DOCSLIB.ORG

Student Teachers' Knowledge and Understanding

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Student Teachers' Knowledge and Understanding View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Wits Institutional Repository on DSPACE STUDENT TEACHERS’ KNOWLEDGE AND UNDERSTANDING OF ALGEBRAIC CONCEPTS: THE CASE OF COLLEGES OF EDUCATION IN THE EASTERN CAPE AND SOUTHERN KWAZULU NATAL, SOUTH AFRICA BY CHRISTIAN MENSAH OSEI A thesis submitted in fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY IN THE SCHOOL OF SCIENCE EDUCATION (MATHEMATICS) OF THE UNIVERSITY OF THE WITWATERSRAND, JOHANNESBURG SUPERVISOR: PROF. P.E. LARIDON 0 ABSTRACT This study is aimed at investigating the knowledge and understanding of algebra amongst final year College of Education students in and around Transkei region of Eastern cape, South Africa. Triangulation methods were used to gather data for the study, which included an algebra test instrument, adapted from the CDMTA and Kaur and Sharon (1994) test instruments, interviews and classroom observations. Six Colleges of Education with a total number of 212 students constituted the sample from the Eastern Cape province and South KwaZulu Natal. Data were collected from August 1997 to July 1998. The motivation for the study was that such an exploratory investigation could contribute significantly to the understanding of some of the principal reasons underlying the poor results in the final schooling examination (the “matric”) of the teaching and learning of mathematics in rural areas of South Africa. Algebra forms a big proportion of the final matric examination in mathematics. The overall results of the study indicate that the conceptual algebraic knowledge and understanding of these College students is weak and fragile. In analysing the algebraic knowledge and understanding of students as evidenced by the data, factors such as language, the nature of mathematics, the philosophy underpinning teaching and learning and textbooks were seen to have played important roles in the conceptions and misconceptions which many of the subjects of the study portrayed. My research clearly shows that College of Education students have misconceptions, poor learning and teaching of algebraic concepts. This suggests that these prospective teachers do not have well developed concepts in algebra. The participants’ knowledge and understanding of algebraic concepts are therefore not good enough to assist learners as far as learning for conceptual understanding is concerned at schools. The results show that much of the knowledge and understanding of algebra came from previous knowledge and understanding gained during high school. Little change had happened during the years spent at the Colleges. The conceptions and misconceptions arose out of the traditional framework of knowledge acquisition rather than through approaches advocated by the newly implemented South African curriculum (Curriculum 2005), which has been revised to a National Curriculum Statement similar to what NCTM (1991) Standards might have envisaged. The lack of pedagogical content knowledge in algebra, shown by these student teachers during their teaching practice lessons reflects a deeper problem pertaining to 1 their future teaching after completion of the courses at Colleges. This is bound to have a cascading effect on teaching and learning in schools, perpetuating the cycle of misconceptions in algebra shown by this study, unless something radical is done about the teaching and learning of algebra at Colleges. The study concludes with recommendations arising out of the results and a number of suggestions for education departments, curriculum implementers, lecturers and future researchers. Strategies are suggested for improving the existing poor state of affairs in the learning and teaching of algebra at Colleges of Education and at secondary schools. These include improvement in algebraic competencies like multiple representations; understanding of basic principles such as: one cannot add unlike terms, checking solutions of equations and inequations. Real life examples should be used to give meaning to the algebraic concept they want to teach where possible. College of Education lecturers should place emphasis on conceptual understanding and correct usage of algebraic notations and symbols. Curriculum developers should include history and relevance of certain algebraic topics in order to create interest and meaning for some of the concepts. Instructional materials in school algebra should be designed specifically for local contexts. Researchers should investigate the cause of misconceptions and misunderstandings of algebraic concepts at high schools and try to address them before they are carried through to the College of Education level. DECLARATION 2 I declare that this thesis is my own unaided work. It is submitted for the degree of DOCTOR OF PHILOSOPHY at the University of the Witwatersrand, Johannesburg. It has not been submitted before for any other degree or examination in any other University. ……………………………………………………………. ……………day of………………………………………… 3 DEDICATION This thesis is dedicated to my late Father, Opayin Kwadwo Nsoah, may his soul rest in perfect peace. 4 ACKNOWLEDGEMENT I would like to thank my supervisor Professor P.E. Laridon for his assistance and support throughout this study. Without his guidance this study would never have reached completion. The study would not have been possible without the permission of the six Colleges of Education and without the cooperation of the lecturers of these institutions. I am especially grateful to the participants from these Colleges for their cooperation. Finally, I would like to thank my family especially my wife Sylvia Nomalinge Osei who has been the source of substantial support and encouragement during the study. To Mr K. Adom-Aboagye and Mr M. C. Snell who helped in proof-reading the drafts before the final presentation to University of the Witwatersrand, Johannesburg; Miss Eleanor Tembo for allowing me access to her internet facility; Mr George Kofi Doe and Mr Baffour Awua-Nsiah for helping me in the use of the computer for the wordprocessing of the study. 5 CONTENTS ABSTRACT i DECLARATION iii DEDICATION IV ACKNOWLEDGEMENTS v CONTENTS VI ABBREVIATIONS AND ACRONYMS XII LIST OF GRAPHS XIII LIST OF FIGURES XIV LIST OF TABLES XV DATA REFERENCES CODES XVI CHAPTER 1: INTRODUCING THE STUDY 1 1.1. INTRODUCTION 1 1.1.1 Historical Knowledge Mathematics 1 1.2. ALGEBRA 3 1.3 ALGEBRA IN SOUTH AFRICAN SCHOOLS 5 1.3.1 Background of OBE and Curriculum 2005 8 6 1.3.2 Reform in mathematics education 9 1.4. CONCEPT DEVELOPMENT 10 1.5. MOTIVATION FOR THE STUDY 12 1.6. RESEARCH FOCUS 20 1.7. RESEARCH QUESTIONS 20 1.8. PROBLEMS AND LIMITATIONS 20 1.9. OUTLINE OF THE STUDY 21 CHAPTER 2: LITERATURE REVIEW 23 2.1. METHOD OF TEACHING ALGEBRA 23 2.1.1. Mathematics as a body of knowledge 23 2.1.2. The traditional Method of Teaching Algebra 24 2.1.3. Learner-centred Education 26 2.1.4. Philosophies of Mathematics 28 2.1.4.1. Logicism 29 2.1.4.2. Formalism 29 2.1.4.3. Platonism 29 2.1.4.4. Fallibilism 30 2.1.4.5. Intuitionism 30 2.1.5 Constructivism 31 2.1.5.1. An Introduction to Constructivism 31 2.1.5.2. Mediation 37 7 2.1.5.3. Constructivism and Fallibilism 39 2.1.5.4. Language and Thought 41 2.1.6. The Investigative Approach 46 2.1.7. Summary 47 2.2. DIFFICULTIES IN THE LEARNING OF ALGEBRA 49 2.2.1. Research on specific difficulties 49 2.2.2. General obstacles in learning algebra 50 2.2.3. Obstacles of an Epistemological nature 51 2.2.4. The obstacles induced by instruction 51 2.2.5. Obstacles associated with learner’s accommodation process 52 2.3. PEDAGOGICAL CONTENT KNOWLEDGE 53 2.3.1. Knowledge about mathematics in culture and society 55 2.3.2. Disposition towards mathematics 55 2.4. PRESERVICE TEACHERS’ SUBJECT-MATTER KNOWLEDGE 56 2.5. MISCONCEPTIONS 57 2.6. CAUSES OF MISCONCEPTIONS AND OTHER COGNITIVE PROBLEMS 58 2.6.1. Misconceptions perpetuated by mathematics teachers 60 2.7. TEACHING 60 2.7.1. Transmission teaching 62 2.7.2. Integrated Teaching 63 8 2.8. THE MATHEMATICS CURRICULUM AND UNDERSTANDING 64 2.9. PROCEDURAL AND CONCEPTUAL UNDERSTANDING 66 2.10. CONCLUSION 67 CHAPTER 3: RESEARCH DESIGN AND PILOT 71 3.1. METHOD 71 3.1.1. Introduction 71 3.1.1.1 Triangulation 72 3.1.2. Methodology 73 3.1.2.1. Research design 73 3.1.2.2. Subjects 75 3.1.2.3. Access and acceptance to Colleges of Education, lecturers and students 76 3.1.2.4. Access to statistical information 77 3.1.2.5. Academic Background 77 3.1.3. Instrumentation 78 3.1.3.1. Rationale 78 3.1.3.2. Algebra test 78 3.1.3.2.1. CDMTA test 78 3.1.3.2.2. Kaur and Sharon test 79 3.1.3.2.3. Explanation of choice of questions from CDTMA 80 3.1.3.2.4. Explanation of choice of questions from Kaur and Sharon (1994) test 81 9 3.1.3.3. Interview 81 3.1.3.4. Observation 83 3.1.4. Procedure 83 3.1.5. Scoring 84 3.2. THE PILOT STUDY 84 3.2.1. Design of the study 84 3.2.1.1. Data collection 84 3.2.1.2. Pilot Interview 86 3.2.1.2.1. Pilot interview format 86 3.2.1.3. Classroom Observation 87 3.2.1.4. Field Notes 87 3.2.2. Analysis and Results of Data from Pilot Study 88 3.2.2.1. Algebra Test Analysis 88 3.2.2.2. Interview Analysis 92 3.2.2.3 Classroom observation 93 3.3. COLLECTION OF DATA FOR FINAL STUDY 94 3.3.1. Sample for Test 94 3.3.2. Sample for Interviews 95 3.3.3. Sample for Observation 95 3.4. VALIDITY 95 CHAPTER 4: ANALYSIS OF DATA AND PRELIMINARY DICUSSIONS 97 10 4.1.
Load more

Recommended publications
  • Algorithmic Factorization of Polynomials Over Number Fields
    Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3).
    [Show full text]
  • Some Properties of the Discriminant Matrices of a Linear Associative Algebra*
    570 R. F. RINEHART [August, SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A LINEAR ASSOCIATIVE ALGEBRA* BY R. F. RINEHART 1. Introduction. Let A be a linear associative algebra over an algebraic field. Let d, e2, • • • , en be a basis for A and let £»•/*., (hjik = l,2, • • • , n), be the constants of multiplication corre­ sponding to this basis. The first and second discriminant mat­ rices of A, relative to this basis, are defined by Ti(A) = \\h(eres[ CrsiCij i, j=l T2(A) = \\h{eres / ,J CrsiC j II i,j=l where ti(eres) and fa{erea) are the first and second traces, respec­ tively, of eres. The first forms in terms of the constants of multi­ plication arise from the isomorphism between the first and sec­ ond matrices of the elements of A and the elements themselves. The second forms result from direct calculation of the traces of R(er)R(es) and S(er)S(es), R{ei) and S(ei) denoting, respectively, the first and second matrices of ei. The last forms of the dis­ criminant matrices show that each is symmetric. E. Noetherf and C. C. MacDuffeeJ discovered some of the interesting properties of these matrices, and shed new light on the particular case of the discriminant matrix of an algebraic equation. It is the purpose of this paper to develop additional properties of these matrices, and to interpret them in some fa­ miliar instances. Let A be subjected to a transformation of basis, of matrix M, 7 J rH%j€j 1,2, *0).
    [Show full text]
  • A Historical Survey of Methods of Solving Cubic Equations Minna Burgess Connor
    University of Richmond UR Scholarship Repository Master's Theses Student Research 7-1-1956 A historical survey of methods of solving cubic equations Minna Burgess Connor Follow this and additional works at: http://scholarship.richmond.edu/masters-theses Recommended Citation Connor, Minna Burgess, "A historical survey of methods of solving cubic equations" (1956). Master's Theses. Paper 114. This Thesis is brought to you for free and open access by the Student Research at UR Scholarship Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. A HISTORICAL SURVEY OF METHODS OF SOLVING CUBIC E<~UATIONS A Thesis Presented' to the Faculty or the Department of Mathematics University of Richmond In Partial Fulfillment ot the Requirements tor the Degree Master of Science by Minna Burgess Connor August 1956 LIBRARY UNIVERStTY OF RICHMOND VIRGlNIA 23173 - . TABLE Olf CONTENTS CHAPTER PAGE OUTLINE OF HISTORY INTRODUCTION' I. THE BABYLONIANS l) II. THE GREEKS 16 III. THE HINDUS 32 IV. THE CHINESE, lAPANESE AND 31 ARABS v. THE RENAISSANCE 47 VI. THE SEVEW.l'EEl'iTH AND S6 EIGHTEENTH CENTURIES VII. THE NINETEENTH AND 70 TWENTIETH C:BNTURIES VIII• CONCLUSION, BIBLIOGRAPHY 76 AND NOTES OUTLINE OF HISTORY OF SOLUTIONS I. The Babylonians (1800 B. c.) Solutions by use ot. :tables II. The Greeks·. cs·oo ·B.c,. - )00 A~D.) Hippocrates of Chios (~440) Hippias ot Elis (•420) (the quadratrix) Archytas (~400) _ .M~naeobmus J ""375) ,{,conic section~) Archimedes (-240) {conioisections) Nicomedea (-180) (the conchoid) Diophantus ot Alexander (75) (right-angled tr~angle) Pappus (300) · III.
    [Show full text]
  • 501 Algebra Questions 2Nd Edition
    501 Algebra Questions 501 Algebra Questions 2nd Edition ® NEW YORK Copyright © 2006 LearningExpress, LLC. All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by LearningExpress, LLC, New York. Library of Congress Cataloging-in-Publication Data: 501 algebra questions.—2nd ed. p. cm. Rev. ed. of: 501 algebra questions / [William Recco]. 1st ed. © 2002. ISBN 1-57685-552-X 1. Algebra—Problems, exercises, etc. I. Recco, William. 501 algebra questions. II. LearningExpress (Organization). III. Title: Five hundred one algebra questions. IV. Title: Five hundred and one algebra questions. QA157.A15 2006 512—dc22 2006040834 Printed in the United States of America 98765432 1 Second Edition ISBN 1-57685-552-X For more information or to place an order, contact LearningExpress at: 55 Broadway 8th Floor New York, NY 10006 Or visit us at: www.learnatest.com The LearningExpress Skill Builder in Focus Writing Team is comprised of experts in test preparation, as well as educators and teachers who specialize in language arts and math. LearningExpress Skill Builder in Focus Writing Team Brigit Dermott Freelance Writer English Tutor, New York Cares New York, New York Sandy Gade Project Editor LearningExpress New York, New York Kerry McLean Project Editor Math Tutor Shirley, New York William Recco Middle School Math Teacher, Grade 8 New York Shoreham/Wading River School District Math Tutor St. James, New York Colleen Schultz Middle School Math Teacher, Grade 8 Vestal Central School District Math Tutor
    [Show full text]
  • The Evolution of Equation-Solving: Linear, Quadratic, and Cubic
    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work.
    [Show full text]
  • Interactive Mathematics Program Curriculum Framework
    Interactive Mathematics Program Curriculum Framework School: __Delaware STEM Academy________ Curricular Tool: _IMP________ Grade or Course _Year 1 (grade 9) Unit Concepts / Standards Alignment Essential Questions Assessments Big Ideas from IMP Unit One: Patterns Timeline: 6 weeks Interpret expressions that represent a quantity in terms of its Patterns emphasizes extended, open-ended Can students use variables and All assessments are context. CC.A-SSE.1 exploration and the search for patterns. algebraic expressions to listed at the end of the Important mathematics introduced or represent concrete situations, curriculum map. Understand that a function from one set (called the domain) reviewed in Patterns includes In-Out tables, generalize results, and describe to another set (called the range) assigns to each element of functions, variables, positive and negative functions? the domain exactly one element of the range. If f is a function numbers, and basic geometry concepts Can students use different and x is an element of its domain, then f(x) denotes the output related to polygons. Proof, another major representations of functions— of f corresponding to the input x. The graph of f is the graph theme, is developed as part of the larger symbolic, graphical, situational, of the equation y = f(x). CC.F-IF.1 theme of reasoning and explaining. and numerical—and Students’ ability to create and understand understanding the connections Recognize that sequences are functions, sometimes defined proofs will develop over their four years in between these representations? recursively, whose domain is a subset of the integers. For IMP; their work in this unit is an important example, the Fibonacci sequence is defined recursively by start.
    [Show full text]
  • Solving Polynomial Equations
    Seminar on Advanced Topics in Mathematics Solving Polynomial Equations 5 December 2006 Dr. Tuen Wai Ng, HKU What do we mean by solving an equation ? Example 1. Solve the equation x2 = 1. x2 = 1 x2 ¡ 1 = 0 (x ¡ 1)(x + 1) = 0 x = 1 or = ¡1 ² Need to check that in fact (1)2 = 1 and (¡1)2 = 1. Exercise. Solve the equation p p x + x ¡ a = 2 where a is a positive real number. What do we mean by solving a polynomial equation ? Meaning I: Solving polynomial equations: ¯nding numbers that make the polynomial take on the value zero when they replace the variable. ² We have discovered that x, which is something we didn't know, turns out to be 1 or ¡1. Example 2. Solve the equation x2 = 5. x2 = 5 2 p x p¡ 5 = 0 (x ¡ 5)(x + 5) = 0p p x = 5 or ¡ 5 p p ² But what is 5 ? Well, 5 is the positive real number that square to 5. ² We have "learned" that the positive solution to the equation x2 = 5 is the positive real number that square to 5 !!! ² So there is a sense of circularity in what we have done here. ² Same thing happens when we say that i is a solution of x2 = ¡1. What are \solved" when we solve these equations ? ² The equations x2 = 5 and x2 = ¡1 draw the attention to an inadequacy in a certain number system (it does not contain a solution to the equation). ² One is therefore driven to extend the number system by introducing, or `adjoining', a solution.
    [Show full text]
  • The Roots of Any Polynomial Equation
    The roots of any polynomial equation G.A.Uytdewilligen, Bergen op Zoomstraat 76, 5652 KE Eindhoven. [email protected] Abstract We provide a method for solving the roots of the general polynomial equation n n−1 a ⋅x + a ⋅x + . + a ⋅x + s 0 n n−1 1 (1) To do so, we express x as a powerseries of s, and calculate the first n-1 coefficients. We turn the polynomial equation into a differential equation that has the roots as solutions. Then we express the powerseries’ coefficients in the first n-1 coefficients. Then the variable s is set to a0. A free parameter is added to make the series convergent. © 2004 G.A.Uytdewilligen. All rights reserved. Keywords: Algebraic equation The method The method is based on [1]. Let’s take the first n-1 derivatives of (1) to s. Equate these derivatives to zero. di Then find x ( s ) in terms of x(s) for i from 1 to n-1. Now make a new differential equation dsi n 1 n 2 d − d − m1⋅ x(s) + m2⋅ x(s) + . + m ⋅x(s) + m 0 n−1 n−2 n n+1 ds ds (2) di and fill in our x ( s ) in (2). Multiply by the denominator of the expression. Now we have a dsi polynomial in x(s) of degree higher then n. Using (1) as property, we simplify this polynomial to the degree of n. Set it equal to (1) and solve m1 .. mn+1 in terms of s and a1 .. an Substituting these in (2) gives a differential equation that has the zeros of (1) among its solutions.
    [Show full text]
  • Chapter 14 Algebraic Fractions, and Equations
    CHAPTER ALGEBRAIC 14 FRACTIONS,AND EQUATIONS AND INEQUALITIES CHAPTER INVOLVING TABLE OF CONTENTS 14-1 The Meaning of an Algebraic FRACTIONS Fraction 14-2 Reducing Fractions to Lowest Terms Although people today are making greater use of decimal fractions as they work with calculators, computers, and the 14-3 Multiplying Fractions metric system, common fractions still surround us. 14-4 Dividing Fractions 1 We use common fractions in everyday measures:4 -inch 14-5 Adding or Subtracting 1 1 1 Algebraic Fractions nail,22 -yard gain in football,2 pint of cream,13 cups of flour. 1 14-6 Solving Equations with We buy 2 dozen eggs, not 0.5 dozen eggs. We describe 15 1 1 Fractional Coefficients minutes as 4 hour,not 0.25 hour.Items are sold at a third 3 off, or at a fraction of the original price. A B 14-7 Solving Inequalities with Fractional Coefficients Fractions are also used when sharing. For example, Andrea designed some beautiful Ukrainian eggs this year.She gave one- 14-8 Solving Fractional Equations fifth of the eggs to her grandparents.Then she gave one-fourth Chapter Summary of the eggs she had left to her parents. Next, she presented her Vocabulary aunt with one-third of the eggs that remained. Finally, she gave Review Exercises one-half of the eggs she had left to her brother, and she kept Cumulative Review six eggs. Can you use some problem-solving skills to discover how many Ukrainian eggs Andrea designed? In this chapter, you will learn operations with algebraic fractions and methods to solve equations and inequalities that involve fractions.
    [Show full text]
  • Analytic Solutions to Algebraic Equations
    Analytic Solutions to Algebraic Equations Mathematics Tomas Johansson LiTH{MAT{Ex{98{13 Supervisor: Bengt Ove Turesson Examiner: Bengt Ove Turesson LinkÄoping 1998-06-12 CONTENTS 1 Contents 1 Introduction 3 2 Quadratic, cubic, and quartic equations 5 2.1 History . 5 2.2 Solving third and fourth degree equations . 6 2.3 Tschirnhaus transformations . 9 3 Solvability of algebraic equations 13 3.1 History of the quintic . 13 3.2 Galois theory . 16 3.3 Solvable groups . 20 3.4 Solvability by radicals . 22 4 Elliptic functions 25 4.1 General theory . 25 4.2 The Weierstrass } function . 29 4.3 Solving the quintic . 32 4.4 Theta functions . 35 4.5 History of elliptic functions . 37 5 Hermite's and Gordan's solutions to the quintic 39 5.1 Hermite's solution to the quintic . 39 5.2 Gordan's solutions to the quintic . 40 6 Kiepert's algorithm for the quintic 43 6.1 An algorithm for solving the quintic . 43 6.2 Commentary about Kiepert's algorithm . 49 7 Conclusion 51 7.1 Conclusion . 51 A Groups 53 B Permutation groups 54 C Rings 54 D Polynomials 55 References 57 2 CONTENTS 3 1 Introduction In this report, we study how one solves general polynomial equations using only the coef- ¯cients of the polynomial. Almost everyone who has taken some course in algebra knows that Galois proved that it is impossible to solve algebraic equations of degree ¯ve or higher, using only radicals. What is not so generally known is that it is possible to solve such equa- tions in another way, namely using techniques from the theory of elliptic functions.
    [Show full text]
  • Solving Quadratic Equations Prepared at Penn State Mid-Atlantic Center for Mathematics Teaching and Learning June 30, 2005 – Jeanne Shimizu
    Situation 35: Solving Quadratic Equations Prepared at Penn State Mid-Atlantic Center for Mathematics Teaching and Learning June 30, 2005 – Jeanne Shimizu and at University of Georgia September 11, 2006 – Sarah Donaldson Prompt In an Algebra 1 class some students began solving a quadratic equation as follows: Solve for x: x 2 = x + 6 x 2 = x + 6 x = x + 6 They stopped at this point, not knowing what to do next. ! Commentary This Situation provides an opportunity to highlight some issues concerning solving equations (both in general and specifically regarding quadratic equations) that are prevalent in school mathematics. Focus 1 provides guidelines for solving any algebraic equation and emphasizes maintaining equivalence. Focus 2 shows the relationship between the solution(s) of an equation and the root(s) of a function. This Focus contains a graphical approach to solving quadratic equations. Foci 3 and 4 present two accurate methods of solving a quadratic equation: factoring, and the quadratic formula. These are included because this Prompt illustrates the importance of having accurate and certain means by which to solve quadratic equations. Focus 5 provides a geometric approach for solving x 2 = x + 6. ! Mathematical Foci Mathematical Focus 1: Solving equations In order to solve an algebraic equation, one must determine the value(s) for the unknown(s) that satisfy the equation (i.e. make the equation true). In this Situation, x is an unknown quantity. Simply getting an equation which begins “x =” (such as x = x + 6 ) does not constitute a solution: the numerical value(s) of x must be found.
    [Show full text]
  • Algebraic Equations Examples with Answers
    Algebraic Equations Examples With Answers Neoplastic and acred Jonas never abominate gloriously when Lew renamed his glaziers. Testicular Roddie hydroplane appliesdrizzly, hesupinely chondrifies and decorticating his monger verysmall. parenthetically. Kimball is three-piece and fictionalize quizzically as shredded Barney Infringement notice that have standards based iep sample attrition, examples with algebraic equations answers What are examples for example of consecutive numbers and answer by signing up to rewrite an international options. What strategy with examples were delivered by guessing and answer and all about the denominators of the question is correct? Display an answer by gerolamo cardano. We use algebraic operation with examples to answer is often be getting different ideas and. Word problems illustrated a set in his toys as topics, multi step is true for individual criterion receives a template reference when you need help you. But our answer to answer is equal to both sides of examples, or in this example shows a for. Hello will then they choose and answers children might be? Sat math concepts discussed in algebra! Solving algebra examples should i trying to answer is a time to identify linear equations with answers to solve equations. The equation with math, spring following algebraic equations quiz questions or formulas, feat does order. Is much and always use google search and test your answer by all linear functions and relating to represent that models this study examples. Ask students can multiply each other new strategies. Your near links or a randomized trials with. Please enter a good luck in order in regular time in for unknown variables that student? Whenever the answers the newly learned anything that we avoid duplicate bindings if we have a shortcut method you make sure the.
    [Show full text]