Teaching and Learning Complex Analysis at Two British Columbia Universities by Peter Danenhower B. Sc., Simon Fraser University, 1977 M. Sc., Simon Fraser University, 1989 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUlREMENTS FOR THE DEGREE OF DOCTOR OF PKiLOSOPHY in the Department of MATHEMATICS AND STATISTICS O Peter Danenhowet 2000 SIMON FRASER UNIVERSm Decemlxr2000 Al1 ri@ rrscrvd This work mry not bc Rproduccd in whole or in part, by photocopy or otbtr m#as, witbout pemtiuion of the author. The author has granted a non- L'auteur a accordé une liace non exciusive licence aüowing the exclusive permettant à la National Li'brary of Canada to Biblioihbcpe nationale du Canada de reproduce, loan, disûi'bute or sel repdm,prêter, dissitmer ou copies of this thesis in microform, vendre des copies de cette thèse sous papa or electronic formats. la forme de microfiche/film, & reproduction sur papier ou sur format électronique. The author retains ownershtp of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protès cette thése. thesis nor substantial extracts from it Ni la thése ni des extraits substantiels may be printed or othenivise de ceiie-ci ne daivmt êûe imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract The objective of the present study was to investigate teaching and leamhg in typical university level complex analysis classes in British Columbia. Thm classes wete studied at two universities. Presentations of material in class by inçtnictots was carefully recorded. A total of 20 subjects were studied during 54 audiotaped interviews of one hour duration. Mnginterviews sîuâents worked on 6 ta 10 questions, recordhg their work on wotksheets. Transctipts wett made of the interviews and these together with the worksheets were analyzed to study sîudent understanding of the many concepts wvered in a beginning cornplex anaiysis cowse. In this thesis we have confuted our analysis to the data wc collected on just one theme: multirepresentations of complex numbers. We fond that students have many misconceptions and difficulties with the basic representations of complex numbers such as, z = x + iy, z = (x, y), z = d,and the symbolic representation, in which z is useâ dimtly. In addition, we studied how well students were able to judge when to shift fiorn one representation to another. Finally, we have examined the data we collected that shows how insufficient understanding of basic material affécts student ability to do problems hmmaterial covered later in the couse. Where possible we have attempted to identify different levels of understanding of the various representatiow. Although we found very little Mathematical Education literature on complex analysis, our analysis of our data supports results reported in the literature on representations of kations and more generai representations. We found that sudents were competent with the z = x + iy and z = re' representations, and most made reasonable shifhg decisions between these two forms, but there is little evidence that students understooâ the symbolic representation of cornplex numbers. in any case, fiom our data we have identified four characteristics of understanding a given reptesentation that are consistent with results reportecl in the literature. We have also studied how well dentswere able to understand the z = (x, y) representation simply as a mathematicai form with certain des. Four stages of understanding of this question were identified. ACKNOWLEDGMENTS 1 am very gratefiil to my supervisor, Dr. Harvey Gerber, for bis wisdom, patience, encouragement, and tireless help. In addition, 1 could not have compleied this thesis without the support of my wife, Laurie Webster. 1 especiaily wish to tbank Dr. Rina Zazkis for her many invaiuable suggestions, encouragement, and enthusiasm about this pmject. Finally, 1 am greatly indebted to the twmty five odd students and three instructors who enthusiastically agreeâ to submit their thinking to public scrutiny and analysis. Unfortunately, 1 cannot name these people for reasons of confidentiality. However, they know who they are, and they have my thanks. TABLE OF CONTENTS Title . i.. Appmval . 11 S.. Absûact 111 Acknowledgements . iv Table of Contents . v List of Tables, Figures and Boxes . x 1 Nature and Goah of this Study . 1 1.1 introduction . 1 1.2 Researc h Rationale 1 1.3 Research Objectives . 3 II Theoretical Framework . 6 2.1 Research Mode1 6 2.1.1 Teaching Mode1 . 6 2.1.2 Learning Mode1 7 2.1.2.1.1 The Representationai Mode1 . 8 2.1.2.1.2 Criticism of the Representationai Model 10 Theoretical Objections 1O Anthropological Objections . 11 Pedagogicai Objections . 12 Philosophical Objections . 13 2.1 Ll.3 Summary of the Representationai Model 14 2.1.2.2 Constmctivism . 15 2.1.2.2.1 Constructivism - The Basic Mode1 . 16 2.12.2.2 Criticisms of Constructivism . 19 2.1 U.3 Comtnictivism - Summary . 21 2.1.2.3 Radical Coastnictivism . 22 2.1 .2.3.l Radical Coastnictivism - The Basic Theory 23 A Brief History of the Development of Radical Constnictivism Radical Constnictivism Radical ComtructivistsModeis of Leaming . 2.1.2.3.2 Radical Constnictivism - Criticism . 2.1.2.3.3 Radicai Constructivism - Summary . 2.1.2.4 Confrey's Modification of Radical Consûuctivism 2.1 .î.S Reification and APOS . 2.1.3 Made1 of Mathematicai Knowledge . 2.2 Expectations before Entering the Study . 39 2.3 Summary . 40 III Literaîure Review . 4 1 3.1 Research in Mathematics Education on Complex Analysis . 4 1 3.2 Related Research 43 3.3 Conclusion 5 1 IV Eistory and Epistemology of the Complex Numbers . 52 4.1 Introduction . 52 4.2 Epistemology of the Complex Numbers and their Representations 4.2.1 The Algebraic Extension Representation: z = x + iy . 4.2.1.1 The Algebraic Extension Representation - Basics . 4.2.1.2 The Algebraic Extension Representation - Meaning . 4.2.1 3 The Aigebraic Extension Representation - Understanding . 4.2.1.4 The Algebraic Extension Representation - Summary . 4.2.2 The Cartesian Vector Representation: z = (x, y) . 4.2.2.1 The Cartesian Vector Representation - Basics . 4.2.2.2 The Cartesian Vector Representation - Meaning . 4.2.2.3 The Cartesian Vector Representation - Understanding . 4.2.2.4 The Cartesian Vector Representation - Summary . 4.2.3 The Polar Representation of Complex Numbers: z = rea . 4.2.3.1 The Polar Representation of Cornplcx Numbcrs - Basics . 4.2.3.2 The Polar Repre~entationof Cornplex Numbcrs - Meaning - 4.2.3.3 The Polar Repeesentation of Complex Numbers - Understanding .. 4.2.3.3 The Polar Repmtation of Complex Numbcrs - summary. 4.2.4 The Symbolic Representation of Complex Numbers . 4.2.4.1 The Symhlic Reprcsentation of Complex Nurnbers - Basics . 4.2.4.2 The Symhlic Representation of Complex Numbers - Meaning .. 4.2.4.3 The Symbolic Representation of Complex Numbff - Understanding. 4.2.4.4 The Symbolic Representation of Complex Numben - Summary . 4.2.5 Geometry, Order and Direction in the Comptex Plane . 4.2.5.1 The Geometry of the Cornpiex Plane . 4.2.5.2 There is No Order (that extends <) in the Cornplex Plane . 4.2.5.3 There is No Direction in the Complex Plane .. 4.2.5.4 The ModuIus . 4.2.5.5 Georneîry - Summary . 4.2.6 Summary of the Epistemology of the Complex Numbers 4.3 The History of the Complex Numbers . 4.3.1 The First Period - Antiquity to 1650 . 4.3.2 The Second Period - 1650 - 1800. 4.3.3 The Third period - The Nineteenth Century . 4.3.4 The History of Complex Numbers - Summary and Conclusion . V Methodology of Data Collection and Amlysis . 5.1 Method of Data CoIIection 5.1.1 Data Collection: Pilot hject - Class 1 5.1.2 Data Coliection: Main Study - Class 2 5.1.3 Data Coilection: Main Study - Class 3. 5.2 Method of Data Analpis . 6.1 Introduction . 6.2 General Considerations . 6.3 Shifting Representations 6.3.1 Interview Questions and Results . 6.3.2 Analysis of Data on Shifting Representations . 6.3.3 Summary of Shifting Representations . 6.4 i is a Unit Vector . 6.4.1 Interview Questions and Results . 6.4.2 Adysis of Question 6 (Class 3, Interview # 1) . 6.4.3 Analysis of Question 2 (Class 3, interview #2) . 6.4.4 Summary of i is an Unit Vector . 6.5 Basic Facts and Calculations . 6.5.1 Cartesian Calculations . 6.5.1.1 Cartesian Calculations: x and y are real . 6.5.1.1.1 interview Questions and Resulîs . 6.5.1.1.2 Analysis of Cartesian Calculations: x and y real 6.5.1.1.3 Conclusion of Cartesian Calculations: x and y real 6.5.1.2 Cartesian Calculations: Cancellation . 6.5.1.2.1 Interview Questions and Resuits . 6.5.1.2.2 Analysis of Cariesian Calculations: Cancellation 6.5.2 Polar Calcuiations 6.5.2.1 Polar Calculations: Interview Questions and Results . 6.5.2.2 Polar Calculations: Analysis 6.5.2.3 Polar Calculations: Summary . 6.5.3 Symbolic Calculations . 6.5.3.1 Symbolic Calculations: interview Questions and Results . 6.5.3.2 Symboiic Calculations: Analysis 6.5.3.3 Symbolic CaIcdations: Conclusion . 6.5.4 Algebraic Methods 6.6 Uneven Ski11 with Basic Material Affects later Wotk . 6.6.1 Class 2, Interview #2, Question 2 6.6.2 Class 2, Interview #2, Question 5 viii 6.6.3 Class 2, interview #2, Question 8 6.6.4 Class 2, interview #4, Question 9 6.6.5 Uneven Skill with Basic Material Affects Later Work - Summary . 6.7 Conclusion of Chapter 6 Wl Summary and Conclusions . 7.1 OveMew 7.1.1 Research Objectives 7.1.2 Theoretical Framework .