Learning difficulties are often associated with Attention-Deficit/Hyperactivity Disorder (ADHD), which has a great impact upon the child’s educational development. During school years, the disorder is often associated with poor academic performance, grade retention, suspension, expulsion (Barkley, 2014; Lahey et al., 2004; Rohde et al., 1999) and difficulties in relationships (Lahey et al., 2004), resulting in a worse quality of life (Klassen et al., 2004). Some authors (Faraone et al., 2001; Mayes et al., 2000) have related this worse performance with the high prevalence of comorbidity between learning disability (LD) and ADHD. Although several theoretical models have been proposed to explain this comorbidity, three of them received greater attention (Biederman et al., 2004; Martinussen et al., 2005; Rhee et al., 2005; Shanahan et al., 2006; Willcutt et al., 2005).  The first one suggests that the two disorders share risk factors in common, that is, there is one (or more) cognitive deficit underlying both the disorders; for example, the working memory and the processing speed (Biederman et al., 2004; Martinussen  et al., 2005; Shanahan et al., 2006; Willcutt et al., 2005). The second one proposes that the presence of one disorder increases the risk for the other, i.e., the three nuclear symptoms of ADHD, inattention, hyperactivity and impulsivity, have a strong impact upon learning (Dupaul and Stoner, 2003). The third model suggests that the comorbidity represents independent disorders (Rhee et al., 2005).  Even though the causes of this comorbidity are not yet clear, there is evidence that, when the two disorders occur together, students have greater attention and academic deficits than when they occur separately (Barkley, 2014).

A set of investigations (Ackerman et al., 1986; Benedetto-Nasho and Tannock, 1999; Kaufmann and Nuerk, 2008; Lindsay et al., 2001; Casas et al., 2009) indicates that the main feature of the calculation problems associated with ADHD is the scarce representation and/or deficient inhibition in the access to the semantic memory of arithmetical facts. These mechanisms determine an overloaded process with interference effects, with a more generalized deficit when the ADHD appears associated with Mathematics Disorder (MD). Sella et al. (2012), comparing students with and without ADHD, concluded that ADHD students showed more difficulty than their peers in  identifying the best counting procedure to use, choosing the easier one, which was the earliest one. Similar results were found by Costa et al. (2012a). Thus, difficulty in storing and/or accessing basic arithmetic facts is identified as a striking feature in students with ADHD. Therefore, it is necessary to teach such students’ strategies that might facilitate access to basic facts from memory.

Recent research  and reviews (Costa et al., 2012a; Costa et al., 2012b; Gersten et al., 2009; Hopkins and Lawson, 2006; Sella et al., 2012; Woodward, 2004) have shown that students with learning difficulties do not advance spontaneously to memory processes, requiring direct and explicit teaching situations that facilitate their acquisition. Moreover, practice, as the sole type of instruction, has proven to be ineffective (Baroody et al., 2009; National Mathematics Advisory Panel, 2008).

In recent years, one widely used method of teaching calculation in classrooms has been based upon the conceptual understanding of the facts, through manipulative materials, and on meaning-based teaching proposals. Miller and Hudson (2007) report that these practices are largely centered upon the student. Woodward (2004) noted that, within this teaching approach, the cognitive load of the curricular activities and materials is very challenging for students with learning difficulties, even more so for  those  with  ADHD.  

It is important to note that the characteristics of the students with ADHD, including memory deficits (Keeler and Swanson, 2001; Kroesbergen and Luit, 2003), difficulty in attending to the main aspects of tasks and a passive approach to concluding tasks (Greenwood et al., 2002; Junod et al., 2006) contribute to increasing the challenges that all students have to face.

A variety of interventions has been tested in order to reduce the academic and social difficulties that often accompany ADHD (Tirado et al., 2004). Consistently, studies (MTA COOPERATIVE GROUP, 1999; Chronis et al., 2006; DuPaul and Stoner, 2003; DuPaul et al., 2006; Raggi and Chronis, 2006) have indicated that the best treatment is based on a more comprehensive approach including the use of medication and behavioral and psycho educational interventions. The two latter approaches are aimed at the student, the parents and the teachers. While, on one hand, there are many studies indicating the efficacy of medication and showing that productivity increases with its use; on the other hand, there are few studies which examine the long-term results of academic interventions (Pfiffner et al., 1998; Raggi and Chronis, 2006).  Most of this research has concentrated on strategies related to handling social behavior and conduct in the classroom, but this is only one aspect of ADHD; another is related to strategies aimed at enhancing academic performance (DuPaul and Stoner, 2003; Iseman and Naglieri, 2011).

To the best of our knowledge, no investigations have attempted to assess efficient strategies for teaching basic arithmetic facts to students with ADHD, despite evidence that this group of students continues to use immature counting procedures up to more advanced grades (Benedetto-Nasho and Tannock, 1999; Costa et al., 2012a; Sella et al., 2012; Zentall, 2007). Thus, the present study is the first to investigate the efficacy of a pedagogical intervention model directed at teaching basic arithmetic facts, as a resource for advancing to memory-based processes.

This is an experimental study using a blind, parallel, randomized, controlled clinical trial. The sample was enrolled from the ADHD Outpatient Clinic at the Child and Adolescent Psychiatric Division of Hospital de Clínicas de Porto Alegre (PRODAH). The research project was approved by the Research Ethics Committee of the Hospital de Clínicas de Porto Alegre, RS, Brazil (project number 07591). Written informed consent from parents or a legal guardian and assent from the child were obtained.

Subjects

Four boys and three girls from the ADHD Outpatient Program were randomly allocated, using a sequential allocation strategy balanced by prognostic factors (Fossaluza et al., 2009), into 2 groups: 1) control group (CG) – two boys and 1 girl aged from  8 to 11 years (M = 9.67), within the average range of intelligence (M = 98.34,  SD = 13.87) based on the WISC-III (2002) vocabulary and cube subtests, and 2) intervention group (IG) – two boys and two girls aged from 8 to 10 years (M = 9), within the average range of intelligence (M = 92.5, SD = 9.57).

The inclusion criteria were: a) attendance at the second to the seventh grade of elementary school; b) diagnosis of inattentive or combined ADHD subtypes, confirmed by the clinical staff according to the DSM-IV criteria (APA, 1994); c) an estimated IQ (WISC-III, 1991) between 80 and 120; d) the use of counting-based procedures and e) not receiving special educational support. Subjects diagnosed as having any comorbid Mood Disorder and Anxiety Disorder were excluded, as they are psychiatric disorders that interfere greatly in scholastic performance.

Mathematical assessments

The participants were assessed in two mathematical measures by two qualified research assistants, both trained in Psychopedagogy and linked to the School of Education at the Federal University of Rio Grande do Sul (UFRGS). The senior investigator (BVD) duly trained the two assistants. The two instruments were always applied in the same session and the children were assessed individually. The tasks used were as follows:

a) Evaluation of the counting procedure and strategy: the sub-item Strategy Windows from the Numeracy Project Assessment (New Zealand, 2007a), which evaluates the strategy employed to resolve addition problems, was used. Strategy Windows consists of 9 tasks with an increasing degree of difficulty. The tasks, arranged one at a time on a sheet of paper, were presented as mathematical stories in which both the parts were greater than 0, and the second was smaller than the first. The investigator read the question, and the child was expected to reply orally as soon as he/she had the answer. The child was informed that he/she could resolve the question in the way he/she found easiest and that it was not permitted to use paper and pencil to avoid the child doing the calculation on paper. To avoid inducing their use, the term “fingers” was not used, but finger counting was allowed. Upon the conclusion of each task, the investigator determined the counting procedure (counting all; counting on the highest) or the memory process used (decomposition or retrieval), based upon the child’s answer and the investigator’s observation (Figure 1). If required, the student was asked if he or she had solved the calculation. At the end of the test, the investigator indicated the predominant memory process or counting procedure and the most advanced counting strategy (fingers, oral or silent) that had been accurately used.

b) Knowledge of basic facts (adapted from Hopkins and Lawson, 2006): the students were requested to answer 38 problems of addition, written in the form a + b in which both the parts were greater than 0, and b, greater than or equal to a. Of the 65 (100%) problems proposed by Hopkins and Lawson (2006), 38 (59%) were chosen. It was decided to execute an abbreviated form of the original proposal, as the assessment was to occur in a single day and the subjects were tired at the end in the pilot study, which could interfere with the results. The problems were presented, one at a time, on a sheet of paper, and the investigator read the problem orally. The students were requested to solve the problems by trying to retrieve the answer from memory. They were told that they could not count on their fingers and should say the number that came to their mind. Memory was considered to have been used when the child answered immediately[1], upon being presented with the question.

Immediately after the intervention (post-test) and two months later (follow-up), the instruments were applied to assess the counting procedure and memory-based processes, as well as the subject’s knowledge of the basic facts. It should be pointed out that 2 subjects failed to attend the second evaluation. Figure 2 illustrates the study design.

The main objective of this study is to evaluate the effectiveness of a basic arithmetic facts education program as a means to advance the use of memory-based processes. The central hypothesis is that the explicit and direct teaching of the basic facts of addition would augment the repertoire of facts the student would be able to access, and that increase would be reflected in the use of a memory-based process.

In the statistical analysis, the variables were described in terms of the mean, median, minimum and maximum standard deviation, and compared within groups over time using the Friedman test and between groups using the Mann-Whitney test. The significance level for statistical tests was 5% (0.050).

The number of basic facts which each subject was capable of automatically retrieving at the three moments of the study (pre-intervention, post-intervention and in the follow-up) are presented in Table 1.

As can be seen, both the intervention group and the control group exhibited a gain in the number of basic facts they were capable of automatically retrieving from the long-term memory between the pre- and post-test. However, the difference was not significant.

It is important to note that the mean percent gain from pre to post-intervention was more than double in the intervention group compared to that of the control group. This improvement compared to the control group was maintained three months afterwards. The increase in the number of basic facts known allowed all the students from the intervention group to advance to using a memory-based counting strategy, a fact not observed in the control group, which continued using the same counting strategy. In the follow-up testing, all the subjects from the intervention group continued to use a procedure based upon memory.

Moreover, the Mann-Whitney test also showed there was no significant difference in the values of the automatically accessed basic facts within the groups.

To our knowledge, this is the first report investigating the efficacy of a pedagogical intervention model aimed at teaching basic facts to students with ADHD. The data from the present study indicate that direct teaching of decomposition added to a time of automation in this new procedure increases the number of basic arithmetic fact that the student  is  capable  of  accessing  automatically.

This finding corroborates the findings of Hopkins and Lawson (2006), suggesting that confidence in automatic retrieval only occurs when the student has a significant number of basic facts in the long-term memory. Thus, the memory process requires the existence of at least some previously stored basic facts to assist the development of others. This would explain typical development, because the more facts the student is able to access automatically, the greater the incentive and satisfaction he/she will feel in engaging tasks and activities involving that skill. Consequently, most basic facts are stored in the memory, and so on successively.

For students with ADHD, the task is even more challenging because they have difficulty representing facts in the memory due to their attention and memory deficits (Kaufmann and Nuerk, 2008; Tannock, 1999). Thus, every time such a student attempts to solve a calculation, he/she needs to use a counting procedure, which is slow and often inaccurate. Consequently, the student is less inclined practice and, so is less capable of representing a larger number of arithmetic facts in the long-term memory. In addition to all this, ADHD students are known to have difficulty engaging in activities in general (Rogers et al., 2009). Thus, there is a real vicious circle of failure and frustration. Moreover, in such cases, practice becomes meaningless, making automatization slow. This is another of the important contributions provided by this study: highlighting the need of students with ADHD to practice the automatization of basic arithmetic facts. Therefore, it is important to find activities that engage such students (Rogers et al., 2009), an idea compatible with the findings of Zentall (1993) that students with ADHD require more instruction time and practice. Fletcher et al. (2009) suggest that an effective intervention program is one that substantially increases the student’s exposure to situations that require mathematical thinking.

It should be noted that, at first glance, the control group showed an improvement, although it was inconsistent, since three months after, the results were lower than in the pre-test. The advance in the control group can be explained by the arithmetic tasks carried out, for example, the number line. Such activities, however, were unable to consolidate the taught knowledge, which was not the case in the intervention group. In the intervention group, there was also a decrease between the post-test and follow-up testing, but, compared with the pre-test, there was an advance. The decrease can be explained in two ways.

The first idea is to highlight the importance of practice. Possibly, after the intervention, the students did not practice anything other than what is commonly proposed in the classroom, which may have led to a diminished performance, as mentioned above.

The second explanation is related to the model of the response to the intervention (RtI) (Fletcher et al., 2009; Gersten   et al., 2009).  This model suggests that one criterion for identifying learning disabilities (LD) should be the lack of a satisfactory response from the student to quality interventions. This criterion helps to distinguish a disorder from a difficulty, since, in the former case, studies (Dowker, 2004; Jordan et al., 2003) have indicated that the deficits are more persistent, which is not so in the case of difficulties. Thus, we may think that the students participating in this study presented a Learning Disorder (LD) as well as ADHD, which would make their response to the intervention even slower.

Research conducted in North America (Miller and Hudson, 2007; Powell et al., 2009) has indicated that in these countries the addition facts are part of the curriculum in preschools and in the first two years of elementary school. Probably then, students with typical typically development advance without any problems during those moments. However, studies in English (Baroody et al., 2009) and in Portuguese (Costa et al., 2012b) have demonstrated how difficult it is for students with difficulties in arithmetic to advance spontaneously from one procedure to another. To make matters worse, mathematics is characterized by a content hierarchy, in which new skills are built on those previously learned. Therefore, when students continue to use immature counting procedures, there is a need to develop teaching strategies suited to that level of knowledge, before moving on to new ones. The conclusion that this aspect is not considered in Brazilian schools provides an important contribution to the understanding of the subsequent arithmetic difficulties seen in this group of students. As they can be expected to have difficulty, for example, in understanding multiplication, a form of knowledge that implies an understanding the part-whole relation, which is not fully formed in this group of students. Furthermore, the lack of automatization in accessing simple addition facts also has an impact on the ability to solve multi-digit calculations, overloading working memory and favoring forgetfulness in the use of number grouping. 

The data from the present study suggest decomposition/composition and commutativity are two difficult to understand principles, mainly for students with MD (Mathematical Disabilities). Similar findings were obtained by Baroody et al. (2009). Thus, understanding and explicitly teaching of these two principles can be crucial for the advance of a process based upon memory. Even so, this study suggests that this challenge can be overcome, depending on availability of suitable tools and opportunities.

While computer-based interventions may be more attractive to children, our main motives for using the present intervention model are twofold. The first is that some studies (Duhon et al., 2012; Ota and DuPaul, 2002) have shown there is no significant difference between computer-based and pencil and paper-based teaching. The second concerns the situation in the Brazilian education system,  in  which  this  study  was  conducted, where using a computer as a teaching tool is not yet part of the educator’s everyday life. Thus, the present teaching model has proven to be a promising path to be replicated in larger samples.

This finding confirms several intervention studies (Fuchs et al., 2005; Fuchs et al., 2006;  Fuchs et al., 2008a;  Fuchs et al., 2008b;  Iseman and Naglieri, 2011; Tournaki, 2003; Woodward, 2006)showing that arithmetical skills are strongly susceptible to teaching. Iseman and Naglieri (2011) analyzed the efficacy of a ten days plan-based teaching with a group of ADHD students and concluded that there was an improvement in the students’ performance, as assessed using math worksheets, and the students showed some knowledge transfer to standardized math tests.

The problem situations presented, being related to daily activities, may also have helped. Studies (Nunes and Bryant, 1997; Orrantia, 2006) have shown that linking mathematical problems to everyday life is capable of facilitating the learning of mathematical procedures and concepts by students. Orrantia (2006) reported that a large part of the arithmetical difficulties stems from the disconnection between the informal knowledge that students develop spontaneously and the formal know-ledge that they learn in schools. Nunes and Bryant (1997) have postulated that this linking exerts a particularly strong influence in children with a low scholastic output. The effects of different teaching materials on students’ performance, as well as the effects of pedagogical intervention time deserve a more detailed analysis in future research.

The main hypothesis is that subjects would advance from a procedure based upon counting to another relying on memory with both direct and explicit teaching of decomposition and moments of practice. This hypothesis was confirmed, indicating that the children in this study responded positively to explicit and direct teaching. The findings suggest two important pedagogical implications: 1) even interventions considered late may lead to changes, indicating that there is a delay in development and not a permanent deficit and 2) the need to pay greater attention to the teaching of basic arithmetic facts in order to facilitate the advance to memory-based processes.

The main methodological limitation of the present study was the sample size. Although the diagnostic criteria for ADHD were carefully controlled, the sample size was small, reducing the possibility of generalizing from the results obtained. Moreover, the small number of students with ADHD prevented comparison between subtypes of ADHD. Finally, the nonexistence of a group without ADHD meant that the comparison of the results found in children with ADHD with those described in children with typical development could only be performed using data obtained from the relevant literature about the latter group (typical development). Another possible limitation is related to the control of the level of ADHD severity. It is likely that the most serious cases, and therefore the most difficult to remedy, were those who gave up or refused to take part in the study. However, it is important to emphasize an extremely innovative aspect of this study, which was to introduce the efficacy test model to the area of education. Further studies will be required to confirm the findings presented herein.

Due to the well-known clinical heterogeneity of ADHD, we cannot suggest that this teaching model would function for all children with ADHD. Nevertheless, a series of procedures can be described, which, as a whole, seem to be valid and should be taken into account in future studies:

1. Direct teaching in adding composition and decomposition using concrete materials, which should be gradually withdrawn.

2. Immediate feedback: during the intervention, students used an unsuitable procedure, because they did not know it was inefficacious.

3. Constant self-monitoring. Several studies (Biederman et al., 2004; Castellanos et al., 2006; Iseman and Naglieri, 2011; Shanahan et al., 2006) showed that students with ADHD use cognitive and metacognitive strategies less efficiently than their peers with typical development. 

4. Time of practice in a determined procedure and not a mechanical practice, devoid of meaning. It should be borne in mind that, in order for the information to be transferred to the long-term memory and, consequently, to consolidate the knowledge, it is of paramount importance that the information be repeated (distributed practice) and organized.

5. The systematization and revision of what was studied, as well as the anticipation of what will be developed, are essential aspects, as students with ADHD have difficulty organizing, maintaining and using new knowledge.

Developing the strategy of quickly and accurately accessing the basic arithmetic facts of addition from the long-term memory is the result of a complex learning process. It should be pointed out that this is a cognitive construction process with different conceptual levels. The present study has shown that it is possible to develop teaching strategies that “destabilize” the student and make him or her advance more quickly to the next level.

In conclusion, it is important to point out that the progress shown by the intervention group suggests the importance and feasibility of executing short, easy-to-apply and low cost pedagogical interventions to improve learning. Moreover, it is important that educators, whether teachers or educational psychologists, pay more attention to the various moments of developing  counting, as each one involves different conceptual levels that should be respected. 

The author(s) have not declared any conflict of interests.