Syracuse University SURFACE Dissertations - ALL SURFACE December 2015 Eliciting Bootstrapping: The Development of Introductory Statistics Students’ Informal Inferential Reasoning via Resampling Jeffrey Allen McLean Syracuse University Follow this and additional works at: https://surface.syr.edu/etd Part of the Education Commons Recommended Citation McLean, Jeffrey Allen, "Eliciting Bootstrapping: The Development of Introductory Statistics Students’ Informal Inferential Reasoning via Resampling" (2015). Dissertations - ALL. 396. https://surface.syr.edu/etd/396 This Dissertation is brought to you for free and open access by the SURFACE at SURFACE. It has been accepted for inclusion in Dissertations - ALL by an authorized administrator of SURFACE. For more information, please contact [email protected]. Abstract Bootstrapping has become an important tool for statisticians, who assert that it is intuitive to novice statistics students. The process of collecting bootstrap resamples was once time demanding, but technology now allows data collection to be performed nearly instantaneously. This study focuses on the construction and development of secondary and tertiary introductory statistics students' (n=68) reasoning about bootstrapping and informal inference. Students engaged in a four-week instructional unit designed as two model development sequences. Through the use of hands-on manipulatives and technology, students constructed and developed reasoning of sampling, the resampling process of bootstrapping, and inference. The focus of my analysis was on the model development of four focus groups of students. Groups of students constructed models of sampling and inference that they used to collect samples, aggregate the samples to form an empirical sampling distribution, and use the aspects of this distribution of samples to make claims about the population from which the samples were drawn. I summarize and categorize groups of students’ models and trace the development of the focus groups’ models throughout the unit. Simulation of data led some students to develop a global view of the randomness of sampling and reason with multiple aspects of empirical sampling distributions to draw inferential claims. Some students applied a multiplicative view of the sample and global view of the sampling process to construct a resampling process similar to bootstrapping, but fell short of constructing the bootstrapping process by not collecting resamples that were equal in size to the original sample. Class discussion of a follow-up activity, similar in structure to the model eliciting activity, encouraged students to consider the value of drawing resamples of equal size to the population and led students to construct the method of bootstrapping. Students then used the bootstrapping process to drawn inferences about one population and to compare two populations of data. The findings of this study contribute to the field of statistics education by examining student thinking while constructing and developing bootstrapping methods, as well as investigating the relationship between this thinking and the drawing of informal inferences. This study demonstrates that it is possible for model development sequences to elicit and develop students’ models of bootstrapping. With the trend in statistics education of moving from the focus on theoretical distributions towards the simulation and analysis of data, these findings have implications towards the design of future introductory statistics curricula. ELICITING BOOTSTRAPPING: THE DEVELOPMENT OF INTRODUCTORY STATISTICS STUDENTS’ INFORMAL INFERENTIAL REASONING VIA RESAMPLING By Jeffrey A. McLean B.S., Applied Mathematics, Rochester Institute of Technology, 2005 M.S., Mathematics, University of Connecticut, 2009 Dissertation Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Education December 2015 © 2015 Jeffrey A. McLean All Rights Reserved v Table of Contents Chapter 1 – Introduction…………………………………………………………………………..1 Aims of Research………………………………………………………………………….6 Repeated Sampling………………………………………………………………………..7 Resampling………………………………………………………………………………..8 Research Questions………………………………………………………………………11 Chapter 2 – Literature Review…………………………………………………………………...12 Informal Inferential Reasoning…………………………………………………………..12 Technology Use in Statistics Education………………………………………………….27 Modeling…………………………………………………………………………………30 Summary…………………………………………………………………………………33 Chapter 3 – Design and Methodology…………………………………………………………...35 Research Design………………………………………………………………………….35 Settings and Participants…………………………………………………………………36 Model Development Sequences………………………………………………………….37 Data Collection…………………………………………………………………………..48 Data Analysis…………………………………………………………………………….51 Chapter 4 – Results……..………………………………………………………………………..55 Pre/Post-Test Results…...………………………………………………………………...57 Models of Sampling and Inference……..………………………………………………..58 Models of Resampling and Inference...………………………………………………...111 Summary of Model Development..……………………………………………... ……..157 vi Chapter 5 – Discussion and Conclusions……………………………………………………….163 Reasoning Developed Moving from Repeated Sampling to Bootstrapping Methods….164 Reasoning Developed about Bootstrapping Methods……………………………….….166 Reasoning Moving from Hands-on Manipulatives to Computer Simulations………….167 Implications for Further Research………………………………………………………168 Limitations……………………………………………………………………………...170 Final Conclusions……………………………………………………………………….173 Appendix A: Pre/Post-test Instrument……………………………………………………….....175 Appendix B: Instructor Debriefing Interview Protocol………………………………………...185 Appendix C: Instructional Unit Activities……………………………………………………...186 References………………………………………………………………………………………216 VITA……………………………………………………………………………………………224 vii List of Tables and Figures Tables Table 1. Intended models in the first model development sequence ...................................... 39 Table 2. Intended models in the second model development sequence .................................. 46 Table 3. Focus group participants .......................................................................................... 51 Table 4. Summary statistics and one-way ANOVA analysis of the pre-test scores between the two settings ............................................................................................................... 57 Table 5. Summary statistics and paired t-test for increase in pre/post-test scores .................. 58 Table 6. Models of resampling ...............................................................................................114 Figures Figure 1. Visual representation of the process of repeated sampling........................................ 8 Figure 2. Visual representation of the process of resampling using bootstrapping methods. 10 Figure 3. Concept of distribution ............................................................................................ 15 Figure 4. Overview of the activities in the two model development sequences. .................... 38 Figure 5. TinkerPlots environment for the Ophelia MXA. ..................................................... 41 Figure 6. Bin of multi-colored beads for the Nike sneakers MXA and MAAs. ..................... 43 Figure 7. TinkerPlots environment for the Nike sneakers MXA. ........................................... 44 Figure 8. Craft sticks used for the Mixed Nuts MEA. ............................................................ 47 Figure 9. TinkerPlots environment for the Mixed Nuts MEA. ............................................... 48 Figure 10. Visual representation of focus group D’s collected samples for Ophelia’s predictions ............................................................................................................. 63 Figure 11. Non-focus groups’ table for the probabilities of guessing the numbers of correct predictions ............................................................................................................. 64 viii Figure 12. Non-focus group’s tree diagram for Ophelia’s eight predictions .......................... 66 Figure 13. Megan’s solution to Paul the Octopus Controversy. ............................................. 68 Figure 14. Focus group A’s dotplots for each student’s 10 sets of Ophelia’s predictions ....... 71 Figure 15. Henry’s dotplot for 50 trials of Ophelia’s predictions (focus group B) ................. 74 Figure 16. George’s dotplot for 50 trials of Ophelia’s predictions (focus group B) ............... 75 Figure 17. Non-focus group student’s table for the number of samples collected for numbers of correct predictions. ........................................................................................... 78 Figure 18. Focus group D’s dotplot of Nike sneaker purchases ............................................. 80 Figure 19. Focus group C’s dotplot of Nike sneaker purchases ............................................. 83 Figure 20. Aggregate data of 105 samples of 20 sneaker purchases. ..................................... 82 Figure 21. Focus group A’s dotplot of Nike sneaker purchases .............................................. 85 Figure 22. Henry’s (focus group B) dotplot for the number of Nike sneaker purchases in samples of 20 after collecting 120 samples. ......................................................... 91 Figure 23. Lena’s (focus group C) sketch for the number of Nike sneaker purchases for 1000 samples .................................................................................................................