Transcript DIFFERENCES BETWEEN STUDENTS WITH AND WITHOUT …

DIFFERENCES BETWEEN STUDENTS WITH
AND WITHOUT LEARNING DISABILITIES
IN MENTAL COMPUTATIONS
Ioannis Demirtsidis & Ioannis Agaliotis
Department of Educational & Social Policy
University of Macedonia
Thessaloniki
Greece
THEORETICAL FRAMEWORK
Mental computations
• Mental computations are processes of carrying out
arithmetic calculations and finding results of
arithmetic operations without the aid of external
devices (Sowder, 1990).
• It has been estimated that mental computations
constitute the 84.6% of all calculations performed by
adults in one day (24 hrs) (Nortcote & McIntosh,
1999).
• Flexibility in mental computations presupposes
the employment of a variety of efficient mental
strategies, the choice of which is informed by
student’s knowledge of number combinations
(Watson, Kelly, & Callingham, 2004).
• A wide variety of mental addition and subtraction
strategies has been identified in the literature,
including: separation, accumulation, bridging of
ten, mixed separation – accumulation, holistic,
counting, and mental image of written algorithm
(e.g. Judy 2007; QCA, 1999).
• The different strategies vary in their speed and
accuracy across problems. Effective strategy choices
allow students to optimize their performance (Luwel
& Verschaffel, 2009).
• A useful framework for analyzing students’ strategy
choice in mental computations is the one proposed
by Lemaire and Siegler (1995), which distinguishes
between four different parameters of strategic
competence: repertoire, frequency, efficiency and
adaptivity.
• Strategy selection depends on cognitive,
metacognitive, and affective factors (Heirdsfield &
Cooper, 2004).
• Students struggling with Mathematics, like students
with Learning Disabilities, face various difficulties
with the development of the above-mentioned
factors, and thus they do not acquire as effective
mental computation strategies as their typical peers
(Torbeyns, Verschaffel, & Ghesquiere, 2004)
• Comparing the strategy choices for mental
computations characterizing students with Learning
Disabilities and those of their typical peers is of great
relevance for the theory and practice of special
mathematics education (Peters, De Smedt, Torbeyns,
Verschaffel, & Ghesquiere, 2014)
• Research question of the present study:
– Are there differences in the repertoire, the
frequency, and the efficiency of strategy use
between Greek 5th and 6th grade students with
Learning Disabilities and their typical peers, when
they execute mentally additions and subtractions?
METHOD
Participants
 60 5th and 6th grade students attending schools
situated in Northern Greece.
 30 typical students (20 males – 10 females)
 30 students with LD (18 males – 12 females)
Measures and Procedures
Questionnaire with 8 additions and 8 subtractions
with numbers bigger than 20 and smaller than 100,
based on a tool used by Lemonidis (2008). Cronbach’s
a = .80
• Problems:
– 23+43, 31+42, 76+22, 84+13
– 17+49, 26+38, 56+25, 58+33
– 48-34, 59-26,78-54, 99-35
– 53-38, 65-47, 72-49, 86-58
• Problems were presented orally
• Students’ efforts were timed
• Students were asked to describe their strategy for
finding the result
• Answers were recorded, transcribed in protocols, and
subjected to qualitative analysis
• Differences between the groups were determined
through MANOVA with Bonferroni correction.
RESULTS
Table 1: Groups’ overall achievement
Group
L.D.
Typical
Low
achievement
0-4 correct
answers
Moderate
achievement
5-8 correct
answers
Good
achievement
9-12 correct
answers
Very good
achievement
16-20 correct
answers
16.7%
26.7%
50%
6.7%
0%
0%
36.7%
63.3%
RESULTS
Table 2: Groups’ achievement in addition
Addition of two
2-digit numbers,
no regrouping
Addition of two
2-digit numbers,
with regrouping
Group
No correct 1-2 correct
answers
answers
3-4 correct
answers
L.D.
3.3%
13.3%
83.3%
Τypical 0%
0%
100%
L.D.
23.3%
50%
6.7%
93.3%
26.7%
Typical 0%
RESULTS
Table 3: Groups’ achievement in subtraction
Group
No correct
answers
1-2 correct
answers
3-4 correct
answers
Subtraction L.D.
of two 2-digit
numbers, no
Typical
borrowing
6.7%
30%
63.3%
0%
3.3%
96.7%
Subtraction L.D.
of two 2-digit
numbers
Typical
with
borrowing
70%
16.7%
13.3%
23.3%
10%
66.7%
RESULTS
Type and frequency of strategies in addition of two 2-digit numbers
without regrouping
L.D.


Typical 


50% mental image of written algorithm
29% separation
66.6% separation
28.3% mental image of written algorithm
3.3% accumulation
Chart 1
L.D.
without L.D.
100%
80%
60%
40%
20%
0%
Count.
Mental image of
written algorithm
Separat.
Accumul.
Bringing of ten
Mixed strategy
Holistic Strategies
Separ. and Accumul.
RESULTS
Time requirements of strategy use in addition of two 2-digit numbers
without regrouping
L.D. + Typical
 most time- consuming strategy: mental image of written
algorithm,
 followed by: separation and accumulation.
Chart 2
L.D.
without L.D.
0:00:26
0:00:22
0:00:17
0:00:13
0:00:09
0:00:04
0:00:00
Count.
Mental image of
written algorithm
Separat.
Accumul.
Bringing of ten
Mixed strategy
Separ. and
Accumul.
Holistic Strategies
RESULTS
Type and frequency of strategies in addition of two 2-digit numbers with
regrouping
L.D.



45.8% mental image of written algorithm
5.8% separation
3.3% mixed separation - accumulation
Typical




58% separation
29% mental image of written algorithm
3.3% mixed separation - accumulation
3.3% holistic
Chart 3
L.D.
without L.D.
70%
60%
50%
40%
30%
20%
10%
0%
Count.
Mental image of
written algorithm
Separat.
Accumul.
Bringing of ten
Mixed strategy Separ.
and Accumul.
Holistic Strategies
RESULTS
Time requirements of strategy use in addition of two 2-digit numbers
with regrouping
L.D.
 most time-consuming strategy: mental image of written algorithm,
 followed by: separation and mixed separation – accumulation.
Typical  most time -consuming strategy: holistic
 followed by: mixed separation – accumulation, mental image
written algorithm, and separation.
Chart 4
L.D.
without L.D.
0:00:52
0:00:43
0:00:35
0:00:26
0:00:17
0:00:09
0:00:00
Count.
Mental image of
written algorithm
Separat.
Accumul.
Bringing of ten
Mixed strategy Separ.
and Accumul.
Holistic Strategies
RESULTS
Type and frequency of strategies in subtraction of two 2-digit numbers
without borrowing
L.D.


62.2% mental image of written algorithm
10% separation
Typical





50.8% separation
25.8% mental image of written algorithm
7.5% accumulation
3.3% bridging of ten
3.3% holistic
Chart 5
L.D.
without L.D.
70%
60%
50%
40%
30%
20%
10%
0%
Count.
Mental image of
written algorithm
Separat.
Accumul.
Bringing of ten
Mixed strategy Separ.
and Accumul.
Holistic Strategies
RESULTS
Time requirements of strategy use in subtraction of two 2-digit numbers
without borrowing
L.D.
 most time-consuming strategy: mental image of written
algorithm,
 followed by: separation
Typical  practically the same time for counting, mental image
written algorithm, separation, accumulation
Chart 6
L.D.
without L.D.
0:00:35
0:00:30
0:00:26
0:00:22
0:00:17
0:00:13
0:00:09
0:00:04
0:00:00
Count.
Mental image of
written algorithm
Separat.
Accumul.
Bringing of ten
Mixed strategy
Holistic Strategies
Separ. and Accumul.
RESULTS
Type and frequency of strategies in subtraction of two 2-digit numbers
with borrowing
L.D.



10.8% mental image of written algorithm
3.3% bridging of ten
3.3% holistic
Typical






31.7% mental image of written algorithm
10.1% separation
7.5% accumulation
6.7% bridging of ten
3.3% holistic
2.5% counting
Chart 7
L.D.
without L.D.
60%
40%
20%
0%
Count.
Mental image of
written algorithm
Separat.
Accumul.
Bringing of ten
Mixed strategy Separ. Holistic Strategies
and Accumul.
RESULTS
Time requirements of strategy use in subtraction of two 2-digit
numbers without borrowing
 most time- consuming strategy: mental image of written
algorithm,
 followed by: bridging of ten and holistic
Typical  most time -consuming strategy: accumulation, bridging of ten
 followed by: mental image written algorithm, counting, separation.
L.D.
Chart 8
L.D.
without L.D.
0:00:43
0:00:39
0:00:35
0:00:30
0:00:26
0:00:22
0:00:17
0:00:13
0:00:09
0:00:04
0:00:00
Count.
Mental image of written
algorithm
Separat.
Accumul.
Bringing of ten
Mixed strategy Separ.
and Accumul.
Holistic Strategies
RESULTS
Table 4: Error types
Operation
Group
Arithmetic
combinations
Place value
Regrouping or
Borrowing
Error
combination
2-digit +
2-digit, no
regrouping
L.D.
12.6%
7.5%
Typical
1%
2%
2-digit +
2-digit,
with
regrouping
L.D.
15.7%
8%
8%
Typical
1%
2.5%
2.5%
2-digit –
2-digit, no
borrowing
L.D.
9.9%
7%
4.2%
6%
Typical
3.4%
3.3%
1%
1%
2-digit –
2-digit,
with
borrowing
L.D.
3.3%
50%
7%
23.3%
Typical
6%
21.66%
9%
3%
Other
operation
1%
1%
RESULTS
Inferential Statistics (MANOVA – Bonferroni .025)
Additions
Accuracy in two 2-digits
no regrouping

No significant difference
Accuracy in two 2-digits
with regrouping
Typical students outperformed
students with L.D. (F (1, 52) = 20.19,
p = 00, η2 = .28),
Type of strategies

Statistically significant difference
between the two groups (F (8, 45) =
5.21, p = .00, λ = .51, η2 = .48).
Time of execution

No significant difference
RESULTS
Subtractions
Accuracy in two 2-digits 
no borrowing
Typical students outperformed
students with L.D. (F (1, 52) =
0.63, p =.002, η2 = .17)
Accuracy in two 2-digits 
with borrowing
Typical students outperformed
students with L.D. (F (1, 52) =
20.70, p =.00, η2 = .28).
Type of strategies

Statistically significant
difference between the two
groups (F (8, 45) = 4.72, p = .00,
λ = .54, η2 = .45).
Time of execution

No significant difference
DISCUSSION
• Additions
• Generally, in adding two 2-digit numbers with or without regrouping
typical students used a wider strategy repertoire than their peers with
L.D.
• In 2-digit + 2-digit additions without regrouping the two groups had no
significant difference in accuracy, probably due to the easiness of the
task.
• In terms of strategy, typical students used mainly the separation strategy
and included in their repertoire the accumulation strategy, whereas
students with LD used mainly the strategy of algorithm’s mental image,
and to a much lesser extent the separation strategy. This is probably due
to difficulties encountered by students with LD in analyzing and
synthesizing numbers, and also in hold the respective information in
their memory.
•The choices LD students probably denote also difficulties in
number sense and number analysis and synthesis (DfES, 2001).
•In terms of speed of strategy use the two groups did not differ
significantly.
•They both needed more time for the strategy of algorithm’s mental
image in comparison to the strategy of separation.
•The finding that students with LD used mainly the more timeconsuming mental image strategy, seems to corroborate the wellestablished difficulty of students with LD in strategy choice and use
(e.g. Learner & Beverley, 2014).
• In accuracy of 2-digit + 2-digit additions with regrouping the two groups
had significant difference, in favor of typical students.
• Main error types of students with LD: place value, retrieval of number
combinations.
• Typical students used mainly the separation strategy, to a lesser extent
the strategy of algorithm’s mental image and to a very low degree the
mixed strategy of separation – accumulation and the holistic strategies.
• Students with LD used mainly the strategy of algorithm’s mental image,
and to a considerably lesser extent the separation strategy and the
strategy separation – accumulation.
• The low use of holistic strategies by Greek students has been found also
in other Greek studies (e.g. Lygouras, 2012).
• Strategy choice by students with LD may be a function of poor
number analysis and synthesis, and ineffective number
combinations retrieval ( Agaliotis, 2011; DfES, 2001).
• In terms of speed in strategy use the two groups did not differ
significantly. They both needed more time for the strategy of
algorithm’s mental image in comparison to the strategy of
separation, and the mixed strategy separation - accumulation.
• Most time-consuming for typical students were the holistic
strategies, as also mentioned by Lygouras (2012). This finding is
in contrast to findings from other educational systems showing an
opposite tendency (e.g. Torbeyns et al. 2009a).
• The finding that students with LD used mainly the more timeconsuming mental image strategy, seems to corroborate the wellestablished difficulty of students with LD in strategy choice and
use (e.g. Learner & Beverley, 2014).
•Subtractions
•Accuracy level of typical students in executing subtractions both
without and with borrowing was significantly higher than the level of
their learning disabled counterparts.
•There were significant differences between the two groups in the
strategies used for subtracting 2-digit numbers from 2-digit numbers
without borrowing. Typical students used mainly separation (half of
them), the algorithm’s mental image (slightly more than ¼ of them),
and to a lesser extent other strategies, like bridging of ten. Students
with LD used in clear majority the algorithm’s mental image and to a
lesser extent separation.
•Use of holistic strategies was kept to a minimum, as found also in
other studies (e.g. Lygouras, 2012). The choices LD students
probably result from difficulties in number sense and number analysis
and synthesis (DfES, 2001).

In terms of speed in the use of the strategies that both groups
employed (separation and algorithm’s mental image), there were
no significant differences. However, for typical students the most
time-consuming strategy was the bridging of ten and the less timeconsuming the holistic strategies (although their use was limited).

There was statistically significant difference in accuracy for
subtraction of 2-digit numbers from 2-digit numbers with borrowing
in favor of typical students.

Both groups used primarily the strategy of mental image of written
algorithm, but typical students used in total 6 different strategies
whereas LD students only 3. Restrictions in strategy use by LD
students were corroborated.

Extended use of algorithm’s mental image by both groups denotes
the difficulty of this mental computation.
DISCUSSION
• Typical students needed more time for the strategies of mental image of
the written algorithm, accumulation, and bridging of ten, and less time
for counting, separation, and holistic strategies. Students with LD
needed more time for the mental image of written algorithm, and less
time for bridging of ten and holistic strategies. The differences for the
common strategies were not statistically significant, but add to the
different picture of the two
groups.


Limitations
There was no differentiation among students with Mathematical
Disabilities, Comorbid Reading and Mathematical Disabilities and
just Reading Disabilities.
 Students were examined only in “choice condition”, meaning that
each could choose the strategy of their preference. According to
Siegler and Lamaire (1995) this condition does not provide data on
adaptivity and efficiency.
 The sample was small.
• Instructional implications
• Instruction of strategies as an independent aim of the curriculum,
especially for LD students
• Systematic use of error analysis in order to specify the nature of
each student’s difficulties
• Identification of lacking prerequisite knowledge and skills, but also
of possible learning style characteristics, that lead certain students
to specific strategy choices.
• Systematic improvement of accuracy and speed in the execution of
operations
• Future research proposals
• Differentiation among LD sub-types in terms of strategy use.
• Investigation of the benefits of providing students with LD with a
small number of reliably used strategies under all circumstance, in
contrast to laying emphasis on developing their flexibility in
strategy choice.
REFERENCES
ِِِAgaliotis, Ι. (2011). Διδασκαλία Μαθηματικών στην Ειδική Αγωγή και Εκπαίδευση: Φύση και εκπαιδευτική
διαχείριση των μαθηματικών δυσκολιών, (27-28). Αθήνα: Εκδόσεις Γρηγόρη.
DfES ( (2001). In Lemonidis, Ch. (2013). Νοεροί υπολογισμοί, Μαθηματικά της φύσης και της ζωής (σελ. 386389). Θεσσαλονίκη: Εκδόσεις Ζυγός.
Hartnett, Judy E. (2007) Categorisation of Mental Computation Strategies to Support Teaching and to
Encourage Classroom Dialogue. In Watson, Jane & Beswick, Kim (Eds.) 30th annual conference of the
Mathematics Education Research Group of Australasia - Mathematics: Essential Research, Essential
Practice, 345-352, Hobart, Tasmania.
Heirdsfield, Ann M., Cooper, Tom J. (2004). Factors affecting the process of proficient
mental addition and subtraction: Case studies of flexible and inflexible computers. Journal of
Mathematical Behavior 23(4), 443-463.
Qualifications and Curriculum Authority (QCA) (1999). The National Numeracy Strategy. Teaching Mental
Calculation Strategies. London: Author.
Lamaire, P., & Siegler, R.S. (1995). In Lemonidis, Ch. (2013). Νοεροί υπολογισμοί, Μαθηματικά της φύσης και
της ζωής (σελ. 99). Θεσσαλονίκη: Εκδόσεις Ζυγός.
Learner, J., & Beverley, J. (2014). Learning Disabilities and Related Disabilities: Strategies for Success. Stamford,
CT: Cengage Learning.
Lemonidis, Ch. (2008). Longitudinal study on mental calculation development in the first two grades of primary
school. International Journal for Mathematics in Education, 1, 47-68.
Lugouras, G. (2012). In Lemonidis, Ch. (2013). Νοεροί υπολογισμοί, Μαθηματικά της φύσης και της ζωής (σελ.
49). Θεσσαλονίκη: Εκδόσεις Ζυγός.
Northcote, M. & McIntosh, A. (1999). In Lemonidis, Ch. (2013). Μαθηματικά της φύσης και της ζωής (σελ. 16).
Θεσσαλονίκη: εκδόσεις ΖΥΓΟΣ.
Peters, G., Smedt, B., Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2014). Subtraction by addition in children
with mathematical learning disabilities. Learning and Instruction, 30, 1-8.
Sowder, J. (1990). Mental Computation and Number Sense. Arithmetic Teacher, 37 (7), 18-20.
Torbeys, J., Verschaffel, L., & Ggesquiere, P. (2004). Strategy Development in Children with Mathematical
Disabilities: Insights from the Choise/No-Choise Method and the Chronoligical-Age/ Abilty-Level-Match
Design. Journal of Learning Disabilities, 37(2), 119-131.
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooten, W. (2009). Conceptualizing, investigating, and enhancing
adaptive expertise in elementary mathematics education. European Journal of Psychology of Education,
24(3), 335-359.
Watson, J.M., Kelly, M.N., & Callingham, R.A. (2004, December). Number sense and errors on mental
computation tasks. Paper presented at the annual conference of the Australian Association for Research in
Education, Melbourne. Available at http://www.aare.edu.au/04pap/alpha04.htm