Matthew Ciancetta - Statistics Students’ Reasoning When Comparing Distributions of Data STATISTICS STUDENTS REASONING WHEN COMPARING DISTRIBUTIONS OF DATA by MATTHEW ALAN CIANCETTA A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS EDUCATION Portland State University ©2007 TABLE OF CONTENTS: List of Tables .iv List of Figures. Global Views of Data. 11 Data set as a Distribution. 12 Comparing Data Sets.
17 Chapter 2: Literature Review and Framework. 21 Intuitive strategies when predicting and making informal inferences when comparing data sets. 22 Acknowledgment, understanding and reasoning about variation when comparing data sets. 41 Reasoning About Distributions.
58 Literature Review Discussion. 75 Initial Framework by Shaughnessy and Colleagues. 77 Lattice Structure Framework By Shaughnessy and Colleagues. 79 Framework by Watson and Moritz.
81 Framework by Bakker and Gravemeijer. 82 Expanded Lattice Structure Framework. 93 Subjects and Data Collection. 99 Task Development: Data Set Comparison Survey .110 Task Development: Interview Protocol .119 i Chapter 4: Framework Refinement, Results and Analysis .121 Refinement of the Expanded Lattice Structure Framework .144 Cross Task Numeric Codes.147 Framework Refinement Summary .154 Survey Results by Group.155 Survey Results: Task 1, the Yellow/Brown task .157 Survey Results: Task 2, the Movie Wait-Time task .165 Survey Results: Task 3, Pink/Black survey task.172 Survey Results: The Pink/Black task – Without descriptive statistics (Task 3) vs.
With descriptive statistics (Task 4) .179 Survey response summary: Pink/Black tasks without and with descriptive statistics .204 Survey Results: Task 5, Ambulance task .205 Survey Results: The Ambulance task – Without descriptive statistics (Task 5) vs. With descriptive statistics (Task 6) .212 Survey response summary: Ambulance tasks without and with descriptive statistics .231 Survey Responses: Cross Task Numeric Codes .237 Analysis of Interviews.237 Background of the six interviewees………………………………………….237 Cross case analysis of interviewees.240 The interviewees’ understandings of statistical terms.241 Responses to task 1: the Yellow/Brown task.251 Responses to Survey task 2: the Movie-Wait-Time task.258 Responses to task 3 and task 4: the Pink/Black task – Without descriptive statistics and With descriptive statistics.266 Responses to task 5 and task 6: the Ambulance task – Without descriptive statistics and With descriptive statistics.296 Summary of Cross Task Numeric Code assignment.323 Chapter 5: Discussion and Conclusion.328 Research Goal: Expand and refine the interpretive framework.334 ii Research Question 2.337 Limitations of the research.342 Implication for future research and teaching.349 Appendix A: Informed Consent forms.358 Appendix B: Text version of survey tasks.361 Appendix C: Detailed survey results .368 iii LIST OF TABLES: Table Page 1. Response strategies for two ‘comparison of data sets’ interviews, by Watson and Moritz. Course enrollment of participants.
Participants’ Major Fields of Study. Statistics backgrounds of the participants. Educational Level breakdown: Counts of each group. Pilot Study results.
Examples of Idiosyncratic type responses. Distribution of Idiosyncratic responses across survey tasks. Examples of Local type responses. The distribution of Local responses across the survey tasks.
Distribution of Transitional responses across the survey tasks. Examples of Transitional-shape type responses. Examples of Transitional-center type responses. Examples of Transitional-variation type responses.
Distribution of Initial-Distributional responses across the survey tasks. Examples of Initial-Distributional: proportional type responses. Examples of Initial-Distributional: initial-global type responses. Examples of Distributional type responses.
Distribution of Distributional responses across the survey tasks. Inter-rater reliabilities for coding the survey tasks. Distribution of Cross Task Numeric Lattice Codes. Statistics backgrounds of the participants.
The distribution of responses coded across framework levels from task 1 (the Yellow/Brown task), for all groups. The distribution of responses coded across framework levels from task 2 (the Movie Wait-Time task), for all groups. The distribution of framework level codes, for responses from task 3 (the Pink/Black task), for all groups. Decisions shifts by group for both Pink/Black tasks.
Distribution of responses from the 1-GS group for the Pink/Black task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the 1-GS group, for the Pink/Black task: Without statistics and With statistics. Distribution of responses from the 1-SE group for the Pink/Black task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the 1-SE group, for the Pink/Black task: Without statistics and With statistics.
Distribution of responses from the 2-GS group for the Pink/Black task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the 2-GS group, for the Pink/Black task: Without statistics and With statistics. Distribution of responses from the 2-SE group for the Pink/Black task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the 2-SE group, for the Pink/Black task: Without statistics and With statistics.
Distribution of responses from the GRAD group for the Pink/Black task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the GRAD group, for the Pink/Black task: Without statistics and With statistics. The distribution of framework level codes, for responses from task 5 (the Ambulance task), for all groups. Decisions by group for the Ambulance tasks: Counts for Without statistics vs.
Distribution of responses from the 1-GS group for the Ambulance task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the 1-GS group, for the Ambulance task: Without statistics and With statistics. Distribution of responses from the 1-SE group for the Ambulance task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the 1-SE group, for the Ambulance task: Without statistics and With statistics.
Distribution of responses from the 2-GS group for the Ambulance task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the 2-GS group, for the Ambulance task: Without statistics and With statistics. Distribution of responses from the 2-SE group for the Ambulance task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the 2-SE group, for the Ambulance task: Without statistics and With statistics.
Distribution of responses from the GRAD group for the Ambulance task: Without statistics vs. Distribution of Level 2 and Level 3 responses, from the GRAD group, for the Ambulance task: Without statistics and With statistics. Overall reasoning levels across groups. Cross Task Numeric Codes.
Background information of interviewees. Interviewees’ decisions and response levels for task 1: the Yellow/Brown task. Interviewees’ decisions and response levels for task 2: the Movie Wait-Time task. Interviewees’ decisions, estimates and response levels for tasks 3 and 4: the Pink/Black task (without statistics) and the Pink/Black task with statistics.
Interviewees’ decisions and response levels for tasks 5 and 6: the Ambulance task (without statistics), and the Ambulance task with statistics. Interviewees’ cross task numeric framework levels. The distribution of responses from task 1 (the Yellow/Brown task), coded across framework levels, for group 1-GS. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 1, for group 1-GS.
The distribution of responses coded across framework levels from task 1 (the Yellow/Brown task), for group 1-SE. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 1, for group 1-SE. The distribution of responses coded across framework levels from task 1 (the Yellow/Brown task), for group 2-GS. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 1, for group 2-GS.
The distribution of responses coded across framework levels from task 1 (the Yellow/Brown task), for group 2-SE. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 1, for group 2-SE. The distribution of responses coded across framework levels from task 1 (the Yellow/Brown task), for group GRAD. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 1, for group GRAD.
The distribution of responses coded across framework levels from task 2 (the Movie Wait-Time task), for group 1-GS. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 2, for group 1-GS. The distribution of responses coded across framework levels from task 2 (the Movie Wait-Time task), for group 1-SE. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 2, for group 1-SE.
The distribution of responses coded across framework levels from task 2 (the Movie Wait-Time task), for group 2-GS. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 2, for group 2-GS. The distribution of responses coded across framework levels from task 2 (the Movie Wait-Time task), for group 2-SE. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 2, for group 2-SE.
The distribution of responses coded across framework levels from task 2 (the Movie Wait-Time task), for group GRAD. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 2, for group GRAD. The distribution of responses coded across framework levels from task 3 (the Pink/Black task), for group 1-GS. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 3, for group 1-GS.
The distribution of responses coded across framework levels from task 3 (the Pink/Black task), for group 1-SE. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 3, for group 1-SE. The distribution of responses coded across framework levels from task 3 (the Pink/Black task), for group 2-GS. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 3, for group 2-GS.
The distribution of responses coded across framework levels from task 3 (the Pink/Black task), for group 2-SE. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 3, for group 2-SE. The distribution of responses coded across framework levels from task 3 (the Pink/Black task), for group GRAD. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 3, for group GRAD.
The distribution of responses coded across framework levels from task 5 (the Ambulance task), for group 1-GS. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 5, for group 1-GS. The distribution of responses coded across framework levels from task 5 (the Ambulance task), for group 1-SE. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 5, for group 1-SE.
The distribution of responses coded across framework levels from task 5 (the Ambulance task), for group 2-GS. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 5, for group 2-GS. The distribution of responses coded across framework levels from task 5 (the Ambulance task), for group 2-SE. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 5, for group 2-SE.
The distribution of responses coded across framework levels from task 5 (the Ambulance task), for group GRAD. The distribution of responses coded at level 2 (transitional) and level 3 (initial distributional) from survey task 5, for group GRAD.424 ix LIST OF FIGURES: Figure Page 1. Two of Gal, Rothschild, and Wagner’s data set comparison tasks. Stem and Leaf Plot of Heights of Students and Basketball Players.
Graphs used in two of the four tasks from Watson and Mortiz’s interview protocol. Item 1: Comparing two samples of measurement data. Item 2: Comparing two samples of measurement data. Hypothetical data generated by students.
Movie Wait-time Task. Meletiou and Lee’s example histograms. Lann and Falk’s data set comparison task. Points Per Game Scored by Two Basketball Players.