Motivating learning in engineering mathematics through online exemplars.

Transcription

1 Motivating learning in engineering mathematics through online exemplars. Richard Browne, Stephen Lee and Chris Robbins. Mathematics in Education and Industry, Monckton House, White Horse Business Park, Trowbridge, Wiltshire, BA14 0XG Abstract This paper focuses on the creation of contextualised examples for teaching mathematics to Engineers. It describes the development of a suite of examples by Chris Robbins at Mathematics in Education and Industry. The work has been supported by a grant from The Royal Academy of Engineering. Every example starts from a clearly stated engineering problem that mathematics has been used to solve. The example exploits the interest that the context provides to motivate more effective learning, by providing a clear explanation of the mathematics that was used in solving the problem, interweaving activities and tasks for the students. The examples will be available online, and will seek where possible to use the advantages of being viewed on a computer to good effect. They will contain some animations that provide rapid access to the relevant context. The paper discusses how the examples can support teachers delivering mathematics within the Advanced Engineering Diploma, particularly: Mathematics teachers who need support with the engineering contextualisation Engineers who need support teaching the mathematical requirements The project is intended to support the Advanced Engineering Diploma in colleges, and the examples are being trialled at a number of schools and colleges across the UK. However, it is highly likely that these examples will have wider value over time, as the materials developed will be of enduring use for those educating future students of engineering at a variety of levels. Background In autumn 2009, Mathematics in Education and Industry (MEI) was contacted by The Royal Academy of Engineering (RAEng) and invited to provide contextualised examples of mathematics applied to problems in engineering. Work began in July The principal target user group is teachers in further education colleges delivering the Engineering Diploma at level 3. The work will provide support for the learning outcomes of both the principal learning (PL) mathematics unit, Mathematical Techniques and Applications for Engineers, and the additional specialist learning (ASL) mathematics unit, Mathematics for Engineering, for the level 3 Engineering Diploma. Development of the materials is being carried out by Chris Robbins who is contracted to MEI on a 0.5 FTE basis. Chris has over 20 years of experience in applying mathematics

2 2 in industry and has for the last 11 years run his own small company. He also has a number of years experience in producing and delivering outreach material on STEM subjects. Once complete, it is intended that the materials will be made available to users through both the RAEng website, and the MEI website, Introduction The methodology underpinning the support materials is essentially new. Consequently it is being refined in the light of experience, as work progresses. This paper describes the basic approach. As required by the RAEng, we have set out to develop resources that relate mathematics required in the Engineering Diploma to engineering contexts that need this mathematics to solve problems. The general approach identified is to start with a clear engineering problem and then demonstrate how an analysis can first show how to describe the problem mathematically and then to use the description in a predictive sense. In this way the initial engagement is with the problem rather than the mathematical methods. The problem statement from the example called Wind Vectors is given below as Figure 1 Our ship is on a mercy mission to supply aid to victims of an island volcanic eruption when there s an unexpected emergency on board. The medical specialist has been taken ill and needs to be replaced. The ship can t stop or turn round as time is critical so the replacement has to be brought on board by helicopter. The ship needs to report the wind direction so the pilot can adjust his course to intercept the ship. This information is also vital to help the pilot control the helicopter as he performs a landing on a moving helipad. How does the ship determine and report the wind? How does the pilot allow for it to rendezvous with the ship? To answer these questions we need to use vectors. Figure 1 Each example is designed to provide material that a teacher can use with a group rather than pre-planned lessons or fully stand-alone student material. However, within an example there will typically be access to online materials that provide worksheets; in addition some of the resources will be interactive and some will provide demonstrations. The provision of the material online allows the possibility of using animation and simulation techniques to introduce and demonstrate a problem in an engaging way. Still shots from typical animations are shown in Figure 2. 2

3 3 Figure 2 In order to support the teachers as fully as possible, the examples contain engineering background material in addition to showing step-by-step use of the mathematics involved. Thus teachers with a predominantly mathematics teaching background or a predominantly engineering teaching background should be equally comfortable in using an example. Syllabus coverage One issue that we have worked on is syllabus coverage. We have done quite a lot of work identifying what mathematics is required in the PL and our aspiration is to cover all of it. This is more difficult as the various awarding organisations specifications are not very clear about the extent of the coverage required. We have developed a list of mathematical topics that we regard as required in the PL, which is considerably more detailed than the learning outcomes of the examination specifications; these latter, for example, include Use algebra to solve engineering problems. We are recording the extent to which the elements of our more detailed list are addressed by the individual examples that we develop. We shall also draw up a table showing the topics that each example covers. A related issue is that of mathematical knowledge that is assumed in the PL syllabus. Very often these are important topics which should have been understood at GCSE but have not been, and consequently present major barriers to students progress. We have included the key aspects of this knowledge in our list of topics. With a number of studies having taken place into using gaming technology as a mechanism for learning, see for example teem (2002), futurelabs (2008) and Mangahigh (2010), we are investigating a way to provide coverage of this material by developing a form of online game that will improve students understanding of and fluency with these critical topics. We are also developing a number of examples that address the more challenging mathematics in the ASL unit. Mathematical pedagogy As well as covering the necessary knowledge, it is critically important that the materials suggest approaches to mathematical pedagogy that reflect current thinking about learning. We have therefore made considerable effort to ensure that the presentation of the mathematical solutions of the problems posed in the materials suggests classroom practice in which students are given opportunities to address rich, challenging problems 3

4 4 for themselves (Ahmed (1987)). We have attempted to encourage teachers to provide opportunities for active learning through discussion, since activities are more effective when they encourage critical, constructive discussion (Swan (2005)). We have also provided audio-visual material for access via a computer that is intended to engage and motivate students by showing aspects of a problem situation that can immediately be recognised as authentic. Support for mathematicians The debate over who should teach mathematics to engineering students, i.e. engineers or mathematicians, is longstanding, with recent perspectives having been given by Ernest (2003) and Mustoe (2006). Where the mathematics teaching for engineers taking level 3 courses in colleges is provided by mathematicians, it is often the case that the teacher, whilst fully familiar with the relevant mathematics, may have little grasp of the range of engineering contexts in which the mathematics is used to solve engineering problems. This lack of expertise may be compounded by teachers preference for textbook material that focuses on the mathematics without providing contextual information. By contrast, the examples developed in this project provide clear links between realistic engineering problems and the mathematics required to solve them. Teachers who make use of the materials will be able to build up a personal portfolio of genuine contexts which will ensure they have a ready answer to the question why are we learning this bit of mathematics? Support for engineers For teachers who are engineers by training, who are aware of a wide range of contexts in which the mathematics in the Engineering Diploma is used, the examples offer a different form of support. This is because they set out the mathematical aspects of the learning in ways that are designed to be highly effective for students. One of MEI s roles in the project has been to ensure that it reflects current thinking about, and excellent classroom practice in, teaching and learning mathematics. Developing effective materials We have given careful thought to ways in which the materials may be made easier and more effective for teachers to use. Part of the solution to this problem has been to make sure that the material for each example is organised in a similar way. For this reason, we have developed a resource structure map that identifies the various kinds of materials that we present for each example, see Figure 3; these include separate materials for teachers and students, with some of the material for students being available throughout, but other material, including hints and solutions, to be held back until the teacher deems appropriate. The students have immediate access audio-visual and other material intended to help them understand the engineering context. We hope to record discussion sessions from trialling that can be made available to users; this could be particularly useful for individual students, or students in very small groups. 4

5 5 Teachers Given to students Immediate Video Engineering background Discussion Audio/video of a group discussing the problem On request Hints Answers Teachers only With permission Solutions Figure 3 - The resource structure map for the examples Within the materials, we have also developed a system of signposting various common situations, including: the initial problem statement; engineering information; information that other material is immediately available; commands that students could or should do something; an invitation to hold a discussion; and hints or tips and comments on the material. Finally, we intend that the material should be made more recognisable through a modest amount of professional design work. This should support the clarity with which the various technical features are recognised by teachers, as well as making the whole collection of examples easier to identify from their appearance. Selecting the examples One question that we have had to address is how to select the engineering contexts that are used as the focus of each of the various examples. The range of possibilities has, of course, been extremely large. In making our selection, we have taken advice from the RAEng, we have selected some examples from industrial contacts, and we have also based some examples on contexts that were familiar to Chris Robbins from his industrial work. We intend that the final collection will be a good reflection of the range of engineering contexts suggested by the level 3 Engineering Diploma specifications. 5

6 6 Wider use of the materials The materials we are developing are designed to support learners who are studying for the level 3 Engineering Diploma. The diploma faces an uncertain future since the new government withdrew central support from all diploma lines of learning in autumn However, the value of the materials is likely to extend beyond any particular type of curriculum structure. Whatever set of post 16 pathways is designed in future, there will always be a need to support learners who wish to study level 3 engineering courses, and mathematics is likely to present particular difficulties to many of these students. The examples we are designing may well also be suitable to motivate mathematical learning for students taking different level 3 engineering courses that are already available. When complete, the materials may also prove useful in university foundation courses, remedial mathematics courses and possibly even first year undergraduate courses. References AHMED A. (1987) Better mathematics: a curriculum development study. HMSO 104, London. ERNEST, P (2003) The Mathematical Attitudes, Beliefs and Abilityof Students, in LTSN Maths Team, Eds Maths for Engineering and Science, Birmingham: Maths, Stats & OR Network, University of Birmingham: 4-5. futurelabs (2008) Games and Learning Policy Recommendations Report. Accessed via _policy_report.pdf (29 January 2011) MANGAHIGH (2010) Mangahigh Research Trial Scotland Results January to March Accessed via (29 January 2011) MUSTOE, L.R. (2006) The Mathematical Background of Undergraduate Engineers International Journal of Elecctrical Engineering Education, 393, SWAN, M. (2005) Improving learning in mathematics: challenges and strategies. DfES Standards Unit, London, UK. teem (2002) Report on the educational use of games. Accessed via (29 January 2011) 6