Using the concrete–representational–abstract sequence to teach math skills to a student with autism spectrum disorder in a general education classroom

Abstract

Objective

This study aims to determine the effectiveness of the concrete-representational-abstract (CRA) sequence presented by the explicit instruction in teaching the skills of basic addition and subtraction, building tens-and-ones to a student with special needs in a general education classroom.

Method

A multiple probe across skills single case research design was used to evaluate the CRA sequence. The research was carried out with a seven-year-old female student diagnosed with autism spectrum disorder (ASD).

Results

It was determined that the participant gained the math skills taught with the CRA sequence, and these skills were displayed two, three, and four weeks after the intervention ended. In addition, it was observed that the student was able to generalize these skills to another environment and person. Social validity findings show that the mother has positive opinions about teaching math skills with CRA sequence.

Conclusion

In this study, it was determined that the CRA sequence could be used effectively in teaching math skills to a student with ASD studying in a general education classroom.

Introduction

Math skills include functional skills necessary for individuals with special needs to live independently. However, individuals with special needs have difficulties in acquiring math skills and in providing permanence and generalization after acquiring them (Gursel 2010). It is possible to say that the math problems that students with special needs are caused by the fact that they are not taught using methods that work for them. To be successful in math teaching, it is necessary to take into account the learning characteristics and needs of individuals and to teach using effective methods (Xin and Tzur 2016). The concrete-representational-abstract (CRA) sequence used in teaching math skills to students with special needs is one of the methods that provide effective results.

CRA sequence

The CRA sequence consists of concrete, representational, and abstract phases that allow students to actively participate in the learning process by interacting with math skills. In the first phase, instruction is carried out with concrete objects. In this phase of instruction, which entails the development of a concrete conceptual understanding (Milton et al. 2019), math skills are modelled with objects. For example, when teaching the number three, the teacher uses three pencils to demonstrate the concept. When teaching with objects is done, he or she moves on to the next phase, teaching by representation. Students create representations of operations using lines, figures, or pictures (Agrawal and Morin 2016). If we proceed from a similar example, teaching can be carried out by drawing three lines in the representative phase. Thus, a transition from the concrete to the abstract phase is realized gradually. The instruction concludes with an abstract phase in which only mathematical symbols and numbers are employed. In addition to conceptual learning, fluency is also developed through abstract instruction (Flores et al. 2014b; Milton et al. 2019).

The CRA sequence aims to provide students with a conceptual understanding of mathematical skills (Zhang et al. 2022) through explicit and systematic instruction using multiple representations (Flores and Hinton 2022). It also includes steps for students to learn why and how to learn the skill. The CRA sequence, which includes the active participation of the teacher and the student, is implemented by following the steps of explicit instruction (Flores and Hinton 2022; Morano et al. 2020; Powell 2015). Explicit instruction is an evidence-based practice in teaching math skills (Riccomini et al. 2017). Explicit instruction is a method in which the active participation of the student is ensured, small and sequential steps are followed, and feedback and clues are frequently used (Rosenshine 2008). In general, the process that starts with the preparation of the student for the lesson is continued by following the stages of being a model, guided applications, and independent applications (Archer and Hughes 2011). It is thought that effective results can be obtained by presenting math skills with explicit instruction, which is an evidence-based practice, and the CRA sequence in educational environments where students with special needs are present.

The majority of students with special needs are educated in general education classrooms. It is known that students with special needs studying in general education classrooms are more likely to experience deficiencies in the skills included in the mathematics curriculum throughout their educational lives (DeSimone and Parmar 2006). CRA sequence emerges as a good option that can contribute to the solution of these problems. There are studies investigating the effectiveness of the CRA sequence in teaching various math skills to students with special needs or at risk for mathematics failure in general education classrooms. Studies show that the CRA sequence is effective in teaching addition by regrouping (Carmack 2011; Miller and Kaffar 2011; Strozier et al. 2015) and subtracting by regrouping (Flores 2009; Flores 2010; Flores et al. 2014a; Mancl et al. 2012; Strozier et al. 2015). Unlike these studies, Yakubova et al. (2016) examined the effectiveness of the CRA sequence in teaching basic addition, subtraction, and number comparison skills. Flores et al. (2014b) presented the basic addition and subtraction operations to students with ASD and developmental disabilities by integrating the Strategic Teaching Model (SIM) into the CRA sequence. In another study, Nar (2018) used the CRA sequence presented with explicit instruction while teaching basic addition to students with intellectual disabilities in general education classrooms. Researches show that the CRA sequence is effective in teaching different types of addition and subtraction. In addition to these, effective results have also been obtained in studies that taught students with learning disabilities (Peterson et al. 1987; Peterson et al. 1989) and students with ASD (Erkaraman et al. 2022) to building ten-and-ones with the CRA sequence.

When the studies are examined, it is seen that there are limited studies on teaching basic addition, basic subtraction, and building tens-and-ones skills to students with special needs studying in general education classes. But the CRA sequence can be used as an effective method to solve the problems experienced by students studying in general education classes regarding the mathematics curriculum. In particular, the fact that the CRA sequence offers mathematical concepts and skills in a flow from concrete to abstract makes the use of this series important in teaching mathematics to students with special needs. For this reason, it would be beneficial to increase the research on the effects of the CRA sequence on students with special needs and to examine the effects of presenting the CRA sequence with different teaching methods and techniques. When we look at the studies conducted with students studying in general education classes, it is seen that there are studies in which the CRA sequence is presented with a video model, explicit instruction, or different strategies. However, these studies are limited in number. With this study, we hope that the findings obtained in the literature regarding the acquisition of math skills for students with special needs in general education classes by using the CRA sequence will be extended.

The aim of this study is to determine the effectiveness of the CRA sequence in teaching basic math skills to a general education student with ASD. It is also hoped that this study will expand the findings of studies investigating the effectiveness of the CRA sequence in Turkey and in the international literature. In line with this purpose, answers were sought to the following questions:

1. Is the CRA sequence presented to the special needs student studying in the general education classroom, using explicit instruction, effective in the student’s acquisition of basic addition, subtraction, and building tens-and-ones skills?

2. If the CRA sequence is effective in teaching math skills, will the student be able to maintain the acquired skills after 2, 3, and 4 weeks?

3. Can the student generalize the acquired skills to practice with another implementer and in another setting?

4. What are the views of the student’s mother on the teaching of math skills with the concrete-representational-abstract sequence presented with the explicit instruction?

Method

Participants and implementers

One child with ASD from a general education classroom in southern Turkey participated in this study. The first researcher visited the principal of special education and rehabilitation centers to explain the purpose of the study. The principal made the arrangements to meet the families of the children who demonstrated the prerequisite skills. Interviews with the two kids’ families took place. One family reported that they could not participate in the study due to COVID-19. The other family stated that they volunteered to participate in the study. After giving detailed information about the purpose and scope of the study to the family that would participate in it, the family signed a form stating that they voluntarily participated in the study. Before the onset of the study, the researchers received ethics committee approval from the University Institutional Review Board (proposal number: 2020-20).

Following and responding to simple instructions, interacting with objects (e.g. following, holding, repositioning), drawing simple shapes, counting backward and forward from 1 to 10 rhythmically, and writing and reading numbers up to 20 are required skills identified for this study. Cansu (the name of the student is different, and "Cansu" was used as a pseudonym in this study), aged 7 years and 5 months, was a female student with ASD. She was diagnosed with atypical autism at the hospital when she was 38 months old. When she reached school age, she was evaluated by the Guidance Research Center (GRCs), which makes educational diagnoses in Turkey, and was diagnosed with autism. She was first directed to a kindergarten with a general education classroom by the GRSc. When her education here was complete, she was directed to primary school by the GRCs as an inclusion student within the scope of general education. In the period when this study was carried out, the Individualized Education Program (IEP) was in the process of being created due to the continuation of home education. She was studying in the first grade within the scope of inclusive education. As she was in the first grade, she was learning to read and write. She also received support training at a special education and rehabilitation center three days a week. Cansu generally showed excited and eager behavior toward the lesson and could react by following verbal instructions. Cansu could maintain her attention for at least 10 min during the activity. She could follow directions for three or more actions and ask questions when she needed help during the activity. She could count backward and forward rhythmically from the desired number between 1 and 10, and could also write and read numbers up to 20. She could draw simple shapes and match numbers with objects.

Except for the generalization sessions, the first researcher was the implementer in all sessions. The implementer graduated from the Education of Mentally Disabled People Program and Elementary Mathematics Education Program with a double major and bachelor’s degree. At the time of the implementation, she was continuing her graduate education (MEd) in special education and was also working as a research assistant at a university. The implementer has practical experience teaching various mathematical skills to students in special education and general education classrooms. The data for the generalization sessions were collected by the mother. The mother is a housewife who is 32 years old.

Setting and materials

Since the study was carried out at a time when schools were closed due to the pandemic, the interventions were made at home. The kitchen, which is the most appropriate space, was set up for work. The worktable was set up so that it could obscure the appliances in the kitchen. The size of the table and chair were arranged considering the student. The variables that could cause the student to become distracted while studying were eliminated. To prevent the student from being distracted, the study materials were set on the table’s edge. To record the study sessions, a camera was also put in the kitchen.

All sessions were typically recorded using a camera and tripod, and the recordings were kept on a computer and a hard disk. A form was employed to identify reinforcers for use in instruction. In the sessions, various materials are used in concrete, representational, and abstract phases. Three-dimensional, touchable, and moving devices, including pencil sharpeners, tapes, straws, and erasers, were utilized in the concrete phase. For addition and subtraction in the representational phase, paper and pencil covered in PVC with an elliptical shape were used, and a worksheet with the drawings of objects was utilized for building tens-and-ones. In the abstract phase, worksheets consisting of mathematical skills were used. In the lessons taught during the concrete and representational phases, mathematical symbols were also used to help prepare students for the abstract phase.

Experimental design

We used a multiple probe across skills single case research design (Gast et al. 2014) to examine the effectiveness of the CRA sequence in teaching math skills to a general education student with ASD. Experimental control was established by the observation that performance levels in the probe sessions held before the first skill being taught significantly improved, following the application of the independent variable, but the levels of other skills to which the independent variable was not applied remained unchanged. Furthermore, this impact was diachronically replicated for the other two skills.

The study’s internal validity was protected by controlling the variables. The implementer informed the student’s family in a conversation not to conduct any study on the target behaviors. For the measurement effect, all sessions were video-recorded, and reliability data was collected. The completion of intervention sessions in a short time eliminated the maturation effect. The fact that the research model does not require frequent probe sessions prevents the testing effect.

Independent variable

The independent variable in this study is the CRA sequence that is presented with explicit instruction. Each phase of the CRA sequence was presented with explicit instruction. At each phase, the flow of modeling, guided, and independent practices—which are the steps of explicit instruction—was followed. Two trials were done with modeling, four were done with guided practices, and five were done with independent practices for each skill.

Dependent variables and data collection

The accuracy performance of the student in basic addition, basic subtraction, and building tens-and-ones was the dependent variable in the present study. This study was carried out during the period when face-to-face education was suspended for approximately 15 months in our country due to the pandemic. During this time, when schools were physically closed because of COVID-19 and teachers were teaching from home, Cansu’s general education teacher taught all the lessons online. As a result of the interviews with Cansu’s teacher and her mother, it was determined that she performed lower than her peers, who showed typically developed in mathematics lessons, and therefore did not want to attend the lessons. Collaboration was established with Cansu’s teacher in order to ensure her participation in mathematics classes and prevent her low performance, and three skills that are included in the general education curriculum and that play a critical role in the teaching of other skills were determined within the scope of this study. In the teaching of basic addition and basic subtraction, the operations were performed with single-digit numbers. In the building tens-and-ones, teaching was carried out with numbers between 10 and 20. During all sessions, permanent product recording was used to collect data on student performance accuracy. At the end of each session, the student’s response sheets were collected. The answers were marked as right or wrong, and the percentage of right answers was figured out.

The performance was defined as accurate or inaccurate for the dependent variables. They were the accurate performances for solving operations correctly and writing the result, addition, and subtraction after the target stimuli are presented. For the building tens-and-ones, it is to say and write the number of tens and ones obtained by correctly dividing the given multiplicity into tens and ones. The inaccurate performance was defined as solving the addition and subtraction operations incorrectly and writing the result incorrectly after the target stimuli are presented. For the building tens-and-ones, it is incorrectly separating the given number into tens and ones as well as saying and writing the number of tens and ones incorrectly.

General procedure

The implementer organized two sessions a day, four days a week. The intervention sessions lasted an average of 10 min. For each skill, three baseline sessions were held before the intervention sessions. The phases of the CRA sequence were presented with the explicit instruction approach. In all concrete, representational, and abstract phases, the implementer first presented information about the skill and prepared the student for teaching, and then introduced the materials to be used. Then the implementer became a model for two different trials of the skill. She used manipulatives, drawings, symbols, and numbers in her modeling. While modeling, the implementer verbally described her thoughts and the procedures she followed. Meanwhile, to stay in touch with the student, she had the student verbally repeat the order she followed (e.g. object counting). After the modeling trials were completed, the implementer performed four trial-guided practices. The student started performing the steps after the implementer presented the target stimuli. The implementer provided verbal or gestural prompts as the student needed. In independent practices, the student was given five different trials. The implementer did not provide feedback to the student regarding any errors.

Implementations across each skill began with concrete instruction. After at least three correct independent practice performances, implementation began with representational instruction. Similarly, when at least three independent practice performances were demonstrated in the representational phase of instruction, the abstract phase was started. Finally, after the abstract instruction, when at least three independent practice performances were performed, the teaching of the skill was completed. The criterion was 80% correct responses for each skill during three consecutive daily probe sessions on each instructional set. The first, second, and third probes were done after the student met the criteria for the target skills.

Baseline sessions

Baseline sessions were conducted before the intervention sessions until stable rates of responses for each skill were present. Abstract-level worksheets for each skill were placed in front of the student. Target stimulus "Look at the operations in front of you and solve the questions." was presented to the student. The implementer collected data as described in the dependent variables and data collection section. At the end of the sessions, the student’s participation behavior was reinforced.

Probe sessions

Two different types of probe sessions—daily and collective—were organized. Collective probe sessions were held after the student met the criteria for each skill, similar to the baseline sessions. During all probe sessions, for each skill, five trials were presented. Daily probe sessions were carried out at the end of each intervention session at the phase of independent practice. Daily probe sessions were organized in the concrete, representational, and abstract phases. Daily probe sessions were conducted until the student met the criterion of 80% correct performance for at least three sequential sessions. The interval between trials was set at 5 s. In the daily probe sessions held in the concrete phase, different manipulatives and different operations were given to the student. The target stimulus, "Look at the operation in front of you and solve the operation using objects" was presented to the student. After all the procedures were completed, the students’ performances were recorded in the data recording form. During the representational phase, daily probe sessions were done with shapes like lines. During the abstract phase, only math symbols were used. At the end of the sessions, the student’s participation behavior was reinforced.

Intervention sessions

The intervention lasted approximately 10 min per session. This time included preparation for teaching, teaching in which the implementer was a model, and guided practices with the student. The usage of the CRA sequence for teaching math skills is examined in the section that follows.

Basic addition

Concrete phase instruction included addition, using objects. First, the implementer prepared the student for the lesson by giving information about the skill. The implementer set the objects on the study table. The student interacted with the objects and chose which one to start the session with. After the implementer put the other objects on the edge of the table, she placed the paper on which the first operation was written on the table. She placed a plate under each number. First, she read the operation and put the objects, counting under the first number. She began by reading the operation and arranging the objects in the correct order, beginning with the first number. She repeated the same steps for the second number in the addition. She then counted the objects on the first and second plates, starting with the plate under the first number, and placed the objects on the third plate. After the counting was complete, the last number it said indicated the result of the operation. The implementer became a model for two distinct operations and presented the student with the new objects and operations. In guided practices, the implementer presented the target stimulus to the student and asked them to solve the operations with objects. Finally, daily probe sessions were performed in independent practices. These steps were carried out similarly in the representative and abstract phases. In the representative phase, the implementer used lines to solve the operations. She drew lines under the first number in the transaction as well as its value and repeated the same steps for the second issue. She found the result of the operation by adding all the lines. In the abstract phase, only symbols were taught without objects or lines. The implementer said that she stored the first number in the operation in her memory. Then she opened her finger as much as the value of the second number. She counted her fingers one by one with the number she kept in her memory and found the result of the operations. At this point, it should be noted that adaptation was carried out in line with the student’s needs at the abstract phase. First of all, in the abstract phase instruction, the student was asked to keep in mind after determining the larger number and to add the smaller number to the number she had in mind with the help of her fingers. Despite knowing the large and small numbers, the student made a mistake in the first three sessions and achieved 0% response accuracy. For this reason, she was asked to keep the first number in mind and to add her finger, which she opened as much as the second number, on the first number. An example of the basic addition steps is in Figure 1.

Figure 1.

Examples of CRA applications for basic addition.

Basic subtraction

In basic subtraction teaching, the teaching at the concrete phase was carried out similarly to basic addition. In the concrete phase, a plate was placed under the numbers in the operation (two in total, one under the first two numbers). The implementer counted the object as much as the value of the first number in the transaction and put it on the plate. Then she counted the second number’s value from the plate and took it out. Finally, she counted the remaining objects on the plate and wrote the last number she said next to the equal sign. In the representative phase, she drew as many vertical lines as the first number. Then she made the subtraction by drawing the second number of horizontal lines on the vertical lines. She counted the vertical lines with no horizontal lines drawn on them and wrote the result of subtraction to the right of the equal sign. Finally, in the abstract phase, the implementer spread her fingers as many as the first number in the transaction. Then she closed her fingers, counting backward by the second number. She wrote the remaining number of fingers next to the equal sign. An example of the basic subtraction steps is in Figure 2.

Figure 2.

Examples of CRA applications for basic subtraction.

Building tens-and-ones

In the concrete phase, the tens and ones were formed from the straws that were left mixed on the table. For example, the first tens are built from 17 straws left scattered on the table. After one "tens" is obtained, it is written in the tens part of the tens-ones table. The remaining "ones" are counted and written in the "ones" part of the table. It indicates which number (17) is obtained from the tens and ones written on the table. In the representative phase, the worksheet with the cube pictures was studied. Like the concrete phase, tens and ones were created by counting the cube pictures, and then the obtained number was written on the table. Finally, in the abstraction phase, the numbers were studied directly, and teachings were made about how many tens and how many ones the numbers consist of. An example of the building tens-and-ones is in Figure 3.

Figure 3.

Examples of CRA applications for building tens-and-ones.

Maintenance and generalization sessions

The maintenance and generalization sessions were conducted similarly to the baseline sessions. Two, three, and four weeks after all skills had been taught, the implementer held the maintenance sessions. In the generalization sessions, it was examined whether math skills could be applied in different settings and with people. For this purpose, the student’s living room was arranged, and the sessions were conducted by the mother. By observing the baseline sessions, the mother received information about the steps. The mother arranged three generalization sessions for each skill after learning the steps.

Procedural reliability and interobserver agreement

Reliability data were collected by two postgraduate students in the field of special education. The observers were trained on the definitions of the dependent and independent variables and the data-collection procedure. To determine the accuracy of the data, the researcher randomly selected 30% of the data forms for each phase per behavior. During each phase, the observer examined the percentage of accurate data for each skill to obtain (IOA) data. The following implementer behaviors were taken into account in the baseline, probe, maintenance, and generalization sessions: (a) having the materials ready, (b) attracting the attention of the student and asking whether she was ready, (c) asking the student to solve the questions in the worksheet, (d) not reacting to the student’s reactions and finally (e) reinforcing the student’s participation behavior at the end of the session. In the intervention sessions, the following behaviors were taken into account: (a) having the materials ready, (b) informing the student about the skill they will study (preparing for teaching), (c) explaining the reinforcement to the student at the end of the session, (d) asking the student whether she is ready for the study, (e) being a model for the implementation of the skill steps, (f) presenting task direction, (g) waiting for the student’s response to perform the skill during the guided practice (5 sec), (h) reinforcing the correct response, and providing clues in case of wrong response or no response, (i) reinforcing the student at the end of the session.

Interobservers calculated IOA data using an "obtained by dividing the number of agreements by the sum of the number of agreements and the number of disagreements" formula (Erbas 2012).Data accuracy for each skill had an IOA of 100%. A second independent observer collected data on procedural reliability by recording the occurrence of all implementation steps, using a checklist for at least 30% of the intervention session. The procedural reliability for each skill was set to 100%.

Social validity

Researchers developed a form to collect social validity data. They then sent this form to experts with doctoral degrees in the field. After the expert opinions were completed, they gave the form its final form. The first researcher conducted semi-structured interviews with the student’s mother about the teaching of mathematics skills, the CRA sequence, and the results of the study. An interview form consisting of 11 questions was prepared. The form was designed to reveal her opinions on the following topics: (a) the taught math skills; (b) the role of math skills in daily life; (c) using the CRA sequence; (d) taking part in the research, and (e) her opinions on both the good and bad experiences she had during the research process. Social validity data were analyzed descriptively.

Data analysis

To determine whether there is a functional relation between the dependent variable and the independent variable, visual analysis and effect size calculation was performed. Visual analysis was carried out by graphing the data obtained regarding the student’s performance in math skills. The Tau-U calculation was used to calculate the effect size. The calculation was carried out using the Web-based calculator created by Vannest et al. (2016). Tau-U values vary between 0 and 1; values between 0 and 0.20 indicate that the effect is very small, values between 0.20 and 0.60 indicate that the effect is moderate, values between 0.60 and 0.80 indicate that the effect is large, and values between 0.80 and 1 indicate that the effect is very large (Vannest and Ninci 2015).

Results

Figure 4 shows the correct response percentages in the data obtained from the collective baseline sessions, probe sessions, intervention sessions, maintenance, and generalization sessions organized for Cansu. Examining the figure shows that the student’s overall response level across skills during the baseline sessions before the intervention was consistently low. The student met the criteria during intervention conditions. The results of the visual analysis show that the CRA sequence is effective for teaching three math skills to a student with ASD in a general education classroom. Within the scope of this study, a functional relation was found between the CRA sequence and math skills (basic addition, basic subtraction, building tens-and-ones).

Figure 4.

Percentage correct performance for Cansu during all probe sessions, maintenance (M) and generalization (G).

Basic addition

When basic addition was the baseline session, Cansu had 0% mean response accuracy with a low and stable trend. Moving on to the concrete phase, it was determined that Cansu showed at least 80% response accuracy in the three interventions. After obtaining stable data in the concrete phase, the representational phase was started. When there was at least 80% response accuracy in the representative phase, the teaching in the abstract phase was started. In the abstract phase, Cansu had 0% response accuracy in the first three intervention sessions. It is seen that Cansu’s response accuracy met the criteria and remained stable after the adaptation. When the collective probe sessions (Probes 1, 2, and 3) conducted after the intervention sessions are examined, it is seen that Cansu maintains the response accuracy that meets the criterion in the basic addition.

Basic subtraction

While Cansu had 0% response accuracy in the baseline and first collective probe sessions, she achieved 100% response accuracy in the three consecutive sessions when the concrete phase of intervention started. After stable data in the concrete phases, it advanced to the representative phase and similarly achieved 100% response accuracy in three consecutive sessions. In the final phase, the abstract one, an average response accuracy of 90% was reached in four sessions. When the collective probe sessions (Probes 2 and 3) conducted after the intervention sessions are examined, it is seen that Cansu maintains the response accuracy that meets the criterion in the basic subtraction.

Building tens-and-ones

Cansu has 0% response accuracy in the baseline and collective probe sessions (Probe 1 and 2) performed before the intervention sessions. When the intervention sessions in the concrete, representational, and abstract phases are examined, it is seen that three sessions in each phase have at least 80% response accuracy. She had 100% response accuracy in all recent collective probe sessions.

Maintenance and generalization

As shown in Figure 4, Cansu continued to do basic addition with 100% response accuracy. Basic subtraction showed 100% response accuracy in the second and fourth weeks and 80% response accuracy in the third week. Finally, she can be seen that the building tens-and-ones exhibits 100% response accuracy across all of the tracking data. When the generalization data is examined, it is seen that Cansu has a response accuracy of at least 80% for all skills in the probes conducted in another environment and with another implementer.

Social validity

The student’s mother expressed positive views on the use of the CRA sequence while teaching math skills. She stated that her child learned mathematical skills and continued to exhibit these skills. She also stated that she would like to teach other mathematical skills with the CRA sequence. The student’s mother reported the following advantages of the study: (a) teaching the three basic math skills in a short time, (b) being inexpensive for teaching, and (c) teaching the skills permanently.

Effect size

The effect size for each skill was calculated utilizing Tau U (Vannest et al. 2016). Data points that do not overlap across stages are combined with an analysis of trends within each phase of the intervention. In the calculation performed for the basic addition, a high effect size (Tau-U = 0.83) was determined. In the same way, both basic subtraction and building tens-and-ones were found to have a high level of effect size (Tau-U = 1 for both skills).

Discussion

The purpose of this study is to examine the effectiveness of the CRA sequence in teaching basic addition, basic subtraction, and building tens-and-ones to a student with special needs in a general education classroom. In the study, the views of the mother of the child with special needs regarding the social validity of the research were also examined. According to results of the research, the student’s performance improved in each math skill taught using the CRA sequence. The intervention, as a result of the visual analysis of the data, had a strong effect on the student’s learning and maintenance of math skills. Also, it was found that she could use the math skills she had learned in different places and with different people. The CRA sequence supports conceptual understanding and provides the acquisition of skills. For this reason, it is important for students to acquire skills with this strategy that supports conceptual understanding in mathematics. It is thought that teaching math skills with this sequence will contribute to the conceptual reasoning of the student about addition and subtraction.

This research has expanded the literature on teaching math skills to special needs students in general education classrooms in various ways. First, the results of the study indicate that the CRA sequence is effective for teaching basic addition. This result supports the findings of some studies conducted on children with special needs (Carmack 2011; Nar 2018; Strozier et al. 2015; Yakubova et al. 2016). These studies show that the CRA sequence is effective in teaching basic addition and addition with regrouping. These results reveal the effectiveness of the CRA sequence in teaching students with special needs at different stages of the addition hierarchy and form the basis for its more widespread use.

Second, the results of this study indicate that the CRA sequence effectively teaches basic subtraction. When the studies using the CRA sequence in teaching subtraction to children with special needs (Ferreira 2009; Flores 2009; Flores et al. 2014a; Mancl et al. 2012; Strozier et al. 2015; Yakubova et al. 2016) are examined, it is seen that similar results are obtained. This study is different in some ways from other studies that use the CRA sequence to teach subtraction. While the effectiveness of the CRA sequence in the teaching of subtraction by regrouping was determined in a group of studies, it is seen that there are a limited number of studies on teaching basic subtraction. In one study on teaching basic subtraction (Yakubova et al. 2016) the CRA sequence was presented with a video model, and in the other (Flores et al. 2014b), the CRA sequence and SIM were used. In this study, it is thought that it is important that the CRA sequence is presented through explicit instruction and that the teachings are effective as a result. It is thought that teaching basic subtraction permanently will contribute to the teaching of subtraction with regrouping. Also, basic subtraction is taught before more complicated subtraction, like taking two-digit numbers apart by regrouping, so the fact that the CRA sequence is effective at teaching basic subtraction is thought to be important.

Third, the results of the current study indicate that the CRA sequence is effective in building tens-and-ones. When the literature is examined, it is seen that there are studies that teach how to building tens-and-ones under the name of place value teaching with the CRA sequence. These studies, conducted with students with learning disabilities (Peterson et al. 1987; Peterson et al. 1989) and students with ASD (Erkaraman et al. 2022), reveal the effectiveness of the CRA sequence. In present study, it is thought to be important to emphasize that it will be taught to building tens-and-ones directly, not under the name of place value. In addition, it has been suggested that these study findings show that the CRA sequence is effective to support the limited number of research findings.

Fourth, the data indicated that the student maintained the acquired skills at the criterion level for 30 days following the intervention. Similar results were obtained in the previous studies examining the effectiveness of the CRA sequence in teaching the mathematical skills studied in the present research. In some studies (Erkaraman et al. 2022; Carmack 2011; Nar 2018; Peterson et al. 1987), the students maintain the skill for 1–2 weeks, while in others (Ferreira 2009; Flores 2009; Peterson et al. 1989; Yakubova et al. 2016), they perform at a level that meets the criteria even after 3–4 weeks. It can be concluded, that using the CRA sequence is effective in helping children both acquire and maintain math skills. Mathematics skills are a prerequisite for another skill. For students with special needs to perform similarly to their peers, they must be fluent and maintain exhibiting prerequisite skills before learning new math skills. For this reason, it is thought that it is important for students with special needs to be able to perform their mathematics skills permanently in terms of educational processes in general education classrooms.

Fifth, the present data show that the student was able to exhibit acquired math skills with another person and in another setting. It was determined that generalization sessions were organized in the limited study in which the CRA sequence was used in teaching addition, subtraction, and building tens-and-ones. As a result of her research on teaching addition with regrouping, Carmack (2011) determined that students were able to generalize their acquired skills to different environments. Nar (2018), on the other hand, held generalization sessions on the commutative property of addition with the students in his study where he taught basic addition. Erkaraman et al. (2022) conducted a study of generalizing to another person and another stimulus for the building tens-and-ones. It is noteworthy that the number of studies in which generalization sessions were organized is low. It is known that students with special needs studying in general education classrooms receive education in different environments (e.g. support training rooms) and from different people. For this reason, studies on generalization are thought to be important.

Sixth, social validity findings showed that the student’s mother had positive views on this study and the CRA sequence. In the studies in which social validity data were collected from students (Carmack 2011; Ferreira 2009) or from both teachers and students (Flores 2009; Nar 2018) or only from teachers (Erkaraman et al.2022), it was determined that there were positive opinions similar to those in the present study. In general, opinions expressed revealed that they were satisfied with the use of the CRA sequence, that it facilitated teaching, and that it was economical. Unlike other studies, in current study, it is thought that it is crucial to interview the mother to determine social validity. Since families are the people who spend the most time with their children, it is thought that it is important to get the opinions of parents.

Implications

In addition to these findings, there are some implications. First, mathematics education programs taught in general education classrooms, for example, in Turkey, include learning outcomes for students to learn many skills in a short time. The student in this study was attending the first year of general education, and it was determined that she had not received systematic training for any skill before. The student learned both the application of the CRA sequence and the three mathematical skills in our country’s mathematics education program (addition in 13 sessions, subtraction in 10 sessions, and building tens-and-ones in 9 sessions) in a short time. These results give us an idea of whether or not the CRA sequence can help students learn math skills quickly and for good, even if they have never had a systematic education before. In addition, it is anticipated that these findings will contribute to the variety of ways teachers might employ to facilitate functional, practical, and maintenance learning. Second, in the teaching of the basic addition process, an adaptation was made to the needs of the student. As a result of the adaptation, it was determined that the 0% response accuracy increased to 40% and 80%. Future research can perform addition by adding the second number on top of the first number in the abstract phase of basic addition teaching. Third, teaching was carried out with tools that can be easily found in every classroom, such as pencils, erasers, and straws. It is thought that the use of easily accessible tools will facilitate the teaching process while providing appropriate mathematics education to all students with and without special needs in general education classrooms. Finally, a student with ASD was taught three basic math skills. Overall, the effectiveness of the CRA sequence has been investigated for students with learning disabilities. In this respect, it is thought that this study will expand the findings in the literature.

Implications for future research

Our suggestions for future research are as follows: (a) the effectiveness of the CRA sequence for different types of addition and subtraction can be investigated; (b) a similar study can be conducted with students in the general education class with different disabilities, such as intellectual disability; (c) the CRA sequence can be presented in combination with other methods or strategies instead of explicit instruction; (d) the effectiveness of this study can be investigated by organizing small group training; (e) social validity data can be determined by collecting social comparison data from typically developing peers, (f) taking into account that the mother has collected data on generalizations, research can be done on how well family members can teach different math skills with the CRA sequence.

Funding Statement

The authors received no financial support for the research, authorship, and/or publication of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.