1. Dyslexia and Dyscalculia Next Steps for All Clare Trott Mathematics Education Centre Loughborough 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 Dyslexics Chinn 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 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)

17. 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_D yscacluia.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. 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".

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 Maths Physics Engineering Economics Human Sciences Business Working memory Language based Reading Understanding text Presentation Number related Number relations Number concepts Number operations Human Sciences Social Science Geography Number related Number relations Number concepts Number operations 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 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.

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 Solve the modified equation Indicate the sign of any constants you introduce What level of R is approached in the long term?

31. 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

32. C = 500 + 20Q - 6Q 2 + 0.6Q 3 Identify the fixed and variable costs C = 500 + 20Q - 6Q 2 + 0.6Q 3

34. Demand = Supply Qd = 25 -0.3P - 0.2P 2 Qs = - 5 + 2P + 0.01P 2 Put demand equal to supply Qd = Qs 25 -0.3P - 0.2P 2 = - 5 + 2P + 0.01P 2 Rearrange 25 - 0.3P - 0.2P 2 = - 5 + 2P + 0.01P 2 0 = 0.21P 2 + 2.3P - 30

35. Given the Lagrangian for the long run cost minimisation problem H = 0.25K+L+h(100 - L 0.5 K 0.5 ) Determine the optimal level of labour(L).

36. H = 0.25K + L + h(100 - L 0.5 K 0.5 ) H = 0.25K + L + 100h - L 0.5 K 0.5 h HKHK HLHL HhHh 0.25 - 0.5 L 0.5 K -0.5 h 0.25 - 0.5 L 0.5 K -0.5 h = 0 0.5 L 0.5 K -0.5 h = 0.25 L 0.5 K -0.5 h = 0.5 (1 ) 1 - 0.5L -0.5 K 0.5 h 1 - 0.5L -0.5 K 0.5 h = 0 0.5L -0.5 K 0.5 h = 1 L -0.5 K 0.5 h = 2 (2) 100 - L 0.5 K 0.5 100 - L 0.5 K 0.5 = 0 L 0.5 K 0.5 = 100 (3) (2)  (1)L -0.5 K 0.5 h = 2 L 0.5 K -0.5 h 0.5 K / L = 4 K = 4LSubstitute in (3) L 0.5 K 0.5 = 100 L 0.5 (4L) 0.5 = 100 2L = 100L = 50, K = 200

37. 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

38. 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

39. 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

40. Independent Samples Test.605.4462.57819.0183.62731.40721.681946.57260 2.55017.288.0213.62731.42259.629666.62489 Equal variances assumed Equal variances not assumed Number of words recalled FSig. Levene's Test for Equality of Variances tdfSig. (2-tailed) Mean Difference Std. Error DifferenceLowerUpper 95% Confidence Interval of the Difference t-test for Equality of Means Independent samples t-test

42. 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

43. A small airline, based at LHR, serves two cities: Oslo and Helsinki. The flying time to Oslo is 2 1 / 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.

44. Helsinki 1Helsinki 2Helsinki 3 Start07.0010.0016.00 Fly time03.00 Land GMT10.0013.0019.00 Time Diff.02.00 + Land local12.0015.0021.00 Turn round00.45 Start local12.4515.4521.45 Fly time03.00 Land local15.4518.4500.45 Time diff.02.00 - Land GMT13.4516.4522.45

45. Oslo 1Oslo 2Oslo 3 Start07.0014.0018.00 Fly time02.15 Land GMT09.1516.1520.15 Time Diff.01.00 + Land local10.1517.1521.15 Turn round00.45 Start local11.0018.0022.00 Fly time02.15 Land local13.1520.1500.15 Time diff.01.00 - Land GMT12.4519.4523.45

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

47. 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 V130 21 10 19 13 V225 15 15 25 13 V330 2215 20 16 V44020 10 22 20 V525 25 12 23 18

48. 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.

49. Mileage L1 L2 L3 L4 L5 V18 9 0 4 3 V20 0 2 7 0 V33 50 0 1 V4188 0 7 10 V51 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

50. 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 V17 8 0 3 2 V20 0 3 7 0 V33 51 0 1 V4177 0 6 9 V50 10 0 5 5

51. 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 L5 V17 6 0 1 0 V22 0 5 7 0 V35 53 0 1 V4175 0 4 7 V50 8 0 3 3 V3 - L4, V4 - L3, V5 - L1, V2 - L2 This leaves L5 and if L1 goes there, there is a zero mileage. From the original mileage matrix, the total mileage is 83 miles.

52. Re-working the Solution Use boxes for instructions Reduce each row of the mileage matrix by its lowest entry

53. Put in each matrix step by step Mileage L1 L2 L3 L4 L5 V130-10 21-10 10-10 19-10 13-10 V225-13 15-13 15-13 25-13 13-13 V330-15 22-1515-15 20-15 16-15 V440-1020-10 10-10 22-10 20-10 V525-12 25-12 12-12 23-12 18-12 Mileage L1 L2 L3 L4 L5 V120 11 0 9 3 V212 2 2 12 0 V315 70 5 1 V4 30 10 0 12 10 V513 13 0 11 6

54. Collect together the steps in an ordered sequence Create a flow chart Place the correct matrix beside each step in the flow chart

55. MATRIX Solution

56. 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

57. 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 + 5yz 2 + x 2 z 3 dx dy dz

58. 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

59. y = x 2 y = -x 2 y = x 3 y = -x 3 y = A/x y = A/x 2

60. 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/

62. 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.

63. 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

64. Student Profile Noel 1 st 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.

65. 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)

66. 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

67. 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?