating system where an object was attached to a spool by string. The object could be raised or lowered by turning a handle attached to the spool.;In each case, students identified variables, selected symbols to represent variables, and generated tables, equations, and graphs of observed functions. Students' equations were often treated as records of the action or relationship of the functions embedded in the dynamic physical models. Some students were reluctant to algebraically manipulate their equations because the record of the action or relationship was lost. Students' graphs, created using an Etch-a-Sketch, were generated via direct translation of the action of a dynamic physical model to the knobs of the Etch-a-Sketch. Several students developed a rate to accomplish the translation. In verbally presented real-world scenarios which differed in context from that of dynamic physical models but which had underlying structures similar to those of functions embedded in dynamic physical models, it was found that students focused on the underlying structure.;The learning environment created by interactive use of the dynamic physical models supported development of instruction through which students' thinking could be challenged and through which students could connect their knowledge of functions generated in one representation to another. Use of the dynamic physical models enabled students to interpret functions as repeated actions with several students constructing generalized interpretation of the process. Instruction incorporating the use of dynamic physical models is indicated and recommended.