Conceptual Knowledge in Introductory Calculus

Paul James White · Michael C. Mitchelmore

178 citations0/0 files storedJournal for Research in Mathematics Education

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Abstract

Responses to word problems involving rates of change were collected on four occasions during and after 24 hours of concept-based calculus instruction given to a group of first-year university students, all of whom had studied calculus in secondary school. The number of students who could symbolize rates of change in noncomplex situations increased dramatically. However, there was almost no increase in the number who could symbolize rates of change in complex items or in items that required modeling a situation using algebraic variables. Detailed analysis revealed three main categories of error, in all of which variables are treated as symbols to be manipulated rather than as quantities to be related. We surmise that the students had developed an “abstract-apart” concept of a variable rather than the “abstract-general” concept that is needed for the successful study of calculus.

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References

1981Concept image and concept definition in mathematics with particular reference to limits and continuity 2013Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis 2006The Learning and Teaching of School Algebra 1989Images and Definitions for the Concept of Function 1983Concept definition, concept image and the notion of function 1993Subject-Matter Knowledge and Pedagogical Content Knowledge: Prospective Secondary Teachers and the Function Concept 1988Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool

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