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Problem-Solving in Undergraduate Mathematics and Computer Aided Assessment View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Birmingham Research Archive, E-theses Repository Problem-Solving in Undergraduate Mathematics and Computer Aided Assessment by Matthew Badger A thesis submitted to The University of Birmingham for the degree of Doctor of Philosophy School of Mathematics The University of Birmingham February 2013 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract Problem solving is an important skill for students of the mathematical sciences, but traditional methods of directed learning often fail to teach students how to solve problems independently. To compound the issue, assessing problem-solving skills with computers is extremely difficult. In this thesis we investigate teaching by problem solving and introducing aspects of problem solving in computer aided assessment. In the first part of this thesis we discuss problem solving and problem-based pedagogies. This leads us, in the second part, to a discussion of the Moore Method, a method ofenquiry- based learning. We demonstrate that a Moore Method course in the School of Mathematics at the University of Birmingham has helped students’ performance in certain other courses in the School, and record the experiences of teachers new to the Moore Method at another U.K. university. The final part of this thesis considers word questions, in particular those involving systems of equations. The work discussed here has allowed the implementation of a range of questions in the computer-aided assessment software STACK. While the programmatic aspects of this work have been completed, the study of this implementation is ongoing. Acknowledgements First and foremost I would like to thank my supervisor, Dr. Christopher Sangwin, for the considerable amount of work that he has put in in assisting me with both the work which comprises this thesis, and the thesis itself. I have also been ably assisted by my co-supervisor, Dr. Dirk Hermans, whose input at points has been invaluable. Both Dr. Sangwin and Dr. Her- mans join a group of teachers and lecturers without whose guidance I would not have found myself in this position. I owe a considerable debt of gratitude to both my examiners for their work in assessing the original version of this thesis, and the valuable suggestions that came from that. Finally, my friends and family have been a great support in helping me through the extended process of pursuing postgraduate education and writing a Ph.D. thesis. I am particularly grateful for the encouragement of those who have been here before: Tom, Angela, James, and Katherine. Contents 1 Introduction 1 I Problem-based Learning 3 2 Problem-solving 4 2.1 Thinking Mathematically ................................ 5 2.2 Problem-solving ..................................... 12 2.2.1 Teaching Problem-solving ............................ 18 3 Problem-based Learning 22 3.1 Directed Learning .................................... 22 3.1.1 Lectures ..................................... 23 3.1.2 Example Classes ................................ 25 3.1.3 The structure of directed learning ....................... 27 3.2 Problem-based Learning ................................ 30 3.2.1 Arguments for and against PBL ......................... 34 4 Problem-based Learning in England and Wales 36 4.1 The HE Mathematics Curriculum Summit and the Problem-solving Project ..... 36 4.2 The Questionnaire .................................... 40 4.2.1 Summary of Questionnaire Responses .................... 41 4.3 Case Study Methodology ................................ 44 4.4 Approaches to Teaching Problem-solving ....................... 45 4.5 Case Studies ...................................... 48 4.6 Durham University .................................... 48 4.6.1 Teaching ..................................... 49 4.6.2 Assessment ................................... 52 4.6.3 Motivations and Development ......................... 52 4.6.4 Problem Selection ................................ 53 4.6.5 Outcomes and the Future ............................ 56 4.7 University of Manchester ................................ 56 4.7.1 Teaching ..................................... 57 4.7.2 Assessment ................................... 59 4.7.3 Motivations and Development ......................... 60 4.7.4 Problem Selection ................................ 61 4.7.5 Outcomes and the Future ............................ 63 4.8 Queen Mary, University of London ........................... 63 4.8.1 Teaching ..................................... 64 4.8.2 Assessment ................................... 66 4.8.3 Motivations and Development ......................... 67 4.8.4 Problem Selection ................................ 67 4.8.5 Outcomes and the Future ............................ 68 4.9 University of Warwick .................................. 69 4.9.1 Teaching ..................................... 70 4.9.2 Assessment ................................... 72 4.9.3 Motivations and Development ......................... 73 4.9.4 Problem Selection ................................ 75 4.9.5 Outcomes and the Future ............................ 76 II The Moore Method 78 5 The Moore Method 79 5.1 Robert Lee Moore and his Method ........................... 79 5.1.1 Moore’s Moore Method ............................. 80 5.2 The Moore Method Today ................................ 85 5.2.1 Course Material ................................. 86 5.2.2 Presentations and Written Solutions ...................... 89 5.2.3 Use in Teaching ................................. 91 5.2.4 Criticisms and Limitations of the Method ................... 92 5.3 Relationship to other PBL, and Research on the Method ............... 94 6 Determining the Effectiveness of the Moore Method at Improving Math- ematics Performance 98 6.1 Introduction ....................................... 98 6.2 The Module ....................................... 99 6.2.1 Teaching and Examination ........................... 100 6.2.2 Self Selection .................................. 102 6.3 Methodology ....................................... 103 6.3.1 The Data ..................................... 103 6.3.2 The Method ................................... 105 6.4 Results .......................................... 108 6.4.1 First Year Results ................................ 108 6.4.2 Second year analysis .............................. 112 6.4.3 Third Year and Mean Results .......................... 119 6.5 Conclusion ........................................ 120 7 Approaches to the Moore Method: Staff Experiences with a Moore Mod- ule 125 7.1 The Module ....................................... 126 7.2 Methodology ....................................... 129 7.3 Analysis of Results ................................... 132 7.3.1 Module Planning and Motivation ........................ 134 7.3.2 Choices of Problems .............................. 140 7.3.3 Events During Class ............................... 144 7.3.4 Benefits to Students ............................... 149 7.3.5 Benefits to Staff ................................. 152 7.3.6 Difficulties and Drawbacks ........................... 154 7.3.7 Changes to the Module ............................. 157 7.3.8 Reflections of Staff ............................... 158 7.4 Conclusions ....................................... 160 III Computer Aided Assessment 163 8 The solution and comparison of equations 164 8.1 STACK and Computer-Aided Assessment ....................... 164 8.2 Validity and Correctness ................................ 166 8.2.1 Validity ...................................... 167 8.2.2 Correctness ................................... 169 8.3 Equivalence of Equations ................................ 170 8.4 Systems of Equations and their Varieties ....................... 172 8.5 Gröbner Bases ..................................... 175 8.5.1 Basic Theory ................................... 175 8.5.2 Hilbert Basis Theorem .............................. 180 8.5.3 Systems of Linear Equations .......................... 182 8.5.4 Systems of Univariate Polynomials ....................... 184 8.6 Systems of Multivariate Polynomials .......................... 192 8.6.1 Term Orders ................................... 192 8.6.2 A Division Algorithm ............................... 195 8.6.3 Gröbner Bases .................................. 198 8.6.4 Comparing Systems of Equations ....................... 202 8.6.5 Limitations of the System ............................ 204 8.7 Conclusion ........................................ 205 9 Applications of Gröbner Bases 206 9.1 Potential Response Trees ................................ 206 9.2 Using Gröbner Bases to assess questions in STACK ................. 209 9.2.1 Alternate Answers ................................ 213 9.2.2 Incorrect Responses ............................... 214 9.3 Testing and Distribution ................................. 217 9.4 Answer Test Code ...................................
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