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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).
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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)
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.
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
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
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)
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)
Understand, represent and interpret function growth patterns • Linear • Exponential • Non-linear polynomial • Rational • Periodic
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.
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
28.30% • 13.67% • 20.19% • 21.62% • 15.58% Point biserial 0.29 Discrimination index 0.47
5.89% • 48.01% • 27.34% • 3.50% • 14.63% Point biserial 0.26 Discrimination index 0.39
a) 24.96% b) 8.27% c) 19.40% d) 20.19% e) 21.14% Point biserial 0.05 Discrimination index 0.14
Algebra and Precalculus Concept Readiness (APCR) Draft Taxonomy • Reasoning Abilities • Quantitative Reasoning • Reasoning with Representations • Functional Reasoning • Covariational Reasoning • Proportional Reasoning • Variable Reasoning
Notations, Conventions and Definitions • Modeling • Measurement • Rate of change • Function concepts • Solving equations • Inequalities • Properties of real numbers