Students' understanding of the concept of chain rule in first year calculus and the relation to their understanding of composition of functions
Abstract
The present study is a follow-up to a study conducted by this author with seven others. That study (Clark et al., 1997) proposed the use of Piaget and Garcia's triad mechanism to describe the development of a schema, specifically with respect to students' understanding of the chain rule. This study was designed to collect data to test this theory. Both quantitative and qualitative methods were employed. Data were collected via a questionnaire on a group of calculus students' (n = 34) knowledge of and skill with function, composition of functions, differentiation and chain rule. The data were analyzed to investigate how their performance on the composition of functions items related to that of the chain rule. No strong correlation is reported. Some of the subjects had experienced a reform calculus course which used computer experiences and cooperative learning. A comparison was made between the two groups, although no overall differences are reported. Follow-up interviews based on the questionnaire responses were conducted with six subjects. These results are positive in the sense that the triad description fits with the data. We present detailed descriptions of the Intra-, Inter-, and Trans-levels of the development of the chain rule schema. Some evidence is presented to support the notion that understanding of composition of functions is key to understanding the chain rule. The type of instruction was a factor in how a student performed on these tasks. The differences between the types seem to be related to the difference between using the chain rule and understanding the chain rule, and an explanation of these differences is offered. The need for collecting differing types of data is established. Two students scored high quantitatively but did not perform well in the interview, while another student failed the quantitative work but demonstrated a high level in the interview. Finally, the use of writing as a pedagogical tool is recommended based on the results.
Degree
Ph.D.
Advisors
Dubinsky, Purdue University.
Subject Area
Mathematics education|Educational psychology
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