Abstract
Data analysis inherently requires problem solving, yet it is the most understudied mathematical skill for individuals with extensive support needs. The current study taught elementary students with extensive support needs (i.e., autism and intellectual disability) to solve math word problems requiring analysis of scaled pictographs through modified schema-based instruction on an iPad. Results of the single-case multiple probe across participants design found a functional relation between the iPad-based math intervention and math problem solving, with a large effect size (Tau-U = .96) confirming visual analysis. In addition, participants were able to generalize problem-solving skills when they were presented with data analysis problems from grade-level social studies textbooks and visual supports were faded. Implications for practice and future research in teaching mathematics to learners with extensive support needs are discussed.
Mathematical problem-solving skills contribute to independence and can lead to greater post-school opportunities (Wei, Yu, Shattuck, & Blackorby, 2015). Unfortunately, findings from a study by Kearns, Towles-Reeves, Kleinert, Kleinert, and Thomas (2011) suggest many students with extensive support needs (i.e., students eligible for special education services under the categories of autism or intellectual disability who participate in alternate-assessments aligned with alternate-achievement standards) are not able to apply known mathematical procedures to problem-solving tasks. From a sample of 12,649 students who took the alternate assessment aligned with alternate-achievement standards, only 32% to 57% could complete computational procedures with or without a calculator (Kearns et al., 2011). Furthermore, only a small percentage of students in that sample (4%-8%) were able to apply computational procedures to solve real-world or routine mathematical word problems.
Data Analysis
Real-world situations that require mathematical problem solving often draw on skills the National Council of Teachers of Mathematics (NCTM; 2000) consider data analysis, or the ability to formulate questions and collect, organize, and display relevant data to answer those questions. Furthermore, statistical reasoning, which involves data analysis, is an imperative skill to be an informed citizen and consumer (NCTM, 2000). We know students with extensive support needs can learn to solve mathematics problems when provided with intensive, high-quality instruction (Spooner, Saunders, Root, & Brosh, 2017). Unfortunately, practitioners only have models for teaching a few of the problem types students may encounter (Spooner, Root, Saunders, & Browder, 2018).
In a meta-analysis of research on teaching mathematics to students with severe disabilities published between 1975 and 2005, Browder, Spooner, Ahlgrim-Delzell, Harris, and Wakeman (2008) found only two studies taught data analysis (i.e., Copeland, Hughes, Agran, Wehmeyer, & Fowler, 2002; Lovett & Haring, 1989). Both Copeland et al. (2002) and Lovett and Haring (1989) taught graphing in the context of self-monitoring with the goal of teaching self-determination rather than mathematics. In an updated review of evidence-based practices for teaching mathematics to students with severe disabilities, Spooner and colleagues (2018) found the majority of studies between 2005 and 2016 focused on number and operations, while just 5% (n = 2) of the studies taught standards related to data analysis (i.e., Browder, Jimenez, & Trela, 2012; Browder, Trela, et al., 2012). Both Browder, Jimenez, and Trela (2012) and Browder, Trela, et al. (2012) used task analytic instruction and a graphic organizer to teach secondary students with disabilities to solve word problems aligned with state standards across the domains of algebra, geometry, measurement, and data analysis.
Data analysis problems in these prior studies involved recording data from a word problem onto a graph and identifying which group or set had more. While these were important first steps in teaching data analysis, these studies have several limitations. First, participants only made one judgment from the graphs, and did not have to determine the quantity of the difference between two amounts, limiting the amount of problem solving that was required. In addition, the task analytic instruction involved a teacher using the task analysis to instruct and assess. It is unknown whether the students would have benefited from using the task analysis to self-monitor progress to increase independence in problem solving. Teaching calculation skills absent from problem solving only shows students how—not when, why, or where—to apply these skills (Browder et al., 2018).
Schema-Based Instruction (SBI) to Teach Problem Solving
According to NCTM (2000), weaving problem solving instruction throughout all mathematical domains may increase student independence. Schemas play a crucial role in problem solving, as they are used to map important information and highlight relationships among quantities from problems to facilitate problem translation and solution (Jitendra, DiPipi, & Perron-Jones, 2002). Key components of SBI include (a) visual representation of mathematical relationship through a schema, (b) a heuristic to aid in memorization of the problem solving process, (c) explicit instruction, and (d) metacognitive strategy instruction to help students monitor their problem solving process. Systematic reviews and meta-analyses have found SBI to be an evidence-based practice for teaching problem solving to students with learning disabilities (Jitendra et al., 2015) and elementary school students with high-incidence disabilities (Peltier & Vannest, 2017).
Traditional SBI has been modified to support students with extensive support needs who are likely to need more intensive supports to overcome additional barriers to problem solving, such as limited procedural and conceptual knowledge and weaknesses in executive functioning and metacognition (Spooner et al., 2017). To support conceptual understanding while decreasing cognitive load and fine motor requirements to draw graphic organizers, modified schema-based instruction (MSBI) uses color coding and visual supports within provided graphic organizers (i.e., schemas). While much of the research on traditional SBI has utilized a mnemonic as a heuristic for problem solving, MSBI provides students with a task analysis. Systematic instruction (e.g., system of least prompts, constant time delay) is added to explicit instruction to further support independence. Finally, to support metacognition, students are taught rules or chants along with hand motions. As such, MSBI includes evidence-based practices for teaching mathematics to students with extensive support needs (i.e., task analysis, graphic organizers, and system of least prompts; Spooner et al., 2018). Recent research has shown technology, another evidence-based practice, can also play an important role within an MSBI instructional package (Spooner et al., 2018).
Using Technology to Overcome Barriers to Problem Solving
One barrier to problem solving students with extensive support needs face is procedural knowledge, including memorization of basic math facts. Spooner et al. (2017) suggest using manipulatives to support procedural knowledge for problems involving whole numbers less than 10 and calculators for quantities greater than 10 or involving decimals and fractions. Experts argue calculators are a form of assistive technology and can decrease cognitive load, thereby providing access to more challenging and grade-aligned content (Bouck & Bouck, 2008; Bouck & Flanagan, 2009). Spooner and colleagues (2018) found technology-assisted instruction (defined as an electronic item, application, or virtual network used intentionally to increase/maintain and/or improve student capabilities; Odom et al., 2015) to be an evidence-based practice for teaching mathematics to students with extensive support needs. Root, Saunders, Spooner, and Brosh (2017) taught middle school students with Down syndrome to solve personal finance problems involving decimals and quantities above 10 (i.e., US$11.50) by using calculators and MSBI. Participants were able to independently solve problems related to finding the final cost of an item or activity when leaving a tip or using a coupon with both scientific and iPhone calculators.
A key factor in the effectiveness of technology-based interventions is the student operation of the device. Ledbetter-Cho, O’Reilly, Lang, Watkins, and Lim (2018) found participant operation of the device in tablet-mediated interventions for teaching academic skills to individuals with autism produced significantly larger effect size estimates. Root and Browder (2017) investigated the role of participant-operated technology in increasing independence and access to mathematical problem solving. Three middle school students with extensive support needs (i.e., autism and intellectual disability) learned to solve algebraic word problems using MSBI with an electronic task analysis. The task analysis was displayed using the SMART Notebook app on an iPad and included both considerate text (i.e., text that enables the reader to gather information with minimal cognitive effort that contains text features that facilitate comprehension; Armbruster & Anderson, 1988; Singer, 1986) and picture supports for each step. Participants were shown how to activate the embedded verbal and specific verbal prompts for each step. For example, when participants touched the number for a step, the verbal prompt was activated telling them what to do for the step (e.g., “Step two says circle the groups”) and when they touched the question mark next to each step, a specific verbal prompt gave them information on how to complete the step (e.g., “Circle the big group and small groups in the problem”). Technology allowed participants to overcome working memory and literacy barriers to problem solving.
Rationale and Purpose
While these findings provide evidence of the effectiveness of technology-based supports (i.e., iPads and calculators) within MSBI to improve mathematical problem-solving skills for students with extensive support needs (i.e., autism and intellectual disability), future research is needed to investigate the effects of MSBI for other domains of mathematics instruction such as data analysis. Furthermore, the question remains how advanced problem solving with multiple operations could be made accessible for individuals with more extensive support needs.
Findings from recent research indicate technology supports can contribute to conceptual and procedural understanding of mathematics problems, and therefore independence in problem solving (e.g., Root & Browder, 2017; Root, Saunders, et al., 2017). However, further research is needed to validate the use of technology to improve mathematical problem solving for students with extensive support needs, particularly with more challenging problems that students will encounter in multiple contexts. Visual displays of data are commonplace in society (i.e., newspapers, social media, and the popular press). In academic settings, data analysis is required to understand and analyze informational text across content areas. Given the importance of data analysis (NCTM, 2000) and the paucity of research on how to teach mathematical problem solving using data analysis to students with extensive support needs (Browder et al., 2008; Spooner et al., 2018), the purpose of this study was to evaluate the effects of MSBI with technology-supports on solving multistep mathematical problems involving data analysis for students with extensive support needs. In addition, researchers sought to measure the degree to which students with extensive support needs (i.e., autism and intellectual disability) would generalize their problem solving when visual supports were faded and they were presented with a data analysis problem from a content-area textbook. We sought to answer the following research questions:
Method
Participants
Approval from the institutional review board was received prior to recruitment. Participants were recruited using teacher nomination and selected based on the following criteria: (a) teacher report of eligibility to participate in state alternate assessment aligned with alternate-achievement standards (AA-AAS) and (b) educational or medical diagnosis of autism. Three students in Grades 4 and 5 participated in the study. During the school year the students were all enrolled in the same multigrade classroom at a private school. They received mathematics instruction aligned with grade-level standards from a noncertified teacher, primarily using the Eureka Math Curriculum (Great Minds, 2018). Researchers administered a pre-assessment to determine participants’ levels of mathematical knowledge (see the “Targeted Mathematics Skill” section).
Scott was a 10-year-old African American male student in the fourth grade with a medical diagnosis of autism level 1 (i.e., requires support) according to the Diagnostic and Statistical Manual of Mental Disorders (5th ed.; DSM-5; American Psychiatric Association, 2013). Standardized assessment information regarding Scott’s cognitive or adaptive functioning was not available, though teachers reported he would have participated in the state’s AA-AAS if he were enrolled in public school. On the pre-assessment, he receptively and expressively identified numbers 0 to 100, compared quantities to determine which was greater, added and subtracted double-digit numbers with and without a calculator, and interpreted simple pictographs, line plots, and bar graphs. He did not interpret or solve problems using scaled pictographs with whole numbers or fractions. He enjoyed talking with peers, showing off items from the classroom or that he had brought from home, and learning about math and science. He sometimes had difficulty transitioning between tasks or when his routine was altered.
Ricky was a 10-year-old Caucasian male in the fourth grade with a combined diagnosis of autism level 1 (i.e., requires support), attention deficit hyperactivity disorder, mild intellectual disability, and obsessive-compulsive disorder from a physician. Standardized assessment information regarding Ricky’s cognitive or adaptive functioning was not available, but his teacher reported previous participation in the state’s AA-AAS when he was enrolled in public school. His behavior on the pre-assessment was similar to Scott, in that he receptively and expressively identified numbers 0 to 100, compared quantities to determine which was greater, added and subtracted double-digit numbers with and without a calculator, and interpreted simple pictographs. Ricky did not interpret or solve problems using line plots, bar graphs, or scaled pictographs with whole numbers or fractions, or solve multiplication or division problems. He enjoyed talking with peers and familiar adults and playing games with his classmates. Standardized assessment information was available on Ricky’s mathematical skills from the Woodcock–Johnson Tests of Achievement, third edition (WJ-III; Woodcock, McGrew, & Mather, 2001). He had an overall mathematical standard score of 66 (1st percentile) on the WJ-III, with relative strengths in calculation (standard score 83, 13th percentile) and weakness in applied problems (standard score 63, 1st percentile).
Kelly was an 11-year-old Caucasian female in the fifth grade with a diagnosis of autism level 2 (i.e., requires substantial support), language impairment, and a mild intellectual disability from a physician. Standardized assessment information regarding Kelly’s cognitive or adaptive functioning was not available, but teachers reported she previously participated in the state’s AA-AAS when enrolled in public school. The pre-assessment measure indicated Kelly expressively and receptively identified numbers 0 to 100, solved problems with all four operations using a calculator, and interpreted simple line plots and bar graphs. Kelly did not solve problems involving multiplication or division without a calculator, or solve problems involving simple or scaled pictographs. Kelly was very friendly and enjoyed meeting new people. She often talked using delayed echolalic phrases from songs or movies. She responded to questions and directions asked by teachers. Kelly also had standardized mathematics information available from a prior administration of the WJ-III. She had an overall mathematical standard score of 60 (<1st percentile), with relative strengths in calculation with a subtest standard score of 81 (10th percentile) and weakness in applied problems with a standard score of 55 (<1st percentile).
Targeted Mathematics Skill
Participants were administered a researcher-developed pre-assessment to (a) determine participants’ level of mathematical knowledge, (b) identify instructional supports that may be needed, and (c) determine which data analysis skills to target. The pre-assessment tool measured students’ ability to (a) skip count by 2s, 5s, 10s, and 100s; (b) multiply single- and double-digit numbers with and without a calculator; (c) add and subtract two- and three-digit numbers with and without a calculator; (d) interpret tables with tally marks, pictographs, scaled pictographs, line plots, bar graphs, and scaled bar graphs; and (e) solve simple comparative word problems both with and without requiring data analysis. The pre-assessment is available from the first author upon request. All participants were able to use the four-function calculator appropriately, and the common area of need across participants was how to interpret and solve problems based on scaled pictographs. These skills align with the following grade-level standards: (a) MAFS.4.MD.2/MAFS.5.MD.2: represent and interpret data, (b) use place value understanding and properties of operations to perform multi-digit arithmetic, and (c) MAFS.5.NBT.2: perform operations with multi-digit whole numbers.
Settings and Interventionists
Sessions for each participant took place 4 days per week and lasted approximately 15 min. Sessions for Scott and Ricky took place at a summer camp for students with autism located in the southeastern United States, while sessions for Kelly took place at an inclusive theater camp. The interventionist worked one on one in a quiet room at each location with the participants, though behavioral technicians were sometimes in the room as well. During the summer none of the participants received regular mathematics instruction. The interventionist (third author) was a Hispanic female enrolled in a graduate program in special education who had completed a course in systematic instruction. The first author trained her in intervention procedures using role-play and modeling.
Materials
During baseline and intervention sessions, participants were provided with electronic worksheets on the GoWorksheet app (Attainment Company, 2018) on an Apple iPad Air. Each worksheet displayed a five-step task analysis, a simple four-function calculator, a scaled pictograph, compare word problem, and graphic organizer (see Figure 1). The scaled pictographs and their corresponding word problems centered on a different theme each session (e.g., zoo, weather, sports, pets). Three scaled pictographs and word problems were developed for 18 themes to ensure participants would not be exposed to the same theme (or problems) more than once. All word problems used a similar question stem of “How many more/fewer . . .” The GoWorksheet app has a text-to-speech option that participants could activate to read all text on the screen. In addition, participants could choose to write on the worksheet using the provided stylus or their finger or type on the worksheet using an on-screen keyboard. During generalization sessions, participants were provided with paper worksheets that had a scaled pictograph from an upper elementary social studies textbook and a corresponding compare word problem (see Figure 1). A simple four-function calculator was provided to participants in all sessions.
Blank electronic worksheet (top) and Scott’s completed generalization worksheet (bottom).
Design and Measurement
A multiple probe across participants design was used to demonstrate a functional relation between the mathematics intervention and the primary dependent variable (Ledford & Gast, 2018). The design adhered to guidelines for single-case research set forth by the What Works Clearinghouse (Kratochwill et al., 2010). There were three experimental conditions: baseline, intervention, and generalization. After the first participant (Scott) showed a clear accelerating trend or improved level for a minimum of two data points, the second participant (Ricky) entered intervention following three consecutive baseline data points. A similar systematic introduction to intervention was used for Kelly. Intervention ceased once participants met mastery criteria of 10 out of 12 critical steps across three sessions (see the “Dependent Variables” section). Generalization sessions were conducted once pre- and post-intervention.
Dependent variables
The primary dependent variable was mathematical problem solving, measured by the number of critical steps of the task analysis completed independently correct. Only steps that were completed independently correct on the first opportunity were graphed. Although the student task analysis (as shown in Figure 1) consisted of five steps, only Steps 1, 2, 4, and 5 were deemed critical for solving the problem (Test & Spooner, 1996; Weng & Bouck, 2014). Table 1 displays the expected (and measured) student response for each critical step in the task analysis. The interventionist took data on the number of steps completed independently correct during each session. Participants solved three problems in each session, for a total of 12 available points in each session. Generalization of problem solving was measured in the same way, with a total of 12 available points across three problems in each probe.
Steps of Task Analysis and Corresponding Expected Student Response.
Note. The dependent variable of mathematical problem solving was measured using the four critical steps of the task analysis (i.e., Steps 1, 2, 4, and 5). Participants completed three problems per session and could earn up to 12 points per session.
Interobserver agreement (IOA) and procedural fidelity
To ensure reliability and fidelity, IOA and procedural fidelity data were collected across all experimental conditions. IOA was evaluated using an item-by-item method and calculated by dividing the total agreed items by the sum of agreed and disagreed items and multiplied by 100. Both in vivo and permanent product (i.e., video) observations were used due to the timing of the study over the summer and in multiple locations. The second observer (i.e., second author) used the same data collection tool as the interventionist for IOA. IOA was calculated for 40% of baseline sessions for Scott (2 out of 5 sessions), 33% of baseline sessions for Ricky (2 out of 6 sessions), and 25% of baseline sessions for Kelly (2 out of 8 sessions). The agreement was 100%. The second observer collected IOA during 25% of all intervention sessions; 20% for Scott (1 out of 5 sessions), 33% for Ricky (2 out of 6 sessions), and 20% for Kelly (1 out of 5 sessions). Agreement was 91% for Scott, 100% for Ricky, and 91% for Kelly.
The second observer used a procedural fidelity checklist to document the degree to which the intervention was implemented consistently and as designed for all sessions in which IOA was collected. This checklist measured (a) provision of prompting and feedback, (b) wait time, and (c) provision of an instructional cue. An item-by-item method was used to evaluate fidelity as well. The mean procedural fidelity for baseline was 100% for all participants. The mean procedural fidelity for intervention was 91% for Scott, 83% for Ricky (range 82%-84%), and 91% for Kelly. Errors in procedural fidelity for Ricky centered on skipping prompt levels (i.e., going directly to a model prompt when a specific verbal was warranted).
Procedures
In all baseline and intervention sessions, participants were provided with the iPad displaying the worksheets on the GoWorksheet app, a stylus, and a calculator. Prior to beginning baseline, a brief (i.e., less than 5 min) materials training session took place that oriented participants to the GoWorksheet app and stylus. The purpose of this training session was to ensure students knew how to (a) use the text-to-speech features, (b) operate the stylus, (c) choose the color of the electronic ink, (d) erase or undo markings, and (e) clear the calculator.
Baseline
Participants were given the worksheets for the three problems on the GoWorksheet app and asked to “Show me how to solve the problem.” Participants were given praise for effort, but no specific feedback as to the accuracy of their responses was provided. If needed, the interventionist did give technical assistance (i.e., activating the text-to-speech features, use of stylus). Participants did not receive any other mathematics instruction during the time of the study.
Intervention
The interventionist taught participants to solve the problems using MSBI and technology-based supports. The first day of intervention was a “model” day, where the interventionist modeled how to use the task analysis to show participants how to use pictographs to solve additive (i.e., compare) word problems with active student participation. The interventionist followed a teaching script to ensure adherence to the MSBI procedure and use of key academic vocabulary terms, similar to the model scripts used in prior MSBI studies (see Browder et al., 2018; Root & Browder, 2017; Root, Browder, Saunders, & Lo, 2017). The model script included information on what pictographs were, how pictographs can be used to gather information, and explained how to solve the problem using the six-step task analysis: (a) read graph title and problem, (b) interpret scale, (c) label graphic organizer, (d) solve, and (e) write answer. Participants were taught to interpret information from a pictograph, write numerical quantities for each column, and then use the relevant numerical quantities to complete the schematic diagram for the compare problem presented in the word problem. Participants then compared the quantities from the pictograph to answer the data analytic question. The model script is available from the first author upon request. No data were collected during the “model” session, as participants did not have an opportunity to make an independent response. Table 1 describes each step of the task analysis and the corresponding expected student response.
Beginning on the second day of intervention, the interventionist used a system of least prompts. The participant was provided with the first worksheet and given the cue, “Show me how to solve this problem.” The system of least prompts was used if the participant failed to make an independent response within 5 s. The prompting hierarchy included a verbal prompt (e.g., activating the text-to-speech function on the iPad to provide a re-read of the step of the task analysis), specific verbal prompt (e.g., providing specific directions on how to complete the step), and a model-retest (e.g., showing the participant how to complete the step and re-presenting the step). Anytime the participant incorrectly completed a step, the interventionist immediately provided error correction in the form of a model (e.g., “My turn. Each apple represents 4 apples, so I am going to count my apples and then multiply the number of apples by 4 in the calculator”) and a retest (e.g., “Your turn. Show me how to find the total number of apples”). This pattern continued until the participant had correctly completed the step. Whenever the participant completed a step independently correct, the interventionist provided specific verbal praise to confirm the step (e.g., “Yes. Each apple represents 4 apples, so you multiplied each apple by 4”).
Generalization
During generalization probes, participants were provided with the generalization worksheet (Figure 1), a pencil, and a calculator, and given the cue to “solve the problem.” They were not given any prompting, feedback, or assistance, other than intermittent praise for staying on task or putting forth effort.
Data Analysis
Researchers used both visual and statistical analysis. Per standard single-case design procedures, data were graphed throughout the study to allow for visual inspection of the trends and to determine the presence of a functional relation between MSBI and mathematical problem solving. Researchers used What Works Clearinghouse guidelines for visual analysis of single-case designs (Kratochwill et al., 2010) across six variables: (a) level, (b) trend, (c) variability, (d) overlap, (e) immediacy of effect, and (f) consistency of data patterns across similar phases. Descriptive analyses of mean and range were calculated for comparison between phases. In single-case research, the term “effect size” is attributed to the amount of improvement seen by an individual that can be ascribed to an intervention. Although the field of special education has not reached a consensus on the most appropriate effect size measures for single-case research, Tau-U is a promising nonparametric measure due to its stability and ability to control for monotonic trends in baseline (Vannest & Ninci, 2015). Tau-U was calculated using an online calculator (Vannest, Parker, Gonen, & Adiguzel, 2016) by first calculating results for each participant and then an aggregate value, weighted by the length of the series. Vannest and Ninci (2015) provide the following guidance in interpreting Tau-U: (a) .20 to .60 indicates a moderate effect, (b) .60 to .80 indicates a large effect, and (c) above .80 is a very large effect.
Social Validity
Single-case research should produce predictable and replicable improvements in “socially important” behavior. In their seminal article, Horner and colleagues (2005) outlined 21 quality indicators for single-case research, including four specific to social validity: (a) dependent variable is socially important, (b) magnitude of change is socially important, (c) implementation is practical and cost effective, and (d) implementation over extended time period, by typical intervention agents, in typical physical and social contexts. Guidelines on how to select and target behaviors to ensure their importance are well established (e.g., Ayllon & Azrin, 1968; Rosales-Ruiz & Baer, 1997), but as Cooper, Heron, and Heward (2007) point out, the ultimate question to be asked is whether or not the behavior change will improve the life experience of the individual. In this study, the primary targeted behavior was mathematical problem solving, which is a behavioral cusp, or a “behavior change that has consequences . . . beyond the change itself, some of which may be considered important” (Rosales-Ruiz & Baer, 1997, p. 537). Bosch and Fuqua (2001) emphasize the consideration of the long-term consequences of acquiring a behavior, rather than simply the potential for change in the immediate environment. Mathematical problem solving is emphasized throughout mathematics curriculum and is required for numerous leisure, vocational, and daily living skills (Browder et al., 2018). The specific skill targeted in the current study included solving data analysis problems involving scaled pictographs using a calculator. Scaled pictographs are seen in other content areas beyond mathematics, such as science and social studies, as well as in popular media such as blog posts, newspapers, and websites. Each participant demonstrated a socially significant magnitude of change in the targeted behavior and demonstrated some ability to generalize to content-area scaled pictographs.
Not only is the skill of solving problems involving scaled pictographs socially valid, but the intervention also taught self-monitoring, which is a pivotal behavior (Koegel, Koegel, & Harrower, 1999). Improvements in pivotal behaviors produce large accompanying improvements in other areas, which shorten the length of time it takes to acquire related skills (Cooper et al., 2007). Self-monitoring and other self-management skills are pivotal because they increase independence and decrease reliance on outside agents to impact behavior change.
Discussion of whether an intervention is practical or effective would be best if it included perceptions of stakeholders, such as teachers, administrators, and students. While we do not have a direct measure of this in the current study, teacher participants in prior studies on MSBI indicate it is an acceptable intervention that can be implemented by natural intervention agents including teachers (Browder et al., 2018) and peers (Ley Davis, 2016). The current study was implemented over a brief summer time period in atypical learning environments by a graduate research assistant (i.e., not a typical intervention agent) and therefore does not meet the fourth social validity quality indicator described by Horner et al. (2005).
Results
Figure 2 shows the effect of MSBI with technology-based supports on mathematical problem solving. The graph shows the number of critical steps of the task analysis performed independently correct across three problems. Not pictured on the graph is the one modeling session at the beginning of each intervention phase, as participants were not given the opportunity to make an independent response and therefore data were not collected.
Graph of number of points earned for critical steps of the task analysis completed independently correct across three problems each session.
During baseline, all participants had a stable pattern of responding in terms of level, trend, and variability. All participants showed a change in level and trend on solving compare problems using pictographs after receiving MSBI, with only one overlapping baseline data point for one participant. Baseline generalization data were consistent across participants, in that scores were at the same level or lower than baseline data. After reaching mastery criteria (completion of 10 out of 12 critical steps independently correct across three sessions), all participants demonstrated generalization to paper-based pictographs from grade-level social studies texts. An overall effect size of .96 was calculated using Tau-U, which can be interpreted as very large (Vannest & Ninci, 2015).
Scott had a descending trend in his baseline, earning an average of 5.2 points (range 7-4) across five sessions. During baseline he did not attend to the task analysis, but was able to use his calculator to interpret the scale on some problems. After beginning intervention, he increased independent responding and was able to reach mastery in five sessions. During the first few intervention sessions he needed prompts to attend to the task analysis and fully read the word problem so that he could set up the equation correctly. Scott increased his generalization score from 3 to 11 points from baseline to post-intervention. Figure 1 displays a photograph of one of Scott’s completed post-intervention generalization probes, during which he drew five marks for the steps of the task analysis and the graphic organizer.
Ricky had a stable trend in baseline, earning an average of 2.3 points (range 0-3) across six sessions. He would usually just write the number of each object he saw on the graph and then go to the next problem and did not attend to the calculator or task analysis. He did not display an immediate jump in responding after beginning intervention, with an initial overlapping data point with baseline. However, from the second intervention session (jump to 7 points) forward he displayed a change in level and trend and reached mastery criteria in five sessions. In intervention he quickly learned how to use the calculator, though the step he commonly missed in the first few intervention sessions was interpreting the scale correctly as he was unfamiliar with multiplication. Ricky’s generalization data demonstrated a meaningful change, as he only earned one point in the baseline generalization session but earned nine (out of 12) points in the post-intervention generalization session.
Kelly had a stable trend in baseline, earning an average of 1.25 points (range 0-3) across eight sessions. During baseline she was sometimes able to label the graphic organizer, reflecting an understanding of what she was solving for, and sometimes she would read the problem as well, but did not use the calculator at all. She had an immediate jump in level and trend once intervention began. Despite a 2-week break in data collection between her third and fourth intervention sessions due to a family vacation, Kelly met mastery criteria in five intervention sessions. She needed prompting and support to set up the subtraction equation correctly and use her calculator to interpret the scale. Her generalization data also demonstrated growth, as she did not earn any points in the baseline generalization probe but earned seven points post-intervention.
Discussion
The purpose of this study was to expand the research on teaching mathematical problem solving to students with extensive support needs in the neglected domain of data analysis, responding to a decade long call for research by experts in the field (see Browder et al., 2008; Spooner et al., 2018). Elementary students with extensive support needs (i.e., autism and intellectual disability) were taught to solve multiple-step mathematics word problems involving data analysis through MSBI with technology-based supports. Results indicate a functional relation between MSBI with technology-based supports and problem solving. Visual analysis was supported by a large effect size.
This is the first study to teach problem solving within a data analysis context to learners with extensive support needs. Data analysis is a skill that inherently requires problem solving and is addressed across multiple content area standards. Two other studies have taught data analysis to students with extensive support needs, but only required students to interpret a graph (see Browder et al., 2012; Browder, Trela, et al., 2012). The current study builds on prior research going beyond simple interpretation of graphs to solving problems involving more complex data sets (i.e., scaled pictograph) using MSBI and measured the degree to which they generalized skills when presented scaled pictographs from content-area textbooks. Prior research has shown MSBI to be effective for teaching single-step additive (Browder et al., 2018) and algebraic problems (Root & Browder, 2017; Root, Henning, & Boccumini, 2018), as well as multiple-step multiplicative problems (Root, Cox, Hammons, Saunders, & Gilley, 2018). All participants in the current study were able to demonstrate some degree of generalization from researcher-created pictographs on an iPad with visual supports (i.e., task analysis and graphic organizer) to scaled pictographs from content area textbooks without stimulus supports. These findings address prior limitations of MSBI studies, providing evidence that learners may be able to internalize strategies once they reach the fluency phase of learning. For example, during his final generalization probe, Scott drew the task analysis for himself from memory and checked off each step as he completed it.
Given the complexity of the targeted mathematics skill, the treatment package capitalized on the potential of technology-based supports. Technology-based instruction has been identified as an evidence-based practice for teaching academics to students with autism (Root, Stevenson, Davis, Geddes-Hall, & Test, 2017; Wong et al., 2015), as well as for teaching mathematics to learners with moderate to severe disabilities (Spooner et al., 2018). Transferring the learning materials from a traditional paper worksheet to an electronic worksheet on the iPad did not change the format, yet increased functionality and accessibility. Participants self-managed technology-based supports (i.e., calculator, speech-to-text options), which has been found to have a significantly positive effect on academic outcomes during tablet-mediated interventions (Ledbetter-Cho et al., 2018). The text-to-speech features decreased reliance on a skilled reader and the option to use the calculator provided students procedural supports to decrease cognitive load and reduce computational errors. Both Kelly and Ricky consistently utilized the text-to-speech options and rarely asked the interventionist for assistance with reading. Finally, the provision of a calculator allowed participants to access and participate in a level of mathematics beyond their procedural abilities. Scott had some basic multiplication facts memorized and was able to skip count by 2s, 5s, and 10s. Ricky and Kelly had very limited multiplication facts memorized and did not demonstrate the ability to use them to interpret the scaled pictograph. All participants used the calculator for this purpose during intervention. It is important to note that simply providing a calculator is not enough for students with extensive support needs; they need to understand how, why, and when to use it.
Limitations and Future Research
One of the contributions of the current study is that it demonstrated MSBI is an effective instructional method for teaching students with extensive support needs to solve mathematics problems involving data analysis and that participants were able to generalize problem solving to pictographs from content area textbooks; however, limitations must be considered. Although acquisition and generalization were measured in this study, the limited summer time frame prevented measurement of maintenance. Furthermore, students were only taught to solve comparative problems from pictographs. Data analysis standards emphasize creating as well as interpreting a variety of graphs, charts, and tables (NCTM, 2000; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). Future research should expand the targeted math content and format of learning materials.
The word problems and pictographs in the current study were formulaic and highly controlled, and therefore do not represent the wide range of problems students would encounter in either academic or real-world contexts. Additional studies could explore generalization to other real-world stimuli, such as visual displays of data in magazines, advertisements, newspapers, or online sources. Finally, the instructional format used in this initial study is not representative of all learning environments, as sessions took place one-on-one between the students and a graduate research assistant. Furthermore, due to the timing of the study it was not possible to obtain additional mathematics information from their school or teacher, such as recent standardized assessment information or mathematics goals. Feasibility of implementation for teachers (i.e., authentic change agents) in education contexts as well as direct measurement of social validity from stakeholders should be a priority of future research. Relatedly, assessment of the acceptability of the procedures and outcomes by individuals with disabilities is needed.
Implications for Practice
Findings from this study offer useful suggestions for improving problem solving for students with extensive support needs. The tablet-mediated math intervention allowed students to control which supports they needed. Teachers and researchers can utilize technology implemented instructional supports such as read alouds to allow students to develop increased independence. Additional technological supports such as calculators can reduce cognitive load and increase procedural accuracy, allowing students the opportunity to develop a conceptual understanding of complex mathematical relationships. Teachers and researchers should consider the use of technological supports to promote student independence and increase the opportunity for students to participate in more complex mathematical problem solving. Instruction that allows students with disabilities to develop mathematical problem solving skills will improve their opportunity to use mathematics in a variety of contexts.
This study also extends prior research on MSBI, suggesting that it is possible to fade supports (i.e., task analysis, graphic organizer, and systematic prompting) while maintaining student performance. All three participants in this study improved their ability to generalize word problem-solving skills to solve scaled pictograph word problems to a novel context (i.e., a social studies textbook) without additional stimulus supports. These findings suggest that teachers and researchers can use MSBI with technology-based supports to promote immediate acquisition of problem-solving skills, and later fade additional supports when students no longer need them. Fading additional supports can promote generalization of skills to novel situations and improve self-determination for students with disabilities.
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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The contents of this article were developed, in part, under a grant from the U.S. Department of Education, H325D140074. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and one should not assume endorsement by the Federal Government.
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