Using research to shape placement and influence instruction
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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

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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

    • R1Use proportional reasoning

    • R2View a function as a process

    • R3Reason 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


Graphical reasoning item:


Functional notation and definitions:


Covariational reasoning item:


Quantitative, functional, representations, modeling


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