m e i . o r g . u k
Curriculum Update
Thinking of offering
Core Maths from
September 2015?
MEI and OCR will be
running free network
events to provide
information about our
Core Maths
qualifications and
introduce teachers to
the free resources
available for teaching
them.
Would you like to host
an event at your
school or college?
Please see our new
Core Maths page for
more information.
GCSE: Progress 8
measure
The DfE has
published information
about the Progress 8
measure, including
how grades from 1-9
GCSEs will be
combined with A*-G
GCSEs.
Click here to view
the document on the
DfE website.
I s s u e
What to believe - friends,
statistics or politicians?
It’s difficult not to know
about the upcoming
parliamentary election, with
every newspaper taking a
different angle every day, and with all
eyes on this week’s Budget
announcement. But when it comes to
deciding which way to vote, how do you
decide? How many people use
statistics to help them to form an
opinion? How many base their
judgement on personal experience?
I revisited this 2013 Telegraph article
following the publication of the Ipsos
MORI Public Understanding of
Statistics Topline Results April 2013.
Tom Chivers, the author of the
Telegraph article, reported:
“One thousand and thirty-four British
adults between 16 and 75 were asked
to choose between the following
statements:
Statistics are more important than my
own experiences or those of my family
and friends in helping me keep track of
how the government is doing
My own experiences or those of my
family and friends are more important
than statistics in helping me keep track
of how the government is doing
Forty-six per cent chose the latter. Just
nine per cent chose the former.”
Click here for the MEI
Maths Item of the Month
M a r c h
4 5
2 0 1 5
These figures are rather alarming taking
into account the narrow experience of
many voters, but, Tom Chivers
explains: “The trouble is, of course, that
people don't trust statistics because
other people use them to hide, rather
than illuminate, the truth. Once you've
been misled once, you're less likely to
trust people again.”
While statistics are extremely valuable,
they are also notorious for being a
means to make false and misleading
arguments or claims. Unless we have
good access to the data and know how
is was obtained, it is important that we
recognise that statistics can
misrepresent what is going on. In this
issue we’ll take a look at how statistics
can be used to represent and to
misrepresent claims.
In this issue

Curriculum Update

March focus: Misleading
statistics

Fast and Furious Problem
Solving

Crash Course: Recursion

Site-seeing with... Claire Baldwin

Teaching Resources:
KS4: Folding and Proof
Core Maths: The Parable of the
Polygons
M4 is edited by Sue Owen, MEI’s Marketing Officer.
We’d love your feedback & suggestions!
Disclaimer: This magazine provides links to other Internet sites for the convenience of users. MEI is not responsible for the availability or content of these
external sites, nor does MEI endorse or guarantee the products, services, or information described or offered at these other Internet sites.
Statistics that changed
the face of nursing
Diagrams
transcend
language
While it isn’t strictly
true to say that
Florence Nightingale
was the first to use
diagrams for
presenting statistical
data, she may have
been the first to use
them for persuading
people of the need for
change.
You can read more
about the evolution of
Nightingale’s
statistical diagrams
on the York
University website.
Robert Kosara’s blog
EagerEyes reflects
on the world of
information
visualisation and
visual communication
of data. His page:
Shining a Light on
Data: Florence
Nightingale provides
useful insight into
Florence
Nightingale’s use of
diagrams to
communicate data to
decision makers who
lacked knowledge of
statistics or
mathematics.
Many of us know of Florence
Nightingale as the founder of nursing as
a profession, but she was also an
accomplished statistician and graph
maker.
Through her work as a nurse at a
British hospital in Turkey in the Crimean
War, Florence Nightingale was a
pioneer in establishing the importance
of sanitation in hospitals. When she left
Turkey after the war ended in 1856, the
hospitals were efficient and well-run
with mortality rates no greater than
civilian hospitals in Britain.
On her return to Britain Nightingale
meticulously gathered data on how
soldiers had died, where and why.
Through her tables of statistics she
discovered that the majority of deaths in
the Crimea were due to poor sanitation
rather than casualties in battle, or poor
food or supplies. Nightingale wanted to
persuade the British government of the
need for better hygiene in hospitals.
She realised though that just looking at
the numbers was unlikely to impress
ministers, and sought a way to get
across her findings using graphics. To
show the number of deaths each month
and their causes, rather than using a
bar graph, Nightingale chose to use a
more arresting graphic that allowed for
easy comparison across the seasons.
For this graphic she used a variation on
the modern pie graph. This polar area
diagram has come to be known as a
“coxcomb.” The circular presentation
uses areas to represent the variation in
the death rate, instead of the length of
radial lines.
The blue wedges, representing death
by sickness, are far bigger than the red
wedges representing wounds. The
black wedges measured from the
centre represent deaths from all other
causes. Click here to view a larger
version of Nightingale’s Diagram of the
Causes of Mortality in the Army in the East.
You can see an animation of this
diagram on the Science News website.
Click here to view this animation.
Clicking on the bar chart tab of the
above interactive diagram clearly
demonstrates the limitations of using a
bar chart to represent the same data.
Florence Nightingale made extensive
use of this type of diagram to present
reports on the conditions of medical
care in the Crimean War to Members of
Parliament and civil servants who would
have been unlikely to read or
understand traditional statistical reports.
Learn more about Florence Nightingale
in Stella Dudzic’s MEI Conference
session on Friday 26 June.
“If the facts don’t fit the
theory, change the
facts.” (Einstein)
Burt’s impact on
educational
testing
Burt’s belief that
educational ability
was usually inherited
by children led to
testing all children in
their final year of
primary education to
see if they had the
academic ability to
attend a grammar
school or if they were
better suited to
technical or
secondary modern
schools.
It is now widely
accepted that it is
impossible to
measure the
proportion of
intelligence that
results from
inheritance and
upbringing.
The Intelligence Fraud
Sir Cyril Burt was a very well known
British psychologist and a leader in the
development of methods of data
analysis. Burt spent much of his career
trying to understand the link between
heredity and intelligence. He believed
that hereditary factors accounted for 85
percent of intelligence and developed a
study of twins reared apart to prove it.
The papers Burt published on the topic
were rarely questioned while he was
alive, but shortly after his death,
colleagues reviewing his work became
suspicious of the study's data.
In the early 1950's, Burt published
results from studies of identical twins.
Correlation coefficients for the IQs
varied between monozygotic (identical)
twins reared apart and dizygotic
(fraternal or non-identical) twins reared
together. Burt’s results showed the IQs
of identical twins reared apart were
much closer than the IQs of nonidentical twins. He concluded that
genetic factors were more important
than environmental factors in
determining intelligence.
However, three years after Burt’s death
attention was drawn to some apparently
puzzling features of his data concerning
the inheritance of intelligence. This led
to accusations of fraud, the major
charge against Burt being that
additional data on kinship correlations
of IQ, especially for identical twins
reared apart, reported to have been
gained over the period 1955 to 1966,
were fabricated. It appeared that Burt
often concealed information on the size
of samples and when they were
collected. The suggestion was that
estimates were adjusted to accord with
Burt’s preconceptions. It was also
claimed that Burt invented some of his
co-workers. Burt's work continues to be
controversial and the subject of
numerous books and journal articles.
You can read more about this
controversy in a Human Intelligence
article.
In his 1991 paper, John Hattie of The
University of Western Australia
describes the concerns of Leslie
Hearnshaw, an established British
historian of education who was writing
an official biography of Burt.
“Hearnshaw concluded that the data
which Burt used for his calculations
were poor and unreliable, that he made
a great many unexplained, careless and
inconsistent adjustments to the raw
scores of his tests with the results that
the figures are quite improbable. He
ultimately applied sophisticated
statistical techniques to scientifically
almost worthless data, with disastrous
results.”
However, Robert Audley said in a 1993
Times Higher Education article that
we should not dismiss all of Burt’s work
out of hand:
“Although some of the evidence he
published in his later years is of
doubtful scientific value, the
contributions he made during a long
professional life remain impressive, and
I believe it is misleading to continue to
hold him up as the icon of scientific
fraud.”
Using data in the
maths classroom
Using Big Data in
education
Using data in the maths
classroom
Here are links to
some publications,
articles and web
pages that you may
find useful:
The new National Curriculum brings a
change in emphasis in using data in
mathematics.
Getting schooled
in the ‘noise’:
learning about
learning using big
data
UK Big Data boost
as Alan Turing
Institute opens in
London
A world full of
data: Statistics
opportunities
across A-level
subjects
Information is
Beautiful: ideas,
issues, knowledge,
data — visualized!
Royal Statistical
Society Teaching
Resources
Significance
magazine
GCSE students will be expected to
interpret, analyse and compare the
distributions of data sets, rather than
creating or contrasting data.
A level students will be expected to use
big data sets, learn to use spreadsheets
and to draw inferences. AS and A level
Mathematics specifications for teaching
from 2017 will require students to work
with large data sets using technology.
Students will also need to be able to
use calculators to analyse subsets of
the large data set. What could this look
like in the classroom and how will it help
students understand statistics better?
MEI Resources
Data Sets
The new Data Sets page on the MEI
website provides data sets for teachers
of statistics to use with their students.
Information is given about the data and
an indication is given of statistical
techniques that may be useful when
working with the data set.
The Data Sets page lists sources that
offer a number of data sets with some
information about the data and
guidance about which statistical
techniques are useful for each data set.
Also listed are useful websites for
working with real data.
Natural history data sets
Chris du Feu has kindly
made data sets available on the MEI
Data Sets page; he has a keen interest
in data about birds and slugs.
Professional Development
Stella Dudzic, MEI Programme Leader
for Curriculum and Resources, and Neil
Sheldon, RSS Vice-President for
Education and Statistical Literacy, will
deliver a plenary about Using
Technology To Explore Large Data
Sets at the MEI Conference in June.
Here are some MEI Conference
sessions relating to statistics and the
use of data: click the link to find out
more about the session and the target
audience:
The new Higher Tier content
Teaching S1/Core Maths statistics
using graphing technology
Statistical insight through
simulation on a spreadsheet
Statistical insight through data
visualisation tools
Resources and Investigations: The
Normal Distribution and Probability
Plots
Resources and Investigations:
Correlation and Hypothesis Tests
Statistical insight through
simulation on a spreadsheet
Statistical insight through data
visualisation tools
Don't believe everything you read
in the papers
Fast and Furious
problem solving
FMSP Problem
Solving posters
You can download
the set of 6 posters in
print-ready PDF
format from the
FMSP website.
MEI and the FMSP
will have stands at the
upcoming ATM
Conference, the MA
Conference in April
and at the STEMtech
Conference &
Showcase at the end
of April - free posters
will be available!
Just as we were going to press, a news
story was published about the soon to
be released film Fast and Furious 7.
In a car chase scene in the film, the
character Dominic Toretto, played by
Vin Diesel, finds himself behind the
wheel of a powerful sports car (a Lykan
HyperSport)
on about the
45th floor of
the tallest of
the five
buildings in the
Etihad Towers complex in Abu Dhabi.
Toretto decides that he can ‘leap’ the
car across the gap into the building
opposite. This video shows what
happens next.
Do you think Toretto could pull this
stunt off if it were for real? The answer
boils down to a fairly simple maths
problem, and as it happens, MEI’s Phil
Chaffé was recently teaching in Abu
Dhabi. Phil was approached afterwards
by a freelance journalist who
challenged Phil to make the calculation,
based on the following assumptions:
The car is a Lykan HyperSport, which
weighs 1,400kg. Let’s assume there is
enough room in the building for it to
reach 100km/h, which it can do in 2.8
seconds.
The distance of the jump is about 50
metres.
How far horizontally would an object
weighing 1400kg travel with an initial
velocity of 100km/h, and how many
metres would it drop vertically for every
metre travelled horizontally? And what
is the formula that would demonstrate
this?
You can read the background to the
stunt, see photographs and read Phil’s
calculations in Jonathan Gornall’s
resulting article in The National, Abu
Dhabi Media's first English-language
publication. There is a graphic
describing the maths behind the
problem—click image to view larger
version.
Charlie Stripp, MEI’s Chief Executive
said: “It's excellent to see MEI
mentioned In this context, and a really
nice application of A level mechanics
and mathematical modelling.”
Mathematics teachers will be pleased to
know that Phil is in the process of
producing a set of problem-solving
enrichment posters for the Further
Mathematics Support Programme,
including one about this stunt. These
will be available in a few weeks’ time.
Crash Course:
Recursion
A maths and
computing puzzle
column written by
Richard Lissaman
This column provides
an introduction to the
programming
language Python
using maths puzzles
as motivation to learn
code!
In this month’s
column we’ll take a
look at an amazingly
powerful feature of
some programming
languages, including
Python, called
recursion.
Take a look at the code on the left of the screengrab below.
In the first line we define a function called sumintegers(n). The next two lines
are simple, they just say that sumintegers(1) is the value 1.
1) The next line is potentially mind-blowing! The else part of the if statement
requires the function to use itself! But we are still in the process of defining the
function!
Think carefully about what happens when sumintegers(4) is taken.
Since 4 ≠ 1, sumintegers goes straight to the else line and so will return 4 +
sumintegers(3).
Now Python needs to start thinking about sumintegers(3). Again since 3 ≠ 1 this
will return 3 + sumintegers(2).
Let’s take stock: sumintegers(4) returns 4 + sumintegers(3) and then Python
calculates that sumintegers(3) is 3 + sumintegers(2).
Python will now work out that sumintegers(2) is 2 + sumintegers(1).
But sumintegers(1) is just 1.
Piecing all this together sumintegers(4) is dealt with as follows
Crash course
February problem –
solution can be
downloaded from the
Monthly Maths web
page, or click the link
above.
4 + sumintegers(3) = 4 + 3 + sumintegers(2) = 4 + 3 + 2 + sumintegers(1) = 4
+ 3 + 2 + 1 = 10
Think about this carefully and you should be able to convince yourself that
sumintegers(n) is the sum of the first n integers, 1 + 2 + 3 +….+ (n – 1) + n.
Crash Course:
Python challenges
Coding resources
Computing At
School has produced
a new resource to
help teachers in
England get to grips
with the new
computing curriculum.
QuickStart
Computing is a
comprehensive,
national programme
designed to help
primary and
secondary teachers to
plan, teach and
assess this brand new
subject. Click here to
find out more and
download resources.
Here is another example. The Fibonacci sequence is defined so that the first two
terms are 0 and 1 and then any subsequent term is the sum of the previous two.
The function fib(n) below returns the nth Fibonacci number. Look how closely the
Python function matches the mathematical definition.
Problems of the month
1) Adapt the code in the first example above to define a function sumpowers(m,n)
which returns the sum of the m th powers of the first n integers, where n and m are
positive integers.
2) The first six rows of a mathematical object called
Pascal’s triangle are shown here.
Each element in the triangle is either 1 (when the
number is at either end of a row) or the sum of the
two numbers immediately above it (when the number
Make it Digital
is not at either end). Can you create a function in
On 12 March 2015 the Python that takes in a positive integer n and prints
BBC launched Make it the first n rows of Pascal’s triangle to the screen?
Digital, a major UKwide initiative. WideFor this problem you might find it useful to recall arrays in Python from the last
reaching content
column. Maybe you can define a function that will produce the nth row of Pascal’s
across TV, radio and
triangle as an array. Here is a reminder about arrays:
online will showcase
how Britain has
helped shape the

To create an empty array in Python called my array use myarray = [].
digital world, raise
awareness among
To add the number 5 to the array use myarray.append(10). This (first)
mainstream audiences 
element of the array is then referred to as myarray[0] and so could be
on why digital matters,
and inspire younger
printed to the screen using print myarray[0]. Further numbers added using
audiences to have a
myarray.append would then be referred to as myarray[1], myarray[2] and
go and get creative
so on.
with digital
technologies.
Site seeing with…
Claire Baldwin
Each month a
different member of
MEI staff will share a
couple of their
favourite resources it might be some
software, a website, a
printable download, a
book, etc.
This month’s
resources are shared
by Claire Baldwin, a
Central Coordinator
for the Further
Mathematics Support
Programme, which is
managed by
MEI. Claire has
specific responsibility
for Higher Education
Liaison and Gender
Participation in
Mathematics within
the FMSP.
One of my favourite
mathematics books in
recent years is
Professor Stewart’s
Cabinet of
Mathematical
Curiosities by
Professor Ian Stewart.
The book presents an
eclectic mix of classical
mathematical puzzles, journeys through
the history of famous theorems, and
even advice on how to identify a fake
coin using a set of scales.
I love the lack of chapters
or structure and the fact
that this miscellany of
mathematics originates
from a notebook which
the author started when
he was 14 years old.
My favourite puzzle involves
placing a long (2m) loop of
string over your wrist, putting
your hand in your pocket
and facing the challenge of
removing the string without
removing your hand from
your pocket (and no, you
can’t undo the string!).
(Diagram opposite is from
IGGY, where you will also
find the solution)
I have just bought the
Claire will be
latest book in the
delivering a session:
series, Professor
“The Participation of
Stewart’s Casebook of
girls in Mathematics
Mathematical
and Further
Mysteries. It contains
Mathematics” at the
many problems that
2nd Annual
you can use in your
STEMtech
classroom to help
Conference &
promote critical thinking
Showcase on
and
to
demonstrate
the beauty of
Wednesday 29th April
mathematics.
at the QEIICC.
When working with students applying
for degree course in mathematics, I am
often asked what different content titles
in the undergraduate course guides
actually mean. For example, what is
‘analysis’? And what would be studied
in ‘linear algebra’? The Further
Mathematics Support Programme
website contains an invaluable
resource, Preparation for
Mathematics, to help answer this
question.
Here, students can see typical
undergraduate content of the first year
of a degree programme and view a
growing number of hyperlinked
resources which show how the content
links to their study of A level
Mathematics or Further Mathematics.
Each resource has a number of tasks
for students to complete, with fully
worked solutions – useful for in-class
extension material or independent
study. In addition the page has more
general advice about preparing to
study a mathematics degree including
recommended reading and guidance on
STEP/AEA/MAT.
Core Maths resources
Thinking of offering
Core Maths from
September 2015?
MEI and OCR will be
running free network
events to provide
information about our
Core Maths
qualifications and
introduce teachers to
the free resources
available for teaching
them.
The events will take
place in the afternoon
or in twilight sessions
and will last
approximately two
hours. We could run
an event at your
school or college – as
well as the
convenience of us
coming to you, OCR
is willing to pay a
small fee to cover
your expenses in
hosting the event.
If you are interested
in hosting an event in
the summer term,
please contact Stella
Dudzic by the end of
March and we’ll try to
schedule it in.
Click here to email
Stella.
New Core Maths teaching and
learning resource
Core Maths qualifications are designed
for students who have achieved grade
C or better in GCSE Mathematics, but
who do not intend to take AS/A level
Mathematics. They enable learners to
strengthen and develop the
mathematical knowledge and skills they
have learnt at GCSE so that they can
apply them to the problems that they
will encounter in their other level 3
courses, further study, life and
employment.
In this issue of M4 you will find a Core
Maths teaching and learning resource
at the end of the magazine, in addition
to the Key Stage 4 resource produced
by Carol Knights. This has been
provided by Terry Dawson, who
develops Critical Maths and Core
Maths resources for MEI
(subscription to which is free). We
have provided the Core Maths resource
in a PowerPoint format for the teacher
to introduce the activity to students, and
in a PDF worksheet format to be used
as a student handout. Both formats can
be downloaded from the Monthly
Maths web page. If you or a colleague
in another teaching department are
planning to deliver Core Maths in your
school or college, you may find this
resource useful.
By registering for the free MEI
resources for Core Maths, you will be
able to access the MEI Introduction to
Quantitative Methods News
Forum, where you can share and
discuss ideas for using the Core Maths
Resources. On this forum there will also
be discussions about the MEI/OCR
Core Maths problems postcards that
will be sent to centres by OCR and also
given out at some events attended by
MEI. You will need to register for the
Core Maths resources to access the
discussions.
Here’s an example of a Core Maths
forum post:
Do the top 1% own 50% of the
world's wealth?
Wealth distribution has been much in
the news recently - if you want to know
where the figures come from and get
some ideas of discussion points to use
with students http://fusion.net/
story/39185/oxfams-misleadingwealth-statistics/ is a good place to
start.
If you would like to subscribe to the
Core Maths resources for the academic
year 2014-15, please complete
the online subscription form. This
subscription is free of charge, will run
until September 2015 and can be
renewed free of charge after that.
New classroom resources
In the following pages are Key Stage 4
and Core Maths teaching and learning
resources:
KS4: Representing Data: Florence
Nightingale, developed by Carol
Knights. Looks at the statistical work
of Florence Nightingale and helps
students consider different
representations and how they can be
misleading.
Core Maths: The Parable of the
Polygons, developed by Terry
Dawson. Uses Game Theory to
investigate segregation in society.
The resources can be downloaded
from the Monthly Maths web page.
Florence Nightingale
Florence Nightingale is well
known as the founder of
modern nursing, particularly for
her work during the Crimean
War in the 1850s.
She is perhaps less well known
for her use of statistics,
although it is precisely this that
underpinned the changes she
instigated within nursing.
Florence Nightingale
In 1854, Nightingale led a team of nurses that
she had trained to care for soldiers wounded in
the Crimean war.
Death rates were very high, and Nightingale
believed that this was largely due to the poor
conditions, which included poor nutrition and a
general lack of hygiene.
She and her team worked to improve diet,
sanitation, morale, and general hygiene
practices.
Florence Nightingale
Her experience in Crimea led Nightingale to
campaign for improved conditions, and she
used statistical diagrams to help display data
she had collected to make a more persuasive
argument.
This short activity looks at the diagrams she
created and some alternative representations.
Florence Nightingale
On the following slide is the most famous of
Nightingale’s diagrams and the key text, in case
it is difficult to read.
Look at the diagrams and spend a few minutes
making sense of them.
• What do the diagrams show?
• What might people misunderstand about the
diagrams?
The key text:
The blue, red and black wedges are each
measured from the centre as the common vertex.
• Blue: preventable disease
• Red: death from wounds
• Black: all other causes
November 1854: black line shows where deaths
from other causes is.
October 1854 & April 1855: black and red
coincide
Understanding the diagrams
The area of each wedge represents the number
of soldiers who died from the 3 causes.
Wedges are overlaid, with blue on the bottom,
then black and then red on top. This means
that some wedges cannot be seen at all. It also
means that it is not possible to see the entire
blue wedge at any time.
Do you think this might mislead people?
Previous diagrams
In an earlier diagram, Nightingale had made the
radius of a wedge proportional to the number of
people in the section.
She decided that this could be misleading.
Why do you think this was?
Previous diagrams
How many times larger does each section look
compared to the smallest one?
Would this be a fair diagram if the sections are
to represent 1000, 2000, 3000 and 6000
soldiers respectively?
Polar Area diagrams
How would you draw a
fair diagram to
represent 1000, 2000,
3000 and 6000
soldiers respectively?
Accurately draw 4
wedges on the diagram
to represent this.
Other diagrams
On the following slides this data set was used to
create the statistical diagrams. Comment on
them.
Blue
Red
1
2
January
2
5
February
3
7
March
4
5
April
5
4
May
6
1
June
Other diagrams
Other diagrams
Polar Area diagrams
Create a Polar area diagram for the data, similar
to the ones that Nightingale created. One colour
wedge should consistently overlay the other
colour wedge.
Which months cause issues in doing this?
Polar Area diagrams
Jan
Feb
Mar
Apr
Jun
May
Area diagrams
Can you also accurately
create a diagram so that all
parts of the area are visible,
i.e. wedges are not overlaid,
but radiate outwards?
Example shown.
Are there advantages to this
diagram?
How difficult is it to create?
Area diagrams
Teacher notes: Florence Nightingale
This month’s edition looks at the statistical work of Florence Nightingale
and then helps students consider different representations and how
they can be misleading.
It would be helpful to print some colour copies of slide 6 to enable
students to look closely at them
Much of the early part of this activity involves students thinking and
discussing.
Slides 12-14 could be missed out with Higher attaining groups.
Teacher notes: Florence Nightingale
Slides 5-8
The diagrams show that as Nightingale and her team continued to
improve hygiene and nutrition, so the rates of death from preventable
diseases decreased. This gave weight to Nightingale’s assertion that
these were fundamental to nursing care.
Potential difficulties with the Polar diagram:
• Some wedges are hidden completely
• Visually, the blue area could look smaller than it should since it is
overlaid with the red and black sectors.
• Similarly for the black area being overlaid with the red.
• No scale, so although we can see that there are far more blue than
black or red, these could be small numbers – which would still be
worth reducing, but it would be more persuasive if there were an
indication of numbers involved.
Teacher notes: Florence Nightingale
Slide 9
If the radius is used instead of the area, it looks far more dramatic a
difference than it actually is. Where a length is doubled, the area is
quadrupled. If a length is tripled, the area is 9 time bigger etc.
Slide 10
This diagram emphasises the visual discrepancies created when using
the radius instead of the area.
On the diagram, the radii are in the ratio 1:2:3:6.
This means that the areas are in the ratio 1:4:9:36
Teacher notes: Florence Nightingale
Slide 11
Copies of the sheet ‘Polar Area Diagrams’ can be used, or students
can construct the diagrams for themselves. 12 sectors are shown,
representing the 12 months of the year that Florence Nightingale used.
The number of soldiers should be proportional to the area of the sector.
Area for 2000 soldiers = 2 x area for 1000 soldiers
π(R2)2 = 2π(R1)2
(R2)2 = 2(R1)2
The radii should be in the ratio 1: √2 : √3 : √6
Approximately 1 : 1.4 : 1.7 : 2.4
A radius of 2cm for the first one will fit in the outlines given.
Teacher notes: Florence Nightingale
Slides 12 -14
The comparative and component bar charts:
Both of these have scales, so it is easy to ascertain values.
It is possibly a little easier to ascertain proportions from a component
bar chart, and a little easier to see trends in the individual items with a
comparative bar chart.
Pie charts:
The data have to be shown on two pie charts in order to compare. It
isn’t clear whether these are drawn to the same scale. No scale given
so the viewer doesn’t know how many people the charts represent.
They do give a good sense of proportion.
Teacher notes: Florence Nightingale
Slides 15-16
Assuming that a radius of 1cm is used for 1:
January
February
March
April
May
June
Blue
1
2
3
4
5
6
Radius
1.00
1.41
1.73
2.00
2.24
2.45
Red
2
5
7
5
4
1
Radius
1.41
2.24
2.65
2.24
2.00
1.00
If red is drawn first and then blue is laid on top, the issues arise when
the blue radius exceeds the red, i.e. for May and June.
If blue is drawn first and then red is laid on top, issues arise for Jan,
Feb, March, and April.
Teacher notes: Florence Nightingale
Slide 17-18
Assuming that the blue is inside and the red is outside, an additional
calculation is required to find the radius for the red section. Since the
outside of the red section encompasses both red and blue, find the
total. The red area is the difference between the total and the blue.
January
February
March
April
May
June
Blue
1
2
3
4
5
6
Radius
1.00
1.41
1.73
2.00
2.24
2.45
Red
2
5
7
5
4
1
Red + Blue Radius
3
1.73
7
2.65
10
3.16
9
3.00
9
3.00
7
2.65
The advantages of this representation are that the whole of each colour
is visible and it is easier to compare the overall totals.
Acknowledgements
Florence Nightingale Photograph and information from
http://en.wikipedia.org/wiki/Florence_Nightingale#Crimean_War
Accessed 10/3/15
Polar Area Diagrams
MEI is a registered charity, number 1058911
Core Maths:
teaching and
learning
resource
Using Game Theory to
Investigate Segregation in
Society
Parable of the Polygons
“Parable of the Polygons” is a website which uses
apps to explore how areas of cities can become
segregated.
Parable of the Polygons
• The site uses a mathematical model to show
that small individual biases can lead to a large
collective bias.
• This activity encourages you to think about the
mathematical model underlying the apps.
Parable of the Polygons
• The starting assumption is that there are two
types of individuals (squares and triangles).
• Each prefers being in a mixed neighbourhood
but is unhappy if too many neighbours are
different.
Parable of the Polygons
• Use the link http://ncase.me/polygons and work
through the site.
• Using a series of apps, you will start to build an
understanding of how certain parameters may
result in segregation.
The underlying message
• Take some time to experience the model in
action and ponder the underlying message from
the site.
• Give some thought to the
assumptions made in constructing
the different apps.
How might these
assumptions limit
any conclusions
made?
Percentage of alike neighbours
• By filling the surrounding
spaces with either
squares, triangles, or
empty spaces, we arrive at
the percentage of alike
neighbours.
How does the
app measure
the percentage
of alike
neighbours?
Percentage of alike neighbours
• For example consider a square near the centre
of the large grid, see the diagram below.
• By filling the surrounding spaces with either
squares, triangles, or empty spaces, we arrive at
the percentage of alike neighbours.
Percentage of alike neighbours
Are all
percentages
possible?
Explain your answer.
Percentage of alike neighbours
• Consider how the possible values for the
percentage of alike neighbours may change
depending on the shape’s position.
• You could start by examining a square in one of
the corners of the large grid, for example (A) in
the diagram below.
Percentage of alike neighbours
• As you did in the previous example, work out all
the possible values for the percentage of alike
neighbours.
Parable of the Polygons
• Now try a square along one of the edges of the
large grid, for example (B).
Percentage of alike neighbours
• Compare the possible percentages you have
just obtained for (A) and (B), to the percentages
you previously obtained for a square in the
central section (C).
Percentage of alike neighbours
• Assume the simulation
was run many times, with
the same setting.
• Which of the three
squares considered
above (A, B, or C) ,
would you expect to
move more frequently on
average?
Percentage of alike neighbours
• The screenshot below suggest that there is 56%
segregation:
Percentage of alike neighbours
• How is this figure calculated?
• Is this a good way to measure segregation?
Percentage of alike neighbours
• The initial problem starts with both shapes
having the same requirements, that is, that at
least 1/3 of their neighbours need to be like
them for them to be happy.
• On the second app, shown in the next screen
shot, you are allowed to change this figure of
1/3 ≈ 33% to other values to investigate its
effect on the level of segregation.
Percentage of alike neighbours
• Is there a maximum level you can set for this
figure beyond which a solution becomes
impossible? Explain why.
Percentage of alike neighbours
• How about if the shapes
showed different levels of
preference?
• For example would it be
possible for a solution where
squares required 75% to be
alike, whilst triangles would be
happy with 1/3?
What constraint
does this place
on the settings?
Wrapping Up
Cartoon by Vi Hart
Teacher notes: Parable of the Polygons
Before using this resource with your students you
will want to check to ensure that you are
comfortable with them viewing the content on the
website and the underlying messages expressed
at the end, under ‘Wrapping up’.
You may find the apps work better in some
browsers than others – they work in Google
Chrome.
Teacher notes: Parable of the Polygons
Game theory is a branch of applied mathematics
which is used in economics, sociology, psychology
and biology. It deals with mathematical modelling
of strategy and behaviour in situations where the
decisions which individuals make are affected by
what others might do.
The 2001 film A Beautiful Mind was based on the
life of John Nash who won the Nobel prize for
economics for his work on game theory.
Mathematical Detail
Slides 10-15
Are all percentages possible?
No. Looking at a position in the middle of the
‘town’ each shape has 8 neighbours. Therefore
there can be 0, 1, 2, 3, 4, 5, 6, 7 or 8 neighbours,
of whom various numbers can be alike. The table
on the following slide gives the possibilities and the
percentages.
% (to 1%)
No. of alike neighbours
1
2
3
4
5
6
7
No. of neighbours
1
100
2
50
100
3
33
67
100
4
25
50
75
100
5
20
40
60
80
100
6
17
33
50
67
83
100
7
14
29
43
57
71
86
100
8
13
25
38
50
63
75
88
8
100
Mathematical Detail
Slides 10-15
Are all percentages possible?
On an edge there are up to 5 neighbours.
In a corner there are up to 3 neighbours.
Mathematical Detail
Slide 18
How is segregation calculated by this App?
It looks as if it is the average of the ‘like
neighbours’ for all of the shapes.
The advantage of this is that ‘partial segregation’ is
included.
An alternative would be to look at how many
shapes had no like neighbours at all. In this case
there are 126 out of 322, which is 39% of shapes
completely segregated.
Acknowledgements
• http://ncase.me/polygons
• https://www.idlethumbs.net/forums/topic/9815parable-of-the-polygons
• https://gbark.wordpress.com/2014/12/12/reblogparable-of-the-polygons
• https://twitter.com/vihartvihart
Accessed 11/03/2015
Using Game Theory to Investigate Segregation in Society
Teacher notes
Before using this resource with your students you will want to check to
ensure that you are comfortable with them viewing the content on the
website and the underlying messages expressed at the end, under
‘Wrapping up’.
Game theory is a branch of applied mathematics which is used in
economics, sociology, psychology and biology. It deals with mathematical
modelling of strategy and behaviour in situations where the decisions which
individuals make are affected by what others might do. The 2001 film A
Beautiful Mind was based on the life of John Nash who won the Nobel prize
for economics for his work on game theory.
MEI is a registered charity, number 1058911
Using Game Theory to Investigate Segregation in Society
“Parable of the Polygons” is a website which uses apps to explore how
areas of cities can become segregated. You may find the apps work better
in some browsers than others – they work in Google Chrome.
The site uses a mathematical model to show that small individual biases
can lead to a large collective bias. This article encourages students to think
about the mathematical model underlying the apps.
The starting assumption is that there are two types of individuals (squares
and triangles), each prefers being in a mixed neighbourhood but is
unhappy if too many neighbours are different.
Use the link http://ncase.me/polygons/ and work through the site. Using a
series of apps, you will start to build an understanding of how certain
parameters may result in segregation.
Take some time to experience the model in action and ponder the
underlying message from the site, then give some thought to the
assumptions made in constructing the different apps and how this may limit
any conclusions made.
MEI is a registered charity, number 1058911
Here are a few questions intended as prompts:
How does the app measure the percentage of alike neighbours? For
example consider a square near the centre of the large grid, see the
diagram below. By filling the surrounding spaces with either, squares,
triangles, or empty spaces, we arrive at the percentage of alike neighbours.
 Are all percentages possible?
 Explain your answer.
Consider how the possible values for the percentage of alike neighbours,
may change depending on the shape’s position. You could start by
examining a square in one of the corners of the large grid, for example (A)
in the diagram below. As you did in the previous example, work out all the
possible values for the percentage of alike neighbours. Now try a square
along one of the edges of the large grid for example (B). Compare the
possible percentages you have just obtained for (A) and (B), to the
percentages you previously obtained for a square in the central section (C)
A
C
B
Assume the simulation was run many times, with the same setting.
 Which of the three squares considered above (A, B, or C) , would you
expect to move more frequently on average?
MEI is a registered charity, number 1058911
The screenshot below suggest that there is 56% segregation:
 How is this figure calculated?
 Is this a good way to measure segregation?
The initial problem starts with both shapes having the same requirements,
that is, that at least 1/3 of their neighbours need to be like them for them to
be happy.
MEI is a registered charity, number 1058911
On the second app, shown in the screen shot below, you are allowed to
change this figure of 1/3 ≈ 33% to other values to investigate its effect on
the level of segregation. In the screenshot below the condition for moving is
set to “I’ll move if less than 50% of my neighbours are like me”
 Is there a maximum level you can set for this figure beyond which a
solution becomes impossible?
 Explain why.
 How about if the shapes showed different levels of preference, for
example would it be possible for a solution where squares required
75% to be alike, whilst triangles would be happy with 1/3?
 What constraint does this place on the settings?
MEI is a registered charity, number 1058911