Mathematics Support for Engineering Undergraduates

Mathematics Support for Engineering Undergraduates M. C. Harrison *

Loughborough University/Mathematics Education Centre, Loughborough, UK

Index Terms: Mathematics, Support, Technology.

Abstract Loughborough University’s Mathematics Education Centre (MEC) in collaboration with Coventry University was designated as a Centre for Excellence in Teaching and Learning (CETL), known as sigma, in 2005 in recognition of their excellence in the delivery of university-wide mathematics support to students from a wide variety of backgrounds. At Loughborough, our particular expertise is in the delivery of mathematics at the school/university interface, the mathematical education of engineers and in the mathematical support of specialists and non-specialists alike for which we aim to provide learning resources of the highest quality. To complement this work we pursue the relevant pedagogic research. This paper describes some of the resources we use to support engineering undergraduates learning mathematics. First of all, these include our Mathematics Learning Support Centre, which provides drop-in facilities with 1-to-1 support and a comprehensive range of paper-based resources available to all students. Second, most of our engineering students are taught mathematics using the HELM (Helping Engineers Learn Mathematics) resources. HELM was a major curriculum development project involving a consortium of five UK universities led by Loughborough that uses interactive technology for teaching and assessment. These high quality resources are now used by students in many other higher education institutions both in the UK and elsewhere. Third, online help is provided via mathcentre. mathcentre, the UK on-line Mathematics Learning Support Centre, provides a range of mathematics resources for engineering students, and others, to help students make the transition from school-level to university-level mathematics. 1. Introduction Mathematics permeates all areas of engineering and it is essential that engineering undergraduates understand many mathematical concepts and learn to apply them successfully to engineering problems. However, teaching mathematics to engineers is not without its difficulties, a major one being the so called ‘mathematics problem’, which has been the subject of numerous research reports [for example (1) and (2)] and has been debated at many conferences [for example (3)]. Loughborough University’s Mathematics Education Centre (MEC) was established in 2002. One of its principal aims is to oversee the teaching of mathematics to engineering undergraduates. As such it must be aware of and react to the differing mathematical backgrounds from which these students come and the problems they may now face in learning university-level mathematics. The Centre houses the University’s highly-regarded Mathematics Learning Support Centre (MLSC) which provides a range of services to all students needing help with basic mathematics or statistics. In 2005, the Mathematics Learning Support Centre, in a collaborative bid with Coventry University, was designated a Centre for Excellence in Teaching and Learning (CETL), now known as sigma [4], by the Higher Education Funding Council for England (HEFCE) in recognition of their longstanding excellence in the delivery of university-wide mathematics support to students from a wide variety of backgrounds. This resulted in a total award of £4.85 million over the period 2005-2010, much of which is being spent directly upon improving the learning infrastructure and opportunities for all students at Loughborough. These include a drop-in centre, interactive classroom equipped with the latest learning technologies, state-of the-art video conferencing facilities, learning spaces and staff. The MLSC particularly aims to help students in the early stages of their University studies as they make the transition from school to university by offering extra resources and tuition over and above that normally provided as part of their degree programme. Some of the resources we use to support engineering undergraduates learning mathematics are described in this paper. Second, we describe the HELM (Helping Engineers Learn Mathematics) learning resources [5]. HELM was a major curriculum development project, involving a consortium of five UK universities led by Loughborough, that uses interactive technology for teaching and assessment. The HELM project produced 1

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high-quality student-centered Workbooks, Computer-Aided Learning (CAL) material closely allied to the workbooks, and a strategy for Computer-Aided Assessment (CAA), which can be used for both formative and for summative module assessment. Most of our engineering students are taught mathematics using the HELM resources and sample materials can be viewed via the HELM website at http://helm.lboro.ac.uk. These high quality resources are now used by students in many other higher education institutions both in the UK and elsewhere; as of May 2008, 132 individuals spanning 91 institutions were using HELM learning resources. Before summarising, we briefly describe the online help all students can access via mathcentre [6]. mathcentre, the UK on-line Mathematics Learning Support Centre, provides a range of mathematics resources for engineering students, and others, to help them make the transition from school-level to university-level mathematics. These resources include quick reference guides, practice and revision materials, video tutorials, workbooks and online practice exercises on many branches of mathematics. 2. Mathematics Learning Support Centre Many UK academics would argue that university engineering mathematics courses start at a higher level than students have been prepared for with a consequent need to provide them with extra mathematics support. Croft et al [7] confirm this in a study, which conducted both a census of academic and academicrelated staff in the School of Mathematics at Loughborough University, most of whom work in the MLSC, and a survey of the students who frequently use it. Their study also reports that students can have the opposite view and be largely unaware of the difficulties they may face in the mathematical content of their engineering programme. Consequently there is a growing need for support in mathematics. At Loughborough mathematics support is available to any student, but particularly those in the early years of study. Consequently, at the start of each academic year we undertake diagnostic testing of incoming students to enable us to try to assess their current mathematical ability in a common core of mathematical skills. This year the paper-based test of previous years was replaced by a computer-based diagnostic test, so that students received their mark and feedback immediately at the end of the test, rather than have to wait for the paper-based test to be marked as in previous years. The diagnostic test results enables us to identify areas where students may need extra support and give them advice as to how and where that may be found. We offer a significant amount of extra mathematics support for engineering students and so we introduce them to the comprehensive range of resources and support available to them in the MLSC, as outlined below: • Drop-in facilities with 1-to-1 support Students can obtain help with their mathematics during timetabled hours on a drop-in basis. This 1-1 support, provided by mathematics lecturers, is above that normally provided as part of their degree programme. They can obtain help with tutorial sheets, past exam questions, lecture notes or transition material; the latter may relate to school mathematics topics, of which they may not have sufficient understanding, may have forgotten or may not even have covered at all due to the huge variety of mathematical backgrounds students can have upon entry. • Workshops Although drop-in support continues in the examination periods, we also offer some exam revision workshops, both for the main examinations and for resit examinations. • Statistics support As well as drop-in support, we offer a Statistics Advisory Service for postgraduates and final year students. We also deliver short courses on statistical topics. • Individual appointments Students can make appointments to see a tutor at a specific time should they so wish. • Additional needs support Specific resources and specialist help is also provided for students with special needs. In particular, we have significant experience of helping students with dyslexia and those who are visually impaired. The Eureka Centre for Mathematical Confidence helps build confidence in those students who lack confidence in their mathematics or statistics, so they can apply it in their engineering studies. • Workrooms Students can simply use the resource centre as a quiet place to work with the resources, both paper-based and online, at hand should they need them. Alternatively small numbers of students may use the smaller study rooms we have for group study. 1


• Paper-based resources As well as an extensive collection of textbooks, there is a comprehensive range of paper-based resources. These include a study skills leaflet and facts and formulae leaflets on both advanced mathematics and mechanics. An ‘Engineering Mathematics First Aid Kit’, in the form of self-study hand-outs on a range of mathematics topics ranging from school to first year engineering level, helps students by guiding them through worked examples before tackling the exercises provided. Hard copies of the HELM Workbooks (see 3 below), now used to teach mathematics and statistics to most of the engineering students at Loughborough, are also provided. • Computer-based resources Many of our resources, including HELM (see 3 below), are now provided online and we make significant use of the mathcentre resources (see 4 below). Some specialist software is also available and students can obtain support with Maple, Matlab, SPSS and Latex. 5000 4500

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Figure 1: MLSC Usage, 2002-3, 2003-4, 2004-5, 2005-6 and 2007-08 (Data compiled by Dr Carol Robinson, Loughborough University) The number of student visits to the MLSC is now significant and Fig. 1 compares the usage (cumulative number of visits) in both semesters over the last six years. So far usage in semester one this year is the highest over the last six years. Feedback from students indicates that one of the main reasons students visit the centre is for the 1-1 support. To promote our activities and encourage student usage, we now distribute a DVD presentation promoting the activities of the MLSC to all new students whose mathematics background suggests to us they may benefit from extra support. This presentation is also available online [8]. A number of other activities combine to enhance the support we offer. Proactive teaching interventions have been used to provide supplementary teaching and support to students, where appropriate. This has been accomplished through small group teaching, additional teaching hours and course-specific drop-in sessions. We are investigating effective and innovative uses of new technologies for both teaching and support. We make podcasts, which support the mathematics and upload them to our Virtual Learning Environment (VLE), we record lectures and upload them to the VLE and we have introduced the use of electronic voting systems (or classroom/student response systems), interactive whiteboards and the audio recording and webdelivery of lectures. Finally, we try to underpin all these activities with an extensive programme of pedagogic research. 1


3. HELM The HELM project [5] grew out of the realization that there was a need for more flexible mathematics learning resources for engineering undergraduates, due in particular to the increasing diversity of university intake standards, who were unable to cope with existing textbooks and struggling to construct accurate and meaningful lecture notes. It used the expertise of its consortium partners and computer technology to enhance the mathematical and statistical education of engineering undergraduates in the UK through the provision of a range of flexible learning resources in the form of workbooks, supplemented by web-delivered interactive lessons, together with an extensive integrated web-delivered CAA (Computer Aided Assessment) regime, to drive student learning [9]. They have been extensively trialled both at Loughborough and elsewhere over a number of years and have been well-received by staff and students alike. Further, the ready availability of the resources enables teaching staff to easily incorporate them within their own teaching, where appropriate, with a possible reduction in workload. 3.1 Workbooks The HELM Student Workbooks and accompanying Guides provide the main student learning resources and amount to around 2800 pages in paper form; these are also available electronically in pdf format. Samples can be obtained from the HELM website and a complete list is given in Appendix 1. In total, 46 workbooks cover the UK engineering mathematics and statistics syllabus. Each Workbook contains: Key points and contents in manageable sizes, Tasks in the form of guided exercises and worked solutions (see Fig. 2), and Engineering examples to contextualise the mathematics. Ample space is provided on the worksheets for students to attempt exercises, which where appropriate guide them through the problems in stages. These are complemented by 2 workbooks of Engineering Case Studies and Applications plus both a Student’s Guide & Tutor’s Guide.

Figure 2: Sample Task The Workbooks can be used in different pedagogic ways. They 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 complete set of 1


Workbooks available. These may then be used to support lectures or for independent learning. Students following this open learning regime can attend lectures or, if they so wish, choose to study, with guidance, the mathematics independently in distance learning mode. They also provide an excellent resource for use in support centres. 3.2 Interactive Lessons Initially we started to develop web-delivered CAL segments associated with the workbooks. These incorporated audio, animation and self-assessment aspects and proved especially useful for supporting students of moderate mathematical ability, and for revision purposes. In all over eighty segments linked to twenty three of the more basic workbooks were produced. However, this work is particularly time intensive and developments in technology mean that future efforts will centre more around the development of podcasts, which can be easily delivered via our VLE. 3.3 Assessment Regime HELM uses an integrated web-delivered CAA (Computer Aided Assessment) regime based on Questionmark Perception [10] for both self-testing and formal assessment. Students are typically tested five times each semester with questions delivered over the web. There are around 5000 HELM CAA questions, most with feedback, designed to match particular mathematical concepts in support of the topics covered by the Workbooks. Questions on relevant specific concepts have been structured into two sets; one is used for formative assessment and the other for summative assessment. Tests can then be then custom-made by choosing questions on the appropriate Workbook topics randomly from the appropriate set. Students can take as many trial (formative) tests as they like, thus encouraging them to engage more in their own learning, before taking a one-attempt summative test. This pattern of study followed by assessment helps motivate students to keep up with their studies, thereby improving achievement and progression. Generally students like flexibility to be able to take tests when they are ready, where they want to and, in the case of the practice tests, as many times as they wish. They dislike however the unforgiving nature of CAA in that for example questions are either right or wrong and marked accordingly, but it is possible to overcome this weakness with extra effort in constructing questions. A major benefit for staff is that CAA reduces the burden associated with continuous assessment. A possible concern associated with flexible access facilitated via web delivery is that of cheating in unsupervised summative testing. 4. mathcentre In the UK in recent years, the increase in the number of students entering university has contributed to a greater variability in the prior knowledge of students starting degree programmes. This can be a significant problem in engineering where students require a large amount of mathematics knowledge and may need extra support to help ‘fill in the gaps’. For example, there has been a significant decline in the number of students studying mechanics in UK schools and a consequent increase in the number of students who are less well prepared in this area than we would like. We use mathcentre [2], the UK on-line Mathematics Learning Support Centre, which was developed by members of the sigma team, along with colleagues from the University of Leeds, the EBS Trust and the Higher Education Academy. mathcentre gives access to a comprehensive but growing range of self-study mathematics resources for engineering students, and students from other disciplines, to help them make the transition from school-level to university-level mathematics. It also provides access to resources for staff engaging in teaching such students. All these resources are freely available and include: • Quick reference guides These leaflets provide support on a range of key mathematics topics ranging from algebra to vectors. The self-study style generally provides a simple explanation of a concept followed by one or more worked examples and then by some exercises with solutions. Appendix 2 shows a sample mechanics leaflet on resolving forces. • Teach-yourself booklets These booklets offer a more comprehensive treatment of some important topics and usually include theory supported by worked examples and practice exercises. 1


• Practice and revision materials Refresher booklets on four important school level topics, Algebra, Differentiation, Integration and Numeracy, can be used by students as part of their preparation for university level engineering mathematics. • Online practice exercises Students can practice basic techniques and test themselves using these online diagnostic tests. • mathtutor mathtutor [11] provides video tutorials with interactive diagnostics, summary text and exercises on a range of important mathematical topics. Although originally on DVD, these resources are now available online. • iPod video segments Short clips from some of the mathtutor video tutorials are available as iPod video segments, which students can download to their video iPod. 5. Summary It is thought that many students entering university are not sufficiently well prepared to undertake the engineering mathematics courses ahead of them. This paper outlines some of the resources used and support available from Lougborough University’s Mathematics Learning Support Centre, a designated Centre for Excellence in Teaching and Learning, to help its engineering undergraduates learning of mathematics. The drop-in 1-1 support is particularly popular with students. Mathematics teaching of our engineers is mostly accomplished by using the HELM Learning Resources. These comprise the HELM Workbooks which can encourage student engagement during lectures, the HELM Interactive Lessons which complement the workbooks and aid understanding, and a web-delivered computer-aided learning regime which facilitates regular testing of large numbers of students and drives student learning. A particular difficulty in helping students make the transition from school to university mathematics is the gaps in their prior knowledge due to their varying backgrounds. We overcome this to a large extent by making use of online resources, particularly those provided by mathcentre, the UK on-line Mathematics Learning Support Centre. REFERENCES [1] [2] [3] [4] [5] [6] [7]

[8] [9]

[10] [11]

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Working Group of London Mathematical Society et al (1995) Tackling the Mathematics Problem, Report – London: The London Mathematical Society, the Institute of Mathematics and its Applications, and the Royal Statistical Society. Institute of Mathematics and its Applications, (1999), Engineering Mathematics Matters, Report, IMA, London. Proceedings of the 4th IMA Conference on the Mathematical Education of Engineers (2003), Loughborough University, Eds. Hibberd, S & Mustoe L.R. IMA sigma, http://www.sigma-cetl.ac.uk/ (accessed 7 May 2008). HELM, http://helm.lboro.ac.uk/ (accessed 7 May 2008). mathcentre, http://www.mathcentre.ac.uk/ (accessed 7 May 2008). Croft, A.C., Perkin, G. and Pell, G., (2007), ''The Mathematics Learning Support Centre at Loughborough University: staff and student perceptions of mathematical difficulties.'', Engineering Education - Journal of the Higher Education Academy Subject Centre, 2(1), 2007, pp. 47-58, ISSN 17500044. MLSC Student Support, http://mlsc.lboro.ac.uk/services.php (accessed 8 May 2008) Harrison, M. C., Palipana, A. S., Pidcock, D. & Ward, J. P., 2006, 'Using Computer Technology to Enhance the Teaching & Learning of Engineering Mathematics', Proceedings of TIME 2006, presented at Conference on Technology and its Integration in Mathematics Education, July 20-23, Dresden, Germany. QuestionMark, http://www.qmark.com/ (accessed 7 May 2008) mathtutor, http://www.mathtutor.ac.uk/ (accessed 7 May 2008)


Appendix 1: HELM Workbooks No. 1 2 3

No. 26 27 28

Title Functions of a Complex Variable Multiple Integration Differential Vector Calculus

29 30 31

7 8 9 10 11 12 13 14 15

Title Basic Algebra Basic Functions Equations, Inequalities and Partial Fractions Trigonometry Functions and Modelling Exponential and Logarithmic Functions Matrices Matrix Solution of Equations Vectors Complex Numbers Differentiation Applications of Differentiation Integration Applications of Integration 1 Applications of Integration 2

16 17

Sequences and Series Conics and Polar Coordinates

41 42

18 19 20 21 22 23 24 25

Functions of Several Variables Differential Equations Laplace Transforms z-transforms Eigenvalues and Eigenvectors Fourier Series Fourier Transforms Partial Differential Equations

43 44 45 46 47 48 49 50

Integral Vector Calculus Introduction to Numerical Methods Numerical Methods of Approximation Numerical Initial Value Problems Numerical Boundary Value Problems Modelling Motion Sets and Probability Descriptive Statistics Discrete Probability Distributions Continuous Probability Distributions The Normal Distribution Sampling Distributions and Estimation Hypothesis Testing Goodness of Fit and Contingency Tables Regression and Correlation Analysis of Variance Nonparametric Methods Reliability and Quality Control Mathematics and Physics Miscellany Engineering Case Studies Student’s Guide Tutor's Guide

4 5 6

1


32 33 34 35 36 37 38 39 40

Appendix 2: mathcentre Mechanics leaflet - Resolving Forces

1


Appendix 2: continued

1
