both papers

paper 1 :
Dyscalculia: Prevalence and identification of students.
Research evidence from a study conducted in Panjim in the state of Goa, India
Abstract
This paper presents the results of a survey conducted in the only three schools with a resource room at the time, in Panjim in the state of Goa, India.The purpose of this survey was to find out the prevalence of Dyscalculia among students. Having done that, the author used a non-standardised dyscalculia test to identify students with dyscalculia in a resource room. This test also threw light on the academic needs of the students with dyscalculia. The prevalence of dyscalculia in the three schools surveyed was found to range from 6% to 19%. The average of all the schools was 11.13%. The prevalence of children with dyscalculia among boys was found to be 6.23%, among girls 8.54% andamong boys and girls 18.61%. A few of the results obtained from the Dyscalculia test were that very few students made mistakes in “Number recognition and regular perception of ‘6’ and ‘9’ correctly.” Majority of the students (more than 75%) had problems in “Knowledge of Number sequencing and recognizing ‘skip-counting’ sequence”. The author found that the 26 students tested had varying degrees of dyscalculia from mild to moderate to acute, based on the percentage of items they got right on the test.
Keywords:Dyscalculia, prevalence, identification, test, checklist, resource room.
1. Introduction

Dyscalculia is a mathematical learning disorder where the mathematical ability is far below expected for a person’s age, intelligence and education. Researchers have found evidence that such a disability exists and because of their findings there is a need to address dyscalculia as an important learning problem in mathematics. Teachers, parents and students are often aware of the fact that there are a number of children with special difficulties in learning mathematics. Pupils with learning disorders may feel uncomfortable in learning situations, unless they are treated in a way that facilitates their learning. In order to give the proper education to a pupil with a learning disorder, it is essential to understand the most suitable ways in which students with learning disabilities can acquire mathematical understanding, which is closely associated with abstract thinking. Dyscalculia is a learning disability that affects the ability to acquire arithmetic skills. It is derived from the Greek generic name, “mathematics difficulty”. Until recently, very little was known about its prevalence, causes or treatment. Children with dyscalculia have difficulty understanding simple number facts and procedures. They tend to operate at average or above average levels in other subjects but demonstrate difficulties in Mathematics. They lack confidence, and practice avoidance strategies often manifesting in behaviour issues and helplessness. Individuals display a mathematics disability when their performance on standardized calculation tests or on numerical reasoning tasks is significantly depressed, given their age, education and intellectual reasoning ability (Mental Disorders IV (DSM IV)). When this loss of ability to calculate is due to cerebral trauma, the condition is called acalculia or acquired dyscalculia. Mathematical learning difficulties that share features with acquired dyscalculia but withoutevidence of cerebral trauma are referred to as developmental dyscalculia.(Hughes, Kolstad&Briggs, 1994). Students who show developmental dyscalculia (DD) have difficulty recalling number facts and completing numerical calculations. They also show chronic difficulties with numerical processing skills such as recognizing number symbols, writing numbers or naming written numerals and applying procedures correctly (Gordon, 1992). They may have low self-efficacy and selective attention difficulties (Gross Tsur, Auerbach, Manor &Shalev, 1996).  It is important to keep in mind that not all students who display low mathematics achievement have Developmental Dyscalculia.
1.1 Prevalence of dyscalculia
Research by Joffee, Fleischner and lately, Brian Butterworth, now acknowledge the recognition of dyscalculia in its own right and Butterworth further estimates that actually 3- 6% of the population are dyscalculic, based upon a proportion of pupils who have specific difficulty in Mathematics despite good performance in other subjects.Students whose only learning disability is developmental dyscalculia have been reported in several studies (Ozols& Rourke,1991).They comprise3% to 6.5% of the general school population,(Gross Tsur, Manor& Shalev1996,Von Aster, 1994), have average or above average intelligence and are found equally among both boys and girls. Dyscalculia is a learning problem which needs appropriate educational provisions. Its identification is complicated because it can be the only difficulty that a child has or it may occur as a part of a broader range of difficulties. It demands appropriate diagnostic procedures and educational interventions.The author believes that there is danger that incorrect diagnosis and classification can lead to helplessness on the part of the learner, in which the learner believes that the difficulties are too serious as to benefit from remediation.This study was conducted to achieve the following objectives:
Objective 1: To find out the prevalence of students with dyscalculia in schools with resource rooms
Objective 2: To identify students with dyscalculia and their academic needs
1.2 Significance of the study
 i. There is a lot of research conducted on dyslexia, but less research has been carried out on dyscalculia in India.
ii. Mathematics is a compulsory subject of school education. Its study has a profound and long lasting effect on the lives of its students. Every student must have an adequate knowledge of mathematics to get on in life. Parents and teachers find it difficult to deal with the problems of children with dyscalculia. They exhibit a certain level of helplessness.
iii. Although children have been given an option to drop mathematics in favour of a vocational subject at the SSC Board Exam, if diagnosed with dyscalculia, parents are reluctant to allow them to do so, because of the importance of the subject.
1.3 Types of dyscalculia
Research by the Dyscalculia Centre, which has been published in SEN Magazine, has shown that there are five different types of dyscalculia – although inevitably many young people suffer from a combination of the types listed below.
Type 1 Dyscalculicsreport significant worries about math.  As a result they feel themselves living in an alien world in which everyone else can grasp math, but they can’t.  Self-doubt becomes so strong that it gets increasingly difficult to persuade the individual that with proper support they might well be able to undertake and understand mathematical calculations.
Type 2 Dyscalculicsalso experience this deep concern but have found strategies for understanding and coping with basic mathematics – yet they feel that they don’t have the automatic grasp that others have and often take twice as much time (or more) to do a mathematics problem as a non-dyscalculic person.
Type 3 Dyscalculicshave a profound difficulty in comprehending and dealing with the concept of time.  Sometimes this issue appears on its own, sometimes in combination with type 1 or 2 dyscalculics.  For such people time itself makes no sense and they are quite unable to estimate “five minutes” or any other time length while questions about timetables and the like are also quite meaningless.
Type 4 Dyscalculics may not always be dyscalculic in the genetic sense, although they display many of the symptoms of dyscalculic people because they have short-term and long-term memory problems.  These students generally have a problem with all sequences – and this, of course, affects their ability to handle mathematics perhaps more than any other subject.
Type 5 Dyscalculicstend not to see numbers as in any way related to the real world.  In one sense most of us have this problem; after all, what is “six”?   We know what six sheep are.  But “six”, alone is close to meaningless.   For such people, mathematics can be learned automatically, but when it gets to issues such as fractions, decimals, and percentages then life gets difficult.
1.4 Signs and symptoms of dyscalculia
Afew of the signs and symptoms a student with developmentaldyscalculiahas are as follows:
 difficulties in holding and processing information in the mind – a
weak working memory which results in: forgetting where you are in
the calculation; failure to carry a number accurately into the next part
of the procedure.
 difficulties in understanding calendars, reading clocks and geometry.
 difficulties in planning and a failure to check one’s answer.
 difficultyunderstanding the information within the problem.
 a weak working memory and difficulties with language and reading
comprehension, will find it difficult to understand information which is
presented in a symbolic or textual manner.
 anxiety relating to mathematics and low self-esteem.
Because mathematics is very developmental, any insecurity or uncertainty in early topics will impact on later topics, hence the need to take intervention back to basics.
1.5 Causes of dyscalculia
Successful intervention is dependent on finding the cause or causes of a problem. Most problems can only be solved if one knows their causes. A viable point of departure would therefore be to ask the question, “What causes dyscalculia?”Mathematical performance deficits, Developmental Dyscalculia, may arisebecause of a wide range of factors, from poor teaching, to low socio-economicstatus, to behavioral attention problems. However, a subset of children with math difficulties, possibly with the most-severe impairments, appears to suffer from a developmental learning disorder that undermines the ability to process basic numerical magnitude information, and that impairment in turn undermines the acquisition of school-level arithmetic skills. This disorder, “primarydevelopmental dyscalculia,” should not be confused with “secondarydevelopmental dyscalculia,” which refers to mathematical deficits stemming from external factors such as those described above. Instead, primary Developmental Dyscalculia is associated with impaired development of brain mechanisms for processing numerical magnitude information and is thus driven by endogenous neurodevelopmental factors. The causes of dyscalculia are still a gray area, and there is little concrete understanding about what causes it or how itcan be prevented.
Hereditary or congenital disorders may be a significant cause, but there is no concrete evidence to prove it as yet. Injury to the brain that affects working memory is seen as a major factor in mental addition disability.
1.6 Assessment and diagnosis of dyscalculia
Recent research has identified the heterogeneous nature of mathematical learning difficulties and dyscalculia, hence it is difficult to identify via a single diagnostic test. Diagnosis and assessment should use a range of measures, a test protocol, to identify which factors are creating problems for the learner. Although on-line tests can be of help, understanding the difficulties will be better achieved by an individual person-to-person diagnostic, clinical interview.
Diagnosis could also involve several tests to determine the presence of a learning disability. Dyscalculia often requires a paper-pencil test. Check lists can also be used for initial testing. Further tests are required to reveal how a person practically applies mathematical concepts. Results are compared with the expected skill, based on educational level and age.
Areas that are assessed during dyscalculia tests include:
skills in adding, subtracting, dividing, multiplying and counting
skills in identifying when to use mathematical operations
skills in organizing objects
skills in measuring time, money, volume, and other quantities
skills in re-checking own work and using alternatives to come up with a correct solution.
2. METHOD
The Lafayette Parish School Dyscalculia check list has 14 items which have to be rated using a four point rating scale and then interpreted. It was given to 15 Mathematics teachers along with a questionnaire to find out the prevalence of dyscalculia in their schools. The entire population of 1757 students was surveyed, belonging to three schools having resource rooms. Percentages were used to report the results. Identificationof students was done by the consistent observation of the students by the teachers. TheDyscalculia test (non-standardised) by Liz Weaverwas used to confirm the identification and to assess the needs of students with dyscalculia. A purposive sample of 26 students from a resource room was used to identify students with dyscalculia and their academic needs.Percentages were used to report the results.
3. Results and discussion
The prevalence of dyscalculia in the three schools surveyed was found to be ranging from 6% to 19%. The average of all three schools was 11.13%.
 Table of prevalence of students with dyscalculia in three schools with resource rooms in Panjim
Standard Total number of students Number of students with dyscalculia Percentage of students with dyscalculia
5 316 21 6.6%
6 247 28 11.3%
7 296 27 9.1%
8 315 26 8.2%
9 290 49 16.9%
10 293 34 11.6%

Graph of prevalence of dyscalculia class-wise in three schools
1. The prevalence of children with dyscalculia among boys was 6.23%
2. The prevalence of children with dyscalculia among Girls was 8.54%
3. The prevalence of children with dyscalculia among boys and girls was 18.61%
Research shows the prevalenceas 5-8% of school age children (Strauss, 2003),
3-6% of population (www.bda-dyslexia.org),6-7% of school age children (www.ldonline.org).Jovanovic G., Jovanovic Z.et.al (2013), conducted a study on, “The frequency of dyscalculia among primary school children”. The results indicated that: The frequency of dyscalculia in the sample was 9.9%. The difference between boys and girls according to the total score on the test was statistically significant (p<0 .005="" p="">From the analysis of the results of the dyscalculia test all the students from the resource room of Our Lady of Rosary H. S. were found to have dyscalculia, but in varying degrees from mild to moderate to acute.
Range of mistakes made in the dyscalculia test by students from standards
 V-VIII
Standard Range of mistakes made
V 70-90%
VI 10-75%
VII 30-60%
VIII 50-80%
The percentage of mistakes made by students of standards V and VIII in the dyscalculia test were more than those made by students of standards VI and VII. This shows that the former students were more dyscalculic than the latter.
ITEM WISE ANALYSIS OF THE DATA COLLECTED FROM THE ADMINISTRATION OF THE DYSCALCULIA TEST
ITEM1: Knowledge of Number sequencing and recognizing “skip-counting” sequence.
All the students tested had knowledge of natural number sequencing, but recognizing skip-counting sequence and then being able to fill in the missing numbers, both forward and backward was a problem for 23 out of 26 (88.46%) students tested. From the selected sample only Saloni and Sanskruti got the item right.
ITEM2: Number recognition.
15 out of 26 (57.69%) students had a problem with number recognition including three from the selected sample viz. Trishna, Tulsi and Reasham. Eight showed ‘6’ and ‘9’ reversals.
ITEM3: Memory of Number sequences.
Eight (30.77%) students had a problem in counting backwards of which only two were from the experimental group viz. Ryanna and Trishna.
ITEM4: Memory of Number sequences.
Four (15.38%) students had a problem with matching the sixes and nines. (reversals).  None of them were from the experimental group.
ITEM5: Pattern recognition and motor coordination.
Seventeen (65.38%) students were unable to complete the pattern. Four students were able to draw only one row correctly. Trishna, Tulsi, Reasham, Ryanna, and Saloni from the selected sample were unable to do it correctly.
ITEM 6: Ability to copy the problems without number reversals and transpositions.
Only one student showed one reversal in copying the problems and three students copied one number in one out of  four  problems incorrectly which is no cause for concern as it is considered to be normal for children who do not have dyscalculia. Three students worked out the problems but did not copy them again.  Twenty students had no problem with aligning the numbers one below the other to add them correctly. Only Saloni copied wrong.
ITEM 7: Skill in adding and subtracting, visual discrimination and mental flexibility in switching between addition and subtraction.
Nine students were not able to visually discriminate and switch between addition and subtraction correctly. Tulsi and Reasham from the sample were not able to get this item right. Six students made a few mistakes which could be due to carelessness. Eleven showed no errors as they solved all the problems correctly. 57.69% students made mistakes in this item.
ITEM 8: Reading comprehension and/understanding of the Mathematics involved.
Four students had no difficulty in understanding and solving the problem correctly. Seven students were able to solve only one part of the problem correctly. Fifteen students were unable to solve the problem showing that they had a difficulty with reading comprehension and understanding the Mathematics involved. All the students from the sample, except Ryanna and Trishna showed this difficulty. 84.61% students made mistakes.
ITEM 9: Understanding the difference between adding and subtracting.
Only five students knew the meaning of subtracting. One of them was Saloni from the sample. All the others (80.77%) could not differentiate between adding and subtracting.
ITEM 10 and 11: Understanding of the concepts of hours and minutes.
Eleven (42.30%) students did not understand the concept of hours and eighteen (69.23%) students failed to understand the concept of minutes. Sanskruti was unable to understand both units of time.
ITEM 12: Understanding of the appropriate use of units of time.
Only ten students (38.46%) out of those tested were not able to understand the appropriate this difficulty.
ITEM 13: Understanding of a multi-step problem.
Thirteen (50%) students including Tulsi and Reasham showed difficulty in understanding the multi-step problem and were unable to solve it correctly.
ITEM 14: Skip counting aloud by 4.
Fifteen (57.69%) students had difficulty in skip counting aloud by 4. It is possible they difficulty.
ITEM 15: Understanding of the differences between units of measurement of length.
Thirteen (50%) students had no understanding of the difference between units of measurement of length including Sanskruti from the selected sample.
ITEM 16: Ability to copy numbers without reversals.
Only six (23.07%) students did not copy the numbers correctly but there were no reversals. Perhaps these students did not understand the question. None from the chosen sample had a difficulty with copying numbers.
ITEM 17: Understanding the nature of the task and adding correctly.
Nine (34.61%) students made mistakes while adding. Only Sanskruti had this difficulty.
ITEM 18: Understanding the nature of the task and subtracting correctly.
Twelve (46.15) students got the answers incorrect including Donea and Sanskruti. Perhaps they did not understand the relation between items 16, 17 and 18.
ITEM 19: Number recognition and regular perception of ‘6’ and ‘9’ correctly.
Only one student (3.85%) had this problem. This shows that the students tested had mild to moderate dyscalculia.
ITEM 20: Capacity for remembering a short sequence of numbers and then reproducing it without reversals or transpositions.
Twelve students could not remember the short sequence of numbers and showed reversals and transpositions. Nine students could remember a short sequence of numbers up to four digits, but could not remember and write the whole sequence correctly.80.77% students had a problem with this item. Tulsi and Reasham had a difficulty in this item.
ITEM 21: Ability to remember and orally repeatthe number sequences without reversals or transpositions.
Twelve (46.15%) students were not able to repeat the number sequences correctly. Eleven students were able to repeat 2-3 of the 4number sequences correctly. Tulsi and Reasham had a difficulty in this ability.
ITEM 22: Ability to remember a picture and then reproduce it without reversals.
None of the students were able to remember the picture and reproduce it correctly. However, they could copy it correctly with the exception of one student who showed a reversal.
ITEM 23: Capacity to identify the operations involved and perform the Math and fill in the blanks with the proper number or operation sign.
Twelve (46.15%) students were unable to identify and perform the operations correctly. Neither were they able to fill in the blanks with the proper operation sign. Donea showed this difficulty.
ITEM 24: Understanding of replacing symbols with numbers and then performing the appropriate operations
Seventeen (65.38%) students were unable to understand the replacement of symbols with numbers and perform the appropriate operations. Donea, Tulsi and Reasham showed this difficulty.
Very few students made mistakes in “Number recognition and regular perception of ‘6’ and ‘9’ correctly.” All the students made mistakes in the “Ability to remember a picture and then reproduce it without reversals.” They could not do so from memory, but were able to copy and reproduce the picture correctly. Only one showed a reversal. Majority of the students (more than 75%) had problems in “Knowledge of Number sequencing and recognizing ‘skip-counting’ sequence”, “Reading comprehension and/understanding of the Mathematics involved”, “Understanding the difference between adding and subtracting”, “Capacity for remembering a short sequence of numbers and then reproducing it without reversals or transpositions” and “Ability to remember and orally repeat the number sequences without reversals or transpositions”.
4. Conclusion
Although the percentage of students with dyscalculia is similar to the results obtained by other researchers, a lot of work needs to be done in this area. The teachers’ consistent observation over a period of time of the common mistakes children with dyscalculia make can go a long way toward correctly ascertaining the prevalence of dyscalculia among children of a particular area. Also the teachers need to be familiarized with the signs and symptoms of children with dyscalculia and given training how to use the checklist mentioned. The non-standardised dyscalculia test used, helped in identification of students with dyscalculia in the resource room. It in a way confirmed that the students were indeed dyscalculic. It also helped in classifying students as mild, moderate or acute in relation to their being dyscalculic, on the basis of the percentage of items they answered correctly. This test could be standardized for future use.
References
Adler, B. (2001). What is dyscalculia? Sweden: Kognitivit Centrum.
Aggarwal, S. M. (1977). A course in teaching of modern mathematics. Delhi: DhanpatRai and Sons.
Bender, W. N. (2005).Differentiating math instruction:Strategies that work for k-8classrooms. Thousand Oaks: Corwin Press.
Best, J. A. (2009).Research in Education (10th edition ed.). New Delhi: PHI Learning.
Beygi, A., &Padakannaya, P. A. (2010).A remediaIntrvention for Addition and subtraction in children with dycalculia.Journal of Educational Psychology, 93, 615-619.
Boosey, J. (n.d.). Dyscalculia and associated mathematical difficulties. Retrieved from http://www.suffolkmaths.co.uk.
Brauldi, A. C. (n.d.). Gardner's theory using multiple inteklligene in the classroom. Retrieved from http://www.springhurst.org/articles/MItheory.htm.
Butler, F. M., Miller, S. P., & Crehan, K. B. (2003). Fraction instruction for students with mathematics disabilities:Comparing two teaching sequences. Learning disabilites reserach and practice, 18(2), 99-111.
Butterworth, B. (1999). The mathematical brain. London: Macmillan.
Butterworth, B. L. (2010). Low numeracy and dyscalculaia:Identification and intervention. ZDM Mathematics education special issue on neuro science and education, 42(6), 527-539.
Butterworth, B., & Varma, S. a. (2011). Dyscalculia:From Brain to Education. Science 332.
Choukade, G. G. (2014). Development of science attitude Improvement programme for adolescents and its effectiveness. Pune: Univeristy of Pune.
De Fries, A. M., & Light, J. G. (1997). A twin study of mathematics disability. Journal of learning disabilites, 30, 617-623.
Elliot, J. L. (2000). Improving test performance of students with disabilites:On district and state assessments. Thousand Oaks: Corwin press Inc.
Fuchs, L. A. (2001). Principles of preventin and Intervention of mahematical difficulties. Learning disabilites reserach and practice, 16(2), 85-95.
Fuchs, L. S., Comptn, D. L., Fuchs, D., Paulsen, K., & Bryant, J. D. (2005). The prevention,identification and cognitive determinants of math difficulty. Journal of educational psychology, 97(3), 495-513.
Gay, L. R. (1976). Educational reserach:Competencies for analysis and application. Ohio: Bell and Howell company.
Geary, D. C. (1993). Mathematical disabilties:Cognition, neuropsychological and genetic components. Psychological Bulletin, 114(2), 345-362.
Geary, D. C. (2000). Mathematical disorders:An overview for educators. Perspectives, 26(3), 6-9.
Geary, D. C. (2004). Mathematics and learning disabilites. Journal of learning disabiites, 37(1), 4-15.
Jovanovic, G., & Jovanovic, Z. E. (2013). The frequency of dyscalculia among primary school children. Psychiatira Damubina, 25(2), 172-174.
Kok Hwee, C. N. (2014). A comparasion of cognitive equations of mathematics leraning process between the Americans and Singaporean students with dyscalculia. Educational reserach International, 3(1).
Kroesbergen, E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and special education, 24(2), 97-114.
Kunsch, C., & Jitendra, A. a. (2007). The effects of peer mediated instruction in mathematics for students with learning problems: A research synthensis. Learning disabilities research and practice, 22(1), 1-12.
Landerl, K., & Bevan, A. A. (1993). Developmental dyscalculia and basic numerical capacities:A study of 8-9 year old students. Science direct , 99-125.
Maccini, P. A. (n.d.). Mathematics strategy instruction(SI) for middle school studnets with learning disabilities. Retrieved November 20th, 2007, from http://www.k8accesscenter.org/training_resources/massini.asp.
Maite, K. T. (2006). An investigation into the leraning difficultesexperienced by Indian school children in learning division algotithm of simple fractions. School sceince, 44(2), 50-57.
Montague, M. (2004, March 21st). Math problem solving for middle school students with disabiities. Retrieved 2008, from Http://www.k8accesscenter.org/training_resources/MathProblemSolving.asp.
Montague, M. (2007). Self regulation and mathematics instruction. Learning disabilites reserach and practice, 22(1), 75-83.
Montis, K. K. (2000). Language Development and concept flexibility in Dyscalculia: A case study. Journal of Reserach in Mathematics Education, 31(5), 541-556.
Nagavalli T. (2015). A Study of Dyscalculic Primary School Children in Salem
            District and  Evaluation of Applicability of Innovative Strategies as Remedial         Measures.Department of Educational Research and Policy Perspectives NCERT New     Delhi.
Patel, R. N. (2012). Teaching and learning mathematics in modern times.Mumbai:Himalaya Publishing house
Peard, R. (2010). Dyscalculia:What is its prevalence?Reserach evidence from case studies. Elsevier Procedia Socail and Behavioural sciences, 8, 106-113.
Sachse, L. (1998). The realtionship between working memeory and mathematical problem solving in children with and without leraning disabilities. Riverside: Unvieristy of California.
Sherman, H. J., & Richardson, L. I. (2005). Teaching children who struggle with mathematics:A systematic approach to analysis and correction. Upper Saddle River,New Jersey: Pearson Education Inc.
Sidhu, K. S. (1995). The teaching of mathematics. New Delhi: Sterling Publishers private limited.
Swanson, H. L. (1998). Experimental intervention reserach on students with learning disabilites:A meta analysis of treatment outcomes. Review of Educational research, 68, 277-321.
Tishler, A. G. (1996). Do some of your students have dyscalculia"Dilemma in MAthematics:learning difficulties in good students. Middle school Journal, 17(3), 12-14.
Uplane, M. M. (2011). Development of text book based computer multimedia software package for school children to enhance their academic achievement in Physics-A study. Pune: Univeristy of Pune.
Vasanthi, R. (1991). Mathematical leraning disability in relation to certain psychological , socail and educational factors. Madras: univeristy of Madras.
Waiganjo, S. (2013). Using ICT to assist Dyscalculia students:Situation analysis for primary schools in Kenya:A case study of Starehe district Nairobi country. International journal of Innovative Education Reserach, 1(1), 74-83.
Walker, D. W. (1989). A comparasion of two methods of teachihg mathematics story problem solving with learning disabled students. National forum of special education journal, 44-51.
Wilson, A. J., Revkin, S. K., & Cohen, L. a. (2006). An open trial assessment of the number race, an adaptive computer game for remediation of dyscalculia. Behavioural and brain functions, 32(4).
Witzwel, B. S., & Mercer, C. D. (2003). Teaching ALgebra to studnets with leraning difficulties:An investigation of a n explicit instruct ion model. Leraning disabilites research and practice, 18(2), 121-131.
Xin, Y. P. (1999). The effect of instruction in solving mathematical word problems for students with learning problems:A meta analysis. The journal of special education, 32(4), 207-225.

WEBSITES
http://www.learning works.org.uk
http://www.dyscalculia.org
http://www.nfer-nelson.co.uk
http://www.mathematicalbrain .com
http://www.dyscalculiainfo.org
http://www.oecd.org/edu/ceri/dyscalculiaprimerandresourceguide.htm
https://litemind.com/what-is-mind-mapping/
http://www.aboutdyscalculia.org
https://www.dyscalculia.me.uk/
https://ldaamerica.org/types-of-learning-disabilities/dyscalculia/
www.dyscalculia-maths-difficulties.org.uk/
www.dystalk.com/topics/5-dyscalculia2







paper 2 :
Dyscalculia: Effectiveness of an intervention programme to teach fractions.
Research evidence from an experimental study conducted with students from a resource room
Abstract
This paper presents the results of an experimental research conducted in a school with a resource room in Panjim in the state of Goa, India. The purpose of this research was to find out the effectiveness of the intervention programme developed by the researcher, using the product development method. A pretest-posttest single group experimental designwas used to investigate the efficacy of the intervention programme. The experimental group consisted of seven students from standard six who regularly attended sessions in the resource room on the topic fractions. A significant improvement was found in the academic achievement of the students. Of the three strategies used in the programme, explicit instruction and visual frameworks were instrumental in increasing the academic achievement of the students. The participants said that they enjoyed learning mathematics with visual frameworks and added that they helped them to remember what they had learnt more easily.
Keywords: Academic achievement, intervention, experimental, explicit instruction, visual frameworks
1. Introduction
Mathematics underachievement can be due to a range of causes, for example, lack of motivation or interest in learning mathematics, low self-efficacy, high anxiety, inappropriate earlier teaching or poor school attendance. It can also be due to generalised poor learning capacity, immature general ability, severe language disorders or sensory processing. Underachievement due to Developmental Dyscalculia has a neuropsychological foundation. The students lack particular cognitive or information processing strategies necessary for acquiring and using arithmetic knowledge. They can learn successfully in most contexts and have relevant general language and sensory processing. They also have access to a curriculum from which their peers learn successfully. One of the primary objectives for working with identified mathematical difficulties is to raise the self-esteem and confidence of the pupil.  Once the pupil understands the origins of their difficulty and the support that is available to them, they can begin to access the mathematical curriculum much more readily.  This in turn can eliminate the associated fear and stress related behaviour.
Dyscalculia is a broad term that includes many different kinds of difficulties in learning mathematics. Dyscalculia is also used for naming general difficulties in learning basic mathematics. Some of the different definitions include:
• Developmental dyscalculia: according to a number of researchers (Kosc, 1974, Shalev& Gross-Tsur, 1993, 2001, Wilson& Dehaene2, 2007) this type of dyscalculia originates from a specific impairment in the brain function. However, this definition has not always been fully acknowledged by the research community. The neuroscience research field has though progressed, where studies are conducted concerning the brain (Wilson &Dehaene, 2007).
• Acalculia: where a person has lost all sense of meaning of numbers or where the person still understands numbers but is unable to perform basic calculations like addition and multiplication, due to a neurological damage.
• Pseudo-dyscalculia: finding math difficult, based on emotional blockage or a confidence problem.
Children with Dyscalculia tend to be of normal intelligence. They will be noticed for their difficulty to do mathematics.  Prof. Butterworth suggests that in fact, dyscalculics are “really good at doing very complicated mathematics but still need help with the basics. Prof. Brian Butterworth (Mathematical Brain) points out that “school mathematics is like a house of cards.”  Each stage of learning depends upon the firm construction of the previous stage. “If a lower level is shaky, the house will eventually fall down.”  Equally if a pupil has missed a particular level then it becomes difficult to build subsequent stages securely.  The gap between what is being taught and the pupil’s knowledge will become wider.

The ‘ Mathematical Brain’ hypothesis implies that there are children who are born with an abnormal brain development which affects their number module just as has been identified for Dyslexia.  Diagnosis of Dyscalculia is currently still very much in its infancy although recently much more research has taken place and the condition has received both educational and media reporting. Just as with any specific learning difficulty, early diagnosis is essential.
Prof. Butterworth estimates that 3-6% of the population is dyscalculic.  In comparison the BDA (British Dyslexic Association) suggest that around 4% of the population is severely dyslexic.    Dyslexics are not necessarily dyscaculic and vice-versa although they do share some similar characteristics.

1.1 How do we recognise Dyscalculia?

Dyscalculia is very specific and currently very difficult to identify accurately.  However, it is important to recognise that just as pupils may have “dyslexic tendencies” so they will have with mathematics difficulties (dyscalculia tendencies).

Dr. Steve Chinn, (a leading expert in dyslexia) defines the difference between dyslexic mathematicians and dyscalculics (National Numeracy Strategy) and suggests that initial observations are the “basic indicator.”   Dyscalculic pupils tend to operate at average or above attainment levels in other subjects but demonstrate difficulties in mathematics and usually associated with number in particular.  Dyscalculic pupils lack confidence and practice avoidance strategies often manifesting in behaviour issues and helplessness.  Dyscalculic pupils will avoid group/partner work and if put in the situation; will rely totally on others for answers.

In their mathematics tasks, Dyscalculic pupils will exhibit difficulties such as:

• Learning and recalling times tables
• Confusion with mathematical symbols
• Unable to estimate an answer
• Misunderstanding place value
• Lack understanding of mathematical vocabulary
• Cannot recognise inverse operations
• Being able to use more than one operation in a task
• Finding any form of sequencing too challenging

Additionally, the presentation of their work may be untidy and calculations incorrectly aligned or equally the presentation may be more important than the calculations, involving considerable time and care, (avoidance strategy).
1.2 Treatment of dyscalculia
Treatments for dyscalculia include the following measures:
Educational Therapy involves modification of teaching approaches to children. One of these is the use of graphing paper to help children present concepts in an organized manner. Simple memorization of the multiplication table may also be modified by explaining the meaning of the numbers.
Software: Calculators and other devicesmay be used to help people with dyscalculia.
Stimulation of the parietal lobe: This has shown improvements in the numerical abilities of a person with dyscalculia. It involves the use of Transcranial Direct Current Stimulation.
Support groups: Help by caregivers and parents is needed to emphasize math skills at home during active play or study period. Parents have a vital role in emphasizing concepts that are learned at school.
Explicit and Systematic Instruction: Explicit instruction carefully constructs interactions between students and their teacher. Teachers clearly state a teaching objective and follow a defined instructional sequence. They assess how much students already know and tailor subsequent instruction, based upon that initial evaluation of student skills. Systematic instruction focuses on teaching students how to learn by giving them the tools and techniques that efficient learners use to understand and learn new material or skills.
Peer Tutoring: It refers to students working in pairs to help one another learn material or practice an academic task. Since explaining a concept to another person helps extend one’s own learning, this practice gives both students the opportunity to better understand the material being studied.
Visual Representations: Broadly defined, they can include manipulatives, pictures, number lines, and graphs of functions and relationships. “Representation approaches to solving mathematical problems include pictorial(e.g. diagramming); verbal(linguistic training); concrete(e.g. manipulatives); and mapping instruction(schema-based)”(Xin&Jitendra 1999).
1.3 Strategies used are as follows:
• Allow use of fingers and scratch paper
• Use diagrams and draw math concepts
• Provide peer assistance
• Suggest use of graph paper
• Suggest use of colored pencils to differentiate problems
• Work with manipulatives
• Draw pictures of word problems
• Use mnemonic devices to learn steps of a math concept
• Use rhythm and music to teach math facts and to set steps to a beat
• Schedule computer time for the student for drill and practice
1.4 Dyscalculia guidance summary
Based on the cognitive principles highlighted above the consensus on
guidelines for effective intervention in dyscalculia can be summarized as follows:
1. Interventions should be personalised according to individual needs.
2. Instructions should be simple and well organised.
3. Initially, abstract concepts should be made concrete.
4. Big concepts should be broke down into smaller parts and introduced step by steps.
5. Ample time should be allowed for students to practice new concepts. Ideally students should be allowed to decide when they are comfortable to move on. Speed should not be emphasised until facts are mastered.
6. Provide pictures, graphs, charts and encourage drawing the problem in order to enable visualization of problems.
7. Provide real life examples relevant to the student’s age and experiences.
8. Provide immediate feedback and opportunities for students to revise their answer.
9. Allow students to communicate about mathematics in multiple ways, be it orally or through journal entries.
10. New vocabulary should be adequately explained.
These guidelines are supported by the Department for Education and Skills (DfES, 2001).
Beygi A., Padakannaya P., and Gowramma (2010) studied “A Remedial Intervention for Addition and Subtraction in Children with Dyscalculia”. Data analysis indicated a significant increase in the subtraction and addition performance after remedial intervention. The article discussed implications for teachers, administrators, researchers, teacher training institutions, and students with learning disabilities.
KaurT.,Kohli T., Batani D., (2008) studied “Impact of various Instructional Strategies for Enhancing Mathematical Skills of Learning DisabledChildren”. The current study tested the comparative efficacy of various strategies on basic mathematical skills of learning disabled children. Learning disabled children were randomly assigned to multimedia, cognitive, eclectic and control conditions. Assessment included the use of IQ, Diagnostic Test of Learning Disability, and pre and post-test administration of the Children with Specific Learning Disabilities in Arithmetic scale. Results indicated that all the tested strategies significantly enhanced basic mathematical skills of learning disabled children. Multimedia, cognitive strategy and eclectic approach can be used for enhancing the mathematical skills of learning disabled children
CONCLUSION:The reported efficacy of an intervention in mathematical learning may depend not only on the type of intervention conducted, but also on the thoroughness of the evaluation procedure. Use of ICT as an intervention for children with dyscalculia was found to be successful. A significant increase in the subtraction and addition performance of children with dyscalculia was observed after remedial intervention. Adaptive software strengthens numerosity processing in children with dyscalculia.
This study was conducted to achieve the following objectives:
1. To develop an intervention programme for students with dyscalculia to increase their academic achievement.
2. To study the effectiveness of the developed intervention programme.
2. Method
Intervention Programme for Students with Dyscalculia, (IPSD) was developed as a treatment variable. The following steps were used.
1) Objective of programme development
In the present study for the development of the IPSD the following objective was decided: To improve the academic achievement of students with dyscalculia in the topic FRACTIONS.Accordingly, the IPSD was designed.
2) Review of related literature regarding development of IPSD.
Before development of IPSD, the theoretical base for each strategy was formed by referring to articles, books and researches.
Theoretical base for IPSD
Sr. No. Strategy selected Theoretical  base Components to be improved
1. Explicit Instruction Kolb’s Learning styles Use materials to suit different learning styles
2. Visual Frameworks Mindmapping Use a mindmap for revision
3. Peer Tutoring Multiple Intelligences Give materials to individual pupils
This exercise helped the researcher in the identification and finalization of the IPSD.
3) Planning for development of IPSD
While planning the IPSD the characteristics of children with dyscalculia were kept in mind. The strategies used were ‘Explicit instruction’, ‘Visual Frameworks’ and ‘Peer tutoring’.
4) Development of  IPSD
During this stage a rough draft of the IPSD was designed. Each of the three chosen strategies was used. The objective of the IPSD was kept in mind.
1. To improve the academic achievement of students with dyscalculia in the topic FRACTIONS.
Feedback for IPSD
Characteristics of dyscalculic students Strategies for improving academic achievement Accepted/rejected
Have difficulty in understanding Mathematics Explicit instruction All five accepted
Preferred learning style is Visual/tactile Visual frameworks All five accepted

After making the rough draft of the IPSD, the identified experts were requested to give their opinion. Altogether five experts were identified. From their feedback, changes were made to the IPSD. 
5) Pilot study and review of the programme.
After developing the IPSD, a pilot study was conducted on four students of VI standard from Santo Minguel High School, Taleigao in the month of June 2016. This was done to test the IPSD consisting of strategies of ‘Explicit and Systematic Instruction’, ‘Visual Representations’ and ‘Peer Tutoring’. The content chosen for this study was ‘Fractions’.
The objectives of the study were as follows:
i. To find out the errors in implementation of the IPSD.
ii. To know the possible difficulties during the implementation of the IPSD
Implementation of the pilot study
The pilot study was conducted to test the IPSD and to obtain insights about the practical difficulties in its actual implementation and to take the necessary precautions while conducting its implementation. The pilot study was conducted on students of VI standard from Santo Minguel High School, Taleigao.
Data collection and analysis of the pilot study
 Sr. No. Sub-unit of fractions Strategy used Previous status Modification after pilot study
1. Meaning of a fraction
1.Explicit& systematic instruction

2.Visual representations

3.Peer tutoring No Worksheets used Worksheets used
2. Fraction on the number line No Worksheets used Worksheets used
3. Types of fractions Formula for conversion No formula for conversion
4. Equivalent fractions Equalising the denominator Cross multiplication
5. Comparing fractions Equalisingthe denominator Cross products
6. Addition and subtraction of fractions No Drill Drill used

After the implementation of the pilot study, the researcher made the suggested changes in the programme and the final draft of the IPSD was prepared.
6) Final draft of the IPSD
With the help of suggestions received from the pilot study the researcher made the final draft of the programme.
7) Implementation of the IPSD
The IPSD was implemented on seven students of standard VI from the resource room of Our Lady of Rosary High School, Dona Paula during the Diwali holidays for a period of three weeks for two hours every day from Monday to Saturday.
Final draft of the intervention programme for students with dyscalculia
SUB-UNIT 1-MEANING OF A FRACTION
Objectives: The pupil will … develop understanding of the meaning of a fraction.
a. explain the meaning of a fraction in own words
b. identify the numerator and denominator of a fraction
c. state the fraction for the given shaded figure
d. shade the given figure according to the given fraction
e. identify the fraction in the given word problem
Materials required: Thermacole cut- outs of circles, squares and rectangles, and collections of flowers, fruits and vegetables. (Photos 1-6 and 10)
Procedure: The researcher used all three strategies to teach Sub-unit 1 of fractions. She used colourful visual representations of fractions made of thermacole to explain explicitly the meaning of a fraction and how to read a fraction. For example, “This is one whole circle. It is divided into four equal parts. Each part is called a fourth. How many fourths are coloured red?” Many examples were solved. Then the researcher asked the students to pair up, and teach the same thing to their peers to use the strategy ‘peer tutoring’. A lot of practice was given in reading fractions after it was explained explicitly with many different examples. Clues to remember were given by reminding students of the various standards such as third, fourth, fifth and so on. This helped them to remember to read the fractions better. Worksheets were given to practice colouring the figure according to the fraction and writing the fraction for the shaded part of the given figure. What fraction of the given numbers is prime was also taught using numbers made of Thermacole and by revising prime and composite numbers. Also the importance of equality of parts for fractions was explained with the appropriate materials, using positive and negative examplars of fractions.
SUB-UNIT2-FRACTIONS ON THE NUMBER LINE
Objectives: The pupil will … develop the skill of representing the given fractions on the number line
a. state the two consecutive whole numbers between which the given    fractions lie
b. plot the given fractions on the number line
Materials required: Strip of paper representing a number line with points marked at equal distances, whole number cards, and fraction cards, ruler and coloured chalk.
Procedure: Representing fractions on the number line was explained by the researcher using the chalk board and coloured chalk. Again, explicit instruction was used. First the researcher explained that a given fraction lies between two consecutive whole numbers. For example the fraction 1/3 lies between 0 and1. Since the denominator of the fraction is 3 the distance between 0 and 1 has to be divided into 3 equal parts. Each part is a third, so we have 0/3 which is zero, 1/3, 2/3, and 3/3 which is 1 whole. If it is 5/3 then it is more than 1, lies between 1 and 2 and the distance between them will be again divided into three equal parts. Also 5/3 is 1 2/3 and can be easily plotted. This was demonstrated by the researcher with a ruler and coloured chalk. Many examples were solved. Thereafter, the researcher used the number line strip and the whole number cards and the fraction cards to mark fractions on the number line. Each student was given a different set of fraction cards to be plotted on the number line while explaining to their peers how it needs to be done. A lot of practice for drawing and plotting fractions on the number line was given.
SUB-UNIT 3-TYPES OF FRACTIONS
Objectives: The pupil will…develop understanding of the different types of fractions.
a. identify the different types of fractions
b. convert improper fractions to mixed fractions
c. convert mixed fractions to improper fractions
Materials required: Different types of fraction cards made of coloured paper.  Cut-outs of apples and apple quarters, pictures of girls and boys, cut-outs of five pooris, operator cards, number cards and flannel board. (Photos 8,9,18)
Procedure: The researcher used many cards of proper fractions and asked students to compare the numerator of each fraction with its denominator. The students correctly pointed out that the numerator is smaller than the denominator. The researcher asked students to plot these fractions on the number line and explained that all proper fractions are less than 1. Then the researcher showed a set of improper fractions. Students compared the numerator of each fraction with the denominator and came to the conclusion that the numerator is greater than the he denominator. The researcher then emphasized that such fractions are called improper fractions and again used the number line to conclude that all improper fractions are greater than 1.Then an identification exercise was carried out with fraction cards. Students identified fractions as proper or improper giving reasons.
Then the researcher used pictures of children and apples to explain how five apples can be shared equally by four children in two different ways. One way is to give one apple each to the four children and divide the fifth apple into four equal parts. So each child will get one whole apple and a quarter of an apple giving a mixed fraction. The other way was to divide all the five apples into four equal parts each and give each child five quarters which is an improper fraction. Similarly division of five pooris among two children was also explained. Later, the children paired up and used the materials to explain the same to each other. The odd student explained to the teacher. From these examples the researcher arrived at the fact that improper and mixed fractions are interconvertible.
Next the researcher went on to explain the conversion of improper fractions to mixed fractions using long division. Since these students made mistakes while writing the mixed number the researcher explained explicitly that it would help them to write the remainder divided by the divisor next to the quotient. The researcher used different coloured chalk to focus on the quotient, remainder and divisor. It was explained that the quotient gives the whole number part and the fraction part is given by how many out of how many (divisor) remain (remainder). The researcher also explained that it is called mixed fraction as it consists of a whole number part and a fraction part. Thereafter, conversion of a mixed fraction to an improper fraction was explained by using number cards and operator cards (+, ×) on the flannel board. The researcher emphasized that the denominator remains the same in the conversion. The numerator is obtained by first multiplying the denominator by the whole number and then adding the numerator to it.  After the use of explicit explanation with visual representation, the researcher asked the students to pair up and explain what they had learned to each other. Then, the students were given worksheets on conversion for practice.
SUB-UNIT 4-EQUIVALENT FRACTIONS
Objectives: The pupil will … develop understanding of equivalent fractions.
a. identify equivalent fractions from the given shaded fractions
b. write equivalent fractions for the given fraction by multiplication or division
c. check whether the given fractions are equivalent
d. reduce the given fractions to simplest form
Materials required: Different square, circular and rectangular cards shaded to represent fractions, fraction cards made of coloured paper, A, B, C, D and operator (×) cards, and flannel board. (Photos 14, 16, 17, 19, 20)
Procedure: The researcher used three square cards representing the fractions ½, 2/4, 3/6 and asked students to identify them. The researcher then asked the students to compare the area occupied by each of these fractions by overlapping the cards. The students concluded that the areas were equal. The researcher explained that such fractions were called equivalent fractions. Then the researcher showed cards representing fractions ¾, 6/8, 9/12, 12/16 and 15/20. Students identified them as equivalent fractions giving reasons. Next the researcher asked students to find out how the other fractions are obtained from the first fraction. Then the researcher explained that equivalent fractions are obtained by multiplying both the numerator and the denominator by the same non-zero number. Many examples were solved by the students to ensure that they understood the method. Similarly the researcher explained how to obtain equivalent fractions from the given fraction by dividing both the numerator and the denominator by the same non-zero number. Emphasis was given on when to use which method. Multiplication is used for fractions in their simplest form. Division is used for those fractions in which the numerator and denominator have some common factors other than 1, since 1 is a factor of all numbers. A lot of drill was done with the help of worksheets. Next the researcher explained how to simplify a fraction to its lowest terms. It was emphasized that the same number (common factor of numerator and denominator) should be used to divide both the numerator and the denominator. This process is continued till the numerator and the denominator have no common factor other than 1. Many examples were solved. The researcher then used peer tutoring, wherein the students solved the given problems on the board one by one while explaining to their peers. Next the researcher explained how to fill in the missing numbers using equivalent fractions. The strategy of cross multiplication was used as the students found the multiplication and division methods difficult during the pilot study especially the division method. This was explained by using A, B, C, D and operator (×) cards. The students then used the material to explain to their peers. The researcher began by explaining that the cross products of two equivalent fractions are equal by using induction. Then the researcher substituted the blank with a letter and explained how to find the value of the variable. Many examples were solved and the students were given a worksheet on fill in the missing number for practice. The worksheet also served as a check for understanding. The same strategy was used to check whether the given pair of fractions was equivalent.
SUB-UNIT 5-COMPARING FRACTIONS
Objectives: The pupil will … develop understanding of comparing the given fractions.
a. compare the given pair of fractions.
b. write the given fractions in ascending/descending order
Materials required:Colourful fraction cards, pictures of fractions (rectangles and circles) and flannel board. (Photos 7, 11, 13, 21)
Procedure: The researcher began with four picture fractions (rectangles of the same size). The students had to identify them as ½, ¼, 1/5. The researcher asked students to identify the greatest and the smallest fraction according to the area occupied by the blue coloured fractions. Then the researcher explained how to order the fractions from small to big. Then the researcher explained using two circular fraction cards representing a cake of ¼ and 1/8. The teacher asked the students. “If I divide a circular cake into 4 equal parts and give you one part or I divide the same cake into eight equal parts and I give you one part when will you get a bigger piece of cake?”  So the students concluded that ¼ is greater than 1/8. Next the researcher presented picture fraction cards of like fractions and explained them by asking students to identify them and to order them. The researcher explained that if the numerators of the unlike fractions are same, greater the denominator smaller the fraction. The cake example was used again and again to improve understanding. Like fractions were explained by using the green fraction picture cards. Again they were compared using area of the green portion. Then the students concluded that if the denominators of the fractions are the same, greater the numerator, greater the fraction. The example of marks in a test was used to help them remember. Getting 9 out of 10 marks is better than getting 5 out of 10 marks. Pairs of fraction cards were given to the students along with cards with < and > symbols. Many examples were given for practice of all three types viz. which is greater, ascending order and descending order. Students solved the problems by arranging the fraction cards on the desk and then explaining to the researcher why they did so. Next the researcher explained how to compare unlike fractions by using two steps. Step1: Find the cross products of the two fractions beginning with the numerator of the first fraction. Step 2: compare the cross products. The same holds good for the fractions. An exercise on “Put an appropriate sign from (<, >, =) between the given fractions was explained. Emphasis was on working out the cross products.
× 0 1 2 3 4 5 6 7 8 9 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11 12
2 0 2 4 6 8 10 12 14 16 18 20 22 24
3 0 3 6 9 12 15 18 21 24 27 30 33 36
4 0 4 8 12 16 20 24 28 32 36 40 44 48
5 0 5 10 15 20 25 30 35 40 45 50 55 60
6 0 6 12 18 24 30 36 42 48 54 60 66 72
7 0 7 14 21 28 35 42 49 56 63 70 77 84
8 0 8 16 24 32 40 48 56 64 72 80 88 96
9 0 9 18 27 36 45 54 63 72 81 90 99 108
10 0 10 20 30 40 50 60 70 80 90 100 110 120
11 0 11 22 33 44 55 66 77 88 99 110 121 132
12 0 12 24 36 48 60 72 84 96 108 120 132 144
 Multiplication tables chart
SUB-UNIT6-ADDITION AND SUBTRACTION OF FRACTIONS
Objectives: The pupil will … develop understanding of addition and subtraction of fractions
a. add like fractions correctly
b. subtract like fractions correctly
c. add unlike fractions correctly
d. subtract unlike fractions correctly
e. solve word problems on addition and subtraction of fractions
Materials required: Picture fraction strips of rectangles and circles, pictures to illustrate the word problems. (Photos 12,15,21)
Procedure: The researcher first explained the meaning of addition of like fractions by using fraction strips and asking questions like: “Into how many equal parts is this strip divided? What fraction of the strip is coloured red? What fraction of the total strip is coloured?” Then the researcher explained that “one-fifth plus two-fifths is equal to three-fifths. Many such examples were solved first by using the strips on the flannel board and then giving more examples to solve of like fractions. The researcher emphasized that when the students read the like fractions they become easier to add. Similarly subtraction of like fractions was explained first using visual representations and then solving many problems by reading aloud the fractions. The researcher also explained the method of adding or subtracting like fractions i. e. add or subtract the numerators and write the answer over the common denominator.
Next the researcher explained how to add and subtract unlike fractions. The steps were clearly explained. Step 1: Find the LCM of the denominators and write it in both the denominators. Step 2: Multiply the first fraction by the LCM to obtain the numerator of the first fraction. Step 3: Multiply the second fraction by the LCM to obtain the numerator of the second fraction. Step 4: Add or subtract the numerators and write the answer over the common denominator. Many such problems were solved by the researcher on the board. Then the students solved one problem each on the board. Then the students explained to each other in pairs the steps in adding and subtracting like and unlike fractions. This was done to help reinforce the steps and fix them in their minds. Addition and subtraction of mixed fractions was also explained. For drill, worksheets on addition and subtraction of like and unlike fractions were given. Word problems were explained by illustrating them on the flannel board with pictures. The researcher focused on words which gave clues for identifying the mathematical operation to be used such as “total” for addition and “which is greater and by how much” for subtraction. Many word problems were solved. A coloured mind map on fractions was explained and given to each student for revision before the post test.
Observations after conducting the intervention programme
SUB-UNIT 1- MEANING OF A FRACTION
Observation: Students enjoyed all these activities and did them correctly. They enjoyed handling the instructional material and playing teacher while explaining to their peers. Some students made language mistakes while doing peer tutoring and were corrected by the researcher. They had much fun colouring the figures according to the given fractions and writing the fraction for the given shaded figures. After completing the worksheet correctly they would ask for more problems to solve. Sub-unit 1 of fractions was comparatively easy for them to understand and learn. Probably they had some previous knowledge of the same.
SUB-UNIT 2- FRACTIONS ON THE NUMBER LINE
Observation: To the researcher’s surprise they picked it up quickly and were ready to do it as an activity on the chalk board. It was heartening to note that all the students did their own individual problems correctly and asked for more problems to solve.
SUB-UNIT 3-TYPES OF FRACTIONS
Observation: The students understood the types of fractions quickly. They enjoyed explaining to each other with the materials given. A few of them had problems with conversions because they made mistakes with the position of the numbers in a mixed number. This was explained again followed by practice to overcome the difficulty. Visual frameworks strategy helped to arouse and sustain the interest of the students in the lesson at hand. Explicit instruction helped the students to understand what they were learning and drill helped to reinforce this learning. The students took a little time to get used to the strategy of Peer tutoring, maybe because of their problems in language skills. However with a little bit of prompting by the researcher, they were able to manage. Most of the students did quite well on the worksheets with an occasional mistake here and there.
SUB-UNIT 4-EQUIVALENT FRACTIONS
Observation: The students understood equivalent fractions quickly. They enjoyed explaining to each other with the materials given why the given fractions were equivalent or not. The method of multiplication was easily understood. In reducing a fraction to its simplest form, a few of them had problems because they made mistakes with the division of the numerator and the denominator by the SAME non-zero number. They did so using different numbers. Short division was also a problem as the researcher found out that it was not formally taught in the lower classes. So the researcher also taught short division and gave a lot of practice in the same till the students agreed that they were confident with it. Fill in the missing numbers was easily understood by the students because of cross multiplication. They did the working which helped them get the right answer. It also helped the researcher to check for errors made by the students. Checking whether the given pair of fractions was equivalent was easily understood by the students using cross multiplication.  They enjoyed explaining cross multiplication to their peers using the A,B,C,D and operator cards.(A×D= B×C).
SUB-UNIT 5- COMPARING FRACTIONS
Observation: Students found it easier to compare like fractions. They made mistakes with fractions having the same numerators. Again the researcher had to remind them of division of the same cake but into different number of equal parts. Comparing unlike fractions was done well by the students because of the method of finding the cross products. This strategy worked much better than the originally planned strategy of ‘equalising the denominators’. A few of them kept asking the researcher the times tables to confirm them. The researcher did not use the peer tutoring strategy as each of them was struggling with their multiplication tables. Retrieving times tables from memory is a problem for most children with Dyscalculia. Some of them would write the whole table to use even one product. Another student had to recite the whole table from the beginning. Another counted in fives on her fingers to get the 5 times table. She did it quite quickly too. The researcher taught the students the nine times table by writing the digits 0,1,2,3,4,5,6,7,8,9 in one column and in reverse order to get the entire table of nine in a column next to the first column. One of the students tried to do the same for the table of seven, where upon the researcher explained that it was a short cut only for the table of 9.They kept asking me if the product they obtained was correct. This is a common problem of children with Dyscalculia. They are not at all confident about the times tables. The researcher decided to give them a card with the tables of zero to twelve. It is as follows. A reverse letter L is used to get the required product. So 7×8=56. Students were excited to use it. They used them to solve their worksheets instead of asking the researcher every time to confirm their times tables. (FIGURE 3.2)
SUB-UNIT 6- ADDITION AND SUBTRACTION OF FRACTIONS
Observation: The students understood addition and subtraction of like fractions easily and were able to solve the given worksheet quickly and correctly.  They found addition and subtraction of unlike fractions difficult and required individual attention to correct their mistakes. With the help of worksheets their confidence in solving these problems increased. They solved addition and subtraction of mixed fractions also. Word problems, especially those that involved subtraction were difficult for them at first but once they gained understanding, they were able to solve them correctly. The students were happy with the mindmap and shared that they found it easier to learn with its help.
2. Results and discussion
Scores obtained in pre-test, post-test and retention test on the topic Fractions
Sr. No. Name of student Pre-test scores(50) Post-test scores(50) Retention test scores (50)
1. Attar Reasham 15 40 36
2. BhomkarSaloni 15 44 38
3. DodmaniTulsi 11 43 39
4. FernandesDonea 10 48 33
5. NaikSanskruti 20 49 40
6. OliTrishna 12 45 41
7. Rodrigues Ryanna 15 42 38
Mean 14 44.43 37.86
Standard Deviation 3.12 2.97 2.47

Observation: For the pre-test the mean was 14 and the standard deviation was 3.12. For the post-test the mean was 44.43 and the standard deviation was 2.97. For the retention test the mean was 37.86 and the standard deviation was 2.47.
Interpretation: Most of the students from the experimental group scored around 14 marks out of 50 in the pre-test, but scored around 44.45 marks in the post test showing a considerable increase because of the effect of the intervention programme. However, they scored around 37.86 in the retention test, showing a decrease, which could be attributed to a short term memory that is characteristic of children with dyscalculia. From the standard deviation values of the three tests it can be concluded that there was a greater spread of scores in the pre-test than in the post-test and retention test.

Graph: Comparison of scores obtained in pre-test, post-test and retention test

Inferential statistics
TABLE
 Comparison of scores of pre-test, post-test ant retention test using matched pair t-value.
Test N Mean SD Treatment variable Test Paired t-value Table value at 0.01 level dF
Pre-test 07 14 3.12

    IPSD Pre - post 20.15 3.71 06
Post-test 07 44.43 2.97 Post -retention 4.47 3.71 06
Retention test 07 37.86 2.47 Pre- retention 18.21 3.71 06
Observation and Interpretation for pre-test and post-test mean scores
Since the obtained t value (20.15) is greater than the table’s’ value with dF equal to 6 at 0.01 level, the difference is significant. Therefore the null hypothesis 1 is rejected.
Interpretation: The above observation reveals the following: There is a significant difference between the mean scores of pre-test and posttest showing significant increase in the academic achievement of the students with dyscalculia in the topic fractions.
This difference between pre-test and post-test mean scores is attributed to the effect of the use of IPSD, developed by the researcher and used as a treatment variable.
Observation and Interpretation for post-test and retention test
Since the obtained t value (4.47) is greater than the table‘t’ value (3.71) with dF equal to 6 at 0.01 level, the difference is significant. Therefore the null hypothesis 2 is rejected.
Interpretation: The above observation reveals that there is a significant difference between the mean scores of posttest and retention test showing significant decrease in the academic achievement of the students with dyscalculia in the topic fractions from the retention test to the post-test. From the above findings, it can be concluded that the posttest and retention test scores differ significantly. This could be attributed to the short term memory of students with dyscalculia. So the researcher decided to carry out the observation and interpretation for pre-test and retention test.
Observation and Interpretation for pre-test and retention test
 Since the obtained t value (18.21) is greater than the table t value (3.71) with dF equal to 6 at 0.01 level, the difference is significant. Therefore null hypothesis 3 is rejected.
Interpretation: The above observation reveals that there is a significant difference between the mean scores of pre-test and retention test showing a significant increase in the academic achievement of students with dyscalculia in the topic fractions from the pre-test to the retention test.  Despite a significant difference between the post-test and retention test, there is still a significant difference between the mean scores of the pre-test and retention test, thereby showing that the effect of the IPSD is retained although not significantly from the post-test as from the pre-test.
Explicit instruction has been found to be especially successful when a child has problems with a specific or isolated skill (Kroesbergen& Van Luit, 2003).The National Mathematics Advisory Panel Report (2008) found that explicit instruction was primarily effective for computation (i.e., basic math operations), but not as effective for higher order problem solving. The meta-analyses and research reviews by Swanson (1999, 2001) and Swanson and Hoskyn (1998) assert that breaking down instruction into steps, working in small groups, questioning students directly, and promoting ongoing practice and feedback seem to have greater impact when combined with systematic “strategies.”
Visual representations, broadly defined, can include manipulatives, pictures, number lines, and graphs of functions and relationships. “Representation approaches to solving mathematical problems include pictorial (e.g., diagramming); concrete (e.g., manipulatives); verbal (linguistic training); and mapping instruction (schema-based)” (Xin&Jitendra, 1999, p. 211). Research has explored the ways in which visual representations can be used in solving story problems (Walker & Poteet, 1989); learning basic math skills such as addition, subtraction, multiplication, and division (Manalo, Bunnell, &Stillman, 2000); and mastering fractions (Butler, Miller, Crehan, Babbitt, & Pierce, 2003) and algebra (Witzel, Mercer, & Miller, 2003).
Peer tutoring works best when students of different ability levels work together (Kunsch, Jitendra, &Sood, 2007). During a peer tutoring assignment, it is common for the teacher to have students switch roles partway through, so the tutor becomes the tutee. Since explaining a concept to another person helps extend one’s own learning, this practice gives both students the opportunity to better understand the material being studied.
3. Conclusion
The IPSD can be effectively used for improving the academic achievement in the topic Fractions of students with varying degrees of dyscalculia in class VI. The developed intervention programme for children with dyscalculia consisting of three strategies, viz. explicit instruction, visual frameworks and peer tutoring was effective.
From the results of the present study it is shown that an intervention programme can be effectively used for the improvement of academic achievement of students with dyscalculia.
The intervention programme helped the students to enjoy Mathematics as a fun activity.
Despite the research work, the fact remains that there still are pupils in school with difficulties when it comes to learning mathematics. These pupils are normally functioning, social, intelligent adolescents with seemingly no disorders. But when it comes to mathematics they no longer feel like intelligent well-functioning persons that they really are and for some of them their whole world falls apart. In lectures I have learned that persons with dyslexia have felt excluded from learning reading, writing and spelling until they received a diagnosis of their disability which opened up a whole new world of understanding to them and also of them. A diagnosis may have the effect to make a person feel understood, and give an explanation to why it is hard to learn a specific matter. A diagnosis also prevents a person from being discarded as “dumb”. The main reason to why research on dyscalculia is conducted is hopefully to give a remedy to the persons with mathematical learning disabilities. A diagnosis of dyscalculia could help the affected children to find a way to not feel excluded from the possibility to understand a core subject like mathematics. To discover dyscalculia at an early stage may prevent these children from
developing a negative self-image, leading to decreasing interest in mathematics, which in turn can affect the interest in other school subjects in a negative way. If the parents are involved and aware of the children’s difficulties, then the extra support from home can help the children in their learning process. A diagnosis could give meaning to the children and their parents and give resources to teachers to better help students with disabilities. Waiting for the problems to solve themselves will probably have the effect of making the children dislike math even more in the future, since it is hard to understand why there is a problem learning math on your own. According to Adler (2001), the best way to teach math to children with dyscalculia is in individual sessions. Further, if trying to teach a child with dyscalculia math of the wrong kind, the situation for the child can even worsen. Today the classroom situation does not make it easy for children with dyscalculia. The classes have many pupils and there is not much time left for individual teaching. In many Swedish upper secondary schools a pupil must have a proper diagnosis to receive special education. Sometimes the school’s economical situation still cannot allow extra individual teaching. We teachers see more and more students who are having severe problems learning mathematics. The scores on mathematical tests are getting lower each year, although the same tests are done every year in the first grade. That the student’s mathematical abilities decrease with the years is an alarming trend that demands immediate attention from the
authorities. There is a need to agree on criteria that will help recognizing and analysing dyscalculia. “Finally, only with better understanding of the nature of developmental dyscalculia can we devise effective ways of helping the millions of our fellow citizens whose lives are blighted by it” (Butterworth,2006). Personally I look forward to the day when I as a teacher
in mathematics have better tools to help all of my students, including the ones with learning disabilities, succeed in mathematics.
References

Adler, B. (2001). What is dyscalculia? Sweden: Kognitivit Centrum.
Aggarwal, S. M. (1977). A course in teaching of modern mathematics. Delhi: DhanpatRai and Sons.
Bender, W. N. (2005). Differentiating math instruction:Strategies that work for k-8 calssrooms. Thousands Oaks: Corwin Press.
Best, J. A. (2009). Research in Education (10th edition ed.). New Delhi: PHI Learning.
Beygi, A., &Padakannaya, P. A. (2010).A remediaIntrvention for Addition and subtraction in children with dycalculia.Journal of Educational Psychology, 93, 615-619.
Boosey, J. (n.d.). Dyscalculia and associated mathematical difficulties. Retrieved from http://www.suffolkmaths.co.uk.
Brauldi, A. C. (n.d.). Gardner's theory using multiple inteklligene in the classroom. Retrieved from http://www.springhurst.org/articles/MItheory.htm.
Butler, F. M., Miller, S. P., & Crehan, K. B. (2003). Fraction instruction for students with mathematics disabilities:Comparing two teaching sequences. Learning disabilites reserach and practice, 18(2), 99-111.
Butterworth, B. (1999). The mathematical brain. London: Macmillan.
Butterworth, B. L. (2010). Low numeracy and dyscalculaia:Identification and intervention. ZDM Mathematics education special issue on neuro science and education, 42(6), 527-539.
Butterworth, B., & Varma, S. a. (2011). Dyscalculia:From Brain to Education. Science 332.
Choukade, G. G. (2014). Development of science attitude Improvement programme for adolescents and its effectiveness. Pune: Univeristy of Pune.
De Fries, A. M., & Light, J. G. (1997). A twin study of mathematics disability. Journal of learning disabilites, 30, 617-623.
Elliot, J. L. (2000). Improving test performance of students with disabilites:On district and state assessments. Thousand Oaks: Corwin press Inc.
Fuchs, L. A. (2001). Principles of preventin and Intervention of mahematical difficulties. Learning disabilites reserach and practice, 16(2), 85-95.
Fuchs, L. S., Comptn, D. L., Fuchs, D., Paulsen, K., & Bryant, J. D. (2005). The prevention,identification and cognitive determinants of math difficulty. Journal of educational psychology, 97(3), 495-513.
Gay, L. R. (1976). Educational reserach:Competencies for analysis and application. Ohio: Bell and Howell company.
Geary, D. C. (1993). Mathematical disabilties:Cognition, neuropsychological and genetic components. Psychological Bulletin, 114(2), 345-362.
Geary, D. C. (2000). Mathematical disorders:An overview for educators. Perspectives, 26(3), 6-9.
Geary, D. C. (2004). Mathematics and learning disabilites. Journal of learning disabiites, 37(1), 4-15.
Jovanovic, G., & Jovanovic, Z. E. (2013). The frequency of dyscalculia among primary school children. Psychiatira Damubina, 25(2), 172-174.
Kok Hwee, C. N. (2014). A comparasion of cognitive equations of mathematics leraning process between the Americans and Singaporean students with dyscalculia. Educational reserach International, 3(1).
Kroesbergen, E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and special education, 24(2), 97-114.
Kunsch, C., & Jitendra, A. a. (2007). The effects of peer mediated instruction in mathematics for students with learning problems: A research synthensis. Learning disabilities research and practice, 22(1), 1-12.
Landerl, K., & Bevan, A. A. (1993). Developmental dyscalculia and basic numerical capacities:A study of 8-9 year old students. Science direct , 99-125.
Maccini, P. A. (n.d.). Mathematics strategy instruction(SI) for middle school studnets with learning disabilities. Retrieved November 20th, 2007, from http://www.k8accesscenter.org/training_resources/massini.asp.
Maite, K. T. (2006). An investigation into the leraning difficultesexperienced by Indian school children in learning division algotithm of simple fractions. School sceince, 44(2), 50-57.
Montague, M. (2004, March 21st). Math problem solving for middle school students with disabiities. Retrieved 2008, from Http://www.k8accesscenter.org/training_resources/MathProblemSolving.asp.
Montague, M. (2007). Self regulation and mathematics instruction. Learning disabilites reserach and practice, 22(1), 75-83.
Montis, K. K. (2000). Language Development and concept flexibility in Dyscalculia: A case study. Journal of Reserach in Mathematics Education, 31(5), 541-556.
Nagavalli T. (2015). A Study of Dyscalculic Primary School Children in Salem
            District and  Evaluation of Applicability of Innovative Strategies as Remedial         Measures.Department of Educational Research and Policy Perspectives NCERT New     Delhi.
Patel, R. N. (2012). Teaching and learning mathematics in modern times.Mumbai:Himalaya Publishing house
Peard, R. (2010). Dyscalculia:What is its prevalence?Reserach evidence from case studies. Elsevier Procedia Socail and Behavioural sciences, 8, 106-113.
Sachse, L. (1998). The realtionship between working memeory and mathematical problem solving in children with and without leraning disabilities. Riverside: Unvieristy of California.
Sherman, H. J., & Richardson, L. I. (2005). Teaching children who struggle with mathematics:A systematic approach to analysis and correction. Upper Saddle River,New Jersey: Pearson Education Inc.
Sidhu, K. S. (1995). The teaching of mathematics. New Delhi: Sterling Publishers private limited.
Swanson, H. L. (1998). Experimental intervention reserach on students with learning disabilites:A meta analysis of treatment outcomes. Review of Educational research, 68, 277-321.
Tishler, A. G. (1996). Do some of your students have dyscalculia"Dilemma in MAthematics:learning difficulties in good students. Middle school Journal, 17(3),  12-14.
Uplane, M. M. (2011). Development of text book based computer multimedia software package for school children to enhance their academic achievement in Physics-A study. Pune: Univeristy of Pune.
Vasanthi, R. (1991). Mathematical leraning disability in relation to certain psychological , socail and educational factors. Madras: univeristy of Madras.
Waiganjo, S. (2013). Using ICT to assist Dyscalculia students:Situation analysis for primary schools in Kenya:A case study of Starehe district Nairobi country. International journal of Innovative Education Reserach, 1(1), 74-83.
Walker, D. W. (1989). A comparasion of two methods of teachihg mathematics story problem solving with learning disabled students. National forum of special education journal, 44-51.
Wilson, A. J., Revkin, S. K., & Cohen, L. a. (2006). An open trial assessment of the number race, an adaptive computer game for remediation of dyscalculia. Behavioural and brain functions, 32(4).
Witzwel, B. S., & Mercer, C. D. (2003). Teaching ALgebra to studnets with leraning difficulties:An investigation of a n explicit instruct ion model. Leraning disabilites research and practice, 18(2), 121-131.
Xin, Y. P. (1999). The effect of instruction in solving mathematical word problems for students with learning problems:A meta analysis. The journal of special education, 32(4), 207-225.

WEBSITES
http://www.learning works.org.uk
http://www.dyscalculia.org
http://www.nfer-nelson.co.uk
http://www.mathematicalbrain .com
http://www.dyscalculiainfo.org
http://www.oecd.org/edu/ceri/dyscalculiaprimerandresourceguide.htm
https://litemind.com/what-is-mind-mapping/
http://www.aboutdyscalculia.org
https://www.dyscalculia.me.uk/
https://ldaamerica.org/types-of-learning-disabilities/dyscalculia/
www.dyscalculia-maths-difficulties.org.uk/
www.dystalk.com/topics/5-dyscalculia