# CLEAR Calculus

## Publications

Collapsing dimensions, physical limitation, and other student metaphors for limit concepts

This study identified basic metaphors used by introductory calculus students to reason about limits and characterized how those metaphors influenced students' understanding of fundamental concepts throughout calculus. Approximation metpahors emerged as a potentially productive approach to supporting powerful student reasoning.

Layers of abstraction: Theory and design for the instruction of limit concepts

This article reviews research literature about students' understanding of limits and various approaches to calculus instruction based on interpretations of this literature. It then outlines the instructional approach guiding the CLEAR Calculus project based on developing the central limit concepts in terms of approximations and error analyses.

Problems and solutions in students’ reinvention of a definition for sequence convergence

This study engaged a pair of students in a "guided reinvention" of the formal definition of sequence convergence. They first constructed as many qualitatively different examples of sequences that converge to 5 and of sequences that do not converge to 5. We then engaged the students in multiple cycles of i) attempting to write a definition that included all of the examples and excluded all of the non-examples, ii) testing their definition against the examples and non-examples, iii) identifying problems with their definition, and iv) attempting to resolve those problems as they revised their definition.

Reinvention six months later: The case of Megan.

The use of dynamic visualizations following reinvention

From intuition to rigor: Calculus students’ reinvention of the definition of sequence convergence

Students’ reinvention of formal definitions of series and pointwise convergence

Part / whole metaphor for the concept of convergence of Taylor series

Strong metaphors for the concept of convergence of Taylor series

Calculus students’ assimilation of the Riemann integral into a previously established limit structure

Student understanding of accumulation and Riemann sums

Approximation as a foundation for understanding limit concepts