Mathematical Modelling and Problem-Solving in Materials Engineering
(interim report)
Paul Hernandez-Martinez
Mathematics Education Centre
Loughborough University
Abstract The introduction of mathematical modelling and problem-solving projects
into the teaching practice of a second year Materials Engineering module at
Loughborough University aims to develop in students the skills that every engineer
should have as a result of a good quality undergraduate education. Modelling is at
the root of every engineering practice and therefore engineering students should
develop mathematical modelling skills as learning-for-life competences. Furthermore,
this is claimed to help raise the attainment and motivation of students to engage with
mathematics as part of their degree. The design of these projects will be made in
conjunction with lecturers from the Materials and the Mathematics department and
two student interns (one from the Materials department and one from the
Mathematics department). An innovative assessment scheme (Adaptive Comparative
Judgement) will be tried and evaluated in order to assess these particular skills (15%
of the module’s overall marks). This innovative scheme allows for peer assessment,
which will supplement the students’ learning of modeling and problem solving skills.
Background and Rationale:
The Mathematics Education Centre at Loughborough University oversees the
teaching of mathematics to most engineering programmes. There is a particular
concern about the mathematics modules in the Materials Engineering degree: the
percentage of students failing the first year module is around 25% and in the second
year is around 35%.
In order to reverse these trends and improve the motivation and engagement of
these students with mathematics, I developed a series of exercises where
mathematics was presented in the context of Materials engineering (e.g. polymers,
crystallography, various mechanics applications related to materials). These were
introduced in the second year module as part of the tutorial sheets, but not directly
assessed in the module. Although these “applied” problems were received well by
students (e.g. some students wrote in the module’s feedback: “Useful to see
mathematics being used in real world scenarios, these problems are helpful”, “Good
way to relate theoretical maths to practical applications”), the fact that they were not
part of the assessment of the module meant that these problems had a low impact in
terms of learning and motivation.
As a further development towards the aim of engaging students with mathematics
and raising the attainment in the second year module, mathematical modeling and
problem solving projects will be introduced during the Autumn of 2012 and this will be
assessed through a groups project (15% of the total marks of the module). The
adoption of mathematical modelling and problem-solving skills in the teaching and
assessment practice of this module intends to develop in students the skills that
every engineer should have as a result of a good quality undergraduate education.
Mathematical modelling is at the root of any engineering practice and therefore
engineering students should develop modelling skills as learning-for-life
competences (Niss & Hojgaard 2011; Niss 2010). It is also expected to improve
students’ attitude towards the subject (Falsetti 2005) and help raise the overall
attainment in the module by promoting deeper understanding of the concepts.
Planned Implementation:
The mathematical modeling projects will be introduced in semester 1 of the academic
year 2012-13 (October to December) and assessed through a group project at the
end of the semester.
The design of the projects and the assessment will be done in the summer of
2012 (July to September) with the help of two third year student interns (one from the
Materials department and one from the Mathematics department). The student
interns will work on designing suitable projects in consultation with Materials lecturers
and Mathematics lecturers. It is hoped that the Materials intern will bring experience
of engineering applications and an informed view of what his colleagues find
engaging (he was asked to gather some data on this before starting his internship)
and the Mathematics intern will bring his knowledge of mathematics to support the
mathematical side of the projects.
It is expected that three sufficiently opened projects will be developed, one for
each of the main topics of the module: Vectors, Differential Equations and Partial
differentiation. Each of these topics is normally covered in 3 weeks and that is the
time that will be given to students to work on each project in groups. Groups of 3 – 4
students will be formed to work together, and these groups will remain the same for
the assessment. Guidance and feedback (formative) will be given to students in
selected lectures or tutorials as they work on the projects. One more project will be
developed in order to be presented in the first lecture of the module, as a way to
introduce the topic of mathematical modeling and explain what is required of
students. Students will work on it for a prescribed time and the lecturer will then give
a plenary, addressing any doubts that students might have.
As an example of how these projects might look like, I previously developed two
projects that were never implemented in practice. These projects might serve as
exemplars or might be further developed into actual projects to be implemented.
These projects are:
Project 1
MANUFACTURING OF CONTAINERS
Suppose that you are working for a company that manufactures containers for
drinking water. They need to design glasses to keep cold water and also hot water
during a summer festival.
They ask you to test different materials and recommend the best one with which to
produce the glasses (remember that costs are also important). You have to use a
mathematical model to support your recommendations. State all your assumptions
and everything that you’ve done to reach your conclusions. Your job depends on the
quality of your report.
Project 2
DESIGN OF JUICE BOTTLES
A company wants to launch two bottles for a new flavour of juice, with the following
designs:
h
g
D2
x
D1
x
x
y
y
D1 is going to be made out of aluminium and D2 is going to be made out of carton.
Both bottles should contain each between 320ml and 340ml of juice (1ml = 1 cm3)
and are to be made out of sheets with dimensions 21 x 25 cm. Aluminium costs
£0.002 per cm2 and carton costs £0.0015 per cm2.
Remember that the cost of production is calculated as:
Cost = fixed costs + (average costs) x number of items produced
The fixed costs of producing bottles of D1 and D2 are, respectively, £500 and £375.
The average costs of producing bottles of D1 and D2 are, respectively, £(cost of
producing one bottle of D1 x number of bottles of D1 produced – 84) and £(cost of
producing one bottle of D2 x number of bottles of D2 produced – 63)
Bottles of D1 can be sold for £0.50 and bottles of D2 can be sold for £0.80, each.
Your task is to:
1) Specify measures for each design of bottle.
2) Produce a prototype of each bottle.
3) Find out how many of each bottle should be produced in order to maximise profits.
4) Present your findings to a group of “investors”.
Summative assessment of the modeling project. The assessment of modeling
projects cannot be subject to a traditional marking scheme. If we are to encourage
creativity and personalised answers to modeling projects, then we cannot expect to
have “similar” answers that can fit into one pre-determined answer. In this way,
students will be able to explore answers, and will have to justify their choices,
evaluate their answers and try to put an argument across, all of which constitute
modeling skills and which we would like to be able to assess. Students should also
be able to compare their work with that of other colleagues and judge the better one.
This is why, in order to assess these project we intend to use Adaptive
Comparative Judgement (ACJ), which is a method for assessing evidence of student
learning that is based on expert judgement rather than mark schemes. In this
method, assessors are presented with pairs of students’ work and asked to decide,
for each pair, which student has demonstrated the greater proficiency in the domain
of interest, in this case, modeling skills. The outcomes of many pairings are then
used to construct a scaled rank order of students. Two aspects of ACJ are of interest
here: it is well suited to assessing creativity and sustained reasoning, and has
potential as a peer-assessment tool.
ACJ has been trialed in a first year Calculus undergraduate module at
Loughborough University with good results (Jones and Alcock 2012), and research
into this method of assessment is ongoing.
Evaluation: Apart from the project assessment described above, instruments
developed by the University of Manchester (Prof Julian Williams) have been
developed to measure students’ capacity to talk about different aspects of modelling
and problem solving and students’ dispositions towards it. These instruments will be
applied at the beginning and towards the end of the first semester, and Prof Williams
will visit Loughborough during that time to do some work with the students in order to
gather further data. These instruments will provide evidence of how students have
developed modelling and problem-solving skills as well as evidence of changing
dispositions towards these.
It is clear from the above that collaborations (with Materials, Mathematics and the
MEC) and activity has been taken place in order to make this a success (the
inclusion of a modeling assessment project in the module was easily approved
because the value it could bring to students’ learning was evident), and that the
adoption of the practice is solidly based and under way. Clearly all this activity will
have impact not only within Loughborough University but also outside, as there are
interesting innovations in theory and practice.
References
Niss, M. (2010) Modeling students’ mathematical competencies. In R. Lesh, P.
Galbraith, C. Haines and A. Hurford (Eds.) ICTMA 13 Study. Springer: New York.
Niss, M. and Hojgaard, T. (Eds.) (2011) Competencies and Mathematical Learning:
Ideas and inspiration for the development of mathematics teaching and learning in
Denmark. Roskilde University: Roskilde.
Falsetti, M.C. (2005) A proposal for improving students’ mathematical attitude based
on mathematical modelling. Teaching Mathematics and its applications, 24(1), 14
– 28.
Jones, I. and Alcock, L. (2012) Summative Peer Assessment of Undergraduate
Calculus using Adaptive Comparative Judgement. In Iannone, P. and Simpson, A.
(Eds.) Mapping University Mathematics Assessment Practices, 63 – 74, Higher
Education Academy.