App-based mathematical intervention for youth with intellectual disabilities: a randomised controlled trial

ABSTRACT

The purpose of the study was to evaluate whether students with intellectual disabilities (ID) can improve their arithmetic skills by participating in an arithmetic intervention programme, theoretically founded on explicit instruction (EI) and administered via an application developed for tablet computers. The intervention study used a randomised controlled trial design (RCT) (n = 30, aged 10-16, 13 females) and lasted for up to 12 weeks. The results show that the intervention group significantly improved in arithmetic fact fluency compared to the controls and the effects remained six months after the intervention. The effects were larger for subtraction than for addition, and this difference remained six months later. These results suggest that mathematics applications based on explicit instruction can be an effective way of teaching arithmetic facts to youth with mild ID.

KEYWORDS:

• Arithmetic
• intellectual disability
• intervention
• explicit instruction
• randomised controlled trial

Introduction

In modern society, basic math skills are a prerequisite for functioning adequately and achieving independence in daily life. These skills are essential for a variety of real-life situations such as shopping and telling time. One group of individuals that struggle to acquire sufficient math knowledge and skills, and thus a high quality of life, are those with intellectual disability (ID) (e.g., Bashash et al., 2003; Faragher & Brown, 2005). ID is defined according to three criteria: 1) the individual must have an IQ of less than 70, 2) the individual must have impaired adaptive behaviour, and 3) these impairments must be present during the developmental period (before age 18) (DSM-5; American Psychiatric Association, 2013; ICD-11; World Health Organization, 2019).

In Sweden, the Act concerning Support and Service for Persons with Certain Functional Impairments (Act 1993:387) declares that individuals with ID are entitled to special support and services. If a child is not expected to achieve the required learning outcomes of the mainstream school due to ID, a place is offered at a special needs school called “compulsory school for pupils with learning disabilities”. Thus, all students enrolled in special needs schools have been diagnosed with ID before admittance. The curriculum of special needs schools in Sweden differs from that of regular schools. Its aim is to provide students with key knowledge and skills which will enable them to function adequately in daily life and achieve independence as adults.

Research shows that the development of basic mathematical skills in most children with ID does not follow the same trajectory as typically developing children (e.g., Brankaer et al., 2013; Cheong et al., 2017). Bashash et al. (2003) examined the counting skills of different age groups of students with moderate ID. They found that none of the seven- to eleven-year-olds could count beyond 10, and only half of them could count to 10. Furthermore, while half of the twelve-to-fifteen-year-olds could count to 50, only 10 percent could count all the way to 100. Brankaer et al. (2013) also found that children with mild ID performed significantly worse on symbolic and non-symbolic number comparison tasks compared to typical children. Further evidence of weak mathematical skills in students with ID has been provided by Sermier Dessemontet et al. (2020). In a sample of seven- to eleven-year-olds with mild or moderate ID, they found that only 12 students had sufficient knowledge and skills to solve simple arithmetic problems supported by pictures, and only 10 students managed to solve simple symbolic arithmetic problems (4 + 5; 6–4) (cf. Baroody, 1999).

In summary, many students with ID experience difficulties in acquiring sufficient mathematical skills, which will likely have negative effects on their quality of life as adults. Thus, it is important to develop intervention programmes and to evaluate their effectiveness on students with ID. In line with this, the purpose of the study was to evaluate whether students with ID can improve their arithmetic skills by participating in an intervention programme, theoretically founded on explicit instruction (EI) and delivered via an application developed for tablet computers.

Arithmetic learning

In first grade, maths instruction usually starts with arithmetic as the students are taught how to solve single-digit arithmetic problems (e.g., 3 + 2; 8 + 6; 5–3; 9–6) (Baroody & Dowker, 2003; Geary, 1994; 2004). However, according to developmental models and studies, students must acquire key conceptual knowledge and procedural knowledge skills before they can begin to learn formal arithmetic (e.g., Baroody & Dowker, 2003; Dowker, 2005). Conceptual knowledge includes knowledge of number symbols (e.g., number words; Arabic numerals), that is, the ability to identify, name and recite them in correct order (Krajewski & Schneider, 2009). Next, the child must understand that each symbol represents a specific quantity, and that they are serially ordered so that each number represents a larger quantity. This knowledge constitutes the foundation for counting ability, that is, being able to establish the total number of objects in a set by assigning one counting word to each object and knowing that the counting word assigned to the last object represents the cardinality of that set. Another important form of conceptual knowledge concerns the relationship between numbers. The child needs to understand that a number (5) can be decomposed into two numbers (2 and 3), and that two or more numbers (1, 2, 3) can be composed into one larger number (6). Understanding that numbers can be represented by other numbers is vital for the conceptual understanding of arithmetic (Geary, 1994; Krajewski & Schneider, 2009). At a higher level of number knowledge is the understanding of the base-10 number system and place value, which are important for solving multidigit problems (e.g., 8 + 14; 16–9) (Dowker, 2005; Geary, 1994).

Key procedural knowledge skills involve knowing when and how to implement different calculation procedures and strategies in an efficient manner (Dowker, 2005; Geary, 2004). Arithmetic developmental research shows that typical children successively develop and utilise various strategies when learning arithmetic (Geary et al., 2004; Siegler, 2000). The main strategies are finger counting, verbal counting, finger support, decomposition, and direct retrieval, where the first two are so-called procedure-based strategies and the remaining three are memory-based strategies. In finger counting, the child uses their fingers (or objects) to represent the numbers in the problem and counts on the fingers to obtain an answer. In verbal counting, the child counts out loud without the support of fingers. Finger support means that the child uses their fingers to represent the numbers but does not actually count the fingers to obtain an answer. The fingers function as memory triggers, facilitating the retrieval of the answer from memory. With the decomposition strategy, the answer to a problem (4 + 3) is derived by first retrieving the answer to a similar problem (3 + 3 = 6) and then adding 1 (6 + 1 = 7). Direct retrieval is a more advanced strategy whereby the child has established associations between certain problems (2 + 4) and answers (6) in their memory, which they can retrieve when confronted with these same problems (Geary et al., 2004; Mulligan & Mitchelmore, 2009; Pound, 1999). Direct retrieval releases more of the child’s limited cognitive capacity, which can be used to perform other operations during complex arithmetic tasks (758 + 363).

As the child’s repertoire of strategies increases, they learn to use them in a flexible manner by selecting the most relevant solution strategy for the problem at hand (Siegler, 2000). However, the general developmental trend in typical children involves a gradual shift from counting strategies to memory-based strategies (Geary, 2004).

Explicit instruction

One approach to teaching mathematics, which has been shown to be effective for several different populations and mathematical domains, is explicit instruction (EI) (e.g., Doabler & Fien, 2013; Ennis & Losinski, 2019; Hudson et al., 2006). The main components of EI are clear instructional objectives, clear modelling, think-aloud, multiple examples, extensive practice, and immediate corrective feedback (Hudson et al., 2006; Hudson & Miller, 2006). When the teacher presents a new concept or procedure, they start with a clear demonstration of how it is performed, while verbally explaining each step. The concept or procedure is then performed with multiple examples, first by the teacher and then by the students. An important aspect of EI is that the student practises the same procedure over and over to achieve arithmetic fluency. During practice, it is important that the student thinks out loud, so that the teacher can determine whether there is a genuine understanding and has the opportunity to provide the immediate corrective feedback.

Previous interventions with ID individuals

To date, four reviews of mathematical interventions aimed at individuals with ID have been published, and all show that interventions tend to have positive effects regardless of the mathematical content (Browder et al., 2008; Hudson et al., 2018; Schnepel & Aunio, 2022; Spooner et al., 2019). The empirical picture demonstrates that EI is an effective instructional method for teaching mathematics to students with ID and that the use of manipulatives is important (Spooner et al., 2019). Moreover, several similar studies have used technology aids and have displayed positive learning effects (Schnepel & Aunio, 2022; Spooner et al., 2019). Although a bulk of studies provide evidence that targeted interventions help students with ID to learn mathematics, few researchers have used RCT designs. Instead, single-case studies have been the dominant form of research design (see Hudson et al., 2018; Spooner et al., 2019).

However, two exceptions to this are provided by Tzanakaki et al. (2014) and Ortega-Tudela and Gómez-Ariza (2006). In the latter, 18 students with Down Syndrome (mean age = 6.52 years) received 15 35-minute computer-assisted teaching sessions over 21 weeks. In the Tzanakaki et al. (2014) study, 19 students with severe ID and five with autism spectrum disorder (mean age = 8.33 years) received between 13 and 33 in-vivo sessions (12 weeks), which amounted to a range of 210-620 min of training per individual. Both studies focused on teaching early numeracy skills (e.g., one-to-one correspondence, stable order, place value) to the students in a one-to-one setting, using the Discrete Trial Teaching (DTT) procedure. DTT is akin to EI, using clear instruction, prompts, and fade-out procedures together with reinforcing correct responses through praise. In both studies, the intervention displayed positive learning effects.

In summary, research indicates that interventions based on EI, delivered either in vivo or in vitro, facilitate mathematical learning in students with ID. However, more RCT research based on EI is required to expand our knowledge concerning what teaching practices are effective for students with ID. As several aspects of EI are resource-intensive to perform (e.g., one-to-one instructions, direct feedback; extensive practice), the use of applications can offer many advantages. More intervention research using applications and EI will hopefully provide special needs educators with evidence-based instructional practices that are effective for teaching mathematics.

Current study

The purpose of the study was to evaluate whether students with ID can improve their arithmetic skills by participating in an arithmetic intervention programme, theoretically founded on EI and administered via an application (“Planetjakten” in English “Chasing Planets”) developed for tablet computers.

The selection of this specific application-based intervention was founded on the following criteria: 1) the application targeted basic arithmetic with the aim of promoting arithmetic fluency; 2) the instructional approach was similar to EI; 3) the application was gamified and targeted typical seven- to eight-year-old children; 4) the application was adaptive to the students’ current knowledge and skill level.; and 5) the application had previously been evaluated on children with mathematical difficulties (MD) (Hassler Hallstedt et al., 2018). The application is developed by Hassler Hallstedt et al. (2018) and is theoretically founded on a combination of instructional design (Tiemann & Markle, 1991) and precision teaching (Johnson & Street, 2012). This has led to a three-phase system being implemented for each level of the application. Every level starts with a modelling phase (i.e., conceptual instruction), where the player is introduced to a new concept (e.g., performing “+1”; the double plus one principle; decomposition strategy) through both audial and visual instruction. The next phase is the guide phase (i.e., conceptual, and procedural instruction), where the player must perform the previously demonstrated function without time constraints and with error correction if needed. In the third phase, the fluency phase (procedural training), the player performs similar tasks but without guidance and with time constraints. Thus, the three-phase system targets both conceptual (phases 1-2) and procedural (phase 3) knowledge and skills to develop arithmetic fluency. So, in a way, it uses EI (e.g., Hudson et al., 2006).

The application has previously been evaluated on second graders with MD, and it demonstrated positive learning effects (Hassler Hallstedt et al., 2018). It is noteworthy that participants with lower IQ (no ID) benefitted more from the intervention than those with higher IQ. With this in mind, the current study set out to test the same intervention on a population of children with mild ID using an RCT design.

It was hypothesised that the students in the intervention group, compared to the active control group, would improve their arithmetic skills, both in terms of the total number of correctly solved problems and the automatic retrieval of arithmetic facts.

Method

Participants

The sample consisted of 30 students, aged 10–16 years, diagnosed with mild ID (IQ 55–70). Since all the students had a diagnosis of ID, which is a requirement to attend special education schools in Sweden, no further evaluation apart from WISC-IV (Wechsler intelligence Scale for Chiildren-IV) matrices (Wechsler, 2003) were performed. Written consent to participate was acquired from each participant’s caregiver prior to testing. The 30 participating students attended six different schools located in five different municipalities. Half of the schools were integrated with the mainstream schools, but none of the participating students were integrated into mainstream school classes.

As the study used an RCT design, the 30 participants, across all schools, were randomised into either the mathematics group (MG) or a writing group (WG). A restricted random assignment was used: all 30 students were ordered randomly in a list and then the first participant was randomly assigned (using random numbers from www.random.org) to the intervention or control condition. Thereafter, the second participant on the list was assigned to the other condition. This procedure was performed to force equal sample size due to the overall small sample, and to ensure that both groups included students from all six schools.

An active control group was used for ethical reasons (to avoid any participant feeling left out); the choice of an RCT was similarly motivated as it helps eliminate systematic biases and is somewhat considered “the golden standard” in intervention research (e.g., Breakwell et al., 2006). Due to general problems with recruiting from a small population (i.e., children with ID), a power analysis was not performed beforehand; instead, the aim was to recruit as many participants as possible. Furthermore, a post-hoc estimation of power was not performed as it is not advisable (see Dziak et al., 2020).

Demographic information and results from the screening tests for the two groups are displayed in Table 1.

Table 1. Descriptive statistics and results of ANOVA for the screening tests by group.

	Intervention Group	Control Group
Measures	M (SD)	M (SD)	Chi-square-test/ANOVA
N and gender distribution (boys/girls)	14 (5/9)	16 (9/7)	$\chi 2(1,N=30)=1.26,p=.261$
Grades (5/6/7/8/9)	2/3/3/5/1	2/1/3/3/7	$\chi 2(4,N=30)=5.89,p=.207$
Age in years	12.93 (1.38)	13.81 (1.72)	F(1, 28) = 2.36, p = .136
Non-verbal intelligence (raw score)	10.21 (3.62)	8.56 (3.93)	F(1, 28) = 1.42, p = .244
Non-verbal intelligence (standard score)	2.71 (1.38)	2.06 (1.06)	F(1, 28) = 2.12, p = .156
Counting and number knowledge	40.71 (2.16)	40.81 (1.72)	F(1, 28) = 0.02, p = .891
Give-N task	4.00 (-)	4.00 (-)	n/a
Sequencing	3.00 (-)	3.00 (-)	n/a
Non-symbolic number comparison (% correct)	96.46 (5.26)	94.47 (5.86)	F(1, 27) = 0.89, p = .354
Non-symbolic number comparison (response time in ms)	1714 (722)	1619 (766)	F(1, 27) = 0.11, p = .740

Chi-square tests and one-way ANOVAs were used to compare the two groups’ scores on the demographic measures and screening tests. The two groups did not significantly differ on any of the measures (all ps > .05), demonstrating that the randomisation was successful.

This research project was approved by the Regional Ethics Committee in Linköping, Sweden (Protocol Number 2016/318-31).

General procedures

All students were tested in rooms that were familiar to them at their respective schools. Most students completed all the tests in one session; three students became too tired to continue, so they finished the tests later the same day. Prior to testing, oral consent was acquired from each participant. The students were also informed that they could ask for a break, as well as opt out of the study completely, whenever they wanted, with no questions asked. All testing was performed by the first author, and all tests, except for two tests, were developed by the authors with inspiration from the literature.

Measurements

Screening tests

Prior to the intervention, screening tests were performed to establish that the two groups did not differ on relevant abilities, and that all students had sufficient skills to be able to use the application.

Non-verbal intelligence

The matrices subtest of the WISC-IV was used to estimate the participants’ non-verbal intelligence (Wechsler, 2003). The task was to identify the missing piece to complete a larger design. For each item there were five response alternatives, and the participant could either give a verbal answer or point to their answer. Stopping criteria were either four incorrect answers in a row, or four incorrect answers in five trials.

Counting and number knowledge

This measure consisted of seven tests. In the first test, the students were asked to count aloud to 20. The second test was to count five up from a given number (3–8; 8–13; 26–31). The students had to start and stop at the correct number. If a student failed to start a problem (e.g., did not know what came after 26), testing stopped. The third test was to count backwards, from 5 and 10 respectively, down to zero.

In the fourth test, a set of dots were displayed on the screen and the students were asked to count them. The following numbers of dots were shown 2, 5, 3, 7, and 9 in that order.

In the fifth test, the participants were asked to first name the following four numbers: 3, 5, 7, and 10, which were shown one at a time on the screen. They were then shown four numbers at once (2, 4, 9, and 11) in random order and were asked to point to each in ascending order.

The sixth test was performed using 10 small cubes that were placed in a straight line, where the tester pointed to the leftmost one and said: “If this is number 1, point to the one that is … ” and then gave the numbers 2, 4, 7 and 9. After four trials, the participant was asked to remove (instead of point to) cube number 3, 8, 5 and 2, in that order.

The last test assessed arithmetic skills, where the tester read out five addition problems (1 + 1; 2 + 1; 4 + 3; 2 + 4; 3 + 2) and four subtraction problems (4 - 2; 5 - 2; 3 - 1; 4 - 3). The participants were asked to say the answer out loud.

For all tasks described above, one point was awarded for each correct response, resulting in a maximum score of 48 for the “counting and number knowledge measure”. Cronbach’s alpha for the seven tests was .91.

Give-N task

In this test, the participants were presented with 20 small cubes and asked to give the tester a certain number of cubes (e.g., “Can you please give me three cubes?”). The participants were asked to give, in order, three, 14, eight, and five cubes. The tester noted the response and returned the cubes to the participant before asking for the next number of cubes. Cronbach’s alpha for the four items was .73.

Sequencing

Here the participants were shown three different sets of numbers (1-4, 3-7, and 5-10) that were not in order and asked to read them out in the correct order starting with the smallest number. Cronbach’s alpha for the three items was .83.

Non-symbolic number comparison

This task was administered using Panamath (v.1.21), developed by Halberda and Feigenson (2008). Two arrays, each containing between five and 21 blue and yellow dots, were presented for 1951ms. The participant then had to decide if more blue or yellow dots had been presented, and then press the key designated for each answer (A or L key). After an answer was given, they pressed the space bar to display the next array. Prior to each trial, a fixation cross was displayed in the centre of the screen. Before starting, the participants were informed that there was no time limit regarding their response but that the arrays would only be presented briefly. Four ratios (1.28; 1.46; 1.75; 2.75) were presented 12 times each. Surface area varied on half the trials, along with dot size to control for confounding variables and ensure attention to numerosity. Response time and accuracy were recorded and used as dependent measures.

Outcome measures

Arithmetic fluency

Two timed tests were used to measure addition and subtraction fluency. The participants had to solve as many addition and subtraction problems as possible during two 60-second rounds. Both rounds increased in difficulty. The addition test began with a block of five problems where both the presented numbers and the answers were single-digit numbers (3 + 1; 5 + 3; 6 + 2; 4 + 5; 7 + 2). The second block included five problems in which the first number and the answer were two-digit numbers (14 + 3; 12 + 5; 17 + 2; 13 + 5; 11 + 8), and the third block included 15 problems that involved 10 transitions (i.e., carrying 5 + 6; 9 + 8; 2 + 9; 6 + 7; 4 + 8; 3 + 7; 6 + 9; 8 + 5; 6 + 8; 7 + 9; 12 + 11; 13 + 8; 16 + 7; 15 + 14; 19 + 4). The subtraction task also began with five single-digit problems with single-digit answers (5–3; 8–4; 6–2; 9–6; 7–5), followed by five larger problems (16–3; 18–7; 13–2; 20–6; 17–5), ending with 15 transition problems (i.e., borrowing; 13–4; 15–6; 19–12; 12–8; 16–9; 19–11; 14–9; 17–15; 13–7; 18–15; 21–4; 22–7; 25–9; 24–6; 29–15). In total, 25 addition and 25 subtraction problems were available, for a possible maximum score of 50. The tests were administered using SuperLab PRO (v.4.5), and the problems were presented in the middle of the computer screen. The participants were told to read the problems themselves and say the answer out loud, and were informed of the time limit. They were not informed of the total amount of problems available in the test. The SuperLab PRO software registered the number of problems displayed and response time for each displayed problem. During performance, the experimenter continually registered the child’s oral answers and registered each error. Two outcome measures were used: 1) the number of correctly performed problems, and 2) the number of correctly performed problems with response times within 3000 ms (i.e., automatic retrieval of arithmetic facts) (Russell & Ginsburg, 1984). Split-half reliabilities for the addition task was r_sh = .60, and for the subtraction task r_sh = .66.

Verbal fluency

The test consisted of four trials. In each one the participants were asked to name as many words as possible in one minute belonging to two semantic categories (animals; foodstuffs) and two phonological categories (words starting with F-sounds and with S-sounds). The tester noted the correct words and kept a timer. The performance score for the four trials were combined into a “verbal fluency score”. Cronbach’s alpha for the four items was .77.

Working memory

In this test the participants had words read aloud to them, and they had to determine whether it was an animal (yes vs. no), while also remembering the words. In the first two trials only one word was used, and then the number of words increased by one every two trials. Testing stopped when the participant failed to remember all the words in two trials of the same length in a row. Split half reliability was r_sh = .47.

Rapid automatic naming (RAN)

A colour naming task was used where the participants were shown dots of various colours on the screen, using PowerPoint (2016) against a white background. Four repetitions were used, with an increasing number of dots presented (three, four, five, six). The participants were asked to name the colour of the dots as quickly as possible, and the test was timed using a stopwatch. Cronbach’s alpha for the four items was .71.

The intervention

Originally, the intervention was planned to stop after 12 weeks regardless of how far the participant had progressed in the application; however, some participants played through all the levels of the application (more information below) in a shorter time. This resulted in the intervention lasting 8–12 weeks. The app is designed to be used for up to 15 min per day, five days a week. It is however possible to quit playing before the 15 min has passed. This, combined with individual difference in skill among the participants, is the main reason for the spread in length of intervention (cf. Hassler Hallstedt et al., 2018). All the participants were accustomed to using tablets for parts of their schoolwork, but they mostly used pen and paper during regular lessons in mathematics and most other subjects.

The applications

The arithmetic application used in the study is developed by Hassler Hallstedt et al. (2018). The theoretical foundation is detailed above. The application consists of 261 levels (i.e., planets). Each planet is focused on a specific area of basic mathematical knowledge (e.g., text-to-symbol – learning that the word “eight” is written numerically as “8” for example). The planets are all constructed in a similar way. They start with instructions, where the narrator explains the topic of the planet (e.g., what happens when you perform +1). This is followed by a few trials where the player is allowed to attempt mental calculation (e.g., “1 + 1 = ?”). If the player completes the trials, they arrive at the last part of the planet, which is a race. If, however, the player did not complete the trials, they return to the instructions, followed by another set of trials. When the player eventually arrives at the race, the player’s character races a non-player character (NPC). To move the character forward, the player must perform mental calculations in a limited amount of time (i.e., before the NPC finishes the race). If a player loses the race, they are allowed to try again. The application is adaptive in that the difficulty is lowered slightly if a player loses the race. To complete a planet, however, the player must finish the race at the highest level of difficulty. Once a player has completed a planet, they can move on to the next one. The planets are set up in such a way that they keep building on what has previously been taught. There is no restriction against going back and replaying the same planet multiple times. While playing, the child does not receive assistance from an adult, since the application itself acts as the teacher. In terms of graphics, the application is primarily aimed at children around seven to eight years old and therefore has a cartoon-like look. In the first session of playing, the player gets to design the look of their avatar to help with immersion and motivation to play the application. At the end of every session of playing, the player can spend the points they have acquired to buy either sweets or fruit to feed “the knowledge monster”, which grows the more it is fed (and it grows more if the player feeds it fruit rather than sweets). The combination of an appealing graphical and gamified experience combined with EI hopefully results in a captivating learning experience, and one which the player wants to return to.

The writing group (WG) used an application called “Trilo Stavar” (in English: “Trilo is Spelling”). This application was developed to help children (aged four to eight) learn and improve their reading and spelling in a gamified manner. The player completes levels by solving increasingly difficult spelling tasks. In the early stages the player is presented with an image of a common object – usually a three- or four-letter word in Swedish, such as “apa” (“monkey”) – and some letters, and the player then must select the correct letters to spell out the word. The application helps by sounding out the letter selected by the player. In the later stages of the game, the difficulty is increased to the level of simple crossword puzzles. As their reward for completing a level, the player gets to hear a little bit more of the story about the game’s central character (Trilo). Unlike Chasing Planets, there is no aspect of time constraint or element of urgency when it comes to solving the puzzles.

Results

Addition and subtraction

To evaluate the effects of the intervention, 2 × 3 mixed ANOVAs were computed, with group (MG vs. WG) and time (pre-test; post-test 1; post-test 2) as independent variables, and an alpha level of p < .05. Descriptive statistics for the outcome measures are displayed in Table 2.

Table 2. Descriptive statistics for the outcome measures by group and time.

	Intervention Group			Control Group
Measures	Pre-test M (SD)	Post-test 1 M (SD)	Post-test 2 M (SD)	Pre-test M (SD)	Post-test 1 M (SD)	Post-test 2 M (SD)
Correct addition calculations	13.07 (4.03)	15.00 (3.84)	15.50 (2.77)	11.94 (5.35)	9.94 (3.71)	10.69 (3.55)
Automatic retrieval of addition facts	5.14 (4.83)	8.79 (4.46)	8.50 (4.35)	4.69 (4.85)	2.69 (3.30)	2.94 (3.32)
Correct subtraction calculations	9.50 (4.33)	12.64 (4.85)	12.71 (4.43)	9.06 (5.06)	7.81 (4.12)	8.43 (3.88)
Automatic retrieval of subtraction facts	3.00 (3.94)	6.79 (4.76)	6.93 (4.73)	2.19 (3.15)	1.38 (2.68)	1.62 (3.01)
Verbal fluency	37.29 (9.66)	37.71 (14.57)	37.57 (12.52)	45.81 (15.89)	53.19 (13.94)	53.75 (13.74)
Working memory	3.04 (0.57)	3.18 (0.46)		2.84 (0.39)	3.28 (0.52)
RAN (response time in seconds)	4.07 (0.28)	4.02 (0.33)		4.67 (1.59)	4.08 (0.78)

The ANOVA performed on the total number of correctly solved addition problems revealed a significant main effect of group, F(1, 28) = 8.57, MSE = 35.21, p = .007, $p=23$, partial η² = .23, but no main effect of time (p= .551). More importantly, a significant interaction effect between group and time was obtained, F(2, 56) = 5.92, MSE = 6.11, p = .005, $p=.17$ partial η² = .17. A test of the simple interaction effect with a pooled error term and adjusted alpha levels (i.e., p = .017) showed that the interaction was due to the MG and WG performing equally on the pre-test measure (p > .017), while the MG outperformed the WG on the post-test 1 measure, F(1, 84) = 12.11, MSE = 15.81, p < .017, and the post-test 2 measure, F(1, 84) = 10.94, MSE = 15.81, p < .017.

Concerning the total number of correctly solved subtraction problems, a main effect of group, F(1, 28) = 5.11, MSE = 44.39, p = .032, $p=.15$ partial η² = .15,emerged, but no effect of time (p = .181). However, a significant interaction effect between group and time was obtained, F(2, 56) = 5.62, MSE = 7.60, p = .006, $p=.17$. partial η² = .17. Again, the interaction was due to the two groups performing equally on the pre-test measure (p > .017), while the MG outperformed the WG on the post-test 1 measure, F(1, 84) = 8.77, MSE = 19.87, p < .017, and the post-test 2 measure, F(1, 84) = 6.87, MSE = 19.87, p < .017.

On the measure of the direct and automatic retrieval of addition facts (i.e., response within 3000 ms), a group effect, F(1, 28) = 9.86, MSE = 37.06, p = .004, $p=.26$, partial η² = .26,emerged, but no main effect of time (p = .448). A significant interaction effect was also obtained on this arithmetic measure, F(2, 56) = 8.98, MSE = 8.07, p < .001, $p=.24.$partial η² = .24.The MG outperformed the WG on the post-test 1 measure, F(1, 84) = 15.66, MSE = 17.73, p < .017, and the post-test 2 measure, F(1, 84) = 13.03, MSE = 17.73, p < .017, while no mean difference was observed on the pre-test measure (p > .017).

The ANOVA computed on the automatic retrieval of subtraction facts displayed a significant main effect of group, F(1, 28) = 10.45, MSE = 31.63, p = .003, $p=.27$partial η² = .27, and time, F(2, 56) = 4.83, MSE = 5.24, p = .012, $p=.15$ partial η² = .15, as well as a significant interaction effect between group and time, F(2, 56) = 9,81, MSE = 5.24, p < .001, $p=.26$ partial η² = .26. A test of the simple interaction effect revealed that the two groups’ mean performances were equal on the pre-test measure (p > .017), while the MG obtained a higher mean score than the WG on the post-test 1 measure, F(1, 84) = 15.57, MSE = 14.04, p > .017, and the post-test 2 measure, F(1, 84) = 14.96, MSE = 14.04, p < .017.

Cognitive measures

On the verbal fluency task, there was an effect of group, F(1, 28) = 8.74, MSE = 459.58, p = .006, $p=.24$partial η²= .24,, and time, F(2, 56) = 3.35, MSE = 47.84, p = .007, $p=.11$partial η² = .11, but no interaction effect between group and time (p = .070).

There was no effect of group on working memory (p = .738), but a significant effect of time, F(1, 28) = 3.37, MSE = 0.22, p = .023, $p=.17$, partial η² = .17, and no interaction between group and time (p = .232).

On the RAN task, there was no effect of group (p = .272) or time (p = .077), nor was any interaction between group and time observed (p = .137).

Discussion

The aim of this study was to examine the effect of an application-based mathematical intervention on children with ID. Contrary to prior intervention research, the present study employed an RCT design, and thereby added evidence regarding the effectiveness of using EI via an application to help youth with ID learn basic arithmetic (Hudson et al., 2018; Spooner et al., 2019).

Consistent with prior intervention research, the MG improved in both addition and subtraction fluency (partial η²= .153 - .271) (Cohen, 1988), as demonstrated by the significant difference between the two groups at post-test (cf. Browder et al., 2008; Hudson et al., 2018; Schnepel & Aunio, 2022; Spooner et al., 2019). This confirms that application-based interventions can be used to help students with ID to improve their arithmetic fluency skills (cf. Ortega-Tudela & Gómez-Ariza, 2006; Spooner et al., 2019). It is also noteworthy that the effect was larger for automatic arithmetic fact retrieval (answers within 3000 ms) than it was for the total number of correctly solved problems. Since arithmetic fact fluency is important for both future success in mathematics and everyday independence, it was the goal of the intervention to not only teach these skills but also to help the student to reach a level of mastery where they are automatic. Our results suggest that the application is effective in this regard for the ID population, and not just for typical developing youth who struggle with mathematics at an early age (cf. Hassler Hallstedt et al., 2018).

It is noteworthy that a greater improvement was found for subtraction compared to addition. This may be due to several factors. One possible explanation is that subtraction is generally considered more difficult than addition, which means that addition is generally taught earlier than subtraction (e.g., Sarama & Clements, 2009). This may lead to special education teachers to focus more on addition over subtraction, due to what may be considered an underestimation of their students’ ability to learn, or in an effort to keep the teaching on a level that all students in the class can follow. The present application-based intervention programme did not differentiate between addition and subtraction; both types of operations were given an even amount of time and focus (Hassler Hallstedt et al., 2018).

In contrast to Tzanakaki et al. (2014) and Ortega-Tudela and Gómez-Ariza (2006), our participants practised mainly arithmetic. The fact that positive results in more advanced skills (arithmetic) were obtained is an important contribution to this domain of research. It may be that ID population has been underestimated when it comes to the goals set by others, especially within mathematics, since this is a research area that is lagging behind compared to other areas of educational research as well as other populations.

The replication of positive results from Hassler Hallstedt et al. (2018) on the population of students with ID is important. First, it suggests that didactics (e.g., EI) administered through an application functions well on the targeted population. This finding suggests that the arithmetic learning of students with ID is not qualitatively different from that of the TD students. Moreover, the results indicate that it is possible to facilitate arithmetic learning in students with ID through an application-based intervention such as Chasing Planets, which is in line with other studies regarding mathematical learning in students with ID (e.g., Spooner et al., 2019).

Perhaps the most important finding, however, was that the intervention group maintained their significantly better performance even six months after the intervention. This was somewhat surprising as most prior studies that include a follow-up observe either a fade-out or a catch-up effect among the participants (e.g., Hassler Hallstedt et al., 2018). The present study did not observe any such effect; thus, the students in the intervention group maintained their arithmetic knowledge and skills they acquired during the intervention period. This long-lasting effect might be due to two vital aspects of the intervention. Firstly, the intervention was based on a three-phase system where the first two phases focused on conceptual understanding (i.e., modelling phase, guide phase), while the final phase focused on fluency training. Secondly, the application was adaptive to the student's current skill level, and the student was not allowed to proceed to the next phase or to a higher level (i.e., planet) until they mastered the previous phase or level. The combination of these two factors ensured that the students possessed the necessary conceptual knowledge before they started the fluency training under time constraints. It also meant that the students possessed the conceptual knowledge and procedural knowledge and skills required to tackle the next highest arithmetic level (i.e., planet). Thus, it seems that this instructional practice enables students to acquire and truly consolidate their new arithmetic knowledge and skills. Furthermore, the lack of any catch-up effect for the active control group suggests that the youth with ID require continuous EI to develop their basic arithmetic skills.

One final important finding is that this study employed a much shorter intervention period (8-12 weeks) compared to Tzanakaki et al. (2014) and Ortega-Tudela and Gómez-Ariza (2006) (12-21 weeks). Thus, it demonstrates that it is possible to obtain substantial learning effects on arithmetic in children with ID after a considerably shorter intervention period. This obviously depends on the instructions being adequate for the mathematical content and the population at hand (see above).

As expected, we found no transfer effects on the general cognitive tasks. The lack of transfer effects is hardly surprising, as nearly all prior intervention studies show that participants improve in the areas that are directly targeted by the intervention – but there are almost no near or far transfer effects (see Brankaer et al., 2013).

Limitations

One limitation is that no arithmetic outcome measure (e.g., 27 + 14; 35–22) outside the scope of the intervention was included to be able to detect potential near transfer effects from the training. Another limitation of this study, in comparison to intervention studies on the typical population, is the relatively low number of participants, which results in low statistical power. However, the issue of low statistical power is due to the ID population itself being small, and it is difficult to recruit participants within it, which meant we did not perform a power calculation but rather tried to recruit as many participants as possible within our time frame. Nevertheless, compared to prior research on the ID population, the present study design (i.e., RCT) and sample size are two important strengths as most prior studies have been single case design studies. Hence, any RCT study within this field is a positive, and adds to the total body of work helping us to better understand how to facilitate individuals with ID in learning basic arithmetic skills.

Conclusions

The results of the present study can be summarised in three main points. Firstly, application-based interventions of the kind used in this study seem to be able to help youth with ID improve their arithmetic fluency skills. Secondly, and perhaps most importantly, no fade-out effect was found six months after the intervention ended. This shows that, even though our intervention was relatively short, the students were able to significantly improve and maintain what they had learned – which may suggest that once obtained, arithmetic fluency is a skill that sticks. However, further studies and replications are needed to validate these findings. Thirdly, and finally, a greater improvement was found in subtraction compared to addition. This difference between operations might be connected to the fact that special education teachers consider subtraction to be too difficult for many students with ID (e.g., Sarama & Clements, 2009), which may lead them to focus more on addition than subtraction. This is problematic as both arithmetic operations must be considered equally important, especially in terms of skills that are needed for everyday independence.

Acknowledgement

We have complied with the ethical standards in the treatment of participants as stated by Scandinavian Journal of Educational Research.

This research project has been approved by the Regional Ethics Committee in Linköping, Sweden (Protocol Number 2016/318-31).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by The Swedish Research Council: [Grant Number 2015-02157].