Nature of the Problem

The Nature of the Problem The inability of most students to apply methods used in a practiced problem to a new situation

During the past seven years we have made substantial changes to our three-semester calculus sequence. We use a variety of teaching methods designed to promote self-discovery of mathematical ideas and cooperation with other students. Despite these changes, we continued to see significant evidence that students were unable to apply the methods used in a practiced problem to a new situation. All our efforts to reform calculus had simply increased what learning theorist Robert Sternberg refers to as the "book smart" intelligence of our students. [Sternberg, R.J., Pathways to Psychology, pgs 266-268, Harcourt Brace & Co., 1997] That is, we provided new and creative ways for them to practice their skills, methods and procedures -- all necessary but unfortunately insufficient for fostering a useful understanding of the material.

What we failed to provide was an environment that allowed students to increase what Sternberg refers to as their "creative intelligence" (i.e., intelligence that allows us to use our knowledge to solve new problems in original ways) or their "street smarts" (i.e., intelligence that allows us to use common sense to find new strategies for solving problems.) [Sternberg, R.J., 1997]

"Prevalent school practices assume, more often than not, that knowledge is individual and self-structured, that concepts are abstract, relatively fixed, and unaffected by the activity through which they are acquired and used, and that Just Plain Folk behavior should be discouraged."[Brown, J.S., Collins, A., Guguid, P, Situated Cognition and the Culture of Learning, "Educational Researcher", Jan-Feb, 1989, pgs 32-42] In their work, Brown, Collins and Duguid compared problem- solving approaches of Just Plain Folk, students and actual practitioners. They found that Just Plain Folks and practitioners reasoned with casual stories, acting on situations, resolving emergent problems, producing negotiable meaning. On the other hand, students reasoned with laws, acting on symbols, resolving well-defined problems, producing fixed meaning. [Brown, et. al, 1989]]

The goal of our project was to increase students' conceptual understanding of first principles in calculus by creating an activity where students could practice solving problems using a Just Plain Folk approach.