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
1. List of Attendees
2. Acronyms used in the report
16
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
MEI Chief Executive
Michelle Richmond
IET Director of Qualifications
1
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
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
Celia Hoyles, NCETM
* MEI staff will be available to demonstrate MEI online learning resources for the Engineering Diploma
2
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.
3
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.
4
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
5
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.
6
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.
7
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.
8
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.
9
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, T 2 = T 1eμβ 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.
10
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.
11
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
12
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.
13
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 online 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,
C(t ) = Ke − kel t − Ke − ka t , 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.
14
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.
15
Appendices
Conference delegates
Jonathan Akester
Mohamad Askari
Anthony Bainbridge
Stephen Baldwin
Vickie Bazalgette
John Begg
Kirsten Bodley
Rod Bond
Richard Browne
Sarah Bucknell
Chris Budd
Tom Button
Alan Cossins
Jenny Cowan
Kevin Coxshall
Sue de Pomerai
Angela Dean
Hugo Donaldson
Stella Dudzic
Nolan Fell
Anna Finn
Bob Francis
Kevin Golden
Keith Gould
Richard Green
Michael Grove
Matthew Harrison
Robert Heathcote
Elise Heighway
Stephen Hibberd
Lee Hopley
Celia Hoyles
Gareth Humphreys
Hal Igarishi
Stephen Lee
Fred Maillardet
Fiona Martland
Helen Meese
Claire Molinario
Dik Morling
Pat Morton
Phil Moxon
Garrod Musto
Roger Porkess
Peter Price
Andrew Ramsay
Parliament Hill School
Kingston University
MEI trustee
University of Warwick, Institute of Education
STEMNET
MEI diploma support
STEMNET
MEI diploma support
MEI
MEI
Bath University/LMS
STEM Programme
NCETM
Diploma Development Manager at QCA
Jersey College for Girls
MEI
Bombardier
IET
MEI
Camden School for Girls
Woodhouse College
MEI diploma support
UWE
Hertfordshire County Council
D&T Association
HE STEM Programme
Royal Academy of Engineering
Queen Elizabeth's Grammar School, Horncastle
MEI
Nottingham University
EEF
NCETM
MBDA UK
Engineering Diploma Development Partnership
MEI
Brighton University
Surrey University
Babcock
IET
Westminster University
MEI diploma support
MEI diploma support
Kingswood School
MEI
Newstead Wood School for Girls
Engineering Council UK
16
Michelle Richmond
Chris Robbins
Aditi Sharma
Sapna Somani
Richard Stedman
Nigel Steele
Alan Stevens
Steve St-Gallay
Charlie Stripp
Christine Townley
Adrian Waller
Jane West
John Williams
David Youdan
Hisham Zakaria
Jacqui Zugg
IET
Grallator
Metronet Rail
Royal Academy of Engineering
York College
Coventry University
IMA
AstraZeneca
MEI
Construction Youth Trust
Thales
MEI diploma support
Gatsby Charitable Foundation
Institute of Mathematics and its Applications
IET
North London Collegiate School
17
Acronyms and other abbreviations used in the report
D&T
EEF
FE
FMSP
HE
IET
IMA
MEI
NCETM
RAEng
STEM
STEMNET
UWE
Design and Technology
Engineering Employers Federation
Further Education
Further Mathematics Support Programme
Higher Education
Institution of Engineering and Technology
Institute of Mathematics and its Applications
Mathematics in Education and Industry
National Centre for Excellence in the Teaching of Mathematics
Royal Academy of Engineering
Science, Technology, Engineering and Mathematics
Science, Technology, Engineering and Mathematics Network
University of the West of England
18
Published by MEI and IET
Available online at:
www.mei.org.uk
www.theiet.org
ISBN 978-0-948186-22-6