AN INQUIRY INTO HIGH SCHOOL STUDENTS' UNDERSTANDING OF LOGARITHMS by Tetyana Berezovski M.Sc, Lviv State University, Ukraine, 199 1 - A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE FACULTY OF EDUCATION OTetyana Berezovski 2004 SIMON FRASER UNIVERSITY Fall 2004 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without permission of the author. APPROVAL NAME Tetyana (Tanya) Berezovski DEGREE Master of Science TITLE An Inquiry into High School Students' Understanding of Logarithms EXAMINING COMMITTEE: Chair Peter Liljedahl _____-________--___ -- Rina Zazkis, Professor Senior Supervisor -l_-______l___l____~-----__ Stephen Campbell, Assistant Professor Member -__-_-___- _-_-- -- Malgorzata Dubiel, Senior Lecturer, Department of Mathematics Examiner Date November 18, 2004 SIMON FRASER UNIVERSITY PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection. The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without the author's written permission. Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright Licence. The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive. W. A. C. Bennett Library Simon Fraser University Burnaby, BC, Canada Simon Fraser University Ethics Approval The author, whose name appears on the title page of this work, has obtained human research ethics approval from the Simon Fraser University Office of Research Ethics for the research described in this work, or has conducted the research as a member of a project or course approved by the Ethics Office. A copy of the approval letter has been filed at the Theses Office of the University Library at the time of submission of this thesis or project. The original application for ethics approval and letter of approval is filed with the Office of Research Ethics. Inquiries may be directed to that Office. Bennett Library Simon Fraser University Burnaby, BC, Canada ABSTRACT The mathematical concept of logarithms plays a crucial role in many aspects of human existence. This study aimed to analyze and describe the issues involved in high school students' understanding of logarithms and to highlight the most common difficulties students face as they develop their understanding. Two general theoretical ideas guided this investigation: mathematical understanding and obstacles. The adapted version of Confrey's model for students' understanding of exponents was applied to investigate students' understanding of logarithms. As a result, a description of students' difficulties with logarithms, and suggestions of possible explanations of the sources of these difficulties were presented. As for teaching practice, I focused on the initial introduction of the logarithms. In the traditional curriculum, logarithms are introduced as exponents. However, historically logarithms were developed completely independently of exponents. Further research will investigate the feasibility and the benefits of the historical approach for teaching. DEDICATION This work is dedicated to my husband, my children, and my friend Lynne: My beloved husband has sacrificed the pleasures of his life for me, so I could work on this research. My loving children havegiven me strength to be theperson that I am. My devotedfried Lynne has inspired me to continue this journey of my life. ACKNOWLEDGEMENTS I would like to thank Dr. Rina Zazkis for inspiring me to complete this thesis. Her knowledge, guidance, unlimited encouragement and positive feedback directed me in this research, and opened a whole path for future study. I would also like to thank Dr. Stephen Campbell, who despite his business and family complications, found time to employ his expertise in the field and offered many helpful suggestions. Gratitude is extended to my other instructors at Simon Fraser University, including Dr. Tom O'Shea, Dr. Len Berggren, Dr. Peter Bonvein, and Dr. Peter Liljedahl. Finally I want to thank my Faculty Associate Norm Ellis for his unconditional support in my perseverance to be a mathematics educator. His determination in directing me back to university will never be forgotten. TABLE OF CONTENTS .. APPROVAL ...11 ABSRACT 111 DEDICATION iv ACKNOWLEDGEMENTS v - TABLE OF CONTENTS .vi. LIST OF TABLES Vlll... LIST OF FIGURES Vlll CHAPTER I: IDENTIFYING THE PROBLEM AND CQNSIDERING A SOLUTION Rationale Personal Motivation About this Study: Beginning my Journey Purpose of the Study Thesis Organization CHAPTER 11: LOGARITHMS IN THE HISTORY OF MATHEMATICS AND IN THE CURRICULUM The Historical Development of Logarithms Genesis of the Napierian Logarithm Curricular Outcomes Importance of Logarithms Conclusion CHAPTER 111: THEORETICAL CONSIDERATIONS Introduction 18 Skemp's Theory of Understanding 18 Sierpinska's Theory of Understanding 2 1 Hiebert and Carpenter's Theory of Understanding 22 Obstacles to Understanding 24 Confrey's Model for Investigation of Student's Understanding of Exponential Functions 26 Conclusion 30 CHAPTER IV: METHODOLOGY Participants and Setting The General Structure of the Unit CHAPTER V: ANALYSIS OF DATA Tasks and Frameworks System of the Interpretive Frameworks for Logarithms Results and Analysis CHAPTER VI: DISCUSSION AND PEDAGOGICAL IMPLICATIONS Discussion of Obstacles Significance of the Study Synopsis of the Pedagogical Considerations Limitations of the Study Suggestions for Future Research Reflections on my Personal Journey Summary APPENDICES A. Logarithms Quiz #1 B. Logarithms Unit Test C. Activity: Multiplying on the Slide Rule REFERENCES vii LIST OF TABLES Table 1 - Table of Values of Powers of 2. 37 Table 2 - Exponential verses Logarithmic Forms 38 Table 3 - Quantitative Summary of Task 1 49 Table 4 - Quantitative Summary of Task 2 5 2 Table 5 - Quantitative Summary of Task 3 56 Table 6 - Table of Strategies Students Used in Their Essays-Explanations 64 LIST OF FIGURES Figure la. Participant's Response IIC. Figure lb. Participant's Response 1IC. Figure 2. Participant's Response 2IC. Figure 3. Participant's Response 1C. Figure 4. Participant's Response 2C. Figure 5. Participant's Response 3C. Figure 6. Essay #l. Figure 7. Essay #2. Figure 8. Essay #3. Figure 9. Essay #4. Figure 10. Essay #5. ... Vlll CHAPTER I: IDENTIFYING THE PROBLEM AND CONSIDERING A SOLUTION Rationale An increasing concern in recent years with regards to the degree of students' learning in mathematics classes, has led to in-depth investigations of specific subject- matters. Mainly, these investigations have covered the three following areas: (1) analysis of student errors, (2) analysis of mathematical and semantic structures, and (3) synthesis of how people construct an understanding of mathematical concepts (Hare1 & Confrey, 1994). Although an understanding of student errors and mathematical structures is very important, the learning entailed in how to teach more effectively depends on an understanding of how students can come to know mathematical concepts; or, more specifically, are able "to specify the operations involved in constructing a mathematical realityu (Steffe, 1988, p. 1 19). Much work in mathematics education and in the research in mathematics education has been done on students' understanding of the fundamental operations of addition and multiplication. However, very limited attention was given to an investigation of students' understanding of the relationship between the two. At the elementary level, the conceptual understanding of the distributive property which relates multiplication and addition was based upon the conceptual understanding of the relationship between additive and multiplicative structures (Campbell & Zazkis, 1994). A detailed analysis was undertaken of the difficulties in general, and the obstacles in particular, that students have to deal with when learning the distributive property. Also, the development of the pre-service elementary school teachers' understanding of the arithmetic sequences through building connections between additive and multiplicative structures was the topic of the research conducted by Zazkis & Liljedahl(2002). At the secondary level, a conceptual understanding of exponential functions was based upon the conceptual understanding of exponents as repeated products of the same multipliers/bases (Confrey, 1994). Viewing multiplication as repeated addition, different mathematical concepts can be introduced in the school curriculum: the concept of ratio and proportion at the