Understanding of function, proof and mathematical modelling in the presence of mathematical computing tools: Prospective secondary school mathematics teachers and their strategies and connections.

摘要

This exploratory study generated ten grounded hypotheses regarding the nature of mathematical understanding of prospective secondary mathematics teachers as they learn and do mathematics in the presence of computer tools. The data for triangulation purposes came from individual interviews, classroom and lab observations, questionnaires, lab reports, mathematics projects, journal entries, and university transcripts of 13 prospective secondary school mathematics teachers voluntarily enrolled in a mathematics education mathematics course and the instructor/investigator. The course was designed to engage the subjects in refining their mathematical understanding of fundamental mathematical ideas that underlies secondary school mathematics programs in a technological world. The description of the study includes discussion of the course goals and materials.;Six of the hypotheses relate to three fundamental mathematics themes--function, proof, and mathematical modelling. The data suggested first that prospective secondary mathematics teachers have a view of function dominated by variations on the rate of change concept and the tendency to prefer and expect functions that resemble simple functions (e.g., linear, sine, quadratic). Second, prospective mathematics teachers recognize complex relationships among variables in the real world but oversimplify the relationships they attempt to mathematize. The models they do construct are usually developed using either simple mathematical operations and personal experience or numerically correct but situationally irrelevant computer-generated fitted functions. They quickly abandon these models in favor of personal expectations when answering questions about the real-world phenomena the functions represent. Third, the reasoning of prospective mathematics teachers in tool-present environments revolves around loose associations and contradictions among various realms of reasoning (e.g., tool results, real world data, personal experience).;The remaining four hypotheses reflect ways in which prospective mathematics teachers use computer tools and draw upon previously learned mathematical ideas as they work within tool-intensive environments. Their strategies are typically inefficient and often mathematically incomplete. They draw upon ideas from their previous mathematical experiences but do not probe the links beyond the surface level.;This work has direct implications for the preparation of mathematics teachers and for the development of curriculum materials at secondary and post secondary levels and suggests directions for future research.