About the HELM Project HELM (Helping Engineers Learn Mathematics) materials wer

About the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a three-year curriculum development project undertaken by a consortium of five English universities led by Loughborough University, funded by the Higher Education Funding Council for England under the Fund for the Development of Teaching and Learning for the period October 2002 September 2005. HELM aims to enhance the mathematical education of engineering undergraduates through a range of flexible learning resources in the form of Workbooks and web-delivered interactive segments. HELM supports two CAA regimes: an integrated web-delivered implementation and a CD-based version. HELM learning resources have been produced primarily by teams of writers at six universities: Hull, Loughborough, Manchester, Newcastle, Reading, Sunderland. HELM gratefully acknowledges the valuable support of colleagues at the following universities and col- leges involved in the critical reading, trialling, enhancement and revision of the learning materials: Aston, Bournemouth & Poole College, Cambridge, City, Glamorgan, Glasgow, Glasgow Caledonian, Glenrothes In- stitute of Applied Technology, Harper Adams University College, Hertfordshire, Leicester, Liverpool, London Metropolitan, Moray College, Northumbria, Nottingham, Nottingham Trent, Oxford Brookes, Plymouth, Portsmouth, Queens Belfast, Robert Gordon, Royal Forest of Dean College, Salford, Sligo Institute of Tech- nology, Southampton, Southampton Institute, Surrey, Teesside, Ulster, University of Wales Institute Cardiff, West Kingsway College (London), West Notts College. HELM Contacts: Post: HELM, Mathematics Education Centre, Loughborough University, Loughborough, LE11 3TU. Email: helm@lboro.ac.uk Web: http://helm.lboro.ac.uk HELM Workbooks List 1 Basic Algebra 26 Functions of a Complex Variable 2 Basic Functions 27 Multiple Integration 3 Equations, Inequalities & Partial Fractions 28 Differential Vector Calculus 4 Trigonometry 29 Integral Vector Calculus 5 Functions and Modelling 30 Introduction to Numerical Methods 6 Exponential and Logarithmic Functions 31 Numerical Methods of Approximation 7 Matrices 32 Numerical Initial Value Problems 8 Matrix Solution of Equations 33 Numerical Boundary Value Problems 9 Vectors 34 Modelling Motion 10 Complex Numbers 35 Sets and Probability 11 Differentiation 36 Descriptive Statistics 12 Applications of Differentiation 37 Discrete Probability Distributions 13 Integration 38 Continuous Probability Distributions 14 Applications of Integration 1 39 The Normal Distribution 15 Applications of Integration 2 40 Sampling Distributions and Estimation 16 Sequences and Series 41 Hypothesis Testing 17 Conics and Polar Coordinates 42 Goodness of Fit and Contingency Tables 18 Functions of Several Variables 43 Regression and Correlation 19 Differential Equations 44 Analysis of Variance 20 Laplace Transforms 45 Non-parametric Statistics 21 z-Transforms 46 Reliability and Quality Control 22 Eigenvalues and Eigenvectors 47 Mathematics and Physics Miscellany 23 Fourier Series 48 Engineering Case Studies 24 Fourier Transforms 49 Student’s Guide 25 Partial Differential Equations 50 Tutor’s Guide Copyright Loughborough University, 2008 Contents Contents 50 50 Tutor's Guide 50.1 Introduction to HELM 2 50.2 HELM Consortium, Triallist Institutions and Individual Contributors 6 50.3 HELM Transferability Project (HELMet) 9 50.4 Usage Modes of the HELM Resources 11 50.5 HELM Uptake and Transferability Project Reports 15 50.6 HELM Workbook Structure and Notation 65 50.7 Issues and Notes for Tutors 68 50.8 HELM Computer Aided Assessment 74 50.9 HELM Electronic Learning Resources 82 50.10 List of Sections in Workbooks 1 to 48 90 50.11 Commentaries on Workbooks 1 to 48 97 50.12 Index of Engineering contexts in Workbooks 1 to 48 134 50.13 Indexes for Workbooks 1 to 48 (electronic version only) Introduction to HELM     50.1 1. The ‘Mathematics Problem’ in the engineering context In recent years the mathematical preparedness of students embarking upon science and engineering degree programmes has been the subject of close scrutiny in numerous research reports [for example 1, 2, 3, 4], and the topic has been debated at many conferences [for example, 5, 6, 7], with some disturbing conclusions. A common theme running through all this work is that these students are, on the whole, significantly mathematically weaker than students coming to university a decade or so ago. Traditionally, students embarking upon engineering degree programmes had to demonstrate a very respectable competence in mathematics and physics through the achievement of good A level grades. However, the recent past has seen a widening of access for understandable financial and social reasons. Academic staffface the conundrum of needing to devote less time (or at least less energy) to teaching and related administration whilst tailoring courses adequately to the needs of the wide range of individual students in their care. In teaching mathematics to engineers, the primary vehicle of transmission still remains the lecture. It is particularly efficient with large groups, although interaction is then especially difficult. Syllabus requirements may, however, encourage lecturers to try to cover too much material thus hindering student understanding. In any case, lectures are not the best place for transmitting a great deal of technical information, especially when students are trying to copy it from an OHP or writing board. Understanding becomes more difficult and many errors of transcription are made. The lecture notes of an average student often bear only passing resemblance to what was actually delivered, which can be crucial to understanding in mathematics. Much material is now available in electronic form, on CD or on the web, and computer aided learning (Interactive Learning Segments) has tremendous potential to assist the learning process. Information can be presented in written format (on screen) which can be supplemented with animation, with commentary and with video. Interaction adds more interest. It can be delivered locally on an intranet network or over the web. It can be recorded on a CD and used by students not wishing (or able) to use an internet connection. However, the development and start-up costs are extremely high, both in time and in resources. There is much anecdotal evidence to suggest that even though there is quite ‘good’ Interactive Learning material available, students prefer not to learn primarily using this approach. What Interactive Learning material is available, it is usually in stand-alone form without accompanying written material, which is a considerable drawback. This has been the Achilles heel of many Interactive Learning projects - failure to adequately link with the curriculum. 2 HELM (2008): Workbook 50: Tutor’s Guide ® 2. The Loughborough experience At Loughborough we have put in place an environment for learning mathematics that we believe will be attractive to the vast majority of undergraduate students, whatever their level and whatever their previous experience. In 1997, funding was made available by the University for the ‘Open Learning Project’ in Mathematics for Engineers which has provided high-quality student-centred workbooks, computer aided learning material closely allied to the workbooks, and a strategy for computer aided assessment which can be used for self-assessment and for module assessment. For students following this open learning regime, lectures are now optional as they can now choose to study, with guidance, the mathematics independently. The success of the Open Learning Project encouraged staffto seek funding to develop further this work resulting in the HELM project (Helping Engineers Learn Mathematics) which was supported by a £250,000 HEFCE FDTL4 grant for the period Oct 2002-Sept 2005. 3. The HELM project The HELM team comprised staffat Loughborough University and four consortium partners in other English universities: Hull, Manchester, Reading and Sunderland. The project’s aims were to consid- erably enhance, extend and thoroughly test Loughborough’s original Open Learning materials. These were to be achieved mainly by the writing of additional Workbooks and incorporating engineering examples and case studies closely related to the mathematics presented, enhancing the question data- banks, upgrading the Interactive Learning segments and adding some more for basic mathematics topics, and promoting widespread trialling. The HELM project’s output consists of Workbooks, Interactive Learning segments, a Computer Aided Assessment regime which is used to help ‘drive the student learning’ and a report on possible modes of usage of this flexible material. The Workbooks may be integrated into existing engineering degree programmes either by selecting isolated stand-alone units to complement other materials or by creating a complete scheme of work for a semester or year or two years by selecting from the large set of Workbooks available. These may be used to support lectures or for independent learning. HELM’s emphasis is on flexibility - the work can be undertaken as private study, distance learning or can be teacher-led, or a combination, according to the learning style and competence of the student and the approach of the particular lecturer. HELM (2008): Section 50.1: Introduction to HELM 3 4. HELM project Workbooks 50 Workbooks are available which comprise: • 46 Student Workbooks (listed in 50.4) written specifically with the typical engineering student in mind containing mathematical and statistical topics, worked examples, tasks and related engineering examples. • A Workbook containing an introduction to dimensional analysis, supplementary mathematical topics and physics case studies. • A Workbook containing Engineering Case Studies ranging over many engineering disciplines. • A Students’ Guide • A Tutor’s Guide (this document) The main project materials are the Workbooks which are subdivided into manageable Sections. As far as possible, each Section is designed to be a self-contained piece of work that can be attempted by the student in a few hours. In general, a whole Workbook typically represents 2 to uploads/Litterature/ tutors-guide.pdf

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