Abstract
The National Council of Teachers of Mathematics (NCTM) emphasizes the teaching of “Big Ideas” in mathematics. This study focuses on the part–part–whole (PPW) relationship as a crucial aspect of word problem solving involving addition and subtraction. This study, conducted in the United States, evaluated the effects of conceptual model-based problem-solving (COMPS) with the concrete-representational-abstract (CRA) sequence on teaching addition and subtraction word problem solving to students with autism spectrum disorder (ASD). Using a multiple-probe design across the participants, the researchers examined the impact of the intervention on students’ performance across three types of word problems: join-in, take-away, and combine. Results demonstrated significant improvements in students’ problem-solving abilities on criterion tests. In addition, students successfully generalized their skills to solve problems with new contexts. The findings of this study provide implications for future research and educational practice.
Keywords
To facilitate effective mathematics teaching and learning, the National Council of Teachers of Mathematics (NCTM) has called for teaching “Big Ideas” in schools (NCTM, 2000). Mathematical big ideas can be defined as “mathematical statements of overarching concepts that are central to a mathematical topic and link numerous smaller mathematical ideas into a coherent whole” (Caldwell et al., 2011, p. 9). Mathematical big ideas draw students’ attention to key concepts, link small facts/ideas together, and connect previously learned ideas to new concepts. As such, teaching big ideas can help students develop a deep understanding of mathematics knowledge (Caldwell et al., 2011).
Addition and subtraction word problem solving is an especially important domain in elementary math because these principles serve as the foundation for learning many other mathematical concepts (Caldwell et al., 2011). Defined as “linguistically presented problems requiring arithmetic solutions,” mathematical word problems provide students with the opportunity to apply basic mathematical knowledge and principles in real-world settings (Zheng et al., 2013, p. 97). By solving real-world problems, students prepare to meet the mathematics problems they may face in their everyday lives, which may improve their ability to function as independent adults (Van de Walle et al., 2010).
The NCTM has established important mathematics big ideas related to different mathematical topics (Caldwell et al., 2011). Ma et al. (2021) identified NCTM big ideas and relevant essential understanding pertaining to addition and subtraction word problem solving, which is summarized in the following five points:
Teaching word problems should include teaching the explicit meaning of the problem and discovering the relationships between numbers in the problems.
Students should be provided with opportunities to solve a variety of addition and subtraction problems. According to Common Core State Standards (the Common Core State Standards Initiative [CCSSI], 2012), addition and subtraction word problems can be classified into types such as “join-in,” (e.g., There were three ladybugs on a leaf. Two more ladybugs joined them. How many ladybugs are there now?); “take-away” (e.g., There were some butterflies. Two butterflies flew away, and now, there are seven butterflies. How many butterflies were there in the beginning?); “combine” (e.g., There are three orange fish and five striped fish. How many fish are in the fish tank?); and “compare” (e.g., Judy has 15 crayons. Judy has three more crayons than Mike. How many crayons does Mike have?). Making connections between these seemingly different problems can help students build conceptual understanding of addition and subtraction (Caldwell et al., 2011).
Although addition and subtraction problems can be categorized into several problem types, they all indicate the same part–part–whole (PPW) relationship. PPW refers to how the two small numbers (the parts) are related to the one big number (the whole) in any addition or subtraction problem (Caldwell et al., 2011). It is worth mentioning that, depending on different problem situations (i.e., join-in, take-away, combine, and compare), each element in the P + P = W model equation may have different denotations (Xin, 2019). For example, in the following join-in problem, “Sammy has 5 pencils. Eddy gave him 2 more pencils. How many pencils does Sammy have in total?”; the beginning amount (five pencils) and the change amount (two more pencils) are the two “parts”; and the ending amount is the “whole” or total. In contrast, in the following comparison problem, “Karen has 5 books. Jenny has 2 books. How many more books does Karen have than Jenny?”; the bigger quantity (the number of books that Karen has) would be the whole, which is made up with the following two parts: the number of books Jenny has (the smaller quantity) and the difference quantity (between Karen and Jenny) (Caldwell et al., 2011, p. 22).
The same PPW relation can be expressed by variants of number sentences. For example, a + b = c or c − b = a are two different ways of expressing the same relation. Providing different forms of model equations or number sentences can help students develop and reinforce conceptual understanding of quantity relations of addition and subtraction, as well as algebraic reasoning.
Teaching algebra readiness in addition and subtraction is important because it not only lays a foundation for algebra but also supports student understanding of the equal sign and helps them understand number relations.
Teaching Word Problem Solving to Students With Autism Spectrum Disorder
According to the NCTM (2000), achieving satisfying performance in word problem solving is one of the primary goals of mathematics instruction. This is challenging for many students, including students with autism spectrum disorder (ASD). Competence in mathematics word problem solving involves aptitude in multiple skills that are common areas of weakness for students with ASD (e.g., language skills and executive functioning) (Das & Janzen, 2004; Minshew et al., 2002; Rockwell et al., 2011). As such, developing interventions targeting word problem solving skills for students with ASD is critical. When educators working with students with ASD develop interventions targeting addition and subtraction word problem solving, they should incorporate the big ideas stated above into their interventions.
In past years, researchers have striven to develop effective methods of helping students with ASD with addition and subtraction word problem solving. For example, by using a multiple-probe design across behaviors (problem types), Rockwell et al. (2011) used schema-based instruction (SBI) which used problem-type-specific schema diagram to teach combine, change, and group addition and subtraction word problem solving to one fourth-grade student with ASD. The result showed that the participant improved their performance in solving all three types of addition and subtraction word problems following instruction and could solve problems with unknowns in different positions.
Later, researchers modified the traditional SBI by incorporating other elements (e.g., using a heuristic task analysis sheet, adding visual support, and delivering instruction with technology support) and conducted a series of studies to investigate the effect of modified SBI (MSBI) on teaching addition and subtraction word problem solving to individuals with ASD (Root & Browder, 2019; Root et al., 2017, 2018, 2019). For example, Root et al. (2017) used the MSBI to teach word problem solving skills to three elementary students with ASD and moderate intellectual disability. This study targeted the compare problem type in which the difference was arranged as the unknown quantity. Employing a multiple probe across participants with an embedded alternating treatment design, the authors found out that the MSBI helped participants with compare problem-solving for both virtual and concrete conditions, and two participants performed more steps in the virtual condition. Furthermore, Root et al. (2018) and Root and Browder (2019) extended the use of MSBI to teach word problem solving that requires algebraic reasoning. Both studies taught participants to solve group word problems with the missing quantity at different positions and to discriminate different problem types (i.e., if a problem was missing the part or the whole). Results indicated a functional relation between the MSBI and the participants’ word problem solving skill and problem-type discriminating skill. Generalizing problem-solving without visual supports was also seen in Root and Browder (2019). By adopting a multiple probe across participants design, Root et al. (2019) evaluated the effect of MSBI with technology supports on students with ASD as they solved word problems involving data analysis. They found that participants could successfully solve multistep compare problems based on pictographs.
Existing studies have addressed some concepts related to the big ideas involving addition and subtraction word problems, such as teaching various problem types and the explicit meaning of problems, as well as algebraic reasoning. However, the existing studies may fall short of addressing the overarching big ideas in additive word problem-solving, that is the PPW relationship among three quantities hidden in all types of addition and subtraction word problems. SBI emphasizes using schematic diagrams that are specific to different problem types (e.g., change, group, and compare) to represent the problem (Xin, 2019). As such, students are taught to categorize a problem into a specific problem type and to “choose a schema that matches the problem type and organize information from the problem onto the schema” (Root & Browder, 2019, p. 5). Most existing studies targeted only one problem type. Studies that involve more than one problem type (Rockwell et al., 2011) also concentrate on teaching students to discriminate problem types and to choose the corresponding schema diagram.
There is a lack of existing research in emphasizing the connections between mathematical ideas and teaching students with ASD about PPW as the unified or cohesive mathematical relationship between quantities across various additive problem types. Awareness of patterns is crucial in mathematical thinking and can facilitate higher level learning and promote generalization (Devlin, 2012). Without learning the “big ideas,” students may only be able to solve familiar problems (Xin, 2019). In fact, limited generalization (e.g., the word problems were written in a predictable format; the problems do not represent a wide range of problems) was reported in the existing intervention research. As such, there is a need for research to explore instructional strategies for students with ASD that emphasize teaching big ideas in word problem solving concerning overarching mathematical relations that connect a range of word problem variations.
Conceptual Model-Based Problem-Solving Instruction
Current curriculum standards emphasize models with mathematics (NCTM Standards for Mathematical Practice), which go above and beyond semantic analysis of word problems. For early grades, “this might be as simple as writing an addition equation to describe a situation” (CCSSI, 2012). Conceptual model-based problem-solving (COMPS) (Xin, 2012), an evidence-based instructional strategy for students with learning disabilities (LDs) or difficulties (Gersten et al., 2009; Witzel et al., 2022; Xin, 2019; Xin et al., 2017, 2020, 2023), is a teaching approach that emphasizes conceptual expression of underlying mathematic relationships discovered from different problem contexts in algebraic equations by involving students in working with a given mathematical model. For instance, part + part = whole is a mathematical model for solving additive word problems and teaching PPW relationships, and Unit rate x # of Units = Product is a mathematical model for solving multiplicative word problems (Xin, 2012). Because there is a lack of intervention research that promotes teaching the PPW big idea in word problem solving involving students with ASD, COMPS would be an alternative approach to fill in this research gap.
Concrete-Representation-Abstract Instructional Sequence
By its nature, COMPS requires an abstract-level and symbolic-level understanding of mathematical concepts. For example, in addition and subtraction word problems, students need to know the meaning of the abstract terminology “part” and “whole,” which might be challenging for students with ASD because they were more successful in understanding visual information and/or processing concrete information (Rourke & Strang, 1978). As such, to effectively carry out the COMPS instruction, it is important to provide necessary scaffolding to support students with ASD in achieving abstract-level thinking that goes beyond what they touch and see.
concrete-representational-abstract (CRA) instructional sequence is a teaching sequence that involves three levels: (a) concrete (e.g., using physical manipulative), (b) representational (e.g., drawing pictures), and (c) abstract (e.g., with numbers only) (Bruner & Kenney, 1965). This graduated instructional sequence provides visual support to students’ learning in mathematics, especially at the concrete and representational levels (Agrawal & Morin, 2016). The CRA instructional sequence has been applied as an evidence-based practice across mathematical areas for students with LDs (Bouck et al., 2018). CRA teaching sequence is considered beneficial for students with ASD because they understand visual information more readily and process concrete information more easily than abstract information (Rourke & Strang, 1978). Considering students’ strength in processing visual information and disadvantages in processing abstract information, the CRA teaching sequence could serve as a bridge helping students with ASD move toward learning the symbolic or abstract PPW model.
The Purpose of This Study
The purpose of the study was to investigate the effect of the COMPS approach, with the support of the CRA teaching sequence, in teaching addition and subtraction word problem solving and an understanding of PPW quantity relationships to elementary students with ASD. We are also interested in knowing whether COMPS could help students generalize word problem solving to other contexts. This study focused on the “join-in,” “take-away,” and “combine” problem types according to the CCSSI classifications. The research questions are as follows:
Method
Participants
Participants were recruited based on teachers’ referrals and the following inclusion criteria: (a) having an educational or medical diagnosis of ASD, (b) receiving mathematics instruction in a general classroom, (c) having no prior experience with COMPS and CRA sequencing, and (d) having word problem solving difficulties as reported by teachers and a score of 60% or lower on a word problem solving screening test. The screening test was the same as the criterion test (described under “Measures and Dependent Variables”) evaluating students’ performance on solving addition and subtraction word problems. Students’ performance was determined by the percentage of problems solved correctly. The score was calculated as the total points earned divided by the total possible earned points. Students who scored 60% or lower were eligible for this study. Three second-grade students with ASD enrolled in two urban elementary schools in the Midwest of the United States of America participated in this study.
Mary was a 7-year-11-month-old White female in second grade with a diagnosis of ASD and language impairment. She received academic instruction in the general education classroom and resource room. She spent 79% of her daytime in general education class and received 30-min supplemental reading and 30-min supplemental math instruction in the resource room every day. Her IQ score was 90 based on Wechsler Intelligence Scale for Children–Fourth Edition (WISC-IV) (Wechsler, 2003). She was an energetic and fun student to be with in a one-on-one situation but became loud and defiant at times in a group setting. Mary showed resistance to environmental change. A visual schedule and self-checklist were helpful to her. English was her first language. Academically, Mary showed below grade level in reading, math, and writing. On the standardized STAR test, her math score was 96 (<10th percentile), and reading score was 74 (<10th percentile). Her score on the screening criterion test indicated her struggle in solving addition and subtraction word problems.
Bill was an 8-year-9-month-old African American male in second grade with a diagnosis of ASD and other health impairment. He received academic instruction in the general education classroom and resource room and spent 80% of his daytime in general education class. His IQ score was not available. Bill enjoyed school, but when he got frustrated with others, he would not communicate with people; instead, he would push or say mean things. He had difficulty transitioning from activity to activity and showed physical aggression at times. English was Bill’s first language. On the standardized NWEA, Measures of Academic Progress (MAP) test, his math Rasch Unit (RIT) score was 133 (second percentile), and his reading RIT was 151 (27th percentile). His teacher reported that Bill needed to work on fluent reading, retelling stories, and reading comprehension. In math, subtraction was especially challenging for him.
Nicolas was an 8-year-2-month-old Hispanic male in second grade with a diagnosis of ASD, language impairment, and cognitive disability. He received academic instruction in the general classroom and the resource room. He spent 30% of his daytime in general education class and received 90-min small group supplemental reading instruction, 60-min small group supplemental math instruction, and 30-min small group supplemental writing instruction in the resource room every day. He had an IQ score of 75 based on the WISC-IV (Wechsler, 2003). Spanish was Nicolas’ first language spoken at home, but he communicated with teachers and peers in English at school. Nicolas was very routine oriented, and he was adversely affected by any unknown change in routine. Nicolas finished tasks better with a visual schedule and self-checklist. His teacher reported that he had severe delay in both reading and math. On the standardized STAR test, his grade level equivalent was 1.6 (the range is 0.0 to 12.9+) for math and 1.0 (the range is 0.0 to 12.9+) for reading.
Settings
This study took place in the elementary schools in which the participants were enrolled and during times that were convenient for the teacher and students. In one school, the study was conducted in a small, quiet room adjacent to the resource room, which was used for storage, copying study materials, and one-on-one working instruction. In the other school, the study was conducted in a quiet corner of the resource room where the students engaged in school-related activities. There were several bookshelves separating the corner from the main activity area of the resource room, which decreased visual and audio distraction.
Design and Measures
Design
A multiple-probe design across participants (Ledford & Gast, 2018) was used in this study to evaluate the functional relationship between the intervention and improvement in participants’ accuracy on addition and subtraction word problem solving. With the multiple-probe design across participants, the intervention is introduced in a time-lagged fashion across at least three participants and a functional relation is indicated if changes in the target skill systematically occur upon administration of the intervention. Also, a multiple-probe design permits researchers to intermittently collect data during the execution of the experimental sessions, which can reduce the testing threats (Ledford & Gast, 2018). The design in this study included baseline phase, intervention phase, and post-test and maintenance assessment phase.
Measures and Dependent Variables
Criterion Test
The primary criterion test included eight one-step addition and subtraction word problems that covered three join-in, three take-away, and two combine problem types. Each problem came with the unknown quantity at a different position. This test included alternative forms of the criterion tests, which are equivalent in problem construction, however, with different numbers and contexts. The sequence of the unknown quantity appearing in word problems was random. This test design reduced the likelihood that students remember the sequence of the word problems presented in each test probe and, therefore, corresponding operations for solution. The order of the various problems was randomized, and problems were not repeated across different forms of the criterion tests. The percentage of problems solved correctly on the criterion test was used as the primary dependent measure. Below is an example of a problem students were asked to solve: “Jose had 10 colored pencils. Jillian gave him some more colored pencils. Now he has 19 colored pencils. How many colored pencils did Jillian give Jose?”
Model With Mathematics Test
The model with mathematics (MWM) test was developed and administered to evaluate students’ ability in recognizing the connection across different forms of equations to represent a contextualized problem (P1 + P2 = W; P2 + P1 = W; W − P1 = P2; and W − P2 = P1). Each test included eight test items. Each item presented four equations, with each equation falling into one of the six variations (i.e., P1 + P2 = W; P2 + P1 = W; W − P1 = P2; W − P2 = P1; P1 + W = P2; and P2 − P1 = W), within which the first four were correct equations that represented the same relationship; however, the last two were incorrect equations. Below is an example:
Luis had six candy bars. Matthew gave him some more candy bars. Now, he has 11 candy bars. How many candy bars did Matthew give Luis? Circle “Yes” if it is a correct equation for solving the problem, circle “No” if it is an incorrect equation for solving the problem
6 + ? = 11 Yes No.
? + 6 = 11 Yes No.
11 − 6 = ? Yes No.
11 + 6 = ? Yes No
In this case, students will circle “Yes” for the first three equations and “No” for the last equation. The MWM tests were adapted from conceptual understanding test items found in participating school’s adopted mathematics textbooks and reviewed by special education and math education experts. They were administered during baseline and post-test phases.
Curriculum-Based Test
To assess the students’ ability to generalize the word problem solving skills learned during the intervention to new contexts, a curriculum-based test (CMT) was administered before and after the intervention. The CMT includes eight problems directly taken from the second-grade mathematics textbook used by the participants’ school district involving much bigger numbers and variously constructed story contexts.
Scoring and Interrater Reliability
For all tests, the percentage of problems solved correctly was used as the dependent variable and calculated as the total points earned divided by the total possible points. For criterion tests and CMT, “solved correctly” was defined as the participants showed correct problem-solving process (math algorithm, math sentence, or equation), correctly mapped PPW (during the intervention and post-test phases), and got the correct numerical answers. Specifically, each problem was worth 1 point, earned only if the mathematical equation, PPW mapping (during the intervention and post-test phases), and final answer were all correct. Labels for the answers were not required considering the challenges in writing of the young children involved. In addition to giving their answers, students were also asked to write down the PPW mathematical equation. For MWM tests, each problem was worth 4 points (each equation worth 1 point), and the correct judgment on each equation needed to be made to get the points. A second rater, naive to the purpose of this study, scored 30% of the criterion tests, 30% of the MWM tests, and 30% of the CMT tests of each experimental phase using the answer key. The interrater reliability was calculated on a point-by-point basis. The total number of agreements was divided by the total number of agreements and disagreements and then multiplied by 100% to obtain the percentage agreement (Kennedy, 2005). The interrater reliability was 100% for the criterion tests, MWM tests, and CMT tests of each experimental phase.
Procedures
The instructor (the first author) implemented the intervention and the assessments with each of the participants one session a day, 5 days a week. After baseline, participant students started the story representation stage. Then, they entered the real problem-solving stage where they were introduced to the join-in problems first, followed by take-away problems, and finally mixed problems (a mixture of join-in, take-away, and combine problems). It should be noted that the participating students were not made aware of neither the problem types nor the instructional sequence of the problem types because the focus of the intervention was on their correct representation of the PPW relationship discovered from various story contexts (e.g., join-in, take-away, and combine). However, based on existing literature (Carpenter et al., 1981), join-in and take-away problems are considered easier than combine problems. As such, we introduced the join-in problems first, followed by take-away problems, then followed by the mixed problems with the intention of enhancing the success experiences of these students who had been historically experiencing constant failure in mathematics.
During the intervention, two criterion tests were administered following participant’s working with the join-in problems for three sessions, two criterion tests following three sessions of the take-away problems, and at least two criterion tests were administered following the mixed problem-solving. For each participant, the intervention phase ended when the participant reached 100% on the criterion test. During the assessment sessions, the instructor read the problems aloud to students because of reading difficulties observed during the baseline assessments. Nevertheless, students were asked to solve problems independently.
Baseline
After all three participants completed one criterion test, one student, Mary, was randomly selected to take three additional equivalent criterion tests. Once Mary’s word problem solving performance showed a stable trend, the intervention was introduced to her. Meanwhile, the other two participants continued with the baseline condition. When Mary’s word problem solving performance on the criterion test showed improvement upon receiving the intervention, the second randomly selected student, Bill, entered the intervention phase after he took four more alternate forms of the criterion tests within which three probes were taken immediately before the intervention started and showed a stable trend. The same sequence was followed until the intervention was administered to all three participants. In addition to the criterion tests, each participant also took one CMT and three MWM tests.
Intervention
The intervention program was adapted from Xin (2012). Teaching was carried out through two phases: the story representation phase and the problem-solving phase.
Teaching Story Representation
For each participant, the intervention started in the context of addition and subtraction word stories in which all the three elements (i.e., PPW) were already known (e.g., I have three apples. You have four apples. We have seven apples altogether). First, the problems were read. Then, the instructor taught the concepts of “part” and “whole,” as well as “part, part make up a whole” (Xin, 2012), followed by teaching students to map the three corresponding quantities in the story to the PPW diagram equation. In this step, the CRA sequence was used to help students understand the concepts of “part” and “whole” and represent the three quantities (i.e., P, P, and W) in the PPW diagram equation to make sense of P + P = W. Specifically, the instructor started by using physical cubes (concrete, C level) to represent the story and teach the concept of PPW. This was followed by teaching the student to map the three numbers in the story to the PPW diagram equation. After the instructor’s teaching, the students were asked to practice story representation and PPW mapping with the physical cubes independently. Once they could correctly complete the task for three consecutive stories with no prompts, they advanced to the representational (R) level. At the R level, the teaching followed the same format as in C level except that physical cubes were replaced with hand-drawn bars. Again, once students could draw bars for story representation and PPW mapping correctly for three consecutive stories with no prompts, they would advance to the abstract (A) level. At the abstract level, students were taught to identify the P, P, and W in the stories directly (without using physical cubes or drawing bars to represent the stories) and map the numbers to the PPW diagram equation.
The main purpose of teaching story representation was to help students understand the underlying PPW relation between the three numbers. This was measured by whether they could identify and map the given numbers into the PPW diagram equation correctly. The CRA only served as a representation mechanism to facilitate the transition to the abstract PPW diagram equation. Although we hoped that the students could successfully identify the PPW at the abstract level, not every participant achieved this level. As such, as long as participants could identify and map the PPW to the diagram equation correctly for three consecutive trials, regardless of what CRA level they achieved, they were advanced to the real problem-solving stage.
Teaching Real Problem-Solving
Following the story representation phase, students proceeded to solve real problems with one quantity unknown (e.g., I have three apples. You have four apples. How many apples do we have altogether?). Students were taught to use “?” to present the unknown quantity in the PPW diagram equation. A four-step RITS sheet (i.e., read and comprehend, identify and map, transform, and solve) adapted from Xin (2012) was used to guide students through the problem-solving process. In Step 1 (read and comprehend) and Step 2 (identify and map), participants were asked to execute at the abstract level; that is, they were asked to identify the quantities pertaining to part, part, and whole, and to map them directly from the word problems into the PPW diagram equation without going through the CRA representational sequences. Nevertheless, physical cubes or hand-drawn bars could be brought back should such scaffolding be deemed necessary to help students understand PPW relationships. In Step 3, after the PPW was correctly mapped to the COMPS diagram, students were taught to transform the COMPS diagram to a true mathematical equation. The diagram (i.e., boxes) was removed, and a real mathematical equation was presented to represent the problem (e.g., 28 + 34 = ?). In Step 4, participants were taught to use the equation to solve for the unknown quantity for the answer. If the whole is unknown, the equation tells that adding the two parts will give the total. If one of the parts is unknown, the equation tells that subtracting the given part from the total will solve for the unknown part. Students then calculated the answers. They were encouraged to use multiple strategies to calculate the answers (e.g., mental math, using fingers, and vertical algorithm). Calculators were allowed to accommodate participants’ skill deficits in calculation. It should be noted that, although the participants had access to the RITS checklist during the intervention phase, the RITS checklist was removed during the assessment sessions.
Post-Test/Maintenance Phase
Each student took at least two consecutive criterion tests immediately after the completion of the intervention as a post-test performance. Maintenance testing occurred 1 week and 2 weeks after termination of the intervention. Besides the criterion test, all students also took three MWM tests and one CMT as part of the generalization assessment.
Results
Figure 1 presents the three participants’ performances on the criterion tests across the baseline, intervention, and post-test/maintenance phases. Visual inspection was used for data analysis to determine the effectiveness of the intervention (Kennedy, 2005). Specifically, the visual inspection focused on dimensions including the level variability and change, which was determined by computing the range between the mean and median values of the data in different phases and the trend variability and direction.
Percentage Correct on Criterion Tests, MWM Tests, and CMT Tests During the Baseline, Intervention, and Post-Test/Maintenance Phases Across the Three Participants.
Baseline Performance
During the baseline phase, the correct percentage of problem-solving on the criterion tests was low and stable across all three participants. The average score on the criterion tests was 37.5% (median = 37.5%) for Mary, 17.5% (median = 25%) for Bill, and 25% (median = 25%) for Nicolas. As for the MWM test, the average score was 28.1% (median = 25%) for Mary, 40.6% (median = 37.5%) for Bill, and 47.8% (median = 37.5%) for Nicolas. The performance on the CMT test was 37.5% for Mary, 25% for Bill, and 50% for Nicolas during the baseline.
Intervention Performance
Mary
During the story representation phase, Mary was independently and correctly able to represent stories and map PPW at the concrete and representational level for at least three consecutive stories, which indicated that she had mastered the PPW relationship among the three numbers (at the representational level). Then, Mary moved to the real problem-solving phase, which took her 10 training sessions. During the real problem-solving phase, Mary needed to draw bars to identify and map PPW in the diagram equation. This was also true for the first six criterion tests. However, during the rest of the criterion tests, Mary was able to solve the problems at the abstract level. Mary scored 87.5% and 75% on Criterion Tests 1 and 2, respectively. She scored 75% and 62.5% on Criterion Tests 3 and 4, respectively. Finally, she scored 87.5% on both Criterion Tests 5 and 6 and eventually reached 100% on Criterion Test 7.
Bill
For at least three consecutive stories during the story representation phase, Bill independently and correctly represented stories and mapped PPW at the concrete and representational levels, indicating that he had mastered PPW relationships among the three numbers (at the representational level). Then, in the real problem-solving phase, Bill had eight training sessions, during which he drew bars to identify and map PPW in the diagram equation. This was also true for the first five criterion tests. During the rest of the criterion tests, Bill solved the problems at the abstract level. Bill scored 75% and 100% on the Criterion Tests 1 and 2. He scored 100% and 85% on Criterion Tests 3 and 4. Then, he scored 87.5% on criterion test Probe 5 and finally 100% on the criterion Probe 6.
Nicolas
During the story representation phase, Nicolas independently and correctly represented stories and mapped PPW into the diagram equation, starting at the concrete. He then moved onto the representational level and finally to the abstract level for at least three consecutive stories. Then, Nicolas moved to the real problem-solving phase, which took him 12 training sessions. During the real problem-solving stage, he solved all the problems at the abstract level. Nicolas scored 75% on the criterion test Probes 1 and 2. He scored 75% on the criterion test Probes 3 and 4. Nicolas reached 87.5% on Criterion Test 5 and then improved to 100% on Criterion Test 6.
Post-treatment and Maintenance Performance
During post-test assessments, all three participants maintained 100% correct, as measured by the criterion tests. As for the maintenance tests that occurred 1 week and 2 weeks after the termination of the intervention, Mary scored 100% and 87.5%, respectively, on the two maintenance criterion tests. Bill scored 75% correct on the first maintenance test and 100% on the second maintenance test. Nicolas scored 100% on the two maintenance tests. All three participants solved the problems at the abstract level in the post-treatment and maintenance.
Generalization Effect
On the MWM tests, Mary increased her performance from an average of 28.1% (median = 25%) in baseline to 84.3% for all three postintervention tests. Bill improved from an average of 40.6% (median = 37.5%) during the baseline to an average of 81.2%, and Nicolas improved his performance from 47.8% (median = 37.5%) to an average of 88.75% during postintervention assessment. The score on the CMT tests across participants was 75% for Mary (an increase of 37.5% from the baseline performance), 100% for Bill (an increase of 25% from baseline performance), and 100% for Nicolas (an increase of 50% from his baseline performance).
Social Validity
A survey including five Likert-scale items was conducted following the intervention. For the statement, “I like doing math word problems,” Mary and Nicolas chose “agree” and Bill chose “strongly agree.” They all chose “agree” for the statements, “Using cubes to represent the problems is helpful for me to understand the COMPS diagram and the word problems” and “Drawing the bars to represent the problems is helpful for me to understand the COMPS diagram and the word problems.” For the statement, “Moving the numbers to the COMPS diagrams is helpful for me to understand the meaning of the quantities in the word problems,” only Bill chose, “agree” while Mary and Nicolas didn’t initially understand it, but after asking for clarification, they both chose “agree.” For the statement, “I enjoy learning the strategies to solve word problems,” all three participants chose “strongly agree.”
Discussion
The results of this study showed that, after the COMPS intervention, all participants demonstrated distinct level changes on the criterion tests, indicated by the nonoverlapping data between the baseline and intervention conditions. Participants’ performance on generalization CMT and relation test MWM also improved upon the intervention.
Understanding the PPW relationship and correctly identifying PPW is a crucial component leading to correct problem-solving of addition and subtraction word problems (Caldwell et al., 2011). However, exploring the underlying pattern and detecting the “part, part, whole” in various word problem contexts might be difficult for students with ASD because they are detail-oriented learners and struggle to construct new concepts from many details (Boutot & Myles, 2011; Minshew et al., 2002). Providing a model could assist learners in identifying which variables to consider among a variety of candidates in new contexts (Lesh et al., 2003; Schwartz et al., 2008). The COMPS approach might serve as a scaffold facilitating the problem representation and solving processes. It might also alleviate students’ memorization burden by pruning off irrelevant information, explicitly informing them of the common PPW structure in all addition and subtraction problems and keeping them from relying on specific rules to solve different types of problems. Students demonstrated an understanding of the PPW relation through their accurate representation of the quantities in the mathematical model equation, regardless of problem situations and the sequence of the quantities (see Figure 2A). It was also noticed that, after learning the “join-in” problems, students could correctly solve most of the “take-away” and “combine” problems which had not been taught at that point on Criterion Tests 1 and 2 (see Figure 2B). This indicated that the COMPS approach might have successfully helped students form the concept of PPW and generalize it across different types of word problems. Furthermore, all three participants improved on their generalization CMT test after the intervention. CMT problems had numbers as big as close to 100. Their schoolteachers indicated that solving word problems within 100 was the state standard for second graders, but the three participants had never dealt with numbers beyond 20 before this study. As such, it would be reasonable to conclude that the COMPS approach could also have helped students solve a wider range of problems appearing in commercial textbooks.
Students’ Work of Problems-Solving and the Mapping of PPW. (A) Shows That Nicolas Could Correctly Map the Three Qualities Into the PPW Diagram. (B) Shows That Mary Could Solve a “Take-Away” Problem Correctly on the Criterion Test Only After “Join-in” Problems Were Taught.
As a generalizable model, the COMPS approach (Xin, 2012) provided participants with multiple ways of expressing the same mathematical relation. According to the NCTM big idea, the same relationship can be represented by more than one expression. Providing different forms of algebraic model equations can help students develop and reinforce the conceptual understanding of complementary quantity relations of addition and subtraction (Caldwell et al., 2011). Although the COMPS PPW model was presented in the form of P + P = W, depending on the position of the unknown quantity, the COMPS approach provided students with opportunities to explore other forms of an equation. For example, in this part-unknown problem, “There were 20 books on the shelf. 2 books were checked out. How many books were left?,” besides the P + P = W model (i.e., 2 + ? = 20 or ? + 2 = 20), students could also see other forms of the equation (e.g., ? = 20 − 2 or 20 − 2 = ?) because that would be the number sentence directly leading them to the correct answer. As such, actively interacting with these different equation variants might enable students to discover different transformations of the PPW relation. The students’ performance on the baseline MWM tests indicated that they did not fully understand the quantity relation. After careful examination of students’ baseline test sheets, we noticed that the students’ choice of the answer on the baseline MWM tests was random. For instance, Mary selected “yes” for all even-numbered questions and “No” for all odd-numbered questions. Students’ performance on MWM tests following the intervention showed that they could recognize and make correct judgment on various equation transformation that the intervention did not explicitly teach. In addition, as an effective tool to bridge concrete-level learning to abstract-level learning, the CRA teaching sequence may have effectively helped students visualize the PPW mathematical relation, leading to their better understanding of the concepts (Roberts & Joiner, 2007). The results showed that after COMSP intervention, all participants correctly identified the PPW from the word problems without concrete and representational level support and could directly map the quantities into the PPW diagram (see Figure 2A).
Limitation and Future Research
There are several limitations in this study. First, this study addressed only join-in, take-away, and combine word problems. The compare problem types were not included. Although linguistically complicated, compare problems also have the PPW underlying structure (Caldwell et al., 2011), so to better evaluate how COMPS facilitates students’ construction of PPW relations in additive word problem-solving, compare problems should not be excluded. According to Xin et al. (2008), COMPS PPW instruction was effective in teaching compare problems to students with LD. Future studies could extend the application of COMPS instruction to students with ASD on solving compare problems to provide more data-driven evidence of the effect of this approach. Second, the MWM test was developed by the first author, and it was first used in this study. Although reviewed by special education and math education experts, the technical adequacy of this test has not been established. Nevertheless, multiple representations of the same mathematical relationship are part of the competency required by school curriculum, and they are in line with NCTM standards (Caldwell et al., 2011), which appears to support the construct validity of this test. On the contrary, as a generalization assessment, the MWM test was administered only during the baseline and the post-test/maintenance phases in this study. In the future, studies could be conducted to use verified MWM as a primary dependent measure to provide further insight about student’s progression toward the construction of the PPW relations across a range of addition and subtraction word problems. Finally, after the completion of the study, the first author followed up with the participating students regarding their learning experiences. It was discovered that two participants had difficulties understanding certain words in some word problems (e.g., a problem on maintenance Probe 1 for Bill and a problem on intervention Probe 4 for Mary), resulting their difficulties in solving the problems (which perhaps contributed to the “dips” in their performance as shown in the line graph in Figure 1). Upon learning about this, the first author provided explanations of the meaning of the words that puzzled the students. Given that, the two students showed eagerness to solve more problems, and in fact, they were able to solve them correctly. For future implementation, teachers may need to administer a vocabulary check to ensure that students understand the word problems they read so that reading comprehension does not serve as a barrier preventing students from accessing mathematics.
Implications for Practice
One aspect of the NCTM big idea pertaining to addition and subtraction word problem solving is, regardless of problem types, PPW is a unified mathematical relation underlying a range of additive word problems. However, this big idea is not well reflected in the existing empirical research. Building a conceptual understanding of the big idea pertaining to PPW relations is important for generalized additive word problem-solving. The findings of this study, consistent with the intervention research involving students with learning difficulties or disabilities, seem to support that the COMPS approach has the potential for educators to help them teach students with ASD who have difficulty accomplishing this goal. This study provides implications for educators in that COMPS could facilitate students’ comprehension of abstract mathematical concepts and relations expressed in algebraic model equation (e.g., P + P = W) and, therefore, enhance generalized word problem solving skills of learners with diverse needs, particularly students with ASD.
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
