1. Students’ Ambiguity Tolerance as a Success Factor in Learning to Reason Statistically Robert H. Carver Stonehill College/Brandeis University June 12, 2007

2. Quick Outline  Genesis of this Research Classroom experience Literature review JSM 2006 presentation  Current project  Invitation to participate  Q&A

3. 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

4. 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

5. 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

6. The inner conflict Per Frenkel-Brunswick: Low ambiguity tolerance   conflict & anxiety in ambiguous situations   rigid adherence to preconceived ideas   failure to process contrary evidence

7. 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

8. 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

9. Ambiguity Tolerance  Measurement Scales Budner,1962 Rydell; Rydell & Rosen 1966 MacDonald, 1970 Norton, 1975 McLain, 1993

10. 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?

11. 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

12. 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

13. 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

14. 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.

15. 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

16. 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

17. 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

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

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

21. Improvement

22. 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

23. 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

24. 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

25. Findings: CAOS Pre-test VariableCoeffSignif Constant9.070.438 Female dummy-1.130.638 AT scale0.0480.537 First year dummy-5.5810.028 Prior course dummy5.2560.032 Math SAT score0.0630.001 F4.890.001 Adj R 2 21.3% A.T. did not have a significant main effect on Pre-test scores

26. Findings:CAOS Post-Test VariableCoeffSignif Constant33.3740.000 CAOS Pre-test score0.5590.000 AT scale0.1100.079 First Year dummy-3.7260.072 Prior course dummy-3.4060.099 F12.290.000 Adj R 2 37.0% AT score has a significant (p < 0.10) effect on Post-Test reasoning score

27. Findings:CAOS Post-Test VariableCoeffSignif Constant-2.5290.751 CAOS Pre-test score0.4370.000 AT scale0.1170.039 Course Cumulative Avg0.4730.000 Prior course dummy-3.9460.035 F19.460.000 Adj R 2 48.9% AT score has a significant (p < 0.05) effect on Post-Test reasoning score

28.  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

29.  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

30. 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

31. 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

32. 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?

33. Q&A/ Discussion  Join me! rcarver@stonehill.edu rcarver@brandeis.edu http://faculty.stonehill.edu/rcarver/

34. References on A.T. Benjamin, A., Riggio, R., & Mayes, B. (1996). Reliability and factor structure of Budner's tolerance for ambiguity scale. Journal of Social Behaviour and Personality, 11, 625-632. Budner, S. (1962). Intolerance of ambiguity as a personality variable. Journal of Personality, 30(1), 29-50. DeRoma, V. M., Martin, K. M., & Kessler, M. L. (2003). The relationship between tolerance for ambiguity and need for course structure. Journal of Instructional Psychology, 30(2), 104-109. Durrheim, K., & Foster, D. (1997). Tolerance of ambiguity as a content specific construct. Personality and Individual Differences, 22(5), 741-750. Feinberg, L., & Halperin, S. (1978). Affective and cognitive correlates of course perfectionism in introductory statistics. Journal of Experimental Education, 46(4), 11-18. Fibert, Z., & Ressler, W. H. (1998). Intolerance of ambiguity and political orientation among israeli university students. The Journal of Social Psychology, 138(1), 33-40. Frenkel-Brunswik, E. (1948). Tolerance of ambiguity as an emotional and perceptual personality variable. Journal of Personality, 18, 108-143. Friedland, N., & Keinen, G. (1991). The effects of stress, ambiguity tolerance, and trait anxiety on the formation of causal relationships. Journal of Research in Personality, 25, 88-107. Furnham, A. (1994). A content, correlational and factor analytic study of four tolerance ambiguity questionnaires. Personality and Individual Differences, 16(3), 403-410. Furnham, A., & Ribchester, T. (1995). Tolerance of ambiguity: A review of the concept, its measurement and applications. Current Psychology, 14(3), 179-199. Grenier, S., Barrett, A.-M., & Ladouceur, R. (2005). Intolerance of uncertainty and intolerance of ambiguity: Similarities and differences. Personality and Individual Differences, 39, 593-600. Johnson, H. L., Court, K. L., Roersma, M. H., & Kinnaman, D. T. (1995). Integration as integration: Tolerance of ambiguity and the integrative process at the undergraduate level. Journal of Psychology and Theology, 23(4), 271-276. Keinen, G. (1994). Effects of stress and tolerance of ambiguity on magical thinking. Journal of Personality and Social Psychology, 67(1), 48-55. Keren, G., & Gerritsen, L. E. M. (1999). On the robustness and possible accounts of ambiguity aversion. Acta Psychologica, 103, 149-172. Kirton, M. J. (1981). A reanalysis of two scales of tolerance of ambiguity. Journal of Personality Assessment, 45, 407-414. Lane, M. S., & Klenke, K. (2004). The ambiguity tolerance interface: A modified social cognitive model for leading under uncertainty. Journal of Leadership and Organizational Studies, 10(3), 69-81. MacDonald, A. P. (1970). Revised scale for ambiguity tolerance: Reliability and validity. Psychological Reports, 26, 791-798. McLain, D. L. (1993). The mstat-i: A new measure of an individual's tolerance for ambiguity. Educational and Psychological Measurement, 53, 183-189. Norton, R. W. (1975). Measurement of ambiguity tolerance. Journal of Personality Assessment, 39(6), 607-619. Wittenburg, K. J., & Norcross, J. C. (2001). Practitioner perfectionism: Relationship to ambiguity tolerance and work satisfaction. Journal of Clinical Psychology, 57(12), 1543-1550.