Buckinghamshire County Council
Progression in Division
Twilight session
14.10.09
Becky Ellers
bellers@buckscc.gov.uk
Buckinghamshire County Council
AIMS
• To consider how to secure pupils’ understanding of
division
• To review the use of models, images and language in
the teaching of division
• To review progression in division in the renewed
framework/update progression grid
• To consider how pupils can be helped to learn division
facts
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Pupils Written Division Strategies:
The role of the number line in
developing children’s intuitive models
The research....
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Why is division hard for children?
Division involves multiplicative reasoning as opposed to
additive reasoning and is therefore harder and is usually
taught after addition and subtraction (Piaget, Grize, Szeminska,
Bangh, 1977, Brown, 1981)
• Additive reasoning
(joining and separating objects)
• Multiplicative reasoning
(sharing, dividing and splitting)
+
=
8 sweets
shared
between
2
children
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Squire and Bryant (2002) From Sharing to
dividing
• Researched whether children can recognise the three
terms, with a particular focus on recognising the quotient
and the divisor from two models of dividing.
• Tested the prediction that young children find it easier to
identify the quotient when the portions are grouped by
the divisor in partitive problems (sharing) and by the
quotient in quotitive problems (grouping).
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Models shown to children
D
Q
I
U
Quotitive
task
O
(grouping)
V
Partitve task
(Sharing)
I
T
S
I
O
E
R
N
T
eg: 12 girls and 4 tables = 3 girls
eg: 12 girls and 4 girls can sit
around each tables = 3 tables
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Results
• In both the partitive and quotitive tasks, one
condition was easier
• There was a considerable difference between
age groups
• In the partitive task (sharing) the easier condition
was grouping by the divisor.
eg: 12 girls and 4 tables = 3 girls
• In the quotitive task (grouping) the easier
condition was grouping by the quotient.
eg: 12 girls and 4 girls can sit around each
tables = 3 tables
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Where now in terms of education, how do we link
research and practice to help children develop their
understanding with division?
• Children appear to find sharing (partitive) tasks easier
but this becomes inefficient with larger numbers,
e.g. 196 Smarties divided by 6 children – on a one for
you, one for me basis this take a while!!!
We therefore need to help children to move from sharing
individual items to groups of items,
e.g. We could do 20 for you, 20 for me etc, we are taking
away groups of the divisor from the dividend
(CHUNKING).
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Research into methods
• Anghileri, J. 2001
The algorithm proves particularly unsuccessful in younger pupils (ages 9-10) in
calculations with a 2-digit divisor where children have only been taught the algorithm
for one digit divisors and they attempt to apply the same rules
•
Anghileri and Beishuizen 1998
Questioned 52 children after completing a division calculation 96 divided by 4. Whilst
many children used counting strategies they were increasingly more efficient a few
children counted in ones and used repeated addition by the divisor to reach 96.
Overall of the 52 pupils questioned, more than twice as many pupils (37%) solved the
problem successfully using a counting strategy than did using a place value
procedure or standard algorithm (17%).
BUT
Those that used a chunking method found their explanation hard due to
complicated or little recording.
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Young children’s intuitive models of division and multiplication
(Mulligan and Mitchelmore,1997)
• Looked at children intuitive
models before formal teaching
of methods.
• Longitudinal study of grades 2
and 3
• Looked at multiplicative
reasoning, one step problems
• Interviews with questions read
out and only 40 cubes
available for the child
DIRECT COUNTING
One-to-many
correspondence
Unitary counting
Sharing
Trial-and-error grouping
REPEATED
SUBTRACTION
Counting backwards
Repeated subtraction
Additive halving
REPEATED ADDITION
Counting forwards
Repeated adding
Additive doubling
MULTIPLICATIVE
OPERATION
Known multiplication facts
Derived multiplication facts
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Results
• Children’s strategies develop over time and they are able to draw on
this repertoire of strategies to solve problems
• Number facts extend
• Children’s most efficient calculation strategies gradually become
more refined and more widely used
However,
‘at least during the early learning period, different problems may be
solved using different intuitive models’
Buckinghamshire County Council
Mulligan and Mitchelmore,1997 had suggestions for
teaching
1. We need to teach multiplication and division
together to draw upon the intuitive models children
use
2. Encourage the use of a range of strategies and
help children refine them over time into an efficient
calculation method
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Research
Part one:
• Testing 90 + children Year 5 children on division
questions
• Test is expected to take about 20 mins but
children will be allowed as long as they like to
complete the task
• Short interview with the teacher about the
method they teach
• Questions will involve both partative and
quotative with and without remainders
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A tent holds 6 children.
How many tents are needed to hold 70
children?
(Paper A)
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So what does all this mean for us?
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222 ÷ 3 =
What knowledge, skills and concepts would children need
in order to be able to do this calculation?
Show the person next to you how you currently teach your
Year 5 children to carry out this calculation.
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How confident would your Year 5 and 6 children in
your school be at answering these questions?
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• Do you think children would find it as difficult to answer
50 ÷5?
• What makes 56 ÷4 a harder question?
• Would asking separately for 300 ÷2 and for two
numbers with a product of 150 make the second
question easier?
• What progression is implied between the Year 2
question and the Year 6 question?
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Task
12 ÷ 4 = 3
Can you write a sentence for this number statement?
Can you draw a picture of you partners sentence?
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Number line division
96 ÷ 6 =
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Number lines and grouping
2
0
2
2
2
4
2
6
8
Slide 2.21
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TASK
0
What division calculation is represented if:
 the step size is 4?
 the right-hand marker represents 18?
 the middle marker represents 6?
Slide 2.22
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Remainders
5
5
5
2
0
5
10
15
17
20
5
17 ÷ 5 = 3
2
5
Slide 2.23
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To divide 81 by 3
81
60 ( 20 lots of 3)
20 x 3
0
7x3
60
21
81
21 lots of 3)
0
Slide 2.24
Slide 4.24
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Discussion point 1
Does this calculation have different answers in
different contexts?
22 ÷ 4
Can you think of a context where the answer
would be:
- rounded to a whole number?
- expressed as a fraction or decimal?
- expressed with a remainder?
Slide 2.25
Types of short division
calculations
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no exchange, no remainder
4)848
no exchange, with remainder
3)635
with exchange, no remainder
7)994
with exchange, with remainder
3)470
empty place at start of quotient
7)287
noughts in the quotient
4)816
8)5608
decimal dividend
5)61.5
3)4.26
Slide 2.26
Slide 4.26
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Summary
• The language ‘divided by’ and images of repeated
addition or division on a number line help to secure
pupils’ understanding of division in KS1 and the early
years of KS2
• It is essential that these early ideas are taught well and
that pupils develop a conceptual and visual framework
linked to the language of division
Slide 2.27
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Summary
• The middle years of KS2 should focus on:
– how to use both factorising and partitioning as
mental strategies for division
– how to record these strategies to support or explain
their thinking
– how and when to express a quotient with a
remainder, or as a fraction or decimal
(a model of a number line is helpful here)
Slide 2.28
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Summary
• Pupils working confidently at level 4 should also be able
to carry out ‘short’ division of a three- or four-digit
number by a single-digit number
Slide 2.29
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