Transcript Assessing Knowledge for Teaching Algebra: Lessons Learned

Mathematical Knowledge for Teaching
Algebra: Validating an Assessment of
Teacher Knowledge
Robert E. Floden
Raven McCrory
Michigan State University
January , 2007
KNOWING MATHEMATICS FOR
TEACHING ALGBRA (KAT) PROJECT
(NSF REC No. 0337595)
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Joan Ferrini-Mundy, Initial PI
Robert Floden, Current PI
Raven McCrory
Mark Reckase
Sharon Senk
Karen Allen & Xuhui Li
GOALS OF KAT PROJECT
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Instrument Design (2004 - 2005)
Develop theoretical constructs, items, test forms
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Assessment Validation (2006 - 2008)
Do the forms measure distinct constructs?
What differences in knowledge affect teacher
responses?
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Sensitivity to Instruction (2007 - 2008)
How do teacher preparation experiences affect
responses?
Item Development
August 2004 - October 2005
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Constructs defined
Item writing workshops with mathematicians,
math educators, secondary teachers
Additional Items written by KAT faculty & GAs
Items reviewed by mathematicians
Items edited by KAT staff
Mathematical Knowledge for Teaching
Algebra
(simplified for assessment design)
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Knowledge of school algebra
algebra in middle and high school
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Advanced mathematical knowledge
related college math, e.g. calculus, abstract algebra
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Teaching knowledge
knowledge of typical errors, canonical uses of school
math, curriculum trajectories, etc.
Sample Item: Identify an Exponential Function
(School Knowledge)
Which of the following situations can be modeled using an exponential
function?
i. The height h of a ball t seconds after it is thrown into the air.
ii. The population P of a community after t years with an increase of n
people annually.
iii. The value V of a car after t years if it depreciates d% per year.
A. i only
B. ii only
C. iii only
D. i and ii only
E. ii and iii only
Pilot Test Results: Identify
Exponential Function
Number of cases Difficulty
Pre-service
teachers
138
0.297
In-service
teachers
287
0.324
Total
431
0.313
Sample Item: Number Systems
(Advanced Knowledge)
For which of the following sets S is the following statement true?
For all a and b in S, if ab = 0, then either a = 0 or b = 0.
i. the set of real numbers
i. the set of complex numbers
iii. the set of integers mod 6
iv. the set of integers mod 5
v. the set of 2x2 matrices with real number entries
A. i only
B. i and ii only
C. i, ii and iv only
D. i, ii, iii and iv only
E. i, ii, iii, iv, and v
Pilot Test Results: Properties of
Number Systems
(Advanced Knowledge)
Number of cases Difficulty
Pre-service
teachers
86
0.151
In-service
teachers
96
0.177
Total
186
0.161
Sample Item: Identifying Student’s Error
in Solving a Linear Equation
(Teaching Knowledge)
A student solved the equation
3(n - 7) = 4 - n
and obtained the solution n = 2.75.
What might the student have done wrong?
Pilot Test Results: Identify Student’s
Error
Number of cases Difficulty
Pre-service
teachers
97
0.760
In-service
teachers
14
0.946
Total
115
0.787
Validation Study Design
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Recruit sample with varying combinations of
constructs
Combination of factor analysis and cluster
analysis to identify empirical sub-scales
Analysis of clustered items to interpret
clusters
Interviews with test-takers to understand
reasoning behind correct and incorrect
responses
Number of Participants in
Validation Studies
Pre-service teachers
In-service teachers
327
492
Math majors
Non-math major
535
283
Total
822 Career stage
818 Major
Next Steps
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Complete interpretation of clusters
Examine results for special samples (e.g.,
very high math knowledge)
Interview study of new sample of teachers
Measure change in assessment results in
selected teacher preparation programs
Questions for the Future
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Will our assessment reveal differences in the effects
among approaches to mathematics teacher
education (pre-service or inservice)?
Will programs vary in their relative impact on the
three components?:
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Knowledge of school algebra
Advanced mathematical knowledge
Teaching knowledge
Will differences in assessment results predict
differences in teaching effectiveness?
Discussion
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What information from our project would be
useful to you in designing courses for
teachers?
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How can we best make our results useful to
teacher educators?
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What do you hope our project can tell you?
Contacts for KAT
Project Information
Xuhui Li, Project Manager,[email protected]
Robert E. Floden, PI, [email protected]
Raven McCrory, Co-PI,[email protected]