Arithmetic Problem-Solving Routine for Students with Autism

Headnote

Abstract: Mathematical problem-solving is a core content ofprimary school education. Therefore, it is necessary to provide all students with a meaningful way of acquisition of problem-solving shills. The objective of this research was to verify the effectiveness of a problem-solving routine based on the Pólya 's model and the use of cognitive strategies combined with systematic and explicit instruction with continuous feedback, to improve additive problem-solving shills of students with autism. Twenty-eight students with autism participated in a quasi-experimental exploratory study. The empirical study consisted offour phases: pretest, intervention, posttest, and maintenance. The results show a significative improvement of the students' performance in the experi-mental group after the intervention, and most of them maintained what they have learned six weeks after the intervention. Between-group effect size during the post-test supports the effectiveness of the intervention.

People with autism may have difficulties relat-ing to others, interest in repetitive activities and following routines, literal interpretation of language and alterations in executive func-tions (APA, 2022; Happé, 1993; Ozon off & Schetter, 2007; Wing & Gould, 1979). How-ever, they usually have good visual memory (Grandin, 1995).

Students with autism often may show difficulties in mathematics, including problem-solving (Bullen et al., 2020). Due to alterations in the theoiy of mind, the abstraction requirements and the lack of reading comprehension, students with autism may present lower performance than their neurotypical developing peers (Polo-Blanco, Suarez-Pinilla et al., 2022; van Vaerenbergh et al., 2022). In addition, these difficulties can lead students with autism and intellectual disabilities to solve only those prob-lems that are familiar to them (Polo-Blanco, Gonzalez-López et al., 2021) or to use strategies in a mechanical way (Bae, 2013; Polo-Blanco et al., 2019).

In recent years, several researchers have developed fruitful lines of work, stating different approaches to promote problem-solving skills in students with autism, for an overview of these findings see (Barnett & Cleary, 2015; Gevarter et al., 2016; King et al, 2016; Root et al., 2021) among others.

Between these effective approaches we can highlight schema-based instruction (SBI), modified schema-based instruction (MSBI) and conceptual models. SBI uses schematic diagrams to facilitate the understanding of the relationships between the statement data (Jitendra et al., 2002). Core components of SBI approach include the identification of the structure of the problem to find the prob-lem type, the use of visual representation (diagram) of this structure to organize the data, joint with explicit instruction on the schema-based problem-solving method (Root et al., 2017). SBI has been successfully employed to teach additive problem-solving to students with autism (Kasap & Ergenekon, 2017; Polo-Blanco, Gonzalez-López et al., 2024; Rockwell et al., 2011).

MSBI approach combines SBI components with enhanced visual supports, a heuristic to perform task analysis, and continuous prompt-ing and feedback (Cox & Root, 2020; Root et al., 2022). Systematic and explicit instruction with continuous feedback are effective strat-egies for students with intellectual disabilities (Schnepel & Aunio, 2022), and explicit instruction is an evidence-based practice for students with autism (Root et al., 2021). MSBI has also been effectively used to teach additive problem-solving to students with autism (Bruno et al., 2021; Root et al., 2017).

The conceptual model-based problem solving (COMPS) methodology consists of two learning stages: problem representation where the student first solves problems without unknown (story grammar); and problem solving where the student tackles problems with unknown (Xin, 2012). This methodology includes the DOTS checklist (Detect-Organize-Transform-Solve) as heuristic and a model equation or schematic diagram that highlights the relationship between the problem data, for example, "Part + Part - Whole" for additive word problems (Xin, 2012; Xin et al., 2008). The implementation of COMPS methodology requires a direct instruction process where ses-sions are divided into two steps: a guided practice and an independent practice (Engelmann, 1980; Xin, 2012). The cognitive heuristic DOTS checklist consists of several steps to guide the student across the problem-solving process: Detect the underlying problem structure; Organ-ize the data using conceptual model diagrams; Transform the diagram into a mathematica] equation; Solve the equation and check the process (Xin, 2012, p. 107).

The DOTS checklist has points in common with general heuristic instructional approaches in the line of Pólya's model (1989), which states four general steps for mathematical problem-solving: (1) understand the problem; 2) devise a plan; 3) execute the plan; 4) examine the obtained result. These steps are intended to be applied jointly with heuristics and ques-tions to guide the student along the process (Pólya, 1989).

The COMPS methodology was compared to a general heuristic five-step problem-solving checklist called SOLVE (Xin et al., 2011). SOLVE means: Search for the question; Organ-ize the information; Look for a strategy; Visual-ize the problem; and Evaluate your answer (Xin, 2012; Xin et al., 2011). A comparison group design was conducted with students with learning disabilities, 15 of them foliowed

COMPS instruction versus 14 participants who foliowed SOLVE instruction, to assess their multiplicative word problem-solving skills. The results obtained confirmed the effectiveness of COMPS methodology, since the COMPS group improved his performance sig-nificantly more than the SOLVE group (Xin et al., 2011).

However, the comparison performed by Grif-fin & Jitendra (2009) between SBI and general strategy instruction -based on the use of heuristic and several strategies built upon Pólya's model (1989)- showed no significative differ-ence between both strategies. Both conditions improved the additive problem-solving skills of the participants, two groups of thirty third-grade students in inclusive mathematics class-rooms, there were three students with learning disability in the SBI group and two students in the GSI group (Griffin & Jitendra, 2009).

The COMPS methodology was initially aimed to students with learning disabilities or difficul-ties in mathematics (Xin, 2012) and has been recently successfully applied to enhance multiplicative problem-solving skills of students with autism (Garcfa-Moya, 2022; Garcfa-Moya et al., 2023; Garcfa-Moya et al., 2024; Polo-Blanco, Bruno et al., 2022; Polo-Blanco, van Vaeren-berghetal., 2022).

In the case of students with autism, previous studies have shown the effectiveness of cognitive strategies, such us heuristics, to improve their problem-solving skills (Barnett & Cleary, 2015; Whitby, 2013). The study conducted by Whitby (2013) employs seven cognitive (read, paraphrase, visualize, hypothesize, estimate, compute, and check) and three meta-cognitive strategies (say, ask and check) from Solve It! Problem Solving Routine (Montague, 2003) with a multiple baseline design across three participants with autism.

Considering the above, cognitive strategies combined with systematic and explicit instruction with feedback seem to be a suitable and effective approach to teach additive problem-solving to students with autism. Therefore, this paper aims to answer the following question:

Could a problem-solving routine with a structureel work-sheet following Pólya's model, and the DOTS sequence be effective to improve the performance of students with autism in additive problem-solving?

Method

This is a quasi-experimental exploratory study consisting of four phases: pretest, intervention, post-test, and maintenance. During the intervention sessions, a problem-solving routine based on an adaptation of Pólya's model (1989) and the DOTS sequence (Xin, 2012) was used.

Setting and Materials

The sessions were carried out individually in a classroom located in the association where the participants usually attend extracurricular activities, or alternatively, in the educational centre where they were enrolled. The classroom was free of distractions and each participant attended a weekly session.

At the beginning of the study the participant was provided with a visual agenda to be employed in all sessions. The agenda con-tained the pictograms corresponding to the activities programmed for each session and an indicator (green arrow) to sign the current activity. The agenda was used along the session so that the participant knew the activity he was doing, the ones he had already com-pleted, and those still to be done.

In addition to the visual agenda, in the pretest, the post-test and maintenance, the fol-lowing materials were used: laminated sheets divided into four rows where each of them has an addition or subtraction operation, coins and euro bills, statements of arithmetic problems, cards with the symbols for addition, subtraction, multiplication and division, a blackboard, a blackboard marker, and paper and pencil.

During the intervention, the experimental group used a structured worksheet as shown in Figure 1. The complementary material to this laminated worksheet consisted of statements of arithmetic problems, statements in pictograms, euro coins and bills, cards with the symbols of basic arithmetic operations, a line of measurement units, a scheme of the sexagesimal system, and paper and pencil. Each section of the worksheet had Velcro for attaching the complementary material provided.

Likewise, the researcher used a record sheet to collect the scores achieved by each student in each of the proposed activities and a video camera to record the sessions.

Participants

The inclusion criteria for the participants were: (1) have a diagnosis of ASD according to the Diagnostic and Statistical Manual of Mental Disorders, 5th edition (APA, 2022); (2) be between 6 and 14 years old; (3) basic knowledge of operations, concept of money, and/or units of measurement; and (4) diffi-culties in problem-solving, detected by their teachers.

This work is part of a larger study with a larger sample of students with autism, who have worked on different mathematical topics such as geometrical skills as well as additive and multiplicative problem-solving (Garcfa-Moya, 2022).

For this part of the study, two experimental groups were created with students with diffi-culties in the same type of arithmetical problems, finding on one hand, 13 students, (EG1) and, on the other hand, 8 students (EG2). Thirteen participants solved one-step additive problems, and eight participants solved two-step problems (an addition or sub-traction and a multiplication), as can be seen in Table 1.

The complete experimental group was composed by 21 students (20 boys and 1 girl) whose characteristics can be seen in Table 2. The mean age of these participants was 10.41 years and an IQ of 82.84 with a Standard devi-ation of 21.626.

The mean age of the 13 participants of the EG1 was 11.11 years andan IQ of 74.08 with a Standard deviation of 19.01. The mean age of the 8 participants of the EG2 was 9.25 years and an IQ of 97.85 with a Standard deviation of 17.99.

To compare the performance of the experimental group, a control group was created. Due to several difficulties during the period between the initial and final evaluation, we collected enough data to match (by age and IQ) only some of the participants in the experimental group with participants in the control group. Then the control group was composed by seven students whose characteristics can be seen in Table 3. The mean age of these participants was 9.75 years and an IQ of 93.57 with a Standard deviation of 14.524.

Tables 4 and 5 show the pairs of participants for every type of problems. In one-step problems, the mean age of the four participants in the experimental group was 9.69 years and an IQ of 72.25 with a Standard deviation of 12.685, and the control group was made up of four students with a mean age of 9.62 years and an IQ of 83 with a Standard deviation of 6.164.

In two-step problems, participants in the experimental group had a mean age of 9.36 years and an IQ of 104 with a Standard deviation of 4.359, and the control group had a mean age of 9.91 years and an IQ of 107.66 with a Standard deviation of 7.371.

The IQ was obtained from the reports of each student or was measured with the Intelli-gence Scale for Children (WISC-V, Wechsler 2015) or LEITER-3 (Roid et al., 2013) depending on whether the participant had usage of functional language.

The families of the participants signed the informed consent for participation in the study, prior agreement with them.

Dependent and Independent Variable

The dependent variable in the pretest, posttest and maintenance was the degree of help that the student needed (0 - not attended; 1 - with big help; 2 - with notable help; 3 - with help; 4 - without help), while in the intervention was the percentage of steps per-formed correctly and autonomously (Root et al., 2022). The independent variable was the problem-solving routine used during the intervention sessions.

Design and Procedure

Pretest. All participants attended an individ-ually problem-solving session of approximately 20 minutes. The type of problems was additive of change (addition and subtraction), and multiplicative of ratio (multiplication), always 2 problems of each type involved.

The researcher only intervened to read the problem statement or to focus partici-pant's attention. Types and number of problems solved by each participant can be seen in Table 1.

Intervention. Experimental group's participants attended to intervention sessions until achieve a stable percentage of steps independ-ently and correctly performed between 80% and 100%. The participants attended at most seven sessions of 20 minutes.

The problem-solving routine foliowed during intervention was supported by a structured worksheet (see Figure 1) following the problem-solving stages of Pólya's model (1989) joint with an adaptation of the DOTS sequence (Xin, 2012). Modifications included a statement in pictograms to provide understanding of the story told by the statement of the problem (Alexander & Dille, 2018; Garcfa-Moya, 2018; Garcfa-Moya, 2022; Garcfa-Moya et al., 2023; Mirenda, 2003) and the inclusion of the fourth Pólya's model stage in each step to guide the student with questions and pro-vide him/her with immediate feedback about his/her progress. Graduated guidance and verbal prompts are used as response prompt-ing strategy, recording only independent responses (Collins et al., 2018).

In step 1 (understand the problem, detect) the participant chooses a statement, read it, and selects the corresponding statement in pictograms, checking if this election is accurate (examine the obtained result). In step 2

(understand the problem, detect, and organ-ize) the participant extracts the information contained in the statement selecting prices or data and completing the box with coins or numbers (organize), checking if this selection is accurate (examine the obtained result). In step 3 (devise a plan, detect) the participant chooses the operation to perform and checks if this operation could solve the problem by reviewing previous completed steps (examine the obtained result). In step 4 (execute the plan, transform, and solve) the participant performs the operation and checks whether the result is consistent with the statement, statement in pictograms, selected data, and the chosen operation. Finally, the researcher reads the problem question, and the participant responds verbally and/or pointing out the result (examine the obtained result).

The researcher only intervened to read the statement of the problem to the students who needed it, to focus their attention by pointing out the step they had to perform and guide them with questions in case of error to correct it immediately.

During the intervention, the control group did not receive any problem-solving instruction.

Post-test. All participants solved the same type and number of problems as they did in the pretest. The researcher's intervention was based on reading the statement and focusing the student's attention.

At the end of the post-test, the families and teacher of the participants of the experimental group answered to a satisfaction questionnaire.

This questionnaire consisted of six questions about the effects of the intervention and a scale with three values: yes, I don't know and no.

Maintenance. Participants of the experi-mental group attended an evaluation session 6 weeks after the intervention. The methodol-ogy foliowed was the same as in the post-test.

Data Analysis

The statistical program SPSS 28.0.0.0 was used to perform Kolmogorov-Smirnov test, Wilcoxon signed rank test on pretest (PR), post-test (PO) and maintenance (M) scores of the experimen-tal group, and the U Mann-Whitney test to com-pare both groups' scores. The effect size ES was also calculated by Cohen's d using this formula: mean of the experimental group - mean of the control group -+- the mean of the Standard devi-ations (Hedges & Olkin, 1985). Positive sign in ES values states a favourable effect on the experimental group.

Internat Validity

The sessions were videotaped and reviewed by an external teacher with experience with students with autism, who was unaware of the study's hypothesis. He randomly selected and reviewed 33.33% of the videos of the pretest, intervention, post-test, and maintenance. The agreement between observers (researcher and external observer) was calculated dividing the number of agreements by the number of sessions reviewed and multiplying by 100. The results obtained were: 100% pretest; 87.5% intervention; 99.47% post-test; and 100% maintenance.

Likewise, the reliability of the procedure was measured. To do this, the external observer randomly selected 33.33% of the videos of the intervention sessions and checked whether the researcher: (1) provided a visual agenda at the beginning of each session; (2) facilitated the manipulative material agreed upon in the sessions; (3) allowed the student to solve the activity independently; (4) con-gratulated the participants on their work at the end of the session; and (5) collected the scores of each student on the record sheet.

The reliability of the procedure was 100% and was calculated using the following formula: number of planned behaviours carried out -i-total number of planned behaviours x 100.

Results

Pretest, Post-Test, and Maintenance of the Experimental Group

Wilcoxon signed rank test shows significant dif-ferences between the scores obtained in the pretest (XPE - 1.15) and post-test (XPO - 3.46) by the 13 participants who worked on one operation problems (Z - 3.314; p - .001). In addition, 7 of the 10 participants who attended the maintenance session achieved the same score as in the post-test, obtaining no significant differences (Z= -1.633; p = .102; XM = 3.2). Figure 2 shows the scores obtained by the participants in each evaluation.

Figure 3 shows the scores obtained by the eight participants who worked on two-step-problems. Wilcoxon signed rank test shows significant differences between the scores of the pretest (XPR - 1.0) and post-test (XPO - 3.63), (Z= 2.636; p = .008). In addition, four of the six participants who attend the maintenance session achieve the same score as in the post-test (Z= .000; p= 1.0; XM = 3.5).

Intervention

Figure 4 shows the percentages of steps per-formed correctly and autonomously achieved by experimental group 1. Most of the participants who worked on one operation problems showed a performance with an increasing and/or constant trend along sessions. Only E18 and E21 participants show a decrease in their progress in the last sessions.

The percentages of steps performed correctly and autonomously obtained by experimental group 2, who worked on two-step-problems can be seen in Figure 5. Most of the participants showed a performance with an increasing and/ or constant trend along sessions.

Comparison with Control Group

Wilcoxon signed rank test shows there are no significant differences between the scores of the pretest and post-test obtained by the stu-dents in the control group, (Z = .000; p = 1.000; XPR = 1.75; XPO = 1.75) in one opera-tion problems, and (Z= -1.342; p = .180; XPR = 3.67; XPO = 2.33) in two-step problems. The effect size (ES) between groups was also calculated, as can be seen in Table 6, showing a positive effect of the intervention.

Figure 6 shows the comparison between the scores of the pretest and post-test of the students who worked on one-step problems. U Mann-Whitney test shows no significant differences between the scores obtained by both groups in the pretest (U = 6.000; p = .317) and the post-test (U = 2.500; p = .082) in one-step problems.

Although the experimental group (XPR = 1) started with a slightly lower mean score than the control group (XPR = 1.75), results show between group effect size of 0.727 during post-test, favourable to the intervention (see Table 6).

Figure 7 shows the between groups compari-son on two-step problems. Again, results show an effect size during post-test favourable to the intervention (ES = 0.323), (see Table 6).

U Mann-Whitney test shows significant differ-ences between the scores obtained by both groups in the pretest in two-step problems (U= .000; p = .034). However, the are no significant differences between the scores obtained by both groups in the post-test, (U = 3.000; p = .487), showing an improvement of the performance of participants on the experimental group after the intervention, with scores above those of the control group. These results should be taken with caution due to the small sample size and the differences between the groups' pretest scores.

Results of the Questionnaire

Table 7 shows the results obtained in the questionnaire answered by the families and teachers of the participants. The 71.43% of the survey respondents consider that the participant has expanded his/her mathematical knowledge, 66.67% claim the participant looks happier from the beginning of the study and 61.90% affirm that the participant is more confident of school mathematical activities. Likewise, only 28.57% of the survey respondents state that the participant has asked them to do activities like those in the study.

Discussion

The results obtained confirm that the problem-solving routine with the structured worksheet following the problem-solving stages of Pólya's model (1989) and an adaptation of the DOTS sequence (Xin, 2012; Xin et al., 2008) has been beneficial for all participants in the experimental group. The participants have improved their performance in problem-solving and most of them have maintained their scores 6 weeks after the intervention. Furthermore, when com-paring the performance of the experimental group with a control group with a higher initial performance, we obtained that the scores of the experimental group have achieved the same level as those of the control group after the intervention. The positive effect size in favour of the experimental group also con-firms the effectiveness of the intervention, however, these findings should be taken with prudence, considering the small sample size.

These findings are in line with the results obtained by other authors with students with learning disabilities (Griffin & Jitendra, 2009; Xin, 2012) and with students with autism (Garcia-Moya, 2018; Barnett & Cleary, 2015; Bruno et al., 2021; Root et al., 2017; Whitby, 2013), where the employment of an approach including a structured sequence and cogni-tive strategies for arithmetic problem-solving increased their performance and problem-solving skills.

In the present study, the DOTS sequence from the COMPS methodology (Xin, 2012) was adapted to be employed jointly with the physical worksheet and systematic and explicit instruction, showing its effectiveness also out-side the framework of the COMPS methodology, where has obtained positive results to improve the multiplicative problem-solving skills of students with autism (Garcfa-Moya et al., 2023; Garcfa-Moya et al., 2024; Polo-Blanco, Bruno et al., 2022; Polo-Blanco, van Vaeren-berghetal., 2022).

Our findings show that suitable adaptations of cognitive strategies combined with augment-ative language (Mirenda, 2003) and instruc-tional supports (Root et al., 2021; Schnepel & Aunio, 2022) can also be effective to teach addi-tive problem-solving to students with autism.

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