Research in Developmental Disabilities 38 (2015) 352–361

Contents lists available at ScienceDirect

Research in Developmental Disabilities

Program of arithmetic improvement by means of cognitive enhancement: An intervention in children with special educational needs ˜ o Dean ˜ o a,*, Sonia Alfonso a, Jagannath Prasad Das b Manuel Dean a b

Department of Evolutionary Psychology, University of Vigo, Campus As Lagoas, 32004 Ourense, Spain Department of Educational Psychology, University of Alberta, 6-123D Education North, Edmonton, Canada

A R T I C L E I N F O

A B S T R A C T

Article history: Received 17 October 2014 Accepted 19 December 2014 Available online 13 January 2015

This study reports the cognitive and arithmetic improvement of a mathematical model based on the program PASS Remedial Program (PREP), which aims to improve specific cognitive processes underlying academic skills such as arithmetic. For this purpose, a group of 20 students from the last four grades of Primary Education was divided into two groups. One group (n = 10) received training in the program and the other served as control. Students were assessed at pre and post intervention in the PASS cognitive processes (planning, attention, simultaneous and successive processing), general level of intelligence, and arithmetic performance in calculus and solving problems. Performance of children from the experimental group was significantly higher than that of the control group in cognitive process and arithmetic. This joint enhancement of cognitive and arithmetic processes was a result of the operationalization of training that promotes the encoding task, attention and planning, and learning by induction, mediation and verbalization. The implications of this are discussed. ß 2014 Elsevier Ltd. All rights reserved.

Keywords: Arithmetic achievement Primary education Cognitive training PASS processes PASS Remedial Program PREP Calculus and mathematical problem solving

1. Introduction The present study aims to improve the arithmetic abilities of the children who present educational support needs associated with low intelligence and children who have learning difficulties throughout their school years. Current studies show that children’s basic knowledge in kindergarten of counting, numbers, and elemental arithmetic are powerful predictors of mathematical achievement in subsequent schooling (Duncan et al., 2007). However, achievement of basic mathematical competence, in addition to domain specific skills, also requires general domain skills that affect learning (Geary, 2011; Mazzocco, Feigenson, & Halberda, 2011; Passolunghi & Lanfranchi, 2012). Among the domain specific mathematical skills (Geary, 2011), the skill of numerical comparison, which addresses the magnitude of the number—its value—has been underscored. Knowledge of numbers requires verbal and symbolized learning of numbers, reading, the place value, numeric sequences, and arithmetic operations with one or more digits (Levine, Jordan, & Huttenlocher, 1992; Siegler & Booth, 2004). There is evidence of the relationship of the ability to understand and represent numerical magnitude with the development of mathematics (De Smedt et al., 2009; Geary, Bow-Thomas, & Yao, 1992).

* Corresponding author at: Campus As Lagoas, 32004 Ourense, Spain. Tel.: +34 988387179. ˜ o). E-mail address: [email protected] (M.D. Dean http://dx.doi.org/10.1016/j.ridd.2014.12.032 0891-4222/ß 2014 Elsevier Ltd. All rights reserved.

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

353

Failures in the basic skills of number comparison, counting, reading and writing numbers, and place position lead to difficulties in solving arithmetic operations (Geary, Hamson, & Hoard, 2000; Landerl, Bevan, & Butterworth, 2004). Among the general domain skills related to learning and also to learning mathematics, the short-term memory, the working memory (Geary, 2011; Passolunghi & Lanfranchi, 2012), and planning (Best, Miller, & Naglieri, 2011; Das, Naglieri, & Kirby, 1994) have been particularly underscored. Deficits in the working memory components of the visual–spatial sketchpad (McLean & Hitch, 1999) and the phonological loop (Geary, 2011) lead to differential learning difficulties in various specific learning areas of mathematics and number representation, problem solving, and the numerical line that are attributed to the visual–spatial sketchpad, whereas other difficulties, such as counting to solve simple addition problems, are attributed to the phonological loop. Students with poor performance in mathematics also perform poorly in executive control tasks and inhibition of irrelevant information (Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Murphy, Mazzocco, Hanich, & Early, 2007; Passolunghi & Siegel, 2004; St Clair-Thompson & Gathercole, 2006). Their central executive is particularly compromised (Bull, Johnston, & Roy, 1999; Geary et al., 2007; Swanson, 1993), with each one of its components (maintaining information in working memory, task switching, and inhibiting the retrieval of irrelevant information) affecting mathematical learning in a different way (Bull & Scerif, 2001; Murphy et al., 2007; Passolunghi, Cornoldi, & De Liberto, 1999; Passolunghi & Siegel, 2004). Poor performance is also observed in mathematical tasks that require planning (Das et al., 1994). Planning is currently considered a general domain skill (Best et al., 2011) that influences mathematical performance at all schooling stages (Garofalo, 1986; Kroesbergen, Van Luit, Naglieri, Taddei, & Franchi, 2010; Naglieri & Das, 1987; Naglieri & Das, 1997b). Planning difficulties are revealed in mathematical disabilities (Naglieri & Gottling, 1995; Naglieri & Gottling, 1997; Naglieri & Johnson, 2000). Planning refers to the use of cognitive strategies to manage goal-oriented behaviors and to the development and implementation of an approach to tasks that are not carried out habitually (Das et al., 1994; Locascio, Mahone, Eason, & Cutting, 2010; Mahone et al., 2002). Planning processes are necessary for students’ decision-making about problem solving, self-monitoring, retrieval and application of the mathematical facts, and assessment of their responses (Das et al., 1994). Planning has been operationalized in the D.N: CAS (Naglieri & Das, 1997a) by means of complex tasks whose resolution requires the elaboration of action steps before their execution, assessing these actions (updating the working memory), avoiding or eliminating non-target behavior (inhibition), and changing the course of action if necessary (Naglieri & Das, 2005). Much of the research on intervention, improvement, and retrieval with poor arithmetic calculators has focused on the development of counting procedures or the retrieval of arithmetic facts due to broadly accepted viewpoint that both disabilities are an explanation and a source of poor mathematical performance (Fuchs et al., 2010). Other explanations of mathematical learning difficulties have included the improvement of general domain skills such as cognitive control (Diamond, Barnett, Thomas, & Munro, 2007), working memory (Holmes, Gathercole, & Dunning, 2009; Klingberg et al., 2005) or the effects of executive functioning (Thorell, Lindqvist, Nutley, Bohlin, & Klingberg, 2009). Research on intervention has also focused on the improvement of academic learning skills through training programs based on cognitive processes. These programs are designed to improve the cognitive development underlying domain specific learning in order to improve both the domain specific and cognitive processes by means of which students learn to interpret, recall, manipulate, and use information (Das, Parrila, & Papadopoulos, 2000). The present study presents the results of a cognitive mathematical intervention program based on the PASS Remedial Program PREP model, for the improvement of mathematics and the underlying cognitive processes. PREP or PASS Remedial Program (Das, 2000; Das, Carlson, Davidson, & Longe, 1997) is a model of academic and cognitive intervention that has been used in research of reading for more than three decades. The first studies (Brailsford, Snart, & Das, 1984; Crawford & Das, 1992; Kaufman & Kaufman, 1979; Krywaniuk & Das, 1976; Spencer, Snart, & Das, 1989) found that simultaneous processing is closely related to reading comprehension, and successive processing to decoding words. In a second analysis, using the PASS Reading Enhancement Program (Carlson & Das, 1997; Das, 1999, 2000; Das, Mishra, & Pool, 1995; Papadopoulos, Das, Parrila, & Kirby, 2003; Parrila, Das, Kendrick, Papadopoulos, & Kirby, 1999), improvement was also observed in word reading, reading comprehension, and cognitive processing strategies (Hayward, Das, & Janzen, 2007). Das et al. (1995) proved the higher efficacy of the complete reading PREP compared to the application of global PREP tasks only and to the PREP bridge tasks. Carlson and Das (1997) reported the efficacy of PREP in an experimental group of 4th grade students in Chapter 1 programs and in a control group in Word Attack and Word Identification. The results showed significant post-intervention improvement with PREP, as well as significant Group  Time interaction effects. Papadopoulos et al. (2003) also reported a long-term effect of the reading results of the PREP through a significant improvement in pseudoword reading in comparison with a control group. Subsequent replication studies have essentially reproduced the original results with children from 3rd, 4th, 5th, and 6th ˜ o, 2014). grade (Boden & Kirby, 1995); in 1st grade (Parrila et al., 1999) and 2nd grade (Ramos, Conde, Alfonso, & Dean Currently, studies with reading PREP successfully analyzed its efficacy to jointly improve reading difficulties and the underlying cognitive weakness, emphasizing its preventive value for reading difficulties (Papadopoulos, Charalambous, Kanari, & Loizou, 2004; Ramos et al., 2014). The results of the above-mentioned PREP programs show that the most effective procedure to teach these cognitive skills includes teaching inference, internalization of principles, mediation, and verbalization designed to allow subjects to

354

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

contribute their own experience to the learning, to gradually take control of the learning process, and to transfer what they learned to new situations. The purpose of the mathematical PASS Remedial Program is to facilitate and promote the development of the cognitive processes that contribute to the acquisition of mathematical skills, improving calculus and problem-solving disabilities. The tasks of the program were designed to facilitate the improvement of deficiencies in simultaneous and successive processing, planning, and attention. The training process provides the students with the opportunity to learn inferentially and to internalize the strategies in the most appropriate way for each student, maximizing generalization and facilitating transference according to the principles of the PREP model. The mathematical program provides training in tasks of basic calculus strategies and solving problem statements and it encourages students to apply them to academic tasks through mediation and verbalization. The present study extends the results of the PREP model in several ways: (a) It extends the use of the PREP model beyond learning to read to mathematics. The mathematical PASS Remedial Program model contributes to the joint development of students’ cognitive processes and arithmetic learning. The contents of the proposed program are expected to have a positive and significant influence on the scores of calculus and cognitive processes. (b) It extends the use of the PREP model also to students with low intellectual capacity (IQ < 75) who may benefit from this teaching system. Prior studies with PREP were carried out with samples of students with learning difficulties. This study follows those carried out previously by Conway (1985), who used global tasks of PREP to improve simultaneous and successive processing in mildly developmentally delayed children and by Brailsford et al. (1984), who emphasized the children’s process of task solution and not the repetition of a sequence of the teacher’s verbal instruction. We expect that the program content, through the structuring of the tasks according to the principles of the PREP model, will influence a cognitive and arithmetic improvement in students with low intelligence, as shown in their post intervention scores in standardized mathematical and cognitive tests. (c) Through the PREP, it favors the identification of the basic processes necessary for better mathematical learning in subjects with low intelligence. Correlational (Das et al., 1994) and empirical studies (Naglieri & Gottling, 1995; Naglieri & Gottling, 1997; Naglieri & Johnson, 2000) of the relations between PASS and mathematics have found univocal relations and influences between mathematical improvement and the weakened cognitive process (mathematics and planning) in students with mathematical learning difficulties. In contrast, in participants with low intelligence, improvement is expected in most of the cognitive processes when there is an intention to improve mathematics and PREP is used. The study will show improvement in various PASS cognitive processes that the current literature considers important for mathematical learning (Geary, 2011). (d) The intervention with the mathematical PREP model will not only improve cognitive and arithmetic functioning, it will also qualify the special mathematical educational needs associated with low intelligence from the joint perspective of specific skills (Geary, 2011) and general domain skills. This approach has been useful to explain mathematical performance and to differentiate groups of students with severe disabilities, low performance, and normal competence (Geary, 2011; Mazzocco et al., 2011; Passolunghi & Lanfranchi, 2012).

2. Method 2.1. Participants The sample was made of 20 students of Primary Education, with low intelligence (IQ < 75) and with no evidence of organic etiology. They all presented generalized global learning difficulties in all the areas. They were students in various public schools of Primary Education from a region of Pontevedra (Spain), who attended classes with their classmates. None were receiving any special educational treatment and they had always attended the same school. Their educational levels of reference ranged between 3rd and 6th grade of Primary Education. Their chronological age ranged from 9;2 to 12;7 years, with a mean of 10;6 years. They were selected from a list of more than 50 cases. Students with an IQ higher than 75 were not considered for this study. The list of participants selected was subsequently consulted with the Guidance services of the corresponding schools, confirming the scores obtained. The 20 selected students were randomly assigned to two groups. One group, made up of 10 children with a mean chronological age of 10;4 years, which included 4 boys and 6 girls, received treatment. The control group, with a mean chronological age of 10;9 years, which included 5 boys and 5 girls, carried on with their normal class activities. There were no significant differences in the general intelligence scores of the two groups, x2(1) = .572, p = .450. 2.2. Measures 2.2.1. General intelligence We used the Standard Progressive Matrices (SPM; Raven, Court, & Raven, 1996). The General Scale or SPM consists of a workbook with six sets (A to E) of 12 elements each. The students should complete complex figures and establish analogies. The split-half reliability rates were generally higher than .90, whereas the test–retest rates ranged between .83 and .90, depending on the characteristics of the samples. With regard to validity, the authors indicate that the concurrent and predictive rates vary with age, possibly with sex, with sample homogeneity, and with the conceptual implications of the definition of the criteria.

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

355

2.2.2. Mathematical achievement We used the Evalu´a-2 (Garcı´a & Gonza´lez, 1996), which is made up of various tests. We only used the Mathematical Learnings, in which the basic acquisitions of the mathematical curriculum are briefly assessed. It includes two subtests: Calculus and Counting, and Problem solving. The Mathematical Learnings test of Evalu´a-2 poses diverse arithmetic problems involving knowledge of numbers below 1000, numerical sequences, value differences of numbers, acquisition of the automatisms of addition and subtraction, comprehension of the problem, and adequate selection of the resolution procedure. The reliability of the battery using the Pearson coefficient presents rates higher than .71 for Evalu´a-2. The correlation of the different subtests with academic achievement was .67 for calculus, and the homogeneity index was .86. 2.2.3. Cognitive processing We used the D.N: CAS (Naglieri & Das, 1997a), battery to measure planning, attention, and encoding of information. 2.2.3.1. Planning. The Planning subtests present tasks that require the children to make decisions in order to solve them. Solving the tasks requires the children to create an action plan, apply it, verify it according to the original goal, and modify it if necessary. Success on the Planning subtests requires children to elaborate an action plan, assess its utility, control its effectiveness, correct or reject an old plan when the task requires a change, and control impulsive performance. The subtests include Matching Numbers, Planned Codes and Planned Connections. 2.2.3.2. Attention. In the Attention subtests, the children must use focal attention to detect a particular stimulus and avoid responding to irrelevant stimuli. In the attention subtests, they must focus their cognitive activity, detect a specific stimulus, and inhibit responses to irrelevant competing stimuli. These subtests always imply examining the characteristics of the stimuli and responding to some characteristics and not to others in a complex situation. The subtests are: Expressive Attention, Number Detection and Receptive Attention. 2.2.3.3. Simultaneous processing. This scale includes tasks that require the perception of the parts of a gestalt, the comprehension of logical-grammatical relations, and the synthesis of the parts into integrated groups, using both verbal and nonverbal content. This takes place through the examination of stimuli during the activity or the recall of the stimuli. To measure this kind of processing, we used the Non-verbal Matrixes, Verbal-Spatial Relations, and Figure Memory subtests. 2.2.3.4. Successive processing. The subtests of this scale require the individual to use the information presented in a specific order that is necessary to understand its meaning, the perception and reproduction of the natural sequence of stimuli, the comprehension of sentences based on syntactic relations, and the articulation of isolated sounds in a consecutive sequence. In the tasks of the successive processing subtests, the individual reproduces a particular sequence of questions about events or responses, which require the correct interpretation from the linearity of the events. The subtests correspond to Word Series, Sentence Repetition, and Sentence Questions. ˜ o, Alfonso, & Ferna´ndez, 2006) was calculated with the split-half Reliability of the D.N: CAS for the Spanish sample (Dean procedure for all the simultaneous and successive subtests (except for speech rate), corrected with the Spearman–Brown formula. For the Planning, Attention, and Speech Rate subtests, we used test–retest. The mean reliability of the sample of 1222 cases for each one of the scales was .90 (Planning), .89 (Attention), .92 (Simultaneous Processing), and .91 (Successive Processing). Construct validity was calculated with confirmatory factor analysis carried out separately in four age groups (5– 7, 8–10, 11–13, and 14–17 years). The model was assessed through various goodness-of-fit and incremental indexes. The results (the goodness of fit and adjusted goodness of fit indexes were all higher than .90, and the root mean square residual values were below .10) indicated a good correspondence between the PASS model and the data for each one of the four age groups. 2.2.4. PASS Remedial Program of arithmetic The PREP-based program for mathematical improvement was developed as an arithmetic recovery program based the PASS processes (planning, attention, simultaneous processing and successive processing) of cognitive functioning (Das et al., 1994). The four cognitive components are differentially related to mathematical skills. The literature has reported relations between measures of mathematical achievement and the PASS cognitive processes (Das et al., 1994) simultaneous ˜ o, 2011), simultaneous and successive processing (Garofalo, 1986; Kroesbergen et al., processing (Iglesias-Sarmiento & Dean 2010; Naglieri & Das, 1987), planning (Ashman & Das, 1980; Joseph & Hunter, 2001; Kirby & Ashman, 1984), and attention (Kroesbergen, Van Luit, & Naglieri, 2003; Warrick, 1989), concluding that successive processing correlates with mathematical performance, but generally at a lower level than simultaneous processing (Das, 1988; Leong, Cheng, & Das, 1985). These predictions are derived from the need of correct simultaneous processing to establish significant numerical units and to integrate them in higher-level numerical units, with one or more numbers, comprehension of the resulting numerical value, as well as of the statements of the verbal arithmetic problems. Successive processing is necessary to analyze the sequence of digits in numbers and the algorithms of calculus operations. The attention process is required to focus on the pertinent dimensions of the problem, inhibit impertinent information, maintain the necessary numerical information to operate with numbers (Kroesbergen et al., 2003; Warrick, 1989) as a function of the goal and of the action regulation,

356

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

according to the decisions made to solve the problem, the cognitive resources to be used, and the strategies employed to perform the operation (planning process) (Das et al., 1994; Kirby & Ashman, 1984). The program aims to improve the cognitive processing skills underlying arithmetic performance by means of inductive teaching (Carlson & Das, 1997). Like the PREP, it is structured so that the tacitly acquired strategies will be susceptible to be used for this purpose (Papadopoulos et al., 2003). It is made up of 10 tasks, each one with three difficulty levels, designed to improve the PASS cognitive processes and the difficulties in numeric skills. The tasks of the program involve a global component without academic content and a curriculum-related bridging component. The global component provides the students with the opportunity to apply their experience, perform the external instruction, and elaborate their own knowledge with the help of internal speech. That is, to inductively internalize tacit problem-solving strategies in their own way, which facilitates transfer (Das et al., 1995). The bridging component involves the same cognitive demands as its matched global component and is closely related to academic content. The increase in the capacity to use the strategies and the knowledge about when to use them occurs during the training (Das et al., 1995). Out of the original tasks included in the PASS Reading Enhancement Program PREP, we selected the ten global tasks. The mathematical bridging tasks were elaborated in correspondence with each one of the global tasks, so that both types of task involved the same cognitive demands. The name of the bridging tasks is the same one as that given by the authors to the global tasks in the original program. The difference is that now, the bridging tasks refer to mathematics. A description of the tasks is presented in Appendix A. 2.3. Procedure The pre- and post-intervention measures were carried out individually following the administration system proposed in the corresponding tests and adapted to the performance instructions. When necessary, the corresponding examples were clarified until the evaluator was sure that the student understood what he or she had to do in each task. The program was applied, facilitating the development of strategies such as rehearsal, sequencing, categorization, relation, seriation, procedures of mental calculus and problem solving, performance monitoring, prediction, review of the prediction, and transcoding of the numerals included in the tasks. The students used these strategies through their experience with the tasks. Instead of these strategies being explicitly taught by the teacher, the children were encouraged to become aware of their use through verbalization (Papadopoulos et al., 2003). The mediation of learning as a collaborative process allows the students to carry out cognitive tasks with the help of others who solve them. Task structure favors this collaboration in various ways: (a) it provides a format for the student to develop the established strategies; (b) it offers scaffolding through a series of prompts that provide the student with the amount of help needed to complete the task successfully. In the case of unsatisfactory responses, the instructor provides aid in diverse phases (not the correct solutions), so that the children can pay attention to certain details or to certain alternative task resolution strategies that help them to become aware of the principles underlying the problem, to distinguish what is essential from what is incidental, and, consequently, to find the correct solution; and (c) it is a monitoring system that helps the teacher to determine when material is too difficult for a child or when a child is able to successfully progress to a more difficult level (Das et al., 1995). The instructor indicates in the response workbook the level of help needed to achieve the correct response and accordingly grants the corresponding points. With the record of the results, it can be seen what each child achieves by himself and what is achieved with the help of other classmates or the instructor. Once task is solved, independently of the prompting level, the students are encouraged to verbalize what they did by themselves to remember its order. The descriptions of the strategies carried out by the students are followed by the instructor’s recognition and practice. The instructor discusses the strategy itself rather than its formal name and describes the one used by the student if the student does not verbalize it (Das et al., 1995). Each global and bridging task is made up of three difficulty levels. A criterion of 80% correct responses is required before a child can proceed to the next level of difficulty. If the criterion was not met, a set of parallel tasks at the same difficulty level was applied. If the child did not obtain the percentage of correct responses with the parallel set, the instructor would provide aids (not the correct solutions) so the child could pay attention to certain details or certain alternative strategies which could allow him to become aware of the principles underlying the problem and, consequently, to find the correct solution. The intervention was carried out in small groups of 3 or 4 students. There were three groups in all, and their members remained together in all the sessions of the program. The program was carried out in 35 sessions, one session per school-day, for each group. In each session, a global and a bridging difficulty level were applied. Mean session duration was 60 min. 2.4. Data analysis We used a pre/intervention/post design, comparing the experimental and control groups. We conducted a 2  2  5 repeated measures ANOVA’s with factors [Group (2: experimental, control)  Moment of measurement (2: pre, post)  Quality of learning (5: planning, attention, simultaneous and successive processing, calculus and problem solving]. As within-factors, we used the moment of measurement and the quality of learning, and as between-factor, the group. We used the SPSS, version 18.0.

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

357

3. Results Table 1 presents the mean score and standard deviation of the variables measuring quality of learning in the experimental and control groups. As can be observed Table 1, there were important oscillations in the mean scores of the variables that make up the quality of learning as a function of the measurement moment and the group. Considering the three study variables simultaneously, the repeated measures ANOVA’s revealed the significant interaction, with a large effect size (Cohen, 1992), of Group  Moment of measurement  Quality of learning, F(4, 72) = 11.385, p < .001, h2p ¼ :387. At the pre-measurement, the participants of the experimental and control groups had a similar performance in Planning, F(1, 18) = .018, p > .05, h2p ¼ :001, Attention, F(1, 18) = 1.152, p > .05, h2p ¼ :060, Simultaneous, F(1, 18) = .171, p > .05, h2p ¼ :009, and Successive processing, F(1, 18) = .479, p > .05, h2p ¼ :026, and Calculus and problem solving, F(1, 18) = .001, p > .05, h2p ¼ :000 (see Fig. 1). After the intervention, at posttest measurement, the experimental group obtained significantly higher mean scores than the control group in Planning, F(1, 18) = 5.117, p < .05, h2p ¼ :221, Attention, F(1, 18) = 4.531, p < .05, h2p ¼ :201, Simultaneous processing, F(1, 18) = 4.878, p < .05, h2p ¼ :213, and Calculus and problem solving, F(1, 18) = 5.980, p < .05, h2p ¼ :249 (see Fig. 1). Analysis of the evolution of each group from pretest to posttest showed that the control group only obtained significant gains in their scores of Simultaneous processing, F(1, 18) = 11.235, p < .01, h2p ¼ :384. The experimental group obtained significant gains with a large effect size (Cohen, 1992) in Planning, F(1, 18) = 42.685, p < .001, h2p ¼ :703, Attention, F(1, 18) = 19.395, p < .001, h2p ¼ :519, Simultaneous processing, F(1, 18) = 52.122, p < .001, h2p ¼ :743, and Calculus, and problem solving, F(1, 18) = 52.484, p < .001, h2p ¼ :745. Combining the factors Group  Moment of measurement, a significant interaction with a large effect size (Cohen, 1992) was produced, F(1, 18) = 50.443, p < .001, h2p ¼ :737. The resulting comparisons of the analysis of variance of this interaction showed that, at pretest, the participants of the experimental (M = 72.46) and control groups (M = 70.54) had similar performances with regard to Quality of learning, F(1, 18) = .112, p > .05, h2p ¼ :006. At posttest, the mean scores obtained by the experimental group (M = 84.84) were significantly greater, F(1, 18) = 4.901, p < .05, h2p ¼ :214, than those obtained by the control group (M = 72.34). Taking into consideration the variables Moment of measurement  Quality of learning, we also obtained a significant interaction, F(4, 72) = 10.861, p < .001, h2p ¼ :376, with a large effect size (Cohen, 1992). This interaction indicates that the mean scores of the participants varied significantly from pretest to posttest in Planning (Mpretest = 75.80; Mposttest = 82.60), F(1, 18) = 26.962, p < .001, h2p ¼ :600, Attention (Mpretest = 82.50; Mposttest = 87.30), F(1, 18) = 12.084, p < .01, h2p ¼ :402, Simultaneous processing (Mpretest = 76.45; Mposttest = 84.65), F(1, 18) = 55.878, p < .001, h2p ¼ :756, and Calculus and problem solving (Mpretest = 40.25; Mposttest = 55.35), F(1, 18) = 27.502, p < .001, h2p ¼ :604. With regard to the main effects of the factors of the study, a main effect with a large effect size (Cohen, 1992) was found for Moment of Measurement, F(1, 18) = 90.611, p < .001, h2p ¼ :834, indicating a significant increase in the participants’ mean scores after the intervention (M = 78.59) with regard to their initial situation (M = 71.50). There was also a main effect with a large effect size (Cohen, 1992) of Quality of learning, F(4, 72) = 69.400, p < .001, h2p ¼ :794, as there were large variations between the mean scores of its variables (MPlanning = 79.20, MAttention = 84.90, MSimultaneous = 80.55, MSuccessive = 82.77, MCalculus and problem-solving = 47.80). No main effect was found for the variable Group, F(1, 18) = 1.630, p > .05, h2p ¼ :083, indicating that the global mean scores obtained by the experimental and control groups were similar (Mexperimental = 78.65, Mcontrol = 71.44).

4. Discussion The goal of the present study was to analyze effect of the cognitive intervention with the mathematical PASS Remedial Program on arithmetic performance and on the cognitive weaknesses of students with low intelligence. We wished to determine the effect of the PREP program of arithmetic on the cognitive and arithmetic processes in students with low intellectual capacity.

Table 1 Descriptive statistics of experimental and control group at pre- and post-intervention in cognitive processes, calculus and problem solving. Quality of learning

Planning Attention Simultaneous processing Successive processing Calculus and problem solving

Experimental Group n = 10

Control Group n = 10

Pretest

Posttest

Pretest

Posttest

M (SD)

M (SD)

M (SD)

M (SD)

75.50 84.70 77.50 84.50 40.10

87.60 93.30 88.70 85.00 69.60

76.10 80.30 75.40 80.50 40.40

77.60 81.30 80.60 81.10 41.10

(8.85) (5.77) (11.38) (14.87) (25.22)

(6.00) (9.61) (7.15) (14.02) (26.94)

(10.80) (11.60) (11.32) (10.64) (25.58)

(12.62) (15.01) (9.13) (9.87) (25.15)

358

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

Fig. 1. Graphic representation of the triple interaction Group  Moment of measurement  Quality of learning.

Both goals were met. The students who followed the program showed significant gains in their calculus and problemsolving scores, as well as in the posttest scores of cognitive processes, as measured by the D.N:CAS battery and by Evalu´a. This gain was significantly greater than that obtained by the control group. This significant improvement in arithmetic and cognitive processes was revealed through the significant interaction of the three variables of the study: Moment of measurement (pre-post), Quality of learning achieved (mathematical and PASS cognitive), and Group (experimentalcontrol). The initial measures of the groups of the study were equivalent; no differences were observed in their scores of calculus and problem solving or in their processes of planning and attention, and simultaneous and successive processing. After the intervention, the experimental group was clearly different from the control group in their arithmetic and cognitive scores, which varied significantly from pretest to posttest. These results are coherent with other findings obtained with the PREP model, in which a significant improvement in reading and cognitive processes was produced (Papadopoulos et al., 2003; Parrila et al., 1999; Ramos et al., 2014) after following the program, and, in addition, an interactive effect of the two variables group and moment of measurement was produced (Carlson & Das, 1997; Das et al., 1995; Ramos et al., 2014). The quality of learning facilitated by the PREP intervention program is highly significant. The students who followed the program improved their cognitive functioning of planning, attention, and simultaneous processing, and their performance in calculus and problem solving, whereas the students who did not follow the program did not improve their arithmetic performance or their cognitive functioning, except for simultaneous processing. The cognitive functioning of the experimental group was considerably higher after the intervention, as well as their arithmetic performance, compared with the control group. This result is important because it is difficult to display better cognitive functioning and competence in tasks in which the participants have not been trained (Das et al., 1995). The improvement was manifest in standardized calculus tests, mathematical problem solving, and cognitive process tests like the Evalu´a and the D.N: CAS battery. This result is also important because there was a significant improvement in the control group in simultaneous processing at the posttest measurement. However, the improvement in the control group did not overshadow that obtained by the experimental group, which was significantly higher.

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

359

These results sustain the utility of considering the PREP model to design mathematical interventions. They also support the PREP as especially indicated for students with special educational needs associated with low intelligence, in view of the functional improvement of learning it promotes and the facilitation for the students to generalize what they learned to new tasks, helping them to learn how to learn. The quality of the cognitive functioning of the experimental group compared with the control group was observed in the planning, attention, and simultaneous processing tests. This improvement means that the joint functioning of the three processes is necessary for mathematical improvement in children with low intelligence, using the PREP model. The correlational studies between mathematics and cognitive processes (Ashman & Das, 1980; Das et al., 1994; Garofalo, 1986; ˜ o, 2011; Joseph & Hunter, 2001; Kirby & Ashman, 1984; Kroesbergen et al., 2003; Warrick, 1989) Iglesias-Sarmiento & Dean underscored the influence of a cognitive process in mathematical performance as a function of school grade. The empirical studies also justified the improvement of learning through the improvement of the promoted cognitive process (Carlson & Das, 1997; Das et al., 1995; Naglieri & Gottling, 1995; Naglieri & Gottling, 1997; Naglieri & Johnson, 2000; Ramos et al., 2014). In contrast, when using PREP training in children with low intelligence, various cognitive processes start to function conjointly: planning, attention, and simultaneous processing, in addition to mathematical learning. The processes improved by PREP concurrently with the tasks are considered important for mathematical performance in general, and particularly, for students with low intelligence. From this perspective, the PREP model identifies the cognitive processes that are necessary for mathematical improvement. This type of intervention allows examining low intelligence in depth and it identifies the processes the subjects already have in order to improve their learning. From the perspective of this study, the PREP model not only identifies and improves cognitive and academic processes, it also qualifies the special educational needs of students with low intelligence. These educational mathematical needs are related to the integration of visual–spatial information, focus of selective and sustained attention, and inhibitional control, as well as to the elaboration of the action steps prior to execution, assessment of these actions (updating working memory), and elimination of non-target behavior (inhibition), as measured by the D.N: CAS (Naglieri & Das, 2005). The improvement of these processes helps making decisions about solving problems, monitoring the action, retrieving and applying the number facts, and assessing the responses (Das et al., 1994), as well as reading numbers, considering their place value, transcoding them, and managing the numerical system more fluently (Fuson & Kwon, 1992; Villarroel et al., 2012). The PREP model is thus coherent with the approach of the convergent study of specific and general domain skills (Geary, 2011; Mazzocco et al., 2011; Passolunghi & Lanfranchi, 2012), considered important to explain mathematical learning difficulties.

Appendix A. Description of bridging tasks Transportation Bridging I. In this task, the child is shown a number made up of various digits written in a single-line matrix strip divided into cells, such that each digit occupies a cell. First, entire strip is shown. Next, the strip is concealed and the child must construct the number, using for this purpose cards with the correct numbers mixed with five distracter cards, or ten cards with natural numbers. Then, the child must read the number formed. Lastly, the instructor directs the child’s attention to the number of digits, how the number can be decomposed, whether or not the digits are repeated, and that name the digits one to one. Transportation Bridging II. Two to five cards with a number written on them are presented to the student. The student must read the numbers and try to memorize them. The numbers are then concealed and the student must repeat them in the correct order, and subsequently say out loud the number formed. Joining Shapes Bridging. Sheets with two or plus rows of numbers placed in column ranging from 0 to 9 are presented to the student. The task consists of joining digits to form numbers, advancing diagonally and following given rules. The student must start at the first digit of the top row and move diagonally until reaching a digit the last row, proceeding diagonally upward again until reaching another digit of the first row. This sequence repeats successively until completing the sheet. The last digit of each number is, in turn, the first digit of the next number. The child must also say the numbers formed out loud. Window Sequencing Bridging. A 2  2 window is used to present digits that form a number. The digits are usually presented one at a time, although they are occasionally presented two at a time. The child must read the digits presented in the window, retain them, join them, and finally say the number they form out loud. Connecting Letters (Numbers) Bridging. A page with five vertical digits on each side is presented. The student is required to follow a line to find which digit on the left side of a page is connected—by superimposed wavy lines—to which digit on the right side of a page. More numbers appear along in these lines when the child follows the lines to join the ends. The child must identify the digits, following the lines visually, and then write them. Related Memory Set Bridging. The task shows two columns of numbers, three numbers in the left column and one number in the right column. The child must identify a number on the left that added to (or subtracted from) the number on the right, gives the number proposed by the instructor. After identifying the number, the child is required to add the other two numbers on the left to (or subtract from) the number on the right, say the result out loud, and say whether this result is higher or lower than the first sum (subtraction) carried out. Matrices Bridging. A five-cell matrix is presented. The matrix is designed as a cross: there is one central cell, with one cell on each of its four sides. Each cell of the matrix contains one number Four of these numbers are related to each other and the other is not. The child should remember the numbers in the correct order and then identify the four related numbers and the unrelated number and explain why the former are related.

360

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

Sentence Verification Bridging. A drawing and a card on which various problems are written are presented to the child. The instructor reads the problems from the card. Only one of the problems corresponds to the drawing; the child is required to choose the correct problem text. Tracking Bridging. This task is an original reading PREP task. It is structured in two parts. The first part contains a map of a commercial center. The child is given a list of tasks, indicating the starting point, and should make a plan to carry out the task, indicating the most efficient path. The second part is organized around a print of a children’s park. The child analyzes everything shown on the print. Then, a card with a written paragraph containing several prompts to locate a hiding place is presented, and the child must discover this hiding place. Shape Design Global-Bridging. This task is an original PREP reading task. The instructor reads a sentence written on a card describing how some animals (2–5) are placed in relation to each other. The child tries to visualize the scene with the animals correctly placed, and should then place the animals correctly as they were described in the scene on the card. Shapes and Objects Bridging. In this task, seven, ten, or thirteen cards are presented. Each card contains a certain type of information. The cards can be grouped into two, three, or four categories as a function of their content. The cards that show the categories are underlined and placed previously on the table. In each set, there is distracter card which does not match any of the indicated categories. References Ashman, A. F., & Das, J. P. (1980). Relation between planning and simultaneous-successive processing. Perceptual and Motor Skills, 51, 371–382. Best, J., Miller, P., & Naglieri, J. A. (2011). Relations between executive function and academic achievement from ages 5 to 17 in a large, representative national sample. Learning and Individual Differences, 21, 327–336. Boden, C., & Kirby, J. R. (1995). Successive processing, phonological coding, and the remediation of reading. Journal of Cognitive Education, 4(2/3), 19–32. Brailsford, A., Snart, F., & Das, J. P. (1984). Strategy training and reading comprehension. Journal of Learning Disabilities, 17, 287–290. Bull, R., & Scerif, G. (2001). Executive functioning as a predictor of children’s mathematical abilities: Inhibition, switching, and working memory. Developmental Neuropsychology, 19, 273–293. Bull, R., Johnston, R. S., & Roy, J. A. (1999). Exploring the roles of the visual–spatial sketch pad and central executive in children’s arithmetical skills: Views from cognition and developmental neuropsychology. Developmental Neuropsychology, 15, 421–442. Carlson, J. S., & Das, J. P. (1997). A process approach to remediating word decoding deficiencies in Chapter 1 children. Learning Disability Quarterly, 20, 93–102. Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159. Conway, R. F. (1985). The information processing model and the mildly developmentally delayed child: Assessment and training (Unpublished doctoral dissertation) Newcastle, Australia: Macquarie University. Crawford, S. A. S., & Das, J. P. (1992). Teaching for transfer: A program for remediation in reading. In J. Carlson (Ed.), Advances in cognitive and educational practice (pp. 73–103). Greenwich, CT: JAI. Das, J. P. (1988). Simultaneous-successive processing and planning. In R. Schmeck (Ed.), Learning styles and learning strategies (pp. 101–129). New York, NY: Plenum. Das, J. P. (1999). PASS Reading Enhancement Program (PREP). Deal, NJ: Sarka Educational Resources. Das, J. P. (2000). PREP: A cognitive remediation program in theory and practice. Developmental Disabilities Bulletin, 28(2), 83–95. Das, J. P., Naglieri, J. A., & Kirby, J. R. (1994). Assessment of cognitive processes: The PASS theory of intelligence. Needham Heights, MA: Allyn & Bacon. Das, J. P., Mishra, R. K., & Pool, J. E. (1995). An experiment on cognitive remediation of word-reading difficulty. Journal of Learning Disabilities, 28, 66–79. Das, J. P., Carlson, J., Davidson, M. B., & Longe, K. (1997). PREP: PASS remedial program. Seattle, WA: Hogrefe. Das, J. P., Parrila, R. K., & Papadopoulos, T. C. (2000). Cognitive education and reading disability. In A. Kozulin & Y. Rand (Eds.), Experience of mediated learning: An impact of Feuerstein’s theory in education and psychology (pp. 274–291). Oxford, UK: Pergamon Press. De Smedt, B., Janssen, R., Bouwens, K., Verschaffell, L., Boets, B., & Ghesquie`re, P. (2009). Working memory and individual differences in mathematics achievement: A longitudinal study from first grade to second grade. Journal of Experimental Child Psychology, 103(2), 186–201. Dean˜o, M., Alfonso, S., & Ferna´ndez, M. J. (2006). El D.N: CAS como sistema de evaluacio´n cognitiva para el aprendizaje [The D.N:CAS as a cognitive assessment system of learning]. In M. Dean˜o (Ed.), Formacio´n del profesorado para atender a las necesidades especı´ficas de apoyo educativo. XXXII Reunio´n Cientı´fica Anual (pp. 159–182). Ourense, Spain: AEDES. Diamond, A., Barnett, S., Thomas, J., & Munro, S. (2007). Preschool program improves cognitive control. Science, 207(318), 1387–1388. Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., et al. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428–1446. Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., et al. (2010). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematics difficulties. Learning and Individual Differences, 20, 89–100. Fuson, K. C., & Kwon, Y. (1992). Korean children’s understanding of multidigit addition and subtraction. Child Developmental, 63, 491–506. Garcı´a, J., & Gonza´lez, D. (1996). Baterı´a Psicopedago´gica Evalu´a. 2-4-6. [Psychology Battery Assess. 2-4-6]. Madrid, Spain: Editorial EOS. Garofalo, J. (1986). Simultaneous synthesis, regulation, and arithmetical performance. Journal of Psychoeducational Assessment, 4, 229–238. Geary, D. C. (2011). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental Behavioral Pediatrics, 32(3), 250–263. Geary, D. C., Bow-Thomas, C. C., & Yao, Y. (1992). Counting knowledge and skill in cognitive addition: A comparison of normal and mathematically disabled children. Journal of Experimental Child Psychology, 54, 372–391. Geary, D. C., Hamson, C. O., & Hoard, M. K. (2000). Numerical and arithmetical cognition: A longitudinal study of process and concept deficits in children with learning disability. Journal of Experimental Child Psychology, 77, 236–263. Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Developmental, 78, 1343–1359. Hayward, D., Das, J. P., & Janzen, T. (2007). Improvement in reading through cognitive enhancement: A remediation study of Canadian First Nations children. Journal of Learning Disabilities, 40(5), 443–457. Holmes, J., Gathercole, S. E., & Dunning, D. L. (2009). Adaptive training leads to sustained enhancement of poor working memory in children. Developmental Science, 12(4).. F9–F15. Iglesias-Sarmiento, V., & Dean˜o, M. (2011). Cognitive processing and mathematical achievement: A study with schoolchildren between fourth and sixth grade of primary education. Journal of Learning Disabilities, 44, 570–583. Joseph, L. M., & Hunter, A. D. (2001). Differential application of a cue card strategy for solving fraction problems: Exploring instructional utility of the cognitive assessment system. Child Study Journal, 31, 123–136. Kaufman, D., & Kaufman, P. (1979). Strategy training and remedial techniques. Journal of Learning Disabilities, 12(6), 416–419. Kirby, J. R., & Ashman, A. F. (1984). Planning skills and mathematics achievement: Implications regarding learning disability. Journal of Psychoeducational Assessment, 2, 9–22.

M.D. Dean˜o et al. / Research in Developmental Disabilities 38 (2015) 352–361

361

Klingberg, T., Fernell, E., Olesen, P. J., Johnson, M., Gustafsson, P., Dahlstro¨m, K., et al. (2005). Computerized training of working memory in children with ADHD: A randomised, controlled trial. Journal of the American Academy of Child and Adolescent Psychiatry, 44, 177–186. Kroesbergen, E. H., Van Luit, J. E. H., & Naglieri, J. A. (2003). Mathematics learning difficulties and PASS cognitive processes. Journal of Learning Disabilities, 36, 574–582. Kroesbergen, E. H., Van Luit, J. E. H., Naglieri, J. A., Taddei, S., & Franchi, E. (2010). PASS processes and early mathematics skills in Dutch and Italian kindergartners. Journal of Psychoeducational Assessment, 28, 585–593. Krywaniuk, L. W., & Das, J. P. (1976). Cognitive strategies in native children: Analysis and intervention. Alberta Journal of Educational Research, 22, 271–280. Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A study of 8–9-year-old students. Cognition, 93, 99–125. Leong, C. K., Cheng, S. C., & Das, J. P. (1985). Simultaneous-successive syntheses and planning in Chinese readers. International Journal of Psychology, 20, 19–31. Levine, S. C., Jordan, N. C., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53, 72–103. Locascio, G., Mahone, E. M., Eason, S. H., & Cutting, L. E. (2010). Executive dysfunction among children with reading comprehension deficits. Journal of Learning Disabilities, 43, 441–454. Mahone, E. M., Hagelthorn, K. M., Cutting, L. E., Schuerholz, L. J., Pelletier, S. F., Rawlins, C., et al. (2002). Effects of IQ on executive function measures in children with ADHD. Child Neuropsychology, 8, 52–65. Mazzocco, M. M., Feigenson, L., & Halberda, J. (2011). Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia). Child Development, 82(4), 1224–1237. McLean, J. F., & Hitch, G. J. (1999). Working memory impairments in children with specific arithmetic learning difficulties. Journal of Experimental Child Psychology, 74, 240–260. Murphy, M. M., Mazzocco, M. M. M., Hanich, L. B., & Early, M. C. (2007). Cognitive characteristics of children with mathematics learning disability (MLD) vary as a function of the cut off criterion used to define MLD. Journal of Learning Disabilities, 40, 458–478. Naglieri, J. A., & Das, J. P. (1987). Construct and criterion related validity of planning, simultaneous and successive cognitive processing tasks. Journal of Psychoeducational Assessment, 4, 353–363. Naglieri, J. A., & Das, J. P. (1997a). Cognitive assessment system. Itasca, IL: Riverside. Naglieri, J. A., & Das, J. P. (1997b). Cognitive assessment system interpretive handbook. Itasca, IL: Riverside. Naglieri, J. A., & Das, J. P. (2005). Planning, attention, simultaneous, successive (PASS) theory: A revision of the concept of intelligence. In D. P. Flanagan & P. L. Harrison (Eds.), Contemporary intellectual assessment (2nd ed., pp. 136–182). New York, NY: Guilford. Naglieri, J. A., & Gottling, S. H. (1995). A cognitive education approach to math instruction for the learning disabled: An individual study. Psychological Reports, 76, 1343–1354. Naglieri, J. A., & Gottling, S. (1997). Mathematics instruction and PASS cognitive processes: An intervention study. Journal of Learning Disabilities, 30, 513–520. Naglieri, J. A., & Johnson, D. (2000). Effectiveness of a cognitive strategy intervention to improve math calculation based on the PASS theory. Journal of Learning Disabilities, 33, 591–597. Papadopoulos, T. C., Das, J. P., Parrila, R. K., & Kirby, J. R. (2003). Children at risk of developing reading difficulties: A remediation study. School Psychology International, 24, 340–366. Papadopoulos, T. C., Charalambous, A., Kanari, A., & Loizou, M. (2004). Kindergarten cognitive intervention for reading difficulties: The PREP remediation in Greek. European Journal of Psychology of Education, 19, 79–105. Parrila, R. K., Das, J. P., Kendrick, M. E., Papadopoulos, T. C., & Kirby, J. R. (1999). Efficacy of a cognitive reading remediation program for at-risk children in grade 1. Developmental Disabilities Bulletin, 27(2), 1–31. Passolunghi, M. C., & Lanfranchi, S. (2012). Domain-specific and domain-general precursors of mathematical achievement: A longitudinal study from kindergarten to first grade. British Journal of Educational Psychology, 82, 42–63. Passolunghi, M. C., & Siegel, L. S. (2004). Working memory and access to numerical information in children with disability in mathematics. Journal of Experimental Child Psychology, 88, 348–367. Passolunghi, M. C., Cornoldi, C., & De Liberto, S. (1999). Working memory and intrusions of irrelevant information in a group of specific poor problem solvers. Memory & Cognition, 27, 779–790. Ramos, A., Conde, A., Alfonso, S., & Dean˜o, M. (2014). Prevencio´n del riesgo de dificultad lectora en estudiantes de primer ciclo de Educacio´n Primaria [Reading difficulty risk prevention in first-year Primary School children]. Aula Abierta, 42(1), 15–21. Raven, J. C., Court, J. H., & Raven, J. (1996). Manual for Raven’s standard progressive matrices. Oxford. England Oxford Psychologists Press. Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75, 428–444. Spencer, F., Snart, F., & Das, J. P. (1989). A process-based approach to the remediation of spelling in students with reading disabilities. Alberta Journal of Educational Research, 35, 269–282. St Clair-Thompson, H. L., & Gathercole, S. E. (2006). Executive functions and achievements in school: Shifting, updating, inhibition, and working memory. Quarterly Journal of Experimental Psychology, 59, 745–759. Swanson, H. L. (1993). Working memory in learning disability subgroups. Journal of Experimental Child Psychology, 56, 87–114. Thorell, L. B., Lindqvist, S., Nutley, B. S., Bohlin, G., & Klingberg, T. (2009). Training and transfer effects of executive functions in preschool children. European Society for Developmental Psychology, 12, 106–113. Villarroel, R., Jime´nez, J. E., Rodrı´guez, C., Bisschop, E., & Peake, C. (2012). Desarrollo del concepto de nu´mero en nin˜os con dificultades de aprendizaje en matema´ticas [Development of the number concept in children with learning difficulties in mathematics]. In J. A. Gonza´lez-Pienda, C. Rodrı´guez, D. A´lvarez, R. Cerezo, E. Ferna´ndez, M. Cueli, T. Garcı´a, E. Tueru, & N. Sua´rez (Eds.), (Coords.), Learning disabilities: Present and future (pp. 560–568). Oviedo, Spain: Ediciones de la Universidad de Oviedo. Warrick, P. D. (1989). Investigation of the PASS model (planning, attention, simultaneous, successive) of cognitive processing and mathematics achievement (Unpublished doctoral dissertation) Ohio State University: Columbus.