The Precalculus Concept Assessment: A Tool for Assessing Students’ Reasoning Abilities and Understandings

Abstract

This article describes the development of the Precalculus Concept Assessment (PCA) instrument, a 25-item multiple-choice exam. The reasoning abilities and understandings central to precalculus and foundational for beginning calculus were identified and characterized in a series of research studies and are articulated in the PCA Taxonomy. These include a strong understanding of ideas of rate of change and function, a process view of function, and the ability to use covariational reasoning to examine and represent how two covarying quantities change together. This taxonomy guided the PCA development and now provides the theoretical basis for interpreting and reporting PCA results. A critical element of PCA's design was to identify the constructs essential for learning calculus and to employ methods to assure that PCA items are effective in assessing these constructs. We illustrate the role that cognitive research played during both the design and validation phases of the PCA instrument. We also describe our Four-Phase Instrument Development Framework that articulates the methods used to create and validate PCA. This framework should also be useful for others interested in developing similar instruments in other content areas. The uses of PCA are described and include (a) assessing student learning in college algebra and precalculus, (b) comparing the effectiveness of various curricular treatments, and (c) determining student readiness for calculus.

Notes

The development of PCA preceded the development of the Calculus Concept Inventory (CCI) by more than 5 years. The CCI had no bearing on the development of the PCA.

The subscores are generated from scoring clusters of items that are associated with each category of the PCA Taxonomy.

The alternate 25-item version contained 15 items from the version of PCA that is the focus of this article. Student scores on these 15 items were correlated highly with students’ overall PCA score. We added 10 new items that were validated using the same procedures as described in this article. Four items assessed students’ covariational reasoning abilities; three items assessed students’ ability to represent rate of change information; and three items assessed students’ understanding of the growth rate of exponential functions. The alternate instrument was intended to better reflect specific program objectives.