Effects of Modified Schema-Based Instruction on Addition and Subtraction Word

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Problem Solving of Students with Autism Spectrum Disorder and Intellectual Disability

A dissertation submitted to the

Graduate School of the University of Cincinnati

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Educational Studies

College of Education, Criminal Justice, and Human Services

July 2, 2019

Amanda G. Buncher

B.S. University of Cincinnati, 2009

M.Ed. University of Cincinnati, 2011

Committee Chair: Dr. Casey Hord, PhD Abstract

According to the United States Department of Education, students diagnosed with an autism spectrum disorder (ASD) make up approximately 7.6% of students with disabilities in the U.S and are being educated within the general education classroom at an increasing rate, requiring educators to modify their current practices to meet the needs of this continually growing population. Impairments in executive functioning central coherence, metacognition, and attention often create barriers for students with ASD in regards to their academic achievement.

Mathematics word problem solving is an area of academic concern for students with ASD, requiring planning and organization (executive functioning) and metacognitive monitoring skills which are often lacking in students with ASD. The majority of strategies for this population target computational skills and math fluency with very few addressing word problem solving.

One promising intervention that has been used recently with this population of students is schema-based instruction (SBI). The purpose of this study was to examine the effectiveness of using SBI to teach word problem solving to students with ASD in the classroom setting.

Intervention effectiveness was analyzed and discussed in addition to educational implications and recommendations for future research.

Keywords: schema-based instruction, mathematics, autism spectrum disorder, word problem solving

i ii Acknowledgements

This has been a long and arduous process, with ups-and-downs along every step of the way. Pursuing a doctoral degree was not an avenue I had ever considered for myself. Through this journey, I have learned that there is more than one way to influence and impact the quality of instruction offered to students with disabilities. I had the pleasure of working directly with these students for over seven years and now I have the privilege of researching best practices for these very same students. The life of an educator never strays far from the students.

I would like to first thank Dr. Casey Hord, my committee chair and graduate mentor, for opening my eyes to the possibilities within the world of special education. Thank you for your continued presence throughout this entire process; I would not have been able to navigate through these past three years without your guidance and support. Your fair judgement and sound advice have lead me in a productive and purposeful direction. In addition, I would like to thank my committee members, Drs. Ana DeJarnette and Heidi Kloos. Your willingness to assist and your advice during planning meetings made all the difference. Thank you all for being open-minded and flexible as I transitioned to a new career, began teaching at the collegiate level, and introduced a new baby to the world.

To my new co-workers at Cincinnati Children’s Hospital Medical Center, including my manager, Erin Riehle, thank you for introducing me to the world of post-secondary transition for students with disabilities, but also for your patience as I completed this major milestone. I cannot wait to see what we accomplish next! To my dear friend and colleague, Christel Murphy, thank you continuing to encourage me through this program and for your constant support. To the students I have taught over the years during my role as a special educator; thank you for always challenging me to think in new ways to achieve the best possible outcomes.

iii I would like to extend my sincere gratitude to the participants of this study, without whom this dissertation would not be possible. It was such a pleasure to get to know you over the course of the study and I wish you all the best as you continue to grow and learn. To the wonderful classroom teacher that made this possible. This study would not have come to fruition if it had not been for her willingness to open her doors to me and change her schedule to meet the needs of the study. I am forever indebted to you, the administration, and classroom aides within the school for making all of this a reality.

Most importantly, I would like to extend my heartfelt thanks to my family, who were truly the catalyst behind my entry into and successful completion from this program. Thank you to my husband, Kelon, for always encouraging me to pursue my dreams, whatever they may be.

Your unwavering support helped me to persevere even when life got challenging. To my children, Ronan and Imogen, you are the lights of my life. You are too young now to realize the work that it took to complete this, but I hope that one day you too will face a challenge head-on and persevere through it. I truly hope that my work will encourage you to never sell yourselves short and be the best possible version of yourselves. To my parents, Ron and Donna Lipps, for providing additional childcare just about every Saturday for the past three years, and for believing in me even when this challenge seemed too big to tackle. To my mom for her unwavering commitment to my children as I navigated the world of full-time employment and doctoral work. To my dad for always expecting the best from me, but always providing support if needed. Thank you to my three sisters, Jordan, Lauren, and Emily Lipps for their “back-up” babysitting, words of encouragement, and much-needed smiles throughout this process. My entire extended family kept me on track and focused these past three years by always checking in whenever we were together. I love and appreciate all of you.

iv Table of Contents

Abstract...... i

Acknowledgements...... iii

Introduction...... 1

Literature Review...... 13

Methodology...... 30

Results...... 59

Discussion...... 99

References...... 110

Appendices...... 133

v LIST OF TABLES

1. Zachariah’s Recent Evaluations ...... … 33

2. Mercedes’ Recent Evaluations...... 34

3. Andres’ Recent Evaluations...... 35

4. Addition Instruction Process...... 47

5. Subtraction Instruction Process...... 50

6. Effects of SBI on Addition Word-Problem Solving: Phase, Means, and Tau………..62

7. Effects of SBI on Subtraction Word-Problem Solving: Phase, Means, and Tau …….64

8. Social Validity Measure Results: Student Participants……………………………….69

9. Social Validity Measure Results: Classroom Teacher ….…………………………....70

10. Zachariah’s Baseline Performance……………...……………………………………..72

11. Zachariah’s Addition Phase Performance……………………………………………..73

12. Zachariah’s Subtraction Phase Performance.………………………………………….75

13. Zachariah’s Maintenance Phase Performance….……………………………………...78

14. Mercedes’ Baseline Phase Performance…………………………………………….....80

15. Mercedes’ Addition Phase Performance………………………………………………82

16. Mercedes’ Subtraction Phase Performance……………………………………………85

17. Mercedes’ Maintenance Phase Performance…………………………………………..87

18. Andres’ Baseline Phase Performance………………………………………………….90

19. Andres’ Addition Phase Performance………………………………………………….92

20. Andres’ Subtraction Phase Performance……………………………………………….94

21. Andres’ Maintenance Phase Performance……………………………………………...97

vi LIST OF FIGURES

1. Participant Progress ...... 60

2. Zachariah’s Subtraction Strategy………………………………………………………...77

3. Mercedes’ Baseline Performance Example……………………………………………...79

4. Mercedes’ Addition Strategy…………………………………………………………….84

5. Andres’ Baseline Performance Example………………………………………………...89

vii Introduction

Foundational skills, such as addition and subtraction, are the building blocks of mathematics proficiency and fluency (National Council of Teachers of Mathematics [NCTM],

2014). The ability to perform basic math computations is necessary for independent living

(Cihak & Foust, 2008) and increases the likelihood an individual will be employed (Brown &

Snell, 2000). These early mathematics skills are one of the strongest predictors of future academic achievement (Duncan et al., 2007) and should be developed in order to improve functional outcomes for students with ASD (John, Dawson, & Estes, 2017). The inability to comprehend and solve basic level addition and subtraction problems can hinder a child’s progress through higher-level mathematics instruction (Confrey, Nguyen, Lee, Panorkou, Corley,

& Maloney, 2012). For example, the successful completion of a multiplication problem relies on an understanding of the basic principles of addition and adding sets or groups (Behr, Harel, Post,

& Lesh, 1994). A word problem with multiple steps (or a chained task) may include aspects of any and/or all of the basic computational procedures (addition, subtraction, multiplication, and division); lack of skill in any/all of those procedures could result in a roadblock to the eventual solving of the problem (Jitendra, Griffin, Deatline-Buchman, & Sczesniak 2007). Even more importantly is the ability to identify the correct operation and relevant information in a problem, which often leads to more profound errors and missteps than an inability to compute (Goldman,

1989). It is that latter that is often lacking or omitted completely from the instruction provided to students with disabilities (Cawley, Parmar, Yan, & Miller, 1998).

With the adoption of The Common Core State Standards in Mathematics (CCSS-M;

National Governors Association for Best Practices & Council of Chief State School Officers,

2010) comes increased expectations for all students, including students with disabilities (King,

1 Lemons, & Davidson, 2016). Currently, students with disabilities in grades three-through-seven display lower math achievement and growth than their peers without disabilities (Schulte &

Stevens, 2015). The disparities in math growth and essential flat-line in achievement level for students with disabilities, with 80% scoring below proficiency (National Assessment of

Educational Progress [NAEP], n.d.) highlights a need for new instructional strategies and more targeted interventions. Specifically, methods that focus more on a conceptual understanding rather than a purely rule-based and computational knowledge of mathematics are needed (Xin,

Jitendra, Deatline-Buchman, 2005).

Students with disabilities, specifically disabilities that affect cognition, working memory, and mathematical understanding, are susceptible to mathematics anxiety (Ashcraft & Krause,

2007). With approximately 80% of students identified with disabilities scoring below proficiency on high-stakes testing in mathematics, there is undoubtedly a feeling of confusion and frustration among this population of students. Feelings of frustration can lead to math avoidance in students who already experience math anxiety (Nelson & Harwood, 2011). Math avoidance can be detrimental to the future aspirations of any individual, as mathematics is essential to all aspects of daily life (i.e. job performance, academic performance, functional independence, and community involvement) (Xin et al., 2005). This cycle of under- performance, anxiety, frustration, and eventual avoidance can be remediated with high quality and purposeful mathematics instruction (National Education Goals Panel, 1997; NCTM, 2000).

Students with Autism Spectrum Disorders

Students diagnosed with ASD have the potential to experience any (or all) of the aforementioned scenarios. Students on the autism spectrum with a co-morbid intellectual disability (ID) may experience higher rates of math anxiety due to impairments that may be

2 present in cognition and executive functioning (Barnett & Cleary, 2015). Students on the autism spectrum with cognitive impairments or ID often experience deficits in executive functioning

(i.e. planning, organization, working memory, mental flexibility, attention, self-monitoring, and impulse control) which can lead to math difficulties (Happe, Booth, Charlton, & Hughes, 2006).

Students with autism and ID may struggle with reading (fluency and comprehension) and language; creating a barrier to solving math problems involving lines of text (Donlan, 2007;

Whitby & Mancil, 2009). While some students with ASD without ID, or high-functioning autism (HFA), perform within an average range on counting, operations, and facts, they may be challenged when they have to apply critical thinking or analytical skills to solve a higher-order problem (Siegel, Goldstein, & Minshew, 1996). Regardless of their level of functioning and development along the spectrum, students with ASD have the potential to experience difficulties with or anxiety surrounding mathematics.

As defined by The American Psychiatric Association (APA) (2013), ASD is a neurodevelopmental disorder characterized by impairment in social communication and restricted or repetitive behaviors and interests. Children with autism may appear to have be lacking the intuitive ability to socialize, have difficulty with verbal and nonverbal communication, and demonstrate a tendency to engage in unusual interests and ways of playing

(Attwood, 2008). More than half of individuals diagnosed with ASD may have a co-morbid diagnosis of ID (APA, 2013) while others possess average or above-average abilities (Chiang,

Tsai, Cheung, Brown, & Li, 2014). The following paragraphs will describe the various characteristics of the disorder.

Social communication. By definition, social communication refers to the sharing of information, thoughts, or ideas with another person (Mundy, Sigman, Ungerer, & Sherman,

3 1986). This includes nonverbal communication, gesturing, and eye contact and refers to the intentional communication with another individual (Fuller & Kaiser, 2019). Many researchers attribute the deficits in theory of mind, or the understanding of the mental states of oneself and others (Premack & Woodruff, 1978) to the deficits in social communication often exhibited in individuals with ASD (Brosnan, Johnson, Grawemeyer, Chapman, Antoniadou, & Hollinworth,

2017). Deficits in social communication are directly related to challenges in other areas, including: academic performance, maintaining appropriate behaviors, and building relationships with others (Bauminger & Kasari, 2000; Koegel, Koegel, & Surratt, 1992).

Adaptive functioning. Adaptive functioning, also referred to as adaptive behavior, include behaviors that allow an individual to be more self-sufficient (Matson & Shoemaker,

2009). A measure of adaptive behavior, such as the Vineland Adaptive Behavior Scales (1st and

2nd editions) often assesses the following: communication, daily living skills (DLS), socialization, and motor skills (Sparrow, Balla, & Cicchetti, 1984; Sparrow, Balla, & Cicchetti,

2005). In terms of adaptive functioning, individuals with ASD demonstrate a deficit in the domain of DLS in relation to their typically developing peers (Matthews, Smith, Pollard, Ober-

Reynolds, Kirwan, & Malligo, 2015). In conjunction with impairments in communication and socialization, a deficit in DLS can limit the number of competitive employment and independent living opportunities available to young adults with ASD (Matthews et al., 2015). Not included in the diagnostic criteria of ASD are maladaptive behaviors, oftentimes exhibited by individuals with ASD (Aman, Lam, & Collier-Crespin, 2003). Maladaptive behaviors such as self-injury, aggression, and uncooperative behaviors have the potential to interfere with daily functioning

(Taylor & Seltzer, 2011).

4 Academic performance. In terms of academic performance, about 3% of students with

ASD are considered intellectually gifted (Charman, Pickles, & Simonoff, 2011), while between

28-70% of individuals with ASD are co-morbidly diagnosed with an ID (Matson & Shoemaker,

2009). To date there is more research focused on the social communication and language capabilities of students on the spectrum rather than the academic performance of these students

(Barnett & Cleary, 2015). One such study identified two subgroups of student’s performance: a higher-functioning group and a lower-functioning group (Stevens, Fein, & Dunn, 2000). Other studies focus specifically on the students who demonstrate high achievement in academics, referred to as hyperlexia and hypercalulia (Estes, Rivera, Bryan, Cali, & Dawson, 2011; Jones et al., 2009). Children who demonstrate higher word recognition skills in comparison to their reading comprehension skills based on their level of intellectual ability are considered hyperlexic

(Silberberg & Silberberg, 1967). Hypercalculia refers to exceptional arithmetic functioning in relation to intellectual ability (Aagten-Murphy, Attucci, Daniel, Klaric, Burr, & Pellicano, 2015).

When studying a group of children with ASD, Grigorenko, Klin, Pauls, Senft, Hooper, and

Volkmar (2002) identified discrepancies in abilities resembling hyperlexia in 20.7% of participants with ASD. In a sample of 100 participants with ASD, Jones and colleagues (2009) found that 16% of participants demonstrated mathematic abilities in line with hypercalculia.

While hyperlexia (20%), hypercalulia (16%) are both possibilities for students with ASD, the larger proportion of students (between 28-70%) are co-morbidly diagnosed with an ID

(Grigorenko et al., 2002; Jones et al., 2009; Maston & Shoemaker, 2009). Students with ID demonstrate differences in the realms of intellectual and adaptive functioning (APA, 2013).

Studies have shown that students with lower Intelligence Quotient (IQ) scores actually demonstrated an increased severity of autistic symptoms, such as higher rates of repetitive and/or

5 ritualistic movements and self-injury (Bartak, & Rutter, 1976) and repetitive and restrictive activities (Deb & Prasad, 1994).

Academic Placement and Evaluation of Students with ASD

Autism is a spectrum disorder that encompasses a range of individuals with a variety of capabilities and deficits (Centers for Disease Control and Prevention [CDC], 2012; Wing, 1997).

Some students with ASD will be required to complete state-standardized tests, based on the

Common Core State Standards, with their peers each year (Knight, Smith, Spooner, & Browder,

2012). Others will complete state standardized tests with individualized accommodations (i.e. read aloud, small group setting, frequent breaks). A third group of students will complete the

Alternate Assessment, an assessment based on state Extended Academic Standards. Students taking the Alternate Assessment may be completing a modified curriculum within their classrooms and are more likely to have a co-morbid diagnosis, such as ID (Thompson,

Quenemoen, Thurlow, & Ysseldyke, 2001).

While students in this third category may not be interested in pursuing post-secondary education, they may have vocational interests. Access to high-quality mathematics instruction leads to more positive math experiences (NCTM, 2000). Positive math experiences increase the likelihood that a student will acquire the skills necessary to obtain and maintain employment in the future (American Association for the Advancement of Science [AAAS], 1990). The ability to find and keep employment strongly influences an individual’s future aspirations, such as independent living and financial stability and overall quality of life (Fleming, Fairweather, &

Leahy, 2013; Gerhardt & Lainer, 2011). The goal of education is to provide students with the knowledge and confidence necessary to become adults who are able to participate in and contribute to society in a way that is within their capabilities. A basic understanding of

6 functional and foundational math concepts as well as mathematical reasoning is crucial to the successful acquisition of a career or entry into a post-secondary training program for students with disabilities (Xin et al., 2005).

Wei, Yu, Shattuck and Blackorby (2017) found a positive correlation between access to science, technology, engineering, and math content (STEM) and the likelihood of pursuing a

STEM-related field in post-secondary education. Even if post-secondary coursework is not the aspiration of a student with ASD, the access and availability of high-quality mathematics instruction will lead to a greater knowledge of content necessary to be successful later in life

(Wilkins, 2000; Wilkins & Ma, 2003). Students with ASD and ID, or other co-morbidity that affects cognition, need access to high quality instruction focused on the transition goals that will help with entry into a future training program, internship, or job (Nuehring & Sitlington, 2003).

Ensuring these skills and experiences happen within the K-12 setting is ideal. Transitions are difficult for students with ASD so it is critical that core academic content be presented with intent and purpose prior to this major shift to post-secondary adult life (Hendricks & Wehman,

2009).

Post-secondary Transition and Outcomes for Students with ASD

Transition throughout the typical school day (i.e. from class to class, activity to activity or teacher to teacher) can be challenging for students with ASD despite the efforts of instructors to provide structure and routine (Hume, Sreckovic, Snyder, & Carnahan, 2014). Due to a difficulty understanding verbal directives, students with ASD may not grasp all information being given during a transition, thus adding to their feeling of anxiety (Mesibov, Shea &

Schopler, 2005). In addition, students with ASD may completely miss a subtle transition signal

(e.g. packing up materials) (Carnahan, Hume, Clarke, & Borders, 2009) or may have difficulty

7 attending to simultaneous stimuli or cues (Marco, Hinkley, Hill, & Nagarajan, 2011). The transition from high school to adult-life can be even more daunting for a student with ASD. A transition of this magnitude could include a new environment in which to live and work, a new set of people to live and work with, and new rules and schedules to follow.

The Individuals with Disabilities Education Act (IDEA) data from 2010, indicates that

419,262 individuals with ASD between the ages of 3-21 received services under Part B of IDEA.

This figure is nearly double that of the 2005 report (IDEA; U.S. Office of Special Education

Programs, 2012). Individuals with ASD make up 6.39% of the population served under IDEA

Part B (receiving services under local public school systems) (U.S. Office of Special Education

Programs, 2012). While the number of children with ASD continues to rise, the employment outcomes for individuals with ASD remains stagnant (Burgess & Cimera, 2014). Several statistics are currently available pertaining to the rate of employment and enrollment in postsecondary education for individuals with ASD. Using a National Longitudinal Study

(NLTS2) sample, Shattuck, Wagner, Narendorf, Sterzing, and Hensley (2011) reviewed service use (such as Vocational Rehabilitation) among young adults with ASD. Based on this sample, the following was discovered: 32% attended postsecondary schools, 6% maintained competitive employment and 21% had no employment or education experiences. In terms of independent living and functioning, 80% of the individuals lived at home with their parents and 40% reported having no friends (Shattuck et al., 2011). Individuals with ASD had the second lowest rate of employment at 63.2% (behind peers with multiple disabilities with 62.5%) since exiting a high school program (Newman et al., 2011). Upon review of a group of 66 young adults with ASD who exited school between 2004 and 2008, Taylor and Seltzer (2011) found that 14% had earned a postsecondary degree, 6% had attained competitive employment, 12% had supported

8 employment, 56% participated in sheltered workshops and 12% had no regular daytime activities. While these statistics differ slightly, they paint a dismal picture for individuals with

ASD transitioning from high school to post-secondary education or employment. Even more reason educators, school personnel, administrators, and researchers should strive to ensure that high quality math instruction is being provided to all students prior to their transition from the K-

12 setting to the adult world. The following paragraphs describe what an ideal instructional program for students with ASD should include.

Mathematics Instruction for Students with ASD

The educational program for children with ASD should include the following: (1) individualized supports and services for students and families, (2) systematic instruction, (3) comprehensible and/or structured environments, (4) specialized curriculum content, (5) a functional approach to problem behaviors, and (6) family involvement (Iovannone, Dunlap,

Huber, & Kincaid, 2003). The following evidence-based strategies for students with cognitive disabilities are a great starting point for building an instructional program for students with ASD:

(a) systematic instruction with prompting (i.e. system of least prompts, constant time delay, model-lead test, and simultaneous prompting) and feedback, (b) task-analytic instruction, and (c) in-vivo, also known as instruction provided in the natural environment (Hudson, Rivera, Grady,

2018).

To date, the research available on interventions for students with ASD highly favors the social communication deficits demonstrated by this population (Plavnik & Ferreri, 2011). Since adaptive behavior and social communication is a prominent area for concern (APA, 2013), it makes sense the majority of studies focus on this aspect. The majority of studies evaluating academic interventions take aim at language arts and reading skills (Bouck, Satsangi, Taber-

9 Doughty, & Courtney, 2014) while the math-oriented studies trail behind (King et al., 2016).

Math intervention studies tend to focus on the ability to compute, rather than problem solve

(Bae, Chiang, & Hickson, 2015). Browder, Spooner, Ahlgrim-Delzell, Harris, and Wakeman

(2008) conducted a meta-analysis to uncover studies focused on teaching mathematics to students with moderate to severe disabilities and discovered that only two out of 68 studies

(published between 1975 and 2005) covered problem-solving. Even then, the strategies that did address word problem solving did not targeted the deficits exhibited by students with ASD, such as executive functioning and metacognition (Browder et al., 2008). It is apparent that due to the dearth of empirical studies in the area and the implicit need related to ASD core-deficits, students with autism would benefit from specially-designed instruction targeting mathematics word- problem solving (Root, Browder, Saunders, & Lo, 2017). The next several paragraphs will describe the word-problem solving process and shed light on a promising practice for teaching word-problem solving to students with ASD.

Word-problem solving. According to Mayer’s 1985 model, word-problem solving consists of four phases, including: problem translation, problem integration, solution planning, and solution execution. Translation involves the construction of meaning from the problem. The student in this case, must understand what is happening in the problem in order to identify what is “known” and “unknown” in the problem (Root et al., 2017). Problem integration requires students to distinguish necessary and unnecessary information within the problem and place the necessary information into a mathematical structure (i.e. number sentence or equation). The third step, solution planning, involves the choice of the correct operation for the problem type.

Finally, a solution is computed during the final phase, problem execution (Mayer, 1985; Root et el., 2017).

10 Teaching students with ASD to solve word problems, in general, is difficult. Students with ASD have difficulty with most aspects of his Mayer’s (1985) 4-phase framework. Deficits in reading and language make problem translation tricky, while working memory issues make integration a challenge for students with ASD. Planning and execution of word problems is nearly impossible if students with ASD have limited math skills (Root et al., 2017).

Schema-based instructional strategy. SBI is an intervention that has been proven effective when teaching word-problem solving to students with learning disabilities (Fuchs,

Fuchs, Prentice, Hamlett, Finelli, & Courey, 2005; Jitendra et al., 2009). When used with students with high incidence disabilities, SBI is considered an evidence-based practice (Gersten,

Chard, Jayanthi, Baker, Morphy, & Flojo, 2009; Jitendra et al., 2015). A schema is a framework often represented with pictures, diagrams, number sentences, or equations to aid students’ organization of their problem-solving processes (Marshall, 1995). Gick and Holyoak (1980) described a schema as a description that can be used to group similar problem types together for easier solving in the future. Regardless of the definition or description chosen, a schema refers to a framework or categorical tool used to make problem solving more efficient.

SBI strategy emphasizes conceptual understanding of the structure of the problem (Xin &

Jitendra, 1999). In other words, a strategy that teaches pattern recognition (Xin, 2008) to help solve similar problems encountered later. There are three essential components of SBI, including: (a) identifying the structure of the problem so one can choose the problem-type, (b) using visual representations of the problem structure, to again assist with the selection of problem-type and also to organize information from the problem, and (c) explicitly teaching how to use the SBI problem-solving method (Jitendra et al., 2015). Several research teams have successfully implemented a modified SBI strategy with ASD-specific supports in an attempt to

11 teach word-problem solving skills to students with ASD (Browder et al., 2018; Rockwell,

Griffin, & Jones, 2011; Root et al., 2017).

While research focused on instructional strategies to promote word-problem solving capabilities in students with ASD is on the rise, there are presently students with ASD sitting in classrooms who could likely benefit from the implementation of an evidence-based strategy, such as SBI, with modifications to specifically target the deficits they encounter as a result of their ASD diagnosis. Special educators and special education researchers should be encouraged to think creatively, to try new things, and to let current trends in research inspire them to investigate a new strategy for their students.

Purpose Statement

The current study will examine the effects of a modified SBI strategy with additional

ASD-specific supports on the addition and subtraction word problem solving abilities of three students with ASD. This research will add to the growing, but limited, research available focused on word-problem solving interventions for students with ASD. The following questions guided this study.

Research Questions

(1) Is SBI an effective strategy for teaching addition “group-type” word problem solving skills to students with ASD?

(2) Is SBI an effective strategy for teaching subtraction “compare-type” word problem solving skills to students with ASD?

(3) Is the SBI strategy with ASD-specific supports a socially valid intervention?

12 Literature Review

Students with Autism Spectrum Disorders

First identified as a disorder by Kanner in 1943, autism was described as a disorder including several distinct characteristics: (a) an inability to develop relationships with others, (b) a delay in speech acquisition, (c) non-communicative speech use, (d) echolalia, (e) repetitive and stereotypical play activities, (f) insistence on routine, (g) good rote memory, (h) lack of imagination, and (i) typically a normal physical appearance. Interestingly, autism is a disorder that has the potential to present itself even after a child of one or two years has already gone through typical development (Kanner, 1943; Lotter, 1966; Rutter, Greenfeld, & Lockyer, 1967).

With the publication of the Diagnostic and Statistical Manual of Mental Disorders Third Edition

(DSM-III) in 1980, autism became its own diagnostic category (APA, 1980). This new description categorized autism as one of a group of pervasive developmental disorders (PDD’s).

A PDD is a developmental disability that pervades all aspects of a child’s life (Tidmarsh &

Volkmar, 2003). The updated DSM-IV definition included the other PDD’s along with autistic disorder, including Rett’s Disorder, Childhood Disintegrative Disorder (CDD), Asperger’s

Disorder (AD), and PDD Not Otherwise Specified (NOS) (APA, 2000). The fifth and most recent revision of the DSM eliminated the various subtypes of PDD and collapsed them all into one single diagnosis of ASD (Lord & Bishop, 2015). This revision was made in large part because research had discovered inconsistency in the diagnoses of the various subtypes

(especially Asperger’s disorder and PDD-NOS) across clinicians (Lord et al., 2012) or across time (Lord, Risi, DiLavore, Shulman, Thurm, & Pickles, 2006).

Students with autism make up approximately 7.6% of students with disabilities in the

United States, according to a United States Department of Education (USDOE) report (USDOE;

13 2014). A recent estimate indicates about one in every 59 children are diagnosed with ASD

(Baio, et al., 2018). Autism refers to a set of neurodevelopmental conditions typically characterized by deficits in executive functioning, social interaction, and communication (APA,

2013). The cognitive and functional profile of each child with an ASD is different, with approximately 50% co-morbidly diagnosed with an intellectual disability (APA, 2013).

Individuals with ASD do commonly share three distinct characteristics, regardless of co- morbidity; these are weak central coherence, impaired executive functioning, and limited or weak theory of mind.

Central coherence. As discussed by Frith (1989), central coherence refers to the tendency of typically developing children and adults to search for meaning. First described by

Barlett in 1932 as a “drive for meaning”, the term “central coherence” was later coined by Frith

(1989). Individuals with autism are considered to have weak central coherence, otherwise known as a detailed-focused processing style. This style of processing makes it difficult for individuals with ASD to see the “big picture” or to “get the gist” of a situation (Happe & Frith,

2006). A weak central coherence can naturally lead to problems with skill generalization. By only focusing on the details of a situation, an individual with ASD would find it challenging to categorize two scenarios as similar if they do not share common key details. Thus making it nearly impossible for skill generalization to occur since scenarios are rarely identical (Happe &

Frith, 2006; Rincover & Koegel, 1975).

Theory of mind. Additionally, individuals with autism have a difficult time separating their own thoughts from the thoughts of others (Baron-Cohen, Leslie, & Frith, 1985), which is oftentimes referred to as a limited theory of mind. Theory of mind (ToM) was originally described as an inability or difficulty in attributing mental states to others and to oneself

14 (Premack & Woodruff, 1978). The ability to represent or understand the mental states of others

(especially desire and belief), allows one to predict and understand the behaviors of others. Thus individuals with ASD, who may have a limited ToM, might struggle with explaining everyday behaviors (Frith, 1994).

ToM typically develops in children between the ages of three and five years old (Morgan,

Mayberry, & Durkin, 2003). Children at this age will start to understand that a person can hold a belief that is different from their own, otherwise known as a false belief (Baron-Cohen, 1991).

ToM deficits help to explain the social communication difficulties often experienced by children with ASD, specifically: relating to others, predicting behaviors, and empathy (Baron-Cohen,

1995b). Some children diagnosed with ASD do develop some ToM skills but to a lesser extent than their peers without ASD (Happe & Frith, 1996; Tager-Flusberg & Sullivan, 1994). Often referred to as “mind-blindedness”, a missing or limited theory of mind does not explain all assets and deficits associated with an ASD diagnosis, though it does shed some light on social communication difficulties experienced by children with ASD.

Executive functioning. The inflexible or perseverative nature oftentimes observed in individuals with autism can be attributed to deficits in executive functioning, a third major commonality found throughout the spectrum disorder. Executive function is the ability to maintain an appropriate problem-solving set to attainment of a future goal; including planning, impulse control, and flexibility of thought and action (Ozonoff, Pennington, & Rogers, 1991). In individuals with autism, who may experience deficits in executive functioning, this can appear as rigidity and inflexibility (Ozonoff et al., 1991). In 1985, Rumsey first compared the performance of a group of adult men with autism and without autism on the Wisconsin Card Sorting Test.

The group of men with autism demonstrated deficiencies in number of categories achieved,

15 completion of a planned task, as well as a higher level of perseveration (Rumsey, 1985). The results all correspond with the traits often attributed to ASD: narrow and repetitive focus of interest, limited goal-orientation or future planning, and difficulty with self-reflection (Ozonoff et al., 1991).

Working memory, one aspect of executive functioning, has received the most attention in the academic literature (John, Dawson, & Estes, 2017). Working memory refers to the ability to hold and manipulate information inside one’s mind (John et al., 2017) and is associated with a range of math skills from basic (e.g. counting) to more complex (e.g. word problem solving)

(Raghubar, Barnes, & Hecht, 2010). Working memory can be divided into four main components using Baddeley’s model: central executive, phonological loop, visuospatial sketchpad, and episodic buffer (Baddeley, 1986). Connections have been made between poor mathematical skills and central executive deficits (Holmes & Adams, 2006; Swanson & Kim,

2007).

Metacognition. Additional impairments in metacognition (Lombardo & Baron-Cohen,

2011) and attention (May, Rinehart, Wilding, & Cornish, 2013) are of also of note when considering the academic abilities of students with ASD. Metacognitive monitoring skills (i.e. knowing what you know) are necessary for daily functioning (Roebers, Krebs, & Roderer, 2014) and are oftentimes lacking in students with ASD (Grainger, Williams, & Lind, 2014). In turn, students with ASD may not possess the confidence to judge what they do and do not know; a fact that some researchers feel is the true root of social difficulties experienced by this population of students (Sawyer, Williamson, & Young, 2014). Research in this area has determined that interventions related to metacognition are effective for students within or slightly below the

16 normal range for mathematical ability (Iuculano et al., 2014; Schneider & Artlet, 2010) which typically includes a large number of students with ASD.

Social communication. The definition of an autism spectrum disorder clearly identifies a deficit in language and social interaction as a determinant for diagnosis in addition to restricted and repetitive behaviors (APA, 2013). While both factors are necessary for a diagnosis, it is the impairments in social communication that have the potential to create challenges in other areas, including: academic performance, relationship-building, and maintaining socially appropriate behaviors (Bauminger & Kasari, 2000; Koegel et al., 1992). As stated previously, one of the main components of social communication is that it is intentional and directed toward another person (Fuller & Kaiser, 2019). Joint attention, behavior regulation, and social initiation are all reliant on social communication skills. Joint attention, or the sharing of attention with another around a point of reference (Vivanti, Fanning, Hocking, Sievers, & Dissanayake, 2017), is a precursor to expressive communication in students with ASD (Fuller & Kaiser, 2019). Studies have shown that early intervention targeting joint attention (Charman, 2003; Mundy, Sigman, &

Kasari, 1990) and social communication (Kasari, Gulsrud, Wong, Kwon, & Locke, 2010) can lead to better long-term expressive language outcomes for students with ASD.

Mathematics Performance of Students with ASD

Increasingly research teams are dedicating man-power to exploring the range of mathematics interventions accessible to students with autism spectrum disorders. The overall mathematics achievement profile of individuals on the autism spectrum is subject to high rates of variability (King et al., 2016). When examining growth trajectories of children identified for special services, Wei, Lenz, and Blackorby (2013) found that students with ASD performed worse on calculation and applied math problems than students with learning disabilities. In

17 2015, Wei, Christiano, Yu, Wagner, and Spiker conducted a longitudinal analysis of elementary children with ASD; the results highlighted the uncertain nature of math achievement for students with ASD, with only 20% exhibiting average or above-average skills in mathematics.

Though mathematics is often considered to be an area of strength for students with ASD

(Chiang & Lin, 2007; Iuculano et al., 2014), there are specific areas that have not yet been assessed, such as basic numerical competencies (Hiniker, Rosenberg-Lee, & Menon, 2015) and problem solving abilities (Bae et al., 2015). King and colleagues (2016) found an extremely targeted range of skills addressed by the interventions included in their review of the literature.

The majority (39%) of interventions targeted computational skills, such as addition and subtraction (Adcock & Cuvo, 2009). Mathematics fluency was addressed in 14% of cases, followed by early numeracy and problem solving which was targeted by only 7% of studies

(Levingston, Neef, & Cihon, 2009; Rockwell et al., 2011). Based on the numbers, it is evident that more research needs to be conducted that specifically targets problem solving because it is the source of the greatest struggle for students with ASD.

Certain characteristics can help explain the challenges with solving word problems experienced by students on the spectrum. Executive functioning deficits, such as planning, organization, working memory, mental flexibility, attention, self-monitoring, and impulse control could lead to mathematics difficulties for students with ASD (Happe et al., 2006). Additionally, reading and language impairments can diminish the overall understanding of mathematics word problems (Donlan, 2007; Whitby & Mancil, 2009), making it difficult for students with ASD to fully make sense of “what is happening in the problem” (Root et al., 2017, p.42). These characteristics are present across the broad spectrum of the disability; exhibited in both individuals with higher functioning autism and more moderate to severe autism.

18 Mathematics performance of students with autism and intellectual disability. It is estimated that as many as half of the individuals diagnosed with ASD have a co-morbid diagnosis of ID (Matson & Shoemaker, 2009; APA, 2013). Students with ID by definition perform below average on intellectual assessments and tasks (APA, 2013). When co-morbidly diagnosed with ASD and ID, an individual typically exhibits some or all of the core deficits of

ASD (deficit in social communication and repetitive or restricted behaviors) while also displaying lower than average performance on academic endeavors. Autism is a spectrum disorder, categorized by a cognitive heterogeneity, which may explain the inconsistent nature of the research surrounding math achievement profiles (Georgiades, Szatmari, & Boyle, 2013). The following paragraphs will detail the highly variable nature of mathematics performance of students with ASD and ID in various domains.

Number sense. Multiple studies have attempted to address basic numeracy skills of students with ASD and ID. Estes and colleagues (2011) found that children, aged nine years, with ASD demonstrated below average basic numeracy skills when compared with their typically developing peers. When analyzing the early number processing skills of students with

ASD, aged four and five years, Titeca, Roeyers, and Desoete (2017) found average early numeracy skills in the group with ASD.

Numerical operations. Numerical operations include all four numerical competencies

(i.e. addition, subtraction, multiplication, and division) and are an important building block to future mathematics success and post-secondary outcomes (Brown & Snell, 2000). A longitudinal analysis performed by Wei and colleagues (2013) suggested that students with ASD perform significantly worse than their counterparts with learning disabilities on calculation.

19 Math reasoning. There are several subsets of mathematical reasoning (e.g., additive, multiplicative, algebraic) that have been identified and reviewed by research teams. Jones and colleagues (2009) found that on average, adolescents, aged 14-16 years, with ASD demonstrated below average math reasoning abilities. Iuculano and colleagues (2014) found average mathematics reasoning abilities in children with ASD when compared to their normed peers.

This discrepancy in abilities may again be due to the diverse nature of the disorder.

Word-problem solving. The word-problem solving capabilities of students with ASD and ID, to date, are scarcely documented in academic literature (Bae et al., 2015). Wei and colleagues (2015) found that children with ASD between the ages of six and nine had below average word-problem solving abilities when compared to their peers. Conversely, Troyb and colleagues (2014) found that students with ASD possessed average problem-solving capabilities.

When compared with students with learning disabilities, Wei and colleagues (2013) found that students with ASD perform worse on applied problems. Some of the challenges associated with word-problem solving may stem from inherent deficits in reading comprehension, often experienced by students with ASD and ID (Whitby & Mancil, 2009). Additionally, the extent of the student’s mathematics vocabulary may also affect their ability to decipher the pertinent information within the problem (Bae et al., 2015).

Mathematics performance of students with high functioning autism. A separate sub- set of individuals with ASD without ID exists and they have commonly been referred to as “high functioning”. Students with HFA tend to score within the average range on standardized mathematics assessments, which is lower than expected based on their cognitive abilities

(Chiang & Lin, 2007; Troyb et al., 2014). In general, students with HFA experience some success in the early grades when rote skills are often emphasized, but experience more

20 difficulties in middle school when the content is focused more on problem solving and higher level thinking (Chiang & Lin, 2007). For example, students with HFA typically display average skills in counting, numerical operations, and facts, but are challenged when asked to think critically or use analytical skills to solve higher-order problems (Siegel et al., 1996). These students also tend to have more success with memorization of vocabulary and more difficulties with arithmetic skills (Williams, Goldstein, Kojkowski, & Minshew, 2008). In critical thinking situations, students with HFA tend to need specialized, strategic interventions in mathematics to address their needs associated with deficits in executive functioning, social interaction, and communication (APA, 2013; Barnett & Cleary, 2015).

Math Interventions for Students with ASD and ID

There is currently a wealth of research on social communication interventions for students with ASD (Barnett & Cleary, 2015) since this is often seen as a core deficit of the disorder. Research on interventions involving language and reading are less common than social communication interventions but are more prevalent than research involving math instruction and intervention for students with ASD (Barnett & Cleary, 2015). Research examining the effectiveness of mathematics interventions for students with ASD is beginning to gain traction, garnering the attention of more research teams (King et al., 2016). Some teams are even utilizing strategies and interventions that have been used for social communication skills (i.e., video modeling) and others that have been used with students with learning disabilities (i.e., schema-based instruction) to introduce basic and complex math concepts to students with ASD

(King et al., 2016). The following pages will identify some of the work that has been done by researchers in the field of special education, specifically highlighting the strategies introduced

21 and research designs utilized to teach and examine the effectiveness of various mathematics interventions.

Video modeling. Video modeling (VM) refers to an instructional strategy that involves the recording of an adult or student peer performing a task or skill. The learner will watch this recorded video with the intention of performing the task or skill in the future (Cihak, Fahrenkrog,

Ayres, & Smith, 2010). An alternative, sometimes referred to as video-based modeling (VBM), provides an exemplar for the learner to view through the integration of core technology into instruction (Ayres & Langone, 2005). Another facet of video modeling is video self-modeling

(VSM) which involves the learner recording himself or herself performing a task or skill and later watching this recorded video with the intention of performing the task or skill (Dowrick,

1999). A third version of VM is point-of-view modeling which records the task performance from the perspective of the learner (i.e. the learner is not in the video) (Shukla-Mehta, Miller, &

Callahan, 2010). The use of video modeling with students with autism has been well documented (Ayres & Langone, 2005; Mason, Rispoli, Ganz, Boles, & Orr, 2012; Shipley-

Benamou, Lutzker, & Taubman, 2002) as an evidence-based practice regarding behavior and academic functioning. Several research teams have explored the use of VM during mathematics instruction or problem solving.

Burton, Anderson, Prater, and Dyches (2013) used a multiple baseline across-participants design to determine whether VSM had an effect on math skill acquisition of adolescents with autism. Four adolescent boys with autism viewed videos of themselves solving math problems involving estimation and making change. The researchers did find a functional relationship between the viewing of the video self-model and math performance for all four participants. Of methodological note was the researchers’ use of intervention fading during the maintenance

22 phase followed by once weekly probes post-maintenance. Additionally, the first author was able to train two para-educators to help with the baseline and training phases. In keeping with single case design procedures, data was analyzed visually and treatment fidelity was monitored with a checklist used by the teacher and para-educators. Social validity was assessed via four-question open-ended surveys given to the para-educators and student participants.

Yakubova, Hughes, and Hornberger (2015) also used a VM intervention to teach word problem solving involving the subtraction of mixed fractions with uncommon denominators. A multiple-probe across students design was used to evaluate the intervention with a one-week follow up maintenance session. The authors used permanent products to review the progress made by participants as well as visual analysis. Both inter-observer and procedural reliability was checked, during 30% of each phase and 20% of the intervention phase, respectively. While

VM is a very powerful tool for students with ASD, other strategies exist to teach new concepts or reinforce previously learned material.

Touch points. The Touch Point strategy is a dot-notation strategy that was first used to teach math skills to students with disabilities in 1973 (Kramer & Krug, 1973). In 1989, Bullock,

Pierce, and McClellan created a curriculum based on the dot-notation method that addressed all four arithmetic operations. Within the Touch Point curriculum, the numbers one through five contain one touch point each and the numbers six through nine contain double touch points in each. Touch points appear as dots within circles inside each of the printed numbers. Students are directed, using an embedded instructional-prompt approach, to touch each point and double point as they count aloud in order to help them with computations (Cihak & Foust, 2008). To date, the Touch Point strategy has been used to successfully teach single-digit addition to students with learning disabilities (Simon & Hanrahan, 2004), mild intellectual disability

23 (Kokaska, 1975), and moderate intellectual disability (Pupo & Hanrahan, 2000). Additionally, this strategy has been used to teach double-digit addition with regrouping (Simon & Hanrahan,

2004) and subtraction (Scott, 1993).

Cihak and Foust (2008) used an alternating treatments design to evaluate the effectiveness of a Touch Point strategy on the acquisition of single-digit addition skills in three elementary students with autism. Baseline data was collected in the form of a pre-test containing ten single-digit addition problems. The intervention phase consisted of several modeling sessions when the instructor demonstrated how to use touch points. Next, the students completed several instructional sessions followed by a test with ten single-digit addition problems.

Criterion was met when a student reached 100% accuracy on the math test for three consecutive sessions. Both inter-observer and procedural reliability were examined throughout all phases of the study. Each observer scored the ten question math tests independently. Additionally, teachers were trained on the strategy using a checklist prior to implementation. The researcher- observer recorded and calculated procedural reliability during each session and found it to be at

98% on average. The researchers found that the Touch Points strategy was effective for students with autism and preferred over the number lines strategy, since all students met criterion faster when using Touch Points.

Concrete representational abstract. The Concrete Representational Abstract (CRA) instructional sequence helps students progress through various representational levels. Students begin by using concrete manipulatives (i.e. Unifix cubes) to solve mathematics problems (i.e., single digit addition). Next, students are taught to use representational drawing (i.e., lines and dots) to solve the same problems. Finally, students learn to solve the problems without any supports (Agrawal & Morin, 2016). CRA is built entirely on the premise of explicit (or direct)

24 instruction; the teacher teaches or models how to solve each type of mathematics problem

(Bouck, Park, & Nickell, 2017). Students must reach 80% accuracy before moving from concrete to representational and again from representational to abstract. The CRA instructional sequence is supported by research to be successful when used with students who have difficulties in mathematics (Bouck, Satsangi, & Park, 2018; Flores, 2010; Mancl, Miller, & Kennedy, 2012).

Stroizer, Hinton, Flores, and Terry (2015) used a multiple baseline across behaviors design to examine the effects of the CRA sequence when teaching students with ASD. Three elementary students with autism were selected to participate in the study, which took place during a four week extended school year session. Teacher reviewers examined computation questions created in the skills areas of: addition with regrouping, subtraction with regrouping, and multiplication facts to five, to ensure content validity prior to implementation. Additionally, the Kauffman Brief Intelligence Test – II (KBIT-II) was used to quickly assess verbal and nonverbal intelligence of participants. Probes were administered to students daily throughout baseline and intervention phases. Criterion to change phases (move from one math concept to the next) was six of nine problems correct. The multiple baseline design allowed for fluidity as students progressed through phases at different points. The researchers cited their short intervention time as the reasoning for the less than optimal criterion levels that were set.

Integrity checklists were completed by observing a digital recording of instruction or administration of mathematics probe. Instructional integrity ranged between 81-97% for the three concept phases. Inter-observer agreement was conducted for 100% of probes conducted and integrity checklists completed, yielding scores of 100% and 97%, respectively. Teachers were given a closed and open-ended questionnaire after the study in order to collect social

25 validity scores. The authors found a functional relationship between the CRA sequence and all three math concepts presented in the four-week period.

Schema-based instructional strategy. One strategy used to help teach math problem solving skills is SBI. In 1995, Marshall described a schema as a framework for solving a problem that is often represented with pictures, diagrams, number sentences, or equations. A schema was later described as having three components: (a) identification of problem structure in order to determine type of problem, (b) visually representing the structure to organize information from problem, and (c) direct instruction on schema-based method (Jitendra & Hoff,

1996; Jitendra et al., 2015). SBI strategy has primarily been used with students with mathematics learning disabilities (Gersten et al., 2009). Recently more researchers have been introducing SBI students diagnosed with autism. The following paragraphs will highlight several separate studies, which made use of SBI to teach mathematics concepts to students on the autism spectrum.

Rockwell and colleagues (2011) found a functional relationship between the use of SBI and the word problem solving abilities of the ten-year old participant with autism. The participant had low-average nonverbal skills and below average language skills. She spent the majority of her day in the general education classroom with four hours of pull-out instruction for math each week and the assistance of an in-class aide during reading. The study took place over a period of eight weeks during the summer in a one-on-one tutoring capacity. The authors employed a multiple probe across behaviors design in order to minimize assessment fatigue.

Over the course of the eight weeks, SBI was used to teach the solving of “group” problems

(combining smaller groups to create larger group), “change” problems (beginning amount, change, ending amount), and “compare” problems (larger amount, smaller amount, difference) to

26 the participant. The author used the mnemonic RUNS (1. Read the problem, 2. Use a diagram,

3. Number sentence, and 4. State the answer) during instructional sessions with the participants.

In addition, the participant was taught how to use the SBI strategy charts designed for each problem type. All instructional sessions were video recorded to assess treatment integrity (for

27% of instructional sessions).

Root and colleagues (2017), used SBI, modified for students with ASD, in order to teach

“compare-type” word problems to three elementary students with ASD and ID. The authors used a multiple probe across participants design with an alternating treatments design embedded within to evaluate the effects of the SBI. Additionally, the authors compared the effects of virtual and concrete manipulatives, with two out of three participants performing more steps within the virtual condition. A functional relationship was demonstrated between the use of the modified SBI and ability to complete “compare-type” word problems. Of specific interest, was the way the authors modified the SBI to meet the needs of elementary-age students with ASD and ID (i.e. manipulative use, systematic prompting and feedback, discrimination training to discern problem type, color-coding documents, and picture representation when possible).

Students were strategically trained on how to identify the problem type, circle the “whats” that were being compared, use the rule that applied to the identified strategy, circle the numbers in the word problem, complete the number sentence based on their prior work, choose the correct symbol, make sets, and solve. A three-day training period was implemented during which time the instructor/author modeled each step of the SBI with active student participation and no data collection.

Similarly, Browder and colleagues (2018) used SBI with embedded effective practices

(i.e. graphic organizers, systematic prompting, and task analysis with pictures) to teach addition

27 and subtraction word problems to students with moderate intellectual disabilities. Students with autism are co-morbidly diagnosed with an ID, in approximately 50% of all diagnoses (APA,

2013), thus making the modifications used by this group of researchers applicable to a study involving students with autism. In this case, the researchers used a multiple probe across student dyads to examine problem discrimination, solving, and generalization skills in eight elementary and middle school students with moderate ID. These authors used a model-test-lead approach when implementing their intervention. The researcher spent two days as the “lead” during the modeling phase, the teacher led for one day in the “testing” phase and the student took the “lead” during one day of leading phase. The leading phase included a system of least prompts with feedback and error correction from the researcher. The researcher used a script during this phase to standardize any language being used. Twelve steps were identified in the problem-solving task analysis and mastery was set at eight out of twelve steps performed independently. Once mastery was reached, the student moved into the maintenance phase, which involved a problem- solving test one week from the completion of their final phase assessment.

Based on the research of the aforementioned teams, it appears that a modified SBI strategy could be used to teach students with ASD and ID a system for solving addition and subtraction math word problems. The previous work done by the research teams (Browder et al.,

2018; Rockwell et al., 2011; Root et al., 2017) provides promising evidence that SBI can be modified to meet the unique needs of students diagnosed with ASD and ID. The current study intends to build off the foundational work of these research teams.

Problem Statement

To date there is limited research available focused on math word problem solving interventions for students with ASD (Barnett & Cleary, 2015, King et al., 2016). The word

28 problem solving capabilities of students with ASD is not well documented, with most studies tending to focus on rote skills and computation (Bae et al., 2015; King et al., 2016). This study will add to the literature currently available in the field by examining the use of a SBI strategy, commonly used with students with learning disabilities, with a small group of students diagnosed with ASD. Secondarily, this study aims to investigate the social validity of ASD-specific supports used in addition to a modified SBI strategy. The following research questions informed this study.

Research Questions

(1) Is SBI an effective strategy for teaching addition “group-type” word problem solving skills to students with ASD?

(2) Is SBI an effective strategy for teaching subtraction “compare-type” word problem solving skills to students with ASD?

(3) Is the SBI strategy with ASD-specific supports a socially valid intervention?

29 Methodology

The purpose of this research was to examine the effectiveness of a modified schema- based instructional strategy when teaching addition and subtraction word problem-solving to elementary students with ASD. A multiple probe design across participants was used to measure intervention effect.

Participants

Three elementary students with ASD between the ages of nine and 11 participated in the study. All participants had a diagnosis of ASD and were academically multiple grade levels behind their peers. All three students were participating in a classroom for students with academic, behavioral, and communication needs. The classroom was the central meeting place for about ten students who were often coming and going from one class to the next, oftentimes with the assistance of a para-educator. The participants spent the majority of their day in this classroom, aside from homeroom, specials, and one course required by the specific school. Two of the participants were male and one was female; one male participant was in the fifth grade while the other male and the female participant were in the fourth grade. Two of the participants

(one female and one male) were Latino and siblings, the third participant was African American.

The two Latino students used English as their primary language at school but spoke both Spanish and English at home. Participants were recruited by the school administer and were referred to this study because they were all struggling with math word problem solving, were able to verbally consent to participate, and were all in the age range of seven-11 years old. Parents of each participant provided verbal and written consent to participate and each participant provided their own verbal and written assent to the researcher prior to the start of the intervention.

30 Records review. Participant records, including Individual Education Programs (IEP’s) and Evaluation Team Reports (ETR’s), were reviewed in order to create a more comprehensive profile of their skills and abilities. The same school psychologist at the current school setting evaluated two of the participants. One participant attended a different school during his most recent ETR and therefore, was evaluated using different measures by a different psychologist.

The psychologist at the current school setting used two different assessments when evaluating two of the participants: the Woodcock Johnson IV Test of Achievement (Schrank, McGrew, &

Mather, 2014) and the Adaptive Behavior Evaluation Scale – 2nd Edition (McCarney & Arthaud,

2006). The psychologist at the alternate school setting used two different assessments to evaluate one participant: the Universal Test of Nonverbal Intelligence (Bracken & McCallum,

1998) and the Vineland Adaptive Behavior Scales Second Edition (Sparrow, et al., 2005).

Descriptions of these assessments can be found below. All physical copies of student records that were obtained by the researcher with school and parent permission were stored in a locked drawer inside the researcher’s locked office at the conclusion of the study.

Woodcock Johnson IV. The Woodcock-Johnson IV Tests of Cognitive Abilities (WJ IV

COG) are used to measure the psychometric intellectual abilities for individuals between the ages of two-90+ (Reynolds & Niileksela, 2015). Originated by Richard Woodcock, the measure is now in its fourth revision and now contains three batteries (cognitive, oral language, and achievement). The WJ IV COG is individually administered and contains 18 tests organized into various clusters based on skill and description (Reynolds & Niileksela, 2015).

Adaptive Behavior Evaluation Scale Revised Second Edition. The Adaptive Behavior

Evaluation Scale Revised Second Edition (ABES-R2) is a measure of adaptive skills that can be used with students experiencing academic or behavioral difficulties (McCarney & Arthaud,

31 2006). The ABES-R2 assesses adaptive skills in ten areas that are categorized under three domains: Conceptual, Social, and Practical. The ABES-R2 can be completed within twenty minutes by anyone familiar with the student (teacher, clinical personnel, school personnel, or parent/guardian) (McCarney & Arthaud, 2006).

Universal Test of Nonverbal Intelligence. The Universal Test of Nonverbal Intelligence

(UNIT) measures a child’s cognitive abilities without using any verbal language. In this way, the assessment offers a fair evaluation of students, regardless of language skills, hearing, or

English language proficiency. The UNIT includes six subtests, including: Symbolic Memory,

Object Memory, Analogic Reasoning, Spatial Memory, Cube Design, and Mazes (Bracken &

McCallum, 1998). This assessment is used with children ages five-through-seventeen.

Vineland Adaptive Behavior Scales Second Edition. The Vineland Adaptive Behavior

Scales Second Edition (Vineland-II) is a measure of personal and social skills. The assessment is meant to be used with individuals who are birth-90 years of age. The Vineland II is organized into three domains, including: Communication, Daily Living, and Socialization (Sparrow et al.,

2005).

Participant characteristics. The three participants that participated in this study had impairments in academic, social, and adaptive functioning. All participants received speech language services and occupational therapy through their school. All participants used verbal language as their primary mode of communication. While some had a more expanded vocabulary; all participants could be heard and understood (through verbal speech) by the researcher. Communication between the researcher and participants was done only through verbal speech, as was standard protocol for this particular classroom.

32 Zachariah. Zachariah was in the fourth grade and the first participant to enter intervention. He had a diagnosis of ASD from a large Midwestern Children’s Hospital Medical

Center at the age of three. Zachariah was an independent worker who liked to “earn” drawing time after successful completion of his work. Zachariah was a very imaginative student who would draw Anime and Manga characters from memory or create entirely new characters. He also had an intense fascination with dinosaurs and would draw incredibly vivid pictures of them with pen and colored pencil. During the study implementation year, Zachariah’s IEP goals were focused on: (a) reading sight words within first grade level passage independently, (b) composing one-three sentences with correct mechanics, (c) completing math problems related to time, money, and subtraction, (d) responding appropriately to “wh” questions, and (e) improving his handwriting and typing skills. Zachariah was given the WJ IV COG (Schrank et al., 2014) and ABES-R2 (McCarney & Arthaud, 2006), during his most recent ETR, the results can be found in Table 1.

Table 1.

Zachariah’s Recent Evaluations

Test Subtests Standard Score Percentile Rank

Woodcock Johnson IV Letter Word 53 <.1 Identification Passage 59 .3 Comprehension Applied Problems 41 <.1 Calculation 72 3

Adaptive Behavior Evaluation Scale – 2nd Edition Conceptual 79 8 Social 84 15

33 Practical 84 15 Adaptive Skills 81 11 Quotient

Mercedes. Mercedes was in the fourth grade and the second participant to enter intervention. She received a diagnosis of ASD from a large Midwestern Children’s Hospital

Medical Center when she was three years old. Mercedes was a classroom helper who enjoyed doing “jobs” for the teacher and helping classmates when possible. She was a very kind and polite student who was known as such throughout the school building. During the study implementation year, Mercedes’ IEP goals were focused on: (a) answering comprehension questions from a first grade level text, (b) composing one-two sentences using proper mechanics,

(c) completing math problems related to time, money, and subtraction, (d) responding appropriately to “wh” questions, and (e) performing self-care tasks (such as tying shoes).

Mercedes was given the WJ IV COG (Schrank et al., 2014) and ABES-R2 (McCarney &

Arthaud, 2006), during her most recent ETR, the results can be found in Table 2.

Table 2.

Mercedes’ Recent Evaluations

Test Subtests Standard Score Percentile Rank

Woodcock Johnson IV Letter Word 53 <.1 Identification Passage 50 <.1 Comprehension Calculation <40 <.1 Applied Problems <40 <.1

Adaptive Behavior Evaluation Scale – 2nd Edition Conceptual 89 24 Social 87 19

34 Practical 92 31 Adaptive Skills 89 23 Quotient

Andres. Andres was in the fifth grade and the third and final participant to enter intervention. He received a diagnosis of ASD from a large Midwestern Children’s Hospital

Medical Center when he was three years old. Andres was an independent worker who enjoyed building things, especially with Legos. Andres was the older brother of Mercedes. He liked to participate in school-wide activities and complete jobs for his teacher. During the study implementation year, Andres’ IEP goals focused on the following: (a) reading and answering comprehension questions, (b) composing a five sentence response to a prompt with proper mechanics, (c) completing math problems related to money and subtraction, (d) self-identifying strategies to use, and (e) developing and using simple, yet complete sentences. Andres was given the Universal Test of Nonverbal Intelligence (UNIT) (Bracken & McCallum, 1998)

Vineland II (Sparrow et al., 2005) and during his most recent ETR, the results can be found in

Table 3. Andres attended a different school than the other two participants when his most recent

ETR was completed and therefore, different assessments were used to evaluate him.

Table 3.

Andres’ Recent Evaluations

Test Subtests Standard Score Percentile Rank Adaptive Level Universal Test of 71 (SD 1.93) Below Nonverbal average Intelligence

Vineland II Communication 51 <1 Low Daily Living Skills 76 5 Low to moderately low Socialization 64 1 Low

35 Adaptive Behavior 62 1 Low Composite

Setting

This study took place in a specialized classroom for students with ASD within a private, parochial elementary school setting in the Midwestern United States. All participants of the study spent the majority of their school day within this specialized classroom, except for specials classes (music, physical education, and art) and religious education class. This specialized classroom provided explicit one-on-one and small group instruction at a slower pace with less sensory stimulation, and a quiet atmosphere to benefit students with ASD.

During the study, participants worked with the researcher each day over a span of 15 days for approximately 20-30 minutes each day. All participants worked with the researcher in a one- on-one capacity; on some days, they worked in a separate science lab (when available) but on most days, they worked at a small table in the corner of the classroom. All baseline, intervention, and maintenance sessions took place in one of these settings with the researcher.

Dependent Measure

Criterion tests. The dependent measure was the number of steps completed independently and correctly on task analysis for each problem on the criterion test. The criterion test consisted of four word problems, two addition word problems and two subtraction word problems, adapted from examples found in the Go Math! Textbook (Houghton Mifflin Harcourt,

2012). The copy of the text used by the researcher was a second grade level text (because this was the student’s level of instruction). Addition word problems were focused on the addition of the same units; participants had to parse out erroneous information found within the problem.

Subtraction problems were all written as “compare-type” problems that required participants to determine “how much more” one individual had than another.

36 The tests administered during baseline, intervention, and maintenance phases were all adapted from the Go Math! Textbook (Houghton Mifflin Harcourt, 2012) and were designed to be similar in level of difficulty. In order to create similar criterion test forms, the researcher altered the names and numbers found within each problem, as well as overall problem context.

These equivalent forms were used across all phases of the study. Example criterion tests can be found in Appendix J.

Task analysis. The researcher used a task analysis to identify the important steps within the word problem-solving process. A task analysis, sometimes called a step analysis, is the process of breaking a complex step into smaller steps that are easier to manage (Anderson, Taras,

& Cannon, 1996). Students are taught to chain the discrete steps of the task analysis together through systematic prompting, modeling, or with graduated guidance. The steps of a task analysis can be taught in clusters, from beginning to end or vice versa (Parker & Kamps, 2010).

For the purposes of this study, participants were taught to complete the steps of the task analysis from beginning to end. Several research teams have demonstrated the effectiveness of using a task analysis to teach a skill to students with disabilities across multiple settings and activities

(Browder et al., 2018; Browder, Jimenez, & Trela, 2012; Root et al., 2017). To create the task analysis, the researcher adapted the work of Root and colleagues (2017), a team that used a modified SBI strategy to teach students with ASD how to solve several different types of word problems using virtual and concrete manipulatives as well as ASD-specific supports (see

Appendix C). The task analysis used for the current study contained the same steps with slightly different visuals (due to availability of images).

Intervention probes. To monitor participant progress, the researcher administered intervention probes containing four problems each throughout the treatment phase of the study.

37 The four problems found on each intervention probe covered only the material that was reviewed that day. For example, the first probe administered during intervention contained addition problems only. Each participant received an addition-only intervention probe for each day they remained in the addition portion of the treatment phase. The researcher gave the participant a criterion test once they felt the participant was ready to progress onto subtraction. After the criterion test, the participant entered into subtraction instruction and received subsequent subtraction intervention probes for each day they remained in treatment. The subtraction probe also contained four word problems, targeting only subtraction. Addition and subtraction intervention probe examples can be found in Appendices H and I, respectively.

Scoring. Criterion tests and intervention probes both contained four problems for consistency with scoring. Each problem was broken down into nine steps on the task analysis.

A participant received one point for each step of the task analysis they completed independently.

If the participant reached a step they could not complete independently, and required a prompt, the researcher provided a prompt, but did not award the participant a point for that step. The participant was then able to move to the next step within the task analysis and could receive a point for completing that step independently. This method of scoring, considered a multiple- opportunity probe, was used to encourage participants with ASD to persevere through the problem and was used by the research team of Root and colleagues (2017). A participant could earn a minimum of zero and a maximum of 36 points on a day they were completing a criterion test or intervention probe. When working through practice problems and scenarios with the researcher, no points were awarded to the participant since a test or probe was not completed.

The researcher used a Microsoft Excel spreadsheet to document which steps of the task analysis the participant completed independently and which they needed prompting to complete.

38 Inter-rater reliability. The researcher scored all steps of the task analysis while each participant was completing the steps. Scoring took place simultaneously so that the researcher could visually examine which steps were completed independently and which steps were not. A

Microsoft Excel document was created by the researcher and used to record participant performance on each step of the task analysis for each problem. This sheet was divided into columns by session, each column labeled with the study phase and date. The worksheet also contained rows that were organized by problem type. There were addition and subtraction sections along the vertical axis of the sheet that were further broken down into individual task analysis steps for each problem. This made data collection timely and efficient which was necessary since the data was recorded while the participant completed each problem. The researcher simply had to type “yes” or “no” into each cell that corresponded with the appropriate problem type on the appropriate day. A blank Microsoft Excel worksheet in this same format was printed and given to the classroom teacher during inter-rater reliability sessions. The classroom teacher sat at the table with the student and researcher to observe and score participants, independently, for 40% of all sessions, exceeding both the minimum (20%) and preferable (33%) set forth by What Works Clearinghouse ([WWC]; Kratochwill et al., 2010;

Kennedy, 2005).

The number of agreements was divided by the numbers of agreements and disagreements and then multiplied by 100 to convert into a percent (Kazdin, 1982). This method, also known as interval agreement, is more precise than the total agreement method because of its ability to compare data at an interval-by-interval basis (Kennedy, 2005). In this case, each problem within each criterion test or intervention probe was considered an interval.

39 The resulting inter-rater reliability was 100%, which surpasses the conventional minimum of

80% inter-rater reliability (Kennedy, 2005).

Design

A modified multiple probe across participants design was used to determine whether a functional relationship existed between the intervention (modified SBI, task analysis, specific

ASD supports) and the participants’ performance on criterion tests (Horner & Baer, 1978). A multiple-baseline design, by definition, aims to establish the efficiency and reliability with which an intervention can alter behavior, across either behaviors, settings, or participants (Baer, Wolf,

& Risley, 1968). Data points are collected for each session in each phase of the multiple- baseline design (Kennedy, 2005). Conversely, a multiple-probe design intermittently collects data throughout the study, thus saving time and effort (Horner & Baer, 1978).

A single subject design was found to be most appropriate due to the nature and severity of each participant’s ASD diagnosis. Odom, Brantlinger, Gersten, Horner, Thompson, and

Harris (2005) went so far as to say that scientific research done in the field of special education is

“hardest-to-do” science because of the complexity of the participant population. Individuals on the autism spectrum possess such a diverse set of skills and abilities that it would be difficult to evaluate an intervention with a large population, for concern that the participants would not all be operating at a similar developmental level (Kennedy, 2005). Choosing to work with only three participants allowed the researcher to limit the scope of the study and target a certain skill set that was not yet mastered. Furthermore, the single subject design was necessary as it allowed for the inclusion of ASD specific supports, which may not have been supported by a large group design (Mesibov & Shea, 2011). Single subject designs conducted in real-world settings have

40 the potential to expand limited resources and improve the usability of research targeting interventions (Jenkins, Price, & Straker, 1998).

The multiple probe design was modified in order to fit the needs of the participants of this study. Addition and subtraction skills were critical to the participants’ success the remainder of the school year. In order to accommodate their need for this content, the researcher did not wait for one participant to complete three data points in treatment before moving another participant into treatment. This process allowed all participants to enter treatment quickly and receive instruction necessary for their success in class (Rizvi & Nock, 2008)

Procedure

Baseline. All participants completed at least three criterion tests during the baseline phase. Zachariah was the first participant to reach a stable baseline; he completed the minimum of three baseline criterion tests before moving into the treatment phase. Mercedes completed four baseline criterion tests before entering treatment. Andres was the third and final participant to enter treatment once a stable baseline had been established after the fifth criterion test. A baseline session typically took no more than 10-15 minutes. The participant was given one baseline criterion test with the SBI and ASD-specific supports (but no specific training on how to use them). Participants were asked to “solve the problems on the page to the best of their ability”. The researcher did not provide assistance or prompting (unless the participant was not attending to the task at all). Participants brought a pencil to the designated area and the researcher provided the criterion test sheet and ASD-supports. All baseline sessions took place consecutively; no probes were used during the baseline phase.

Treatment. The treatment phase consisted of two distinct instructional categories: addition and subtraction. The researcher started each participant in the addition phase of

41 treatment; providing instruction, practice with teacher, and an intervention probe. A criterion test was administered once the participant was ready to move onto the subtraction phase. The same format was followed for the subtraction phase of treatment. At the completion of both addition and subtraction phases, the participant was given a criterion post-test.

Maintenance. One week after completing the treatment phase, participants were given a criterion test to assess the maintenance of the targeted skills. A final maintenance criterion test was given three weeks after completion of the treatment phase. During maintenance sessions, the researcher placed all materials on the table in front of the participant and used the same type of prompting (if needed) that was used during the treatment phase.

Schedule. The researcher spent approximately 20-40 minutes with each student during each session. Criterion testing sessions were shorter in length since there was no new content being introduced. Some participants required less time than others did due to different processing speeds and attention capacities. The researcher visited the classroom five days a week (unless there was a holiday or school closure due to weather) for approximately 15 school days (about three weeks). The study was designed to be participant-driven; once the participant was performing adequately on an intervention probe they progressed to the next instructional phase, subsequent intervention probes, and criterion tests.

Materials. During the baseline phase, all intervention materials were provided for the participants (i.e., SBI number sentence mat, task analysis checklist, ten-frame mat, and rule cards) but no instruction was provided on how to use the materials. The researcher used multiple classroom materials (flat circle counters, dry erase markers, erasers, and pencils) during intervention as well as some researcher-created materials (participant task analysis checklist, ten frame mat, schema-based number sentence graphic, rule cards, criterion tests, intervention

42 probes, and practice problem sheets). All ASD-specific supports, tests, probes, and practice sheets were created using Microsoft Word. To record data, the researcher used a Microsoft

Excel document on a personal laptop computer and the second-rater used a printout of this same

Microsoft Excel document to score for inter-rater reliability. All documents used can be found in the Appendix.

ASD-specific supports. As previously mentioned, several specific supports for individuals with ASD were created for this study. The participant task analysis checklist, found in Appendix C, was adapted from the previous work of Root and colleagues (2017). The researcher used the same task analysis steps but created a separate template with images found through an electronic web search. This task analysis checklist provided a visual for the participants to organize the steps of the problem, since it paired text with pictures. The SBI number sentence graphic and ten-frame mat, found in Appendices A and B were also adapted from the materials used by Root and colleagues (2017). The ten-frame mat, found in Appendix

B, was designed for “Step 8” of the task analysis, “Make Set”. Participants were instructed to place flat circle counters into the ten frames based on the numbers within the problem. At the bottom of the ten-frame grid were two ovals: one with the word “total” and one with the word

“difference”. Participants were advised to pull their counters into one of the ovals (depending on operation) after they counted and lined them up appropriately on the ten-frame. The SBI number-sentence graphic (Appendix A) was used mainly during “Steps 3, 5, 6, & 7” of the task analysis checklist. The researcher modeled how and where to place appropriate numbers into the graphic as well as operation, answer, and label. All ASD-specific supports (i.e. participant task analysis checklist, ten-frame mat, rule cards, and schema-based number sentence graphic) were

43 laminated for durability. Participants used the dry erase markers to write on the laminated sheets.

Intervention

While it has been deemed an evidence-based practice for students with high incidence disabilities (Gersten et al., 2009), SBI may also provide a “roadmap” for students with ASD and

ID who may have difficulty with executive functioning skills like planning and organization

(Root et al., 2017). SBI gives students an alternate visual format to categorize information into when solving a mathematics problem. The use of these schematic diagrams aids the learner in recognizing the underlying structure of a problem (Jitendra et al., 2009). SBI highlights the relationship between pertinent elements of the problem structure that are required to solve the problem (Hegarty & Kozhevnikov, 1999). The ability to chunk multiple elements of a problem into a single schema make SBI an effective strategy for students with ASD who may experience working memory deficits (Kalyuga & Sweller, 2004). The intervention chosen for this study was a combination of a modified SBI strategy with ASD-specific visual supports.

Modeled after the intervention package used by Root and colleagues (2017), this study aimed to reach similar conclusions with different participants and problem types. Specifically, the current study used the nine task analysis steps used by Root and colleagues (2017). The text of each step matched exactly, while the visuals were replicated to the greatest degree possible through an electronic image search. The ten-frame mat was similar, but not identical since the current study included both addition and subtraction problems it contained an oval for the “total” and for the “difference”. Also different was the placement of the “rule”. The previous study placed the rule at the top of the ten-frame mat, while the current study created two separate “rule cards” for participants to choose between. The SBI number sentence mat used in the current

44 study was formatted identically to the template used by the previous researchers. However, the previous authors placed the ten-frame mat and SBI number sentence mat onto the same sheet.

Root and colleagues (2017) also dedicated a portion of their study to the comparison of virtual manipulatives and concrete manipulatives used in conjunction with the modified SBI and ASD supports. The current study did not analyze the differences between manipulatives as the researcher did not have access to the same programming required to implement the virtual manipulatives.

Addition. Following the baseline phase, participants entered the addition portion of the treatment phase. All three participants had a stronger grasp on addition concepts so it was introduced first in order to boost participant confidence.

Instruction. During the first addition session with each participant, the researcher spent about three minutes explaining the various study materials (i.e., task analysis checklist, ten frame, SBI number sentence graphic, rule cards, and flat circle counters) available for participant use. Next, the researcher placed a worksheet with two addition word problems in front of the participant. The researcher then taught the participant, through direct instruction and modeling, how to use the materials provided to solve the problem. The researcher walked through the problem step-by-step with the participant, marking each step on the checklist as it was completed

(see Appendix C for checklist). Table 4 outlines what a typical interchange looked like during the addition phase of intervention.

All addition word problems were modeled in the same fashion. The first three sentences included information regarding context, participants, and objects to be counted (including some erroneous information). The fourth and final sentence asked “How many do they have in all?” or

“How many do they have altogether?” For this study only “group type” addition word problems

45 were used. With “group-type” problems, the two small groups are known and the combined larger group, or total, is unknown. An example of a group problem used for this study would be

“Jason has six toy cars. Juan has three toys cars. Max has three toy trucks. How many toy cars do they have in all?” The group problems used in this study included erroneous information as well, as the classroom teacher identified this as an area of concern. In the example above, the information not needed to solve the problem would be “three toy trucks”. Since the problem is asking, “How many toy cars do they have in all?” the participant was tasked with recognizing and excluding the unnecessary information and including only the information relevant to the problem. The researcher modeled how to look at the final sentence of the word problem to determine which “what” they would be adding together and circle it with their pencil. Next, the participants were instructed to circle those same “whats” in sentences one, two, and three. If they found another object that did not match the “what” label, they could cross it out with their pencil.

When it came time to decide which rule to use (Step 4), the researcher would hold up the two laminated rule cards (see Appendix D) and ask the participant which rule they would be applying. The researcher would say, “Which rule do we use for this problem?” “Addition?”

(while pointing to the addition card) or “Subtraction?” (while pointing to the subtraction card).

If the participant chose the addition card (or had to be prompted), the researcher would read the addition rule with the participant: “Look at the labels, find the same, add them all together”.

While reading this rule the researcher would point to different areas of the problem (like the labels) simultaneously to provide a visual cue.

46 Table 4.

Addition Instruction Process

Step in Task Analysis Action Dialogue

Step 1: Read the Problem R: Points to first step on R: What is Step 1? checklist. P: Read the problem. R: Would you like me to read the problem or will you read the problem? Either participant or researcher reads the problem aloud.

Step 2: Circle the “what” R: Points to next step on R: What is Step 2? checklist. P: Circle the “whats”. P: Circles the “whats” on worksheet with pencil. May also cross out the “whats” they do not need.

Step 3: Write your label R: Points to next step on R: What is Step 3? (what) checklist. P: Write your label. R: Where do we write our P: Writes the chosen label label? (what) on the line on the SBI P: On the line. number sentence visual, using dry erase marker.

Step 4: Use my rule R: Points to next step on R: What is Step 4? checklist. P: Use my rule. R: Holds up two cards: one R: Which rule are we with the addition rule and one following: addition or with the subtraction rule for subtraction? Look back in participant to choose. the problem if you need to. P: Chooses addition rule. R: Reads through addition R: Look at the labels, find the rule with participant. same, add them altogether! P: May choose to read aloud or recite rule with researcher.

Step 5: Circle the numbers R: Points to next step on R: What is Step 5? checklist. P: Circle the numbers. P: Circles the numbers on the worksheet needed to solve the

47 problem. May also cross out the numbers they do not need.

Step 6: Fill-in number R: Points to next step on R: What is Step 6? sentence checklist. P: Fill-in the number P: Fills in the SBI number sentence. sentence visual with the numbers necessary to solve the problem (filling in the first two squares).

Step 7: + or - R: Points to next step on R: What is Step 7? checklist. P: Plus or minus. P: Fills in a (+) within the R: Are we adding or circle on the SBI number subtracting? sentence mat. P: Adding!

Step 8: Make Set R: Points to the next step on R: What is Step 8? checklist. P: Make your sets. P: Grabs ten-frame mat and circle counters to make their sets. The first row depicts the first group and the second row is the second group.

Step 9: Solve and write your R: Points to next step on R: What is Step 9? answer checklist. P: Solve and write your P: Looks at sets on ten frame answer. and pulls all circle counters R: How are you going to into the “total” oval. solve this problem? P: Counts all counters in the P: Add them all together. “total” oval. P: Writes their answer on the SBI number sentence visual.

After the initial practice problem was completed together, the researcher prompted the participant to solve the second practice problem. The researcher noted the participant’s level of independence when solving the problem and provided prompting (if needed) to help the participant succeed. If necessary, the researcher replicated the process for solving the first practice problem with the participant. The researcher did not record participant performance on

48 practice problems. Only performance on the intervention probes and criterion tests was recorded for data analysis.

Assessment. Once the practice problems were completed, the researcher presented an addition intervention probe to the participant. The intervention probe contained four addition word problems (similar to the practice problems). The participants were instructed to use their resources to complete all four word problems. Prior to starting the probe, the participants were reminded of the resources available to them (i.e., SBI number sentence visual, task analysis checklist, ten-frame mat, circle counters, and rule cards). Most participants requested that the researcher read the problems aloud to them as opposed to reading the problems on their own.

Their classroom teacher taught participants to have problems read aloud to them if needed.

Participants spent anywhere between 20-40 minutes working with the researcher on practice problems and then completed an intervention probe. After the initial day of addition instruction, the researcher used their discretion to determine how much addition instruction participants would need the following day. For example, one participant needed only to practice one problem with the researcher to be prepared for the intervention probe while another participant needed to complete both practice problems with the researcher before moving on. A participant was able to progress to the next phase after demonstrating improvement within the addition phase of treatment. All participants spent at least two days in the addition phase before receiving a criterion test. The criterion test given at the conclusion of the addition phase consisted of two addition and two subtraction word problems (similar to the baseline criterion test).

Prompting. Due to the nature and severity of each participant’s diagnosis, the researcher provided prompting to task if needed. Prompts were both verbal and gestural in nature. The

49 researcher gave prompts such as: “What comes next?” or “Stay focused.” if a participant’s attention was waning. A participant was not penalized a point on the task analysis if they were prompted back to task. However, if a participant required prompting in order to complete a task, then they did not receive the point for that particular step (since they were not able to complete it independently).

Subtraction. Once a participant progressed out of the addition phase of treatment and completed the criterion test, they started the subtraction phase of treatment. The process for instructing, assessing, and prompting during this phase of treatment was very similar to the process during the addition phase.

Instruction. On the first day of subtraction instruction for each participant, the researcher spent about three minutes reviewing all of the materials available for participant use

(i.e., task analysis checklist, rule cards, ten-frame mat, SBI number sentence mat, and flat circle counters). The researcher then gave the participant a worksheet with two subtraction word problems printed on it. Through direct instruction and modeling, the researcher demonstrated how to solve the first subtraction problem. Table 5 outlines a typical interchange between researcher and participant during the subtraction portion of treatment.

Table 5.

Subtraction Instruction Process

Step in Task Analysis Action Dialogue

50 Step 2: Circle the “what” R: Points to next step on R: What is Step 2? checklist. P: Circle the “whats”. P: Circles the “whats” on worksheet with pencil.

Step 4: Use my rule R: Points to next step on R: What is Step 4? checklist. P: Use my rule. R: Holds up two cards: one R: Which rule are we with the addition rule and one following: addition or with the subtraction rule for subtraction? Look back in participant to choose. the problem if you need to. P: Chooses subtraction rule. R: Reads through subtraction R: Big number, small rule with participant. number, difference between P: May choose to read aloud the two! or recite rule with researcher.

Step 5: Circle the numbers R: Points to next step on R: What is Step 5? checklist. P: Circle the numbers. P: Circles the numbers on the worksheet needed to solve the problem.

Step 7: + or - R: Points to next step on R: What is Step 7? checklist. P: Plus or minus. P: Fills in a (-) within the R: Are we adding or circle on the SBI number subtracting? sentence mat. P: Subtracting!

51 Step 8: Make Set R: Points to the next step on R: What is Step 8? checklist. P: Make your sets. P: Grabs ten-frame mat and circle counters to make their sets. The first row depicts the first group and the second row is the second group.

Step 9: Solve and write your R: Points to next step on R: What is Step 9? answer checklist. P: Solve and write your P: Looks at sets on ten frame answer. and draws lines matching the R: How are you going to counters in the top frame to solve this problem? those in the bottom frame. P: Match the sets and P: Pulls all counters down to compare! the “difference” oval that do not have a match. P: Counts all counters in the “difference” oval. P: Writes their answer on the SBI number sentence visual.

All subtraction problems were modeled in the same fashion. The first two sentences included information on the context, participants, and objects to be counted in the problem. For this study only, “compare-type” subtraction problems were used. “Compare-type” problems are solved with subtraction, specifically the differences between groups. An example of a compare- type word problem used in this study would be: “Joe has seven eggs. Stephanie has five eggs.

How many more eggs does Joe have than Stephanie?” During the instructional period, the researcher mentioned how the subtraction “compare-type” problems differed from the addition

“group-type” problems because they were comparing two sets of items. The researcher also pointed out the words “more than” within the subtraction problems and indicated that these words often appear with comparison type problems, but ultimately to use their math knowledge and critical thinking skills to make an informed decision (Karp, Bush, & Dougherty, 2015).

52 When it came time to use their rule (Step 4), the researcher would pull out two rule cards.

The researcher would say, “Are we using the addition rule?” (while pointing to the addition card) or “Are we using the subtraction rule?” (while pointing to the subtraction card). If the participant chose the subtraction rule (or needed prompting to do so), the researcher then read through the subtraction rule with the participant, “Big number, small number, difference between the two”. While reading the subtraction rule card, the researcher pointed to different numbers in the problem (i.e., the big number and the smaller number) to provide a visual cue in addition to the verbal cue.

After the first subtraction practice problem was completed together, the researcher prompted the participant to solve the second practice problem independently. Again, the researcher observed the participant’s level of independence and provided prompting accordingly.

The participant was expected to try the second practice problem on his or her own. The researcher allowed them to work independently until they reached a step they could not complete. Then intervention was provided to help the participant be successful. Performance was only recorded for data analysis on intervention probes and criterion tests, not for practice problems.

Assessment. Once the practice problems were completed, the researcher gave a subtraction intervention probe to the participant. The intervention probe contained four subtraction word problems (similar to the practice problems). Before starting the probe, the researcher reviewed all the available materials with the participants (i.e., SBI number sentence visual, task analysis checklist, ten-frame mat, circle counters, and rule cards). The participants were instructed to use their resources to complete the intervention probe. Again, most

53 participants requested the researcher to read problems aloud to them. One participant wanted to re-read the problem aloud after the researcher read it aloud once.

Each instructional session lasted anywhere from 20-40 minutes. After the first day of subtraction instruction, the researcher determined how much instruction would be necessary for the next day, as was done during the addition phase. All participants spent at least two days in the subtraction phase before receiving a criterion test. The criterion test given at the conclusion of the subtraction phase consisted of two addition and two subtraction word problems (similar to the baseline criterion test).

Prompting. Similar to the addition phase, the researcher provided both verbal and gestural prompts to keep participants on task. Points were not deducted on the task analysis, if a prompt-to-task was used, such as “What comes next?” or “Stay focused.” If the researcher provided prompting to help the participant complete a step in the task analysis, then a point was not awarded for that step. However, participants could earn points for the remaining steps on the task analysis, even if they needed prompting to complete a step.

Maintenance

After all three participants finished the subtraction phase of intervention and completed their criterion test; the researcher vacated the classroom for one week. A maintenance probe

(same format as a criterion test) was administered to each participant after one week with intervention. This maintenance probe consisted of four word problems: two addition and two subtraction, modeled after the criterion and baseline tests. The researcher then vacated the classroom for two more weeks. At three weeks post-intervention, the researcher administered one more maintenance probe to all three participants. Scores were recorded in the same manner as they were recorded during intervention.

54 Social Validity

Oftentimes in the context of educational research, there are stakeholders other than the participants and researchers who, if not directly involved in the research process, may be affected indirectly by the consequences of the intervention introduced. These individuals could be teachers, other students, para-educators, school administrators, or related service personnel within the school (Kennedy, 2005). Social validity, or the evaluation of the importance, effectiveness, appropriateness, and/or satisfaction experienced by various stakeholders as a result of the intervention, is an important measure in the realm of educational research (Kennedy,

2005). Social validity was first introduced to the field of applied behavior analysis by Kazdin

(1977) and Wolf (1978) after its previous induction into other fields (i.e., business, psychotherapy, and medicine). Three approaches to social validity have since been developed: subjective evaluation, normative evaluation, and sustainability (Kennedy, 2005). This study aimed to determine the perceptions of various stakeholders (i.e., participants and classroom teacher) in regards to multiple elements of the intervention package, with a subjective evaluation measure. The following paragraphs will describe the subjective evaluation measures used to gage the perceptions of participants and classroom teacher.

Participants. At the conclusion of the three-week post-intervention maintenance probe, all participants were given a one-page social validity measure (SVM) created by the researcher

(Appendix F). This measure contained six total items: five yes/no questions and one free response question. Questions posed focused on the usefulness of various materials (i.e., SBI number sentence mat, ten-frame mat, and task analysis checklist) throughout the intervention period. All yes/no style questions contained a picture of a smiling face (paired with yes) and a frowning face (paired with no) as an additional accommodation to participants. The researcher

55 read each question aloud to each participant and pointed to each material the questions were referencing, for clarity. Participants used a pencil to circle and write their own answers.

Teacher. At the conclusion of the maintenance phase, a 13-item SVM was given to the classroom teacher. The teacher measure included 12 Likert-style questions and one free- response question (see Appendix G). The teacher was asked about the usefulness of materials, effectiveness and efficiency of the intervention, and independence and engagement levels of students. Likert-style questions were based on a scale of one-six (one being completely disagree and six being completely agree). A final free response question elicited any additional feedback or comments the teacher may have had regarding the overall intervention and experience.

Data Analysis

As is common with most single subject designs, this study employed the use of visual analysis to evaluate the results of the intervention. Single case designs differ from traditional group designs in that they are dynamic. Data is collected prior to and following the intervention with a large group design as opposed to the continuous nature of data collection with a single- case design (Kennedy, 2005). The data is fluid and the researcher has the capability to change experimental strategies and design throughout the intervention process (Gast, 2010). For this reason, it is necessary to use visual analysis in an ongoing manner to recognize shifts and trends in the data as they are happening and to adjust practices accordingly (Gast, 2005).

Visual analysis. In order to analyze the results of the modified SBI intervention, visual analysis was conducted. As is custom with multiple baseline designs, the first participant entered the treatment phase after reaching a somewhat stable baseline level (Kennedy, 2005). Each participant after this started treatment as soon as possible. This study was categorized as a modified multiple probe design across participants because it did not require each participant to

56 remain in the baseline phase for a certain number of data points. Since this intervention was introduced in the middle of the school year, it was paramount that the participants receive the instruction as quickly and efficiently as possible. It was not considered an ethical choice to withhold the intervention from participants for an extended period of time (Rizvi & Nock, 2008).

For this reason, each participant entered into treatment within days of the first participant entering treatment. Similarly, the participants were not required to remain in the addition or subtraction phases until a certain number of data points was reached. They remained in those phases until they started to show improvement (which often happened after the first session).

The researcher brought their personal laptop to each session to record participants’ correct and incorrect answers, placing them into a Microsoft Excel document. Next, each participant’s total score out of 36 was plotted on a graph within the Microsoft Excel document. Visual analysis of this graph took place after each session throughout the study’s duration (Lane & Gast, 2014).

Level, trend, and variability of data were considered during the visual analysis process.

On a graph, the level refers to the average of the data within a given condition or phase

(Kennedy, 2005). The level is normally calculated as the mean or median. By visually analyzing the level, one is able to compare patterns between different phases of an experiment

(Kennedy, 2005). Trend is synonymous with the “line-of-best-fit” that can be drawn through the data of a particular phase. When analyzing trend, one must consider the slope (upward or downward inclination) and magnitude (size or extent) of the best-fit line. Slope can be categorized as positive or negative and magnitude can be described as high, medium, or low

(Kennedy, 2005). Finally, variability is the extent to which individual data points deviate from the “line-of-best-fit” or overall trend of the data. Variability is used to evaluate data patterns that occur within-phase and can be estimated as high, medium, or low (Kennedy, 2005).

57 Tau-U. In addition to visually analyzing the data, Tau-U was calculated to determine the effect size of the intervention. Tau-U is a measure of effect size that is considered to be a

“distribution free” way to examine the results of visual analysis (Parker & Vannest, 2012).

Kendall’s tau for nonoverlap between groups, also known as Taunovlap, is a measure of the percentage of non-overlapping data minus overlapping data (Parker, Vannest, Davis, & Sauber,

2011). Tau-U is a measure that extends Taunovlap in order to control for monotonic trend (an undesirable positive baseline trend) (Parker et al., 2011). In addition to being distribution-free,

Tau-U is nonparametric and suitable for data with any scale or distribution shape (Parker &

Vannest, 2012). Tau-U has strong statistical power of 91%-95%, which makes it suitable for shorter data series, as it does not require the minimum of four to six data points per series as nonparametric measures require (Bowman-Perrott, Davis, Vannest, Williams, Greenwood, & Parker,

2013). Since Tau-U is based on data non-overlap between phases, it is consistent with visual analysis (Parker & Vannest, 2012).

58 Results

The purpose of this study was to investigate the effects of a modified SBI strategy on the word-problem solving capability of three elementary students with ASD. “Group-type” and

“compare-type” problems were targeted by the chosen intervention. The usefulness of ASD- specific support materials was also examined during this study. Figure 1 provides a visual depiction of each participant’s progress over the course of the study. The findings of the study are outlined in the following pages and organized by research question.

59 60 Research Question 1

Is SBI an effective strategy for teaching addition “group-type” word problem-solving skills to students with ASD?

The first research question focused on examining the effectiveness of using a modified

SBI strategy with students with ASD as they were taught to solve addition “group-type” word problems. Table 6 shows the results from the multiple-probe across participants, specifically highlighting baseline, addition intervention, and maintenance phases. Baseline levels for all participants were low, with an increase occurring at the onset of intervention for each specific problem type. Once the addition intervention was introduced, performance on addition problems improved while performance on subtraction problems remained stagnant. Performance on the subtraction problems improved at the onset of the subtraction intervention. During each session that included an intervention probe or criterion test, the participants had the opportunity to earn

36 points on the task analysis (i.e. nine points per problem for four problems). Table 6 shows mean total scores for all phases including Tau-U Effect Size calculation. Tau baseline and intervention contrasts for each participant were calculated using a free online Tau calculator found at www.singlecaseresearch.org. Based on the results the effect size for each participant was considered large, suggesting that there was an effect on the addition word problem solving capabilities of participants with ASD who were given the modified-SBI intervention.

61 Table 6.

Effects of SBI on addition word problem solving: Phase Means and Tau

Participant Baseline Addition Maintenance Tau Effect Size Intervention Zachariah 10 32 35 1.0 = Large

Mercedes 12 31 33 1.0 = Large

Andres 9 31 34 1.0 = Large

Visual analysis of addition phase. Consideration was given to level, trend, and variability when visually analyzing all data recorded during the addition phase of intervention.

Additionally, level, trend, and variability were analyzed, individually, across all phases of the study. The following paragraphs detail the visual analysis of each participant’s progress during the addition phase of treatment.

Zachariah. During baseline there was low to moderate variability in performance, with a mean of (M=10). Upon introduction of the intervention during the addition phase of treatment, there was an immediate increase in the level of the dependent variable, with a mean of (M = 32).

The trend during the addition phase was stable, with two data points at 31 and 33, respectively.

Administration of the first criterion test probe led to an immediate decrease in level of the dependent variable (to 25 total points) which was expected since the criterion test contained addition and subtraction problems. Between baseline and the addition phase of treatment,

Zachariah made an average gain of 22 points, with a large effect size (Tau = 1.0). Table 6 outlines Zachariah’s performance during the baseline and addition phases of the study.

Mercedes. Mercedes’ performance during baseline maintained a low variability, with a mean of (M= 12). Once the intervention was introduced during the addition phase of treatment, the level of the dependent variable increased immediately, with a mean of (M= 31). The trend

62 during the addition phase of treatment was stable, with two data points at 27 and 35. The variability could be because Mercedes was absent after the first day of addition intervention, followed by a school holiday, before resuming for her second day of addition intervention. Once the first criterion test probe was administered, there was an immediate decrease in level of the dependent variable (to 24 total points) which was expected since the criterion test contained both addition and subtraction word problems. Mercedes made an average gain of 19 points between baseline and the addition intervention, with a large effect size (Tau = 1.0).

Andres. During baseline there was low to moderate variability in Andres’ performance with a mean of (M = 9). The introduction of the intervention resulted in an immediate increase in the level of the dependent variable, with a mean of (M = 31). The trend during the addition phase of treatment was fairly stable, with two data points at 28 and 34. The two data points recorded were scheduled around a school holiday, which may have led to some of the variability.

After the administration of the criterion test, containing both addition and subtraction word problems, the level of the dependent variable immediately decreased (to 23 total points), which was again, expected. Andres made an average gain of 22 points between baseline and addition intervention, with a large effect size (Tau = 1.0).

Summary of addition phase. All participants demonstrated positive, stable trends in the total points they received on intervention probes between the baseline and addition phase of treatment. Experimental control was demonstrated for all participants based on the following:

(a) the data changed in relation to the intervention introduced (i.e., the level and trend of addition data increased with the introduction of the addition intervention), (b) at the completion of a phase of treatment (i.e., addition) the level and trend of the data decreased for concepts not yet introduced (i.e., subtraction), and (c) once the subtraction phase was introduced, the level and

63 trend of the data increased to reflect the instruction that had been given (Kennedy, 2005). A return to baseline was not demonstrated (or expected) since the intervention was not withdrawn.

Research Question 2

Is SBI an effective strategy for teaching subtraction “compare-type” word problem- solving skills to students with ASD?

The second research question examined the effectiveness of using a modified-SBI to teach subtraction “compare-type” word problems to students with ASD. Table 7 shows the results of the multiple-probe across participants design, specifically highlighting baseline, subtraction intervention, and maintenance phases. The Tau Effect Size was again calculated by using the free online calculator at www.singlecaseresearch.org. The Tau Effect Size was 1.0, which is considered a large effect size.

Table 7.

Effects of SBI on subtraction word problem solving: Phase Means and Tau

Participant Baseline Subtraction Maintenance Tau Effect Size Intervention Zachariah 10 35 35 1.0 = Large

Mercedes 12 33 33 1.0 = Large

Andres 9 34 34 1.0 = Large

Visual analysis of the subtraction phase. Once again, consideration was given to level, trend, and variability when visually analyzing all data collected during the subtraction phase of treatment. The following paragraphs will detail the visual analysis of each participant’s progress during the subtraction phase of treatment.

Zachariah. After completion of the first criterion test probe at the conclusion of the addition phase of treatment, the level of the dependent variable had dropped to 25. The

64 following day, Zachariah entered into the subtraction phase of treatment. Once the subtraction intervention was introduced the level of the dependent variable immediately increased with a mean of (M = 35). The trend during the subtraction phase of treatment was stable, with data two data points at 34 and 35. After two days of instruction with follow-up intervention probes,

Zachariah was ready to take his second criterion test. This test also included both addition and subtraction word problems. He scored 36 points on this criterion test. Zachariah made an average gain of 25 points between his original baseline and subtraction intervention, with a large effect size (Tau = 1.0).

Mercedes. Mercedes’ dependent variable level dropped to 24 on the criterion test administered at the conclusion of the addition phase of treatment. The next day, she entered into the subtraction phase of treatment and remained there for two days. As soon as the subtraction intervention was introduced the level of the dependent variable increased, with a mean of (M =

33). The trend during the subtraction phase of treatment was stable, with two data points at 33 and 33. Mercedes took her second criterion test after completing two days of subtraction instruction and follow-up intervention probes. She scored 33 points on this criterion test.

Mercedes made an average gain of 21 points between her original baseline and subtraction intervention, with a large effect size (Tau = 1.0).

Andres. After completing the first criterion test probe at the completion of the addition phase of treatment, Andres’ dependent variable level dropped to 23 points. The following day he entered into the subtraction phase of treatment. The introduction of the subtraction intervention led to an immediate increase in level of the dependent variable with a mean of (M = 34). He spent two days in this phase, receiving subtraction instruction and intervention. The trend during the subtraction phase was stable, with two data points at 33 and 35. Andres completed his

65 second criterion test probe at the conclusion of the subtraction intervention and scored 33 points on this probe. Andres made an average gain of 25 points between his original baseline and subtraction intervention, with a large effect size (Tau = 1.0).

Summary of subtraction phase. All participants demonstrated a positive, stable trend in the total points they received on intervention probes and criterion tests between the baseline, addition, and subtraction phases of treatment. Experimental control was demonstrated for all participants based on the following: (a) the data changed in relation to the intervention introduced (i.e., the level and trend of subtraction data increased with the introduction of the subtraction intervention), and (b) the level of dependent variable remained high once participants were introduced to the addition and subtraction intervention (Kennedy, 2005). Again, a return to baseline levels was not observed because the intervention was not withdrawn.

Research Question 3

Is the SBI strategy with ASD-specific supports a socially valid intervention?

The third and final research question examined the social validity of the additional ASD- specific supports paired with the SBI intervention. This study made use of various visual aids and supports designed for students with ASD. The ASD-specific supports created for and used throughout treatment and maintenance phases were the participant task analysis checklist, ten- frame mat, SBI number sentence graphic, and rule cards. As discussed previously, these supports were adapted from those created by the research team of Root and colleagues (2017). A review of the participant and classroom teacher’s answers to the SVM’s were conducted to determine whether the intervention package was socially valid. The following paragraphs will further examine each support by analyzing the qualitative responses recorded by the researcher, which can be found in Tables 8 and 9.

66 Participant task analysis checklist. This checklist (as shown in Appendix C) was used by each participant during the addition and subtraction phases of treatment was well as during maintenance. This checklist included both written text and visuals to illustrate each step of the nine-step task analysis. All participants felt that the task analysis checklist was useful and a tool that they could use when solving mathematics word problems. Additionally, the classroom teacher felt that the task analysis checklist was something she could utilize in her classroom with little effort.

Ten-frame mat. The ten-frame mat (as shown in Appendix B) was used by each participant during the addition and subtraction phases of treatment as well as during maintenance. The ten-frame mat consisted of two rows of ten squares, one row colored red and one row colored blue. Near the bottom of the ten-frame mat were two ovals: one colored green with the word “total” within and one colored red with the word “difference” within. All participants indicated that this was a useful tool to use when solving word problems. The classroom teacher also felt that the ten-frame mat was a worthwhile tool that benefited all participants.

Schema-based number sentence graphic. The SBI number sentence graphic was also used by each participant during the addition and subtraction phases of treatment and during maintenance. The number sentence graphic (Appendix A) consisted of three squares (two for the separate operands and one for the result (sum or difference). A small circle figure sat in between the two squares, this circle was meant for the operator (either addition or subtraction). All participants indicated that this graphic was helpful when solving word problems. The classroom teacher too felt that the number sentence graphic was easy to use and required little training.

67 Rule cards. The rule cards were referred to specifically during step four of the task analysis for both addition and subtraction phases of treatment. Each rule was printed on a separate half sheet of 9.5 x 11 inch paper using both written text and visual pairings. A rule card was created for the addition and the subtraction rules separately. The rule cards are displayed in

Appendix D. All participants felt that the rule cards were useful as they solved addition and subtraction math word problems and their classroom teacher shared the sentiment.

Social validity analysis. Based on the responses to the SVM’s and participant improvement over the course of the study, it appears that the modified SBI with ASD-specific supports were socially valid. All participants felt that the visual supports helped them solve word problems over the course of the study. The classroom teacher agreed that the supports were useful for her students and a great addition to her daily mathematics instruction (with a few tweaks). When asked if they felt like they could now solve math problems by themselves, two out of three participants answered “no”. This could be due in part to the mathematics anxiety experienced by these individual participants. It is quite possible that even though they did find the intervention helpful, they still do not feel confident enough to navigate and solve a word problem completely independent of any adult support. All three participants answered positively when asked whether they like solving math word problems. This is hopefully an indication that their feelings about math are shifting.

68 Table 8.

Social Validity Measure Results: Student Participants

Social Validity Item Participant Name

Zachariah Mercedes Andres

The task analysis Yes Yes Yes checklist helped me solve my math problems.

The ten-frame mat Yes Yes Yes helped me solve my math problems.

The SBI number Yes Yes Yes sentence graphic helped me solve my math problems.

I feel like I can solve No Yes No math problems by myself.

I like solving math Yes Yes Yes word problems. Note. SVM items were presented to participants in a yes/no format. The “yes” was accompanied by a smiling face visual and the “no” was accompanied by a frowning face visual.

The classroom teacher responded that she “mostly agreed” and “completely agreed” to all

SVM prompts except for one. When asked if she felt that she would have enough time in her day to use this intervention (i.e., was it an effective and efficient strategy) she answered that she

“slightly agreed”. While this answer still agrees with the statement, it does indicate a slight difference in opinion. The classroom teacher raises an excellent concern that comes along with any multi-factored intervention: Will there be enough time to use it? A good intervention is one that is both effective and efficient. While it is challenging to create a robust intervention that can

69 be implemented within a short amount of time, it is not sufficient to give up hope that it is possible. Considering this information will only make future interventions better.

Table 9.

Social Validity Measure Results: Classroom Teacher

Social Validity Item Rating

I feel that the number sentence mat was useful 5 for my students.

I feel that I would use the number sentence 6 mat within my own instruction.

I feel that the ten-frame grid was useful for 5 my students.

I feel that I would use the ten-frame grid 5 within my own instruction.

I feel that the task-analysis checklist was 5 helpful for my students.

I feel that I would use the checklist within my 5 own instruction.

I feel that I could use this intervention as a 5 whole package within my own instruction.

I feel that this intervention met the current 5 needs of my students.

I feel that I would have enough time in my 4 day to use this intervention (i.e. it was an effective and efficient strategy).

I feel that I could teach this strategy to a 5 teacher’s aide or other instructor to use with students.

I feel that my students have gained some 5 independence in regards to solving math word problems as a result of this intervention.

70 I feel that my students maintained an 5 appropriate level of engagement throughout the intervention.

Please offer any additional feedback you may No response. have in regards to the intervention, materials, or instructional methods.

Note. The Classroom Teacher SVM was presented as a Likert-style rating scale (1 = completely disagree, 2 = mostly disagree, 3 = slightly disagree, 4 = slightly agree, 5 = mostly agree, and 6 = completely agree).

Individual Participant Progress

The following section does not directly answer any of the identified research questions.

Instead, if offers an in-depth look into the individual progress of each participant. Within this section, the progress of each participant is analyzed and related to the overall outcomes of the study. Anecdotal thoughts and notations from the researcher are included to add some richness to the data set (Creswell, 2012).

Zachariah. Zachariah entered the baseline phase of the study with confidence. It was immediately clear that he possessed the skills needed to solve basic addition and subtraction problems. When presented with the “group-type” word problem with erroneous information, he felt a bit uneasy, but still worked hard to solve the problem. He asked questions if he was not sure of what to do and did not hesitate to say, “I don’t know.” Of the three participants,

Zachariah entered baseline at the highest level of academic functioning and understanding.

Zachariah’s baseline performance on each step within the task analysis can be viewed in Table

10. It appears that Zachariah was slightly more comfortable solving the addition “group-type” problems, completing anywhere from two-six items correctly per question. Whereas he completed no more than two steps of the subtraction “compare-type” problems correctly and independently. During the baseline phase, Zachariah consistently completed “Step 1” (“Read the

71 Problem Aloud”) independently. In this case, Zachariah typically asked the researcher to read the problem aloud to him.

Table 10.

Zachariah’s Baseline Performance

Day Test Type Question Question Type Question Question Test Total Number Total Steps Points (out Points (out Missed of 36) of 9) 1 Addition 6 2, 3, and 4

2 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 3 Addition 2 2, 3, 4, 5, 1 Baseline 6, 8, and 9 Criterion Test 4 Subtraction 2 2, 3, 4, 6, 7, 8, and 9

1 Addition 2 2, 3, 4, 5, 6, 8, and 9

2 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 2 Baseline 3 Addition 2 2, 3, 4, 5, Criterion 6, 8, and 9 Test 4 Subtraction 2 2 3, 4, 6, 7, 8, and 9

1 Addition 6 2, 3, 4,

2 Subtraction 2 2, 3, 4, 6, Baseline 8, and 9 3 Criterion 3 Addition 2 2, 3, 4, 5, Test 6, 8, and 9 4 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 12

72 Zachariah responded immediately to the intervention at the start of the addition phase.

His average baseline score of three correct steps per addition problem jumped to an average of eight correct steps per addition problem on addition intervention probes and criterion tests within the treatment phase. Based on his task analysis results, found in Table 11, Zachariah quickly learned the nine-step intervention process. During his two instructional days with addition intervention probes, it appears that he struggled the most with Steps Three and Four (“Write

Your Label [what] and “Use My Rule”). It is quite possible that these two steps did not come naturally to Zachariah, as they had not been a part of his traditional mathematics instruction.

Once he took the first criterion test at the end of the addition phase of treatment, he scored nine and eight points respectively on the two addition problems, while scoring four and four points respectively on the subtraction problems. This was an expected result since the subtraction instruction had not been introduced.

Table 11.

Zachariah’s Addition Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points Points Missed (out of 36) (out of 9) 1 Addition 7 3, 4 (VP)

Addition 2 Addition 7 2, 4, (VP) 1 Intervention Probe 3 Addition 8 4 (VP)

4 Addition 9 31

1 Addition 8 3 (VP) Addition 2 Intervention 2 Addition 8 3 (VP) Probe 3 Addition 8 3 (VP)

73 4 Addition 9 33

1 Addition 9

2 Subtraction 4 4, 6, 7, 8, and 9 (VP) Criterion 3 Test 3 Addition 8 2 (GP)

4 Subtraction 4 4, 5, 7, 8, and 9 (VP) 25 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

By the end of the first day of subtraction instruction Zachariah had already made progress toward mastery of the skill. He scored four correct steps per subtraction problem on his last criterion test prior to entering the subtraction phase of intervention. Once he received the subtraction instruction, his level immediately jumped to eight correct steps per subtraction problem. His average for correct steps per subtraction problem for the subtraction phase of intervention was 8.6 steps (rounded to eight full steps) for all subtraction intervention probes and criterion tests. Once again, Zachariah was able to learn this process rather seamlessly. His performance on each step of the task analysis, during the subtraction phase, can be seen in Table

12. Based on this information it is apparent that the one step that Zachariah needed some additional teaching on was “Step 6” (“Fill-In Number Sentence”). Zachariah needed to be prompted twice to remember to write the larger number in the first box of his SBI number sentence mat and the smaller number in the second box when subtracting. He quickly learned from his mistake, and by the final criterion test of the subtraction phase, he scored nine points on each problem within the test, for a score of 36. During both criterion tests administered within

74 the subtraction phase, Zachariah scored nine points on all addition problems. This was expected since he had already completed addition instruction.

Table 12.

Zachariah’s Subtraction Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points (out Points (out Missed of 36) of 9) 1 Subtraction 8 7 (VP)

2 Subtraction 9 Subtraction 3 Subtraction 8 6 (put 1 Intervention small Probe number first, VP)

4 Subtraction 9 34

1 Subtraction 9

2 Subtraction 9 Subtraction 3 Subtraction 9 2 Intervention Probe 4 Subtraction 8 6 (put small number first, VP) 35

1 Addition 9

2 Subtraction 8 8 (put Criterion 3 incorrect Test number of counters out,

75 distracted, VP)

3 Addition 9

4 Subtraction 9 35

1 Addition 9

2 Subtraction 9 Criterion 4 Test 3 Addition 9

4 Subtraction 9 36 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

Of note during the subtraction phase of the intervention was Zachariah’s ability to think creatively to problem solve. During the first subtraction intervention probe, he placed the smaller number in the first box of the SBI number sentence mat and the larger number in the second box. He started to perform the steps of the task analysis so methodically that he wanted to set this problem up in the same way he set up the addition problem. After reviewing the subtraction rule with him again (“Big Number, Small Number, Difference Between the Two”) he quickly realized and corrected his mistake. When completing his subtraction intervention probe the following day, he was able to correct his own error when he placed the smaller number first.

On the third day of the subtraction phase of treatment, Zachariah initiated a new strategy. When he read the words “how many more” within the problem, he audibly said “Oh”. When asked what he noticed, he quickly picked up his pencil and drew a subtraction sign above the words

“how many more” within the problem text. This sign served as a reminder for himself to put the larger number ahead of the smaller number when subtracting. See Figure 2 for a depiction of the strategy Zachariah used when solving subtraction problems for the remainder of the study. The

76 researcher and Zachariah had a discussion after he initiated his novel strategy regarding the use of “key words”. Following the discussion, Zachariah had a clear understanding that the words

“how many more” do not always indicate the need to use subtraction (Karp et al., 2015) and that he should carefully read the problem to make an informed decision.

Figure 2. Zachariah’s Subtraction Strategy

At the completion of the subtraction phase of treatment, the researcher left the school site for a week’s time. After one week, all participants were given a criterion test to determine whether they were able to maintain the skills learned. On this first criterion test, Zachariah earned a total of 35 out of 36 points, missing just one point on “Step 6” (“Fill-In Number

Sentence”) of a subtraction problem because he put the smaller number before the larger number on his SBI number sentence mat. This result was a positive indication that Zachariah had maintained the skill he learned throughout in treatment for one week. Results from Zachariah’s progress during the maintenance phase can be found in Table 13.

Once all three participants completed the first maintenance criterion test the researcher left the school site for two weeks’ time. Two weeks after the first maintenance test (or three weeks after the completion of the treatment phase), each participant was given a second maintenance criterion test. On this second criterion test, Zachariah earned 36 out of 36 total points. Zachariah was very confident in his abilities and proud of his accomplishments at the closure of the final phase of the study. This result indicates that Zachariah was able to maintain the skills he learned in treatment for at least three weeks. Of note during the maintenance phase was Zachariah’s ability to move through the correct steps of problem solving without needing to

77 look at the task analysis checklist after each step. He had internalized the process and therefore was able to complete several steps in a succession without referring to the checklist for support.

This indicates that the process has the potential to be internalized and later applied appropriately in word problem solving scenarios.

Table 13.

Zachariah’s Maintenance Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points Points Missed (out of 36) (out of 9) 1 Addition 9

2 Subtraction 8 6 (put small 1 Criterion number (1 week Test first, VP) out ) 3 Addition 9

4 Subtraction 9 35

1 Addition 9

2 2 Subtraction 9 Criterion (3 weeks Test out) 3 Addition 9

4 Subtraction 9 36 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

Mercedes. Mercedes entered the baseline phase of the study with many questions. She wanted to know what was coming next and what to expect when she was finished. She was highly focused on her routine and schedule and liked to know how long she would be working and how many problems she would be expected to complete. Once she learned the typical

78 routine for each session, she was able to relax a bit more in order to better focus on the problem solving. It was apparent that she had some basic addition solving capabilities from the start of this phase. However, she wanted to apply addition rules to each problem (even if the problem required subtraction). See Figure 3 for Mercedes’ performance on one baseline intervention probe.

Figure 3. Mercedes’ Baseline Performance Example

Mercedes’ performance on each step within the task analysis, during the baseline phase, can be viewed in Table 14. She was definitely more comfortable solving addition “group-type” problems than she was with “change-type” subtraction problems. Mercedes completed four

79 baseline criterion tests over a span of four days. Within those four days, she completed the same four steps (one, five, six, and seven) correctly when given an addition problem. She was able to

“Read the Problem” or in her case, she would ask the researcher to read the problem aloud,

“Circle the Numbers”, “Fill-In the Number Sentence” and “Choose (+) or (-)” for each addition problem presented during the baseline phase. When given a subtraction problem she completed no more than two steps correctly and independently.

Table 14.

Mercedes’ Baseline Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points (out Points (out Missed of 36) of 9) 1 Addition 4 2, 3, 4, 8, and 9

2 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 Criterion 1 Test 3 Addition 4 2, 3, 4, 8, and 9

4 Subtraction 1 2, 3, 4, 5, 6, 7, 8, and 9 11

1 Addition 4 2, 3, 4, 8, and 9

2 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 Criterion 2 Test Addition 4 2, 3, 4, 8, 3 and 9

4 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 12

80 1 Addition 4 2, 3, 4, 8, and 9

2 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 Criterion 3 Test 3 Addition 4 2, 3, 4, 8, and 9

4 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 12

1 Addition 4 2, 3, 4, 8, and 9

2 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 Criterion 4 Test 3 Addition 4 2, 3, 4, 8, and 9

4 Subtraction 2 2, 3, 4, 6, 7, 8, and 9 12

Mercedes responded immediately to the introduction of the intervention during the addition phase, though her level of performance did not increase as quickly as Zachariah’s level increased. Her average baseline score of four correct steps per addition problem moved to an average of 7.6 (rounded down to seven) full correct steps per addition problem. Based on the task analysis results, found in Table 15. Mercedes learned the nine-step intervention process at a slower rate than Zachariah. During her first addition intervention probe, she answered six, seven, seven, and seven steps correctly, respectively. She seemed to struggle the most with

“Steps 2 and 3” (“Circle the ‘whats’ and “Write Your Label”) during this first day. Again, these steps may not have been part of her traditional mathematics instruction. Mercedes possessed a strong knowledge of the fundamental processes involved in addition even during baseline, as she

81 consistently performed “Steps 5, 6, and 7” correctly and independently. Once she learned the process and had a few days to practice, her scores moved into the eight-nine correct step range for the subsequent addition intervention probe and criterion test. At the end of the addition phase of treatment, she scored nine and eight points, respectively, on the addition questions found in the criterion test, while scoring three and four points, respectively, on the subtraction problems.

This was an expected result since she had not been introduced to the subtraction intervention yet.

Table 15.

Mercedes’ Addition Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points Points Missed (out of 36) (out of 9) 1 Addition 6 2, 3, 6 (VP)

Addition 2 Addition 7 2, 3 (VP) 1 Intervention Probe 3 Addition 7 2, 3 (VP)

4 Addition 7 2, 3 (VP) 27

1 Addition 8 5 (VP)

Addition 2 Addition 9 2 Intervention Probe 3 Addition 9

4 Addition 9 35

1 Addition 9

2 Subtraction 3 3, 4, 6, 7, Criterion 8, and 9 3 Test (VP)

3 Addition 8 2 (distracted,

82 circled wrong item, GP)

4 Subtraction 4 4, 6, 7, 8, and 9 (VP) 24 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

Of note during the addition phase of treatment was Mercedes’ ability to persevere and develop strategies that worked for her. During the first addition instructional session, she was constantly second guessing which information was needed and which was erroneous. By the second day of instruction, she had developed a strategy that she would use intermittently throughout the remainder of the study (see Figure 4). After reading the first problem along with the researcher, she initially circled all of the numbers within the problem. She reviewed her work and decided that she needed to erase the circle around the third number, which corresponded with “toy fire trucks” since the first two numbers she circled corresponded with

“toy cars”. She then proceeded to put an “X” through the erroneous information about “toy fire trucks”. Figure 4 depicts her work on this problem and the following problem. When solving the second problem she chose not to put an “X” through the erroneous information, but instead audibly exclaimed “No plums!” when circling necessary information. By the criterion test day,

Mercedes was beginning to “Circle the Whats” automatically (Step 2) as soon as the problem was read, without having to refer to her task analysis checklist.

83 Figure 4. Mercedes’ Addition Strategy

Mercedes’ level of performance on task analysis steps for subtraction problems immediately rose from three or four correct steps per problem to six correct steps on her first problem after receiving subtraction instruction. Her average for correct steps per subtraction problems for the subtraction phase of treatment was 8.3 steps (rounded to eight full steps) for all subtraction intervention probes and criterion tests. By this point, Mercedes understood the nine- step intervention process; she just needed to solidify her subtraction skills. Her performance on each step of the task analysis, during the subtraction phase, can be seen in Table 16. On four occasions, Mercedes needed prompting to “Use her Rule” (Step 4) because she did not always have a grasp on which problems required addition and which required subtraction. Because she was more comfortable with addition, she tended to choose the addition rule when she was unsure. Mercedes would have benefited from more direct instruction surrounding this topic towards the beginning of the treatment phase to really sharpen her decision-making process.

Mercedes scored between eight and nine points for all subtraction problems on both criterion tests given at the end of the subtraction phase. She scored no lower than eight points for all

84 addition problems within the criterion tests, which was expected since she had already received addition instruction.

Table 16.

Mercedes’ Subtraction Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points (out Points (out Missed of 36) of 9) 1 Subtraction 7 4, 9 (VP)

2 Subtraction 9 Subtraction 3 Subtraction 8 6 (put 1 Intervention small Probe number first, VP)

4 Subtraction 9 33

1 Subtraction 9

2 Subtraction 9

3 Subtraction 8 8 (put incorrect Subtraction number of 2 Intervention counters, Probe VP)

4 Subtraction 7 4, 6 (distracted, put small number first, VP) 33

1 Addition 8 9 (started Criterion 3 matching Test as if to

85 subtract, VP)

2 Subtraction 8 4 (chose addition rule, VP)

3 Addition 8 4 (chose subtraction rule, VP)

4 Subtraction 9 33

1 Addition 9

2 Subtraction 9 Criterion 4 Test 3 Addition 9

4 Subtraction 9 36 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

Once all participants completed their final criterion test at the conclusion of the subtraction phase, the researcher left the school site for one week. Upon return, all participants were given another criterion test to determine their level of maintenance. Mercedes’ performance level dropped slightly between the conclusion of the subtraction phase and maintenance phase from 36 total points to 32 total points. She was highly distracted during this session because her schedule had been changed due to a school assembly in the morning. She desperately wanted to finish her work so that she could enjoy a preferred task. She rushed through several problems, making careless errors (i.e., trying to subtract when she had set up an addition problem, forgetting to put all counters out, and putting the smaller number before the larger when subtracting).

86 Mercedes was given a second maintenance criterion test two weeks later (or three weeks from the completion of the treatment phase). On the second criterion test, her score reached 34 total points, which was within her previous range of performance during treatment. She was much less distracted during this session and only missed two steps throughout the entire test.

Mercedes’ maintenance phase performance can be viewed in Table 17. The fact that Mercedes was able to reach her treatment phase levels is promising and once again indicates that the skills learned have the ability to be maintained past the completion of instruction.

Table 17.

Mercedes’ Maintenance Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points Points Missed (out of 36) (out of 9) 1 Addition 7 4, 9 (started to, VP)

2 Subtraction 8 8 (forgot top row of 1 counters, Criterion (1 week VP) Test out ) 3 Addition 9

4 Subtraction 8 6 (put smaller number first, VP) 32

1 Addition 9 2 Criterion (3 weeks Test 2 Subtraction 9 out)

87 3 Addition 8 4 (said subtract, VP)

4 Subtraction 8 6 (VP) 34 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

Andres. Andres entered the baseline phase of this study a bit distracted and with no consistent problem-solving process. Andres possessed the skills needed to solve basic addition and subtraction problems, but was very confused about when to “choose” addition and when to

“choose” subtraction. Andres was very easily distracted, requiring quite a bit of prompting to get his work started. Like his sister, Andres was also very focused on his routine, as long as he knew what to expect next and how long he was working he was very amenable. Unlike the other two participants, Andres’ performance on the task analysis steps did not present any predictable patterns. He did not answer the same steps correctly with any consistency. His baseline performance on the task analysis steps is presented in Table 18. For most baseline problems, regardless of problem type, he would write an addition problem out numerically but draw a subtraction visual (see Figure 5). While this did not lead him to correct responses on the task analysis steps, it did support that he remembered some processes he was taught previously. He was just unsure of when it was appropriate to use each.

88 Figure 5. Andres’ Baseline Performance Example

Andres never scored more than four points on any given problem during baseline.

Andres was the last participant to enter the treatment phase, so he completed five baseline criterion tests. During that time, the steps he completed correctly with the most consistency were

“Step 1 and 5” (“Read the Problem” and “Circle the Numbers”). These two steps were the most straightforward of the nine, so it makes sense that he completed these with the highest degree of consistency and accuracy. Andres demonstrated a bit more stability during his last two days of baseline, when he earned ten and eleven points, respectively.

89 Table 18.

Andres’ Baseline Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points (out Points (out Missed of 36) of 9) 1 Addition 4 2, 3, 4, 8, and 9

2 Subtraction 2 1, 2, 3, 4, 6, 7, and 9 Criterion 1 Test 3 Addition 2 2, 3, 4, 5, 6, 8, and 9

4 Subtraction 1 1, 2, 3, 4, 6, 7, 8, and 9 9

1 Addition 1 2, 3, 4, 5, 6, 7, 8, and 9

2 Subtraction 0 1, 2, 3, 4, 5, 6, 7, 8, and 9 Criterion 2 Test 3 Addition 1 1, 2, 3, 4, 6, 7, 8, and 9

4 Subtraction 0 1, 2, 3, 4, 5, 6, 7, 8, and 9

1 Addition 2 2, 3, 4, 5, 6, 8, and 9 Criterion 3 Test 2 Subtraction 4 2, 3, 4, 6, and 7

90 3 Addition 4 2, 3, 4, 8, and 9

4 Subtraction 4 2, 3, 4, 6, and 7 14

1 Addition 2 2, 3, 4, 5, 6, 8, and 9

2 Subtraction 4 2, 3, 4, 6, and 7 Criterion 4 Test 3 Addition 4 2, 3, 4, 8, and 9

4 Subtraction 1 1, 2, 3, 4, 6, 7, 8, and 9 11

1 Addition 2 2, 3, 4, 5, 6, 8, and 9

2 Subtraction 1 2, 3, 4, 5, 6, 7, 8, and Criterion 9 5 Test 3 Addition 4 2, 3, 4, 8, and 9

4 Subtraction 3 2, 3, 4, 6, 7, 8, 10

Andres also responded positively to the intervention at the start of the addition phase.

His average baseline score of 2.6 (rounded to two full steps) per addition problem jumped to an average of 7.8 (rounded to seven full steps) per addition problem on addition intervention probes and criterions tests within the treatment phase. Based on his task analysis results for the addition phase of treatment, found in Table 19, Andres quickly learned the nine-step intervention process.

During this first addition instruction session, he required a higher level of verbal prompting,

91 mostly when he reached “Steps 2, 3, and 5” (“Circle the ‘Whats’”, “Write your Label (what)”, and Circle the Numbers”). More than likely, Andres was not taught these steps as part of his traditional math instruction, so they were novel to him. Once he practiced and understood the sequence of events, his performance level increased. By his second day of intervention, he scored between eight and nine points for each problem. Andres scored eight points on both addition items on the criterion test and three and four points on both subtraction items on the test.

This result was expected because he had not yet been introduced to the subtraction intervention.

Of note during the addition phase was Andres’ preference for subtraction. By viewing

Table 19, it is apparent that Andres missed “Step 4” (“Use my Rule”) on three separate occasions during the administration of the criterion test. For some reason, at this instant Andres decided to only choose the subtraction rule, regardless of prompting methods used by the researcher. This is an action he would carry out for the duration of the study, with almost ritualistic adherence. It became a part of his routine and he got enjoyment from it (which made it very difficult to reverse).

Table 19.

Andres’ Addition Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points Points Missed (out of 36) (out of 9) 1 Addition 6 3, 4, 5 (VP)

Addition 2 Addition 8 3 (VP) 1 Intervention Probe 3 Addition 7 2, 4 (VP)

4 Addition 7 3, 4 (VP) 28

2 1 Addition 9

92 2 Addition 8 9 (distracted, VP) Addition Intervention 3 Addition 9 Probe 4 Addition 8 8 (distracted, VP) 34

1 Addition 8 4 (chose subtraction, VP)

2 Subtraction 3 4, 5, 6, 7, 8, and 9 Criterion (VP) 3 Test 3 Addition 8 4 (chose subtraction, VP)

4 Subtraction 4 3, 6, 7, 8, and 9 (VP) 23 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

Due to his recent perseverance on subtraction, it was expected that Andres would perform well during this phase. Andres’ task analysis performance levels for subtraction jumped immediately from an average of three correct steps per subtraction problems on his last criterion test to an average of 8.5 (rounded to eight full steps) per subtraction problem on both intervention probes and criterion tests. On the very first problem presented on the first day of subtraction instruction, Andres performed eight out of nine steps correctly. Andres missed several steps during this phase due to a lack of focus. Additionally, on three occasions he missed

“Step 6” (“Fill-In Number Sentence”) because he placed the smaller number in the first box on

93 the SBI number sentence mat and the larger number in the second box, as if to add. The researcher provided verbal prompts to remedy the mistake.

Once the addition problems were re-introduced at the administration of the first criterion test, Andres again reverted to his preference for subtraction. Out of the four addition problems presented between two criterion tests, Andres missed “Step 4” (“Use my Rule”) on three occasions. He would move on with the problem after being prompted by the researcher to choose addition. On one occasion he audibly stated, “I don’t want to add”. Every time he required a prompt to make the correct decision both the addition and subtraction rules were reviewed in the hopes that he would come to understand the difference between the two and eventually choose the correct operation based on the problem. Andres’ performance during the subtraction phase of intervention can be seen in Table 20.

Table 20.

Andres’ Subtraction Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points (out Points (out Missed of 36) of 9) 1 Subtraction 8 6 (distracted, VP) 2 Subtraction 9 Subtraction 3 Subtraction 7 2, 6 1 Intervention (distracted, Probe put small number first, VP)

4 Subtraction 9 33

2 1 Subtraction 9

94 2 Subtraction 9

Subtraction 3 Subtraction 9 Intervention Probe 4 Subtraction 8 6 (puts small number first, VP) 35

1 Addition 8 4 (wants to subtract, VP)

2 Subtraction 9

Criterion 3 Addition 8 4 (picks 3 Test subtraction rule, VP)

4 Subtraction 8 6 (puts small number first, VP) 33

1 Addition 9

2 Subtraction 9

3 Addition 8 4 (wants to pick subtraction, Criterion 4 VP) Test 4 Subtraction 8 7 (distracted, puts incorrect operation, VP) 34 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

95 One week after the completion of the final criterion test, all participants were given another criterion test to determine if they maintained the skill for one week. On his first maintenance criterion test, Andres earned 34 out of 36 total points. He missed “Step 4” (“Use my Rule”) for each addition problem on the test because he wanted to “do subtraction”.

However, this result falls within the range of scores he was receiving during the treatment phase, thus indicating that he had maintained the skill learned for a week. Andres’ maintenance phase performance can be seen in Table 21.

Two weeks after the first maintenance test (or three weeks after the completion of treatment), all participants completed a second maintenance criterion test. On his second criterion test, Andres’ score dropped from 34 to 33 total points out of 36. He did miss “Step 4” on one of the addition problems but completed it correctly for the other. During this session,

Andres was very distracted and placed the numbers from the problem in the incorrect order on his SBI number sentence mat. The researcher provided a verbal prompt to help him fix the mistake and get him re-focused on the task. Andres’ performance significantly improved over the course of the study. He had the most unpredictable and unstable baseline performance and did not appear to have a clear grasp on either addition or subtraction. As soon as the intervention was introduced, he had a routine to follow that helped him to focus in order to problem solve.

Andres did not feel confident enough to fade his use of the task analysis checklist at any point of the study. This was not an expectation and did not affect his performance or results in any way.

The fact that Zachariah and Mercedes felt that they could perform several steps without looking at the checklist was a phenomenon that the researcher did not expect.

96 Table 21.

Andres’ Maintenance Phase Performance

Day Test Type Question Question Question Question Test Total Number Type Total Steps Points Points Missed (out of 36) (out of 9) 1 Addition 8 4 (chooses subtraction, VP)

1 2 Subtraction 9 Criterion (1 week Test out ) 3 Addition 8 4 (chooses subtraction, VP)

4 Subtraction 9 34

1 Addition 8 4 (chooses subtraction, VP)

2 Subtraction 7 6, 7 2 (switches Criterion (3 weeks number Test out) order from problem, VP) 3 Addition 9

4 Subtraction 9 33 Note. VP = Verbal Prompt, GP = Gestural Prompt, PP = Physical Prompt

Overall Outcomes Summary

In reviewing the individual performance of each participant, it is apparent that the modified SBI strategy with specific-ASD supports had a positive effect on their ability to solve addition “group-type” and subtraction “compare-type” math word problems. Each participant

97 demonstrated immediate improvement in his or her accuracy and level of independence from the initial administration of the intervention. The increased levels of performance carried throughout all phases of the study and continued into the post-study maintenance phase. Participants developed a “sense of flow” and strategic planning when presented with each subsequent word problem throughout the course of the study. Qualitatively, the participants expressed positive feelings towards the intervention package. The participants stated that the intervention helped them to solve math word problems although two participants did still indicate concern with the thought of solving math problems on their own. The classroom teacher agreed that all aspects of the SBI strategy were useful and effective tools. She felt that she would have no problem incorporating the various aspects of the modified SBI intervention into her daily mathematics routine. She did indicate some concern surrounding the length of time it took to administer the intervention. She was not sure that she would be able to implement the intervention completely given the amount of time she has allotted for mathematics each day. All stakeholders expressed overall approval of the intervention and quantitative results demonstrated a positive impact on participant performance.

98 Discussion

The results of this modified multiple probe across participants study suggest that the modified SBI strategy with ASD-specific supports was an effective intervention for teaching addition “group-type” and subtraction “compare-type” word problems to the three elementary participants with ASD involved in this study. The results of the present study are similar to others that have investigated the effectiveness of the SBI strategy with students with ASD and ID

(Browder et al., 2018; Rockwell et al., 2011; Root et al., 2017). The present study drew heavily on the work of Root and colleagues (2017), modifying their ASD-specific supports and problem types to fit this participant population’s needs. This study focused on “group-type” addition word problems and “change-type” subtraction word problems specifically to meet the participants at their levels of mathematical performance.

Effectiveness of Modified SBI with ASD Supports

Through phase-by-phase visual analysis and Tau-U measurement, the SBI strategy intervention with ASD specific supports was determined to be effective. The results support that all three participants improved their word problem solving capabilities in terms of accuracy and independence. Participant performance increased from between two-12 correct task analysis steps during the baseline phase to 27-31 correct task analysis steps at the onset of the addition phase of treatment. The transition from the first criterion test to the subtraction phase of treatment resulted in an increase in participant performance from a range of 23-25 correct task analysis steps to a range of 33-34 correct task analysis steps. This suggests a functional relationship between the intervention and participants’ performance on addition “group-type” and subtraction “compare-type” word problems found within the criterion tests. Additionally, all three participants were able to maintain this level of performance for three weeks post-

99 intervention. Participant performance during the maintenance phase ranged from 32-36 total points on the task analysis, which was very similar to their performance throughout both the addition and subtraction phases of treatment.

All three participants demonstrated the ability to independently use the task analysis checklist to assist with problem organization and planning. All participants enjoyed using the dry-erase markers to “check” each individual box on the task analysis checklist, which made the intervention more motivating. The visuals paired with text on the checklist were a necessary feature for participants who were not confident reading word problems independently. For this reason, participants were given the option to ask for the problem to be read aloud to them in step one. Two out of three participants were able to consistently choose the correct rule (either addition or subtraction) to use with each problem. One participant, no matter the strategy used, preferred to “choose” the subtraction rule during step four because he had created a routine around it. With a prompt from the researcher, this participant was able to quickly move on and complete the appropriate operation.

The participants quickly learned the nine-step process and would organize their space at the table to best maximize their workflow. The intervention package included four different laminated sheets (task analysis checklist, ten-frame mat, SBI number sentence mat, and rule cards) that had to be accessed at different points of the intervention process. The participants learned to organize their space to maximize efficiency throughout the course of the intervention.

Even though the participants were able to modify their space to accommodate the intervention materials, this study shed light on the fact that this intervention package may contain too many loose materials to manage.

100 Connection to the Literature

This intervention package was identified as a possible resource for students with ASD and ID after its successful implementation by several research teams (Browder et al., 2018;

Rockwell et al., 2011; Root et al., 2017). Specifically, the current study drew heavily from the work of Root and colleagues (2017) and their analysis of the use of modified SBI with ASD- specific supports to teach three students with ASD how to solve “group-type”, “compare-type”, and “change-type” word problems. All ASD-specific supports created for the current study were modeled directly after the materials used by Root and colleagues (2017). These materials and the intervention were chosen for this current study to fit the unique needs of the study’s participant population. The SBI strategy in conjunction with ASD-specific supports (i.e., task analysis checklist, ten-frame mat, SBI number sentence mat, and rule cards) seemed to support the participants throughout the word-problem solving process, specifically assisting them with planning and organization (John et al., 2017; Ozonoff et al., 1991). The pairing of ASD-specific supports with the SBI strategy may have helped the participants more readily and seamlessly understand the underlying structure of the problem (Jitendra et al., 2009) while also creating a plan to solve it (Hegarty & Kozhevnikov, 1999; Kalyuga & Sweller, 2004). As the study progressed, participant level of independence increased, indicating an improved perception of the word-problem solving process.

An important factor within the instructional design was the multiple-opportunity probe

(as described in Root et al., 2017) feature, which allowed participants the chance to perform each step of the task analysis with prompting from the researcher, if needed. Since word-problems are essentially chained tasks, the successful completion of one phase of the problem depends entirely on the successful completion of the previous phase (Jitendra et al., 2007). The participants of the

101 current study struggled with foundational mathematics concepts in addition to exhibiting deficits in executive functioning (including working memory), metacognition, and social communication. During baseline the participants would become frustrated when they “knew” they were not completing a step correctly but just needed to move on to complete the problem.

Once the intervention was introduced, the attitude of participants changed dramatically and they soon gained confidence, as they understood that the researcher would provide prompting to assist them through the problem if needed. For data collection purposes, the task analysis steps that required prompting from the instructor were not awarded points, but participants appreciated the availability of supports regardless. The level of prompting needed by each participant decreased throughout the course of the study and remained low even in baseline (see Figure 1).

Challenges with Intervention Processes

As with the implementation of any intervention, there were a few missteps and learning experiences along the way. Fortunately, the progress made by the participants outweighed the challenges they faced. The researcher addressed two of the three noted challenges with strategic re-teaching. The third challenge was one of space and materials that inherently led to some questions about efficiency that could be addressed in future research. The three documented challenges are described briefly in the following paragraphs.

Challenges with choosing the correct rule. One participant in particular had a preference for the subtraction rule out of a ritual he created. Though the other two participants did not demonstrate a preference for the subtraction rule, they did exhibit a slight degree of hesitancy when asked to “use their rule”. This uncertainty lasted throughout the early stages of the addition phase until the rules became solidified in their minds. This level of understanding was established around the end of the addition phase of treatment prior to moving into the

102 subtraction phase for Mercedes and Zachariah. It is difficult to say if Andres ever reached this level of understanding due to his proclivity for choosing the subtraction rule. Mercedes and

Zachariah had several instances after the addition phase when they made a mistake and chose the wrong rule, usually due to distraction or because they were working at a hurried pace. It is apparent now that all participants would have benefitted from more instruction on each rule prior to the start of the intervention. It was not until halfway through the study that the participants formulated a true understanding of each rule.

Challenges with number order. At one point throughout the course of the study, each participant placed the smaller number prior to the larger number when solving a subtraction word problem. This phenomenon was not isolated to the early stages of the study, it happened throughout all phases. The participants did not have a clear grasp on the difference between comparing (when subtracting) and grouping together (when adding) that was critical to the successful completion of the subtraction “compare-type” and addition “group-type” problems chosen for the purposes of this study. An understanding of this concept would have led to less errors surrounding number order. Two out of three participants (Zachariah and Mercedes) required only a verbal prompt (i.e., reading the subtraction rule card aloud) to realize their mistake and work to correct it. These two participants would get into a routine when completing their criterion tests and intervention probes that they would at times forget to “think through” the entire process, thus leading to error. For example, when solving the problem “Violet has two dogs. Ivy has three dogs. How many more dogs does Ivy have than Violet?” Zachariah and

Mercedes would write each number (in the order presented in the problem) onto their SBI number sentence mat. When subtracting, it was necessary to reverse the order of the operands from the problem so that the larger number would come first in the number sentence. The

103 participants missed this step and were provided a reminder of the subtraction rule “Big Number,

Small Number, Difference Between the Two”. These types of mistakes could be easily reversed by reminding the participants to “slow down”. One participant, Andres, made this error three times after the intervention had been introduced. When the researcher reviewed the subtraction rule with him, he seemed confused when the researcher asked him to identify the larger number.

Andres possessed some gaps in his mathematical content knowledge regarding number sense or experienced momentary lapses, which caused a breakdown in his overall performance. On all other occasions after the implementation of the intervention, he was able to perform the task correctly.

Challenges with materials. At times, managing four laminated sheets, flat counters, dry erase markers, an eraser, a printed criterion test or intervention probe, and pencil on one table was a challenge. At one point, Mercedes pulled up a second chair to the table so that she could spread all of her laminated sheets out flat and move between chairs to access all of the documents. Regardless of intervention effectiveness, this indicates the need to reduce some of the materials or consolidate them in some way. The results suggest that all aspects of the modified SBI strategy with specific ASD-supports were crucial to the overall success of the intervention. Since the separate aspects of the intervention were not evaluated in isolation, it is impossible to determine which materials could be eliminated based solely on this study.

Additionally, while the participants did enjoy using the dry-erase markers to fill in pertinent information on the SBI number sentence mat and “check-off” finished items on the task analysis checklist, the markers did create a bit of a distraction for some participants. At one point Andres’ dry-erase marker was running low on ink and he continued to perseverate on this fact until a new marker was retrieved for him. Andres was already prone to distraction, so it was

104 very important to make sure that all markers were changed out intermittently to ensure legibility and to decrease distraction.

Limitations and Future Research

The small sample size was the first limitation of the current study. Due to the nature of the single-subject design, a small and selective group of participants is ideal (Kennedy, 2005). It was necessary to find a group of students with ASD who were performing around the same academic level in order to compare the intervention’s effectiveness across participants. To strengthen the external validity of results, future studies should focus on identifying a larger pool of possible participants.

The modified-SBI intervention was introduced solely by the researcher during each session with each participant. The researcher was also the administrator of all intervention probes and criterion tests. Ideally, an independent teacher or test administrator would be used throughout all phases of a future study. The participants of this study became very comfortable working with the researcher throughout all phases of the study. It would be beneficial for future studies to examine participant performance when different adults/researchers introduce various aspects of the intervention package.

This study utilized one-on-one teaching and prompting throughout all phases. One-on- one teaching is a natural element found within single-subject designs. Most students tend to perform better when given individualized attention, instruction, and prompting. The satisfactory performance of the participants could have been due in part to the one-on-one nature of the study. Future studies should try to incorporate other forms of instruction (i.e. small group or whole group) in addition to (or instead of) one-on-one instruction to examine whether participant performance is affected.

105 This study’s main purpose was to evaluate the effectiveness of the modified SBI intervention with ASD-specific supports. However, due to the multi-factored nature of the intervention, it was difficult to determine which components were the most effective. Since the

ASD-specific supports were woven into the overall fabric of the SBI intervention, it was not possible to separate the results during data analysis. Future studies should try to separate out the various components of the multi-faceted intervention in order to analyze each individually to determine which was the most effective.

Implications

Theoretical. As research focused on mathematics performance of students with ASD continues to develop, it is paramount that the studies focused on intervention effectiveness do not lose traction or support. Specifically, more research is needed that focuses on teaching word- problem solving to students with ASD and ID. There is currently a trend to focus on only functional mathematics for students with more profound needs (i.e. moderate ASD or ASD and

ID) (Cihak & Grim, 2008) in both the research and practical realms. As noted previously, mathematical knowledge is one of the greatest predictors of future success (Duncan et al., 2007).

Having a functional understanding of foundational mathematics is necessary for future employment, training, or independent living opportunities (Brown & Snell, 2000). Special education researchers should continue to broaden their focus in order to assess the functional abilities as well as academic achievement of students with more significant needs. The current study adds to the current literature on mathematics word-problem solving interventions for students with ASD and ID.

SBI is considered an evidence-based practice for students with high incidence disabilities, such as learning disabilities (Gersten et al., 2009). With the number of students diagnosed with

106 ASD on the rise it makes logical sense to employ the use of strategies that have been proven effective that could be adapted for use with this population. With approximately (40%) of students with ASD spending the majority of their instruction (i.e., >80%) in the general education classroom (King et al., 2016; USDOE, 2014), it is imperative they receive instruction that meets their diverse needs. However, research in this field needs to guide the way so that special educators, student support personnel, and related service providers can make appropriate and informed decisions based on empirical evidence. The research teams of Browder and colleagues (2018), Rockwell and colleagues (2011), and Root and colleagues (2017) used the existing research base to determine the potential advantage of using a modified SBI with students with ASD. Special education researchers need to continue to push the envelope and think creatively to devise “workable” solutions for students with ASD in schools that may be experiencing budget limitations and staff shortages.

Practical. Students with ASD are receiving instruction in the general education classroom more than they ever have before (Sansoti & Powell-Smith, 2008). Even if students with more moderate or severe ASD are not spending as much time in the general education setting, they are still expected to master academic content standards and participate in some type of high-stakes testing (whether it be the traditional or alternate assessment) (Knight et al., 2012).

High quality mathematics instruction is essential for all students, including students with ASD

(regardless of their placement along the spectrum) (Hudson et al., 2018). However, special educators are often mandated to use the curriculum purchased by their district and they may have little flexibility in the additional materials they are able to purchase (Leachman, Albares,

Masterson, & Wallace, 2016). Access to recent empirical studies evaluating intervention effectiveness is critical for educators who are interested in improving their practice (Borg, 2010;

107 Furlong, Menter, Munn, Whitty, Hallgarten, & Johnson, 2014). Special educators should feel confident in their abilities to modify and adapt instructional materials and methods to meet the needs of all students (similar to the work of the researchers in the aforementioned studies).

Additionally, special educators and those working directly with students with ASD need to search for ways to help this population effectively plan and organize scenarios in order to sharpen their problem-solving abilities (Rockwell et al., 2011; Root et al., 2017). It has been documented that executive functioning (including working memory) inhibits the organizational capabilities of students with ASD (John et al., 2017; Ozonoff et al., 1991). This knowledge should be applied to the problem-solving techniques introduced to this population of learners.

Additionally, students with ASD struggle with metacognition (Grainger et al., 2014) and social communication (Kasari et al., 2010). Interventions chosen need to account for these deficits and provide accommodations accordingly. Providing visual cues to signal a shift or change

(Dettmer, Simpson, Myles, & Ganz, 2000), embedding prompting strategies into instruction

(Ayres & Cihak, 2010; Hudson et al., 2018, and using technology (like VM) (Ayres & Langone,

2005) are all strategies that have been supported by research for use with students with ASD.

The challenge now is to incorporate the various strengths and deficits exhibited by students with

ASD into an intervention package tailored to meet their unique needs.

Conclusion

The current study’s participants made gains in their word-problem solving abilities, comparable to the participants in previous studies, such as the work of Root and colleagues

(2017). This study differed from the previous study in several ways, including: (a) problem type

(specifically chosen to fit the participants’ needs), (b) exclusion of virtual manipulatives (not available to researcher), and (c) shortened study duration (due to time constraints of school site)

108 but still reached the same conclusions of improved word-problem solving ability. The positive results of this study suggest that the modified SBI with ASD-supports is a promising strategy for students with ASD despite problem type and even without virtual manipulatives. Additionally, it appears that this intervention package has the ability to produce results in a shorter amount of time than previously expected. Several participants of this study unexpectedly began to fade their use of the task analysis checklist as they started to internalize the problem-solving process.

This phenomenon indicates the importance of not only fading the intervention but also evaluating the importance of each component of the intervention in an effort to maximize its effectiveness for each individual student.

While word-problem solving is just one facet of mathematics instruction, the art of problem solving (in general) is necessary throughout all realms of independent functioning.

Being a “problem-solver” is a skill that each independent and successful adult possesses. So how do students become independent problem-solvers? SBI provides a great start; students must first understand the underlying structure of the problem prior to solving it (Jitendra & Hoff,

1996). This notion can be applied to any problem-solving scenario (not just math). However, it is almost equally important to also possess the skills needed to organize and plan the problem’s solution (Ozonoff et al., 1991). The current study presented the participants with a modified SBI strategy with ASD-specific supports embedded to (1) help them recognize underlying problem structure and (2) map out a plan to solve it. This small, single-case study affected three participants with ASD in one aspect of their lives. The benefits of providing tailored problem- solving instruction to students with ASD across all aspects of their lives are countless. The opportunities are endless and the importance of delivering this instruction is paramount, as the number of students with ASD transitioning to adulthood increases each year.

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132 Appendices

Appendix A: Schema-Based Instruction Number Sentence Mat

133 Appendix B: Ten Frame Mat Difference

Total

Group 2 Group Group 1 Group

134 Appendix C: Task Analysis Check List

1. Read the problem 2. Circle the “what” Apple 3. Write your How many ____? label “what” 4. Use my rule

5. Circle the 5 numbers 6. Fill-in number sentence 7. + or -

8. Make Set

9. Solve & Write Answer

135 Appendix D: Rule Cards Addition Rule

Look at the labels How many apples?

Find the same He has 5 apples. She has 3 apples. He has 2 carrots. Add them all 5 apples + 3 apples = 8 apples together

Subtraction Rule

Big Number

Small Number

Difference Larger Number – Smaller Number = ____ between the 2

136 Appendix E: Task Analysis: Expected Student Responses

Step Expected Student Response 1. Read the problem Teacher reads problem aloud or student attempts to read it

2. Circle the “whats” in the problem Find and circle the items (units) within the problem being changed

3. Find label in the problem Find the question mark and then find the label (i.e., the word that comes after “how many”); put the word in number sentence above underline

4. Use my rule Stated rules for both addition (“first group plus second group, equals total”) and subtraction (“larger number minus smaller number equals difference”) 5. Find how many Circle the numbers in the problem 6. Fill-in number sentence Insert circled numbers into the number sentence

7. + or - Determine whether subtracting or adding and put appropriate symbol into the circle within the number sentence

8. Make sets Used manipulatives to demonstrate the problem on ten frame

9. Solve and write answer Used manipulatives to determine total (pushing all manipulatives to the total box) or difference (pushing those manipulatives without a match to the difference box); and wrote the number of manipulatives in the final box of the number sentence

137 Appendix F: Social Validity Measure (Student)

Social Validity Measure (student)

Name: ______

Date: ______

1. Using the number sentence mat helped me solve my math problems.

Yes No

2. Using the ten-frame grid helped me solve my math problems.

Yes No

3. Using the checklist helped me solve my math problems.

Yes No No

4. I feel like I can solve my math problems by myself.

Yes No

5. I like solving math word problems.

Yes No

6. What did you like about our math lessons together?

______

138 Appendix G: Social Validity Measure (teacher)

Social Validity Measure (teacher)

Name: ______

Date: ______

Please respond by rating each statement below.

(1 = completely disagree, 2 = mostly disagree, 3 = slightly disagree, 4 = slightly agree,

5 = mostly agree, 6 = completely agree).

1. I feel that the number sentence mat was useful for my students.

1 2 3 4 5 6

2. I feel that I would use the number sentence mat within my own instruction. 1 2 3 4 5 6

3. I feel that the ten-frame grid was useful for my students. 1 2 3 4 5 6

4. I feel that I would use the ten-frame grid within my own instruction. 1 2 3 4 5 6

5. I feel that checklist was helpful for my students. 1 2 3 4 5 6

6. I feel that I would use the checklist within my own instruction. 1 2 3 4 5 6

7. I feel that I could use this intervention as a whole package within my own instruction. 1 2 3 4 5 6

139 8. I feel that this intervention met the current needs of my students. 1 2 3 4 5 6

9. I feel that I would have enough time in my day to use this intervention (i.e. it was an effective and efficient strategy). 1 2 3 4 5 6

10. I feel that I could teach this strategy to a teacher’s aide or other instructor to use with students. 1 2 3 4 5 6

11. I feel that my students have gained some independence in regards to solving math word problems as a result of this intervention. 1 2 3 4 5 6

12. I feel that my students maintained an appropriate level of engagement throughout the intervention. 1 2 3 4 5 6

Please offer any additional feedback you may have in regards to the intervention,

materials, or instructional methods.

______

140 Appendix H: Addition Intervention Probe (Example)

Name: ______

Date: ______

1. Maria has 5 rings. Sally has 3 rings. Anabelle has 4 necklaces. How many rings do they have in all?

2. Jason buys 6 bananas. Terry buys 1 banana. Angelina buys 2 oranges. How many bananas do they buy altogether?

3. Brett has 4 marbles. Alex has 4 marbles. Helen has 2 jacks. How many marbles do they have in all?

4. Louisa has 5 stickers. Isabella has 2 stickers. Matthew has 7 erasers. How many stickers do they have altogether?

141 Appendix I: Subtraction Intervention Probe (Example)

Name: ______

Date: ______

1. Paul has 9 crayons. His friend Tricia has 3 crayons. How many more crayons does Paul have than Tricia?

2. Nathan bought 5 flowers. His friend Zeke bought 2 flowers. How many more flowers did Nathan buy than Zeke?

3. Hank has 3 erasers. Natalie has 7 erasers. How many more erasers does Natalie have than Hank?

4. Joe has 7 eggs. Stephanie has 5 eggs. How many more eggs does Joe have than Stephanie?

142 Appendix J: Criterion Test (Example)

Name: ______

Date: ______

1. Jordan has 9 books. Lauren has 4 books. Lydia has 5 balls. How many books do they have in all?

2. Emily has 7 pictures. Her brother John has 9 pictures. How many more pictures does John have than Emily?

3. Ron finds 8 acorns at the park. Matt finds 3 acorns at the park. Jimmy finds 2 sticks at the park. How many acorns do they find altogether?

4. Mario made 6 cards. Lena made 8 cards. How many more cards did Lena make than Mario?

143