Students’ Ambiguity Tolerance as a Success Factor in Learning to Reason Statistically

Students’ Ambiguity Tolerance as a Success Factor in Learning to Reason Statistically

Robert H. CarverStonehill College/Brandeis UniversityJune 12, 2007

Quick Outline

Genesis of this Research Classroom experience Literature review JSM 2006 presentation

Current projectInvitation to participateQ&A

Genesis of the ResearchSome 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

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

CAOS post-test

807060504030

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80

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60

50

40

30

CAOSPre

CA

OSPost

MaleFemale

Gender

Post vs. Pre-test Scores

CAOS post-test

807060504030

90

80

70

60

50

40

30

CAOSPre

CA

OSPost

MaleFemale

Gender

Post vs. Pre-test Scores

Improvem

ent

Measuring AT

Independent Measures & variables: Abiguity Tolerance:

McLain’s 22 question instrument 7-point Likert Scales

Max score for extreme tolerance = 74Min 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% 1st year) Math SAT Prior Stat Education (37% had some) Course cumulative average Attendance

Findings: CAOS Pre-test Variable Coeff Signif

Constant 9.07 0.438

Female dummy -1.13 0.638

AT scale 0.048 0.537

First year dummy -5.581 0.028

Prior course dummy 5.256 0.032

Math SAT score 0.063 0.001

F 4.89 0.001

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 0.559 0.000

AT scale 0.110 0.079

First Year dummy -3.726 0.072

Prior course dummy -3.406 0.099

F 12.29 0.000

Adj R2 37.0%

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 0.437 0.000

AT scale 0.117 0.039

Course Cumulative Avg 0.473 0.000

Prior course dummy -3.946 0.035

F 19.46 0.000

Adj R2 48.9%

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! [email protected] [email protected]

http://faculty.stonehill.edu/rcarver/

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