mathematics and dyscalculia
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A lecture by Professor Kirby on desconstructing difficulties in mathematics- what is dyscalculia?TRANSCRIPT
Mathematics and
Dyscalculia
Professor Amanda Kirby
Amanda Kirby copyright
Dyscovery Centre 2011
You had a tax bill to pay on January
29th for £3500 which you failed to
pay. For every day after that you
have to pay 5% of this amount on
top.
It is now February 11th.
Estimate how much you need to pay
all together.
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Which contains more milk?
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DCD
ODD
Conduct disorder
Specific language impairment
Pragmatic language impairment
PDDNOS
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We all have a bit of “it”
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How do children learn
mathematics?
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Learning about maths in the real
world• Shape concepts
• Number correspondence
• Ordinality
• Money
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Mathematics
• Abstract subject
• Need for symbolic
understanding
• Estimation
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Are there more difficulties
today?
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• Less opportunity for
early multisensory
experiences
• More emphasis on
mental mathematics
• Alternatives to doing
tasks- parents may find it
harder to support
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Where do you use maths in
your life?
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Difficulties extend to other
areas:• Poor management of money
• Errors doing simple calculations
• Difficulty understanding a series of
commands- 2 to the left/1 to the right
• Difficulty keeping score in a game
• Harder to play strategic games like chess
• Estimation – cooking, measuring
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What is dyscalculia?
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Dyscalculia
• Specific difficulties in numeracy skills
• 6%
• …but overlaps with Dyslexia and DCD
• Is this children who are working at Level 1
at age 7, or Level 3 at age 11, have some
degree of mathematical difficulty?
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Characteristics
• Same strategies as younger children but
are error prone
• Slow at calculating and counting
• Difficulty retrieving number facts
• Do not know their tables
• Poor at monitoring their counting
• Problems switching between different
strategies when completing mathematical
problems Amanda Kirby copyright
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Difficulties despite:
• Adequate
• Appropriate
Teaching and exposure
Compared with age and cognitively matched
peers.
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Different terms used :
315.1 Mathematics DisorderAs measured by a standardized test that is given individually, the patient's mathematical ability is substantially less than you would expect considering age, intelligence and education. This deficiency materially impedes academic achievement or daily living.If there is also a sensory defect, the mathematics deficiency is worse than you would expect with it.
DSM1V (APA)
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Commonest maths problems
• memory for arithmetical facts
• difficulty include word problem solving
• representation of place value
• the ability to solve multi-step
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What are you doing when
you are looking at this?
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Need for :
• Prior knowledge
• Estimation
• Understanding
• Visual skills
• Time to do it
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Some children have more than
one problem as well ..• Working memory difficulties
• Spatial representation
• Language
• Motor
• Executive functioning- planning and
checking
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What is a persistent problem?
• The children with mathematical difficulties
use almost exclusively counting- based
strategies, while those without such
difficulties children were more likely to use
retrieval or derived fact strategies
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What are core maths
concepts?
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Core primary mathematical
competenciesNumerosity
Ability to accurately determine the quantity of sets up to 3 or 4 items, or events, without counting
Ordinality
Implicit understanding of “more than” & “less than” for comparison of sets of 3 to 4 items
Counting
Nonverbal system for enumerating small sets of items & implicit knowledge of counting principles (1 to 1 correspondence)
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Simple arithmeticSensitivity to increases (addition) and decreases (subtraction) in the quantity of
small sets of items
EstimationInexact estimation of relative quantity, magnitude, or size
Geometry
manipulating shapes, visualising
Geary (2007): Child Dev. 78(4)
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Which lessons and how will
mathematics be involved?
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Difficulties can impact on
different lessons
Science- weights, measures, recording, using tools, estimation, calculations
Sport- sequences of instructions, direction, timing, understanding the plan/teams
Geography- map reading, recording, orientation, graphs
History- time lines
CDT- measuring, tool usage, planning, language
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Doing a mathematics problem
John walked down the street to the shops to
buy a DVD . He had £20. The decided to
buy one for £11.99 and then buy an apple
for 50p. How much change will he have.
He needs £7.20 for the train home, will he
have enough money?
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How do you do this?
What skills do you need?
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Mathematical anxiety?
• Many people develop anxiety about
mathematics, which can be a distressing
problem in itself, and also inhibits further
progress in the subject (Fennema, 1989;
Hembree, 1990; Ashcraft, Kirk and Hopko,
1998).
• This is rare in young children (Wigfield and
Meece, 1988) and becomes much more
common in adolescence. Amanda Kirby copyright
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These all overlap?
Dyslexia
Receptive
languageAuditory
processing
Visual
processing
SLI difficultiesDCD
Motor difficultiesEF
ADHD
reading spelling maths writing
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Dyscalculia commonly overlaps
with:
• DCD
• Dyslexia
• Specific language difficulties
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Dyscalculia and…..
• Dyslexia-Miles (1993) found that 96% of a
sample of 80 nine-to-twelve-year- old
dyslexics had were unable to recite the 6x,
7x and 8x tables without stumbling.
• Dyspraxia/DCD-spatial and copying (Piek)
• Speech and language disorders- verbal
concepts- counting-related concepts, such
as the fact that the last item in a count
sequence indicates the number of items
in the set. Amanda Kirby copyright
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What are the underlying
cognitive skills needed for
mathematics?
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Vision and perception
Language and
comprehension
Sequencing
Fine motor skills
Gross motor skills
Working memory
Focus and attention
Sufficient time
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What are the underlying cognitive
skills needed for mathematics?
• Vision- acuity, perception, sequencing
• Hearing-hearing, filtering, perception
• Language skills -receptive, inference
• Motor skills –for rulers/protractors
• Planning skills - EF; working memory,
focus and attention
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Lets go through each of these
components for learning....
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Focus and inattention
as rated by classroom teachers, is a
significant of Gr. 1 & Gr. 3 children‟s in
three key areas
• Fact fluency
• Computation
• Story problems
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Working Memory is:
A limited-capacity cognitive system that
allows us to hold and manipulate
information “on-line” for a few seconds.
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WM required for..
Keeping track of a conversation
• Who said what..to whom
• who asked what…
Keeping track of a game
• Whose turn is next…
• What cards have
• already gone…
• What stage is the game
• at….Amanda Kirby copyright
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Impact of poor working
memoryDifficulty with:
Mental arithmetic
• will tend to use „finger counting‟,
• need visual representation
Retrieval of math facts in word problems will tend to use
• immature strategies („count all‟, „count on from largest”)
Arithmetic procedures, such as „carrying‟ &„borrowing‟
Ignoring irrelevant information in word problems
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Working memory processes are important
predictors of performance on national
curriculum achievement scores:
Verbal WM
• Literacy: vocabulary, reading
comprehension,written expression
Quantitative literacy: math
Visual-Spatial WM
• Science
• Quantitative literacy: math(Gathercole & Pickering, 2000; Jarvis & Gathercole, 2003)
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Signs of visual/VP difficulties
• Copying from the board
• Needing to reread text
• Avoiding reading out in class
• Moving in a moving environment
• Poor writing – spacing , not on the lines
• Words bouncing
• Spelling errors- longer words
• Not getting their “abc ”
• Geometry
• Not being able to select key words and sense easily and quickly
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Planning and executive
functioning difficulties
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EF and Specific Learning Difficulties
• ADHD – all ADHD children have EF impairment to varying degrees (Barkley 2001)
• ASD – Pennington and Ozonoff (1996) found children performed 1 SD below control group on EF tasks
• DCD – children impaired on tests of working memory (Alloway & Temple, 2007)
• Dyslexia – studies have found WM deficits that compound their phonological problems (Wolf 2010)
• Dyscalculia – Askenazi & Henik (2010) found evidence of specific EFDs in university students with „pure‟ dyscalculia
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Components
–Activation- organising , prioritising
tasks, time estimation, initiation..
procrastinate
–Focus-sustaining and shifting..
Reading over and over
–Effort-regulating
alertness..completing tasks, sleep
pattern (can’t shut off)
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Components
– Emotion-managing frustrations and modulating emotions..keeping things in perspective
– Memory- using working memory and accessing recall.. what has just been said, remembering a sequence
– Action- monitoring and regulating self action..impulsive, not considering the context, can’t adjust pace
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Seeing the process as well as
the product• Do a problem on a
Tablet PC using
software like
Camtasia, and then
upload the process
to a website- see the
process in action
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Visual perceptual difficulties ...
In mathematics• Estimation
• Manipulation
• Orientation
• Sequencing
• Problem
solving
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e.g. Using a number line
• To compare integers, plot the points on
the number line, the number farther to the
right is the larger number
• Compare 1 and -3
___________
*_______*_______________-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
8
Since 1 is to the right of -3, 1 > -3 or
Since -3 is to the left of 1, -3 < 1Amanda Kirby copyright
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Time concepts/estimation
When do you need to know about
time passing?
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Linguistics
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The linguistic difficulties (Tannock)
Word problem Problem type Arithmetic
operation
John had three marbles and then Nina gave him
six more. How many does he have?
change addition
John had some marbles. Then he gave six
marbles to Nina. Now John has three marbles.
How many marbles
did John have in the beginning?
change addition
John has three marbles. Nina has six marbles.
How many marbles do they have altogether?
combine Addition
John and Nina have nine marbles altogether.
John has three marbles. How many marbles
does Nina have?
combine subtraction
John has nine marbles. Nina has six marbles.
How many marbles does John have more than
Nina?
compare subtraction
John has nine marbles. He has six more marbles
than Nina. How many marbles does Nina have?
compare subtraction
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Fine motor difficulties
How will this impact on
mathematics?
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Too many words
Don‟t understand the words
Too little time
Can‟t show workings
Work very untidy
Not sure of the process- where to start
Not sure what calculation to do
Know the calculation but make an error
Misread it
Need to reread it several times before starting
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How can you help?
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“interventions that focus on the
particular components with which an
individual child has difficulty are likely
to be more effective than those which
assume that all children's arithmetical
difficulties are similar”
(Weaver, 1954; Keogh, Major, Omari, Gandara and
Reid, 1980).Amanda Kirby copyright
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GeneralFind
specific areas
Practice sufficiently
Adapt
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Help needs to be:
• in different formats- multisensory
• regular ( at least 3 times per week
minimum)
• reinforced
• some people process very slowly.. So rote
learning may be counterproductive
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“the child's need is just as much
to unlearn his incorrect rule as
it is to learn the correct rule.”
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TIE
Individual
Environment
TASK
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Keep cognitive
load at an optimal level
When cognitive load is reduced &/or
supported to a level that is within the
individual‟s capacity, that person will be
better able to benefit from instruction
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What works….
• Learning concrete operations to give
confidence
• Metacognition
– Knowing what one does not know
– Reflection
– Planning
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First.... create a resource kit
• Dominos
• Playing Cards
• Egg cartons
• Coins and buttons
• Different shapes
• Number cards
• Language of maths cards
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Rehearsal cards
• mathematical facts (5 x 4 =20,
6X2=12;4x3=12)
• Each individual child is given a small set of
cards to practice each day under adult
supervision.
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Different children need different
approaches
• e.g clapping and dancing as the
children chant or sing multiplication
tables, with the aim of assisting those
children who, for example, have
better motor than verbal memories.
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Different children need
different approaches
e.g. Learning numbers adding to
12....
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Multiple methods to solve
problems• Encourage multiple methods to carry out
problems (e.g. to solve arithmetic
problems with a number line, a calculator,
concrete objects such as blocks, in written
form,Numicon etc.)
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Part
Part
Whole
7
7
Spatial
Recognize
sets of
objects in
patterned
1 and 2 more
1 and 2 less
Anchors
5 and 10
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1 2 3 4 5 6
1 2 3 4 5 6
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100 10 1
100 10 1
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Counting activities
• These include general practice in counting
objects.
• They also include practice in grouping
objects, especially into tens.
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HOW MANY?
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Playing with dice
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TURNING WORDS INTO
PICTURES
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Specific sessions
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Create a session
• 2-3 minutes practicing counting skills.
• 2 minutes revising individual known facts.
• 10-12 minutes practicing derived fact
strategies building on known facts.
• 2 minutes playing with big numbers or
working on a problem.
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Visual clues
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The understanding of maths
Joan has 12 sweets. She has twice as many
sweets as Leela. How many has Joan
got?
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The language of mathematics
Highlight key words in a maths question
Provide a glossary or checklist
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Check the language of
mathematics
+, add, and, plus, in addition, more
-, take away, minus, less, subtract
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More / Less /Same in their language is a
benchmark for the teacher that changes over time.
Key Language Development:
Terms being used
Player 1st 2nd 3rd 4th Final
Ms.
Brooks
Mr.
Adams
Game Facts:
1. Ms. Brooks won by 15 points.
2. In one quarter they scored the same number of points.
3. In another quarter, Mr. Adams scored 10 more points.
4. In yet another quarter, Mr. Adams was outscored by 5 points.
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Taller,
Shorter
Up
down
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Getting harder
Harry Potter has asked his friend Hermione for a potion to turn them and their friend Ron into birds. (The flying car is in for repairs, so they need to make the trip to Diagon Alley as birds.)
Diagon Alley is 9 miles away and a dose of Hermione's potion lasts 50 minutes. They only have enough potion for one dose each. If they can go 24 miles an hour as birds, and they start at 4:30 p.m., can they get to Diagon Alley and back to Hogwarts again before the potion runs out at 5:20 p.m.? If so, how much time will they be able to spend in the Alley?
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Breaking down the problem
What exactly is the question you need to answer?
• What do you need to know- what can you ignore
• Can you draw a picture or rewrite the question in your own words
• Is there a formula you need to answer this- have you the skills to do this
• Have you checked your results
• Have you been given enough time
• Have you got the answer write but written it down wrong
• Are you anxious….
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45 minutes
5 minutes
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Estimation
Visual skill/EFskills
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“I‟m thinking of a number between ….”
“I am thinking of a number between 25 and 72”
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If difficult in visualising or hearing it
• Need to feel and see it
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If difficult in visualising or hearing it
• Need to feel and see it
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Visual prompts
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Could you build an abacus?
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http://countdown.luc.edu/Amanda Kirby copyright
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Use poems etc as reminder tools
Find your number.Look right next door.4 or less just ignore.
5 or more, add 1 more.Amanda Kirby copyright
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Give rules e.g. number squares
• 1 more is 1 square to its right
• 10 more is 1 square down
• 1 less is 1 square to its left
• 10 less is 1 square up
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If planning is an issue
• Check the student understands the expected outcome
• Providing worked model answers with the processes as examples ( e.g. Camtasia)
• Provide steps to complete- a scaffold
• Talk through what is required by the student- what skills are needed, how will points be apportioned in a test
• Provide a glossary of terms to refer to
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http://www.readwritethink.org/files/resources/i
nteractives/essaymap/
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Using colour coding
• The elements of a question
• Signs being used
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The tools of maths
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Scissors
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Recording maths
• Use set symbols for science
• Larger maths paper with bigger squares
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http://www.mathsphere.co.uk/resources/MathSphereFreeGraph
Paper.htm
Larger graph paper
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Offer a choice of pens and pencils
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Functional mathematics
• Handling money
• Filling in a form
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How do you make maths
real?
What could you do with
cookery to teach maths
concepts?
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Using ICT and other
resources
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Maths mania
Topics covered include:
• Numeracy with number questions on the four rules, decimals, fractions etc
• Angles, what they are and how they differ
• Telling the time, with time intervals of days, months and years
• Measures includes length, volume and mass
• Shape and Space
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“Maths Circus”
Topics covered include:
• Spatial awareness
• 3-dimensional geometry
• Addition and multiplication
• Investigating direction and angle, compass bearings, time and vectors
• Positive and negative numbers
• Planning a sequence of moves in advance, and executing them later on
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Math talk
MathTalk & Scientific Notebook is a bolt on bundle for Dragon NaturallySpeaking
• for creating, editing and typesetting mathematics and scientific equations by voice.
• The program has learning modules for algebra, trigonometry, calculus, statistics and video demos demonstrating how to use features.
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Playing Bingo
Bingo card maker :http://www.teach-nology.com/web_tools/materials/bingo/
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Resources
• http://www.teachingideas.co.uk/maths/cont
ents02problems.htm
ICT in Maths;Alison Clark-Jeavons :ISBN: 1
85539 191 0,Exciting
Precision maths
(http://www.johnandgwyn.co.uk/home.html)
Useful links to other sites:
http://www.mad4maths.com/parents/links/
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• http://www.woodlands-
junior.kent.sch.uk/maths/index.html
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Ten ticks
http://www.10ticks.co.uk/s_help.aspx
http://www.math.com/students/tools.html
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CDROM
Steve Chinn
What do you do when you can‟t learn times tables
Mental Maths Olympics Year 4.
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Reading materials
• El-Naggar (1996) and Poustie (2001) on mathematical
difficulties in general;
• Kay and Yeo (2003) on mathematical difficulties
associated with dyslexia National Numeracy Strategy
such as Guidance to Support Pupils with Special
Educational Needs in the Daily Mathematics Lesson
(DfES 05451/2001) and Including All Children in the
Literacy Hour and Daily Mathematics Lesson (DfES
0465/2002).
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Further reference materials
• http://www.dcsf.gov.uk/research/data/uplo
adfiles/RR554.pdf
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Bringing maths into other
curricular areas• Fishing game- for numbers
• Adding and subtracting- crawling to the
end of the room
• Cookery
• Gardening
• CDT
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Conclusions
• There are different reasons for someone
finding mathematics difficult
• It is necessary to “ tease” out the reasons
to find the right approach to help
• Sometimes the child needs confidence in
order to become competent
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Vision and perception
Language and
comprehension
Sequencing
Fine motor skills
Gross motor skills
Working memory
Focus and attention
Sufficient time
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Big Math Programme
components • (1) use of numbers, involving counting procedures and principles, the use of numbers as labels
(e.g. house numbers), and the different ways in which numbers may be represented
• (2) shape, involving not only recognition and naming of shapes, but exploration of their
characteristics (e.g number of sides and angles), symmetry, and ways of partitioning them into
other shapes
• (3) measurement, involving comparison, seriation, and iteration (repeated use of a measurement
unit) with regard to a wide variety of quantities: length, weight, capacity, area, time, temperature
and money
• (4) working with numbers, including grouping of objects, adding and subtracting, and the
relationships between sets and their subsets
• (5) patterns, involving the systematic repetition of elements in the context of number, shape,
colour, and sound (e.g. rhythm). Children copy patterns; extend them, e.g. adding 2 repeatedly to
make 1, 3, 5, 7...); describe them; and create their own.
• (6) spatial relationships, involving describing and mapping positions and routes. Amanda Kirby copyright
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