m e i . o r g . u k
Curriculum Update
New A levels in
Mathematics and
Further Mathematics
for teaching from
September 2017
New A levels in
Mathematics and
Further Mathematics
are due for first
teaching from
September 2017. MEI
has created a
narrated PowerPoint
to lead you through
the changes – this is
available on our
website and on our
YouTube channel.
You will also find
Ofqual’s timelines
for GCSE and AS and
A level change useful;
there is further
information in the
Ofqual blog where
you will also find the
opportunity to sign up
to receive the Ofqual
newsletter.
MEI is working on
new specifications in
Mathematics and
Further
Mathematics. If you
would like to trial
some of the new
assessment
questions we are
working on, please
contact Keith Proffitt.
I s s u e
4 6
A p r i l - M a y
2 0 1 5
Engaging with your students
Piloted and
trialled in the
“Three types of feedback are
post-16
essential…student to teacher…teacher
sector,
to student ….between students”.
Improving
(Hodgen & Wiliam, 2006). This quote
Learning in
featured in an MEI Conference session
Mathematics
last year: Assessment for Learning
provides ideas for any maths teacher
with examples from GCSE and A
who wants to make lessons interesting
level. This year Debbie Barker and
and engaging for their students. This
Simon Clay offer ‘Planning for
project was part of the Department for
Assessment for learning’ with a
Education and Skills' response to the
mixture of discussion, trying out some
Smith Report and offers practical and
classroom activities and reflecting upon
effective ways to improve learning in
the implications for planning, with
mathematics.
examples from across key stages 3, 4
and 5. See also the CfBT Education
You can access and read more about
Trust report: Assessment for learning: the resources, comprising teaching
effects and impact
sessions and professional development
sessions to help support the teaching,
The TLA (Teaching and
on the National STEM Centre website,
Learning Academy)
the Secondary Maths ITE and on Mr
summarises Malcolm
Barton’s maths website.
Swan’s research and
development with
colleagues at the University
In this issue
of Nottingham into more effective ways

Curriculum Update
of teaching and learning mathematical
concepts and strategies. Collaborative

April-May focus: Student
learning in mathematics: A challenge
Engagement; Girls in Maths;
to our beliefs and practices (Swan,
MEI Conference
M. 2006), stated that: “The research

Crash Course: while command
concluded that student-centred,
collaborative and discussion based

Site-seeing with... Terry Dawson
approaches to learning were more

KS4 Teaching Resource: What
effective than more traditional
do you see?
transmission methods, especially in the
development of conceptual
understanding of mathematics.”
Click here for the MEI
Maths Item of the Month
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.
Engaging students
Developing
independent
mathematicians
‘Engaging Students, Developing
Confidence, Promoting
Independence’
Although aimed at
teachers of KS1 and
2, Jennie Pennant’s
2013 nrich article
Developing a
Classroom Culture
That Supports a
Problem-solving
Approach to
Mathematics offers
all teachers practical
ways to investigate
aspects of your
classroom culture.
This 2011 nrich article by
Charlie Gilderdale and
Alison Kiddle outlines
ways in which maths teachers might
make their lessons, teaching approach
and the learning environment more
engaging for students. They encourage
teachers to ask themselves the
following questions:
It also offers
suggestions to help
you develop the
culture further so that
students are
encouraged to
develop as
independent
mathematicians with
strong problemsolving skills.
“Generally, in a strong
problem-solving
environment the
teacher needs to be
doing around 30% of
the talking and the
students 70%.”
How do we develop positive attitudes
towards mathematics and learning
mathematics?
How do we develop confident
learners who are able to work
independently and willing to take risks?
How do we develop good
communicators - good at listening,
speaking and working purposefully in
groups?
The Further
Mathematics
Support Programme
aims to promote
participation in advanced level
mathematics to all students who would
benefit from taking the qualifications,
especially girls. In her MEI Conference
session, Claire Baldwin will discuss the
FMSP’s October 2014 briefing
document that summarised the key
findings from the FMSP/IoE Literature
Review. This document outlines the
interim findings from the recent gender
case studies, which aim to identify and
share good practice in promoting
participation in advanced level
mathematics by girls. For example:
“A key strategy for engaging girls is to
provide opportunities for checking
understanding with friends and quiet
conversations with the teacher.”
How do we develop students who
have appropriate strategies when they
get stuck?
How do we develop lessons that
maintain the complexity whilst making
the mathematics accessible?
How do we develop students' ability
to make connections (e.g. see/utilise
different aspects of mathematics in one
context, see applications in other
areas)?
How do we develop critical learners
who value and utilise differences (e.g.
different approaches/ routes to
solution)?
The Maths Hubs,
operational since
September 2014, have
a remit to drive up the quality of
teaching and learning in mathematics
through the sharing of good practice
across local areas and nationally. One
of the Maths Hubs’ priority areas is a
focus on strategies for increasing
participation in AS/A level Mathematics
and Further Mathematics, especially for
girls. For more information on local
Maths Hubs, click here.
Engagement with
STEM subjects
Engagement in the
Classroom
nrich suggests that
there are four levels
at which students/
teachers can be
engaged in the
process of STEM
engagement:
1.Raise awareness of
general
connections across
subjects
2.Active reference in
lessons to timetabled
curriculum links
across departments
3.Use of crosscurricular tasks in the
standard curriculum
4.Cross-curricular
days and projects
You can read about
these four levels in
more detail on the
nrich website.
The National STEM
Centre offers support
to schools that
includes access to
high-quality
teaching resources:
UK’s largest collection
of teaching resources
for STEM subjects for
use with students
from early years to
post-16.
The paper stresses that it is necessary
to:
In their 2013 NFER
Thinks paper
Improving Young
People’s
Engagement with
Science,
Technology,
Engineering and
Mathematics
(STEM), Suzanne Straw (Deputy Head
of NFER’s Centre for Evidence and
Evaluation) and Shona MacLeod
(Deputy Head of NFER’s Centre for
Evidence and Evaluation) present
NFER’s research evidence about what
works to encourage further engagement
in, and take-up of, STEM subjects.
Engage pupils at an early age and at
key transition points
Focus teaching on practical activities,
set in real-life contexts and offer good
quality enrichment and enhancement
activities
Link teaching to careers in STEM
Make clear links across and between
the STEM subjects
Support teachers
Straw and MacLeod say that in addition
to these measures, teachers, lecturers,
schools and colleges should adopt “a
holistic approach which combines a
number, or all, of the elements below is
at the heart of successful practice. “
Diagram from Straw, S. and MacLeod, S. (2013). Improving Young People’s Engagement
with Science, Technology, Engineering and Mathematics (STEM) (NFER Thinks: What the
Evidence Tells Us). Slough: NFER.
On the following pages you’ll find information about some of the sessions at the
upcoming MEI 2015 Conference that explore ways in which teachers and lecturers can improve their students’ engagement in and take-up of mathematics.
The MEI 2015 Conference is taking place 25-27 June in the stunning new conference facilities at the University of Bath.
MEI Conference:
engaging students
Girls & Maths:
issues and
solutions
(Claire Baldwin)
09:00 - 10:00
Saturday 27 June
Based on recent
publications and
research evidence,
this session will
discuss 'what works
for girls' and consider
the myths and facts
surrounding the issue
of why girls
participate in A level
Mathematics and
Further Mathematics
at lower rates than
boys. We will
consider the
resources produced
by the FMSP, which
aim to generate
awareness of the
ongoing need to
promote post-16
participation in
mathematics to girls.
Improving student attitudes to
maths (Charlie Stripp)
Questioning techniques at A level
(Mohammed Basharat & Nick Thorpe)
13:45 - 14:45 Saturday 27 June
17:30 - 18:30 Thursday 25 June
If we are to improve maths education,
and the ability of our population to be
able to use mathematics confidently
and effectively in work and life, we need
to overcome the idea that 'you're either
good at maths or you are not', and that
you are somehow born with a fixed
ability to do maths.
This session will look at a variety of
questioning techniques which can be
used in A level Mathematics. We will
focus on how questions can be used to
stimulate discussion in the classroom
and be accessible to all students.
Connecting learning to the world of
work (Geoff Wake)
13:15 - 14:30 Thursday 25th June
16:00 - 17:00 Friday 26 June
MaSciL is an EU funded project that is
developing approaches to teaching and
learning that connects maths and
science to the world of work using
enquiry methods. Participants will
explore KS3 support materials.
Problem solving at KS3&4
(Phil Chaffé)
13:45 - 14:45 Saturday 27 June
This session will be hands-on with
plenty of problems to solve. You will get
the chance to reflect on the problem
solving skills that you are using and
how the careful selection of appropriate
problems can develop a student's
problem solving toolkit.
Tricks and Tips (Jo Morgan)
11:45 - 12:45 Saturday 27 June
FMSP resources:
Encouraging Girls
to Take Mathematics
Information About
Girls in Mathematics
In this highly interactive session we'll
look at a variety of mathematical
methods, shortcuts and tricks. We'll
debate whether these methods create
barriers to learning mathematical
concepts or have the potential to make
mathematics accessible to all.
Active learning in Core 1-4
(Jo Sibley)
This very active session is a distillation
of the 'A level Active!' CPD day run by
the FMSP. The session will focus on
continuous assessment and feedback
in the A level classroom in the light of
rapidly increasing A level numbers.
Rich Starting Points at GCSE
(Carol Knights)
13:15 - 14:30 Thursday 25 June
During this session we will explore
some of my favourite 'Rich Starting
Points' gathered from the last 20+
years, and consider how they can be
used to challenge current GCSE
students to think (more).
Active learning with GeoGebra
(Tom Button)
16:00 - 17:00 Friday 26th June
This hands-on session will focus on
student-centred activities that help
students enhance their understanding
of A level Core Maths concepts through
using GeoGebra. All the activities are
designed to be used on the computer or
tablet version of GeoGebra.
View more sessions here.
Crash Course:
command
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 the while
command which
allows us to keep
applying a loop of
code until some
condition is met. This
is really useful when
dealing with iterative
processes for which
we anticipate some
behaviour eventually
but we don’t quite
know when!
Crash course March
problem – solution
can be downloaded
from the Monthly
Maths web page, or
click the link above.
The code below demonstrates how a while loop works (the output from the code is
shown immediately below the code itself).
1) After a variable called alpha has been defined to be 0 (notice that this is defined
to be 0.0, in Python 2 this is important - the decimal point means that Python
knows we’d like it to take non-integer values). The while statement checks whether
alpha squared is less than 2. This is of course true (at the moment) and so the
code below is carried out and alpha become 0.00001.
Now the program checks whether the square of 0.00001 is less than 2. This is still
true and so a further 0.00001 is added to alpha. This will continue until alpha
reaches a value which squares to a number not less than 2 (i.e. greater than or
equal to 2). At this point the code in the while loop is no longer carried out and the
program moves on to the next instruction.
2)This means that value alpha reaches before leaving the while loop squares to a
number greater than or equal to 2 and the alpha - 0.00001 squares to a number
less than 2.
Therefore we can conclude that, for that value of alpha
alpha - 0.00001 < √2 < alpha
Here is another interesting example using the while function. It’s well known that
the harmonic series
1 1 1 1
1      ...
2 3 4 5
diverges.
This means that given any value λ by including enough terms in the summation
above, its total will exceed λ.
Crash Course:
Collatz Conjecture
Problem of the
month
We can find how many terms are required for the summation to exceed 10 as
follows:
The Collatz
conjecture states
that, for any choice of
positive integer for the
first term x0, the series
defined as follows:
If xn is even then
xn + 1 = xn /2.
If xn is odd then
xn+1 = 3xn + 1
includes the value 1.
(i.e. it eventually
reaches the value 1,
after which it will be
periodic with values
1, 4, 2, 1, 4, 2, 1, 4, 2,
1,…)
For example
If x0 = 13 this
sequence is:
13, 40, 20, 50, 5,
16, 8, 4, 2, 1, 2, 4,
1, 2, 4,…
X9 = 1.
For which starting
integer x0 with
1 ≤ x0 ≤ 200 does the
sequence above
take longest to
reach the value 1?
For this value of x0
what is the smallest
n such that xn = 1?
It’s interesting to look at how many terms are required for the harmonic series to
exceed each of the values from 1 to 10:
Site seeing with…
Terry Dawson
Terry Dawson, a
Curriculum Developer
for MEI, shares his
favourite resources
this month. Terry was
involved in using
Professor Sir Timothy
Gowers's ideas to
inform a Critical
Maths curriculum
based on students
engaging with realistic
problems and
developing skills of
analysing problems
and thinking flexibly to
solve them.
Modelling Epidemics
One of my favourite resources at the
moment is the epidemic modelling
tool on the nrich website. Using this
resource, students can explore some
of the factors which influence the rate at
which an epidemic spreads.
I love this resource because it offers so
many possibilities for learning. You
could ask the students to work in
groups and simply give them the task
“investigate how changing some of the
parameters affect the outcome of the
epidemic”. This would allow the
students to decide on how best to
divide the labour, what factors to
investigate, and how to proceed. Or you
could add a little more structure, run the
model through on the default setting
and then look at the results. Follow this
with some prompts like:
Do you think it would be exactly the
same if I ran the model again?
What do you think would change if I
made the probability of infection higher/
lower?
The OCR Level 3
Certificate in
Quantitative
Reasoning (MEI)
includes a Critical
Maths component.
MEI Critical Maths
resources are freely
available, thanks to
DfE funding.
After setting the parameters of the
epidemic using the configuration data
bar, click on the start process button to
make the model go through one cycle.
Terry will be delivering A summary of the results can be
Critical Maths
obtained by clicking on the results bar:
sessions at the MEI
Conference:
Learning through
Discussion; Don't
believe everything
you read in the
papers; Fermi
Estimates, and
Adapting resources
to suit your class.
What do you think would change if I
made the probability of death higher/
lower?
Similar questions could be asked of the
other parameters. You could then give
the task of investigating a particular
parameter to each pair. The students
could share their findings with others at
the end the session. You could then ask
them to speculate on what would
happen if two parameters were
changed at the same time, or look at
real life cases such as Ebola and Flu.
Similar mathematical structures can be
used for modelling things such as
YouTube clips going viral, a computer
virus, Twitter followers and Facebook
friends. There are some interesting
TED Talks on this. If you are interested
in getting students to understand a little
more about the model, then Motivate
Maths has some great ideas.
MEI Resources
MEI Conference
Exhibition
(Friday 26 June)
Delegates can
discover what support
these exhibiting
organisations offer to
mathematics
teachers:
ASDAN Education
ATM
bksb
Cambridge
University Press
Casio Electronics
Chartwell-Yorke
Collins
Core Maths Support
Programme
Eduqas/WJEC
FMSP
Hodder Education
IMA
Integral Resources
Mathspace UK
MEI
Moravia Europe
National STEM
Centre
NCETM
NST Travel Group
OCR
The OR Society
Oxford University
Press
Pearson
Science Studio
ScienceScope
Tarquin Books
UKMT
MEI 2013 and 2014 Conference
session resources
Here are links to resources and
handouts from some of previous MEI
Conference sessions that are relevant
to the theme of increasing engagement
with students:
Introducing problem solving into the
KS4 classroom - Phil Chaffe
Using problem solving to develop
mathematical thinking in post GCSE
students - Phil Chaffe
Neriage: a problem solving approach
to learning - Terry Dawson
The pros and cons of using contexts
in the teaching of mathematics - Sue
Hough & Steve Gough
Assessment for Learning with
examples from GCSE and A level Debbie Barker & Simon Clay
Trialling a different approach to
GCSE resit - Sue Hough & Steve
Gough
Problem Solving Post-16 - Tim
Gowers, Stella Dudzic, Charlie Stripp
and Terry Dawson
Free MEI and FMSP Resources
MEI Critical Maths
resources
Designed for post-16
students at level 3; especially useful for
Core Maths classes, enabling students
to think about real problems using
maths. Many of the resources start by
engaging the students in giving an initial
opinion and then encourage them to
think more deeply and to evaluate their
initial thoughts.
FMSP Resources
These include KS4 and
Post-16 problem-solving
resources, online GCSE and A level
revision sessions, Senior Team
Mathematics Challenge past materials,
GCSE and A level Enrichment and
Extension resources, Real World
mathematics, and resources for
promoting mathematics.
New KS4 classroom resource
In the following pages is a Key Stage 4
teaching and learning resource:
Rich Starting Points in KS4 maths Christina Williams
KS4: What do you see? developed by
Carol Knights. This edition looks at a
range of visual stimuli and asks ‘What
do you see?’ Many of the activities are
visual representations of proof,
including sums of infinite series and
algebraic equivalence.
Assessment for Learning at A level Simon Clay & Debbie Barker
The resource can be downloaded from
the Monthly Maths web page.
Using multilink cubes in the
secondary classroom - Debbie Barker
Problem solving at KS4 - Debbie
Barker
What do you see? (1)
Watch what happens.
Start again
What do you see? (1)
Describe what happens.
How many dots are added each time?
What do you see? (2)
Watch what happens; it starts with a square of
dots.
Start again
What do you see?(2)
Can you describe what happens?
Would the same thing happen if you
started with a larger square?
Or a smaller square?
Can you write this algebraically?
Start with any
square: a²
a
a
Take one
away: a²-1
a
a
Move the top row to
become a column…
a
a
Move the top row to
become a column…
a
a
Move the top row to
become a column…
a
a
Move the top row to
become a column…
a-1
a
Move the top row to
become a column…
a-1
a
Move the top row to
become a column…
a-1
a+1
Move the top row to
become a column…
a-1
a+1
(a-1)(a+1)
a-1
a+1
a²-1
=
(a-1)(a+1)
What do you see? (3)
Watch what happens; the initial shape is a
square.
The shape
removed is
a square.
Start again
What do you see? (3)
Can you describe what happens?
Would the same thing happen if you
with a larger square?
Or a smaller square?
Can you write this algebraically?
What do you see? (4)
Watch what happens.
Start again
What do you see? (4)
What would happen if it kept on going?
Describe what happens in words.
Can you describe what happens using
numbers?
1
2
1
2
1
8
1
32
1
16
What
should
these be?
1
4
Start again
What do you see? (4)
Does this help you to find the sum of the
following:
1 1 1 1 1
     ... 
2 4 8 16 32
What do you see? (5)
Watch what happens.
Start again
What do you see? (5)
Describe what happens in words.
What would happen if it kept on going?
Can you describe what happens using
numbers?
1
9
1
3
1
27
1
81
1
81
1
9
1
27
1
3
What do you see? (5)
Does this help you to find the sum of the
following:
1 1 1 1
 
  ... 
3 9 27 81
What do you see? (5)
Can you think of a similar geometrical proof to
find the sum of the infinite series:
1 1
1
1



 ... 
4 16 64 256
Teacher notes: What do you see?
This edition looks at a range of visual stimuli and asks ‘What do you
see?’ Many of the activities are visual representations of proof,
including sums of infinite series and algebraic equivalence.
The pedagogical focus for these activities is on generating discussion
amongst small groups of students in order that they arrive at an
understanding. At various times, the teacher might circulate and listen
in unobtrusively to discussions, ask probing questions, or draw out
conflicting or differing responses from groups to contribute to a whole
class plenary. The activities could be used as a series of starter
activities.
These are some of the approaches highlighted within the AfL session
run by Simon Clay and Debbie Barker at the MEI 2014 Conference.
Additionally, these are strategies which many girls find particularly
supportive – but that’s not to suggest that boys don’t find them
supportive too!
Teacher notes: What do you see? (1)
This activity shows the sum of consecutive odd numbers.
Show the sequence to students and ask
them to describe what they see.
Make it ‘girl friendly’:
Ask students to discuss
it with a friend before
giving an answer
At KS3 or 4, students could express in
words that adding odd numbers always gives a square number.
At a higher level students might use notation for the sum of a series.
Either: 1+3+5+…+(2n-1) =n2
Or:
𝑛
𝑖=1 2𝑛
− 1 = 𝑛2
Teacher notes: What do you see? (2)
This activity shows that a²-1 = (a-1)(a+1)
Show the sequence to students and ask
them to describe what they see.
A specific size of square is shown, ask
students to consider what would happen
with larger or smaller starting squares.
They might like to sketch them out to check,
so some squared paper might be useful.
Make it ‘girl friendly’:
Ask students to “think,
pair, share”: give students
silent thinking time before
a short discussion with a
partner and then others.
Teacher notes: What do you see? (3)
This activity shows that a²-b2 = (a-b)(a+b)
Show the sequence to students and ask
them to describe what they see.
A generic square is shown and a smaller
generic square is removed.
Make it ‘girl friendly’:
Ask students to work
with a partner; you
circulate, listening in and
asking questions quietly
to pairs.
Can students convince themselves that the rectangle ‘on top’ will always
fit ‘at the side’?
Encourage students to write this algebraically.
Teacher notes: What do you see? (4)
This activity shows the sum of the reciprocals of powers of 2.
Show the sequence to students and ask
them to describe what they see.
Transitions are timed – no mouse click
needed.
Make it ‘girl friendly’:
Show the sequence two
or three times through.
Accept and validate all
responses.
Students in KS3 and 4 may describe the
series in words.
Ask how much of the white square is used each time. The second part of
the sequence also shows the fractional values.
Teacher notes: What do you see? (5)
This activity shows the sum of the reciprocals of powers of 3.
Show the sequence to students and ask
them to describe what they see.
Transitions are timed – no mouse click
needed.
Make it ‘girl friendly’:
Ask students to
• Convince themselves
• Convince a friend
• Convince the class
Students in KS3 and 4 may describe the
series in words.
Ask how much of the white square is used each time. The second part of
the sequence also shows the fractional values.
The third part of the activity asks if students can think of something
similar for reciprocals of powers of 4. An example shown on the
following slides shows that the sum is 1/3.
Teacher notes: What do you see? (5)
In this case, divide the square into quarters,
colour one quarter dark and two light.
Teacher notes: What do you see? (5)
With the remaining space, divide it into quarters,
colour one quarter dark and two light.
Teacher notes: What do you see? (5)
With the remaining space, divide it into quarters,
colour one quarter dark and two light.
Teacher notes: What do you see? (5)
For every one dark section there will be two light
sections.