Precalculus Concept Assessment (PCA)

Developed by Marilyn Carlson, Michael Oehrtman, Nicole Engelke

Purpose To assess essential knowledge that mathematics education research has revealed to be foundational for students’ learning and understanding of the central ideas of beginning calculus.
Format Pre/post, Multiple-choice
Focus Mathematics Content knowledge (rate of change, function, process view of functions, covariational reasoning)
Level Intro college

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Example question from the PCA (Carlson, Oehrtman, and Engelke, 2010).

Gold Star Validation
This is the highest level of research validation, corresponding to all seven of the validation categories below.

Research Validation Summary

Based on Research Into:

  • Student thinking

Studied Using:

  • Student interviews
  • Expert review
  • Appropriate statistical analysis

Research Conducted:

  • At multiple institutions
  • By multiple research groups
  • Peer-reviewed publication

The PCA developers did research around the understandings and abilities that are foundational to learning  calculus, and developed a taxonomy based on their findings. They used the PCA taxonomy to write open-ended questions and tested them with students. They then interviewed students about their answers and revised the questions. Multiple-choice answers, based on students' written answers, were developed and tested. The cycle of testing the questions with students, interviewing students, and revising the questions and answer choices was repeated eight times. While iteratively developing questions, scoring rubrics were created and reviewed by experts in math. These rubrics were used to refine the answer choices. Statistical analysis of reliability found reasonable results. The PCA has been used with thousand of students at over 40 institutions and the results published in one peer-reviewed paper.  

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Typical Results

From (Carlson, Oehrtman, and Engelke 2010):

The final 25-item version of the PCA has been administered to students upon completion of
college algebra and precalculus courses taught in small classes with different teachers at a large public university and nearby community college. The mean score for 550 college algebra students was 6.8 with standard deviation of 3.2. For 902 pre-calculus students, the mean score was 10.2 with standard deviation of 4.1.

While the PCA was not designed for use as a placement exam, it performs reasonably well in this regard. We administered the PCA as a pretest to 248 students entering first-semester introductory calculus courses with six different instructors at our institution and compared final course grades with PCA scores. The most discriminating cut-off score was 50%. Specifically, 77% of the students scoring 13 or higher on the 25-item test passed the first-semester calculus course with a grade of C or better while 60% of the students scoring 12 or lower failed with a D or F or withdrew from the course. 

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