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  Current project  Invitation to participate  Q&A

Genesis of the Research  Some observations from the classroom…  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  Focus on variation as a central theme  Studies on activities, techniques, topics  Relatively little work on variation among learners

Learners Vary!  Variation among learners 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. context- dependent Relationship to rigidity, uncertainty tolerance, openness

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

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 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

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.

A. 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. B. This sample is too small to draw any conclusions about the relationship between income level and amount of recycling for adults in the US C. This result indicates that earning more money influences people to recycle more than people who earn less money. CAOS post-test

A. 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. B. This sample is too small to draw any conclusions about the relationship between income level and amount of recycling for adults in the US C. 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 Median29.4 mm Std. Dev.5.0 mm Minimum12.0 mm Maximum33.4 mm CAOS post-test

Mean 26.8mm Median29.4 mm Std. Dev.5.0 mm Minimum12.0 mm Maximum33.4 mm CAOS post-test

Mean 26.8mm Median29.4 mm Std. Dev.5.0 mm Minimum12.0 mm Maximum33.4 mm CAOS post-test

Improvement

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 Reliability: Cronbach’s alpha = 0.897

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. Measuring AT

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% 1 st year)  Math SAT  Prior Stat Education (37% had some)  Course cumulative average  Attendance

Findings: CAOS Pre-test VariableCoeffSignif Constant Female dummy AT scale First year dummy Prior course dummy Math SAT score F Adj R % A.T. did not have a significant main effect on Pre-test scores

Findings:CAOS Post-Test VariableCoeffSignif Constant CAOS Pre-test score AT scale First Year dummy Prior course dummy F Adj R % AT score has a significant (p < 0.10) effect on Post-Test reasoning score

Findings:CAOS Post-Test VariableCoeffSignif Constant CAOS Pre-test score AT scale Course Cumulative Avg Prior course dummy F Adj R % AT score has a significant (p < 0.05) effect on Post-Test reasoning score

 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 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  Not significant Gender, cohort, section, MathSAT Summary of Key Findings

Discussion  Main Findings: Ambiguity Tolerance may have a positive main effect Low A.T. likely to be surmountable  Caveats: CAOS scales measure several aspects of statistical thinking Small sample Substantial unexplained variance Measurement issues: effort, engagement

Discussion  Implications: 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  Note: Course averages not explained by AT

Discussion/Invitation  Research directions: 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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