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Enhancing Mathematical Conceptual Understanding through Research-Based Assessment Tools

This study emphasizes the insufficient achievement of precalculus and calculus courses in fostering a deep conceptual understanding of fundamental functional ideas. It discusses the development and implementation of assessment tools like the Precalculus Concept Assessment (PCA) and the Algebra and Precalculus Concept Readiness (APCR) taxonomy, which aim to evaluate reasoning abilities and mathematical understanding. Supported by NSF funding, the findings highlight the importance of aligning curricula with standards and promoting conceptual knowledge through research-backed instructional strategies.

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Enhancing Mathematical Conceptual Understanding through Research-Based Assessment Tools

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  1. Using Research to Shape Placement and Influence Instruction Bernard L. Madison Director, MAA Placement Testing Program (PTP) University of Arkansas PTP materials available from MAA/Maplesoft Placement Testing Suite Dissemination of CCR and new APCR development supported by NSF (DUE-1122965)

  2. Theses • Precalculus courses are not achieving their potential in conveying conceptual understand of fundamental functional ideas. • Calculus courses are not achieving their potential in conveying conceptual understand of fundamental functional ideas.

  3. Developments based on Research Results • The Precalculus Concept Assessment (PCA): A Tool for Assessing Students’ Reasoning Abilities and Understandings by M. Carlson, M. Oehrtman, and N. Engelke, Cognition and Instruction, 28(2), 113–145, 2010 • Calculus Concept Readiness (CCR) 2010 Paper by M. Carlson, R. West & BLM. . To appear International Journal for Research in Undergraduate Mathematics Education, Projected Vol. 1, Issue 2, 2015. • Algebra and Precalculus Concept Readiness (APCR) 201213

  4. Alignment with CCSSM • Algebraic thinking Grades 1-12 • Computing with unknowns • Functions from grade 8 • Process view of a function • Conceptual understanding emphasized • Standards of mathematical practice • Modeling in many contexts

  5. Calculus Concept Readiness (CCR)Taxonomy • Reasoning abilities • R1 Use proportional reasoning • R2 View a function as a process • R3 Reason about and represent how two quantities change together (covariational reasoning)

  6. Understand and use the following concepts or ideas • Quantity • Variable • Slope/Constant rate of change • Average rate of change • Function composition • Function inverse • Function translations (horizontal and vertical shifts)

  7. Understand, represent and interpret function growth patterns • Linear • Exponential • Non-linear polynomial • Rational • Periodic

  8. Understand central ideas of trigonometry • Angle measure • Radian as a unit of measure • Sine and cosine functions as covariation • Sine and cosine functions in right triangles • Reduction from any circle to a unit circle.

  9. Other abilities • Solve equations • Represent and interpret inequalities • Represent and interpret absolute value inequalities • Use and solve systems of equations • Understand and use function notation to express one quantity in terms of another

  10. 28.30% • 13.67% • 20.19% • 21.62% • 15.58% Point biserial 0.29 Discrimination index 0.47

  11. 5.89% • 48.01% • 27.34% • 3.50% • 14.63% Point biserial 0.26 Discrimination index 0.39

  12. a) 24.96% b) 8.27% c) 19.40% d) 20.19% e) 21.14% Point biserial 0.05 Discrimination index 0.14

  13. Algebra and Precalculus Concept Readiness (APCR) Draft Taxonomy • Reasoning Abilities • Quantitative Reasoning • Reasoning with Representations • Functional Reasoning • Covariational Reasoning • Proportional Reasoning • Variable Reasoning

  14. Notations, Conventions and Definitions • Modeling • Measurement • Rate of change • Function concepts • Solving equations • Inequalities • Properties of real numbers

  15. Graphical reasoning item:

  16. Functional notation and definitions:

  17. Covariational reasoning item:

  18. Quantitative, functional, representations, modeling

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