Introducing Indivisibles into Calculus Instruction

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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.