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PAGE 1 1 TEACHING STUDENTS WITH AUTISM TO SOLVE ADDITIVE WORD PROBLEMS USING SCHEMA BASED STRATEGY INSTRUCTION By SARAH B. ROCKWELL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMEN T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012 PAGE 2 2 2012 Sarah B. Rockwell PAGE 3 3 To all of the children with autism who h ave inspired me to do this work PAGE 4 4 ACKNOWLEDGMENTS First, I would like to thank my docto ral committee Doctors Cynthia Griffin, Brian Iwata, Hazel Jones, and Thomasenia Adams, for always being willing to listen to my ideas, answer my questions, and guide my inquiry. Next, I thank my amazing husband, Alex Rockwell, who enco uraged me to pursue m y doctoral degree. Without his support love and understanding this would not have been possible. I also thank our beautiful son, Devin, who tolerated coun tless hours with a rotating cast of volunteer babysitters so that I could complete my research and w rite my dissertation and who made me smile and laugh even when I thought the stress might overtake me Next, I offer my sincere and profound appreciation to my pa rents Michele and Walter Belew; my in laws Nora, Skip and Peter Rockwell ; and my friends Jen nifer Reardon, Martyna Levay, Valerie Walters, Marin Smillov, Tim Gavey, Michelle Neilson, and Judith and Bob Kendall who volunteered their time and stretched their patience caring for Devin while I completed my dissertation. I also offer my sincere thanks to my friends Sarah Key DeLyria Aile Montoya, Judith Kendall, and Megan Jobes who volunteered for the tedious job of proofreading my dissertation Finally, I owe my most heartfelt gratitude to my participants and their families for making this study poss ible PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBREVIATIONS ................................ ................................ ........................... 10 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INT RODUCTION ................................ ................................ ................................ .... 12 Language and Executive Functioning Profiles of Children with Autism .................. 13 Impact of Language and Executive Functioning on Mathematics Performance ...... 14 Need for Research Based Mathematics Instruction for Students with Autism ........ 16 Purpose and Research Questions ................................ ................................ .......... 18 Definitions of Terms ................................ ................................ ................................ 19 Problem Type Definitions ................................ ................................ ................. 20 Unknown Quantities and Algebr aic Reasoning. ................................ ............... 21 Delimitations and Limitations of the Study ................................ .............................. 22 2 REVIEW OF THE LITERATURE ................................ ................................ ............ 23 Schema Based Instruction and Additive Word Problem Solving ............................. 23 Purpose of the Literature Review ................................ ................................ ............ 26 Literature Review Search Procedures ................................ ................................ .... 27 Synthesis of SBI Research ................................ ................................ ..................... 28 SBI with Representative Diagrams ................................ ................................ ... 29 SBI with Self Monitoring ................................ ................................ ................... 31 SBI with Multi Step Problems ................................ ................................ ........... 35 SBI with Additional Instruction ................................ ................................ .......... 37 SBI for Individuals with Developmental Disabilities ................................ .......... 40 Synthesis of Schema Broadening Research ................................ .......................... 44 Implications for Future Research ................................ ................................ ............ 47 3 METHOD ................................ ................................ ................................ ................ 49 Participants ................................ ................................ ................................ ............. 49 Inclusion and Exclusion Criteria ................................ ................................ ....... 49 Participant Informati on for Daniel ................................ ................................ ..... 50 Participant Information for Justin ................................ ................................ ...... 51 Independent Variable ................................ ................................ .............................. 53 PAGE 6 6 Setting ................................ ................................ ................................ .............. 53 Treatment ................................ ................................ ................................ ......... 53 Treatment Integ rity ................................ ................................ ........................... 59 Dependent Variable ................................ ................................ ................................ 59 Formative Assessment ................................ ................................ ..................... 6 0 Probes ................................ ................................ ................................ .............. 60 Reliability and Validity Evidence ................................ ................................ ....... 62 Data Collection and Analysis ................................ ................................ .................. 64 4 RESULTS ................................ ................................ ................................ ............... 65 Treatment Integrity and Inter Rater Reliability ................................ ........................ 65 Break Use ................................ ................................ ................................ ............... 65 ................................ ................................ ... 66 Group Problems ................................ ................................ ............................... 66 Change Problems ................................ ................................ ............................. 67 Compare Problems ................................ ................................ .......................... 67 Generalization to Problems with Irrel evant Information ................................ .... 69 Generalization to Problems with Unknowns in the Initial Position .................... 70 Generalization to Problems with Unknowns in the Medial Position. ................. 72 Maintenance ................................ ................................ ................................ ..... 72 ................................ ................................ ....... 73 ................................ ................................ ... 76 Generalization to Problems with Irrelevant Information ................................ .... 77 Generalization to Problems with Unknowns in the Initial Position .................... 80 Generalization to Problems with Unknowns in the Medial Position .................. 82 Maintenance ................................ ................................ ................................ ..... 84 ................................ ................................ ........ 84 Satisfaction Data ................................ ................................ ................................ ..... 87 Summary of Results ................................ ................................ ................................ 89 5 DISCUSSION ................................ ................................ ................................ ......... 91 SBI for Students with Autism ................................ ................................ .................. 91 Participant Profiles and Impact on Problem solving Performance .................... 91 Shortcomings of Traditional Problem solving Instruction ................................ .. 94 Benefits of SBI ................................ ................................ ................................ .. 98 Limitations ................................ ................................ ................................ ............. 105 Implications ................................ ................................ ................................ ........... 107 APPENDIX A STUDY SUMMARY TABLE ................................ ................................ .................. 110 B SAMPLE LESSON SCRIPT ................................ ................................ .................. 121 C SAMPLE LESSON CHECKLIST ................................ ................................ ........... 129 PAGE 7 7 D SAMPLE PRACTICE SHEETS WITH HYPOTHETICAL RESPONSES AND SCORING ................................ ................................ ................................ ............. 131 E SAMPLE PROBES WITH HYPOTHETICAL RESPONSES AND SCORING ....... 137 F SATISFACTION SCALES ................................ ................................ ..................... 149 G INFORMED CONSENT DOCUMENTS ................................ ................................ 151 LIST OF REFERENCES ................................ ................................ ............................. 154 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 161 PAGE 8 8 LIST OF TABLES Table page A 1 Summary Table ................................ ................................ ................................ 110 PAGE 9 9 LIST OF FIGURES Figure page 1 1 Group diagram ................................ ................................ ................................ .... 20 1 2 Change diagram ................................ ................................ ................................ 21 1 3 Compare diagram ................................ ................................ ............................... 21 4 1 Points earned by Daniel on probes and practice sheets ................................ ..... 74 4 2 Points earned by Justin on probes and practice sheets. ................................ .... 85 PAGE 10 10 LIST OF ABBREVIATION S ASD Autism Spectrum Disorder AYP Adequate Yearly Progress IEP Individualized Education Program NCLB No Child Left Behind Act SBI Schema based Strategy Instruction PAGE 11 11 Abstract of Dissertation Presented to the Graduate School of th e University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TEACHING STUDENTS WITH AUTISM TO SOLVE ADDITIVE WORD PROBLEMS USING SCHEMA BASED STRATEGY INSTRUCTION By Sarah B. Rockwell August 20 12 Chair: Cynthia C. Griffin Major: Special Education Students with autism often struggle with mathematical word problem solving due to executive dysfunction and communication impairment. The purpose of this study is to provide evidence of the efficacy o f using schema based strategy instruction (SBI) to improve the additi on and subtraction word problem solving performance of elementary school students with autism. A first grade student with autism and a sixth grade student with autism were taught to use s chematic diagrams to solve three types of addition and subtraction word problems based on the semantic structure of the problems. A multiple probes across behaviors single case design was used with solving each of the three problem types treated as a separ ate behavior This design was replicated across problems were analyzed prior to and following SBI. Finally, partic i pants and their parents completed s atisfaction scales regar ding SBI Results indicated that problem solving performance improved following SBI, improvements were maintained over time, and participants and their parents were satisfied with the SBI. Obs ervations of changes in problem solving performance suggest that SBI may res ult in increased use of problem solving strategies and self monitoring. PAGE 12 12 CHAPTER 1 INTRODUCTION Autism Spectrum Disorders (ASDs) are becoming increasingly prevalent within the school aged population. In 2002, one in 152 children were diagnosed w ith an ASD (Croen, Gre ther, Hoogstrate, & Selvin, 2003 ). By 2006, the number of children diagnosed with an ASD had increased to one in 110 (Centers for Disease Control and Prevention, 2009). Due to the increasing prevalence of ASDs among school aged childr en, it is important to consider the impact of characteristic and secondary features Diagnostic and Statistical Manual of Mental Disorders, Fourth Edition, Text Revision (DSM IV TR; American Psychiatric Asso ciat ion, 2000), autism is characterized by communication and social difficulties, as well as repetitive and ritualistic interests and behaviors. In addition, executive dysfunction is considered to be a secondary characteristic of autism such that Attentio n Deficit Hyperactivity Disorder (ADHD) and autism cannot be comorbidly diagnosed. These two disorders are characterized by differences in both the severity and profile of executive dysfunction, as well as disparate diagnostic criteria and treatment regime ns demonstrating the uniqueness of the disorders (Pennington & Ozonoff, 2006). Intellectual disabilities are also typical of individuals with autism. According to Ghaziuddin (2000), approximately 75% of individuals with autism also have intellectual disabi lities as characterized by a full scale intelligence quotient of less than 70. However, compared to individuals with intellectual disabilities who do not have autism, those with autism are more likely to have visual strengths as noted by a performance or nonverbal intelligence quotient in the average range. PAGE 13 13 Language and Executive Functioning Profiles of Children with Autism According to Nation and Norbury (2005), language and reading comprehension difficulties associated with autism are related and inclu de difficulties integrating information, accessing prior knowledge, resolving anaphoric references, and monitoring comprehension. Several studies have assessed the language and comprehension profiles of children with autism. For instance, Frith and Snowlin g (1983) conducted multiple experiments to assess the reading ability and comprehension skills of children with autism as compared to ability matched readers without autism. One of their findings indicated that children with autism had difficulty making us e of semantic cues when comprehending sentences. Other researchers found that compared to typically developing children, children with high functioning autism had comprehension and recall deficits, such as difficulty making inferences (Norbury & Bishop, 20 02) The authors hypothesized that pragmatic language deficits associated with autism interfered with ) found that students with autism have difficulty resolving anaphori c references when reading and were able to improve their reading comprehension when given cues to resolve these references Executive dysfunction may include difficulties with planning, organization, switching cognitive set, and working memory (Zentall, 2 007). Switching cognitive set involves the ability to quickly change focus from one activity to another and working memory involves the ability to hold information in immediate awareness while manipulating that information ( Geary, Hoard, Nugent, & Byrd Cra ven, 2008 ). For instance, being able to remember the numerical information contained in a mathematics word problem while also determining which computational operations to perform to PAGE 14 14 solve the problem requires working memory. Researchers have found that ch ildren with autism are differentially impaired in executive functioning in the areas of vigilance, working memory, planning, and set shifting when compared to typically developing children and children with ADHD (Corbett, Constantine, Hendren, Rocke, & Ozo noff, 2009). When compared to ability matched and language matched children, children with autism were found to be differentially impaired in the areas of working memory, planning, set shifting, and inhibitory control (Ozonoff, Pennington, & Rogers, 1991; Hughes, Russel, Robbins; 1994). Additional research has explored the working memory abilities of children with autism. According to Gabig (2008), children with autism have greater difficulties with ve rbal working memory than do age matched controls. These difficulties become more pronounced as vocabulary and language processing demands increase. Compared to ability matched c ontrols, children with autism a re less likely to use verbal rehearsal strategies leading to poor verbal working memory (Joseph, Steel e, Meyer, & Tager Flusberg, 2005 ) and are less likely to use organized search strategies leading to poor spatial working memory (Steele, Minshew, Luna, & Sweeney, 2007). Impact of Language and Executive Functioning on Mathematics Performance According to Do nlan (2007), one of the areas in which language impairment problems requires that students use semantic mapping to determine the relationships between known and unknown quantit ies to determine the correct operation to use to solve for the unknown (Christou & Phillip pou, 1999 ). The failure of students with autism to use semantic cues (Frith & Snowling, 1983) has important implications for teaching PAGE 15 15 word problem solving because it will likely make semantic mapping more difficult for these students. In addition, language comprehension deficits impact the ability of students to solve word problems. In a longitudinal study, Jordan, Hanich, and Kaplan (2003) compared children with read ing disabilities, reading and math disabilities, and math disabilities. They found that language comprehension deficits led to poorer performance on word problems for students with reading disabilities compared to those with math disabilities only. In addi tion, Cowan, Donlan, Newton, and Lloyd (2005) found that children with language impairments had greater difficulty solving word problems than did typical children due to phonological working memory and language deficits. Executive dysfunction, particularl y in the areas of sustained attention and working atics progress and word problem solving performance (Zentall, 2007). For i nstance, Zentall and Ferkis (199 3) found that children with executive dysfunction had difficul ty filtering and manipulating relevant informa tion to facilitate word problem solving due to working memory and reading difficulties. Working memory deficits may also lead to the use of immature problem solving strategies and procedures and difficulty inh ibiting irrelevant information leading to errors in problem solving (Geary, Hoard, Nugent, & Byrd Craven, 2008 ). In fact, working memory deficits may underlie many mathematics difficulties (Geary, Hoard, Byrd Craven, Nugent, & Numtee, 2007). Geary and his colleagues compared first grade students with mathematics difficulties to average and low achieving first graders. They found that the students with math difficulties scored one standard deviation lower than PAGE 16 16 did the other students on measures of working me mory and that these deficits in working memory partially or fully mediated def icits in mathematics cognition. Need for Research Based Mathematics Instruction for Students with Autism Problem solving is a primary goa l of mathematics instruction and a critic al process for successful functioning in an increasingly technological and mathematically oriented society (National Council of Teachers of Mathematics [NCTM], 2002). Moreover, federal legislation has mandated that schools be come increasingly accountable f or the academic progress of all students. The No Child Left Behind Act (NCLB 2002 ) requires that schools demonstrate Adequate Yearly Progress (AYP) toward proficiency of all students on assessments of reading and mathematics. Additionally, the Individuals with Disabilities Education Improvement Act ( IDEA; Assistance to States for the Education of Children with Disabilities 2004) requires that students with disabilities have access to and make progress in the ge neral education curriculum. NCLB also require s that states assess and document the progress of at least 95% of students with disabilities according to statewide standards Although some students with disabilities are assessed using alternate standards, the majority of these students are evaluated aga inst the same high academic standards as their non disabled peers. Within the area of mathematics, the focus of the general education curriculum and of AYP assessments is increasingly on conceptual understanding and problem solving rather than on computati onal proficiency (Bottge, 2001). According to Bottge helping students become proficient problem solvers has proven challenging. S tudents with disabilities advance at a slower rate than their non disabled peers resulting in ever increasing discrepancies an d limited proficiency on tests of mathematics competence. For instance, on the National Assessment of Educational Progress, only 40 % of fourth PAGE 17 17 grade students who took the assessment in Florida were proficient on complex mathematical problem solving tasks ( Institute of Educational Statistics, 2007 ). Therefore, fostering successful problem solving performance in mathematics for students with disabilities, including those with autism is critical. Despite the need to identify scientifically based practices for teaching mathematical problem solving to students with autism, few studies have attemp ted to improve the word problem solving performance of these students. However, a body of research has evaluated the efficacy of interventions aime d at improving the mat h problem solving performance of students with math difficulties and learning difficulties. In their meta analysis, Gersten et al. (2009) reviewed 42 studies aimed at improving ematics performance. The stu dies focused on problem solving used various interventions including instruction in general problem solving procedures or heuristics, multiple strategy instruction, peer assisted instruction, direct instruction, and instruction using visual representations. Results of the meta analysis indica ted that instruction using visual representations resulted in large effect sizes, particularly when visual representations were paired with heuristics and direct instruction. Schema based strategy instruction (SBI) is an intervention characterized by visu al representations and direct instruction for teaching students to solve mathematical word problems. The results from several studies have demonstrated the efficacy of using SBI to im prove the additive word problem solving performance of students with lear ning disabilities and low performing students in resource and inclusive settings (e.g., Griffin & Jitendra, 2009; Jitendra et al., 2007b; Jitendra et al. 1998 ). SBI may hold particular promise for students with autism due to the difficulties these childre n experience with PAGE 18 18 communication, executive functioning, and working memory. Semantic mapping during word problem solving may be facilitated through the use of visual supports like those included in SBI (Mesibov & Howley, 2003; Tissot & Evans, 2003). In add ition, s chema induction and rule automiz ation have been found to reduce cognitive load (Cooper & Sweller, 1987), and schema induction following SBI has been found to reduce working memory demands and cognitive load (Capiz zi, 2007). Finally, the problem sol ving heuristic component of SBI, which includes verbal rehearsal and a mnemonic, provides support for attention and organization (Deshler, Alley, Warner, & Schumaker, 1981). Despite the potential benefits of using SBI to teach students with autism to solve additive word problems, only one such study has been identified. This preliminary, but promising, study conducted by Rockwell, Griffin, and Jones ( 2011 ) examined SBI with a fourth grade student with autism. Therefore, further research on using SBI with st udents with autism is warranted. Purpose and Research Questions This study wa s designed to evaluate the use of SBI to teach additive word problem solving to students with autism. Research questions included (a) W ill students with autism demonstrate improve ment in their ability to solve one step additive word problems with the unknown in the final position following SBI? (b) W ill chang es in generalize to problems with the unknown in the initial position? (c) Will chang es in generalize to problems with the unknown in the medial position? problem solving generalize to proble ms with irrelevant information? ( e problem solving be mainta ined eight weeks following the intervention? (f) How do the problem solving behaviors of students with autism change following SBI? (g) How do PAGE 19 19 stu dents with autism perceive SBI? and (h) How do parent s of students with autism perceive SBI? Definitions of Te rms S CHEMA BASED STRATEGY INSTR UCTION ( SBI ) An intervention that i nvolves using direct instruction to teach students to discriminate types of mathematical word problems found in the theoretical literature based on their semantic structure and to use schem atic diagrams to facilitate problem solving (Willis & Fuson, 1988). D IRECT INSTRUCTION Teacher directed, explicit approach to instruction involving teacher modeling, guided practice, independent practice, and continuous teacher feedback (Rosenshine & Ste vents, 1986) S CHEMATIC DIAGRAMS Visual representations of the semantic st ructure of group change, and compare word problems by visually depicting the relationships among the known quantities and the unknown quantity for which students must solve. A DDI TIVE WORD PROBLEMS T hose word problems that require addition or subtraction to achieve a problem solution. G ENERAL PROBLEM SOLVING HEURISTICS A sequence of steps u sed to encourage st udents to monitor their problem solving process. For instance, a four s tep heuristic represented by the mnemonic FOPS has been used to ensure that students f ound the information in the problem, o rganized the information using a diagram, p lanned to solve the problem, and s olved the problem (e.g., Jitendra et al. 2007 b ). PAGE 20 20 Probl em Type Definitions Group problems consist of two smaller groups, termed parts which are combined to form a larger group, or all Consider the story situation: Jen bought 12 apples and 15 oranges. Jen bought 27 pieces of fruit. In this problem the two pa rts are 12 apples and 15 oranges while the all is 27 pieces of fruit. Part Part All Figure 1 1. Group diagram adapted from Willis and Fuson (1988). Change problems consist of a beginning amount, change amount indicated by an action, and an ending am ount. The change amount in the problem can indicate addition story situation: J en had 12 oranges. She ate 3 of them. Now she has 9 oranges. This is 3 diagram can be adapted to reflect the larger beginning amount and smaller ending amount by increasing the width of the box representing the beginning amount Now, consider the story situation: Jen had 12 oranges. She picked 14 more. Now she has 29 a beginning amount of 12 oranges change amount of +14 oranges and ending amount of the change diagram can be adapted to reflect the smaller beginning amount and larger ending amount by inc reasing the width of the box representing the ending amount PAGE 21 21 Figure 1 2. Change diagram adapted from Willis and Fuson (1988) Compare problems consist of a larger amount, a smaller amount, and a difference, which results from comparing the small er and larger amounts. Compare problems can be written in two forms. Consider the story situation: Jen has 22 apples. She has 15 oranges. Jen has 7 more apples than oranges. This story situation can also be written: Jen has 22 apples. She has 15 oranges. J en has 7 fewer oranges than apples. In both cases, 22 apples is the larger amount, 15 oranges is the smaller amount, and 7 oranges is the difference. Figure 1 3. Compare diagram adapted from Willis and Fuson (1988). Unknown Quantities and Algebr aic Reasoning. represent and analyze mathematical situations and structures using algebraic symbols p. 37). Additive word problems include two known quantities and one unknown quantity for which s tudents must solve. When this unknown quantity is located in the final position of the schematic diagram (i.e., the all in group problems, ending in change problems, and difference in compare problems), the schematic diagram can be translated into a number sentence that can be solved without algebraic reasoning. When the unknown quantity is in the Larger Amount Smaller Amount Difference ending change beginning PAGE 22 22 initial or medial position in the schematic diagram, the number sentence derived from the schematic diagram can be solved only through the application of algebrai c reasoning. Delimitations and Limitations of the Study This study wa s intended to investigate the use of SBI to teach students with autism who have no comorbid intellectual disabilities to solve one step additive word problems. This study d id not address the use of other problem solving interventio ns. It also did not address the use of SBI with students with comorbid intellectual disabilities or with disabilities other than auti sm. Furthermore, this study did not address multiplicative word problems, mult i step word problems, or other types of mathematical problem solving. This study has several limitations. First, the inclusion only of students with autism but without comorbid intellectual disabilities, the use of a single subject design, and the inclusi on of only one step additive word problems limit the generalizability of the findings. Generalizability is further limited because participants wer e recruited from the Center for Autism and Related Disabilities at the University of Florida which serves ch ildren with autism in North Central Florida. Furthermore, because SBI was conducted by the researcher rather than by classroom teachers in a school setting the feasibility of training teachers to conduct SBI with fidelity could not be determined. Social validity was also reduced because typical intervention agents did not conduct the SBI in a typical instructional setting PAGE 23 23 CHAPTER 2 REVIEW OF THE LITERATURE Schema Based Instruction and Additive Word Problem Solving Additive word pr oblem solving requires that students be able to apply their knowledge of whole number operations while simultaneously manipulating information presented in written form (Verschaffel, Greer, & De Corte, 2007). Students who struggle with this type of problem solving may do so for a variety of reasons. Geary and Hoard (2005) present a conceptual framework for viewing mathematics difficulties that takes into account the supporting competencies and underlying systems that support successful mathematics performan ce. According to the authors children may experience mathematics difficulties due to weaknesses in conceptual understanding or in procedural knowledge. Such weaknesses may lead to difficulties with actual problem solving. In their model, the central execu tive provides the attention control and working memory needed to carry out procedures for successful problem solving. In addition, the language system and visuospatial system allow students to represent and manipulate information in order to develop concep tual knowledge and carry out procedures. Because students with autism have deficits in the language system and central executive, these students will likely have difficulties building concepts for whole number operations and carrying out procedures necessa ry for word problem solving. To addre ss the word problem solving difficulties of students with autism, it may be helpful to consider ways to mediate deficits of the language system and central executive, and to directly teach necessary conceptual and proce dural knowledge. SBI is a promising intervention because it addresses deficits of the language system and central executive while directly teaching conceptual and procedural PAGE 24 24 knowledge. The schematic diagrams included in SBI provide support for the language system by allowing students to visually represent the semantic information presented in word pro blems. In addition, the problem solving heuristics included as part of SBI provide procedural supports for students. Finally, by teaching the critical features of different problem types, SBI helps students develop conceptual knowledge needed to efficiently solve word problem types. SBI is a schema knowledge mediated method of teaching mathematical word proble m solving. All schema knowledge mediated instruction has its basis in research on analogical problem solving from the field of cognitive psychology. Gick and Holyoak (1983) define analogical problem solving as applying the solution strategy of a known problem to a novel problem with similar underlying struct ure. Accomplishing this transfer of solution strategy requires that an individual be able to delete irrelevant dissimilarities between problems while preserving relevant commonalities. After exposure to multiple similar problems, individuals are able to de velop a schema, or mental representation, of a generalized problem solution strategy for that type of problem. This generalized problem schema then acts as a mediator that facilitates transfer to analogous problems leaving additional cognitive capacity ava ilable to allow the individual to handle novel features of problems (Cooper & Sweller, 1987; Gick & Holyoak). To determine whether instruction in abstract problem schemas would facilitate transfer to novel problems, Gick and Holyoak (1983) conducted a stud y with college students. Findings suggest that when participants were provided with just one analog problem from which to abstract the generalized problem schema, additional instruction was not helpful. However, when participants were provided with two pro blems from PAGE 25 25 which to abstract the generalized problem schema, they benefited from provision of verbal and diagrammatic representations of the generalized problem schema. Provision of verbal and diagrammatic representations resulted in higher quality schema and more successful solution of analogous problems. In a similar study, Chen and D ae h ler (1989) sought to determine whether explicit instruction in problem schema would facilitate transfer to novel problems for six year old children. They found that expli cit instruction in problem schema did lead to improved transfer to analogous problems; however, children continued to apply the learned schema even when problems were not analogous. Chen and D ae h ler concluded that children might need additional instruction to prevent misapplication of learned schema to dissimilar problems. To extend these findings, Chen (1999) conducted a study with elementary school students to assess whether exposure to multiple problems with disparate procedural features would help child ren build higher quality and broader container must be filled using several smaller containers without any water being spilled or left in a smaller container. He compared the sche ma formation of children who were given several problems that could be solved using the same combination of smaller containers to those given problems that required different combinations of smaller containers. He found that exposure to multiple problems w ith varying procedural features such that each problem required a different combination of smaller containers resulted in higher quality schemas and facilitated transfer to a wider variety of problems. Taken together, these three studies of analogical prob lem solving indicate that both children and adults can benefit fro m multiple exposures to problem solving tasks PAGE 26 26 and explicit instruction in generalized problem solution schema when solving problems. Due to the promise of research on schema induction in ana logical problem solving, additional studies were conducted to assess applicability to mathematical word problem solving. First, studies with children and studies involving computer simulations of word problem solving were conducted to determine the schemat ic structures underlying one step addi tive word problems (Verschaffel et al., 2007). These studies revealed 14 underlying schemas for one step additive word problems that were condensed into three primary categories of word problems: combine or group, chan ge, and compare. Additional research has demonstrated that children develop understanding of these three primary additive schemas gradually as they mature. For instance, Carpenter olving change problems, to combine problems, and finally to compare problems. In contras t, Christou and Phillippou (1999 schemas progressed from combine, to change, to compare. In both of these studies children were first able to solve problems from a given schema in which unknowns were located in the final position (missing sum, missing difference) and were later able to solve problems with the unknown in the initial position (missing addend). Both se ts of researchers also hypothesized that instruction in problem schemas might help students move along the trajectory of schema development more quickly. Purpose of the Literature Review Due to the contribution of schema theory to t he field of mathematics problem solving instruction, this review will focus on studies of schema based instruction to enhance mathematical problem solving The intent of the review is to synthesize the research base on using schema based instruction to improve additive word probl em PAGE 27 27 solving skills The goal is to determine instructional features that lead to successf ul schema induction and problem solving transfer on additive word problems, and to determine which instructional features can be adapted for use with students with auti sm. The question guiding this inquiry is: Which features of interventions that enhance schema induction and additive word problem solving can be successfully applied to a sc hema based mathematical problem solving intervention with students with autism? In addressing this question, the unique mathematical difficulties of children with autism as well as instructional strategies that have generally been found to be effective with these students and others with disabilities will be considered Relevant research studies will be reviewed and major conceptual, methodological, and design/measurement issues will be discussed. In conclusion, implications for future research will be addressed Literature Review Search Procedures Due to the contribution of schema theory to t he field of mathematics problem solving instruction, this review will focus on studies of SBI to enhance mathematical problem solving. In identifying relevant research for this review, several criteria were used. First, only studies that utilized SBI to teach the three types of additive word problems were formally reviewed. Also, in an effort to review more recent research, only studies published since 1983, when Gick and Holyoak published their seminal study on analogical problem solving, were include d. Finally, only studies published or submitted for publication in refereed journals were reviewed to ensure that rigorous research was included. First, literature was obtained through searches of the following databases: Academic Search Premiere, Profess ional Development Collection, PsychInfo, and Psychological and Behavioral Sciences Collection. Keyword searches were conducted PAGE 28 28 using combinations of the terms schema, problem solving, and mathematics These searches yielded 18 empirical studies. Seven of t hese studies, which used SBI to teach additive word problem solving, were included in this review. Three studies were excluded because they used SBI to teach multiplicative word problem solving to middle school students. An additional seven studies used sc hema broadening instruction, which uses verbal representations to teach problem types derived from third grade basal curricula, rather than SBI. These studies will be discussed separately. Because databases were found to be incomplete, an ancestral search of the reference lists of the published literature was performed. This search yielded one additional empirical study that met the criteria for inclusion in this review. Three additional studies that were known to the author were also included. The final sa mple contained 12 empirical studies, of which one employed a case study design; four a single subject research design; and seven a group experimental, quasi experimental, or pre experimental design. A co mplete summary of SBI and schema broadening studies i ncluded in this review can be found in Appendix A Synthesis of SBI Research As discussed previously, SBI consists of using direct instruction to teach students to discriminate types of mathematical word problems based on their semantic structure, using s chematic diagrams to facilitate problem solving. Although all 1 2 studies included direct instruction, schematic diagrams, and problem types obtained from the theoretical literature in their definitions of SBI, they differed in the specific procedures they used to implement SBI and the populations with whom they implemented the intervention. In two studies (Fuson & Willis, 1989; Willis & Fuson, 1988), the focus was on using highly representative diagrams and whole class instruction to teach average and high PAGE 29 29 achieving students. In three of the studies (Jitendra & Hoff, 1996; Jitendra et al., 1998; Jitendra, Griffin, Deatline Buchman, & Sczesniak, 2007a ), a self monitoring component was added to the intervention, which was primarily used to teach students with high incidence disabilities. Another two studies (Jitendra et al., 2007b; Griffin & Jitendra, 2009) extended SBI to teaching students in inclusive settings to solve multi step problems. SBI in two other studies (Fuchs et al., 2008; Xin Wiles, & Lin 2008) involving students with high incidence disabilities was combined with other types of instruction to achieve greater gains in mathematics achievement. In the final t hree studies ( Jitendra, George, Sood, & Price, 2010; Neef, Nelles, Iwata, & Page, 2003; Roc kwell, Griffin, & Jones, 2011 ), SBI was used with individuals with significant disabilities. SBI with Representative Diagrams In two descriptive studies Willis and Fuson (1988) and Fuson and Willis (1989) sought to demonstrate that SBI could be used to tea ch single step group, change get more, change get less, and compare story problems to high and average achieving second graders using whole group instruction. Each problem type and its diagram was introduced and taught separately. For group and compare pr oblems, the researchers developed diagrams that were highly representative of the relationships among the magnitudes of the quantities in the word problems However, the diagram used for change problems represented only the semantic structure of the word p roblem, rather than the relationships among the magnitudes of the quantities. The researchers taught teachers to conduct the SBI with 43 students in the 1988 study and then 76 students in the 1989 study. The results of both studies indicated that SBI when implemented with fidelity could assist young children in solving complex addition and subtraction story problems. Results also indicated that children had more PAGE 30 30 success solving problems in which the final quantity in the diagram was unknown. The researcher s hypothesized that this differential difficulty might have been due to the increased algebraic reasoning required to solve problems in which the unknown was not the final quantity. Unfortunately, results of the Fuson and Willis (1989) study indicated that teachers implemented SBI with differential fidelity leading to outcomes for students in different classrooms that could not be explained by the effects of SBI alone. Results of these two studies indicate that SBI could be a valuable method for teaching yo ung children to solve word problems. In addition, they indicate that students could benefit from whole group instruction using SBI, provided that the instruction is implemented with fidelity. Ensuring that teachers implement SBI with fidelity may require o ngoing training and support rather than just the initial training provided by Fuson and Willis (1989). Despite their promising results, the studies have several limitations. First, the primary use of descriptive statistics in these studies provided some e vidence of the practical significance of the intervention, but did not indicate statistical significance. Using significance testing would have allowed the authors to minimize that possibility that their results were due to chance and allowed them to attri bute the experimental effects to the SBI. The decision to use descriptive statistics was likely based on the small sample sizes included in these studies and on the number of outcomes in which the researchers were interested; however, it is still a signifi cant limitation. Another limitation of the studies is the lack of a control or comparison group thus limiting the certainty with which observed effects can be attributed to SBI. PAGE 31 31 The applicability of these studies to teaching additive word problem solving to students with autism is questionable because these studies did not include students with disabilities. However, there are elements of these studies that hold promise for working with students with autism. For instance, the inclusion of schematic diagram s that clearly represent the relationships among the quantities in the word problems provide visual support that may help facilitate semantic mapping for students w ith autism (Tissot & Evans, 2003 ). Furthermore, instructing students in one problem type at a time may be beneficial because it allows students to practice to mastery before introducing new information ( Truelove, Holoway Johnson, Mangione & Smith, 2007) Finally, separating change get more and change get less problems may help students determine which operation to use when solving word problem s. SBI with Self Monitoring In three studies, Jitendra and colleagues (1996, 1998, 2007a) sought to determine whether SBI could be used to improve the ability of students with high incidence disabilities to solve single step group, change, and compare word problems. SBI in these studies consisted of using direct instruction to teach students to use schematic diagrams and to use a problem solving mnemonic to self monitor their performance. Students received in struction in all problem types at once; however, they first received problem schemata instruction using story situations in which all quantities were given and then received problem solving instruction in which one quantity was unknown. Jitendra and Hoff (1996) conducted SBI one to one with three students with learning disabilities. Results of an adapted multiple probes across students single case design indicated no change in level from baseline to just after problem schemata instruction; however, all thr ee participants demonstrated improvement following problem PAGE 32 32 solving instruction and maintained improvement over time. These results suggest that exposure to schema types alone was not sufficient to improve problem solving performance and that explicit instr uction in using schematic diagrams to solve problems was necessary t o effect improvement in problem solving performance. Interviews with students indicated that they enjoyed using schematic diagrams, found them useful, and would recommend SBI to friends. I n order to replicate these findings with larger numbers of students, Jitendra et al. (1998) conducted a n experimental study with 34 students with mathematics disabilities or at risk for math failure in which they compared SBI and traditional basal instruct ion. Participants were randomly assigned to receive either SBI or basal instruction in small groups from trained graduate students. Results indicated that althoug h both groups improved from pretest to generalization. T he SBI group improved more than did th e basal instruction group with their performance approaching that of a normative sample of third graders In addition, students in the SBI group consistently rated their instruction higher than did the basal instruction group. In order to determine the fea sibility of having teachers conduct SBI in classroom settings, Jitendra et al. (2007a) conducted two design experiments or flexible studies conducted prior to formal experimental studies that are intended to provide a deeper understanding of the realities of independent variable implementation in classroom settings (Gersten, Baker, & Lloyd, 2000). The first included 38 students attending two low ability third grade classrooms, their two teachers, and the special education teacher. The second design experim ent involved 56 students in two heterogeneously grouped thi rd grade classrooms and their two teachers. Results indicated that teachers PAGE 33 33 in both experiments implemented SBI with high fidelity and that students in both design studies improved their problem so lving performance following SBI; however, effect sizes were larger in the first experiment than in the second. In addition gains in performance were more evident for low achieving students in the second study than for average or high achieving students. Fi nally, student responses to a strategy satisfaction questionnaire administered to participants in the first experiment indicated that students with learning disabil ities rated SBI higher than low achieving students. Taken together, the results of these th ree studies indicate that SBI with a self monitoring component holds particular promise for teaching additive word problem solving to students with high incidence disabilities and students with poor math performance. They also suggest that it may be benefi cial to include separate components of instruction to address schema induction and problem solving. When conducted in small groups, SBI may be more effective than traditional basal instruction (Jitendra et al., 1998). The design experiments (Jitendra et al 2007a) showed that, when provided with training and ongoing support, teachers could effectively implement SBI in their classrooms with high levels of fidelity. These design experiments also allowed the researchers to adapt the SBI in response to teacher concern. This ultimately led to a more externally valid intervention in experiment two that included problems with information presented in tables, graphs, and pictographs; and instruction in mathematical vocabulary and writing descriptions of problem sol ution strategies. suggest that SBI is a socially valid intervention. PAGE 34 34 Despite the promising results of these studies, they have several limitations. First, results in the Jit endra and Hoff (1996) study should be interpreted with caution because the use of only one probe following problem schemata instruction and of just two probes at maintenance precludes the establishment of stable trend and level at these times. In the Jiten dra et al. (1998) study, intervention effects may have been confounded by small differences in IQ between groups and by missing IQ scores for some participants. Also, the results of the design experiments (Jitendra et al. 2007a) should be interpreted with caution due to classroom differences prior to intervention and the lack of control or comparison groups, which make it difficult to determine whether the effects found were due to SBI or to other factors such as history or maturation. Furthermore, the agg regation of data across different mild disability categories in the Jitendra et al. studies does not allow for the statistical analysis of differential effectiveness based upon disability category. With regards to external validity, all three studies have limited generalizability due to small sample size. Finally, the inclusion of only one step additive word problems may not be representative of the variety of problem types found in the general education curriculum. Despite these methodological flaws, the studies provide insight into how SBI may be adapted for use with students with autism. First, the use of one to one or small group instruction may be particularly beneficial for students with disabilities (Truelove et al., 2007). The inclusion of a self mo nitoring component also has applicability to students with autism because research suggests that self monitoring may help students with autism increase desirable behaviors and decrease competing behaviors (Shabani, Wilder, & Flood, 2001). Finally, the sepa ration of problem schemata instruction and PAGE 35 35 problem solving instruction may be helpful because it allows students to achieve mastery at semantically mapping the critical features of word problems to schematic diagrams before requiring that they engage in pr oblem solving. This may benefit students with autism, in particular, by ensuring the solid formation of problem schema thereby reducing working memory demands associated with word problem solving (Capizzi, 2007). SBI with Multi Step Problems Researchers ha ve also conducted quasi experimental studies to demonstrate the effectiveness of SBI in inclusive, general education classrooms (Griffin & Jitendra, 2009; Jitendra et al., 2007b). In these studies, SBI was extended to include two step, as well as one step, additive word problems. Following problem schemata instruc tion and problem solving instruction using one step word problems, students were taught to simultaneously use two schematic diagrams to solve two step word problems. The effectiveness of SBI was co mpared to general strategy instruction (GSI), which involve d teaching general problem solving heuristics found in the mathematics basal text. Classroom teachers conducted all instruction in a whole group format. In the Jitendra et al. study instruction was conducted for 25 minutes each day, while in the Griffin and Jitendra study instruction was conducted once per week for 100 minutes. The two studies included 154 students in inclusive, general education classrooms. Participants were assigned to either SBI or GSI using random assignment with matching based on initial math problem solving performance. Results of the Jitendra et al. (2007b) study indicated statistically significant effects and medium effect sizes in favor of SBI over GSI on all measures, inclu ding a statewide achievement test. Results of the Griffin and Jitendra (2009) study indicated that both the SBI and GSI groups PAGE 36 36 demonstrated similarly improved problem solving performance over time; however, follow up tests indicated that the SBI group acqu ired word problem solving skills more quickly than did the GSI group. It is possible that main effects favoring SBI over GSI in this study because instruction was conducted only once per week. Taken together, the results of these two studies indicate that classroom teachers can successfully conduct SBI with whole groups of students with high fidelity when provided with adequate training, minimal ongoing support, and scripted materials. The results also suggest that SBI benefits students with and without di sabilities in inclusive classrooms. Furthermore, the findings indicating differential effects favoring SBI on a nationally standardized and statewide test of achievement (Jitendra et al., 2007b) are promising because these tests assess a variety of math sk ills not limited to problem solving and represent an authentic measure of generalization. Moreover, because SBI resulted in mo re rapid acquisition of problem solving skills than did general strategy instruction in the Griffin and Jitendra (2009) study, it is possible that SBI may have potential as a short term intervention to help struggling students catch up to more capable peers. Finally, because teachers were asked to study rather than read scripts, the use of scripted lessons did not limit the flexibili ty or personal style with which teachers conducted lessons Although the results of these studies suggest that SBI can be used effectiv ely to improve the math problem solving performance of students in inclusive settings, they have several limitations. Des pite the inclusion of students with disabilities in these studies, the small numbers of students with learning disabilities (n=9) and minimal analysis of the differential progress of students with learning disabilities make the PAGE 37 37 results related to different ial effectiveness of SBI for students with learning disabilities less clear. Additional studies should assess further the effectiveness of SBI for students with disabilities in inclusive settings. Moreover, because the authors did not match based on readin confounding variable could have impacted the internal validity of the findings. Furthermore, despite the inclusion of two step word problems, this study did n ot address the range of problem solving tasks found in the general education. Finally, the use of whole class instruction and scripted lessons may have limited the extent to which teachers could differentiate instruction to meet student needs. These studies also have important im plications for instructing students with autism. SBI as operationalized in these studies addressed a wider range of problem types while continuing to include components that may benefit students with autism (e.g., visual supports, direct instruction, self monitoring). In addition, these studies clearly demonstrate that teachers can effectively implement SBI in inclusive classroom settings with minimal support. SBI therefore has the potential to increase access to the general education curriculum and incre ase time spent in the general education classroom for students with disabilities including those with autism. SBI with Additional Instruction Xin et al. (2008) conducted a study using a multiple probes across participants single case design to determine w hether combining direct instruction in using schematic diagrams with explicit instruction in story problem grammar would improve the pre algebraic concept form ation and additive word problem solving performance of three fourth and fifth grade students wit h math difficulties. Story problem grammar instruction was aimed at helping students understand the part part whole nature of additive word PAGE 38 38 problems. The authors developed cue cards for the different types of additive word problems to help students identif y sentences or parts of sentences containing relevant information and to help students map this information to part part whole diagrams. The authors hypothesized that by helping students understand the underlying part part whole structure of additive word problems; algebraic reasoning and operation choice would be facilitated. performance on experimenter developed word problem solving probes and equation solving probes, and on pro bes derived from the KeyMath R/NU (Connolly, 2007 ). Although the results of this study are promising, the administration of the KeyMath R/NU and equation solving probes at pre and post test only may not have been adequate to control for threats to interna l validity in the within subjects single case design. Because SBI with and without story problem grammar instruction was not compared in this study it is unclear that story problem grammar instruction provides an added benefit. Future research may seek to evaluate the differential impact of SBI with and without story problem grammar instruction. In their 2008 experimental study, Fuchs et al. reconceptualized SBI as a means of providing preventative tutoring to students with mathematics and reading difficul ties within the secondary tier of a response to intervention (RtI) model. RtI is defined by Vaughn and Fuchs (2003) as a three tier model for identifying and remediating learning problems. The primary tier includes implementation of research based general instructional programs and screening of all students by the general education teacher to identify those at risk of learning problems. These at risk students then receive short PAGE 39 39 term small group tutoring accompanied by progress monitoring within the secondar y tier. Students who fail to respond to this intensive tutoring are referred for special education evaluation as part of the tertiary tier, while those who respond are returned to the primary tier. Students with mathematics and reading difficulties were r andomly assigned to either continue in their general education curriculum using a math basal text or receive SBI. In this study, SBI included instruction in the three additive problem schema as well as additional schema broadening instruction, which involv ed instruction in four superficial transfer features (e.g. includes irrelevant information, uses 2 digit operands, has missing information in the first or second position, presents information in charts, graphs, or pictures). Results indicated statisticall y significant differences and large effect sizes favoring students who received SBI over those who stayed in the general curriculum on problem solving measures. The results of this study suggest that SBI may have potential as a secondary intervention for s truggling students within and RtI model. However, t he use of one to one tutoring rather than small group tutoring is a limitation because within RtI secondary preventative interventions are generally conducted in small groups of three to six students ( Vaug hn & Fuchs, 2003 ). Also, t he lack of a maintenance measure makes it uncertain whether the tutoring intervention led to continued success in the gener al curriculum, which the authors suggest is the goal of secondary preventative tutoring within the RtI fram ework Furthermore, because SBI with and without schema broadening instruction was not compared, it is not clear that the additional instruction in PAGE 40 40 superficial transfer features led to improved outcomes. Future research may seek to evaluate the differentia l impact of SBI with and without schema broadening instruction. The additional features added to SBI in these studies have important implications for working with students with autism. For instance, research has shown that students with autism benefit fro m story grammar instruction and cues to support reading problem grammar and the use of cue cards to help students identify relevant information within word problems may be helpful to students with autism. It is also possible that this instruction may facilitate algebraic reasoning and operation choice for students with autism. However, the use of a part part whole diagram to represent all additive word problems may pr ovide less semantic support for students with autism than the provision of schematic diagrams particular to each problem type. Students will also benefit from explicit instruction for generalization (Cowan & Allen, 2007) and may therefore benefit from inst ruction on transfer features. SBI for Individuals with Developmental Disabilities Despite the evident promise of using SBI to improve additive word problem solving performance, only three such studies have been identified that included participan ts with si gnificant disabilities In the first, Neef, Nelles, Iwata, & Page (2003) used a behavioral approach to teach two males ages 19 and 21 with below average intellectual abilities to solve change word problems Although the authors did not identify their study as using SBI, their intervention included the critical components S BI, including direct instruction, use of diagrams to represent problems and use of a problem type from the theoretical literature Their study was unique in that it addressed precurrent s kills for problem solving. Precurrent skills are the underlying skills that an individual must PAGE 41 41 successfully and simultaneously execute in order to complete a complex task successfully in this case identifying the initial value, change value, o peration, an d ending value in change problems The participants received instruction on each precurrent skill separately before being instructed in solving the problem. A multiple baseline across behaviors single case design was used wherein each precurrent skill repr esented a separate behavior. Probes were conducted at baseline and following training on each precurrent skill. Results of the Neef et al. (2003) study indicated that both participants improved in their ability to identify components of change story probl ems from baseline to post training on each component. In addition, following training on all precurrent skills, both participants improved in their ability to solve change problems. These results suggest that teaching students with developmental disabiliti es to identify the component parts of word problems could effectively improve their ability to solve novel problems. However, b ecause only two participants were included in this study, causality can be inferred, but generalization is limited due to a lack of replications. Additional research will be needed to verify the results of these findings, determine whether problem solving gains are maintained over time, and extend these results to school aged children with developmental disabilities. In another stu dy, Jitendra et al. (2010) conducted a case study to illustrate the use of SBI with students with emotional behavioral disorders (EBD). The ir case study included a fourth grade student and a fifth grade student. Both of these students were diagnosed with EBD, were attending a special school for students with behavioral problems, were struggling in mathematics, and were receiving mathematics instruction PAGE 42 42 from the same special education teacher. SBI i n this study included a problem solving heuristic, addresse d one and two step problems, and was delivered daily by the special education teacher over a 20 week period. Students were administered expe rimenter developed word problem solving pre a nd post tests, and word problem solving fluency probes. Both students improved th eir performance on word problem solving fluency probes as instruction progressed. In addition, both students showed gains from pre to post test. The authors hypothesized that SBI was effective with these students with EBD because it focuses on conceptual understanding and links procedures to that understanding, involves systematic instruction, and uses continuous progress monitoring to inform instruction. Although the results of this case study provide some insight into the use of SBI with stud ents with EBD, they should be interpreted with caution. Because of the lack of experimental control in a case study, it is not clear that improvements noted during and following SBI were due to the intervention and not to a confounding variable such as mat uration. In addition, the inclusion of only two participants limits the generalizability of the findings to other students with EBD. Additional research using larger numbers of students with EBD and using more rigorous designs will need to be conducted to provide convincing evidence of the efficacy of using SBI with this population. The third study was conducted by Rockwell et al. ( 2011 ) to evaluate the efficacy of using SBI for improv ing the additive word problem solving performance of a fourth grade stude nt with autistic disorder and low average intellectual abilities who was fully included in the general education classroom, but was performing below grade level in mathematics SBI in this study used highly representative diagrams paired with a four PAGE 43 43 step p roblem solving mnemonic for self monitoring. In addition, problem types were taught separately and instruction in each problem type included problem schema ta instruction and then problem solving instruction. Results of a multiple probes across problem type s design indicated that the participant improved word problem solving performance across all three additive problem types when unknowns were in the final location. These gains were maintained over time and were generalized to problems requiring algebraic r easoning to solve for an unknown quantity not located in the final position ; however, the participant demonstrated greater difficulty using algebraic reasoning to solve change problems The authors hypothesized that this may have been due to the less repre sentative nature of using a single diagram to represent change get more and change get less problems. As in the Jitendra et al. (2010) study, t here are several limitations to this study. The inclusion of only one participant limits the generalizability of the findings. In addition, the use of a multiple probes across behaviors design may not have been optimal because solving the three types of additive word problems were related behaviors. As a result, the participant did increase her performance on change problems slightly following instruction in solving group problems and using self monitoring. Finally, because this study included only single step addition and subtraction story problems, it may not adequately represent the range of problem solving tasks f ound in the general education curriculum. These three studies have important implications for using SBI to teach additive word problem solving to students with autism. Results of the Neef et al. (2003) study suggest that individuals with developmental dis abilities, and perhaps students with PAGE 44 44 autism, may benefit from a behavioral approach to problem solving instruction that involves task analysis and discrete training in precurrent skills The Jitendra et al. (2010) case study indicates that students with be havior problems may benefit from the systematic instruction and continuous progress monitoring included in SBI. This is particularly promising given that the behavioral problems exhibited by many students with autism can make it difficult for them to benef it from classroom instruction (Mesibov & Howley, 2003). The Rockwell et al. ( 2011 ) study provides preliminary evidence that SBI may be an effective intervention for students with autism. In addition, it suggests that using highly representative schematic d iagrams and a self monitoring mnemonic may be beneficial for students with autism. Furthermore, this study provides some evidence that teaching each problem type separately and instructing for schema induction and problem solving separately may be effectiv e with students with autism. Finally, it is possible that teaching change get more and change get less problems separately using more highly representative diagrams would help facilitate algebraic reasoning. Synthesis of Schema Broadening Research Accordin g to Fuchs, Fuchs, Hamlett, and Appleton (2002), s ch ema broadening instruction involves teaching the problem solution rules for problem ty pes commonly found in the third grade mathematics curriculum using verbal representations of the problem solution sche ma Four problem types were obtained through discussions with teachers and a review of the basal mathematics textbook. These include shopping list, buying bags, half, and pictograph problems These problem types include single and multi step problems and both additive and multiplicative operations. Schema b roadening instruction also involves explicit instruction in superficial problem features PAGE 45 45 (e.g. different format, different keyword, additional question, larger proble m solving context) and transfer Fuc hs and colleagues (2002; 2003a; 2003b; 2004a, 2004b, 2004c, 2006) have conducted a line of research using quasi experimental designs to evaluate the efficacy of using schema broadening instruction to improve the mathematics word problem solving performance of third grade students. These studies included a total of 1,973 part icipants ranging from typically achieving students to students with mild disabilities. The studies compared schema broadening instruction delivered by researchers or teachers using scrip ted lessons to instruction delivered by teachers using the basal textbook, and one study in which computer assisted instruction was used (Fuchs et al., 2002). They also explored the relative contribution of both schema instruction and transfer instruction (Fuchs et al., 2003a; Fuchs, Fuchs, Finelli, Courey, & Hamlett 2004c). In addition, schema broadening instruction has been combined with self regulated learning strategies (Fuchs, Fuchs, & Prentice, 2004b; Fuchs et al., 2003b), problem sorting activities ( Fuchs et al., 2004a), and real life problem solving strategy instruction (Fuchs et al., 2006). Problem solving performance in these studies was assessed using researcher developed outcome measures. These outcome measures assessed immediate transfer using problems like those used for instruction, near transfer using problems with varying superficial features, and far t ransfer using real life problem solving contexts that combine multiple problem schema and varying superficial features. In all studies, stati stically significant effects favoring schema broadening instruction over comparison groups were found. Effect sizes were often large, sometimes surpassing three standard PAGE 46 46 deviations. In addition, the inclusion of specific instruction on transfer features re sulted in greater gains than did instruction in problem schema alone. Furthermore, self regulated learning strategies, problem sorting activities, and real life problem solving strategy instruction all were found to enhance the efficacy of schema broadenin g instruction. Overall, the findings of the seven studies on using schema broadening instruction indicate that this intervention holds promise for enhancing problem solving skills. The use of problem types found in the general curri culum improves the external validity of the intervention. Also, the use of measures with increasing demands for transfer mirror s the types of tasks that students will be exposed to in the general education curriculum. Moreover, the use of primarily intact, heterogeneously grouped classrooms demonstrates the feasibility of conducting the schema broadening intervention in general education classroom s. The studies also indicate that schema broadening instruction can be implemented with high treatment fidelity by both researchers and teachers, particularly when teachers are provided with scripted lessons. Finally, schema broadening instruction has been used effectively in a variety of settings. It has been used with students with disabilities in pull out setting s, with at risk students in pull out settings, and with students of varying ability in inclusive settings. Although these studies demonstrate that schema broadening instruction has great potential, there are limitations that merit discussion For instance although using problems drawn from the general education curriculum enhances the external validity of the intervention, the way these problems were grouped for instruction is not consistent PAGE 47 47 with schema theory which specifies specific problem types within additive and multiplicative formats ( Chri s tou & Philippou, 1999). The theoretical basis of the intervention would be strengthen ed if existing schema theory were used to categorize problem types from the general education curriculum. In addition, because o nly researcher developed assessments were used, it is unclear whether the effects of the intervention would generalize to classroom assessments or standardized assessments used to document AYP The value of schema broadening instruction would be more evide nt if standardized measures were used in addition to researcher developed measures. Limitations aside, t he research on schema broadening instruction has implications for working with students with autism. The inclusion of explicit transfer instruction may be particularly beneficial for students with autism who typically have difficulty generalizing skills and require explicit instruction to do so (Cowan & Allen, 2007). However, the lack of maintenance assessments in all of these studies is concerning, parti cularly when considering that students with autism tend to have difficulty with maintenance as well as generalization (Kendall, 1989). In addition, the use of verbal instruction and verbal representations of problem schema may be less effective for student s with autism who tend to benefit from the use of visual supports (Mesibov & Howley, 2003; Tissot & Evans, 2003). Implications for Future Research Although there are several studies providing evidence to support the use of SBI to im prove the additive word problem solving performance of students with high incidence disabilities and students without disabilities, additional research is needed to establish SBI as an evidence based practice for teaching mathematical word problem solving to PAGE 48 48 children with develop mental disabilities. Future studies should seek to replicate the resu lts of the Rockwell et al. (20 11 ) study using larger numbers of participants and more rigorous designs. In addition, information from the Rockwell et al. study can be used to develop and evaluate more effective SBI procedures for use with students with autism. Finally, methods that have been used in SBI studies that may hold promise for students with autism should be evaluated with this population. This study seeks to address some of the i dentified limitations in the research base by using a multiple probes across behavior s design replicated across participants to ensure internal validity, assessing generalization to problems with irrelevant information, and assessing maintenance of treatme nt effects over an extended period of time to improve social validity. The proposed study will also improve upon the procedures u sed in the Rockwell et al. (2011 ) study by teaching students to adapt the change diagram to reflect the larger beginning amount in the Willis and Fuson (1988) study. In addition, explicit instruction for generalization will involve teaching students to use the part part whole structure of the word problems as an aid to algebraic reasoning when determining which operation to use as in the Xin et al. ( 2008 ) study Generalization instruction will also address superficial transfer features like those inc luded in the Fuchs et al. (2008 ) study (i.e inclu des irrelev ant information has the unknown in the initial or medial position) PAGE 49 49 CHAPTER 3 METHOD Participants P articipants were recruited through the Center for Autism and Related Disabilities (CARD). Initially, families of six students with autism agreed to allow their children to participate, however, only two students met the inclusion criteria. These two study participants will be referred to by the pseudonyms Daniel and Justin. Inclusion and Exclusion Criteria In order to be included in this study, st udents had to be in grades one through six, have a diagnosis of autistic disorder as indicated in a report from a qualified professional, and have nonverbal cognitive abilities within two standard deviations of the mean as indicated in a report from a qual ified professional. They also had to participate in grade level Annual Yearly Progress (AYP) assessments as indicated by their Individual Education Programs (IEPs), but be performing below grade level on mathematics as measured by previous AYP assessments or other standardized tests of addressing mathematics problem solving. Furthermore, students had to be able to decode at a second grade reading level and be able to perform add ition and subtraction computations involving two digit numbers without regrouping with at least 90% accuracy as assessed using curriculum based measures of second grade oral reading accuracy and addition and subtraction computation accuracy obtained from t he Basic Skill Builders Series ( Beck, Conrad, & Anderson, 1997 ). Students were excluded from participation in this study if they had a concomitant diagnosis of mental retardation, PAGE 50 50 were unable to communicate verbally, or were unable to attend to one to one group instructional sessions lasting 30 minutes. Participant I nformation for Daniel Daniel is a seven year old male who has a diagnosis of autistic disorder from a developmental behavioral pediatrician. A school psychologist employed by the local school district conducted his most recent psychoeducational evaluation in March 2009. intellectual and academic abilities could not be completed. to addres s his behavioral issues, including attending to classroom instruction and activities, remaining in his seat, refraining from talking in class without permission, and transitioning from one activity to another with the rest of the class. Daniel attends a re gular education first grade class where he is assisted by a paraprofessional and participates in regular assessments with accommodations. He also receives language therapy and occupational therapy each for 30 minutes each oom teacher, classroom assessments, and his IEP, he is performing at or above grade level in most areas of the curriculum; however, he struggles with mathematics word problem solving and written expression includes a goal related to answering open ended questions. Objectives for this goal address oral language, solving mathematics word problems, answering reading comprehension questions, and responding to writing prompts. At the time data ed a chapter on addition and subtraction that included additi on and subtraction word problem solving. Word problem solving instruction consisted primarily of associating keywords with operations. On the chapter test that included multiple choice computatio n questions and open ended word PAGE 51 51 problem solving tasks, Daniel scored a 90%; however, item analysis revealed that he responded incorrectly to all of the word problems. On screening assessments obtained from the Basic Skill Builders Series (Beck et al., 1997 ) and conducted as part of t his study, Daniel read a second grade passage with 98% accuracy and performed two digit addition and subtraction without regrouping with 100% accuracy Daniel is a cheerful and friendly boy. He loves to watch Disney Pixar movie s and enjoys reenacting scenes from them using toys. He also enjoys drawing and coloring. He draws pictures of the characters from his favorite movies, recreates visual displays that can be found in his classroom, and writes in various fonts when completin g schoolwork and homework. He has an acute visual memory and is able to duplicate pictures he has seen only a few times with great detail. Daniel is also able to benefit from verbal instructions. He often repeats things that his teacher has taught him in c lass while completing his homework. He is able to follow one and two step verbal instructions. Although Daniel uses speech as his primary means of communication, his speech is often formulaic and echolalic. He is able to make his wants and needs known, bu t struggles to answer even simple questions unless he is given choices. Pa rticipant I nformation for Justin Justin is a 12 year old male who has a diagnosis of autistic disorder from a developmental behavioral pediatrician. A team of evaluators at the Univ ersity of Florida conducted his most recent psychoeducational evaluation in July 2011. Results of that range while his verbal cognitive abilities are within the imp aired range. On executive functioning measures, J ustin performed in the average range on tasks assessing sustained attention, inhibition of competing behaviors, and set shifting. However, he PAGE 52 52 performed in the impaired range on tasks assessing verbal working memory. His ability to perform mathematics calculations correctly and fluently falls within the average range. However, his ability to solve applied problems falls within the impaired range. Justin attends regular edu cation sixth grade classes for langua ge arts, mathematics, science, and social studies. He also attends a special education reading class, has a study hall period, and participates in regular assessments with accommodations. He is assisted throughout the school day by a paraprofessional who m he shares with another student with autism. He also receives language therapy for 60 minutes each week and receives consultative services from an occupational therapist. On previous assessments of AYP, Justin performed below grade level in both reading and mathematics problem solving On screening assessments obtained from the Basic Skill Builders Series (Beck et al., 1997 ) and conducted as part of this study, Justin read a second grade passage with 97% accuracy and performed two digit addition and subtraction without regrouping with 100% accuracy Justin is a kind, gentle boy who is eager to please. He smiles and claps his hands when he is told that he has done something well. H e enjoys playing Wii Sports with his having predictable routines in his life and especially enjoys school because of its structure. Justin is able to benefit from verba l instructions and is able to follow multi step verbal directions. His primary means of communication is speech and he is able to make his wants and needs clearly known. He is also able to answer open ended questions if given ample time to formulate his re sponse. Although Justin communicates PAGE 53 53 verbally, he is often frustrated by his inability to quickly formulate responses to egain his composure. Despite requiring these breaks, Justin is reluctant to take breaks when they are offered. He seems to want to complete his tasks quickly and without breaks and please the adults with whom he is working. Independent Variable Setting A ll instruction took place during 30 min. sessions conducted Monday through Friday from 3:00 author conducted all instruction with individual participants. For Daniel instruction was conducte d after he had completed his school day and finished his assigned homework. For Justin, instruction was conducted after he had finished participating in various activities during his summer vacation including a part time summer school session, language the rapy, occupational therapy, and a church based summer camp. Treatment Lesson checklist s and scripts were developed based on those used in previous SBI studies ( Griffin & Jitendra, 2009; Rockwell et al., 2011 ) to ensure that all instructional components we re addressed during each session. Lessons included an average of 8 (range 7 10) additive story situations or word problems. Lessons that involved problem sorting included an additional 6 word problems for sorting. Behavioral consequences were an important component of the treatment. Although the specific behavioral consequences employed were individualized for each participant, some behavioral consequences were employed universally. Verbal praise PAGE 54 54 was provided intermittently for on task behavior and for problem completion. In addition, participants were allowed up to two, 3 minute breaks during each instructional session. Breaks were provided non uest. When participants paused in their work or in answering a question for longer than a 5 second wait time, the researcher gestured toward visual supports of the RUNS steps or schematic diagrams to prompt the participant When a participant completed a schematic diagram incorrectly or performed a computation incorrectly during guided or independent practice, the researcher prompted him to check his work. For Daniel, whose behaviors were more severe, food item s were paired with verbal praise to reinforce on task behavior on a variable schedule. Additional behavioral accommodations included the use of an air disk to allow Daniel to move while staying seated, and proximity and touch to encourage on task behavior. Daniel sometimes researcher place a hand firmly on his back or shoulder while he was working. Although Justin did not need additional reinforcement f or on task behavior, he often needed additional reassurance about his performance. When Justin became with a pat on the shoulder or back. PAGE 55 55 Treatment was divided into three phases with each phase addressing one of the three problem types identified by Carpenter and Moser (1984). Word problem types were addressed in this study based on the sequence proposed by Christou and Philippou (1999 ): first group, then change, and finally compare problem types. Prior to the start of instruction, baseline problem solving and generalization probes were conducted. Instruction was initiated after stability of data had been established on baseline probes. The SBI intervention procedures used in this study were based on those used in previous SBI studies (e.g., Jitendr a et al., 2007 a, Jitendra et al., 2007b Rockwell et al., 2011). Prior to beginning the SBI, the participant learned a four step procedure for problem solving adapted from that used by Jitendra et al. using the mnemonic RUNS. The RUNS steps consist of: (1) Read the problem, (2) Use a diagram, (3) Number sentence, and (4) State the answer. The researcher taught the participant to write the RUNS mnemonic prior to solving a problem, and to check off the steps as he completed them. The researcher modeled writin g the RUNS mnemonic and checking off the problem solving steps throughout the teacher modeling portions of SBI. The researcher also instructed the participant to write the RUNS mnemonic and check off the problem solving steps during the guided practice por tions of SBI, and reminded him to engage in these behaviors as needed during independent practice. The researcher included the RUNS mnemonic to encourage the participants to self monitor their problem solving behavior; which can benefit students with autis m by increasing desirable behaviors and decreasing competing behaviors (Shabani et al., 2001). PAGE 56 56 As in previous SBI studies, this study used direct instruction ( Rosenshine & Stevens 1986) consisting of modeling with think alouds, guided practice, and indepe ndent practice throughout the SBI. Modeling with think alouds involved the researcher in solving problems for the participant while telling him the thought processes she used to solve the problems (Examples of researcher think alouds are found in the sa mpl e lesson script in Appendix B ). Guided practice involved the participant in solving problems while the researcher verbally guided him through the steps and thought processes required to arrive at a correct answer. The researcher gradually reduced the amoun t of verbal guidance she provided as the participant demonstrated his ability to solve the problems more independently. Independent practice invo lved the participant in problem solving with the researcher providing feedback regarding any missed steps after he had solved each problem. The researcher introduced and taught each problem type separately starting with group problems, moving on to change problems, and finally teaching compare problems. Instruction in solving each problem type first involved instr uction using story situations in which all components of the problem consisted of known quantities in order to facilitate schema induction. The researcher used the direct instruction sequence to teach each participant to identify the critical components of each type of word problem. For group problems this involved teaching participants to identify the two distinct parts and the all obtained from putting those parts together. For change problems, the researcher taught participants to identify the beginning, the change, and the action word obtained. For compare problems, the researcher taught each participant to identify the larger PAGE 57 57 amount, the smaller amount, the comparative word or phrase used to compare them, and the difference obtained The researcher then taught participants to put the critical components into the diagram for the problem type being taught. When teaching each participant to use the change diagram, the researcher also taught him to adapt the sizes of the ending and beginning This was intended to help participants visually represent the structure and the relationship between addition and sub traction in change problems. According to Van de Walle, Karp, and Bay Williams (2010), understanding the relationship between addition and subtraction is critically important conceptual knowledge for solving additive word problems. If the action word indic the researcher taught the participant to make the ending box larger, but if the action box larger. This allowed the change diagram to clearly represent which quantit y in the problem was the largest making the relationship among the quantities more visually evident. Next, the researcher taught the participants to translate their diagrams into number sentences, write the answer, and label that answer with a noun. For ex ample, if a group story noun, vegetables. The researcher then exposed each participant to story situations representing all three problem types and helped him to practice determining whether or not each problem fit the schema being currently taught, one of the previously taught schemas, or a schema that had not been taught yet. These sorting activit ies were based on those used in the Fuchs et al. (2004a) study to help reinforce discrimination of the PAGE 58 58 schema types. Once each participant demonstrated schema induction on formative assessments, the researcher applied the direct instructional sequence to t eaching that participant to solve for an unknown all, ending, or difference using problem situations Once the participant had demonstrated his ability to solve problems of the type being taught on formative assessments, the researcher initiated a probe ph ase. When stability of data on probes had been established, the researcher began the next phase of instruction by teaching the next problem type following instruction in solving group and change problems or by initiating generalization probes to assess for spontaneous generalization following instruction in solving compare problems. The researcher initiated generalization instruction for Daniel, the participant who did not demonstrate spontaneous generalization on generalization probes. Generalization lesso ns were not necessary for Justin because he performed at ceiling on generalization probes, indicating spontaneous generalization. Generalization instruction for Daniel included one lesson each on generalizing to solving problems with unknowns in the initia l position, solving problems with unknowns in the medial position, and solving problems with irrelevant information. These were brief, 15 minute lessons wherein the researcher modeled the use of schematic diagrams with generalization problems. The research er presented six generalization problems and modeled how to solve these problems using think alouds. Guided practice and independent practice were not included in the generalization mini lessons. The researcher administered a generalization probe immediate ly following each lesson. Had Daniel not demonstrated mastery on this probe, the researcher would have initiated a full generalization lesson PAGE 59 59 performance on generalization probes f ollowing each mini lesson The researcher c onducted a final set of problem solving and generalization probes with each participant 8 weeks later to assess maintenance of treatment effects over time. Treatment Integrity Analysis of the fidelity of independe nt variable implementation is a desirable component for single subject research in special education (Horner et al., 2005). All treatment sessions were video taped. Two doctoral students familiar with SBI directly observed (i.e., Doctoral Student T) or vie wed tapes (i.e., Doctoral Student J) of 32% of lessons within and across phases and participants. These doctoral students used detailed lesson checklists to evaluate the percentage of instructional components addr essed during the sessions. A sample checkli st can be found in Appendix C. According to Peterson, Homer, and Wonderlich (1982), ideal assessment of treatment integrity occurs continuously and accounts for problems with observational data such as reactivity and observer drift. Assessing treatment int egrity through direct observation ensured that fidelity of implementation was continuously assessed and that lack of fidelity wa s addressed promptly. The additional assessment of treatment integrity using videotapes ensured that high levels of treatment in tegrity were not obtained due to reactivity. Dependent Variable Researcher developed materi als were used to assess problem solving progress throughout this study. Practice sheets were used as a formative assessment of student progr ess during instruction. Problem solving probes were used to evaluate treatment effects using a multiple probes across behaviors design with the solving of each PAGE 60 60 problem type (i.e., change, group, and compare ) treated as separate behaviors. This design was replicated across two par ticipants. Formative Assessment During training on each problem type, the researcher assessed participant progress using a formative assessment that was administered at the end of each treatment session at the table where instruction was conducted. These p ractice sheets consisted of two problems of the type taught. During story situation instruction, practice sheets included story situations in which all three quantities were known. During problem solving instruction, practice sheets included problem situat ions in which the final quantity was unknown. Each of the problems was worth a possible three points: o ne point for using the correct schematic diagram, one point for writing the correct numbe r sentence to solve the problem, and one point for correctly com puting the answer. These scoring procedures were derived from those used by Jitendra and Hoff (1996) and Willis and Fuson (1988). The researcher used student scores on practice sheets administered at the end of each training session for making decisions ab out the pace of instruction. Only after the direct instruction sequence had been completed and a participant had earned all six possible points on at least one practice sheet did the researcher progress from story situation to problem solving instruction o r from problem solving instruction to a probe phase. Appendix D contains sample practice sheets for each problem type with hypothetical student responses and scores. Probes To evaluate the effectiveness of SBI, problem solving and generalization probes w ere administered to participants at baseline, following instruction on each problem type following generalization instruction, and eight weeks later to assess maintenance PAGE 61 61 of treatment effects. Baseline probe sessions and probe sessions following instructi on problem solving behaviors before and after SBI could be analyzed and compared. During each probe phase, the research er administered as many problem solving probes as were necessary to obt ain stability of data as well as one generalization probe addressing each type of generalization task to assess for spontaneous generalization During probe phases, no training sessions took place. Instead, the participants completed one problem solving pr obe and one generalization probe during the time scheduled for training sessions. Participants were given the following instructions prior to beginning each probe: some math word problems I want you to solve by yourself. Remember to use what you have learned to solve these problems. If you have trouble reading a problem, let me know and I will help you Do Participants were given up to 20 min. to complete each probe Each of the problems was worth a possible t wo points. One point was awarded for writing the correct number sentence to solve the problem. Assuming this number sentence wa s correct, an additional point could be earned for correctly computing the answer. These scoring procedures are derived from those used by Jitendra and Ho ff (1996) and Willis and Fuson (1988). See Appendix E for sample problem solving and generalization probe s with hypothetical student responses and scores. Problem solving probes Problem solving probes consisted of nine item experimenter developed tests. These probes were adapted from probes used in a previous SBI study (Rockwell et al., 2011). Each problem solving probe presented three PAGE 62 62 story problems from each problem type for a total of nine problems. Problem solving probes included story problems with t he unknown quantity located in the final position in the schematic diagram: the all in group problems, the ending in change problems, and the difference in compare problems Generalization probes also consisted of three story problems from each problem typ e for a total of nine problems. These probes include items used in generalization probes in the Rockwell et al. study. Because three generalization tasks were targeted in this study, three types of generalization probes were developed. Generalization prob es Generalization probes assessing generalization to problems containing irrelevant information included problems with unknowns in the final position but provided text based information on a fourth, irrelevant quantity. Generalization probes assessing gen eralization to problems with the unknown in the initial position involved story problems with the unknown quantity located in the initial position in the schematic diagram: the first part in group problems, the beginning quantity in change problems, and th e larger quantity in compare problems. Generalization probes assessing generalization to problems with the unknown in the medial position included story problems with the unknown quantity located in the medial position in the schematic diagram: the second part in group problems, the change quanitity in change problems, and the smaller quantity in compare problems. Reliability and Validity Evidence According to Crocker and Algina (2006) it is desirable to use measurement instruments that have evidence of pr oducing reliable scores and are valid assessments of the intended construct. PAGE 63 63 Inter rater agreement The researcher scored all of the probes and practice sheets. Two other doctoral students who were familiar with SBI independently score d 30% of the probes ( i.e., Doctoral Student M) and 30% of the practice sheets (i.e., Doctoral Student J) within and across phases and participants. Doctoral Student M was blind to the phases from which probes were sampled. Agreement was calculated on a point by point basis suc h that each probe could have a maximum of 18 agreements and each practice sheet could have a maximum of six agreements The total number of agreements w as divided by the total number of agreements and disagreements and then multiplied by 100 to obtain the percent agreement. Equivalent forms reliability The problem solving forms were pilot tested with a fourth grade student with autism (Rockwell et al., 2011). Because differences between the probes would have led to instability of the data, the stability o f the data from the Rockwell et al. study provide some evidence that the probes yielded equivalent scores. Validity evidence based on test content Dr. Cynthia Griffin, an expert in SBI, reviewed the problem solving probes to ensure that the problems conta ined the critical features of the schema intended. Validity evide nce based on response processes Results of the Rockwell et al. (2011 ) study indicated that a fourth grade student with autism was able to respond to problem solving probes and accompanying i nstructions as intended. Thi s student, who read at a second grade level, was able to read the problems included on the probes, identify the corresponding problem types following SBI, and respond to the problems in the space provided. PAGE 64 64 Data Collection and A nalysis A single case, multiple probes across behaviors design was used in this study (Kennedy, 2005). The researcher chose a multiple probes design because the use of probe sessions rather than continual assessment of the dependent variable minimizes test fatigue. In addition, the multiple probes design allows for the demonstration of experimental control following an instructional intervention that cannot be easily removed for a return to baseline. The length of baselines and implementation of the interve ntion was staggered across the three problem types for a total of three demonstrations of experimental control per participant and six total demonstrations of experimental control. Data from probes and practice sheets were graphed following each training o r probe session to facilitate decision making. Instructional phases were initiated only after stability, trend, and level were established on all previous baselines. In addition, participants were videotaped while completing probes at baseline and followi ng instruction on all problem types. The researcher viewed these tapes and problem solving behaviors (e.g., reading the problem aloud, underlining parts of the problem, using diagrams or drawings to solve the problem) were coded. A second doctoral student viewed 46% of these tapes within and across probe phases and coded behaviors as a validity check. Problem solving behaviors observed during probe sessions conducted at baseline were compared to those conducted following SBI to determine if and how particip problem solving behaviors changed following SBI. Finally, participants and their parents completed Likert type satisfaction questionnaires during the week maintenance probe sessions were conducted to assess the social validity of the SBI (Kennedy, 2 005). Samples of these questionn aires can be found in Appendix F PAGE 65 65 CHAPTER 4 RESULTS Treatment Integrity and Inter Rater Reliability Doctoral Student T directly observed 7% of lessons to assess treatment integrity. Treatment integrity for these lessons ave raged 98.5% (range 97% to 100%). Doctoral Student J viewed videotapes of 25% of lessons. Treatment integrity for these lessons averaged 98% (range 94% to 98%). A total of 32% of lessons were assessed for treatment integrity. Overall treatment integrity ave raged 98% (range 94% to 100%). Doctoral Student M scored 30% of probes within and across participants and phases. Inter scorer reliability as assessed on a point by point basis averaged 99% (range 88% to 100%). Doctoral Student J scored 30% of practice s heets within and across participants and phases. Inter rater reliability as assessed on a point by point basis was 100% for all probes scored. All discrepancies in scoring were resolved through discussion. Break Use During the first instructional session, the researcher offered Daniel two break cards. He used both of these breaks. During the breaks, the researcher set a timer for three minutes and engaged Daniel in stretching activities. Daniel engaged in these activities willingly, but then took several mi nutes after the break ended to transition back explained that he prefers to complete all of his work quickly and then have a long period of free time. She also explained t hat Daniel has great difficulty transitioning back to work after free time making short breaks more problematic than helpful. After that session, the researcher did not offer Daniel breaks. He did ask to use the restroom during two PAGE 66 66 additional sessions and was given a break to do so. He was able to transition back to work after using the restroom without difficulty. Justin was given two break cards at the beginning of each session, but rarely used these breaks. He requested breaks during 6 instructional sess ions. During these breaks, the researcher set a timer for 3 minutes and engaged Justin in stretching activities. Justin willingly participated in these activities and smoothly transitioned back to work when the breaks were finished. He used both break card s during five of the six instructional sessions in which he requested breaks The sessions during which Justin asked for breaks all occurred following a day at summer school. Problem Solving Performance During the first set of baseline probes, Da niel was unable to solve any of the problems on problem solving or generalization probes. For each problem he wrote two number sentences that were members of the same single digit fact family. Because led the space allotted for student work beneath each word problem. Daniel appeared aware that completing the task required him to use mathematics. He also seemed to make an effort to finish the task in an acceptable manner. Group Problems Daniel responded quickly to instruction in solving group problems. He immediately performed at ceiling on practice sheets. On problem solving probes following instruction in solving group problems, he was able to correctly solve all group problems. His average performance on group problems improved from 0/6 at baseline to 6/6 following instruction on solving this problem type. PAGE 67 67 Change P roblems Following instruction on solving problems improved from an average of 0/6 to 1.33/6. This improvement was the result of overgeneralization of the group schema to change problems. Daniel used the group diagram to solve change problems. This allowed him to inadvertently obtain correct number sentences and answers for change problems wherein Although he used the incorrect diagram, he obtained a correct answer because the problem required addition as would a group problem with an unknown all. Daniel responded quickly to instruction in solving change problems. He immediately performed at ceilin g on practice sheets. On problem solving probes following change instruction he was able to correctly solve all but one of the change problems. His average performance on change problems improved from 0.67/6 at baseline to 5.67/6 following instruction on s olving change problems. Compare Problems Although Daniel showed the same overgeneralization of the group problem schema to compare problems as he did to change problems, this did not allow him to obtain correct number sentences or answers to compare prob lems. As a result his ability to solve compare problems did not improve following instruction in solving group problems. However, Daniel demonstrated some improvement in his ability to solve compare problems following instruction in solving change problems (from an average of 0/6 prior to change instruction to 1.33/6). Daniel was not observed using group or PAGE 68 68 een working with Daniel on studying for a unit test in mathematics and had been reviewing key words meant to subtract and his mother had been reinforcing this with him. It is therefore possible that the improvement in compare problem solving performance was due to classroom instruction in key words and review of that instruction at home. This keyword instruction likely did not result in Daniel correctly solving all compare prob lems correctly Daniel did not attain mastery during instruction as quickly with compare problems as he did with group and change problems. It appeared he had difficulty distinguishing between group and compare problem types, perhaps due to similarities between the the compare diagram. The group and compare diagrams both consist of one large rectangle and two smaller rectangles. In the group diagram the larger rectangle is below the smaller rectangles; however, in the compare diagram the larger rectangle is above the smaller rectangles. If one were to flip the group diagram upside down it would look very similar to the compare diagram with the exception that all of the rectangles in the group diagram are solid whereas the rectangle representing the difference in compa re problems is dashed. It is therefore not surprising that Daniel had some difficulty distinguishing between these diagrams and thought that the compare diagram was a group diagram that had been turned upside down. Daniel required an additional lesson PAGE 69 69 with problem sorting activities and independent practice to demonstrate mastery on performance on compare problems improved from 0.07/6 at baseline to 5.33/6 following instructi on. Generalization to P roblem s with Irrelevant I nformation Spontaneous generalization to problems with irrelevant information was not observed. Following instruction on solving group problems, Daniel overgeneralized the group problem schema to all problem s with irrelevant information. However, rather than excluding the irrelevant information, he treated these problems as story situations in which all of the components of the problem were known problems. He also tended to use the quantities in the order in which they were given in the problem. Following instruction on solving change problems, he began to apply the schema corresponding to the problem correctly; however, rather than excluding the irrelevant information he continued to treat these problems as s tory situations in which all of the components of the problem were known quantities. He also continued to place the quantities into the schematic diagram in the order in which they were given in the word problem without considering whether those quantities represented that particular component of the problem. This led Daniel to create number sentences that represented inequalities window. He counts 42 boys, 4 teachers, and 36 gi rls on the playground. How many number sentence 42+4=36. Sometimes Daniel did not seem to notice the disparity, but at other times he would try to change one of the quantities in the original number sentence or change the order in which quantities were placed into the diagrams to correct an inequality. For instance, PAGE 70 70 but 5 scouts get sick. The campground can hold 23 tents. How many scouts are going would hold true. One exception to the lack of spontaneous generalization was seen in the generalization probe given during session 17, which read, students to the museum. If 16 students get sick and it costs $5 per student to enter the change problem was a monetary amount rather than a numeral Daniel appeared to realize that this monetary amount did not apply and was able to correctly solve the problem. Due to his lack of spontaneous generalization, Daniel was provided with one 15 minute lesson on solving problems with irrelevant information. During this lesson, the researcher modeled identifying the important information and underlining it and then problems with irrelevant information improved to ceiling level f or all problems types. This resulted in an average improvement from 0/6 to 6/6 for group problems, 0.5/6 to 6/6 for change problems and 0/6 to 6/6 for compare problems. Generalization to Problems with Unknowns in the I nitia l P osition Following instruction in solving group problems, Daniel tended to over generalize the group problem schema with an unknown all to every problem. As a result, Daniel was unable to solve any group problems with the unknown in the initial position because he continued to add the g iven quantities together. For instance, when given the word PAGE 71 71 parts. He then used the number sentence 15+47=52. However, this overgeneralization of the group schema with an unknown all to every problem allowed Daniel to problems and for compare problems with t he unknown in the initial position. For Now there are 21 students on the bus. How many st used a group diagram with 7 and 21 as the two parts resulting in the correct number sentence and answer 7+21=28. When given the following compare problem with an 12 pigs outside in the mud. If there are 32 more pigs in the barn than in the mud, how many pigs are in the the correct number sentence and answer 12+32=44. This over ge neralization of the group schema with an unknown all allowed Daniel to obtain the correct answer to these problems by chance, not because he had an understanding of how to solve change or compare problems with unknowns in the initial position. As a result, performance on change and compare problems with the unknown in the initial position improved from 0/6 at baseline to 3.33/6 and 4.67/6 respectively following instruction on solving group problems. After one 15 minute lesson, during which the researcher modeled how to identify and solve problems with the unknown in the initial position, Daniel was able to correctly solve all problems on subsequent probes assessing generalization to problems with unknowns in the initial position. When compar ing the probes assessing generalization to problems with unknowns in the initial position PAGE 72 72 administered prior to generalization instruction to those administered after gro up problems, from 2.5/6 to 6/6 for change problems, and from 3.5/6 to 6/6 for compare problems. Generalization to P roble ms with Unknowns in the Medial P osition. Spontaneous generalization to problems with the unknown in the medial position was not seen. R ather, Daniel tended to overgeneralize the group problem schema to these problems as though he was solving for an unknown all. This resulted in incorrect number sentences and answers for all problems. However, after one 15 minute lesson, during which the r esearcher modeled how to solve for an unknown in the medial position, Daniel was able to correctly solve all problems on subsequent probes assessing generalization to problems with unknowns in the initial position. His performance on group, change, and com pare problems with the unknown in the medial position improved from 0/6 prior to generalization instruction to 6/6 following generalization instructions. Maintenance performance on group, change, and compare problems with unknowns located in the final position remained at ceiling level on these probes indicating that treatment effects from the SBI were maintained over time. He also demonstrated maintencance of treatment effects on generalization probes involving problems with uknowns in the initial or medial position as his performance remained at ceiling level save one computational information did not remain at ceiling level as he crossed out relevant information and PAGE 73 73 included irrelevant information on one change problem. However, his performance on these problems still remained well above what they were prior to generalization instruction. Figure 4 is a graph of problem solving probes, generalization probes, and practice sheets. Data are graphed for each session so that trend and level can be visualized. Danie Problem Solving Behaviors Prior to receiving SBI, Daniel engaged in s imilar behaviors on all problem solving and generalization probes adm inistered. On the first problem solving probe, Daniel responded to problems without reading them or waiting for the researcher to read them aloud. This behavior continued despite the re searcher prompting Daniel to read the while reading the problem aloud to him beginning with the 6 th problem on the first problem solving probe and continuing for all pr oblems on subsequent problem solving and generalization probes conducted prior to SBI. After she had finished reading the problem, the researcher gave Daniel back the pencil so that he could show his work. The researcher made this accomodation in an effort to obtain a more valid assessment problem solving performance. Despite this accommodation, Daniel did not use the information provided in the problem prior to SBI. Instead, immediately after getting the pencil he quickly wrote horizontal numbe r sentences, usually two number sentences from the same fact family for each problem. The majority of the time these fact families consisted of single digit numbers that were not included in the word problem just read. Daniel verbally recited the number se ntences while writing them. PAGE 74 74 Change Problems Figure 4 1 Points earned by Daniel on probes and practice sheets. Compare Problems Group Problems Follow Up Generalization Baseline Instruction Maintenance PAGE 75 75 For instance, as he was writing 3+3=6 and 6 Six minus three equals three number sentences and sometimes noticed these mistakes and self corrected verbally and by erasing and re writing all or part of the number sentence.While Daniel worked on probes, the researcher provided in termittent verbal and edible reinforcers for on task Daniel would smile and eagerly e at the edibles when reinforcers were delivered. Following SBI, Daniel waited for the researcher to finish readi ng each word problem on problem solving and generalization probes before trying to solve it. As a result, the researcher no longer implemented th pencil while reading the problem aloud. After hearing the problem, Daniel did not immediately start writing. Instead he quietly looked up from the probe to the RUNS and schematic diagram cards placed on the table above t he probe for several seconds before beginning to talk and write. During the majority of problems Daniel used self talk that included SBI vocabulary (i.e., group problem, change get more, comparing words, put together all). When solving compare problems, Da niel was observed to circle the comparing w ords in the problem. On problem solving probes, Daniel was not observed making any additional marks on the word problems or using schematic diagrams to solve the problems. Instead he wrote correct, vertical number sentences to solve the problems. On generlization probes including irrelevant information, Daniel crossed out the irrelevant information in each problem and circled the comparing words in compare problems. Again, he did not use schematic diagrams to solve the problems, but instead PAGE 76 76 wrote the correct vertical number sentence for each problem. On generalization probes with the unknowns in the initial and medial positions, Daniel continued to circle the comparing words in compare problems. He also used schemat ic diagrams to solve these problems, writing vertical number sentences only after completing the diagram. Although Daniel continued to make errors when solving problems post SBI, he was able to self correct these mistakes consistently. After completing a n umber sentence, Daniel often did not write his answer on the answer line and did not include a label with his answer. The researcher again provided intermittent verbal and edible reinforcers for on task behavior and Daniel responded to these reinforcers by smiling and eagerly eating the edibles. problem solving behaviors changed dramatically following SBI. Following the intervention, he began to use the information contained in the word problems in his solutions as evidenced by his willingness to listen to the problem before starting his work, his use of number sentences containing quantities from the word problems, and knowledge and use of the schema for group, ch ange, and compare problems also improved following SBI as evidenced by his use of SBI vocabulary, schematic diagrams, monitor his problem solving increased following SBI as evidenced by his mo re consistent self correction of errors. Problem Solving Performance During the first set of baseline probes, Justin was able to successfully solve several of the group and change problems. He earned an average of 3/6 points on both of these probl em types. However, he only correctly solved one compare problem, PAGE 77 77 earning an average score of 0.5/6 on these problems. Justin appeared to be aware that the problems he was given were additive word problems. However, his strategy for determining which opera tion to use to solve the problem seemed to be based on his chosen operatio n to the two quantities included in the problem. Justin responded quickly to instruction on solving each type of word problem. He immediately performed at ceiling on practice sheets and continued to perform at this level throughout instruction on each prob lem type. Following instruction on solving group change and compare problems declined. He oblems. Following instruction in solving change problems, Justin continued to refuse to attempt solving compare problems on group or change line for compare problems Following instructio n on solving each type of word problem, level for the remainder of instruction instruction from 3/6, 1.7/6 and 0.6/6 fo r group, change, and compare problems respectively to 6/6 for each problem type following instruction on solving that type of word problem. Generalization to P robl ems with Irrelevant Information Prior to receiving SBI, Justin was able to successfully sol ve only one problem con taining irrelevant information. For the majority of problems, Justin appeared to choose an operation either addition or subtraction at random. After reading a PAGE 78 78 then proceeded to apply his chosen operation to all three quantities included in the problem. The change problem that he correctly solved contained two quantities that referred to a count of items and one quantity that referred to a co st (i.e., A teacher is taking 28 students to the zoo. If 6 students get sick and each zoo ticket costs $5, how many students did the teacher take to the zoo?). He appeared to apply the same strategy to this problem as to others, but omitted the monetary am ount, chose the correct operation presumably by chance, and was therefore able to obtain the correct answer. Following instruction on solving group problems, Justin began to apply the group schema to solving group problems with irrelevant information and refused to solve the answer line for these problems. For two of the group problems he added all three quantities together, but for one of the group problems, in which the re levant quantities were both 2 digit numbers and the irrelevant quantity was a 1 digit number, he omitted the irrelevant quantity and obtained the correct answer (i.e., Jim looks out the window. He counts 42 boys, 4 teachers, and 36 girls on the playground. How many children are on the playground?). Following change instruction, Justin demonstrated understanding that there was came agitated and started to cry, repeatedly PAGE 79 79 your diagrams. If you want to, y reassurance, Justin was able to calm down and complete the generalization probe, crossing out the irrelevant information in each problem. He applied the group problem schema correctly to group problems containing irrelevant information and his average performance on these problems improved from 1/6 to 6/6. Justin applied the change problem schema to change and compare problems containing irrelevant information. His average performance improved from 1/6 t o 6/6 for change problems and from 0/6 to represents spontaneous generalization of the change schema to problems containing irrelevant information. His slight improvement in solving compare problems is likely due to change rather than due to spontaneous generalization. Justin correctly solved the 15 students in the sand box. How many more studen ts are on the monkey bars than in as the beginning amount and 15 as the change amount. This led Justin to obtain the correct number sentence and answer (27 15=12) despite u sing the wrong schematic diagram. Following instruction in solving compare problems, Justin continued to cross out crossed out the irrelevant information. Justin also began to correctly apply the group, change, and compare schemas correctly to problems containing irrelevant information. This resulted in his average performance on group and change problems containing PAGE 80 80 irrelevant information remaining at ceiling level, wh ile his performance on compare problems containing irrelevant information improved form 0.67/6 to 6/6. G eneralization to Problems with U n knowns in the Initial Position Prior to receiving SBI, Justin was able to successfully solve several problems with the unknown in the initial position of the schematic diagram. For the majority of problems, Justin appeared to choose an operation either addition or subtraction at quantities included in the problem. As a result, Justin was able to correctly solve two group problems, one change problem, and two compare p roblems; presumably by chance. Fol lowing instruction on solving group problems, Justin began to apply the group schema to solving group problems with the unknown in the initial position. When applying the group schema to these problems, he did not seem to realize that the initial part was the unknown and instead completed the diagram as though both parts were known and the all was the unknown resulting in incorrect answers on all group problems leading to a decline from 4/6 at baseline to 0/6 following group instruction. Justin also refused spends $20 on a new doll. Now she has $38. How much money did Kim have in her obtaining the correct number sentence and answer (20+38=58) to this problem presumably by chance. As a result his performance on change problems was PAGE 81 81 maintained at 2/6 while his performance on compare problems declined from 4/6 to 0/6 following instruction in solving group problems. Following change instruction, Justin demonstrated understanding that the unknown was in a different location in the diagram. When he read the first problem (A teacher is planning to take some students to the museum. 9 students cancelled. Now 50 students are going to the museum. How many students was the teacher planning to take?) Justi are doing a great job. Put things in the diagram where you think they belong and then reassured him. He then correctly completed the problems with the unknown in the initial position of the change diagram with a larger beginning amount. He then proceeded to correctly solve the remaining group and change problems using the correct diagrams and placing the unknowns in the initial position. As a result, his average performance on group and change problems improved from 2/6 to 6/6 following change instruction. On this generalization probe, when solving compare problems with an unknown in the initial position, Justin used the change diagram. However, he completed the change diagram for these p roblems as though he was solving for an unknown ending. As a result, schema sometimes resulted in the correct number sentence and answer for a compare PAGE 82 82 74 cows in the pasture. If there are 22 more cows in the barn than in the pasture, how beginning amount and +22 as the change resulted in the correct number sentence and answer 74+22=96. Following instruction in solving compare problems, Justin correctly applied the group, change, and compare schemas. He also continued to demonstrate an understanding of the location of the unknown by correctly completing schematic diagrams and by orally labeling each quantity as representing a particular part of the performance on group and change problems remaining at ceiling level and his performance on compare problems improved form 2.67/6 to 6/6 following instruction on solving compare problems. Generalization to P r oblems with Unknowns in the Medial Position Prior to receiving SBI, Justin was able to successfully solve all problems with the unknown in the medial position of the schematic diagram. Justin read the first three use subtraction for all of the remaining problems without reading them aloud. As a result, Justin was able to obtain correct number sentences and answers for all of the problems. However, his ability to obtain t he correct answer did not appear to be due to understanding, but rather to his assumption that the problems were difficult thus requiring subtraction rather than addition. Following instruction on solving group problems, Justin refused to solve any problem s with the unknown in the medial position. PAGE 83 83 declined from 6/6 to 0/6 following instruction o n solving group problems. Following change instruction, Justin demonstrated understanding that the unknown was in a different location in the diagram. When he read the first problem (A teacher is taking 25 students to the museum. Some students cancelled. Now 21 students are going to the museum. How many students canceled?) Justin pointed to the and subsequent group and change problems with the unknowns in the medial position of the diagram using the correct schematic diagram for each problem. His average performance on group and change problems improved from 3/6 to 6/6 following instruction on so lving change problems. Following change instruction, Justin began to apply the change schema to compare problems with an unknown in the medial position. However, he completed the change diagram for these problems as though he was solving for an unknown end ing. Using this strategy, Justin was able to correctly solve amount and 32 as the change, Justin obtained the correct number sentence and answer (89 32=57) despite using the incorrect diagram. As a result he earned 2/6 points on compare problems with the unknown in the medial position following instruction on solving change problems. Following instru ction on solving compare problems, Justin correctly applied the group, change, and compare schemas to problems with unknowns in the medial PAGE 84 84 position. He also continued to demonstrate an understanding of the location of the unknown by correctly completing sc hematic diagrams and by orally labeling each problems remaining at ceiling lev with unknowns in the medial position improved form 2.67/6 to 6/6 following instruction on solving compare problems. Maintenance Maintenance probes were conducted eight weeks after SBI was completed. Ju group, change, and compare problems with unknowns located in the final position remained at ceiling level on these probes indicating that treatment effects from the SBI were maintained over time. He also demonstrated maintencance of t reatment effects on generalization probes involving problems with irrelevent information and problems with uknowns in the initial or medial position as his performance remained at ceiling level on these probes as well. Figure 4 2 rformance on probes and practice sheets Data are graphed for each session so that trend and level can be visualized. Problem Solving Behaviors Prior to receiving SBI, Justin tended to work quickly and quietly on all probes. When the researcher o He would write a vertical addition or subtraction number sentence for each word PAGE 85 85 Figure 4 2. Points earned by Justin on probes and practice sheets. PAGE 86 86 Following SBI, Justin would read the problems aloud to him. He appeared to follow along with his eyes as she read the problems. After the resear cher read each problem, Justin pointed to a phrase in the problem and verbalized which critical component that phrase represented and to which problem type the word problem belonged. For instance, he would point to the words how many more birds than in a p 5 children got sick in another probes with problems containing irrelevant information, Justin would cross out the irrelevant information in the word problem. He would then draw the corresponding schematic diagram, complete the diagram correctly, and write the number sentence corresponding to the diagram. He would then write his answer and a label on the answer line. When Justin made errors in his diagram or number sentence he would say intermittent verbal praise for on task behavior to which Justin responded by smiling and repeating the praise. Prior to SBI, Justin used information contained in the problem to determine a solution. However, his solution strategy did not allow him to obtain correct answers consistently. He appeared to read the problem, make comments indicating that he write number sentences using quantities contained in the word problems. Following SBI, Justin continued to use the information contained in the problems to determine a PAGE 87 87 solution; however, his use of the strategies he learned in SBI allowed him to consistently obtain correct answers on word problems. He used the RUNS mnemonic to self monitor his problem solving, used critical problem features to determine to which schema a problem belonged, and used schematic diagrams to help formulate his number sentences. Justin used self correction both before and after SBI to correct computational errors; however, following SBI he also self corrected errors i n schema use and operation choice. Satisfaction Data Both participants and their mothers completed satisfaction scales following SBI. Copies of these s cales can be found in Appendix F On the participant satisfaction scale, Dani el indicated feeling positiv e about solving math word problems and about using diagrams to solve word problems. He also indicated that he thought other chi ldren might also feel positive about using diagrams to solve word problems. On the other hand, Daniel indicated feeling neutral a bout using the RUNS steps and indicated that othe r children might feel negative about using the RUNS steps. This is consistent with his reference to, but failure to write down or completely follow the RUNS steps following SBI. Justin, on the other hand, in dicated feeling positively about solving math word problems, using the RUNS steps, and using diagrams. He indicated that he thought other children would feel neutral about using the RUNS steps and using diagrams to solve word problems. This is consistent w ith his use of the RUNS steps and of diagrams when solving problems following SBI. strongly that SBI addressed important skills, was a valuable use of instructional time, PAGE 88 88 was enjoy able for Daniel, and resulted in i problem solving performance. She also agreed strongly that she would recommend SBI to others and to solve word problems following SBI and was neutral regarding his use of the RUNS steps following SBI. On the open provided evidence of spontaneous generalization of SBI to the school setting. She indicated that on th e mathematics unit test that was completed after SBI, Daniel correctly answered all four of the open ended addition and subtraction word problems included. She said that his paraprofessional reported that he used vocabulary from SBI while completing these items and used a schematic diagram for one of the problems. skills, was a valuable use of instructional time, and was enjoyable for Justin. She also agreed that she would recom mnemonic and diagrams when solving word problems, and regarding improvements in his abilities to solve problems. On the open ended section of the questionnaire, she indicated that because it was summer break she had not had an opportun ity to observe problem solving performance or strategies outside of SBI sessions and therefore was not certain whether Justin would cont inue to show improvement or use school teacher and indicated that she was grateful for being able to observe most of the SBI sessions as she hopes to be able to use SBI with her own students in the future. PAGE 89 89 Summary of Results Although Justin and Daniel are children with Autistic Disorder, their profiles are ve ry different. Daniel is a first grade student with obvious visual strengths. He also has a number of behaviors that can make it challenging to work with him. For instance, Daniel needed frequent prompting and reinforcement to stay seated during instruction, had difficulty answering questions due to echolalic speech, and often seemed more interested in drawing pictures than in completing mathematics word problems. In addition, because Daniel was a first grader he had minimal experience with solving word problems. At the time of this study, he had recently completed his first classroom unit on word problem solving which inclu ded only keyword instruction. Justin, on the language more flexibly than does Daniel. During the study Justin exhibited fewer behaviors that interfere d with instruction and w a s very eager to please adults. As a sixth grader, Justin h ad been exposed to word problem solving instruction numerous times and likely to a variety of word problem solving strategies. Despite having been exposed to some keyword instruction prior to SBI, Daniel did not seem to have any understanding of word problems other than to know that they were a mathematics task. He did not use any information from the word problems in his solutions and instead wrote number sentences for unrelated fact families. His performance for each problem type was at basal level on the first set of probes and remained at near basal level until after instruction on each problem type. Justin, on the other hand, appeared to have a strategy for solving additive word problems. He se emed to use sentence complexity as an indicator of which operation to use and was able to correctly solve approximately half of the word problems on initial probes. On PAGE 90 90 subsequent baseline probes, his performance declined as a result of his refusal to solve problems not belonging to schemas already addressed in SBI. problem solving probes reached ceiling level and was maintained at that level up to eight weeks following instruction. Although both participan ts achieved mastery on the word problems targeted in this study, Justin did so in fewer lessons than did Daniel. In addition, Daniel required additional lessons to help him generalize SBI to problems with irrelevant information and with the unknown in the initial or medial positions. Justin, on the other hand, demonstrated spontaneous generalization to these types of problems after receiving SBI on group and change problems Justin also was observed to follow the SBI procedures more fully when completing pr obes after SBI. He used the RUNS mnemonic and schematic diagrams consistently to solve each problem. Daniel, on the other hand, did not write out the RUNS mnemonic, though he was observed to visually refer to it while solving problems. In addition, Daniel only used schematic diagrams when solving problems with unknowns in the initial and medial positions. Both participants used self talk that included vocabulary taught as part of the SBI, and demonstrated increased use of self correction and self monitoring following SBI. PAGE 91 91 CHAPTER 5 DISCUSSION SBI for Students with Autism Although researchers have studied a variety of me thods for improving the problem solving performance of students with mathematics difficulties or mild disabilities (Gersten et al., 2009), a literature search revealed few studies that included students with autism. The purpose of this study was to explore the use of SBI with students with autism and contribute to the small body of literature that has addressed ways to teach mathematical word problem solving to students with autism. Because this study is primarily a population expansion, it is important to consider how the characteristic features of autism may have responses to it. Although t he interaction between SBI and the characteristic features of behaviors and support from the literature can provide some insight into how the characteristic features of autism and the SBI intervention may interact. Participant Profiles and Impact on Problem solving Performance Daniel and Justin both have diagnoses of A utistic D isorder from developmental behavioral pediatricians. They both display difficulties with language and communication that are typical of autistic disorder ( American Psychiatric Associat ion, 2000). Although both participants use oral language as their primary means of communication, their language is impaired compared to other children of the same age. Dani el speaks primarily in echolalic phrases that he uses to convey his wants and needs. He only answers questions when given choices or when he can answer by pointing to an object or picture, or by drawing his own picture. Although Justin uses language more f lexibly PAGE 92 92 than does Daniel, his language is still severely impaired compared to other children his age. He can follow simple two step instructions, but has difficulty with complex multi step instructions. He can sometimes answer open ended questions without being provided with choices or visual cues. However, he often becomes frustrated by his inability to formulate his thoughts into language. He also has difficulties with complex language comprehension skills such as making inferences and predictions In add ition, both Daniel and Justin exhibit difficulties with executive functioning that often accompany ASD ( American Psychiatric Associat ion, 2000). For instance, Daniel has difficulties attending to classroom instruction (sustained attention), refraining from getting out of his seat or calling out without permission (inhibition), and transitioning psychoeducational evaluation, he too has difficulties with executive functioning part icularly within the area of verbal working memory. Language impairment and executive dysfun c tion can both contribute to difficulties with mathematics problem solving. Children with language impairments can be expected to have difficulties with mathematic al word problem solving because solving problems requires that students use semantic mapping to determine the relationships between known and unknown quantities to determine the correct operation to use to solve for the unknown (C hristou & Phillippou, 1999 ). The failure of students with autism to use semantic cues (Frith & Snowling, 1983) has important implications for teaching word problem solving because it will likely make semantic mapping more difficult for these students. Executive dysfunction, includ ing deficits in working memory and sustained attention, can compound these difficulties by making it difficult for children to PAGE 93 93 plan a solution strategy and filter and manipulate relevant information in word pro blems (Zenta l l, 2007; Zentall & Ferkis, 1993). Working memory deficits may also lead to the use of immature problem solving strategies and procedures and difficu lties inhibiting irrelevant informati on leading to errors in problem solving (Geary et al., 2008). Geary and Hoard (2005) suggest that diffi culties in mathematics problem solving can be due to weaknesses in the underlying systems and supporting competencies that underlie mathematics problem solving. For students with autism, deficits in the language system and central executive may lead to dif ficulties building concepts for whole number operations and carrying out procedures necessary for word problem solving problems prior to starting SBI suggest that their language i mpairment and executive dysfunction may have been impacting their ability to solve problems. Daniel attempt ed to solve word problems without reading the problem or allowing the researcher to read the problem to him. It is possible that this impulsivity was the result of executive Even when the researcher offered him reminders to read the problem before starting to work, Daniel appeared to be unable to inhibit his impulsive behavior and attend to the word problem. When the researcher implemented an intervention to prevent this impulsivity, Daniel still did not make use of the information contained in the word problem. This may have been due to difficulties attending to the researcher or to difficulties making use o f the semantic information contained in the word problems, but it is likely that language impairment and executive PAGE 94 94 When solving problems prior to SBI, Justin seemed to rely on h is perception of the complexity of the syntax of the word problem to choose which operation to use. On those problems that he perceived contained more complex syntax, he subtracted. On problems that he perceived contain ed less complex syntax, he added. Thi s strategy suggests that Justin may have had difficulty making use of the semantic information contained in word problems. It is possible that these difficulties were due to language impairments or verbal working memory deficits, but it is likely that both may have contributed to his difficulty arriving at an effective solution strategy. Shortcomings of Traditional Problem solving Instruction Daniel and Justin had both received mathematical word problem solving instruction in their public school classrooms prior to starting SBI. The county in which Daniel and Justin reside used the state adopted mathematics series Harcourt Math Florida ( Harcourt School Publishers, 2002 ). Starting in kindergarten, this math series encourages students to draw pictures to repre sent mathematical word problems. The workbooks accompanying the text include pages with pictures of different objects and students are instructed to either draw more objects or cross some of the objects out as the teacher reads aloud change problems involv ing those objects. For instance, the worksheet might include a box with a picture of 5 fish in it. The teacher will then read the draw 2 more fish, count the total number of fish, and write 7 on the answer line. The same worksheet might include a box with a eats 4 cookies. How many cookies does Amy h cross out 4 of the cookies, count the remaining cookies, and write 4 on the answe r line. PAGE 95 95 These activities assume that students possess the prerequisite language and verbal working memory skills to keep in mind what t he teacher read, understand whether the representation without additional instruction. By first grade, students are given written problems with accompanying pictures they can then alter. By second grade they are given space in which to draw their own pictorial representations to accompany word problems. In addition to teaching students to create pictorial representations for word problems, the mathematics text also introd uces keyword instruction starting in first grade, wherein students are taught that the keywords more and together indicate addition, and the keywords less fewer and more than indicate subtraction. Starting in second grade, studen ts are taught a general p roblem solving heuristic that includes the following steps: (1) read the problem carefully, (2) look for clues, (3) choose a strategy, and (4) check your work. Keyword instruction, use of pictorial representations, and the general problem solving heuristic are reintroduced and reinforced in the texts for grades three to five. Additional keywords and types of pictorial representations are introduce d for multiplicative word problem solving starting in grade three. Presumably, both Daniel and Justin had been exposed t o the mathematical word problem solving instruction included in the mathematics basal text prior to SBI. As a first grader, Daniel was likely introduced to pictorial representations in kindergarten. According to his mother, his class had just comp leted a chapter that reinforced the use of pictorial representations and introduced keywords for additive word problems when this study commenced. As a sixth grader, Justin had likely been taught to use pictorial PAGE 96 96 representations to assist in solving mathem atical word problems for the past six years. He had also likely been taught to identify keywords and to choose an operation based on those keywords when solving additive word problems for several years. Justin had also likely been taught to use the general problem solving heuristic. Despite t his prior additive word problem solving instruction, both Justin and Daniel exhibited difficulty solving word problems on probes prior to SBI. Daniel seemed completely unaware that the word problem contained informatio n relevant to obtaining a correct answer as evidenced by his use of two unrelated number sentences from the same fact family to solve problems. He did however seem to be aware that the word problems were a mathematics task. He did not use either the pictor ial representations or the keyword instruction that he was exposed to in school. Justin appeared to have more understanding of how to solve additive word problems than did Daniel. This is to be expected given the greater number of exposur es he would have h ad to problem solving instructions in over six years of public schooling as compared to Daniel who had been in school for less than two years Justin seemed to be aware that in order to solve the problem he needed to first read it. He also seemed to know t hat the syntax of the problem held clues as to whether to add or subtract. However, he seemed to rely on his strategy included in the mathematics basal text or using semanti c mapping. Justin did not attempt to draw pictorial representations to assist with problem solving. Initially it may seem surprising that neither Daniel nor Justin used the strategies they had presumably learned in school to solve word problems. However, i t is possible that the strategies included in the mathematics basal text are not optimal for students PAGE 97 97 with disabilities or mathematics difficulties. For instance, the basal text relies heavily on keyword instruction. According to Van de Walle et al. (2010 ) keyword instruction can be misleading because some problems do not contain keywords, and even those problems that do contain keywords may not always require the operation that keyword has been linked to during instruction. Keyword instruction is also ine ffective because it fails to take into account the underlying meaning and semantic structure of the problem and fails to teach the relationships between addition and subtraction. The basal mathematics textbook also encourages students to use pictorial repr esentations to solve word problems. Although visual representations can be helpful in teaching problem solving (Gersten et al., 2009), this strategy is only effective when students use visual representations that accurately reflect the underlying structure and relationships among quantities in the word problem (Hegarty & Kozhevnikov, 1999). In addition, although problem solving heuristics can be helpful to students (Gersten et al.), the heuristic included in the basal mathematics textbook does not provide a way for students to systematically approach the problem (Van de Walle et al. ). Instead, the heuristic directs them to choose from among the less than optimal strategies to which they have been exposed. Students with autism, in particular may flounder whe n using a problem solving heuristic that does not provide a systematic approach to problem solving. When asked to complete multi step activities, students with autism benefit from structure in the form of explicit and systematic instructions that provide t hem with information regarding what steps to take, the order in which to do those steps, and what behaviors constitute completion of each step (Mesibov & Howley, 2003). Bec ause the general problem solving heuristic included in the basal textbook does not p rovide the specific behaviors PAGE 98 98 expected during each problem solving step, it is unlikely to be helpful to students with autism. Benefits of SBI SBI provides supports that may address several of the weaknesses i dentified in the basal textbook ap proach to tea ching word problem solving. SBI includes a problem solving heuristic that represents a systematic approach to solving word problems. SBI also includes visual representations, an effective approach to improving mathematical problem solving performance (Gers ten et al., 2009). However, unlike the basal text that expects students to create their own pictorial representation for each individual problem, SBI teaches students three specific diagrams that visually represent the underlying semantic structure of the word problems (Jitendra et al. 2007b). SBI combines visual representations, direct instruction, and a problem solving heuristic that supports a systematic approach to solving word problems. As such, SBI holds great promise as a means of supporting the pro blem solving performance of students for whom traditional classroom instruction has not been effective (Gersten et al., 2009; Jitendra et al., 2007b; & Van de Walle et al., 2010 ). SBI may be particularly beneficial to students with autism because it seems to provide scaffolding to support the language system. Visual representations make use of strengths in the visuospatial system of students with autism while helping them to compensate for difficulties with language (Mesibov & Howley, 2003; Tissot & Evans, 2003). The diagrams used in SBI provide a visual representation of the semantic structure of the three types of additive word problems. Various studies of SBI have used different diagrams to represent the three additive word problem schemas. The diagrams used in this and the Rockwell et al. (2011) study were adapted from those used PAGE 99 99 previously in an effort to most clearly represent the relationships among the quantities in each problem type. The diagrams used in this study clearly differentiate the larger q uantity from the two smaller quantities in each problem type. For change problems, in which the larger quantity can be either the beginning or the ending students were taught to adapt the diagram to clearly represent the larger quantity. The researcher ch ose to use diagrams that explicitly differentiated the larger quantity so that instruction could focus on the conceptual knowledge that addition results in a sum that is larger than the addends while subtraction results in a difference that is smaller than the minuend. Thus, the diagrams used in this study not only provided a visual representation of the semantic structure of additive word problems, but also provided a visual support that could be used to teach the underlying relationships between addition subtraction problems (Van de Walle et al., 2010 ). However, Daniel seemed to have some difficulty distinguishing the group and compare diagrams. It is possible that use of a different shape for the compare diagram would help students to more easily differentiate the two diagrams in the future. SBI may also help to reduce working memory demands and thereby may provide support for the central executive of students with autism (Cooper & Sweller, 1987) First, SBI includes a problem solving heuristic represented in this study by the mnemonic RUNS. The RUNS mnemonic provides support for planning and organization (Deshler et al., 1981). Participants were taught to check off the RUN S steps as they were completed. This allowed participants to approach each problem systematically without skipping or omitting steps and without having to rely on verbal working memory to keep PAGE 100 100 track of their problem solving plan. The RUNS heuristic, is pai red with a specific problem solving strategy (i.e. using schematic diagrams); which allows students to know precisely what behaviors are expected during each step of the problem solving process. This may provide them with the structure necessary to plan, i nitiate, and complete the activity, in this case word problem solving independently and with increased confidence (Mesibov & Howley, 2003). SBI also focuses on teaching the three schemas underlying the majority of single step additive word problems. In tr aditional instruction, students often focus on the superficial features of word problems and approach each problem as requiring a unique solution (Fuchs et al., 2003a). In SBI, on the other hand, students are taught to look for the critical features that i ndicate to which schema a problem belongs and then choose from several known and rehearsed solution strategies. Because each problem does not require a unique solution and because the solution strategies are known, working memory loads are reduced (Capizzi 2007). Although SBI provides many supports that may be beneficial to students with autism, additional behavioral supports are needed to lead to an effective intervention package. The participants in this study both seemed to benefit from intermittent rei nforcement for on task behavior. Reinforcers were chosen individually for each participant. It is likely that students with autism in general will benefit from reinforcement for on task behavior as a support for executive functioning. Both participants als o seemed to benefit from prompting and corrective feedback that included gestures and references to visual supports. Prompting and corrective feedback are a necessary component of guided practice, but the additional use of gestures and visual supports may help scaffold the language of students with autism. Justin benefited from being PAGE 101 101 allowed to request noncontingent breaks; however, Daniel had great difficulty transitioning back to instruction following breaks. Therefore, the provision of noncontingent brea ks may be beneficial for some, but not all, students with autism. individual needs. For instance, Justin was provided with verbal reassurance when he became agitated. Daniel was provided with various means of sensory stimulation such a physical touch and an air disk on which to sit. It may be important for intervention allow them to most benefit from instruction. The results of this study indicate that SBI was effective in improving the word problem solving performance of two participants with autism. The participants in this study both ma de measureable gains in problem solving perfo rmance following SBI. level on problem solving probes following instruction in solving each type of word problem. Although he did not demonstrate spontaneous generalizati on, he was able to reach ceiling level on generalization probes with minimal instruction. Justin also made significant gain s in his problem solving performance following SBI. While his baseline performance was above basal le vel, an analysis of his problem solving behaviors while completing probes suggests that he was getting the correct answers to problems due more to chance than to the use of an effective problem solving strategy. Following SBI, problem solving probes improved to ce iling level. He also demonstrated spontaneous generalization requiring no specific instruction to reach PAGE 102 102 ceiling level on generalization probes. Both Daniel and Justin maintained improvements in their problem solving performance eight weeks following the c ompletion of SBI. The improvements in problem solving performance Justin and Daniel demonstrated following SBI suggest that SBI may be an effective interve ntion for improving the problem solving performance of student s with autism. Gains in problem solvin g performan ce were evident following only four weeks of instruction al time This suggests that students with autism who recei ve SBI can acquire word problem solving to spontaneous generalization to target generalization problems, and both participant maintenance of improvements in problem solving performance eight weeks after completing SBI. This evidence of maintenance and generalization is particularly promising given the challenges of programming for skill generalization and maintenance for indiv iduals with developmental disabilities (e.g. Koegel & Rincover, 1977; Stokes & Baer, 1977). Daniel and Justin also engaged in behaviors that lend support to the theory that SBI effectively makes use strengths in the visuospatial systems of students with au tism while providing support for weaknesses in their language systems and central executives. Following SBI, Daniel was able to inhibit his impulsivity and attend to the researcher while she read problems aloud to him. He also referred to the RUNS steps an d the schematic diagrams when solving problems. Th e s e behavior s suggest that he used these visual supports to scaffold his organization and planning while solving PAGE 103 103 problems. Daniel also engaged in self talk about the critical components of word problems, an d circled some critical components and crossed out irrelevant information. This suggests that Daniel made use of schema knowledge when solving word problems and lends support to the assumption language system to allow him to make use of the semantic information contained in the word problems. One of the most interesting behaviors Daniel displayed while solving probes following SBI was selective use of schematic diagrams. He did not use schematic diagrams at a ll when the unknown quantity was located in the final position of the schematic diagram. However, he used schematic diagrams to solve all problems wherein the unknown quantities were located in the initial or medial position of the schematic diagram. Accor ding to Christou and Phillipou (1999) it is more difficult for children to solve additive word problems with the unknowns located in the initial or medial position than in the final position. The researchers hypothesized that these problems were more diffi cult to solve because they required more complex algebraic the initial or medial position, but not with problems with unknowns in the final position, suggests that the sch ematic diagrams may have provided him with necessary scaffolding to support his algebraic reasoning and to help make clear the relationship between addition and subtraction. When solving problems follow ing SBI, Justin was observed using the RUNS steps as instructed. He wrote the mnemonic RUNS before starting to solve each problem and crossed each letter off as he completed that step. Justin also indicated on satisfaction PAGE 104 104 scales that he enjoyed using the RUNS steps. According to Mesibov and Howley (2003) a n explicit heuristic like RUNS can provide structure that helps students with use of the RUNS mnemonic and his positive feelings about it suggest that this heuristic may n ot only have may also have allowed him to solve word problems with greater independence and confidence. Following SBI, Justin also used schematic diagrams consistently to solve all problems and reported having posi tive feelings about using schematic diagrams. While solving problems and completing diagrams he engaged in self talk about the critical problem features and sometimes circled critical problem features or crossed out irrelevant information. ent use of diagrams suggests that schematic diagrams may have functioned as an effective visual support for his language system and may have helped him to manipulate and use the semantic information contained in word problems. Not only did Justin use diag rams consistently, but he also generalized the use of schematic diagrams to problems with unknowns in the initial and medial position without any additional generalization instruction. This lends support to the theory that the diagrams may have sufficientl between addition and subtraction to allow him to spontaneously engage in algebraic reasoning without teacher modeling of the process. According to Geary and Hoard (2005) the language system and centra l executive are both critical in developing the problem solving behaviors prior to SBI indicated that he did not have an adequate understanding of the PAGE 105 105 relationship between addition and subtracti on needed for additive word problem solving. Despite six years of public school instruction, Justin was still unable to make use of the semantic information contained in word problems or of the conceptual relationship between addition and subtraction when generalization of schematic diagrams to solving problems with unknowns in the initial and medial position suggest that SBI may have improved his conceptual understanding of the relationship between addition and s ubtraction and allowed him to make use of the semantic information contained in word problems. Limitations Although this study suggests that SBI hold s promise for teaching additive word problem solving to students with autism, there are several limitations to this study. The purpose of this study was to evaluate the use of SBI to teach additive word problem solving to students with autism. Only one previous study was identified (Rockwell et al., 2011) that addressed the use of SBI with children with autism. Both this study and the Rockwell et al. study included students who had diagnoses of Autistic D isorder but did not have comorbid intellectual disabilities. These participants, while having language impairments characteristic of Autistic Disorder, still us ed language as their primary means of communication. Finally, all the participants were included in regular education classrooms and participated in AYP assessments. The results of this study and of the Rockwell et al., study may not generalize to children with different profiles of intellectual, language, and academic functioning. In addition, because this study included only single step addition and subtraction story problems, it only addressed a subset of skills needed for successful mathematical word p roblem solving. Finally, SBI was conducted one to PAGE 106 106 by a researcher familiar with the intervention. This study did not evaluate whether teachers in typical classroom settings could successfully implement SBI with students with autism. This further limits the generality of the findings. A further limitation of this study is the inclusion of only two participants. Originally, the researcher planned to include four participants. However, due to unforeseen difficulties with recruiti ng, only two participants were included. Because a multiple probes across behaviors design was employed wherein solving each type of word problem was treated as a separate behavior, there were three demonstrations of experimental control for each participa nt for a total of six demonstrations of experimental control. As a result of using a multiple probes across behaviors design, the number of participants did not impact the internal validity of the results (Kennedy, 2005). However, the inclusion of only two participants limits the external validity of the findings. Ideally, a population expansion would include at least six and preferably more participants. According to Horner et al. (2005) replication across multiple participants is necessary to ensure the e xternal validity of the results of single subject research. Furthermore, because this study looked only at problem solving as an outcome measure, it is not possible to draw data based conclusions about whether SBI improved the conceptual knowledge or algeb raic reasoning, reduced the working memory demands, or increased levels of inhibitory control or sustained attention of the participants. Although the results on outcome measures observed behaviors and literature may suggest outcomes other than changes i n additive word problem solving performance, conclusions drawn about the impact of SBI on other o utcomes are theoretical in nature and tentative at best. PAGE 107 107 Implications Future research designed to replicate and expand this study is indicated to contribute t o the evidence on using SBI to teach mathematical word problem solving to students with autism. To enhance external validity future studies may wish to replicate this study using larger numbers of participants and participants with more varying academic, i ntellectual, and language profiles. To allow conclusions to be drawn about other outcome variables, future studies of SBI with students with autism may wish to include additional dependent variables such as measures of attention, on task behavior, working memory, conceptual knowledge, computation accuracy and fluency, or algebraic reasoning. Once the results have been verified through replication with students with autism with varying profiles, teachers should be trained to conduct the SBI with small groups of students to ensure that teachers can successfully implement the intervention with fidelity in typical educational contexts. Additional studies may also be designed to teach children with autism to solve multiplicative word problems, multi step word pro blems, and word problems that include information beyond the text in charts and graphs. Future studies may also wish to use instruments currently approved to assess AYP in order to ensure that improvements in mathematical word problem solving generalize to high stakes assessments. Furthermore, it is possible that schematic diagrams may have the potential to benefit students with autism in other areas of the curriculum. For instance, schematic diagrams may help students with autism to develop writing or read ing comprehension skills. Additional research may wish to examine the impact of using schematic diagrams representing various text structures on the written expression and reading PAGE 108 108 comprehension of students with autism. For instance, studies could assess wh ether the use of plot diagrams would improve the narrative text reading comprehension or written expression of students with autism. Studies may also wish to evaluate the impact of using schematic diagrams to develop the conceptual knowledge of students wi th autism in other areas of the curriculum. For instance, schematic diagrams might be used to represent cause and effect relationships in science or history courses. Although future research is needed to verify the results of this study and expand its exte rnal validity, the results of this and the Rockwell et al. (2011) study suggest that SBI may have important practical implications. First, SBI may be an effective means of teaching students with autism to solve mathematics word problems. It appears that SB I may provide support for the language systems and central executives of students with autism, thereby facilitating their ability to solve word problems. In addition, the schematic diagrams used in SBI provide visual supports that not only support problem solving, but may also support conceptual understanding of the underlying relationships between addition and subtraction. Because previous studies (Griffin & Jitendra, 2009; Jitendra et al., 2007a; Jitendra et al., 2007b; Jitendra et al., 2010) have demonst rated that teachers in classroom settings can effectively implement SBI with students with mild disabilities and math difficulties, it is likely that teachers will be able to implement SBI in classrooms with students with autism. Furthermore, visual suppor ts and highly structured activities provide needed support for students with autism in classroom settings (Mesibov & H owley, 2003; Tissot & Evans, 20 0 3 ). SBI provides visual supports through t he use of schematic diagrams and provides structure through the use of the PAGE 109 109 RUNS heuristic. Therefore, SBI may be well suited to supporting the problem solving performance of students with autism in classroom settings. Although the research base on using SBI with students with autism is still small, many studies have b een conducted that indicate that SBI is an effective means of teaching word problem solving to students with out disabilities, with mild disabilities and those at risk of mathematics failure. It may be valuable for textbook developers to consider using ele ments of the SBI approach when developing word problem solving instruction. For instance, mathematics textbooks typically use keyword instruction, which focuses on superf icial features of word problems, despite the misleading nature of keywords. Focusing o n the underlying structure of word problems may better facilitate schema induction successful problem solving, and conceptual knowledge of all students. Furthermore, SBI may have particular promise for the tier 2 and 3 intervention materials included with textbooks as part of the RtI model. As a direct instruction approach that can be used with small groups or individual students, SBI may be ideal as an additional intervention for struggling students. Finally, it may be helpful for academic standards write rs to consider SBI and schema theory when writing mathematics standards. Understanding the underlying schematic structure of word problems can facilitate transfer of problem solving skills to novel problems and can encourage the development of conceptual k nowledge. Standards developers may therefore wish to include schema induction as an objective. PAGE 110 110 APPENDIX A STUDY SUMMARY TABLE Table A 1: Summary Table Author/Date Purpose Participants Procedures Fuchs, Fuchs, Finelli, Courey, & Hamlett, 2004c Asses s the added value of an expanded version of schema broadening instruction including additional transfer featur es to enhance real life problem solving strategies 351 students from 24 third grade classrooms in 7 urban schools. Students identified as low, ave rage, or high performing. Classrooms were randomly assigned to general classroom instruction, schema broadening instruction, or expanded schema broadening instruction Control consisted of teacher delivered instruction from the textbook. Schema broadening instruction involved instruction in solution rules for four problem types from the textbook and three superficial transfer features. Expanded schema broadening instruction added three additional superficial transfer features. Research assistants conducted scripted lessons for treatment groups. Pre and post testing included four measures with varying degrees of transfer. Results: Two factor mixed model ANOVAs conducted with teacher as the unit of analysis on pre test and improvement scores. Results indicat ed that groups were comparable prior to treatment. For transfer 1 and transfer 2, the two treatment groups improved significantly more than did the control group. For transfer 3 and transfer 4, the expanded schema broadening instruction group improved more than the schema broadening instruction group, which improved more than the control group. Results were similar for students with disabilities. Effect sizes large for significant contrasts. PAGE 111 111 Table A 1 Continued Author/Date Purpose Participants Procedures Fuchs, Fuchs, Finelli, Courey, Hamlett, Son es, & Hope, 2006 Assess the contribution of instruction in strategies to promote real life problem solving by comparing the effectiveness of schema broadening instruction wit h and without real life problem solving strategies 445 students in 30 third grade classrooms from 7 schools in an urban district. Classrooms randomly assigned to control, schema broadening instruction (SBI) or SBI plus real life problem solving strategies (SBI RL) Teachers implemented all instruction, using the textbook in the control condition and scripted lessons in the treatment conditions. SBI involved teaching four problem types from the textbook and four superficial transfer features. The SBI RL condition added instruction in five strategies for solving real life problems. Student s were pre and post tested using immediate, near, and far transfer measures. Results: ANOVAs were conducted. Results indicated group comparability. On immediate transfer, near transfer, and questions tw o and four of far transfer, both SBI groups outgrew the control group. Effect sizes were large, often over 3 standard deviations. On question one of the far transfer measure all groups grew comparably. On question three of the far transfer measure the SBI RL group outgrew the SBI and control groups. Again, effect sizes were large, hovering at approximately two standard deviations. Fuchs, Fuchs, Hamlett, & Appleton, 2002 Assess efficacy of schema broadening tutoring for improving mathematics problem solving pe rformance of fourth grade students with mathematics disabilities. Compare the treatment effects of schema broadening tutoring to computer assisted tutoring. 40 fourth grade students with mathematical disabilities from three schools in a southeastern city a ssigned to schema broadening tutoring, schema broadening tutoring and computer assisted tutoring, computer assisted tutoring, and control groups Teachers used the textbook with the control groups. In schema broadening tutoring a research assistant used scr ipted lessons to teach schemas for four problem types obtained from the textbook and four transfer features to small groups of students. Computer assisted tutoring involved guided practice using real life problems and motivational scoring. Pre and post te sting using story problems, transfer story problems, and real life problems. Results: Chi sq u a re analysis and ANOVAs indicated group comparability before treatment. ANOVAs indicated statistically significant treatment effects on both the story problems an d transfer story problems. On both measures, students in schema broadening tutoring conditions outgrew students in the computer assisted and control conditions. On the transfer story problem students in the computer assisted condition also outgrew students in the control condition. Effect sizes comparing the schema broadening tutoring and control group for these measures were large. PAGE 112 112 Table A 1. Continued Author/Date Purpose Participants Procedures Fuchs, Fuchs, & Prentice, 2004b Assess the differential effects of sche ma broadening instruction with self regulated learning strategies (SRL) on complex mathematics problem solving tasks for students without disability risk (NDR), at risk of mathematics disability (MDR), at risk of mathematics and reading disabilities (MDR/R DR), and at risk of reading disability (RDR) 201 students who met criteria for inclusion in the NDR, MDR, RDR, or MDR/RDR category. These students were from 16 of the classrooms included in the previous study (Fuchs et al., 2003b) and were in the control g roup or the schema broadening instruction plus SRL group. For descriptions of conditions, see Fuchs et al., 2003b. Pre and post test measures included immediate and near transfer. A three factor ANOVA assessed each performance dimension (conceptual underp innings, computation, and labeling) for these measures. Exploratory analyses using subtests of the TerraNova were conducted to generate hypotheses about the relative contribution of reading difficulty and mathematics difficulty to the responsiveness of stu dents with MDR/RDR. Results: On the immediate transfer measure for conceptual underpinnings the NDR, RDR, and MDR students outgrew the MDR/RDR students; and for computation and labeling the NDR students outgrew the RDR, MDR, and MDR/RDR students. On the n ear transfer measure on conceptual underpinning the MDR/RDR students improved less than the NDR students; and on computation and labeling the MDR/RDR, MDR, and RDR students improved less than the NDR students. Across all measures and performance dimensions the treatment group outgrew the control group. Computation deficits accounted for more of the variance in responsiveness of students with MDR/RDR compared to reading comprehension deficits (1.5% versus 21%). PAGE 113 113 Table A 1. Continued Author/Date Purpose Participants Proce dures Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, Hosp, & Jancek, 2003a Assess the contribution of explicitly teaching for transfer in a schema broadening intervention that combined instruction in problem solution rules with instruction in transfer 375 students in 24 third grade classrooms at 6 schools in a southeastern urban school district. Students identified as at, above, or below grade level. Classes randomly assigned to control, solution rules, partial solution rules plus transfer, full solution r ules plus transfer. Teachers used the math text book with the control group. In treatment conditions, teachers assisted research assistants who provided instruction using scripted lessons. Solution rules involved teaching the schemas for four problem ty pes f rom the textbook. Transfer instruction involved teaching four superficial problem features. Pre and post testing was conducted using transfer, near transfer, and far transfer measures. Results: ANOVAs indicated group comparability before treatm ent. ANOVAs indicated that for immediate and near transfer treatment groups outgrew the control group. For far transfer the two groups receiving transfer instruction significantly outg rew the control group. Effect sizes for significant post hoc contrasts w ere large (ranging from 0.78 to 2.25). Results were not mediated by initial achievement status. Students with disabilities did best in full solution plus transfer and worst in partial solution plus transfer. Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, & Schroeter, 2003b Examine the contribution of self regulated learn ing strategies (SRL) on problem solving improvement when combined with schema broadening instruction 395 students in 24 3 rd grade classrooms from 6 schools, designated as having high, avera ge, or low math achievement. Each class randomly assigned to control, schema broadening instruction, or schema broadening instruction/SRL Schema broadening instruction taught solution rules for 4 problem types obtained from the textbook and 4 transfer fea tures. SRL involved having students score their work, chart their progress, set goals, and report examples of transfer. Teachers used scripted lessons to conduct instruction. Pre and post tests included immediate, near, and far transfer. A self regulation questionnaire was given at post test. Results: ANOVAs indicated that groups were comparable prior to treatment. For immediate trans fer and near transfer with high achieving students, the schema broadening plus SRL group improved more than the schema broa dening group, which improved more than the control group. For nea r transfer with average and low achieving students both treatment groups improved more than the control group. For far transfer, the schema broadening plus SRL group outgrew the other two gro ups. On the student questionnaire, the schema broadening plus SRL group indicated increased levels of self regulation compared to the other groups. With respect to students with disabilities, ANOVAs indicated that for immediate transfer both treatment grou ps outperformed the control group, for near transfer the schema broadening plus SRL group outperformed the control group, but there were no treatment effects for far transfer. Effect sizes for significant contrasts were moderate to large. PAGE 114 114 Table A 1. Continued Author/Date P urpose Participants Procedures Fuchs, Fuchs, Prentice, Hamlett, Finelli, & Courey, 2004a Assess whether guided practice in sorting problems into schemas might add value to the schema broadening intervention 366 students from 24 third grade classrooms a t 6 schools in an urban southeastern school district. Students ide ntified as high, average or low achieving. Classes randomly assigned to contrast, schema broadening, or schema broadening plus sorting Teachers conducted instruction for the contrast groups using the textbook. In the treatment conditions, teachers assisted research assistants who provided instruction using scripted lessons. Schema broadening instruction consisted of teaching four problem types from the textbook and four superficial problem fe atures for transfer. Sorting instruction involved presenting sample problems and asking students to identify the problem type and transfer type. Pre and post testing with immediate, near, and far transfer measures evaluated for problem solving proficiency and problem type schema and transfer type schema knowledge. Results: ANOVAs indicated that groups were comparable prior to the intervention. Results of ANOVAs indicated main effects of condit ion on all measures. On problem solving and problem type sche ma knowledge, both treatment groups improved more than the contrast group. On transfer type schema knowledge, the schema broadening plus sorting group improved more than the schema broadening group, which improved more than the control group. Results of th e regression indicated that schema development accounted for more variance in problem solving p erformance than initial problem solving level. For students with disabilities, there were main effects of condition on problem solving and transfer type schema w ith both treatment groups outperforming the control group. For problem type schema, there was no main effect of condition. Effect sizes were large, oft en over three standard deviations. PAGE 115 115 Table A 1. Continued A u thor/Date Purpose Participants Procedures Fuchs, Seet haler, Powell, Fuchs, Hamlett, & Fletcher, 2008 Assess the efficacy of using SBI as a secondary preventative tutoring protocol within tier two of a response to intervention model to ad dress mathematical word problem solving difficulties of third grade stud ents with math and reading difficulties 42 students with mathematics and reading difficulties from 29 classrooms in eight schools in an urban southeastern school district were randomly assigned to either continue in their general education curriculum (cont rol) or receive schema broadening tutoring The tier two schema broadening tutoring consisted of instruction in three problem schemas (e.g. total, difference, and change) and four superficial transfer features (e.g. includes irrelevant information, uses 2 digit operands, has missing information in the first or second position, presents information in charts, graphs, or pictures). University students conducted the one to one tutoring sessions using scripted lessons. Pre and post testing involved multiple me asures of foundational skills and word problem solving. Results: ANOVAs applied to pre test scores indicated g roup comparibility prior to treatment. Results of ANOVAs on improvement scores indicated no main effect of treatment for addition and sub traction fact retrieval, double digit addition and subtraction, Simple Algebraic Equations, WRAT Reading, and KeyMath Problem Solving. Significant effects and large effect sizes favoring the schema broadening tutoring were found for WRAT Arithmetic, Jordan Fuson & Willis, 1989 Determine whether regular classroom teachers could implement SBI sufficiently t o allow problem solving performance to improve. To determine if student proble m solving would improve following teacher implementation of the schematic diagram intervention. 76, second graders in 3 ability grouped (1 average achieving, 1 high/average achieving, 1 high achieving) classrooms at two schools in a small city near Chicago Intervention in schema identification, diagram choice/labeling, and problem solution conducted by teachers following an in service. Intervention addressed put together, change, and compare problems. Classroom observations to determine facility with whic h teachers implemented the intervention. Pre and post testing using a 24 item researcher developed test scored for diagram drawing and labeling, solution strategy choice, and correct answer. Results: Observations indicated that teachers implemented the interventions with varying facility. Despite variance in the quality of instruction, descriptive statistics indicated that all students made gains in choosing the correct diagram, labeling the diag ram correctly, choosing the correct solution strategy, and arriving at the correct answer. Student ability and quality of instruction may both influence effectiveness of instruction using schematic diagrams on problem solving. PAGE 116 116 Table A 1. Continued Author/Date Purpose Participants Procedures Griffin & Jitendra, 2009 Assess differ ential effects and maintenance effects of SBI instruction and general strategy instruction (GSI). Asses s the influence of word problem solving instruction on computation skills 60 students in 3 inclusive 3 rd grade classrooms in an elementary school in a co llege town in FL. 4 teachers (3 general, 1 special education) Assignment to conditions based on initial SAT 9 scores. Two instructional groups within each condition. SBI consisted of instruction in using schematic diagrams and self monitoring to solve one and two step additive story problems involving group, change, and compare schemas. GSI consisted of strategies taught in math basal text. Intervention conducted for 100 minutes one day per week. Results: ANOVA and Chi square tests indicated group equiva lency. ANCOVA indicated that students in both groups made statistically significant gains and maintai ned those gains on word problem solving word problem solving fluency, and computation fluency, but no effect based on group assignment was noted. However, further analysis indicated that the SBI group acquired problem solving skills more quickly than did the GSI group. Group differences decreased over time. Jitendra, Griffin, Deatline Buchman & Scesniak, 2007a Two design studies to evaluate the efficacy of SBI to solve one step group, change, and compare addition and subtraction story problems in classroom settings before conducted formal experimental studies. Study 1: 38 students in 2 low ability 3 rd grade classes 2 general & 1 special education teacher. Study 2: 56 students in 2 hetero geneously grouped 3 rd grade classes, 2 teachers In service training for teachers. Whole class instruction conducted by teachers with support from researchers. Pre a nd post testing on word problem solving criterion referenc ed test, word problem solving fluency test, and basic mathematics calculation fluency test. Student satisfaction questionnaire Results: Study 1: Repeated measures ANOVA indicated statistically significant main e ffects for time on word problem solving cri terion referenced tes t and word problem solving fluency test. Moderate to large effect sizes. Similar results for low achieving and learning disabled students. Study 2: Repeated measures ANOVA indicated statistically significant main e ffects for time on w ord problem solving fluency test and basic mathematics computation fluency test. Small to moderate effect sizes. Gains following schema based instruction were more apparent for the low performing students. Treatment fidelity high for both studies (93%, 98% ). PAGE 117 117 Table A 1. Continued Author/Date Purpose Participants Procedures Jitendra, George, Sood, & Price, (2010) To describe how SBI was used to improve the additive word problem solving of two students with Emotional Behavioral Disorders 4 th grade student with severe lear ning disability and emotional disturbance, 5 th grade students with behavioral disorders Following pre testing using a curriculum based measure, students received SBI in solving one and two step additive word problems from their trained special education t eacher for 45 minutes daily for 20 weeks. Results: Both participants showed gains on expe rimenter developed word problem solving fluency probes as instruction progressed, and showed gains from pre to post test on expe rimenter developed word problem solvi ng tests. Jitendra, Griffin, Haria, Leh, Adams, & Kaduvettoor, 2007b Assess the differential effects, maintenance effects, and generalization effects of SBI and general strategy instruction (GSI). 94 students in 5 inclusive 3 rd grade classrooms from an e lementary school in a northeastern school district. 6 teachers (5 general, 1 special education) Assignment to conditions based on initial SAT 9 scores. 3 instructional groups within each condition. SBI consisted of instruction in using schematic diagrams t o solve one and two step additive story problems involving group, change, and compare schemas. GSI consisted of strategies taught in math basal text. Intervention conducted for 100 minutes one day per week. Results: ANOVA and Chi square tests indicated g roup equivalency. ANCOVAs of word problem solving and SAT 9 post test scores indicated statistically significant effects and medium effect sizes in favor of SBI over GSI. ANCOVA applied to PSSA s cores found statistically significant effects in favor of SBI over GSI. Analysis of group effects based on disability status, English language learner status, and Title I status resulted in statistically significant covariates for the word problem solving maintenance test and one subtest of the SAT 9 in favor of SBI PAGE 118 118 Table A 1. Continued Author/Date Purpose Participants Procedures Jitendra, Griffin, McGoey, Gardill, Bhat, & Riley, 1998 Compare effectiveness of SBI and traditional basal instruction on the ability of students with and at risk of disabilities to solve story p roblems. Examine maintenance and generalization. 34 elementary students from four public school classrooms in a northeastern and southeastern school district (25 with mild disabilities, 9 at risk) Random assignment to groups. Teachers conduct traditional b asal instruction. Researchers conduct schema based instruction to small groups using diagrams to teach group, change, and compare problem types. Pre post maintenance, and generalization testing using 15 item exp erimenter designed word problem solving t ests. Student interviews to assess value of instruction. Results: ANOVA indicated no between group differences at pre test. ANOVAs indicated statistically significant main effects favoring the SBI group for post and maintenance testing. Statistically sig nificant interaction between group and test time on generalization where SBI group improved at a greater rate than did the traditional basal instruction group. Jitendra & Hoff, 1996 Determine if students with learning disabilities would improve in solving simple one step story problems following SBI. Three students attending third or fourth grade at a northeastern private elementary school for students with learning disabilities Adapted multiple probes across students design with probes administered at bas eline, following problem schemata instruction using story situations, following schema intervention using story problems, and two to three weeks later to assess maintenance. Schema based instruction used diagrams to teach group, change, and compare problem types. Results: No change in level from baseline to just after problem schemata instruction, increased level following intervention using story problems and schematic diagrams, and maintenance level at or near the levels seen following the intervention p hase. PAGE 119 119 Table A 1. Continued A uthor/Date Purpose Participants Procedures Neef, Nelles, Iwata, & Page, 2003 Evaluate effects of teaching students with developmental disabilities the precurrent skills of identifying the component parts of change problems on their abi lity to solve addition and subtraction story problems. Two young adult males (ages 19 and 23) with below average intellectual ability (IQ: 46 and 72) as measured by the Wechsler Adult Intelligence Scale Multiple baseline across behaviors design with each p recurrent skill representing a separate behavior. Problem solving probes consisting of 5 change story problems conducted at baseline and following training on each precurrent skill. Instruction was conducted one to one by researchers using a teaching to ma stery. Results: Participants demonstrated improved ability to identify components of change story problems following training sessio ns. After all precurrent skills had been trained, participants improved in their ability to solve addition and subtraction story problems. Rockwell, Griffin, & Jones, 2011 P rovide preliminary evidence of the efficacy of using SBI to teach additive word problem solving to an elementary student with autism. A fourth grade student with autistic disorder and low average nonverba l cognitive abilities. Multiple probes a cross behaviors design. Problem solving probes consisting of six story problems were conducted at baseline and following training on each problem type. SBI was conducted one to one by researchers. Results: The part icipant demonstrated improved ability to solve each type of additive story problem following SBI. After one lesson on generalizing to problems with unknowns in the initial or medial position, the participant was able to successful ly solve such problems. Pr oblem solving gains were maintained six weeks following the intervention. PAGE 120 120 Table A 1. Continued Author/Date Purpose Participants Procedures Willis & Fuson, 1988 Assess efficacy of SBI to teach average and high achieving second graders to additive story problems. Assess which problem types were within the development 43 second grade students in two ability grouped (one high achieving, one average achieving) public school classrooms in a city near Chicago. Intervention in schema identif ication, diagram drawing/labeling, and problem solution conducted by researchers to teach put together, change, and compare problem types. Pre and post testing using a 10 item researcher developed test scored for diagram drawing and labeling, solution st rategy choice, and correct answer. Results: Descriptive statistics indicate increases in percentages of students choosing correct diagram and correctly labeling it from pre to post test. Confusions with put together and compare problems noted. Statistica lly significant improvements in choice of solution strategy and correct answer were found using t tests. ANOVA revealed that problems involving an unknown in the first position were significantly more difficult for students. Xin, Wiles, & Lin, (2008) Dete rmine if combining SBI with instruction in story problem grammar would improve pre algebraic concept form ation and additive word problem solving performance of students with math difficulties 2 fourth grade students with learning disabilities and math diff iculties. A fifth grade student with math difficulties Used a multiple probes across participants and problem types design with probes conducted at baseline, following part part whole lessons, following additive compare problem lessons, and to assess maint enance. Researchers conducted instruction in 20 35 min sessions 3 days week. Story problem grammar instruction used cue cards to help students identify and map relevant information to diagrams Results: Results indicated that the intervention was effective developed word problem solving probes and equation solving probes, and on probes derived from the KeyMath R/NU PAGE 121 121 APPENDIX B SAMPLE LESSON SCRIPT Intervention: Change Lesson 1 Script Materials Word Probl em solving (RUNS) Poster Group Diagram Poster Compare Diagram Poster 2 Copies of Change problems 4.1 4.6 on note cards 2 Copies of Compare problems 4.1 4.2 on note cards 2 Copies of Group problems 4.1 4.2. on note cards 1 Dry erase board 1 Dry erase pen 1 Copy of Change Practice Sheet 1 2 Pencil s Teacher: So far, you have learned to use the RUNS steps to solve one type of ( Display Word Problem solving (RUNS) poster. Point t o each step on the word problem solving (RUNS) poster and have student name the step. ) Tell me each step when I point to it. Student : R Read the problem; U Use a diagram; N Number sentence; S State the answer. Teacher: We have also learned a diagram to help us solve one type of problem What type of problem did we learn the diagram for? Student : Group problems. Teacher: Right, we learned the group diagram. Can we solve all problems with the group diagram? Student : No, we can only solve group problems. Teacher: iagram can only be used to solve group problems. Today, we are going learn about the diagram for change problems. A change problem has a beginning, a change, and an ending. The beginning, change, and ending all describe the same thing/object. Our change pr oblems can change in two ways. They can get more, or they can get less. Whether the change is to get more or get less, the change is always an action. ( Display the change diagram poster. ) Where on this change diagram do you think the beginning should go? Student : ( Point to first circle. ) Teacher: do you think the change amount should go? Student : ( Point to middle box. ) PAGE 122 122 Teacher: Right, the change amount goes in the middle box. If th e change means get more, what operation could I put in this circle to show get more mean add or subtract? Student : Add Teacher: Right, if the change is get more that means add. So when the change is get more I would put a plus sign and the change amount in the Student : Subtract Teacher: Right, if the change is get less that means subtract. So when the change is get less I put a minus sign and the change amount in the circle. Wher e does the ending amount go in this change diagram? Student : ( Point to circle on right. ) Teacher: ( Display Change problem 4.1 ) I will use my RUNS steps and the change diagram to help me solve change problems. First I will write my RUNS steps on my dry erase board. Now, we are ready for Step I: Read the problem (Point to the first checkbox on the problem solving checklist) Follow along as I read. ( Read the problem aloud. ) Tammy likes to paint pictur es. She has painted 8 pictures so far. If she paints 3 more pictures, she will have 11 pictures. Now I can check off the R. The next step is Use a Diagram. I have two diagrams. ( Point to each diagram and name it. ) I need to decide if this is a change probl em, a group problem or neither. I see that this whole problem is talking about the same thing beginning 8 pictures. Then I see a change She paints 3 more pictures. I know this is a change because it is an a can see that this is a change get more because she paints more. So my change is plus 3 paintings. This problem also has an ending 11 pictures. So, this is a change problem. I will draw my change diagram on the dry erase board and fill in the numbers. Now I am ready for Step 3: Number sentence I can use my diagram to write my number sentence. 8 + 3 = 11. Finally, I will state the answer. 11 talking about another problem. ( Display Change problem 4.2 ). First, I will write RUNS on board to help me remember my steps. The first step is read the problem. Follow along as I read. ( Read the problem aloud ). Ryan had $10. He spent $6 on melons for a picnic. He has $4 left. I read the problem. The U stands for Use a diagram. I need to decide which diagram I will use. I can tell that this is a change problem because it talks about just one thing also see that it has a beginning Ryan had $10; a nd a change he spent $6. The word spent tells me that this is a change get less $6. The problem also has an ending $4 left. So, now I will draw my change diagram and put my numbers in. The next step is number sentence. I just use my diagram to hel p me. 10 6 = 4. Now I can PAGE 123 123 state my answer 4. Exactly so I will write $4. problem. ( Display group problem 4.1 ) First I will write RUNS on my board. R stands for read the problem. Follow along with me as I read. Sandra has 4 pairs of blue jeans. She also has 5 pairs of khakis. Sandra has 9 pairs of pants. Now I need to use a diagram. I must decide which diagram to use. I do not think that this is a compare problem. It talks about more than one thing jeans, khakis, and pants; and it does not have an action that shows a change. Let me see if this could be a group problem. Jeans and khakis could be the small parts, and since jeans and khakis are both types of pants, pants would be the all. So this is a group problem. It has small par ts jeans and khakis; and a big all pants. I will draw a group diagram on my board and put the numbers in. Now I will write my number sentence. 4 + 5 = 9. Now I can state my answer. 9 jeans. I labeled my answer. Here is another problem. ( Display compa re problem 4.1 ) First I will write RUNS on my board. R stands for read the problem. Follow along with me as I read. James has 28 marbles. Joe has 52 marbles. James has 24 fewer marbles than Joe has. Now I will use a diagram. I must decide which diagram to use. I do not think that this is a change problem. It talks about more than one thing shows a change. Let me see if this could be a group problem. ould be the small parts. If I put all. This problem does not talk about the all. So this is not a group problem either. I do not have a diagram to solve this problem yet. I will move onto the next problem. Now I want you to help me solve some problems. Here is the first one. ( Display change problem 4.3 ) What should I do first? Student : Write RUNS on your board. Teacher: Exactly Write RUNS. Now what should I do? Student : Read the problem Teacher: Good. Follow along with me as I read. John has 15 toy cars. He gets 7 more for his birthday. Now John has 22 toy cars. Now what should we do? Student : Use a diagram. Teacher: We need to decide which diagram to use. Do you think this is a chang e problem? Student : Yes Teacher: How do you know? Student : It talks about one thing shows a change gets more toy cars. Teacher: thing and h as an action that shows a change. Now I will draw the change diagram. What goes in this first circle at the beginning ? PAGE 124 124 Student : 15 toy cars Teacher: Yes, at the beginning, John has 15 toy cars. What goes in this middle box for the change? Student : 7 toy ca rs Teacher: 7 toy cars is the amount of change. Hmm, I still need to know if this Student : John got more. So we need a plus. We need to add. Teacher: Yes, it says John got more cars, so that is adding. W hat goes in this last circle at the ending? Student : 22 toy cars. Teacher: Right, John had 22 cars at the ending. I used a diagram, now what should I do? Student : Number sentence. Teacher: What number sentence should I write? Student : 15 + 7 = 22 Teacher: Very good, you used the diagram to make that number sentence. Now what do we do? Student : State the answer. Teacher: What should I write? Student : 22 toy cars. Teacher: next problem. ( Di splay group problem 4.2 ) What should I do first? Student : Write RUNS on your board. Teacher: Exactly Write RUNS. Now what should I do? Student : Read the problem Teacher: Good. Follow along with me as I read. Jan bought 12 carrots and 9 tomatoes at th e store. She bought 21 vegetables at the store. Now what should we do? Student : Use a diagram. Teacher: We need to decide which diagram to use. Do you think this is a change problem? Student : No Teacher: How do you know? Student : It talks about more tha n one thing carrots, tomatoes and vegetables. And there is no action. Teacher: more than one thing and has no action that shows a change. Do you think this a group problem? Student : Yes Teacher: How do you know? Student : It has two small parts carrots and tomatoes. Those are both vegetables. It has the all vegetables. Teacher: a big all. Now I will draw the group diagram. What goes in this part? Student : 12 carrots Teacher: What goes in this part? PAGE 125 125 Student : 9 tomatoes Teacher: Yes, 12 carrots and 9 tomatoes are the small parts. What goes in the all? Student : 21 vegetables. Teacher: Right, carrots and tomatoes are bo th vegetables. So that is the all. What now? Student : Number sentence. Teacher: What number sentence should I write? Student : 12 + 9 = 21 Teacher: Very good, you used the diagram to make that number sentence. Now what do we do? Student : State the answer. T eacher: What should I write? Student : 21 vegetables. Teacher: Display change problem 4.4. What should I do first? Student : Write RUNS on your board. Teacher: Exactly Write RUNS. Now what should I do? Student : Read the problem. Teacher: Good. Follow along with me as I read. Donovan collects words. He has 55 words in his collection. If he gives 27 of his words away he will still have 28 words. Now what should we do? Student : Use a diagram. Tea cher: We need to decide which diagram to use. Do you think this is a change problem? Student : Yes Teacher: How do you know? Student : It talks about one thing shows a change gives away words. Teacher: e right, this is a change problem because it talks about one thing and has an action that shows a change. Now I will draw the change diagram. What goes in this first circle for the beginning. Student : 55 words Teacher: Yes, at the beginning, Donovan has 55 words. What goes in this middle box for the change? Student : 27 words Teacher: 27 words is the amount of change. Hmm, I still need to know if this Student : Donovan gave words away. So he got less. We ne ed a minus sign to subtract. Teacher: Yes, it says Donovan gave words away, so that is subtracting. What goes in this last circle at the ending? Student : 28 words. Teacher: Right, Donovan had 28 words at the ending. I used a diagram, now what should I do? Student : Number sentence. PAGE 126 126 Teacher: What number sentence should I write? Student : 55 27 = 28 Teacher: Very good, you used the diagram to make that number sentence. Now what do we do? Student : State the answer. Teacher: What should I write? Student : 28 wor ds. Teacher: problem. ( Display compare problem 4.2 ) What should I do first? Student : Write RUNS on your board. Teacher: Exactly Write RUNS. Now what should I do? Student : Read the probl em Teacher: Good. Follow along with me as I read. John has 15 toy cars. Mark has 7 toy cars. John has 8 more toy cars than Mark has. Now what should we do? Student : Use a diagram. Teacher: We need to decide which diagram to use. Do you think this is a c hange problem? Student : No Teacher: How do you know? Student : It talks about more than one thing And there is no action that shows a change. Teacher: more than one thing and has no action that shows a change. Do you think this is a group problem? Student : No Teacher: How did you know that? Student : Well, it sort of has two small parts you put those together you Teacher: the next problem. ( Display change problem 4.5 ) What should I do first? Student : Write RUNS on your board Teacher: Exactly Write RUNS. Now what should I do? Student : Read the problem Teacher: Good. Follow along with me as I read. Lydia had 26 flowers on the roof. She p lanted 58 more flowers. Now Lydia has 84 flowers on the roof. Now what should we do? Student : Use a diagram. Teacher: We need to decide which diagram to use. Do you think this is a change problem? Student : Yes Teacher: How do you know? PAGE 127 127 Student : It talks about one thing shows a change plants more. Teacher: thing and has an action that shows a change. Now I will draw the change diagram. What goes in this first circle at the beginning? Student : 26 flowers Teacher: Yes, at the beginning, Lydia had 26 flowers. What goes in this middle box for the change? Student : 58 flowers Teacher: 58 flowers is the amount of change. Hmm, I still need to know if this Student : Lydia planted more. So we need a plus. We need to add. Teacher: Yes, it says Lydia planted more flowers, so that is adding. What goes in this last circle at the ending? Student : 84 flowers Teacher: Right, Lydia had 84 flowers at the ending. I used a diagram, now what should I do? Student : Number sentence. Teacher: What number sentence should I write? Student : 26 + 58 = 84 Teacher: Very good, you used the diagram to make that number sentenc e. Now what do we do? Student : State the answer. Teacher: What should I write? Student : 84 flowers. Teacher: Display change problem 4.6. What should I do first? Student : Write RUNS on you r board. Teacher: Exactly Write RUNS. Now what should I do? Student : Read the problem Teacher: Good. Follow along with me as I read. There are 9 cakes at the bakery. Customers buy 7 of the cakes. There are 2 cakes left at the bakery. Now what should we do? Student : Use a diagram. Teacher: We need to decide which diagram to use. Do you think this is a change problem? Student : Yes Teacher: How do you know? Student : It talks about one thing cakes. And there is an action that shows a change customers buy cakes. Teacher: thing and has an action that shows a change. Now I will draw the change diagram. What goes in this first circle at the beginning? Student : 9 cakes Teacher: Yes, at the b eginning, there are 9 cakes at the bakery. What goes in this middle box for the change? PAGE 128 128 Student : 7 cakes Teacher: 7 cakes is the amount of change. Hmm, I still need to know if this Student : Customers bou get less. We need a minus for subtract. Teacher: Yes, it says bought 7 cakes. That means there are fewer cakes at the bakery. So this is a subtraction problem. What goes in this last circle at the e nding? Student : 2 cakes. Teacher: Right, there are 2 cakes in the bakery at the ending. I used a diagram, now what should I do? Student : Number sentence. Teacher: What number sentence should I write? Student : 9 7 = 2 Teacher: Very good, you used the diag ram to make that number sentence. Now what do we do? Student : State the answer. Teacher: What should I write? Student : 2 cakes. Teacher: use the RUNS steps with the group and change diagram s. Now I would like for you to try to complete the R U N S steps for two change problems. On this piece of paper are two problems. Do your best to complete the R U N S steps using the change diagram. Tell me when you are finished. ( Give student Change Prac tice Sheet 1 and a pencil. Provide access to RUNS poster and Change Diagram poster. Allow up to 10 minutes to complete the worksheet. Score worksheet for correct number sentence and computation. Provide corrective feedback ). You did a great job using the change diagram and the RUNS steps to solve problems. Tomorrow we will work on some more problems together and you will do some more problems by yourself. PAGE 129 129 APPENDIX C SAMPLE LESSON CHECKLIST Change Lesson 1 Checklist ____Review RUNS ____Review Group Proble ms/Diagram ____Student identifies that group problems talk about more than one thing ____Student identifies that group problems have small parts ____Student identifies that the small parts are put together to make one big all ____Introduce Change Problems/ Diagram ____Change problems talk about one thing ____Change problems have a beginning, change, and ending ____Model with think alouds using RUNS and Change Diagram 6.1 _____Read problem aloud _____Identify critical features of change problems that match diagram _____Draw and fill out diagram _____Write number sentence _____Write answer with label ____Model with think alouds using RUNS and Change Diagram 6.2 _____Read problem aloud _____Identify critical features of change problems that match diagram _____Draw and fill out diagram _____Write number sentence _____Write answer with label ____Model using RUNS and using Group Diagram with Group Problem 6.1 _____Read problem aloud _____Identify critical features of group problems that match diagram _____Draw and fill out diagram _____Write number sentence _____Write answer with label ____Model using RUNS and not using a diagram with Compare Problem 6.1 PAGE 130 130 _____Read problem alou d _____Identify lack of critical group or change problem features ____Have student help using RUNS and Change Diagram 6.3 _____Have student identify the steps to be followed _____Teacher prompts praise s provides corrective feedback as needed ___ __Read problem aloud _____Identify lack of critical features of change problems that match diagram _____Draw and fill out diagram _____Write number sentence _____Write answer with label ____Have student help using RUNS and Group Diagram with Group pro blem 6.2 _____Have student identify the steps to be followed _____ Teacher prompts, praises, provides corrective feedback as needed _____Read problem aloud _____Identify lack of critical features of group problems that match diagram _____Draw and fill out diagram _____Write number sentence _____Write answer with label ____Have student help using RUNS and Change Diagram 6.4 _____Have student identify the steps to be followed _____ Teacher prompts, praises, provides corrective feedback as needed _____Read problem aloud _____Identify lack of critical features of change problems that match diagram _____Draw and fill out diagram _____Write number sentence _____Write answer with label ____ Administer Change Practice Sheet 1 ____Provide corrective feedback/praise ____Preview next lesson ____Provide intermittent reinforcement for on task behavior ____Allow up to 2 movement breaks during lesson PAGE 131 131 APPENDIX D SAMPLE PRACTICE SHEETS WITH HYPOTHETICAL RESPONSES AND SCORING Group Story Situation Practice S heet 1. There are 11 bath towels and 7 dish towels in the linen closet. There are 18 towels in the linen closet. 11 7 18 11+7=18 Answer: _______ 18 towels ______________________________________________ Scoring: 1_ C orrect diagram, 1 C orrect numbe r sentence, 1_ Correct computation 2. There are 16 puppies and 13 kittens at the pet store. There are 29 animals at the pet store. 16 13 29 16+13=19 Answer: __________ 19 animals ___________________________________________ Scoring: 1_ C orrect diagr am, 1 C orrect number sentence, 1_ Correct computation PAGE 132 13 2 Group Problem solving Practice Sheet 1. Jose has 46 legos and 53 wooden blocks. How many blocks does Jose have? 46 53 ? 46+53= 98 Answer: _____ 98 blocks ____________________________________ _____________ Scoring: 1 C orrect diagram, 0 C orrect number sentence, 0 Correct computation 2. There are 51 girls and 46 boys in the third grade. How many children are in the third grade? 51 46 ? 51+46=97 Answer: ___ 97 children ______________ ____________________________________ Scoring: 1 C orrect diagram, 1 C orrect number sentence, 1 Correct computation PAGE 133 133 Change Story Situation Practice Sheet 1. There are 77 apples on a tree. Some people pick 32 apples. Now there are 45 apples on the tree. 77 32=45 Answer: _______ 45 apples _______________________________________________ Scoring: 1_ C orrect diagram, 1 C orrect number sentence, _1 Correct computation 2. Jack has $23 in his bank account. He gets $50 for his birthday. Now h e has $73 in his bank account. 23+ 50=73 Answer: _____ $73 __________________ ___________________________________ Scoring: 1_ C orrect diagram, 1 C orrect number sentence, _1 Correct computation 45 77 32 73 23 +50 PAGE 134 134 Change Problem solving Practice Sheet 1. Ju lie has 42 silly bands. She buys 25 more silly bands. How many silly bands does Julie have now? 42+25=67 Answer: _________ 67 silly bands __________________________________________ Scoring: 1 C orrect diagram, 1 C orrect number sentence, 1 Correct computation 2. There were 6 7 roses in the garden. The gardener picked 2 1 roses to put in a vase. How many roses are in the garden now? 67+21=88 Answer: _________ 88 roses ______________________________________________ Scoring: 0 C orrect diagram, 0 C orrect number sentence, 0 Correct computation Compare Story Situation Practice Sheet ? 42 +25 ? 67 +21 PAGE 135 135 1. There are 19 bath towels and 7 dish towels in the linen closet. There are 12 more bath towels than dish towels in the linen closet. 19 5. 19 7=12 Answer: ____ 12 more dish towels __ _____________________________________ Scoring: 1_ C orrect diagram, 1 C orrect number sentence, 1_ Correct computation 2. There are 16 puppies and 13 kittens at the pet store. There are 3 fewer kittens than puppies at the pet store. 16 5. 16 13=3 Answer: ____ 3 fewer kittens __________________________________________ Scoring: 1_ C orrect diagram, 1 C orrect number sentence, _1 Correct computation Change Problem solving Practice Sheet 7 13 12 3 PAGE 136 136 1. Mrs. S mith baked 29 cookies. She also baked 18 cupcakes. How many more cookies than cupcakes did Mrs. Smith bake? 29 5. 29 18=11 Answer: ______ 11 more cookies __________________________________________ Scoring: 1_ C orrect diagram, 1 C orrect nu mber sentence, 1_ Correct computation 2. At the animal shelter there are 39 dogs and 17 cats. How many fewer cats are there than dogs at the animal shelter. 39 5. 39 17=23 Answer: ______ 23 fewer cats __________________________________________ Scoring: 1_ C orrect diagram, 1 C orrect number sentence, 0_ Correct computation 18 17 ? ? PAGE 137 137 APPENDIX E SAMPLE PROBES WITH HYPOTHETICAL RESPONSES AND SCORING Problem solving Probe 1. A scout troop leader is taking 14 scouts to the movies. Three scouts cancelle d. How many scouts are going to the movie now? 14 3 = 11 ANSWER : ___ 11 scouts ____________________________________________ Scoring: 1 Correct number sentence, 1 Correct computation 2. There are 10 crickets in the shed and 16 crickets outside How many more crickets are outside than in the shed? 16 5. 16 10 = 6 ANSWER : _____ 6 crickets ____________ ____________________________ Scoring: 1 Correct number sentence, 1 Correct computation 3. You spend $15 on games and $12 on balls. How mu ch money did you spend? 15 12 ? 15 + 12 = 27 ANSWER : _____ 27 dollars _________________________________________ Scoring: 1 Correct number sentence, 1 Correct computation 10 ? 14 3 ? PAGE 138 138 4. Paul picks 14 apples from one tree. He picks 11 apples from another. How many a pples did he pick? 14 11 ? 14 + 11 = 25 ANSWER : _____ 25 apples _________________________________________ Scoring: 1 Correct number sentence, 1 Correct computation 5. There are 12 chicks in the shed and 16 chicks outside. How many fewer chicks are in the shed than outside? 16 5. 16 12 = 2 ANSWER : _____ 2 chicks ___________________________________________ Scoring: 1 Correct number sentence, 0 Correct computation 6. There are 12 chicks at the farm. If 17 more chicks are born, how many chicks ar e at the farm now? 12 + 17 = 29 ANSWER : ___ 29 chicks ____________________________________________ Scoring: 1 Correct number sentence, 1 Correct computation ? ? +17 12 12 PAGE 139 139 7. Mrs. Smith buys 15 carrots and 13 potatoes. How many vegetables did she buy? 15 13 ? 15 + 13 = 27 ANSWER : _____ 27 vegetables _____________________________________ Scoring: 1 Correct number sentence, 0 Correct computation 8. Joe has 38 baseball cards. Tom has 26 baseball cards. How many more baseball cards does Joe have than Tom? 38 5. 38 26 = 12 ANSWER : _____ 12 baseball cards __________________________________ Scoring: 1 Correct number sentence, 1 Correct computation 9. There are 24 children going on the field trip. If 13 more children decide to go, how many children will b e going on the field trip then? 24 + 13 = 37 ANSWER : ___ 37 children _____ _____________________________________ Scoring: 1 Correct number sentence, 1 Correct computation ? 26 ? +13 24 PAGE 140 140 Generalization Probe Irrelevant Information 1. A teacher is taking 28 students to the museum. If 6 students get sick and it costs $5 per person to enter the museum, how many students went to the museum? 28 6=22 ANSWER : ____ 22 students _______________________________________ Scoring: 1 Correct number sentence, 1 Correct computation 2. There are 18 students on the monkey bars, 6 students on the swings, and 5 students in the sand box. How many more students are on the monkey bars than in the sand box? 18 5. 88 6 = 13 ANSWER : ____ 13 more students ______ ______________________________ Scoring: 0_ Correct number sentence, 0_ Correct computation 3. Mrs. Jones spent $15 on a shirt, $34 on a dress, and $7 on lunch. How much money did she spend on clothing? 15 3 4 ? 15 + 3 4 = 48 ANSWER : ___ $48 _____________ ________________________________ Scoring: 1_ Correct number sentence, 0_ Correct computation 28 6 ? ? 5 PAGE 141 141 4. Peg bought 11 apples, 5 pears, and 2 loaves of bread. How many pieces of fruit did Peg buy? 11 5 ? 11+5 15 ANSWER : _____ 15 pieces of fruit _____________ ___ _____ ______________ Scoring: 1 Correct number sentence, 0 Correct computation 5. There are 57 cows in the pasture, 15 cows in the barn and, and 6 horses in the stable. How many more cows are in the pasture than in the barn? 5. 57 5. 57 15 = 4 2 ANSWER : ____ 42 more cows _____________________________________ Scoring: 1 Correct number sentence, 1 Correct computation 6. Kelly has $38 in her piggybank. She gets $20, some books, and a bicycle for her birthday. How much money does Kelly have now? 38+20=58 ANSWER : ___ $58 ____________________________________________ Scoring: 1 Correct number sentence, 1 Correct computation ? +20 38 ? 15 PAGE 142 142 7. Jim looks out his class window. He sees 23 cars, 12 trucks, and 5 trees in the parking lot. How many vehicl es does Jim see? 23 12 ? 23+12=35 ANSWER : ____ 35 vehicles _______________________________________ Scoring: 1 Correct number sentence, 1 Correct computation 8. pages. How many 78 5. 78 52=26 ANSWER : __ 26 fewer pages _____________________________________ Scoring: 0 Correct number sentence, 0 Correct computation 9. There are 62 cows and 3 7 pigs on a farm. If 23 more cows are born, how many cows will be on the farm? 62 + 2 3 = 85 ANSWER : ___ 85 cows _______________________________________ Scoring: 1 Correct number sentence, 1 Correct computation ? +23 62 ? 52 PAGE 143 143 Generalization Probe Initial Position 1. Many children ride the school bus home. At the first stop 7 children get off the bus. Now there are 21 students on the bus. How many children ride the bus home? 21+7=28 ANSWER : ____ 28 children _______________________________________ Scoring: 1_ Correct number s entence, 1_ Correct computation 2. There are some boys and 43 girls in the first grade. If there are 94 children in first grade, how many boys are there? ? 43 94 94 43=61 ANSWER : ____ 61 boys _____________________________________________ Scoring: 1_ Correct number sentence, 0_ Correct computation 3. Mr. Smith bought pants and a shirt. The shirt cost $15. If he spent $47, how much did the pants cost? ? 15 47 47 15=32 ANSWER : ____ $32 ____________________________________________ Scoring: 1_ Corr ect number sentence, 1_ Correct computation ? 7 21 PAGE 144 144 4. A farmer has some pigs and 42 cows. The farmer has 96 animals. How many pigs does he have? ? 42 96 96 42=54 ANSWER : ____ 54 pigs __________________________________________ Scoring: 1_ Correct number sen tence, 1_ Correct computation 5. There are some pigs in the barn and some 12 pigs outside in the mud. If there are 32 more pigs in the barn than in the mud, how many pigs are in the barn? ? 5. 32 12 = 20 ANSWER : __ 20 pigs _________________________________________ Scoring: 0_ Correct number sentence, 0_ Correct computation 6. Kim has some money in her piggybank. She spends $20 on a new doll. Now she has $38. How much money did Kim have in her piggybank? 38 + 20=58 ANSWER : ____ $58 ____________________________________________ Scoring: 1_ Correct number sentence, 1_ Correct computation 32 12 ? 20 38 PAGE 145 145 7. There are some goldfish and 23 guppies in a fish tank. If t here are 32 fewer guppies than gold fish in the tank how many goldfish are there? ? 1. 23+32=55 ANSWER : __ 55 goldfish __________________________________________ Scoring: 1_ Correct number sentence, 1_ Correct computation 8. Jim can type faster than Pete. Pete can type 72 words a minute. If Pete can type 11 words fewer than Jim, how many words can Jim type in a minute? ? 2. 72+11=83 ANSWER : __ 83 pages _____________________________________________ Scoring: _1_ Correct number sentence, 1_ Correct computation 9. Ellen has r ead a lot of books this year. If she reads 12 more books, she will have read 75 books. How many books has she already read? 75 12=63 ANSWER : ___ 63 books _______________________________________ Scoring: 1_ Correct number sentence, 1_ Correc t computation 32 23 11 72 75 +12 ? PAGE 146 146 Generalization Probe Medial Position 1. There were 22 children in little league. Some more children signed up. Now there are 37 children in little league. How many children signed up? 37+22=59 ANSWER : ___ 59 children ___ ____________________________________ Scoring: 0_ Correct number sentence, 0_ Correct computation 2. There are 24 students in M There are 3 more students in M s. Lee Lee 24 5. 24 3=21 ANSWER : __ 21 children _________________________________________ Scoring: 1_ Correct number sentence, 1_ Correct computation 3. A mom bought her son a bike and a helmet for his birthday. The bike cost $78. If she spe nt $99, how much did the scooter cost? 78 ? 99 99 78=21 ANSWER : ____ $21 ____________________________________________ Scoring: 1_ Correct number sentence, 1_ Correct computation 37 +? 22 3 ? PAGE 147 147 4. There are 38 children on a soccer team. If there ar e 17 boys on the team, how many girls are on the team? 17 ? 38 38 17=21 ANSWER : ____ 21 girls __________________________________________ Scoring: 1_ Correct number sentence, 1_ Correct computation 5. 78 first graders rid e the bus to school. If 23 more first graders ride the bus than get dropped off, how many first graders get dropped off? 78 5. 78 23=55 ANSWER : __ 55 first graders __ _____________________________________ Scoring: 1_ Correct number s entence, 1_ Correct computation 6. Kelly has 32 silly bands. She uses her allowance to buy some more silly bands. Now she has 67 silly bands. How many silly bands did Kelly buy? 67 32=35 ANSWER : ___ 35 silly bands ___________________________ _________ Scoring: 1_ Correct number sentence, 1_ Correct computation 67 +? 32 23 ? PAGE 148 148 7. 63 children bought lunch today. If there are 96 children eating lunch in the school cafeteria, how many of them brought lunches from home? 63 ? 96 96 63=36 ANSWER : ____ 3 6 children __________________________________________ Scoring: 1_ Correct number sentence, 1_ Correct computation 8. The apple tree has 53 pieces of fruit on it. The orange tree ha s 21 fewer pieces of fruit than the apple tree. How many pieces of fruit does the orange tree have? 53 5. 53 21 = 32 ANSWER : __ 32 pieces of fruit ____________________________________ Scoring: 1_ Correct number sentence, 1_ Correct computation 9. 78 students tried out for the high school footb all team. After cuts, 56 students were on the team. How many students got cut? 78 56=22 ANSWER : ____ 22 students ____ _________________________________ Scoring: 1_ Correct number sentence, 1_ Correct computation 78 ? 56 21 ? PAGE 149 149 APPENDIX F SATISFACTION SCALE S Student Satisfaction Scale Circle the face that mathes each statement. How I feel about solving math word problems How I feel about using the RUNS steps to solve math word problems How other kids might feel about using the RUNS steps t o solve math word problems How I feel about using diagrams to solve math word problems How other kids might feel about using diagrams to solve math word problems? PAGE 150 150 Parent Satisfaction Scale Please place an X in the box indicating the extent to which you agree or disagree with each of the following statements about Schema Based Strategy Instruction (SBI). Strongly Agree Agree Neutral Disagree Strongly Disagree SBI addresses important skills for my child SBI was a valuable use o f instructional time My child seemed to enjoy SBI My child uses the RUNS mnemonic when solving math word problems My child uses schematic diagrams when solving math word problems My child has shown improvement in solving math word p roblems implement SBI with my child I would recommend SBI to others Please provide any additional comments regarding SBI: ______________________________________________________________________ _____________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ _______________________________________________ _______________________ ______________________________________________________________________ PAGE 151 151 APPENDIX G INFORMED CONSENT DOCUMENTS The following informed consent and assent documents were approved by the on July 28, 2010 and renewed for use through July 28, 2012. This study is Protocol #2010 U 0649. Child Consent Script Investigator: Hi! My name is Sarah Rockwell and this is (other doctoral student) We are students at the University of Florida. We are w orking on a special project to help us learn more about how to teach children to solve math word problems. For the next nine weeks, we will be working with you every day to help you learn to solve math word problems. We are also going to ask you to solve some word problems on your own. Is it OK for us to teach you to solve word problems? (Child's response) Is it OK if we ask you to solve some word problems on your own? Do you have any questions about what we want you to do? (Child's response) Investigator: I need to tell you one more thing. After we get started, if you decide that you don't want to be in this study any more, let me know. It is okay if you decide not to be in the study. Is it okay to start now? (Child's response) PAGE 152 152 Parent Informed Consent Letter September 16, 2010 Dear Parent/Guardian: I am a doctoral candidate in the School of Special Education, School Psychology, and Early Childhood Studies at the University of Florida. I will be conducting a study to eva luate the effectiveness of an approach to teaching children with autism to solve addition and subtraction word problems. This approach is called Schema Based Strategy Instruction and involves teaching children to use diagrams to solve math word problems. The procedures used in this study are as follows. First, your child will participate in screenings to determine if he/she is able to perform addition and subtraction computations and read word problems. Then your child will participate in pre testing con ducted one to one using nine item problem solving tests. Next, I and another doctoral student will conduct one to one lessons to teach your child to use diagrams to solve addition and subtraction word problems. This instruction will take place during 30 mi nute sessions conducted daily for approximately nine weeks. Instruction and assessments will take place after school at a time that is convenient for you. The Center for Autism and Related Disabilities (CARD) has agreed to provide space for this study, or you may choose a location that will be more convenient for you. Instruction will address three types of addition and subtraction word problems. Your child will be tested using the same nine item tests after learning each to solve each type of problem. Afte r instruction, you and your child will complete short questionnaires asking your perceptions of the instruction. Finally, your child will participate in follow up testing eight weeks later conducted one to one using the same nine item tests. I will need your permission to obtain background information about your child from existing school records. Specifically, I would like to have access to current scores from standardized tests of achievement and tests of learning aptitude. The information will be used to addition, I will need your permission to videotape lessons and assessments conducted with your child. I and another doctoral student will view these vide otapes in order to collect data on the behaviors. When not in use, the videotapes will be stored in a secured and locked cabinet. I will also need your permiss ion to conduct individual problem solving assessments with your child. identifying information about your child will be reported. Finally, I will need your per mission to conduct satisfaction questionnaires with you and your child. The results of these questionnaires will be used to ensure that instruction was enjoyable for and beneficial to your child and no information about you or your child will be reported. If you agree to allow your child to participate in this study, both you and your child retain the right to withdraw consent for participation at any time without penalty. This will be explained to your child. No compensation will be given to your child f or participation in this project. In addition, no risks or benefits to you or your child are anticipated as a result of PAGE 153 153 participation in this study. If you should have any questions about your child's participation, please feel free to call me at (352) 284 6000 or contact my faculty supervisor, Dr. Cynthia Griffin at (352) 273 4265. We would be happy to talk to you about the project. Questions or University of Flor ida, Gainesville, FL 32611 2250; phone: (352) 392 0433. Sincerely, Sarah B. Rockwell, M.Ed. Doctoral Candidate School of Special Education, School Psychology, and Early Childhood Studies University of Florida _ _ _ _ _ _ _ _ _ _ _ _ Please return this section as soon as possible. I have received a copy of the letter from Sarah B. Rockwell describing the study of using schema based strategy instruction to teach addition and subtrac tion word problem solving to students with autism. I have read the procedures described. I agree to allow my child to participate in the project. Student's Name __________________________ Parent's or Guardian's Signature ______________________________ 2nd Parent/Witness Signature __________________________ Date: ________________________ In addition, I give my permission for Sarah B. Rockwell or another doctoral student to obtain my child's school records to collect my child's most recent scores on tests o f academic achievement and learning aptitude. (Please check "yes" or "no") ___________ yes _____________ no I also give my permission for Sarah B. Rockwell or another doctoral student to videotape mathematics lessons and assessments with my c hild. (Please check "yes" or "no") ___________ yes _____________ no ability to read word problems and perform addition and subtraction computatio ns. (Please check "yes" or "no") ___________ yes _____________ no mathematics progress using nine item problem solving tests. (Please check "yes" or "no") ___________ yes _____________ no I also give my permission for Sarah B. Rockwell or another doctoral student to assess my and my (Please check "yes" or "no") ___________ yes _____________ no PAGE 154 154 LIST OF REFERENCES American Psychiatric Association (2000). Diagnostic and statistical manual of mental disorders (4 th ed., Text Revision). Washington, DC: Author. Algina, J., & Crocker, L. (2006). Introduction to classical and modern test theory. Independence, KY: Cengage Learning. Assistance to States for the Education of Children with Disabilities, 34 C.F.R. Part 300.39(3)(i)(ii). Beck, R., Conrad, D., & Anderson, P. (1997). Basic Skill Builders Longmont, CO: S opris West. Bottge, B. A. (2001). Reconceptualizing mathematics problem solving for low achieving students. Remedial and Special Education, 22, 102 112. Capizzi, A. M. (2007). Mental efficiency as a measure of schema formation: An investigation in elementa ry math problem solving. (Doctoral dissertation, Vanderbilt University, 2006). Retrieved January 20, 2009, from ProQuest Dissertation & Thesis Full Text database. (Publication No. AAT 3237020). Carpenter, T. P. & Moser, J. M. (1984). The acquisition of add ition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179 202. Centers for Disease Control and Prevention (2009). Prevalence of autism spectrum disorders Autism and developmental disabilities monit oring network, 2006. Morbidity and Mortality Weekly Report 58, 1 24. Retrieved January 27, 2010, from http://www.cdc.gov/mmwr/ Journal of Educational Psychology, 91, 703 715. Chen, Z. & Daehler, M. W. (1989). 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Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of m athematical learning difficulties and disabiliites (pp. 219 244). Baltimore, MD: Paul H. Brookes Publishing Co. Zentall, S. S. & Ferkis, M. A. (1993). Mathematical problem solving for youth with ADHD, with and without learning disabilities. Learning Disabi lity Quarterly, 16, 6 18 PAGE 161 161 BIOGRAPHICAL SKETCH Sarah B. Rockwell has 16 years of experience working with children with autism and other developmental disabilities. While in high school, she logged over 300 hours volunteering in the homes and classrooms of children with disabilities. While working on her undergraduate degree, she continued her volunteer work in early intervention and pre kindergarten classes for children with disabilities. She also worked as substitute teacher in special education classes a nd as a Personal Care Assistant and Behavior Technician for Special Friends, Inc., a company providing services for children with the Medicaid waiver due to their disability status. After receiving her undergra duate degree in special e ducation with a focus on early childhood at the University of Florida in 2004, Sarah began working as a teacher in a pre kindergarten class for students with disabilities. She later taught students in grades two thru five with Autism and worked as an educational diagnostician and consultant at the Multidisciplinary Diagnostic and Training Program in the department of Pediatric Neurology at the University of Florida. Sarah was awarded her m ducation with a focus on reading disabilities from the Universi ty of Florida in 2007. She immediately began working as a full time doctoral student focusing her energy on mathematics instruction for students with autism. When Sarah became a mother in January 2010, she developed an interest in baby wearing and breastfe eding education. She is a trained baby wearing educator and would like to combine her interests in special education, child development, and baby wearing by conducting future research on the role of baby wearing in sensory integration, muscle tone and so cial and communication behavior of children with disabilities. 