What's the Tip? Using a Schematic Diagram to Support Life Skills Math

Headnote

Abstract : Researchers have repeatedly found schematic diagrams, both as part of Modified Schema Based Instruction and outside of this problem-solving approach, effective for teaching students with disabilities, including intellectual and developmental disabilities (IDD), to solve math word problems. The current study was a systematic replication of Bouck, Long, et al. (2021) research examining students with disabilities finding the total bill inclusive of tip. This conceptual replication examined the impact of systematic changes to the previous intervention materials (i.e., color-coded schematic diagram) on student progression through the learning stages in trained and unfamiliar (i.e., simulated) experiences. In this single case design study, researchers examined three middle school students' ability to accurately and independently calculate the total bill, with tip, using a color-coded schematic diagram and taught via the system of least prompts and explicit instruction. The researchers found two main results: (a) like previous results, the intervention package was effective with regards to students acquiring and maintaining independent and accurate responses; and (b) extending previous findings, the revised intervention was efficient (i.e., required fewer sessions) and supported student generalization during real-world simulations.

What's the Tip? Using a Schematic Diagram to Support Life Skills Math

Researchers suggested learning occurs across stages for students with disabilities, with particular emphasis on intellectual and developmental disabilities (IDD): acquisition, fluency, maintenance, and generalization (Alberto & Troutman, 2009; Collins, 2012; Shurr et al., 2019). When students first start to learn a task, skill, or concept, they are within the acquisition stage; this stage is defined as lower independent accuracy and increasingly higher accuracy with support (Collins, 2012; Snell & Brown, 2011). Once students demonstrate independent success at a rate around 60%, or 100% with support, they move into the stage of fluency (Shurr et al., 2019).

Within the fluency stage, students show they can engage in the task, skill, or concept both effectively (i.e., independent, accurate) as well as efficiently (i.e., decreased time and effort; Collins, 2012; Snell & Brown, 2011). From there, students move into the maintenance learning stage, where they are asked to show their ability to perform the task, skill, or concept without prior instruction (Alberto & Troutman, 2009). Maintenance also addresses at students' ability to continually

engage in this skill over time (Shurr et al., 2019). The final stage of learning involves generalization, which educators can observe across the other stages of learning (Shurr et al., 2019). With generalization, students are asked to show they can engage in a task, skill, or concept across different people, places, materials, or contexts (Collins, 2012). Scholars contend the goal of learning is for students to maintain what they learn each year and generalize learned skills and concepts to their everyday lives (Laski et al., 2023; Marini & Genereux, 2013).

The learning stages framework for students suggests a focus solely on the acquisition of skills is insufficient for students, particularly in math (Schnepel & Aunio, 2022; Shurr et al., 2019). Focusing on students" capability to be both effective and efficient (i.e., fluent), and to maintain this proficiency without additional instruction, is crucial for the educational and lifelong success of students with disabilities (Shurr et al., 2019). Despite the importance of fluency and maintenance, attention in math research and practice for students with disabilities, particularly IDD, historically has been on acquisition and, to a lesser extent, generalization (Spooner et al., 2019). Fewer researchers have

attended to the fluency and maintenance stages of math learning for students with disabilities (e.g., Lafay et al., 2019; Park et al., 2020; Spooner et al, 2012). Within the limited research base, researchers have examined interventions to support the learning stages in math for students with disabilities, particularly IDD, including manipulative-based instructional sequences (Bouck & Park, 2020; Bouck, Park, et al., 2021;

Bouck, Shurr, et al., 2020), explicit instruction (Park et al., 2020; Root, 2019), simultaneous prompting (Jimenez & Saunders, 2019), schematic diagrams (Bouck & Long, 2020; Bouck, Long, et al., 2021; Root, Cox, et al., 2018), and the system of least prompts (SLP). Many interventions reflect a package of strategies (e.g., Bouck & Park, 2020; Long et al., 2021).

Researchers have examined the learning stages in math via schematic diagrams interventions (Bouck & Long, 2020). Schematic diagrams have a solid research base for students with disabilities (Bouck et al., 2024; Bouck & Long, 2020; Jitendra et al., 2016; Polo-Blanco et al., 2024; Root et al., 2021).

Schematic diagrams can be used in a variety of ways, including as part of an intervention called schema-based instruction (SBI), which is an evidence-based practice for students with learning disabilities (Jitendra et al., 2016),

and an adapted version for students with IDD referred to as modified schema-based instruction (MSBI; Root et al., 2017; Root et al., 2023). MSBI builds upon SBI, adding supports for students with IDD such as reading problems aloud, problem solving task analyses, and color-coded schematic diagrams to teach students to identify problem types (e.g., percent of change), represent the math relationship in the correct schematic diagram, and follow a problem-solving routine using explicit instruction (Browder et al., 2018; Root et al., 2017; Root et al., 2023).

Yucesoy-Ozkan et al. (2022) concluded MSBI as an emerging EBP and Clausen et al. (2021) determined MSBI was not an EBP but resulted in large positive effects for participating students.

While some researchers explored MSBI with attention to more academically oriented math, such as data analysis and algebraic word problems (e.g., Root, Cox, et al., 2018; Root, Henning, et al., 2018), others examined MSBI with a focus on life skills or applied math such as percent of change problems in the context of purchasing (e.g., Root, Cox, Davis, et al., 2020; Root, Cox, Saunders, et al., 2020). Root, Cox,

Davis, et al. (2020) examined the efficacy of MSBI to support middle school students calculating cost after a coupon and determining if students had sufficient money. They found a functional relation between the intervention and students completing the steps to solve the problems but the three participants required a more explicit task analysis to support independence (Root, Cox, Davis, et al., 2020). Root, Cox,

Saunders, et al. (2020) also explored MSBI to support three middle school students in solving problems involving finding the final price after receiving a discount or adding a tip. They found a functional relation between the intervention and students' independent accuracy in completing task analysis steps. Three participants were less successful generalizing to percent of change problems outside the context of purchasing (Root, Cox, Saunders, et al., 2020).

While research groups have invested significant attention to MSBI and shown its efficacy (e.g., Root, Cox, Davis, et al., 2020; Root, Cox, & Gonzalez, 2018; Root, Cox, Saunders, et al., 2020; Root, Henning, & Boccumini, 2018),

other researchers have explored the use of schematic diagrams for students with disabilities outside of MSBI, meaning without explicitly teaching different types of schemas (e.g., Bouck & Long, 2020; Bouck, Long et al., 2021). Bouck and Long (2020) taught middle school students with disabilities to solve word problems involving finding the total cost after discounts. Using a schematic diagram, explicit instruction, and the SLP, the researchers found a functional relation between the intervention package and the three students' accuracy in finding the final cost. The students also met independence criteria on the task analysis steps but only two students generalized to a real world simulation. Bouck, Long,

etal. (2021) similarly examined the effectiveness of using a schematic diagram and the SLP to aid middle school students with disabilities in problemsolving tasks. These tasks involved calculating the total bill with a tip and, later, extending this skill to compute the total bill after tax.

Researchers provided explicit instruction using the SLP to teach students how to utilize the schematic diagram. Focusing mastery criteria on acquisition alone, they observed a consistent improvement in accuracy for all four students,

suggesting a functional relation between the intervention and improved performance. The four participants demonstrated variable performance on generalization (i.e., faded supports) and maintenance tasks.

Current Study

Teaching students with disabilities to solve math word problems-both academically oriented and life skills focused-is important. The literature supports the use of schematic diagrams, both as part of MSBI and outside of this problem-solving approach, for student use in solving math word problems (e.g., Bouck, Long, et al., 2021; Bouck & Long 2020; Clausen et al.,

2021; Yucesoy-Ozkan et al., 2022). The current study was a systematic replication of Bouck, Long, et al. (2021), seeking to both replicate prior research (i.e., replicating other studies to support identification of EBPs) as well as extend (i.e., adding color to schematic diagram) and focus across the learning stages (e.g., Root, Cox, Davis, et al., 2020; Travers et al., 2016). In this study, the researchers sought to answer the following research questions: (a) Are secondary students with disabilities able to acquire the skill of solving problems finding the total restaurant bill with tip when taught via a schematic diagram,

explicit instruction, and the SLP? (b) Are secondary students with disabilities able to independently complete the task analysis steps of solving problems finding the total restaurant bill with a tip with a schematic diagram? (c) Are secondary students with disabilities able to maintain their accuracy and independence in solving problems finding the total bill inclusive of a tip when explicit instruction does not proceed? (d) Are secondary students with disabilities able to generalize solving problems on paper and pencil with a schematic diagram to simulated contexts? and (e) What are student perceptions of using schematic diagrams to solve life skills-based math word problems?

Method

Systematic Replication

In this study, researchers sought to systematically replicate-with purposeful changes-a prior study that examined use of a schematic diagram, modeling, and a SLP to support students' acquisition of problem-solving accuracy (Bouck, Long, et al., 2021; see Table 1). Both studies included middle school students with disabilities and used a multiple probe across participants design. The independent and dependent variables were intentionally manipulated to systematically test for potential differences. In the current study, the intervention procedures and materials were adjusted to decrease student disinterest and increase student independence. Researchers hypothesized the changes would lead to students acquiring, maintaining, and generalizing the targeted math skill of finding the total cost including tip.

The procedures were modified to reduce the number of problems students engaged with each session, from nine (i.e., two modeled, two guided, five independent) to five problems (i.e., one modeled, one guided, three independent). Explicit instruction was not faded during intervention sessions in the current study. The researchers also focused on food bills containing a whole number, as opposed to a decimal as presented in the Bouck, Long, et al. (2021) study. Finally, the researchers added color to the schematic diagram and coordinated color within the schematic diagram to color within the problems, as suggested by Root et al. (2022) as an additional scaffold. To approximate student progress through the learning stages, researchers added a second dependent variable of independence to the mastery criteria. Maintenance data were collected in both studies, but the current study allowed students to use the schematic diagram if desired, rather than removing. In addition, researchers explored generalization through a simulated restaurant bill rather than generalizing to problems involving tax (Bouck, Long, et al., 2021).

Participants

Three sixth-grade middle school students with disabilities participated in this study. Students were recruited through their special education teacher, who sent home consent forms with students enrolled in the special education program for students with intellectual disability. Students were included in the study if they had parental

TABLE 1

Study Dimensions Held Constant and Intentionally Varied Between Studies

Dimension

Authors et al. (2021)

Current study

1. Participants

2. Setting

3. Intervention·

4. Outcome Measures ·

5. Research Design 6. Analyses

7. Findings

N(DD) = 4 Ages: 14, 14, 14, 14 Grades: 7", 7, 32, 7 Gender: 3 male, 1 female Disability: ID/ADHD, ASD, LD, ID Race: 4 white IQ Scores: 63, NR, 71, 60 Interventionists: white female faculty, Asian female GRA Geographic location: Rural small town, midwestern USA School Type: Public middle School Session Location: Hallway Group Size: 1:1 Strategies: Schematic diagram, explicit instruction with fading, and SLP. Materials: Schema, data collection form, learning sheets, probes, and calculator.

Procedure: Two model, two guided, five independent problems. a a

Primary: Accuracy writing or stating the total bill.

Food bill contained quantities with decimal

values.

Additional: Independence (task analysis steps completed without SLP), range reported. a (1) Fill in % number /\

(2) Fill in bill $ O

(3) Calculate % A 100

(4) Calculate пр O 100 x O (5) Transfer O in schema

(6) Transfer O) in schema (7) Ааа O + O

(8) Write answer

Maintenance: Removed schematic diagram.

Generalization:

(1) Faded supports (no schema)

(2) Problems involving tax

Concurrent multiple probe across participants

Primary: Visual analysis Effect Size: Tau-U

Accuracy: Functional relation between IV and primary DV (accuracy).

Independence: Not systematically analyzed.

N (IDD) = 3 Ages: 12,12,13 Grades: 6", 6", 6" Gender: 1 male, 2 female Disability: ID, ID, OHI Race: 1 Black, 2 white IQ Scores: 61, 59, NR

Interventionists: white female faculty, white female GRA

Geographic location: Rural small town, midwestern USA

School Type: Public middle school

Session Location: Separate classroom

Group size: 1:1

Strategies: Schematic diagram, explicit instruction, and SLP.

Materials: Color coded schematic diagram and word problems, learning sheets, probes, calculator.

Procedure: One model, one guided, three independent problems. - a

Primary: Accuracy writing or stating the total bill.

Food bill contained whole number quantities.

Secondary: Independence (number of task analysis steps completed without SLP) graphed.

(1) Fill in % number A

(2) Fill in bill $ O

(3) Calculate % /\ 100

(4) Calculate tip O 100 x O (5) Transfer O in schema

(6) Transfer O in schema (7) Add O + O

(8) Write answer

Maintenance: Schematic diagram available.

Generalization:

(1) Faded supports (black and white vs. colored schematic diagram)

(2) Simulated restaurant bill

Concurrent multiple probe across participants

Primary: Visual analysis Effect Size: Tau

Accuracy: Functional relation between IV and primary DV (accuracy).

Independence: All students increased independence

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192 / Education and Training in Autism and Developmental Disabilities-June 2025

Dimension Authors et al. (2021)

Current study

Generalization:

Generalization:

(1) Tax Problems: Two students immediately (1) Faded Supports (black and white

generalized to tax problems, all four at 100% accuracy on generalization to tax

problems two weeks after IV.

(2) Faded Supports (no schema): Three stu- (2) Simulated Bill: Two students accurate

dents maintained performance (80%100%), one student dropped from 100%

to 60%

Maintenance: All four students maintained

80% - 100% accuracy.

schema): Three students accurate and independent (90% -100%).

and independent (80% -100%), one student variable (60%-100%).

Maintenance: Two students maintained at 100% accuracy, one student initially dropped (30%) and then achieved 100% accuracy.

Note. Variables intentionally varied marked with an asterisk · and specific elements manipulated are underlined and italic. List of acronyms: DD = developmental disabilities; IDD = intellectual and developmental disabilities; ID = intellectual disability, ADHD = attention deficit hyperactivity disorder; ASD = autism spectrum disorder; GRA = graduate research assistant; NR = not reported; OHI = other health impairment; USA =

United States of America; SLP = system of least prompts.

consent, assented to participate, and the teacher indicated a possible benefit from one-on-one additional math instruction. All participants attended the same math class led by a White female special education teacher with 15 years of experience. Although no specific curriculum was used in the math class, the students did not receive instruction on percent of change problems during the intervention.

Danny. Danny was a White, male, sixthgrade student. He turned 12 years of age during the study. Danny's eligibility was intellectual disability (ID), and his full-scale IQ on the WISC-IV was reported as 61. He received his math instruction from the special education teacher in the mild intellectual disability program, although he attended general education classes for science, social studies, and electives. Researchers administered the KeyMath (Connolly, 2007) during the fall semester to provide a standardized score of math abilities. On the KeyMath, Danny showed relative strengths in Mental Computation and Estimation (raw score = 9; grade equivalent = 2.3) and Addition and Subtraction (raw score = 8; grade equivalent = 1.8) and significant difficulties with Numeration (raw score = 6; grade equivalent = K.5), Geometry (raw score = 7; grade equivalent = K.2), and Data Analysis (raw score = 5; grade equivalent = <k.0).>

Mindy. Mindy, a White female student, was also in the sixth grade and, like Danny, turned 12 years old during the study. Mindy's eligibility was ID, and her reported fullscale IQ was 59. Like Danny, Mindy attended science, social studies, and electives in the general education setting. She was in the same math class as Danny, taught within the mild intellectual disability program. Mindy's performance on all KeyMath subtests suggested a need for intensive intervention, with raw scores in Numeration (7), Geometry (9), Data Analysis and Probability (3), and Mental Computation and Estimation (1) all falling within kindergarten equivalents and Addition and Subtraction (4) equivalent with first grade. Her math IEP goal involved adding double-digit numbers with regrouping with 80% accuracy.

Tamara. Tamara, a Black, female, sixthgrade student, turned 13 years of age during the study. Tamara's eligibility was otherwise health impaired, and she had been receiving services since early intervention. She struggled with anxiety and engaged in limited spoken exchanges, often preferring to nod or shake her head. When she did speak, it was often a whisper. Tamara was prone to crying when she felt overwhelmed or did not know what to do. Similarly to the other two participants, Tamara's performance on the KeyMath subtests supported the need for intensive intervention. Her raw scores of 6 on the Numeration subtest and 4 on the Mental Computation subtest were both K.8 equivalence and her Mental Computation and Estimation raw score was 4, converting to a 1.2

grade equivalent. Her math IEP goal was to add two-digit numbers with regrouping with 80% accuracy. While she did not have a formal identification of ID, she was placed in a special education program designed for students with ID (and referred to as such within her IEP, given the state identified programs and certified teachers per disability eligibility categories) and performed similarly in many aspects to her peers in that class.

Setting

The setting of the study was a rural midwestern small town. The school housed about 500 students across grades sixth through eighth; the district also had one high school, one primary school, and one intermediate elementary school. The school district's population was predominantly White, and about onethird of students were free and reduced lunch eligible. When working with students, researchers met one-on-one in another classroom. The classroom was empty and had multiple tables situated around the room. Researchers sat next to students to deliver the intervention. The teacher was not present within the research study space but was just down the hall in their classroom.

Materials

For this study, researchers used a schematic diagram, a data collection form involving the task analysis steps, learning sheets, probes, and a graphing calculator. In the schematic diagram, a particular shape corresponded to specific elements within the problems (e.g., the amount spent represented a square).

Researchers adapted the schematic diagram from the prior study (Bouck, Long, et al., 2021) by associating a unique color to each shape and adding the color, as relevant to the numbers in the problem (e.g., tip percentage in red to connect with red triangle tip amount went in; see Figure 1). Researchers laminated the schematic diagram so students could write on it with dry erase markers and erase; both were presented to students each session.

Researchers also adapted the data collection form from a prior study (Bouck, Long, et al., 2021). The data collection form presented the task analysis steps to solve a problem, allowing researchers to collect both student accuracy data (i.e., correct or incorrect) as well as independence data in completing each step (i.e., no SLP or steps of SLP needed). While students did not have access to the task analysis steps, researchers provided instruction following the steps. Researchers could further collect the amount and type of prompting students needed for each probe. Researchers also provided students with a graphing calculator to use for the problems for all phases. Researchers ensured students knew how to use the graphing calculator prior to baseline data collection.

Researchers used learning sheets to deliver the schematic diagram intervention via explicit instruction. The researcher-created learning sheets consisted of three parts: one problem involving finding the total bill with the tip when eating out at a restaurant that researchers modeled, one problem involving finding the total bill with the tip when eating out at a restaurant that researchers guided (i.e., provided prompts and cues as needed) as students tried solved, and three problems involving finding the total bill with the tip when eating out at a restaurant that students solved (i.e., the probe). It was during the independent problems that researchers applied the SLP. Researchers created the learning sheets to be unique; none were repeated.

Problems were also not repeated across the entire study. Each problem has a unique food bill amount attached to one of three tip amounts: 15%, 20%, and 18%. Researchers randomly selected the 15%, 20%, and 18%

1.) Your food bill at a restaurant was $13. You want to leave a 15% tip. What is the total amount you will pay?

2.) Your food bill at a restaurant was $27. You want to leave an 18% tip.

'What is the total amount you will pay?

3.) Your food bill at a restaurant was $86. You want to leave a 20% tip. What is the total amount you will pay?

Figure 1. Student Materials-Problems and Color-Coded Schematic Diagram. Note. Pictured left is an example of a learning sheet with three color coded word problems. Learning sheets were printed on white computer paper and were similar in all three phases. The Schematic diagrams (pictured right) were laminated and

available to students (with dry-erase markers) during intervention and maintenance phases.

for the modeled and guided problems but the three problems comprising the probe had all three, in a random order. The amount of the food was a whole number; researchers selected whole numbers for which a terminating decimal in the hundredths (i.e., pennies) existed when the tip was applied. An example problem included: "Your food bill at a restaurant was $44 [purple font]. You want to leave а 20% [red font] tip. What is the total amount you will pay?" As noted, during the intervention, researchers selected the probe to be the three problems students solved. During baseline and maintenance phases, students were given similar probes, but no modeling or guided phase preceded. During the first generalization, researchers gave students three problems without any color cues and provided the schematic diagram without color.

For the second form of generalization, researchers presented the three problems on a simulated restaurant bill receipt (available upon request from first author) and students were given the option of selecting the tip amount before solving (15%, 20%, 18%).

Experimental Design

Researchers selected a concurrent multiple baseline replicated across participants single case design study. Within this design, each student started baseline sessions at the same time but entered intervention within a systematically staggered fashion. Each student completed at least three baseline sessions,

consistent with the guidelines for quality indicators set by Cook et al. (2014). When the first student-Danny-had a zero-celerating or decelerating trend in baseline and stable baseline data (i.e., 80% of each student's baseline data falling within 25% of their median), they began their first intervention session. After one intervention session, in which the student received explicit instruction and the SLP to use the schematic diagram to solve three problems involving finding the total bill with the tip, the second student entered intervention if the first student's accuracy on the first intervention probe was greater than any of their accuracy scores during baseline. All subsequent students entered similarly. All students stayed in intervention until they met mastery criteria, which was defined as completing at least two consecutive sessions with 100% accuracy and over 90% independence. After intervention, students entered a maintenance phase and then the generalization phase.

Independent and Dependent Variables

Researchers examined one independent variable during this study: an intervention package

composed of the schematic diagram taught via explicit instruction and the SLP. Researchers used explicit instruction (i.e., modeling and guiding) before students independently attempted to solve three word problems involving finding the total bill with tip using the schematic diagram while receiving feedback from the researcherimplemented SLP. The researchers modeled the problems following the task analysis steps. The SLP involved a sequence of gesture (i.e., point at the schematic diagram), indirect verbal (i.e, "What do you do next?"), direct verbal (e.g., "Put the tip amount in the red triangle"), and model prompts (ie, showing the student how the step is done). Researchers used а 10-5 wait time to administer prompts initially and between each subsequent prompt (Shepley et al, 2019). Researchers examined two dependent variables for the study: student independent accuracy (i.e., responding with the correct answer for the total bill in writing or verbally without prompting on that step out of three problems) and student independence (ie, total number of seven task analysis steps completed without prompting across the three problems for a total of 21 steps per probe).

Procedures

The study consisted of baseline, intervention, maintenance, and generalization phases. The study was implemented by a researcher with experience exploring math interventions and supports for students with disabilities and a doctoral student who was a former special education teacher. All interventionists were White females. The interventionists both worked with all three participants across all phases; interventionists were not assigned to student or by phases. The first author developed the intervention and trained others to deliver with fidelity. Students worked with researchers once a week for one to two sessions per visit. Sessions generally lasted 1520 min per student and occurred across three months, inclusive of spring break.

Baseline. During baseline sessions, students solved three word problems in which they found the total restaurant bill with tip. Researchers provided each student with the color-coded schematic diagram, an erasable marker, an

eraser, and a calculator. During baseline probes, researchers provided students no instruction prior to and no SLP during. The researchers provided only encouragement (e.g., "thank you for working hard") but not confirmatory or corrective feedback.

Intervention. For intervention sessions, researchers taught students how to use the schematic diagram to solve finding the total bill with tip problems set at a restaurant via explicit instruction and the SLP. On each intervention probe, students were given access to the schematic diagram (and the tools to use- markers, eraser), a calculator, a learning sheet, and a probe. During intervention probes, researchers employed the SLP with a 10 s wait time on all steps except writing or verbalizing the answer (i.e., students could answer this step incorrectly). This step was not counted in the eight task analysis steps per problem. The researchers provided no response prompting or reinforcement feedback for students completing the steps of the task analysis. As with baseline, they did provide encouragement (e.g., "thank you for working hard").

During the modeling portion of explicit instruction for each intervention session, researchers demonstrated and verbally narrated how to use the schematic diagram to solve the problems and find the total bill with the tip at the restaurant following the task analysis steps. The researcher started by noting that each shape corresponded to one part-and only one part-of the problem, each shape was color-coded, and some of those colors were represented within the problem. Following the task analysis, the researcher modeled reading the problem and then placing values within the problem in their corresponding shapes in the schematic diagram. For example, with the problem "Your food bill at a restaurant was $44 [purple font]. You want to leave a 20% [red font] tip. What is the total amount you will pay?," the researcher modeled placing the food bill amount (i.e., $44- in purple font in the problems) in both purple circles. Then, the researcher noted the tip amount, in red font in the problem, and placed that value (e.g., 20) in the red triangle on the schematic diagram. Next, the researcher modeled computing the percentage (20% by

dividing the value 20 by 100, as depicted in the schematic diagram) and multiplying that decimal (i.e., .2) by the food bill (i.e., $44), resulting in $8.80 (or 8.8), using the calculator. This value-$8.80-was placed in both blue rhombuses. Finally, the researcher modeled, using the calculator, adding the blue rhombus value ($8.80 in this case) to the original food bill ($44), depicted by the purple circle, to get the final bill of $52.80, which was written into the green square.

Maintenance. Researchers collected one maintenance probe per week for 2 weeks following the last intervention session. Maintenance data collection session procedures mirrored those used in baseline sessions. Researchers did not provide the SLP and did not provide explicit instruction prior to the probe. The schematic diagram and calculator were provided as students worked to solve three total restaurant bill with tip written word problems.

Generalization. Researchers collected two forms of generalization. The first form of generalization involved three word problems that mirrored baseline and maintenance but color cues were removed from the schematic diagram and word problems; students completed two of these probes. Students were given access to a calculator. The second form of generalization was a simulated situation in which researchers presented the students with a fake restaurant bill in which a food total was presented. Students selected the tip amount they wanted to give- 15%, 18%, or 20%-and found the total bill. The color schematic diagram was available for student use. Each of these generalization sessions involved three simulated restaurant checks. Students were given access to a calculator on a phone, which was more aligned with an actual restaurant situation. For both generalization types, researchers collected accuracy and independence data, implementing the SLP as needed if students used the schematic diagram. However, no instruction preceded either generalization session.

Interobserver Agreement and Treatment Fidelity

Researchers collected interobserver agreement (IOA) data for student accuracy across all phases and independence in intervention and generalization. IOA was collected live for at least 20% of sessions in each phase. For IOA, a second researcher collected the same data and both researchers compared their agreements. Researchers computed IOA by dividing the total number of agreements by the total possible amount to agree and multiplying by 100 for each student. Researchers determined IOA for accuracy for all three students was 100% across all phases. Independence IOA for Danny was 100% for intervention, 100% for maintenance, and 97.9% for general ization. For Mindy, independence IOA was 100% for all phases. Tamara's independence and maintenance IOA were 100%, and her generalization IOA was 95.8%.

Researchers employed a checklist to assess treatment fidelity during the intervention phase. The treatment fidelity checklist included such elements as: (a) provided students with a colorcoded schematic diagram, (b) provided students with a calculator, (c) read the problems to students, (d) modeled one problem and guided one problem with students prior to the probe, (e) implemented the SLP with 10-sec wait time when students failed to initiate or initiated incorrectly for all but writing down the answer step of the task analysis, and (f) implemented the SLP in the order of gesture, indirect verbal, direct verbal, and model. Researchers collected treatment fidelity for 100% of intervention sessions. When the second researcher was present for IOA, they also collected treatment fidelity. Treatment fidelity was 100% for each student; IOA for treatment fidelity was 100% for each student.

Social Validity

At the end of the study, researchers asked students questions for social validity. Researchers asked the following questions to students oneon-one after the last generalization session: (a) Do you think the schematic diagram helped you to solve the finding the total bill problems? Why or why not? (b) Did you enjoy using the schematic diagram to help you solve the problems? Why or why not? (c) Do you think the colored-coded shapes and colors in the word problems were helpful? Why or why not? (d) Do you think you would solve similar