What's the problem?
Common errors in Decision
Maths
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MEI conference 2008
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What they do wrong (D1)
What’s the answer?
I really wish I knew! Here’s a brief guide to
some of the techniques which I have found
useful
Some Students
„ find it difficult to give reasons or explain logic
„ don't know the packing algorithm
„ don’t know the network algorithms
„ can’t spell (Dijkstra, Kruskal ……)
„ don’t draw neatly or label diagrams fully
„ don't understand about rejecting numbers in simulations
„ find Linear programming difficult– especially getting constraints
from words
„ shade the wrong side of the line (some can’t draw a straight line)
„ Mix up the x and y axes
„ Waste a lot of time solving simultaneous equations instead of
reading solutions from an accurate graph
What they do wrong (D2)
Some Students
„ find it difficult to give reasons or explain logic
„ don’t know the network algorithms
„ Know the network algorithms but forget which does
what
„ don’t draw neatly or label diagrams fully
„ Can’t do decision trees
„ Don’t know the difference between a switching circuit
and a logic (or Boolean) circuit
„ Forget to subtract surplus variables and add artificial
variables
They under-estimate the rigour required
You under-estimate the rigour required
They don't show working (there is little credit for just
writing the right answer)
They are not used to learning facts and processes
They are not used to explaining ideas in maths
They get confused by the number of algorithms
General points
DO
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seek clarification if you are unsure about anything
Put diagrams on a computer – it will save you hours in the
long term and avoids spending time drawing on the board
every lesson
Use lots of colours (and encourage them to). Colour coding
diagrams makes them easy to read and memorable
Play games
Insist they learn which algorithm is which
Make sure you know how the algorithms are applied and their
uses in the “real” world
Use context - where in industry are these techniques used
Work through as many past paper questions as you can with
the students
1
It works for me
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It works for me
Treat the course as a series of modules
Assess at the end of the modules
Try to get a cascade chart from a local firm
(builders) or an example of where LP is used
Talk to the business studies teachers about CPA
Be picky – make them write every detail of the
solutions
Use correct names for algorithms and insist that
they do too
Use short presentations so they get used to
explaining ideas.
Accept
solution
Yes
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Encourage students to write their own
algorithms
Power points can be very useful in Decision
Maths
Learning Decision through active learning,
group work, discussion and problem solving
is much more effective than working from
textbooks
Teach algorithms last!
The Modelling Cycle
Exam advice
Real life Problem
No
Make simplifying
assumptions
Review
„
Marks
‰
‰
Compare the
solution with reality
– is it realistic?
Interpret the
solution in terms of
the original problem
‰
Define variables and
decide on the
mathematical
techniques to be used
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B
D1
37
D2
20
M
A
15
20
20
32
From M1A1
to M1 A5
Solve the mathematical
problem
Exam advice
„
B – independent
M – method
A – accuracy (dependent on method marks)
Most marks (70%+) are for clearly demonstrating
use of the algorithms, very few of the marks in the
exam go on answers.
Examiners are trying to give as many marks as
possible
Students often show signs of running out of time
There are ‘extension’ marks on every paper –
usually at the end of long questions
Poor setting out and illegible handwriting makes it
very hard to give marks; clear working is absolutely
essential.
Examiners report
D1 – graphs and networks
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Candidates do not know about Eulerian/Hamiltonian cycles
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Most candidates were very poor at showing what they were
doing. Many did not show their selections, without which it cannot
be inferred that Prim is being applied. Others failed to indicate
their order of node selection.
Most candidates were unable to give a practical application for a
minimum connector.
The majority did not seem to understand what was meant by "a
practical application", and many gave as their answer the order in
which they had selected arcs. Of those that did understand, few
of their imagined applications could be said to be "connecting" in
any way.
„
„
2
Examiners report
D1 – networks
„ The instruction "… explain how your working …" caused
difficulties. All that was required was a reference to working
values, or to shortest paths to neighbouring vertices.
„ This was well done, although all the usual faults were to be
seen. Apart from those producing solutions without clear
evidence of the use of the algorithm (especially correct
working values and only correct working values
8
49
49
6
31
7
5 25
26 25
31
2
12
45
45
12
3
4
14
14
15
15
1
Examiners report
D1 – simulation
„ apparently believing that using two-digit random numbers
leads to an increase in "accuracy".
„ A substantial minority of candidates thought that the reliability
could be improved by using 3-digit random numbers.
„ The vast majority of candidates clearly did not read the
question. They saw a set of probabilities and set off in a
knee-jerk routine, simulating a supposed probability
distribution with 6 outcomes, despite the fact the probabilities
did not add up to 1, and despite the fact that nothing much
made sense thereafter.
Examiners reports
D1 - CPA
„ Not many candidates knew what a resource
histogram is.
0
Examiners reports
D1 Linear programming
„ How do we persuade candidates properly to define their
variables?
„ Some very tiny graphs were seen.
„ Many candidates showed weaknesses in interpretation in these
parts. Good candidates answered them quickly and efficiently.
Examiners reports
D1 algorithms
„ All that was required was the observation that
the algorithm was missing a stopping
condition – candidates did not need to
provide such a condition.
3
Examiners reports
D2 networks
„ Surprisingly few candidates were able to make the correct
deduction that the answer is 11 or 12.
„ A large number didn’t realise that the solution had to lie
between the upper and lower bounds
„ Many candidates came adrift here by incorrectly applying the
technique to the original network, instead of to the complete
network of shortest distances. The resulting "lower bound" is
bigger than the upper bound of found in part (iv). Candidates
who found themselves in this situation, without exception,
were unconcerned or oblivious to the problem, and went on to
make ridiculous comments in part (vi).
Examiners reports
D2 Decision analysis
„ the concept of utility completely passed most of the
candidates by.
„
Most candidates were able to score heavily
on this question, but a substantial minority
failed to produce a correct tree at the
beginning of part (i).
Examiners reports
D2 logic
„ Not all of those who gave the correct answer could justify it,
and there were even some who produced correct arguments
having given an incorrect answer.
„ The truth table work was well done. Most attempting it had 8
rows to their tables. Many had their entries completely correct
and some just made the odd slip.
Examiners reports
D2 LP
„ Choosing the wrong pivot always leads to a negative element
appearing in the last column. It is that which the ratio test,
when applied correctly, avoids. Candidates making this error
carried on with their negative RHS, blissfully unaware that
there was a problem.
There are many useful resources on the MEI
online resources [ www.mei.org.uk ]
„ You need a password to access the
resources.
„ If you haven’t already done so, register with
your local Further Maths Centre
[ www.fmnetwork.org.uk ] and this will enable
you to get a free password for staff use.
„
4
Decision Maths – common errors and how to avoid them
What's the problem?
•
•
•
•
•
•
They under-estimate the rigour required
You under-estimate the rigour required
They don't show working (there is little credit for just writing the right answer)
They are not used to learning facts and processes
They are not used to explaining ideas in maths
They get confused by the number of algorithms
D1 Some Students
•
•
•
•
•
•
•
•
•
•
find it difficult to give reasons or explain logic
don't know the packing algorithm
don’t know the network algorithms
can’t spell (Dijkstra, Kruskal ……)
don’t draw neatly or label diagrams fully
don't understand about rejecting numbers in simulations
find Linear programming difficult– especially getting constraints from words
shade the wrong side of the line (some can’t draw a straight line)
Mix up the x and y axes
Waste a lot of time solving simultaneous equations instead of reading solutions from an accurate
graph
D2 Some Students
•
•
•
•
•
•
•
find it difficult to give reasons or explain logic
don’t know the network algorithms
Know the network algorithms but forget which does what
don’t draw neatly or label diagrams fully
Can’t do decision trees
Don’t know the difference between a switching circuit and a logic (or Boolean) circuit
Forget to subtract surplus variables and add artificial variables
DO
• seek clarification if you are unsure about anything
• Put diagrams on a computer – it will save you hours in the long term and avoids spending time
drawing on the board every lesson
• Use lots of colours (and encourage them to). Colour coding diagrams makes them easy to read and
memorable
• Play games
• Insist they learn which algorithm is which
• Make sure you know how the algorithms are applied and their uses in the “real” world
• Use context - where in industry are these techniques used
• Work through as many past paper questions as you can with the students
• Treat the course as a series of modules
• Assess at the end of the modules
• Try to get a cascade chart from a local firm (builders) or an example of where LP is used
• Talk to the business studies teachers about CPA
• Be picky – make them write every detail of the solutions
• Use correct names for algorithms and insist that they do too
• Use short presentations so they get used to explaining ideas.
• Encourage students to write their own algorithms
• Power points can be very useful in Decision Maths
• Learning Decision through active learning, group work, discussion and problem solving is much more
effective than working from textbooks
• Teach algorithms last!
Exam Advice
• Marks
o B – independent
o M – method
o A – accuracy (dependent on method marks)
• Most marks (70%+) are for clearly demonstrating use of the algorithms, very few of the marks in the
exam go on answers.
• Examiners are trying to give as many marks as possible
• Students often show signs of running out of time
• There are ‘extension’ marks on every paper – usually at the end of long questions
• Poor setting out and illegible handwriting makes it very hard to give marks; clear working is
absolutely essential.
• There are many useful resources on the MEI online resources [ www.mei.org.uk ]
• You need a password to access the resources. If you haven’t already done so, register with your
local Further Maths Centre [ www.fmnetwork.org.uk ] and this will enable you to get a free password
for staff use.