Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS
WITH DYSCALCULIA
A Research presented to the Faculty of College of Teacher Education
Pamantasan ng Lungsod ng Marikina
In Partial Fulfillment of the Course
Action Research in Mathematics
College of Teacher Education
Major in Mathematics
September 2014
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Table of Contents
Title
…………………………………………..
1
Acknowledgment
…………………………………………..
4
Abstract
…………………………………………..
5
CHAPTER I: Introduction
…………………………………………..
7
Background of the Study
…………………………
9
Statement of the Problem
…………………………
12
…………………………………………..
13
Significance of the Study
………………………….
13
Scope and Delimitation
………………………….
14
Definition of Terms
………………………….
15
…………
17
Hypothesis
CHAPTER II: Review of Related Literature and Studies
Theoretical Framework
…………………………..
17
Conceptual Framework
…………………………..
20
Related Literature (Foreign)
…………………………..
21
Related Studies (Foreign)
………………………….
42
CHAPTER III: Research Methodology
………………………….
91
Method of the Research
………………………….
91
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Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Locale of the Study
………………………….
92
Respondents of the Study
………………………….
93
Instrumentation
…………………………………...
94
Data Gathering
……………………………………
95
CHAPTER IV: Presentation, Analysis, and Interpretation of Data …
96
CHAPTER V: Summary, Conclusions, and Recommendations
…
102
Summary
……………………………………………
102
Conclusions
……………………………………………
105
…………………………………..
Recommendations
APPENDIX
107
……………………………………………………. 108
…………………………………..
108
CURRICULUM VITAE
…………………………………………...
112
BIBLIOGRAPHY
……………………………………………
123
……………………………
123
SURVEY QUESTIONNAIRE
References and Bibliography
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Acknowledgment
This is a pleasure to express our sincere gratitude to all those who made
this thesis possible. First and foremost, we would have not finished this thesis
without the support of our adviser, Prof Ma. Teresa R. Abadam, who has always
been there for us whenever We need her, the encouragement she gave to keep
us going and her care to empower us which never fails all the time. Ma’am
Abadam, you taught us things beyond our understanding. Thank you for treating
us with respect and being a friend throughout our time of doing this thesis. You
really are a wonderful adviser. To you ma’am, we give you lots of sincere thanks
and respect. Thank you. To Professor Edgardo Canda who shared his valuable
time and gave us helpful information to finish this study. Thank you. To our
friends who supported us in our research work. We appreciated all the time and
advice you gave to us. Thank you. Especially, we would like to give special
thanks to our beloved family for their patient love, unflagging belief, and
dedication during the time of doing thesis and throughout our life. To all of you,
thanks for supporting us and always being there for us.
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Abstract
This study aimed to determine the effective teaching strategies in teaching
student with dyscalculia. Specifically the researcher aimed to answer the
following questions. What is the profile of the respondents and is there a
significance differences in terms age, gender, length of teaching, number of
seminars/Training attended. What are the different teaching strategies used by
teachers in teaching mathematics to students with dyscalculia?
Among the
different strategies which are commonly used by teachers to students with
dyscalculia? What are common problems encountered by teachers in teaching
mathematics to students with dyscalculia? What are the recommendations of
teachers in teaching Mathematics to students with dyscalculia?
Based on the findings of the study, the researchers found out that A
teacher with more seminar attended, experience and has a lot of length of time
in the service knows more effective teaching strategies in teaching mathematics
to students with dyscalculia. In addition, there is no significance difference when
it comes to gender.
The teachers use Direct Instruction, Collaboration,
Teaching with Manipulative, Experimentation, Questioning and Repetitions in
teaching mathematics to students with dyscalculia. The teacher commonly uses
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
experimentation and collaboration in teaching students with dyscalculia in
teaching students with dyscalculia. The teacher encountered many problems in
teaching mathematics to students with dyscalculia, this are the following: Short
attention, Often complete task/activity, Forgot the lesson, Repeating the
discussion more than twice, You have to prepare lots of exercises about the
subjects, Lack of knowledge of the disabilities of the child, Provide seminars
about teaching children with dyscalculia, They have difficulties in retaining
information/numbers, Short attention span, Some students do not want to repeat
the lesson, Some students hate math. Make the lesson interesting and base on
their needs, Some students are having a hard time understanding the lesson,
Some of them are not participating during the discussion, Negative outlook
towards math, Low confidence in terms of solving math problems, Some
students with dyscalculia are not receiving enough support and guidance from
their family, Can’t understand the importance of learning math, Too distract to
study, and Can’t see the relationship of learning math to real life situation.
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Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
CHAPTER I
Introduction
This chapter discusses the background of the study, the statement of the
problem, the hypothesis, the significance of the study, the scope and delimitation,
and definition of terms.
Mathematics is the study of topics such as quantity (numbers), structure,
space, and change. There is a range of views among mathematicians and
philosophers as to the exact scope and definition of mathematics.
Mathematicians seek out patterns and use them to formulate new
conjectures. Mathematicians resolve the truth or falsity of conjectures by
mathematical proof. When mathematical structures are good models of real
phenomena, then mathematical reasoning can provide insight or predictions
about nature. Through the use of abstraction and logic, mathematics developed
from counting, calculation, measurement, and the systematic study of the shapes
and motions of physical objects. Practical mathematics has been a human
activity for as far back as written records exists. The research required to solve
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mathematical problems can take years or even centuries of sustained inquiry.
As we speak, students are sitting in their various math classes tapping
their fingers impatiently, daydreaming and complaining aloud "I'm
not going to
use this stuff ever again in life!" They might be correct when it comes to their
specific responsibilities in the workplace, but not how they accomplish these
responsibilities. Sure, they might never graph linear equations, determine slope
and y-intercept or solve radical equations in a workday, but the cool math skills
they acquired while completing these problems will last a lifetime. Math shows
you that you can reach a desirable result if you will follow a certain series of
steps in a particular order, and complete each step without making an error. If
you find an error in your process, you can start over, making sure to alter your
methods at the moment you messed up the first time. Life doesn't allow you to
redo anything most of the time, but when it comes to stuff you do over and over
on a consistent basis, you're allowed to change things in between attempts.
For instance, consider what you do every morning to get ready for work or
school. If your process consists of waking up, getting ready, having breakfast and
going to work, you must complete each step successfully to develop a routine. If
you miss one step, your entire process will be thrown out of sync, compromising
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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your chances of satisfactorily getting everything else done.
Background of the Study
Mathematical skills are fundamental to independent living in a numerate
society, affecting educational, employment opportunities and thus socioeconomic status. An understanding of how concepts of numeracy develop, and
the manifestation of difficulties in the acquisition of such concepts and skills, is
imperative.
But many students, despite a good understanding of mathematical
concepts, are inconsistent at computing. They make errors because they
misread signs or carry numbers incorrectly, or may not write numerals clearly
enough or in the correct column. These students often struggle, especially in
primary school, where basic computation and "right answers" are stressed. Often
they end up in remedial classes, even though they might have a high level of
potential for higher-level mathematical thinking.
One fairly common difficulty experienced by people with math problems is
the inability to easily connect the abstract or conceptual aspects of math with
reality. Understanding what symbols represent in the physical world is important
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to how well and how easily a child will remember a concept. Holding and
inspecting an equilateral triangle, for example, will be much more meaningful to a
child than simply being told that the triangle is equilateral because it has three
equal sides. And yet children with this problem find connections such as these
painstaking at best.
Some students have difficulty making meaningful connections within and
across mathematical experiences. For instance, a student may not readily
comprehend the relation between numbers and the quantities they represent. If
this kind of connection is not made, math skills may be not anchored in any
meaningful or relevant manner. This makes them harder to recall and apply in
new situations.
For some students, a math disability is driven by problems with language.
These children may also experience difficulty with reading, writing, and speaking.
In math, however, their language problem is confounded by the inherently difficult
terminology, some of which they hear nowhere outside of the math classroom.
These students have difficulty understanding written or verbal directions or
explanations, and find word problems especially difficult to translate.
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A far less common problem -- and probably the most severe -- is the
inability to effectively visualize math concepts. Students who have this problem
may be unable to judge the relative size among three dissimilar objects.
This disorder has obvious disadvantages, as it requires that a student rely
almost entirely on rote memorization of verbal or written descriptions of math
concepts that most people take for granted. Some mathematical problems also
require students to combine higher-order cognition with perceptual skills, for
instance, to determine what shape will result when a complex 3-D figure is
rotated.
All of the difficulties mentioned a while ago are called dyscalculia.
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 due to their findings there is
a need to address dyscalculia as an important educational issue in mathematics
According to the Learning Disabilities Association of Minnesota(LDAM,
2005),pupils with dyscalculia may have complications computing and calculating
problems, identifying patterns in numbers, comprehending ideas and the
language of mathematics, and mastering methods and facts of mathematics.
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They may also have difficulty with problem solving skills and understanding
spatial representation concepts.
With this in mind, the researchers aim to present the effective teaching
strategies of Mathematics teachers to students with dyscalculia at St Joseph’s
College Quezon City SPED Department.
Statement of the Problem
This study aimed to determine the effective teaching strategies in teaching
student with dyscalculia.
Specifically the researcher aimed to answer the following questions:
1
2
Demographic profile of the respondents in terms of the following:
1.1
Age;
1.2
Educational Attainment
1.3
Length of teaching;
1.4
Number of seminars/Training attended.
What are the common teaching strategies applied in teaching mathematics
to students with dyscalculia?
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3
What are the feedbacks of the respondents in terms of most effective in
teaching Mathematics to students with dyscalculia?
4
What are the common problems encountered by teachers in teaching
mathematics to students with dyscalculia?
Hypothesis
The following are the null hypothesis of the study:
1. There is no different teaching strategies use by teachers in teaching
mathematics to students with dyscalculia.
2. There is no teaching strategy commonly used by teachers in teaching
mathematics to students with dyscalculia.
3. There are no common problems encountered by teachers in teaching
students with dyscalculia.
4. The study assumed that there is no significant difference among
Mathematics Teachers in the used of effective teaching strategies in terms
of gender, age, length of service and number of seminars and trainings
attended.
Significance of the study
The study aimed to benefit the following:
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Mathematic Teachers. This study will serve as a source of information to the
teachers to provide them information about effective strategies to students
encountering Math difficulties. This information may help identify students who
have problems in Mathematics.
Pre-Service Teachers. The study will be helpful to the graduating students who
will eventually become future teachers and can use the result of the study to suit
their needs.
Students. The study will be able to provide the students with to make used of the
different strategies to overcome Math difficulties.
Future Researchers. The proposed study can be used as guide and references.
Administrators. That the study will be able to provide programs to students
encountering Math problems and support provided to Mathematics teachers.
For Parents. That the study will provide information to assist their children at
home.
Scope and delimitation
This study only focused on determining the effective teaching strategies in
teaching Mathematics to students with dyscalculia. This study will be conducted
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
at St Joseph’s College of Quezon City, located at E. Rodriquez Avenue, Quezon
City who served students with different exceptionality particularly students with
dyscalculia.
Definition of Terms
Actions and Games- Including activities and games in your teaching strategy
will not only make it fun, but also interactive. It will open up the child and he/she
will increase self-effort. Use of actions (for example: using fingers for
multiplication) is another effective way to teach.
Chunking- Make a 'bundle' or 'team' of simpler steps and go through them one
by one.
Computer Time- Make use of the various math learning resources for children
with dyscalculia, available on the Internet. A lot of teaching resources are also
available in the form of software and CDs. It makes learning enjoyable and very
effective. You will find children eagerly waiting for the next 'computer time'.
Dyscalculia- is difficulty in learning or comprehending arithmetic, such as
difficulty in understanding numbers, learning how to manipulate numbers, and
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learning math facts. (en.wikipedia.org/Dyscalculia)
Flashcards- Write down one or two parts or steps of a complex concept or
problem on one flash card. Have the child read them, and once they are learned,
shuffle all the cards and ask him/her to arrange them in a sequence much like a
game.
Imagery- Children have a very active imagination. So in general, they remember
better when they can picturize or imagine something.
Learning disability- is a classification including several areas of functioning in
which a person has difficulty learning in a typical manner, usually caused by an
unknown factor or factors.
Math Anxiety- a feeling of tension, apprehension, or fear that interferes with
math performance. (Ashcraft 2002)
Mathematics Teachers- a person or thing that teaches something; especially: a
person whose job is to teach students about certain subjects
Simplification- Breaks down complex mathematical concepts into smaller and
easier parts. This will enable the child to understand faster and much better. For
this, you could use:
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Teaching strategies- method that a teacher uses to help students to accomplish
mastery of concepts.
CHAPTER II
Review of Related Literature and Studies
This chapter presents the related literature and studies that have bearing
on the present study to be conducted. This chapter also includes the theoretical
framework and the conceptual framework of the study.
Theoretical Framework
The study is based on the theory of Edward Thorndike.
Specifically, this study was guided by Edward Thorndike’s law of exercise, which
states that bonds between stimuli and responses are strengthened through
frequent and vigorous exercise. (Lefrancois 2000)
Law of Effect
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The law of effect was described by Thorndike in 1898. It holds that
responses to stimuli that produce a satisfying or pleasant state of affairs in a
particular situation are more likely to occur again in the same or similar situation.
Conversely, responses that produce a discomforting, annoying, or unpleasant
effect are less likely to occur again in a similar situation.
Thorndike’s second law is the law of exercise: “Any response to a situation
will, all other things being equal, is more strongly connected due to the number of
times it has been connected with that situation and to the average vigor and
duration of the connection” (Elliot et al, 1996).
Thorndike contended that these two laws can account for all behavior, no
matter how complex: thinking to mere secondary consequences of the laws of
exercise and effect.” Thorndike analyzed language as a set of vocal responses
learned because parent reward some of a child’s sounds but not others. The
rewarded ones are then acquired and the non-rewarded ones are unlearned,
following the law of effect.
These laws are important in understanding learning, especially in relation
to operant conditioning. However their status is controversial: particularly in
relation to animal learning, it is not obvious how to define a “satisfying state of
affairs” or an “annoying state of affairs” independent of their ability to induce
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instrumental learning, and the law of effect has therefore been widely criticized
as logically circular. In the study of operant conditioning, most psychologists have
therefore adopted B. F. Skinner’s proposal to define a reinforcer as any stimulus
which, when presented after a response (Microsoft Encarta Reference Library,
2004). On that basis, the law of effect follows tautologically from the definition of
is enforcer.
The law of effect, or influences of reinforcement, requires active
recognition by the subject. Since the effects presumably feedback to strengthen
an associative bond between a response and a stimulus, some mechanism or
principle of realization is needed for the subject to recognize whether the
reinforcement was satisfying or not. This problem, which still plagues
reinforcement theory, revolves around the need for the mediation of responseprocedure effects. Is some postulation of consciousness needed to adequately
deal with the judgmental realization order to act on reinforcement effect?
Thorndike suggested that this explanation is not supported, Thorndike’s
principles of repetition and reinforcement, in accounting for learning, are
accepted.
In his influential paper of 1970, Herrnstein proposed a quantitative
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Pamantasan ng Lungsod ng Marikina
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relationship between response rate (B) and reinforcement rate (Rf):
B = k Rf / (Rf0 + Rf)
Where k and Rf0 are constants. Herrnstein proposed that this formula, which he
derived from the matching law he had observed in studies of concurrent
schedules of reinforcement, should be regarded as a quantification of
Thorndike’s law of effect.
Conceptual Framework
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INPUT
• Respondents (High
School
MathematicsTeach
ers of St. Joseph
who are exposed in
teaching students
with dyscalculia)
• Demographic
Profile
• Age
• Length of
Teaching
Experience
• Highest Possible
Degree
• Trainings and
seminar attended
with regards to
Mathematics
• Latest Related
Researches
PROCESS
OUTPUT
• Different Teaching
Strategies
• Formulate
Questionaire
• Analysis and
Interpretation of
collected data
• Effective Teaching
Strategies
• Recommendations
of the respondents
• Improvement of the
Researcher
Related Literature
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Foreign
Teaching Strategies
Institutions of higher learning across the nation are responding to political,
economic, social and technological pressures to be more responsive to students’
needs, and are more concerned about how well students are prepared to
assume future societal roles. Faculty are already experiencing the pressure to
lecture less, to make learning environments more interactive, to integrate
technology into the learning experience, and to use collaborative learning
strategies when appropriate.
Some of the more prominent strategies are outlined below (Felder 2003).
Case Method. Providing an opportunity for students to apply what they
learn in the classroom to real-life experiences has proven to be an effective way
of both disseminating and integrating knowledge. The case method is an
instructional strategy that engages students in active discussion about issues
and problems inherent in practical application. It can highlight fundamental
dilemmas or critical issues and provide a format for role playing ambiguous or
controversial scenarios.
Course content cases can come from a variety of sources. Many faculties
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have transformed current events or problem reported through print or broadcast
media into critical learning exercises that illuminate the complexity of finding
solutions to critical social problems. The case study approach works well in cooperative learning or role playing environments to stimulate critical thinking and
awareness of multiple perspectives.
Discussion. There are a variety of ways to stimulate discussion. For
example, some faculties begin a lesson with a whole group discussion to refresh
the student’s list critical points or emerging issues, or generate a set of questions
stemming from the assigned reading(s). These strategies can also be used to
help focus large and small group discussions.
Obviously, a successful class discussion involves planning on the part of
the instructor and preparation on the part of the students. Instructors should
communicate this commitment to the students on the first day of class by clearly
articulating course expectations. Just as the instructor carefully plans the learning
experience, the students must comprehend the assigned reading and show up
for class on time, ready to learn.
Active Learning. Meyers and Jones (1993) define active learning as a
learning environments that allow “students to talk and listen, read, write, and
reflect as they approach course content through problem-solving exercises,
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informal small groups, stimulations, case studies, role playing, and other
activities- - all which require students to apply what they are learning” (p. xi).
Many studies show that learning is enhanced when students become actively
involved in the learning process. Instructional strategies that engage students in
the learning process stimulate critical thinking and a greater awareness of other
perspectives. Although there are times when lecturing is the most appropriate
method for disseminating information, current thinking in college teaching and
learning suggest that the use of a variety of instructional strategies can positively
enhance student learning.
Obviously, teaching strategies should be carefully matched to the teaching
objectives of a particular lesson. Assessing or grading students’ contributions in
active learning environments is somewhat problematic. It is extremely important
that the course syllabus explicitly outlines the evaluation criteria for each
assignment whether individual or group. Students need and want to know what is
expected of them.
Cooperative Learning is a systematic pedagogical strategy that
encourages small groups of students to work together for the achievement of a
common goal. The term ‘Collaborative Learning’ is often used as a synonym for
co-operative learning when, in fact, it is a separate strategy that encompasses a
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broader range of group interactions such as developing learning communities,
stimulating student/faculty discussions, and encouraging electronic exchanges
(Bruffee, 1993). Both approaches stress the importance of faculty and student
involvement in the learning process.
When integrating co-operative or collaborative learning strategies into a
course, careful planning are essential and critical to the achievement of a
successful co-operative learning experience understanding how to form groups,
ensure positive interdependence, maintain individual accountability, resolve
group conflict, develop appropriate assignments and grading criteria, and
manage active learning environments.
Integrating Technology. Today, educators realize that computer literacy
is an important part of a student’s education. Integrating technology into a course
curriculum when appropriate is proving to be valuable for enhancing and
extending the learning experience for faculty and students. Many faculties have
found electronic mail to be a useful way to promote student/student or
faculty/student communication between class meetings. Others use list serves or
on-line notes to extend topic discussions and explore critical issues with students
and colleagues, to increase student understanding of difficult concepts.
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What is dyscalculia?
Many of us have had some difficulty with mathematical concepts at one
time or another. This investigator had difficulty with geometric concepts and in
our society, struggling with mathematical concepts is often over looked, much as
with dyslexia years ago. Today, if someone is having difficulty reading, it is
usually addressed and intervention strategies and remediation plans are put in
place. This is not always the case for someone who struggles with mathematical
concepts. People who have difficulty with mathematical concepts may not know
what mathematical operation to use when completing an exercise. They may
not be able to count change for a parking meter. Mathematics is all around us,
whether following the speed limit on street signs, calculating a restaurant bill, or
keeping score at a football game. Imagine not being able to recognize numbers
or knowing what they mean. Welcome to the world of dyscalculia. Someone
who has dyscalculia faces these challenges every day.
According to the Learning Disabilities Association of Minnesota (LDAM)
(2005), pupils with dyscalculia may have complications computing and
calculating problems, identifying patterns in numbers, comprehending ideas
and the language of mathematics, and mastering methods and facts of
mathematics.
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Math Anxiety
It is a challenge to differentiate between a student who is simply
struggling in mathematics and a student who suffers from dyscalculia. Perhaps a
pupil may have a lack of motivation in mathematics or hasn’t been taught the
basic skills that are essential. A difficulty in mathematics could also be attributed
to a pupil’s fear of calculations or arithmetic. An individual’s performance on a
test and the understanding of mathematical concepts can be affected by these
deficiencies. A pupil who suffers from math anxiety can be fearful of
mathematics beyond childhood and in to adulthood. This can be very frightening
and stressful for a pupil, or adult, who may avoid mathematics altogether.
If a
pupil received poor instruction when introduced to mathematics, it can also be a
challenge. Poor teaching methods can also increase a pupil’s anxiety towards
mathematics.
If the pupil was pre disposed to a lack of understanding
mathematical concepts from the beginning due to poor instruction, this can
develop some very unpleasant thoughts from the child’s point of view towards
mathematics.
Beilock’s study (ascitedin Sparks, 2011) found that female elementary
teachers who had extreme math anxiety were more likely to pass it on to their
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female pupils. This resulted in lower test scores for the female pupils, which
affected their performance in mathematics and decreased their confidence in
their math ability. Ansari conducted some experiments on adults who had high
math anxiety. He found their ability to identify a change in numerical magnitude
was much lower than average (Sparks, 2011).
Someone who does not have
the capability to recognize a difference in numerical magnitude quickly is
considered dyscalculia. Dehaene (1997) believes pupils may have had a “false
start” which can lead to more fears. Pupils may have been taught there is no
real meaning or purpose to learning mathematics.
Therefore, they believe
mathematics is a subject they will never understand.
Educators can help
children overcome these fears by sharing personal situations they had with
anxiety and what they did to overcome it (Wadlington&Wadlington, 2008). They
can also assist them in understanding that mathematical operations have an
intuitive meaning.
Deficits of dyscalculia
A weakness in number sense is also common in adolescents who portray
traits of dyscalculia. Making sense of numbers, number fluency, and being able
to
perform
mathematics
in
your
head,
all
involve
number
sense
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(Gersten&Chard, 1999).
Butterworth, a professor at The Institute of Cognitive Neuroscience and
Psychology at the University College of London, believes that dyscalculia
students suffer from a lack of numerosity or numerical magnitude.
“The
capacity to represent and mentally manipulate numerosities is the key to
learning arithmetic” (Iuculano, Tang,Hall, &Butterworth, 2008, p. 669). A set
containing any number of items has numerosity and an array of squares or dots
would be a representation of numerosity.
Sub-types of Dyscalculia
Whether innate or learned, dyscalculia needs to be addressed and
remedial actions must be taken. Findings conclude that there may be different
types of dyscalculia and this makes it extremely difficult to design and
implement anyone specific program to address the disorder. In Rousselle and
Noel’s study(as cited in Spinney, 2009) children were asked to analyze two sets
of objects (for example, 5 sticks and 7 sticks) and had no problem comparing
the two and choosing the larger group, but when asked to circle the greater
number(5 or7), they were unable to do so.
Their ability to detect the
relationship between groups of objects proves a functional ANS. According to
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Spinney(2009), they concluded the problem was not the ANS itself, which may
work fine in some dyscalculia children; it was the ability to map symbols that are
numerical onto the ANS. With these findings, they believe some children who
have dyscalculia may have a damaged ANS and in other children it may be
complete. Since all dycalculics had been considered to be deficient in their
ANS, this study complicates the way to handle this disorder. Therefore, subtypes like these will make it very challenging to pinpoint the cause of dyscalculia
and develop specific screening methods for children.
Wadlington&Wadlington (2008) describe three subtypes of dyscalculia;
semantic, procedural, and visual-spatial. Semantic refers to difficulty with
memorization. For a pupil who suffers from semantic dyscalculia, it is best to
provide the pupil with visual aids. Procedural is difficulty with procedures and it
may be challenging for a pupil to follow steps when solving a problem. Finally,
visuo- spatial is difficulty with spatial representation concepts, such as place
value, which can be troublesome for a pupil.
Scientists continue to look for the root cause of the problem, but with
these sub-types of dyscalculia it may take longer than expected. This becomes
particularly problematic when coming up with programs to screen children.
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Currently most tests for dyscalculia rely “on the discrepancy between the child’s
IQ or general cognitive abilities and their scores in mathematics” (Spinney,
2009). This can be problematic since it may not fully recognize dyscalculia
tendencies and certainly will not identify which subtype exists. In conclusion,
scientists hope that one day each sub-type will be addressed.
Connection to other learning disabilities?
Approximately6%of the pupil population suffers from dyscalculia, which
is about the same number as dyslexics (LDAM, 2005). In many cases,
educators are just not aware of dyscalculia and therefore it is never addressed.
Educators may believe the pupil is just having difficulty in mathematics, but
more investigation is being implemented to identify and evaluate pupils who
suffer from dyscalculia. And until recently there hasn’t been much emphasis on
dyscalculia. More investigation has been done in the areas of dyslexia. If an
educator or parent detects a student is struggling in reading, a special
education teacher is notified and a team of school professionals meet to
discuss plans for remediation. A qualified professional may also be referred so
the pupil may be tested and appropriately diagnosed. A less common learning
disabilities dysgraphia, which causes a pupil to have difficulty with writing. If an
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educator or parent suspects a pupil is having difficulty with writing abilities, the
special education department is notified. If it seems like a severe case, the
family doctor should be notified who may bring in an occupational therapist to
administer tests to the pupil. Though dyslexia and dysgraphia are considered
language disabilities, they can also affect a pupil’s performance in mathematics.
According toWadlingtonandWadlington (2008) thesedisabilities can
prevent someone from learning concepts and vocabulary as well as the ability
to use operations and symbols. LDAM (2005) agrees, stating that people who
have disabilities in reading may also encounter problems in mathematics. If a
pupil is struggling with processing information, the pupil is also likely to have
difficulty with mathematical concepts. They also believe that dyscalculia is as
complicated and involved as dysgraphia and dyslexia. One study found that as
many as 17%of dyscalculia children are also dyslexic.
Fingeragnosiais
another
disability that may be
connected
with
dyscalculia. Someone who suffers from fingeragnosia cannot recognize or
name his or her own fingers (for example, finger vs. thumb).
One study
involved a male who was 58years old and had a stroke. After the cerebral
accident he suffered some language loss, had difficulty speaking, and had
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fingeragnosia. Discriminating between right and left was also a struggle. He
had a hard time with number concepts, phone numbers, and calculating money.
This is referred to as secondary dyscalculia which can happen from a deficit in
attention, language, or memory (Ardila, Concha,&Rosselli, 2000).
The outcome is a deficit in calculation. Prior to the accident, he was as
successful businessman who had no trouble with mathematical calculations.
The association between fingers and counting can be dated back to when we
were children. According to Ardilaet al. (2000), “a strong relationship between
numerical knowledge and fingergnosis(another name for agnosia) begins to
become evident and some commonality in brain activity or anatomy can be
expected. “Therefore, there might be a connection between the two.
Identifying dyscalculia
It is difficult to identify a pupil as dyscalculia. Pupils who are diagnosed
as dyscalculia may not necessarily be deficient in other areas. A pupil who is
dyscalculia can be healthy, intelligent, well-behaved, and perform well in other
subject areas. In fact, some pupils may excel in writing. Even though the pupil
works hard in the classroom and is always compliant and prepared, dyscalculia
can cause failure in mathematics (Montis, 2000). One particular pupil
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diagnosed with dyscalculia had an IQ that was above average, but had difficulty
reading an analogue clock and calculating her restaurant bill. The pupil was
somewhat relieved to find out why she struggled so much in her everyday life
but nevertheless very distraught. Children with a deficiency in the same areas
can receive one-to-one assistance or tutoring, but may still have trouble
understanding the mathematical world surrounding them. The use of concrete
materials, such as Cuisenaire rods and base ten blocks, can also be a struggle
for the pupil. Mathematical complications like these can be very frustrating and
perplexing to an educator. Ansari believes fluency in mathematics is crucial for
one to succeed socially and in terms of employment (Spinney, 2009).
Furthermore, it was reported by the British government in October of2008 that
dyscalculiamake100, 000 pounds (equivalent to $155,037in the U.S., as of
October3, 2011) less in in come over a lifespan.
Diagnosing Dyscalculia
Geary,(as cited in Wadlington and Wadlington, 2008) who has done
much investigation on the topics of mathematical disorders and disabilities,
believes there are no specific benchmarks in diagnosing dyscalculia.
Anywhere between 3-8% of school-age children have difficulty calculating and
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computing problems, understanding mathematical language, and mastering
facts of mathematics; all signs of dyscalculia. Ages of these pupils range from
6-14years. The United Kingdom Department for Education and Skills (as cited
in Rosselli, Matute, Pinto,&Ardila,2006)states that dyscalculia is present in
children who cannot comprehend number facts, concepts, and procedures.
What does an educator do if a student is displaying these signs? The
same procedure can be followed when diagnosing other disabilities. The pupil
can be referred to an educational psychologist and the special education
department. It is also crucial for parents to be involved. They should attend
meetings regarding this disability so that they may receive helpful tips for
remediation. Unfortunately, there are some limitations. Many professionals are
not aware of dyscalculia. After teaching13years in the public schools, it was
only by chance this investigator heard of dyscalculia. This investigator asked
the special educator and colleagues at her school if they knew anything about
dyscalculia. Most had not even heard of dyscalculia.
Educators should be trained to seek out deficiencies in mathematics and
how to detect if it is a mathematical disability. In the classroom, an educator
can get an idea of how a student is performing by a baseline assessment. If
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the student is struggling with mathematics, ongoing formal and informal
assessments are strongly encouraged.
Wadlingtonand Wadlington (2008)
believes formal and informal assessments are also necessary and should be
continuous. Formal assessments that are written should be easy for the pupil
to read and provide good spacing. Extra time, when completing the
assessment, is also advised. Informal assessments, such as observations,
should happen in an atmosphere where the pupil can feel at ease.
Observations can help an educator pinpoint deficiencies in mathematical skills.
An interview conducted by the educator is another method to gain some insight
into how pupils measure any strengths and weaknesses they may have in
mathematics. Pupils may also communicate and share helpful strategies that
enhance their performance (Wadlington&Wadlington, 2008).
Assessments
such as these can be very beneficial to the educator when planning instruction
that is appropriate for dyscalculic pupils. If the pupil is having difficulty on the
informal and formal assessments, a team of experts is advised. This team can
consist of a school psychologist, teachers, parents, and mathematicians. The
team can focus on intervention strategies and determine appropriate
placement for the pupil in the school.
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Screening Methods
One specific screening method is The Dyscalculia Screening Quiz.
This free, on-line quiz asks a series of questions regarding mathematical
concepts. For example, the test-taker is asked if he or she is able to make
change and tell time.
If the test taker has difficulty with concepts such as these, the test
concludes that the test-taker shows signs of dyscalculia, and should be tested
further for this disability or another learning disability. It also suggests websites
that may give some helpful tips on coping with dyscalculia. Silbert (2011), a
specialist in education, has awebsite(http://drlindasblog.com/about/dr-lindasilbert-150x185/) which also provides a dyscalculia screening quiz. There are
two sets of questions parents can ask their child. If the test-taker struggles with
mathematical operations (adding, subtracting, multiplying, and dividing),
number sense, keeping track of time and remembering mathematical formulas
and concepts, then the child maybe dyscalculic. The test- taker is again asked
questions regarding mathematical skills. If the child has difficulty with direction,
making change, sequencing events, and remembering the correct formation of
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numbers on an analog clock, then it is advised for parents to call a phone
number provided by the website for recommendations on what to do next. The
website also provides a free dyscalculia kit containing fun crafts, songs, and
games teaching mathematical concepts.
Treatments
Once a pupil demonstrates signs of dyscalculia and is evaluated, there
are accommodations and instructional methods an educator can provide. Some
of these may come as complete packages with a screening program as stated
previously. Michaelson (2007) suggests colored overlays which can help reduce
the glare for a pupil. It may be difficult for a pupil to see clearly when the black
print is on white paper. Using page breaks and bullet points can also help the
pupil visually. A particular font for pupils to read is also recommended. A sans
serif font, such as Arial or Tahoma, is an easier font for pupils with dyscalculia to
read,(Michealson, 2007).
Other methods which can be beneficial to dyscalculic pupils are the use
offline readers, which highlight the selected text; and the use of separate multistep problems which can be presented in small manageable steps (Michealson,
2007). Since the pupil can have difficulty with procedures, it may be helpful for
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the problems to be broken down. Large wall posters of basic concepts and flow
diagrams are also suggested. When planning lesson, educators need to explain
the meaning and importance of mathematics rather than focusing on
memorization. If the mathematical concept is explained to the pupil, the pupil
may understand it more clearly. The use of concrete materials and mathematical
manipulative, such as Cuisenaire rods and bean counters, can help pupils with
meaning. It is also crucial that educators accentuate skills that are practical
(Wadlington&Wadlington, 2007), such as telling time and counting money for
lunch. These particular skills incorporate mathematics and are not only beneficial
to the pupil in the present but the future as well.
Since dyscalculic pupils may need more time to solve problems,
educators have to consider this as they plan a lesson. It is also important that
time management and organization skills are taught.
These particular
accommodations can be very beneficial to the pupil. Wadlington and Wadlington
(2008) suggest other methods of instruction as well. Pupils who are dyscalculic
should always be seated in the front so their focus is centered on the educator.
Pupils should receive step by step instruction with specific concepts broken
down and additional time for them to complete a problem.
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Educators should provide appropriate textbooks and materials, taking into
account the needs of the pupil.
Another recommendation is to photocopy
specific problems from the mathematics text. Too much information on a page,
such as charts, tables, and diagrams, can confuse the pupil. By eliminating
these, it is easier for the pupil to read the text. Michaelson believes that even
though photocopying can be very time consuming for the educator, it will be
beneficial to the pupil. If possible, educators could implement small group
instruction into their lesson plans. If the educator has extra help in the
classroom, such as paraprofessional or parent volunteer, small group
instruction could enhance mathematical skills. The paraprofessional or parent
could focus on specific math concepts that give the pupil trouble.
Parental involvement is absolutely necessary for proper treatment of
dyscalculia. Parents and educators must communicate on a weekly basis so
parents can reinforce concepts taught in the classroom at home with their
children. With the use of beans or blocks, parents ‘can have their children
perform addition and subtraction problems and other skills in mathematics. A
mathematics tutor is also recommended for the pupil to receive one-to-one
assistance.
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Another treatment program which can be used is by Learning Link
Technologies. It provides an ‘At Home Program ‘which is a 12 month program
of exercises to treat dyscalculia and other disabilities. According to Harp, who
developed the program, treating dyscalculia involves balancing both sides of
the brain.
The dominance is found in the right hemisphere; therefore
dyscalculic children do not use the left side of their brain.
The program
provides exercises that will help balance both sides of the brain.
There are also free helpful tips for dyscalculic children given on the
website. Using colored pencils and blank sheets of paper when completing
mathematical exercises are just a few of the suggestions.
The right
hemisphere stays focused and busy when using color while the left
hemisphere works.
Using blank sheets of paper when doing mathematics
helps children from becoming distracted. Exercises for brain building, such as
playing baseball and martial arts, are also recommendations that can help
children with dyscalculia. Playing cards and board games can help good brain
building as well.
However, the ‘At Home Program’ does have a fee. To receive the12
month program on DVD, the cost is $70 a month, and to receive it on-line for a
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year, it is $50 a month. The program was developed by Harp, a learning
disability specialist and educator. Harp founded the Harp Institute in California
which has helped thousands of children with learning problems. The program
was so successful; she created a website to help children at home.
Living with Dyscalculia
How does an adult cope with dyscalculia on a daily basis? Fortunately,
with the help of technology today, many adults living with dyscalculia rely on
calculators and computers. When it comes to financial decision-making, they
may need assistance. For example, a dyscalculic may need help when buying a
car or a house. Since these purchases involve mathematics (calculating a
loan), a dyscalculic cannot figure this out alone. There are numerous websites
someone
with
dyscalculia
can
turn
to.
One
such
website
is
www.dyscalculiaforum.com. On this particular website, people struggling with
dyscalculia can relate to other dyscalculics and discuss strategies that are
helpful.
The
website
was
designed
for
dyscalculics
to
share
whattheyknowabout dyscalculia and inform others of this learning disability. The
Dyscalculia
Centrein
England
also
has
a
website
http://www.dyscalculia.me.uk/index.htm) which presents knowledge pertaining
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to dyscalculia. There are resources to help aid a child who suffers from this
learning disability.
Related Studies
Foreign
Mathematical skills are fundamental to independent living in a numerate society,
affecting educational opportunities, employment opportunities and thus socioeconomic status. An understanding of how concepts of numeracy develop, and
the manifestation of difficulties in the acquisition of such concepts and skills, is
imperative. The term Dyscalculia is derived from the Greek root ‘dys’ (difficulty)
and Latin ‘calculia’ from the root word calculus - a small stone or pebble used for
calculation. Essentially it describes a difficulty with numbers which can be a
developmental cognitive condition, or an acquired difficulty as a result of brain
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injury.
Dyscalculia is a specific learning difficulty that has also been referred to as
‘number blindness’, in much the same way as dyslexia was once described as
‘word blindness’. According to Butterworth (2003) a range of descriptive terms
have been used, such as ‘developmental dyscalculia’, ‘mathematical disability’ ,
‘arithmetic learning disability’,
‘number fact disorder’ and ‘psychological
difficulties in Mathematics’.
The Diagnostic and Statistical Manual of Mental Disorders, fourth edition
(DSM-IV ) and the International Classification of Diseases (ICD) describe the
diagnostic criteria for difficulty with Mathematics as follows:
Von Aster (2000) wrote about the state of dyscalculia, describing a study made at
the Department of Child and
Adolescent Psychiatry in Zürich, defining three subtypes of dyscalculia; The
Verbal subtype was found in children with the largest difficulty when counting,
especially when counting mentally, not using pen and paper. Subtraction was the
most problematic operation but also remembering methods of counting. Children
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with the Verbal subtypes also had other learning disabilities. In nine of eleven
children in the study, Von Aster found similar dyslexic conditions, and six of the
children also had Attention-Deficit/Hyperactivity Disorder (ADHD). The Arabic
subtype of dyscalculia included children with problems reading Arabic numbers
out loud and writing them after hearing the numbers, but these children had no
further learning disabilities. The third group of dyscalculia, called the Pervasive
subtype, included disabilities with most mathematical thinking, writing, spelling
and the children also possessed emotional and behavioral problems.
The latest study found refuting the existence of dyscalculia as a phenomenon
was made by Sjöberg in 2006. His opinion was that the results from previous
studies were inconclusive when pointing towards dyscalculia as an indication of
difficulties in mathematics. The idea that the pupils had not put enough effort and
time into their work with math was a more likely scenario. Sjöberg claimed that
the research results showing that 6 % of compulsory school pupils suffered from
dyscalculia were incorrectly interpreted. Sjöberg presented his conclusions in a
thesis where he studied 200 pupils from grade five in the Swedish compulsory
school to the second year of education in the upper secondary school during a
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six year period. Of these students, 13 were having specific mathematical
problems. Material was gathered regularly in which pupils filled in questionnaires.
A total of 100 classroom observations were made and 40 of them were video
recorded. On two occasions Sjöberg conducted in-depth interviews with the 13
pupils who had specific mathematical problems. Sjöberg noticed other
components that were possibly affecting the pupil’s low understanding of math
like a low work rate during the lessons, disturbance in the classroom
environment, large groups and also emotional stress in test situations. Some
students claim to get blackouts during tests due to experiencing a high stress
level.
The results of the study did not refute the existence of dyscalculia as a
phenomenon, but the findings made Sjöbergdraw the conclusion that diagnosing
dyscalculia should be exercised cautiously, if at all. All the students in his study
finished math studies in upper secondary school with satisfying results. This led
Sjöberg to the conclusion that in order to make a diagnose properly the whole
environment has to be examined, not only the pupil with the assumed
mathematical learning disorder.
Unlike Sjöberg; Shalev, Gross-Tsur et al. (2000) suggested prevalence of 3–6 %
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school children with dyscalculia. The range was interpreted as a consequence of
different definitions of developmental dyscalculia. “To determine prevalence we
must develop a scientific and clinical consensus as to what constitutes a learning
disability and which definition best describes the problem.” (Shalev, Gross-Tsur
et al., 2000). The definitions of dyscalculia were not precise and Shalev, GrossTsur et al. referred to different options like; “a specific, genetically determined
learning disability in a child with average intelligence” and “a learning disability in
mathematics, the diagnosis of which is established when arithmetic performance
is substantially below that expected for age, intelligence and education”.
The first study of dyscalculic prevalence was made by Koscin Bratislava in 1974.
Different mathematical assignments of basic character were used. Kosc
regarded the children with results below the 10th percentile as dyscalculic, which
was 6.4 % of the 375 fifth-graders participating in the study. Several other studies
have been made in other countries showing similar figures; In Germany by
Badian (1983), Klauer (1992), in Switzerland by Von Aster (1997), in England by
Lewis, Hitch and Walker (1994). The margin of errors in these studies were
affected by reading difficulties, dyslexia, ADHD and other disabilities, since the
children with these difficulties had a tendency to show poorer arithmetical skills
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than those only having dyscalculia. Shalev and Gross-Tsur (2000) acknowledged
that other authors emphasized other reasons than brain dysfunctions for
dyscalculia; lower social and economic status, mathematical anxiety, large
classes and less well thought-through curricula and teaching.
Kadosh4 (2007) discovered through tests that a specific part of the brain was
associated with automatic magnitude processing5. Kadosh was using functional
Magnetic Resonance
Imaging (fMRI) in order to show that the intraparietal sulcus (IPS), illustrated in
figure A, had a role in the ability to recognize numbers. In order to cope with
ambiguous results regarding left and right side of the brain, a trans cranial
magnetic stimulation (TMS) was used to block the energy activation of the brain
on one side at a time. When blocking the right side of the brain, the test persons
appeared to obtain dyscalculic behavior, as opposed to the persons that were
being tested without the blocking, where there was no appearance of dyscalculic
behavior. The experimenters also took into consideration that other areas of the
brain that were communicating at the same time could be affected by the TMS.
But according to careful studies of the fMRI no other areas showed any activity
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during the automatic magnitude processing.
However, the researchers did not claim to have found the cause of dyscalculia
after this test, since it was not made clear whether developmental dyscalculia
and the effects produced through TMS could be classified as identical deficits.
The experimental procedures were carefully described in the report. It is
important to notice that only five subjects were tested. This study confirmed the
results of earlier studies, one of them made by Butterworth (2006), also using
fMRI and identifying the IPS as the center responsible for the handling of number
information.
Figure A: Right side of the brain and the intraparietal sulcus (IPS) Rubinstein and
Henik (2009) regarded the IPS findings as surprising. The reasons they stated
were heterogeneous behavioral deficits, comorbidity7 and number processing
represented in more than one brain area. Developmental disorders often
generate multiple problems, according to Rubinstein and Henik. How is it
possible to sort out the dyscalculia? Rubinstein and Henik focused on three
different views; single restricted biological deficit, cognitive deficits due to
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instances
of
biological
damage
and
neurocognitive
damage
causing
developmental dyscalculia which might result in 3 several, not related, behavioral
disorders. Rubinstein and Henik considered brain dysfunction as a possible
reason for dyscalculia and stated that evidence has been found that IPS was
involved in attention and related cognitive processes. Injury in the IPS area could
thus result in dyscalculia. The authors recommended future researchers to
examine the whole brain, since developmental disorders were considered
heterogeneous.
Focusing
on
single
brain-behavior
deficits
exclusively
could
prevent
understanding of the variety of deficits connected to developmental dyscalculia
and MLD, yet standardized tests of arithmetic computation could be a helpful
screening tool to detect and separate dyscalculia from MLD, according to
Rubinstein and Henik (2009).Geary et al. (2009) referred to other authors (GrossTsur, Manor, &Shalev, 1996; Kosc, 1974; Ostad, 1998; Shalev, Manor, & GrossTsur, 2005) when discussing predictions of the percentage of children diagnosed
with a mathematical learning disorder, which was found to be in the range from 5
to 10 %. It was regarded critical on multiple levels that a diagnosis should be
made at an early stage. The reason to that was for measures to be taken early in
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order to help the child develop their mathematical abilities in the future. The early
stages of mathematical learning were essential for the later outcome in the
education. The authors pointed out that the causes for mathematical learning
disorders were still under investigation, even if some conclusions had been
drawn about the main areas contributing to the disorder. Comparisons between
normally achieving children and children with a MLD have been made.
The latter group’s counting strategies were less mature, their understanding of
counting was not fully developed and they had continuous difficulties learning
math and recalling basic arithmetic facts stored in the long-term memory.
Working memory has come to develop a central role in the area of MLD.
Children with MLD probably have a dysfunction in the basic recognition of
numbers and their magnitude (Geary et al.,2009). The Number Sets Tests were
designed by Geary et al. (2009) to measure the speed and correctness of the
basic number and quantity recognition in children. The further developed test
was meant to serve as a quick screening to find sensitivity for numbers and
predict MLD. Participants in the test were 228 children from kindergarten, first,
second and third grade. In the analysis were proficiency scores, IQ in the first
year, working memory and mathematical cognition test scores taken into
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consideration. The IQ scores were measured using the test by Wechsler
Abbreviated Scale of Intelligence (WASI;Wechsler, 1999). Examples of
understanding numbers are given in figure B.
Figure B: A part of the Number Sets Test (Geary et al., 2009) Number estimation
was assessed through number lines; a blank line with two endpoints; 0 and 100.
The assignment was to mark on the line where the number presented should be
placed. The score was defined as the absolute difference between the marking of
the number and the correct position of the number. The overall score was
calculated as the mean of these differences across the trials. Measuring the
children’s counting ability was made by letting the child look at a puppet counting
red and blue chips. The child had to tell if the puppet had made a correct
calculation, and not double-calculated.
Geary et al. (2009) noted that children with MLD made errors throughout the test
when the first chip was counted twice like “one – one – two – three – etc.”. The
score for this part of the test was calculated as a percentage of the number of
times the child successfully identified wrong calculations. To further asses the
children’s addition strategies, a mix of simple and complex addition problems
was shown one at a time. Each problem should be solved as quickly and
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correctly as possible. The child could use any strategy to get the answer, but
without pen and paper and the answer was to be told out loud. Geary et al.
(2009) classified the trials into six different categories; specifically, counting
fingers, fingers8, verbal counting, retrieval, decomposition9 or other/mixed
strategy. The percentage of correct direct-retrieval trials for simple problems was
correlated with the mathematics achievement scores, and used in the analysis by
Geary et al. (2009). The Working Memory Test Battery for Children10 was
composed by nine subtests in order to assess “the central executive11,
phonological loop and visual spatial sketchpad” (Geary et al., 2009). The tests
had six items from one to six to one to nine, where four were to be remembered
to get to the next level. To get to the next level after that, the numbers
remembered increased by one. The test ended when the child failed three times
in a row. The central executive was investigated through three different tests;
Listening Recall, Counting Recall and Backward Digit Recall. The idea of the first
test was to let the child listen to a sentence, then determine if the sentence was
true or false and then repeat the last word in a series of sentences. The second
test then demanded of the child to count a set of dots on a card, maximum
seven, and then remember the number of counted dots at the end of a series of
cards. The third test, Backward Digit Recall is a standard format backward digit
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span12 (Geary et al., 2009). The Phonological loop was examined with Digit
Recall, Word List Recall, and Nonword List Recall, for instance by repeating
words spoken by the experimenter in the same order. Series of words were
presented to the child in the Word List Matching task. Then the words were
presented again, maybe in another order, and the child had to decide whether
the second list had the same order as the first list. The Visual spatial sketchpad
was tried with other span tasks13, like Block Recall. A board was set up with nine
one-sided numbered blocks in a “random” arrangement; the numbers could only
be seen by the experimenter. Then the experimenter touched series of blocks
and the child should repeat the order. Mazes Memory task was conducted like
this; the child got a picture of a maze with more than one solution. A picture of an
identical maze with one solution drawn was presented. The child’s task was to
replicate the solution when the picture was removed. After a successful trial, the
maze was increased with one wall. The tests were conducted every term for
each age group, except for the children in kindergarten who were tested one time
in spring. The location of the tests was at the children’s schools, most of the time
in a quiet place. Some of the children’s tests were executed on the university
campus or in a mobile testing van. A score below the 15th percentile on the
mathematics achievement test characterized the child as having a MLD. Results
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between the 15th and 30th percentiles were being considered as “low achieving”
(Geary et al., 2009).
Pupils with results above the 30th percentile were considered normally achieving.
No comments were made on high achieving children in the test. To find the most
accurate measure to predict third grade mathematics achievement, all measures
were compared in 4 independent regressions. The central executive and
sensitivity measures predicted 25 % in the variation; the number line scores
predicted 27 %. The number line measure appeared to somewhat assess the
children’s intuitive understanding of numerical quantity. The researchers claimed
that their results provided initial support for their hypothesis. Competencies
assessed by the sensitivity score were unrelated to reading achievement and
achievement in general or to IQ or working memory (Geary et al., 2009). Of the
non-MLD children in third grade 96 % were correctly identified in first grade.
According to Geary et al. (2009) the Number Sets Test was a potential screening
tool for discovering children prone to MLD at an early stage. The results from the
test in first grade could be used to detect 67 % of children risking MLD at the end
of third grade and correctly identify nearly 90 % of non-MLD children.
Benefits of early identification of children at risk for MLD would be early remedial
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services and low costs. The authors emphasized that the Number Sets Test was
not yet ready for use as a diagnostic tool since the test was not normed and it
could not be used for making predictions later than the third grade. To enable
further tests, the authors were willing to provide the test on request.
DSM-IV 315.1
‘Mathematics Disorder’
Students with a Mathematics disorder have problems with their math skills. Their
math skills are significantly below normal considering the student’s age,
intelligence, and education.
As measured by a standardized test that is given individually, the person's
mathematical ability is substantially less than you would expect considering age,
intelligence and education. This deficiency materially impedes academic
achievement or daily living. If there is also a sensory defect, the Mathematics
deficiency is worse than you would expect with it. Associated Features:
Conduct disorder
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Attention deficit disorder
Depression
Other Learning Disorders
Differential Diagnosis: Some disorders have similar or even the same
symptoms. The clinician, therefore, in his/her diagnostic attempt, has to
differentiate against the following disorders which need to be ruled out to
establish a precise diagnosis.
WHO ICD 10 F81.2
‘Specific disorder of arithmetical skills’
Involves a specific impairment in arithmetical skills that is not solely explicable
on the basis of general mental retardation or of inadequate schooling. The
deficit concerns mastery of basic computational skills of addition, subtraction,
multiplication, and division rather than of the more abstract mathematical skills
involved in algebra, trigonometry, geometry, or calculus.
However it could be argued that the breadth of such a definition does not
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account for differences in exposure to inadequate teaching methods and / or
disruptions in education as a consequence of changes in school, quality of
educational provision by geographical area, school attendance or continuity of
teaching staff. A more helpful definition is given by the Department for Education
and Skills (DfES, 2001) A condition that affects the ability to acquire arithmetical
skills. Dyscalculic learners may have difficulty understanding simple number
concepts, lack an intuitive grasp of numbers, and have problems learning
number facts and procedures. Even if they produce a correct answer or use a
correct method, they may do so mechanically and without confidence.’
Blackburn (2003) provides an intensely personal and detailed description of the
dyscalculic experience, beginning her article:
“For as long as I can remember, numbers have not been my friend. Words are
easy as there can be only so many permutations of letters to make sense.
Words do not suddenly divide, fractionalize, have remainders or turn into
complete gibberish because if they do, they are gibberish. Even treating numbers
like words doesn’t work because they make even less sense. Of course numbers
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have sequences and patterns but I can’t see them. Numbers are slippery.”
Public understanding and acknowledgement of dyscalculia arguably is at a level
that is somewhat similar to views on dyslexia 20 years ago.
Therefore the
difference between being ‘not good at Mathematics’ or ‘Mathematics anxiety’ and
having a pervasive and lifelong difficulty with all aspects of numeracy, needs to
be more widely discussed.
The term specific learning difficulties describes a
spectrum of ‘disorders’, of which dyscalculia is only one. It is generally accepted
that there is a significant overlap between developmental disorders, with multiple
difficulties being the rule rather than the exception.
Investigating brain asymmetry and information processing, Hugdahl and
Westerhausen (2009) claim that differences in spacing of neuronal columns and
a larger left planum temporal result in enhanced processing speed. They also
state that the evolution of an asymmetry favoring the left hand side of the brain is
a result of the need for lateral specialization to avoid ‘shuffling’ information
between hemispheres, in response to an increasing demand on cognitive
functions. Neuroimaging of dyslexic brains provides evidence of hemispherical
brain symmetry, and thus a lack of specialization. McCrone (2002) also argues
that perhaps the development of arithmetical skills is as artificial as learning to
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read, which may be problematic for some individuals where the brain ‘evolved for
more general purposes’.
Study of triple code
Dehaene (1992) and Dehaene.and Cohen (1995, 1997) suggest a ‘triple-code’
model of numerosity, each code being assigned to specific numerical tasks. The
analog magnitude code represents quantities along a number line which requires
the semantic knowledge that one number is sequentially closer to, or larger or
smaller than another; the auditory verbal code recognizes the representation of a
number word and is used in retrieving and manipulating number facts and rote
learned sequences; the visual Arabic code describes representation of numbers
as written figures and is used in calculation. Dehaene suggests that this is a
triple processing model which is engaged in mathematical tasks.
Historically, understanding of acquisition of numerical skills was based on
Piaget’s pre-operational stage in child development (2 – 7 years). Specifically
Piaget argues that children understand conservation of number between the
ages of 5 – 6 years, and acquire conservation of volume or mass at age 7 – 8
years.
Butterworth (2005) examined evidence from neurological studies with
respect to the development of arithmetical abilities in terms of numerosity – the
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number of objects in a set.
Research evidence suggests that numerosity is
innate from birth (Izard et al, 2009) and pre-school children are capable of
understanding simple numerical concepts allowing them to complete addition and
subtraction to 3. This has significant implications as “….the capacity to learn
arithmetic – dyscalculia – can be interpreted in many cases as a deficit in the
child’s concept of numerosity” (Butterworth, 2005).
Butterworth provides a
summary of milestones for the early development of mathematical ability based
on research studies.
Geary and Hoard (2005) also outline the theoretical pattern of normal early years
development in number, counting, and arithmetic compared with patterns of
development seen in children with dyscalculia in the areas of counting and
arithmetic.
Counting
The process of ‘counting’ involves an understanding of five basic principles
proposed by Gelman and Gallistel (1978):

one to one correspondence - only one word tag assigned to each
counted object
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
stable order - the order of word tags must not vary across counted
sets

cardinality - the value of the final word tag represents the quantity
of
items counted

abstraction - objects of any kind can be counted

order-irrelevance - items within a given set can be counted in any
sequence
In conjunction with learning these basic principles in the early stages of
numeracy, children additionally absorb representations of counting ‘behavior’.
Children with dyscalculia have a poor conceptual understanding of some aspects
of counting rules, specifically with order-irrelevance (Briars and Siegler, 1984).
This may affect the counting aspect of solving arithmetic problems and
competency in identifying and correcting errors.
Arithmetic
Early arithmetical skills, for example calculating the sum of 6 + 3, initially may be
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computed verbally or physically using fingers or objects, and uses a ‘counting-on’
strategy. Typically both individuals with dyscalculia and many dyslexic adults
continue to use this strategy when asked to articulate ‘times tables’ where they
have not been rote-learned and thus internalized. Teaching of number bonds or
number facts aid the development of representations in long term memory, which
can then be used to solve arithmetical problems as a simple construct or as a
part of more complex calculation. That is to say the knowledge that 6 + 3 and 3 +
6 equal 9 is automatized.
This is a crucial element in the process of decomposition where computation of a
sum is dependent upon a consolidated knowledge of number bonds.
For
example where 5 + 5 is equal to 10, 5 + 7 is equal to 10 plus 2 more. However
this is dependent upon confidence in using these early strategies; pupils who
have failed to internalize such strategies and therefore lack confidence tend to
‘guess’.
As ability to use decomposition and the principles of number facts or
bonds becomes automatic, the ability to solve more complex problems in a
shorter space of time increases. Geary (2009) describes two phases of
mathematical competence: biologically primary quantitative abilities which are
inherent competencies in numerosity, ordinality, counting, and simple arithmetic
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enriched through primary school experiences, and biologically secondary
quantitative abilities which are built on the foundations of the former, but are
dependent upon the experience of Mathematics instruction (Appendix 2).
In the same way that it is impossible to describe a ‘typical’ dyslexic profile, in that
individuals
may
experience
difficulties
with
reading,
spelling,
reading
comprehension, phonological processing or any combination thereof, similarly a
dyscalculic profile is more complex than ‘not being able to do Mathematics’.
Geary and Hoard (2005) describe a broad range of research findings which
support the claim that children with dyscalculia are unable to automatically
retrieve this type of mathematical process. Geary (1993) suggests three possible
sources of retrieval difficulties:
‘….a deficit in the ability to represent phonetic/semantic information in longterm memory…….. and a deficit in the ability to inhibit irrelevant associations
from entering working memory during problem solving (Barrouillet et al.,
1997). A third potential source of the retrieval deficit is a disruption in the
development or functioning of a ……cognitive system for the representation
and
retrieval
of
arithmetical
knowledge,
including
arithmetic
facts
(Butterworth, 1999; Temple & Sherwood, 2002).’
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Additionally responses tend to be slower and more inaccurate, and difficulty at
the most basic computational level will have a detrimental effect on higher
Mathematics skills, where skill in simple operations is built on to solve more
complex multi-step problem solving.
Emerson (2009) describes difficulties with number sense manifesting as severely
inaccurate guesses when estimating quantity, particularly with small quantities
without counting, and an inability to build on known facts. Such difficulty means
that the world of numbers is sufficiently foreign that learning the ‘language of
Mathematics’ in itself becomes akin to learning a foreign language.
Behavioral
Competence in numeracy is fundamental to basic life skills and the
consequences of poor numeracy are pervasive, ranging from inaccessibility of
further and higher education, to limited employment opportunities: few jobs are
completely devoid of the need to manipulate numbers.
Thus developmental
dyscalculia will necessarily have a direct impact on socio-economic status, selfesteem and identity.
Research by Hanich et al (2001) and Jordan et al (2003) claim that children with
mathematical difficulties appear to lack an internal number line and are less
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skilled at estimating magnitude.
This is illustrated by McCrone (2002) with
reference to his daughter:
“A moment ago I asked her to add five and ten. It was like tossing a ball to a
blind man. “Umm, umm.” Well, roughly what would it be? “About 50…or 60”,
she guesses, searching my face for clues. Add it up properly, I say. “Umm,
25?” With a sigh she eventually counts out the answer on her fingers. And
this is a nine-year old.
The problem is a genuine lack of feel for the relative size of numbers. When
Alex hears the name of a number, it is not translated into a sense of being
larger or smaller, nearer or further, in a way that would make its handling
intuitive. Her visual spatial abilities seem fine in other ways, but she
apparently has hardly any capacity to imagine fives and tens as various
distances along a mental number line. There is no gut felt difference between
15 and 50. Instead their shared “fiveness” is more likely to make them seem
confusingly similar.”
Newman (1998) states that difficulty may be described at three levels:

Quantitative dyscalculia - a deficit in the skills of counting and
calculating
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
Qualitative
dyscalculia
-
the
result
of
difficulties
in
comprehension of instructions or the failure to master the skills
required for an operation. When a student has not mastered the
memorization of number facts, he cannot benefit from this stored
"verbalizable information about numbers" that is used with prior
associations to solve problems involving addition, subtraction,
multiplication, division, and square roots.

Intermediate dyscalculia – which involves the inability to
operate with symbols or numbers.
Trott and Beacham (2005) describe it as:
“a low level of numerical or mathematical competence compared to
expectation. This expectation being based on unimpaired cognitive and
language abilities and occurring within the normal range. The deficit will
severely impede their academic progress or daily living. It may include
difficulties recognizing, reading, writing or conceptualizing numbers,
understanding numerical or mathematical concepts and their inter-
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relationships.
It follows that dyscalculics may have difficulty with numerical operations, both
in terms of understanding the process of the operation and in carrying out the
procedure. Further difficulties may arise in understanding the systems that
rely on this fundamental understanding, such as time, money, direction and
more abstract mathematical, symbolic and graphical representations.”
Butterworth (2003) states that although such difficulties might be described at the
most basic level as a condition that affects the ability to acquire arithmetical
skills, other more complex abilities than counting and arithmetic are involved
which include the language of Mathematics:
 understanding number words (one, two, twelve, twenty …), numerals
(1, 2, 12, 20) and the relationship between them;
 carrying out mental arithmetic using the four basic arithmetical
operations – addition, subtraction, multiplication and division;
 completing written multi-digit arithmetic using basic operations;
 solving ‘missing operand problems’ (6 + ? = 9);
 solving arithmetical problems in context, for example handling money
and change.
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Trott (2009) suggests the following mathematical difficulties which are also
experienced by dyslexic students in higher education:
Arithmetical
• Problems with place value
• Poor arithmetical skills
• Problems moving from concrete to abstract
Visual
• Visual perceptual problems reversals and substitutions e.g. 3/E
or +/x
• Problems copying from a sheet, board, calculator or screen
• Problems copying from line to line
• Losing the place in multi-step calculations
• Substituting names that begin with the same letter, e.g.
integer/integral, diagram/diameter
• Problems following steps in a mathematical process
• Problems keeping track of what is being asked
• Problems remembering what different signs/symbols mean
• Problems remembering formulae or theorems
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Memory
• Weak short term memory, forgetting names, dates, times,
phone numbers etc.
• Problems remembering or following spoken instructions
• Difficulty listening and taking notes simultaneously
• Poor memory for names of symbols or operations, poor
retrieval of vocabulary
Reading
• Difficulties reading and understanding Mathematics books
• Slow reading speed, compared with peers
• Need to keep re-reading sentences to understand
• Problems understanding questions embodied in text
Writing
• Scruffy presentation of work, poor positioning on the page,
changeable handwriting
• Neat but slow handwriting
• Incomplete or poor lecture notes
• Working entirely in pencil, or a reluctance to show work
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General
• Fluctuations in concentration and ability
• Increased stress or fatigue
However a distinction needs to be drawn between dyscalculia and maths phobia
or anxiety which is described by Cemen (1987) as ‘a state of discomfort which
occurs in response to situations involving mathematics tasks which are perceived
as threatening to self-esteem.’ Chinn (2008) summarizes two types of anxiety
which can be as a result of either a ’mental block’ or rooted in socio-cultural
factors.
’Mental block anxiety may be triggered by a symbol or a concept that
creates a barrier for the person learning maths. This could be the
introduction of letters for numbers in algebra, the seemingly irrational
procedure for long division or failing to memorise the seven times
multiplication
facts.
[...]
Socio-cultural
maths
anxiety
is
a
consequence of the common beliefs about maths such as only very
clever (and slightly strange) people can do maths or that there is only
ever one right answer to a problem or if you cannot learn the facts you
will never be any good at maths.’
According to Hadfield and McNeil (1994) there are three reasons for
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Mathematics anxiety: environmental (teaching methods, teacher attitudes and
classroom experience), intellectual (influence of learning style and insecurity over
ability) and personality (lack of self confidence and unwillingness to draw
attention to any lack of understanding). Findings by Chinn (2008) indicate that
anxiety was highest in Year 7 (1st year secondary) male pupils, which arguably is
reflective of general anxiety associated with transition to secondary school.
Environmental
Environmental factors include stress and anxiety, which physiologically affect
blood pressure to memory formation. Social aspects include alcohol consumption
during pregnancy, and premature birth / low birth weight which may affect brain
development.
Isaacs, Edmonds, Lucas, and Gadian (2001) investigated low
birth-weight adolescents with a deficit in numerical operations and identified less
grey matter in the left IPS.
Assel et al (2003) examined precursors to mathematical skills, specifically the
role of visual-spatial skills, executive processing but also the effect of parenting
skills as an environment influence.
The research measured cognitive and
mathematical abilities together with observation of maternal directive interactive
style. Findings supported the importance of visual-spatial skills as an important
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early foundation for both executive processing and mathematical ability. Children
aged 2 years whose mothers directed tasks as opposed to encouraging
exploratory and independent problem solving, were more likely to score lower on
visual–spatial tasks and measures of executive processing.
This indicates the
importance of parenting environment and approach as a contributory factor in
later mathematical competence.
1.3 Assessment
Shalev (2004) makes the point that delay in acquiring cognitive or attainment
skills do not always mean a learning difficulty is present. As stated by Geary
(1993) some cognitive features of the procedural subtype can be remediated and
do not necessarily persist over time. Difficulties with Mathematics in the primary
school are not uncommon; it is the pervasiveness into secondary education and
beyond that most usefully identifies a dyscalculic difficulty.
A discrepancy
definition stipulates a significant discrepancy between intellectual functioning and
arithmetical attainment or by a discrepancy of at least 2 years between
chronologic age and attainment.
However, measuring attainment in age
equivalencies may not be meaningful in the early years of primary age range, or
in the later years of secondary education.
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Wilson et al (2006) suggest that assessment of developmental symptoms should
examine number sense impairment. This would include:
‘reduced understanding of the meaning of numbers, and a low performance
on tasks which depend highly on number sense, including non-symbolic
tasks (e.g. comparison, estimation or approximate addition of dot arrays), as
well as symbolic numerical comparison and approximation’.
They add that performance in simple arithmetical calculation such as subtraction
would be a more sensitive measure, as addition and multiplication is more open
to compensatory strategies such as adding or counting on, and memorization of
facts and sequences.
Assessment instruments
As yet there are few paper-based dyscalculia specific diagnostic.
Existing
definitions state that the individuals must substantially underachieve on
standardized tests compared to expected levels of achievement based on
underlying ability, age and educational experience. Therefore assessment of
mathematical difficulty tends to rely upon performance on both standardized
mathematical achievement and measurement of underlying cognitive ability.
Geary and Hoard (2005) warns that scoring systems in attainment tests blur the
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identification of specific areas of difficulty:
‘Standardized achievement tests sample a broad range of arithmetical and
mathematical topics, whereas children with MD often have severe deficits in
some of these areas and average or better competencies in others. The
result of averaging across items that assess different competencies is a level
of performance […] that overestimates the competencies in some areas and
underestimates them in others.’
Von
Aster
(2001)
developed
a
standardized
arithmetic
test,
the
Neuropsychological Test Battery for Number Processing and Calculation in
Children, which was designed to examine basic skills for calculation and
arithmetic and to identify dyscalculic profiles. In its initial form the test was used
in a European study aimed at identifying incidence levels (see section 1.4).
It
was subsequently revised and published in English, French, Portuguese,
Spanish, Greece, Chinese and Turkish as ZarekiR, This test is suitable for use
with children aged 7 to 13.6 years and is based on the modular system of
number processing proposed by Dehaene (1992).
Current practice for assessment of dyscalculia is referral to an Educational
Psychologist. Trott and Beacham (2005) claim that whilst this is an effective
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assessment method where students present with both dyslexic and dyscalculic
indicators, it is ineffective for pure dyscalculia with no co-morbidity. Whilst there
is an arithmetical component in tests of cognitive ability such as the Weschler
Intelligence Scale for Children (WISC) and the Weschler Adult Intelligence Scale
(WAIS), only one subtest assesses mathematical ability. Two things are needed
then:
an accurate and reliable screening test in the first instance, and a
standardized and valid test battery for diagnosis of dyscalculia.
Standardized tests
A review of mathematical assessments was conducted through formal
psychological test providers Pearson Assessment and the Psychological
Corporation.
The following describe tests that are either fully available or have
limited availability, depending upon the qualifications of the test user.
Test of Mathematical Abilities-Second Edition (TOMA-2)

Administration time: 60-90 minutes

Standard scores percentiles, and grade or age equivalents providing a
Mathematics quotient

Age Range: 8 to 18.11 years
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Five norm-referenced subtests, measuring performance in problems and
computation in the domains of vocabulary, computation, general Information
and story problems. An additional subtest provides information on attitude
towards Mathematics.
Reliability coefficients are above .80 and for the Math Quotient exceed .90.
Wide Range Achievement Test 4 (WRAT 4)

Administration time: approximately 35-45 minutes for individuals ages
8 years and older

Standard scores percentiles, and grade or age equivalents providing a
Mathematics quotient

Age Range: 5 to 94 years
Measures ability to perform basic Mathematics computations through
counting, identifying numbers, solving simple oral problems, and calculating
written mathematical problems. Reliability coefficients are above .80 and for
the Math Quotient exceed .90.
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1.
Wechsler Individual Achievement Test - Second UK Edition (WIAT-II UK)

Administration: Individual - 45 to 90 minutes depending on the age of
the examinee

Standard scores percentiles, and grade or age equivalents providing a
Mathematics quotient

Age Range: 4 to 16 years 11 months. Standardized on children aged 4
years to 16 years 11 months in the UK. However, adult norms from the U.S
study are available from 17 to 85 years by simply purchasing the adult scoring
and normative supplement for use with your existing materials.
Measures ability in numerical operations and mathematical reasoning. Strong
inter-item consistency within subtests with average reliability coefficients
ranging from .80 to .98.
1.
Mathematics Competency Test
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
Purpose: To assess Mathematics competency in key areas in order to
inform teaching practice.

Range: 11 years of age to adult

Administration: 30 minutes – group or individual
Key Features:

Australian norms

Provides a profile of mathematical skills for each student

Identifies weaknesses and strengths in Mathematics skills

Open ended question format

Helpful in planning further teaching programs

Performance based on reference group or task interpretation
Assessment Content:

Using and applying Mathematics

Number and algebra

Shape and space

Handling data
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Provides a quick and convenient measure of Mathematics skills, a skills
profile as well as a norm-referenced total score. The skills profile allows
attainments to be expressed on a continuum from simple to complex, making
the test suitable for a wide range of purposes and contexts, in schools,
colleges, and pre-employment. The test utilizes 46 open-ended questions,
presented in ascending order, and is easy to score.
Strong reliability with internal consistency of 0.94 for the full test
Validated against 2 tests with a correlation co-efficient of 0.83 and 0.80
Working memory as an assessment device
Working Memory (WM) can be described as an area that acts as a storage space
for information whilst it is being processed. Information is typically ‘manipulated’
and processed during tasks such as reading and mental calculation. However
the capacity of WM is finite and where information overflows this capacity,
information may be lost. In real terms this means that some learning content
delivered in the classroom is inaccessible to the pupil, and therefore content
knowledge is incomplete or ‘missing’. St Clair-Thompson (2010) argues that
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these gaps in knowledge are ‘strongly associated with attainment in key areas of
the curriculum’.
Alloway (2001) conducted research with 200 children aged 5 years, and claims
that working memory is a more reliable indicator of academic success. Alloway
used the Automated Working Memory Assessment (AWMA) and then re-tested
the research group six years later. Within the battery of tests including reading,
spelling and Mathematics attainment, working memory was the most reliable
indicator.
Impairment,
Similarly recent findings with children with Specific Language
Developmental
Coordination
Disorder
(DCD),
Attention-
Deficit/Hyperactivity Disorder, and Asperger’s Syndrome (AS) also support these
claims.
Alloway states that the predictive qualities of measuring WM are that it tests the
potential to learn and not what has already been learned. Alloway states that ‘If
a student struggles on a WM task it is not because they do not know the answer;
it is because their WM ‘space’ is not big enough to hold all the information’.
Typically, children exhibiting poor WM strategies under-perform in the classroom
and are more likely to be labeled ‘lazy’ or ‘stupid’.
She also suggests that
assessment of WM is a more ‘culture fair’ method of assessing cognitive ability,
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as it is resistant to environmental factors such as level of education, and socioeconomic background. The current version of AWMA has an age range of 4 to
22 years.
In a review of the literature on dyscalculia, Swanson and Jerman (2006) draw
attention to evidence that deficits in cognitive functioning are primarily situated in
performance on verbal WM. Currently there is no pure WM assessment for adult
learners, however Zera and Lucian (2001) state that processing difficulties
should also form a part of a thorough assessment process.
Rotzer et al (2009)
argue that neurological studies of functional brain activation in individuals with
dyscalculia have been limited to:
‘…….number and counting related tasks, whereas studies on more general
cognitive domains that are involved in arithmetical development, such as
working memory are virtually absent’.
This study examined spatial WM processes in a sample of 8 – 10 year old
children, using functional MRI scans. Results identified weaker neural activation
in a spatial WM task and this was confirmed by impaired WM performance on
additional tests. They conclude that ‘poor spatial working memory processes
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may inhibit the formation of spatial number representations (mental number line)
as well as the storage and retrieval of arithmetical facts’.
Computerized assessment
The Dyscalculia Screener (Butterworth, 2003) is a computer-based assessment
for children aged 6 – 14 years that claims to identify features of dyscalculia by
measuring response accuracy and response times to test items. In addition it
claims to distinguish between poor Mathematics attainment and a specific
learning difficulty by evaluating an individual’s ability and understanding in the
areas of number size, simple addition and simple multiplication. The screener
has four elements which are item-timed tests:
1. Simple Reaction Time
Tests of Capacity:
2. Dot Enumeration
3. Number Comparison (also referred to as Numerical Stroop)
Test of Achievement:
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4. Arithmetic Achievement test (addition and multiplication)
Speed of response is included to measure whether the individual is responding
slowly to questions, or is generally a slow responder.
The Mathematics Education Centre at Loughborough University began
developing a screening tool known as DyscalculiUM in 2005 and this is close to
publication. The most recent review of development was provided in 2006 and is
available
from
http://Mathematicstore.gla.ac.uk/headocs/6212dyscalculium.pdfThe screener is
now in its fourth phase with researchers identifying features as:

Can effectively discriminate dyscalculia from other SpLDs such as
Asperger’s Syndrome and ADHD

Is easily manageable

Is effective in both HE and FE

Can be accommodated easily into various screening processes

Has a good correlation with other published data, although this
data is competency based and not for screening purposes

Can be used to screen large groups of students as well as used
on an individual basis
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1.4 Incidence
The lack of consensus with respect to assessment and diagnosis of dyscalculia,
applies equally to incidence. As with dyslexia, worldwide studies describe an
incidence ranging from 3% to 11%, however as there is no formalized method of
assessment such figures may be open to interpretation.
Research by Desoete et al (2004) investigated the prevalence of dyscalculia in
children based on three criterion: discrepancy (significantly lower arithmetic
scores than expected based on general ability), performance at least 2 SD below
the norm, and difficulties resistant to intervention. Results indicated that of 1,
336 pupils in 3rd grade (3rd class) incidence was 7.2% (boys) and 8.3% (girls),
and of 1, 319 4th grade (4th class) pupils, 6.9% of boys and 6.2% of girls.
Koumoula et al. (2004) tested a sample population of 240 children in Greece
using the Neuropsychological Test Battery for Number Processing and
Calculation in Children, and a score of <1.5 SD was identified in 6.3%of the
sample. Findings by Von Aster and Shalev (2007) in a sample population of 337
Swiss children reported an incidence of 6.0 % using the same assessment
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method and criterion.
Mazzocco and Myers (2003) used multiple tests of
arithmetic skills (Key Math Subtests, Test of Early Math Ability, and Woodcock-
Johnson Revised Math Calculations) together with a criterion of persistent
diagnosis across more than one school year.
Incidence rates for 3 rd grade
children fell between 5% and to 21%.
Findings from cross-cultural studies indicate that incidence is more prevalent in
boys than girls, the risk ratio being 1.6 to 2.2. In terms of co-morbidity with other
specific learning difficulties, studies by Gross-Tsur et al (1996), Barbaresi et al
(2005) and Von Aster and Shalev (2007) provide evidence of a coexisting
reading difficulty, the percentages across all three studies falling at 17%, 56.7%
and 64%.
Additionally, a greater number of children with dyscalculia exhibit
clinical behavior disorders than expected.
Barbaresi et al (2005) investigated the incidence of Mathematics learning
disorder among school-aged children, via a population-based, retrospective, birth
cohort study. The research study used a population sample of all children born
between 1976 and 1982.
Data was extracted from individually administered
cognitive and achievement tests together with medical, educational, and
socioeconomic information. Findings identified a cumulative incidence rate of
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Mathematics disorder by age 19 years within a range of 5.9% to 13.8%. The
results suggest that dyscalculia is common among school children, and is
significantly more frequent among boys than girls.
This level of incidence
reflects a similar incidence of dyslexia, which is identified as being between 4%
and 10% of the population.
1.5 Intervention
At a neurological level, St Clair-Thompson (2010) states that remediation of WM
would enhance performance in academic progress. She suggests that memory
strategy training and practice in memory tasks are effective intervention tools.
This might include adjustments to the teaching environment such as repetition of
material in a variety of formats, breaking down tasks into smaller units, and use
of memory techniques. Research into the use of computer programs such as
‘Memory Booster’ (Leedale et al, 2004) whilst demonstrating improved WM
performance, does not confirm that they can enhance or improve academic
attainment (St Clair-Thompson et al, 2010; Holmes et al, 2009).
Wilson et al (2006) developed and trialed software designed to remediate
dyscalculia, called ‘The Number Race’. The underlying rationale of this system is
the presence of a "core deficit" in both number sense and accessing such a
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sense through visual symbolic representation. The program claims to remediate
difficulties using mathematical problems which are adaptive to the age and ability
level of the child. The software was piloted with a small sample of 7–9 year old
French children with mathematical difficulties, for 30 minutes a day over 5 weeks.
Children were tested pre and post intervention on tasks measuring counting,
transcoding, base-10 comprehension, enumeration, addition, subtraction, and
symbolic and non-symbolic numerical comparison.
Whilst the sample exhibited
increased performance on core number sense tasks such as subtraction
accuracy, there was no improvement in addition and base-10 comprehension
skills. However this is the first step in a series of clinical trials to build on this
program.
Sharma (1989) argues that Mathematics should be considered as a separate,
symbolic ‘language’ system and teaching should reflect this. Specifically, that
terminology, vocabulary and syntax of mathematical language must be taught
strategically to ensure understanding of mathematical concepts, to underpin
learning of mathematical methods.
Sharma also makes the point that
consideration should be given to inclusive teaching principles, methods and
materials to address difficulties at every level. She suggests five critical factors
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in delivering the Mathematics curriculum effectively:
1.
Assessment of mathematical knowledge and strategies used by
the learner to determine teaching methodology.
2.
Assessment
and
identification
of
learning
style
(whether
quantitative or qualitative) and recognition that this is unique to the
individual. For example quantitative learners may favor learning
the procedural aspect of Mathematics, and to deduce answers
from having learned general mathematical principles. Qualitative
learners are more dependent upon seeing
parallels and
relationships between elements.
3.
Assessment of seven ‘pre-Mathematics’ skills:
 Sequencing
 Direction and laterality
 Pattern recognition
 Visualization
 Estimation
 Deductive reasoning
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 Inductive reasoning
4.
Specific teaching of mathematical language and syntactical
variations, for example that 33 – 4 is the same as ‘subtract 4 from
33’ and 4 less than 33’.
5.
A systematic approach to the introduction and teaching of new
mathematical concepts and models.
A detailed discussion of these factors is available in Appendix 4. The consensus
on guidelines for effective intervention can be summarized as follows:
1. Enable visualization of Mathematics problems.
Provide pictures,
graphs, charts and encourage drawing the problem.
2. Read questions / problems aloud to check comprehension. Discuss
how many parts / steps there may be to finding the solution.
3. Provide real life examples.
4. Ensure that squared / graph paper is used to keep number work and
calculation.
5. Avoid fussy and over-detailed worksheets; leave space between
each question so that pupils are not confused by questions that seem
to merge together.
6. Teach over-learning of facts and tables, using all senses and in
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particular rhythm and music. Warning: meaningless repetition to
learn facts off by heart does not increase understanding.
7. Provide one-to-one instruction on difficult tasks. If a pupil does not
understand, re-frame and re-word the question / explanation
8. Use a sans serif font in minimum 12 point.
9. Provide immediate feedback and provide opportunities for the pupil
to work through the question again. Encourage opportunities to see
where an error has occurred.
10. In early stages of Mathematics teaching, check that the pupil has
understood the syntactical variations in Mathematics language.
Encourage the pupil to verbalize the problem stages, for example:
‘To do this I have to first work out how many thingies there are and
then I can divide that number by the number of whatsits to find out
how many each one can have.’
11. Allow more time to complete Mathematics work.
12. Ask the pupil to re-teach the problem / function to you.
Whilst Sharma (1989) highlights the language of Mathematics as key in the
building of foundation skills, critically, in the NCCA Report (2005) only 17.2% or
primary teachers identified the use of Mathematics language as an effective
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strategy in the teaching of Mathematics skills, and only 10.7% reported linking
Mathematics activities to real life situations. Butterworth (2009) suggests four
basic principles of intervention:
• Strengthen simple number concepts
• Start with manipulables and number words
• Only when learner reliably understands relationship between number
words and concrete exemplars, progress to numeral symbols
• Structured teaching program designed for each learner
Technological aids tend to be limited to tool such as calculators, which include
talking calculators and enlarged display screens, buttons and keypads.
There
are a plethora of computer programs (Appendix 5) on the market which claim to
improve the underlying cognitive skills associated with reading, spelling and
number.
However caution should be exercised with regard to computerized
training. Owen et al (2010) researched the efficacy of brain training exercises
conducting an online study with more than 11,000 participants.
Whilst
performance of all participants in improved over time on the experiment, retesting on the initial performance tests indicated that ‘these benefits had not
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generalized, not even when the training tests and benchmark tests involved
similar cognitive processes’.
CHAPTER III
RESEARCH METHODOLOGY
This chapter deals with the methods of research used whether it may be
historical,
descriptive,
and
experimental
or
a
case
study.
The
techniques used under Descriptive Research Method as well as
t h e data gathering tools and analytical tools used will be further explained in this
chapter as well as the methods used in developing the software and for
evaluation.
A. The Method of the Research
The
researchers
have
used
the
Descriptive
Research
Method wherein the study is focused on present situations. It
involves
the
recording,
description,
analysis
and
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presentation of the present system, composition or processes
of phenomena. Under the Descriptive Research Method, the
technique used is the Survey Method, which is otherwise known
as normative survey. The results and findings of the study
should always be compared with the standards. W ith the survey
method, researchers are able to statistically study the specific
areas
where
the
proponents
must
concentrate.
Findings
regarding the common practices being done and the methods
which are commonly adopted by the teachers are obtained with
the use of the survey method.
B. Locale of the Study
The study was conducted at St. Joseph College, 295 E. Rodriguez Sr.
Blvd Quezon City.
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History of St. Joseph’s College of Quezon City
St. Joseph’s College of Quezon City was founded 75 years ago as St.
Joseph’s Academy by Dutch Franciscan Sisters. Situated along España
Extension, the school admitted its first primary school pupils in 1932 and drew
children from the rapidly growing communities of New Manila, Kamuning and
San Juan. Under the leadership of its first school directress, Mother
MagdalaVerhuizen, the academy opened the high school department the
following year. During the Japanese occupation, the school was closed down, the
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Dutch sisters interned in Los Baños, and the buildings were used as a minimilitary hospital by the Japanese army and later by the US military. SJA officially
became St. Joseph’s College of Quezon City in 1948 with the opening of the
college department which offered programs in education, liberal arts, secretarial
science and music.
C. Respondents of the study
The researcher used purposive sampling in choosing their respondents.
The researchers conducted a survey on 5 teachers with specific experience,
knowledge, skills, and exposure on teaching Mathematics to15 students with
Dyscalculia at St. Joseph College Quezon City SPED Department.
D. Instrumentation
Statistical Treatment
The Likert scale was used to interpret items in the questionnaire. These
responses were based on the list of effective teaching strategies on teaching
Mathematics. There were instances that the respondents are asked to rate the
strategies. The range and interpretation of the five-point scale are shown in Table
2.
The Five-point Likert Scale
Scale
Range
Interpretation
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5
4.20-5.00
4
3.40-4.19
3
2.60-3.39
2
1.80-2.59
1
1.00-1.79
Most Effective
Effective
Moderately Effective
Less Effective
Not Effective
Percentage was used to express values between zeros to 1.
P = N/T
Where:
N - Number of response
T - Total number of response
Weighted mean was used to measure the general response of the survey
samples, whether they agree to a given statement or not.
The formula in computing weighted mean is as follows:
W= fx/xt
Where:
w- weighted mean
f – Weight given to each
response
x – Number of responses
xt – total number of responses
The survey result was analysed with the use of statistical approach and Microsoft
Excel spread sheets.
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E. Data Gathering
The researchers formulate a survey questionnaire based on the effective
teaching strategies in teaching mathematics. The researchers make a contact
and give the survey questionnaire with the potential 5 teachers with specific
experience, knowledge, skills, and exposure on teaching Mathematics to 5
students with Dyscalculia at St. Joseph College Quezon City SPED Department.
The feedback had been received from period August 3 – 14, 2014. The results
have been organized at Microsoft excel spreadsheet with the code that has been
developed that measures the attitudes from the data of the survey results.
CHAPTER IV
Presentation, Analysis, and Interpretation of Data
This chapter presents the data gathered from the respondents, analysis and
interpretation.
Table 1. Demographic profile of the respondents in terms of the following:
Age
Gender
Educational
attainment
A
22
Female
Bse-Math
B
20
Female
Bachelor of
Science in
Special
Education
C
25
Male
BEEDSped
D
24
Male
BSEdMath
E
66
Female
MA in
Mathematics
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Length Of
Teaching
Numbers of
Seminars
Attended
3 Years
1 Years
5 Years
4 years
41 Years
5
5
10
8
More than a
Thousand
Table 1.1 AGE
AGE
20-29
30-49
50-59
60-69
Total
Number of Respondents
4
0
0
1
5
Percentage
80%
0%
0%
20%
100%
The table shows that 80 % (4) of the respondents are in the stage range of 20 –
29 which are still young and newly exposed in teaching students with dyscalculia,
and 20% which is 1of the respondents at the age between 60 – 69 yrs. has a lot
of experiences and taught students with dyscalculia. It shows that only few
teachers had the patience and techniques to teach students with dyscalculia.
Table 1.2 Educational attainment
Educational attainment
Number of respondents
Percentage
Bachelor of Science in
2
40 %
Special Education
BSEd-Math
3
60%
Total
5
100%
The table shows that the range of numbers of SPED major (40%) and
Mathematics major (60%) who are teaching students with dyscalculia are not far
from each other’s.
The table only shows that in teaching students with dyscalculia, you don’t only
need the ability in teaching and solving mathematics, you also need to
understand students with this kind of difficulty and that’s why you need to
collaborate with SPED major to modify/acquire teaching techniques for students
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with dyscalculia.
Table 1.3 Length of teaching
Length of teaching
1-10
11-20
21-30
31-40
41-50
Number of respondents
4
0
0
0
1
Percentage
80%
0%
0%
0%
20%
The table shows that 80 % of the respondents had a 1 to 10 years in experienced
in teaching students with dyscalculia. And only 20% had stay in the field of
teaching students with dyscalculia with the range of year for 41- 50 years.
Table 1.4 Numbers of seminars/trainings attended
Numbers of seminars
attended.
1 to 20
More than 20
Total
Number of respondents
4
1
5
80%
20%
100%
The table shows that 80 % of the respondents had attended seminars but they
are still now in the field of teaching students with difficulty in math but had
experiences in teaching.
The table shows that only few teachers can handle and has a patience to teach
students with dyscalculia.
Table 2.Common Teaching Strategies applied in teaching Mathematics to
students with dyscalculia.
Teaching Strategies
1. Experimentation
2. Collaboration
3.
4. Repetitions
Number of response
5
5
Percentage
20.83%
20.83%
4
16.67%
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5. Direct Instruction
6. Questioning
7. Teaching with
Manipulative
8. Mnemonic
Total
3
3
2
12.5%
12.5%
8.3%
2
24
8.3%
100%
The table only shows that you need to try all the possible strategies which you
think are effective in teaching students with dyscalculia. But what are really
mentioned that has a high percentage are the experimentation and collaboration
which gathered 20.83% both.
Table 3. Feedback of the respondents in terms of most effective in teaching
mathematics to students with dyscalculia
Scale
5
4
3
2
1
Teaching Strategies
Direct Instruction
1. Explaining concept
Rating Scale
4.21 – 5.00
3.41 – 4.20
2.61 – 3.40
1.81 – 2.60
1.00 – 1.80
Description
Most Effective
Effective
Moderately Effective
Less Effective
Not Effective
Numbers of
respondents
answered
1
2
3 4 5
3
2. Modeling procedures
3. Guiding students
1
Weighted Description
mean
2
3
2
1
3
3.4
4
4.6
Collaboration
1. Small group
1
4
3.2
2. Peer tutoring
1
1
3
3
Moderately
Effective
Effective
Most
Effective
Moderately
Effective
Moderately
Effective
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3. Think pair share
1
1
2
1
3.2
Teaching with
Manipulative
1. Using graph/chart
2. Using geometric
shapes and
manipulative
3. Jigsaw Puzzle
2
1
2
2
3
2
3
4
1.
2.
3.
1.
2.
3.
Questioning
Questioning choices
Scaffolding
3 Steps interview
Repetitions
Activities after
teaching concepts
Solve a certain kind
of problem (practice)
Practice
Effective
Most
Effective
4.6
4.6
Experimentation
1. Solve novel problem
2. Learning Center
3. Real life applications
Moderately
Effective
2
1
3
4
3.6
3.8
5
Effective
Effective
Most
Effective
4
4
3.8
Effective
Effective
Effective
5
1
5
5
4
1
4
1
1
3
1
4
Most
Effective
Effective
3.8
4.4
4.8
Most
Effective
Most
Effective
Mnemonic
1. Using symbols
4 1
4.2
Effective
2. Using visual
1 4
Most
representations
4.8
Effective
3. Enhance
3 2
Most
meaningfulness
Effective
(concrete meaningful
examples)
4.4
The table shows that Modeling procedures, Guiding students, Using graph/chart,
Using geometric shapes and manipulative, Jigsaw Puzzle, Solve novel problem,
Learning Center, Real life applications, Questioning choices, Scaffolding, 3 Steps
interview, Activities after teaching concepts, Enhance meaningfulness (concrete
meaningful examples), Practice, Using symbols, Using visual representations
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and Solving a certain kind of problem (practice) are the effective and most
effective teaching strategies in teaching mathematics to student with dyscalculia.
Table 4. Problems respondents encountered by teachers in teaching
Mathematics to students with dyscalculia.
A.
1. Short attention
2. Often complete task/activity
3. Forgot the lesson
4. Repeating the discussion more than twice
B.
1. You have to prepare lots of exercises about the subjects.
2. Lack of knowledge of the disabilities of the child.
3. Provide seminars about teaching children with dyscalculia.
C.
1. They have difficulties in retaining information/numbers.
2. Short attention span.
3. Some students do not want to repeat the lesson.
4. Some students hate math. Make the lesson interesting and base on their
needs.
5. Some students are having a hard time understanding the lesson
D.
1. Some of them are not participating during the discussion.
2. Negative outlook towards math.
3. Low confidence in terms of solving math problems
4. Poor comprehension in word problems.
5. Some students with dyscalculia are not receiving enough support and
guidance from their family.
E.
1. Can’t understand the importance of learning math
2. Too distract to study
3. Can’t see the relationship of learning math to real life situation.
The table shows that there are many and different problems teachers will
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encountered to the students with dyscalculia.
CHAPTER V
Summary, Conclusions and Recommendations
This chapter presents the summary, conclusions and recommendations.
Summary
This study attempted to assess the effective teaching strategies in
teaching mathematics to students with dyscalculia.
Specifically, it sought answers to the following questions:
1. Demographic profile of the respondents and is there a significance
differences in terms of the following:
1.1
Age;
1.2
Educational Attainment;
1.3
Length of teaching;
1.4
Number of seminars/Training attended.
2. What are the common teaching strategies applied in teaching mathematics to
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students with dyscalculia?
3. What are the feedbacks of the respondents in terms of most effective in
teaching Mathematics to students with dyscalculia?
4. What are the common problems encountered by teachers in teaching
mathematics to students with dyscalculia?
This study made used the Descriptive Research Method wherein the study
is focused on present situations. It involves the recording, description, analysis
and the presentation of the present system, composition or processes of
phenomena. Under the Descriptive Research Method, the technique used is the
Survey Method, which is otherwise known as normative survey. The results and
findings of the study should always be compared with the standards. With the
survey method, researchers are able to statistically study the specific areas
where the proponents must concentrate. Findings regarding the common
practices being done and the methods which are commonly adopted by the
teachers are obtained with the use of the survey method.
The researcher used purposive sampling in choosing their respondents.
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The researchers conducted a survey on 5 teachers with specific experience,
knowledge, skills, and exposure on teaching Mathematics to15 students with
Dyscalculia at St. Joseph College Quezon City SPED Department.
The Likert scale was used to interpret items in the questionnaire. These
responses were based on the list of effective teaching strategies on teaching
Mathematics. There were instances that the respondents are asked to rate the
strategies.
The researchers formulate a survey questionnaire based on the effective
teaching strategies in teaching mathematics. The researchers make a contact
and give the survey questionnaire with the potential 5 teachers with specific
experience, knowledge, skills, and exposure on teaching Mathematics to 5
students with Dyscalculia at St. Joseph College Quezon City SPED Department.
The feedback had been received from period August 3 – 14, 2014. The results
have been organized at Microsoft excel spreadsheet with the code that has been
developed that measures the attitudes from the data of the survey results.
80 % (4) of the respondents are in the stage range of 20 – 29 which are
still young and newly exposed in teaching students with dyscalculia, and 20%
which is 1of the respondents at the age between 60 – 69 yrs. has a lot of
experiences and taught students with dyscalculia. It shows that only few teachers
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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had the patience and techniques to teach students with dyscalculia. Numbers of
SPED major (40%) and Mathematics major (60%) who are teaching students
with dyscalculia are not far from each other. Only few teachers can handle and
has a patience to teach students with dyscalculia. Modeling procedures, Guiding
students, Using graph/chart, Using geometric shapes and manipulative, Jigsaw
Puzzle, Solve novel problem, Learning Center, Real life applications, Questioning
choices, Scaffolding, 3 Steps interview, Activities after teaching concepts,
Enhance meaningfulness (concrete meaningful examples), Practice, Using
symbols, Using visual representations and Solving a certain kind of problem
(practice) are the effective and most effective teaching strategies in teaching
mathematics to student with dyscalculia. There are many problems encountered
by the teachers when they teach mathematics to students with dyscalculia.
Conclusions
The following conclusions were attained based on the findings of the study:
1. A teacher with more seminar attended, experience and has a lot of length of
time in the service knows more effective teaching strategies in teaching
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mathematics to students with dyscalculia. In addition, there is no significance
difference when it comes to gender.
2. The teachers use Direct Instruction, Collaboration, Teaching with
Manipulative, Experimentation, Questioning and Repetitions in teaching
mathematics to students with dyscalculia.
3. The teacher commonly uses experimentation and collaboration in teaching
students with dyscalculia in teaching students with dyscalculia.
4. The teacher encountered many problems in teaching mathematics to
students with dyscalculia, this are the following: Short attention, Often complete
task/activity, Forgot the lesson, Repeating the discussion more than twice, You
have to prepare lots of exercises about the subjects, Lack of knowledge of the
disabilities of the child, Provide seminars about teaching children with
dyscalculia, They have difficulties in retaining information/numbers, Short
attention span, Some students do not want to repeat the lesson, Some students
hate math. Make the lesson interesting and base on their needs, Some students
are having a hard time understanding the lesson, Some of them are not
participating during the discussion, Negative outlook towards math, Low
confidence in terms of solving math problems, Some students with dyscalculia
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are not receiving enough support and guidance from their family, Can’t
understand the importance of learning math, Too distract to study, and Can’t
see the relationship of learning math to real life situation.
Recommendations
1. Mathematics Teachers should employ different teaching strategies which
were found effective based on the researches done.
2. Pre-Service Teachers should attend trainings and seminars on teaching
principles and strategies for more effective teaching.
3. Students must be taught strategies on how they would be able to acquire and
remember mathematical skills and concepts.
4. Future Researchers to replicate same research but more on how students
with dyscalculia experiences in dealing with mathematics.
5. Administrators should implement and encourage teachers to use the
teaching strategies as part of faculty development program. In addition,
administrator should employ trainings and seminars on teaching principles and
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strategies in handling children with dyscalculia.
6. Parents should provide extra time and provides additional exercises to
enhance math skills, care and understanding for their children with dyscalculia.
APPENDIX
SURVEY QUESTIONAIRE
1.
Name: (optional)
1.1 Age:
1.2 Educational Attainment:
1.3 Educational Attainment:
1.4 Length of teaching:
1.5 Number of seminars attended:
2. List down common teaching strategies you used in teaching Math to students
with dyscalculia
1.
2.
3.
4.
5.
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3. Kindly check among the different teaching strategies you find most effective in
teaching math to children with dyscalculia.
Legend:
5- Most effective
4- Effective
3- Moderately effective
2- Less effective
1- Not effective
Teaching Strategies
1
2
3
4
5
Direct Instruction
1. Explaining concept
2. Modeling procedures
3. Guiding student
Collaboration
1. Small group
2. Peer tutoring
3. Think Pair share
Teaching with
Manipulatives
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Pamantasan ng Lungsod ng Marikina
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1. Using graph/chart
2. Using geometric
shapes, and
manipulative
3. Jigsaw puzzles
Experimentation
1. Solve novel
problem prior to
presenting
concepts/lessons
2. Learning centers
3. Real life
applications
Questioning
1. Questioning
choices
2. Scaffolding
3. 3 steps Interview
Repetitions
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1. Activities after
teaching concepts
2. Solve a certain
kind of problem
(practice)
3. Practice
Mnemonic
1. Using symbols
2. Using Visual
representations
3. Enhance
meaningfulness
(concrete
meaningful
examples)
4. What are the common problems you encountered in teaching Math to
students with dyscalculia?
1.
2.
3.
4.
5.
5. What are your recommendations/suggestions for future Math teachers in
teaching students with dyscalculia?
1.
2.
3.
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4.
5.
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Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
LOE A. BALORO
BACKGROUND
09305976728
Gender: Male
LoeBaloro.jobs180.com
baloro06@yahoo.com
baloro21.lb@gmail.com
Birth date: June 6, 1995
Birth place: Soob Albuera Leyte
14 b. Bangkaan St.
Concepcion I,
Marikina City
Citizenship: Filipino
Religion: Roman Catholic
Height: 5’5
Weight: 145 lbs.
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
WORK EXPERIENCE
Home Tutor (2009-Present)
EDUCATION
Soob Central Elementary School
2001-2007
Dr. Geronimo B. Zaldivar Memorial School of Fisheries
2007-2009
Valeriano E. Fugoso Memorial High School
2009-2011
Pamantasan ng Lungsod ng Marikina
2011-2015
AWARDS
Most Outstanding Students of Albuera District (2002, 2004, 2005, 2006)
Valedictorian Soob Central Elementary School (2007)
Most Industrious Students of SCES (2002, 2004, 2005, 2006)
Most Diligent Students of SCES (2005, 2007)
Mercury Drug Awardee Best in Science (2007)
Mercury Drug Awardee Best in Mathematics (2007)
Most Outstanding Boy scouts Leader of the year (2006)
Boy scouts of the Philippines awardee Region VIII (2006)
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
116
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Most Outstanding Leader of the year Area IV KAMMIP (2007)
Mathematician of the year DGBZMSF (2007-2008, 2008-2009)
Top 1 DGBZMSF 1st year (2008)
Top 1 DGBZMSF 2nd year (2009)
Artist of the year DGBZMSF (2009)
Dancer of the year DGBZMSF (2008, 2009)
Zaldivarians of the year DGBZMSF (2007-2008, 2008-2009)
Most Outstanding Leader of DGBZMSF (2008-2009)
Mayor Sixto Barte Dela Victoria Leadership awardee (2005, 2006, 2009)
Most Loyal Students of DGBZMSF (2009)
Division Damath Champion (2006, 2007)
Division Sci Dama 1st Runner up (2009)
Regional Sudoku Competition 3rd Runner up Region VIII (2009)
MTAP Math Challenge District Champion (2005, 2007, 2008, 2009)
MTAP Math Challenge Division Champion (2008, 2009)
MTAP Math Challenge Regional 1st Runner up (2008)
MTAP Math Challenge Regional Champion (2009)
MTAP Math Challenge National Qualifier (2009)
References:
Dr. Euegenio S. Adrao
093988321602
Head, Mathematics Dept. PLMAR
Prof. Jeannette G. De Jesus
Dean, College of Teachers Education
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
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Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
09165677280
Mr. Ferdinand G. Raymundo
09105976728
Teacher IV, DGBZMSF
Cerdeña, Leonardo III B.
Peru St. L18 B7 Greenheights Subd.
Concepcion Uno, Marikina City
09104510173
Birthday: May 1, 1995
Age: 19
Birthplace: Marikina City
Gender: Male
Status: Single
Citizenship: Filipino
Religion: Roman Catholic
Language Spoken: Filipino and English
Name of Father: Leonardo Cerdeña Jr.
Occupation: Poultry Worker
Name of Mother: Norlyn Cerdeña
Occupation: Factory Worker
In Case of Emergency: Lynette Cerdeña
Contact Number: 09154820783
Educational Attainment:
School
Tertiary: Pamantasan ng Lungsod ng Marikina (1st year)
Secondary: Pantay national High School
Primary: Talaga Elementary School


Skills and Interests:
Reading books
Computer Literate
Year
2011- 2012
2010-2011
2006-2007
 Playing guitar
 Loves to cook
Character References:
Occupation
Address
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
118
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Mrs. Avelene Velgado
Teacher in Pantay National High School
Antipolo
City
Prof. Jeannette De Jesus Dean of College of Education
Marikina City
Hernan M. Derecho
Customer Interaction Associate at TELUS International
Philippines
(+63) 09052715243 | hernan.derecho@yahoo.com | 26
years old | Marikina, National Capital Region
Experi
ence
8 years
Previo
us
Directory Assistant Agent
Eperformax Contact Center
Educat Pamantasan ng Lungsod ng Marikina
ion
Bachelor in Secondary Education Major in
Mathematics (2015)
Polytechnic University of the Philippines
Bachelor of Science in Mathematics
Nation
ality
Filipino
Experience
Dec 2009 Present
(4 years 6 months
)
Customer Interaction Associate
TELUS International Philippines | National Capital Reg,
Philippines
This position is responsible for responding to customer
inquiries and concerns. Explain company products/services
and the ability to recommend various products/services to
meet the customer’s needs. Ensures customers receive
efficient and courteous service. Work is performed under
direct supervision.
Apr 2008 - Jul
2009
(1 year 3 months )
Directory Assistant Agent
Eperformax Contact Center | National Capital Reg, Philippines
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
119
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Jan 2006 - Jan
2006
(1 month)
service crew (Counter and dining)
Jollibee Foods Corporation
Education
2015
Pamantasan ng Lungsod ng Marikina
Bachelor in Secondary Education
Major
Mathematics
Skills
Excellent Computer Skills, Excellent Communication Skills, Excellent
Interpersonal Skills
Additional Info
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
120
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
EDUCATION:
School
Year Graduated
Honors, Awards and Extra Curricular Activities
Pamantasan ng Lungsod ng Marikina Undergraduate ( 4th Year )
Intercollegiate Quiz bee Champion
Quiz bowl 2013 1st Placer
Malan day National High School
2005
Valedictorian
President Gloria Macapagal-Arroyo Award of Outstanding Achievement
Manuel B. Villar Jr. Excellence Award
GawadTalino Award
Excellent in Science
Excellent in Mathematics
Excellent in Social Studies
Supreme Student Government (Vice President 2004-2005)
Boy Scout of the Philippines
Math Club President (2004-2005)
Malanday Elementary School
2001
Alvin Quesada
42 Mabuhay Street Nangka, Marikina City
Contact # 09074273488
Email Add: sniper_gwp2@yahoo.com
Professional Summary
Responsible employer. Passionate and motivated, with a drive for excellence.
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
121
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Looking for a position in a fast grow company.
Skills
∙ Operation and control
∙ Computer Literate
∙ Mathematics
∙ Trouble Shooting
∙ Food Production
∙ Repairs
Experience
Factory Worker
GTGF – Banaba
∙ Package finished product and prepare them for shipments
∙ Prepare quality check on products
∙ Rotate through all the tasks required in a particular production process
∙ Swipe or otherwise clean work area.
Education
Tertiary Level: Bachelor in Secondary Education Major in Mathematics
Pamantasan ng Lungsod ng Marikina, Marikina City
Expected Graduation April 2015
Joenelle Jover Sandagon
joenellesandagon.jobs180.com
PERSONAL INFORMATION
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
122
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Address: Balubad, Nangka Marikina City NCR 1808 Philippines
Birthday: February 1, 1995
Gender: Male
Marital Status: Single
Nationality: Filipino
CONTACT INFORMATION
Phone:
0936-669-6201
Email: junel.sandagon@yahoo.com
Social Network:
http://www.facebook.com/junel.sandagon
WORK EXPERIENCE
Home tutor
2012 October – present
Specialization: Education/Mathematics
EDUCATIONAL QUALIFICATIONS
Primary: Nangka Elementary School
Y.G.
2001-2007
Secondary: Nangka High School
Y.G.
2007-2011
Tertiary: Pamantasan ng Lungsod ng Marikina (PLMar)
Course: Bachelor of Secondary Education Major in Mathematics
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
123
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Y.G.
2011-present
Field of Study: Education/Teaching/Training
ACHIEVEMENTS
Champion in Essay Writing Contest in Science - NHS (2008)
MTAP Math Challenge Qualifier (2008-2010)
P.O.O of Student Council - NHS (2010-2011)
First Honorable Mention - Nangka High School
Local Quiz bowl 2013 Champion - PLMar
REFERENCES
Dr. Eugenio S. Adrao
Company: Pamantasan ng Lungsod ng Marikina
Phone: 093988321602
Prof. Jeannette G. De Jesus
Company: Pamantasan ng Lungsod ng Marikina
Phone: 09165677280
MANUELITO PADUA VILORIA
#73 Malaya St. Malanday, Marikina City
Contact #: 0998-544-4595
Educational Attainment:
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
124
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City
Tertiary
:
2011-Present
PAMANTASAN NG LUNGSOD NG MARIKINA
Concepcion Uno, Marikina City
Bachelor of Secondary Education
Major in Mathematics
Vocational :
2007-2009
INFOTECH Institute of Arts and Sciences
Dela Paz, Pasig City
Computer Science and Technology
Secondary :
2000-2004
RIZAL HIGH SCHOOL-MANGGAHAN
Manggahan, Pasig City, Philippines
Elementary :
MANGGAHAN ELEMENTARY SCHOOL
1993-2000
Manggahan, Pasig City, Philippines
Employment Record:

DRUG CHECK PHILIPPINES
Drug Test Representative
2009-2011
Rosario, Pasig City
EFFECTIVE TEACHING STRATEGIES IN MATHEMATICS TO STUDENTS WITH DYSCALCULIA
125
Pamantasan ng Lungsod ng Marikina
Brazil St., Greenheights Subd., Phase I, Concepcion Uno, Marikina City

LIFE FORCE DRUG STORE
Pharmacist Assistant
2004-2005
Makati City
Personal Information:
Age
:
27 yrs. old
Date of Birth
:
March 25, 1987
Civil Status
:
Single
Religion
:
Roman Catholic
Father’s Name
Mother’s Name
:
:
Manuel Viloria
MerlyViloria
Character References:
Mrs. Jeannette Guillermo-De
Jesus
Dean, College of Education
Pamantasan ng Lungsod ng
Marikina
Mr. Eugenio S. Adrao
Head, Mathematics Department
Pamantasan ng Lungsod ng
Marikina
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