Manipulatives, Mastery and Calculation

Report
13.1.15
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To understand what is a manipulative
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To understand the mastery approach
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To consider a progressive calculation policy
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To know ways you can use manipulatives in
your own setting
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Manipulatives are all practical bits of equipment
that children can pick up and manipulate to help
them get to grips with the very abstract notions
of numbers, the relationships between them and
the ways in which they work in the number
system.
Some examples include Multilink cubes, Dienes
apparatus, counters, place value counters, bead
strings, Cuisenaire rods, sticks divided into 10
equal sections and also those that use numerals
such as place value cards, hundred squares, digit
cards, dice, dominoes and so on.
Bead strings
Dienes
Cuisenaire Rods
Numicon
Place value counters Place value arrow cards
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In 2012 the OECD’s PISA study found that
Shanghai and Singapore topped the table,
with students scoring the equivalent of nearly
3 years of schooling above most other OECD
counties.
The top performing countries all use a
mastery curriculum approach.
The findings highlight that our children’s
current achievements in mathematics are not
the best they can be.
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A mastery approach is about closing the gap
between our highest and lowest achievers, also
about raising achievement for all.
The aim is to teach every concept or skill in a
way that promotes understanding and problem
solving so it is not a collection of memorised
techniques but a coherent body of
interconnected knowledge that can be flexibly
applied to solve problems in unfamiliar
contexts.
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Deepening conceptual understanding through the
use of physical and pictorial representations (Very
effective in countries such as Singapore & Netherlands)
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Developing pupils’ communication, through
explicitly teaching pupils’ to discuss mathematics
through grammatically correct full sentences with
accurate vocabulary (A key priority in Asian countries)
Encouraging pupils to think like mathematicians
through giving them opportunities to seek patterns
and rules, and ask and answer questions
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A cumulative curriculum, with sufficient time
for every child to access age appropriate
concepts and skills.
Involves supporting and challenging pupils
through depth.
Involves purposeful planning that considers
the use of different manipulatives and
representations.
All children working on the same learning
intention at the same time.
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A policy that has been updated to mirror
the new national curriculum expectations. It
was designed in collaboration by different
maths experts.
It is a progressive approach to teaching
mathematics across a school. Each school
can then supplement strategies and
resources at their own discretion.
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Maths Leaders from each school will meet
to discuss the implementation of the policy,
including misconceptions, hesitations and
learning implications.
The next step will be to raise people’s
understanding of the pedagogical choices
teachers have and reasoning behind certain
aspects such as using a blank number line
and teaching grouping and not sharing.
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Fractions:
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Division:
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Cuisenaire rods for fractions
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Have a go!
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“The bottom number is the denominator. That tells
you how many equal parts it is divided up in to.
The top number is the numerator, the fraction part”
Hajra
“6/4 is equivalent to 1 2/4 and 1 ½” Holly
“When the numerator matches the denominator
that means it’s one whole” Arafath
“2 ¼ that’s a mixed fraction” Pitro
“ If the white rod is ¼ the dark green rod is six lots
of a ¼, that means it is 6/4” Ellie
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Place Value Counters for Division
Teach explicitly vocabulary for the unit then build in
opportunities and time for conversations about the maths
using the correct vocabulary.
Calculate division mentally. At first the children needed to use
jottings to support their thinking. This was over several
lessons until all children had a clear understanding. Children
were challenged through applying the skills to real life word
problems.
Design a series of lessons with a clear learning intention for
each.
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Other examples 369 ÷ 3 = 488 ÷ 4 =
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Have a go!
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The children were set this calculation as a problem
to discuss before I started any direct teaching. The
children worked in pairs and used manipulatives.
Responses included…
“Use the counter to show the dividend”
“We can make one group of 6 with the
hundreds but what do we do now?”
“We could exchange the ten like we do in
subtraction”
“Zero is the place holder”
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Daniel and Alhosna who are Year 4 working
just below MARE could solve and explain how
to solve.
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“First I made 832 with my place
value counters. I knew I had to
make groups of 8 because 8 is
the divisor, 832 is the dividend,
the big number. With the
hundreds, I had one group of 8. I
wrote that to start the answer. I
then looked at the tens, I only
had 3 so I had to write zero as a
place holder. I then exchanged 3
tens for 30 units making it 32. I
then put the units into groups of
8. I checked with my times table
knowledge. Each time I had four
groups. The quotient or answer
is 104.”
Tyreke Year 4 (Working at ARE)
Children used the place value
counters to illustrate their
understanding of division with
larger numbers
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To begin my learning intention include using the
manipulatives and reasoning. Can I use
manipulatives to…. Can I use my reasoning skills
to….
As children master a skill it is their decision when
to move away from manipulatives, then on to
pictorial representation and then the abstract.
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446 ÷ 4 =
Children first discussed in pairs if this calculation would have
a remainder and how they knew. Children used their
reasoning skills to estimate and predict even before they got
out the place value counters.
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Children have become more enthusiastic about maths. It
has supported all children with their speaking and
listening as they feel empowered by their deeper
understanding of number. Some of our reluctant speakers
are now so engaged they want to talk about their maths all
day long!
With a high number of children with EAL and new arrivals
to England we have found it has supported the children to
demonstrate their understanding even when they can’t yet
verbalise it.
Using images and manipulatives has supported SEN
children, those who are just below MARE and reluctant
mathematicians. They are now more engaged, talk more
about their learning and show their understanding more
clearly.
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“The counters help me see what is happening” – Kelsie 1B – 2B
“The place value counters help me to subtract when you have to take
a ten from the tens column and then put it in the units. I like
changing the ten in to units” Tyreke 2A – 3C
“I get Dienes out and pick them up. I count in ones, the sticks in
tens and the squares in hundreds. It helps me remember. I can add
and take away big numbers now.”– Aiden P8 – 1A
“When I have a tricky calculation I use the place value counters to
help me get the numbers in my mind. It helps me remember” – Ellie
2A – 3A
“They help me (Cuisenaire rods) add and subtract fractions because
the rods are different colours and you can see. I liked finding things
out by myself and then talking to my friends about it” – Sophie 2A –
3A
“My Maths is
on fire!”
Tyreke
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“First I made 832 with my place
value counters. I knew I had to
make groups of 8 because 8 is
the divisor, 832 is the dividend,
the big number. With the
hundreds, I had one group of 8. I
wrote that to start the answer. I
then looked at the tens, I only
had 3 so I had to write zero as a
place holder. I then exchanged 3
tens for 30 units making it 32. I
then put the units into groups of
8. I checked with my times table
knowledge. Each time I had four
groups. The quotient or answer
is 104.”
Tyreke Yr4 (Working at MARE)

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