Transcript Shades of Gray: Ambiguity Tolerance & Inferential Thinking

Students’ Ambiguity
Tolerance as a Success
Factor in Learning to
Reason Statistically
Robert H. Carver
Stonehill College/Brandeis University
June 12, 2007
Quick Outline
 Genesis of this Research
Classroom experience
 Literature review
 JSM 2006 presentation
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 Current project
 Invitation to participate
 Q&A
Genesis of the Research
 Some observations from the classroom…
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Learning statistics is difficult in many ways
Intro Stats can activate profound emotional
responses
“but usually I like/I dislike math classes…”
 Stat Ed literature
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Focus on variation as a central theme
Studies on activities, techniques, topics
Relatively little work on variation among
learners
Learners Vary!
 Variation among learners
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Prior coursework
Level of effort—motivation, capacity, etc.
Aptitude
Attitudinal orientation (Schau, et al.)
Myers-Briggs (BTI)
Other personality/emotional characteristics
Ambiguity Tolerance
 Frenkel-Brunswik, Else (1948)
 Ambiguity Tolerance Construct:
Some are stimulated by ambiguity, some are
threatened
 Personality trait vs. preferred process
 Stable personality attribute vs. contextdependent
 Relationship to rigidity, uncertainty tolerance,
openness
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The inner conflict
Per Frenkel-Brunswick:
Low ambiguity tolerance
 conflict & anxiety in ambiguous situations
 rigid adherence to preconceived ideas
 failure to process contrary evidence
Statistical Thinking
 Statistical thinking requires simultaneous
consideration of variation within one sample and
among possible samples.
 Statistical methods provide a means of making
decisions in inherently ambiguous situations, relying
on incomplete information.
 Inference requires a leap of faith—a ready embrace
of ambiguity
Contrast with ’Ambiguity’ in
Decision Theory
 Ambiguity as a property of the situation or
state of knowledge
 Ambiguity as property or proclivity of the
thinker
Ambiguity Tolerance
 Measurement Scales
Budner,1962
 Rydell; Rydell & Rosen 1966
 MacDonald, 1970
 Norton, 1975
 McLain, 1993
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Questions
 Do students with high AT have an advantage in
learning to think statistically?
 Do students with low AT tend to “shut down” when
presented with instruction in inferential reasoning
and techniques?
OR
 Do students with low AT welcome statistical
thinking as a way to cope with ambiguity?
Methods
Sample:
 85 undergraduates enrolled in 4 sections over 2
semesters
 Differences among sections
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Technology: Minitab vs. SAS
Normal, Learning Community, Honors
 Informed consent
 Credit & incentives
 Course-embedded data collection
Methods
Dependent variable:
 Score on Comprehensive Assessment of
Outcomes for a first course in Statistics
(CAOS) post-test
Developed by Web ARTIST Project
(U.Minnesota and Cal Poly) team
 Pre- and Post-test
 40 items
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Purpose of CAOS test
The CAOS test was designed to provide an instrument
that would assess students’ statistical reasoning
after any first course in statistics. Rather than focus
on computation and procedures, the CAOS test
focuses on statistical literacy and conceptual
understanding, with a focus on reasoning about
variability.
ARTIST project, University of Minnesota
CAOS post-test
Illustrative question:
Researchers surveyed 1,000 randomly selected adults
in the US. A statistically significant, strong positive
correlation was found between income level and
the number of containers of recycling they typically
collect in a week. Please select the best
interpretation of this result.
CAOS post-test
A.
B.
C.
We cannot conclude whether earning more money causes
more recycling among US adults because this type of
design does not allow us to infer causation.
This sample is too small to draw any conclusions about the
relationship between income level and amount of recycling
for adults in the US
This result indicates that earning more money influences
people to recycle more than people who earn less money.
CAOS post-test
A.
B.
C.
We cannot conclude whether earning more money causes
more recycling among US adults because this type of
design does not allow us to infer causation.
This sample is too small to draw any conclusions about the
relationship between income level and amount of recycling
for adults in the US
This result indicates that earning more money influences
people to recycle more than people who earn less money.
CAOS post-test
A study examined the length of a certain species of
fish from one lake. The plan was to take a random
sample of 100 fish and examine the results.
Numerical summaries on lengths of the fish
measured in this study are given.
Mean
26.8mm
Median
29.4 mm
Std. Dev.
5.0 mm
Minimum
12.0 mm
Maximum
33.4 mm
CAOS post-test
Mean
26.8mm
Median
29.4 mm
Std. Dev.
5.0 mm
Minimum
12.0 mm
Maximum
33.4 mm
CAOS post-test
Mean
26.8mm
Median
29.4 mm
Std. Dev.
5.0 mm
Minimum
12.0 mm
Maximum
33.4 mm
CAOS post-test
Post vs. Pre-test Scores
90
Gender
Male
Female
80
CAOSPost
70
60
50
40
30
30
40
50
CAOSPre
60
70
80
CAOS post-test
Post vs. Pre-test Scores
90
Gender
Male
Female
80
CAOSPost
70
60
50
40
30
30
40
50
CAOSPre
60
70
80
Measuring AT
Independent Measures & variables:
 Abiguity Tolerance:
 McLain’s 22 question instrument 7-point Likert
Scales
 Max
score for extreme tolerance = 74
 Min score for extreme intolerance = - 58
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Reliability: Cronbach’s alpha = 0.897
Measuring AT
Selected items:
 I don’t tolerate ambiguous situations well.
 I’m drawn to situations which can be
interpreted in more than one way.
 I enjoy tackling problems which are complex
enough to be ambiguous.
 I find it hard to make a choice when the
outcome is uncertain.
Covariates
Other explanatory factors and controls tested:
 Score on CAOS Pre-test
 Section controls
 Cohort (55% 2006; 45% 2007)
 Gender dummy (49% female; 51% male)
 First-year student dummy (61% 1st year)
 Math SAT
 Prior Stat Education (37% had some)
 Course cumulative average
 Attendance
Findings: CAOS Pre-test
Variable
Constant
Coeff
Signif
9.07
0.438
Female dummy
-1.13
0.638
AT scale
0.048
0.537
-5.581
0.028
Prior course dummy
5.256
0.032
Math SAT score
F
0.063
4.89
0.001
0.001
First year dummy
Adj R2
21.3%
A.T. did not have a significant main effect on Pre-test scores
Findings:CAOS Post-Test
Variable
Coeff
Signif
Constant
33.374
0.000
CAOS Pre-test score
AT scale
First Year dummy
Prior course dummy
0.559
0.110
-3.726
-3.406
0.000
0.079
0.072
0.099
F
Adj R2
12.29
37.0%
0.000
AT score has a significant (p < 0.10) effect on Post-Test reasoning score
Findings:CAOS Post-Test
Variable
Coeff
Signif
Constant
-2.529
0.751
CAOS Pre-test score
AT scale
Course Cumulative Avg
Prior course dummy
0.437
0.117
0.473
-3.946
0.000
0.039
0.000
0.035
F
Adj R2
19.46
48.9%
0.000
AT score has a significant (p < 0.05) effect on Post-Test reasoning score
Summary of Key Findings
 AT non-significant in predicting pre-test
scores
Suggests that the pre-test does not measure
ambiguity tolerance
 Significant findings re: prior coursework,
academic preparation (though not much
explanatory power), Math SAT
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Summary of Key Findings
 AT is significant in predicting Post-Test
scores
 Also significant
Pre-Test score
 Prior statistics coursework (but negative)
 First year dummy
 Course results
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 Not significant
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Gender, cohort, section, MathSAT
Discussion
 Main Findings:
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Ambiguity Tolerance may have a positive main effect
Low A.T. likely to be surmountable
 Caveats:
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CAOS scales measure several aspects of statistical
thinking
Small sample
Substantial unexplained variance
Measurement issues: effort, engagement
Discussion
 Implications:
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An individual’s orientation toward ambiguity
can affect his/her success with statistical
reasoning.
Tolerance of ambiguity construct may provide
a motivation for success
 Course pedagogy may address A.T. directly
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 Note:
Course averages not explained by AT
Discussion/Invitation
 Research directions:
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Can these results be replicated, especially in larger
samples?
Would the results hold up with different measures of
statistical reasoning?
Do other personality or personal style variables shape
success in statistical reasoning?
How can we structure pedagogy to address personality
variation among learners?
Does A.T. affect application of statistical reasoning in
practice?
Q&A/ Discussion
 Join me!
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[email protected]
[email protected]
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http://faculty.stonehill.edu/rcarver/
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