DIFFERENCES BETWEEN STUDENTS WITH AND WITHOUT LEARNING DISABILITIES IN MENTAL COMPUTATIONS

1. 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

2. 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).

3. 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).

4. 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.

5. 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)

6. 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?

7. 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

8. 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.

9. RESULTS Table 1: Groups’ overall achievement

10. RESULTS Table 2: Groups’ achievement in addition

11. RESULTS Table3: Groups’ achievement in subtraction

12. RESULTS Type and frequency of strategies in addition of two 2-digit numbers without regrouping L.D.  50% mental image of written algorithm  29% separation Typical  66.6% separation  28.3% mental image of written algorithm  3.3% accumulation Chart 1

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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: bridging of ten and holistic Typical  most time -consuming strategy: accumulation, bridging of ten  followed by: mental image written algorithm, counting, separation. Chart 8

20. RESULTS Table 4: Error types

21. 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

22. RESULTS Subtractions Accuracy in two 2-digits  Typical students outperformed no borrowing students with L.D. (F(1, 52) = 0.63, p =.002, η2 = .17) Accuracy in two 2-digits  Typical students outperformed with borrowing 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

23. 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.

24. 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 time-consuming mental image strategy, seems to corroborate the well-established difficulty of students with LD in strategy choice and use (e.g. Learner & Beverley, 2014).

25. 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).

26. 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 time-consuming mental image strategy, seems to corroborate the well-established difficulty of students with LD in strategy choice and use (e.g. Learner & Beverley, 2014).

27. 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).

28. 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 time-consuming 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.

29. 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.

30. 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 toSiegler and Lamaire (1995) this condition does not provide data on adaptivity and efficiency. • The sample was small.

31. 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.

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