The uses of mathematics in the engineering workplace

Report of the MEI/IET joint conference

December 2010

Published by MEI and IET on 10 December 2010

Available online at:

www.mei.org.uk

www.theiet.org

ISBN 978-0-948186-22-6

Report of the MEI/IET joint conference

The uses of mathematics in the engineering workplace Is the mathematics curriculum sufficiently influenced by the way the subject is

used in the workplace? How can mathematics teachers and engineers improve mutual

understanding?

Held on 1 June 2009 at Savoy Place, London

Contents page

Background 1 The rationale for the conference The conference programme How delegates were identified for invitation

Executive summary 4

Contributions from the principal speakers 5 Lee Hopley Matthew Harrison Rod Bond Celia Hoyles

Summary of work on exemplars 10 Automating data monitoring led by Angela Dean Using capstans led by Helen Meese Collecting rain in gutters led by Chris Robbins Drug concentrations in patients’ bodies led by Steve St Gallay

Action plan for IET and MEI 15 Strategies to encourage take up of Further Mathematics and engineering

Appendices 16 1. List of Attendees 2. Acronyms used in the report

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Background

The rationale for the conference

The origins of this conference lie in the synergy between MEI and the IET. Mathematics in Education and Industry (MEI) is a curriculum development body, promoting approaches to school mathematics that make the subject interesting in its own right and foster skills and knowledge that are relevant to its use in the workplace. The Institution of Engineering and Technology (IET) is one of the world’s leading professional societies for the engineering and technology community. The IET has more than 150,000 members in 127 countries and has offices in Europe, North America and Asia-Pacific. The Institution provides a global knowledge network to facilitate the exchange of knowledge and ideas and promotes the positive role of Science, Technology, Engineering and Mathematics in the world. Through the Education 5-19 Department, the Institution supports the teaching of STEM subjects with a wide range of resources, activities and training for teachers. The potential value of a working relationship between the two organisations was recognised by IET Fellow Dr. Anthony Bainbridge when he became an MEI Trustee in 2005. An exploratory meeting was held at which it was agreed to hold a joint conference in May 2007 about attracting the best students of mathematics into engineering. Following the success of the 2007 conference, MEI and the IET decided to collaborate to offer a second joint conference on 1 June 2009. This was designed to consider whether the mathematics curriculum is sufficiently influenced by the way the subject is used in the workplace and how mathematics teachers and engineers can improve mutual understanding. The approach we adopted to addressing these questions was to invite delegates to take part in activities designed to engage their attention on four particular uses of mathematics. We then asked them to design attractive materials that could be produced to exemplify the particular use of mathematics for teachers and learners at schools and colleges. We intend to put some energy into constructing innovative exemplar materials based on the activities presented at the conference, with a view to encouraging their widespread use in STEM classrooms.

Charlie Stripp Michelle Richmond MEI Chief Executive IET Director of Qualifications

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The conference programme

MEI/IET joint conference

The uses of mathematics in the engineering workplace

1 June 2009 at Savoy Place, London

Is the mathematics curriculum sufficiently influenced by the way the subject is used in the workplace? How can mathematics teachers and engineers improve mutual understanding?

10.00–10.30 Registration and coffee*

10.30–10.40 Welcome from IET and MEI

10.40–10.45 Introduction to the day Andrew Ramsay, Engineering Council

10.45–11.05 The importance of skills in the engineering workplace Lee Hopley, EEF

11.05–11.25 The importance of mathematics in the 14-19 Diploma in Engineering

Matthew Harrison, RAEng

11.25–11.35 Introduction to the breakout group sessions Rod Bond, FMN

11.35–12.45 Breakout group session 1

12.45–1.45 Lunch*

1.45–2.05 Mathematics in the workplace: symbolism and meaning Celia Hoyles, NCETM

2.05–3.00 Breakout group session 2

3.00–3.30 Report back from the breakout groups

3.30–4.00 Discussion of questions raised by breakout groups

4.00 Tea and depart

* MEI staff will be available to demonstrate MEI online learning resources for the Engineering Diploma

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Conference delegates Working together, IET and MEI set out to invite an appropriate mix of people from the worlds of engineering and mathematics education. Most of those invited fell into one of a number of categories.

• Significant stakeholders from both academic and industrial branches of the engineering community, including people with an interest in educational issues

• Key contacts in the engineering institutions and other key central bodies • Engineers working in industry • Mathematics educators with an interest in engineering • People supporting diploma teachers with level 3 mathematics

The organisers were very pleased with the outcome, and would like to thank the delegates most warmly for their contributions. Delegates put genuine intellectual effort into the day, and the breakout sessions in particular. The organisers hope to do justice to this effort as the ideas from the conference are followed up.

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Executive summary Introduction The conference considered ways in which mathematics is used in the engineering workplace and how increasing awareness of its use could be harnessed in seeking to improve mathematics education. One question is whether the mathematics curriculum is sufficiently influenced by the way the subject is used in the workplace. Related to that is the question how mathematics teachers and engineers can improve mutual understanding.

Setting the scene Three contributions set the scene for a day of lively discussion and interaction. First, Lee Hopley of the Engineering Employers Federation set out the importance of skills, especially mathematical skills, for the engineering workplace. Though engineering is facing difficult conditions we can be optimistic about its significance in future. Engineering has a big role to play in finding the solutions to societal challenges, leading to an increase in demand for more highly skilled workers. Matthew Harrison observed that the Engineering Diploma is important because engineering businesses now seek engineers with both technical understanding and enabling skills. The Engineering Diploma develops both. Importantly, the compulsory mathematics unit requires learners to solve engineering problems. The optional ASL mathematics unit requires learners to use and apply their mathematics in engineering contexts, thus making them more capable and effective as developing young engineers. Celia Hoyles described the work of The Techno-mathematical literacies project, which aimed to investigate and characterise the mathematical needs of employees in modern workplaces. The way to improve understanding in such contexts is not to provide mathematics teaching, but rather to make more visible some of the mathematics on which the processes in the factories depend. The project was successful in working with employees to enable them to appreciate some of the key concepts underlying their work.

The four workshop activities Rod Bond introduced the workshop activities by describing his work with ambassadors from industry and commerce. The enthusiasm and flair of these ambassadors in presenting the mathematics they use in their work is inspiring to learners, and shows that mathematics is essential to the success of these powerful role models. Rod explained that the breakout group sessions would start with a presenter outlining an activity using mathematics that is part of their job. Delegates were asked to develop ideas about how to produce a learning resource for diploma students that explains this activity and the mathematics that it involves - an exemplar.

Taking things forward The IET and MEI will take forward key ideas and proposals raised at the conference. Doing this will involve extending existing plans and developing new strands of activity, encompassing the worlds of both education and industry.

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Setting the scene - contributions from the principal speakers

The importance of skills in the engineering workplace

Lee Hopley - Head of Economic Policy at the Engineering Employers Federation

The context for engineering employers Skills are very important to EEF members. EEF is a manufacturers’ organisation, with six thousand member companies in engineering, manufacturing and technology based businesses. It provides a range of services to twenty thousand other companies. The engineering industry contributes around 70 billion pounds to the UK economy every year. Directly it employs over a million people in the sector and many more indirectly. It has a more highly skilled workforce than the economy as a whole. Over half the people employed in the engineering sector are qualified to level 3 or above. Over the past decade the rise of low cost competition from around the world has really pushed engineering companies to raise their game and look for new ways to add value. Conditions are difficult at the moment. The UK and indeed the world economy have got a number of fairly substantial challenges that we need to rise to and clearly engineering has potentially a substantial role to play in finding solutions to them. The question for the UK and for the engineering sector in the UK is will we be able to meet these challenges with our own domestic capabilities and what does this mean for our skill needs.

Skills that are key to the sector’s success Core engineering skills are clearly central to our members’ skill needs but engineering companies need a much broader range of skills than simply science and engineering. Look at the sort of occupations involved in managing supply chains. Forecasting demand, the stock control that that requires and managing lean manufacturing techniques, for example, all require a very broad range of higher level skills; with that in mind the issue of talent management has not gone away with this recession. We can’t overstate how important it is for young people to have a good foundation in key skills, such as maths, science and technology; these are really the building blocks for future participation and achievement in the skills and qualifications that engineering companies depend on now and will indeed depend on in the future.

How education and training can address skill needs A survey we did last Spring showed that around 40% of engineering companies were recruiting for apprentices and that a similar proportion expected that demand to increase in the next 5 years. Similarly the introduction of adult apprenticeships has also been a really valuable programme for employers and employees looking to gain new skills and formal qualifications. Nevertheless companies have in the past struggled to recruit young people on to apprenticeship programmes. We should also be looking beyond the advanced apprenticeships and on to higher apprenticeships at level 4 and there’s certainly demand going forward from both individuals and employers for higher level apprenticeships. In schools, one of the most significant of recent developments is the introduction of the engineering diplomas. These new qualifications supported by industry provide another potential route to higher level skills which are directly applicable and relevant to the

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engineering sector. EEF has been supportive of the diploma. The content looks good but success will really depend on the consortia delivering the diploma to a high standard and working closely to ensure that young people receive a learning experience that demonstrates the relevance of what they are studying to real life applications. There is still progress to be made in some other areas. We are not seeing the hoped for improvements in numbers progressing to higher education, studying subjects like science, maths and engineering. We need more young people from the UK to progress in these subjects. There are signs that employers are working a bit more closely with higher education institutions in terms of curriculum development and also in developing short courses for their existing employees.

In conclusion Engineering is clearly facing some difficult conditions at the moment but that’s not to say that we can’t be optimistic about its role in future. We have some fairly significant societal challenges ahead of us and engineering has a big role to play in finding the solutions to those challenges. That’s also going to lead to an increase in demand for more highly skilled workers of all ages. Some progress has been made in recent years and that shouldn’t be understated but at the same time we can’t be complacent. The competition, not just for engineering business, but for also for talent is international and we fail to recognise that at our peril.

The importance of mathematics in the 14-19 Diploma in Engineering

Matthew Harrison - Director of Education Programmes at The Royal Academy of Engineering

The Engineering diploma The Engineering diploma is important because we need a steady supply of engineers and technicians to meet the huge challenges of our increasingly technological society. As a nation we have been good at providing a supply of such people to date, with a very welcome rise in the number of apprenticeships recently. Engineering is a career that offers something tangible to young people, so the diploma, which offers a progression towards engineering as a career, is very important to society as a whole. It’s good that 14-year olds can get involved in a structured programme where they can understand what they will do for two years with a clear decision point about their future after that. The diploma provides an authentic engineering education, an important part of which is the inclusion of mathematics. As part of my role as the director of Education Programmes for the Royal Academy of Engineering, I chaired the Southwark and Lambeth Partnership, a consortium set up to provide a pilot for the Engineering Diploma, for nearly three years. The partnership included seven schools, two FE Colleges, two local authorities, London South Bank University, the Royal Academy of Engineering and was supported by Gatsby’s Technical Enhancement Programme, Transport for London, and Tubelines. It is really important that employers are part of the consortium, and the diploma needs deep employer engagement. The diploma is available at three levels. At level 1 it provides a toe in the water, principally for students who are less academic. At level 2, it offers a challenging programme, two days a week, alongside GCSEs in core subjects, including mathematics. It is proving popular with learners. At level 3, the requirements include principal learning and options (which include broad and deep engineering topics, or courses outside engineering), as well as generic learning such as functional skills and a project. There is also a requirement for work experience, and employers are key to enabling this to be authentic.

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The diploma compares with the more established BTEC Diploma by having fewer modules, almost twice as much time for its principal learning, a much larger project and a more challenging workload.

Why mathematics is important The RAEng report, Educating Engineers for the 21st Century, RAEng, June 2007, based on the views of 500 UK engineering companies, stated that: ‘Engineering businesses now seek engineers with abilities and attributes in two broad areas - technical understanding and enabling skills. The first of these comprises: a sound knowledge of disciplinary fundamentals; a strong grasp of mathematics; creativity and innovation; together with the ability to apply theory in practice. The second is the set of abilities that enable engineers to work effectively in a business environment: communication skills; teamworking skills; and business awareness of the implications of engineering decisions and investments.’ The Engineering Diploma has been constructed closely to match these expectations. The three engineering science units, one mathematics unit and three engineering practice units reflect the need for technical understanding. The two contextual units reflect the need for enabling skills. The mathematics unit, a month’s study, requires learners to solve engineering problems using:

• Algebraic methods • Trigonometric Methods • Elementary Calculus Techniques

Students should also be able to use Statistical Methods to display engineering data. This gives a firm foundation, but is not sufficient for a student who wants to go on to an advanced apprenticeship or an engineering degree. For this reason, an additional specialist unit in Mathematics for Engineering has been developed and accredited by OCR. It requires a considerable amount of advanced mathematics to be covered. More importantly, it requires learners to learn to use and apply their mathematics in engineering contexts, thus making them more capable and effective as developing young engineers. The course is designed to develop students’ self-efficacy and confidence.

The importance of exemplars Being confident that you can use and apply the skills you have learnt is to have a sense of self-efficacy. This is what employers want when they seek technical understanding coupled with enabling skills. Self-efficacy requires ‘practices that deliver authentic experience relevant to the student’s domain of activity’. The mathematics exemplars that the RAEng is developing are authentic because they are grounded in current engineering practice. Because the exemplars are about engineering, even though they are intended to exemplify the use of mathematics, they reinforce learners’ principal motivation for learning. Learners can see how others use mathematics effectively and attempt to emulate their practice.

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Introduction to the breakout group sessions

Rod Bond – Area Coordinator, Further Mathematics Support Programme

Bringing reality to the mathematics curriculum I work with teachers and students and have a mission to get the students to take a greater interest in engineering, science and technology. The most successful way I know is to work with ambassadors from industry and commerce, whose enthusiasm and flair in presenting the mathematics they use in their work is inspiring to learners, showing them that mathematics is essential to the success of these powerful role models.

Effective workshops Key factors for exciting, effective workshops for young people are:-

• Getting the level right and providing a clear and interesting activity • Getting the structure correct – young people enjoy active sessions • Ensuring materials are clear and attractive • Being enthusiastic

Today’s task In your breakout groups you will work in teams. Your presenter will outline an application of an activity using mathematics that is part of their job. Your task is to develop ideas about how to produce a learning resource for diploma students that explains this activity and the mathematics that it involves. We shall call this resource an exemplar. It is vital that the exemplar is plainly authentic in the way it describes the workplace activity, interesting in itself and inspirational in terms of its effect on mathematics learning.

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Mathematics in the workplace: symbolism and meaning

Celia Hoyles - Director of the National Centre for Excellence in the Teaching of Mathematics

The Techno-mathematical literacies project The project aimed to investigate and characterise the mathematical needs of employees in modern workplaces. This often involved interpreting computer outputs. It focused on the intermediate skills level eg team leaders in manufacturing. The way to improve understanding in such contexts is not to provide mathematics teaching, but rather to make more visible some of the mathematics on which the processes in the factories depend. The project was successful in working with employees to enable them to appreciate some of the key concepts underlying their work.

Statistical process control One example of the work we did was to help people in the automotive sector with statistical process control. The whole of this work depended on the use of the normal distribution. A particular component should be fitted in a certain position, but this will vary to an extent. It is critical to be able to distinguish between random variation and a significant change. In the industry we worked with, all points measured as falling outside the control limits have to be investigated and explained. Cp and Cpk are process capability indices: Cp a measure of spread; Cpk a measure of spread and position. However, these measures in the factory are used as single number evaluations of how well the process is running. The statistical background is often poorly understood, and some workers did not accept that the measures were actually based on the data – rather they thought these measures might be related to an arbitrary target. The algebraic definitions of Cp and Cpk were not readily understood by workers.

Unveiling the meaning of the indices By using what became known as Technology Enhanced symbolic Boundary Objects (TEBOs), the project team enabled the workers to focus on the important mathematical features of the indices, so that they began to understand what these meant. It took months to produce the tool that finally made this possible, so that workers could use understanding and not rules of thumb. For learning to take place it was crucial that the contexts in which we described the necessary concepts were completely authentic.

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Summary of the delegates’ work on exemplars Chairs of the breakout group discussions were asked to ensure that their groups considered how to use the mathematics and the context presented to them to produce ideas for an exemplar of how mathematics is used in the engineering workplace. In particular, chairs were asked to invite their groups to include all kinds of technology and media as means of making the exemplar more immediate and compelling.

Using capstans

Led by Helen Meese - Babcock International Group Helen introduced the question of how a heavy load may be controlled using only a modest force by a rope passing over a capstan. The contexts for this include refuelling at sea and controlling a heavy object that is deployed from an aircraft at the end of a cable.

Rapporteur - Jamie Curry We would like to start by using video to really engage the students. The Capstan is used to control a refuelling hose deployed from one ship to another, and also on some aircraft, and both would form excellent contexts for the initial video. Then we recommend exploring aspects of the model in the classroom. One very important idea is to get students to collect suitable experimental data using a simple capstan constructed by winding rope round a drum. This will help them build a basic understanding of the components and also the equation. This lesson will follow on from others in which the basic mathematics that they will need will have been covered including basic logarithms and some basic work on friction - blocks sliding on slopes for example. Then we can introduce the topic without overwhelming the students as they will understand the meaning of the symbols and the meaning of the mathematics. The equation that actually governs the situation, 2 1eT T μβ= is that the load is equal to the product of the tension at the other end of the rope and the exponential of µβ where µ is the coefficient of friction and β is the angle that you wound the rope around the capstan. What we wanted the students to look at is rearranging the equation and taking natural logs of both sides and coming up with something that they could take away and investigate in an experimental way. One of the ideas we came up with is if we use a newton meter at one end we could then quite easily change the angles so that students could investigate how that angle changes the tension they need on the other end in order to carry a constant weight at one end. We thought it a very rich task and there is a great deal we could actually do with it.

Students could do modelling on spreadsheets, and experiment with these, gaining some IT knowledge. If you do an experiment like this you could actually end up with a design graph as long as you know the ratio of loads you want to use, the ratio of tensions on either side of the capstan. You would then very quickly be able to pick a value for the angle you have wound it through. We would want to end up by using what the students have developed, the design graph to look at problems about winding - one of the ones we looked at was if we had a baby holding up an aircraft how many times did we need to wind the rope round the capstan to permit that? Although this sounds completely implausible, it would bring in discussions about how plausible the model is, and whether there are actual physical limitations to the model; this is a very important discussion to be having.

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Collecting rain in gutters

Led by Chris Robbins - Grallator

Rapporteur - Pat Morton The group considered a simple task about rainfall and guttering which Chris Robbins set up beautifully. I recommend that you go through the slides that he has provided (see http://www.mei.org.uk/files/pdf/ietmei/ChrisRobbinsPresentation.pdf ) because there are lots of really useful references about where to get interesting real life data. First of all we felt that this was a very rich task which could be used to tackle level 2 and level 3 content in both mathematics and engineering. The task was too good to use in an enrichment session because you could get some real hard core teaching out of this and it would be a shame to waste it. We felt that we wanted to create a resource pack to use alongside this because guttering probably doesn’t catch a 16 year old’s imagination, but once they start thinking about rainfall, water-capture, water-management, flooding, and avoiding waste of precious natural resources, you can engage them. For this reason, one of the things we would like in the resource pack is some sort of video which might already be around from the BBC or YouTube. We wanted the students to be able to communicate the mathematics in the engineering environment so an initial activity would be to start to get the students really engaged in this task and to begin to seek the mathematics. We wanted all of the resource packs to be available online so that they would be easily accessible. The only thing that we couldn’t imagine getting online were the off-cuts of the guttering; we felt that we would have to go to a DIY store for those. We looked at the task and we looked at the mathematics in it and we found things from basic ‘count the squares to find the area’ through to differential calculus looking at flows. We started on a dynamic system where we had water going into the gutter and water out of the gutter We found a very rich environment which would go well into post-16 mathematics and could lead to Further Mathematics as well. So for those students who are studying Mathematics and Further Mathematics there was something there. Equally well there was material for those students who were doing the engineering diploma, not doing mathematics as an additional study, perhaps only having a C grade or less at GCSE, there was work there that they could do. We would like to provide structured worksheets in the resource pack to help teachers address any diverse ability range that they may find. We felt that this would be a long task, possibly a week or two weeks of a student’s work programme and that you could diversify in different ways. The diversification of this task was quite important and if you were in a school environment and you were teaching perhaps a level 2 diploma you could get a lot of cross curricular work out of this, and equally well if you were an engineering level 3 student. If you looked at your curriculum map of the nine units you could find connections to do with moulding, manufacturing, building regulations, the history of products, how easy it would be to manufacture, and costing. We felt that teachers would engage with this problem without any problem at all. However there will be a need for well-crafted resources that teachers can pick up and use in such a way that they are concentrating on the pedagogy rather than having to research the content. Finally, we felt that what we were doing when we were using this task, although its context is relatively straightforward, is taking a practical problem and building a mathematical model of it, providing an exemplar of mathematical modelling.

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Automating data monitoring

Led by Angela Dean – Bombardier Transportation Angela presented a scenario in which data from a sensor on a train had to be analysed to monitor the performance of a particular component. She described the issue of automating the data analysis by constructing an algorithm. The algorithm needs to recognise accurately significant changes in the data that should be regarded as indicating an event has occurred that requires action.

Rapporteur - Chris Budd Most of us are going to go home by train, at least I am, and it would be very nice for that train to arrive without breaking down on the way. Trains are full of sensors as we’ve just learnt and Bombardier are interested in how they can take data from those sensors and interpret these in order that they can work out if a train is going to break down and if they need to send a technician or not. So the question that will be posed to the students doing this problem is how can you take real engineering data and reach a really useful conclusion that will help inform the people operating the trains as to whether to tell their train engineers to fix it or not. This activity is an object lesson in the fact that if you look at data on their own as a mathematician you are not really going to make very much progress. It is very important when using data in an engineering context for mathematicians and engineers to work together which is one of the main points we want to get out of this exercise. First of all we would start the students off by telling them what the story is here to help them get their heads round what’s going on. Then we’d get them to plot lots of dots on a piece of paper and first of all we had to identify the axis. This axis corresponds to charging pressure and these are a series of measurements of the pressure within a turbocharger, which is feeding hot, pressurised air to the engine, at high-speed conditions. One of the first things we can do after plotting it is to ask the students if they can spot anything interesting - as you can see there are gaps in various places and those gaps probably represent week-ends because they are 7 days apart so that’s a nice thing to ask. So they’ve got the data and then the question is what story can you learn from the data?

We start by doing a bit of mathematics and the first bit of mathematics the students have to do when they see any data is to identify the outliers and maybe try to explain them. What are the outliers, can you just eliminate them, are they just errors in the sensor? So the first thing you do is identify the difference between outliers and real data. The second thing is to try to identify some things that the data appear to convey are happening. Just looking at the data roughly you can see something happening - there is some sort of steady operating condition then it jumps, steady, jumps, jumps, jumps again and so on. So you start to tell a story from

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the data so that the students have to start to look at the data and interpret the story and at this point you bring in the engineer. The engineer can say ‘Well yes that corresponds to the train operating in a normal state so that’s fine. This bit here is not exactly normal but it’s OK. There is a problem, but the train is functioning. However, that’s where the train is unable to carry on and you need to call out the technicians’. Such comments immediately get the dialogue going between the mathematics and the engineering. The next question is ‘OK now we know what’s going on from the engineering point of view, how can the mathematics help the engineer?’ And these are the questions the students have to answer. The answer is they need to be able to find when the jumps are, when the train is operating away from it’s normal state, which gives you a bit of opportunity to work out things like the mean and standard deviation from the data here, and then you can have a discussion about how you identify a jump. A jump is obviously when you get a change in the data which is more than, say, one standard deviation away from the mean but, because of the outliers, that change has to be sustained not only over one data point but over several data points. So that gives you a way of identifying a specific objective thing that you can tell the engineer. I think that would be a very useful point to get to in the lesson and then you can discuss how you might automate that in terms of, say, a moving average and working out differences in the data points. The first thing is do make sure when you look at the data that you can represent these and understand what all the bits mean; secondly look at the story and emphasise the fact that mathematics and engineering work hand in hand, and then identify lots of opportunities for mathematics, leading finally to the recommendation that we make to the train company.

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Drug concentrations in patients’ bodies

Led by Steve St Gallay - AstraZeneca

Rapporteur - Jacqui Zugg We had a very good introduction from Steve explaining how as a chemist in the pharmaceutical industry he gets a lot of data on what drugs do and looks at what dosages to use and how often a patient needs a dose in order to get the required effect. This involved lots of really horrible formulae to start with, but we found a lot of very exciting mathematics in looking at the effects of various patterns and quantities. Basically you’ve got the formula for the plasma concentration, the plasma in the blood, the minimum concentration to actually do some good. Because of the exponential functions and also because later on the analysis actually uses sequences and series, we felt that it is something that students can really only access when they have done the core 3 of A level Mathematics, so it would be best to use this exemplar with, say, further mathematicians in year 12 or single mathematicians in year 13. We were looking at making sure they had met exponentials particularly. We would start off with either a real person coming in to talk about the situation or else a video introduction which explains the problem. Although some of the equations might look a little scary, when their meaning was explained the situation was so much more accessible. I really think it is important to have someone in the industry explaining that and having an on-line resource bank of materials to draw on. Then, when presenting the problem you could actually simplify things quite a lot. If we go back to the equation with the constants,

elC( ) e e ak t k tt K K− −= − , you could just start off with adjusting the constants rather than all the more detailed variables and the information pack would have information about those constants. We felt that it is actually very important that the students could relate to the detailed meaning of the equation. To help this we could get some information about the legal drugs they are likely to use, for example paracetamol or ibruprofen, and when they actually got the data and started to look at these they could start off by looking at the initial single dose and changing some of the parameters and seeing what different outcomes there were. The work would be mostly graphic and so the students would get a feel for using the equations but also they could see how changes affect the graphs. We could also export the data into a spreadsheet and use a graphical programme like Autograph so having had an introduction giving students a chance to get a feel of what changing the parameters does on a single dose, we would then build the model up to look at multiple doses. We talked about creating misunderstanding, about people thinking the more they took, the better it was for them. Another aspect that we could look at that came out of the graph was that the area under the graph effectively represented the amount of toxicity in the body and you could look at areas in the graph and discuss that as an aspect as well; you could extend the problem that way and perhaps in the beginning when the problem is introduced the parameters which one had to work with, and the limitations and things you had to take account of could be listed. That is something you could go back to along the different stages of the problem so you remind the students that they are actually working to a design specification which would make a bit more of a real life application. Another thing about creating misunderstandings is dealing with the toxicity and looking at some of the misconceptions about what effects you might predict; some of these things are counter intuitive, but you could address some of these issues. The multi-dose model actually ended up with quite a long sequence which would be very frightening to a student but was a geometric sequence which condensed down into something surprisingly simple. We felt that this offered quite a good learning opportunity in applying real mathematics in a school setting. There was a lot of mileage in this problem and perhaps once students had got to the stage where they had looked at the multi-dose model you could actually then ask what parameters you would need to create a drug that had to produce particular effects.

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Action plan for IET and MEI This section sets out IET and MEI plans and strategies to take forward the ideas and proposals raised at the conference. In terms of publicising and disseminating the outcomes of the conference, this report will be put on both the IET and MEI websites. We will be delighted if other organisations offer to provide links to the report from pages within their own sites.

Education Through the Further Mathematics Support Programme (FMSP), MEI will work with schools and colleges to support level 3 mathematics in diplomas. The support provided will include mentoring teachers and providing online learning resources for learners that include significant engineering contexts. MEI will take the ideas developed during the conference, and will seek to develop exciting exemplars of mathematics being used in the workplace that take advantage of the opportunities provided by the electronic environment in which they are stored. This programme of work is to be funded by The Royal Academy of Engineering. The IET will consider how the materials developed by MEI may link with the IET Faraday programme, once the reformatting of the Faraday resources and website are complete in early 2011, and what opportunities exist to exchange materials and cooperate on the improvement of existing resources and the development of others . The exchange of learning resources between the two organisations may strengthen the offering from Faraday, particularly at level 2 and 3 in schools and Further Education Colleges; whereas, drawing on the Faraday resources may enable MEI to better reach Key Stage 3 audiences. MEI will develop a programme of enrichment events across England involving collaboration between the FMSP and industrial partners. These events will be intended to inspire students to take up engineering courses at university. The IET will look to support such events by helping to promote them through the IET’s links to schools through its Education Partners, Local Networks and other awareness raising and dissemination routes. It will also help MEI to identify organisations involved with STEM Enhancement and Enrichment with whom they may wish to collaborate on the development and delivery of such events.

Industry MEI will seek suitable opportunities to develop courses for mathematics in the workplace. Such courses would involve use of MEI’s extensive online learning resources to back up learning. MEI will seek to support people at work in acquiring further qualifications such as A Level Further Mathematics. A pilot involving suitable companies would give the idea substance and encourage MEI to widen participation. IET will help promote such collaboration through its Business Partners network. The national presence of the FMSP will support such developments. IET and MEI will consider how best to support and promote mathematics learning in the engineering workplace through established and new networks and communities.

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Appendices

Conference delegates

Jonathan Akester Parliament Hill SchoolMohamad Askari Kingston UniversityAnthony Bainbridge MEI trusteeStephen Baldwin University of Warwick, Institute of Education Vickie Bazalgette STEMNETJohn Begg MEI diploma supportKirsten Bodley STEMNETRod Bond MEI diploma supportRichard Browne MEISarah Bucknell MEIChris Budd Bath University/LMSTom Button STEM ProgrammeAlan Cossins NCETMJenny Cowan Diploma Development Manager at QCA Kevin Coxshall Jersey College for GirlsSue de Pomerai MEIAngela Dean BombardierHugo Donaldson IETStella Dudzic MEINolan Fell Camden School for GirlsAnna Finn Woodhouse CollegeBob Francis MEI diploma supportKevin Golden UWEKeith Gould Hertfordshire County CouncilRichard Green D&T AssociationMichael Grove HE STEM ProgrammeMatthew Harrison Royal Academy of Engineering Robert Heathcote Queen Elizabeth's Grammar School, HorncastleElise Heighway MEIStephen Hibberd Nottingham UniversityLee Hopley EEFCelia Hoyles NCETMGareth Humphreys MBDA UKHal Igarishi Engineering Diploma Development PartnershipStephen Lee MEIFred Maillardet Brighton UniversityFiona Martland Surrey UniversityHelen Meese BabcockClaire Molinario IETDik Morling Westminster UniversityPat Morton MEI diploma supportPhil Moxon MEI diploma supportGarrod Musto Kingswood SchoolRoger Porkess MEIPeter Price Newstead Wood School for Girls Andrew Ramsay Engineering Council UK

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Michelle Richmond IETChris Robbins GrallatorAditi Sharma Metronet RailSapna Somani Royal Academy of Engineering Richard Stedman York CollegeNigel Steele Coventry UniversityAlan Stevens IMASteve St-Gallay AstraZenecaCharlie Stripp MEIChristine Townley Construction Youth TrustAdrian Waller ThalesJane West MEI diploma supportJohn Williams Gatsby Charitable FoundationDavid Youdan Institute of Mathematics and its Applications Hisham Zakaria IETJacqui Zugg North London Collegiate School

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Acronyms and other abbreviations used in the report D&T Design and Technology EEF Engineering Employers Federation FE Further Education FMSP Further Mathematics Support Programme HE Higher Education IET Institution of Engineering and Technology IMA Institute of Mathematics and its Applications MEI Mathematics in Education and Industry NCETM National Centre for Excellence in the Teaching of Mathematics RAEng Royal Academy of Engineering STEM Science, Technology, Engineering and Mathematics STEMNET Science, Technology, Engineering and Mathematics Network UWE University of the West of England

Published by MEI and IET Available online at: www.mei.org.uk www.theiet.org ISBN 978-0-948186-22-6