Introducing Indivisibles into Calculus Instruction

PI的Bronislaw Czarnocha和Vrunda Prabhu将进行为期两年的教学实验，将“不可分”的直觉引入微积分教学，并通过黎曼结构将其与标准教学方法相结合。他们计划密切研究接受这一指导的学生在概念和计算方面的发展。作为这个项目的一部分，PI将调查：(1)将线条的直觉（不可分割的）转化和发展成精确的数学概念的过程是什么？(2)引入基于不可分的W-C结构、该结构与标准黎曼结构的相互作用，以及将两者结合起来的教学，是否能加强学生对定积分概念的理解？在拟议的项目中，他们将在六所院校指导和协调微积分教学，通过测试、论文和临床访谈评估学生的表现，收集不同评估结果的数据，并分析数据，为改进教学策略做准备。作为一个总体框架，他们将采用维果茨基的理论，强调最近发展区的认知过程，这对教学和评估有重要影响。学生将在课堂活动的引导下，体验数学思维的过程，并向老师展示数学思维的内容。教师的作用将是在对问题概念的精确数学理解的基础上，为他们的自发发展提供系统化。他们的自发思维过程所采取的方向，将在课程中进行调查。该项目是各方面之间的平衡：(1)学习者的自然直觉vs.学术指导，(2)计算掌握vs.概念理解，(3)教学和研究。

The PI's Bronislaw Czarnocha and Vrunda Prabhu, will conduct two year-long cycles ofteaching experiments to introduce the intuition of "indivisibles" into calculus instruction andintegrate it with standard ways of instructing via Riemann construction. They plan to studyclosely the conceptual and computational development of students who receive this instruction. ItAs part of this project, the PI's will investigate:(1) What is the process of transformation and development of the intuition of lines (theindivisibles) into a precise, mathematical concept?(2) Does the introduction of the W-C construction based on indivisibles, the interplay of thatconstruction with the standard Riemann construction, and the instruction that integrates both,strengthen students' understanding of the concept of the definite integral?In the proposed project, they will conduct and coordinate Calculus instruction at sixinstitutions, assess students' performance via tests, essays and clinical interviews, collect dataresulting from the different assessment and analyze the data in preparation for a refinement of theteaching strategy. As a general framework they will adopt Vygotsky's theory with its emphasison the cognitive processes in the zone of proximal development, which dictates importantconsequences on the instruction and assessment. Students will be guided by the activities of theclass to enable them to experience the process of mathematical thinking and to reveal its contentto the teacher. The role of the instructor will be to provide systematization of their spontaneousdevelopment grounded in the precise mathematical understanding of the concept in question. Thedirections their spontaneous thought processes take, will be under investigation during the course.The project is a balance between various aspects: (1) the natural intuition of the learner vs.the academic instruction, (2) computational mastery vs. conceptual understanding, (3) teachingand research.