Teaching Mathematics – a guide for postgraduates and teaching assistants Bill Cox and Michael Grove Teaching Mathematics – a guide for postgraduates and teaching assistants BILL COX Aston University and MICHAEL GROVE University of Birmingham Published by The Maths, Stats & OR Network September 2012 © 2012 The Maths, Stats & OR Network All rights reserved www.mathstore.ac.uk ISBN 978-0-9569171-4-0 Production Editing: Dagmar Waller. Design & Layout: Chantal Jackson Printed by lulu.com. ii iii Contents About the authors vii Preface ix Acknowledgements xi 1 Getting started as a postgraduate teaching mathematics 1 1.1 Teaching is not easy – an analogy 1 1.2 A teaching mathematics checklist 1 1.3 General points in working with students 3 1.4 Motivating and enthusing students and encouraging participation 8 1.5 Preparing for teaching 12 2 Running exercise classes 19 2.1 Exercise and problem classes 19 2.2 Preparation for exercise classes 20 2.3 Starting the session off 23 2.4 Keeping things going 24 2.5 Explaining to students 26 2.6 Working through problems on the board 28 2.7 Maintaining a productive working atmosphere 29 3 Supervising small discussion groups 33 3. 1 Discussion groups in mathematics 33 3.2 Benefits of small discussion groups 33 3.3 The mathematics of small group teaching 34 3.4 Preparation for small group discussion 36 3.5 Keeping things moving and engaging students 37 4 An introduction to lecturing: presenting and communicating mathematics 39 4.1 Introduction 39 4.2 The mathematics of presentations 39 4.3 Planning and preparing the presentation 41 4.4 Giving the presentation 44 4.5 Use of resources and media 51 4.6 Maintaining a good learning environment in the session 55 iv 5 Assessing student work and providing feedback 65 5.1 Assessing student knowledge and understanding 65 5.2 The mathematics of assessment 68 5.3 Marking student work 69 5.4 A detailed example 72 5.5 Feedback on students’ work 74 6 Some frequently asked questions 77 6.1 Your questions answered? 77 6.2 Small group teaching (exercise classes, discussion groups and presentations) 77 6.3 Marking and feedback on student work 86 6.4 Anything else related to your teaching duties 90 Appendix 1: Marking scheme for binomial expansion question 93 Appendix 2: Student solutions to binomial expansion question 95 Bibliography and References 97 v Teaching Mathematics – a guide for postgraduates and teaching assistants vi Teaching Mathematics – a guide for postgraduates and teaching assistants About the authors Bill Cox Bill Cox is Visiting Senior Lecturer in Mathematics at Aston University, and has been a Consultant for the Maths, Stats and OR Network of the Higher Education Academy for over ten years. He has a professional life-time of experience of teaching mathematics at university, at all levels, in pure and applied mathematics. He has written several books on teaching and learning of mathematics and generic aspects of teaching, and has contributed to the annual workshops on supporting postgraduates new to teaching mathematics since they were instigated by Michael Grove. Contact: [email protected] Michael Grove Michael Grove is currently Director of the National HE STEM Programme at the University of Birmingham. He teaches mathematics at university level, primarily to first year students, and is involved with providing mathematics support to undergraduate students as they make the transition to university. He has delivered an annual national series of workshops to support postgraduate students new to teaching mathematics within higher education for over five years. Contact: [email protected] vii Teaching Mathematics – a guide for postgraduates and teaching assistants viii Teaching Mathematics – a guide for postgraduates and teaching assistants Preface This book is aimed at postgraduate students who assist in the teaching of mathematics-rich subjects - in which we include statistics and more broadly Science, Technology, Engineering and Mathematics (STEM) subjects generally. It is also suitable for others that assist to a more or less limited extent in teaching, such as research fellows, and, as they are known in the United States of America for example, graduate teaching assistants. Such assistants (which we will now refer to as postgraduates throughout the remainder of this book) have an important role in supporting undergraduate students in learning mathematics, but until recently, have rarely received training to support them in this role – certainly this is true within the United Kingdom where both authors are based. Following an initiative by one of us (MG), a series of one-day workshops were initiated across the UK, entitled Supporting Postgraduates who Teach Mathematics. These have been run by the Maths, Stats & OR Network of the Higher Education Academy. At these events experienced lecturers offered advice, guidance and support to postgraduates who have teaching duties. This book has evolved from our experience in running these events, along with colleagues, since 2005. Our hope is that it forms a useful guide for any postgraduate heading out for that next tutorial, or facing the next pile of marking. This book focuses mainly upon the discipline-based aspects of the duties postgraduates might reasonably be expected to undertake. It is perhaps complemented by more generic material such as Morss and Murray (2005). There will be inevitable overlap with such material, but this will only serve to emphasize the importance of the messages we are trying to convey. There are several key messages or themes that underpin the entire content of this book that are worth highlighting here. We cannot tell you how you should teach, there are too many variables that make each teaching session unique, but we can share our experience and suggest ideas that you might like to try. It is up to you to implement and evaluate these, if they work as intended that is ideal, if they don’t, don’t necessarily give up but adapt them according to needs of you and your students. Similarly, you will have your own experiences of being taught as an undergraduate student, both good and bad, build upon them; implement the ideas and approaches that you found helpful, avoid those that weren’t. Talk to other colleagues, both fellow postgraduates and more experienced members of teaching staff; teaching is an activity where you can learn much from others. Finally, try to engage the students directly in each teaching activity you undertake; this is perhaps our core message. Effective teaching involves a dialogue with the students: listen to what they have to say, question them, and only then explain. Their feedback is critical to not only helping you develop as a teacher, but also for ensuring that the teaching sessions are productive and meet their needs and expectations. Duties assigned to Postgraduates Surveys of postgraduate students that have been undertaken through the Maths, Stats and OR Network have shown that the teaching duties usually assigned to postgraduates in mathematics fall mainly into three areas: ix Teaching Mathematics – a guide for postgraduates and teaching assistants 1. Small group teaching such as leading seminars and discussion. 2. Exercise and problem classes. 3. Marking and providing feedback on student work. Apart from these main duties there are other activities that some postgraduates are exceptionally asked to undertake. These include actual course design and marking examinations, however such instances of postgraduates undertaking these duties are rare. Should you find yourself, however, participating in such teaching, material on these more advanced activities can be found in the book Teaching Mathematics in Higher Education – the Basics and Beyond (Cox, 2011, referred to as TMHEBB from now on). The exact definition of what is meant by ‘small group teaching’ is one that varies between different institutions, however, in the majority of cases it is used to describe an interactive session in which the tutor is responsible for a small group of students, normally less than 30. In this book we will take small group teaching to be any of the following: exercise classes, small discussion groups, or examples classes What each of these teaching sessions have in common is their purpose in encouraging students to interact with both the tutor and each other, usually in the solving of mathematical problems. Such sessions are not solely intended to encourage the simple ‘recall’ of facts, but to develop more fundamental concepts such as application and problem solving. The terminology in small group teaching in mathematics is also not universally agreed, but usually by an exercise class we mean one in which students work through problems together with help and support available from an expert tutor or teacher. This sort of class is dealt with in Chapter 2. By small discussion groups, covered in Chapter 3, we are considering ‘real’ group teaching where the objective is to develop interactions between students and tutors to facilitate learning. It is the interplay between students and tutors that provides a greater range of learning activity. By example classes we mean the situation where you work through solutions to problems, taking a more leading role in guiding students through their solutions. Normally, postgraduates will be working for a full-time member of staff who will set the context of the teaching duties and provide supporting materials such as exercise or problem sheets. They may also moderate the marking you are asked to undertake. In general a postgraduate may appear to be directed and guided by the lecturer, supervised in their duties and therefore expected to take minimum responsibility. However, in terms of supporting students to learn, postgraduates are at the forefront and in direct contact with students when they are actively working and trying to understand the lecture and course materials.