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Algebraic Functions, Computer Programming, and the Challenge of Transfer Algebraic Functions, Computer Programming, and the Challenge of Transfer The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Schanzer, Emmanuel Tanenbaum. 2015. Algebraic Functions, Computer Programming, and the Challenge of Transfer. Doctoral dissertation, Harvard Graduate School of Education. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:16461037 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA ! ! ! Algebraic Functions, Computer Programming, and the Challenge of Transfer Emmanuel Tanenbaum Schanzer Jon Star Karen Brennan Kathi Fisler A Thesis Presented to the Faculty of the Graduate School of Education of Harvard University in Partial Fulfillment of the Requirements for the Degree of Doctor of Education 2015 © 2015 Emmanuel Schanzer All Rights Reserved For Lakshyan and Shelley. For Toggle and Fritz. For Nori. Acknowledgements This work would not have been possible without the tireless support of Shriram Krishnamurthi, Kathi Fisler, and Matthias Felleisen. Thank you for believing in me all these years, and for always making sure I was thinking three steps ahead. Danny Yoo, thank you for pouring so much of your life into WeScheme. Special thanks go to Emma Youndtsmith and Rosanna Sobota, who took a chance after college to join me on this journey. It is a privilege to work with all of you. Many have contributed to the work that went into this thesis, and I want to acknowledge the efforts of the teachers and students in this study, as well as the hundreds who I have learned from over the years. I especially want to thank Mrs. Bidwell, Wrenn Goodrum, Mrs. Maybray, Herbert Woodel and Gregory Morrisett – the great teachers in my life, who taught me to see the world through the exacting, playful, and piercing eyes of an educator. I am grateful to David Perkins, Chris Dede, and Kurt Fischer each served on various committees along the way. You have all shaped so much of my thinking in this dissertation, and pointed me in the right direction. Above all else, I want to thank Jon Star. Thank you for keeping my feet on the ground, for poking holes in my thoughts until I truly understood them, and for your infinite patience with me. Table of Contents ACKNOWLEDGEMENTS........................................................................................................................................II! TABLE.OF.CONTENTS.........................................................................................................................................III! ABSTRACT................................................................................................................................................................V! CHAPTER.1:.INTRODUCTION.............................................................................................................................1! CHAPTER.2:.LITERATURE.REVIEW..................................................................................................................5! TRANSFER. 5! THE.ALGEBRAIC.DOMAIN. 21! RICH.TASKS.FOR.ALGEBRA. 26! THE.PROMISE.OF.TECHNOLOGY. 32! CHAPTER.3:.THE.BOOTSTRAP.CURRICULUM.............................................................................................55! THE.SHOULDERS.OF.GIANTS. 56! INTRODUCING.BOOTSTRAP. 59! CHAPTER.4:.CONCEPTUAL.FRAMEWORK....................................................................................................80! PROCEDURAL.KNOWLEDGE. 81! CONCEPTUAL.KNOWLEDGE. 81! ATTITUDINAL.FACTORS. 84! RESEARCH.QUESTIONS. 85! CHAPTER.5:.RESEARCH.METHODS................................................................................................................86! DESIGN. 86! DATA.SET. 86! MEASURES. 88! ANALYSIS. 91! CHAPTER.6:.RESULTS.........................................................................................................................................94! FIDELITY. 94! TEACHER.INTERVIEWS. 97! PREE.AND.POSTETESTS. 134! CHAPTER.7:.DISCUSSION................................................................................................................................143! RESEARCH.QUESTIONS. 144! LIMITATIONS. 156! IMPLICATIONS. 157! CONCLUSION. 166! APPENDICES.......................................................................................................................................................168! APPENDIX.A:.OBSERVATION.PROTOCOL. 168! APPENDIX.B:.EXAMPLES.OF.FORMATIVE.PROGRAMMING.ASSESSMENTS. 169! APPENDIX.C:.TRANSFER.OUTCOMES.(PRETEST). 170! APPENDIX.D:.INTERVIEW.PROTOCOL. 175! REFERENCES.......................................................................................................................................................178! VITA.......................................................................................................................................................................190! Abstract Students' struggles with algebra are well documented. Prior to the introduction of functions, mathematics is typically focused on applying a set of arithmetic operations to compute an answer. The introduction of functions, however, marks the point at which mathematics begins to focus on building up abstractions as a way to solve complex problems. A common refrain about word problems is that “the equations are easy to solve - the hard part is setting them up!” A student of algebra is asked to identify functional relationships in the world around them - to set up the equations that describe a system- and to reason about these relationships. Functions, in essence, mark the shift from computing answers to solving problems. Researchers have called for this shift to accompany a change in pedagogy, and have looked to computer programming and game design as a means to combine mathematical rigor with creative inquiry. Many studies have explored the impact of teaching students to program, with the goal of having them transfer what they have learned back into traditional mathematics. While some of these studies have shown positive outcomes for concepts like geometry and fractions, transfer between programming and algebra has remained elusive. The literature identifies a number of conditions that must be met to facilitate transfer, including careful attention to content, software, and pedagogy. This dissertation is a feasibility study of Bootstrap, a curricular intervention based on best practices from the transfer and math-education literature. Bootstrap teaches students to build a video game by applying algebraic concepts and a problem solving technique in the programming domain, with the goal of transferring what they learn back into traditional algebra tasks. The study employed a mixed-methods analysis of six Bootstrap classes taught by math and computer science teachers, pairing pre- and post-tests with classroom observations and teacher interviews. Despite the use of a CS-derived problem solving technique, a programming language and a series of programming challenges, students were able to transfer what they learned into traditional algebra tasks and math teachers were found to be more successful at facilitating this transfer than their CS counterparts. Chapter 1 Introduction Many disciplines rely on functions as a foundational concept for thinking about abstraction: a physicist uses functions to describe the movement of a projectile, a chemist to model reactions, and a biologist to reason about population growth. As a gateway concept for so many disciplines, it is no surprise that the course where students are first introduced to functions – algebra – is a key component of standardized tests across the United States. Algebra is a graduation requirement for high schools in many school districts (Loveless, 2008), and studies have found that student performance in algebra is the greatest predictor of future income out of all high school classes (Rose & Betts, 2004). For years, researchers have pointed to functions as an essential concept in math education (Breslich, 1928; Schaaf, 1930), and modern reforms have called functions a “unifying idea in mathematics” (National Council of Teachers of Mathematics, 1989, p. 154). If algebra is thought of as a gateway class, it is a gate students often struggle to pass through. Our collective fear of algebra is well known, with the phrase “a train leaves Chicago, traveling east at 60mph…” teasing uneasy laughter from most Americans. A recent study (Fong, Jaquet, & Finkelstein, 2014) of California students found that 44% repeated Algebra 1, and that percentage is even higher for traditionally underserved groups. With Algebra 1 being a graduation requirement, the students who finish high school are by definition more likely to be higher performing math students than their peers. Even among this group, however, a 2012 National Science Board report found that only 76% of graduating seniors even completed Algebra 2, despite graduation requirements in 32 states that mandate a third year of mathematics. Many people see math and computer programming as closely related, and researchers have long viewed programming as a promising domain in which to learn mathematical concepts (Feurzeig, 1969; Khan, 1996; Papert, 1972; Resnick, 2009). The approach taken by many of these studies relies on the achievement of transfer: a student’s understanding of a particular
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