Education and Training in Autism and Developmental Disabilities, 2011, 46(2), 214 –219
© Division on Autism and Developmental Disabilities
Arithmetic Operations and Attention in Children with
Intellectual Disabilities
Aleksandra Djuric-Zdravkovic, Mirjana Japundza-Milisavljevic, and
Dragana Macesic-Petrovic
University of Belgrade
Abstract: This paper is aimed at depicting the quality of functions of some of the aspects of attention in children
with mild intellectual disabilities and their influence on the mastering of arithmetic operations, including
addition and subtraction. The sample used in this study encompasses 60 pupils, both males and females. The
criteria used in the selection of examinees included the IQ level of the students which ranged from 50 – 69,
calendar age from 12 to 14 years, school age which involved the inclusion of pupils attending grades five to
seven of primary school in Serbia. To evaluate the quality of attention in our study we used the Trail Making
Test form A and the Double Letter Cancellation Test, whilst a Criterion-referenced test was used to evaluate how
well the specified body of knowledge, in this case arithmetic operations i.e. addition and subtraction were learned
by the pupils. The implications of the study pertain to the proposal of implementing specific, creative activities
and exercise during play, concrete contents, demonstration, experiments and teaching resources susceptible for
teaching arithmetic.
The unsatisfactory results achieved by pupils
during the assessment of their knowledge of
mathematics at all levels of education, demonstrate a need for continuous examination of
parameters that can influence the process of
its comprehension (OECD, 2006).
Throughout the world, at the beginner level
of mathematical education at school age, children most often begin with learning addition
and subtraction and continue with this concept to the end of primary school, building on
their knowledge (Canobi, 2004). Addition
and subtraction represent the building blocks
of the majority of future mathematical notions
and for this reason it is important for a child
to understand their basic concepts (Robinson
& Dube, 2009).
Addition, as an arithmetic operation,
should be understood by a child as the process
of accumulation of certain elements, while the
arithmetic operation subtraction is presented
Correspondence concerning this article should
be addressed to Aleksandra Djuric-Zdravkovic, Department of Oligofrenology, University of Belgrade,
Faculty for Special Education and Rehabilitation,
2 Visokog Stevana, Belgrade, SERBIA. Email:
adjuriczdravkovic@yahoo.com
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to the child as the separation of elements
(Gilmore & Spelke, 2008; Nunes, Bryant, Hallett, Bell, & Evans, 2009).
Children with mild intellectual disabilities
(MID) have difficulties in utilizing arithmetic
operations, involving addition and subtraction, even once they comprehend their fundamental concepts. For example, because of
difficulties in the generalization process, these
children cannot apply the learnt arithmetic
knowledge if the question is set up differently
from the conditions given in those questions
that were practiced (Butler, Miller, Lee, &
Pierce, 2001).
In children with MID the importance of
mastering fundamental mathematical knowledge is emphasized, including the ability to
utilize arithmetic operations involving addition and subtraction to resolve a series of
problems in everyday situations. Taking into
consideration the importance of socialization
for children with MID the application of addition and subtraction enables shopping
(price comparison, understanding of weights
and measurements, value of money) and later
on household budget management (planning, saving), etc. (Rosenberg, Westling, &
McLeskey, 2008).
Education and Training in Autism and Developmental Disabilities-June 2011
One of the most relevant neuropsychological functions important for successful mastering of program content for mathematics is
surely attention (Djuric-Zdravkovic, 2006; Japundza-Milisavljevic, 2009). The study of attributes pertaining to the attention of children
with MID is inflicted by its manifestative signs,
which are particularly emphasized in the
teaching process.
In children with MID various attention
functions are disturbed. It is characterized by
short-livedness, it is unselective and distinctively oscillatory. Attention deficits in children
with MID, are amongst other things, considered fundamental cognitive deficits, which
hinder the quality of adoption of mathematical knowledge (Macesic-Petrovic, 1998; according to Djuric-Zdravkovic, 2006).
In that sense, the aim of this paper is to give
an overview of the quality of function of some
of the components of attention in children
with MID and their influence on the mastering of the content of arithmetic operations,
including addition and subtraction.
Apprehension of the characteristics of cognitive organization and more accurate determination of the primary deficit of attention
with the purpose of improving the adoption of
arithmetic knowledge would lead to the individualization of the approach taken in working with this population of children. The results obtained in this study should represent a
basis for planning and appropriate directing
of the educational process. Planning and organization of individual educational programs
in working with these children, regardless of
the variety of their forms and modalities, must
be founded on the knowledge of the developmental rhythm of certain structures and functions and they must be harmonized with that
rhythm.
Method
Participants
The sample used in this study consisted of 60
pupils of both sexes. The criteria used in selecting the examinees included the intelligence quotient of the pupils which ranged
from 50 to 69, that was assessed using the
WISC scale for the assessment of intellectual
capabilities, the calendar age involved exam-
inees from 12 to 14 years of age, and the
school age which encompassed pupils attending grades 5 to 7 at primary schools in Serbia.
Within the aforementioned age group we evaluated arithmetic operations: addition and
subtraction up to one thousand and one million (thirty pupils were assesses from each
calendar age group), they showed no evidence of neurological, psychological, sensory,
distinct emotional and combined disturbances. In the sample there were a somewhat
larger number of male pupils (56.7%) in relation to the female examinees (of which
there were 43.3%).
Materials and Procedure
In the assessment of the quality of attention in
our study we used the following:
a) Trail Making Test - form A – the test
measures the direction and sustainment
of visual attention, visual processing
speed, visuospatial and visuomotor competence of the examinees. In the study we
assessed the processing speed and the
number of mistakes made on the stimuli.
This test was initially part of the Army
Individual Test Battery (1944), and was
afterwards incorporated into the Halstead-Reitan Battery (Reitan & Wolfson,
1985; Tombaugh, 2004).
b) Double Letter Cancellation Test (Diller
et al., 1974) – for the assessment of attention agility. The test involves discrimination of the letters E and C, which are
distributed 105 times into six rows with 52
letters in one row (Wang, Huang, &
Huang, 2006). In the study we assessed
the processing speed and number of mistakes made.
In the assessment of the knowledge of the
content of arithmetic operations: addition
and subtraction, we used the criterion referenced test, especially constructed for the requirements of this study. Pupils that were 12
years of age (attending grade 5 of primary
school) solved questions involving arithmetic
operations: addition and subtraction up to the
number one thousand, whilst the pupils that
are 14 years of age (grade 7 of primary school)
Arithmetic Operations and Attention
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215
solved questions involving addition and subtraction up to the number one million.
The results of the criterion referenced test
were distributed into three categories:
●
●
●
Mastered the program requirements fully
(⫹)
Partially mastered the program requirements (⫹⫺)
Did not master the program requirements
(⫺)
The other data required for the study were
gathered by analyzing the student records of
the pupils involved in the study.
The processing of the obtained data, entered into the database of SPSS, was completed using the descriptive and parameter
statistic methods. Of the available statistical
proceedings and measures we used: frequency, percentage, arithmetic mean, standard deviation, Student’s t-test, ␹2 test, Pearson’s Coefficient of Linear Correlation (r),
contingency coefficient (c).
The study was conducted on the territory of
Belgrade, Serbia, at primary schools for children with mild intellectual disabilities. Testing
was implemented in continuity, without time
intervals, individually, with each student privately. At the end of the school year a criterion
referenced test was issued, once all of the
program content on arithmetic was presented
in its entirety.
TABLE 1
Average Results by the Examinees on the Trail
Making Test–form A
Time in Seconds
AM
SD
min.
max
I (25%)
II (50%)
III (25%)
120.86
131.22
25 sec.
720 sec.
134–720 (n ⫽ 10)
51–133 (n ⫽ 40)
25–49 (n ⫽ 10)
Number of
Mistakes
4.04
6.74
0
33
1–33 (n ⫽ 35)
0 (n ⫽ 25)
r ⫽ ⫹0.40 (level 0.01).
ber of mistakes) within the test there is no
statistically significant correlation (see Table
2). Examinees were divided up into three categories for a better and easier overview of the
results. The first category (I) includes the examinees with the worst results (25%), the second category (II) encompasses the examinees
whose results fall into the intermediary values
of our sample (50%), and the third category
(III) encompasses the pupils who completed
the test the quickest and with the least number of mistakes (25%).
In analyzing Table 3, we observed that the
examinees encompassed in our sample do not
achieve the necessary 75% comprehension
level of program content at any of the levels of
education, which is necessary for the program
Results
Table 1, shows there is a low level of positive
correlation (r ⫽ ⫹ 0.40, level of 0.01) between
the two variables (time of work and number of
mistakes). Thus we can conclude with 99%
certainty that the examinees that do their test
quicker make fewer mistakes.
In accordance with the given achievements,
examinees are distributed into three categories: first category (I) encompasses examinees
that obtained the worst results on the Trail
Making Test (25%), second category (II) encompasses examinees whose results are found
within the boundaries of intermediary values
in our sample (50%), and the third category
(III) is made up of examinees with the best
results on the TMT - form A - (25%).
Between the given variables (time and num-
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TABLE 2
Average Results of the Examinees on the Double
Letter Cancellation Test
Double Letter Cancellation Test
AM
SD
min.
max
I (25%)
II (50%)
III (25%)
Time in Seconds
Number of
Mistakes
206.52
122.74
49 sec.
800 sec.
228–800 (n ⫽ 10)
137–227 (n ⫽ 41)
49–134 (n ⫽ 9)
9.64
9.55
0
63
12–63 (n ⫽ 11)
4–11 (n ⫽ 31)
0–3 (n ⫽ 18)
No significant correlation.
Education and Training in Autism and Developmental Disabilities-June 2011
TABLE 3
Success of Examinees on the Test on the Level of
Comprehension of Arithmetic Operations:
Addition and Subtraction
Arithmetic Operations:
Addition and Subtraction
Grade five (to 1000)
Grade seven
(to 1 000 000)
⫺
29.44%
⫹⫺
⫹
8.33% 62.23%
44.45% 19.44% 36.11%
to be considered appropriate and adapted to
the capabilities of children with MID (Bojanin, 2002, according to Djuric-Zdravkovic,
2007).
Testing of the correlation between the variables in Table 4 has shown a statistically significant difference of relations on the tests for
assessing the quality of the developedness of
attention and the comprehension of content
of arithmetic operations involving addition
and subtraction at the studied levels of education.
Discussion
Positive results at school by pupils can hardly
be expected if we do not devote more attention to effectuating the conditions for adequate stimulation of attention development
through its components, which lie at the basis
of adopting arithmetic knowledge. The results
of this study confirm the previously written
fact, whilst the conclusions made in earlier
studies in this field express similar views
(Djuric-Zdravkovic, 2006; Japundza-Milisavljevic, 2009). Researchers are proving that chil-
dren with MID often have problems sustaining
their attention whilst solving mathematical
equations, which contributes to the difficulties
arising in acquiring, recognizing and generalizing new arithmetic skills and knowledge
(Heward, 2008).
The results obtained in our study show that
the examinees enrolled in grade 5 are most
successful at completing tasks which require
one-step adding of hundreds and tenths,
while they are distinctly unsuccessful in solving textual problems where they are required
to apply both arithmetic operations. Pupils in
grade seven comprehend tasks which involve
long addition over 1000, while they are quite
unsuccessful in resolving problems which involve deciding on the use of one or both
arithmetic operations. The data from this
study that point to problems with mathematical conclusions and the application of concepts for resolving mathematical problems by
children with MID to a great extent coincide
with the data obtained in other studies (Butler
et al., 2001; Heward, 2008).
The focus of stimulating the development
of attention in children with MID should be
implemented through an effective teaching
design where emphasis would be placed on
strengthening the direction and sustainment
of attention by way of specific, creative activities and exercises during play, arithmetic experiments, practical situations from everyday
life and through various arithmetic problems
(Geary, 2004; Heward, 2008; Rosenberg et al.,
2008). Specific creative activities and exercises
as outlined above would involve: directing attention, distinguish the important stimuli
from the unimportant, indicating the significant details of an arithmetic problem, stimu-
TABLE 4
Correlation of the Quality of the Developedness of Attention with the Comprehension of the Content of
Arithmetic Operations Involving Addition and Subtraction
Comprehension of the Teaching Material on
Addition and Subtraction
Double Letter Cancellation Test/processing speed
Double Letter Cancellation Test/accuracy
Trail Making Test/processing speed
Trail Making Test/accuracy
␹2
␹2
␹2
␹2
⫽
⫽
⫽
⫽
14.23
18.22
16.54
12.87
df
df
df
df
⫽
⫽
⫽
⫽
4c
4c
4c
2c
⫽
⫽
⫽
⫽
⫹0.33
⫹0.36
⫹0.35
⫹0.31
(level
(level
(level
(level
Arithmetic Operations and Attention
/
0.01)
0.01)
0.01)
0.05)
217
lating to the end of the task, strengthening
resistance to distractions, executive control,
sustaining of attention, etc.
When working with children with MID extra care should be taken that they are stimulated during exercises to resolve concrete
problems from everyday life, whenever that is
possible. In that way the pupils would gradually understand the role of mathematics. If
possible, it would be a good idea to offer as
many ways as possible to resolve the given
tasks.
Long-term studies of children with MID
show that comprehension of the content of
arithmetic operations including addition and
subtraction had achieved the best possible results by way of the program Multisensory
mathematics that engages all the senses.
TouchPoints within the scope of this program
familiarizes the pupils with MID with the fact
that each number is a realistic value and that
by manipulation including summation and
dissociation, addition and subtraction are
learned, as a representative quantity (Scott,
1993).
Studies show a significant improvement in
the quality of multiple capabilities and knowledge, amongst others attention and knowledge of arithmetic operations by children with
MID, with the use of information technology
and appropriate software which support the
adoption of this type of program contents
(Wehmeyer, Smith, Palmer, & Davies, 2004).
With consideration for the specific needs of
children with MID, in learning arithmetic operations there is mention of the possibility of
activating numerous cognitive functions
through the use of natural and manufactured
objects, through various demonstrations, as
well as the possibility of implementing “Cool
Math” – an alternative method of learning
mathematical content through the combination of fun, games, arts, etc (Whiten &
Whiten, 2009).
Finally, we can conclude that there is a prevailing viewpoint that one treatment approach
cannot resolve the problem of a child with an
attention deficit when attempting to comprehend mathematical content, thus the combination of various therapeutic methods is a
necessity. It is useful to use all of the therapeutic methods in practice that have proven
to be effective (Montague, 2008).
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Received: 8 February 2010
Initial Acceptance: 2 April 2010
Final Acceptance: 22 July 2010
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