Dyslexia and Dyscalculia Next Steps for All Clare Trott Mathematics Education Centre Loughborough...

Dyslexia and Dyscalculia Next Steps for All

Clare Trott

Mathematics Education CentreLoughborough University

•Maths support for dyslexic and dyscalculic students

•Approx. 20 students per week, one-to-one basis

•Referred from ELSU or DANS

•All registered dyslexic or dyscalculic

•All have some element of maths in their course, which they struggle with as a result of their SpLD

Dyslexia

support

Mathematics

support

Characteristics of Dyslexia

A marked inefficiency in working or short-term memory-Problems retaining the meaning of text

-Failure to marshall learned facts effectively in exams

-Disjointed written work or omission of words

Inadequate phonological processing skills-Affects the acquisition of phonic skill in reading

and spelling,

-Affects comprehension

Difficulties with motor skills or coordination-Particularly difficulty with automatising skillsE.G. listening and taking noted

simultaneously

Visual processing problems-Affecting reading, especially large strings of

text

From: Dyslexia in Higher Education: policy, provision and practice. The National Working Party on Dyslexia in Higher Education (1999)

Mathematical Difficulties Dyslexics Experience

• Poor arithmetical skills

• Poor short term memory

• Poor long term memory for retaining number facts and procedures, leading to poor numeracy skills

• Reading the words that specify the problem

• Slow reading, mis-reading or not understanding what has been read

• Substituting names that begin with the same letter e.g. integer/integral, diameter/diagram

• Remembering and retrieving specialised mathematical vocabulary

• Problems associating the word, symbol and function

• Slow information processing means few notes. example 1, example 9, and nothing in between

• Poor working speed

• Problems sequencing complex instructions, and past/future events

• Presentation of work on the page

• Inadequate documentation of method

• Visual perception and reversals E.G. 3/E or 2/5 or +/x

• Copying errors from line to line

• Errors when transferring between mediums

• Frequently loss of place when scrolling on screen

• Reluctant to try new work, More inclined to omit questions

• Difficulty learning theorems and formulae

• Mathematical procedures, sequences of operations

• Holding various aspects of a problem in mind and combining them to achieve a final solution

• In multi-step problems, frequently lose their way, omit sections

• Overload occurs more frequently, forced to stop

•These difficulties occur in non-dyslexics as well, but it is a matter of how many of these problems and how severe and persistent they are.

•Chinn and Ashcroft noticed a change in levels of performance when word problems are introduced.

Mathematics for Dyslexics Chinn and Ashcroft (1997)

They define two types of mathematical learners

1) Inchworms - work step by step, and relying on well rehearsed procedures

2) Grasshoppers - an intuitive feel for a problem, adopting an overall view

For dyslexics:

• Sequential, formula orientated inchworms with poor STM are at high risk of failure in maths

• Equally, inaccurate intuitive grasshoppers are at risk

DyscalculiaStatistics

According to current estimates (Butterworth (1999)) • about 10% of the population are dyslexic (4% severe, 6% mild/moderate)

• of these 40% have some degree of difficulty with maths

• additionally 4 to 6% is dyscalculic only.

• There is currently no accepted definition of dyscalculia

• A number of different definitions exist

• Numerically based• Cognitive based• Neuroscience based

• The DSM-IV document, used by educational psychologists, defines Mathematics disorder in term of test scores:

"as measured by a standardised test that is given individually, the person’s mathematical ability is substantially less than would be expected from the person’s age, intelligence and education. This deficiency materially impedes academic achievement or daily living"

Two Important Features 1. Mathematical level compared to expectation

"most dyscalculic learners will have cognitive and language abilities in the normal range, and may excel in non-mathematical subjects".

Butterworth (1999)

2. Impedance of academic achievement and daily living

"Dyscalculia is a term referring to a wide range of life long learning disabilities involving math… the difficulties vary from person to person and affect people differently in school and throughout life".

The National Center for Learning Disabilities, http://www.ld.org/LDInfoZone/InfoZone_FactSheet_Dyscacluia.cfm, Access: 22/10/03

More precise specification (Mahesh Sharma)  “Dyscalculia is an inability to conceptualise numbers, number relationships (arithmetical facts) and the outcomes of numerical operations (estimating the answer to numerical problems before actually calculating).”  

The emphasis here being on conceptualisation rather than on the numerical operations

The National Numeracy Strategy The DfES (2001)

 " Dyscalculia is a condition that affects the ability to acquire arithmetical skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures. Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence."  

• Currently used by the BDA.

• Perhaps more applicable to education in the early years

• In H.E. emphasis is less on basic computation and more on the application and understanding of skills and techniques

Butterworth (2002)

Effective problem solving:

"One of the things that distinguishes people who are good at maths, have effective 'mathematical brains', is an ability to see a problem in different ways. This is because they understand it. This, in turn, allows the use of a range of different procedures to solve it and to select the one that will be most effective in this particular task".

Key Points

• Mathematical ability substantially less than expectation

• “Impedes academic achievement or daily living”

• Inability to conceptualise

• Failure to understanding number concepts and relationships

For as long as I can remember, numbers have not been my friend. Words are easy as there can be only so many permutations of letters to make sense. Words do not suddenly divide, fractionalise, have remainders or turn into complete gibberish because if they do, they are gibberish. Even treating numbers like words doesn't work because they make even less sense. Of course numbers have sequences and patterns but I can't see them.Numbers are slippery.

J. Blackburn (2003)

Mathematics Support for students with

dyslexia and dyscalculia

Dyslexia and no dyscalculia

Dyslexia and dyscalculia

No dyslexia and

dyscalculia

Mathematically able

Mathematical difficulties

•Working memoryLanguage based•Reading•Understanding text•Presentation

Moving from concrete to

abstract

MathsPhysics

Engineering

EconomicsHuman Sciences

Business

•Working memoryLanguage based•Reading•Understanding text•Presentation

Number related•Number relations•Number concepts•Number operations

Human Sciences

Social ScienceGeography

Number related•Number relations•Number concepts•Number operations

Human SciencesSocial Science

Geography

Framework for Dyslexic and Dyscalculic students

Case study 1 – KateMaths

• Dyslexic, not dyscalculic

• No problems with basic number (95th percentile)

• Difficulties • With word recognition• Speed and accuracy of reading• Very slow handwriting• Poor spelling• Weak auditory memory

• Poor short term memory with retrieval of phonological information (1st percentile)

• Frequently loses her place

With no influence from other factors, the decrease ina variable R over a given small time interval isobserved to be proportional to both the length of thetime interval and the initial value at the start of theinterval. Write down a conservation law for thechange in R over a typical time interval, Hence obtaina differential equation for R as a function of time t.The differential equation should indicate that withoutthe influence of other factors, the level of decreaseswith time.

In order to increase the level of R, anotherfactor Q is introduced. The amount of Q per unittime is a constant. It has been observed thatthe effect of the factor Q is to increase the rateof change of R with t by an amount that isproportional to both Q and the differencebetween R and another factor P. Modify thedifferential equation by including an extra termTo take account of Q, Solve the modifiedequation. Indicate the sign of any constants youintroduce. What level of R is approached in thelong term?

• With no influence from other factors, the decrease in a variable R over a given small time interval is observed to be proportional to both the length of the time interval and the initial value at the start of the interval

• Write down a conservation law for the change in R over a typical time interval

• Hence obtain a differential equation for R as a function of time t

• The differential equation should indicate that without the influence of other factors, the level of R decreases with time. In order to increase the level of R, another factor Q is introduced. The amount of Q per unit time is a constant.

• It has been observed that the effect of the factor Q is to increase the rate of change of R with t by an amount that is proportional to both Q and the difference between R and another factor P

• Modify the differential equation by including an extra term to take account of Q

• Solve the modified equation

• Indicate the sign of any constants you introduce

• What level of R is approached in the long term?

Case Study 2 – NickEconomics

• dyslexic, not dyscalculic

• no problems with basic number

• difficulties in generalisation, in translating from concrete to abstract

• slow processing speed

• poor sequencing ability

• short term memory is weaker for symbolic material

C = 500 + 20Q - 6Q2 + 0.6Q3

Identify the fixed and variable costs

C = 500 + 20Q - 6Q2 + 0.6Q3

Given demand and supply functions: Qd = 25 -0.3P - 0.2P

2

Qs = - 5 + 2P + 0.01P2

find the equilibrium position.

Demand = Supply

Qd = 25 -0.3P - 0.2P2 Qs = - 5 + 2P + 0.01P2

 Put demand equal to supply

Qd = Qs 25 -0.3P - 0.2P2 = - 5 + 2P + 0.01P2

Rearrange 

25 - 0.3P - 0.2P2 = - 5 + 2P + 0.01P2

 0 = 0.21P2 + 2.3P - 30

2

2

0.2

P=

- ± -4x xP=

2x

P=-18.6

-302.3 1

0.2

2,7.67

1

2.3

- ±b b

a2

ca-4

Given the Lagrangian for the long run cost minimisation problem

H = 0.25K+L+h(100 - L0.5K0.5)

Determine the optimal level of labour(L).

H = 0.25K + L + h(100 - L0.5K0.5)

H = 0.25K + L + 100h - L0.5K0.5h 

HK

HL

Hh

0.25 - 0.5 L0.5K-0.5h

0.25 - 0.5 L0.5K-0.5h = 0

0.5 L0.5K-0.5h = 0.25

L0.5K-0.5h = 0.5 (1)

1 - 0.5L-0.5K0.5h

1 - 0.5L-0.5K0.5h = 0

0.5L-0.5K0.5h = 1

L-0.5K0.5h = 2 (2)

100 - L0.5K0.5

100 - L0.5K0.5 = 0

L0.5K0.5 = 100 (3)

(2) (1) L-0.5K0.5h = 2 L0.5K-0.5h 0.5

K / L = 4 K = 4L Substitute in (3) L0.5K0.5 = 100L0.5(4L)0.5 = 1002L = 100 L = 50, K = 200

Case Study 3 – MariaPsychology

• Dyslexic and Dyscalculic

• very poor numerical skills, problems with basic concepts

• difficulty seeing numbers inter-relationships

• 0.4 percentile for numeracy

• acutely anxious

Maths Anxiety Behaviors

• Making yourself small, hunching up or hiding inside a jumper, trying not to be noticed• Considerable self-doubt, feeling that your contributions have no value• Rarely believing you have a correct method or solution• Panic• Hating being watched• Looking at the paper and pen or calculator and not wanting to touch or try things out

• Working only in pencil, so mistakes can be quickly erased, always assuming you will make mistakes • Feeling threatened by mathematical vocabulary, not knowing the “right” words to use. Never talking about maths, except to say can’t do it.” • Poor history of maths in school• Avoiding maths, hoping it will go away

Group Statistics

11 17.7273 2.86674 .86435

10 14.1000 3.57305 1.12990

ConditionMnemonic condition

No mnemonic condition

Number of words recalledN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

.605 .446 2.578 19 .018 3.6273 1.40721 .68194 6.57260

2.550 17.288 .021 3.6273 1.42259 .62966 6.62489

Equal variancesassumed

Equal variancesnot assumed

Number of words recalledF Sig.

Levene's Testfor Equality ofVariances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of theDifference

t-test for Equality of Means

Independent samples t-test

Independent Samples Test

2.578 19 .018Equal variancesassumed

Number of words recalledt df Sig. (2-tailed)

Case Study: Liam Transport Management

• Dyscalculic

• Weak working memory– Holding information during calculation

• Difficulties sequencing

• Problems with mathematical calculation– Unsure of basic operations– Use of inappropriate strategies

• Non-verbal reasoning

A small airline, based at LHR, serves two cities: Oslo and Helsinki. The flying time to Oslo is 21/4 hours and to Helsinki is 3 hours. There should be 3 return flights a day to each city and the turn-round time must be at least 40 minutes, but not more than 1 hour.

Construct a schedule.

Helsinki 1 Helsinki 2 Helsinki 3

Start 07.00 10.00 16.00

Fly time 03.00 03.00 03.00

Land GMT 10.00 13.00 19.00

Time Diff. 02.00 + 02.00 + 02.00 +

Land local 12.00 15.00 21.00

Turn round 00.45 00.45 00.45

Start local 12.45 15.45 21.45

Fly time 03.00 03.00 03.00

Land local 15.45 18.45 00.45

Time diff. 02.00 - 02.00 - 02.00 -

Land GMT 13.45 16.45 22.45

Oslo 1 Oslo 2 Oslo 3

Start 07.00 14.00 18.00

Fly time 02.15 02.15 02.15

Land GMT 09.15 16.15 20.15

Time Diff. 01.00 + 01.00 + 01.00 +

Land local 10.15 17.15 21.15

Turn round 00.45 00.45 00.45

Start local 11.00 18.00 22.00

Fly time 02.15 02.15 02.15

Land local 13.15 20.15 00.15

Time diff. 01.00 - 01.00 - 01.00 -

Land GMT 12.45 19.45 23.45

L

O H

07.0010.0016.00 12.00

15.0021.00

12.4515.4521.45

13.4516.4522.45

07.0014.0018.0010.15

17.1521.15

11.0018.0022.00

12.4519.4523.45

AllocationA haulage company has vehicles in 5 locations and 5 vehicles (V1… V5) are required in a further 5 locations (L1… L5). Given the mileages matrix below, which vehicles should be sent where? (minimise mileage).

Mileage L1 L2 L3 L4 L5V1 30 21 10 19 13V2 25 15 15 25 13V3 30 22 15 20 16V4 40 20 10 22 20V5 25 25 12 23 18

Objective: to allocate vehicles to locations to minimise total mileage.

Algorithm:

Step 1 Reduce each row and column of the mileage matrix by its lowest entry.

V 1 is 10 miles from L1, so if it were to be assigned any other location the opportunity cost is the entry minus 10 miles - do this for each row and then each column.

Mileage L1 L2 L3 L4 L5

V1 8 9 0 4 3

V2 0 0 2 7 0

V3 3 5 0 0 1

V4 18 8 0 7 10

V5 1 11 0 6 6

Examine rows and columns with one zero.

If V1 goes to L3 this prevents V4 from doing so

and V4 has no other zeros

Step 2 Draw lines through the least number of rows and columns to delete all the zero entries. Subtract the lowest uncovered entry from all uncovered entries. Add its value to any number covered by 2 lines.

Mileage L1 L2 L3 L4 L5V1 7 8 0 3 2V2 0 0 3 7 0V3 3 5 1 0 1V4 17 7 0 6 9V5 0 10 0 5 5

Examine the zero entries: If V1 goes to L3, V3 goes to L4 but V4 also goes to L3, so repeat to Step 2.

Mileage L1 L2 L3 L4 L5V1 7 6 0 1 0V2 2 0 5 7 0V3 5 5 3 0 1V4 17 5 0 4 7V5 0 8 0 3 3

V3 - L4, V4 - L3, V5 - L1, V2 - L2This leaves L5 and if L1 goes there, there is a zero mileage.

From the original mileage matrix, the total mileage is 83miles.

Re-working the Solution

• Use boxes for instructions

Reduce each row of the mileage matrix by its lowest entry

• Put in each matrix step by step

Mileage L1 L2 L3 L4 L5V1 30-10 21-10 10-10 19-10 13-10V2 25-13 15-13 15-13 25-13 13-13V3 30-15 22-15 15-15 20-15 16-15V4 40-10 20-10 10-10 22-10 20-10V5 25-12 25-12 12-12 23-12 18-12

Mileage L1 L2 L3 L4 L5V1 20 11 0 9 3V2 12 2 2 12 0V3 15 7 0 5 1

V4 30 10 0 12 10V5 13 13 0 11 6

• Collect together the steps in an ordered sequence

• Create a flow chart

• Place the correct matrix beside each step in the flow chart

R educe each row of the m atrix by itslow es t entry.

E xam ine row s and co lum ns w ith one zero.

D raw lines through the leas t num ber o f row s andco lum ns to de le te a ll the zero entries .

E xam ine the zero entries

From the orig ina l m ileage m atrix,the to ta l m ileage

R educe each co lum n of the m atrix by itslow es t entry.

A dd its value to any num ber co vered by 2 lines

S ubtrac t the low es t uncovered entryfrom a ll uncovered entries .

is there aso lu tion

no

yes

MATRIX

MATRIX

MATRIX

MATRIX

MATRIX

MATRIX

MATRIX

Solution

Facilitating mathematics learning for dyslexia/dyscalculia

• Break up large sections of text • Sans serif fonts (E.G. Arial) easier for dyslexics

to read• Left justify text, easier to read• Remove unnecessary words• Photocopy and reorder mathematics texts• Use of a VTM line reader to highlight a specific

line and reduce line to line copying errors

• Supplement missing notes• Use coloured overlays or coloured paper to reduce

the glare from black type on white paper• Procedural flow diagrams or tree diagrams E.G.

partial differentiation• Break down multi-step problems into small,

manageable steps• Colour variables in different colours• Use of colour to highlight. E.G. a triple integration,

can be done in 3 colours

3xz + 5yz2 + x2z3 dx dy dz

• Colour cells on Excel or SPSS spreadsheets to facilitate greater clarity

• Edit down tables of data and statistical output

• Provide “memory aids”, such as large wall posters. Use of card indexes and “card carrying” cases. E.G. one theorem per card

• “Gallery" of graphs showing various functions, transformations or plots

y

0

10

-4 -3 -2 -1 0 1 2 3 4

y

-30

0

30

-4 -3 -2 -1 0 1 2 3 4

y

-10

5

-4 -3 -2 -1 0 1 2 3 4

y

-15

0

15

-6 -4 -2 0 2 4 6

y

0

15

-6 -4 -2 0 2 4 6

y

-30

0

30

-6 -4 -2 0 2 4 6

y = x2 y = -x2

y = x3 y = -x3

y = A/x y = A/x2

• Graph functions – helpful to SEE the functions they are considering

• Mind maps for with more extended work (projects). Use colour, and show connections

• Centimetre square paper when requiring rows and columns E.G. matrices

• Designing graph paper specific to the students needs http://incompetech.com/beta/plainGraphPaper/

http://incompetech.com/beta/plainGraphPaper/

Go through the work at the students own pace

Progress is often slow and frequent revision

is necessary. The same ground may need to be

covered many times. However, by providing the

student with appropriate strategies and a

framework they can relate to, it is possible for the

dyslexic or dyscalculic student to grow in

confidence, become independent in their learning

and, above all, to succeed.

Task• In small groups:

• Look at the student profile – his strengths and weaknesses

• Look at the mathematical problem he has to understand

• Try to identify the difficulties he might face• Try to find ways to help the student access,

process and understand the problem, as well as help him to achieve a solution to this and similar problems

Student Profile

Noel • 1st year, Business Studies• Background

– attended a small school – received much support from staff and family– not aware, at this stage, of difficulties – successfully applied to University to study Business– immediately started to experience difficulties,

particularly in relation to lectures, coursework and his mathematics module

– Following a diagnostic interview, he was referred for screening and then to the Educational Psychologist.

Strengths

• Level of knowledge (67th percentile)

• Verbal reasoning (53rd percentile)

• Non-verbal reasoning (58th percentile)

• Verbal expressive (73rd percentile)

• Alertness to visual detail (69th percentile)

• Spatial problem solving (61st percentile)

Weaknesses• Auditory Working Memory (9th percentile)

– Sequencing, poor memory for lengthy instructions, holding information in memory during mathematical calculations

• Processing speed (8th percentile)– Sequencing and processing of visual information, poor copying

of symbols at speed• Reading

– word recognition and word recognition rate (6th percentile)• Spelling (10th percentile)• Writing (1st percentile)

– Messy and slow speed• Arithmetic (16th percentile)• Phonological awareness and retrieval (7th percentile)• Visual disturbance

A factory can produce either product A or product B. 7 litres of raw material and 5 hours of labour are needed to produce 1kg of product A and 13 litres of raw material and 3 hours of labour are needed to produce 1kg of product B. Profits per kg are £4 for product A and £7 for product B. Each day up to 60 labour hours are available and up to 182 litres of raw material are available. What quantities of each product should the factory make in order to maximise their profits?