mathematics and dyscalculia

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

It is now February 11th.

Estimate how much you need to pay

all together.

Which contains more milk?

Conduct disorder

Specific language impairment

Pragmatic language impairment

PDDNOS

We all have a bit of “it”

How do children learn

mathematics?

Learning about maths in the real

world• Shape concepts

• Number correspondence

• Ordinality

• Money

Mathematics

• Abstract subject

• Need for symbolic

understanding

• Estimation

Are there more difficulties

today?

• Less opportunity for

early multisensory

experiences

• More emphasis on

mental mathematics

• Alternatives to doing

tasks- parents may find it

harder to support

Where do you use maths in

your life?

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

What is dyscalculia?

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?

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

Difficulties despite:

• Adequate

• Appropriate

Teaching and exposure

Compared with age and cognitively matched

peers.

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)

Commonest maths problems

• memory for arithmetical facts

• difficulty include word problem solving

• representation of place value

• the ability to solve multi-step

What are you doing when

you are looking at this?

Need for :

• Prior knowledge

• Estimation

• Understanding

• Visual skills

• Time to do it

Some children have more than

one problem as well ..• Working memory difficulties

• Spatial representation

• Language

• Motor

• Executive functioning- planning and

checking

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

What are core maths

concepts?

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)

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)

Which lessons and how will

mathematics be involved?

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

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?

How do you do this?

What skills do you need?

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

These all overlap?

Dyslexia

Receptive

languageAuditory

processing

Visual

processing

SLI difficultiesDCD

Motor difficultiesEF

reading spelling maths writing

Dyscalculia commonly overlaps

• DCD

• Dyslexia

• Specific language difficulties

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

What are the underlying

cognitive skills needed for

mathematics?

Vision and perception

Language and

comprehension

Sequencing

Fine motor skills

Gross motor skills

Working memory

Focus and attention

Sufficient time

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

Lets go through each of these

components for learning....

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

Working Memory is:

A limited-capacity cognitive system that

allows us to hold and manipulate

information “on-line” for a few seconds.

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

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

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)

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

Planning and executive

functioning difficulties

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

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)

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

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

Visual perceptual difficulties ...

In mathematics• Estimation

• Manipulation

• Orientation

• Sequencing

• Problem

solving

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

Since 1 is to the right of -3, 1 > -3 or

Since -3 is to the left of 1, -3 < 1Amanda Kirby copyright

Time concepts/estimation

When do you need to know about

time passing?

Linguistics

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

compare subtraction

John has nine marbles. He has six more marbles

than Nina. How many marbles does Nina have?

compare subtraction

Fine motor difficulties

How will this impact on

mathematics?

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

How can you help?

“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

GeneralFind

specific areas

Practice sufficiently

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

“the child's need is just as much

to unlearn his incorrect rule as

it is to learn the correct rule.”

Individual

Environment

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

What works….

• Learning concrete operations to give

confidence

• Metacognition

– Knowing what one does not know

– Reflection

– Planning

First.... create a resource kit

• Dominos

• Playing Cards

• Egg cartons

• Coins and buttons

• Different shapes

• Number cards

• Language of maths cards

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.

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.

Different children need

different approaches

e.g. Learning numbers adding to

12....

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.)

Spatial

Recognize

sets of

objects in

patterned

1 and 2 more

1 and 2 less

Anchors

5 and 10

1 2 3 4 5 6

100 10 1

NUMICON www.numicon.comAmanda Kirby copyright

Counting activities

• These include general practice in counting

objects.

• They also include practice in grouping

objects, especially into tens.

HOW MANY?

Playing with dice

TURNING WORDS INTO

PICTURES

Specific sessions

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.

Visual clues

The understanding of maths

Joan has 12 sweets. She has twice as many

sweets as Leela. How many has Joan

The language of mathematics

Highlight key words in a maths question

Provide a glossary or checklist

Check the language of

mathematics

+, add, and, plus, in addition, more

-, take away, minus, less, subtract

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

Brooks

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.

Taller,

Shorter

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?

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….

45 minutes

5 minutes

Estimation

Visual skill/EFskills

“I‟m thinking of a number between ….”

“I am thinking of a number between 25 and 72”

If difficult in visualising or hearing it

• Need to feel and see it

If difficult in visualising or hearing it

• Need to feel and see it

Visual prompts

Could you build an abacus?

http://countdown.luc.edu/Amanda Kirby copyright

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

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

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

http://www.readwritethink.org/files/resources/i

nteractives/essaymap/

Using colour coding

• The elements of a question

• Signs being used

The tools of maths

Scissors

Recording maths

• Use set symbols for science

• Larger maths paper with bigger squares

http://www.mathsphere.co.uk/resources/MathSphereFreeGraph

Paper.htm

Larger graph paper

Offer a choice of pens and pencils

Functional mathematics

• Handling money

• Filling in a form

How do you make maths

What could you do with

cookery to teach maths

concepts?

Using ICT and other

resources

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

“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

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.

Playing Bingo

Bingo card maker :http://www.teach-nology.com/web_tools/materials/bingo/

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/

• http://www.woodlands-

junior.kent.sch.uk/maths/index.html

Ten ticks

http://www.10ticks.co.uk/s_help.aspx

http://www.math.com/students/tools.html

Steve Chinn

What do you do when you can‟t learn times tables

Mental Maths Olympics Year 4.

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).

Further reference materials

• http://www.dcsf.gov.uk/research/data/uplo

adfiles/RR554.pdf

Bringing maths into other

curricular areas• Fishing game- for numbers

• Adding and subtracting- crawling to the

end of the room

• Cookery

• Gardening

• CDT

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

Vision and perception

Language and

comprehension

Sequencing

Fine motor skills

Gross motor skills

Working memory

Focus and attention

Sufficient time

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

mathematics and dyscalculia

mathematics deficiency

mathematics disorderas

right difficulty

arithmetical facts difficulty

counting problems

simple calculations

different strategies

number facts

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