CTP0412 – Fordbridge Community School
Developing the use of models and images to support progression and proficiency in subtraction and division
methods throughout the school
Claire Nicholson of Fordbridge Community School, Birmingham
Abstract/Summary
The initial aim of the project was to produce a calculation policy for subtraction and division, as these were both felt to be the weakest operations within all the
partnership schools. Research was carried out into the teaching of division but changes in current practice and methods led to great debate about which calculation
methods to include in the division policy. This will now therefore be completed outside of this project.
This project culminated in the development of a routes through calculation policy for subtraction, which has been shared and implemented successfully in the schools
involved. The policy considers the use of models and images, websites and other resources, vocabulary and appropriate questions, as well as clearly defining the
methods that children will be using. Looking at a range of documents and current practices in teaching of calculation methods was invaluable in producing this effective
policy.
The initial impact shows that there are improvements to the children’s awareness and confidence in the concept of subtraction, as well as having methods that they can
more confidently and successfully independently draw upon. Over time, the aim is that progression between methods will be more successful, leading to greater
confidence and proficiency for all children.
Background
Four primary schools, Fordbridge, Coleshill Heath, Yorkswood and St Mary and St Margaret’s, worked in collaboration with an improvement agent, Donna Wright, on
this project. All four schools belong to the North Solihull Collaborative, Unity, and have previously worked together successfully. Most recently, following a Unity Lead
Moderators course for English and Mathematics, the four project leads have become lead moderators for mathematics. Two of the four schools have also worked
together in a collaborative project, ‘Learning in a Maths Network’, with Donna Wright and ten other schools from Unity. During this project, representatives from each
school worked on developing an area of mathematics in their classroom and across school, feeding back and sharing good practice with their colleagues on a termly
basis.
CTP0412 – Fordbridge Community School
All four schools already had a routes through calculation policy in place but these policies where dated and needed reviewing to ensure consistent and progressive use
through school. Due to amalgamations and turnover in staff within these schools, recent monitoring has shown a need to revisit this practice. When reviewing the
calculation policies, the appropriate models and images will be explicitly exemplified alongside each step to aid effective teaching and learning.
Aims of the Collaborative Teacher Project
The aim of this project was to achieve proficiency, consistency and confidence in calculation methods across each school. All the schools involved have routes through
calculation policies that need reviewing to ensure consistent and progressive use through school. The intention was to produce a new policy that outlined the methods
to be used, as well as the models and images that can be effectively implemented alongside these, leading to proficiency and confidence in the methods that children
use.
Details of those involved in the Collaborative Teacher Project
Claire Nicholson, Fordbridge Community Primary School
Julia Bayman, Yorkswood Primary School
Sophie Miles, St Mary & St Margaret’s CE Primary School
Deb Wallis, Coleshill Heath Primary School
Improvement Agent: Donna Wright, Solihull School Improvement and Advisory Service
A description of the Collaborative Teacher Project
The project began with research on two levels; reviewing current practice within schools, to form a baseline, and a review of relevant current research and findings in
the field. Each school baseline was formed by representatives carrying out a work scrutiny, looking at consistency and progression in methods across the school.
Questionnaires were also designed and given to pupils and staff to gauge opinions about confidence and proficiency in calculation methods. Information was then
collated, forming a broad picture both within and across the schools. A review of research was carried out into current good practice and pedagogical research. The
representatives from the schools then worked collaboratively alongside our improvement agent to begin to develop routes through calculation policies. We decided to
complete the subtraction policy and introduce this to staff first rather than trying to introduce two new policies together. Once the subtraction policy was complete
each representative shared the policy first with the Senior Leadership Team at each school and then with the rest of the staff. Representatives then monitored its use
and impact. Finally each representative completed a further work/planning scrutiny to explore the impact the introduction of the new policy had had on the teaching
and learning of subtraction within each school.
What has been learned from the project
Fordbridge Community Primary School
The development of the policy has been useful in recognising that consistency in approach to teaching calculation methods is important to provide proficiency. This is
something that will take time to establish with the new policy and to embed consistent practice. The project has also been useful in thinking more about the range
CTP0412 – Fordbridge Community School
models and images that are used when teaching calculation methods and in ensuring that children have clear mental images of what they are calculating when
subtracting. A greater range of resources are now being implemented across school and evidence of this is seen within weekly planning.
St Mary & St Margaret’s
The development of the policy has enabled staff to re-visit the way they teach subtraction and to ensure that throughout school there is a consistent approach to the
methods the children are shown and the models and images that are used to support them. Staff are implementing the new policy and more evidence of the use of a
range of models and images is evident in planning and during learning walks. Children are encouraged to use equipment during lessons. Staff are also more aware of
the progression children need to go through to become proficient in subtraction.
Yorkswood Primary School
As mentioned earlier, over the last few years, a change in staff has meant that a need to re-launch the calculation policy has been necessary. This project provided a
great opportunity for the school to up date it and ensure all staff were aware. This policy has been fully implemented into school and staff have had the opportunity to
work with it while planning to address the progression in their particular year groups. Staff have received the policy well and have found a great benefit to the inclusion
of examples of models and images and suggested links to websites. The policy is now showing through in the work that children are doing in class and children are more
secure in carrying out subtraction calculations where before, this was identified as a weakness.
In all schools, children’s confidence and proficiency in calculation methods has developed well, which was the main aim of the project. Over a greater period of time, we
expect to see more successful progression from one method to another, with this consistent approach in place across the schools.
Impact on teachers’ practice
Teachers are clearer about the methods, vocabulary and models that need to be used at each phase of progression. The policy has helped to clarify how one method
links and advances to another. Teachers are more regularly planning for a range of models and images when teaching mathematics, not just in subtraction, which is
helping pupils to have clearer mental images about what they are learning. The children are becoming more independent in accessing and using such resources for
themselves to support their own learning when they feel appropriate.
Impact on others
A review of practice by the project leads was completed in each school at the start and end of the project to help evaluate its impact. This included pupil questionnaires,
work scrutinies and observing teaching and learning.
Pupil questionnaires, carried out in years 2, 4 and 6, indicate that the project has improved awareness of the concept of subtraction. Nearly all children could give a
good explanation of what subtraction is, which wasn’t the case at the start of the project. Less than half of children in year two could accurately explain subtraction at
the start of the project, with over three-quarters of children giving good explanations now. The policy includes the key vocabulary to be used in each phase of
progression to ensure that teachers are encouraging the use of this greater range of mathematical vocabulary. ‘Take away’ was most commonly associated with
subtraction at the start of the project and although this is still the case, vocabulary has developed with words and phrases such as ‘minus’ and ‘difference between’ far
CTP0412 – Fordbridge Community School
more commonly used and understood by pupils. Other developments, such as more readily recognising that it is the inverse of addition and knowing language such as
‘reduce’ and ‘decrease’ have also been noted.
When given questions to answer independently, children are far more confident at making attempts, and the majority of children are using appropriate methods,
whereas we found before that many children reverted to immature and inefficient methods, such as using lots of tally marks. Children’s accuracy in answering
subtraction questions independently has increased and nearly all are consistent with methods outlined in the new policy.
Advice to teachers who may want to try something similar
The timescale of the milestones set out needed to be adjusted over the course of this project. If doing a similar project, we would advise thinking carefully about the
time that each step will take. We felt that spending longer than initially planned on the development of our policy was an important factor in the success of the project
and if the project had been rushed, adjustments may have been needed at a later date, perhaps limiting the impact.
References and resources produced or used
New Routes Through Subtraction Policy produced (See Appendix A that follows)
In creating this a number of sources used, including:
Mathematical Vocabulary Document produced (DfEE)
Models and Images for Addition and Subtraction Facts (DfES)
Pitch and Expectations Documents
New Draft Curriculum
APP documents
Range of other websites and calculation policies.
CTP0412 – Fordbridge Community School
APPENDIX A - New Routes Through Subtraction Policy
ROUTES THROUGH CALCULATION - SUBTRACTION
Stage
A
FKS
Vocabulary
Recording
Images and Resources
Websites and Links
Possible Questions
Children should be taught subtraction through a range of practical activities relating to ‘real life’ situations with great emphasis on the language and vocabulary of
subtraction. As for addition, children should be experiencing very visual approaches e.g. number lines and hundred squares. Children should experience counting songs and
rhymes. Expose children to the subtraction symbol alongside practical activities, with more confident children recording themselves. Children should be able to count back in
ones using resources e.g. a number track to help them. Children should be confident subtracting at least up to ten objects.
take (away), leave
Children to use concrete objects to
Five fat sausages frying in a Interactive bead string:
There are five cubes in this box. I am
how many are
experience subtraction, using the relevant
pan
http://www.ictgames.com/bri taking out two of the cubes. How
left/left over?
vocabulary.
Ten green bottles hanging
lliant_beadstring_with_colour many cubes are left in the box?
how many have
on the wall
.html
gone?
Evidence through photographs.
Counting and subtraction of
Here are some cups and some
one less, two less…
Cubes, counters, animals,
aliens:
saucers. Find out whether there are
ten less…
Pupils may record pictorially, with a greater
real life objects
http://www.primaryonline.co. more cups, or more saucers, or the
how many fewer
emphasis on use of language.
uk/sitetour/pol/planetstaken. same number.
is… than…?
html
How did you find out that there are
difference between
Adaptable number line, draw
more cups?
subtract, fewer,
on own jumps:
count back,
http://www.ictgames.com/nu Ella has three plums.
cheap, costs less,
mberLine.html
David has two plums.
cheaper
Who has fewer plums, Ella or David?
spend, spent
How many more does Ella have?
pay
change
Choose two cards from this set.
12345
Which of your two numbers is more?
Which number is less?
Take away one flower from this set of
7 flowers.
How many flowers are there now?
CTP0412 – Fordbridge Community School
B
KS1
Consolidate informal recording of practical subtractions and introduce the use of number sentences. Consolidate the use of number tracks for subtraction and introduce a
number line, where numbers are marked points instead of spaces. Annotate the number lines to show jumps on or back. Children should experience counting on and back
and understand that they have the same effect. Children should be made aware through practical examples, that when subtracting the order of the numbers is important.
Children should experience subtraction problems from a variety of contexts, not just taking away e.g. find the difference between…. Children should be confident
subtracting numbers within twenty, including subtracting a single digit number from a small two digit number.
minus
Coins
Interactive bead string:
Amy has these coins in her purse.
how much less is…?
Dienes
http://www.ictgames.com/bri
subtraction
Cubes
lliant_beadstring_with_colour
one hundred less
Counters
.html
tens boundary
Everyday objects
Simple subtraction problems
inverse
using images to support:
http://www.topmarks.co.uk/
Flash.aspx?f=TakeAway
Using images, taking away
How much is in Amy’s purse?
footballers:
http://www.ictgames.com/so Amy spends 10p. How much does
she have left?
ccer_subtraction.html
Number track:
http://www.ictgames.com/nu Jon is nine years old today.
Sarah is twelve years old today.
mberLineBoxes/index.html
How many years older than Jon is
Interactive number line,
Sarah?
counting back:
http://www.ictgames.com/nu
mberlineJumpMaker/index.ht
ml
Adaptable number line, draw
on own jumps:
http://www.ictgames.com/nu
mberLine.html
Number lines and empty
number lines to count on:
http://www.topmarks.co.uk/
Flash.aspx?f=Subtractdiffere
nce
Difference ITP:
http://www.mathsframe.co.u
k/resources/ITP_Difference.a
spx#
Ben puts 15 buttons on a table.
He hides some of them under his
hand.
How many buttons is Ben hiding?
CTP0412 – Fordbridge Community School
C
KS1
Use a number line to count on or back to find a difference between two numbers. Children should be using empty number lines and making appropriate jumps. Children
should be encouraged to apply known facts and place value to help them choose efficient jumps e.g. finding the next multiple of ten, partitioning and subtracting multiples
of ten etc. Children should experience subtracting TU – U and TU – TU, including bridging the tens boundary. Children should understand addition and subtraction as
inverses and be able to use inverses to check. Children should experience partitioning numbers in different ways in preparation for decomposition methods e.g. 18 = 10 + 8
but also 9 + 9, 14 + 4 etc.
Draft new curriculum proposes that children will use column subtraction in year 2 without exchange.
minus
how much less is…?
subtraction
one hundred less
tens boundary
inverse
Coins
Dienes
Cubes
Counters
Everyday objects
Counting up to find a
difference game:
http://www.ictgames.com/co
untonconvict.html
Adaptable number line, draw
on own jumps:
http://www.ictgames.com/nu
mberLine.html
Number lines and empty
number lines to count on:
http://www.topmarks.co.uk/
Flash.aspx?f=Subtractdiffere
nce
Balancing problems, using an
empty number line:
http://www.mathsframe.co.u
k/resources/Difference_unma
rked_number_line.aspx
Counting back on empty
number lines:
http://www.mathsframe.co.u
k/resources/Add_and_Subtra
ct_on_a_Number_Line.aspx
Counting on to 100:
http://www.mathsframe.co.u
k/resources/Subtraction_Usin
g_a_Number_Line_.aspx
There are 29 children. 5 children are
painting.
How many children are not painting?
82 – 45 =
Put a number in the box to make this
correct.
38 – c = 11
Write the number which is 11 less
than 40.
Write the answer.
75 – 43 =
CTP0412 – Fordbridge Community School
D
LKS2
Number lines should be continued to be used as children progress to using the expanded column method, initially without exchanging e.g. 46 – 23. Then exchanging should
be introduced when children are confident with this method, ensuring initially there is only one exchange required e.g. 274 – 157. Children should continue to use the
expanded column method to reinforce the value of digits and process of exchanging. Children should be working with three-digit numbers and calculations that use different
numbers of digits e.g. HTU – TU. Children should also experience decimals in the context of money.
hundreds boundary
Number lines and empty
Lewis makes a call from a telephone
less/least expensive 245- 132 =
200 + 40 + 5
number lines to count on:
box. He has £2 in coins. He uses
exchange
- 100 + 30 + 2
http://www.topmarks.co.uk/
these five coins to make the call.
decrease
100 + 10 + 3 = 113
Flash.aspx?f=Subtractdiffere
reduce
nce
Modelling column
How much money has he got left
483-289 =
subtraction:
from the £2?
Dienes apparatus to model
http://www.mathsframe.co.u
Emphasise and
exchanging
k/resources/Column_Subtract Chen has £9.10. He wants to buy a
model place
ion.aspx (please use
value when
game which costs £11.50. How much
Place value cards
exchanging.
exchange, not borrow
more does he need to save?
Coins
though!)
Cubes
Work out the difference between 147
Counters
and 205.
Calculate 137 – 65.
CTP0412 – Fordbridge Community School
E
UKS2
Consolidate the expanded column method with exchanging and model the compact column method alongside this. When first introducing, there should be one exchange
being made e.g. 274 – 157 and then children should be extended to thinking about further exchanges. Progress to then including exchanging when children need to
consider zero as a place holder e.g. 307 – 139. Children should be able to apply their skills to subtracting numbers with up to 5-digits and decimals, with the same number
of decimal places at this stage.
units boundary,
Compact column method with one exchange: Place value grids / cards
A shop sells three types of sunglasses.
tenths boundary
Links to measure and
discount
money
Dienes apparatus
Compact column method with more than one
exchange:
Use place value vocabulary
e.g. 17 tens subtract 8
tens gives 9 tens.
Compact column method with decimals (with
the same number of decimal places):
What is the difference in price
between the most expensive and least
expensive sunglasses?
A tree is 15.85 m tall. The house it
stands next to is 11.39 m high. How
much taller is the tree than the
house?
At the supermarket my shopping bill
came to £92.46. Of this £38.83 was
for shopping I bought for my mother.
How much did I spend on my
shopping?
Calculate 2006 – 289.
Calculate 1025 – 336.
Calculate 6247 – 2752.
Calculate 8.52 – 7.78.
Calculate 13.6 – 2.8
CTP0412 – Fordbridge Community School
F
UKS2
Children should continue to use the compact column method and confidently be able to subtract numbers with different numbers of digits, including different numbers of
decimal places e.g. 14.24 – 8.7. Children should know that decimal points should line up with each other.
Consolidate
vocabulary
14.24 – 8.7 in columns
Place value grids / cards
Links to measure and
money
A shop sells scarves and hats.
Ben buys one of the scarves and the
£4.50 hat.
How much change does he get from
£20? Show your working.
Emily buys two scarves and a hat.
What is the most she could pay?
I have cycled 4.6 km of a 72.5 km
cycle ride. How much further do I
have to cycle?
Calculate 15.05 – 14.84.
Calculate 8.6 – 3.75.