1. Adaptive Computer Assisted Instruction (CAI) for Students
with Dyscalculia
(Learning Disability in Mathematics)
A. Brunda J. Bhavithra
Senior Lecturer, Lecturer,
Department of Computer Science & Engineering, Department of Computer Science & Engineering,
Dr. Mahalingam College of Engineering and Dr. Mahalingam College of Engineering and
Technology, Pollachi, Tamilnadu, India. Technology, Pollachi, Tamilnadu, India.
rambru@drmcet.ac.in bavi@drmcet.ac.in
ABSTRACT General Terms
Dyscalculia refers to a persistent difficulty in the learning or Experimentation
understanding of number concepts (e.g. 4 > 5), counting
principles (e.g. cardinality – that the last word tag, such as Keywords
“four”, stands for the number of counted objects), or arithmetic Dyscalculia, Learning Disability (LD), Multi dimensional
(e.g. remembering that 2+ 3 = “5”). These difficulties are often Learning Algorithm, Educational Software, E-learning, Adaptive
called a mathematical disability. Learning-disability (LD) E-learning.
students face difficulties in processing and retaining information
and thus have problems in time keeping up with classroom
1. INTRODUCTION
Mathematics difficulties are widespread in the United States as
instruction. It follows that students with dyscalculia may have well as in other industrialized nations [13]. The consequences of
difficulty with numerical operations, both in terms of such difficulties are serious and can be felt into adulthood. Low
understanding the process of the operation and in carrying out math achievement is especially pronounced in students from low-
the procedure. Further difficulties may arise in understanding income households. Children with weaknesses in basic arithmetic
abstract mathematical concepts, symbolic and graphical may not develop the conceptual structures required to support the
representations. The lack of specialized teacher training, as well learning of advanced mathematics. Although competence in high-
as unavailability of effective study material and traditional LD- level math serves as a gateway to myriad careers in science and
friendly curricula further worsen the educational problems technology, many students never reach this stage. Some children
Learning-Disabled student must challenge. Here we develop a gradually learn to avoid all things involving math and even
E-learning and an Adaptive E-learning tool, that trains develop math anxieties or phobias. In recent years, there has been
dyscalculia children on an entertaining numerical comparison growing interest in children with mathematics difficulties, in part
task, by presenting problems adapted to the performance level of because of the relatively large body of research on children‟s
individual child and the adaptation was implemented using mathematical cognition more generally [3].
Multidimensional Learning Algorithm to constantly adapt the
Research on children with mathematics difficulties has been
difficulty of the problem to the Childs difficulty level. This influenced by the wealth of studies in phonetic based reading
Multimedia tool for students with math learning disabilities difficulties, studies that have led directly to the development of
gives students an opportunity to maximize learning and skill evidence-based methods for determining who is going to need
improvement through Educational software. Multimedia tutoring support and how to provide help [5, 6, 7]. As in reading,
provides students the opportunity to proceed at a self-guided longitudinal research is critical for understanding how math
pace with immediate feedback. difficulties develop and change over time. Difficulties with
number combinations reflect basic weaknesses in number sense,
Categories and Subject Descriptors such as the ability to grasp counting principles or to manipulate
K.3.1 [Computers and Education]: Computer Uses in quantities mentally .There may also be underlying problems in
Education– Computer-assisted instruction (CAI). working memory, spatial representation, and attention.
K.4.2 [Computer and Society]: Social Issues – Handicapped
Persons/ Special Needs. Longitudinal research on subgroups of mathematics difficulties is
beginning to bear fruit for the development of instructional
methods [9]. For example, Fuchs, and Prentice (2004)
Permission to make digital or hard copies of all or part of this work for demonstrate that children with mathematics difficulties who are
personal or classroom use is granted without fee provided that copies are good readers respond differently to instruction than children with
not made or distributed for profit or commercial advantage and that mathematics difficulties who are also poor readers [6]. This
copies bear this notice and the full citation on the first page. To copy investigation was concerned with the emergence of mathematics
otherwise, to republish, to post on servers or to redistribute to lists,
requires prior specific permission and/ or a fee.
difficulties. If children‟s learning needs can be identified early on,
A2CWiC 2010, September 16-17, 2010, India. we may be able to design interventions that prevent failure in
Copyright © 2010 978-1-4503-0194-7/10/0009… $10.00. math. The importance of different components of „„n umber

2. sense‟‟ to mathematics achievement is not well understood, years are less researched and understood. Fortunately, attention is
although the aforementioned research on mathematics difficulties now being directed to helping students who struggle learning
in elementary school is suggestive [2]. basic mathematics skills, mastering more advance mathematics
(e.g., algebra), and solving math problems. [4, 5, 7] explains in
Although no two researchers define number sense in exactly the detail about math disability, the sources that cause such a
same way [5, 6, 7], most agree that the ability to subitize small disability, and how a math disability impacts students at different
quantities, to discern number patterns, to compare numerical grade levels.
magnitudes and estimate quantities, to count, and to perform
simple number transformations are key elements of number sense 2.1 Number Sense
in young children [7]. Most children develop fundamental number In mathematics education, number sense can refer to “an intuitive
sense before they receive formal instruction in elementary school, understanding of numbers, their magnitude, relationships, and how
although there is significant variation by social class and cognitive they are affected by operations” [5]. Some definitions emphasize
ability. Even infants appear to be sensitive to small numbers and an ability to work outside of the traditionally taught algorithms, e.g.,
number transformations. Preschool children learn basic counting “a well organized conceptual framework of number information
principles and can perform addition and subtraction calculations. that enables a person to understand numbers and number
These foundational aspects of number sense are important to the relationships and to solve mathematical problems that are not
“higher order” mathematical thinking that results from formal bound by traditional algorithms”.
education.
The following are the components of Number sense: Counting,
1.1 Objective Number knowledge, Arranging Numbers, Big and Small Numbers,
To design e-learning and adaptive learning tool for students with Simple Addition and Subtraction [6, 7].
Dyscalculia by presenting problems adapted to the performance
level of the individual child. Department of education and 2.2 Multidimensional Learning Algorithm
professional organization should increase their efforts and The multidimensional learning algorithm constantly adapts the
continue to support the dissemination of research based practices difficulty of the program to the child's performance level [2].
especially given the goals of “No Child Left Behind”. Adaptation was implemented using three dimensions of difficulty,
which were based on our instructional principles and our
2. LITERATURE REVIEW knowledge of the key determinants of performance in basic
A learning disability (LD) is a neurological disorder that affects numerical cognition in adults and children. The three adaptive
the brain's ability to receive process, store and respond to dimensions are: Distance, Speed, Conceptual complexity.
information. The term learning disability is used to describe the
seeming unexplained difficulty a person of at least average 2.2.1 Distance Dimension
intelligence has in acquiring basic academic skills. Studies show The first dimension, “distance”, increases difficulty of the
that mathematics content is especially challenging for students numerical comparison by decreasing the numerical distance (as
who have learning disabilities. Learning-disabled (LD) students measured by the Weber ratio) between the two compared
face difficulties in processing and retaining information and thus quantities [4]. This dimension is designed to adapt to the precision
have problems time keeping up with classroom instruction [2]. of the children's quantity representation and to push children to
progressively increase this precision.
For this reason, it appears that learning disabled students may
vastly benefit from self-paced computer-assisted instruction. The 2.2.2 Speed Dimension
purpose of this project is to review literature that: (1) clarifies The second dimension, “speed”, implements an increasingly short
problematic areas of instruction for LD students, (2) discusses deadline by which the child must respond. This is designed to
successful teaching strategies that can be applied to software, and increase speed and automaticity of to quantity representations, and
(3) reviews optimal characteristics of current motivational to encourage more efficient calculation and eventually memory
educational software. This information allows the formation of a recall of simple number facts. At the lower end of this dimension,
general overview on the design of multimedia / software for there is no deadline, so that if children are particularly slow at a
students with math learning disabilities. task, they will still be able to succeed.
Recently [12, 13], increased attention has focused on students 2.2.3 Conceptual Complexity
who demonstrate challenges learning mathematics skills and The third dimension, “conceptual complexity”, is a composite
concepts that are taught in school across the grade levels. dimension which is designed to move children along a
Beginning as early as preschool, parents, educators, and pedagogical progression which teaches them about number
researchers are noticing that some students seem perplexed symbols and elementary arithmetic. Difficulty is increased in two
learning simple math skills that many take for granted. For ways: 1) by decreasing the ratio of non-symbolic to symbolic
example, some young children have difficulty learning number information available to make a choice between the two quantities
names, counting, and recognizing how many items are in a group. on the “choice screen”, and 2) by introducing addition and
In fact [13], we know that that 5% to 8% of school-age children subtraction at higher levels.
are identified as having a math disability. Research on These steps were designed to cement links between symbolic and
understanding more completely what a math disability means and non-symbolic representations of number, and to increase
what we can do about it in school have lagged behind similar understanding and of and fluency of access to simple arithmetical
work being done in the area of reading disabilities. Compared to facts. However the dimension includes some other aspects, such
the research base in early reading difficulties [8], early difficulties as restricting magnitude range at times, and adding hazards to the
in mathematics and the identification of math disability in later board.

3. Table 1: Conceptual Complexity Table
Non Symbolic: Range Dot
Symbolic Addition
Symboli Verbal Restriction? Fading Subtraction
Levels : Arabic Required Instructional goal
c (dot (Spoken (Numbers Present? Required?
(Digits) ?
clouds) numbers) 1-5 only) (Duration)
Attention to and
1 Yes No No Yes No No No Processing of small non
symbolic quantities
Attention to and
2 Yes No No No No No No Processing of large non
symbolic quantities
Link small non symbolic
3 Yes Yes Yes Yes No No No quantities to symbolic
codes
Link large non symbolic
4 Yes Yes Yes No No No No quantities to symbolic
codes
Increase reliance on
5 Yes Yes Yes Yes Yes No No
symbolic codes
Further Increase reliance
6 No Yes Yes Yes Yes No No
on symbolic codes
Require complete reliance
7 No Yes Yes No No No No
on symbolic codes
Require complete reliance
8 No No Yes Yes No No No
on Arabic codes
Attention towards exact
9 No No Yes Yes No Yes Yes
quantity
Comprehension and
10 No No Yes No No Yes Yes fluency of small addition
problems
Comprehension and
11 No No Yes No No Yes Yes fluency of large addition
problems
Comprehension and
12 No No Yes Yes No No Yes fluency of small
subtraction problems
Comprehension and
13 No No Yes No No No Yes fluency of large
subtraction problems
Distinguishing between
14 No No Yes No No Yes Yes
addition and subtraction
3. EXISTING SYSTEM 3.1 Limitations of Existing System
Mathematical difficulties are widespread in all industrialized The following are the limitations of the existing system:
nations. Elementary school and Kindergarten students with 1.Instruction principles not relevant to the remediation of
learning disabilities often struggle to learn math. They have Dyscalculia. 2. Same set of questions presented in same order.
trouble in counting, naming numbers, remembering numbers etc. 3. Children‟s get uninterested in using these tools. 4. Designed
Children with Learning disabilities, particularly Dyscalculia, have only to particular group of children. 5. Not adaptive to children‟s
less “Number Sense” [5]. Children with weakness in basic performance level.
arithmetic may not develop the conceptual structures required to
support learning of advanced mathematics. 3.2 Need for the Proposed System
Many training institutions are not teaching scientifically based
Competence in high level math serves as a gateway to a numerous practices. Beyond an emphasis on the dissemination of research
careers in Science and technology; many students never reach this based practices, teacher preparation programme should infuse
stage. Some children gradually learn to avoid all things involving information about screening and formative assessment procedures,
math and even develop math anxieties [3]. specific content area instruction methodologies and methods of
E-Learning tool is available in the market to enhance the key individual and small group instruction into curricular for all
elements of Number Sense in young children. The areas include: educators, not just for special educators. Towards that end,
Counting, Number Knowledge, Number transformation, Dot Department of education and professional organization should
enumeration, Number Patterns. increase their efforts and continue to support the dissemination of

4. research based practices especially given the goals of “ No Child 5.2.1 Counting
Left Behind”. In counting children can learn counting by clicking on the
particular number. On clicking, they get the corresponding
4. PROPOSED SYSTEM number of objects and their representation in English on the
In modern societies, computers have become so ubiquitous that screen (see Figure 1). Children can also select Autoplay to play
computer-aided instruction is now low-cost, and can be used in automatically till twenty five.
either the home or the school environment. The use of computer
aided instruction also allows us to capitalize on the fascination
that children have for computer games, which makes it easier to
provide intensive training on exercises which might otherwise
become boring for them.
Adaptive Tutoring train‟s children on numerical task, by
presenting problems adapted to the performance level of the
individual child. The tool uses an algorithm to adapt to an
individual child‟s ability and provide intensive training in an
entertaining context. This approach for remediation of Dyscalculia
provides intensive training in number sense.
The instruction principles may be equally pertinent to the
instruction of mathematics for younger non-Dyscalculia children.
The most important design principle was that of enhancing quality
representation or number sense, cementing the links between
representations of number, conceptualizing and automizing
arithmetic, and maximizing motivation.
Figure 1. Counting Numbers
A multidimensional learning algorithm constantly adapts the
difficulty of the tool to the child‟s performance level. Adaptation 5.2.2 Arranging Numbers
can be implemented using three dimensions of difficulty. The In arranging children learn the number sequence by moving the
Distance dimension increases difficulty of numerical comparison. numbers to the correct boxes. They can also select autoplay to
The Speed dimension implements an increasingly short deadline move the number automatically to appropriate boxes (see Figure
by which the child must respond. 2).
The third dimension “Conceptual Complexity” which teaches the
children about number symbols and elementary arithmetic.
Difficulty is increased by decreasing the ratio of symbolic and
non-symbolic information and by introducing addition and
subtraction at higher levels.
5. IMPLEMENTATION
The Computer Assisted Instruction system includes two modules:
E-Learning and Adaptive E-Learning. The Students of age 6-7
registers and logins to the system, the system displays the menu
for E-learning and Adaptive Tutoring.
Through E- Learning, the students can learn the basics of
mathematics. Students can undergo tests through Adaptive
Tutoring. The adaptive tutoring contains 14 different levels of test.
Response time is calculated at each level and a report is generated.
A general report is generated at the end of 14th level.
Figure 2. Arranging Numbers
5.1 Login Module
Each user has to login using their Name, Age and School. After 5.2.3 Number Knowledge
logging in they will be taken to a menu where they can select E- In Number knowledge children comes to know the number names
Learning or Adaptive Learning. The registered details will be used and representation. Here they have to move the golden balls to the
to generate reports in each of the 14 levels and a consolidated appropriate position in the 5x5 Matrix given or select autoplay to
report. place the numbers automatically (see Figure 3).
5.2 The E-Learning Module 5.2.4 Simple Addition and subtraction
In E-Learning module students can learn the basics of Here children learn simple and basic addition by a method called
mathematics like counting, number knowledge, and number line addition and line subtraction. To add a number, move to the
names, simple Addition and subtraction. Students can select any right on the number line (see Figure 4). To subtract, move to the
one of the above mentioned basics using the E-Learning Menu. left on the number line (see Figure 5).

5. difficulty of the program to the child's performance level. It
contains 14 levels. A Child will be taken to the next level only if
he clears the current level else he will be taken to the E-Learning
tool.
Level 1: Questions will be presented in the form of dot clouds and
there is number restriction of 1 to 5 (see Figure 6);
Level 2: Questions will be presented in the form of dot clouds and
there is no number restriction;
Level 3: Includes both dot cloud and numbers with a restriction 1
to 5;
Level 4: Includes both dot cloud and Numbers with no number
restriction;
Level 5: Includes a new concept called Dot Fading in which the
question fades in 4 seconds. Children have to select the correct
option after fading. This concept is used to enhance the memory
of the children (see Figure 7);
Figure 3. Number Knowledge Level 6: Uses Dot fading but with fading time of 1 second;
Level 7: Used to provide complete reliance on symbolic codes.
Questions contain Arabic digits;
Level 8: Used to provide complete reliance on Arabic digits;
Level 9: Used to provide Attention towards exact quantity;
Level 10: Provides Comprehension and fluency of small Addition
problems;
Level 11: Provides Comprehension and fluency of larger Addition
problems;
Level 12: Provides Comprehension and fluency of small
subtraction problems;
Level 13: Provides Comprehension and fluency of larger
subtraction Problems (see Figure 8);
Level 14: Provides questions to Distinguishing between addition
and subtraction;
6. RESULTS AND DISCUSSION
We and others have suggested that dyscalculia may involve
impairment in quantity representation or its access via symbolic
Figure 4. Simple Addition representations .In order to enhance number sense; we firstly
selected number comparison as the primary task of the software.
Number comparison is a simple task which draws heavily on
quantity representation, and which produces activity in the area of
the brain thought to underlie a neuronal code for numerical
quantity, the horizontal intra-parietal sulcus (HIPS).
The difficulty of the task and degree of associated brain activity is
modulated by numerical distance in adults and children.
Dyscalculia children and children who are at risk for
mathematical under-achievement perform slowly or inaccurately
in numerical comparison.
Our comparison task included varying levels of numerical
distance, thus allowing the software to adapt to the current level of
precision of the child's quantity representation. We also included
an adaptable response deadline to encourage faster, increasingly
automatic access to quantity representation.
The software was also designed to emphasize the association
Figure 5. Simple Subtraction between representations of number and space, which are known to
be closely linked. One previous highly successful number sense
5.3 The Adaptive E-Learning Module intervention achieved this by capitalizing on the key features of
The Adaptive Tutoring tool trains children on an entertaining board games, in which the number/space link is concretized as
numerical comparison task, by presenting problems adapted to the playing pieces are moved along the board; the distance of their
performance level of the individual child. This tool uses a moves being enumerated or estimated numerically by children.
multidimensional learning algorithm to constantly adapt the

6. level. The consolidated report contains mark and time of all the
levels.
The software is tested by nine children with mathematical
learning difficulties. The results indicate that the software adapts
well to varying levels of initial knowledge and learning speeds.
Feedback from children, parents and teachers was positive. A
companion article [6] describes the evolution of number sense and
arithmetic scores before and after training.
The following graph (see Figure 9) shows the score of a tested
student (age 7). The graph exposes that the student performs good
in Level 1, 5, 9, 10 and 14. He performs poor in level 4 and 6.
Overall statistics says that most of the students find difficulty in
dot fading (level 6) and no number restriction (level 2, level 4).
Figure 6.Report for Level 1
Figure 9.Graph exposing a student performance in different
levels of test
Figure 7.Adaptive E-Learning Level 5 (Dot Fading 4 sec) 7. CONCLUSION AND FUTURE SCOPE
This Project describes the cognitive and algorithmic principles
underlying the development of software for dyscalculia. The
software is based on current understanding of the cerebral
representation of number and the hypotheses that dyscalculia is
due to a "core deficit" in number sense or in the link between
number sense and symbolic number representations. The design
of the software was based on several instructional principles
relevant to the remediation of Dyscalculia.
Our comparison task included varying levels of numerical
distance, thus allowing the software to adapt to the current level of
precision of the child's quantity representation. We also included
an adaptable response deadline to encourage faster, increasingly
automatic access to quantity representation. Children‟s confidence
in their mathematical ability improved. Profiles generated at each
level showed the performance of children across different
dimensions. The software may have applications to the general
instruction of number sense for normal children at younger age (3-
6 yrs).
Figure 8.Adaptive E-Learning Level 13
The software tool used to investigate different causes and
The performance of the software was evaluated by Adaptive subtypes of dyscalculia. The software tool may be useful for
learning module .A report is generated at the end of each level and remediation of dyscalculia for children aged 7-8 and under. Few
consolidated report containing results of all the level is generated. aspects of software tool: speed deadlines, complexity, sound
The report contains percentage of marks and time in particular feedback, characters were found entertaining. The results indicate

7. that the software adapts well to varying levels of initial knowledge [4] Bradley S. Witzel, Christine J. Ferguson, and Dale S. Brown,
and learning speeds. Feedback from children, parents and teachers 2007, Developing early Number sense for students with
was positive. The tool may also be useful for general instruction disabilities, LD Online.
of normal preschool children. The learning algorithm reported is [5] David Kaplan, Leslie Nabors Ola´h, Nancy C. Jordan, and
highly general, and may be applied in other domains.Further this Maria N.Locuniak, 2006, Number sense Growth in kinder-
work can be extended using Touch Screen implementation, garten: A longitudinal Investigation of children at risk for
Children with age 5-7 have less or no knowledge in computer mathematics difficulties. Child Dev , Jan-Feb, 77(1), 153-75.
operation, Tool with Voice recognition for interactive learning,
Dyscalculia Assistant which is a talking calculator will be an [6] Daniel B.Berch, 2008, A Remedial Teaching program to help
appropriate tool for people with Dyscalculia. The synthesized children with mathematical disability.
voice output of a talking calculator provides feedback to the user [7] Griffin, Sharon, 2004, Building number sense with number
that helps them identify any input errors. Additionally, hearing the worlds, Early Childhood Research Quarterly, 19(1), 173-180.
calculated answer can provide a check against the transposition of
[8] Mary Rack, 2005, Learning Disabilities: A Handbook for
numbers commonly reversed in reading by people with Dyslexia
Instructors and Tutors, Sabbatical Project, Fall 2005.
or Dyscalculia.
[9] Regina G. Richards, 2008, Strategies to Facilitate math
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