On doing and undoing in general and applied precalculus-calculus tracks
Haile Gilroy^ & Isabel Harris*
^McNeese State University, Lake Charles, Louisiana, USA
*Rice University, Houston, Texas, USA
Embedded Files
Full Paper (CalcConf3 Proceedings)
Slides from CalcConf3 Coming soon!
This project was inspired by our experience that students find Calculus much more challenging than Precalculus. We investigate the possibility that this challenging transition arises in part because of the differences in reasoning required by the textbook tasks students are expected to complete. We call inverse reasoning processes doing (forward thinking) and undoing (backward thinking). This work will be presented at The Learning and Teaching of Calculus Across Disciplines 2 -- The Third Calculus Conference in Milan, Italy, June 2025.
Sections Analyzed
Sections Analyzed
Applied Precalculus
Dennette & Kaskosz (2020)
Ch 1, Sec 1-5
Ch 2, Sec 1-3
Ch 3, Sec 1-4
Ch 4, Sec 1-4
Ch 5, Sec 1-5
Ch 6, Sec 1-3
Ch 7, Sec 1-4
Ch 8, Sec 1-5
Total # of Examples: 329
This textbook is open-source, and can be found here.
General Precalculus
Sullivan (2018)
Ch 1, Sec 1-4
Ch 2, Sec 1-6
Ch 3, Sec 1-5
Ch 4, Sec 1-6
Ch 5, Sec 1-9
Ch 6, Sec 1-6
Ch 7, Sec 1-7
Ch 8, Sec 1-5
Ch 9, Sec 1-7
Ch 10, Sec 2-7
Ch 11, Sec 1-8
Ch 12, Sec 1-4
Ch 13, Sec 1-3
Total # of Examples: 953
Applied Calculus
Hughes-Hallett (2014)
Ch 1, Sec 1-10
Ch 2, Sec 1-5
Ch 3, Sec 1-5
Ch 4, Sec 1-8
Ch 5, Sec 1-6
Ch 6, Sec 1-7
Ch 7, Sec 1-3
Ch 8, Sec 1-6
Ch 9, Sec 1-7
Ch 10, Sec 1-3
Total # of Examples: 467
General Calculus
Stewart (2016)
Ch 1, Sec 1-5
Ch 2, Sec 1, 7, 8
Ch 3, Sec 1-4, 6-8
Ch 4, Sec 1, 3, 7, 9
Ch 5, Sec 1-5
Ch 6, Sec 5
Ch 7, Sec 1
Ch 8, Sec 4, 5
Ch 9, Sec 1, 2, 4-6
Ch 11, Sec 2
Ch 14, Sec 1, 3, 7
Total # of Examples: 326
Examples of Doing and Undoing Processes
Examples of Doing and Undoing Processes
Evaluating <=> Solving
Evaluating (Doing)
Solving (Undoing)
Simplifying <=> Expanding
Simplifying (Doing)
Expanding (Undoing)
Composition <=> Decomposition
Composition (Doing)
Decomposition (Undoing)
Forward Conversion <=> Backward Conversion
Forward Conversion (Doing)
Backward Conversion (Undoing)
Strategize and Execute (Doing & Undoing)
Precalculus Example. Notice that the doing and undoing processes are structured as different prompts within the task.
Calculus Example. Notice that here, the undoing process is implied in the first statement of the solution. Unlike the precalculus example, students must grapple with "undoing" before "doing", but only "doing" is explicitly prompted.
All examples shown are real tasks we analyzed from the textbooks.
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