Monthly
Maths
I s s u e
NCETM to lead
new ‘MESH’
initiative
In Dec 2013 the
government
pledged £11m to
allow the
development of a
national network
of around 30
‘mathematics
education
strategic
hubs’ (MESH).
National
coordination will
be provided by
the NCETM with
each hub led by a
teaching school,
providing support
to all schools in
the area, across
all areas of maths
education. Hubs
will include local
university
faculties, area
representatives of
national initiatives
(e.g. the Further
Mathematics
Support
Programme and
the Core Maths
Support
Programme),
subject
associations and
appropriate local
employers.
3 5
Curriculum Changes
In this issue we will look at the recent and
upcoming reforms and developments in
curriculum, specifically in mathematics. As
well as outlining what these changes are,
we will tell you how MEI can provide
support to teachers through resources,
advice and professional development.
As part of its curriculum reform policy
the Government is introducing a slimmeddown national curriculum for 5- to 16year-olds to be taught in maintained
schools from 2014. Following public
consultations (see MEI 2013 response)
DfE has published the final statutory
version of the new national curriculum to
be taught in maintained primary and
secondary schools from Sep 2014.
Key stage 3
‘National curriculum in England:
mathematics programme of study - key
stage 3’ details the statutory programme
of study and attainment targets for
mathematics at key stage 3, to be taught
in England from Sep 2014. Examples of
formal written methods for addition,
subtraction, multiplication and division can
be seen at Mathematics Appendix 1.
Key stage 4
GCSEs are being reformed “so they
provide a strong foundation for further
academic and vocational study.” In
February 2013 the Secretary of State for
Education set out the Government’s
policy on reforms to qualifications at the
end of Key Stage 4:
Click here for the MEI
Maths Item of the Month
www.mei.org.uk
F e b r u a r y
2 0 1 4
“GCSEs will be comprehensively
reformed, to ensure that pupils have
access to qualifications that set
expectations that match and exceed
those in the highest-performing
jurisdictions. The reformed GCSEs will
remain universal qualifications
accessible, with good teaching, to the
same proportion of pupils as currently
sits GCSE exams at the end of Key
Stage 4.”
The new GCSE sets higher expectations;
demanding more from all students and
providing further challenges. GCSE
mathematics: subject content and
assessment objectives sets out the
learning outcomes and content coverage
required for GCSE maths specifications
in mathematics.
Reforms to key stage 4 mathematics are
subject to a full public consultation on
draft programmes of study before being
finalised to align with new GCSEs.
You can read MEI’s responses to the
recent consultations at the links below:
DfE consultation on GCSE subject
content (08/13)
Ofqual GCSE consultation (09/13)
Ofqual's Dec 13 GCSE technical
consultation (01/14)
DfE KS4 Mathematics and English
consultation (02/14)
A new MEI teaching resource is at the end of this
bulletin. Click here to download it from our website.
Disclaimer: This newsletter 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.
Timetable of reforms
MEI response
and curriculum
development
As a champion for
mathematics
education, MEI
responds to
consultations by
central
government, public
bodies or other
organisations
whose work
impacts on our
strategic goals and
priorities, taking
into account both
our stakeholders'
and our own
considered and
informed views.
We also invite
comment on many
of our recent
Position Papers;
summaries of
outcomes for MEI
consultations can
be found in the
Responses
section.
The Curriculum
section of the MEI
website contains
information about
curriculum
development work
that we are
involved in. You
will also find the
timetable of
reforms (opposite)
in this section, with
updates as they
become available.
Date
KS3 and GCSE
Core Maths
A level
Early adopter schools and
colleges identified and
individual plans agreed
April 2014
June 2014 First full sitting of linear
GCSEs - no modular
option
July 2014
Aug 2014
Sept 2014 New National Curriculum
for KS1-3 (remains
disapplied for Maths in
Years 2 and 6)
Specifications for new
Mathematics GCSEs
teaching from 2015 in
schools
Early adopter schools and
colleges identified and
individual plans agreed
Outline qualification
information available for
schools and colleges
Outline qualification
information available for
schools and colleges
Qualifications available to
schools for planning and
‘early adopter’ teaching
Oct 2014
Nov 2014 November GCSE is for
resit only; students cannot
be entered for the first time
July 2015
Sept 2015 First teaching of new
Mathematics GCSEs
Sept 2016
Summer
2017
Sept 2017
Review of initial
implementation - early
teaching continues to June
2016 with awards in
Summer 2016
Qualifications introduced
Specifications for new
widely
Mathematics A levels
teaching from 2016 in
schools
First teaching of new
Mathematics A levels
First assessments that will
count towards level 3 maths
performance measure
First cohort with revised
GCSE begins Core Maths
16-18 mathematics
New
Professional
Development
from MEI
New National
Curriculum:
opportunities,
practicalities and
realities
To enable 11-16 or
11-19 departments
and students to
gain the most
benefit from the
opportunities
afforded by the
imminent
curriculum
changes.
The course
consists of two
separate days,
with Day 1 (April)
focusing on KS3
and Day 2
(September)
focusing on
KS4. Read more
on the course
web page.
Course Leader
Carol Knights is
also leading a
session at MEI’s
annual
conference:
‘Preparing for the
new National
Curriculum at
KS3&4’.
As part of its
curriculum reform
policy, ‘Reforming
qualifications and
the curriculum to better prepare pupils
for life after school’, the DfE wishes to
increase the involvement of universities in
the design and development of more
rigorous A levels, and to encourage more
16- to 18-year-olds to take up
mathematics and science subjects.
advanced calculation and modelling will
form the key content of Core Maths. The
qualifications will also count as the maths
element of the new TechBacc, to be
introduced from September 2014, to
recognise high performance by students
in vocational education. The DfE has
published a policy statement on Core
maths qualifications for post-16
students.
DfE’s plans to reform A-levels, and their
specific plans for mathematics, can be
read in the transcript of the Institute of
Education 2013 open lecture on A level
reforms delivered by Elizabeth Truss.
The ‘TechBacc’ provides an alternative to
the A level study
route for post-16
education. The
technical
baccalaureate
measure will be aimed at “ambitious,
talented students who want to pursue a
technical career. It will give young people
the opportunity to be stretched and
demonstrate their personal best”. The
group most likely to opt for qualifications
included in the TechBacc measure are
those who choose to study advanced
vocational qualifications having already
achieved a grade C or above in GCSE
maths.
MEI has responded to Ofqual's 2013-14
consultation on regulatory requirements
concerning proposed changes to A levels
for teaching from Sept 2015. There are
to be consultations about A level
Mathematics and Further Mathematics
before the requirements are finalised for
new A levels in these subjects for
teaching from Sept 2016. ALCAB, the A
Level Content Advisory Board has a
mathematics subject panel which has
started meeting.
Core Maths
Courses in ‘Core Maths’ will target 16-18year-olds who have been successful at
GCSE (grade C or better) but who don’t
want to study A level or AS maths. Post16 students taking A Levels in a range
of non-maths subjects will develop the
skills needed to solve the mathematical
problems arising in their studies, or in
employment and everyday life.
Topics such as statistics, probability,
Technical Baccalaureate
The Technical Baccalaureate
performance table measure records
achievement of students taking advanced
(level 3) programmes which include a
Department for Education-approved Tech
Level qualification, a level 3 maths
qualification and an extended project.
This measure will be introduced
alongside the teaching of approved
occupational qualifications from Sep
2014. The measure will first be reported
in the 2016 performance tables, due to be
published Jan 2017.
MEI’s 16-18 curriculum
development work
Professional
development
(KS3, KS4, 16+)
MEI offers
professional
development to
support the
changing
curriculum, in the
form of short
courses at venues
across the country,
or as bespoke
courses for whole
maths
departments.
Please see
opposite for more
information.
MEI Conference
2014 offers
sessions relating
to the Introduction
to Quantitative
Methods
specification and
the National
Curriculum at
KS3&4.
Also available are
workshops in
introducing
problem solving
into the KS4
classroom; using
problem solving to
develop
mathematical
thinking in post
GCSE students,
and the pros and
cons of using
contexts in the
teaching of
mathematics.
Quantitative Methods
Professional Development for IQM
MEI has developed a new course,
Introduction to Quantitative Methods,
in association with OCR and accredited
by Ofqual for first teaching from
September 2014.
There will be a day of training for each of
IQM, S1 and D1 at MEI’s annual
conference, taking place at Keele
University 26-28 June 2014. Please
contact MEI’s Programme Leader for
Curriculum, Stella Dudzic, with any
queries about professional development
for the Quantitative Methods
qualifications.
The Introduction to Quantitative
Methods unit (IQM) is designed for
students who have at least grade C in
GCSE Mathematics and who would
benefit from continuing with some
mathematics but who do not need or want
to do A level
Mathematics. It
will be particularly
useful for
students studying
subjects such as
geography,
business, chemistry, biology and
sociology.
Students who are successful in the
examination for the unit will be able to
gain a Level 3 Certificate in Quantitative
Methods. It can also be combined with
the S1 and D1 units to make an AS in
Quantitative Methods. AS Quantitative
Methods will count towards the Technical
Baccalaureate performance measure
reported in 2016.
Resources for Quantitative Methods
The course is resourced and supported
by MEI; please visit the Quantitative
Methods page on the MEI website. Draft
sample resources are available free of
charge by using a guest log-in on our
Integral website. We are now completing
the resources for the unit. Information
about how to access the full resources
will be available in 2014.
Critical Maths
MEI is also developing a
curriculum to promote
mathematical problem
solving, with DfE
funding, to investigate
how Professor Sir
Timothy Gowers's ideas
for teaching mathematics
to non-specialists might inform a
curriculum that could become the basis of
a new course for students who do not
currently study mathematics post-16.
The curriculum will be based on students
engaging with realistic problems and
developing skills of analysing problems
and thinking flexibly to solve them. In
addition to trialling resources with
schools, we have given some thought to
how such a curriculum might be
assessed. The draft curriculum content
and assessment report are available on
the Critical Maths web page of the MEI
website.
Curriculum developer Terry Dawson has
written an article about this exciting
project (see next page) and has also
developed a teaching and learning
resource that is included at the end of the
newsletter.
A problem-solving
approach to learning
Terry Dawson
Critical Maths
Role
Terry Dawson is a
Curriculum
Developer for MEI
MEI has been developing a maths
curriculum which is significantly different
from anything currently available. Critical
Maths was inspired by a blog from
Professor Sir Timothy Gowers and it has
steadily evolved into something quite
unique through extensive consultation
with learned societies, universities,
employers and organisations such as the
CITB and the national skills academies
for health and construction.
Background
Terry worked in
the steel industry
before completing
a BEd (Hons) in
Secondary
Mathematics in
1991. He has
taught in
secondary schools
for over 21 years,
including 13 years
as Head of
Mathematics, and
2 years as an
Assistant
Headteacher and
Local Authority
Consultant. He
joined MEI in April
2013.
His work on
Curriculum
Development
includes an active
involvement in the
Cre8ate Maths
project and many
STEM related
projects. Terry
has led CPD
sessions for
teachers and
subject leaders.
The aims for the curriculum include
developing the mathematical and
statistical knowledge and skills that
students need to become educated
citizens in the context of today’s society.
At the heart of the
curriculum is a
teaching approach
which involves
discussing the
answer to a question which requires the
student to give an opinion, justify a choice
or decision, or explain someone’s
reasoning. In some respects, this is
similar to the approach used in many
other subjects such as R.E. and English.
Mathematics begins to appear as the
students attempt to explain their opinion,
or justify their decision.
This has a number of
similarities to the
Japanese approach to
teaching through problem
solving referred to as
“neriage” by Professor
Akihiko Takahashi in his
paper “beyond show and tell”.
There are
several key
concepts/
skills which
are
developed throughout the curriculum
such as the ability to produce quick
estimates from limited information,
questioning the validity of data presented
to support an argument, and
understanding risk and its role in
influencing our decisions.
To get an idea of some of the curriculum
content take a look at the article “twenty
tips for interpreting scientific claims”
from nature magazine.
With the kind assistance of volunteer
schools, some early classroom trials were
carried out in the summer of 2013 and
this enabled us to refine the approach
and the resources a little. Further trials
are taking place throughout this year, and
we would welcome any schools who
would like to take part in this curriculum
development.
We would like to think that the end result
of this project will be a unique and
credible level 3 qualification which will
engage those students with a
Mathematics GCSE grade C or better
who chose not to pursue an AS/A level in
mathematics.
Terry Dawson will be leading a session
at the MEI Conference 2014: Neriage: a
problem solving approach to learning.
If you would like to trial some of the
developing resources, please contact
MEI’s Programme Leader for Curriculum,
Stella Dudzic.
Medical Screening
The government has a screening program for a potentially fatal
medical condition which is thought to affect 1 in 1000 of the
population.
100 000 people are tested.
The test gives the correct result in 98 out of
100 patients.
People will naturally worry if they have a
result which suggests they have the condition,
but how much of this concern is misplaced?
Where do we start?
• How worried should someone be if they have a positive
result?
– Extremely worried,
– Very worried,
– A little worried,
– Not at all worried.
• Why?
• Do you think many will get a FALSE positive result?
Explain why.
Some calculations that might help
(i) How many of the 100 000 people tested would you
expect to have the condition?
(ii) How many of the 100 000 people tested would you
expect to NOT have the condition?
(iii) How many people would be expected to get a positive
test result?
False Positives
• A ‘false positive’ is a result which suggests a person has
a condition when they actually don’t have it at all
• Fill in the ‘tree diagram’ on the following slide, with
numbers, to see how many people will receive genuine
positive results and how many will receive false positive
results
• Remember that the test result is correct 98% of the time:
– 98% of the time it says you do have it when you do
– 98% of the time it says you don’t have it when you
don’t
False Positives
Have the
medical
condition
???
Number tested
100000
Don't have the
medical
condition
???
Test Positive
???
Test Negative
???
Test Positive
???
Test Negative
???
False Positives
Have the
medical
condition
100
Number tested
100000
Don't have the
medical
condition
99900
Test Positive
98
Test Negative
2
Test Positive
1998
Test Negative
97902
False Positives
• What percentage of the people who have received
positive results actually have the condition?
• Does this surprise you?
• Should doctors use this test?
• Is it better to worry people who don’t really have the
condition, or to miss people who do have it?
Hit and Run
The police receive a report of a hit and run accident involving a
taxi. Although it is dark, an eye witness is 90% sure it was a
green cab.
There are 1000 taxis in the city, 10 are green and the rest are
red.
Is it more likely the accident involved
a red or green taxi?
Hit and Run
Involved ??
Green ??
Taxis ???
Not
involved ??
Involved ??
Red ???
Fill in the blanks:
• Eye witness is 90% sure
• There are 1000 taxi’s in the city
• 10 are green and the rest are red
Not
involved ??
Hit and Run
Involved 9
Green 10
Not involved
1
Taxis 1000
Involved 99
Red 990
Not involved
891
Is it more likely the accident involved a red or green taxi?
Drug Testing
In the USA, 20 000 air traffic controllers
undergo random drug testing.
The test is good but not perfect:
• 96% of those who use drugs test positive
• 93% of those that do not use drugs test negative
Based on previous figures the Federal Airline Authority believe
that 99% of air traffic controllers are drug free.
Do you think it’s likely that people identified as positive by the
test are guilty of taking drugs?
Drug Testing
How is this problem different to the previous ones?
Assuming that the Federal Aviation Authority are correct – that
99% of pilots are drug free – what percentage of those testing
positive are actually drug free?
Complete the tree diagram on the next slide and discuss your
results.
Drug Testing
??%
Number
expected to be
innocent of
taking drugs
???
??%
??%
Number tested
20000
??%
Number
expected to be
guilty of taking
drugs
???
??%
??%
Number expected
to test Positive
???
Number expected
to test Negative
???
Number expected
to test Positive
???
Number expected
to test Negative
???
Drug Testing
99%
Number
expected to be
innocent of
taking drugs
19800
7%
93%
Number tested
20000
1%
Number
expected to be
guilty of taking
drugs
200
96%
4%
Number expected
to test Positive
1386
Number expected
to test Negative
18414
Number expected
to test Positive
192
Number expected
to test Negative
8
Lie Detector Test
A TV show uses a lie detector test to try to establish
which one of two people is telling the truth.
Assume that just one of the two people in the dispute is
not being honest, and it is equally likely to be either
guest.
The American Polygraph Association claim the tests are
89% accurate for a single issue response.
Out of 1000 shows, how many people will be
falsely accused of lying?
How many guests will receive the wrong result?
Lie Detector Test
• Draw out a diagram to help you ascertain how
many inaccurate results there are likely to be.
• What will happen on the shows if one or other or
both of the results are inaccurate?
Lie Detector Test
Truthful
1000
Pass Lie Detector
890
Fail Lie Detector
110
Number tested
2000
Not Truthful
1000
Pass Lie
Detector
110
Fail Lie Detector
890
Down’s Syndrome Occurrence
In an article in 2009, a “top doctor” called for changes to
the pre-natal screening for Down’s Syndrome.
To consider the concerns expressed by the doctor here are
the most recent figures from the Office for National
Statistics which show that there were approximately
730,000 babies born in the UK in 2012 and gives the
approximate age of their mothers.
2012 Live Births In England and
Wales
All ages Under 20 20-24
25-29
30-34 35-39 40-44 45 and over
729,674 33,815 132,456 202,370 216,242 114,797 28,019
1,975
As stated in the article, screening for Down’s Syndrome is
currently offered to all prospective mothers.
Down’s Syndrome Occurrence
The current test has a false positive rate of about 3% according to
information from the NHS.
To complicate matters further the chances of having a baby with
Down’s Syndrome increases with the age of the mother.
(Figures from the NHS)
• 25 years of age has a risk of 1 in 1,250
• 30 years of age has a risk of 1 in 1,000
• 35 years of age has a risk of 1 in 400
• 40 years of age has a risk of 1 in 100
• 45 years of age has a risk of 1 in 30
How many false positives are there likely to be for the different age
groups?
Use this evidence to decide whether you think pre-test counselling is a
good idea, explaining your response.
Down’s Syndrome Occurrence
Births:
The current test has a false positive rate of about 3%
Chances of having a baby with Down’s Syndrome
increases with the age of the mother.
• 25 years of age has a risk of 1 in 1,250
• 30 years of age has a risk of 1 in 1,000
• 35 years of age has a risk of 1 in 400
• 40 years of age has a risk of 1 in 100
• 45 years of age has a risk of 1 in 30
Teacher notes: It’s a Risk
Pupils will be familiar with situations in which ‘risk’ is used, such as the
use of ‘Lie Detector tests’ on day-time TV shows and athletes testing
positive for the use of banned substances, but may be less familiar with
the impact of test accuracy in medical tests.
Through these activities, pupils will gain a better understanding of false
positives and how to interpret ‘risk'.
Teacher notes: It’s a Risk
There are 5 activities with a decreasing amount of support and
structure.
It is important to give students time to think, discuss and absorb. Many
of the results will challenge what they believe or suspect to be true.
Attaining an understanding of why there are so many false negatives is
helpful.
It is suggested that students work in pairs or small groups in order to
discuss their initial thoughts and then to make sense of the outcomes.
Activities could all be tackled within a lesson or each could be used as
a starter activity in a series of lessons.
Teacher notes: Medical Screening
Where do we start
Most students in the trials said very or extremely worried. The cause of this
worry was generally stated as the 98% accuracy of the test.
Some calculations that might help
• Leave a little time between each question for the students to think and
respond.
• Students should attempt to produce a figure by calculation… the answers
will be confirmed in the next section ‘False Positives’
False Positives
• Of the 2096 positive results, only 98 are genuine – that’s less than 5%
• Why, with such an accurate test, are there so many false positives?
• Because 98% of small amount (in this case 100) < 2% of much larger
amount (in this case 99900).
• This is a key point so take some time to ensure that the students fully
appreciate it.
Teacher notes: Hit and Run
Eye witness testimony is notoriously inaccurate. In this case, the eye-witness
is 90% sure it was a green taxi, which means that there’s a 90% chance of it
being one of the green taxis and a 10% chance of it being one of the red taxis.
This is similar in structure to the ‘Medical Screening’ question, where there are
a lot more items in the ‘non-target’ group. In this case there’s a lot more red
taxis than green ones.
Out of the 108 taxis that could have been involved, 9 are green and 99 are red,
so it’s far more likely that the taxi involved was actually red.
Teacher notes: Drug Testing
What is different in this problem? The probabilities are conditional this
time. 96% for correctly identifying a drug user, but only 93% chance of
correctly identifying a non-user.
Ask the students to fill in the blanks in the diagram and discuss the
outcomes.
Pose the question: “If an air traffic controller has a positive result, how
likely are they to actually have taken drugs?”
Of the 1578 likely to test positive, only 192 are likely to have done.
That means approximately 88% of those testing positive in this test are
actually innocent.
Teacher notes: Lie Detector Test
This is a relatively straight forward problem.
If students have tried the previous problems with some structure, then
this would be a good one to let them try without teacher input and
without being given a blank tree diagram.
In the 1000 shows, there are likely to be 110 false negatives and 110
false positives.
This would mean that on some shows the following could happen:
• If one result is accurate and one is erroneous, then either both
people are found to be telling the truth or both people are found to
be lying
• Occasionally, the results are completely the wrong way round – the
liar is found to be telling the truth and the honest person is found to
be lying
Teacher notes: Down’s Syndrome
Sensitivity must be shown with this content; this activity is simply about
identification of a condition and allowing prospective mothers to be prepared.
Students should be encouraged to explore the information for themselves and
justify their responses to the questions. It might be helpful to print out copies of
the information slide for students to refer to more easily. Working with a partner
or in a small group would be helpful for students to share the workload and to
interpret their answers.
If students struggle to get started, encourage them to think about the following:
• Number of Down’s Syndrome babies expected for each age group
• Number of non-Down’s Syndrome babies expected for each age group
Then consider how many babies would fall into each of the following categories
for each age group:
Correct test result
Down’s baby
Non-Downs baby
Incorrect test result
Teacher notes: Down’s Syndrome
Numbers
Risk of Down's (1 in ……..)
Expected Number of Children born with Down's
Expected Number of children Without Down's
Test False indication rate
False Positives
False Negatives
Correct positives
Correct negatives
Probability of a positive result being false
All
ages
729,674
Under
20
33,815
20-24
25-29
30-34
35-39
40-44
45+
132,456
202,370
216,242
114,797
28,019
1,975
1144
1250
27
33,788
1250
106
132,350
1250
162
202,208
1000
216
216,026
400
287
114,510
100
280
27,739
30
66
1,909
1014
1
26
32,774
3971
3
103
128,380
6066
5
157
196,142
6481
6
210
209,545
3435
9
278
111,075
832
8
272
26,907
57
2
64
1,852
3%
21856
34
1110
706,674
97.5%
97.5%
97.5%
96.9%
92.5%
75.4%
47.3%
Teacher notes: Down’s Syndrome
•
•
•
Looking at the results it’s possible to conclude that mothers below 35 years
of age are much more likely to get a false positive result.
However, about half the children with Down’s Syndrome are born to
mothers aged under 35; this is because they make up the largest proportion
of mothers.
Perhaps mothers should be made aware of the probability of a false
positive, for their age group, and they can then make an informed choice of
whether they want the test.
References
American Polygraph Association accessed 3-2-2014
http://www.polygraph.org/section/resources/polygraph-validity-research
Lie Detector Test accessed 3-2-2014
http://www.dailymail.co.uk/femail/article-1203070/Jeremy-Kyle-nearly-killed-Thehorrifying-story-womens-decision-daytime-TVs-notorious-show.html
Down’s Syndrome article accessed 3-2-2014
http://www.dailymail.co.uk/health/article-1223406/Doctor-calls-mothers-opt-DownsSyndrome-test.html
Data from the ONS accessed 3-2-2014
http://www.ons.gov.uk/ons/taxonomy/index.html?nscl=Births+by+Mother%27s+Area+of+
Residence#tab-data-tables
NHS information accessed 3-2-2014
http://www.nhs.uk/news/2013/06June/Pages/New-Downs-syndrome-blood-test-morereliable.aspx
http://www.nhs.uk/Conditions/Downs-syndrome/Pages/Causes.aspx