Transcript Using Research to Influence Placement and Instruction

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
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•
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•
•
•
•
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
Percent
Correct
Discrimination
0.62
0.47
0.33
0.47
0.73
0.51
0.37
Identify the interval where one function exceeds the other from the graph of two functions
0.31
0.13
Identify the statement that best describes a given linear relationship of two variables
0.54
0.32
Identify the average rate of change of a function over an interval from a graph
0.54
0.38
Choose an exponential model of growth of a beanstalk
0.27
0.27
0.18
Identify how two exponential growth functions differ
0.61
0.36
0.18
Identify an expression of the area of an expanding circular oil spill
0.51
0.44
0.24
Choose the meaning of G(m+5) knowing G(m)
0.47
0.66
0.35
Choose the value of a composite function from a table
0.31
0.39
0.26
Identify how the slope of a ladder changes
0.71
0.58
0.37
Identify an equation that models distance on a map
0.41
0.26
Identify the inverse of a given exponential function
0.46
0.33
Identify expression that gives distance between two points on a number line
0.47
0.28
Identify the behavior of a function at a point of inflection
0.51
0.31
Identify the behavior of rational function
0.36
0.16
Choose an expression of the relative position of two runners on a track
0.29
0.08
Choose an expression of a function that is translated vertically and horizontally
0.24
0.09
Given sine of an angle identify the cosine of the angle
0.14
0.08
Identify specified point of intersection of a circle and a parabola
0.15
0.07
Identify the range of the sine function over an interval
0.29
0.14
Identify formula of a translated trigonometric function
0.14
0.05
Choose an expression of the vertical distance on a circle centered at origin in terms of arc length
0.17
0.08
Identify coordinates of a point on a circle centered at the origin in terms of cosine and sine
0.29
0.18
Identify the relationship between two functions connected by a parameter
0.51
0.61
0.38
0.32
0.32
0.63
0.43
0.34
0.43
0.49
0.15
0.14
0.36
0.24
0.16
0.29
Point
biserial
0.28 Identify an inference for a population from a sample using proportional reasoning
0.29 Choose an expression of the area of circle as a function of its circumference
a.
b.
c.
d.
e.
28.30%
13.67%
20.19%
21.62%
15.58%
Point biserial 0.29
Discrimination index 0.47
a)
b)
c)
d)
e)
Point biserial 0.26
Discrimination index 0.39
5.89%
48.01%
27.34%
3.50%
14.63%
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:
Suppose R(m) represents the number of acres of land needed to support feeding m
number of cattle. Which statement best reflects what R ! 1 (2200) represents?
1. The number of acres of land needed to support 2200 cattle.
2. The number of cattle that can be supported on 2200 acres of land.
3. The number of cattle supported is 2200.
4. One divided by the number of acres of land needed to support feeding 2200
cattle.
5. There is not enough information to determine what R ! 1 (2200) represents.
Covariational reasoning item:
Quantitative, functional, representations, modeling