ONLINE ASSESSMENT OF GRAPH THEORY A thesis submitted for the degree of Doctor of Philosophy By Justin Dale Hatt College of Engineering, Design and Physical Sciences Brunel University August 2016 Abstract The objective of this thesis is to establish whether or not online, objective questions in elementary graph theory can be written in a way that exploits the medium of computer-aided assessment. This required the identification and resolution of question design and programming issues. The resulting questions were trialled to give an extensive set of answer files which were analysed to identify whether computer delivery affected the questions in any adverse ways and, if so, to identify practical ways round these issues. A library of questions spanning commonly-taught topics in elementary graph theory has been designed, programmed and added to the graph theory topic within an online assessment and learning tool used at Brunel University called Mathletics. Distracters coded into the questions are based on errors students are likely to make, partially evidenced by final examination scripts. Questions were provided to students in Discrete Mathematics modules with an extensive collection of results compiled for analysis. Questions designed for use in practice environments were trialled on students from 2007 – 2008 and then from 2008 to 2014 inclusive under separate testing conditions. Particular focus is made on the relationship of facility and discrimination between comparable questions during this period. Data is grouped between topic and also year group for the 2008 – 2014 tests, namely 2008 to 2011 and 2011 to 2014, so that it may then be determined what factors, if any, had an effect on the overall results for these questions. Based on the analyses performed, it may be concluded that although CAA questions provide students with a means for improving their learning in this field of mathematics, what makes a question more challenging is not solely based on the number of ways a student can work out his/her solution but also on several other factors that depend on the topic itself. I Acknowledgements My experience of academic study and research within the United Kingdom has indeed been exhausting and challenging. However, I am very grateful for this experience and especially for all of the support I have received during this time. There are many people I wish to thank for this extensively long experience… also, this is the one part of the thesis where I can speak casually, so I am taking advantage of this opportunity while I can. I must thank the academic staff within the Department of Mathematics at Lakehead University. The academic support I received was incredible and I was treated with great care throughout my undergraduate degree. I especially wish to thank Dr. Liping Liu, who introduced me to the Atlantic Association for Research in the Mathematical Sciences (AARMS), which eventually led me to the opportunity to relocate to the United Kingdom to conduct postgraduate research. I must thank Dr. Anthony and Mrs. Sarah Aldous, whom I met at AARMS, as well as Dr. Manti Mendi for supporting me in my move to the United Kingdom and my transition into a postgraduate research career. Their support has been excellent and I am especially grateful to Dr. and Mrs. Aldous for providing me a place to stay whenever needed, especially more recently when I visited them in Singapore. You will probably notice that I am playing to a particular timeline here, so that is why it is fitting now to thank my Ph.D. supervisor, Dr. Martin Greenhow, for putting up with me all of this time. This has not been such a straightforward Ph.D. degree to complete and neither of us knew what was going to come out of this kind of research. However stressful or difficult the experience may have appeared to be at times, the experience of conducting research and understanding its benefits are absolutely incredible and I am very happy to have been able to use my mathematics training and also a bit of my educational training to create such useful and practical research in mathematics. I want to thank Bloomsbury Central Baptist Church for their incredible support of me since I joined them in 2007. It was quite the stretch for me to travel from Uxbridge in London’s Zone 6 to Holborn in Zone 1 just to attend a church, but to then find fellow Canadians in attendance, along with so many academics… from the Reverend Doctors of Theology to a former President of a postsecondary II institution to medical doctors, Masters students, etc., was simply too good to pass up. The community atmosphere has been incredible and although some difficult times have occurred, there is still a great sense of community care between everyone. I am especially grateful to the ministers, Rev. Dr. Ruth Gouldbourne and Rev. Dr. Simon Woodman, for their extended support, both spiritually and financially (for help in attending an international conference several years ago now), and former minister, Rev. Dr. Simon Perry, for his casual, relaxed view in understanding our place in the world and how God works through us within society. I also wish to thank Tim Jones, Dave and Sandy Porter, Peter How, Elsie Martin, Rosel Schmidt, Ruth Johnson, Barbara Collett, Chris Green, Helen Walsh, Howard Brown, the Filipino Community, Daniel and Bryant Taffou, and especially Ben Haden for the friendship, kindness, and incredible support all of them have provided me within the church during this time. Speaking of Canadians, I really have to thank the London Expat Canadian Meetup community of expatriate Canadians living in the United Kingdom. I have made so many friends through my involvement with this group as co-organiser and the friendships have been absolutely excellent. I especially wish to thank the organiser, Dave Mathews, along with Vicky Whaley, Johnson Tse, Cynthia Fisher, and so many others (Sorry for not being able to mention all of you!) for all of their kindness and gratitude in keeping me calm and sane during this experience as a Canadian living overseas. I also want to thank other people back home, including Darlene Robertson, Theresa Flanagan, Tiffany Johnson, Kimberly Johnson, Misty-Dawn Anderson, Travis Gagné, Pat Norman, Shelley Sowers, Heidi Conrod, Conrad Cline, Shannon Neathway, Helen Woodcock, Sydney Linton, Dominique Scott, Deanna Hanley-Luevano, Kim Lomax, Miranda Jones, Tracy Spear, Jen Campbell, Liz Monks, Jillian Monaghan, Penny Cooke, Stacey Tucker, Jenna Martell, and any others I have forgotten to mention (Again… sorry!) for the help and support provided to me back in Canada when I greatly needed it. During my time in employment as a secondary school teacher, I have made some rather good friends whom I wish to thank. I want to thank Joanne Wright and Jo Franklin for their help in adjusting to teaching in a new educational environment, Ian Tay, Rob Mathers, Hilary Downey, Helene Ng Wheeler, III Stephanie Cuthbertson, Dave Hinchcliffe, Günter Morson, and Cameron Mitchell, with whom I worked outside London. That should be most everyone. Sorry to those who I forgot to mention… again; it’s a Canadian thing to be overly apologetic and, ironically, I’m not apologising for that. IV Table of Contents Chapter 1 Introduction............................................................................................... 1 1.1 Background and Motivation ..................................................................................... 1 1.2 Computer-Aided Assessment and Learning ............................................................... 4 1.2.1 Definitions .................................................................................................................... 4 1.2.2 Software Applications .................................................................................................. 8 1.3 Design of Questions ............................................................................................... 11 1.3.1 Features of QuestionMark Perception ....................................................................... 11 1.3.2 Question Types ........................................................................................................... 14 1.3.2.1 Multiple-Choice Questions ............................................................................................. 14 1.3.2.2 Numerical Input Questions ............................................................................................ 16 1.3.2.3 Word Input Questions .................................................................................................... 17 1.4 Analysis of Questions ............................................................................................. 18 1.5 Research Questions and Hypothesis ....................................................................... 18 1.6 A Note About The References ................................................................................. 21 Chapter 2 Design of Template Codes for Graph Theory in Mathletics ........................ 22 2.1 Question and Template Design: An Introduction ..................................................... 22 2.2 Simple Network ..................................................................................................... 23 2.3 Digraphs ................................................................................................................ 25 2.4 Labelled Digraph .................................................................................................... 26 2.5 Vertex Colouring ...................................................................................................