Teaching Number to
Post-16 GCSE
students
s.hough@mmu.ac.uk
s.j.gough@mmu.ac.uk
MEI conference 2015
Problem Solving
1 1

4 2
Do you think you have got
this right?
Explain why.
Year 11
%
Fractions correct
N=50
Target
grade C
%
Correct
Target
Grade
D/E
Project
83%
57%
Control
72%
30%
Pupils may attempt to solve a
problem by
• Engaging solely with the numbers
• Attempting to make sense of the
problem
GCSE resit cohort Mathematical issues
Memory of formal methods can be poor
Use of misremembered strategies –
‘doing something with the numbers’
Desire to use informal methods – these
tend not to have been developed to a
more efficient, ‘of worth’ level
GCSE resit cohort – attitudinal issues
Repeated failures in the
examination
Lack of confidence + belief
‘Oh no not this again’
GCSE resit cohort – Some
Teacher dilemmas
How to cover the syllabus in 9 months
(or less)
How to cope with the variety of methods
students may bring
Insanity: doing the same
thing over and over
again and expecting
different results.
Albert Einstein
Realistic Mathematics Education
(RME)- the Dutch approach to teaching
maths
• Well researched activities encourage
students to move from informal to formal
representations of maths
• Use of context is sustained throughout
• Use of models to support student
development
• Progress towards formal notions seen as a
long term process
RME, MMU and Post 16 GCSE resit
• MMU have worked on RME related projects
for the last 13 years
• ‘Making Sense of Maths’ series published
by Hodder, written by MMU with support
from the Dutch
• Has the potential to offer students a
different approach to GCSE resit
Nuffield Funded GCSE RESIT
Project
• 4 project and 4 control classes in three sites
• Intervention consists of Number and
Algebra modules
• Data includes: pre and post tests, attitude
data, post test video interviews, teacher
interviews
Today’s session
• Working with the material
- the bar model
- the ratio table
• Students discussing their post test
strategies
Post test interview 1
• Wirral Metropolitan College, adults class
• Student A
Examples of the Material
• Use of context
• Use of models
The Download Bar
This is an image of a download bar on a computer.
1) When did you last see something like this?
2) What information is there in the window?
3) How long has the program been loading so far?
4) How many megabytes (MB) are there left to load?
The Download Bar
On Worksheet N1, work out the total installation time for each
of the 9 bars.
1. Work out the total installation time for
the program below:
0
4
0%
25%
? minutes
100%
2. Now work out the total installation time
for this program:
0
9
0% 10%
? minutes
100%
Reverse Percentage Calculations
At Shoo, there is a “25% off everything” sale. A pair of boots cost
£63.75 in the sale. What would the original price have been?
Draw a bar to represent this information
Buying ribbon
Louisa is a dress maker. She uses ribbon to edge her
garments as a way of making her designs stand out.
Louisa buys a sample of the ribbon shown below:
a)It costs her £ 1.16 for 80cm of the spotty ribbon. Draw
a picture to represent this information and make it look
realistic
b)What else do you know? Mark on your picture some
other quantities of this ribbon, which you would know
the cost of.
Sharing Food
The café where Ruby works specialise in making large rectangular
pizzas. These are popular with lunch time diners who often order one of
the large pizzas to share. Masood and Tim are students. They meet for
a pizza in the café once a week. On one occasion they order the
rectangular cheese and tomato pizza shown below costing £10.35.
Tim is not as hungry as Masood, so they cut the pizza into 9 slices, Tim
eats 4 slices and Masood eats 5 slices
Draw a picture to show how they might share out the pizza
Share out the cost of the pizza between Masood and Tim
in the ratio 5 : 4
Can this be true?
I’ve got an 80 waist
How can this be true?
You need to sort that out.
Mine’s only 40
Extending the Number Line
Unlike many countries, in the UK we still use two different units for
measuring length. Both of these can be seen on a standard tape
measure.
The tape shows that 6 inches is roughly the same as 15 cm.
What do you think 12 inches is in centimetres?
The Number bar
Daniel drew the number bar above to represent the tape measure
1. Make a copy of this bar and mark four other points on it that
you know in both centimetres and inches.
2. Mark is 5 ft and 3 inches. Alexandra is 1 metre and 72
centimetres. Use the 30cm to12 inches rule of thumb to decide
who do you think is the taller of the two?
Recipes
Helen decides to use the ratio table
below to work out the ingredients
needed for different numbers of
pancakes.
Pancakes
Makes 8 pancakes
Copy the table and fill in four more
columns showing the ingredients
needed for different numbers of
pancakes
Ingredients
125 g plain flour, sifted
1 medium size egg, beaten
300 ml milk
a little oil for frying
Best buy
In the supermarket, Helen and Nisha are buying ingredients for
other pancakes and trying to make sure that they get the ‘best
buy’
Which do you think is the best buy?
Best Buy
To work out which is the best
buy, they decide to use a ratio
table. They begin by doubling to
see if this helps them to compare
the two buys. After a few
minutes, they both have exactly
the same thing on their notepads.
It looks like this:
Can you decide from here which is the ‘best
buy’?
From the original table above, Helen and Nisha then work in
slightly different ways to make a comparison easier. Below is
their working.
Explain carefully what each of them has done and how you can now
see which is the best buy.
Starting the ratio table
For the next questions, draw your ratio table and fill in four other sets of
values that you know to be true.
1) A hare travelling at top speed
can run 260 metres in 12 seconds.
What else do you know? (30 secs)
2) 8 metres of ribbon costs £4.80
What else do you know? (30 metre
3) 5 miles is roughly 8 km. What
else do you know? (120 miles)
4) To make pink paint Joel uses a
ratio of 4 parts red to 10 parts white.
What else do you know? (30 RED)
5) A printing machine can produce
24 copies in 30 seconds. What else
do you know? (95 SECS)
Menu
Back
Forward Cont/d
Idea
Vocabulary Q 1
6) A multipack of crisps contains 15
bags. What else do you know? (12
MULTIPACKS)
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Post test interview 2
• Salford City College
• Student B
Post test interview 3
• Salford City College
• Student 3
Some early findings
• Easier for the adults to take on the
approach than the 17 year olds
• Time is required to develop the approaches
• Students are able to recognise the potential
for using the bar and the ratio table to
answer questions in a variety of topics, and
do so several weeks later.
• More difficult to have impact in the Algebra
module
Some early recommendations
• Teachers need support to take on this
approach in the way it is intended
• Students who narrowly missed a grade C
may do as well to be taught topics in the
‘usual’ way
• Students who are at Grade E or below
could really benefit from this approach if it is
used over a two year course
Access to materials
• Hodder ‘Making Sense of Maths’ series
‘Fair Shares’ book for Number resources
‘All Things Equal’ for Algebra resources
About MEI
• Registered charity committed to improving
mathematics education
• Independent UK curriculum development body
• We offer continuing professional development
courses, provide specialist tuition for students
and work with industry to enhance mathematical
skills in the workplace
• We also pioneer the development of innovative
teaching and learning resources