The Effects of Problem-Based Instruction on the Geometric Knowledge of High School Students

全国数学教师委员会的《学校数学课程和评价标准》要求学校数学教学的范式从以技能为基础的教学转变为以问题为基础的教学。本提案描述了一个研究项目，其中将研究基于问题的教学对高中几何学生的概念和思维过程的影响。在这个为期三年的项目中，将设计一种以学生解决问题为重点的教学模式，适合不同能力的学生。教学的效果将从两个不同但相互作用的角度进行评估——使用解决问题的过程和特定几何概念的知识。要研究的几何问题解决过程包括以下能力：(a)从经验证据中进行归纳，(b)使用演绎证据进行归纳，(c)处理反向思路，(d)灵活地重新调整思路，以及(e)将几何情况可视化。要研究的几何概念包括(a)证明，(b)无穷，(c)相似和同余，(d)面积，周长和体积，(e)平行，(f)变换等概念。数据收集将分为三部分，包括(a)对个别学生进行教学实验，以评估特定的教学方法；(b)完整课堂的教学实验，以评估教学性质和课堂互动的变化；(c)状态测试，以记录开发概念和过程。本项目由教学材料开发计划共同资助。

The Curriculum and Evaluation Standards for School Mathematics of the National Council of Teachers of Mathematics calls for a shift in paradigm in the teaching of school mathematics from skills-based instruction to problem-based instruction. This proposal describes a research project in which the effects of problem-based instruction on the concepts and thought processes of high school geometry students will be studied. Over the course of the proposed three-year project, a teaching model which focuses on student solving of problems and which is accessible to students of varying abilities will be designed. The effects of the instruction will be assessed from two contrasting yet interactive perspectives--use of problem-solving processes and knowledge of particular geometric concepts. The geometric problem-solving processes to be investigated include the abilities to (a) generalize from empirical evidence, (b) generalize using deductive evidence, (c) deal with reverse trains of thought, (d) flexibly refocus a train of thought, and (e) visualize a geometric situation. The geometric concepts to be investigated include the concepts of (a) proof, (b) infinity, (c) similarity and congruence, (d) area, perimeter, and volume, (e) parallelism, and (f) transformations. Data collection will be three-fold, including (a) teaching experiments with individual students to assess particular instructional approaches; and (b) teaching experiments with intact classes to assess the changing nature of instruction and classroom interactions; and (c) status testing to document developing concepts and processes. This project is jointly funded by the Instructional Materials Development Program.