Inaugural Conference
Dyscalculia and
Maths Learning Difficulties
+−×÷= ?
Conference Chair: Steve Chinn
19th June 2009
Holiday Inn, Bloomsbury (nr. Euston Station)
London
For all teachers of numeracy and maths, SENCOs and Learning Support teachers,
LA inclusion and numeracy support teams and Educational Psychologists
www.dyscalculia-maths-difficulties.org.uk
Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
Making maths real…what works?
Dyscalculia Inaugural Conference London, June 2009
From Counting to Calculating
The use of concrete materials and structural
equipment to develop number sense and understanding, helps children with low numeracy and dyscalculia, to move from rote counting in ones as a main strategy to internalised cognitive models.
as a main strategy, to internalised cognitive
models
Jane E Emerson M Sc
Jane E. Emerson M.Sc.
Director, Emerson House,
London, UK.
www.emersonhouse.co.uk
This session will demonstrate how pupils can move out of the counting trap into calculating using essential facts to derive answers and to apply universal strategies with understanding.
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©Jane Emerson 2009
Learning Works® [email protected]
Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
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3 Mats should be used to show HTU HTU HTU
Place ‘flats’:
Hundreds here.
Place ‘longs’:
Tens here.
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What are Concrete Manipulatives?
Real materials that can be touched, felt, moved.
Place ‘ones’ Units here.
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• Discrete materials: objects such counters, cubes, plastic animals, glass nuggets, 1p coins etc.
• Continuous materials: Cuisenaire rods, Stern blocks, Base Ten equipment
• Semi‐Abstract Visuo‐Spatial models: Dice,100 squares, number lines, Slavonic Abacus
• Abstract Models: Empty or partially empty number lines
©Jane Emerson 2009
Learning Works® [email protected]
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Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
What do we mean by Structural Equipment?
Base Ten created by Dienes
1000s, 100s flats,10s longs, units.
• Some researchers point out that some learners do not perceive the same as the teaching point
same as the teaching point.
• Learners may think the 1000 cube represents 600 because they can see 6 faces of 100 ones per face.
• In fact, ten flat yellows are represented in one thousand cube.
• Stern Dual Board
For place value
Tens and Units
• Stern: building to 20 staircase model
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The Stern Counting Board • To introduce size comparisons and early maths language • The counting board can be used from the start to introduce an idea of relative size of the numbers in relation to each other.
• Language of one more, one less, bigger than /smaller than
• Young children can use it as a puzzle to discover how each number is one block bigger than the last: number sense of 3<4 etc.
• www.mathsextra.com 11
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Counting Track with beads for location activities
• Counting beads can facilitate the setting up of an internal number line when combined with location activities
• Find 9, find 18 etc.
• Colour change at 10 can aid advanced subitizing as position of 11 for example becomes possible.
©Jane Emerson 2009
Learning Works® [email protected]
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Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
What do we mean by Number Sense? (N.S.)
What is a Slavonic Abacus?
• The uses of the Slavonic Abacus has been extensively written about by Tandi Clausen‐May
• This was described in 1998 by Eva Grauberg
• It provides a counting model marking the 5s
model, marking the 5s boundaries
• This enables children to count rows of tens but also to subitize, not only less than 5 but also more than 5 aided by the colour change.
• Having a basic feel for numbers: quantities the fiveness of 5
• And for number comparisons: 6>5
• Those with poor N.S. can find estimating quantities very difficult: may be wildly inaccurate guesses
• Those with poor N.S. find it difficult to estimate very small quantities <5 without counting, that is, to subitize.
• If 2+2=4 then 2+3 must = ‘just one more’. Those with poor N.S. treat each problem as a new one. They do not derive new facts from known facts easily.
• Numbers are foreign territory
• Language of maths may be like a foreign language
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How do we get students to learn by using concrete materials?
Enquiry Maths? Discovery Maths?
• Paul Cobb in 1999 spoke of Enquiry maths, where an approach involving understanding and finding solutions that had meaning for a student personally was encouraged.
• This is not enough for dyscalculics who need to be This is not enough for dyscalculics who need to be
guided in a multi sensory way to experience the concrete material and structural equipment in order to make sense of what it conveys.
DYSCALCULICS NEED A PROCESS OF ACTIVE GUIDED DISCOVERY
• Not just by looking at them.
• Not by taking the teachers interpretation as the right conceptual action or thought.
ACTIVE GUIDED DISCOVERY
• Active: by touching materials, moving them, talking about them, as a visuo‐motor experience leading to conceptual actions, with a two way conversation, guided by the teacher who responds to the child’s comments about the materials as the child makes a discovery which takes him forward to his next step at his level of proximal development.
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©Jane Emerson 2009
Learning Works® [email protected]
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Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
Where is the maths
brain anyway?
EXPOSURE TO MATERIALS AND IMAGES FOR VISUALISATION WHEN NOT PRESENT
Is this a bad day at the hairdressers?
Conceptual action
Visuo motor
experience
• LOOKING
• THINKING
• TALKING
•SEEING
•TOUCHING
•MOVING
Or is this the state
of the Dyscalculia research?
Constructing
•CONCEPTS
•IMAGES
in the Maths
Brain
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What are these sensory‐motor/conceptual actions? Paul Cobb 1990:
• Sensory motor actions: using the senses to experience the materials by seeing them, touching them, moving them in a similar way to multi‐sensory teaching for dyslexics learning synthetic phonics.
• Conceptual actions are when you see and then think in words when you look at materials: for example the thousand cube, as 10 hundreds piled up, or 100 tens, depending on what conceptual act takes place.
• ‘A
A process of constructing increasingly sophisticated conceptual process of constructing increasingly sophisticated conceptual
actions’. ( Paul Cobb ’90) :
At Emerson House, this is underpinned by a belief that the child must internalise the process for themselves by talking. The teacher accepts the child’s words and helps the child to express their understanding in increasingly sophisticated ways, through a dialogue.
• To make a connection between school and everyday situations, word problems are used at every stage, to bridge the gap between artificial settings over to more everyday thinking by using realistic situations and transparent words whenever possible.
Making counting real by working on bead tracks or with bead strings.
• Children perform sensory motor actions by experiencing them and their sensori‐motor actions in moving them as they count. • Building up their sense of quantities and locations leads to conceptual actions as they regard the track or b d
beads.
• Dialogue serves to introduce the language: 9 is just before 10, 19 is just before 20,11 is just after 10 etc.
• Name a number, find the bead of that number etc.
• Find a bead and describe its location.
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©Jane Emerson 2009
Learning Works® [email protected]
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Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
Rods for components
Cuisenaire Rods
A Numicon Component Box
• Rods can be used as continuous materials when children are ready to move from d
f
discrete dice patterns • They can be used to demonstrate both whole numbers and parts of wholes.
‘The story of 6’
6 can be seen to be made of:
• 1 and 5
• 2 and 4
• 3 and 3
• 4 and 2
• 5 and 1
• Children can derive the facts from the key (known) fact of 3+3
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The Doubles Pyramid: seeing moving touching, looking, thinking, talking, and reasoning.
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From Counting in Ones to Calculation.
• The component model teaches the key facts of the dot patterns first • Doubles for the even numbers: 8=4+4
• Near doubles for the odd N
d bl f th dd
numbers: 7=3+4
• Reasoning: 7 is made from 3 and 4 so 3+4=7 so 7‐3=4 and 7‐
4=3
• Post it notes can take away one of the numbers to leave the other visible.
• Reasoning from 10+10=20 or 5+5=10
• So each step must be 2 more or 2 les
more or 2 les
• Child can place ones on each step to notice spaces either side
• If 5+5=10 then 6+6 must be 2 more:12
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EMERSON HOUSE MATHEMATICS
CARDS FOR MATHS GAMES:
PATTERNS
1 – 10
[CIRCLES]
©Jane Emerson 2009
Learning Works® [email protected]
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Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
Semi Abstract Representations
for Doubles and near Doubles
Semi Abstract Representations for
Doubles and near Doubles Plus and Minus
Doubles/Evens:
4=2+2 so 4‐2=2
6=3+3 so 6‐3=6
8=4+4 so 8‐4=4
10 5 5 10 5 5
10=5+5 so 10‐5=5
• To reason for near doubles:
• If 2+2=4
• Then 2+3=‘just one more’ so must be 5.
• This is highly diagnostic when a young child with possible dyscalculia is first seen.
• Can they reason at all about number?
Near Doubles/Odd
9=5+4 so 9‐5=4 9‐4=5
7=4+3 so 7‐4=3 7‐3=4
5=3+2 so 5‐3=2 5‐2=3
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EMERSON HOUSE MATHEMATICS
CARDS FOR MATHS GAMES:
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EMERSON HOUSE MATHEMATICS
CARDS FOR MATHS GAMES:
PATTERNS
PATTERNS
Semi‐abstract images
a)
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1
2
0
1
3
1
1 – 10
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[CIRCLES]
[CIRCLES]
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Essential KEY facts of the counting numbers to 10. 4
1
2
5
2
3
6
2
3
7
3
4
8
3
4
9
4
5
10
4
5
5
7 = 4+3 so 7‐3=4 and 7‐4=3 leads to finger‐free calculations.
b)
1
1
2
0
1
3
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2
4
?
2
5
?
3
One is made of..Two is made of..Three is made of two and….
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?
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7
?
4
9
8
?
4
?
5
10
?
5
?
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Missing addend work: 7=4+? 4+?=7 ?+3=7 varying language.
©Jane Emerson 2009
Learning Works® [email protected]
Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
Stage ONE: Conceptual Stage
7
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7
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5
7
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Grouping Model of Multiplication and Division to demonstrate the connection.
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e.g) Here are 9 counters. How many 3s do you think you can
build out of 9?
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KEY FACT
7
1
• The triad presentation can be useful for recording concrete representations in an
abstract way.
• These triads explore the rest of the components for 7.
7
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2
?
3 threes
e.g) Here’s 20 counters. How many 5s do you think you can build out of 20?
‘I can build 4 fives so there are 4 fives in 20’.
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The Tables universal strategy.
Key facts 1x the table, 10x the table and 5x the table being studied.
Counting in Fives with Cuisenaire Rods
on Numicon Track
• Track to 100 but several tracks can be used to count several hundreds.
• Many children who can rote count have a Many children who can rote count have a
hazy idea beyond 100.
• Step-count in fours
• Step-count from different starting points
4,8,12
or 20,24,28
• Ask one-step questions: What is one more 4
than 20?
• Ask two-steps questions: What are two more
4s than 24? (28+4 revises bridging)
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©Jane Emerson 2009
Learning Works® [email protected]
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Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
Concrete materials to teach Bridging.
Concrete materials to teach Bridging.
• Bridging forward 9+3= (9+1)+2= 12
• Bridging back for numbers close on a number line
12 ‐3 =(12‐2)‐1=9
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Bus word problems
Semi‐Abstract representation of Bridging to the next tens number
•
9+5= 10+4= 14 1
4
Ten on the top and ten on • Ten people on the top the bottom
deck, and ten on the bottom deck.
• 58+6= (58+2)+4= 64
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• 5 people get off.
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• How many left on the bus?
• 5 more get off.
• How many left?
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• Then 10 others get on? How many on the bus then?
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©Jane Emerson 2009
Learning Works® [email protected]
Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
More Word Problems. People on the Bus.
Counting in Tens on Track with Numicon track and with Cuisenaire orange 10 rods inserted.
• The bus is full upstairs with 10 people.
• Downstairs 4 more get on.
• How many more can get on before the bottom deck is full?
• No standing room!
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Partitioning with concrete materials on a place value mat Building over 100
Numicon boxes with Base Ten Materials
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• Once children are calculating comfortably up to 100
they can study the numbers over 100
• Base 10 can be used to build 100s in tens to emphasise the repeating 10s
• Revise adding 1 or 10 for 100,101,110, 120,121, etc.
• A 100 metre stick can be exchanged for 100s.
57 is partitioned into:
• 50 and 7
• 5 tens and 7 units
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©Jane Emerson 2009
Learning Works® [email protected]
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Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
Concrete Demonstration of Formal Addition
without exchange
Demonstration of addition with exchange with Stern Structured Apparatus for 49+3
‘You must start with the units,’ mantra to prepare for exchange.
• 4 units and 3 units = 7
• 3 tens and 2 tens = 5 tens=50
• 50+7=57
• Vary language of addition to add, plus etc. • I built 4 tens and 9 units (sensori‐motor)
• I can see 49
(cognitive action in words)
(cognitive action in words)
• Add 3 more units
• 49+1=50 so swap 10 units for 1 ten block
• That leaves 2 units • 49+3 =(49+1)+2=52
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Spinner Games with Dice patterns
Area Model for beyond the tables:
13x10: Can be built with rods:
orange to total = 130
10x10=100
10 ten rods or
Base Ten 100 square
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Spin and Win, e.g. by formulating an addition, subtraction, times or division to gain extra points: i t
9=5+4 or 2x9=18 etc. 3x10=30
(3 ten
rods)
Cards could indicate which operation to apply depending on level of child.
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©Jane Emerson 2009
Learning Works® [email protected]
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Making Maths Real
Jane Emerson
Dyscalculia & Maths Learning Difficulties
Inaugural Conference – 19th June 2009
8
The Facts of 10
4
1
8
2
10
7
3
5
6
9
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©Jane Emerson 2009
Learning Works® [email protected]
Every school is different and
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The Trouble with Maths
A pragmatic course for helping learners who
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411 & 12 November 2009
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