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Maths support for dyslexic and dyscalculic students Approx. 20 students per week, one-to-one basis

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

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Maths support for dyslexic and dyscalculic students Approx. 20 students per week, one-to-one basis

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  1. Dyslexia and Dyscalculia Next Steps for AllClare TrottMathematics Education CentreLoughborough University

  2. 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

  3. Dyslexia support Mathematics support

  4. 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

  5. Difficulties with motor skills or coordination -Particularly difficulty with automatising skills E.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)

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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 DyslexicsChinn and Ashcroft (1997)

  12. 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

  13. Dyscalculia • Statistics • 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.

  14. There is currently no accepted definition of dyscalculia • A number of different definitions exist • Numerically based • Cognitive based • Neuroscience based

  15. 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"

  16. Two Important Features • 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)

  17. 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

  18. 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

  19. 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."

  20. 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

  21. 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". Butterworth (2002)

  22. Key Points • Mathematical ability substantially less than expectation • “Impedes academic achievement or daily living” • Inability to conceptualise • Failure to understanding number concepts and relationships

  23. 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)

  24. Mathematics Support for students with dyslexia and dyscalculia Dyslexia and no dyscalculia Dyslexia and dyscalculia No dyslexia and dyscalculia Mathematically able Mathematical difficulties • Working memory • Language based • Reading • Understanding • text • Presentation Moving from concrete to abstract • Working memory • Language based • Reading • Understanding • text • Presentation • Number related • Number • relations • Number • concepts • Number • operations • Number related • Number • relations • Number • concepts • Number • operations Human Sciences Social Science Geography Maths Physics Engineering Economics Human Sciences Business Human Sciences Social Science Geography Framework for Dyslexic and Dyscalculic students

  25. Case study 1 – Kate Maths • 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

  26. 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 decreases with time.

  27. 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?

  28. 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 • Henceobtain 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.

  29. 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 • Solvethe modified equation • Indicate the sign of any constants you introduce • What level of R is approached in the long term?

  30. Case Study 2 – Nick Economics • 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

  31. C = 500 + 20Q - 6Q2 + 0.6Q3 Identify the fixed and variable costs C = 500 + 20Q - 6Q2 + 0.6Q3

  32. 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

  33. 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).

  34. H = 0.25K + L + h(100 - L0.5K0.5) H = 0.25K + L + 100h - L0.5K0.5h H K H h H L 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 = 100 L0.5(4L)0.5 = 100 2L = 100 L = 50, K = 200

  35. Case Study 3 – Maria Psychology • Dyslexic and Dyscalculic • very poor numerical skills, problems with basic concepts • difficulty seeing numbers inter-relationships • 0.4 percentile for numeracy • acutely anxious

  36. 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

  37. 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

  38. Independent samples t-test Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means 95% Confidence Interval of the Difference Mean Std. Error F Sig. t df Sig. (2-tailed) Difference Difference Lower Upper Number of words recalled Equal variances .605 .446 2.578 19 .018 3.6273 1.40721 .68194 6.57260 assumed Equal variances 2.550 17.288 .021 3.6273 1.42259 .62966 6.62489 not assumed

  39. 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

  40. 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.

  41. 07.00 14.00 18.00 07.00 10.00 16.00 L 10.15 17.15 21.15 12.00 15.00 21.00 13.45 16.45 22.45 12.45 19.45 23.45 H O 12.45 15.45 21.45 11.00 18.00 22.00

  42. Allocation A 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 L5 V1 30 21 10 19 13 V2 25 15 15 25 13 V3 30 22 15 20 16 V4 40 20 10 22 20 V5 25 25 12 23 18

  43. 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.

  44. 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

  45. 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 L5 V1 7 8 0 3 2 V2 0 0 3 7 0 V3 3 5 1 0 1 V4 17 7 0 6 9 V5 0 10 0 5 5

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