Whither Decision?
The attached discussion paper regarding the future of Decision Mathematics was
prepared in spring 2008. Events have since taken their course, and it now seems that
candidates for A level post 2011 may not be exposed to Decision Maths at all. This is
somewhat inconsistent with what I understand to be one of the aims of the A level
review – The applications in AS/A level mathematics should have a greater emphasis
on problem-solving, modelling, and mathematical communication, building on the
emphasis in the new Key Stage 4 Programme of Study, and be defined in the criteria.
I argue in the paper that it is in Decision Maths, much more so that in any other area
of mathematics, where that aim is addressed.
The paper has not been circulated widely until now, and most responses have been
directly concerned with the process of the A level review, rather than specifically with
the place of DM within the new structure. Here, though, is one response:
I strongly approve your motives
(Yes I disagree with some of what you say, BUT) I wholeheartedly support your
overall message.
It's taken so long to get 'Decision Maths' accepted and it would silly in today's society
to loose it.
All I'd want to add to your article is that the maths used in 'Decision Maths' is
probably the most widespread application of maths in today's world - especially in
industry, planning, commerce, business, management etc. The relevance to the
development, and greater use of technology is obvious. As is, as you say, the ability
to do something a computer can't do - which is design the model. 'Decision maths' has
a breadth and flexibility of application that is impressive and forward looking.
Please join in this debate, posting your comments in the Secondary Forum on the
National Centre for the Excellence in the Teaching of Mathematics (NCETM) web
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Keith Parramore,
University of Brighton
November 2008
Whither Decision?
April 2008, Keith Parramore
There will shortly, one way or another, be (yet) another review of A level
mathematics. This document represents my thoughts on some of the issues relating to
Decision Mathematics.
Much of my input in this area of late has been through the MEI specification, so some
of the detail is written from that standpoint. Whilst accepting that there will be (and
should be) differences between specifications, I would also hope that there can be
agreement on significant aspects.
Why Decision?
My first justification for DM is mathematical modelling. In putting emphasis on
modelling, I would not wish it to be thought that I have no regard for other elements
of the mathematics syllabus, particularly in relation to the other applied areas. Apart
from anything else, I think that it is desperately sad that we are producing so few
individuals who have any feel for how their physical world operates. On the other
hand, I think that it is excellent that so many have had their mathematical educations
extended by the need to study statistics.
But we all model − that is arguably the defining characteristic of consciousness. If I
were to ask you to consider rearranging the furniture in a room you would not start to
move things. You would start to move mental images in an abstract model of the
room. I would argue that to be the essence of mathematics. Yet paradoxically we do
hardly any teaching of it within mathematics. I see plenty of evidence of modelling
skills being developed in other subject areas − but not in mathematics.
How is that? Because in mathematics we teach models not modelling! The
overwhelming majority of the population we stuff full of the real numbers − a model
(which, arguably, is just a mathematical convenience and bears no relation to the real
world − but let’s not go there now!). Of those that survive this, some go on to be
schooled in Newtonian mechanics, which consists of but one model − that force,
acceleration, velocity and position may be regarded as vectors, and that F  dtd  mv  .
Students do no modelling − they just apply this model. In statistics there are more
models − Bernoulli, binomial, Poisson, normal, … − but my point is the same.
Given this, it is hardly surprising that, even by the time good mathematicians are
graduating, there are still very few of them who have good mathematical modelling
skills.
And of course, it is difficult!
Decision Maths offers the possibility of a greater focus on modelling. It was there in
much greater measure when we had coursework. That proved to be unsustainable, but
we still examine modelling. Where else, for instance, do students have to think about
what their variables should be?
My second justification lies in the context of Decision Maths − “real world”
problems. This applicability is self-evident, and was also more pointed when we had
coursework − when students had to find their own applications of the techniques. It
might be argued that, in retreating to assessment by examination, we have lost sight of
this. There is a tendency for examination questions to become ever more recondite.
Indeed, some of the syllabus areas (in some specifications) give this tendency a head
start. For instance (hobby-horse), the Hungarian algorithm is an opaque algorithm
which converts one intractable problem, allocation, into another, line-covering. But
more on that below.
What is the content?
So Decision Maths is about the modelling of real world problems, often in a business
context. The techniques taught in solving the resulting mathematical problems are
algorithms. These can seem to constitute something of a rag-bag. But there is a
consolidating theme which runs through the subject − constrained optimisation. This
is transparent in linear programming, but is also sitting below most of the major
subject areas. Shortest path, critical path, matching, allocation, transportation and
dynamic programming problems are all easily formulated as linear programming
problems, as are many, many other real world problems. The minimum connector
problem is more difficult to formulate, but it is certainly constrained optimisation.
Similarly with the TSP. Simulation is all about trying to find optimal values.
The current MEI Decision Maths (which I examine) was designed around this central
theme, and then packaged into definable areas, with careful regard to appropriate
levels. Thus in the AS there are sections on:
Graph Theory
elementary, yet often very useful in modelling
Algorithms
to lay the technical foundation
Networks
minimum connector and shortest path − for their
immediate applicability
Linear Programming
the “glue” − but only in graphical form at AS
CPA
exactly right for AS
There are two A2 papers in MEI, wherein the constrained optimisation is further
developed. In D2 the theoretical basis is represented by the simplex algorithm. This
includes the beautiful mathematics whereby the basic algorithm is used to find an
initial feasible solution to a problem in which the origin is not feasible − real
“bootstrapping”! In D2C the candidates have a computer available for the
examination, and use an LP package to solve real problems. In both papers other
sections hang off this central trunk − further networks, decision analysis and logic in
D2, yet more networks, computer-based simulation (using a spreadsheet) and
recurrence relations in D2C.
How is it delivered?
The huge benefit of Decision Maths is that students can come to it fresh − it does not
need the algebraic baggage required by mechanics and statistics. Students can make
immediate contact with easily-understood problems, and can start modelling.
Surprisingly often students with poor technical skills can show good modelling skills
− but perhaps that’s not so surprising since there are many examples of great
mathematicians who fall into that category.
I also see two big disbenefits:

the lack of explicit coherence, in specifications, texts, teaching and assessing

the tedium in having students apply by hand algorithms developed for
computer implementation.
With regard to the latter point D2C has been extremely successful in showing what
can be done. Regrettably the take-up has been small, and the approach has not been
used in other subject areas which could benefit, particularly in statistics. Technology
has not developed as quickly as I had anticipated when we launched D2C. By
comparison, within a few years of Cockcroft reporting (1982?) that schools should
provide access to calculators, they were commonplace. That has not been the case
with laptops, at least not in the UK, and we are only now seeing prices begin to fall.
We are yet some years away in secondary education from being able routinely and
easily to provide class sets of full keyboard/full screen machines for LP and
spreadsheet work.
The way forward?
We shall have a new structure forced upon us by QCA following their evaluation of
the current trials. I think that is very bad news, but that it makes it all the more
important for us to plan our content and delivery very carefully.
I guess that we will all be forced into just two modules, a Decision Maths AS for
many, and an A2 for a few. I want to argue, as strongly as I can, for the Decision
Maths contents of the new specifications each to demonstrate greater coherence, and
each to embrace computing capability.
Structure
I think that we must embrace a strong discipline in choosing the content of our
specifications. I have heard colleagues defend the inclusion of some topics by
reference to students’ supposed enjoyment of them. I don’t want to be a killjoy, but I
really do not think that is good enough. What is the point of learning an algorithm
which is of no use, even if it is teachable and examinable? (Hobby- horse again − I’m
thinking Hungarian, but there are other examples. I’d even argue that Dynamic
Programming comes into the same category.)
I think that we need together to agree criteria against which to construct our contents,
and I would suggest the following:
 accessibility
is it at the right level for the unit?
 applicability
is it actually used?
 appropriateness
does it sit coherently within the material?
I’ve constructed an attempt at using these criteria to select content for two units in the
Appendix below.
Computing
I really do think that we must practise what we preach, and that we must embrace
computing power in both modules. We can’t do it yet with laptops, so we’ll have to
make do with powerful calculators. The algorithms we need are now well within
calculator capability, including simplex and a spreadsheet. But we need to move fast
to make sure that what we need is available in time. I think that we need to agree a
list, and then either commission someone to produce the goods (Adrian Oldknow?) or
get the manufacturers to submit tenders.
Again, I’ve had a go at listing what I’d like to see in the Appendix.
Conclusion
What do you think?
Please contribute your thoughts by, posting your comments in the Secondary Forum
on the National Centre for the Excellence in the Teaching of Mathematics (NCETM)
web portal.
Click here for a link to the Secondary Forum.
Please note that if you are not a member of the NCETM portal, you will need to
register to post your comment.
Adopting criteria does not need agreed action, but doing something about computing
capability does. If there is a degree of agreement on the need then we will need to
agree on what is wanted. So please include a wish list, and we will see if we can
achieve some consensus.
Keith Parramore
School of Computing and Mathematical Sciences
University of Brighton
k.parramore@brighton.ac.uk
Appendix
AS level
Algorithms
get them started on thinking algorithmically
to include some packing, some sorting, Prim, Kruskal and
Dijkstra
(The latter could be packaged under a “Networks” heading.)
Graphical LP
essential introduction to constrained optimisation
Basic Simplex
making it algorithmic
CPA
ticks all of the boxes, including being expressible as an LP
problem
Simulation
For me simulation just has to be in. It scores so highly on
applicability.
It is the technique which practising
mathematicians use more that any other − and it is concerned
with optimising under constraints.
Computing capability: sorting algorithms
finding a shortest path (using Dijkstra and LP) and
programmed for use in a CPA context
Prim (probably not Kruskal!)
an LP package delivering straightforward Simplex
a spreadsheet − for simulation.
A2 level
Further Simplex
Two-stage, and/or “big M” − beauty and power combined
LP applications
Loads of them − matching; allocation (who needs the
Hungarian when it’s so easy to formulate as an LP?);
transportation; set covering; network flows; etc,etc.
Integer programming Simple approach to solving IPs − (towards branch and bound).
Lots of use in solving problems, including such things as the
use of indicator variables in (eg) investment decisions.
Networks
TSP; route inspection.
Also, just because we can do it with LP doesn’t mean to say
that we shouldn’t bother with useful/transparent algorithms,
such as alternating paths for matchings and “max flow = min
cut” for networks. Also in for modelling capability.
Further simulation
more advanced stuff − simulating queues, epidemics, etc.
estimating the number of repetitions needed for a required
level of accuracy
Computing capability: Full LP capability (this is actually very easy to provide).
Spreadsheet as above.
Some topics I have excluded:
(a1 − accessibility
( − yes
a2 − applicability
 − no
a3 − appropriateness)
?? − iffy)
Graph theory
a1 
a2 ??
a3 
Logic
a1 
a2 ??
a3 
Recurrence relations
a1 
a2 
a3 
Decision analysis
a1 
a2 
a3 
Dynamic programming
a1 
a2 ??
a3 
Hungarian algorithm
a1 ??
a2 
a3 
Transportation algorithm
a1 
a2 
a3 
Game theory
a1 
a2 ??
a3 
Note 1:
I’ve marked the Hungarian algorithm as “iffy” under “accessibility”
because, although they can do it, they don’t know what they’re doing!
Note 2:
The transportation algorithm is an application of dual simplex, but that is
well beyond A2!
Note 3:
Game theory questions can be posed under LP applications.